Section 3 QCD Topics

#### **Chapter 5**

## Double Pole Method in QCD Sum Rules for Vector Mesons

*Mikael Souto Maior de Sousa and Rômulo Rodrigues da Silva*

#### **Abstract**

The QCD Sum Rules approach had proposed by Shifman, Vaishtein Zakharov Novikov, Okun and Voloshin (SVZNOV) in 1979 and has been used as a method for extracting useful properties of hadrons having the lowest mass, called as ground states. On the other hand, the most recent experimental results make it clear that the study of the excited states can help to solve many puzzles about the new XYZ mesons structure. In this paper, we propose a new method to study the first excited state of the vector mesons, in particular we focus our attention on the study of the *ρ* vector mesons, that have been studied previously by SVZNOV method. In principle, the method that we used is a simple modification to the shape of the spectral density of the SVZNOV method, which is written as "pole + continuum", to a new functional form "pole + pole + continuum". In this way, We may obtain the *ρ* and the *ρ*ð Þ 2*S* masses and also their decay constants.

**Keywords:** QCD Sum Rules, double pole, light quarks, vector mesons

#### **1. Introduction**

The successful QCD sum rules was created in 1977 by Shifman, Vainshtein, Zakharov, Novikov, Okun and Voloshin [1–4], and until today is widely used. Using this method, we may obtain many hadron parameters such as: hadrons masses, decay constant, coupling constant and form factors, all they giving in terms of the QCD parameters, it means, in terms of the quark masses, the strong coupling and nonperturbative parameters like quark condensate and gluon condensate.

The main point of this method is that the quantum numbers and content of quarks in hadron are presented by an interpolating current. So, to determine the mass and the decay constant of the ground state of the hadron, we use the two-point correlation function, where this correlation function is introduced in two different interpretations. The first one is the OPE's interpretation, where the correlation function is presented in terms of the operator product expansion (OPE).

On the phenomenological side we can be written the correlation function in terms of the ground state and several excited states. The usual QCDSR method uses an ansatz that the phenomenological spectral density can be represented by a form "pole + continuum", where it is assumed that the phenomenological and OPE spectral density coincides with each other above the continuum threshold. The continuum is represented by an extra parameter called *s*0, as being correlated with the onset of excited states [5].

In general, the resonances occurs with ffiffiffiffi *s*0 p lower than the mass of the first excited state. For the *ρ* meson spectrum, for example, the ansatz "pole + continuum" is a

good approach, due to the large decay width of the *ρ(2S)* or *ρ(*1450*)*, which allows to approximate the excited states as a continuum. For the *ρ* meson [6], the value of ffiffiffiffi *s*0 p that best fit the mass and the decay constant is ffiffiffiffi *s*0 <sup>p</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>2</sup> *GeV* and for the *<sup>φ</sup>(*1020*)* meson the value is 1*.*41 *GeV*. We note that the values quoted above for ffiffiffiffi *s*0 p are about 250 MeV below the poles of *ρ(*1450*)* and *φ(*1680*)*. One interpretation of this result is due to the effect of the large decay width of these mesons.

Novikov et al. [1], in a pioneering paper, proposed, for the charmonium sum rule, that the phenomenological side with double pole ("pole + pole + continuum") and ffiffiffiffi *s* 0 0 <sup>p</sup> <sup>¼</sup> <sup>4</sup> *GeV*, where *<sup>s</sup>* 0 <sup>0</sup> is the new parameter that takes in to account the second "pole" in this ansatz. In this way, thi value is correlated with the threshold of pair production of charmed mesons. Using this ffiffiffiffi *s* 0 0 p value and the Sum Rule Momentum at *<sup>Q</sup>*<sup>2</sup> <sup>¼</sup> 0, they presented the first estimate for the gluon condensate and a very good estimated value for *η<sup>c</sup>* meson, that is about 3 *GeV,* while the experimental data had shown 2.83 *GeV* for this meson [1, 2, 4].

In general, by QCDSR, the excited states are studied in "pole + pole + continuum" ansatz with *<sup>Q</sup>*<sup>2</sup> <sup>¼</sup> 0 [1, 2], as we can see in the spectral sum [7], the Maximum Entropy [8] and Gaussian Sum Rule with "pole + pole + continuum" ansatz [9] approaches. There are studies on the *ρ(*1*S,* 2*S)* mesons [8, 10, 11], nucleons [7, 12], *ηc(*1*S,* 2*S)* mesons [2], *ψ(*1*S,* 2*S)* mesons [1, 13] and *ϒ(*1*S,* 2*S)* mesons [14]. In this paper, we obtain the *ρ(*1*S,* 2*S)* mesons masses and their decay constants taking the "pole + pole + continuum" ansatz in QCD sum rules,

#### **2. The two point correlation function**

As it is known, to determinate the hadron mass and the decay constant in QCD sum rules, we may use the two-point correlation function [3], that is given by

$$\Pi\_{\mu\nu} = i \int d^4 \mathbf{x} e^{iq\mathbf{x}} \left< \mathbf{0} \middle| T \{ j\_\mu(\mathbf{x}) j\_\nu^\dagger(\mathbf{0}) \middle| \mathbf{0} \right>},\tag{1}$$

where, on the OPE side, this current density for *qq* vector mesons has the following form:

$$j\_{\mu}(\mathbf{x}) = \delta\_{ab}\overline{q}\_a(\mathbf{x})\chi\_{\mu}q\_b(\mathbf{x}),\tag{2}$$

where the subscribe index *a* and *b* represents the color index. Now, using Eq. (2) in Eq. (1) we have

$$\Pi\_{\mu\nu} = i\delta\_{ab}\delta\_{cd} \int d^4 \mathbf{x} e^{iq\mathbf{x}} \left< \mathbf{0} \middle| T \{ \overline{q}\_a(\mathbf{x}) \chi\_\mu q\_b(\mathbf{x}) \left[ \overline{q}\_c(\mathbf{0}) \chi\_\nu q\_d(\mathbf{0}) \right]^\dagger \middle| \mathbf{0} \right> . \tag{3}$$

Evaluating Eq. (3) in terms of the OPE [15], which can be written by a dispersion relation, where this relation depends on the QCD parameters, the correlation function takes the form:

$$
\Pi^{\rm OPE}\_{\mu\nu}(q) = \left(q\_{\mu}q\_{\nu} - q^{2}\mathbf{g}\_{\mu\nu}\right)\Pi^{\rm OPE}(q^{2}),\tag{4}
$$

with:

$$
\Pi^{OPE}(q^2) = \int\_{s\_0^{\text{min}}}^{\infty} ds \frac{\rho^{OPE}(s)}{s - q^2} + \Pi^{nonPer}(q^2), \tag{5}
$$

*Double Pole Method in QCD Sum Rules for Vector Mesons DOI: http://dx.doi.org/10.5772/intechopen.97421*

Note that

$$\rho^{\text{Pert}}(\mathfrak{s}) = \frac{\text{Im}\left[\Pi^{\text{OPE}}(\mathfrak{s})\right]}{\pi},\tag{6}$$

and <sup>Π</sup>*nonPert <sup>q</sup>*<sup>2</sup> ð Þ is the term that add the condensates contributions, besides, *<sup>s</sup> min* 0 is the minimum value of the *<sup>s</sup>* parameter to have an imaginary part of the <sup>Π</sup>*Pert*ð Þ*<sup>s</sup>* .

On the phenomenological side, the interpolating current density may be written considering just hadronic freedom degrees, it means, inserting a complete set of intermediate states among the operator, where they are the creation and annihilation describes by the interpolating current density. In this way we can use the following operator algebra

$$
\left< \mathbf{0} \middle| \, j\_{\mu}(\mathbf{0}) \middle| V(q) \right> = f m \varepsilon\_{\mu}(q), \tag{7}
$$

where *f* and *m* are, respectively, the decay constant and the mass of the meson and *ϵμ*ð Þ*q* is a unitary polarizing vector. So, substituting Eq. (7) in Eq. (1) after some intermediate mathematical steps we get:

$$
\Pi^{\rm Phen}\_{\mu\nu}(q) = \left(q\_{\mu}q\_{\nu} - q^{2}g\_{\mu\nu}\right)\Pi^{\rm Phen}(q^{2}).\tag{8}
$$

The Invariant part of Eq. (8), <sup>Π</sup>*Phen <sup>q</sup>*<sup>2</sup> ð Þ, is given by the following dispersion relation:

$$
\Pi^{\mathrm{Phen}}(q^2) = \int\_{s\_0^{\mathrm{min}}}^{\infty} ds \frac{\rho^{\mathrm{Phen}}(s)}{s - q^2},\tag{9}
$$

where *<sup>ρ</sup>Phen*ðÞ¼ *<sup>s</sup> <sup>f</sup>* 2 *<sup>δ</sup> <sup>s</sup>* � *<sup>m</sup>*<sup>2</sup> ð Þþ *<sup>ρ</sup>cont:* ð Þ*s* . Thus, we have the Eq. (9) written as follow:

$$\Pi^{Phen}\left(q^2\right) = \frac{f^2}{m^2 - q^2} + \int\_{s\_0}^{\infty} ds \frac{\rho^{out.}\left(s\right)}{s - q^2}.\tag{10}$$

Note that, we can introduce a minimum number of parameters in the calculus by the approach *ρcont:* ðÞ¼ *<sup>s</sup>* <sup>Θ</sup>ð Þ *<sup>s</sup>* � *<sup>s</sup>*<sup>0</sup> *<sup>ρ</sup>OPE*ð Þ*<sup>s</sup>* , using this in Eq. (10) we get:

$$\Pi^{\rm Plen}\left(q^2\right) = \frac{f^2}{m^2 - q^2} + \int\_{s\_0}^{\infty} ds \frac{\rho^{\rm OPE}(s)}{s - q^2},\tag{11}$$

so, *s*<sup>0</sup> can be understood as a parameter indicating that for *s* values greater than *s*<sup>0</sup> there is only contribution from the continuum, it means, *s*<sup>0</sup> is called a continuum threshold.

Note that, by the Quark-Hadron duality we can develop the two-point correlation function in both different interpretation that are equivalent each other. I.e., we can match the correlation function by de OPE, Eq. (5), with the correlation function by the Phenomenological side, Eq. (10), through the Borel transformation.

#### **3. Borel transformation**

To macht the Eqs. (5) and (10) is not that simple, because in the OPE side the calculations of the all OPE terms is almost impossible, in this way, at someone moment

#### *Quantum Chromodynamic*

we must truncate the series and, beyond this, guaranties its convergence. However, for the truncation of the series to be possible, the contributions of the terms of higher dimensions must be small enough to justify to be disregarded in the expansion.

Thereby, for both descriptions to be in fact equivalent, we must suppress both the contributions of the highest order terms of the OPE and the contributions of the excited states on the phenomenological side. It is can be done by the Borel transformation that is define as follow:

$$B[\Pi(Q^2) = \Pi[M^2] = \lim\_{\substack{Q^2, n \to \infty \ \infty}} \frac{\left(Q^2\right)^n}{n!} \left(-\frac{\partial}{\partial Q^2}\right)^n \Pi(Q^2), \tag{12}$$

$$\frac{Q^2}{n} = M^2$$

where *<sup>Q</sup>*<sup>2</sup> ¼ �*q*<sup>2</sup> is the momentum in the Euclidian space and *<sup>M</sup>*<sup>2</sup> is a variable rising due to Borel transformation application and it is called *Borel Mass*.

Because of this, we can determine a region of the M2 space in which both the highest order contributions from OPE and those from excited states are suppressed, so that the phenomenological parameters associated with the hadron fundamental state can be determined. Therefore, we must determine an interval of M2 where this comparison is adequate, enabling the determination of reliable results. This interval is called Borel Window.

At the Phenomenological side, we introduce some approximations when we assume that the spectral density can be considering as a polo plus a continuum of excited states. So, we must suppress the continuum contributions for the result to be sufficiently dominated by de pole.

#### **4. The double pole method**

This method is consisted by the assumption that the spectral density at the phenomenological side can be given like [16]:

$$\rho^{\rm Phcn}(\mathfrak{s}) = \left(f\_1\right)^2 \delta\left[\mathfrak{s} - (m\_1)^2\right] + \left(f\_2\right)^2 \delta\left[\mathfrak{s} - (m\_2)^2\right] + \rho^{\rm OPE}(\mathfrak{s})\Theta(\mathfrak{s} - \mathfrak{s}\_0'),\tag{13}$$

where *m*<sup>1</sup> and *f* <sup>1</sup> are, respectively, the ground state meson mass and decay constant, *m*<sup>2</sup> and *f* <sup>2</sup> are, respectively, the first excited state meson mass and decay constant, beyond this, we include a new parameter *s* 0 <sup>0</sup> marking the onset of the continuum states. As we can see in **Figure 1**, the parameters Δ and **Δ**<sup>0</sup> consists in a gap among the ground and first excited states and among the first excited and the continuum states respectively. They are defined by the decay width of these states.

Note that, inserting Eq. (13) in Eq. (9) we get the following two-point phenomenological correlation function:

$$\Pi^{\rm Planck}(q^2) = \frac{\left(f\_1\right)^2}{\left(m\_1\right)^2 - q^2} + \frac{\left(f\_2\right)^2}{\left(m\_2\right)^2 - q^2} + \int\_{l\_0'}^\infty ds \frac{\rho^{\rm OPE}(s)}{s - q^2},\tag{14}$$

Applying the Borel transformation in Eqs. (14) and (5) we get:

$$\Pi^{\rm Phen}(\mathsf{M}^2) = \left(f\_1\right)^2 e^{-\left(\mathsf{m}\_1\right)^2/\mathsf{M}^2} + \left(f\_2\right)^2 e^{-\left(\mathsf{m}\_2\right)^2/\mathsf{M}^2} + \int\_{s\_0'}^{\mathsf{so}} ds \rho^{\rm OPE}(s) e^{-s/\mathsf{M}^2},\tag{15}$$

#### **Figure 1.**

*On the left side it is seen the double pole ansatz,* **Δ** *and* **Δ**<sup>0</sup> *represent the Gaps among the ground, first excited and continuum states. On the right side it is seen the mass spectra for the* **ρ** *meson and its resonances [16–18].*

$$\Pi^{\rm OPE}(\mathbf{M}^2) = \int\_{s\_0^{\rm min}}^{s\_0'} ds \rho^{\rm OPE}(s) e^{-s/\mathcal{M}^2} + \int\_{s\_0'}^{\infty} ds \rho^{\rm OPE}(s) e^{-s/\mathcal{M}^2} + \Pi^{\rm cond}(\mathbf{M}^2). \tag{16}$$

By the Quark-Hadron duality we have <sup>Π</sup>*Phen <sup>M</sup>*<sup>2</sup> � � <sup>¼</sup> <sup>Π</sup>*OPE <sup>M</sup>*<sup>2</sup> � �, thus:

$$\left(\left(f\_1\right)^2 e^{-\left(m\_1\right)^2/M^2} + \left(\left(f\_2\right)^2 e^{-\left(m\_2\right)^2/M^2} = \int\_{s\_0'^{\min}}^{s\_0'} ds \rho^{\rm OPE}(s) e^{-s/M^2} + \Pi^{\rm cond}\left(M^2\right). \tag{17}$$

The contribution of the resonances is given by:

$$\text{CR} = \int\_{s'\_0}^{\infty} ds \rho^{\text{OPE}}(s) e^{-s/M^2}. \tag{18}$$

To develop Eq. (17) let us make a variable change taking *<sup>M</sup>*�<sup>2</sup> <sup>¼</sup> *<sup>x</sup>*, so we write:

$$\left(\left(f\_1\right)^2 e^{-\left(m\_1\right)^2 \mathbf{x}} + \left(\left(f\_2\right)^2 e^{-\left(m\_2\right)^2 \mathbf{x}}\right) = \int\_{s\_0'^{\text{min}}}^{s\_0'} ds \rho^{\text{OPE}}(s) e^{-\epsilon \mathbf{x}} + \Pi^{\text{cond}}(\mathbf{x}).\tag{19}$$

Now, taking de derivative of Eq. (19) with respect to *x* we get:

$$-\left(m\_1f\_1\right)^2e^{-\left(m\_1\right)^2/M^2} - \left(m\_2f\_2\right)^2e^{-\frac{\left(m\_2\right)^2}{M^2}} = \frac{d}{d\mathfrak{x}}\Pi^{OPE}(\mathfrak{x}),\tag{20}$$

where, now, we are considering

$$
\Pi^{OPE}(\mathbf{x}) = \int\_{\epsilon\_0'^{\rm min}}^{\epsilon\_0'} d\mathbf{s} \rho^{OPE}(\mathbf{s}) e^{-\epsilon \mathbf{x}} + \Pi^{cond}(\mathbf{x}).\tag{21}
$$

We observe that the Eqs. (19) and (21) form a equations system in *x* variable. In this system we can make a new change of variables as follow:

$$A(\mathbf{x}) = \left(\int f\_1\right)^2 e^{-\left(m\_1\right)^2 \mathbf{x}} \text{ and } B(\mathbf{x}) = \left(\int f\_2\right)^2 e^{-\left(m\_2\right)^2 \mathbf{x}},\tag{22}$$

this way we get the following system:

$$A(\mathfrak{x}) + B(\mathfrak{x}) = \Pi^{OPE}(\mathfrak{x}).\tag{23}$$

$$-(m\_1)^2 A(\varkappa) - (m\_2)^2 B(\varkappa) = \frac{d}{d\varkappa} \Pi^{OPE}(\varkappa) \tag{24}$$

Solving the above system of equation for *A(x)* and *B(x)* we have:

$$A(\mathbf{x}) = \frac{\frac{d}{d\mathbf{x}}\Pi^{OPE}(\mathbf{x}) + \Pi^{OPE}m\_2^2}{m\_2^2 - m\_1^2}. \tag{25}$$

$$B(\mathbf{x}) = \frac{\frac{d}{d\mathbf{x}}\Pi^{OPE}(\mathbf{x}) + \Pi^{OPE}m\_1^2}{m\_1^2 - m\_2^2}. \tag{26}$$

Note that, Eqs. (25) and (26) presents information about the hadron masses and their coupling constants, to eliminate de coupling constants we have to take the derivative of Eq. (25) and then dividing the result by the own Eq. (25) and the same procedure with Eq. (26). Thus, we have:

$$m\_1 = \sqrt{-\frac{\frac{d}{dx}\Pi^{OPE}(\varkappa)m\_2^2 + \frac{d^2}{dx^2}\Pi^{OPE}}{\frac{d}{dx}\Pi^{OPE}(\varkappa) + \Pi^{OPE}m\_2^2}},\tag{27}$$

$$m\_2 = \sqrt{-\frac{\frac{d}{dx}\Pi^{OPE}(\mathbf{x})m\_1^2 + \frac{d^2}{dx^2}\Pi^{OPE}}{\frac{d}{dx}\Pi^{OPE}(\mathbf{x}) + \Pi^{OPE}m\_1^2}}.\tag{28}$$

This way we have the both solutions coupling each other. On the other hand, what we are looking for are mass solutions for the ground state and its first excited state independent each other. To do so, we take the second derivative of the Eq. (19) with respect to *x* and the result we divide by Eq. (23) for decoupling of the *m*<sup>1</sup> mass. Note that the same procedure can be done for Eqs. (19) and (24) for decoupling of the *m*<sup>2</sup> mass. So, for the *m*<sup>2</sup> mass we have:

$$m\_2^4 = \frac{\frac{d^\beta}{dx^\beta} \Pi^{OPE}(\varkappa) + m\_1^2 \frac{d^2}{dx^2} \Pi^{OPE}(\varkappa)}{\frac{d}{dx} \Pi^{OPE}(\varkappa) + \Pi^{OPE}(\varkappa) m\_1^2}. \tag{29}$$

Substituting Eq. (29) in Eq. (28) we obtain the following polynomial equation:

$$m\_2^4 a + m\_2^2 b + c = 0,\tag{30}$$

where *a, b* and *c* are respectively:

$$a = -\left(\frac{d}{d\boldsymbol{\omega}}\boldsymbol{\Pi}^{\rm OPE}(\boldsymbol{\omega})\right)^2 + \boldsymbol{\Pi}^{\rm OPE}(\boldsymbol{\omega})\frac{d^2}{d\boldsymbol{\omega}^2}\boldsymbol{\Pi}^{\rm OPE}(\boldsymbol{\omega}),\tag{31}$$

$$b = -\left(\frac{d^2}{d\mathbf{x}^2} \Pi^{\rm OPE}(\mathbf{x})\right) \frac{d}{d\mathbf{x}} \Pi^{\rm OPE}(\mathbf{x}) + \Pi^{\rm OPE}(\mathbf{x}) \frac{d^3}{d\mathbf{x}^3} \Pi^{\rm OPE}(\mathbf{x}),\tag{32}$$

$$\mathcal{L} = \left(\frac{d^3}{d\boldsymbol{\kappa}^3} \boldsymbol{\Pi}^{OPE}(\boldsymbol{\kappa})\right) \frac{d}{d\boldsymbol{\kappa}} \boldsymbol{\Pi}^{OPE}(\boldsymbol{\kappa}) - \frac{d^2}{d\boldsymbol{\kappa}^2} \boldsymbol{\Pi}^{OPE}(\boldsymbol{\kappa}).\tag{33}$$

*Double Pole Method in QCD Sum Rules for Vector Mesons DOI: http://dx.doi.org/10.5772/intechopen.97421*

Note that, for the *m*<sup>1</sup> mass, following the same procedure we get the other polynomial equation like Eq. (30). Thus, solving the polynomial equation, given by Eq. (30), and the same for *m*<sup>1</sup> mass, the physical solutions the represent *m*<sup>1</sup> like the ground state mass and *m*<sup>2</sup> like the first excited state mass are given by:

$$m\_1 = \sqrt{-\frac{b + \sqrt{\Delta}}{2a}}\,,\tag{34}$$

$$m\_2 = \sqrt{-\frac{b-\sqrt{\Delta}}{2a}}\ .\tag{35}$$

These results can be developed to obtain the masses of the ground state and its first excited state for any *qq* vector meson, also, we can calculate their coupling constants using the masses estimated in the Eqs. (25) and (26).

#### **5. Results for the** *ρ* **meson**

For the *<sup>ρ</sup>* meson we use the <sup>Π</sup>*OPE*ð Þ *<sup>x</sup>* given by the Feynman diagrams that to be seen in [citar o greiber]. In this way we have:

$$\rho^{\rm OPE}(\mathbf{s}) = \frac{1}{4\pi^2} \left(\mathbf{1} + \frac{a\_\mathbf{s}}{\pi}\right) \tag{36}$$

and

$$\Pi^{cond}(\mathfrak{x}) = \mathfrak{x}\left(\frac{1}{12}\left\langle \frac{a\_{\mathfrak{s}}}{\pi}G^{2}\right\rangle + 2\,\,m\_{q}\langle \overline{q}q\rangle\right) - \mathfrak{x}^{2}\frac{112}{18}\pi a\_{\mathfrak{s}}\langle \overline{q}q\rangle^{2},\tag{37}$$

where *α<sup>s</sup>* is the strong coupling constant, *mq* is the light quark mass, *<sup>α</sup><sup>s</sup> <sup>π</sup> <sup>G</sup>*<sup>2</sup> � � is the gluon condensate, h i *qq* is the quark condensate and *s min* <sup>0</sup> <sup>¼</sup> <sup>4</sup> *<sup>m</sup>*<sup>2</sup> *q*.

Following [19], for the *ρ* meson we use the parameters: *α<sup>s</sup>* ¼ 0*:*5, *mq* ¼ ð Þ <sup>6</sup>*:*<sup>4</sup> � <sup>1</sup>*:*<sup>25</sup> *MeV*, h i *qq* ¼ �ð Þ <sup>0</sup>*:*<sup>24</sup> � <sup>0</sup>*:*<sup>01</sup> <sup>3</sup> *GeV*<sup>3</sup> , *<sup>α</sup><sup>s</sup> pi <sup>G</sup>*<sup>2</sup> D E <sup>¼</sup> ð Þ <sup>0012</sup> � <sup>0</sup>*:*<sup>004</sup> *GeV*<sup>4</sup> at the renormalization scale *μ* ¼ 1 *GeV*.

Using the mass of the *<sup>ρ</sup>*ð Þ¼ <sup>3</sup>*<sup>S</sup>* <sup>1</sup>*:*<sup>9</sup> *GeV* [16], we get ffiffiffiffi *s* 0 0 <sup>p</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>9</sup> *GeV*, but in this case, the decay constant of the excited state is bigger than the decay constant of the ground state, that way the sum rules fails. Furthermore, the maximum value of the ffiffiffiffi *s* 0 0 p parameter is 1.66 *GeV*. That way, the excited state decay constant is a bit lower than the ground state decay constant. The minimum value for ffiffiffiffi *s* 0 0 p is 1.56 *GeV*, because ffiffiffiffi *s* 0 0 <sup>p</sup> � *<sup>m</sup><sup>ρ</sup>*ð Þ <sup>2</sup>*<sup>S</sup>* reaches the value of 100 *MeV.*

In this way, we can find the Borel window where the QCDSR is valid. In this case, the Borel window is shown in **Figure 2** and it is calculated by the ratio between Eqs. (17) and (18) for a given ffiffiffiffi *s* 0 0 <sup>p</sup> value and considering the *<sup>ρ</sup>*ð Þ <sup>1</sup>*S*, 2*<sup>S</sup>* masses given by [16]. We can see that to have a good accuracy on our results we have to evaluate the QCDSR in a range of 0.8 ≤ M ≤ 2.3, where the pole contribution is bigger than 40%.

In **Figure 3** we display the masses of the *ρ*ð Þ 1*S*, 2*S* mesons as function of the Borel mass for three different values of the ffiffiffiffi *s* 0 0 p parameter that are: 1.66 *GeV* (polygonal blue point), 1.61 *GeV* (red dot-dashed line), 1.56 *GeV* (diagonal green cross point) and the grey lines representing the masses of the ground state and the first excited state for the *ρ* meson according to [16]. We can see that for the first

**Figure 2.**

*The red dashed line represents the pole contribution as function of the Borel mass. Note that, for* ffiffiffiffi *s*0 **0** <sup>p</sup> <sup>¼</sup> **1***:***66** *GeV and the ρ*ð Þ **1***S*, **2***s meson masses given by [16], the range where the QCDSR is 8*GeV ≤ *M* ≤ *2.3*GeV*, where the polo contribution is bigger than 40%.*

#### **Figure 3.**

*The masses of the <sup>ρ</sup>*ð Þ **<sup>1</sup>***S*, **<sup>2</sup>***<sup>S</sup> mesons as function of the Borel mass for three different values of the* ffiffiffiffi *s*0 **0** p *parameter where 1.66* GeV *is given by polygonal blue point, 1.61* GeV *by red dot-dashed line, 1.56* GeV *by diagonal green cross point and the grey lines representing the masses of the ground state and the first excited state for the ρ meson according to [16].*

excited state the mass average is closely to the experimental data for the *ρ*ð Þ 2*S* , where its mass is about 1.45 *GeV* [16].

For the ground state, in **Figure 3**, we show that the mass average of the *ρ*ð Þ 1*S* is about 750 MeV, also pretty close to that one seen in [16], that is about 775 *MeV* for the experimental data.

*Double Pole Method in QCD Sum Rules for Vector Mesons DOI: http://dx.doi.org/10.5772/intechopen.97421*

#### **Figure 4.**

*On the left side, we have the value of the decay constant for the ρ*ð Þ **1***S meson about* ð Þ **203** þ **5** *MeV. The values of the* ffiffiffiffi *s*0 **0** p *parameter are: 1.66* GeV *(polygonal blue dot line), 1.61* GeV *(red dash-dotted line) and 1.56* GeV *(diagonal green cross dot line), the grey dashed line is the experimental value [16] that is 220* MeV. *On the right side, we have the value of the decay constant for the ρ*ð Þ **2***S meson about* ð Þ **186** þ **14** *MeV for the same values of the* ffiffiffiffi *s*0 **0** p *parameter are: 1.66* GeV *(polygonal blue dot line), 1.61* GeV *(red dash-dotted line) and 1.56* GeV *(diagonal green cross dot line).*

To evaluate the decay constants, we use the experimental masses given by [16]. For the *ρ*ð Þ 1*S* we use 0.775 *GeV* and for the *ρ*ð Þ 2*S* we use 1.46 *GeV*. In **Figure 4**, we display the decay constant for the *ρ*ð Þ 1*S* and *ρ*ð Þ 2*S* mesons as function of the Borel mass for three different values of the ffiffiffiffi *s* 0 0 p parameter.

In **Figure 4**, on the left side, we have an average for the decay constant of the *ρ*ð Þ 1*S* about 203 ð Þ � 5 *MeV*, note that the maximum value for the decay constant is that one where ffiffiffiffi *s* 0 0 <sup>p</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>66</sup> *GeV* (polygonal blue point), the grey dashed line represents the experimental data [16] for the *ρ*ð Þ 1*S* decay constant that is 220 *MeV*. On the right side, we have an average for the decay constant of the *ρ*ð Þ 2*S* about 186 ð Þ þ 14 *MeV*, where we considered uncertainty with respect to ffiffiffiffi *s* 0 0 <sup>p</sup> parameter at *<sup>M</sup>* <sup>¼</sup> <sup>2</sup> *GeV*.

Furthermore, it is interesting note that in Ref. [20] we can see another way to extract the experimental decay constant of the *ρ*� from semileptonic decay, *τ*� ! *ρ*�*ντ*. Note that in Ref. [19].

#### **6. Conclusions**

In this work, we made a little revision about the QCD Sum rules method and presented a new method for calculation of the hadronic parameters like mass and decay constant [19] as function of the Borel mass.

We show that the double pole method on QCDSR consists in a fit on the interpretation of the correlation function by the phenomenological side, where the relations dispersion is now presented with two poles plus a continuum of excited states, being these two poles representing the ground state and the first excited state.

For the ρ(1S, 2S) mesons we had a good approximation for the calculations of these masses comparing with the experimental data on the literature. Beyond that, for the decay constant of the *ρ(2S)* meson we had a good prediction like it is seen in [19] where *<sup>f</sup> <sup>ρ</sup>*ð Þ <sup>2</sup>*<sup>S</sup>* <sup>¼</sup> ð Þ <sup>182</sup> � <sup>10</sup> MeV.

Our intention with this work consists on the studying of the vector mesons testing the accuracy of the double pole method and apply this method to analyze others kind of mesons such as scalar mesons.

### **Acknowledgements**

We would like to thank Colégio Militar de Fortaleza and the Universidade Federal de Campina Grande by technical and logistical support.

### **Conflict of interest**

The authors declare no conflict of interest.

### **Author details**

Mikael Souto Maior de Sousa<sup>1</sup> \* and Rômulo Rodrigues da Silva<sup>2</sup>


\*Address all correspondence to: mikael.souto@ufrr.br

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Double Pole Method in QCD Sum Rules for Vector Mesons DOI: http://dx.doi.org/10.5772/intechopen.97421*

#### **References**

[1] V.A. Novikov, L.B. Okun, M.A. Shifman, A.I. Vainshtein, M.B., Voloshin and V.I. Zakharov, Phys. Rept. 41, 1, 1978.

[2] V.A. Novikov, L.B. Okun, M.A. Shifman, A.I. Vainshtein, M.B., Voloshin and V.I. Zakharov, Phys. Lett. B 67, 409, 1977.

[3] M.A. Shifman, A.I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 385, 1979.

[4] M.A. Shifman, A.I. Vainshtein, M.B. Voloshin and V.I. Zakharov, Phys. Lett. B 77, 80, 1978.

[5] P. Colangelo and A. Khodjamirian, in *At the frontier of particle physics*, ed. by M. Shifman. arXiv:hep-ph/0010175, Vol. 3, pp. 1495-1576.

[6] L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rept. 127, 1,1985.

[7] J.P. Singh, F.X. Lee, Phys. Rev. C 76, 065210, 2007. arXiv:nucl-th/0612059

[8] P. Gubler and M. Oka, Prog. Theor. Phys. 124, 995, 2010. arXiv:1005.2459 [hep-ph]

[9] D. Harnett, R.T. Kleiv, K. Moats and T.G. Steele, Nucl. Phys. A 850, 110, 2011. arXiv:0804.2195 [hep-ph]

[10] A.P. Bakulev and S.V. Mikhailov, Phys. Lett. B 436, 35, 1998. arXiv:hepph/9803298

[11] A.V. Pimikov, S.V. Mikhailov and N. G. Stefanis, arXiv:1312.2776 [hep-ph]

[12] K. Ohtani, P. Gubler and M. Oka, AIP Conf. Proc. **1343**, 343, 2011. arXiv: 1104.5577 [hep-ph]

[13] P. Gubler, K. Morita and M. Oka, Phys. Rev. Lett. 107, 092003, 2011. arXiv:1104.4436 [hep-ph]

[14] K. Suzuki, P. Gubler, K. Morita and M. Oka, arXiv:1204.1173 [hepph]

[15] W. Greiner, S. Schramm and E. Stein. Quantum Cromodynamics. Ed Springer, 2002.

[16] Particle Data Group: J. Beringer et al., Phys. Rev. D, 86, 010001, 2012.

[17] S. Godfrey, N. Isgur, Phys. Rev. D, 32, 189,1985.

[18] D. Ebert, R.N. Faustov, V.O. Galkin, Phys. Rev. D, 79, 114029, 2009. arXiv: 0903.5183 [hep-ph]

[19] M. S. Maior de Sousa and R. Rodrigues da Silva. Braz. J. Phys, 46, 730-739, 2016. DOI: 10.1007/ s13538-016-0449-9

[20] D. Becirevic, V. Lubicz, F. Mescia and C. Tarantino, JHEP 0305, 007, 2003. arXiv:hep-lat/0301020

#### **Chapter 6**

## Application of Einstein's Methods in a Quantum Theory of Radiation

*Richard Joseph Oldani*

#### **Abstract**

Einstein showed in his seminal paper on radiation that molecules with a quantum-theoretical distribution of states in thermal equilibrium are in dynamical equilibrium with the Planck radiation. The method he used assigns coordinates fixed with respect to molecules to derive the A and B coefficients, and fixed relative to laboratory coordinates to specify their thermal motion. The resulting dynamical equilibrium between quantum mechanical and classically defined statistics is critically dependent upon considerations of momentum exchange. When Einstein's methods relating classical and quantum mechanical statistical laws are applied to the level of the single quantum oscillator they show that matrix mechanics describes the external appearances of an atom as determined by photon-electron interactions in laboratory coordinates, and wave mechanics describes an atom's internal structure according to the Schrödinger wave equation. Non-commutation is due to the irreversibility of momentum exchange when transforming between atomic and laboratory coordinates. This allows the "rotation" of the wave function to be interpreted as the changing phase of an electromagnetic wave. In order to describe the momentum exchange of a quantum oscillator the Hamiltonian model of atomic structure is replaced by a Lagrangian model that is formulated with equal contributions from electron, photon, and nucleus. The fields of the particles superpose linearly, but otherwise their physical integrity is maintained throughout. The failure of past and present theoretical models to include momentum is attributed to the overwhelming requirement of human visual systems for an explicit stimulus.

**Keywords:** Einstein's quantum theory, matrix mechanics, wave mechanics, momentum exchange, conservation laws, non-commutation, wave function, Schrödinger wave equation, Lagrangian

#### **1. Introduction**

Two possibilities are available in the literature for describing the interaction of matter and radiation, classical theory and nonrelativistic quantum theory. Classical theory explains the continuous aspects of electromagnetic radiation, Maxwell's laws, and the theory of heat. Quantum theory explains the Planck radiation law of black body radiation, the discrete nature of observables, and the statistical properties of matter. A third possibility that has remained relatively obscure as an alternative derives from Einstein's 1917 paper "Quantum theory of radiation" and includes aspects of both theories [1]. He shows there that as a consequence of the conservation of momentum the velocity distribution of molecules emitting black

body radiation *quantum mechanically* are in dynamical equilibrium with the *classically* derived Maxwell-Boltzman velocity distribution due to thermal exchange. The internal energy distribution of the molecules demanded by quantum theory is then in strict conformance with the emission and absorption of radiation. Because the link between quantum mechanical and classical properties of matter is statistically defined and applies to material systems rather than individual atoms Einstein's theory is considered to be unfinished. A description of atomic structure using his methods is sought after here as a way to fulfill these ideas.

The authors of nonrelativistic quantum theory adopted Einstein's ideas for the A and B coefficients, which are determined by the classical field effect resonance, and described the discrete transfer of energy from a radiation beam to an atomic state; but they neglected the effect of momentum exchange required by the conservation of momentum. The momentum of a photon E/c causes an atom or molecule to recoil in the direction of the beam when it is absorbed and in the opposite direction when it is emitted. Nonrelativistic quantum mechanics places primary importance on the observable properties of radiation in the form of energy measurements while ignoring the more subtle effects of momentum which are more difficult to observe. Consequently the Schrödinger wave equation is formulated continuously without provision for transmitting the discontinuous impulses of photons. The relationship between classical and quantum mechanical statistics that Einstein had carefully constructed breaks down so that instead of a gradual evolution of ideas in which classical and quantum concepts develop together a complete break from classical theory occurred. In the absence of an underlying classical foundation different interpretations of quantum mechanics developed which use methods drawn from facts that are supported by experiment specific to that model alone, but show no relationship to each other. No model has emerged that can account for all the facts. In the following discussion we shall see that the reason no single model of quantum mechanics is able to explain all of the experimental facts, discrete and continuous, yet they concern the same topic is that each one addresses a different aspect of the *same* physical phenomenon; the interaction between matter and radiation.

#### **2. Matrix mechanics**

#### **2.1 Historical perspectives**

The Bohr model of the atom gives the quantum rule for changes in energy state *E*<sup>2</sup> � *E*<sup>1</sup> ¼ *hν*, but says nothing about the processes of emission and absorption. Improved understanding of radiation came gradually as experimental techniques improved. Einstein's 1917 paper marks the beginning of quantum mechanics since all subsequent research on the absorption, emission, and dispersion of radiation is based upon it [2]. Through the use of thought experiments and results obtained in an earlier paper on Brownian motion he showed how the microscopic structure of matter is able to influence matter macroscopically. The induced absorption of black body radiation occurs continuously due to random inputs of momentum from thermal collisions and radiation, while induced and spontaneous emission occurs discretely according to the Bohr frequency rule for changes in state and is directed along an infinitesimal solid angle consistent with the photon's recoil momentum E/ c. A dynamic equilibrium is thereby created between the thermal energy absorbed by molecules and the quantum mechanical emission of radiation.

Although the A and B coefficients of Einstein's radiation theory have been incorporated into nonrelativistic quantum mechanics the transition of energy from a classical thermal origin to the discrete energy states of atoms and molecules

#### *Application of Einstein's Methods in a Quantum Theory of Radiation DOI: http://dx.doi.org/10.5772/intechopen.97734*

creates discontinuities that are not accounted for by the Schrödinger wave equation. A vastly improved knowledge of the mechanical properties of photons due to momentum in the astronomical sciences, molecular manipulation, optical tweezers, and laser cooling; technological advances has not translated into an understanding of how to incorporate momentum into the equations of quantum mechanics. The momentum of light is treated separately from energy, and Einstein's theory of radiation is the only one that makes explicit use of it when describing absorption and emission. To see why this is true it will be necessary to examine the historical origins of quantum mechanics.

The dynamic equilibrium between classical and quantum mechanical statistical laws that exists for black body radiation is closely related to the phenomenon of dispersion. Dispersion is the continuous change in the angle of refraction of different frequencies of light by a prism or other medium. Although light disperses continuously across the entire spectrum, at certain specific frequencies characteristic of the medium, it is completely absorbed forming lines. When Bohr introduced his theory of electron orbitals he immediately recognized the possibility that the discrete lines of atomic spectra are related to the discrete lines in dispersion phenomena [3]. Other researchers, in particular Debye and Sommerfeld, were also inspired by that possibility and a series of papers appeared that tried to explain the discrete and continuous properties of dispersion by introducing classically inspired modifications of the electron orbitals [2, 4, 5]. However, when experiments revealed that the characteristic frequencies of anomalous dispersion coincide with the frequencies of the spectral lines it was evident that orbiting electrons could not account for both and a complete break from classical theory was necessary. Ladenburg was the first to suggest how the new quantum theory would appear by following Einstein's reasoning leading to the A and B coefficients [5–7]. This enabled him to equate two theoretical expressions, the energy absorbed/emitted by N classical resonators and the energy absorbed/emitted by N<sup>0</sup> quantum atoms. By obtaining a statistical balance between classical and quantum mechanical energy exchange he satisfied the conservation of energy, but not that of momentum. Four years later Kramers reinterpreted Ladenburg's results by using the Bohr model of the atom as a multiply periodic system of virtual oscillators [2, 5, 8, 9]. In that model a quantum mechanical variable X is described with a classical Fourier series, where A(n, n- τ) is the quantum analog of the classical amplitude, n indicates the electron orbital number, and τ assumes integral values to denote positive or negative transitions [9].

$$X = \sum\_{\tau} A(n, n - \tau) \exp\left[2i\pi\nu(n, n - \tau)t\right], \tau = \mp 1, \mp 2, \dots \tag{1}$$

The Bohr-Kramer method distanced itself from that of Einstein in an important way. Einstein argued that momentum conservation is what distinguishes classical properties observed in laboratory coordinates from quantum mechanical properties observed in atomic coordinates. The discrete and continuous properties of matter are thereby separated from each other physically. In the interpretation by (1), on the other hand, matter-radiation interactions are described exclusively in laboratory coordinates. Fields are described classically by means of Fourier series while quantization is imposed on the field energy. Quantization is thereby understood to be a localization of energy even though the fields extend to infinity and are therefore diffuse. The concept of photon momentum, a property whose displacement in time is *directional*, is replaced by a wave model that is *isotropic* and treats emission as a spherically symmetric process with no net momentum transfer, and processes that are *reversible* in time.

Once Kramer had reinterpreted Einstein's quantum theory of radiation with fictitious harmonic oscillators Heisenberg was able to use it to formulate a theory of quantum mechanics that reconciles the continuity of radiation fields with the discrete energy states of an atom [2, 7, 9]. The complex sets of mathematical rules that he used to describe the frequencies and intensities of spectral lines, may be expressed in the form of a matrix.

$$\sum\_{k} (p\_{nk}q\_{km} - q\_{nk}p\_{km}) = \begin{cases} i\hbar \, for \, n = m \\ 0 \, for \, n \neq m \end{cases} \tag{2}$$

Each matrix element represents a pair of energy states of the type (1) with the observable properties, frequency and intensity, of an electromagnetic wave. The complete matrix has an infinite number of components and corresponds in its entirety to one of the dynamic variables; the coordinates, the momenta, or the velocities of the particles. The matrix products do not commute as they do in classical theory. When *n* ¼ *m* the elements are diagonal and the value of the equation is equal to iћ. For non-diagonal elements, *n* 6¼ *m*, and its value is zero.

The quantum mechanical reformulation of the classical Fourier series (1) and (2) is further simplified into its more familiar form by replacing the summed elements with single terms.

$$\mathbf{pq} - \mathbf{qp} = \text{i}\hbar \tag{3}$$

The momentum **p** and position **q** are not numbers; but rather arrays of quantities, or matrices. Each component of the matrix is a Fourier series associated with any two of an infinite number of orbits. Because the orbits may extend to infinity both in space and in time exchanges of momentum are delocalized.

#### **2.2 Classical interpretation of matrix mechanics**

After three successive modifications from Ladenburger to Kramers to Heisenberg, Einstein's theory is scarcely recognizable. Mathematical modifications that dilute its physical content are given by the Eqs. (1)–(3). Very little remains of Einstein's carefully crafted relationship between classical and quantum mechanical variables despite the fact that all three reinterpretations and the Eqs. (1) through (3) claim to describe the same physical phenomenon, the interaction between matter and radiation. The theories differ dramatically because the directional properties of emitted radiation due to recoil momentum have been replaced by virtual harmonic oscillators which emit energy isotropically as spherical waves and are reversible in time. The balance between thermal energy and radiative energy maintained by momentum exchange depends on oscillators that absorb thermal energy classically and emit energy quantum mechanically directed along an infinitesimal solid angle with momentum E/c. Virtual oscillators that emit isotropically disrupt the delicate balance between classical and quantum mechanical statistical principles which Einstein had so carefully constructed.

The advantage of using energy rather than momentum in a theory of radiation is its ease of use. Energy is defined as a magnitude, which is easier to describe mathematically, to measure, and to calculate. The advantage of momentum, on the other hand, is that its description provides a more accurate picture of a system's time evolution. Position coordinates are assigned to particles relative to a system of reference in order to specify the direction and magnitude of momentum. The conservation of momentum may then be applied and used to interpret observable

#### *Application of Einstein's Methods in a Quantum Theory of Radiation DOI: http://dx.doi.org/10.5772/intechopen.97734*

phenomena. The Ptolemaic planetary system, for example, introduced fictitious epicycles in violation of the conservation of momentum, but continued to be used for a thousand years because it successfully reproduced what was observed. If astronomers had understood the universal properties of momentum they would have immediately rejected a theory that suggests massive objects could reverse motion in empty space.

Einstein used atomic coordinates fixed with respect to a molecule to derive his A and B coefficients describing momentum exchange during the absorption and emission of energy. The linear momentum of molecules due to thermal impulses is described by introducing a second coordinate system defined with respect to the black body container, that is, in laboratory coordinates. The momentum exchange between the opposing external and internal forces of molecules creates a dynamic equilibrium and allows a clear separation between classical and quantum observables respectively. In contrast, the Bohr-Kramers method describes all observables, discrete and continuous, externally with respect to laboratory coordinates. From Heisenberg's perspective there was no need to treat the discrete spectral lines due to atomic orbitals and the continuous observables due to dispersion phenomena differently, concluding that [10], "Quantum mechanics [is] founded exclusively upon relationships between quantities which are in principle observable."

Dispersion phenomena are observed and measured in laboratory coordinates, and not in the coordinates of an atom. They are given by off-diagonal elements of matrices *n* 6¼ *m* where elements above the diagonal refer to changes in frequency due to energy absorption and elements below the diagonal refer to frequency changes due to energy emission. The elements represent the continuously variable resonances of radiation with an atom's valence electrons. The energy of an absorption offsets the energy of an emission except for a difference in phase so a value of zero is obtained for Eq. (2). On the other hand, the diagonal elements of matrices for *n* ¼ *m* are real eigenvalues representing ground state energy levels. Absorption results in stimulation to a higher orbital and the subsequent emission of a photon upon decay according to the Bohr frequency condition. The off-diagonal interactions due to continuous momentum exchanges are governed by the Compton equation pλ = h. Each matrix element is a photon-electron interaction obtained by resolving the Fourier series (1) into its individual components. It is hypothesized that the complete matrix array expresses the conservation of momentum. Heisenberg mistakenly believed that matrices describe atomic structure, but as Einstein showed atomic structure must be described by internally defined coordinates in the unobservable space–time of an atom. To compare atomic and laboratory coordinates a transformation of coordinates must be performed. Transformations may be visualized with the assistance of the electron oscillator shown in the figure.

#### **2.3 Non-commutation**

To see how non-diagonal and diagonal elements differ we introduce the idea of an electron oscillator in the figure below. If an electron is raised from the ground state j1⟩ to an excited state j2⟩ and then returns a photon is irreversibly emitted. This is shown schematically in the figure below, where 1 and 2 denote the states and arrows refer to transitions. On the left the energy of an electron increases and then decreases, while on the right the reverse occurs. Each arrow represents one-half cycle of the electron oscillator. If the arrows are used to describe off-diagonal matrix elements, they refer to different atoms. If the elements are diagonal they refer to the same atom. It is a simple way of comparing the laboratory coordinates of matrix elements, as determined by photons, to coordinates of atomic structure determined by electron shells during the absorption and emission of radiation. Although the

final state of the quantum system differs the two processes are identical when described in terms of energy differences.

Now consider what happens when the same two energy exchanges are analyzed in terms of the momentum. Using Compton's equation for the momentum of a photon, p = h/λ, the first exchange may be expressed:

$$p\_{12}\lambda\_{12} - p\_{21}\lambda\_{21} = 0\tag{4}$$

Angular momentum increases by an amount ђ when the electron is excited and is then reduced by the same amount when the atom returns to its ground state j1⟩. Thus this type of photon emission ends up with the atomic system in its ground state.

However, when the order of the electron transitions is reversed on the right of the figure we see by the following expression that a description of momentum exchange gives a different result.

$$p\_{21}\lambda\_{21} - p\_{12}\lambda\_{12} = \hbar \tag{5}$$

The electron begins in an excited state j2⟩, reverts to the ground state j1⟩ by emitting a photon, and is excited once again. Thus the final state of the atomic system has an angular momentum that is greater than the ground state by an amount ђ. In both cases (4) and (5) a photon is emitted, but because the order of the physical variables changed the angular momentum of the atomic system described by (5) is greater than (4) and the physical variables do not commute. Noncommutation is interpreted as the irreversibility of momentum when transforming between atomic and laboratory coordinates.

#### **3. Wave mechanics**

#### **3.1 Historical perspectives**

Einstein introduced the founding principles of wave mechanics with concepts from his 1905 papers on special relativity and the photoelectric effect which de Broglie extended to material particles. He also provided the stimulus which led to completion of these ideas in a series of papers on the quantum theory of gases by showing that the same statistics Bose had applied to light quanta could also be used to describe emission from a monatomic ideal gas [11]. This led directly to the further development of wave mechanics by Schrödinger and the introduction of the wave function who openly acknowledged his indebtedness to Einstein in a letter [12]. "By the way, the whole thing would not have started at present or at any other time (I mean as far as I am concerned) had not your second paper on the degenerate gas directed my attention to the importance of de Broglie's ideas." His papers also

stimulated Dirac to write the first paper on quantum electrodynamics introducing the concept of second quantization [13]. Despite the implicit dependence of their theories upon his ideas none of them heeded his advice about momentum [1]. "Most important appears to me the result about the momentum transferred to the molecule by incoming and outgoing radiation." If they had followed Einstein's logic a more coherent description of quantum mechanics would have emerged.

#### **3.2 Physical interpretation of the wave function**

The concept of electron oscillator may be used to describe the rotation of the wave function of half-integer spin particles [14]. Excitation consists of the rotation of an electron's wave function through 2π radians during the *absorption* of one complete cycle of an electromagnetic wave. Decay corresponds to a second rotation of 2π radians during the *emission* of a complete wave cycle. In other words, a complete electron cycle, excitation and decay, consists of two wave function rotations, or 4π radians, and two cycles of an electromagnetic wave, where rotation refers to a change in phase of the electromagnetic field rather than a change in physical space. The electron begins its cycle during energy absorption by entering into a superposition state with a photon's sinusoidal electromagnetic fields and it exits the superposition state when the photon is released. The completed rotation consists of one cycle of an electron oscillator and two cycles of a wave. Thus changes in state can be viewed variously as the excitation and decay of an electron, photon creation and annihilation, superposition of fields, or cycling of a wave; depending upon which physical aspect of the phenomenon one chooses to describe. We use imaginary numbers to describe the transformation of coordinates from the atom to ordinary space so that it is possible to describe the rotation of a wave function mathematically.

The transfer of photon momenta to molecules in induced absorption and emission was predicted theoretically by Einstein and has been verified macroscopically by experiments of many types. It has also been verified microscopically by recent experiments with ultracold three-level artificial atoms which support the idea that momentum is a necessary parameter for the description of emission processes [15]. In the quantum Zeno effect frequent measurements arrest the progress of a "quantum jump". The measurements are equivalent to momentum exchange thereby confirming the earlier hypothesis that photon momenta need to be included in theories of the stimulated absorption and emission of radiation. An incoming photon transfers a momentum þ*E=c* to an atom in the ground state and superposes its fields with an electron's fields. When it exits the superposition state it transfers recoil momentum �*E=c* to the atom and is expelled. The induced absorption and emission momenta are applied at different locations, the ground state electron shell and the excited state electron shell; and they are directed in opposite directions. Taking momentum into account during the time evolution of absorption and emission processes suggests that the electron oscillator cycles at discrete points in space due to momentum exchange and discontinuously in time.

In the wave mechanical view emission occurs by discrete energy exchange, but momentum exchange is either undetectable or does not occur; a situation that is refuted by the Einstein theory of radiation and cannot be sustained by experiment. The Schrödinger wave equation must be reformulated to reflect the discontinuous spatial coordinates and asymmetry of time necessary for momentum exchange.

#### **3.3 Lagrangian model**

The matrix mechanical observables of matter-radiation interactions are described in laboratory coordinates, while wave mechanical properties of matter are described in atomic coordinates. Both describe the same characteristic, the steady states of an atom, but they approach them from different points of view; external and internal. The Einstein theoretical model of matter-radiation interactions adopts both points of view simultaneously within a single material system, as the dynamic equilibrium between external and internal forces. To describe the radiative processes of a single atom a wave equation is needed that includes photons, describes the time evolution of the wave function, and explains how discrete exchanges of momentum can occur during stimulated absorption and emission. Finally, in order to be in agreement with special relativity theory it must be symmetric in the space and time coordinates.

It is possible to formulate a relativistic wave equation by taking the action integral of a Lagrangian *<sup>S</sup>* <sup>¼</sup> <sup>Ð</sup> *Ldt*. Dirac has previously advised on the proper use of the Lagrangian in quantum mechanics [16], "we ought to consider the classical Lagrangian not as a function of the coordinates and velocities but rather as a function of the coordinates at time t and the coordinates at time t + dt." Following Dirac's initiative we let the coordinates at time t and at time t + dt denote electron shells corresponding to the states j1⟩ and j2⟩ respectively. Next, "We introduce at each point of space-time a Lagrangian density, which must be a function of the coordinates and their first derivatives with respect to x,y,z, and t corresponding to the Lagrangian in particle theory being a function of coordinates and velocities. The integral of the Lagrangian density over any (four-dimensional) region of space-time must then be stationary for all small variations of the coordinates inside the region, provided the coordinates on the boundary remain invariant"; where the "fourdimensional region of space-time" refers to the area between electron shells and "the coordinates on the boundary" refers to the electron shells. Absorption initiates from the steady state j1⟩ with coordinates r1 = (x1,y1,z1) and time t1, and it finalizes at j2⟩ with coordinates r2 = (x2,y2,z2) and time t2; where r1 and r2 denote electron shells. The Lagrangian density within the four-dimensional space–time region bounded by the electron shells is a function of the coordinates and their first derivatives *L ϕi*, *ϕ<sup>i</sup>*,*<sup>μ</sup>* � �. The conditions are satisfied by an action integral of the Lagrangian density.

$$\mathcal{S}[\phi\_i(t)] = \int\_{r\_1}^{r\_2} \int\_{t\_1} L\left(\phi\_i, \phi\_{i,\mu}\right) d^3 \mathbf{x} dt = \mathbf{h} \tag{6}$$

The action is a functional, a function of the values of coordinates on the *discrete* boundaries of the space–time surfaces r1 and r2 which are in turn functions of the *continuous* space–time variables of the fields within the surface. The discrete space– time variables assigned to the limits of integration describe electron shells and the continuous space–time variables of the Lagrangian density describe electromagnetic fields. Thus the photon is represented as a four-dimensional localization of field within the electron shells. Momentum exchange occurs when a photon makes contact with a point on the electron shell whether by absorption or emission. Even though complementarity denies the simultaneous presence of wave and particle properties in free space, they are present in atomic space when a photon's sinusoidal fields are localized within electron shells.

#### **3.4 Physical model of the atom**

If the photon is created as an independent entity when energy is absorbed; then quantum mechanics refers to not two, but three bodies. It presumes that the three

*Application of Einstein's Methods in a Quantum Theory of Radiation DOI: http://dx.doi.org/10.5772/intechopen.97734*

field sources are loosely bound within a conservative, or frictionless system, that they are free to interact with each other, and that each of the three particles contributes to the atomic system independently. For the related case of three particles with gravitational fields no general closed form solution is possible [17]. Gravitationally bound three-body systems result in chaos for nearly all initial conditions. It should not be surprising therefore that a physical system consisting of three electromagnetic field sources; electron, photon, and nucleus; also has an indeterminate outcome. To obtain the equations of motion for an electromagnetic three-body problem when the only knowledge available about the particles is their field properties, we need to obtain a series of partial solutions, which are the different mathematical models. Because an exact solution is not possible for the dynamic evolution of a three body system all solutions are considered approximations.

The three-body model of atomic structure may be described formally by introducing a wave-like, physically independent field source ε, the localized photon, into our description of excited atomic states. The modified Hamiltonian is now given by,

$$\mathbf{H} = \mathbf{T} + \mathbf{e} + \mathbf{V} \tag{7}$$

where T refers to an electron, ε represents a "captured" photon, and V represents the nucleus. Each of the three field sources (or particles), possesses a unique vector field; that is, a well-defined field geometry, while the plus and minus signs indicate that the superposition of fields is linear. The Eq. (7) contains the essence of quantum mechanics as a three-body conservative system in real space, as opposed to nonrelativistic descriptions in abstract space. The equations revert to their classical two-body form when the influence of ε is negligible.

#### **4. Conclusion**

If momentum is not taken into account the structure of an atom and its observable properties may be described in the same space. In other words, we can plot the motion of a hydrogen atom's electron in the same space as the motion of the nucleus. If momentum is included a single space–time no longer suffices. When a photon interacts with an atom its linear momentum is transformed into angular momentum and an electron is excited. The angular momentum can no longer be described in laboratory coordinates and instead is expressed in atomic coordinates. All matter has internal and external aspects that are described in distinct coordinate systems. The idea of internal and external properties of matter is as old as science itself having first been expressed by Socrates and Aristotle; however, by introducing Eq. (6) it is proposed as a universal property of matter. Only Einstein fully grasped the need for distinct coordinate systems to describe matter through his theories of the photoelectric effect and Brownian motion. He concluded his quantum theory of radiation by stating [1], "For a *theoretical* discussion such small effects [due to momentum] should be considered on a completely equal footing with the more conspicuous effects of a radiative *energy* transfer, since energy and momentum are linked in the closest possible way." His advice was not fully appreciated due to an inability to visualize the time evolution of a radiating atom.

The conscious mind requires mental images to be able to understand and describe natural phenomena. "For Plato says that we would be engaging in futile labor if we tried to explain these phenomena without images that speak to the eyes." [18]*.* The need for visual images forms the foundation of classical physics and is the source and origin of science itself. All stages of formulating a theory; whether

#### *Quantum Chromodynamic*

observation, analysis, or experiment; is intimately connected to the visual system. In fact the visual cortex is so dominant an area of the brain that when blindness occurs it processes tactile and auditory sensory data instead. Visualization was an important factor during the derivation of quantum mechanics and as well of scientific theory in the past. The need to visualize explains why Heisenberg insisted on a theory of "observables", and it also explains why wave mechanics quickly became more popular. It also accounts for the fact that none of the mathematical models explicitly includes the photon.

### **Author details**

Richard Joseph Oldani Clymer, NY, USA

\*Address all correspondence to: oldani@juno.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Application of Einstein's Methods in a Quantum Theory of Radiation DOI: http://dx.doi.org/10.5772/intechopen.97734*

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[2] van der Waerden B. L. *Sources of Quantum Mechanics* North -Holland, Amsterdam, 1967), p. 4.

[3] Bohr, N. On the constitution of atoms and molecules Phil Mag **26** (151), 1–24. DOI:10.1080/ 14786441308634955

[4] Sommerfeld, A. "Quantum theory of spectral lines" Ann Phys **51**, 125. (1916)

[5] Taltavullo Jordi, M. "Rudolf Ladenburg and the first quantum interpretation of optical dispersion" *Eur Phys J*. H **45**, 123 (2020).

[6] Ladenburg, R.. "The quantum theoretical meaning of the number of dispersions electrons" Z. Phys. **4**: 451–468 (1921).

[7] Jammer, M. The Conceptual Development of Quantum Mechanics (Tomash, 1989)

[8] Kramers, H. "The law of dispersion and Bohr's theory of spectra" Nature **113**, 573 (1924).

[9] J. Mehra & H. Rechenberg,*The Historical Development of Quantum Theory,* Vol. II (NY: Springer, 1982-1988), p. 172.

[10] Heisenberg W. (1925), *Z Phys* **33**, 879 in B.L. van der Waerden (ed.), *Sources of Quantum Mechanics*, (Dover, 1968), p. 261.

[11] Einstein, A. *Sitz Preus Akad d. Wiss* (1924), p. 261; (1925), p. 3.

[12] Schrödinger, E. (1926) in M. Jammer,*The Conceptual Development of Quantum Mechanics* 2nd ed. (NY: Tomash, 1989), p. 258.

[13] Bromberg, J. in C. Weiner (ed.), *History of Twentieth Century Physics*, (NY: Academic, 1977), p. 147.

[14] Schiff, L.I. *Quantum Mechanics*, (NY: 1968), p.205.

[15] Minev, Z., Mundhada, S., Shankar, S. *et al.* "To catch and reverse a quantum jump mid-flight." Nature **570**, 200 (2019). https://doi.org/10.1038/ s41586-019-1287-z

[16] P.A.M. Dirac, Phys Zeit Sow **3**, 1 (1933).

[17] Barrow-Green J. in Gowers, T.; Barrow-Green, J.; Leader, I. (eds.) The Princeton Companion to Mathematics, (Princeton U. Press, 2008), p. 726.

[18] Theon of Smyrna, 1st century

### *Edited by Zbigniew Piotr Szadkowski*

Quantum chromodynamics is a quantum field theory that describes strong interactions between quarks and gluons. It is the SU(3) gauge theory of the current Standard Model for elementary particles and forces. The book contains several chapters on such topics as quark mixing, the double pole method, the inter-nucleon up-to-down quark bond and its implications for nuclear binding, and more. Readers should be fluent in advanced mathematics and quantum physics. Knowledge of quantum electrodynamics is welcome.

Published in London, UK © 2021 IntechOpen © kasezo / iStock

Quantum Chromodynamic

Quantum Chromodynamic

*Edited by Zbigniew Piotr Szadkowski*