**2. Resonant spontaneous bremstrahlung of an electron scattered by a nucleus in the field of a pulsed light wave**

We consider in this section the problem of spontaneous bremsstrahlung (SB) of an electron scattered by a nucleus in the external field of a pulsed light wave (see Fig. 1). Studying of SB when an electron is scattered by a nucleus or by an atom in presence of an external electromagnetic field has had a long-standing interest. Analytic expressions for the radiation spectrum of SB in a plane monochromatic wave in the nonrelativistic case have been derived by Karapetian & Fedorov (1978) for any atomic potential field in the Born approximation and by Zhou & Rosenberg (1993) for a short-range potential in the low-frequency approximation. Resonant SB of a nonrelativistic electron scattered by a nucleus in a plane-wave field was studied by Lebedev (1972). Borisov et al. (1980) considered resonant SB, which accompanies collisions of ultrarelativistic electrons for the case of large transferred momenta. In the general relativistic case the problem of electron-nucleus SB in the field of a plane monochromatic wave was studied by Roshchupkin (1985). It should be noted that the theory of SB in presence of an external field is also developed in Lötstedt et al. (2007); Schnez et al. (2007). They contain important numeric calculations for the case of a strong field. Nonresonant SB in a pulsed field was considered by Lebed' & Roshchupkin (2009). Resonant SB of an electron scattered by a nucleus in the field of a pulsed light wave was studied in the general relativistic case (Lebed' & Roshchupkin (2010)).

Fig. 1. Feynman diagrams of electron-nucleus SB in the field of a pulsed light wave. Incoming and outgoing double lines correspond to the Volkov functions of an electron in initial and final states; inner lines designate the Green function of an electron in a pulsed field. Wavy lines correspond to four-momenta of spontaneous photon and "pseudophoton" of nucleus recoil.

#### **2.1 Amplitude of resonant spontaneous bremsstrahlung**

The process of electron-nucleus SB in a pulsed light field (2) in the Born approximation on interaction of an electron with a nucleus, which corresponds to the transition of an electron from an initial state with the four-momentum *pi* = (*Ei*, **p***i*) into a final state with the four-momentum *pf* = *Ef* , **p***<sup>f</sup>* , is described by two Feynman diagrams (Fig. 1).

The *S*-matrix element is given by

4 Will-be-set-by-IN-TECH

The relativistic system of units, where ¯*h* = *c* = 1 and the standard metric for 4-vectors (*ab*) =

**2. Resonant spontaneous bremstrahlung of an electron scattered by a nucleus in**

We consider in this section the problem of spontaneous bremsstrahlung (SB) of an electron scattered by a nucleus in the external field of a pulsed light wave (see Fig. 1). Studying of SB when an electron is scattered by a nucleus or by an atom in presence of an external electromagnetic field has had a long-standing interest. Analytic expressions for the radiation spectrum of SB in a plane monochromatic wave in the nonrelativistic case have been derived by Karapetian & Fedorov (1978) for any atomic potential field in the Born approximation and by Zhou & Rosenberg (1993) for a short-range potential in the low-frequency approximation. Resonant SB of a nonrelativistic electron scattered by a nucleus in a plane-wave field was studied by Lebedev (1972). Borisov et al. (1980) considered resonant SB, which accompanies collisions of ultrarelativistic electrons for the case of large transferred momenta. In the general relativistic case the problem of electron-nucleus SB in the field of a plane monochromatic wave was studied by Roshchupkin (1985). It should be noted that the theory of SB in presence of an external field is also developed in Lötstedt et al. (2007); Schnez et al. (2007). They contain important numeric calculations for the case of a strong field. Nonresonant SB in a pulsed field was considered by Lebed' & Roshchupkin (2009). Resonant SB of an electron scattered by a nucleus in the field of a pulsed light wave was studied in the general relativistic case (Lebed'

(a) (b)

Fig. 1. Feynman diagrams of electron-nucleus SB in the field of a pulsed light wave. Incoming and outgoing double lines correspond to the Volkov functions of an electron in initial and final states; inner lines designate the Green function of an electron in a pulsed field. Wavy lines correspond to four-momenta of spontaneous photon and "pseudophoton"

The process of electron-nucleus SB in a pulsed light field (2) in the Born approximation on interaction of an electron with a nucleus, which corresponds to the transition of an electron from an initial state with the four-momentum *pi* = (*Ei*, **p***i*) into a final state with the

, is described by two Feynman diagrams (Fig. 1).

*<sup>i</sup> pf p*

*f q*

*k*c

*Ze*

*q*

*k*c

*<sup>i</sup> pf p*

**2.1 Amplitude of resonant spontaneous bremsstrahlung**

 *Ef* , **p***<sup>f</sup>* 

*i q*

*<sup>a</sup>*0*b*<sup>0</sup> <sup>−</sup> **ab** will be used throughout this paper.

**the field of a pulsed light wave**

& Roshchupkin (2010)).

of nucleus recoil.

four-momentum *pf* =

*Ze*

*q*

$$\begin{split} S\_{fi} &= -ie^2 \int d^4 \mathbf{x}\_1 d^4 \mathbf{x}\_2 \vec{\psi}\_f \left( \mathbf{x}\_2 \, \middle| \, A \right) \left[ \tilde{\gamma}\_0 A\_0 \left( \left| \mathbf{x}\_2 \right| \right) \mathbb{G} \left( \mathbf{x}\_2 \mathbf{x}\_1 \, \middle| \, A \right) \hat{A}' \left( \mathbf{x}\_1 \, k' \right) + \\ &+ \hat{A}' \left( \mathbf{x}\_2 \, k' \right) \mathbb{G} \left( \mathbf{x}\_2 \mathbf{x}\_1 \, \middle| \, A \right) \tilde{\gamma}\_0 A\_0 \left( \left| \mathbf{x}\_1 \right| \right) \right] \psi\_i \left( \mathbf{x}\_1 \, \middle| \, A \right) . \end{split} \tag{6}$$

Here, *<sup>ψ</sup><sup>i</sup>* (*x*1|*A*) and *<sup>ψ</sup>*¯*<sup>f</sup>* (*x*2|*A*) are wave functions of an electron in initial and final states in the field (2), and *G* (*x*2*x*<sup>1</sup> |*A* ) is the Green function of an intermediate electron in the field of a pulsed light wave (2). Hereafter, expressions with hats above mean scalar products of corresponding four-vectors with the Dirac *γ*˜-matrices. In the amplitude (6) *A*<sup>0</sup> �� � � **x***j* � � � � is the Coulomb potential of a nucleus, and *A*� *μ* � *xj*, *k*� � is the four-potential of a spontaneously radiated photon. They have the following forms

$$A\_0 \left( \left| \mathbf{x}\_j \right| \right) = \frac{\mathbf{Z}e}{\left| \mathbf{x}\_j \right|},\tag{7}$$

$$A'\_{\mu} \left( \mathbf{x}\_{j}, k' \right) = \sqrt{\frac{2\pi}{\omega'}} \varepsilon\_{\mu}^{\*} \exp \left( ik' \mathbf{x}\_{j} \right), \quad j = 1, 2. \tag{8}$$

Here, *ε*∗ *<sup>μ</sup>* and *k*� = (*ω*� , **k**� ) are the polarization four-vector and the four-momentum of a spontaneous photon, *k*� *xj* = *ω*� *tj* − **k**� **x***j*.

The SB amplitude of an electron scattered by a nucleus in the field of a moderately strong pulsed wave (6) in the general relativistic case was derived early (Lebed' & Roshchupkin (2009)). This amplitude may be presented in the following form

$$\mathbf{S}\_{f\bar{1}} = \sum\_{l=-\infty}^{\infty} \mathbf{S}\_{l\nu} \tag{9}$$

where *Sl* is the process partial amplitude with emission or absorption of |*l*| laser-wave photons, that is

$$S\_{l} = -i\frac{Ze^{3}\sqrt{\pi}}{\sqrt{2\omega' E\_{f}E\_{l}}}\overline{u}\_{f}\left[B\_{li}\left(\tilde{\gamma}\_{0}, \mathfrak{E}^{\*}\right) + B\_{lf}\left(\mathfrak{E}^{\*}, \tilde{\gamma}\_{0}\right)\right]u\_{i}.\tag{10}$$

Here, the functions *Bli* (*γ*˜0,*ε*ˆ <sup>∗</sup>) and *Bl f* (*ε*ˆ ∗, *γ*˜0) correspond to the diagrams of electron-nucleus SB in Fig. 1; *ui*, *u*¯*<sup>f</sup>* are the Dirac bispinors.

Let us consider the diagram (a):

$$B\_{li}\left(\tilde{\gamma}\_{0},\mathcal{E}^{\*}\right) = \sum\_{r=-\infty}^{\infty} \frac{2\omega r \tau^{2}}{\mathbf{q}^{2} + q\_{0}\left(q\_{0} - 2q\_{z}\right)} \int\_{-\infty}^{\infty} d\tilde{\xi} \frac{\Lambda\_{l+r}\left(\tilde{\xi}\right) \left[\dot{q}\_{l} + m + \tilde{\xi}\hat{k}\right] \Lambda\_{-r}\left(\tilde{\xi}\right)}{q\_{l}^{2} - m^{2} + 2\xi\left(kq\_{l}\right) + i0},\tag{11}$$

where the four-vector *q* = (*q*0, **q**) is the transferred four-momentum, *qi* is the four-momentum of an intermediate electron for the diagram (a) (Fig. 1)

$$\begin{cases} q = p\_f - p\_i + k' + lk, \\ q\_l = p\_i - k' + rk\_\prime \\ q\_f = p\_f + k' + (l + r) \, k; \end{cases} \tag{12}$$

The form of integral functions (18) is considerably simplified for the case of a circular

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 113

= exp

It is obvious from Eqs. (11), (13) that the essential range of the integration variable *ξ* is

We emphasize, that dependence of the integrand denominator in Eq. (11) on the integration variable expresses consequence of accounting of the field pulsed character. The similar correction is absent in the monochromatic wave case, thus the resonant infinity in the

Fulfillment of the energy-momentum conservation law for components of a process of the second order caused a phenomenon when a virtual intermediate particle becomes real – that is, on-shell. Thus, the resonant divergence occurs in the process's amplitude. The energy-momentum conservation law for QED processes in a pulsed light field does not fulfill strictly. This peculiarity is inessential when nonresonant processes are studied. On the contrary, the energy-momentum nonconservation in the case of resonant SB of an electron

scattered by a nucleus in a pulsed light field results following resonance conditions

 *kqj* 

(it follows from consideration of Eqs. (11), (24)). Therefore, the four-momentum of an

It is convenient to set down expressions which determine *qi*, *<sup>f</sup>* and *q* (12) in following form for

,

,

*pi* + *rk* = *qi* + *k*�

*q <sup>f</sup>* + *rk* = *pf* + *k*�

These laws are fulfilled for only the values *r* > 0 under the conditions (25).

resonances occur only when these photons propagate nonparallel to each other.

Eqs. (26)-(27) represent the four-momentum conservation laws for the diagrams' vertices.

It is easy to ascertain that if a spontaneous photon propagates in the same direction as a photon of an external field, the conditions (25) cannot be satisfied simultaneously with the conservation laws (26) or (27) because the transit amplitude equals zero in this case. Therefore,

*q*2 *<sup>j</sup>* <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

intermediate electron occurs near the mass shell.

the both amplitudes (a) and (b) (Fig. 1), respectively

<sup>|</sup>*ξ*<sup>|</sup> <sup>1</sup> *ωτ* *irχqi pi* · *J*−*<sup>r</sup> γqi pi φ*�

. (23)

<sup>−</sup><sup>1</sup> the integrals in Eqs. (13)

∗, *γ*˜0) may be easily obtained from

� 1. (24)

*ωτ* , *<sup>j</sup>* <sup>=</sup> *<sup>i</sup>*, *<sup>f</sup>* . (25)

*<sup>q</sup>* <sup>=</sup> *pf* <sup>−</sup> *qi* <sup>+</sup> (*<sup>l</sup>* <sup>+</sup> *<sup>r</sup>*) *<sup>k</sup>*; (26)

*<sup>q</sup>* <sup>=</sup> *<sup>q</sup> <sup>f</sup>* <sup>−</sup> *pf* <sup>+</sup> (*<sup>l</sup>* <sup>+</sup> *<sup>r</sup>*) *<sup>k</sup>*. (27)

∗ will be performed.

polarization of an external light wave:

determined by the condition

**2.2 Resonance conditions**

*L*−*<sup>r</sup> χqi pi* , *γqi pi φ*� , 0

In view of quick oscillation of the integrand under |*ξ*| � (*ωτ*)

amplitude of SB of an electron scattered by a nucleus in a light field occurs.

are small. Note that the expression of the amplitude *Bl f* (*ε*ˆ

Eqs. (11), (13)-(21), if the replacements *qi* → *q <sup>f</sup>* , *γ*˜0 ↔ *ε*ˆ

*q <sup>f</sup>* is the four-momentum of an intermediate electron for the diagram (b) (Fig. 1). The integral functions Λ*l*+*r*, Λ−*<sup>r</sup>* are specified as

$$\begin{cases} \Lambda\_{l+r}\left(\vec{\xi}\right) = \tilde{\gamma}\_0 \int\_{-\infty}^{\infty} d\phi \cdot L\_{l+r}\left(\phi\right) \cdot \exp\left\{i q\_0 \tau \phi - i \left(\vec{\xi}\omega \tau\right) \phi\right\}, \\\ \Lambda\_{-r}\left(\vec{\xi}\right) = \int\_{-\infty}^{\infty} d\phi' \cdot F\_{-r}\left(\phi'\right) \cdot \exp\left\{i \left(\vec{\xi}\omega \tau\right) \phi'\right\}. \end{cases} \tag{13}$$

The integration variables in Eqs. (13):

$$
\phi = \frac{\varrho}{\omega \tau}, \quad \phi' = \frac{\varrho'}{\omega \tau}. \tag{14}
$$

The integral functions *F*−*<sup>r</sup>* (*φ*� ), *Ll*<sup>+</sup>*<sup>r</sup>* (*φ*) in Eqs. (13) are stepless depended on the integration variables (14), and are determined as

$$\begin{array}{l} F\_{-r}\left(\phi'\right) = \mathfrak{E}^\* \cdot L\_{-r}\left(\chi\_{q\_l p\_{l'}} \gamma\_{q\_l p\_i}\left(\phi'\right), \mathfrak{f}\_{q\_l p\_i}\left(\phi'\right)\right) + \\ + \left(\mathfrak{e}\_+ b\right) \cdot \mathfrak{g}\left(\phi'\right) L\_{-r+1}\left(\chi\_{q\_l p\_{l'}} \gamma\_{q\_l p\_i}\left(\phi'\right), \mathfrak{f}\_{q\_l p\_i}\left(\phi'\right)\right), \end{array} \tag{15}$$

where

$$b = \frac{1}{4} \eta\_0 m \left( \frac{\mathfrak{E}^\* \hat{k} \bar{\gamma}}{(kp\_{\bar{l}})} + \frac{\tilde{\gamma} \hat{k} \mathfrak{E}^\*}{(kq\_{\bar{l}})} \right), \tag{16}$$

$$e\_{+} = e\_{\chi} + i\delta e\_{y\prime} \tag{17}$$

$$\begin{split} & \mathcal{L}\_{-r} \left( \chi\_{q\_l p\_{l'}} \gamma\_{q\_l p\_{l}} \left( \phi' \right), \mathfrak{f}\_{q\_l p\_{l}} \left( \phi' \right) \right) = \\ & = \frac{1}{2\pi} \int d\varphi \exp \left\{ i \left[ \gamma\_{q\_l p\_l} \left( \phi' \right) \sin \left( \varphi - \chi\_{q\_l p\_l} \right) + \mathfrak{f}\_{q\_l p\_l} \left( \phi' \right) \sin 2\varphi + r\varphi \right] \right\}. \tag{18} \end{split} \tag{18}$$

The arguments of functions (18) are defined by the expressions

$$\tan \chi\_{q\_l p\_l} = \delta \frac{(e\_y Q\_{q\_l p\_l})}{(e\_x Q\_{q\_l p\_l})}, \quad Q\_{q\_l p\_l} = \frac{q\_{\bar{l}}}{(k q\_{\bar{l}})} - \frac{p\_{\bar{l}}}{(k p\_{\bar{l}})},\tag{19}$$

$$\gamma\_{q\_l p\_l} \left( \phi' \right) = \eta\_0 g \left( \phi' \right) \cdot m \sqrt{\left( e\_x Q\_{q\_l p\_l} \right)^2 + \delta^2 \left( e\_y Q\_{q\_l p\_l} \right)^2} \,\tag{20}$$

$$\beta\_{q\_l p\_l} \left( \phi' \right) = \frac{1}{8} \left( 1 - \delta^2 \right) \eta\_0^2 g^2 \left( \phi' \right) m^2 \left[ \frac{1}{(k q\_l)} - \frac{1}{(k p\_l)} \right]. \tag{21}$$

Expressions for integral functions *Ll*<sup>+</sup>*<sup>r</sup>* (*φ*) ≡ *Ll*<sup>+</sup>*<sup>r</sup>* � *χpf qi* , *γpf qi* (*φ*), *βpf qi* (*φ*) � may be easily obtained from the appropriate expressions (18)-(21) after following replacements of indices and four-momenta: −*r* → *l* + *r*, *qi* → *pf* , *pi* → *qi*.

Functions *Ln* (*χ*, *γ*, *β*) determine probabilities of multiphoton processes produced by the presence of a strong external field. Note that properties of these functions were studied by Roshchupkin et al. (2000) in detail. Thus, they may be represented as series in integer-order Bessel functions, i.e.

$$L\_{\mathfrak{n}}\left(\chi,\gamma,\beta\right) = \exp\left(-i n \chi\right) \sum\_{s=-\infty}^{\infty} \exp\left(2i s \chi\right) \cdot \mathfrak{J}\_{\mathfrak{n}-2s}\left(\gamma\right) \cdot \mathfrak{J}\_{s}\left(\beta\right). \tag{22}$$

The form of integral functions (18) is considerably simplified for the case of a circular polarization of an external light wave:

$$L\_{-r}\left(\chi\_{q\_i p\_i}, \gamma\_{q\_i p\_i}\left(\boldsymbol{\phi}'\right), 0\right) = \exp\left(i r \chi\_{q\_i p\_i}\right) \cdot I\_{-r}\left(\gamma\_{q\_i p\_i}\left(\boldsymbol{\phi}'\right)\right). \tag{23}$$

It is obvious from Eqs. (11), (13) that the essential range of the integration variable *ξ* is determined by the condition

$$|\xi| \lesssim \frac{1}{\omega \tau} \ll 1. \tag{24}$$

In view of quick oscillation of the integrand under |*ξ*| � (*ωτ*) <sup>−</sup><sup>1</sup> the integrals in Eqs. (13) are small. Note that the expression of the amplitude *Bl f* (*ε*ˆ ∗, *γ*˜0) may be easily obtained from Eqs. (11), (13)-(21), if the replacements *qi* → *q <sup>f</sup>* , *γ*˜0 ↔ *ε*ˆ ∗ will be performed.

We emphasize, that dependence of the integrand denominator in Eq. (11) on the integration variable expresses consequence of accounting of the field pulsed character. The similar correction is absent in the monochromatic wave case, thus the resonant infinity in the amplitude of SB of an electron scattered by a nucleus in a light field occurs.

#### **2.2 Resonance conditions**

6 Will-be-set-by-IN-TECH

*q <sup>f</sup>* is the four-momentum of an intermediate electron for the diagram (b) (Fig. 1). The integral

*ωτ* , *<sup>φ</sup>*� <sup>=</sup> *<sup>ϕ</sup>*�

, *γqi pi* (*φ*�

*ϕ* − *χqi pi*

� , *Qqi pi* <sup>=</sup> *qi*

*exQqi pi*

� *χpf qi*

obtained from the appropriate expressions (18)-(21) after following replacements of indices

Functions *Ln* (*χ*, *γ*, *β*) determine probabilities of multiphoton processes produced by the presence of a strong external field. Note that properties of these functions were studied by Roshchupkin et al. (2000) in detail. Thus, they may be represented as series in integer-order

> ∞ ∑ *s*=−∞

, *γqi pi* (*φ*�

*kε*ˆ ∗ (*kqi*)

> � + *βqi pi* � *φ*� �

*dφ* · *Ll*<sup>+</sup>*<sup>r</sup>* (*φ*) · exp {*iq*0*τφ* − *i*(*ξωτ*) *φ*},

}.

), *Ll*<sup>+</sup>*<sup>r</sup>* (*φ*) in Eqs. (13) are stepless depended on the integration

), *βqi pi* (*φ*�

�

) � +

*e*+ = *ex* + *iδey*, (17)

) �

), *βqi pi* (*φ*�

(*kqi*) <sup>−</sup> *pi*

(*kqi*) <sup>−</sup> <sup>1</sup>

�<sup>2</sup> + *<sup>δ</sup>*<sup>2</sup> �

(*kpi*)

*eyQqi pi* �2

(*kpi*)

, *γpf qi* (*φ*), *βpf qi* (*φ*)

�

exp (2*isχ*) · *Jn*−2*<sup>s</sup>* (*γ*) · *Js* (*β*). (22)

*ωτ* . (14)

, (15)

��. (18)

, (19)

, (20)

. (21)

may be easily

�

, (16)

sin 2*ϕ* + *rϕ*

(13)

) · exp {*i*(*ξωτ*) *φ*�

functions Λ*l*+*r*, Λ−*<sup>r</sup>* are specified as ⎧ ⎪⎪⎨

⎪⎪⎩

The integration variables in Eqs. (13):

variables (14), and are determined as

The integral functions *F*−*<sup>r</sup>* (*φ*�

*L*−*<sup>r</sup>* � *χqi pi*

<sup>=</sup> <sup>1</sup> 2*π* � 2*π*

Bessel functions, i.e.

0

where

Λ*l*+*<sup>r</sup>* (*ξ*) = *γ*˜0

<sup>Λ</sup>−*<sup>r</sup>* (*ξ*) <sup>=</sup> �

*F*−*<sup>r</sup>* (*φ*�

, *γqi pi* (*φ*�

*dϕ* exp � *i* � *γqi pi* � *φ*� � sin �

*γqi pi* � *φ*� � = *η*0*g* � *φ*� � · *m* ��

*βqi pi* � *φ*� � <sup>=</sup> <sup>1</sup> 8 � <sup>1</sup> <sup>−</sup> *<sup>δ</sup>*<sup>2</sup> � *η*2 <sup>0</sup> *<sup>g</sup>*<sup>2</sup> � *φ*� � *m*2 � 1

Expressions for integral functions *Ll*<sup>+</sup>*<sup>r</sup>* (*φ*) ≡ *Ll*<sup>+</sup>*<sup>r</sup>*

and four-momenta: −*r* → *l* + *r*, *qi* → *pf* , *pi* → *qi*.

*Ln* (*χ*, *γ*, *β*) = exp (−*inχ*)

� ∞

−∞

*dφ*� · *F*−*<sup>r</sup>* (*φ*�

*<sup>φ</sup>* <sup>=</sup> *<sup>ϕ</sup>*

∞

−∞

) = *ε*ˆ

+ (*e*+*b*) · *g* (*φ*�

<sup>∗</sup> · *L*−*<sup>r</sup>* � *χqi pi*

*<sup>b</sup>* <sup>=</sup> <sup>1</sup> 4 *η*0*m* � *ε*ˆ ∗ ˆ *kγ*˜ (*kpi*) <sup>+</sup> *<sup>γ</sup>*˜ <sup>ˆ</sup>

), *βqi pi* (*φ*�

� *eyQqi pi* �

� *exQqi pi*

The arguments of functions (18) are defined by the expressions

tan *χqi pi* = *δ*

) *L*−*r*+<sup>1</sup>

) � =

� *χqi pi*

> Fulfillment of the energy-momentum conservation law for components of a process of the second order caused a phenomenon when a virtual intermediate particle becomes real – that is, on-shell. Thus, the resonant divergence occurs in the process's amplitude. The energy-momentum conservation law for QED processes in a pulsed light field does not fulfill strictly. This peculiarity is inessential when nonresonant processes are studied. On the contrary, the energy-momentum nonconservation in the case of resonant SB of an electron scattered by a nucleus in a pulsed light field results following resonance conditions

$$
\alpha\_j^2 - m^2 \lesssim \frac{\left(k\eta\_j\right)}{\omega \tau}, \quad j = i, f. \tag{25}
$$

(it follows from consideration of Eqs. (11), (24)). Therefore, the four-momentum of an intermediate electron occurs near the mass shell.

It is convenient to set down expressions which determine *qi*, *<sup>f</sup>* and *q* (12) in following form for the both amplitudes (a) and (b) (Fig. 1), respectively

$$\begin{cases} p\_i + rk = q\_i + k', \\ q = p\_f - q\_i + (l+r)k; \end{cases} \tag{26}$$

$$\begin{cases} q\_f + rk = p\_f + k', \\ q = q\_f - p\_f + (l+r)k. \end{cases} \tag{27}$$

Eqs. (26)-(27) represent the four-momentum conservation laws for the diagrams' vertices. These laws are fulfilled for only the values *r* > 0 under the conditions (25).

It is easy to ascertain that if a spontaneous photon propagates in the same direction as a photon of an external field, the conditions (25) cannot be satisfied simultaneously with the conservation laws (26) or (27) because the transit amplitude equals zero in this case. Therefore, resonances occur only when these photons propagate nonparallel to each other.

*f p*

Fig. 2. Resonant electron-nucleus SB in the field of a pulsed light wave.

*i p*

. (32)

*<sup>m</sup>*<sup>2</sup> . (33)

is of the order of magnitude:

*<sup>f</sup>* . Within the field range specified

. (35)

*EiEf*

∼ *η*<sup>0</sup> � 1. (34)

2 2 *<sup>i</sup> q m*|

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 115

The difference for the other diagram (Fig. 1(b)) is concluded in the both replacement of the intermediate electron four-momentum (*qi* → *q <sup>f</sup>*) and interchange of sequence of first order processes. Thus, an electron is scattered by a nucleus with absorption or emission of *r* wave photons, and then it spontaneously emits a photon with the four-momentum *k*� with |*l* + *r*|

As it was pointed above, the integral functions (18) are determined by the integer-order Bessel functions (23) for the case of a circularly polarizated external wave. It is not difficult to verify that for given type polarization under the resonance conditions the arguments of the Bessel

> *φ*� · *u ur* · <sup>1</sup> <sup>−</sup> *<sup>u</sup> ur*

, *ur* = 2*r* ·

Consequently, within the range of fields specified by Eq. (5) the first resonance, that is, the resonance with *r* = 1, provides the main contribution to the resonant cross section, when the Bessel function has the largest value. This implies that the Compton scattering of a light wave by an initial electron is mainly due to absorption of one photon of an external field.

*γpf qi* (*φ*) ∼ *γ*<sup>0</sup> 1. Thus, scattering of an intermediate electron by a nucleus in a pulsed wave

Interference of the resonant amplitudes (which correspond to direct and exchange diagrams)

by Eq. (5) the condition of interference between direct and exchange resonant amplitudes is

**n**� × **n** = **n**� **n** · *rω* · *κ<sup>i</sup>* + *κ <sup>f</sup>* 

*γpf qi* (*φ*)

*<sup>i</sup>* = *ω*�

(*kpi*)

= 2*r* · *η*0*g*

) (*kqi*)

*<sup>u</sup>* <sup>=</sup> (*kk*�

*γqi pi φ*�  *k*c

*Ze*

*q*

wave photons absorption.

Equations (32)-(33) imply

written as:

 **v***<sup>f</sup>* − **v***<sup>i</sup>*

functions (20) may be represented as

*γqi pi φ*� 

Here, *u*, *ur* are the relativistic invariant parameters

However, the argument of the Bessel function *Jl*<sup>+</sup>*<sup>r</sup>*

implies the equality of their resonant frequencies, i.e. *ω*�

 **n** − **n**� + **v***<sup>f</sup>* × **v***<sup>i</sup>*

field under these conditions is generally a multiphoton process.

Taking Eq. (25) into account, we can use Eqs. (26), (27) for a moderately strong field (5) to find the frequency of a spontaneous photon in the resonance (the resonant frequency) for the both direct and exchange amplitudes (Figs. 1(a) and 1(b), respectively). Within zeroth order with respect to the small parameter (*ωτ*) <sup>−</sup><sup>1</sup> the resonant frequency is specified:

$$
\omega\_{\rm res}' \equiv \omega\_{\rm j}' = r\omega\_{\rm j} \frac{1}{1 \pm d\_{\rm j}}, \quad j = i, f. \tag{28}
$$

where the signs "+" and "–" correspond to index values *i* and *f* , respectively,

$$d\omega\_{\dot{j}} = \omega \cdot \frac{\kappa\_{\dot{j}}}{\kappa\_{\dot{j}}'} \quad d\_{\dot{j}} = r \, (nn') \cdot \frac{\omega}{\kappa\_{\dot{j}}'} \, \tag{29}$$

$$\mathbf{x}\_{\circ} = E\_{\circ} - \mathbf{n} \mathbf{p}\_{\circ}, \quad \mathbf{x}\_{\circ}^{\prime} = E\_{\circ} - \mathbf{n}^{\prime} \mathbf{p}\_{\circ}. \tag{30}$$

$$
\hbar \mathfrak{n} = \frac{k}{\omega} = (1, \mathfrak{n}) , \quad \mathfrak{n}' = \frac{k'}{\omega} = \left(1, \mathfrak{n}'\right) . \tag{31}
$$

It is obvious from Eq. (29), that within a rather broad range of electron energies and scattering angles we have *dj* � 1 (except an ultrarelativistic electron with the energy <sup>∼</sup> *<sup>m</sup>*2/*ω*, moving within a narrow cone close to the direction of the momentum of a spontaneous photon). Therefore, resonances are mainly observed when the frequency of a spontaneous photon is multiple to *ω<sup>j</sup>* (29).

Eqs. (28)-(31) for the resonant frequency imply that we may separate four characteristic domains of the frequency *ωj*: the nonrelativistic case, *ω<sup>j</sup>* ∼= *ω*; the limiting case of ultrarelativistic energies, when an electron moves within a narrow cone related to a photon of an external field *ω<sup>j</sup>* � *ω*; an ultrarelativistic electron moves within a narrow cone with a spontaneous photon, *ω<sup>j</sup>* � *ω*; otherwise, *ω<sup>j</sup>* ∼ *ω*. Here, we consider resonant frequencies in detail.

The four-momentum conservation law (26) and the function *F*−*<sup>r</sup>* explicit form (15) result that this function represents the amplitude of such process: an electron with the four-momentum *pi* absorbs *r* photons of the external wave and emits a photon with four-momentum *k*� . This process was considered by Nikishov & Ritus (1979) in the case of a plane monochromatic wave, and by Narozhniy & Fofanov (1996) in the case of a pulsed light wave. The expression for the transferred four-momentum *q* (see the second equality in Eq. (26)) shows that the quantity *Ll*<sup>+</sup>*<sup>r</sup> χpf qi* , *γpf qi* (*φ*), *βpf qi* (*φ*) defines the amplitude of scattering of an intermediate electron with the four-momentum *qi* by a nucleus in the field of a light wave with absorption or emission of |*l* + *r*| wave photons. In the nonrelativistic limiting case this process was studied by Bunkin & Fedorov (1966). Denisov & Fedorov (1967) considered this process in the general relativistic case. The process when an electron scattered by a nucleus in a pulsed light wave was studied by Lebed' & Roshchupkin (2008).

Consequently, if the interference between the direct and the exchange amplitudes is absent, the process of resonant electron-nucleus SB in the field of a light wave effectively decomposes into two consecutive processes of the first order: emission of a photon with the four-momentum *k*� by an electron in a pulsed light wave and scattering of an electron by a nucleus in a pulsed light wave (see Fig. 2).

Fig. 2. Resonant electron-nucleus SB in the field of a pulsed light wave.

The difference for the other diagram (Fig. 1(b)) is concluded in the both replacement of the intermediate electron four-momentum (*qi* → *q <sup>f</sup>*) and interchange of sequence of first order processes. Thus, an electron is scattered by a nucleus with absorption or emission of *r* wave photons, and then it spontaneously emits a photon with the four-momentum *k*� with |*l* + *r*| wave photons absorption.

As it was pointed above, the integral functions (18) are determined by the integer-order Bessel functions (23) for the case of a circularly polarizated external wave. It is not difficult to verify that for given type polarization under the resonance conditions the arguments of the Bessel functions (20) may be represented as

$$\gamma\_{q\_l p\_l} \left( \phi' \right) = 2r \cdot \eta\_0 g \left( \phi' \right) \cdot \sqrt{\frac{\mu}{\mu\_r} \cdot \left( 1 - \frac{\mu}{\mu\_r} \right)}. \tag{32}$$

Here, *u*, *ur* are the relativistic invariant parameters

$$
\mu = \frac{(kk')}{(kq\_i)}, \quad \mu\_r = 2r \cdot \frac{(kp\_i)}{m^2}.\tag{33}
$$

Equations (32)-(33) imply

8 Will-be-set-by-IN-TECH

Taking Eq. (25) into account, we can use Eqs. (26), (27) for a moderately strong field (5) to find the frequency of a spontaneous photon in the resonance (the resonant frequency) for the both direct and exchange amplitudes (Figs. 1(a) and 1(b), respectively). Within zeroth order with

> 1 1 ± *dj*

> > *nn*� · *ω κ*� *j*

*<sup>j</sup>* = *Ej* − **n**�

**p***j*

1, **n**� 

*<sup>ω</sup>* <sup>=</sup>

*<sup>j</sup>* = *rω<sup>j</sup>*

, *dj* = *r*

, *κ*�

*<sup>ω</sup>* <sup>=</sup> (1, **<sup>n</sup>**), *<sup>n</sup>*� <sup>=</sup> *<sup>k</sup>*�

It is obvious from Eq. (29), that within a rather broad range of electron energies and scattering angles we have *dj* � 1 (except an ultrarelativistic electron with the energy <sup>∼</sup> *<sup>m</sup>*2/*ω*, moving within a narrow cone close to the direction of the momentum of a spontaneous photon). Therefore, resonances are mainly observed when the frequency of a spontaneous photon is

Eqs. (28)-(31) for the resonant frequency imply that we may separate four characteristic domains of the frequency *ωj*: the nonrelativistic case, *ω<sup>j</sup>* ∼= *ω*; the limiting case of ultrarelativistic energies, when an electron moves within a narrow cone related to a photon of an external field *ω<sup>j</sup>* � *ω*; an ultrarelativistic electron moves within a narrow cone with a spontaneous photon, *ω<sup>j</sup>* � *ω*; otherwise, *ω<sup>j</sup>* ∼ *ω*. Here, we consider resonant frequencies in

The four-momentum conservation law (26) and the function *F*−*<sup>r</sup>* explicit form (15) result that this function represents the amplitude of such process: an electron with the four-momentum *pi* absorbs *r* photons of the external wave and emits a photon with four-momentum *k*�

process was considered by Nikishov & Ritus (1979) in the case of a plane monochromatic wave, and by Narozhniy & Fofanov (1996) in the case of a pulsed light wave. The expression for the transferred four-momentum *q* (see the second equality in Eq. (26)) shows

intermediate electron with the four-momentum *qi* by a nucleus in the field of a light wave with absorption or emission of |*l* + *r*| wave photons. In the nonrelativistic limiting case this process was studied by Bunkin & Fedorov (1966). Denisov & Fedorov (1967) considered this process in the general relativistic case. The process when an electron scattered by a nucleus in

Consequently, if the interference between the direct and the exchange amplitudes is absent, the process of resonant electron-nucleus SB in the field of a light wave effectively decomposes into two consecutive processes of the first order: emission of a photon with the four-momentum *k*� by an electron in a pulsed light wave and scattering of an electron by a

, *γpf qi* (*φ*), *βpf qi* (*φ*)

a pulsed light wave was studied by Lebed' & Roshchupkin (2008).

where the signs "+" and "–" correspond to index values *i* and *f* , respectively,

*κj κ*� *j*

<sup>−</sup><sup>1</sup> the resonant frequency is specified:

, *j* = *i*, *f* , (28)

, (29)

, (30)

defines the amplitude of scattering of an

. (31)

. This

respect to the small parameter (*ωτ*)

multiple to *ω<sup>j</sup>* (29).

that the quantity *Ll*<sup>+</sup>*<sup>r</sup>*

 *χpf qi*

nucleus in a pulsed light wave (see Fig. 2).

detail.

*ω*� *res* ≡ *ω*�

*ω<sup>j</sup>* = *ω* ·

*<sup>n</sup>* <sup>=</sup> *<sup>k</sup>*

*κ<sup>j</sup>* = *Ej* − **np***<sup>j</sup>*

$$
\gamma\_{q\_l p\_i} \left( \phi' \right) \sim \eta\_0 \ll 1. \tag{34}
$$

Consequently, within the range of fields specified by Eq. (5) the first resonance, that is, the resonance with *r* = 1, provides the main contribution to the resonant cross section, when the Bessel function has the largest value. This implies that the Compton scattering of a light wave by an initial electron is mainly due to absorption of one photon of an external field. However, the argument of the Bessel function *Jl*<sup>+</sup>*<sup>r</sup> γpf qi* (*φ*) is of the order of magnitude: *γpf qi* (*φ*) ∼ *γ*<sup>0</sup> 1. Thus, scattering of an intermediate electron by a nucleus in a pulsed wave field under these conditions is generally a multiphoton process.

Interference of the resonant amplitudes (which correspond to direct and exchange diagrams) implies the equality of their resonant frequencies, i.e. *ω*� *<sup>i</sup>* = *ω*� *<sup>f</sup>* . Within the field range specified by Eq. (5) the condition of interference between direct and exchange resonant amplitudes is written as:

$$\left(\mathbf{v}\_f - \mathbf{v}\_i\right)\left(\mathbf{n} - \mathbf{n}'\right) + \left(\mathbf{v}\_f \times \mathbf{v}\_i\right)\left(\mathbf{n}' \times \mathbf{n}\right) = \left(\mathbf{n}'\mathbf{n}\right) \cdot \frac{r\omega \cdot \left(\mathbf{x}\_i + \mathbf{x}\_f\right)}{E\_i E\_f}.\tag{35}$$

*<sup>F</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

∞

−∞

**2.4 Cross-section of spontaneous bremsstrahlung**

Here, the function erf(2*φ* + *iβ*/2) is the error function.

Berestetskii et al. (1982)) for the spontaneous photon frequency (37):

*dwl* <sup>=</sup> *<sup>Z</sup>*2*e*6*<sup>π</sup>* 2*ω*�*Ef Ei*

> *θ* = ∠ **p***i*,**p***<sup>f</sup>* �

of scattered particles *vi* = |**p***i*| /*Ei*. Thus, we derive

*dσ<sup>l</sup> dω*�*d*Ω�*d*Ω*<sup>f</sup>*

interval *dω*� within the solid angle *d*Ω�

× *u*¯*<sup>f</sup> Miui* 2 · *τ T*

*I* (*q*0, *β*) = *τ*

from the consideration:

<sup>2</sup> exp{*iχqi pi* } · *<sup>γ</sup>qi pi* (0) · *<sup>ε</sup>*<sup>ˆ</sup>

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 117

Let us calculate the differential probability during the entire time of electron-nucleus SB in a pulsed light field from the amplitude, Eqs. (9)-(10), (41)-(43) in standard manner (see

> ∞ ∑ *l*=−∞

*dφ* · *Jl*<sup>+</sup><sup>1</sup> (*φ*) exp {*i*(*q*0*τ* + 2*β*) *φ*}

*dw* =

· *u*¯*<sup>f</sup> Bliui* 2 ·

Here, *T* is some comparatively large (*T* � *τ*) interval of the observation time. The energy-momentum conservation law for SB of an electron scattered by a nucleus in a pulsed light field does not fulfill strictly, however, under the condition (1) the essential range of integration is converged. Energies of a final electron are negligibly differ from the values, which are specified by the strict energy conservation law. We exclude small scattering angles

The differential cross section of SB of an electron scattered by a nucleus in the field of a pulsed light wave is obtained by means of division of the probability per unit time per a flux density

> ∞ ∑ *l*=−∞

where *dσ<sup>l</sup>* is the partial cross section of a process with a spontaneous photon in the frequency

 **p***f* |**p***i*|

> erf

*dσ* =

emission (*l* > 0) or absorption (*l* < 0) of wave photons. It may be written in the form

*π* (*ωτ*) 2

*dφ* · *J* 2 *<sup>l</sup>*+<sup>1</sup> (*φ*)

<sup>=</sup> *<sup>Z</sup>*2*e*6*ω*�

*T* /2*τ*

−*T*/2*τ*

(2*π*) <sup>2</sup> **q**<sup>4</sup> *ω*

∗ + (*e*+*b*). (42)

. (43)

2*φ* + *iβ* 2 + 1 

*dwl*, (44)

<sup>|</sup>**p***i*|(*ωτ*) � 1. (46)

*dσl*, (47)

(48)

, and a final electron within the solid angle *d*Ω*<sup>f</sup>* with

2 ×

*Mi* = *γ*˜0 (*q*ˆ*<sup>i</sup>* + *m*) *F*ˆ. (49)

exp{−*β*2/2} 64 (*kqi*)

> 2*φ* + *iβ* 2 + 1 2 ,

<sup>6</sup> . (45)

 erf

*d*<sup>3</sup> *pf d*3*k*� *T* (2*π*)

Here, **v***<sup>j</sup>* = **p***j*/*Ej* is the electron velocity before (*j* = *i*) and after (*j* = *f*) scattering. The quantity involved in the right-hand side of Eq. (35) is small compared with the unity. Therefore, this equality is satisfied when directions of motion of photons (a spontaneous photon and a photon of an external field) or electrons (before and after scattering) are close to each other. It follows from Eq. (35) and from the fact that resonances vanish, when direction of spontaneous photon motion is close to direction of external field photon motion, that resonant amplitudes, which correspond to the processes shown on Figs. 1(a) and 1(b), interfere when an electron is scattered on the small angles, i.e.

$$\theta = \angle \left( \mathbf{v}\_{i,} \mathbf{v}\_{f} \right) \sim \left( 1 - \mathbf{n} \mathbf{v}\_{i} \right) \cdot \left( \omega / \left| \mathbf{v}\_{i} \right| \to\_{i} \right) \ll 1. \tag{36}$$

Hereinafter, we consider the resonance of one diagram. We assume that the spontaneous photon frequency is equal

$$
\omega' \approx \omega\_{\rm res}' = \omega\_{\rm i}'.\tag{37}
$$

#### **2.3 Amplitude integration**

Let us study the process of resonant SB of an electron scattered by a nucleus in a pulsed light field at the expense of only one photon absorption, i.e. *r* = 1. The condition (24) allows to simplify the integration in Eq. (11)

$$\int\_{-\infty}^{\infty} d\xi \frac{\exp\left\{i\xi\omega\tau\left(\phi'-\phi\right)\right\}}{q\_i^2 - m^2 + 2\xi\left(kq\_i\right) + i0} = \frac{\exp\left\{-2i\beta\left(\phi'-\phi\right)\right\}}{2\left(kq\_i\right)} i\pi\left(\text{sgn}\left(\phi'-\phi\right) - 1\right). \tag{38}$$

Eq. (38) contains the relevant parameter, which determines resonant electron-nucleus SB in the field of a pulsed light wave:

$$
\beta = \frac{q\_i^2 - m^2}{4\left(kq\_i\right)} \omega \tau.\tag{39}
$$

As it can be seen from Eq. (39), values of the parameter *β* are defined by process kinematics and external pulsed-wave properties. This parameter specifies how closely the four-momentum of an intermediate electron coincides with the value on the mass shell under resonance conditions for electron-nucleus SB in the field of a pulsed light wave.

The subsequent analysis will be performed for the particular form of the envelope function of the pulsed light wave four-potential. We choose the Gaussian function:

$$\lg\left(\frac{\varrho}{\omega \tau}\right) = \exp\left\{-\left(\frac{2\varrho}{\omega \tau}\right)^2\right\} = \exp\left\{-\left(2\phi\right)^2\right\}.\tag{40}$$

Under the condition (34) the function *F*−*<sup>r</sup>* (*φ*� ) (15) in the amplitude may be expanded in the Taylor series. We may keep only linear terms with respect to the parameter *η*0. For the envelope function (40), after simple computation we obtain the amplitude of resonant SB of an electron scattered by a nucleus in a pulsed light field:

$$\mathcal{B}\_{li}\left(\tilde{\gamma}\_{0},\hat{\varepsilon}^{\*}\right) = \frac{2\pi \cdot \tilde{\gamma}\_{0}\left(\mathfrak{d}\_{\hat{l}} + m\right)\hat{\mathbf{f}}}{\mathbf{q}^{2} + q\_{0}\left(q\_{0} - 2q\_{z}\right)} \cdot \frac{-i\omega\tau\sqrt{\pi}}{4\left(kq\_{\hat{l}}\right)} \exp\{-\frac{\beta^{2}}{4}\} \cdot I\left(q\_{0},\beta\right),\tag{41}$$

$$\hat{F} = -\frac{1}{2} \exp\{i\chi\_{q\_i p\_i}\} \cdot \gamma\_{q\_i p\_i}(0) \cdot \mathcal{E}^\* + (\varepsilon\_+ b) \,. \tag{42}$$

$$I(q\_0, \boldsymbol{\beta}) = \text{tr} \int\_{-\infty}^{\infty} d\boldsymbol{\phi} \cdot \boldsymbol{J}\_{l+1}(\boldsymbol{\phi}) \exp\left\{i\left(q\_0 \boldsymbol{\tau} + 2\boldsymbol{\beta}\right)\boldsymbol{\phi}\right\} \left(\text{erf}\left(2\boldsymbol{\phi} + \frac{i\boldsymbol{\beta}}{2}\right) + 1\right). \tag{43}$$

Here, the function erf(2*φ* + *iβ*/2) is the error function.

#### **2.4 Cross-section of spontaneous bremsstrahlung**

10 Will-be-set-by-IN-TECH

Here, **v***<sup>j</sup>* = **p***j*/*Ej* is the electron velocity before (*j* = *i*) and after (*j* = *f*) scattering. The quantity involved in the right-hand side of Eq. (35) is small compared with the unity. Therefore, this equality is satisfied when directions of motion of photons (a spontaneous photon and a photon of an external field) or electrons (before and after scattering) are close to each other. It follows from Eq. (35) and from the fact that resonances vanish, when direction of spontaneous photon motion is close to direction of external field photon motion, that resonant amplitudes, which correspond to the processes shown on Figs. 1(a) and 1(b), interfere when

Hereinafter, we consider the resonance of one diagram. We assume that the spontaneous

Let us study the process of resonant SB of an electron scattered by a nucleus in a pulsed light field at the expense of only one photon absorption, i.e. *r* = 1. The condition (24) allows to

Eq. (38) contains the relevant parameter, which determines resonant electron-nucleus SB in

*<sup>i</sup>* <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

As it can be seen from Eq. (39), values of the parameter *β* are defined by process kinematics and external pulsed-wave properties. This parameter specifies how closely the four-momentum of an intermediate electron coincides with the value on the mass shell under

The subsequent analysis will be performed for the particular form of the envelope function of

the Taylor series. We may keep only linear terms with respect to the parameter *η*0. For the envelope function (40), after simple computation we obtain the amplitude of resonant SB of

<sup>2</sup> 

= exp − (2*φ*) 2 

<sup>−</sup>*iωτ*√*<sup>π</sup>*

<sup>4</sup> (*kqi*) exp{− *<sup>β</sup>*<sup>2</sup>

*res* = *ω*�

<sup>2</sup> (*kqi*) *<sup>i</sup><sup>π</sup>*

*ω*� ≈ *ω*�

*<sup>i</sup>* <sup>−</sup> *<sup>m</sup>*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>ξ</sup>* (*kqi*) <sup>+</sup> *<sup>i</sup>*<sup>0</sup> <sup>=</sup> exp {−2*i<sup>β</sup>* (*φ*� <sup>−</sup> *<sup>φ</sup>*)}

*<sup>β</sup>* <sup>=</sup> *<sup>q</sup>*<sup>2</sup>

resonance conditions for electron-nucleus SB in the field of a pulsed light wave.

 − 2*ϕ ωτ*

the pulsed light wave four-potential. We choose the Gaussian function:

∼ (1 − **nv***i*) · (*ω*/|**v***i*| *Ei*) � 1. (36)

*sgn*

<sup>4</sup> (*kqi*) *ωτ*. (39)

) (15) in the amplitude may be expanded in

*<sup>i</sup>*. (37)

*φ*� − *φ* − 1 

. (38)

. (40)

<sup>4</sup> } · *<sup>I</sup>* (*q*0, *<sup>β</sup>*), (41)

an electron is scattered on the small angles, i.e.

photon frequency is equal

**2.3 Amplitude integration**

∞

−∞

simplify the integration in Eq. (11)

*q*2

the field of a pulsed light wave:

*<sup>d</sup><sup>ξ</sup>* exp {*iξωτ* (*φ*� <sup>−</sup> *<sup>φ</sup>*)}

*g ϕ ωτ* = exp

Under the condition (34) the function *F*−*<sup>r</sup>* (*φ*�

*Bli* (*γ*˜0,*ε*ˆ

an electron scattered by a nucleus in a pulsed light field:

<sup>∗</sup>) <sup>=</sup> <sup>2</sup>*<sup>π</sup>* · *<sup>γ</sup>*˜0 (*q*ˆ*<sup>i</sup>* <sup>+</sup> *<sup>m</sup>*) *<sup>F</sup>*<sup>ˆ</sup> **<sup>q</sup>**<sup>2</sup> <sup>+</sup> *<sup>q</sup>*<sup>0</sup> (*q*<sup>0</sup> <sup>−</sup> <sup>2</sup>*qz*) ·

*θ* = ∠ **v***i*,**v***<sup>f</sup>* 

Let us calculate the differential probability during the entire time of electron-nucleus SB in a pulsed light field from the amplitude, Eqs. (9)-(10), (41)-(43) in standard manner (see Berestetskii et al. (1982)) for the spontaneous photon frequency (37):

$$dw = \sum\_{l=-\infty}^{\infty} dw\_{l\prime} \tag{44}$$

$$dw\_{l} = \frac{Z^{2}e^{6}\pi}{2\omega' \mathbf{E}\_{f}\mathbf{E}\_{l}} \cdot \left| \vec{u}\_{f} \mathbf{B}\_{li} u\_{l} \right|^{2} \cdot \frac{d^{3}p\_{f} d^{3}k'}{T\left(2\pi\right)^{6}}.\tag{45}$$

Here, *T* is some comparatively large (*T* � *τ*) interval of the observation time. The energy-momentum conservation law for SB of an electron scattered by a nucleus in a pulsed light field does not fulfill strictly, however, under the condition (1) the essential range of integration is converged. Energies of a final electron are negligibly differ from the values, which are specified by the strict energy conservation law. We exclude small scattering angles from the consideration:

$$\theta = \angle \left( \mathbf{p}\_i \mathbf{p}\_f \right) \gg \sqrt{\frac{\omega}{|\mathbf{p}\_i| \left( \omega \tau \right)}} \ll 1. \tag{46}$$

The differential cross section of SB of an electron scattered by a nucleus in the field of a pulsed light wave is obtained by means of division of the probability per unit time per a flux density of scattered particles *vi* = |**p***i*| /*Ei*. Thus, we derive

$$d\sigma = \sum\_{l=-\infty}^{\infty} d\sigma\_{l\prime} \tag{47}$$

where *dσ<sup>l</sup>* is the partial cross section of a process with a spontaneous photon in the frequency interval *dω*� within the solid angle *d*Ω� , and a final electron within the solid angle *d*Ω*<sup>f</sup>* with emission (*l* > 0) or absorption (*l* < 0) of wave photons. It may be written in the form

$$\begin{split} \frac{d\boldsymbol{\sigma}\_{l}}{d\boldsymbol{\omega}^{\prime}d\Omega^{\prime}d\Omega\_{f}} &= \frac{Z^{2}e^{6}\boldsymbol{\omega}^{\prime}\boldsymbol{\pi}\left(\boldsymbol{\omega}\boldsymbol{\pi}\right)^{2}}{\left(2\pi\right)^{2}\mathbf{q}^{4}} \frac{\left|\mathbf{P}\_{f}\right|}{\left|\mathbf{p}\_{i}\right|} \frac{\exp\{-\boldsymbol{\beta}^{2}/2\}}{64\left(kq\_{l}\right)^{2}} \times\\ \times \left|\boldsymbol{\bar{\boldsymbol{\eta}}\_{f}M\_{i}\boldsymbol{\omega}\_{i}}\right|^{2} \cdot \frac{\pi}{T} \int\_{-T/2\pi}^{T} d\boldsymbol{\phi} \cdot \boldsymbol{J}\_{l+1}^{2}\left(\boldsymbol{\phi}\right) \left|\boldsymbol{\text{erf}}\left(2\boldsymbol{\phi} + \frac{i\boldsymbol{\beta}}{2}\right) + 1\right|^{2}, \end{split} \tag{48}$$
 
$$M\_{i} = \tilde{\gamma}\_{0}\left(\boldsymbol{\vartheta}\_{l} + \boldsymbol{m}\right)\hat{\boldsymbol{\epsilon}}. \tag{49}$$

Fig. 3. Shape of the first resonant peak in the cross section of electron-nucleus SB in a pulsed

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 119

In the frame of subsequent analysis we are to estimate the magnitude of the resonant width. For this purpose we consider the case when the four-momentum of an intermediate photon

<sup>4</sup> · (*a*1/*a*2)

(*kqi*) *m*

(erf(*φ*) + 1)

−*ρ*

It is important to underline that the relationship for the function (53) under the condition (56) turns into the standard resonant expression (57), which is usually used in the Breit-Wigner prescription. It is convenient to represent the resonant peak profile *Pres* in the form (57) to compare obtained results with corresponding ones for the case of a monochromatic wave. Note, that in the monochromatic wave case the resonant infinity in the cross section is eliminated by radiative corrections introducing into the Green function. The Breit-Wigner

�<sup>2</sup> + <sup>4</sup>*m*2Γ<sup>2</sup>

1

*<sup>φ</sup>* exp �

<sup>−</sup>*φ*<sup>2</sup> �

*τ*

� *q*2 *<sup>i</sup>* − *<sup>m</sup>*<sup>2</sup>

where the width Γ*τ*, caused by the pulsed character of an external wave, is equal to:

<sup>Γ</sup>*<sup>τ</sup>* <sup>=</sup> <sup>2</sup> <sup>√</sup>*a*<sup>2</sup>

> *ρ* �

−*ρ*

2*ρ* � + *ρ* �

<sup>4</sup> (*kqi*) *ωτ* � 1. (56)

, (57)

*ωτ* , (58)

<sup>2</sup> *dφ*, (59)

⎞

⎠ . (60)

(erf(*φ*) + 1) *dφ*

light field (*<sup>ρ</sup>* <sup>=</sup> <sup>3</sup>). The dashed line represents the Gaussian function: exp(−*β*2/2).

*β* = � *q*2 *<sup>i</sup>* <sup>−</sup> *<sup>m</sup>*2�

*Pres* <sup>≈</sup> *<sup>π</sup>*

*<sup>a</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> 2*ρ*

2erf �<sup>√</sup>

⎛ ⎝√

**2.4.1 Resonant width**

occurs near the mass shell:

Thus, we can easily write

*<sup>a</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup>

and the coefficients *a*<sup>1</sup> and *a*<sup>2</sup> are specified by

<sup>2</sup> <sup>−</sup> <sup>1</sup> 4 <sup>√</sup>*πa*1*<sup>ρ</sup>*

It is taken into account that *d*<sup>3</sup> *pf* = **p***f Ef dEf d*Ω*<sup>f</sup>* and *d*3*k*� = *ω*�2*dω*� *d*Ω� . It is important to note that the main contribution into the sum (47) is given by the processes with emission (absorption) of |*l*| *γ*<sup>0</sup> number of wave photons within the range of a moderately strong field for electron relativistic energies *Ei*, *<sup>f</sup> m* . Therefore, the energy contribution of stimulated photons may be neglected (|*l*| *ω*/*Ei*, *<sup>f</sup> η*0*m*/*Ei*, *<sup>f</sup>* � 1) in Eq. (12). Thus, it is easy to sum over all possible partial processes of electron scattering by a nucleus (47).

If polarization effects are not of interest, then averaging over polarizations of an initial electron and summation over polarizations of a final electron and a spontaneous photon are to be done. Performing the relevant procedures of averaging and summation, we derive the general relativistic expression for the resonant differential cross section of electron-nucleus SB in a pulsed light field in the case of electron large-angle scattering (46)

$$\frac{d\sigma\_{\rm res}}{d\Omega'} = \frac{1}{\pi^2} \cdot \frac{E\_i \kappa\_i^2 \left| \mathbf{q}\_i \right| u}{\left( n n' \right)^2 \left| \mathbf{p}\_i \right| \left( 1 + u \right)} \cdot P\_{\rm res} \cdot d\sigma\_{\rm s} d\mathcal{W}^{(1)}.\tag{50}$$

Here,

$$d\sigma\_s = 2Z^2 r\_\varepsilon^2 \frac{\left| \mathbf{p}\_f \right| m^2}{\left| \mathbf{q}\_i \right| \mathbf{q}^4} \left( m^2 + E\_f q\_{i0} + \mathbf{p}\_f \mathbf{q}\_i \right) d\Omega\_f \tag{51}$$

is the differential cross section of scattering of an intermediate electron with the four-momentum *qi* by a nucleus in a wave field; *re* is the classical electron radius.

$$dW^{(1)} = \frac{a\eta\_0^2 m^2}{4E\_l} \left\{ 2 + \frac{u^2}{1+u} - \frac{4u}{\mu\_1} \left( 1 - \frac{u}{\mu\_1} \right) \right\} \cdot \frac{du}{\left( 1+u \right)^2} \tag{52}$$

is the probability that an electron with the four-momentum *pi* absorbs one photon from an external field and emits a photon with the four-momentum *k*� . The function *Pres* in Eq. (50) has the form

$$P\_{\rm res} = \frac{\pi \left(\omega \tau\right)^2}{64 \left(kq\_{\rm i}\right)^2} \cdot P\_{\rm res}^{\beta} \tag{53}$$

$$P\_{\rm res}^{\beta} = \exp\{-\beta^2/2\} \cdot \frac{1}{2\rho} \int\_{-\rho}^{\rho} d\phi \cdot \left| \text{erf}\left(\phi + \frac{i\beta}{2}\right) + 1 \right|^2,\tag{54}$$

$$
\rho = \mathcal{T}/\mathsf{T}.\tag{55}
$$

Here, the parameter *ρ* is the relation between the observation time and the pulse duration, its value is determined by conditions of the concrete experiment. Thus, if an external field is represented as electromagnetic pulses abiding one by one, then the parameter *ρ* assumes sense of the ratio of a distance between the nearest-neighbor pulses to the characteristic pulse duration. Dependence of the function *Pres* (53) on the parameter *β* (39) defines magnitude and shape of the resonant peak in the cross section of an electron-nucleus SB process in a pulsed light field (see Fig. 7).

Fig. 3. Shape of the first resonant peak in the cross section of electron-nucleus SB in a pulsed light field (*<sup>ρ</sup>* <sup>=</sup> <sup>3</sup>). The dashed line represents the Gaussian function: exp(−*β*2/2).

#### **2.4.1 Resonant width**

12 Will-be-set-by-IN-TECH

to note that the main contribution into the sum (47) is given by the processes with emission (absorption) of |*l*| *γ*<sup>0</sup> number of wave photons within the range of a moderately strong field

photons may be neglected (|*l*| *ω*/*Ei*, *<sup>f</sup> η*0*m*/*Ei*, *<sup>f</sup>* � 1) in Eq. (12). Thus, it is easy to sum over

If polarization effects are not of interest, then averaging over polarizations of an initial electron and summation over polarizations of a final electron and a spontaneous photon are to be done. Performing the relevant procedures of averaging and summation, we derive the general relativistic expression for the resonant differential cross section of electron-nucleus SB in a

*<sup>i</sup>* |**q***i*| *u*

is the differential cross section of scattering of an intermediate electron with the

is the probability that an electron with the four-momentum *pi* absorbs one photon from an

64 (*kqi*)

*ρ* 

−*ρ*

Here, the parameter *ρ* is the relation between the observation time and the pulse duration, its value is determined by conditions of the concrete experiment. Thus, if an external field is represented as electromagnetic pulses abiding one by one, then the parameter *ρ* assumes sense of the ratio of a distance between the nearest-neighbor pulses to the characteristic pulse duration. Dependence of the function *Pres* (53) on the parameter *β* (39) defines magnitude and shape of the resonant peak in the cross section of an electron-nucleus SB process in a pulsed

2

*dφ* · erf *φ* + *iβ* 2 + 1 2

<sup>2</sup> · *<sup>P</sup><sup>β</sup>*

<sup>2</sup> <sup>|</sup>**p***i*|(<sup>1</sup> <sup>+</sup> *<sup>u</sup>*)

*<sup>m</sup>*<sup>2</sup> + *Ef qi*<sup>0</sup> + **<sup>p</sup>***<sup>f</sup>* **<sup>q</sup>***<sup>i</sup>*

*Ef dEf d*Ω*<sup>f</sup>* and *d*3*k*� = *ω*�2*dω*�

*d*Ω�

. Therefore, the energy contribution of stimulated

· *Pres* · *<sup>d</sup>σsdW*(1)

· *du* (1 + *u*)

*ρ* = *T*/*τ*. (55)

. It is important

. (50)

*d*Ω*<sup>f</sup>* (51)

. The function *Pres* in Eq. (50)

*res*, (53)

<sup>2</sup> (52)

, (54)

 **p***f* 

*Ei*, *<sup>f</sup> m*

*<sup>π</sup>*<sup>2</sup> · *Eiκ*<sup>2</sup>

2 *e* **p***f m*2

 2 +

(*nn*�)


four-momentum *qi* by a nucleus in a wave field; *re* is the classical electron radius.

*u*2 <sup>1</sup> <sup>+</sup> *<sup>u</sup>* <sup>−</sup> <sup>4</sup>*<sup>u</sup> u*1 <sup>1</sup> <sup>−</sup> *<sup>u</sup> u*1

*Pres* <sup>=</sup> *<sup>π</sup>* (*ωτ*)

2*ρ*

all possible partial processes of electron scattering by a nucleus (47).

pulsed light field in the case of electron large-angle scattering (46)

*dσres <sup>d</sup>*Ω� <sup>=</sup> <sup>1</sup>

*dW*(1) <sup>=</sup> *αη*<sup>2</sup>

*Pβ*

*dσ<sup>s</sup>* = 2*Z*2*r*

0*m*<sup>2</sup> 4*Ei*

external field and emits a photon with the four-momentum *k*�

*res* <sup>=</sup> exp{−*β*2/2} · <sup>1</sup>

It is taken into account that *d*<sup>3</sup> *pf* =

for electron relativistic energies

Here,

has the form

light field (see Fig. 7).

In the frame of subsequent analysis we are to estimate the magnitude of the resonant width. For this purpose we consider the case when the four-momentum of an intermediate photon occurs near the mass shell:

$$
\beta = \frac{\left(q\_i^2 - m^2\right)}{4\left(kq\_i\right)} \omega \tau \ll 1. \tag{56}
$$

Thus, we can easily write

$$P\_{\rm res} \approx \frac{\pi}{4} \cdot \frac{(a\_1/a\_2)}{\left(q\_i^2 - m^2\right)^2 + 4m^2\Gamma\_\tau^2},\tag{57}$$

where the width Γ*τ*, caused by the pulsed character of an external wave, is equal to:

$$
\Gamma\_{\pi} = \frac{2}{\sqrt{a\_2}} \frac{(kq\_i)}{m} \frac{1}{\omega \tau'} \tag{58}
$$

and the coefficients *a*<sup>1</sup> and *a*<sup>2</sup> are specified by

$$a\_1 = \frac{1}{2\rho} \int\_{-\rho}^{\rho} \left( \text{erf}\left(\phi\right) + 1\right)^2 d\phi,\tag{59}$$

$$a\_2 = \frac{1}{2} - \frac{1}{4\sqrt{\pi}a\_1\rho} \left(\sqrt{2}\text{erf}\left(\sqrt{2}\rho\right) + \int\_{-\rho}^{\rho} \phi \exp\left(-\phi^2\right) \left(\text{erf}\left(\phi\right) + 1\right) d\phi\right). \tag{60}$$

It is important to underline that the relationship for the function (53) under the condition (56) turns into the standard resonant expression (57), which is usually used in the Breit-Wigner prescription. It is convenient to represent the resonant peak profile *Pres* in the form (57) to compare obtained results with corresponding ones for the case of a monochromatic wave. Note, that in the monochromatic wave case the resonant infinity in the cross section is eliminated by radiative corrections introducing into the Green function. The Breit-Wigner

to direction of the initial electron momentum the resonant frequency falls within the interval:

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 121

This frequency reaches its minimum and maximum when a spontaneous photon is emitted

*<sup>i</sup>* <sup>≤</sup> *<sup>ω</sup>* · *<sup>κ</sup><sup>i</sup>*

*nn*� · *ω*� *κi*

> 2 · cos *θ*�

> > **k**, **p***i*, *<sup>f</sup>*

> > > 1 + *μ*<sup>2</sup> *τ*

*res*. The solid angle which corresponds to spontaneous photon

*resdϕdt*. Then the resonant cross section (50) assumes the

· *dW*(1)

*<sup>d</sup>ω*�

<sup>−</sup><sup>1</sup> � 1. Since the resonance angular width is

*Ei* − |**p***i*|

. (65)

� 1. (66)

*res*<sup>2</sup> <sup>+</sup> *<sup>C</sup>*<sup>2</sup> *τ* 

, (68)

*<sup>ω</sup>*� <sup>|</sup>**p***i*<sup>|</sup> , (69)

<sup>0</sup>*ωτ*. (70)

*res* with an accuracy up to the term of the first

. (67)

*<sup>i</sup>* in Eq. (67)

*dσ<sup>S</sup>* (*qi*). (71)

*dσ<sup>S</sup>* (*qi*), (72)

. (73)

*<sup>i</sup>* − cos *θ*�

*<sup>ω</sup>* · *<sup>κ</sup><sup>i</sup>*

*ur* = 2*r* ·

+ (2*m*Γ*τ*)

 **k**� , **p***i*, *<sup>f</sup>* 

*res* <sup>=</sup> *Ei* <sup>−</sup> (*ω*/*ω*�

*μτ* <sup>=</sup> <sup>Γ</sup>*<sup>R</sup>* Γ*τ* = <sup>√</sup>*a*<sup>2</sup> <sup>6</sup> *αη*<sup>2</sup>

For the resonant angles that are not too close to zero and *π* we can expand cos *θ*�

The invariant parameters (33) assume the form

2 2

> *θ*� *<sup>i</sup>*, *<sup>f</sup>* = ∠

cos *θ*�

into the Taylor series near the resonant angle *θ*�

<sup>4</sup>*π*<sup>2</sup> ·

*<sup>i</sup>* − *θ*�

*dϕ* · *d* (*t*/*y*) 1 + (*t*/*y*)

infinity because of integral fast convergence). Finally, we derive

*dW*(1) = *αη*<sup>2</sup>

*<sup>d</sup>σres* <sup>=</sup> *Eiκ<sup>i</sup>*

0 *nn*� *<sup>m</sup>*<sup>2</sup> 2*Eiκ<sup>i</sup>*

2 (*nn*�) *m* |**p***i*| Γ*<sup>τ</sup>*

*<sup>i</sup>* − (*m* − *i*Γ*R*)

Here we introduced the notations

order with respect to *t* = *θ*�

following form

where

emission is written as *d*Ω� = sin *θ*�

*<sup>d</sup>σres* <sup>=</sup> <sup>1</sup>

Here, *y* = *m*Γ*<sup>τ</sup>* (1 + *μτ*)/(*ω*� |**p***i*| sin *θ*�

 *q*2 *Ei* <sup>+</sup> <sup>|</sup>**p***i*<sup>|</sup> <sup>≤</sup> *<sup>ω</sup>*�

along direction of the initial electron motion and in opposite direction, respectively.

*<sup>m</sup>*<sup>2</sup> , *<sup>u</sup>* <sup>∼</sup><sup>=</sup>

<sup>2</sup> =

Taking the radiation width into account, we may represent the resonant denominator (57) as

) · *κ<sup>i</sup>* <sup>|</sup>**p***i*<sup>|</sup> , *<sup>C</sup><sup>τ</sup>* <sup>=</sup> *<sup>m</sup>*Γ*<sup>τ</sup>*

<sup>2</sup> · *Eiκ<sup>i</sup>* (*nn*�)|**p***i*| Γ*<sup>τ</sup>*

very small, we may integrate the expression (71) with respect to the azimuthal angle *dϕ*, and with respect to *d* (*t*/*y*) within the limits from zero to +∞ (we extend the integration limits to

> 1 + *μ*<sup>2</sup> *τ*

 <sup>1</sup> <sup>−</sup> <sup>2</sup>*<sup>u</sup> u*1 · <sup>1</sup> <sup>−</sup> *<sup>u</sup> u*1

Derived expressions (72)-(73) for the resonant cross section are valid within the range of field intensities (5) when an electron scatters into the large angles *θ* � *ω*/|**p***i*|. Spontaneous

*res*) ∼ (*ωτ*)

 1 + *μ*<sup>2</sup> *<sup>τ</sup>m*

· *dW*(1)

2*ω*� |**p***i*|

, *θi*, *<sup>f</sup>* = ∠

*ωκ<sup>i</sup>*

broadening prescription is concluded in addition of the imaginary part of the electron mass, that is *m* → *m* − *i*Γ*R*. Here, the radiation width is specified

$$
\Gamma\_R = \frac{1}{3} a \eta\_0^2 \frac{\sigma\_c(q\_i)}{\sigma\_T} \cdot \frac{(kq\_i)}{m},\tag{61}
$$

where *σ<sup>c</sup>* (*qi*) is the total cross section of the Compton scattering of an external field photon by an intermediate electron with the four-momentum *qi* (it is the most probable way out of an electron from an intermediate state), *σ<sup>T</sup>* is the cross section of the Thomson scattering.

The resonant width (58) providing by finite time of particle-field interaction is so-called transit width. In real experiments the transit width value is generally determined by geometry of an experiment and linear sizes of space where a particle interacts with an external field. It can be seen from Eq. (58) that the transit width is specified by the pulse duration and process kinematics. Influence of the pulse duration on the resonant behavior of the electron-nucleus SB cross section was discussed by Schnez et al. (2007). The electromagnetic pulse duration has to be longer than the lifetime of an intermediate electron state. Otherwise, an electron will not have enough time to interact with a wave. Thus,

$$
\pi \gtrsim \frac{1}{\Gamma\_R}.\tag{62}
$$

Requirements (62), (58) implies that values of the quantity *ωτ* have to satisfy the following condition:

$$
\omega \tau \gtrsim \frac{1}{a\eta\_0^2} \frac{\omega m}{(k\eta\_i)}.\tag{63}
$$

Comparison of the resonance widths for the pulse duration values (63) implies that Γ*<sup>τ</sup>* ∼ Γ*<sup>R</sup>* within a sufficiently broad range of electron energies and scattering angles. Consequently, the both radiation and transit widths have to be simultaneously considered in resonant SB study. An exception is the case of ultrarelativistic energies when

$$\frac{1}{a\eta\_0^2} \frac{\omega m}{(k\eta\_i)} \lessapprox \omega \tau \ll \frac{1}{a\eta\_0^2}.\tag{64}$$

In this case Γ*<sup>τ</sup>* � Γ*<sup>R</sup>* and the expressions for the resonant differential cross section of electron-nucleus SB in a pulsed field (50)-(52), (57)-(60) are correct without radiation width accounting.

It should be pointed that we excluded other causes of the resonant peak widening from consideration. Thus, we assume that the Doppler broadening of the resonance (the real electron bunch is not monochromatic) and broadening caused by collisions of electrons are negligible.

#### **2.4.2 Range of relativistic energies**

In this section we consider the range of electron relativistic energies: *Ei m*. Here we eliminate the case when ultrarelativistic electrons are moving within a narrow cone with a spontaneous photon or an external field photon from consideration. Then |*di*| � 1 (it follows from Eq. (29)). Therefore, the resonant frequency *ω*� *<sup>i</sup>* (28) in this case is of the order of the external field frequency. Depending on the spontaneous photon emission angle with respect to direction of the initial electron momentum the resonant frequency falls within the interval:

$$
\omega \cdot \frac{\kappa\_{\bar{i}}}{E\_{\bar{i}} + |\mathbf{p}\_{\bar{i}}|} \le \omega\_{\bar{i}}' \le \omega \cdot \frac{\kappa\_{\bar{i}}}{E\_{\bar{i}} - |\mathbf{p}\_{\bar{i}}|}. \tag{65}
$$

This frequency reaches its minimum and maximum when a spontaneous photon is emitted along direction of the initial electron motion and in opposite direction, respectively.

The invariant parameters (33) assume the form

$$
\mu\_r = 2r \cdot \frac{\omega \kappa\_i}{m^2}, \quad \mu \cong (nn') \cdot \frac{\omega'}{\kappa\_i} \ll 1. \tag{66}
$$

Taking the radiation width into account, we may represent the resonant denominator (57) as

$$\left(q\_i^2 - \left(m - i\Gamma\_R\right)^2\right)^2 + \left(2m\Gamma\_\tau\right)^2 = \left(2\omega'\left|\mathbf{p}\_i\right|\right)^2 \cdot \left[\left(\cos\theta\_i' - \cos\theta\_{\rm res}'\right)^2 + \mathcal{C}\_\tau^2\right].\tag{67}$$

Here we introduced the notations

14 Will-be-set-by-IN-TECH

broadening prescription is concluded in addition of the imaginary part of the electron mass,

where *σ<sup>c</sup>* (*qi*) is the total cross section of the Compton scattering of an external field photon by an intermediate electron with the four-momentum *qi* (it is the most probable way out of an electron from an intermediate state), *σ<sup>T</sup>* is the cross section of the Thomson scattering.

The resonant width (58) providing by finite time of particle-field interaction is so-called transit width. In real experiments the transit width value is generally determined by geometry of an experiment and linear sizes of space where a particle interacts with an external field. It can be seen from Eq. (58) that the transit width is specified by the pulse duration and process kinematics. Influence of the pulse duration on the resonant behavior of the electron-nucleus SB cross section was discussed by Schnez et al. (2007). The electromagnetic pulse duration has to be longer than the lifetime of an intermediate electron state. Otherwise, an electron will not

> *<sup>τ</sup>* <sup>1</sup> Γ*R*

Requirements (62), (58) implies that values of the quantity *ωτ* have to satisfy the following

Comparison of the resonance widths for the pulse duration values (63) implies that Γ*<sup>τ</sup>* ∼ Γ*<sup>R</sup>* within a sufficiently broad range of electron energies and scattering angles. Consequently, the both radiation and transit widths have to be simultaneously considered in resonant SB study.

(*kqi*) *ωτ* �

In this case Γ*<sup>τ</sup>* � Γ*<sup>R</sup>* and the expressions for the resonant differential cross section of electron-nucleus SB in a pulsed field (50)-(52), (57)-(60) are correct without radiation width

It should be pointed that we excluded other causes of the resonant peak widening from consideration. Thus, we assume that the Doppler broadening of the resonance (the real electron bunch is not monochromatic) and broadening caused by collisions of electrons are

In this section we consider the range of electron relativistic energies: *Ei m*. Here we eliminate the case when ultrarelativistic electrons are moving within a narrow cone with a spontaneous photon or an external field photon from consideration. Then |*di*| � 1 (it follows

external field frequency. Depending on the spontaneous photon emission angle with respect

*ωm* (*kqi*)

> 1 *αη*<sup>2</sup> 0

*ωτ* <sup>1</sup> *αη*<sup>2</sup> 0

*<sup>m</sup>* , (61)

. (62)

. (63)

. (64)

*<sup>i</sup>* (28) in this case is of the order of the

that is *m* → *m* − *i*Γ*R*. Here, the radiation width is specified

have enough time to interact with a wave. Thus,

An exception is the case of ultrarelativistic energies when

from Eq. (29)). Therefore, the resonant frequency *ω*�

1 *αη*<sup>2</sup> 0 *ωm*

condition:

accounting.

negligible.

**2.4.2 Range of relativistic energies**

<sup>Γ</sup>*<sup>R</sup>* <sup>=</sup> <sup>1</sup> 3 *αη*<sup>2</sup> 0 *σ<sup>c</sup>* (*qi*) *σT* · (*kqi*)

$$\boldsymbol{\theta}\_{\mathrm{i},f}^{\prime} = \angle \left( \mathbf{k}^{\prime}, \mathbf{p}\_{\mathrm{i},f} \right), \quad \boldsymbol{\theta}\_{\mathrm{i},f} = \angle \left( \mathbf{k}, \mathbf{p}\_{\mathrm{i},f} \right), \tag{68}$$

$$\cos \theta\_{\rm res}^{\prime} = \frac{E\_{\rm i} - (\omega / \omega^{\prime}) \cdot \kappa\_{\rm i}}{|\mathbf{p}\_{\rm i}|}, \quad \mathbf{C}\_{\rm \tau} = \frac{m \Gamma\_{\rm \tau} \sqrt{1 + \mu\_{\tau}^{2}}}{\omega^{\prime} |\mathbf{p}\_{\rm i}|}, \tag{69}$$

$$
\mu\_{\tau} = \frac{\Gamma\_R}{\Gamma\_{\tau}} = \frac{\sqrt{a\_2}}{6} a \eta\_0^2 \omega \tau. \tag{70}
$$

For the resonant angles that are not too close to zero and *π* we can expand cos *θ*� *<sup>i</sup>* in Eq. (67) into the Taylor series near the resonant angle *θ*� *res* with an accuracy up to the term of the first order with respect to *t* = *θ*� *<sup>i</sup>* − *θ*� *res*. The solid angle which corresponds to spontaneous photon emission is written as *d*Ω� = sin *θ*� *resdϕdt*. Then the resonant cross section (50) assumes the following form

$$d\sigma\_{\rm res} = \frac{1}{4\pi^2} \cdot \frac{d\boldsymbol{\varrho} \cdot \boldsymbol{d}\left(t/\boldsymbol{y}\right)}{1 + \left(t/\boldsymbol{y}\right)^2} \cdot \frac{E\_i \kappa\_i}{\left(nn'\right)\left|\mathbf{p}\_i\right|\Gamma\_\Gamma\sqrt{1+\mu\_\tau^2}m} \cdot d\mathcal{W}^{(1)}d\sigma\_S\left(q\_i\right). \tag{71}$$

Here, *y* = *m*Γ*<sup>τ</sup>* (1 + *μτ*)/(*ω*� |**p***i*| sin *θ*� *res*) ∼ (*ωτ*) <sup>−</sup><sup>1</sup> � 1. Since the resonance angular width is very small, we may integrate the expression (71) with respect to the azimuthal angle *dϕ*, and with respect to *d* (*t*/*y*) within the limits from zero to +∞ (we extend the integration limits to infinity because of integral fast convergence). Finally, we derive

$$d\sigma\_{\rm res} = \frac{E\_i \kappa\_i}{2 \left(nn'\right) m \left| \mathbf{p}\_i \right| \Gamma\_\tau \sqrt{1 + \mu\_\tau^2}} \cdot d\mathcal{W}^{(1)} d\sigma\_S \left(\mathbf{q}\_i\right) \tag{72}$$

where

$$dW^{(1)} = a\eta\_0^2 \left( nm'\right) \frac{m^2}{2E\_i \kappa\_i} \left\{ 1 - \frac{2u}{u\_1} \cdot \left( 1 - \frac{u}{u\_1} \right) \right\} d\omega'. \tag{73}$$

Derived expressions (72)-(73) for the resonant cross section are valid within the range of field intensities (5) when an electron scatters into the large angles *θ* � *ω*/|**p***i*|. Spontaneous

Fig. 4. Ratio *Rres* (76) as a function of the initial velocity for electron momentum preset orientations in initial (*θ<sup>i</sup>* = 163◦, *ϕ<sup>i</sup>* = 0◦) and final (*θ <sup>f</sup>* = 150◦,*ϕ<sup>f</sup>* = 0◦) states and

Fig. 5. Ratio *Rres* (76) as a function of the azimuthal angle of a spontaneous photon for electron momentum preset orientations in initial and final states and the spontaneous photon fixed polar angle: *θ<sup>i</sup>* = 163◦, *θ <sup>f</sup>* = 150◦, *θ*� = 120◦. Solid line, *ϕ<sup>i</sup>* = *ϕ<sup>f</sup>* = 90◦; dashed line,

correspond to spontaneous photons emission just within this plane (solid line). In the case when a final electron scatters in another way the peak position in distribution over the azimuthal angle is specified by both initial and final azimuthal angles. The value of the ratio of the resonant differential cross section of electron-nucleus SB to the ordinary bremsstrahlung cross section as a function of the azimuthal angle may be changed in two orders of magnitude.

*ϕ*� = 60◦.

*ϕ<sup>i</sup>* = 90◦, *ϕ<sup>f</sup>* = 320◦.

spontaneous photon fixed orientation: solid line, *θ*� = 120◦, *ϕ*� = 10◦; dashed line, *θ*� = 120◦,

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 123

photon frequency and emission angle with respect to the initial electron momentum are unambiguously related to each other by Eq. (69), where the spontaneous photon frequency is chosen from the interval (65).

Note, that the conventional cross section *dσ*<sup>∗</sup> of electron-nucleus bremsstrahlung (in external field absence) may be factorized as a product of the cross section *dσ<sup>S</sup>* (*pi*) of electron-nucleus elastic scattering (see (51)) and the probability *dW<sup>γ</sup>* of photon emission

$$d\sigma\_\* = d\sigma\_{\mathcal{S}} \cdot d\mathcal{W}\_{\mathcal{Y}'} \tag{74}$$

$$d\mathcal{W}\_{\boldsymbol{\gamma}} = \frac{\boldsymbol{\alpha}}{4\pi^{2}} \cdot \left\{ \mathbf{q}^{2} - \left(\mathbf{n}'\mathbf{q}\right)^{2} \cdot \frac{m^{2}}{\kappa\_{i}'\kappa\_{f}'} \right\} \cdot \frac{d\boldsymbol{\omega}'}{\boldsymbol{\omega}'\kappa\_{i}'\kappa\_{f}'} \cdot d\Omega', \quad \mathbf{q} = \mathbf{p}\_{f} - \mathbf{p}\_{i}.\tag{75}$$

Let us calculate the ratio of the resonant cross section (72) to the conventional cross section of electron-nucleus bremsstrahlung (74) (in absence of an external field). At that we take into account the resonant relation (69) between spontaneous photon frequency and emission angle

$$R\_{\rm res} = \frac{d\sigma\_{\rm res}}{d\sigma\_{\ast}/d\Omega'} = f\_1 \cdot \pi^2 \eta\_0^2 \frac{\omega \cdot \tau}{\sqrt{1 + \mu\_\tau^2}} \left(\frac{m}{|\mathbf{p}\_i|}\right)^2,\tag{76}$$

where the function *f*<sup>1</sup> ∼ 1 and has a rather cumbersome form:

$$f\_1 = \frac{\sqrt{a\_2} \mathbf{x}\_f^\prime}{2 \left| \mathbf{p}\_i \right|} \frac{1 - \left( nm^\prime \right) \frac{m^2}{\kappa\_i \kappa\_i^\prime} \left( 1 - \left( nm^\prime \right) \frac{m^2}{2 \kappa\_i \kappa\_i^\prime} \right)}{4 \sin^2 \left( \frac{\theta}{2} \right) - \left( \cos \theta\_f^\prime - \cos \theta\_i^\prime \right)^2 \frac{m^2}{\kappa\_i \kappa\_f^\prime}}. \tag{77}$$

Let us choose for calculation the laser field characteristic according to SLAC experiments (Bula et al. (1996)): laser-wave frequency, *ω* = 2.35 eV; laser pulsewidth, *τ* = 1.5 ps; field strength in a pulse peak, *<sup>F</sup>*<sup>0</sup> <sup>=</sup> <sup>6</sup> · 109 V/cm; ratio between observation time and laser pulse width, *ρ* = 5. Fig. 4 displays the ratio of the resonant differential cross-section of electron-nucleus SB to the cross section of bremsstrahlung in absence of an external field (76) as a function of the electron velocity.

Eq. (76) and Fig. 4 show that within the range of electron relativistic energies the resonant differential cross section of electron-nucleus SB, when the scattered electron ejection angle is detected simultaneously with the spontaneous photon emission angle, may be five orders of magnitude greater than the corresponding cross section in external field absence. Within the range of electron ultrarelativistic energies this ratio decreases drastically: *Rres* ∼ (*m*/*Ei*) <sup>2</sup> <sup>→</sup> 0.

The ratio (76) as a function of the spontaneous photon azimuthal angle is of interest from a perspective of experimental testing of obtained results. In the actual experiment usually the radiation detection over the azimuthal angle is technically implemented significantly easier than over the polar angle. Fig. 5 displays lg *Rres* (76) as a function of the spontaneous photon azimuthal angle.

Fig. 5 shows that the ratio (76) may change its order of magnitude with the azimuthal angle value. This dependence is characterized by presence of two maxima in distribution over the azimuthal angle. Thus, when the final electron azimuthal angle coincides with the initial electron angle (it is scattering in the plane of the vectors (**k**, **p***i*)) the maxima in distribution 16 Will-be-set-by-IN-TECH

photon frequency and emission angle with respect to the initial electron momentum are unambiguously related to each other by Eq. (69), where the spontaneous photon frequency is

Note, that the conventional cross section *dσ*<sup>∗</sup> of electron-nucleus bremsstrahlung (in external field absence) may be factorized as a product of the cross section *dσ<sup>S</sup>* (*pi*) of electron-nucleus

0

) *<sup>m</sup>*<sup>2</sup> *κiκ*� *i* 

Let us choose for calculation the laser field characteristic according to SLAC experiments (Bula et al. (1996)): laser-wave frequency, *ω* = 2.35 eV; laser pulsewidth, *τ* = 1.5 ps; field strength in a pulse peak, *<sup>F</sup>*<sup>0</sup> <sup>=</sup> <sup>6</sup> · 109 V/cm; ratio between observation time and laser pulse width, *ρ* = 5. Fig. 4 displays the ratio of the resonant differential cross-section of electron-nucleus SB to the cross section of bremsstrahlung in absence of an external field (76) as a function of the

Eq. (76) and Fig. 4 show that within the range of electron relativistic energies the resonant differential cross section of electron-nucleus SB, when the scattered electron ejection angle is detected simultaneously with the spontaneous photon emission angle, may be five orders of magnitude greater than the corresponding cross section in external field absence. Within the

The ratio (76) as a function of the spontaneous photon azimuthal angle is of interest from a perspective of experimental testing of obtained results. In the actual experiment usually the radiation detection over the azimuthal angle is technically implemented significantly easier than over the polar angle. Fig. 5 displays lg *Rres* (76) as a function of the spontaneous photon

Fig. 5 shows that the ratio (76) may change its order of magnitude with the azimuthal angle value. This dependence is characterized by presence of two maxima in distribution over the azimuthal angle. Thus, when the final electron azimuthal angle coincides with the initial electron angle (it is scattering in the plane of the vectors (**k**, **p***i*)) the maxima in distribution

range of electron ultrarelativistic energies this ratio decreases drastically: *Rres* ∼ (*m*/*Ei*)

*ωτ* 1 + *μ*<sup>2</sup> *τ*

1 − (*nn*�

*<sup>f</sup>* − cos *θ*� *i* <sup>2</sup> *<sup>m</sup>*<sup>2</sup> *κ*� *i κ*� *f*

Let us calculate the ratio of the resonant cross section (72) to the conventional cross section of electron-nucleus bremsstrahlung (74) (in absence of an external field). At that we take into account the resonant relation (69) between spontaneous photon frequency and emission angle

*<sup>d</sup>*Ω� <sup>=</sup> *<sup>f</sup>*<sup>1</sup> · *<sup>π</sup>*2*η*<sup>2</sup>

1 − (*nn*�

4 sin2 *θ* 2 − cos *θ*�

· *<sup>d</sup>ω*� *ω*�*κ*� *i κ*� *f* · *d*Ω�

*dσ*<sup>∗</sup> = *dσ<sup>S</sup>* · *dWγ*, (74)

 *m* |**p***i*|

) *<sup>m</sup>*<sup>2</sup> 2*κiκ*� *i* 

<sup>2</sup>

, **q** = **p***<sup>f</sup>* − **p***i*. (75)

, (76)

. (77)

<sup>2</sup> <sup>→</sup> 0.

elastic scattering (see (51)) and the probability *dW<sup>γ</sup>* of photon emission

chosen from the interval (65).

*dW<sup>γ</sup>* <sup>=</sup> *<sup>α</sup>*

<sup>4</sup>*π*<sup>2</sup> ·

*f*<sup>1</sup> =

electron velocity.

azimuthal angle.

*Rres* <sup>=</sup> *<sup>d</sup>σres dσ*<sup>∗</sup> 

where the function *f*<sup>1</sup> ∼ 1 and has a rather cumbersome form:

<sup>√</sup>*a*2*κ*� *f* 2 |**p***i*|

**<sup>q</sup>**<sup>2</sup> <sup>−</sup> **n**� **q** <sup>2</sup> · *<sup>m</sup>*<sup>2</sup> *κ*� *i κ*� *f*

Fig. 4. Ratio *Rres* (76) as a function of the initial velocity for electron momentum preset orientations in initial (*θ<sup>i</sup>* = 163◦, *ϕ<sup>i</sup>* = 0◦) and final (*θ <sup>f</sup>* = 150◦,*ϕ<sup>f</sup>* = 0◦) states and spontaneous photon fixed orientation: solid line, *θ*� = 120◦, *ϕ*� = 10◦; dashed line, *θ*� = 120◦, *ϕ*� = 60◦.

Fig. 5. Ratio *Rres* (76) as a function of the azimuthal angle of a spontaneous photon for electron momentum preset orientations in initial and final states and the spontaneous photon fixed polar angle: *θ<sup>i</sup>* = 163◦, *θ <sup>f</sup>* = 150◦, *θ*� = 120◦. Solid line, *ϕ<sup>i</sup>* = *ϕ<sup>f</sup>* = 90◦; dashed line, *ϕ<sup>i</sup>* = 90◦, *ϕ<sup>f</sup>* = 320◦.

correspond to spontaneous photons emission just within this plane (solid line). In the case when a final electron scatters in another way the peak position in distribution over the azimuthal angle is specified by both initial and final azimuthal angles. The value of the ratio of the resonant differential cross section of electron-nucleus SB to the ordinary bremsstrahlung cross section as a function of the azimuthal angle may be changed in two orders of magnitude.

where the function *f*<sup>2</sup> ∼ 1 and has the form

*f*<sup>2</sup> =

<sup>√</sup>*a*<sup>2</sup> 2

<sup>1</sup> <sup>−</sup> (1/2) sin2 *<sup>θ</sup>*�

*<sup>f</sup>* − cos *θ*� *i*

<sup>2</sup> . (86)

 cos *θ*�

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 125

4 sin2 (*θ*/2) <sup>−</sup>

Fig. 6. Dependence of *Rres* (85) on the polar angle of a spontaneous photon for a nonrelativistic electron with the initial velocity *vi* = 0.1. Orientations of the electron momentum in initial (*θ<sup>i</sup>* = 163◦, *ϕ<sup>i</sup>* = 90◦) and final (*θ <sup>f</sup>* = 150◦, *ϕ<sup>f</sup>* = 320◦) states are fixed. The azimuthal angle, corresponding to emission of a spontaneous photon, is (solid line)

Fig. 6 shows the dependence of quantity *Rres* (85) on the polar angle of spontaneous photon emission for a nonrelativistic electron with the initial velocity *vi* = 0.1. Fig. 6 shows that for the case of electron kinetic energies of several kiloelectronvolts the resonant differential SB cross section may be 5–6 orders of magnitude greater than the corresponding cross section of bremsstrahlung in external field absence when the angle of spontaneous photon emission is detected simultaneously with the ejection angle of an electron scattered into the large angle.

**2.4.4 Range of ultrarelativistic energies of electrons moving within a narrow cone with a**

In this section we consider an ultrarelativistic electron that moves (in initial or final states) within the narrow cone related to an external field photon. Therefore, the quantities *κi*, *<sup>f</sup>* (30)

Taking these relations into account and using Eqs. (28)-(31) we find that the resonant

<sup>2</sup> (*nn*�) ·

 *m Ei*, *<sup>f</sup>* <sup>2</sup>

 1 + *δ*<sup>2</sup> *i*, *f* 

frequencies are much less than the external field frequency. They are given by:

*<sup>i</sup>*, *<sup>f</sup>* = *rωi*, *<sup>f</sup>* , *ωi*, *<sup>f</sup>* =

· *<sup>m</sup>*2/2*Ei*, *<sup>f</sup>* , *<sup>δ</sup>i*, *<sup>f</sup>* <sup>=</sup> *<sup>θ</sup>i*, *<sup>f</sup>* · *Ei*, *<sup>f</sup>* /*m*. (87)

· *ω* � *ω*. (88)

*ϕ*� = 60◦, (dashed line) *ϕ*� = 160◦, and (dash-dotted line) *ϕ*� = 270◦.

**photon from the wave**

in Eqs. (28)-(31) may be written as

*κi*, *<sup>f</sup>* = 1 + *δ*<sup>2</sup> *i*, *f* 

*ω*�

#### **2.4.3 Range of nonrelativistic electron energies**

In this section we assume that initial and final electron energies are small in comparison with the light speed: *Zα* � *vi*, *<sup>f</sup>* � 1. It follows from Eqs. (28)-(31) that resonant frequencies for nonrelativistic electrons are given by

$$
\omega\_{i,f}' = r\omega \left(1 + \mathbf{v}\_{i,f} \left(\mathbf{n}' - \mathbf{n}\right)\right) \cong r\omega. \tag{78}
$$

Thus, resonances occur when the spontaneous photon frequency is multiple to the external field frequency. The condition of interference between direct and exchange resonant amplitudes (35) is written as

$$\left(\mathbf{v}\_f - \mathbf{v}\_i\right)\left(\mathbf{n} - \mathbf{n}'\right) = 2r \cdot \left(nn'\right) \cdot \frac{\omega}{m} \ll 1,\tag{79}$$

and, consequently, interference appears when an electron scatters into the small angels *θ* ∼ *ω*/*mvi* � 1.

The resonant cross section in the case when a nonrelativistic electron scatters into the large angles is obtained from Eq. (50):

$$d\sigma\_{\rm res} = \frac{1}{2\left(nn'\right)v\_i\Gamma\_\tau\sqrt{1+\mu\_\tau^2}}dW^{(1)}d\sigma\_{\rm s} \tag{80}$$

where

$$d\mathcal{W}^{(1)} = \frac{1}{2} a \eta^2 \left( n n' \right) \cdot \left\{ 1 - \frac{2u}{u\_1} \cdot \left( 1 - \frac{u}{u\_1} \right) \right\} d\omega',\tag{81}$$

$$\frac{u}{u\_1} = \left( n n' \right) \frac{\omega'}{2\omega},$$

$$d\sigma\_s = (2Z)^2 r\_e^2 \frac{\left| \mathbf{p}\_f \right|}{|\mathbf{q}\_i|} \frac{m^4}{\mathbf{q}^4} d\Omega\_f.\tag{82}$$

The resonant frequency of a spontaneous photon depends on the emission angle of this photon with respect to the initial electron momentum and lies within a narrow interval:

$$
\omega \left( 1 - 2v\_i \cos^2 \left( \theta\_i / 2 \right) \right) \le \omega\_{\text{res}}' \le \omega \left( 1 + 2v\_i \sin^2 \left( \theta\_i / 2 \right) \right). \tag{83}
$$

The transit width Γ*<sup>τ</sup>* (58) and the radiation width Γ*<sup>R</sup>* (61) in the nonrelativistic limit are given by

$$
\Gamma\_{\mathsf{T}} = \frac{2}{\sqrt{a\_{\mathsf{T}}}} \frac{1}{\mathsf{\tau}'} , \quad \Gamma\_{\mathsf{R}} = \frac{1}{3} a \eta^2 \omega . \tag{84}
$$

We may write the ratio of the resonant cross section (50) to the corresponding conventional nonrelativistic cross section of electron-nucleus bremsstrahlung as

$$R\_{\rm res} = f\_2 \cdot \pi^2 \eta\_0^2 \frac{\omega \tau}{\sqrt{1 + \mu\_\tau^2}} v\_i^{-3} \,\,\,\,\,\tag{85}$$

where the function *f*<sup>2</sup> ∼ 1 and has the form

18 Will-be-set-by-IN-TECH

In this section we assume that initial and final electron energies are small in comparison with the light speed: *Zα* � *vi*, *<sup>f</sup>* � 1. It follows from Eqs. (28)-(31) that resonant frequencies for

> **n**� − **n**

Thus, resonances occur when the spontaneous photon frequency is multiple to the external field frequency. The condition of interference between direct and exchange resonant

and, consequently, interference appears when an electron scatters into the small angels *θ* ∼

The resonant cross section in the case when a nonrelativistic electron scatters into the large

 1 + *μ*<sup>2</sup> *τ* *dW*(1)

 *dω*�

1 + 2*vi* sin2 (*θi*/2)

<sup>∼</sup><sup>=</sup> *<sup>r</sup>ω*. (78)

� 1, (79)

*dσs*, (80)

**<sup>q</sup>**<sup>4</sup> *<sup>d</sup>*Ω*<sup>f</sup>* . (82)

*αη*2*ω*. (84)

*<sup>i</sup>* , (85)

, (81)

. (83)

 1 + **v***i*, *<sup>f</sup>*

 **n** − **n**� = 2*r* · *nn*� · *ω m*

*<sup>d</sup>σres* <sup>=</sup> <sup>1</sup>

*dσ<sup>s</sup>* = (2*Z*)

with respect to the initial electron momentum and lies within a narrow interval:

 ≤ *ω*�

*Rres* <sup>=</sup> *<sup>f</sup>*<sup>2</sup> · *<sup>π</sup>*2*η*<sup>2</sup>

<sup>Γ</sup>*<sup>τ</sup>* <sup>=</sup> <sup>2</sup> <sup>√</sup>*a*<sup>2</sup> 1

nonrelativistic cross section of electron-nucleus bremsstrahlung as

2 *r* 2 *e* **p***f* |**q***i*|

The resonant frequency of a spontaneous photon depends on the emission angle of this photon

The transit width Γ*<sup>τ</sup>* (58) and the radiation width Γ*<sup>R</sup>* (61) in the nonrelativistic limit are given

We may write the ratio of the resonant cross section (50) to the corresponding conventional

0

*res* ≤ *ω*

*<sup>τ</sup>* , <sup>Γ</sup>*<sup>R</sup>* <sup>=</sup> <sup>1</sup>

*m*4

3

*ωτ* 1 + *μ*<sup>2</sup> *τ v*−<sup>3</sup>

2 (*nn*�) *vi*Γ*<sup>τ</sup>*

**2.4.3 Range of nonrelativistic electron energies**

*ω*� *<sup>i</sup>*, *<sup>f</sup>* = *rω*

 **v***<sup>f</sup>* − **v***<sup>i</sup>*

*dW*(1) <sup>=</sup> <sup>1</sup>

*u u*1 = *nn*� *ω*� 2*ω* ,

*ω*  2 *αη*<sup>2</sup> *nn*� · <sup>1</sup> <sup>−</sup> <sup>2</sup>*<sup>u</sup> u*1 · <sup>1</sup> <sup>−</sup> *<sup>u</sup> u*1

<sup>1</sup> <sup>−</sup> <sup>2</sup>*vi* cos2 (*θi*/2)

nonrelativistic electrons are given by

amplitudes (35) is written as

angles is obtained from Eq. (50):

*ω*/*mvi* � 1.

where

by

$$f\_2 = \frac{\sqrt{a\_2}}{2} \frac{1 - (1/2)\sin^2\theta'}{4\sin^2\left(\theta/2\right) - \left(\cos\theta'\_f - \cos\theta'\_i\right)^2}.\tag{86}$$

Fig. 6. Dependence of *Rres* (85) on the polar angle of a spontaneous photon for a nonrelativistic electron with the initial velocity *vi* = 0.1. Orientations of the electron momentum in initial (*θ<sup>i</sup>* = 163◦, *ϕ<sup>i</sup>* = 90◦) and final (*θ <sup>f</sup>* = 150◦, *ϕ<sup>f</sup>* = 320◦) states are fixed. The azimuthal angle, corresponding to emission of a spontaneous photon, is (solid line) *ϕ*� = 60◦, (dashed line) *ϕ*� = 160◦, and (dash-dotted line) *ϕ*� = 270◦.

Fig. 6 shows the dependence of quantity *Rres* (85) on the polar angle of spontaneous photon emission for a nonrelativistic electron with the initial velocity *vi* = 0.1. Fig. 6 shows that for the case of electron kinetic energies of several kiloelectronvolts the resonant differential SB cross section may be 5–6 orders of magnitude greater than the corresponding cross section of bremsstrahlung in external field absence when the angle of spontaneous photon emission is detected simultaneously with the ejection angle of an electron scattered into the large angle.

#### **2.4.4 Range of ultrarelativistic energies of electrons moving within a narrow cone with a photon from the wave**

In this section we consider an ultrarelativistic electron that moves (in initial or final states) within the narrow cone related to an external field photon. Therefore, the quantities *κi*, *<sup>f</sup>* (30) in Eqs. (28)-(31) may be written as

$$
\kappa\_{i,f} = \left(1 + \delta\_{i,f}^2\right) \cdot m^2 / 2\mathcal{E}\_{i,f}, \quad \delta\_{i,f} = \theta\_{i,f} \cdot \mathcal{E}\_{i,f} / m. \tag{87}
$$

Taking these relations into account and using Eqs. (28)-(31) we find that the resonant frequencies are much less than the external field frequency. They are given by:

$$
\omega\_{i,f}' = r\omega\_{i,f}, \quad \omega\_{i,f} = \frac{\left(1 + \delta\_{i,f}^2\right)}{2\left(nn'\right)} \cdot \left(\frac{m}{E\_{i,f}}\right)^2 \cdot \omega \ll \omega. \tag{88}
$$

When an ultrarelativistic initial electron moves within the narrow cone with a spontaneous photon and scatters on the large angle *θ* � *ω*/*Ei* we may use Eqs. (50)-(52) to find the resonant cross section. In this case, it is convenient to represent the resonant denominator

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 127

<sup>2</sup> = *m*<sup>4</sup>

�� *x* − *δ*� *i* 2 �2 + *y*<sup>2</sup> � · *<sup>u</sup>*<sup>2</sup> (1 + *u*)

*<sup>u</sup>* · *<sup>m</sup>*<sup>2</sup> <sup>−</sup> 1, *<sup>y</sup>* <sup>=</sup> <sup>2</sup> (<sup>1</sup> <sup>+</sup> *<sup>u</sup>*) <sup>Γ</sup>*<sup>τ</sup>*

� *nn*� � · *ωEi*

, and integration should be performed with respect to the azimuthal angle,

<sup>=</sup> <sup>1</sup> 2 + 1 *<sup>π</sup> arctg* � *<sup>x</sup>*

1, if *u* < *u*1, 0.5, if *u* = *u*1, *y* · *u*/*π* (*u* − *u*1), if *u* > *u*1. <sup>2</sup> , (93)

(97)

(98)

<sup>2</sup> . (99)

. (100)

� 1 + *μ*<sup>2</sup> *τ <sup>u</sup>* · *<sup>m</sup>* . (94)

*<sup>m</sup>*<sup>2</sup> . (95)

· *dW*(1) · *<sup>d</sup>σ<sup>S</sup>* (*qi*). (96)

*y* �

��

· *dW*(1) *dWpi* (*k*�)

· *du* (1 + *u*)

in the following form

where

� *m*2/2*E*<sup>2</sup> *i* � *dϕdδ*� *i* 2

and *δ*� *i*

Here,

resonant cross section:

The probability is given by

� *q*2

*<sup>i</sup>* − (*m* − *i*Γ*R*)

Here, the invariant parameters *u*, *u*<sup>1</sup> are given by

<sup>Υ</sup> (*xy*) <sup>=</sup> <sup>1</sup>

*dW*(1) <sup>=</sup> *αη*<sup>2</sup> · *<sup>m</sup>*<sup>2</sup>

*Rres* <sup>=</sup> *<sup>d</sup>σ*(1) *res dσa*

*π*

Υ (*xy*) =

4*Ei* · � 2 +

*<sup>x</sup>* <sup>=</sup> *<sup>u</sup>*<sup>1</sup> *u*

2 �2

<sup>+</sup> (<sup>1</sup> <sup>+</sup> *<sup>u</sup>*) <sup>Γ</sup><sup>2</sup>

*τ* � 1 + *μ*<sup>2</sup> *τ* �

*<sup>u</sup>* <sup>∼</sup><sup>=</sup> *<sup>ω</sup>*�

*<sup>d</sup>σres* <sup>=</sup> <sup>Υ</sup> (*xy*) · *qi*<sup>0</sup>

�∞

� *x* − *δ*� *i* 2 �2 + *y*<sup>2</sup>

⎧ ⎨ ⎩

0

+ (2*m*Γ*τ*)

*Ei* <sup>−</sup> *<sup>ω</sup>*� , *<sup>u</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup>

*m*Γ*<sup>τ</sup>* � 1 + *μ*<sup>2</sup> *τ*

> *dδ*� *i* 2

and at the resonance point *u*<sup>1</sup> = *u* this function takes the following limiting values:

is a smoothed step function. In regions far from the resonance |*u*<sup>1</sup> − *u*| � 2 (1 + *u*) (Γ*τ*/*m*)

*u*2 <sup>1</sup> <sup>+</sup> *<sup>u</sup>* <sup>−</sup> <sup>4</sup>*<sup>u</sup> u*1 � <sup>1</sup> <sup>−</sup> *<sup>u</sup> u*1

<sup>=</sup> <sup>Υ</sup> (*xy*) · *Ei*

*m*Γ*<sup>τ</sup>* � 1 + *μ*<sup>2</sup> *τ*

We consider ratio of the resonant cross section (96) to the conventional cross section of electron-nucleus bremsstrahlung in the case when an ultrarelativistic electron moves within the narrow cone with a photon producted in bremsstrahlung and scatters on the large angles. Using the results obtained by Baier et al. (1973) we may write the following expression:

Now Eqs. (50)-(52), (93) are to be taken into account, the solid angle is to be written as *d*Ω� =

<sup>2</sup> within the limits from zero to +∞. Thus, we derive the following expression for the

From Eq. (88) follows that the condition of interference between direct and exchange resonant amplitudes implies that *δ<sup>i</sup>* = *δ<sup>f</sup>* and *θ<sup>i</sup>* ∼= *θ <sup>f</sup>* , that is, initial and final electrons form the equal angles with the external field photon momentum and are located on different sides of this photon momentum. Also, it can be seen from (35) that *<sup>θ</sup><sup>i</sup>* <sup>∼</sup> *<sup>ω</sup>m*2/*E*<sup>3</sup> *<sup>i</sup>* � 1. When an ultrarelativistic initial electron moves within the narrow cone with an external field photon and scatters into the large angle *<sup>θ</sup><sup>i</sup>* � *<sup>ω</sup>m*2/*E*<sup>3</sup> *<sup>i</sup>* the resonant cross section is derived from Eq. (50) under the condition (87):

$$d\sigma\_{\rm res} = \frac{\left(1 + \delta\_{\rm i}^{2}\right)m}{4\left(nn'\right)E\_{\rm i}\Gamma\_{\rm r}\sqrt{1 + \mu\_{\rm r}^{2}}} \cdot d\mathcal{W}^{(1)} \cdot d\sigma\_{\rm S}\left(q\_{\rm i}\right) \,. \tag{89}$$

Here, the spontaneous photon resonant frequency is given by Eq. (88) with value *r* = 1, and the angle of spontaneous photon emission is not close to direction of initial electron motion. Ratio of the resonant cross section (89) to the conventional cross section of electron-nucleus bremsstrahlung may be derived from Eq. (76) with respect to Eq. (87):

$$R\_{\rm res} = f\_3 \cdot \pi^2 \eta\_0^2 \frac{\omega \tau}{\sqrt{1 + \mu\_\tau^2}} \left(\frac{m}{E\_i}\right)^2,\tag{90}$$

where the function *f*<sup>3</sup> ∼ 1 and has a rather cumbersome form.

It may be easily estimated that for the pulsed field parameters *ω* = 2.35 eV, *τ* = 1.5 ps, *<sup>F</sup>*<sup>0</sup> <sup>=</sup> <sup>6</sup> · 109 V/cm, *<sup>ρ</sup>* <sup>=</sup> 5 and the electron energy *Ei* <sup>=</sup> 5 MeV the resonant cross section is of the order of the ordinary cross section when the angle of spontaneous photon emission is detected simultaneously with the ejection angle of an electron scattered on the large angle.

#### **2.4.5 Range of ultrarelativistic energies of electrons moving within a narrow cone with a spontaneous photon**

We suppose that an ultrarelativistic electron (an initial or a final one) moves within the narrow cone with a spontaneous photon. Then the quantities *κ*� *<sup>i</sup>*, *<sup>f</sup>* (30) may be written in an analogous to Eq. (87) form, where

$$
\delta\_{\mathbf{i},f} \rightarrow \delta\_{\mathbf{i},f}' = \theta\_{\mathbf{i},f}' \cdot \mathbf{E}\_{\mathbf{i},f} / m. \tag{91}
$$

Here, depending on the electron energy we may deal with one of two possible situations. It is provided that *<sup>m</sup>* � *Ei*, *<sup>f</sup>* � *<sup>m</sup>*2/*ω*, than resonant frequencies fall within the interval *ω* � *ω*� *<sup>i</sup>*, *<sup>f</sup>* � *Ei*, *<sup>f</sup>* and are given by

$$
\omega\_{i,f}' = r\omega\_{i,f}, \quad \omega\_{i,f} = \frac{2\left(nn'\right)}{\left(1+\delta'^2\_{i,f}\right)} \cdot \left(\frac{E\_{i,f}}{m}\right)^2 \cdot \omega. \tag{92}
$$

It was demonstrated by Roshchupkin (1985) that resonances do not occur for energies *Ei*, *<sup>f</sup>* � *m*2/*ω*. It is obviously that direct and exchange resonant amplitudes may interfere with each other only when initial and final electrons move within the narrow cone with a spontaneous photon, so *δ*� *<sup>i</sup>* = *δ*� *f* .

When an ultrarelativistic initial electron moves within the narrow cone with a spontaneous photon and scatters on the large angle *θ* � *ω*/*Ei* we may use Eqs. (50)-(52) to find the resonant cross section. In this case, it is convenient to represent the resonant denominator in the following form

$$\left(q\_i^2 - \left(m - i\Gamma\_R\right)^2\right)^2 + \left(2m\Gamma\_\tau\right)^2 = m^4\left[\left(\mathbf{x} - \boldsymbol{\delta}\_i^{\prime 2}\right)^2 + y^2\right] \cdot \frac{\boldsymbol{u}^2}{\left(1 + \boldsymbol{u}\right)^2} \tag{93}$$

where

20 Will-be-set-by-IN-TECH

From Eq. (88) follows that the condition of interference between direct and exchange resonant amplitudes implies that *δ<sup>i</sup>* = *δ<sup>f</sup>* and *θ<sup>i</sup>* ∼= *θ <sup>f</sup>* , that is, initial and final electrons form the equal angles with the external field photon momentum and are located on different sides

ultrarelativistic initial electron moves within the narrow cone with an external field photon

 1 + *μ*<sup>2</sup> *τ*

Here, the spontaneous photon resonant frequency is given by Eq. (88) with value *r* = 1, and the angle of spontaneous photon emission is not close to direction of initial electron motion. Ratio of the resonant cross section (89) to the conventional cross section of electron-nucleus

0

It may be easily estimated that for the pulsed field parameters *ω* = 2.35 eV, *τ* = 1.5 ps, *<sup>F</sup>*<sup>0</sup> <sup>=</sup> <sup>6</sup> · 109 V/cm, *<sup>ρ</sup>* <sup>=</sup> 5 and the electron energy *Ei* <sup>=</sup> 5 MeV the resonant cross section is of the order of the ordinary cross section when the angle of spontaneous photon emission is detected simultaneously with the ejection angle of an electron scattered on the large angle.

**2.4.5 Range of ultrarelativistic energies of electrons moving within a narrow cone with a**

We suppose that an ultrarelativistic electron (an initial or a final one) moves within the narrow

*<sup>i</sup>*, *<sup>f</sup>* = *θ*�

Here, depending on the electron energy we may deal with one of two possible situations. It is provided that *<sup>m</sup>* � *Ei*, *<sup>f</sup>* � *<sup>m</sup>*2/*ω*, than resonant frequencies fall within the interval

> 1 + *δ*�<sup>2</sup> *i*, *f* ·

It was demonstrated by Roshchupkin (1985) that resonances do not occur for energies *Ei*, *<sup>f</sup>* � *m*2/*ω*. It is obviously that direct and exchange resonant amplitudes may interfere with each other only when initial and final electrons move within the narrow cone with a spontaneous

)

 *Ei*, *<sup>f</sup> m*

2

*ωτ* 1 + *μ*<sup>2</sup> *τ*  *m Ei* <sup>2</sup>

*<sup>i</sup>* � 1. When an

*<sup>i</sup>* the resonant cross section is derived from

· *dW*(1) · *<sup>d</sup>σ<sup>S</sup>* (*qi*). (89)

, (90)

*<sup>i</sup>*, *<sup>f</sup>* (30) may be written in an analogous

· *ω*. (92)

*<sup>i</sup>*, *<sup>f</sup>* · *Ei*, *<sup>f</sup>* /*m*. (91)

of this photon momentum. Also, it can be seen from (35) that *<sup>θ</sup><sup>i</sup>* <sup>∼</sup> *<sup>ω</sup>m*2/*E*<sup>3</sup>

 1 + *δ*<sup>2</sup> *i m*

4 (*nn*�) *Ei*Γ*<sup>τ</sup>*

bremsstrahlung may be derived from Eq. (76) with respect to Eq. (87):

where the function *f*<sup>3</sup> ∼ 1 and has a rather cumbersome form.

cone with a spontaneous photon. Then the quantities *κ*�

*Rres* <sup>=</sup> *<sup>f</sup>*<sup>3</sup> · *<sup>π</sup>*2*η*<sup>2</sup>

*δi*, *<sup>f</sup>* → *δ*�

*<sup>i</sup>*, *<sup>f</sup>* <sup>=</sup> *<sup>r</sup>ωi*, *<sup>f</sup>* , *<sup>ω</sup>i*, *<sup>f</sup>* <sup>=</sup> <sup>2</sup> (*nn*�

and scatters into the large angle *<sup>θ</sup><sup>i</sup>* � *<sup>ω</sup>m*2/*E*<sup>3</sup>

*dσres* =

Eq. (50) under the condition (87):

**spontaneous photon**

*<sup>i</sup>*, *<sup>f</sup>* � *Ei*, *<sup>f</sup>* and are given by

*ω*�

to Eq. (87) form, where

*ω* � *ω*�

photon, so *δ*�

*<sup>i</sup>* = *δ*� *f* .

$$\chi = \frac{u\_1}{u} + \frac{(1+u)\,\Gamma\_\tau^2 \left(1+\mu\_\tau^2\right)}{u \cdot m^2} - 1, \quad y = \frac{2\left(1+u\right)\,\Gamma\_\tau \sqrt{1+\mu\_\tau^2}}{u \cdot m}.\tag{94}$$

Here, the invariant parameters *u*, *u*<sup>1</sup> are given by

$$
\mu \cong \frac{\omega'}{E\_i - \omega'}, \quad \mu\_1 = 2 \left( m' \right) \cdot \frac{\omega E\_i}{m^2}. \tag{95}
$$

Now Eqs. (50)-(52), (93) are to be taken into account, the solid angle is to be written as *d*Ω� = � *m*2/2*E*<sup>2</sup> *i* � *dϕdδ*� *i* 2 , and integration should be performed with respect to the azimuthal angle, and *δ*� *i* <sup>2</sup> within the limits from zero to +∞. Thus, we derive the following expression for the resonant cross section:

$$d\sigma\_{\rm res} = \Upsilon(\mathbf{x}y) \cdot \frac{q\_{i0}}{m\Gamma\_{\tau}\sqrt{1+\mu\_{\tau}^{2}}} \cdot d\mathcal{W}^{(1)} \cdot d\sigma\_{\rm S} \left(q\_{i}\right) \,. \tag{96}$$

Here,

$$\Upsilon(xy) = \frac{1}{\pi} \int\_0^\infty \frac{d\delta\_i^{s2}}{\left(\chi - \delta\_i^{s2}\right)^2 + y^2} = \frac{1}{2} + \frac{1}{\pi} \operatorname{arctg}\left(\frac{\chi}{y}\right) \tag{97}$$

is a smoothed step function. In regions far from the resonance |*u*<sup>1</sup> − *u*| � 2 (1 + *u*) (Γ*τ*/*m*) and at the resonance point *u*<sup>1</sup> = *u* this function takes the following limiting values:

$$\mathbf{Y}\left(\mathbf{x}\mathbf{y}\right) = \begin{cases} 1, & \text{if } u < u\_{1\prime} \\ 0.5, & \text{if } u = u\_{1\prime} \\ y \cdot u / \pi \left(u - u\_{1}\right), & \text{if } u > u\_{1}. \end{cases} \tag{98}$$

The probability is given by

$$d\mathcal{W}^{(1)} = a\eta^2 \cdot \frac{m^2}{4E\_l} \cdot \left\{ 2 + \frac{u^2}{1+u} - \frac{4u}{u\_1} \left( 1 - \frac{u}{u\_1} \right) \right\} \cdot \frac{du}{\left( 1+u \right)^2}. \tag{99}$$

We consider ratio of the resonant cross section (96) to the conventional cross section of electron-nucleus bremsstrahlung in the case when an ultrarelativistic electron moves within the narrow cone with a photon producted in bremsstrahlung and scatters on the large angles. Using the results obtained by Baier et al. (1973) we may write the following expression:

$$R\_{\rm res} = \frac{d\sigma\_{\rm res}^{(1)}}{d\sigma\_a} = \mathbf{Y}\left(\mathbf{xy}\right) \cdot \frac{E\_i}{m\Gamma\_\tau \sqrt{1+\mu\_\tau^2}} \cdot \frac{d\mathcal{W}^{(1)}}{d\mathcal{W}\_{p\_i}\left(\mathbf{k'}\right)}.\tag{100}$$

 ˃) ˄)

Fig. 7. Photoproduction of an electron–positron pair on a nucleus in a pulsed light wave.

Consequently, the four-momentum of an intermediate particle appears near the mass surface

It is convenient to write Eqs. (103), which define the four-momenta *q* and *q*±, for amplitudes

*ki* <sup>+</sup> *rk* <sup>=</sup> *<sup>q</sup>*<sup>−</sup> <sup>+</sup> *<sup>p</sup>*+,

*ki* <sup>+</sup> *rk* <sup>=</sup> *<sup>p</sup>*<sup>−</sup> <sup>+</sup> *<sup>q</sup>*+,

Eqs. (105)-(106) represent the four-momentum conservation laws for the diagrams vertices

which a resonance can be observed (the resonant frequency) from the Eq. (105). Within the

Within the region of moderately strong fields (5) the energy conservation law (*q*<sup>0</sup> ≈ 0) may be

Therefore, it follows from Eq. (107) that within the moderately strong fields region resonances are possible only for ultrarelativistic positron *p*<sup>+</sup> (diagram (a), Fig. 7) and electron *p*<sup>−</sup> (diagram (b), Fig. 7), if they move within a narrow cone with the incident *γ*-ray photon *ki*. In

, *<sup>W</sup>*<sup>±</sup> <sup>=</sup> *<sup>m</sup>*<sup>2</sup>

*<sup>r</sup><sup>ω</sup>* ·

 1 + *δ*<sup>2</sup> *i*± 

*<sup>δ</sup>i*<sup>±</sup> = *<sup>θ</sup>i*<sup>±</sup> · (*E*±/*m*), *<sup>θ</sup>i*<sup>±</sup> = ∠ (**k***i*, **<sup>p</sup>**±) � 1. (111)

*<sup>i</sup>* <sup>≡</sup> *<sup>r</sup><sup>ω</sup>* · (*np*∓)

(*niq*±)

*n* = *k*/*ω* = (1, **n**), *ni* = *ki*/*ω<sup>i</sup>* = (1, **n***i*). (108)

*ω<sup>i</sup>* ≈ *E*<sup>−</sup> + *E*+. (109)

Taking into account the condition (104) we will obtain the initial photon frequency *ωres*

*p*

*q*

*<sup>q</sup> <sup>p</sup>*

*i k*

*<sup>q</sup>* <sup>=</sup> *<sup>p</sup>*<sup>−</sup> <sup>−</sup> *<sup>q</sup>*<sup>−</sup> <sup>+</sup> (*<sup>l</sup>* <sup>+</sup> *<sup>r</sup>*) *<sup>k</sup>*; (105)

*<sup>q</sup>* <sup>=</sup> *<sup>p</sup>*<sup>+</sup> <sup>−</sup> *<sup>q</sup>*<sup>+</sup> <sup>+</sup> (*<sup>l</sup>* <sup>+</sup> *<sup>r</sup>*) *<sup>k</sup>*. (106)

<sup>−</sup><sup>1</sup> for the diagrams (a) and (b) (see

, (107)

<sup>2</sup> (*nni*) , (110)

*<sup>i</sup>* for

*Ze*

*i k*

*q*

*Ze*

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 129

*p*

under the resonant conditions.

Fig. 7), we obtain

written as

where

(a) and (b) in Fig. 7, respectively, as

*p*

*q*

(Fig. 7) that, in view of the condition (104), hold only for *r* > 0.

*ωres <sup>i</sup>* = *ω*<sup>±</sup>

zeroth order with respect to the small parameter (*ωτ*)

this case resonant frequencies (107) take the form

*ω*±

*<sup>i</sup>* <sup>=</sup> *<sup>E</sup>*<sup>±</sup>

1 − *W*±/*E*<sup>±</sup>

Here, *dWpi* (*k*� ) is the probability that an electron with the four-momentum *pi* emits a photon with the four-momentum *k*� . For electron energies *<sup>m</sup>* � *Ei* � *<sup>m</sup>*2� *ω* the expression (100) may be written as

$$R\_{\rm res} = \frac{\sqrt{a\_2}}{8} \pi \frac{\eta\_0^2 \omega \tau}{\sqrt{1 + \mu\_\tau^2}} \cdot \frac{1}{\ln \left( E\_i / m \right)}. \tag{101}$$

If the considered process characteristics satisfy the conditions (64), than the parameter *μτ* � 1 (70) and the resonant shape is specified by the laser pulse duration. Eq. (101) implies, when the ultrarelativistic electron energy grows, the resonant cross section decreases drastically.

## **3. Resonant photoproduction of an electron-positron pair on a nucleus in the field of a pulsed light wave**

The most general computations of the resonant Coulomb electron-positron pair photoproduction (CPP) on a nucleus in the field of an electromagnetic plane wave was performed by Roshchupkin (1983). Borisov et al. (1981) studied the resonant CPP in the special case of ultrarelativistic electron and positron energies where the incident photon and the wave photon fly toward each other. The work of Lötstedt et al. (2008) in which resonant cross sections were calculated for strong external fields should also be noted. The resonant CPP in the pulsed light wave was studied in detail in the work of Lebed' & Roshchupkin (2011).

We consider the photoproduction of an electron-positron pair on a nucleus in a pulsed light field (2). The interaction of an electron and positron with a nucleus is considered in the first order of the perturbation theory (the Born approximation). Note that CPP is a crossed channel of bremsstrahlung due to electron scattering by a nucleus. Spontaneous bremsstrahlung of an electron scattered by a nucleus in a pulsed light field was studied early. In consideration of the known calculation procedure we may obtain the amplitude of CPP process on a nucleus in the field of a moderately strong pulsed wave from the expressions (9)-(18) by the following replacement:

$$p\_- \to p\_{f'} \quad p\_+ \to -p\_{i'} \quad k\_i \to -k',\tag{102}$$

where *p*−, *p*+, *ki* are the four-momenta of an electron, a positron and an initial photon, respectively. For CPP on a nucleus *q* = (*q*0, **q**) is the four-vector is the transferred momentum, *q*<sup>−</sup> and *q*<sup>+</sup> are the four-momenta of an intermediate electron and an intermediate positron (for the diagrams on Fig. 7 (a) and (b), respectively). These quantities are expressed by the relationships:

$$\begin{cases} q = p\_- + p\_+ - k\_i + lk\_\prime\\ q\_- = k\_i + rk - p\_{+\prime} \\ q\_+ = k\_i + rk - p\_- \end{cases} \tag{103}$$

#### **3.1 Resonance conditions**

Let us consider the resonances that occur when an intermediate particle reaches the mass shell. The conditions of resonant CPP on a nucleus in a pulsed light field is determined by the relationship

$$q\_{\pm}^2 - m^2 \lesssim \frac{(kq\_{\pm})}{\omega \tau}.\tag{104}$$

Fig. 7. Photoproduction of an electron–positron pair on a nucleus in a pulsed light wave.

Consequently, the four-momentum of an intermediate particle appears near the mass surface under the resonant conditions.

It is convenient to write Eqs. (103), which define the four-momenta *q* and *q*±, for amplitudes (a) and (b) in Fig. 7, respectively, as

$$\begin{cases} k\_l + rk = q\_- + p\_{+, \prime} \\ q = p\_- - q\_- + (l + r) \, k \end{cases} \tag{105}$$

$$\begin{cases} k\_i + rk = p\_- + q\_{+\prime} \\ q = p\_+ - q\_+ + (l+r)k. \end{cases} \tag{106}$$

Eqs. (105)-(106) represent the four-momentum conservation laws for the diagrams vertices (Fig. 7) that, in view of the condition (104), hold only for *r* > 0.

Taking into account the condition (104) we will obtain the initial photon frequency *ωres <sup>i</sup>* for which a resonance can be observed (the resonant frequency) from the Eq. (105). Within the zeroth order with respect to the small parameter (*ωτ*) <sup>−</sup><sup>1</sup> for the diagrams (a) and (b) (see Fig. 7), we obtain

$$
\omega\_i^{\rm res} = \omega\_i^{\pm} \equiv r\omega \cdot \frac{\left(np\_{\mp}\right)}{\left(n\_i q\_{\pm}\right)}\tag{107}
$$

$$n = k/\omega = (1, \mathbf{n}) , \quad n\_{\mathbf{i}} = k\_{\mathbf{i}}/\omega\_{\mathbf{i}} = (1, \mathbf{n}\_{\mathbf{i}}) \,. \tag{108}$$

Within the region of moderately strong fields (5) the energy conservation law (*q*<sup>0</sup> ≈ 0) may be written as

$$
\omega\_{\rm i} \approx E\_{-} + E\_{+}.\tag{109}
$$

Therefore, it follows from Eq. (107) that within the moderately strong fields region resonances are possible only for ultrarelativistic positron *p*<sup>+</sup> (diagram (a), Fig. 7) and electron *p*<sup>−</sup> (diagram (b), Fig. 7), if they move within a narrow cone with the incident *γ*-ray photon *ki*. In this case resonant frequencies (107) take the form

$$
\omega\_{i}^{\pm} = \frac{E\_{\pm}}{1 - W\_{\pm}/E\_{\pm}}, \quad W\_{\pm} = \frac{m^2}{r\omega} \cdot \frac{\left(1 + \delta\_{i\pm}^2\right)}{2\left(m i\_{i}\right)},\tag{110}
$$

where

22 Will-be-set-by-IN-TECH

. For electron energies *<sup>m</sup>* � *Ei* � *<sup>m</sup>*2�

<sup>0</sup>*ωτ* � 1 + *μ*<sup>2</sup> *τ*

If the considered process characteristics satisfy the conditions (64), than the parameter *μτ* � 1 (70) and the resonant shape is specified by the laser pulse duration. Eq. (101) implies, when the ultrarelativistic electron energy grows, the resonant cross section decreases drastically.

**3. Resonant photoproduction of an electron-positron pair on a nucleus in the field**

The most general computations of the resonant Coulomb electron-positron pair photoproduction (CPP) on a nucleus in the field of an electromagnetic plane wave was performed by Roshchupkin (1983). Borisov et al. (1981) studied the resonant CPP in the special case of ultrarelativistic electron and positron energies where the incident photon and the wave photon fly toward each other. The work of Lötstedt et al. (2008) in which resonant cross sections were calculated for strong external fields should also be noted. The resonant CPP in the pulsed light wave was studied in detail in the work of Lebed' & Roshchupkin

We consider the photoproduction of an electron-positron pair on a nucleus in a pulsed light field (2). The interaction of an electron and positron with a nucleus is considered in the first order of the perturbation theory (the Born approximation). Note that CPP is a crossed channel of bremsstrahlung due to electron scattering by a nucleus. Spontaneous bremsstrahlung of an electron scattered by a nucleus in a pulsed light field was studied early. In consideration of the known calculation procedure we may obtain the amplitude of CPP process on a nucleus in the field of a moderately strong pulsed wave from the expressions (9)-(18) by the following

*p*<sup>−</sup> → *pf* , *p*<sup>+</sup> → −*pi*, *ki* → −*k*�

where *p*−, *p*+, *ki* are the four-momenta of an electron, a positron and an initial photon, respectively. For CPP on a nucleus *q* = (*q*0, **q**) is the four-vector is the transferred momentum, *q*<sup>−</sup> and *q*<sup>+</sup> are the four-momenta of an intermediate electron and an intermediate positron (for the diagrams on Fig. 7 (a) and (b), respectively). These quantities are expressed by the

> *q* = *p*<sup>−</sup> + *p*<sup>+</sup> − *ki* + *lk*, *q*<sup>−</sup> = *ki* + *rk* − *p*+, *q*<sup>+</sup> = *ki* + *rk* − *p*−.

Let us consider the resonances that occur when an intermediate particle reaches the mass shell. The conditions of resonant CPP on a nucleus in a pulsed light field is determined by the

<sup>±</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup> <sup>&</sup>lt;<sup>∼</sup> (*kq*±)

⎧ ⎨ ⎩

*q*2

*Rres* =

<sup>√</sup>*a*<sup>2</sup> <sup>8</sup> *<sup>π</sup> <sup>η</sup>*<sup>2</sup>

) is the probability that an electron with the four-momentum *pi* emits a photon

· <sup>1</sup> ln (*Ei*/*m*) *ω* the expression (100) may

. (101)

, (102)

*ωτ* . (104)

(103)

Here, *dWpi* (*k*�

be written as

(2011).

replacement:

relationships:

relationship

**3.1 Resonance conditions**

with the four-momentum *k*�

**of a pulsed light wave**

$$
\delta\_{i\pm} = \theta\_{i\pm} \cdot \left(\mathbf{E}\_{\pm}/m\right), \quad \theta\_{i\pm} = \angle\left(\mathbf{k}\_{i\prime}\mathbf{p}\_{\pm}\right) \ll 1. \tag{111}
$$

Here, the functions *Bl*<sup>−</sup> (*γ*˜0,*ε*ˆ*i*) and *Bl*<sup>+</sup> (*ε*ˆ*i*, *<sup>γ</sup>*˜0) correspond to the CPP diagrams in Figs. 7(a)

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 131

2*ωτ*<sup>2</sup> **<sup>q</sup>**<sup>2</sup> + *<sup>q</sup>*<sup>0</sup> (*q*<sup>0</sup> − <sup>2</sup>*qz*)

*dφLl*<sup>+</sup>*<sup>r</sup>* (*φ*) exp {*i*(*q*0*τ* + 2*β*) *φ*} ·*γ*˜0 (*q*ˆ<sup>−</sup> + *m*) ×

*sgn*

−2*iβφ*�

<sup>−</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

With allowance of the four-momentum conservation law (that is the first Eq. in (105)), the

the amplitude of the production of an electron-positron pair with the four-momenta *q*<sup>−</sup> and *p*<sup>+</sup> by a photon with the four-momentum *ki* in a pulsed light field through *r* wave photons absorption. This process was considered by Nikishov & Ritus (1979) in the case of a plane monochromatic wave, and by Narozhny & Fofanov (1997) in the case of a pulsed light wave. With allowance of the transferred four-momentum *q* (see the second equality in (105))

intermediate electron with the four-momentum *q*<sup>−</sup> by a nucleus in a pulsed light field with

Consequently, if the interference between direct and exchange amplitudes is absent, the process of resonant CPP on a nucleus in a pulsed light field effectively decomposes into two consecutive processes of the first order. The distinction for the diagram (b) on Fig. 7 is concluded in replacement of the four-momentum of an intermediate electron *q*<sup>−</sup> → −*q*<sup>+</sup>

Integral functions (18) are determined by the integer-order Bessel functions (23) for the case of a circularly polarized external wave. For circular polarization of a wave under resonance conditions the arguments of the Bessel functions (20) for CPP on a nucleus may be represented

> 1 + *z*+ *z*+*zr*

, *zr* = 2*r* ·

*z*+*zr* − (1 + *z*+)

(*kki*)

2

∼ *η*<sup>0</sup> � 1. (122)

, (120)

*<sup>m</sup>*<sup>2</sup> . (121)

absorption or emission of |*l* + *r*| photons of the wave (Lebed' & Roshchupkin (2008)).

)

), *βq*<sup>−</sup> *<sup>p</sup>*<sup>+</sup> (*φ*�

*<sup>β</sup>* <sup>=</sup> *<sup>q</sup>*<sup>2</sup>

· *<sup>i</sup><sup>π</sup>* 2 (*kq*−)

*φ*� − *φ* − 1 ,

) and *Ll*<sup>+</sup>*<sup>r</sup>* (*φ*) are defined by relations (15)-(21) with the replacement

×

<sup>4</sup> (*kq*−) *ωτ*. (119)

(15) under resonance conditions defines

*γ*˜0 defines the amplitude of scattering of an

(118)

and 7(b), respectively

Here, functions *F*−*<sup>r</sup>* (*φ*�

matrix function *F*−*<sup>r</sup>*

the quantity *Ll*<sup>+</sup>*<sup>r</sup>*

(102).

as

*Bl*<sup>−</sup> (*γ*˜0,*ε*ˆ*i*) =

*dφ*� *F*−*<sup>r</sup> φ*� exp

*χq*<sup>−</sup> *<sup>p</sup>*<sup>+</sup> , *γq*<sup>−</sup> *<sup>p</sup>*<sup>+</sup> (*φ*�

*χp*−*q*<sup>−</sup> , *γp*−*q*<sup>−</sup> (*φ*), *βp*−*q*<sup>−</sup> (*φ*)

= 2*r* · *η*0*g*

becomes a classical one under resonance conditions (see Eqs. (32)-(33)).

 *φ*� ·

(*kq*−) <sup>≈</sup> *<sup>E</sup>*<sup>+</sup>

*γq*<sup>−</sup> *<sup>p</sup>*<sup>+</sup> *φ*� 

*ω<sup>i</sup>* − *E*<sup>+</sup>

It was expected for this part of the amplitude that the Bunkin-Fedorov quantum parameter

× ∞

× ∞

and change of sequence of first order processes.

*γq*<sup>−</sup> *<sup>p</sup>*<sup>+</sup> *φ*� 

where the invariant parameters *z*+ and *zr* are defined by

*<sup>z</sup>*<sup>+</sup> <sup>=</sup> (*kp*+)

−∞

−∞

∞ ∑ *r*=−∞

Hence the resonances are possible only for the electron (positron) energies above some threshold value *<sup>W</sup>*±: *<sup>E</sup>*<sup>±</sup> <sup>&</sup>gt; *<sup>W</sup>*<sup>±</sup> <sup>∼</sup> *<sup>m</sup>*2/*ω*.

Using Eqs. (110) it is easy to obtain the positron energy at resonance:

$$E\_{+} = \frac{1}{2} \left\{ 1 \pm \sqrt{1 - \frac{\omega\_{i}^{th}}{\omega\_{i}}} \right\} \cdot \omega\_{i\nu} \tag{112}$$

where *ωth <sup>i</sup>* is the threshold frequency of an incident *γ*-ray photon,

$$
\omega\_i^{\rm th} = \frac{2m^2}{\omega \left(1 - \cos \theta\_i\right)}, \quad \theta\_i = \angle \left(\mathbf{k}, \mathbf{k}\_i\right). \tag{113}
$$

As we see from Eq. (113), the threshold energy of an initial photon appreciably depends on its orientation relative to wave propagation direction. Thus, the threshold energy is minimal when an incident photon propagates towards the wave. In the opposite case, when an initial photon moves parallel to external field photons, no resonances are observed. Note that the electron energy can be obtained from Eq. (112) by reversing the sign in front of the square root. It follows from Eq. (112), that the energies of produced electron and positron near the threshold (*ω<sup>i</sup>* <sup>−</sup> *<sup>ω</sup>th <sup>i</sup>* � *<sup>ω</sup>th <sup>i</sup>* ) are equal *<sup>E</sup>*<sup>+</sup> <sup>=</sup> *<sup>E</sup>*<sup>−</sup> <sup>∼</sup><sup>=</sup> *<sup>ω</sup>th <sup>i</sup>* /2. If, alternatively, the frequency of an incident *<sup>γ</sup>*-ray photon is great (*ω<sup>i</sup>* � *<sup>ω</sup>th <sup>i</sup>* ) then electron and positron energies differ considerably (*E*<sup>+</sup> <sup>=</sup> *<sup>ω</sup><sup>i</sup>* <sup>−</sup> *<sup>ω</sup>th <sup>i</sup>* /4 <sup>≈</sup> *<sup>ω</sup>i*, *<sup>E</sup>*<sup>−</sup> <sup>≈</sup> *<sup>ω</sup>th <sup>i</sup>* /4).

The condition of interference of resonant amplitudes, that is *ω*<sup>+</sup> *<sup>i</sup>* = *ω*<sup>−</sup> *<sup>i</sup>* , assumes the form

$$(np\_{-})\left(n\_{i}q\_{-}\right) = \left(np\_{+}\right)\left(n\_{i}q\_{+}\right).\tag{114}$$

Using the energy conservation law (109) and Eq. (110) we derive that the interference of resonant amplitudes appears when an electron *p*<sup>−</sup> and a positron *p*<sup>+</sup> propagate within a narrow cone with an incident *<sup>γ</sup>*-ray photon *ki*, with *<sup>δ</sup>i*<sup>−</sup> = *<sup>δ</sup>i*<sup>+</sup> and *<sup>θ</sup>i*<sup>−</sup> ∼ *<sup>ω</sup>*/*E*−.

Below, we will consider the resonance of one diagram. We will assume that the initial photon frequency is

$$
\omega\_{\rm i} \approx \omega\_{\rm i}^{\rm res} = \omega\_{\rm i}^{-}.\tag{115}
$$

#### **3.2 Resonant amplitude**

The amplitude of CPP on a nucleus in a pulsed light field under resonance conditions (107) has the form

$$\mathcal{S}^{(\pm)} = \sum\_{l=-\infty}^{\infty} \mathcal{S}\_l^{(\pm)},\tag{116}$$

where *S*(±) *<sup>l</sup>* is the partial amplitude, which corresponds to processes with emission (*l* > 0) or absorption (*l* < 0) of laser-wave |*l*| photons

$$S\_{l}^{(\pm)} = -i \frac{Ze^{3}\sqrt{\pi}}{\sqrt{2\omega\_{l}E\_{-}E\_{+}}} \overline{u}\_{-} \left[B\_{l-}\left(\tilde{\gamma}\_{0\prime}\pounds\_{i}\right) + B\_{l+}\left(\pounds\_{i\prime}\tilde{\gamma}\_{0}\right)\right]u\_{+}.\tag{117}$$

24 Will-be-set-by-IN-TECH

Hence the resonances are possible only for the electron (positron) energies above some

�

As we see from Eq. (113), the threshold energy of an initial photon appreciably depends on its orientation relative to wave propagation direction. Thus, the threshold energy is minimal when an incident photon propagates towards the wave. In the opposite case, when an initial photon moves parallel to external field photons, no resonances are observed. Note that the electron energy can be obtained from Eq. (112) by reversing the sign in front of the square root. It follows from Eq. (112), that the energies of produced electron and positron near the

*<sup>i</sup>* /4).

Using the energy conservation law (109) and Eq. (110) we derive that the interference of resonant amplitudes appears when an electron *p*<sup>−</sup> and a positron *p*<sup>+</sup> propagate within a

Below, we will consider the resonance of one diagram. We will assume that the initial photon

The amplitude of CPP on a nucleus in a pulsed light field under resonance conditions (107)

∞ ∑ *l*=−∞

*S*(±)

*<sup>l</sup>* is the partial amplitude, which corresponds to processes with emission (*l* > 0) or

*<sup>i</sup>* = *ω*<sup>−</sup>

*<sup>ω</sup><sup>i</sup>* <sup>≈</sup> *<sup>ω</sup>res*

*S*(±) =

<sup>1</sup> <sup>−</sup> *<sup>ω</sup>th i ωi*

⎫ ⎬

<sup>⎭</sup> · *<sup>ω</sup>i*, (112)

*<sup>i</sup>* /2. If, alternatively, the frequency

*<sup>i</sup>* . (115)

*<sup>l</sup>* , (116)

*<sup>u</sup>*¯<sup>−</sup> [*Bl*<sup>−</sup> (*γ*˜0,*ε*ˆ*i*) + *Bl*<sup>+</sup> (*ε*ˆ*i*, *<sup>γ</sup>*˜0)] *<sup>u</sup>*+. (117)

*<sup>i</sup>* , assumes the form

*<sup>i</sup>* ) then electron and positron energies differ

*<sup>i</sup>* = *ω*<sup>−</sup>

(*np*−) (*niq*−) = (*np*+) (*niq*+). (114)

, *θ<sup>i</sup>* = ∠ (**k**, **k***i*). (113)

Using Eqs. (110) it is easy to obtain the positron energy at resonance:

*<sup>E</sup>*<sup>+</sup> <sup>=</sup> <sup>1</sup> 2

*ωth*

*<sup>i</sup>* � *<sup>ω</sup>th*

of an incident *<sup>γ</sup>*-ray photon is great (*ω<sup>i</sup>* � *<sup>ω</sup>th*

*<sup>i</sup>* is the threshold frequency of an incident *γ*-ray photon,

*<sup>i</sup>* <sup>=</sup> <sup>2</sup>*m*<sup>2</sup>

⎧ ⎨ ⎩ 1 ±

*ω* (1 − cos *θi*)

*<sup>i</sup>* ) are equal *<sup>E</sup>*<sup>+</sup> <sup>=</sup> *<sup>E</sup>*<sup>−</sup> <sup>∼</sup><sup>=</sup> *<sup>ω</sup>th*

*<sup>i</sup>* /4 <sup>≈</sup> *<sup>ω</sup>i*, *<sup>E</sup>*<sup>−</sup> <sup>≈</sup> *<sup>ω</sup>th*

narrow cone with an incident *<sup>γ</sup>*-ray photon *ki*, with *<sup>δ</sup>i*<sup>−</sup> = *<sup>δ</sup>i*<sup>+</sup> and *<sup>θ</sup>i*<sup>−</sup> ∼ *<sup>ω</sup>*/*E*−.

The condition of interference of resonant amplitudes, that is *ω*<sup>+</sup>

threshold value *<sup>W</sup>*±: *<sup>E</sup>*<sup>±</sup> <sup>&</sup>gt; *<sup>W</sup>*<sup>±</sup> <sup>∼</sup> *<sup>m</sup>*2/*ω*.

where *ωth*

threshold (*ω<sup>i</sup>* <sup>−</sup> *<sup>ω</sup>th*

frequency is

has the form

where *S*(±)

**3.2 Resonant amplitude**

absorption (*l* < 0) of laser-wave |*l*| photons

*S*(±)

*<sup>l</sup>* <sup>=</sup> <sup>−</sup>*<sup>i</sup> Ze*3√*<sup>π</sup>*

<sup>√</sup>2*ωiE*−*E*<sup>+</sup>

considerably (*E*<sup>+</sup> <sup>=</sup> *<sup>ω</sup><sup>i</sup>* <sup>−</sup> *<sup>ω</sup>th*

Here, the functions *Bl*<sup>−</sup> (*γ*˜0,*ε*ˆ*i*) and *Bl*<sup>+</sup> (*ε*ˆ*i*, *<sup>γ</sup>*˜0) correspond to the CPP diagrams in Figs. 7(a) and 7(b), respectively

$$\begin{split} \mathcal{B}\_{l-}\left(\tilde{\gamma}\_{0},\mathcal{E}\_{l}\right) &= \sum\_{r=-\infty}^{\infty} \frac{2\omega r \tau^{2}}{\mathbf{q}^{2} + q\_{0}\left(q\_{0} - 2q\_{z}\right)} \cdot \frac{i\pi}{2\left(kq\_{-}\right)} \times \\ &\times \int\_{-\infty}^{\infty} d\phi \mathcal{L}\_{l+r}\left(\phi\right) \exp\left\{i\left(q\_{0}\tau + 2\beta\right)\phi\right\} \cdot \tilde{\gamma}\_{0}\left(\phi\_{-} + m\right) \times \\ &\times \int\_{-\infty}^{\infty} d\phi' \mathcal{F}\_{-r}\left(\phi'\right) \exp\left\{-2i\beta\phi'\right\} \left(\operatorname{sgn}\left(\phi' - \phi\right) - 1\right), \\ &\phi = \frac{q\_{-}^{2} - m^{2}}{4\left(kq\_{-}\right)} \omega\tau. \end{split} \tag{119}$$

Here, functions *F*−*<sup>r</sup>* (*φ*� ) and *Ll*<sup>+</sup>*<sup>r</sup>* (*φ*) are defined by relations (15)-(21) with the replacement (102).

With allowance of the four-momentum conservation law (that is the first Eq. in (105)), the matrix function *F*−*<sup>r</sup> χq*<sup>−</sup> *<sup>p</sup>*<sup>+</sup> , *γq*<sup>−</sup> *<sup>p</sup>*<sup>+</sup> (*φ*� ), *βq*<sup>−</sup> *<sup>p</sup>*<sup>+</sup> (*φ*� ) (15) under resonance conditions defines the amplitude of the production of an electron-positron pair with the four-momenta *q*<sup>−</sup> and *p*<sup>+</sup> by a photon with the four-momentum *ki* in a pulsed light field through *r* wave photons absorption. This process was considered by Nikishov & Ritus (1979) in the case of a plane monochromatic wave, and by Narozhny & Fofanov (1997) in the case of a pulsed light wave. With allowance of the transferred four-momentum *q* (see the second equality in (105)) the quantity *Ll*<sup>+</sup>*<sup>r</sup> χp*−*q*<sup>−</sup> , *γp*−*q*<sup>−</sup> (*φ*), *βp*−*q*<sup>−</sup> (*φ*) *γ*˜0 defines the amplitude of scattering of an intermediate electron with the four-momentum *q*<sup>−</sup> by a nucleus in a pulsed light field with absorption or emission of |*l* + *r*| photons of the wave (Lebed' & Roshchupkin (2008)).

Consequently, if the interference between direct and exchange amplitudes is absent, the process of resonant CPP on a nucleus in a pulsed light field effectively decomposes into two consecutive processes of the first order. The distinction for the diagram (b) on Fig. 7 is concluded in replacement of the four-momentum of an intermediate electron *q*<sup>−</sup> → −*q*<sup>+</sup> and change of sequence of first order processes.

Integral functions (18) are determined by the integer-order Bessel functions (23) for the case of a circularly polarized external wave. For circular polarization of a wave under resonance conditions the arguments of the Bessel functions (20) for CPP on a nucleus may be represented as

$$\gamma\_{\mathfrak{q}-p\_{+}}\left(\phi'\right) = 2r \cdot \eta\_{\mathfrak{0}}\left(\phi'\right) \cdot \frac{1+z\_{+}}{z\_{+}z\_{r}} \sqrt{z\_{+}z\_{r} - \left(1+z\_{+}\right)^{2}}\tag{120}$$

where the invariant parameters *z*+ and *zr* are defined by

$$z\_{+} = \frac{(kp\_{+})}{(kq\_{-})} \approx \frac{E\_{+}}{\omega\_{i} - E\_{+}} , \quad z\_{r} = 2r \cdot \frac{(kk\_{i})}{m^{2}} . \tag{121}$$

It was expected for this part of the amplitude that the Bunkin-Fedorov quantum parameter becomes a classical one under resonance conditions (see Eqs. (32)-(33)).

$$
\gamma\_{q\_-p\_+}\left(\phi'\right) \sim \eta\_0 \ll 1.\tag{122}
$$

performing of corresponding averaging and summation procedures and considering that

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 133

is the differential cross section of scattering of an intermediate electron with the

*z*+*z*<sup>1</sup>

is the probability of production of an electron-positron pair with the four-momenta *q*<sup>−</sup> and *p*<sup>+</sup> by the an incident photon with the four-momentum *ki* at the expense of one wave photon absorption. We can perform integration in Eq. (129) over the azimuthal angle *dϕaz* and *dδ*<sup>2</sup>

<sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>z</sup>*+)

*z*+*z*<sup>1</sup>

Within the kinematical region of resonance, CPP on a nucleus in external field absence was investigated by Baier et al. (1973). It was concluded that amplitudes (a) and (b) (see Fig. 7) have poles within different regions of pair emission angles, therefore, they do not interfere.

(*p*+*q*−) by an incident *γ*-ray photon with the four-momentum *ki*. We express the resonant

<sup>4</sup>*m*Γ*<sup>τ</sup>* (<sup>1</sup> <sup>+</sup> *<sup>z</sup>*+) · *dW*(1)

(*kq*−)

<sup>=</sup> *<sup>ω</sup><sup>i</sup>*

 2 *<sup>π</sup>* · <sup>1</sup> *ωτ* ·

The transit width Γ*<sup>τ</sup>* of the resonance was introduced here. It has the form

Γ*<sup>τ</sup>* =

2

2

 1 + *δ*<sup>2</sup> *i*+ 

<sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>+</sup> *<sup>z</sup>*+)

· *Pres* · *<sup>d</sup>σ<sup>s</sup>* (*q*−) *dW*(1)

*<sup>m</sup>*<sup>2</sup> <sup>+</sup> *<sup>E</sup>*−*q*0<sup>−</sup> <sup>+</sup> **<sup>p</sup>**−**q**<sup>−</sup>

*pairdδ*<sup>2</sup>

<sup>−</sup> <sup>2</sup> <sup>+</sup> (<sup>1</sup> <sup>+</sup> *<sup>z</sup>*+)

*<sup>i</sup>*<sup>+</sup> → *dβ* is to be carried out. The parameter *β* (119) under resonance

· *<sup>d</sup>σ<sup>s</sup>* (*q*−) *dW*(1)

*dσpair* = *dWki* (*p*+, *q*−) · *dσ<sup>S</sup>* (*q*−), (134)

*pair dWki* (*p*+, *q*−)

(*p*+, *q*−) is the probability of production of an electron-positron pair

*z*+

2

· *dz*<sup>+</sup> (1 + *z*+)

. (132)

*pair*. (133)

. (135)

*<sup>m</sup>* . (136)

*<sup>i</sup>*+*dϕaz*. (129)

*d*Ω<sup>−</sup> (130)

<sup>2</sup> (131)

*i*+.

*d*Ω<sup>+</sup> =

Here,

*dW*(1) *pair* = *α*

At that replacement *dδ*<sup>2</sup>

We derive consequently

where *q*<sup>−</sup> = *ki* − *p*+; *dWki*

conditions assumes the form

*m*2/2*E*<sup>2</sup> + *dδ*<sup>2</sup>

*<sup>i</sup>*+*dϕaz* we derive

*<sup>d</sup>σ<sup>s</sup>* (*q*−) <sup>=</sup> <sup>2</sup>*Z*2*<sup>r</sup>*

4 (1 + *z*+)

*z*+*z*<sup>1</sup>

<sup>2</sup>*π*<sup>2</sup> ·

*m*2*ω<sup>i</sup> z*+

> <sup>|</sup>**p**−| *<sup>m</sup>*<sup>2</sup> <sup>|</sup>**q**−| **<sup>q</sup>**<sup>4</sup>

2 *e*

2

*<sup>β</sup>* <sup>=</sup> *ωτ* 2

*<sup>d</sup>σ*(±) <sup>1</sup>*res* =

At that, the cross section is factorized, i.e.

cross section (133) in terms of ordinary one (134),

*Rres* <sup>=</sup> *<sup>d</sup>σ*(±) 1*res dσpair* *π* 2 · *ωτ* <sup>2</sup> · *<sup>ω</sup><sup>i</sup> m*2*z*<sup>1</sup>

*<sup>d</sup>σ*(±) <sup>1</sup>*res* <sup>=</sup> <sup>1</sup>

four-momentum *q*<sup>−</sup> by a nucleus, and

*η*2 0*m*<sup>2</sup> 4*ω<sup>i</sup>*

Consequently within the field range, specified by Eq. (5), the first resonance, that is, the resonance with *r* = 1, provides the main contribution to the resonant cross section, when the Bessel function has the largest value. This implies that the single-photon production of an electron–positron pair in a pulsed field proceeds mainly through absorption of one external field photon. However, the argument of the Bessel function *Jl*<sup>+</sup>*<sup>r</sup> γp*−*q*<sup>−</sup> (*φ*) is of the order of a magnitude *γp*−*q*<sup>−</sup> (*ϕ*) ∼ *γ*<sup>0</sup> 1, i.e. it saves the quantum nature. Thus, scattering of an intermediate electron by a nucleus in the field of a moderately strong pulsed wave is a multiquantum process.

We perform the subsequent analysis for the case of wave circularly polarization (*δ* = ±1) at expense of one wave photon absorption, i.e. *r* = 1. In view of the envelope function (40), after simple manipulations we obtain the amplitude (118) in the form

$$B\_{l-}\left(\tilde{\gamma}\_{0},\hat{\mathbb{H}}\_{i}\right) = \frac{2\pi \cdot \tilde{\gamma}\_{0}\left(\hat{\mathbb{H}}\_{-}+m\right)\hat{\mathbb{H}}}{\mathbf{q}^{2}+q\_{0}\left(q\_{0}-2q\_{z}\right)} \cdot \frac{-i\omega\tau^{2}\sqrt{\pi}}{4\left(kq\_{-}\right)}\exp\left\{-\frac{\beta^{2}}{4}\right\}\cdot I\left(q\_{0\prime}\beta\right),\tag{123}$$

$$\hat{F} = -\frac{1}{2} \exp\{i\chi\_{q-p\_+}\} \cdot \gamma\_{q-p\_+}\left(0\right) \cdot \hat{\varepsilon}\_i + \left(\left(e\_x + i\delta e\_y\right)b\right),\tag{124}$$

$$I(q\_0, \boldsymbol{\beta}) = \int\_{-\infty}^{\infty} d\boldsymbol{\phi} \cdot \boldsymbol{J}\_{l+1} \left(\boldsymbol{\phi}\right) \exp\left\{i\left(q\_0 \boldsymbol{\tau} + 2\boldsymbol{\beta}\right) \boldsymbol{\phi}\right\} \left(\boldsymbol{\text{erf}}\left(2\boldsymbol{\phi} + \frac{i\boldsymbol{\beta}}{2}\right) + 1\right). \tag{125}$$

Here, erf(2*φ* + *iβ*/2) is the error function.

#### **3.3 Resonant cross section**

The differential cross section of CPP on a nucleus in a pulsed light field may be easily obtained by standard mode (Berestetskii et al. (1982)) from the amplitude, Eqs. (116)-(117), (123)-(125)

$$d\sigma^{(\pm)} = \sum\_{l=-\infty}^{\infty} d\sigma\_l^{(\pm)} \, , \tag{126}$$

where *<sup>d</sup>σ*(±) *<sup>l</sup>* is the partial cross section of CPP on a nucleus in a pulsed light field with emission (*l* > 0) or absorption (*l* < 0) of |*l*| wave photons.

Under resonance conditions and for ultrarelativistic electron and positron energies, the energy contribution from external pulsed field photons may be neglected. Therefore, the resonant cross section (126) may be summed over all possible partial processes. Thus, the differential cross section of CPP on a nucleus in a pulsed light field with the positron energy in the interval [*E*+, *E*+ + *dE*+] within the solid angle [Ω+, Ω+ + *d*Ω+] and the final electron within the solid angle [Ω−, Ω<sup>−</sup> + *d*Ω−] assumes the form

$$\frac{d\sigma\_{\rm 1res}^{(\pm)}}{dE\_{+}d\Omega\_{+}d\Omega\_{-}} = \frac{Z^{2}e^{6}}{(2\pi)^{2}} \frac{|\mathbf{p}\_{-}| \left|\mathbf{p}\_{+}|}{\omega\_{i}\mathbf{q}^{4}} \left|\bar{u}\_{-}M\_{-}u\_{+}\right|^{2} \cdot \mathcal{P}\_{\rm res} \tag{127}$$

$$M\_- = \tilde{\gamma}\_0 \left( \mathfrak{f}\_- + m \right) \hat{\mathcal{F}}.\tag{128}$$

In Eq. (127) the function *Pres* is defined by the expression (53), where the replacement *qi* → *q*<sup>−</sup> has to be performed. We don't take polarization effects into consideration. After performing of corresponding averaging and summation procedures and considering that *d*Ω<sup>+</sup> = *m*2/2*E*<sup>2</sup> + *dδ*<sup>2</sup> *<sup>i</sup>*+*dϕaz* we derive

$$d\sigma\_{1\text{res}}^{(\pm)} = \frac{1}{2\pi^2} \cdot \frac{m^2 \omega\_{\text{i}}}{z\_{\text{+}}} \cdot P\_{\text{res}} \cdot d\sigma\_{\text{s}}\left(q\_{-}\right) \, d\mathcal{W}\_{\text{pair}}^{(1)} d\delta\_{\text{i}+}^2 d\varphi\_{a\text{z}}.\tag{129}$$

Here,

26 Will-be-set-by-IN-TECH

Consequently within the field range, specified by Eq. (5), the first resonance, that is, the resonance with *r* = 1, provides the main contribution to the resonant cross section, when the Bessel function has the largest value. This implies that the single-photon production of an electron–positron pair in a pulsed field proceeds mainly through absorption of one external

of a magnitude *γp*−*q*<sup>−</sup> (*ϕ*) ∼ *γ*<sup>0</sup> 1, i.e. it saves the quantum nature. Thus, scattering of an intermediate electron by a nucleus in the field of a moderately strong pulsed wave is a

We perform the subsequent analysis for the case of wave circularly polarization (*δ* = ±1) at expense of one wave photon absorption, i.e. *r* = 1. In view of the envelope function (40), after

<sup>2</sup> exp{*iχq*<sup>−</sup> *<sup>p</sup>*<sup>+</sup> } · *<sup>γ</sup>q*<sup>−</sup> *<sup>p</sup>*<sup>+</sup> (0) · *<sup>ε</sup>*ˆ*<sup>i</sup>* <sup>+</sup> *ex* <sup>+</sup> *<sup>i</sup>δey*

The differential cross section of CPP on a nucleus in a pulsed light field may be easily obtained by standard mode (Berestetskii et al. (1982)) from the amplitude, Eqs. (116)-(117), (123)-(125)

> ∞ ∑ *l*=−∞

Under resonance conditions and for ultrarelativistic electron and positron energies, the energy contribution from external pulsed field photons may be neglected. Therefore, the resonant cross section (126) may be summed over all possible partial processes. Thus, the differential cross section of CPP on a nucleus in a pulsed light field with the positron energy in the interval [*E*+, *E*+ + *dE*+] within the solid angle [Ω+, Ω+ + *d*Ω+] and the final electron within the solid


In Eq. (127) the function *Pres* is defined by the expression (53), where the replacement *qi* → *q*<sup>−</sup> has to be performed. We don't take polarization effects into consideration. After

*<sup>ω</sup>i***q**<sup>4</sup> <sup>|</sup>*u*¯−*M*−*u*+<sup>|</sup>

*<sup>M</sup>*<sup>−</sup> <sup>=</sup> *<sup>γ</sup>*˜0 (*q*ˆ<sup>−</sup> <sup>+</sup> *<sup>m</sup>*) *<sup>F</sup>*ˆ. (128)

*<sup>d</sup>σ*(±)

*<sup>l</sup>* is the partial cross section of CPP on a nucleus in a pulsed light field with

*dφ* · *Jl*<sup>+</sup><sup>1</sup> (*φ*) exp {*i*(*q*0*τ* + 2*β*) *φ*}

*dσ*(±) =

<sup>=</sup> *<sup>Z</sup>*2*e*<sup>6</sup> (2*π*) 2 <sup>−</sup>*iωτ*2√*<sup>π</sup>*

<sup>4</sup> (*kq*−) exp{− *<sup>β</sup>*<sup>2</sup>

 erf 2*φ* + *iβ* 2 + 1 

 *b* 

*<sup>l</sup>* , (126)

<sup>2</sup> · *Pres*, (127)

*γp*−*q*<sup>−</sup> (*φ*)

<sup>4</sup> } · *<sup>I</sup>* (*q*0, *<sup>β</sup>*), (123)

, (124)

. (125)

is of the order

field photon. However, the argument of the Bessel function *Jl*<sup>+</sup>*<sup>r</sup>*

simple manipulations we obtain the amplitude (118) in the form

**<sup>q</sup>**<sup>2</sup> <sup>+</sup> *<sup>q</sup>*<sup>0</sup> (*q*<sup>0</sup> <sup>−</sup> <sup>2</sup>*qz*) ·

*Bl*<sup>−</sup> (*γ*˜0,*ε*ˆ*i*) <sup>=</sup> <sup>2</sup>*<sup>π</sup>* · *<sup>γ</sup>*˜0 (*q*ˆ<sup>−</sup> <sup>+</sup> *<sup>m</sup>*) *<sup>F</sup>*<sup>ˆ</sup>

emission (*l* > 0) or absorption (*l* < 0) of |*l*| wave photons.

*<sup>d</sup>σ*(±) 1*res dE*+*d*Ω+*d*Ω<sup>−</sup>

*<sup>F</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

∞

−∞

Here, erf(2*φ* + *iβ*/2) is the error function.

angle [Ω−, Ω<sup>−</sup> + *d*Ω−] assumes the form

*I* (*q*0, *β*) =

**3.3 Resonant cross section**

where *<sup>d</sup>σ*(±)

multiquantum process.

$$d\sigma\_s \left( q\_- \right) = 2Z^2 r\_\varepsilon^2 \frac{|\mathbf{p}\_-| \, m^2}{|\mathbf{q}\_-| \, \mathbf{q}^4} \left( m^2 + E\_- q\_{0-} + \mathbf{p}\_- \mathbf{q}\_- \right) d\Omega\_- \tag{130}$$

is the differential cross section of scattering of an intermediate electron with the four-momentum *q*<sup>−</sup> by a nucleus, and

$$dW\_{pair}^{(1)} = a \frac{\eta\_0^2 m^2}{4\omega\_i} \left\{ \frac{4\left(1 + z\_+\right)^2}{z\_+ z\_1} \left(1 - \frac{\left(1 + z\_+\right)^2}{z\_+ z\_1}\right) - 2 + \frac{\left(1 + z\_+\right)^2}{z\_+} \right\} \cdot \frac{dz\_+}{\left(1 + z\_+\right)^2} \tag{131}$$

is the probability of production of an electron-positron pair with the four-momenta *q*<sup>−</sup> and *p*<sup>+</sup> by the an incident photon with the four-momentum *ki* at the expense of one wave photon absorption. We can perform integration in Eq. (129) over the azimuthal angle *dϕaz* and *dδ*<sup>2</sup> *i*+. At that replacement *dδ*<sup>2</sup> *<sup>i</sup>*<sup>+</sup> → *dβ* is to be carried out. The parameter *β* (119) under resonance conditions assumes the form

$$\beta = \frac{\omega \tau}{2} \left[ 1 - \frac{(1 + z\_+)^2}{z\_+ z\_1} \left( 1 + \delta\_{i+}^2 \right) \right]. \tag{132}$$

We derive consequently

$$d\sigma\_{1\text{res}}^{(\pm)} = \sqrt{\frac{\pi}{2}} \cdot \frac{\omega \tau}{2} \cdot \frac{\omega\_{\text{i}}}{m^2 z\_1} \cdot d\sigma\_{\text{s}}\left(q\_{-}\right) d\mathcal{W}\_{\text{pair}}^{(1)}.\tag{133}$$

Within the kinematical region of resonance, CPP on a nucleus in external field absence was investigated by Baier et al. (1973). It was concluded that amplitudes (a) and (b) (see Fig. 7) have poles within different regions of pair emission angles, therefore, they do not interfere. At that, the cross section is factorized, i.e.

$$d\sigma\_{pair} = d\mathcal{W}\_{k\_i} \left(p\_{+\prime}q\_{-}\right) \cdot d\sigma\_{\mathcal{S}\_-} \left(q\_{-}\right),\tag{134}$$

where *q*<sup>−</sup> = *ki* − *p*+; *dWki* (*p*+, *q*−) is the probability of production of an electron-positron pair (*p*+*q*−) by an incident *γ*-ray photon with the four-momentum *ki*. We express the resonant cross section (133) in terms of ordinary one (134),

$$R\_{\rm res} = \frac{d\sigma\_{\rm 1res}^{(\pm)}}{d\sigma\_{\rm pair}} = \frac{\omega\_{\rm i}}{4m\Gamma\_{\rm \tau}\left(1 + z\_{+}\right)} \cdot \frac{d\mathcal{W}\_{\rm pair}^{(1)}}{d\mathcal{W}\_{\rm k\_{\parallel}}\left(p\_{+}, q\_{-}\right)}.\tag{135}$$

The transit width Γ*<sup>τ</sup>* of the resonance was introduced here. It has the form

$$
\Gamma\_{\pi} = \sqrt{\frac{2}{\pi}} \cdot \frac{1}{\omega \tau} \cdot \frac{(kq\_{-})}{m}. \tag{136}
$$

a positron - Bhabha (1938), the scattering of an electron by a muon - by Bhabha (1938) and Massey & Corben (1939). The detailed consideration of nonresonant scattering of an electron

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 135

We underline that the Bunkin–Fedorov quantum parameter *γ*<sup>0</sup> (4) is the main one which determines multiphoton processes in leptons nonresonant scattering. However in the case of leptons resonant scattering the influence of the quantum parameter *γ*<sup>0</sup> does not appear (it becomes a classical one due to resonance conditions and possess the values in order to *η*0), thus the classical parameter *η*<sup>0</sup> (3) determines multiphoton processes. Therefore study of lepton by a lepton resonant scattering is carried out within the intensity range (5), that is within the framework of the first order of the perturbation theory with respect to an external

The electron mass *me* is considerably less than the muon one *m<sup>μ</sup>* (*me* � *mμ*), therefore the

The classical parameters *η*0*<sup>μ</sup>* and *η*0*<sup>e</sup>* are defined by Eqs. (3), where replacements *m* → *m<sup>μ</sup>* and *m* → *me* are to be performed. Hereinafter we consider resonances for direct Feynman diagrams of scattering type exceptionally (Fig. 8). Exchange diagrams for identical leptons and annihilation diagrams of scattering of a lepton by an antilepton are outside of attention. Such a problem statement is possible due to fact that resonances for direct diagrams of scattering type and resonances for exchange (annihilation) diagrams within the intensity range (5) occur within essentially different nonoverlapping kinematical regions (Roshchupkin & Voroshilo (2008)). For direct scattering amplitude within the fields range (5) the process of lepton by a lepton resonant scattering occurs when leptons scatter forwards into the small angles in the frame of the reference related to the center of inertia of initial particles and effectively decomposes into two processes of the first order similar to the Compton scattering

*<sup>ψ</sup>*¯ *<sup>p</sup>*�

(*x* |*A* ) are the wave functions of initial and final leptons in the field of a pulsed light

Here, *Dμμ*� (*x*<sup>1</sup> − *x*2) is the Green function of an intermediate free photon; *ψpj* (*x* |*A* ) and

The amplitude of scattering of a lepton *l*<sup>1</sup> (with the mass *m*<sup>1</sup> and the four-momentum *p*1) by a lepton *l*<sup>2</sup> (with the mass *m*<sup>2</sup> and the four-momentum *p*2) in a pulsed light field may be represented as a sum of partial components with emission (*l* > 0) and absorption (*l* < 0) |*l*|

> ∞ ∑ *l*=−∞

*δ* (*qx*) *δ qy* 

*S* =

<sup>4</sup> *ie*<sup>2</sup>

*E*1*E*2*E*� 1*E*� 2 <sup>2</sup> (*x*<sup>2</sup> <sup>|</sup>*<sup>A</sup>* ) *<sup>γ</sup>*˜*μ*�

*η*0*<sup>μ</sup>* � *η*0*e*. (140)

*ψp*<sup>2</sup> (*x*<sup>2</sup> |*A* )

*Sl*, (142)

*δ* (*q*<sup>0</sup> − *qz*) *Dls*. (143)

. (141)

by a muon in a pulsed light field was performed by Padusenko et al. (2009).

corresponding classical parameters (3) satisfy the following condition as well

The *S*-matrix element for a direct amplitude (see Fig. 8) is given by

*<sup>S</sup>* <sup>=</sup> *ie*<sup>2</sup> *<sup>d</sup>*4*x*1*d*4*x*2*Dμμ*� (*x*<sup>1</sup> <sup>−</sup> *<sup>x</sup>*2)<sup>×</sup>

<sup>1</sup> (*x*<sup>1</sup> <sup>|</sup>*<sup>A</sup>* ) *<sup>γ</sup>*˜*μψ<sup>p</sup>*<sup>1</sup> (*x*<sup>1</sup> <sup>|</sup>*<sup>A</sup>* )

*Sl* <sup>=</sup> (2*π*)

2 

laser field.

of a wave by a lepton.

*ψ*¯ *p*� *j*

wave photons:

× *ψ*¯ *p*�

wave (2), respectively (*j* = 1, 2).

It is obvious from Eq. (136) that the transit width is specified by the pulsed field frequency and duration as well as by the particle energy and process kinematics. We underline that when CPP on a nucleus in the field of a plane monochromatic wave is studied the divergence in the differential cross section is eliminated by introducing of radiative corrections into the Green function of an intermediate particle according to the Breit-Wigner prescription as usual. It is concluded in addition of the imaginary part of the electron or positron mass: *m* → *m* − *i*Γ*R*. Here, the radiation width of resonance Γ*<sup>R</sup>* is introduced phenomenologically. It has the form

$$
\Gamma\_R = \frac{1}{3} \alpha \eta\_0^2 \cdot \frac{\sigma\_c \left(q\_-\right)}{\sigma\_T} \cdot \frac{\left(kq\_-\right)}{m} \,. \tag{137}
$$

where *σ<sup>c</sup>* (*q*−) is the total cross section of the Compton scattering of an external field photon by an intermediate electron with the four-momentum *q*<sup>−</sup> (it is the most probable channel of an electron escape from an intermediate state), and *σ<sup>T</sup>* is the Thompson cross section. Comparison of resonant widths (136) and (137) ascertains that the transit width exceeds the radiation one if laser pulse parameters satisfy the condition

$$
\omega \tau < \frac{3}{a\eta\_0^2} \cdot \frac{\sigma\_c \left(q\_{-}\right)}{\sigma\_T}. \tag{138}
$$

Moderately strong fields of optical frequencies and the picosecond range of widths meet the inequality (138). The titanium-sapphire laser (Ti:Sapphire) or the solid-state laser based on aluminum-yttrium garnet Y2Al5O12 with neodymium Nd admixtures (Nd:YAG) can be used as sources of such pulsed fields. Titanium-sapphire lasers have a broad lasing band (700-1100 nm) and a wide range of pulse duration (10 ps –10 fs) due to various choices of pulse compression. The PICAR picosecond Nd:YAG laser (designed at the International Educational-Scientific Laser Center of the Moscow State University named by M.V. Lomonosov) appropriate field characteristics to be achieved through the combined action of active-passive mode locking and a negative feedback (Gorbunkov et al. (2005)).

Ratio of cross-sections (135) is simplified considerably in the logarithmic approximation:

$$R\_{\rm res} = \frac{\pi}{8} \sqrt{\frac{\pi}{2}} \cdot \eta\_0^2 \omega \tau \cdot \left[ \ln \frac{E\_+}{m} \right]^{-1} \,. \tag{139}$$

Let us estimate the ratio of the cross sections (139) for PICAR picosecond Nd:YAG laser with additional amplifiers with parameters *η*<sup>0</sup> ≈ 0.1, *λ* = 1064 nm (*ω* = 1.17 eV), *τ* = 25 ps. An incident *<sup>γ</sup>*-ray photon with an energy near the threshold value (113) *<sup>ω</sup><sup>i</sup>* <sup>=</sup> <sup>5</sup> · <sup>10</sup>5*<sup>m</sup>* <sup>=</sup> 255 GeV propagates towards the pulsed laser wave. We obtain the following ratio of cross-sections: *Rres* ≈ 40. Consequently, the resonant cross-section of CPP on a nucleus in a pulsed light field may exceed the corresponding one in external field absence by an order of magnitude.

#### **4. Resonant scattering of a lepton by a lepton in the pulsed light field**

Study of various processes of leptons scattering in an external electromagnetic fields is one of the fundamental directions of QED. Cross sections of basic scattering processes in the external field absence were obtained in the middle of the twentieth century. Thus, the scattering of an electron by an electron was considered by Möller (1932), the scattering of an electron by 28 Will-be-set-by-IN-TECH

It is obvious from Eq. (136) that the transit width is specified by the pulsed field frequency and duration as well as by the particle energy and process kinematics. We underline that when CPP on a nucleus in the field of a plane monochromatic wave is studied the divergence in the differential cross section is eliminated by introducing of radiative corrections into the Green function of an intermediate particle according to the Breit-Wigner prescription as usual. It is concluded in addition of the imaginary part of the electron or positron mass: *m* → *m* − *i*Γ*R*. Here, the radiation width of resonance Γ*<sup>R</sup>* is introduced phenomenologically. It has the form

> *σ<sup>c</sup>* (*q*−) *σT*

where *σ<sup>c</sup>* (*q*−) is the total cross section of the Compton scattering of an external field photon by an intermediate electron with the four-momentum *q*<sup>−</sup> (it is the most probable channel of an electron escape from an intermediate state), and *σ<sup>T</sup>* is the Thompson cross section. Comparison of resonant widths (136) and (137) ascertains that the transit width exceeds the

> 3 *αη*<sup>2</sup> 0 ·

Moderately strong fields of optical frequencies and the picosecond range of widths meet the inequality (138). The titanium-sapphire laser (Ti:Sapphire) or the solid-state laser based on aluminum-yttrium garnet Y2Al5O12 with neodymium Nd admixtures (Nd:YAG) can be used as sources of such pulsed fields. Titanium-sapphire lasers have a broad lasing band (700-1100 nm) and a wide range of pulse duration (10 ps –10 fs) due to various choices of pulse compression. The PICAR picosecond Nd:YAG laser (designed at the International Educational-Scientific Laser Center of the Moscow State University named by M.V. Lomonosov) appropriate field characteristics to be achieved through the combined action

· (*kq*−)

*σ<sup>c</sup>* (*q*−) *σT*

−<sup>1</sup>

*<sup>m</sup>* , (137)

. (138)

. (139)

<sup>Γ</sup>*<sup>R</sup>* <sup>=</sup> <sup>1</sup> 3 *αη*<sup>2</sup> 0 ·

*ωτ* <

of active-passive mode locking and a negative feedback (Gorbunkov et al. (2005)).

**4. Resonant scattering of a lepton by a lepton in the pulsed light field**

*Rres* <sup>=</sup> *<sup>π</sup>* 8 *π* <sup>2</sup> ·*η*<sup>2</sup> <sup>0</sup>*ωτ* · ln *<sup>E</sup>*<sup>+</sup> *m*

Ratio of cross-sections (135) is simplified considerably in the logarithmic approximation:

Let us estimate the ratio of the cross sections (139) for PICAR picosecond Nd:YAG laser with additional amplifiers with parameters *η*<sup>0</sup> ≈ 0.1, *λ* = 1064 nm (*ω* = 1.17 eV), *τ* = 25 ps. An incident *<sup>γ</sup>*-ray photon with an energy near the threshold value (113) *<sup>ω</sup><sup>i</sup>* <sup>=</sup> <sup>5</sup> · <sup>10</sup>5*<sup>m</sup>* <sup>=</sup> 255 GeV propagates towards the pulsed laser wave. We obtain the following ratio of cross-sections: *Rres* ≈ 40. Consequently, the resonant cross-section of CPP on a nucleus in a pulsed light field may exceed the corresponding one in external field absence by an order of magnitude.

Study of various processes of leptons scattering in an external electromagnetic fields is one of the fundamental directions of QED. Cross sections of basic scattering processes in the external field absence were obtained in the middle of the twentieth century. Thus, the scattering of an electron by an electron was considered by Möller (1932), the scattering of an electron by

radiation one if laser pulse parameters satisfy the condition

a positron - Bhabha (1938), the scattering of an electron by a muon - by Bhabha (1938) and Massey & Corben (1939). The detailed consideration of nonresonant scattering of an electron by a muon in a pulsed light field was performed by Padusenko et al. (2009).

We underline that the Bunkin–Fedorov quantum parameter *γ*<sup>0</sup> (4) is the main one which determines multiphoton processes in leptons nonresonant scattering. However in the case of leptons resonant scattering the influence of the quantum parameter *γ*<sup>0</sup> does not appear (it becomes a classical one due to resonance conditions and possess the values in order to *η*0), thus the classical parameter *η*<sup>0</sup> (3) determines multiphoton processes. Therefore study of lepton by a lepton resonant scattering is carried out within the intensity range (5), that is within the framework of the first order of the perturbation theory with respect to an external laser field.

The electron mass *me* is considerably less than the muon one *m<sup>μ</sup>* (*me* � *mμ*), therefore the corresponding classical parameters (3) satisfy the following condition as well

$$
\eta\_{0\mu} \ll \eta\_{0\epsilon}.\tag{140}
$$

The classical parameters *η*0*<sup>μ</sup>* and *η*0*<sup>e</sup>* are defined by Eqs. (3), where replacements *m* → *m<sup>μ</sup>* and *m* → *me* are to be performed. Hereinafter we consider resonances for direct Feynman diagrams of scattering type exceptionally (Fig. 8). Exchange diagrams for identical leptons and annihilation diagrams of scattering of a lepton by an antilepton are outside of attention. Such a problem statement is possible due to fact that resonances for direct diagrams of scattering type and resonances for exchange (annihilation) diagrams within the intensity range (5) occur within essentially different nonoverlapping kinematical regions (Roshchupkin & Voroshilo (2008)). For direct scattering amplitude within the fields range (5) the process of lepton by a lepton resonant scattering occurs when leptons scatter forwards into the small angles in the frame of the reference related to the center of inertia of initial particles and effectively decomposes into two processes of the first order similar to the Compton scattering of a wave by a lepton.

The *S*-matrix element for a direct amplitude (see Fig. 8) is given by

$$\begin{array}{l} S = ie^2 \int d^4 \mathbf{x}\_1 d^4 \mathbf{x}\_2 D\_{\mu\mu'} \left( \mathbf{x}\_1 - \mathbf{x}\_2 \right) \times \\ \times \left[ \bar{\psi}\_{p\_1'} \left( \mathbf{x}\_1 \, \middle| \, A \right) \bar{\gamma}^{\mu} \psi\_{p\_1} \left( \mathbf{x}\_1 \, \middle| \, A \right) \right] \left[ \bar{\psi}\_{p\_2'} \left( \mathbf{x}\_2 \, \middle| \, A \right) \bar{\gamma}^{\mu'} \psi\_{p\_2} \left( \mathbf{x}\_2 \, \middle| \, A \right) \right]. \end{array} \tag{141}$$

Here, *Dμμ*� (*x*<sup>1</sup> − *x*2) is the Green function of an intermediate free photon; *ψpj* (*x* |*A* ) and *ψ*¯ *p*� *j* (*x* |*A* ) are the wave functions of initial and final leptons in the field of a pulsed light wave (2), respectively (*j* = 1, 2).

The amplitude of scattering of a lepton *l*<sup>1</sup> (with the mass *m*<sup>1</sup> and the four-momentum *p*1) by a lepton *l*<sup>2</sup> (with the mass *m*<sup>2</sup> and the four-momentum *p*2) in a pulsed light field may be represented as a sum of partial components with emission (*l* > 0) and absorption (*l* < 0) |*l*| wave photons:

$$S = \sum\_{l=-\infty}^{\infty} S\_{l\nu} \tag{142}$$

$$S\_{I} = \frac{(2\pi)^{4}ie^{2}}{2\sqrt{E\_{1}E\_{2}E\_{1}'E\_{2}'}}\delta\left(q\_{X}\right)\delta\left(q\_{Y}\right)\delta\left(q\_{0} - q\_{z}\right)D\_{ls}.\tag{143}$$

Here, *<sup>ε</sup>*ˆ<sup>±</sup> is the compression of four-vectors *<sup>ε</sup>*<sup>±</sup> <sup>=</sup> *ex* <sup>±</sup> *<sup>i</sup>δey* with the Dirac's *<sup>γ</sup>*˜ *<sup>ν</sup>*-matrices. The expression for the function *Gs<sup>ν</sup>* (*ϕ*2) is ensued from Eqs. (148)-(149) by following indices replacement: 1 → 2, *l* − *s* → *s*, *l* − *s* ± 1 → *s* ± 1, *l* − *s* ± 2 → *s* ± 2 and by the index *ν* omission

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 137

*<sup>κ</sup><sup>j</sup>* <sup>=</sup> *Ej* <sup>−</sup> **np***<sup>j</sup>*

**<sup>n</sup>** <sup>=</sup> **<sup>k</sup>** |**k**|

There are the integral functions *Ll*−*<sup>s</sup>* (*φ*1), *Ls* (*φ*2) which determine probability of emission

 *ex gj* 2 + *δ*<sup>2</sup> *ey gj* 2

, *gj* <sup>=</sup> *<sup>p</sup>*�

Before performing of integration of the function *Dls* (145) over the variable *ζ* we remind that the subject of studying is the resonant character of amplitude behavior caused by quasi discrete structure: charged particle + plane electromagnetic wave. It is obvious that the resonant character of lepton-lepton scattering occurs when the denominator of the function *Dls* approaches zero. We should underline that the possibility of lepton-lepton resonant scattering in a pulsed light field is provided by the both energy (with accuracy *q*<sup>0</sup> 1/*τ* � *ω*) and momentum conservation laws fulfillment. Thus, the squared four-momentum of an

shell, i.e. an intermediate virtual photon becomes a real one. In this case the correction to the intermediate photon squared four-momentum in the denominator of the expression (145) is caused by the external field pulsed character and is essential through integration of the

function *Dls* (145) over the variable *ζ*. Hence, the following correlation is valid

*q*�2 <sup>1</sup> *kq*� 1 

*j κ*� *j* − *pj κj*

> 1 *κ*� *j* − 1 *κj*

<sup>1</sup> vanishes. It implies that the considered particle falls within the mass

*ωτ* . (157)

*κ*� *<sup>j</sup>* = *E*�

Here, **n** is the unit vector along the direction of external wave propagation

and absorption of external wave photons in Eqs. for *G<sup>ν</sup>*

(*j* = 1 for *n* = *l* − *s*, *j* = 2 for *n* = *s*) with the arguments

*tgχ<sup>j</sup>* = *δ*

 *ey gj* 

 *ex gj*

*γ*0*<sup>j</sup> φj* = *η*0*<sup>j</sup> φj* · *mj ω*

> *βj φj* = <sup>1</sup> <sup>−</sup> *<sup>δ</sup>*<sup>2</sup> *η*2 *j φj <sup>m</sup>*<sup>2</sup> *j* 8*ω*

,

*<sup>j</sup>* − **np**� *j*

*<sup>l</sup>*−*<sup>s</sup>* (*φ*1) and *Gs<sup>ν</sup>* (*ϕ*2) the following expressions are

. (151)

. (152)

*<sup>l</sup>*−*<sup>s</sup>* (*φ*1), *Gs<sup>ν</sup>* (*φ*2). They have the form

*<sup>j</sup>* − *nϕ*� *j* (153)

, (154)

, (155)

. (156)

*<sup>j</sup>* in functions *<sup>G</sup><sup>ν</sup>*

also. By means of *κ<sup>j</sup>* and *κ*�

*Ln φj* ≡ *Ln χj*, *γ*0*<sup>j</sup> φj* , *β<sup>j</sup> φj* <sup>=</sup>

= <sup>1</sup> 2*π* 2 *π* 0 *dϕ*� *<sup>j</sup>* exp *i γ*0*<sup>j</sup> φj* sin *ϕ*� *<sup>j</sup>* − *χ<sup>j</sup>* + *β<sup>j</sup> φj* sin 2*ϕ*�

intermediate photon *q*�

denoted

Fig. 8. The Feynman diagram of direct amplitude of scattering of a lepton *l*<sup>1</sup> by a lepton *l*<sup>2</sup> in the field of a pulsed light wave. External incoming and outgoing double lines correspond to the wave functions of leptons in initial and final states in the field of a plane wave (the Volkov functions), and an inner dashed line corresponds to a Green function of a free photon.

Here, the arguments of delta-functions are the four-vector *q* = (*q*0, **q**) components

$$q = p\_1' + p\_2' - p\_1 - p\_2 + lk.\tag{144}$$

The function *Dls* in Eq. (143) has the form

$$D\_{ls} = \sum\_{s=-\infty}^{\infty} \int\_{-\infty}^{\infty} \frac{\left(\overline{u}\_{p\_1'} \Lambda\_{l-s}^{\nu} \left(\zeta\right) u\_{p\_1}\right) \left(\overline{u}\_{p\_2'} \Lambda\_{\text{sV}} \left(\zeta\right) u\_{p\_2}\right)}{q\_1'^2 + 2\zeta \left(k q\_1' \right) + i0} d\zeta. \tag{145}$$

Here, *q*� <sup>1</sup> is the four-vector of an intermediate photon

$$q\_1' = p\_2' - p\_2 + sk = p\_1 - p\_1' + (l - s) \,\, k\_\prime \tag{146}$$

and functions Λ*<sup>ν</sup> <sup>l</sup>*−*<sup>s</sup>* (*ζ*), <sup>Λ</sup>*s<sup>ν</sup>* (*ζ*) are represented by

$$\begin{cases} \Lambda\_{l-s}^{\nu}(\boldsymbol{\zeta}) = \mathop{\pi} \int\_{0}^{\infty} d\boldsymbol{\phi}\_{1} \cdot \mathbb{G}\_{l-s}^{\nu}(\boldsymbol{\phi}\_{1}) \cdot \exp\left\{i\frac{q\_{0}+q\_{z}}{2}\tau\boldsymbol{\phi}\_{1}\right\} \cdot \exp\left\{-i\left(\boldsymbol{\zeta}\boldsymbol{\omega}\tau\right)\boldsymbol{\phi}\_{1}\right\},\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{(147)}\\ \Lambda\_{\text{SV}}\left(\boldsymbol{\zeta}\right) = \mathop{\pi} \int\_{-\infty}^{\infty} d\boldsymbol{\phi}\_{2} \cdot \mathbb{G}\_{\text{SV}}\left(\boldsymbol{\phi}\_{2}\right) \cdot \exp\left\{i\left(\boldsymbol{\zeta}\boldsymbol{\omega}\tau\right)\boldsymbol{\phi}\_{2}\right\}. \end{cases} \tag{147}$$

Functions *G<sup>ν</sup> <sup>l</sup>*−*<sup>s</sup>* (*φ*1) in Eq. (147) have the form

$$\begin{split} &G^{\nu}\_{l-s}\left(\boldsymbol{\Phi}\_{1}\right) = a^{\nu}L\_{l-s}\left(\boldsymbol{\Phi}\_{1}\right) + \eta\_{01}\left(\boldsymbol{\Phi}\_{1}\right)\frac{m\_{1}}{4\omega\kappa\_{1}}\hat{\gamma}^{\nu}\hat{k}\left[\mathbb{E}\left.L\_{l-s+1}\left(\boldsymbol{\Phi}\_{1}\right) + \mathbb{E}\_{l}L\_{l-s-1}\left(\boldsymbol{\Phi}\_{1}\right)\right] + \\ &+ \eta\_{01}\left(\boldsymbol{\Phi}\_{1}\right)\frac{m\_{1}}{4\omega\kappa\_{1}^{\nu}}\left[\mathbb{E}\left.L\_{l-s+1}\left(\boldsymbol{\Phi}\_{1}\right) + \mathbb{E}\_{l}L\_{l-s-1}\left(\boldsymbol{\Phi}\_{1}\right)\right]\hat{k}\hat{\gamma}^{\nu} + \\ &+ \left(1 - \delta^{2}\right)\eta\_{01}^{2}\left(\boldsymbol{\Phi}\_{1}\right)\frac{m\_{1}^{2}}{8\omega^{2}\kappa\_{1}\kappa\_{1}^{\nu}}k^{\nu}\hat{k}\left[\left(L\_{l-s+2}\left(\boldsymbol{\Phi}\_{1}\right) + L\_{l-s+2}\left(\boldsymbol{\Phi}\_{1}\right)\right)\right], \end{split} \tag{148}$$

$$a^{\nu} = \tilde{\gamma}^{\nu} + \left(1 + \delta^2\right) \eta\_{01}^2(\phi\_1) \frac{m\_1^2}{4\omega^2 \kappa\_1 \kappa\_1'} \mathbf{k}^{\nu} \hat{\mathbf{k}},\tag{149}$$

*η*0*<sup>j</sup>* � *φj* � = *η*0*<sup>j</sup>* · *g* � *φj* � . (150) 30 Will-be-set-by-IN-TECH

1 *p*c

1 *q*c

<sup>2</sup> *p* <sup>2</sup> *p*c

Fig. 8. The Feynman diagram of direct amplitude of scattering of a lepton *l*<sup>1</sup> by a lepton *l*<sup>2</sup> in the field of a pulsed light wave. External incoming and outgoing double lines correspond to the wave functions of leptons in initial and final states in the field of a plane wave (the Volkov functions), and an inner dashed line corresponds to a Green function of a free photon.

*<sup>l</sup>*−*<sup>s</sup>* (*ζ*) *up*<sup>1</sup>

*q*�2

<sup>2</sup> − *p*<sup>2</sup> + *sk* = *p*<sup>1</sup> − *p*�

� *i q*<sup>0</sup> + *qz* <sup>2</sup> *τφ*<sup>1</sup>

� � *u*¯ *p*� 2

<sup>1</sup> + 2*ζ* (*kq*�

Here, the arguments of delta-functions are the four-vector *q* = (*q*0, **q**) components

<sup>1</sup> + *p*�

*q* = *p*�

� *u*¯ *p*� 1 Λ*<sup>ν</sup>*

The function *Dls* in Eq. (143) has the form

Here, *q*�

and functions Λ*<sup>ν</sup>*

⎧ ⎪⎪⎪⎪⎨

Λ*ν*

<sup>+</sup>*η*<sup>01</sup> (*φ*1) *<sup>m</sup>*<sup>1</sup>

*<sup>l</sup>*−*<sup>s</sup>* (*ζ*) <sup>=</sup> *<sup>τ</sup>*

<sup>Λ</sup>*s<sup>ν</sup>* (*ζ*) = *<sup>τ</sup>* �

4*ωκ*� 1

⎪⎪⎪⎪⎩

Functions *G<sup>ν</sup>*

*Gν*

+ � <sup>1</sup> <sup>−</sup> *<sup>δ</sup>*<sup>2</sup> � *η*2 *Dls* =

�∞

−∞

∞

−∞

*<sup>l</sup>*−*<sup>s</sup>* (*φ*1) <sup>=</sup> *<sup>a</sup><sup>ν</sup>Ll*−*<sup>s</sup>* (*φ*1) <sup>+</sup> *<sup>η</sup>*<sup>01</sup> (*φ*1) *<sup>m</sup>*<sup>1</sup>

<sup>01</sup> (*φ*1) *<sup>m</sup>*<sup>2</sup>

∞ ∑ *s*=−∞

<sup>1</sup> is the four-vector of an intermediate photon

*<sup>d</sup>φ*<sup>1</sup> · *<sup>G</sup><sup>ν</sup>*

*<sup>l</sup>*−*<sup>s</sup>* (*φ*1) in Eq. (147) have the form

1 8*ω*2*κ*1*κ*� 1 *k<sup>ν</sup>* ˆ

*a<sup>ν</sup>* = *γ*˜ *<sup>ν</sup>* +

*q*� <sup>1</sup> = *p*�

�∞

−∞

*<sup>l</sup>*−*<sup>s</sup>* (*ζ*), <sup>Λ</sup>*s<sup>ν</sup>* (*ζ*) are represented by

*<sup>l</sup>*−*<sup>s</sup>* (*φ*1) · exp

*dφ*<sup>2</sup> · *Gs<sup>ν</sup>* (*φ*2) · exp {*i*(*ζωτ*) *φ*2} .

4*ωκ*<sup>1</sup>

[*ε*ˆ−*Ll*−*s*+<sup>1</sup> (*φ*1) <sup>+</sup> *<sup>ε</sup>*ˆ+*Ll*−*s*−<sup>1</sup> (*φ*1)] <sup>ˆ</sup>

� 1 + *δ*<sup>2</sup> � *η*2

*η*0*<sup>j</sup>* � *φj* � *γ*˜ *<sup>ν</sup>* ˆ

*<sup>k</sup>* [(*Ll*−*s*+<sup>2</sup> (*φ*1) + *Ll*−*s*+<sup>2</sup> (*φ*1))],

= *η*0*<sup>j</sup>* · *g*

<sup>01</sup> (*φ*1) *<sup>m</sup>*<sup>2</sup>

� *φj* �

1 4*ω*2*κ*1*κ*� 1 *k<sup>ν</sup>* ˆ

1 *p*

<sup>2</sup> − *p*<sup>1</sup> − *p*<sup>2</sup> + *lk*. (144)

�

<sup>1</sup> + (*l* − *s*) *k*, (146)

· exp {−*i*(*ζωτ*) *φ*1} ,

*<sup>k</sup>* [*ε*ˆ−*Ll*−*s*+<sup>1</sup> (*φ*1) + *<sup>ε</sup>*ˆ+*Ll*−*s*−<sup>1</sup> (*φ*1)] +

*dζ*. (145)

(147)

(148)

*k*, (149)

. (150)

Λ*s<sup>ν</sup>* (*ζ*) *up*<sup>2</sup>

<sup>1</sup>) + *i*0

�

*kγ*˜ *<sup>ν</sup>*+

Here, *<sup>ε</sup>*ˆ<sup>±</sup> is the compression of four-vectors *<sup>ε</sup>*<sup>±</sup> <sup>=</sup> *ex* <sup>±</sup> *<sup>i</sup>δey* with the Dirac's *<sup>γ</sup>*˜ *<sup>ν</sup>*-matrices. The expression for the function *Gs<sup>ν</sup>* (*ϕ*2) is ensued from Eqs. (148)-(149) by following indices replacement: 1 → 2, *l* − *s* → *s*, *l* − *s* ± 1 → *s* ± 1, *l* − *s* ± 2 → *s* ± 2 and by the index *ν* omission also. By means of *κ<sup>j</sup>* and *κ*� *<sup>j</sup>* in functions *<sup>G</sup><sup>ν</sup> <sup>l</sup>*−*<sup>s</sup>* (*φ*1) and *Gs<sup>ν</sup>* (*ϕ*2) the following expressions are denoted

$$\begin{cases} \kappa\_j = E\_j - \mathbf{n} \mathbf{p}\_{j'} \\ \kappa\_j' = E\_j' - \mathbf{n} \mathbf{p'}\_j. \end{cases} \tag{151}$$

Here, **n** is the unit vector along the direction of external wave propagation

$$\mathbf{n} = \frac{\mathbf{k}}{|\mathbf{k}|}. \tag{152}$$

There are the integral functions *Ll*−*<sup>s</sup>* (*φ*1), *Ls* (*φ*2) which determine probability of emission and absorption of external wave photons in Eqs. for *G<sup>ν</sup> <sup>l</sup>*−*<sup>s</sup>* (*φ*1), *Gs<sup>ν</sup>* (*φ*2). They have the form

$$\begin{aligned} \mathcal{L}\_{\mathfrak{n}}\left(\boldsymbol{\phi}\_{\dot{\boldsymbol{\upbeta}}}\right) & \equiv \mathcal{L}\_{\mathfrak{n}}\left(\chi\_{\boldsymbol{\upbeta}}, \gamma\_{0\dot{\boldsymbol{\up}}}\left(\boldsymbol{\phi}\_{\dot{\boldsymbol{\up}}}\right), \boldsymbol{\upbeta}\_{\dot{\boldsymbol{\up}}}\left(\boldsymbol{\upphi}\_{\dot{\boldsymbol{\up}}}\right)\right) = \\ & = \frac{1}{2\pi} \int\_{0} d\boldsymbol{\upphi}\_{\dot{\boldsymbol{\up}}}' \exp\left\{i\left[\gamma\_{0\dot{\boldsymbol{\up}}}\left(\boldsymbol{\upphi}\_{\dot{\boldsymbol{\up}}}\right)\sin\left(\boldsymbol{\upphi}\_{\dot{\boldsymbol{\up}}}' - \chi\_{\dot{\boldsymbol{\up}}}\right) + \boldsymbol{\upbeta}\_{\dot{\boldsymbol{\up}}}\left(\boldsymbol{\upphi}\_{\dot{\boldsymbol{\up}}}\right)\sin 2\boldsymbol{\upphi}\_{\dot{\boldsymbol{\up}}}' - n\boldsymbol{\upphi}\_{\dot{\boldsymbol{\up}}}'\right]\right\} \end{aligned} \tag{153}$$

(*j* = 1 for *n* = *l* − *s*, *j* = 2 for *n* = *s*) with the arguments

$$
\gamma\_{0\dot{\jmath}}\left(\phi\_{\dot{\jmath}}\right) = \eta\_{0\dot{\jmath}}\left(\phi\_{\dot{\jmath}}\right) \cdot \frac{m\_{\dot{\jmath}}}{\omega} \sqrt{\left(e\_{\rm x}g\_{\dot{\jmath}}\right)^2 + \delta^2 \left(e\_{\rm y}g\_{\dot{\jmath}}\right)^2},\tag{154}
$$

$$
tg \chi\_{\bar{j}} = \delta \frac{\left(e\_{\bar{y}} g\_{\bar{j}}\right)}{\left(e\_{\mathbf{x}} g\_{\bar{j}}\right)}, \quad g\_{\bar{j}} = \frac{p\_{\bar{j}}'}{\kappa\_{\bar{j}}'} - \frac{p\_{\bar{j}}}{\kappa\_{\bar{j}}}.\tag{155}$$

$$\left(\boldsymbol{\beta}\_{\dot{\boldsymbol{\gamma}}}\left(\boldsymbol{\phi}\_{\dot{\boldsymbol{\gamma}}}\right) = \left(1 - \delta^2\right) \eta\_{\dot{\boldsymbol{\gamma}}}^2\left(\boldsymbol{\phi}\_{\dot{\boldsymbol{\gamma}}}\right) \frac{m\_{\dot{\boldsymbol{\gamma}}}^2}{8\omega} \left[\frac{1}{\kappa\_{\dot{\boldsymbol{\gamma}}}'} - \frac{1}{\kappa\_{\dot{\boldsymbol{\gamma}}}}\right].\tag{156}$$

Before performing of integration of the function *Dls* (145) over the variable *ζ* we remind that the subject of studying is the resonant character of amplitude behavior caused by quasi discrete structure: charged particle + plane electromagnetic wave. It is obvious that the resonant character of lepton-lepton scattering occurs when the denominator of the function *Dls* approaches zero. We should underline that the possibility of lepton-lepton resonant scattering in a pulsed light field is provided by the both energy (with accuracy *q*<sup>0</sup> 1/*τ* � *ω*) and momentum conservation laws fulfillment. Thus, the squared four-momentum of an intermediate photon *q*� <sup>1</sup> vanishes. It implies that the considered particle falls within the mass shell, i.e. an intermediate virtual photon becomes a real one. In this case the correction to the intermediate photon squared four-momentum in the denominator of the expression (145) is caused by the external field pulsed character and is essential through integration of the function *Dls* (145) over the variable *ζ*. Hence, the following correlation is valid

$$q'^2\_1 \lesssim \frac{\left(kq'\_1\right)}{\omega\tau}.\tag{157}$$

process when an intermediate real photon is absorbed by a lepton *l*<sup>2</sup> with *s*� = |*s*| + *l* external

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 139

Remind that integral functions (153) are determined by the integer-order Bessel functions for the case of a circularly polarizated external wave. It is not difficult to verify that for this polarization under the resonance conditions (157) the arguments of the Bessel functions (154)

. (163)

. (164)


� = 0. (165)

. (166)

sin *θ<sup>i</sup>* � 1, (167)

<sup>1</sup> , (168)

*Gs<sup>ν</sup>* (*φ*2) = *g* (*φ*2) · *G*1*ν*. (169)

� · *η*0*<sup>j</sup> φj u us*� · <sup>1</sup> <sup>−</sup> *<sup>u</sup> us*� 

<sup>−</sup> <sup>1</sup> <sup>=</sup> *<sup>κ</sup><sup>j</sup>*

lepton *l*<sup>2</sup> emits equally the only one external wave photon is realized, i.e.:

*s*

*κ*� *j*

� = |*s*| = 1, *l* = *s* + *s*

introduce also the unit vectors along the directions of initial and final momenta **n***<sup>f</sup>* and **n***<sup>i</sup>*

momentum **p** = **p**<sup>1</sup> = −**p**<sup>2</sup> and after scattering changes only the direction: |**p**�

**<sup>n</sup>***<sup>f</sup>* <sup>=</sup> **<sup>p</sup>**� |**p**� |

The region of resonant scattering is to be defined. We use the frame of reference related to a center of initial particles inertia, that is **p**<sup>1</sup> + **p**<sup>2</sup> = 0. In this frame the particle relative

With expressions (157) consideration it is easy to verify that in view of chosen direction of intermediate photon motion the resonance occurs if leptons scatter into the small angles in

<sup>=</sup> *<sup>θ</sup>res* <sup>=</sup> <sup>2</sup> *<sup>ω</sup>*

where *θ<sup>i</sup>* = ∠ (**n**, **n***i*) is the angle between the directions of wave propagation and the initial

Meanwhile the resonance for exchange (annihilation) amplitude occurs in the essentially

Thus, we expand the Bessel functions (148) as series in order of *γ*0*<sup>j</sup>* ∼ *η*0*<sup>j</sup>* � 1 and keep the summands proportional to the first order of the parameter *η*0*j*. Under the condition (165) we

*<sup>l</sup>*−*<sup>s</sup>* (*φ*1) <sup>=</sup> *<sup>g</sup>* (*φ*1) · *<sup>G</sup><sup>ν</sup>*

, **<sup>n</sup>***<sup>i</sup>* <sup>=</sup> **<sup>p</sup>**



It was expected, that for processes of resonant lepton-lepton scattering the influence of the Bunkin-Fedorov quantum parameter does not reveal, in opposite the nonresonant case. Since *γ*0*<sup>e</sup>* ∼ *η*0*<sup>e</sup>* � 1 (see Eq. (163)), then the most probable case when a lepton *l*<sup>1</sup> absorbs and a

− 1, *us*� ≡ 2*s*

� · *ωκ<sup>j</sup> m*2 *j*

wave photons emission.

may be represented as

*γ*0*<sup>j</sup> φj* = 2*s*

*u* ≡

the frame of reference related to a center of inertia:

relative momentum *p*.

obtain:

*θ* = ∠ **n***<sup>f</sup>* , **n***<sup>i</sup>* 

different kinematical region (see Roshchupkin & Voroshilo (2008)).

*Gν*

 *kpj* 

 *kp*� *j*

The condition (157) determines such a kinematical region, which is accepted to name the resonant one. In the case of external field modeling as a plane monochromatic wave there is the intermediate particle squared four-momentum alone in a process amplitude denominator. Therefore, when a denominator is equal zero the resonant divergence occurs. It is eliminated by radiative corrections introducing into the Green function according the Breit–Wigner prescription. But now there is an addition in a denominator, caused by the laser wave pulsed character. Thus, the divergence in the process amplitude disappears.

Finally, the function *Dls* (145) assumes the form:

$$D\_{ls} = \sum\_{s=-\infty}^{\infty} \frac{i\pi\omega\tau^2}{\left(kq\_1'\right)} \left(\vec{u}\_{p\_\mu'}\Delta\_{l-s}^\nu u\_{p\_\mu}\right) \left(\vec{u}\_{p\_c'}\Delta\_{s\nu} u\_{p\_c}\right),\tag{158}$$

where integral functions Δ*<sup>ν</sup> l*−*s* , Δ*s<sup>ν</sup>* are defined by following expressions

$$\begin{cases} \Delta\_{l-s}^{\nu} = \int\_{0}^{\infty} d\phi\_{1} \cdot \mathbf{G}\_{l-s}^{\nu} \left(\phi\_{1}\right) \cdot \exp\left\{i\left(\frac{q\_{0}+q\_{z}}{2}\tau + 2\beta\_{\tau}\right)\phi\_{1}\right\},\\ \begin{array}{c} -\infty\\ \Delta\_{\text{SV}} = \int\_{-\infty}^{\infty} d\phi\_{2} \cdot \mathbf{G}\_{\text{SV}}\left(\phi\_{2}\right) \cdot \exp\left\{-2i\beta\_{\tau}\phi\_{2}\right\}\left(\operatorname{sgn}\left(\phi\_{1}-\phi\_{2}\right)-1\right);\\ \end{array} \end{cases} \tag{159}$$

$$\mathcal{B}\_{\mathsf{T}} \equiv \frac{{\boldsymbol{q}}\_{1}^{\mathsf{T}}}{4 \left(k \boldsymbol{q}\_{1}^{\mathsf{T}}\right)} \boldsymbol{\omega} \boldsymbol{\tau}. \tag{160}$$

Here *βτ* is the relevant parameter which is defined by the both resonant scattering kinematics and external pulsed wave characteristics.

#### **4.1 Resonance conditions**

In this section we analyze in detail the case when an intermediate photon falls within the mass shell. Inner line discontinuity at the Feynman diagram appears and the studying process is effectively decomposes into two consecutive processes of the first order: a lepton *l*<sup>1</sup> with the four-momentum *p*<sup>1</sup> emits a real photon with the four-momentum *q*� <sup>1</sup> at the expense of external wave photons absorption, then a real photon is absorbed by a lepton *l*<sup>2</sup> with external wave photons emission or vice versa.

Generally speaking owing to condition (157) the squared four-momentum of an intermediate photon is founded within the very narrow region near zero. We will show below that this region depends on initial four-momenta of scattered particles and their scattering angles. However, the given region has to be taken into consideration in the denominator of the resonant amplitude exceptionally (145). Thus, the four-momentum conservation laws for resonant diagram vertexes may be written as two equalities:

$$\left|p\_1 + \left|s\right|k = p\_1' + q\_{1'}'\right.\tag{161}$$

$$p\_2 + q\_1' = p\_2' + \mathbf{s'} \cdot \mathbf{k}.\tag{162}$$

The equality (161) expresses the four-momentum conservation law in the process when an intermediate real photon is emitted by a lepton *l*<sup>1</sup> at the expense of |*s*| external wave photons absorption. The equality (162) corresponds to the four-momentum conservation law in the 32 Will-be-set-by-IN-TECH

The condition (157) determines such a kinematical region, which is accepted to name the resonant one. In the case of external field modeling as a plane monochromatic wave there is the intermediate particle squared four-momentum alone in a process amplitude denominator. Therefore, when a denominator is equal zero the resonant divergence occurs. It is eliminated by radiative corrections introducing into the Green function according the Breit–Wigner prescription. But now there is an addition in a denominator, caused by the laser wave pulsed

> � � *u*¯ *<sup>p</sup>*� *e* Δ*sνupe* �

� *q*<sup>0</sup> + *qz*

*dφ*<sup>2</sup> · *Gs<sup>ν</sup>* (*φ*2) · exp {−2*iβτφ*2} (*sgn* (*φ*<sup>1</sup> − *φ*2) − 1);

<sup>2</sup> *<sup>τ</sup>* <sup>+</sup> <sup>2</sup>*βτ*

� *φ*1 � ,

� *ωτ*. (160)

, Δ*s<sup>ν</sup>* are defined by following expressions

� *i*

1 4 � *kq*� 1

Here *βτ* is the relevant parameter which is defined by the both resonant scattering kinematics

In this section we analyze in detail the case when an intermediate photon falls within the mass shell. Inner line discontinuity at the Feynman diagram appears and the studying process is effectively decomposes into two consecutive processes of the first order: a lepton *l*<sup>1</sup> with the

wave photons absorption, then a real photon is absorbed by a lepton *l*<sup>2</sup> with external wave

Generally speaking owing to condition (157) the squared four-momentum of an intermediate photon is founded within the very narrow region near zero. We will show below that this region depends on initial four-momenta of scattered particles and their scattering angles. However, the given region has to be taken into consideration in the denominator of the resonant amplitude exceptionally (145). Thus, the four-momentum conservation laws for

*p*<sup>1</sup> + |*s*| *k* = *p*�

<sup>1</sup> = *p*�

The equality (161) expresses the four-momentum conservation law in the process when an intermediate real photon is emitted by a lepton *l*<sup>1</sup> at the expense of |*s*| external wave photons absorption. The equality (162) corresponds to the four-momentum conservation law in the

*p*<sup>2</sup> + *q*�

<sup>1</sup> + *q*�

<sup>2</sup> + *s*

, (158)

<sup>1</sup> at the expense of external

<sup>1</sup>, (161)

� · *k*. (162)

(159)

character. Thus, the divergence in the process amplitude disappears.

*iπωτ*<sup>2</sup> � *kq*� 1 � � *u*¯ *<sup>p</sup>*� *μ*Δ*ν <sup>l</sup>*−*sup<sup>μ</sup>*

*<sup>l</sup>*−*<sup>s</sup>* (*φ*1) · exp

*βτ* <sup>≡</sup> *<sup>q</sup>*�<sup>2</sup>

∞ ∑ *s*=−∞

*<sup>d</sup>φ*<sup>1</sup> · *<sup>G</sup><sup>ν</sup>*

four-momentum *p*<sup>1</sup> emits a real photon with the four-momentum *q*�

resonant diagram vertexes may be written as two equalities:

Finally, the function *Dls* (145) assumes the form:

where integral functions Δ*<sup>ν</sup>*

⎧ ⎪⎪⎪⎪⎨

Δ*ν <sup>l</sup>*−*<sup>s</sup>* <sup>=</sup>

<sup>Δ</sup>*s<sup>ν</sup>* = � ∞

and external pulsed wave characteristics.

⎪⎪⎪⎪⎩

**4.1 Resonance conditions**

photons emission or vice versa.

*Dls* =

*l*−*s*

�∞

−∞

−∞

process when an intermediate real photon is absorbed by a lepton *l*<sup>2</sup> with *s*� = |*s*| + *l* external wave photons emission.

Remind that integral functions (153) are determined by the integer-order Bessel functions for the case of a circularly polarizated external wave. It is not difficult to verify that for this polarization under the resonance conditions (157) the arguments of the Bessel functions (154) may be represented as

$$\gamma\_{0j}\left(\phi\_{\dot{\gamma}}\right) = 2s' \cdot \eta\_{0j}\left(\phi\_{\dot{\gamma}}\right) \sqrt{\frac{u}{u\_{s'}} \cdot \left(1 - \frac{u}{u\_{s'}}\right)}.\tag{163}$$

$$
\mu \equiv \frac{\left(kp\_{\rangle}\right)}{\left(kp\_{\rangle}^{\prime}\right)} - 1 = \frac{\kappa\_{\rangle}}{\kappa\_{\rangle}^{\prime}} - 1, \quad \mu\_{\mathbf{s}^{\prime}} \equiv 2\mathbf{s}^{\prime} \cdot \frac{\omega \kappa\_{\rangle}}{m\_{\rangle}^{2}}.\tag{164}
$$

It was expected, that for processes of resonant lepton-lepton scattering the influence of the Bunkin-Fedorov quantum parameter does not reveal, in opposite the nonresonant case. Since *γ*0*<sup>e</sup>* ∼ *η*0*<sup>e</sup>* � 1 (see Eq. (163)), then the most probable case when a lepton *l*<sup>1</sup> absorbs and a lepton *l*<sup>2</sup> emits equally the only one external wave photon is realized, i.e.:

$$s'=|s|=1, \quad l=s+s'=0.\tag{165}$$

The region of resonant scattering is to be defined. We use the frame of reference related to a center of initial particles inertia, that is **p**<sup>1</sup> + **p**<sup>2</sup> = 0. In this frame the particle relative momentum **p** = **p**<sup>1</sup> = −**p**<sup>2</sup> and after scattering changes only the direction: |**p**� | = |**p**|. We introduce also the unit vectors along the directions of initial and final momenta **n***<sup>f</sup>* and **n***<sup>i</sup>*

$$\mathbf{n}\_f = \frac{\mathbf{p}'}{|\mathbf{p}'|}, \quad \mathbf{n}\_i = \frac{\mathbf{p}}{|\mathbf{p}|}. \tag{166}$$

With expressions (157) consideration it is easy to verify that in view of chosen direction of intermediate photon motion the resonance occurs if leptons scatter into the small angles in the frame of reference related to a center of inertia:

$$
\theta = \angle \left( \mathbf{n}\_{f'} \,\mathbf{n}\_{\dot{l}} \right) = \theta\_{\text{res}} = 2 \frac{\omega}{|\mathbf{p}|} \sin \theta\_{\dot{l}} \ll 1,\tag{167}
$$

where *θ<sup>i</sup>* = ∠ (**n**, **n***i*) is the angle between the directions of wave propagation and the initial relative momentum *p*.

Meanwhile the resonance for exchange (annihilation) amplitude occurs in the essentially different kinematical region (see Roshchupkin & Voroshilo (2008)).

Thus, we expand the Bessel functions (148) as series in order of *γ*0*<sup>j</sup>* ∼ *η*0*<sup>j</sup>* � 1 and keep the summands proportional to the first order of the parameter *η*0*j*. Under the condition (165) we obtain:

$$\mathcal{G}\_{l-s}^{\nu} \left( \phi\_1 \right) = \mathcal{g} \left( \phi\_1 \right) \cdot \mathcal{G}\_{1\prime}^{\nu} \tag{168}$$

$$\mathcal{G}\_{\rm sv} \left( \phi\_2 \right) = \mathcal{g} \left( \phi\_2 \right) \cdot \mathcal{G}\_{\rm lv}. \tag{169}$$

**4.2 Resonant cross-section**

*l*<sup>1</sup> by a lepton *l*2. Thus,

*dW*

<sup>×</sup> exp

*<sup>d</sup>σl*1*l*<sup>2</sup> *res*

∞

*dξ*

2*φ* +

−∞

*H* =

× erf

*f*<sup>0</sup> =

× 2 + *d*2 *f i* 

*<sup>T</sup>* <sup>=</sup> *dw* <sup>=</sup> *<sup>e</sup>*<sup>4</sup> 2 (2*π*)

<sup>−</sup>*β*<sup>2</sup> *τ* 2 

simple transformations we derive

Here, the function *H* has the form

2*p*4*E*1*E*<sup>2</sup>


*iβτ* 2 + 1 *g φ*� exp

Here, the function *f*<sup>0</sup> is determined by

<sup>2</sup>*df ihf i* <sup>|</sup>*p*|(*E*<sup>1</sup> <sup>+</sup> *<sup>E</sup>*2)

*E*1*E*<sup>2</sup> + **p**2

(*E*1− |**p**| cos *θi*) (*E*2+ |**p**| cos *θi*)

·

2 *ξω* <sup>+</sup> *<sup>E</sup>* <sup>=</sup> *<sup>τ</sup>*<sup>2</sup>

*ω* (*ωτ*) 2

*θ*2*θ*<sup>2</sup> *res* · *u*¯ *p*� 1 *Gν* <sup>1</sup>*up*<sup>1</sup>

∞

∞

∞

−∞

−∞

pulsed light wave into the elementary solid angle may be represented as

2 *e* 4*πm*<sup>2</sup> *em*<sup>2</sup> 1*m*<sup>2</sup> 2

+ 4*df ihf i* |**p**|(*E*<sup>1</sup> + *E*2)

−∞

*<sup>d</sup>σl*1*l*<sup>2</sup> *res <sup>d</sup>*Ω� <sup>=</sup> *<sup>r</sup>*

(*E*<sup>1</sup> − |**p**| cos *θi*) (*E*<sup>2</sup> + |**p**| cos *θi*)

*<sup>d</sup>*Ω� <sup>=</sup> *<sup>e</sup>*4*<sup>E</sup>*

In view of finite duration of an external pulsed light field there is a sense to define the differential probability over all the observation time *T* in the process of scattering of a lepton

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 141

<sup>2</sup> *d*<sup>3</sup> *p*� (2*π*) 3

Using the expressions for the amplitude (174)-(178) and performing uncomplicated computations we obtain the differential probability per time unit and per volume unit:

> (*ωτ*) 2

*θ*2*θ*<sup>2</sup> *res* · *u*¯ *p*� 1 *Gν* <sup>1</sup>*up*<sup>1</sup>

The differential cross section we obtain from Eq. (181) by division by a density of the scattered particles flux *<sup>j</sup>* <sup>=</sup> <sup>|</sup>**p**|/*E*. The integration of the differential cross section over *<sup>d</sup>*3*P*� should be

> 1*E*� 2 **p**� *<sup>d</sup>*Ω� *dE*�

where *d*Ω� is the elementary solid angle of particles scattering, and introduce a new integration dimensionless variable: *dE*� → *dξ* (*ξ* = *q*0/*ω*, *E*� = *ξω* + *E*, *dE*� = *ωdξ*). After

*dξdφdφ*�

The differential cross section of resonant scattering of nonpolarized leptons in the field of a

*p*4*E*1*E*<sup>2</sup>

2 + 2 +

−*i*(*ξωτ* + 2*βτ*) *φ*�

*η*2 01*η*<sup>2</sup>

 .

 *<sup>u</sup>*¯ *<sup>p</sup>*� 2 *G*1*νup*<sup>2</sup>  2 exp

*ξω* <sup>+</sup> *<sup>E</sup> <sup>g</sup>* (*φ*) exp {*i*(*ξωτ* <sup>+</sup> <sup>2</sup>*βτ*) *<sup>φ</sup>*}×

 erf

*d*2 *f i* 

*d*3*P*� (2*π*)

<sup>3</sup> . (180)

 2 ×

*<sup>E</sup>*� , (182)

<sup>−</sup>*β*<sup>2</sup> *τ*/2

<sup>2</sup>*φ*� <sup>−</sup> *<sup>i</sup>βτ* 2 + 1 .

<sup>02</sup> · *f*<sup>0</sup> · *fres*. (185)

 ×

*E*1*E*<sup>2</sup> + **p**2

(*E*1− |**p**| cos *θi*) (*E*2+ |**p**| cos *θi*)

(181)

· *H*. (183)

(184)

(186)

 *<sup>u</sup>*¯ *<sup>p</sup>*� 2 *G*1*νup*<sup>2</sup>

*dW* = |*S*|

1*E*� 2*T* ·

*d*<sup>3</sup> *p*� = *E*�

2 *δ P*� *x δ P*� *y δ q*<sup>0</sup> − *P*� *z d*<sup>3</sup> *p*� *d*3*P*� .

<sup>3</sup> *<sup>p</sup>*4*E*1*E*2*E*�

· |*I<sup>τ</sup>* (*q*+*τ*)|

performed via the delta-functions. We present *d*<sup>3</sup> *p*� as

where the matrices *G<sup>ν</sup>* <sup>1</sup> and *G*1*<sup>ν</sup>* have the following form

$$\mathbf{G}\_{1}^{\nu} = (-1)\frac{\gamma\_{01}}{2} \exp\left(i\chi\_{1}\right)\hat{\gamma}^{\nu} + \frac{\eta\_{01}m\_{1}}{2\omega\varkappa\_{1}} \left[k^{\nu}\boldsymbol{\varepsilon}\_{-} - \boldsymbol{\varepsilon}\_{-}^{\nu}\hat{k}\right] + \frac{\eta\_{01}m\_{1}}{4\omega} \left(\frac{1}{\kappa\_{1}^{\prime}} - \frac{1}{\kappa\_{1}}\right)\boldsymbol{\varepsilon}\_{-}\boldsymbol{\hat{k}}\boldsymbol{\gamma}^{\nu},\tag{170}$$

$$G\_{1\upsilon} = \frac{\gamma\_{02}}{2} \exp\left(-i\chi\_2\right)\tilde{\gamma}\_{\upsilon} + \frac{\eta\_{02}m\_2}{2\omega\kappa\_2} \left[k\_{\upsilon}\mathbb{1}\_{+} - \varepsilon\_{+\upsilon}\hat{k}\right] + \frac{\eta\_{02}m\_2}{4\omega} \left(\frac{1}{\kappa\_2^{\upsilon}} - \frac{1}{\kappa\_2}\right)\mathbb{1}\_{+}\hat{k}\tilde{\gamma}\_{\upsilon}.\tag{171}$$

The resonant region of scattering angles in the frame of reference related to a center of inertia is determined as

$$|\theta - \theta\_{\rm res}| \lesssim \frac{\theta\_{\rm res}}{\omega \tau} \ll \theta\_{\rm res} \tag{172}$$

and expressions for the parameter *βτ* (160) assumes the form

$$
\beta\_{\rm 7} = \frac{1}{2} \omega \tau \left( 1 - \frac{\theta}{\theta\_{\rm res}} \right) \lesssim 1. \tag{173}
$$

Finally, the resonant amplitude of a lepton *l*<sup>1</sup> scattered by a lepton *l*<sup>2</sup> in the field of a pulsed electromagnetic moderately strong wave of a circularly polarization in the frame of reference related to a center of inertia takes the form

$$S = S\_0 \cdot Y\_{\tau\nu} \tag{174}$$

where

$$S\_0 = \frac{i\pi^{3/2}e^2\hat{M}}{\mathbf{p}^2\sqrt{E\_1E\_2E\_1'E\_2'}}\delta\left(P\_x'\right)\delta\left(P\_y'\right)\delta\left(E'-E-P\_z'\right),\tag{175}$$

$$
\hat{M} = \left(\tilde{\mathfrak{u}}\_{p\_1'} \mathbf{G}\_1^{\vee} \mathfrak{u}\_{p\_1}\right) \left(\tilde{\mathfrak{u}}\_{p\_2'} \mathbf{G}\_{1\vee} \mathfrak{u}\_{p\_2}\right). \tag{176}
$$

The function Υ*<sup>τ</sup>* in Eq. (174) is represented by

$$\mathbf{Y}\_{\mathsf{T}} = \frac{\omega \tau}{\theta \cdot \theta\_{\mathrm{res}}} \exp\{-\frac{\beta\_{\mathsf{T}}^{2}}{4}\} \cdot I\_{\mathsf{T}}\left(q\_{+}\pi\right). \tag{177}$$

Here, *I<sup>τ</sup>* (*q*+) is the integral function:

$$I\_{\mathsf{T}}\left(q\_{+}\tau\right) = \tau \int\_{-\infty}^{\infty} d\phi \cdot g\left(\phi\right) \cdot \exp\left\{i\left(\frac{q\_{0} + q\_{z}}{2}\tau + 2\beta\_{\mathsf{T}}\right)\phi\right\} \cdot \left[\text{erf}\left(2\phi + \frac{i\beta\_{\mathsf{T}}}{2}\right) + 1\right].\tag{178}$$

In Eqs. (177) and (178) the parameter *βτ* is determined by the expression (173). We underline, that presence of three delta-functions in the resonant amplitude (174)-(178) is considered as realizing of three following conservation laws:

$$P\_x' = 0, \quad P\_y' = 0, \quad E' - E = P\_{z'}' \tag{179}$$

where **P**� = *P*� *<sup>x</sup>*, *P*� *<sup>y</sup>*, *P*� *z* is the momentum of the inertia center after scattering, *E* and *E*� are particle total energies before and after scattering, correspondingly.

#### **4.2 Resonant cross-section**

34 Will-be-set-by-IN-TECH

*<sup>k</sup>νε*ˆ<sup>−</sup> <sup>−</sup> *<sup>ε</sup> ν* − ˆ *k* 

*<sup>k</sup>νε*ˆ<sup>+</sup> <sup>−</sup> *<sup>ε</sup>*+*<sup>ν</sup>* <sup>ˆ</sup>

The resonant region of scattering angles in the frame of reference related to a center of inertia

Finally, the resonant amplitude of a lepton *l*<sup>1</sup> scattered by a lepton *l*<sup>2</sup> in the field of a pulsed electromagnetic moderately strong wave of a circularly polarization in the frame of reference

> exp{− *<sup>β</sup>*<sup>2</sup> *τ*

<sup>2</sup> *<sup>τ</sup>* <sup>+</sup> <sup>2</sup>*βτ*

*<sup>y</sup>* = 0, *E*� − *E* = *P*�

is the momentum of the inertia center after scattering, *E* and *E*� are

In Eqs. (177) and (178) the parameter *βτ* is determined by the expression (173). We underline, that presence of three delta-functions in the resonant amplitude (174)-(178) is considered as

 *φ* · erf 2*φ* +

*ωτ*

*k*  <sup>+</sup> *<sup>η</sup>*01*m*<sup>1</sup> 4*ω*

<sup>+</sup> *<sup>η</sup>*02*m*<sup>2</sup> 4*ω*

 1 *κ*� 1 − 1 *κ*1 *ε*ˆ− ˆ

 1 *κ*� 2 − 1 *κ*2 *ε*ˆ<sup>+</sup> ˆ

*S* = *S*<sup>0</sup> · Υ*τ*, (174)

*z* 

<sup>4</sup> } · *<sup>I</sup><sup>τ</sup>* (*q*+*τ*). (177)

*iβτ* 2 + 1 

*<sup>z</sup>*, (179)

*E*� − *E* − *P*�

� *θres*, (172)

1. (173)

, (175)

. (178)

. (176)

*kγ*˜ *<sup>ν</sup>*, (170)

*kγ*˜*ν*. (171)

<sup>1</sup> and *G*1*<sup>ν</sup>* have the following form

<sup>|</sup>*<sup>θ</sup>* <sup>−</sup> *<sup>θ</sup>res*<sup>|</sup> *<sup>θ</sup>res*

2*ωκ*<sup>1</sup>

2*ωκ*<sup>2</sup>

*βτ* <sup>=</sup> <sup>1</sup> 2 *ωτ* <sup>1</sup> <sup>−</sup> *<sup>θ</sup> θres* 

<sup>2</sup> exp (*iχ*1) *<sup>γ</sup>*˜ *<sup>ν</sup>* <sup>+</sup> *<sup>η</sup>*01*m*<sup>1</sup>

and expressions for the parameter *βτ* (160) assumes the form

*<sup>S</sup>*<sup>0</sup> <sup>=</sup> *<sup>i</sup>π*3/2*e*2*M*<sup>ˆ</sup> **p**2 

*E*1*E*2*E*� 1*E*� 2 *δ P*� *x δ P*� *y δ* 

*M*ˆ = *u*¯ *p*� 1 *Gν* <sup>1</sup>*up*<sup>1</sup> *u*¯ *p*� 2 *G*1*νup*<sup>2</sup> 

<sup>Υ</sup>*<sup>τ</sup>* <sup>=</sup> *ωτ θ* · *θres*

> *i*

*<sup>x</sup>* = 0, *P*�

*q*<sup>0</sup> + *qz*

<sup>2</sup> exp (−*iχ*2) *<sup>γ</sup>*˜*<sup>ν</sup>* <sup>+</sup> *<sup>η</sup>*02*m*<sup>2</sup>

related to a center of inertia takes the form

The function Υ*<sup>τ</sup>* in Eq. (174) is represented by

*dφ* · *g* (*φ*) · exp

*P*�

particle total energies before and after scattering, correspondingly.

realizing of three following conservation laws:

Here, *I<sup>τ</sup>* (*q*+) is the integral function:

∞

−∞

*I<sup>τ</sup>* (*q*+*τ*) = *τ*

where **P**� =

 *P*� *<sup>x</sup>*, *P*� *<sup>y</sup>*, *P*� *z* 

where the matrices *G<sup>ν</sup>*

*<sup>G</sup>*1*<sup>ν</sup>* <sup>=</sup> *<sup>γ</sup>*<sup>02</sup>

is determined as

where

<sup>1</sup> <sup>=</sup> (−1) *<sup>γ</sup>*<sup>01</sup>

*Gν*

In view of finite duration of an external pulsed light field there is a sense to define the differential probability over all the observation time *T* in the process of scattering of a lepton *l*<sup>1</sup> by a lepton *l*2. Thus,

$$dW = \left| \mathbf{S} \right|^2 \frac{d^3 p'}{\left(2\pi\right)^3} \frac{d^3 P'}{\left(2\pi\right)^3}. \tag{180}$$

Using the expressions for the amplitude (174)-(178) and performing uncomplicated computations we obtain the differential probability per time unit and per volume unit:

$$\begin{split} \frac{d\mathcal{W}}{T} &= dw = \frac{e^{4}}{2\left(2\pi\right)^{3}p^{4}E\_{1}E\_{2}E\_{1}^{\prime}E\_{2}^{\prime}T} \cdot \frac{\left(\omega\tau\right)^{2}}{\theta^{2}\theta\_{\rm res}^{2}} \cdot \left| \left(\bar{u}\_{p\_{1}^{\prime}}\mathcal{G}\_{1}^{\prime}u\_{p\_{1}}\right) \left(\bar{u}\_{p\_{2}^{\prime}}\mathcal{G}\_{11}u\_{p\_{2}}\right) \right|^{2} \times \\ &\times \exp\left(-\beta\_{\rm \tau}^{2} \left/2\right> \cdot \left| I\_{\rm \tau}\left(q\_{\rm \tau}\tau\right) \right|^{2} \delta\left(P\_{\rm \chi}^{\prime}\right) \delta\left(P\_{\rm \chi}^{\prime}\right) \delta\left(q\_{0}-P\_{\rm \chi}^{\prime}\right) d^{3}p^{\prime}d^{3}P^{\prime}. \end{split} \tag{181}$$

The differential cross section we obtain from Eq. (181) by division by a density of the scattered particles flux *<sup>j</sup>* <sup>=</sup> <sup>|</sup>**p**|/*E*. The integration of the differential cross section over *<sup>d</sup>*3*P*� should be performed via the delta-functions. We present *d*<sup>3</sup> *p*� as

$$d^3 p' = E\_1' E\_2' \left| \mathbf{p'} \right| d\Omega' \frac{dE'}{E'},\tag{182}$$

where *d*Ω� is the elementary solid angle of particles scattering, and introduce a new integration dimensionless variable: *dE*� → *dξ* (*ξ* = *q*0/*ω*, *E*� = *ξω* + *E*, *dE*� = *ωdξ*). After simple transformations we derive

$$\frac{d\sigma\_{\rm res}^{l\_1 l\_2}}{d\Omega'} = \frac{e^4 E}{2p^4 E\_1 E\_2} \cdot \frac{\omega \left(\omega \tau\right)^2}{\theta^2 \theta\_{\rm res}^2} \cdot \left| \left(\vec{u}\_{p\_1'} G\_1^{\nu} u\_{p\_1}\right) \left(\vec{u}\_{p\_2'} G\_{1\nu} u\_{p\_2}\right) \right|^2 \exp\left(-\beta\_\tau^2/2\right) \cdot H. \tag{183}$$

Here, the function *H* has the form

$$\begin{split} H &= \int\_{-\infty}^{\infty} d\xi \frac{|I\_{\mathsf{T}}\left(\xi\omega\tau\right)|^{2}}{\xi\omega + E} = \pi^{2} \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \frac{d\xi d\phi d\phi'}{\xi\omega + E} g\left(\phi\right) \exp\left\{i\left(\xi\omega\tau + 2\beta\_{\mathsf{T}}\right)\phi\right\} \times \\ &\times \left(\text{erf}\left(2\phi + \frac{i\beta\_{\mathsf{T}}}{2}\right) + 1\right) g\left(\phi'\right) \exp\left\{-i\left(\xi\omega\tau + 2\beta\_{\mathsf{T}}\right)\phi'\right\} \left(\text{erf}\left(2\phi' - \frac{i\beta\_{\mathsf{T}}}{2}\right) + 1\right). \end{split} \tag{184}$$

The differential cross section of resonant scattering of nonpolarized leptons in the field of a pulsed light wave into the elementary solid angle may be represented as

$$\frac{d\sigma\_{\rm res}^{l\_1 l\_2}}{d\Omega'} = r\_\varepsilon^2 \frac{4\pi m\_e^2 m\_1^2 m\_2^2}{p^4 E\_1 E\_2} \eta\_{01}^2 \eta\_{02}^2 \cdot f\_0 \cdot f\_{\rm res}.\tag{185}$$

Here, the function *f*<sup>0</sup> is determined by

$$\begin{split} f\_{0} &= \left[ \frac{2d\_{f}h\_{fi}\left|\mathbf{p}\right| \left(E\_{1} + E\_{2}\right)}{\left(E\_{1} - \left|\mathbf{p}\right|\cos\theta\_{l}\right)\left(E\_{2} + \left|\mathbf{p}\right|\cos\theta\_{l}\right)} \right]^{2} + \left[ 2 + \frac{d\_{fi}^{2}\left(E\_{1}E\_{2} + \mathbf{p}^{2}\right)}{\left(E\_{1} - \left|\mathbf{p}\right|\cos\theta\_{l}\right)\left(E\_{2} + \left|\mathbf{p}\right|\cos\theta\_{l}\right)} \right] \times \\ &\times \left[ 2 + \frac{d\_{fi}^{2}\left(E\_{1}E\_{2} + \mathbf{p}^{2}\right) + 4d\_{fi}h\_{fi}\left|\mathbf{p}\right|\left(E\_{1} + E\_{2}\right)}{\left(E\_{1} - \left|\mathbf{p}\right|\cos\theta\_{l}\right)\left(E\_{2} + \left|\mathbf{p}\right|\cos\theta\_{l}\right)} \right]. \end{split} \tag{186}$$

and

**<sup>n</sup>***iτf i*

scattering angle *θ* ), and finally derive

where function *F* (*ρ*) is determined by

cross-sections coincide each with other.

*<sup>d</sup>σl*1*l*<sup>2</sup> *res dϕ<sup>f</sup>*

the parameter *βτ* 1 within the resonant region).

= 16*πr* 2 *e η*2 01*η*<sup>2</sup> 02 *m*2 *em*<sup>2</sup> 1*m*<sup>2</sup> 2

*F* (*ρ*) =

∞

−∞

Fig. 9. The dependence or the differential cross-section of scattering of an electron by an electron (an electron by a positron) in a pulsed light field (195) (in units of respective

line), *V* = 0.6 (dotted line), and *V* = 0.9 (dash-dotted line) are represented.

cross-sections in an external field absence) on the initial polar angle when an azimuthal angle is fixed *ϕ<sup>i</sup>* = *π*/4 and value of the parameter *ρ* = 2. The external laser wave frequency amounts to the value *ω* = 2.35 eV, the pulse duration is equal to *τ* = 1.5 ps, the field strength in a pulse peak *<sup>F</sup>*<sup>0</sup> <sup>=</sup> <sup>6</sup> · 109 V/cm. The cases of particles relative velocities *<sup>V</sup>* <sup>=</sup> 0.2 (solid

Let us consider the ratio of the derived resonant differential cross section (195) to the differential cross section of scattering of the same leptons in an external field absence for such processes: scattering of an electron by an electron, scattering of an electron by a positron, scattering of an electron by a muon. Figs. 9, 10 show the dependencies of the considered ratio on the initial polar angle *θi*. We should underline that under scattering of both an electron by an electron and an electron by a positron within the small angles range (172) the respective

In accordance with the Figs. 9, 10 we consider that within the broad range of particles velocities the resonant cross sections of scattering of an electron by an electron (an electron by a positron, an electron by a muon) in a pulsed light field exceed the corresponding differential

Here, the function *f* (*ρ*, *βτ*) is specified by Eq. (192). The limits of the integration in Eq. (196) are extended over the infinity owing to the integral quick convergence (though the values of

� 0. We perform the integration over the parameter *βτ* (173) (instead the

· *f*<sup>0</sup> · *F* (*ρ*), (195)

*dβτ* · *f* (*ρ*, *βτ*). (196)

*E*1*E*<sup>2</sup> |*p*| 4 (*ωτ*) *θ*2 *res*

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 143

The following designations are used in Eq. (186)

$$h\_{fi} = (\mathbf{e}\_{\mathbf{x}} \mathbf{n}\_i) \cos \chi\_{fi} + \delta \left(\mathbf{e}\_{\mathbf{y}} \mathbf{n}\_i\right) \sin \chi\_{fi\nu} \tag{187}$$

$$d\_{fi} = 2\left(\mathbf{n}\tau\_{fi}\right)\sqrt{\left(\mathbf{e}\_x\tau\_{fi}\right)^2 + \left(\mathbf{e}\_y\tau\_{fi}\right)^2},\tag{188}$$

$$
tg \chi\_{fi} = \delta \frac{\left(\mathbf{e}\_{y} \cdot \boldsymbol{\tau}\_{fi}\right)}{\left(\mathbf{e}\_{x} \cdot \boldsymbol{\tau}\_{fi}\right)}\,\tag{189}$$

$$\pi\_{fi} = \frac{\mathbf{n}\_f - \mathbf{n}\_i}{\left| \mathbf{n}\_f - \mathbf{n}\_i \right|}. \tag{190}$$

The function *fres* in Eq. (185) has the form

$$f\_{\rm res} = \left(\frac{\omega \tau}{\theta\_{\rm res}^2}\right)^2 \cdot f\left(\rho\_\prime \beta\_\mp\right),\tag{191}$$

$$f\left(\boldsymbol{\rho}, \boldsymbol{\beta}\_{\tau}\right) = \exp\left(-\beta\_{\tau}^{2}/2\right) \cdot \frac{1}{\rho} \int\_{-\rho}^{\rho} d\phi \, g^{2}\left(\phi\right) \left| \text{erf}\left(\phi + \frac{i\beta\_{\tau}}{2}\right) + 1 \right|^{2}.\tag{192}$$

We underline that the dependence of the function *fres* on the parameter *βτ* (173) determines resonant peak magnitude and shape. It is easy to notice that when leptons scatter into the resonant angle *θ* � *θres* than the parameter *βτ* becomes equal zero (see Eq. (173)). At that the function *fres* (191) possesses the finite value as opposed to the plane monochromatic wave case when *fres* → ∞ is correct.

The significant issue is the influence of the pulse finite duration on the cross section resonant behavior. The pulse duration has to exceed the time required for the Compton scattering of an external field photon by each of leptons *l*<sup>1</sup> and *l*2. If this condition is not satisfied than particles do not have time to interact with a wave under the resonance conditions. Consequently, the following correlation for the pulse duration is valid:

$$
\omega \tau \gtrsim \frac{1}{\alpha \eta\_{0j}^2} \frac{E\_j}{\kappa\_j}. \tag{193}
$$

Thus, experimental treatment of resonant scattering of a lepton by a lepton may be verified in the fields created by picosecond pulsed lasers which generate the radiation within the optical frequencies range. Such scientific facilities are employed in SLAC National Accelerator Laboratory (Bula et al. (1996); Burke et al. (1997)) research centers and also in the frame of the FAIR project (Bagnoud et al. (2009)).

We can integrate the differential cross section (185) within the narrow range of scattering angles near the resonance (172). Under the resonance conditions the vector *τf i* (190) may be represented as

$$
\pi\_{fi} \approx \frac{1}{\theta \text{res}} \left( \mathbf{n}\_f - \mathbf{n}\_i \right) \tag{194}
$$

and **<sup>n</sup>***iτf i* � 0. We perform the integration over the parameter *βτ* (173) (instead the scattering angle *θ* ), and finally derive

$$\frac{d\sigma\_{\rm res}^{l\_1 l\_2}}{d\varphi\_f} = 16\pi r\_\varepsilon^2 \eta\_{01}^2 \eta\_{02}^2 \frac{m\_e^2 m\_1^2 m\_2^2}{E\_1 E\_2 \left| p \right|^4} \frac{\left(\omega \tau\right)}{\theta\_{\rm res}^2} \cdot f\_0 \cdot F\left(\rho\right), \tag{195}$$

where function *F* (*ρ*) is determined by

36 Will-be-set-by-IN-TECH

 **e***y***n***<sup>i</sup>* 

**e***xτf i* 2 + **e***yτf i* 2

 **e***<sup>y</sup>* · *τf i* 

 **e***<sup>x</sup>* · *τf i*

*<sup>τ</sup>f i* <sup>=</sup> **<sup>n</sup>***<sup>f</sup>* <sup>−</sup> **<sup>n</sup>***<sup>i</sup>* **n***<sup>f</sup>* − **n***<sup>i</sup>* 

<sup>2</sup>

*ρ* 

*dφg*<sup>2</sup> (*φ*)

 erf *φ* + *iβτ* 2 + 1 2

−*ρ*

We underline that the dependence of the function *fres* on the parameter *βτ* (173) determines resonant peak magnitude and shape. It is easy to notice that when leptons scatter into the resonant angle *θ* � *θres* than the parameter *βτ* becomes equal zero (see Eq. (173)). At that the function *fres* (191) possesses the finite value as opposed to the plane monochromatic wave

The significant issue is the influence of the pulse finite duration on the cross section resonant behavior. The pulse duration has to exceed the time required for the Compton scattering of an external field photon by each of leptons *l*<sup>1</sup> and *l*2. If this condition is not satisfied than particles do not have time to interact with a wave under the resonance conditions. Consequently, the

> *ωτ* <sup>1</sup> *αη*<sup>2</sup> 0*j*

Thus, experimental treatment of resonant scattering of a lepton by a lepton may be verified in the fields created by picosecond pulsed lasers which generate the radiation within the optical frequencies range. Such scientific facilities are employed in SLAC National Accelerator Laboratory (Bula et al. (1996); Burke et al. (1997)) research centers and also in the frame of the

We can integrate the differential cross section (185) within the narrow range of scattering angles near the resonance (172). Under the resonance conditions the vector *τf i* (190) may

> *<sup>τ</sup>f i* <sup>≈</sup> <sup>1</sup> *θ*res **n***<sup>f</sup>* − **n***<sup>i</sup>*

*Ej κj*

 *ωτ θ*2 *res* sin *χf i*, (187)

, (189)

. (190)

· *f* (*ρ*, *βτ*), (191)

. (193)

, (194)

, (188)

. (192)

*hf i* = (**e***x***n***i*) cos *χf i* + *δ*

*tgχf i* = *δ*

 **n***τf i*

*fres* =

 <sup>−</sup>*β*<sup>2</sup> *<sup>τ</sup>*/2 · 1 *ρ*

The following designations are used in Eq. (186)

The function *fres* in Eq. (185) has the form

case when *fres* → ∞ is correct.

FAIR project (Bagnoud et al. (2009)).

be represented as

*f* (*ρ*, *βτ*) = exp

following correlation for the pulse duration is valid:

*df i* = 2

$$F(\rho) = \int\_{-\infty}^{\infty} d\beta\_{\tau} \cdot f\left(\rho, \beta\_{\tau}\right). \tag{196}$$

Here, the function *f* (*ρ*, *βτ*) is specified by Eq. (192). The limits of the integration in Eq. (196) are extended over the infinity owing to the integral quick convergence (though the values of the parameter *βτ* 1 within the resonant region).

Fig. 9. The dependence or the differential cross-section of scattering of an electron by an electron (an electron by a positron) in a pulsed light field (195) (in units of respective cross-sections in an external field absence) on the initial polar angle when an azimuthal angle is fixed *ϕ<sup>i</sup>* = *π*/4 and value of the parameter *ρ* = 2. The external laser wave frequency amounts to the value *ω* = 2.35 eV, the pulse duration is equal to *τ* = 1.5 ps, the field strength in a pulse peak *<sup>F</sup>*<sup>0</sup> <sup>=</sup> <sup>6</sup> · 109 V/cm. The cases of particles relative velocities *<sup>V</sup>* <sup>=</sup> 0.2 (solid line), *V* = 0.6 (dotted line), and *V* = 0.9 (dash-dotted line) are represented.

Let us consider the ratio of the derived resonant differential cross section (195) to the differential cross section of scattering of the same leptons in an external field absence for such processes: scattering of an electron by an electron, scattering of an electron by a positron, scattering of an electron by a muon. Figs. 9, 10 show the dependencies of the considered ratio on the initial polar angle *θi*. We should underline that under scattering of both an electron by an electron and an electron by a positron within the small angles range (172) the respective cross-sections coincide each with other.

In accordance with the Figs. 9, 10 we consider that within the broad range of particles velocities the resonant cross sections of scattering of an electron by an electron (an electron by a positron, an electron by a muon) in a pulsed light field exceed the corresponding differential

**+**

*f k*

*i p*

*f* 

) is the wave function of a photon (8);

) is the Green function of an electron in the

�∗


*<sup>ν</sup>e<sup>μ</sup>* · *<sup>u</sup>*¯ *pf <sup>T</sup>νμ*

*T*(*j*)*νμ l*−*l*� ,*l*�

*f i upi*

⎞ ⎠ ⎞ , (200)

⎠, (201)

<sup>0</sup> � 1, *ϕ*<sup>0</sup> = *ωτ*, (199)

*<sup>f</sup> p <sup>i</sup> k*

*i p*

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 145

Fig. 11. The Feynman diagram of the Compton effect in the field of a pulsed light wave.

where *pf* = (*Ef* , **p***f*) and *k <sup>f</sup>* = (*ω<sup>f</sup>* , **k***f*) are four momenta of an outgoing electron and a

is considered through this section. This condition allows both to carry out the decomposition with respect to the small parameter and to neglect the interference of contributions of the

(**p***i*,<sup>⊥</sup> + **<sup>k</sup>***i*,<sup>⊥</sup> − **<sup>p</sup>***<sup>f</sup>* ,<sup>⊥</sup> − **<sup>k</sup>***<sup>f</sup>* ,⊥)*δ*(*pi*,<sup>−</sup> + *ki*,<sup>−</sup> − *pf* ,<sup>−</sup> − *<sup>k</sup> <sup>f</sup>* ,−)*<sup>e</sup>*

*T*(*j*)*νμ l*−*l*�

,*l*� <sup>+</sup> *<sup>η</sup>*<sup>2</sup> 0 ⎛ ⎝*T*(*j*)*νμ*

where *j* = *e*, *d*; indices *d*, *e* are concerned to direct and exchange diagrams; *B* is the normalization factor; **<sup>p</sup>***i*,⊥, **<sup>k</sup>***i*,⊥, **<sup>p</sup>***<sup>f</sup>* ,⊥, **<sup>k</sup>***<sup>f</sup>* ,<sup>⊥</sup> are the projections of corresponding vectors on the wave polarization plane; *pi*,<sup>−</sup> = *Ei* − *pi*,*z*, *ki*,<sup>−</sup> = *<sup>ω</sup><sup>i</sup>* − *ki*,*z*, *pf* ,<sup>−</sup> = *Ef* − *pf* ,*z*, *<sup>k</sup> <sup>f</sup>* ,<sup>−</sup> = *<sup>ω</sup><sup>f</sup>* − *<sup>k</sup> <sup>f</sup>* ,*<sup>z</sup>* are differences between zeroth components of the corresponding four momentum and its projection on direction of wave propagation; *q*, *f* are four momenta of an intermediate particle, which conform to direct and exchange diagrams on Fig. 11, at that under the four momenta

**<sup>q</sup>**<sup>⊥</sup> = **<sup>p</sup>***i*,<sup>⊥</sup> + **<sup>k</sup>***i*,⊥, *<sup>q</sup>*<sup>−</sup> = *pi*,<sup>−</sup> + *ki*,−; **<sup>f</sup>**<sup>⊥</sup> = **<sup>p</sup>***i*,<sup>⊥</sup> − **<sup>k</sup>***<sup>f</sup>* ,⊥, *<sup>f</sup>*<sup>−</sup> = *pi*,<sup>−</sup> − *<sup>k</sup> <sup>f</sup>* ,−. (202)


The summands in Eq. (201), proportional to the zeroth degree of *η*<sup>0</sup> determine the amplitude of the Compton effect in external field absence (Klein & Nishina (1929)). The summands, proportional to the first degree of the parameter *η*0, determine the corrections (for them

summands, proportional to the second degree of the parameter *η*0, determine the corrections

<sup>0</sup> assumes the form

0,0 + ∑ *l*,*l*�∈(|*l*−*l*�

*<sup>f</sup> p <sup>i</sup> k*

The case when a laser field intensity meets the following condition

*η*2 <sup>0</sup> *<sup>ϕ</sup>*−<sup>1</sup>


photon; *γ*˜ *<sup>ν</sup>* (*ν* = 0, 1, 2, 3) are the Dirac matrices; *Aμ*(*kir*�

*e<sup>μ</sup>* is the photon polarization four-vector; G(*r*,*r*�

pulsed wave anterior and posticous parts.

The amplitude (197) accurate within terms <sup>∼</sup> *<sup>η</sup>*<sup>2</sup>

0,0 + *η*<sup>0</sup> ∑ *l*,*l*�∈(|*l*−*l*�

field (2).

*Sf i* <sup>≈</sup> *<sup>B</sup>δ*(2)

⎛ ⎝*T*(*j*)*νμ*

conservation laws we have

*Tνμ f i* = ∑ *j*


*q*

Fig. 10. The dependence of the differential cross-section of scattering of an electron by muon in a pulsed light field (195) (in units of respective cross-sections in an external field absence) on the initial polar angle when an azimuthal angle is fixed *ϕ<sup>i</sup>* = *π*/4 and the value of the parameter *ρ* = 2. The cases of particles relative velocities *V* = 0.2 (solid line), *V* = 0.6 (dotted line) and *V* = 0.9 (dash-dotted Line) are represented.

cross sections in an external field absence within the whole polar angles range. Hereby, the greatest exceeding appears for the case of particles small relative velocities (*V* = 0.2), at that the exceeding reaches into five orders of the magnitude (for scattering of an electron by an electron (positron)), and two orders for scattering of an electron by a muon. Also there is a suppression of the resonant cross section in the case of leptons high relative velocities within the range of the initial polar angles *θ<sup>i</sup>* ≈ 60◦.

#### **5. Resonant scattering of a photon by an electron in the pulsed laser field**

Oleinik (1967) specified resonances in the Compton effect in the field of a plane monochromatic wave for the first time, but his studies had a rather fragmentary form (see also Belousov (1977)). The resonance of direct and exchange diagrams in the general relativistic case for the field of a weakly intensive plane monochromatic electromagnetic wave was considered by Voroshilo & Roshchupkin (2005). Scattering of a photon by an electron in a pulsed light field for the direct diagram resonance in the range of weak fields (5) was studied in work Voroshilo et al. (2011).

#### **5.1 Process amplitude**

The amplitude of scattering of a photon with the four momentum *ki* = (*ωi*, **k***i*) by an electron with the four momentum *pi* = (*Ei*, **p***i*) in an pulsed field (2) (Fig. 11) is given by the expression

$$\mathcal{S}\_{fi} = \mathcal{S}\_{fi}^{(d)} + \mathcal{S}\_{fi}^{(e)} \, \, \, \, \tag{197}$$

$$S\_{fi}^{(d)} = -ie^2 \int d^4r d^4r' \bar{\Psi}\_{p\_f}(r) \hat{\gamma}^\mu \mathcal{G}(r,r') \hat{\gamma}^\nu \Psi\_{p\_l}(r') A\_\mu^\*(k\_f r) A\_\nu(k\_l r'), \quad S\_{fi}^{(e)} = S\_{fi}^{(d)} \Big(k\_f \leftrightarrow -k\_i\Big), \tag{198}$$

38 Will-be-set-by-IN-TECH

Fig. 10. The dependence of the differential cross-section of scattering of an electron by muon in a pulsed light field (195) (in units of respective cross-sections in an external field absence) on the initial polar angle when an azimuthal angle is fixed *ϕ<sup>i</sup>* = *π*/4 and the value of the parameter *ρ* = 2. The cases of particles relative velocities *V* = 0.2 (solid line), *V* = 0.6 (dotted

cross sections in an external field absence within the whole polar angles range. Hereby, the greatest exceeding appears for the case of particles small relative velocities (*V* = 0.2), at that the exceeding reaches into five orders of the magnitude (for scattering of an electron by an electron (positron)), and two orders for scattering of an electron by a muon. Also there is a suppression of the resonant cross section in the case of leptons high relative velocities within

Oleinik (1967) specified resonances in the Compton effect in the field of a plane monochromatic wave for the first time, but his studies had a rather fragmentary form (see also Belousov (1977)). The resonance of direct and exchange diagrams in the general relativistic case for the field of a weakly intensive plane monochromatic electromagnetic wave was considered by Voroshilo & Roshchupkin (2005). Scattering of a photon by an electron in a pulsed light field for the direct diagram resonance in the range of weak fields (5) was studied

The amplitude of scattering of a photon with the four momentum *ki* = (*ωi*, **k***i*) by an electron with the four momentum *pi* = (*Ei*, **p***i*) in an pulsed field (2) (Fig. 11) is given by the expression

*f i* <sup>+</sup> *<sup>S</sup>*(*e*)

)*A*∗

*<sup>μ</sup>*(*k fr*)*Aν*(*kir*�

*f i* , (197)

*f i* <sup>=</sup> *<sup>S</sup>*(*d*) *f i* 

*k <sup>f</sup>* ↔ −*ki*

 , (198)

), *S*(*e*)

*Sf i* <sup>=</sup> *<sup>S</sup>*(*d*)

)*γ*˜ *<sup>ν</sup>*Ψ*pi*(*r*�

Ψ¯ *pf* (*r*)*γ*˜*μ*G(*r*,*r*�

**5. Resonant scattering of a photon by an electron in the pulsed laser field**

line) and *V* = 0.9 (dash-dotted Line) are represented.

the range of the initial polar angles *θ<sup>i</sup>* ≈ 60◦.

in work Voroshilo et al. (2011).

 *d*4*rd*4*r*�

**5.1 Process amplitude**

*S*(*d*) *f i* <sup>=</sup>−*ie*<sup>2</sup>

Fig. 11. The Feynman diagram of the Compton effect in the field of a pulsed light wave.

where *pf* = (*Ef* , **p***f*) and *k <sup>f</sup>* = (*ω<sup>f</sup>* , **k***f*) are four momenta of an outgoing electron and a photon; *γ*˜ *<sup>ν</sup>* (*ν* = 0, 1, 2, 3) are the Dirac matrices; *Aμ*(*kir*� ) is the wave function of a photon (8); *e<sup>μ</sup>* is the photon polarization four-vector; G(*r*,*r*� ) is the Green function of an electron in the field (2).

The case when a laser field intensity meets the following condition

$$
\eta\_0^2 \lesssim \varphi\_0^{-1} \ll 1, \quad \varrho\_0 = \omega \,\mathsf{r}, \tag{199}
$$

is considered through this section. This condition allows both to carry out the decomposition with respect to the small parameter and to neglect the interference of contributions of the pulsed wave anterior and posticous parts.

The amplitude (197) accurate within terms <sup>∼</sup> *<sup>η</sup>*<sup>2</sup> <sup>0</sup> assumes the form

$$S\_{fi} \approx B\delta^{(2)}(\mathbf{p}\_{i,\perp} + \mathbf{k}\_{i,\perp} - \mathbf{p}\_{f,\perp} - \mathbf{k}\_{f,\perp})\delta(p\_{i,-} + k\_{i,-} - p\_{f,-} - k\_{f,-})e\_{\nu}^{\prime \ast}e\_{\mu} \cdot \vec{u}\_{p\_f}T\_{f\mathbf{i}}^{\mu\mu}u\_{p\_i\nu} \quad (200)$$

$$T\_{fi}^{\nu\mu} = \sum\_{j} \left( T\_{0,0}^{(j)\nu\mu} + \eta\_0 \sum\_{l,l' \in (|l-l'| + |l'| = 1)} T\_{l-l',l'}^{(j)\nu\mu} + \eta\_0^2 \left( T\_{0,0}^{(j)\nu\mu} + \sum\_{l,l' \in (|l-l'| + |l'| = 2)} T\_{l-l',l'}^{(j)\nu\mu} \right) \right), \tag{201}$$

where *j* = *e*, *d*; indices *d*, *e* are concerned to direct and exchange diagrams; *B* is the normalization factor; **<sup>p</sup>***i*,⊥, **<sup>k</sup>***i*,⊥, **<sup>p</sup>***<sup>f</sup>* ,⊥, **<sup>k</sup>***<sup>f</sup>* ,<sup>⊥</sup> are the projections of corresponding vectors on the wave polarization plane; *pi*,<sup>−</sup> = *Ei* − *pi*,*z*, *ki*,<sup>−</sup> = *<sup>ω</sup><sup>i</sup>* − *ki*,*z*, *pf* ,<sup>−</sup> = *Ef* − *pf* ,*z*, *<sup>k</sup> <sup>f</sup>* ,<sup>−</sup> = *<sup>ω</sup><sup>f</sup>* − *<sup>k</sup> <sup>f</sup>* ,*<sup>z</sup>* are differences between zeroth components of the corresponding four momentum and its projection on direction of wave propagation; *q*, *f* are four momenta of an intermediate particle, which conform to direct and exchange diagrams on Fig. 11, at that under the four momenta conservation laws we have

$$\mathbf{q}\_{\perp} = \mathbf{p}\_{i,\perp} + \mathbf{k}\_{i,\perp}, \quad q\_{-} = p\_{i,-} + k\_{i,-}; \quad \mathbf{f}\_{\perp} = \mathbf{p}\_{i,\perp} - \mathbf{k}\_{f,\perp}, \quad f\_{-} = p\_{i,-} - k\_{f,-}.\tag{202}$$

The summands in Eq. (201), proportional to the zeroth degree of *η*<sup>0</sup> determine the amplitude of the Compton effect in external field absence (Klein & Nishina (1929)). The summands, proportional to the first degree of the parameter *η*0, determine the corrections (for them |*l* − *l* � | + |*l* � | = 1 is valid) specified by participation of one wave photon in the process. The summands, proportional to the second degree of the parameter *η*0, determine the corrections

where **n***<sup>f</sup>* is the directive unit vector for the final photon emission.

<sup>±</sup>1(*pf* , *<sup>q</sup>*−1) = <sup>±</sup>*y*0(*pf* , *<sup>q</sup>*−1)

 *ε*ˆ (∓) *<sup>k</sup><sup>ν</sup>* <sup>−</sup> <sup>ˆ</sup> *kε* (∓)*ν* + 1 (*kpf*) <sup>−</sup> <sup>1</sup>

<sup>±</sup><sup>1</sup> in (203), (204) are determined by:

<sup>−</sup>*g*2(*pf* , *<sup>q</sup>*), tan *<sup>χ</sup>* <sup>=</sup> *<sup>δ</sup>*(*gey*)

The parameter *β* ≡ *β* (*q*−1) which corresponds to the resonant process with *l*

<sup>1</sup> <sup>−</sup> *<sup>u</sup>*˜ 1 + *u*˜ 

(one field photon emits in the beginning, and one photon absorbs at the end of the process)

*ωi*/*ωi*,res − 1

Here, the invariant parameter *u*˜ and the frequency *ωi*,res, which corresponds to the resonant

where **n***<sup>i</sup>* is the unit vector along the propagation direction of incident photon. We rewrite this

(*Ei* − *ω*)

We consider that the correlation *ω* � *m* is valid in the region *υ<sup>i</sup>* = |**p***i*|/*Ei* � *ω*/*m* � 1 (it is the nonrelativistic case, which also corresponds to the rest frame of an final electron) and

*<sup>m</sup>* (<sup>1</sup> <sup>−</sup> cos ˜

*θ*) , ˜

*<sup>ω</sup>i*,res <sup>=</sup> (*kpi*)

(*Ei* − *ω*)/*m* +

1 − *u*<sup>1</sup> +

˜

, 0 <sup>≤</sup> *<sup>u</sup>*˜ <sup>≤</sup> *<sup>u</sup>*1, *<sup>u</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup>(*kpi*)

*Ei* − *ω* − ([**p***<sup>i</sup>* − **k**]**n***i*)

(*Ei* − *ω*)

<sup>2</sup> *<sup>e</sup>*

Here, quantities *<sup>ε</sup>*(±) <sup>=</sup> *ex* <sup>±</sup> *<sup>i</sup>δey*; *<sup>y</sup>*0(*pf* , *<sup>q</sup>*), *<sup>χ</sup>* <sup>≡</sup> *<sup>χ</sup>*(*pf* , *<sup>q</sup>*) are the kinematical parameters

<sup>∓</sup>*i<sup>χ</sup>* · *<sup>γ</sup>ν*<sup>+</sup>

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 147

(*kq*−1)

(*gex*) , *<sup>g</sup>* <sup>≡</sup> *<sup>g</sup>*(*pf* , *<sup>q</sup>*) = *pf*

*<sup>ω</sup><sup>i</sup>*

*<sup>ω</sup>i*,res <sup>−</sup> <sup>1</sup>

<sup>2</sup> /*m*<sup>2</sup> <sup>+</sup> *<sup>u</sup>*<sup>1</sup> <sup>−</sup> <sup>1</sup> · cos ˜

*θ***<sup>S</sup>** = ∠(**S**, **n***i*), **S** = **p***<sup>i</sup>* − **k**. (216)

 sin<sup>2</sup> ˜ *θ***S**

<sup>2</sup> /*m*<sup>2</sup> <sup>+</sup> *<sup>u</sup>*<sup>1</sup> <sup>−</sup> <sup>1</sup>

 *ε*ˆ (∓)ˆ *kγ<sup>ν</sup>* .

(*kpf*) <sup>−</sup> *<sup>q</sup>*

(*kq*)

. (212)

*<sup>m</sup>*<sup>2</sup> , (213)

, (214)

*θ***S**

*<sup>θ</sup>* <sup>=</sup> <sup>∠</sup>(**k**, **<sup>n</sup>***i*) <sup>≈</sup> *<sup>π</sup>* <sup>−</sup> ˜

, (215)

*θ***S**. (217)

(210)

. (211)

� = −1, *l* = 0

Bispinor matrices *M<sup>ν</sup>*

*M<sup>ν</sup>*

 2 (*kq*−1)

**5.2.1 Resonance conditions for the direct diagram**

*β ϕ*0

*<sup>ω</sup>i*,res <sup>=</sup> *mu*<sup>1</sup>

2

<sup>1</sup> <sup>−</sup> cos ˜ *θ* ≈ *<sup>ω</sup>* 1 + *ω*

<sup>=</sup> <sup>1</sup> 2

*<sup>u</sup>*˜ <sup>=</sup> (*kki*) (*piki*)

+ *m* 4

*y*0(*pf* , *q*) = *mη*

**5.2 Resonant kinematics**

may be written in the form:

maximum, are determined by:

expression as

where

obtain:

*<sup>ω</sup>i*,res <sup>=</sup> *<sup>ω</sup>*

<sup>1</sup> <sup>−</sup> *<sup>ω</sup>*/*<sup>m</sup>*

specified by participation of two wave photons in the process (for them |*l* − *l* � | + |*l* � | = 2 is valid).

In the case of a plane monochromatic wave the resonance is associated with the fact that an intermediate particle falls within the mass shell: *q*<sup>2</sup> = *m*2, *f* <sup>2</sup> = *m*2. The corrections to the Compton effect probability, which are specified by processes with one wave photon participation, are the nonresonant. They are proportional to the second order of the parameter *η*<sup>0</sup> and, therefore, are small in comparison with the Compton effect probability. But among processes with two wave photons participation there are such ones, which may have the resonant behavior. The both resonance of direct diagram through an electron intermediate state and resonance of exchange diagram through a positron intermediate state permit the processes with *l* � = −1, *l* = 0. The resonance of the exchange diagram through an electron intermediate state permits the process with *l* � = 1, *l* = 0. These processes may have resonant character in the case of a pulsed field (2) (Voroshilo et al. (2011)).

The expressions for *T*(*d*,*e*)*νμ l*−*l*� ,*l*� in Eq. (201) for resonant processes have the form:

$$T\_{1,-1}^{(d)\nu\mu} \approx \frac{\pi\omega}{8(kq\_{-1})} \cdot I(\beta\_{-1}(q\_{-1}), l\_\*) \left[M\_1^{\nu}(p\_{f'}q\_{-1})\left(\hat{q}\_{-1} + m\right)M\_{-1}^{\mu}(q\_{-1}, p\_i)\right],\tag{203}$$

$$T\_{\mp 1, \pm 1}^{(\varepsilon)\nu\mu} \approx \frac{\pi\omega}{8(kf\_{\mp 1})} \cdot I(\mathcal{S}\_{\mp 1}(f\_{\mp 1}), l\_\*) \left[M\_{\pm 1}^{\nu}(p\_{f'}f\_{\mp 1})\left(f\_{\mp 1} + m\right)M\_{\mp 1}^{\mu}(f\_{\mp 1}, p\_i)\right]. \tag{204}$$

Here

$$I(\boldsymbol{\beta}\_{l'}, l\_\*) = \frac{\pi}{4(kq\_{l'})} \left( \text{erfi}\left(\frac{\sqrt{2}}{2}\left(\boldsymbol{\beta}\_{l'} - \frac{l\_\*\,\boldsymbol{\varrho}\_0}{4}\right)\right) + i\right) \exp\left\{-\frac{\boldsymbol{\varrho}\_0^2 l\_\*^2 + 8\left(\boldsymbol{\beta}\_{l'} - l\_\*\,\boldsymbol{\varrho}\_0/4\right)^2}{16}\right\},\tag{205}$$

where erfi(z) is the error function of imaginary argument; *βl*� is the resonant parameter:

$$
\beta\_{l'}\left(q\_{l'}\right) = \frac{q\_{l'}^2 - m^2}{4(kq\_{l'})}\omega\tau. \tag{206}
$$

Exactly the parameter *βl*� determines the process behavior character. Thus, the values *βl*� 1 correspond to the resonant behavior. The opposite case *βl*� � 1 corresponds to the nonresonant one. Under the values *βl*� � 1 the function *I*(*βl*� , *a*) has the following asymptotic form:

$$I(\beta\_{l'}, l\_\*) \approx \sqrt{\frac{\pi}{2}} \frac{1}{\beta\_{l'}} \exp\left\{-\frac{1}{32} l\_\*^2 \varphi\_0^2\right\}.\tag{207}$$

In Eqs. (203)-(206) the quantities *<sup>q</sup>*−<sup>1</sup> = *pi* + *ki* − *<sup>k</sup>*, *<sup>f</sup>*∓<sup>1</sup> = *pi* − *<sup>k</sup> <sup>f</sup>* ± *<sup>k</sup>* correspond to the "strict" four momentum conservation law (like the monochromatic wave case, when summands <sup>∼</sup> *<sup>η</sup>*<sup>2</sup> 0 are neglected); the quantity *l*<sup>∗</sup> are the invariant parameter which are determined from the following equation:

$$p\_i + k\_i + l\_\*k = p\_f + k\_f.\tag{208}$$

It follows from Eq. (205) that <sup>|</sup>*l*∗| <sup>∼</sup> *<sup>ϕ</sup>*−<sup>1</sup> <sup>0</sup> . Consequently, in the zero-order approximation with respect to the parameter *ϕ*−<sup>1</sup> <sup>0</sup> the frequency of a scattered photon is amount:

$$
\omega\_f \approx \omega\_f^{(0)}, \quad \omega\_f^{(0)} = \frac{(p\_i k\_i)}{E\_i + \omega\_i - \left( [\mathbf{p}\_i + \mathbf{k}\_i] \mathbf{n}\_f \right)} \tag{209}
$$

where **n***<sup>f</sup>* is the directive unit vector for the final photon emission.

Bispinor matrices *M<sup>ν</sup>* <sup>±</sup><sup>1</sup> in (203), (204) are determined by:

$$\begin{split} M\_{\pm 1}^{\nu}(p\_f, q\_{-1}) &= \pm \frac{y\_0(p\_f, q\_{-1})}{2} e^{\mp i \chi} \cdot \gamma^{\nu} + \\ \frac{1}{4} + \frac{m}{4} \left( \frac{2}{(kq\_{-1})} \left[ \mathfrak{E}^{(\mp)} k^{\nu} - \hat{k} \mathfrak{E}^{(\mp) \nu} \right] + \left[ \frac{1}{(kp\_f)} - \frac{1}{(kq\_{-1})} \right] \mathfrak{E}^{(\mp)} \hat{k} \gamma^{\nu} \right). \end{split} \tag{210}$$

Here, quantities *<sup>ε</sup>*(±) <sup>=</sup> *ex* <sup>±</sup> *<sup>i</sup>δey*; *<sup>y</sup>*0(*pf* , *<sup>q</sup>*), *<sup>χ</sup>* <sup>≡</sup> *<sup>χ</sup>*(*pf* , *<sup>q</sup>*) are the kinematical parameters

$$g\_0(p\_f, q) = m\eta \sqrt{-g^2(p\_f, q)}, \quad \tan \chi = \frac{\delta(ge\_f)}{(ge\_X)}, \quad g \equiv g(p\_f, q) = \frac{p\_f}{(kp\_f)} - \frac{q}{(kq)}.\tag{211}$$

#### **5.2 Resonant kinematics**

40 Will-be-set-by-IN-TECH

In the case of a plane monochromatic wave the resonance is associated with the fact that an intermediate particle falls within the mass shell: *q*<sup>2</sup> = *m*2, *f* <sup>2</sup> = *m*2. The corrections to the Compton effect probability, which are specified by processes with one wave photon participation, are the nonresonant. They are proportional to the second order of the parameter *η*<sup>0</sup> and, therefore, are small in comparison with the Compton effect probability. But among processes with two wave photons participation there are such ones, which may have the resonant behavior. The both resonance of direct diagram through an electron intermediate state and resonance of exchange diagram through a positron intermediate state permit the

� = −1, *l* = 0. The resonance of the exchange diagram through an electron

<sup>1</sup> (*pf* , *<sup>q</sup>*−1)(*q*ˆ−<sup>1</sup> <sup>+</sup> *<sup>m</sup>*) *<sup>M</sup><sup>μ</sup>*

 ˆ *<sup>f</sup>*∓<sup>1</sup> + *<sup>m</sup>*

,*l*� in Eq. (201) for resonant processes have the form:

<sup>±</sup>1(*pf* , *<sup>f</sup>*∓1)

*<sup>l</sup>*� <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

Exactly the parameter *βl*� determines the process behavior character. Thus, the values *βl*� 1 correspond to the resonant behavior. The opposite case *βl*� � 1 corresponds to the nonresonant one. Under the values *βl*� � 1 the function *I*(*βl*� , *a*) has the following asymptotic

In Eqs. (203)-(206) the quantities *<sup>q</sup>*−<sup>1</sup> = *pi* + *ki* − *<sup>k</sup>*, *<sup>f</sup>*∓<sup>1</sup> = *pi* − *<sup>k</sup> <sup>f</sup>* ± *<sup>k</sup>* correspond to the "strict" four momentum conservation law (like the monochromatic wave case, when summands <sup>∼</sup> *<sup>η</sup>*<sup>2</sup>

are neglected); the quantity *l*<sup>∗</sup> are the invariant parameter which are determined from the

<sup>0</sup> the frequency of a scattered photon is amount:

*<sup>f</sup>* <sup>=</sup> (*piki*) *Ei* + *ω<sup>i</sup>* −

 *M<sup>ν</sup>*

 *M<sup>ν</sup>*

> + *i* exp − *ϕ*2 0*l* 2

where erfi(z) is the error function of imaginary argument; *βl*� is the resonant parameter:

*<sup>β</sup>l*� (*ql*�) <sup>=</sup> *<sup>q</sup>*<sup>2</sup>

*π* 2 1 *βl*� exp − 1 32 *l* 2 ∗*ϕ*2 0 

*<sup>β</sup>l*� <sup>−</sup> *<sup>l</sup>*∗*ϕ*<sup>0</sup> 4

*I*(*βl*� , *l*∗) ≈

� = 1, *l* = 0. These processes may have resonant

 *M<sup>μ</sup>*

−1(*q*−1, *pi*)

∓1(*f*∓1, *pi*)

<sup>∗</sup> + <sup>8</sup> (*βl*� − *<sup>l</sup>*∗*ϕ*0/4)

16

<sup>4</sup>(*kql*�) *ωτ*. (206)

*pi* + *ki* + *l*∗*k* = *pf* + *k <sup>f</sup>* . (208)

[**p***<sup>i</sup>* + **k***i*]**n***<sup>f</sup>*

<sup>0</sup> . Consequently, in the zero-order approximation with

2

. (207)

, (209)

, (203)

. (204)

, (205)

0

� | + |*l* � | = 2 is

specified by participation of two wave photons in the process (for them |*l* − *l*

valid).

processes with *l*

The expressions for *T*(*d*,*e*)*νμ*

1,−<sup>1</sup> <sup>≈</sup> *πω*

<sup>∓</sup>1,±<sup>1</sup> <sup>≈</sup> *πω*

4(*kql*�)

*T*(*d*)*νμ*

*T*(*e*)*νμ*

*<sup>I</sup>*(*βl*� , *<sup>l</sup>*∗)= *<sup>π</sup>*

following equation:

It follows from Eq. (205) that <sup>|</sup>*l*∗| <sup>∼</sup> *<sup>ϕ</sup>*−<sup>1</sup>

*<sup>ω</sup><sup>f</sup>* <sup>≈</sup> *<sup>ω</sup>*(0)

*<sup>f</sup>* , *<sup>ω</sup>*(0)

respect to the parameter *ϕ*−<sup>1</sup>

Here

form:

intermediate state permits the process with *l*

*l*−*l*�

 erfi √<sup>2</sup> 2 

character in the case of a pulsed field (2) (Voroshilo et al. (2011)).

<sup>8</sup>(*kq*−1) · *<sup>I</sup>*(*β*−1(*q*−1), *<sup>l</sup>*∗)

<sup>8</sup>(*k f*∓1) · *<sup>I</sup>*(*β*∓1(*f*∓1), *<sup>l</sup>*∗)

#### **5.2.1 Resonance conditions for the direct diagram**

The parameter *β* ≡ *β* (*q*−1) which corresponds to the resonant process with *l* � = −1, *l* = 0 (one field photon emits in the beginning, and one photon absorbs at the end of the process) may be written in the form:

$$\frac{\beta}{\varphi\_0} = \frac{1}{2} \frac{1 - \tilde{\mathfrak{u}}}{[1 + \tilde{\mathfrak{u}} \left(\omega\_{\hat{\mathfrak{u}}}/\omega\_{\hat{\mathfrak{u}}, \text{res}} - 1\right)]} \left(\frac{\omega\_{\hat{\mathfrak{u}}}}{\omega\_{\hat{\mathfrak{u}}, \text{res}}} - 1\right). \tag{212}$$

Here, the invariant parameter *u*˜ and the frequency *ωi*,res, which corresponds to the resonant maximum, are determined by:

$$\vec{u} = \frac{(k k\_l)}{(p\_i k\_i)}, \quad 0 \le \vec{u} \le u\_{1'} \quad u\_1 = \frac{2(k p\_i)}{m^2},\tag{213}$$

$$\omega\_{i, \text{res}} = \frac{(kp\_i)}{E\_i - \omega - \left( [\mathbf{p}\_i - \mathbf{k}] \mathbf{n}\_i \right)} \Big|\_{\text{\(\,\,k\)}} \tag{214}$$

where **n***<sup>i</sup>* is the unit vector along the propagation direction of incident photon. We rewrite this expression as

$$\omega\_{i, \text{res}} = \frac{m u\_1}{2} \frac{(E\_i - \omega)/m + \sqrt{(E\_i - \omega)^2/m^2 + u\_1 - 1} \cdot \cos \tilde{\theta}\_{\mathbf{S}}}{1 - u\_1 + \left(\left(E\_i - \omega\right)^2/m^2 + u\_1 - 1\right) \sin^2 \tilde{\theta}\_{\mathbf{S}}},\tag{215}$$

where

$$
\tilde{\theta}\_{\mathbf{S}} = \angle(\mathbf{S}, \mathbf{n}\_{i}), \quad \mathbf{S} = \mathbf{p}\_{i} - \mathbf{k}. \tag{216}
$$

We consider that the correlation *ω* � *m* is valid in the region *υ<sup>i</sup>* = |**p***i*|/*Ei* � *ω*/*m* � 1 (it is the nonrelativistic case, which also corresponds to the rest frame of an final electron) and obtain:

$$\omega\_{\text{i,res}} = \frac{\omega}{1 - \omega/m \left(1 - \cos \tilde{\theta}\right)} \approx \omega \left(1 + \frac{\omega}{m}(1 - \cos \tilde{\theta})\right), \quad \tilde{\theta} = \angle(\mathbf{k}, \mathbf{n}\_{\text{i}}) \approx \pi - \tilde{\theta}\_{\text{S}}.\tag{217}$$

The cases close to realization of the condition *u*˜ = 1 (˜

<sup>2</sup> (<sup>1</sup> <sup>−</sup> *<sup>u</sup>*˜)

**5.2.2 Resonance conditions for the exchange diagram**

<sup>=</sup> <sup>1</sup> 2

*υ*� =

under limitations on parameter *υ*� and, hence, angle ˜

� *υ*� �

� *kk <sup>f</sup>* �

� *pik <sup>f</sup>*

*<sup>υ</sup>*� <sup>&</sup>gt; <sup>1</sup>(*u*<sup>1</sup> <sup>&</sup>gt; <sup>1</sup>) <sup>0</sup> <sup>≤</sup> ˜

**<sup>h</sup>**<sup>∓</sup> = ∠(**h**∓, **n***f*):

�, *h*<sup>∓</sup> = (*h*<sup>∓</sup>

*u*˜ 2

photon, which correspond to the resonant maximum, are defined by:

*β ϕ*0 ≈ 1

parameters *β*<sup>∓</sup> ≡ *β* (*f*±1) have the form:

*β*∓ *ϕ*0

For the processes with *l*

through an electron (*l*

**h**∓ and the opening angle *θ*�

cos *θ*�

has to be met.

**<sup>h</sup>**<sup>∓</sup> = *h*<sup>∓</sup> 0 �� �**h**∓� � � � �

Under the condition (220) the resonant parameter assumes the form:

� *ω<sup>i</sup>*

� = 1) and a positron (*l*

<sup>1</sup> <sup>−</sup> *<sup>ω</sup><sup>i</sup> ωi*,res

*<sup>ω</sup>i*,res <sup>−</sup> <sup>1</sup>

*υ*� ± 1

*<sup>f</sup>* ,res�

where the upper sign is concerned to an electron intermediate state, the lower sign is

It follows from Eq. (223) that the resonance via positron intermediate state can be observed

Equating the expressions (209), (223) we obtain that under the exchange diagram resonance directions of a scattered photon correspond to the condition of the resonant maximum; these directions lie on the surface of a cone (see Fig. 13); axis of the cone coincides with the vector

± 1 �

*<sup>f</sup>* ,res <sup>=</sup> (*kpi*)

*θ*�

**<sup>S</sup>** <sup>≤</sup> *<sup>α</sup>*<sup>0</sup> and *<sup>π</sup>* <sup>−</sup> *<sup>α</sup>*<sup>0</sup> <sup>≤</sup> ˜

<sup>1</sup> <sup>−</sup> *<sup>ω</sup><sup>f</sup>* /*ω*(∓)

concerned to a positron one; the invariant parameter *<sup>υ</sup>*� and the frequencies *<sup>ω</sup>*(∓)

�, *<sup>ω</sup>*(∓)

*θ*�

Thus, the four vector has to be a spatially similar one (*h*∓)<sup>2</sup> <sup>≤</sup> 0, i.e. the inequality

<sup>1</sup>(<sup>1</sup> <sup>−</sup> *uu*˜ <sup>1</sup>) <sup>∓</sup> <sup>2</sup>*u*˜1*u*<sup>1</sup> <sup>+</sup> *<sup>u</sup>*<sup>2</sup>

by:

because the frequency of a resonant photon in these cases has to be infinite, but it is impossible to put into practice. Therefore, the condition of the direct diagram resonance is determined

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 149

� � � � <sup>∼</sup> <sup>1</sup> *ϕ*0

�

, 0 < *u*˜ <

� = ±1, *l* = 0 which permit the resonance of the exchange diagram

⎛

(*Ei* − **p***i***n***f*)(*υ*� ± 1)

**<sup>S</sup>** = ∠(**S**, **n***f*):

*θ*�

<sup>0</sup> , **h**∓)=(*kpi*)[*pi* + *ki*] − (*piki*)[*k* ± *pi*] . (225)

<sup>1</sup> ≤ 0, (226)

<sup>⎝</sup><sup>1</sup> <sup>−</sup> *<sup>ω</sup><sup>f</sup> <sup>ω</sup>*(∓) *f* ,res

*θ***<sup>S</sup>** = *α*0) also have to be excluded,

� 1. (220)

1, *<sup>u</sup>*<sup>1</sup> <sup>&</sup>gt; 1. (221)

⎠ , (222)

. (223)

**<sup>S</sup>** ≤ *π*. (224)

*<sup>f</sup>* ,res of a final

� *u*1, *u*<sup>1</sup> < 1;

� = −1) intermediate states the resonant

⎞

Therefore, in this case the resonant frequency is closely approximated to the laser field frequency.

In range where the correlation *ω*/*m* � *υ<sup>i</sup>* < 1 is valid (it is the ultrarelativistic case) we derive:

$$
\omega\_{i, \text{res}} \approx \frac{mu\_1}{2} \frac{E\_i/m + \sqrt{E\_i^2/m^2 - 1} \cos \tilde{\theta}\_{\mathbf{S}}}{1 - u\_1 + \left(E\_i^2/m^2 - 1\right) \sin^2 \tilde{\theta}\_{\mathbf{S}}}
$$

In the ultrarelativistic case (*u*<sup>1</sup> > 1, *Ei*/*m* > *m*/*ω* � 1) under (*m*/*ω*) <sup>√</sup>*u*<sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; ˜ *θ***<sup>S</sup>** � 1 (˜ *<sup>θ</sup>***<sup>S</sup>** <sup>≈</sup> ˜ *θ***p**<sup>i</sup> = ∠(**p***i*, **n***i*)) we obtain:

$$
\omega\_{i, \text{res}} \approx \frac{u\_1 E\_i}{1 - u\_1 + \left(E\_i / m\right)^2 \tilde{\theta}\_{\mathbf{S}}^2}. \tag{218}
$$

.

Fig. 12 demonstrated dependence of the resonant frequency on the angle ˜ *θ***<sup>S</sup>** for different energies of an electron.

Fig. 12. The dependence of ratio of the resonant frequency of an ingoing photon to the laser field frequency *<sup>ω</sup>i*,res/*<sup>ω</sup>* (215) from the angle ˜ *θ***<sup>S</sup>** (216) under *ω*/*m* = 10−<sup>5</sup> for different energies of an ingoing electron.

The resonance of the amplitude, which corresponds to the direct diagram, is feasible only when the condition *u*˜ < 1 is satisfied, so that for the values *u*<sup>1</sup> > 1 the angle ˜ *θ***<sup>S</sup>** is restricted by the interval:

$$a\_0 < \tilde{\theta}\_{\mathbf{S}} < \pi, \quad a\_0 = \arccos \frac{E\_i - \omega}{|\mathbf{S}|}. \tag{219}$$

42 Will-be-set-by-IN-TECH

Therefore, in this case the resonant frequency is closely approximated to the laser field

In range where the correlation *ω*/*m* � *υ<sup>i</sup>* < 1 is valid (it is the ultrarelativistic case) we derive:

 *E*2

*E*2

1 − *u*<sup>1</sup> + (*Ei*/*m*)

0° 45° 90° 180°

Fig. 12. The dependence of ratio of the resonant frequency of an ingoing photon to the laser

The resonance of the amplitude, which corresponds to the direct diagram, is feasible only

*θ***<sup>S</sup>** < *π*, *α*<sup>0</sup> = arccos

when the condition *u*˜ < 1 is satisfied, so that for the values *u*<sup>1</sup> > 1 the angle ˜

*α*<sup>0</sup> < ˜

135°

*Ei* − *ω*

*θ***<sup>S</sup>** (216) under *ω*/*m* = 10−<sup>5</sup> for different

 *R***S**

<sup>|</sup>**S**<sup>|</sup> . (219)

*θ***<sup>S</sup>** is restricted by

*<sup>i</sup>* /*m*<sup>2</sup> <sup>−</sup> 1 cos ˜

2 ˜ *θ*2 **S**

/ 10 *<sup>i</sup> E m*

/ 20 *<sup>i</sup> E m*

/ 5 *<sup>i</sup> E m*

/ 1.5 *<sup>i</sup> E m*

 sin<sup>2</sup> ˜ *θ***S** .

*<sup>i</sup>* /*m*<sup>2</sup> − <sup>1</sup>

*θ***S**

<sup>√</sup>*u*<sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>&</sup>lt; ˜

. (218)

*θ***<sup>S</sup>** � 1

*θ***<sup>S</sup>** for different

*Ei*/*m* +

<sup>1</sup> <sup>−</sup> *<sup>u</sup>*<sup>1</sup> <sup>+</sup>

*<sup>ω</sup>i*,res <sup>≈</sup> *<sup>u</sup>*1*Ei*

Fig. 12 demonstrated dependence of the resonant frequency on the angle ˜

*<sup>ω</sup>i*,res <sup>≈</sup> *mu*<sup>1</sup>

*θ***p**<sup>i</sup> = ∠(**p***i*, **n***i*)) we obtain:

0

field frequency *<sup>ω</sup>i*,res/*<sup>ω</sup>* (215) from the angle ˜

energies of an ingoing electron.

the interval:

100

200

300

*X X <sup>i</sup>*,res

energies of an electron.

2

In the ultrarelativistic case (*u*<sup>1</sup> > 1, *Ei*/*m* > *m*/*ω* � 1) under (*m*/*ω*)

frequency.

(˜ *<sup>θ</sup>***<sup>S</sup>** <sup>≈</sup> ˜ The cases close to realization of the condition *u*˜ = 1 (˜ *θ***<sup>S</sup>** = *α*0) also have to be excluded, because the frequency of a resonant photon in these cases has to be infinite, but it is impossible to put into practice. Therefore, the condition of the direct diagram resonance is determined by:

$$\left|1 - \frac{\omega\_{\bar{l}}}{\omega\_{\bar{l}, \text{res}}}\right| \sim \frac{1}{\varphi\_0} \ll 1. \tag{220}$$

Under the condition (220) the resonant parameter assumes the form:

$$\frac{\beta}{\varphi\_0} \approx \frac{1}{2} \left( 1 - \mathfrak{a} \right) \left( \frac{\omega\_{\rm i}}{\omega\_{\rm i, \rm res}} - 1 \right), \quad 0 < \mathfrak{a} < \left\{ \begin{array}{ll} u\_1, & u\_1 < 1; \\ 1, & u\_1 > 1. \end{array} \right. \tag{221}$$

#### **5.2.2 Resonance conditions for the exchange diagram**

For the processes with *l* � = ±1, *l* = 0 which permit the resonance of the exchange diagram through an electron (*l* � = 1) and a positron (*l* � = −1) intermediate states the resonant parameters *β*<sup>∓</sup> ≡ *β* (*f*±1) have the form:

$$\frac{\beta\_{\overline{\mp}}}{\varphi\_0} = \frac{1}{2} \frac{v' \pm 1}{\left[v' \left(1 - \omega\_f/\omega\_{f, \text{res}}^{(\mp)}\right) \pm 1\right]} \left(1 - \frac{\omega\_f}{\omega\_{f, \text{res}}^{(\mp)}}\right),\tag{222}$$

where the upper sign is concerned to an electron intermediate state, the lower sign is concerned to a positron one; the invariant parameter *<sup>υ</sup>*� and the frequencies *<sup>ω</sup>*(∓) *<sup>f</sup>* ,res of a final photon, which correspond to the resonant maximum, are defined by:

$$v' = \frac{\begin{pmatrix} k k\_f \\ p\_i k\_f \end{pmatrix}}{\begin{pmatrix} p\_i k\_f \end{pmatrix}}, \quad \omega\_{f, \text{res}}^{(\mp)} = \frac{(k p\_i)}{(E\_i - \mathbf{p}\_i \mathbf{n}\_f) \begin{pmatrix} v' \pm 1 \end{pmatrix}}. \tag{223}$$

It follows from Eq. (223) that the resonance via positron intermediate state can be observed under limitations on parameter *υ*� and, hence, angle ˜ *θ*� **<sup>S</sup>** = ∠(**S**, **n***f*):

$$
v' > 1(u\_1 > 1) \quad 0 \le \tilde{\theta}'\_{\mathsf{S}} \le u\_0 \quad \text{and} \quad \pi - u\_0 \le \tilde{\theta}'\_{\mathsf{S}} \le \pi. \tag{224}$$

Equating the expressions (209), (223) we obtain that under the exchange diagram resonance directions of a scattered photon correspond to the condition of the resonant maximum; these directions lie on the surface of a cone (see Fig. 13); axis of the cone coincides with the vector **h**∓ and the opening angle *θ*� **<sup>h</sup>**<sup>∓</sup> = ∠(**h**∓, **n***f*):

$$\cos\theta\_{\mathbf{h}^{\mp}}' = h\_0^{\mp} / \left| \mathbf{h}^{\mp} \right| \,, \quad h^{\mp} = (h\_0^{\mp}, \mathbf{h}^{\mp}) = (kp\_i) \left[ p\_i + k\_i \right] - (p\_i k\_i) \left[ k \pm p\_i \right] . \tag{225}$$

Thus, the four vector has to be a spatially similar one (*h*∓)<sup>2</sup> <sup>≤</sup> 0, i.e. the inequality

$$
\tilde{u}\_1^2 (1 - \tilde{u}u\_1) \mp 2\tilde{u}\_1 u\_1 + u\_1^2 \le 0,\tag{226}
$$

has to be met.

Fig. 14 shows the resonant region of final photon frequency values *ω<sup>i</sup>* (in units of the initial electron energy *Ei*), which is determined by the system of the equations and the inequalities

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 151

*θ*), *θ* = 163◦.

resonance of direct diagram (line thickness <sup>1</sup>

Fig. 14. Resonant region of frequencies *ωi*(*α*) of an ingoing photon (in units of the ingoing electron energy *Ei*), which is determined by the system of equations and inequalities (215),

We consider the case when the conditions of the direct diagram resonance (220) are realized. Thought it is accompanied by the exchange diagram resonance, but its contribution may be

1. when an initial photon is emitted out of the strictly defined and narrow region of an initial

2. when the total probability is obtained, since the contribution to the total probability from

The differential probability is obtained by standard mode (Berestetskii et al. (1982)). After averaging over initial particle polarizations and summation over final particle polarizations

, *u*˜1)*f*(*u*, *u*˜1) − *g*(*u*�

photon directions when the exchange diagram resonance occurs (see Fig. 14);

the exchange diagram is <sup>∼</sup> (*ωτ*)−<sup>1</sup> � 1 and, therefore, it may be neglected.

and also the integration over frequencies *ω<sup>f</sup>* and the azimuthal angle *ψ*� = ∠

 *f*(*u*�

final photon emission we obtain the differential probability:

<sup>2</sup> *<sup>P</sup>*res (*β*)

(*ωτ*)

resonance of exchange diagram through electron state; resonance of exchange diagram through positron state.

*θ*)(*m*/*E*) when *ω* = 2.36 eV,

0 *K* );

*du*�

, *u*˜1)*g*(*u*, *u*˜1)

**<sup>e</sup>***x*, **<sup>k</sup>***<sup>f</sup>* ,<sup>⊥</sup>

(<sup>1</sup> <sup>+</sup> *<sup>u</sup>*�)<sup>2</sup> *<sup>τ</sup>*. (232)

 of a

(215), (228), (230), as a function of the parameter *<sup>α</sup>* = (*<sup>θ</sup>* <sup>−</sup> ˜

*<sup>θ</sup>***<sup>S</sup>** <sup>≈</sup> ˜

*Ei* <sup>=</sup> 48.0 GeV (since *Ei* � *<sup>ω</sup>*, then ˜

(228), (230).

*dWres*

*f i* <sup>≈</sup> <sup>2</sup>*e*4*η*<sup>4</sup>

0*m*<sup>2</sup> *πωiEiVu*˜1

**5.3 Resonant probability for the direct diagram**

neglected in the following cases:

The invariant parameter *u*˜1 is equal to:

Fig. 13. Geometry of emission of an outgoing photon in the case of occurrence of the exchange diagram resonance.

From the inequality (226) we derive the following condition on the initial photon frequency; at this frequency the exchange diagram resonance through the electron intermediate state occurs:

$$\begin{cases} \frac{\omega\_f}{1 + \sqrt{\mu\_1 \tilde{\mu}}} \le \omega\_{i, \text{res}} \le \frac{\omega\_f}{1 - \sqrt{\mu\_1 \tilde{\mu}}}, & \tilde{u} < u\_1^{-1};\\\omega\_{i, \text{res}} \ge \frac{\omega\_f}{1 + \sqrt{\mu\_1 \tilde{\mu}}}, & u\_1^{-1} < \tilde{u} < u\_1. \end{cases} \tag{228}$$

Here, the function *f* has the form

$$f = \frac{1 - v\_i \cos \theta}{1 - v\_i \cos \tilde{\theta}}'\tag{229}$$

where *θ* = ∠(**k**, **k***i*), ˜ *θ* = ∠(**k**, **p***i*).

For a positron intermediate state the resonance occur under the condition that the initial photon frequency exceeds a certain threshold value:

$$
\omega\_{i, \text{res}} \ge \frac{\omega\_f}{\sqrt{\mu\_1 \tilde{\mu}} - 1}, \quad \mu\_1^{-1} < \tilde{\mu} < \mu\_1. \tag{230}
$$

Values of initial photon frequencies meet the condition of the direct diagram resonance *ω<sup>i</sup>* = *ωi*,res (214). They are founded within the frequencies interval (228); the exchange diagram resonance through an electron intermediate state occurs under these frequencies. Consequently, the direct diagram resonance is always accompanied by the exchange diagram resonance through an electron intermediate state, and within the region

$$
u\_1 > 1, \quad 1/u\_1 < \text{if} < 1\tag{231}$$

through a positron intermediate state also.

44 Will-be-set-by-IN-TECH

*<sup>u</sup>*˜1 <sup>=</sup> <sup>2</sup>(*pki*)

*<sup>f</sup>* **n n**

**h** *R* B

*Z* c

res *R*c

*z*

*x*

*ωf*

*ωi*,res ≥

*ωi*,res ≥

resonance through an electron intermediate state, and within the region

<sup>1</sup> <sup>+</sup> <sup>√</sup>*u*1*u*˜ <sup>≤</sup> *<sup>ω</sup>i*,res <sup>≤</sup>

*ωf* <sup>1</sup> <sup>+</sup> <sup>√</sup>*u*1*u*˜

**h** *Z* B

From the inequality (226) we derive the following condition on the initial photon frequency; at this frequency the exchange diagram resonance through the electron intermediate state

> *<sup>f</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>υ</sup><sup>i</sup>* cos *<sup>θ</sup>* <sup>1</sup> <sup>−</sup> *<sup>υ</sup><sup>i</sup>* cos ˜

For a positron intermediate state the resonance occur under the condition that the initial

Values of initial photon frequencies meet the condition of the direct diagram resonance *ω<sup>i</sup>* = *ωi*,res (214). They are founded within the frequencies interval (228); the exchange diagram resonance through an electron intermediate state occurs under these frequencies. Consequently, the direct diagram resonance is always accompanied by the exchange diagram

*ωf* <sup>√</sup>*u*1*u*˜ <sup>−</sup> <sup>1</sup>

*ωf* <sup>1</sup> <sup>−</sup> <sup>√</sup>*u*1*u*˜

, *u*−<sup>1</sup>

*θ*

, *u*−<sup>1</sup>

Fig. 13. Geometry of emission of an outgoing photon in the case of occurrence of the

**h** *R* Bc

*<sup>m</sup>*<sup>2</sup> . (227)

**h e** <sup>B</sup>

*y*

, *u*˜ < *u*−<sup>1</sup> <sup>1</sup> ;

(228)

<sup>1</sup> < *u*˜ < *u*1.

*u*<sup>1</sup> > 1, 1/*u*<sup>1</sup> < *u*˜ < 1 (231)

, (229)

<sup>1</sup> < *u*˜ < *u*1. (230)

The invariant parameter *u*˜1 is equal to:

exchange diagram resonance.

Here, the function *f* has the form

where *θ* = ∠(**k**, **k***i*), ˜

⎧ ⎪⎨

⎪⎩

*θ* = ∠(**k**, **p***i*).

photon frequency exceeds a certain threshold value:

through a positron intermediate state also.

occurs:

Fig. 14 shows the resonant region of final photon frequency values *ω<sup>i</sup>* (in units of the initial electron energy *Ei*), which is determined by the system of the equations and the inequalities (215), (228), (230), as a function of the parameter *<sup>α</sup>* = (*<sup>θ</sup>* <sup>−</sup> ˜ *θ*)(*m*/*E*) when *ω* = 2.36 eV, *Ei* <sup>=</sup> 48.0 GeV (since *Ei* � *<sup>ω</sup>*, then ˜ *<sup>θ</sup>***<sup>S</sup>** <sup>≈</sup> ˜ *θ*), *θ* = 163◦.

Fig. 14. Resonant region of frequencies *ωi*(*α*) of an ingoing photon (in units of the ingoing electron energy *Ei*), which is determined by the system of equations and inequalities (215), (228), (230).

### **5.3 Resonant probability for the direct diagram**

We consider the case when the conditions of the direct diagram resonance (220) are realized. Thought it is accompanied by the exchange diagram resonance, but its contribution may be neglected in the following cases:

1. when an initial photon is emitted out of the strictly defined and narrow region of an initial photon directions when the exchange diagram resonance occurs (see Fig. 14);

2. when the total probability is obtained, since the contribution to the total probability from the exchange diagram is <sup>∼</sup> (*ωτ*)−<sup>1</sup> � 1 and, therefore, it may be neglected.

The differential probability is obtained by standard mode (Berestetskii et al. (1982)). After averaging over initial particle polarizations and summation over final particle polarizations and also the integration over frequencies *ω<sup>f</sup>* and the azimuthal angle *ψ*� = ∠ **<sup>e</sup>***x*, **<sup>k</sup>***<sup>f</sup>* ,<sup>⊥</sup> of a final photon emission we obtain the differential probability:

$$d\mathcal{W}\_{f^{\mathrm{I}}}^{\mathrm{res}} \approx \frac{2e^{4}\eta\_{0}^{4}m^{2}}{\pi\omega\_{i}\mathbb{E}\_{i}V\tilde{\boldsymbol{u}}\_{1}} \left(\omega\,\mathrm{\boldsymbol{\tau}}\right)^{2}P\_{\mathrm{res}}\left(\boldsymbol{\beta}\right)\left[f(\boldsymbol{u}',\boldsymbol{\tilde{u}}\_{1})f(\boldsymbol{u},\boldsymbol{\tilde{u}}\_{1}) - g(\boldsymbol{u}',\boldsymbol{\tilde{u}}\_{1})g(\boldsymbol{u},\boldsymbol{\tilde{u}}\_{1})\right] \frac{d\boldsymbol{u}'}{(1+\boldsymbol{u}')^{2}}\boldsymbol{\boldsymbol{\tau}}.\tag{232}$$

where *W*<sup>1</sup> is the total probability of the intermediate state decay in a weakly intensive field;

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 153

ln (1 + *u*˜1) +

<sup>≈</sup> 8.51 *e*2*η*<sup>2</sup> <sup>0</sup> (*ωτ*)

When the condition (199) is met the appraisal value of ratio is equal to <sup>Γ</sup>imp/Γ<sup>R</sup> <sup>≥</sup> 103 � 1. Therefore, the width specified by the field pulsed character is the major one and the radiation

After the integration over the invariant parameter *u*� we derive the total probability of

<sup>1</sup> + (4 + 10*u*˜1 + 8*u*˜

<sup>0</sup> (*ωτ*)<sup>2</sup> *π*2

Ratio of the total probability (239) to the total probability of the Compton effect in external

where *T* is the observation time (*T τ*), which is determined by conditions of the concrete

(1 − *u*˜) *u*1

Fig. 16 demonstrates the ratio of the resonant probability of scattering of a photon by an electron in the field of a pulsed wave to probability of the Compton effect as a function of parameters *u*˜, *u*<sup>1</sup> within the resonant peak (*β* = 0) under *τ*/*T* = 1, *η*<sup>0</sup> = 0.05. It can be seen from Fig. 16 that the resonant probability may exceed considerably the probability of the Compton effect in external field absence. This fact becomes apparent particularly in the case *u*<sup>1</sup> � 1 (but it should be noticed here, that in view of infrared divergence the formulae (239),

 <sup>1</sup> <sup>−</sup> <sup>2</sup> *<sup>u</sup>*˜ *u*1 <sup>1</sup> <sup>−</sup> *<sup>u</sup>*˜ *u*1

<sup>0</sup> <sup>∼</sup> *<sup>u</sup>*<sup>1</sup> � 1 within the range of optical frequencies *Ei*/*<sup>m</sup>* � *<sup>m</sup>*/*<sup>ω</sup>* <sup>∼</sup> 105

1 2 + 8 *u*˜1

*u*˜1 *F* (*u*˜1) <sup>−</sup> <sup>1</sup> 2 (1 + *u*˜1)

(*ωτ*)2*P*res (*β*) [*F*(*u*˜1)*f*(*u*, *<sup>u</sup>*˜1) <sup>−</sup> *<sup>G</sup>*(*u*˜1)*g*(*u*, *<sup>u</sup>*˜1)] *<sup>τ</sup>*, (239)

2 <sup>1</sup> + 2*u*˜ 3 <sup>2</sup> . (237)

. (238)

<sup>1</sup>)ln(1 + *u*˜1)

[*F*(*u*˜1)*f*(*u*, *u*˜1) − *G*(*u*˜1)*g*(*u*, *u*˜1)]

<sup>0</sup>). Within the region *u*<sup>1</sup> 1 this effect disappears.

 .

*<sup>u</sup>*˜1*F*(*u*˜1) ,

. (242)

(240)

(241)

Γimp ΓR

the function *F* (*u*˜1) is defined by

widening may be neglected.

*Wres*

, *<sup>u</sup>*˜1) = *<sup>u</sup>*˜1

4*u*˜1(1 + *u*˜1)<sup>2</sup>

field absence is expressed as

≈ *τ*

When *u*<sup>1</sup> � 1 we derive

Under conditions *η*<sup>2</sup>

0

*G*(*u*�

*Wres f i* wCompt

experiment.

<sup>=</sup> <sup>1</sup>

*f i* <sup>≈</sup> <sup>2</sup>*e*4*η*<sup>4</sup>

*g*(*u*�

This ratio equals

*<sup>F</sup>*(*u*˜1) =

<sup>1</sup> <sup>−</sup> <sup>4</sup> *u*˜1 − 8 *u*˜2 1

photon-electron scattering under the direct diagram resonance

, *<sup>u</sup>*˜1) *du*�

−4*u*˜1 − 8*u*˜

*<sup>T</sup> <sup>P</sup>*res (*β*) · *<sup>R</sup>*(*u*, *<sup>u</sup>*˜1), *<sup>R</sup>*(*u*˜, *<sup>u</sup>*1) = <sup>2</sup>*η*<sup>4</sup>

<sup>0</sup> (*ωτ*)<sup>2</sup> *π*2

*<sup>R</sup>*(*u*˜, *<sup>u</sup>*1) <sup>≈</sup> <sup>4</sup>*η*<sup>4</sup>

(241) are correct within the region *u*<sup>1</sup> *η*<sup>2</sup>

for the ratio of probabilities is correct *<sup>R</sup>* <sup>∼</sup> 103.

(<sup>1</sup> <sup>+</sup> *<sup>u</sup>*�)<sup>2</sup> <sup>=</sup>

2 <sup>1</sup> − 5*u*˜ 3

0*m*<sup>2</sup> *πωiEiVu*˜1

Here,

$$\begin{aligned} u &= \frac{(kk\_i)}{(qk\_i)}, \\ u' &= \frac{(kk\_f)}{(qk\_f)}, \\ \tilde{u}\_1 &= \frac{u\_1}{1-\tilde{u}}, \\ u\_{\text{Test}} &= \frac{\tilde{u}}{1-\tilde{u}}. \end{aligned} \tag{233}$$

at that 0 ≤ *u* ≤ *u*˜1, 0 ≤ *u*� ≤ *u*˜1, 0 ≤ *u*˜� ≤ *u*˜1. In Eq. (232) *P*res (*β*) is the function, which determines the resonant profile (see Fig. 15). It is obtained by

$$P\_{\text{res}}\left(\boldsymbol{\beta}\right) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} |I\_1(\boldsymbol{\beta}, l\_\*)|^2 d\left(\boldsymbol{\varphi}\_0 l\_\*\right). \tag{234}$$

We determine the resonance width at a half of the probability maximum (see Fig. 15). The

Fig. 15. Dependence of the function *P*res (234), which determines the resonant profile, on the resonant parameter *β* (221).

width which corresponds to the resonant parameter *β* is equal to Δ*β* ≈ 3.40. Therefore, the width specified by the field pulsed character is obtained by

$$
\Gamma\_{\rm imp} = \frac{\Delta\left(q^2 - m^2\right)}{4m} = \Delta\beta \frac{\tilde{\mu}\_1}{2} \frac{m}{q\_0} \approx 1.70 \frac{m\tilde{\mu}\_1}{q\_0}.\tag{235}
$$

We compare the resonance width specified by the field pulsed character (235) with the radiation width:

$$
\Gamma\_R = \frac{q\_0}{m} \mathcal{W}\_1 = \frac{e^2 m}{4\sqrt{\pi}} \eta\_0^2 F \left(\tilde{u}\_1\right) \, , \tag{236}
$$

where *W*<sup>1</sup> is the total probability of the intermediate state decay in a weakly intensive field; the function *F* (*u*˜1) is defined by

$$F(\vec{u}\_1) = \left(1 - \frac{4}{\vec{u}\_1} - \frac{8}{\vec{u}\_1^2}\right) \ln\left(1 + \vec{u}\_1\right) + \frac{1}{2} + \frac{8}{\vec{u}\_1} - \frac{1}{2\left(1 + \vec{u}\_1\right)^2}.\tag{237}$$

This ratio equals

(233)

46 Will-be-set-by-IN-TECH

*<sup>u</sup>* <sup>=</sup> (*kki*) (*qki*) ,

*<sup>u</sup>*� <sup>=</sup> (*kk <sup>f</sup>*) (*qk <sup>f</sup>*) ,

*<sup>u</sup>*˜1 <sup>=</sup> *<sup>u</sup>*<sup>1</sup> 1 − *u*˜ ,

at that 0 ≤ *u* ≤ *u*˜1, 0 ≤ *u*� ≤ *u*˜1, 0 ≤ *u*˜� ≤ *u*˜1. In Eq. (232) *P*res (*β*) is the function, which

1 − *u*˜ ,


 *P*res *C*

2

 res *P* 0

res  <sup>1</sup> <sup>0</sup> 2 *P*

*d* (*ϕ*0*l*∗). (234)

*C*

<sup>≈</sup> 1.70*mu*˜1 *ϕ*0

. (235)

<sup>0</sup>*F* (*u*˜1), (236)

*<sup>u</sup>*res <sup>=</sup> *<sup>u</sup>*˜

∞

−∞

We determine the resonance width at a half of the probability maximum (see Fig. 15). The

1.0


Fig. 15. Dependence of the function *P*res (234), which determines the resonant profile, on the

width which corresponds to the resonant parameter *β* is equal to Δ*β* ≈ 3.40. Therefore, the

<sup>4</sup>*<sup>m</sup>* <sup>=</sup> <sup>Δ</sup>*<sup>β</sup> <sup>u</sup>*˜1

We compare the resonance width specified by the field pulsed character (235) with the

*<sup>W</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup>*2*<sup>m</sup>* 4 <sup>√</sup>*<sup>π</sup> <sup>η</sup>*<sup>2</sup>

2 *m ϕ*0

*<sup>q</sup>*<sup>2</sup> <sup>−</sup> *<sup>m</sup>*<sup>2</sup>

<sup>Γ</sup>*<sup>R</sup>* <sup>=</sup> *<sup>q</sup>*<sup>0</sup> *m* 0.5

'*C*

determines the resonant profile (see Fig. 15). It is obtained by

width specified by the field pulsed character is obtained by

<sup>Γ</sup>imp <sup>=</sup> <sup>Δ</sup>

resonant parameter *β* (221).

radiation width:

*<sup>P</sup>*res (*β*) <sup>=</sup> <sup>1</sup>

2*π*

Here,

$$\frac{\Gamma\_{\rm imp}}{\Gamma\_{\rm R}} \approx \frac{8.51}{e^2 \eta\_0^2(\omega \tau)} \frac{\tilde{u}\_1}{F\left(\tilde{u}\_1\right)}.\tag{238}$$

When the condition (199) is met the appraisal value of ratio is equal to <sup>Γ</sup>imp/Γ<sup>R</sup> <sup>≥</sup> 103 � 1. Therefore, the width specified by the field pulsed character is the major one and the radiation widening may be neglected.

After the integration over the invariant parameter *u*� we derive the total probability of photon-electron scattering under the direct diagram resonance

$$\mathcal{W}\_{fi}^{\rm res} \approx \frac{2e^4 \eta\_0^4 m^2}{\pi \omega\_i \underline{E}\_i V \tilde{u}\_1} (\omega \tau)^2 \text{Pres } (\beta) \left[ \mathcal{F}(\tilde{u}\_1) f(u, \tilde{u}\_1) - \mathcal{G}(\tilde{u}\_1) \mathcal{g}(u, \tilde{u}\_1) \right] \tau,\tag{239}$$

$$\begin{split} G(u',\vec{u}\_1) &= \int\_0^{\vec{u}\_1} g(u',\vec{u}\_1) \frac{du'}{(1+u')^2} = \\ &= \frac{1}{4\vec{u}\_1(1+\vec{u}\_1)^2} \left( -4\vec{u}\_1 - 8\vec{u}\_1^2 - 5\vec{u}\_1^3 + (4+10\vec{u}\_1 + 8\vec{u}\_1^2 + 2\vec{u}\_1^3)\ln(1+\vec{u}\_1) \right). \end{split} \tag{240}$$

Ratio of the total probability (239) to the total probability of the Compton effect in external field absence is expressed as

$$\frac{W\_{fi}^{\text{res}}}{\text{W}\_{\text{Compt}}} \approx \frac{\tau}{T} \mathbb{P} \text{res}\left(\boldsymbol{\beta}\right) \cdot \mathbb{R}(\boldsymbol{u}, \vec{u}\_1), \quad \boldsymbol{\mathcal{R}}(\vec{\boldsymbol{u}}, \boldsymbol{u}\_1) = \frac{2\eta\_0^4 (\omega \tau)^2}{\pi^2} \frac{\left[F(\vec{\boldsymbol{u}}\_1) f(\boldsymbol{u}, \vec{u}\_1) - G(\vec{\boldsymbol{u}}\_1) g(\boldsymbol{u}, \vec{u}\_1)\right]}{\vec{\boldsymbol{u}}\_1 F(\vec{\boldsymbol{u}}\_1)},\tag{241}$$

where *T* is the observation time (*T τ*), which is determined by conditions of the concrete experiment.

When *u*<sup>1</sup> � 1 we derive

$$R(\vec{u}, u\_1) \approx \frac{4\eta\_0^4 (\omega \tau)^2}{\pi^2} \frac{(1 - \tilde{u})}{u\_1} \left(1 - 2\frac{\tilde{u}}{u\_1} \left(1 - \frac{\tilde{u}}{u\_1}\right)\right). \tag{242}$$

Fig. 16 demonstrates the ratio of the resonant probability of scattering of a photon by an electron in the field of a pulsed wave to probability of the Compton effect as a function of parameters *u*˜, *u*<sup>1</sup> within the resonant peak (*β* = 0) under *τ*/*T* = 1, *η*<sup>0</sup> = 0.05. It can be seen from Fig. 16 that the resonant probability may exceed considerably the probability of the Compton effect in external field absence. This fact becomes apparent particularly in the case *u*<sup>1</sup> � 1 (but it should be noticed here, that in view of infrared divergence the formulae (239), (241) are correct within the region *u*<sup>1</sup> *η*<sup>2</sup> <sup>0</sup>). Within the region *u*<sup>1</sup> 1 this effect disappears. Under conditions *η*<sup>2</sup> <sup>0</sup> <sup>∼</sup> *<sup>u</sup>*<sup>1</sup> � 1 within the range of optical frequencies *Ei*/*<sup>m</sup>* � *<sup>m</sup>*/*<sup>ω</sup>* <sup>∼</sup> 105 for the ratio of probabilities is correct *<sup>R</sup>* <sup>∼</sup> 103.

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Fig. 16. Ratio of the resonant probability of scattering of a photon by an electron in the field of a pulsed wave to the probability of the Compton effect in external field absence (241) as a function of the parameter *u*<sup>1</sup> (213) in the resonance peak (*β* = 0) under *τ*imp/*T* = 1, *η* = 0.05.

### **6. Conclusions**

Performed studies of resonant QED processes in a pulsed light field result:


The results can be tested in the experiments on verification of quantum electrodynamics in presence of strong fields (SLAC and FAIR).

#### **7. References**

Bagnoud, V., Aurand, B., Blazevic, A., Borneis, S., Bruske, C., Ecker, B., Eisenbarth, U., Fils, J., Frank, A., Gaul, E., Goette, S., Haefner, C., Hahn, T., Harres, K., Heuck, H. M., Hochhaus, D., Hoffmann, D., Javorková, D., Kluge, H., Kuehl, T., Kunzer, S., Kreutz, M., Merz-Mantwill, T., Neumayer, P., Onkels, E., Reemts, D., Rosmej, 48 Will-be-set-by-IN-TECH

<sup>1</sup> *u* 0.1

0.02

12

0 0.004 0.008 0.06

*u*

<sup>1</sup> *u* 0.01

*R R*

20

28

0.1

*u*

1.0

character of an external field.

presence of strong fields (SLAC and FAIR).

**6. Conclusions**

**7. References**

120

200

280

0.5

Performed studies of resonant QED processes in a pulsed light field result:

higher than the corresponding cross section in external field absence.

Fig. 16. Ratio of the resonant probability of scattering of a photon by an electron in the field of a pulsed wave to the probability of the Compton effect in external field absence (241) as a function of the parameter *u*<sup>1</sup> (213) in the resonance peak (*β* = 0) under *τ*imp/*T* = 1, *η* = 0.05.

1. The QED processes of the second order in a pulsed light field may occur under resonant conditions when the four-momentum of an intermediate particle lies near the mass surface. 2. The resonant behavior of the cross-section is specified by characteristics of the laser pulse. The resonant infinity in the process amplitude is eliminated by accounting for the pulsed

3. The differential cross section of the resonant process may be several orders of magnitude

The results can be tested in the experiments on verification of quantum electrodynamics in

Bagnoud, V., Aurand, B., Blazevic, A., Borneis, S., Bruske, C., Ecker, B., Eisenbarth, U.,

Fils, J., Frank, A., Gaul, E., Goette, S., Haefner, C., Hahn, T., Harres, K., Heuck, H. M., Hochhaus, D., Hoffmann, D., Javorková, D., Kluge, H., Kuehl, T., Kunzer, S., Kreutz, M., Merz-Mantwill, T., Neumayer, P., Onkels, E., Reemts, D., Rosmej,

*u*

0 0.4 0.8

1 2 *<sup>u</sup> <sup>u</sup>*

1.0

1 *u*

2.0

0

4.0

6.0

4

8

ln *R*

ln *R*

O., Roth, M., Stoehlker, T., Tauschwitz, A., Zielbauer, B., Zimmer, D. & K.Witte (2009). Commissioning and early experiments of the PHELIX facility, *Appl. Phys B.* 100(1): 137–150.


**6** 

*Chile* 

**Cold Atoms Experiments: Influence of Laser** 

Following a number of initial experiments in a magneto optical trap published by us in the period 2008-2009 (Olivares et al, 2008, 2009), there has been an increase in activity in the field. A brief review of the experimental methods can be found in (Olivares, 2007, 2008; Milonni, 2010). The physical details needed to obtain a cloud of cold atoms were described by Metcalf (1999). We will survey the literature and make a thorough discussion of the conditions that permit a stable cloud. We will outline our approach to the construction of the magneto optical trap. Our experiments are based on the construction described by Wieman et al. (1998) and Rapol et al. (2001). We followed the guidelines given in these articles but used state of the art equipment to obtain reliable results in our initial attempts to obtain a cloud of cold atoms. The only initial exception was a self made optical glass cell that was considered inexpensive. Subsequently, it was replaced by a more technically advanced cell that permitted us to improve the observational capability of the system. We will describe an experiment that proved the stability of the cloud and the optical method to vary the laser intensity of the pump and trap beams. We will study the influence of laser intensity imbalance on cloud formation and give values for the threshold intensity of each laser that

Saturated absorption spectroscopy is a simple technique to measure the narrow-line atomic spectral feature limited only by the natural linewidth, that is typically 6 MHz or less (Milonni, 2010). A strong laser beam called the pump beam is directed through an optical cell that contains a vapour as shown in Fig.1. A small part of the pump beam used as a probe beam is sent through the cell in the counter-propagating direction and detected by a

**1. Introduction** 

supports cloud formation.

simple photodiode.

**2. Description of saturated absorption spectroscopy** 

pump beam vapour cell

Fig. 1. Basic setup for saturation absorption spectroscopy.

probe beam

**Intensity Imbalance on Cloud Formation** 

Ignacio E. Olivares and Felipe A. Aguilar

*Universidad de Santiago de Chile* 

