**The Gluon Emission Model for Vector Meson Decay\***

### D. White

*Dept. of Biological, Chemical and Physical Sciences, Roosevelt University, Chicago, Illinois USA* 

#### **1. Introduction**

32 Will-be-set-by-IN-TECH

228 Measurements in Quantum Mechanics

Shimony, A. (n.d.). Historical philosophica, and physical inquiries into the foundations of quantum mechanics, *In Sixty-Two Years of Uncertainty*, Plenum, New York.

Sakurai, J. J. (n.d.). *Modern Quantum Mechanics, (revised edition)*, Addison-Wesley, 1999.

Scarani, V. & Gisin, N. (2001). *Phys. Rev. Lett.* 87: 117901. Seevinck, M. & Svetlichny, G. (2002). *Phys. Rev. Lett.* 89: 060401.

Werner, R. & Wolf, M. (2000). *Phys. Rev. A.* 61: 062102.

Zuckowski, M. & Brukner, C. (2002). *Phys. Rev. Lett.* 88: 21041.

Svetlichny, G. (1987). *Phys. Rev. D* 35: 3066. Werner, R. F. (1989). *Phys. Rev. A* 40: 4277.

> Quantum Chromodynamics (QCD), the heart of the so-called Standard Model, was developed along the lines of the most successful theoretical structure in all of physics, namely, Quantum Electrodynamics (QED), which represents the interactions between the electron and the electromagnetic field, photons serving as the mediator between the two entities. QCD, therefore, contains many objects which are analogous to those within QED. There are the quarks, for example, which come in six "flavors" (up, down, strange, charm, bottom, top), serving as the analogous construct to the electron and its cousins, the muon and the tauon. Plexiformation, of course, always accompanies the proceeding by analogy to any theory, and so we find that quarks must come not only in flavors, but also in "colors" (three total), and they must be fractionally charged (one or two thirds of the electron charge in absolute value). What mediates the quark and the strong field, analogous to the photon in QED, is the gluon. In QED all quantum events are described by a coupling between the electron and the photon, called the fine structure constant of magnitude approximately 0.007. Regarding the coupling between quarks and the strong field responsible for hadronization … the production of hadron pairs in a colliding beams experiment, for example … the situation in QCD is quite different. Work continues at present in the field of high-energy physics to determine the precise nature of the quark-gluon coupling, but one overarching behavior pattern of such coupling, called the strong coupling, is that it is *not* a constant. Rather, it varies generally as the reciprocal of the natural logarithm of the energy wrapped up in the colliding beams.

> In the work which follows we will have occasion to investigate the phenomenon of vector meson formation and decay in accord with a QCD model, called the Gluon Emission Model (GEM), first developed by F. E. Close in the 1970s. The GEM follows rigorously the precepts of QED proper, the *only* QCD quantity entering into the calculations being the strong coupling parameter, which replaces the fine structure constant in the relevant places. The GEM thus provides for a self-contained formalism that follows the constructs of QED essentially as closely as is possible at the present time. As we will see, even the precise form for the strong coupling parameter may be determined within the GEM, the valid range

<sup>\*</sup> Much work presented in this Chapter is taken from D. White, "GEM and the Y(1S)", *The Journal of Informatics and Mathematical Sciences*, Vol. 2, Nos. 2 & 3 (2010), pp. 71 – 93.

The Gluon Emission Model for Vector Meson Decay 231

where me represents the electron mass of 0.511 Mev, so that 2me = 1.022 Mev., s represents the strong coupling parameter, given by s = 1.2[ln(mv/50 Mev)]-1, mρ represents the mass of the ρ-meson, mv represents the mass of the vector meson with designate "v", and qi represents the charge of the relevant quark type(s) "i" to undergo the spin-flip to form the vector meson under consideration. As mentioned above, the qi involved in ρ formation, are qu = 2/3 and qd = -1/3, where "u" designates an "up quark" and "d" designates a "down quark". Only qs = -1/3, where "s" designates a "strange quark", is involved in the formation of the kaon branch of the , whereas qu, qd, and qs are all involved in the formation of the K\*(892). In addition, as we will see below, the qi mainly associated with the J(3097) is actually qs, and that associated with the *Y*(1S) is actually qc = 2/3, where "c" is the designate

Focusing now upon Eq. 3 above, A and Λ can be determined simultaneously by utilizing the appropriate qi and mv associated with the ρ and mesons in conjunction with their published widths (see, for example, pdg.lbl.gov; "Meson Table" (2004)). The result4 is that A ≈ 1960 Mev and Λ ≈ 50 Mev. What is most interesting about the above result is the extremely small value for Λ as per the GEM applied to the ρ and the , as the accepted value5 for Λ as of 1996 is around 290 Mev. Nevertheless, in QCD, Λ is considered to be an arbitrary parameter, so no "rules" laid forth within QCD itself are violated by such result. Hence, the

Now, in the asymptotically free regions of energy space we expect the ratio of a vector meson's electron/positron partial width (Γee) to its hadronic width to be approximately (/s), where is the fine structure constant = (1/137.036), due to electromagnetic, rather than strong coupling at the pair vertex. At the *Υ*(1S) energy (mΥ = 9460 Mev according to the 2004 "Meson Table") from the 2004 "Meson Table" on p. 86, Γee = 1.31 Kev, while the *theoretical* hadronic width, as per the GEM, of the *Υ*(1S) is, as we will see below, assuming single gluon emission, 41 Kev. (As we will in addition see below, the GEM requires that the resonant state at the *Υ*(1S) energy be characterized by an essentially instantaneous transition from its root bb\* state (b represents the bottom quark, while b\* represents the bottom antiquark) to an excited cc\* state (c represents the charm quark, while c\* represents the charm

Setting (from the general form for s described above) 0.2284 = B[ln(9460/50)]-1, we obtain B

With the strong coupling parameter defined as above, it presumably valid over the entire range of energy from the ρ energy to that of the *Υ*(1S), the GEM width formula for vector

Γ<sup>v</sup> ≈ (1960 Mev) (mρ/mv)3(Σi(qi)4) [ln(mv/50 Mev)]-1 (5)

s|Υ energy ≈ (41/1.31) = 0.2284 (6)

s ≈ 1.2[ln(Q/50 Mev)]-1 (7)

for the "charm quark".

GEM formulation of Γv becomes:

anti-quark) before its decay.)

≈ 1.2; hence, the GEM determines that

mesons takes the form, then, of

Thus,

**2. The constants, A and Λ, and the specific form of <sup>s</sup>**

Γ<sup>v</sup> ≈ (s /2*π*)(10,042)(2me)(mρ/mv)3(Σi(qi)4) (4)

being the range of energy encompassed by the known vector mesons themselves. Let us begin by reviewing a central feature of QED:

In all quantum systems in which natural decay occurs between an excited level and the ground state, the absorption cross-section goes as1

$$\sigma(\alpha) \equiv \text{Ka|V|^2 (1/m)^2 (1/\alpha)L(\alpha)}\tag{1}$$

where K is a constant, ω represents photon frequency, V 2 represents the square of the matrix element descriptive of the photon emission process, the system has mass m, L(ω) is a Lorentz Amplitude with a peak at ω = ω0 and with a width Γ, and = (1/137.036) represents the fine structure constant. Assuming "asymptotic freedom", i.e., that we may ignore the masses of the decay products (light hadron pairs) in relation to the total energy involved in the system under investigation, we may employ Eq. 1 to predict the width of vector mesons by making the following substitutions to take us from a general quantum electrodynamics (QED) to a specific quantum chromodynamics (QCD) process:

We substitute for the photon frequency the gluon energy Q0. We evaluate the right hand side of Eq. 1 at a specific vector meson mass, mv, i.e., Q0 = m = mv. (Hence, the associated Lorentz Amplitude equals unity.) We require V 2 to be proportional to i(qi)4, where qi = quark charge (in units of electron charge magnitude) associated with the quarks comprising the relevant vector meson. The above criterion is consistent with a spin-spin interaction2 proportional to qi 2, where i denotes quark flavor, giving rise to spin-flip transitions, and the sum is required only in the case of the , as it comprises both the up quark (u) of charge qu = 2/3 and the down quark (d) of charge qd = -1/3. We postulate V 2 to be proportional to only i(qi)4, i.e., the precise form of the interaction is universal to all vector mesons in their ground states, except for quark charge differences.

We replace by s, the strong coupling parameter, which has the well-known form from QCD gauge invariance theories of3:

$$\alpha\_s = \text{B} [\ln(\text{Q}\_0/\Lambda)]^{\text{-1}} \tag{2}$$

where B is a constant and is a parameter, called the QCD scale factor, to be determined. Again, we emphasize that commensurate with the above replacements is that we must assume that the initial energy involved in the formation of a given vector meson is extremely high, i.e., in the "asymptotically free" region of energy space, where the masses of emerging hadron pairs as decay products can be neglected. Accordingly, then, we find in terms of the above ansatz (normalizing to the )

$$\Gamma\_{\mathbf{v}} = A(\mathbf{m}\_{\flat}/\mathbf{m}\_{\mathbf{v}})^{\odot}(\Sigma\_{\mathbf{i}}(\mathbf{q}\_{\flat})^{4})[\ln(\mathbf{m}\_{\mathbf{v}}/\Lambda)]^{\cdot 1} \tag{3}$$

where v represents the width of a given vector meson, v, A is a constant to be determined, and , the QCD scale factor, is to be determined, as well.

The constants, A and Λ, may be determined (see Section 2 below for the determination of the values of A and Λ) by simultaneously fitting the width of the ρ and the width of the kaon branch of the to the form of Eq. 3 above, and B may be determined by evaluating <sup>s</sup> at the *Y*(1S) energy through the utilization of the experimentally determined partial width associated with the *Y*(1S) → e+e decay in conjunction with the GEM-theoretical hadronic width of the *Y*(1S) (see Section 2 below). In conventional terms we will find that the hadronic width of any vector meson may be expressed as the following:

230 Measurements in Quantum Mechanics

being the range of energy encompassed by the known vector mesons themselves. Let us

In all quantum systems in which natural decay occurs between an excited level and the

matrix element descriptive of the photon emission process, the system has mass m, L(ω) is a Lorentz Amplitude with a peak at ω = ω0 and with a width Γ, and = (1/137.036) represents the fine structure constant. Assuming "asymptotic freedom", i.e., that we may ignore the masses of the decay products (light hadron pairs) in relation to the total energy involved in the system under investigation, we may employ Eq. 1 to predict the width of vector mesons by making the following substitutions to take us from a general quantum electrodynamics

We substitute for the photon frequency the gluon energy Q0. We evaluate the right hand side of Eq. 1 at a specific vector meson mass, mv, i.e., Q0 = m = mv. (Hence, the associated

qi = quark charge (in units of electron charge magnitude) associated with the quarks comprising the relevant vector meson. The above criterion is consistent with a spin-spin

transitions, and the sum is required only in the case of the , as it comprises both the up quark (u) of charge qu = 2/3 and the down quark (d) of charge qd = -1/3. We postulate V

to be proportional to only i(qi)4, i.e., the precise form of the interaction is universal to all

We replace by s, the strong coupling parameter, which has the well-known form from

where B is a constant and is a parameter, called the QCD scale factor, to be determined. Again, we emphasize that commensurate with the above replacements is that we must assume that the initial energy involved in the formation of a given vector meson is extremely high, i.e., in the "asymptotically free" region of energy space, where the masses of emerging hadron pairs as decay products can be neglected. Accordingly, then, we find in

where v represents the width of a given vector meson, v, A is a constant to be determined,

The constants, A and Λ, may be determined (see Section 2 below for the determination of the values of A and Λ) by simultaneously fitting the width of the ρ and the width of the kaon branch of the to the form of Eq. 3 above, and B may be determined by evaluating <sup>s</sup> at the *Y*(1S) energy through the utilization of the experimentally determined partial width

width of the *Y*(1S) (see Section 2 below). In conventional terms we will find that the

hadronic width of any vector meson may be expressed as the following:

vector mesons in their ground states, except for quark charge differences.

2 (1/m)2(1/ω)L(ω) (1)

2 represents the square of the

2 to be proportional to i(qi)4, where

2

2, where i denotes quark flavor, giving rise to spin-flip

s = B[ln(Q0/)]-1 (2)

v = A(m/mv)3(i(qi)4)[ln(mv/)]-1 (3)

decay in conjunction with the GEM-theoretical hadronic

(ω) = KV

where K is a constant, ω represents photon frequency, V

(QED) to a specific quantum chromodynamics (QCD) process:

Lorentz Amplitude equals unity.) We require V

interaction2 proportional to qi

QCD gauge invariance theories of3:

associated with the *Y*(1S) → e+e-

terms of the above ansatz (normalizing to the )

and , the QCD scale factor, is to be determined, as well.

begin by reviewing a central feature of QED:

ground state, the absorption cross-section goes as1

$$
\Gamma\_{\rm v} \approx (\mathbf{u}\_{\rm s} / 2\pi)(10,042)(2\,\mathrm{m}\_{\rm e})(\mathbf{m}\_{\rm p} / \mathbf{m}\_{\rm v})^{3}(\Sigma\_{\rm i}(\mathbf{q}\_{\rm i})^{4})\tag{4}
$$

where me represents the electron mass of 0.511 Mev, so that 2me = 1.022 Mev., s represents the strong coupling parameter, given by s = 1.2[ln(mv/50 Mev)]-1, mρ represents the mass of the ρ-meson, mv represents the mass of the vector meson with designate "v", and qi represents the charge of the relevant quark type(s) "i" to undergo the spin-flip to form the vector meson under consideration. As mentioned above, the qi involved in ρ formation, are qu = 2/3 and qd = -1/3, where "u" designates an "up quark" and "d" designates a "down quark". Only qs = -1/3, where "s" designates a "strange quark", is involved in the formation of the kaon branch of the , whereas qu, qd, and qs are all involved in the formation of the K\*(892). In addition, as we will see below, the qi mainly associated with the J(3097) is actually qs, and that associated with the *Y*(1S) is actually qc = 2/3, where "c" is the designate for the "charm quark".

#### **2. The constants, A and Λ, and the specific form of <sup>s</sup>**

Focusing now upon Eq. 3 above, A and Λ can be determined simultaneously by utilizing the appropriate qi and mv associated with the ρ and mesons in conjunction with their published widths (see, for example, pdg.lbl.gov; "Meson Table" (2004)). The result4 is that A ≈ 1960 Mev and Λ ≈ 50 Mev. What is most interesting about the above result is the extremely small value for Λ as per the GEM applied to the ρ and the , as the accepted value5 for Λ as of 1996 is around 290 Mev. Nevertheless, in QCD, Λ is considered to be an arbitrary parameter, so no "rules" laid forth within QCD itself are violated by such result. Hence, the GEM formulation of Γv becomes:

$$
\Gamma\_{\rm v} \approx \text{(1960 MeV) (m\_p/m\_v)} \text{(}\Sigma\_{\rm i}(\text{q}\_{\rm i})\text{4) [In(m\_v/50 MeV)]}\text{1}\tag{5}
$$

Now, in the asymptotically free regions of energy space we expect the ratio of a vector meson's electron/positron partial width (Γee) to its hadronic width to be approximately (/s), where is the fine structure constant = (1/137.036), due to electromagnetic, rather than strong coupling at the pair vertex. At the *Υ*(1S) energy (mΥ = 9460 Mev according to the 2004 "Meson Table") from the 2004 "Meson Table" on p. 86, Γee = 1.31 Kev, while the *theoretical* hadronic width, as per the GEM, of the *Υ*(1S) is, as we will see below, assuming single gluon emission, 41 Kev. (As we will in addition see below, the GEM requires that the resonant state at the *Υ*(1S) energy be characterized by an essentially instantaneous transition from its root bb\* state (b represents the bottom quark, while b\* represents the bottom antiquark) to an excited cc\* state (c represents the charm quark, while c\* represents the charm anti-quark) before its decay.)

Thus,

$$a\_s \mid\_{\text{Y energy}} \approx a \; (41/1.31) = 0.2284\tag{6}$$

Setting (from the general form for s described above) 0.2284 = B[ln(9460/50)]-1, we obtain B ≈ 1.2; hence, the GEM determines that

$$
\alpha\_s \approx 1.2 [\ln(\text{Q/50 MeV})]^{\frac{1}{4}} \tag{7}
$$

With the strong coupling parameter defined as above, it presumably valid over the entire range of energy from the ρ energy to that of the *Υ*(1S), the GEM width formula for vector mesons takes the form, then, of

The Gluon Emission Model for Vector Meson Decay 233

spin interaction proportional to qx2, proceeds straight along the dictates of standard QED,

For comparison, immediately below we present the FD associated with the same X meson, assumed to exist in the realm of asymptotic freedom, decaying into an electron-positron

Fig. 2. Basic Feynman Diagram for Conventional Vector Meson Formation & Decay into an

The only fundamental difference between Figure 1 and Figure 2 is that in Figure 2 ζ2 starts out as a gluon and ends up as a virtual photon at the right hand vertex, at which point the coupling, of course, is now . Hence, all in the width calculation associated with Figure 1 is the same in Figure 2, except that s in Eq. 10 is replaced by . Of note, too, and we shall return to the point made here, Figure 2 represents rigorously a straight-forward calculation in QED, again, given the stated mechanism for the formation of the resonance state. However, it is also important to note that Figure 2 applies only to vector mesons existing in the realm of "asymptotic freedom", i.e., to the J(3097), the *Y*(1S), and "Toponium", or the

Immediately below we will view the detailed FDs required by the GEM to describe the widths of the ρ, the , the K\*(892) … a very interesting case, as the K\*(892) is not conventionally thought of as a vector meson per se, though it is of the spin one variety …

Although the width of the ρ (and the ) as determined by the GEM is guaranteed to be a match to experiment by construction, the ρ is a good place to start with the elucidation of the application of the GEM to the various spin one mesons because of the simplicity involved. Let us begin by viewing Figure 3 below … the FD associated with the formation

In Figure 3 ζ1 represents a virtual photon created at the e+e- vertex which transmutes to a gluon, which, in turn, is absorbed by the [ququ\* + qdqd\*] combination; ζ2 represents the emitted gluon, which converts to pion pairs. The application of Eq. 10 results in the

Γρ ≈ (s /2*π*)(10,042)(2me)(Σi(qi)4) ≈ (s /2*π*)(10,042)(2me)(17/81) (12a)

s = 1.2[ln(776/50)]-1 = 0.4376 (12b)

except for the replacement of by s at the hh\* vertex.

Electron/Positron Pair via the GEM

the J(3097), the *Y*(1S), and the "T".

following for the hadronic width of the ρ:

and decay of the ρ meson.

pair.

"T" meson.

**The ρ-meson** 

where

$$
\Gamma\_\mathrm{v} \approx \begin{pmatrix} 16 \text{\textdegree } \mathrm{MeV} \text{ } \mathrm{MeV} \text{ } \mathrm{(m}\_\mathrm{p}/\mathrm{m}\_\mathrm{v})^3 \text{(}\Sigma\_\mathrm{i}(\mathrm{q}\_\mathrm{i})^4 \text{)} \text{ } \alpha\_\mathrm{s} \tag{8}
$$

Expressing Eq. 8 in more conventional form, we have:

$$
\Gamma\_{\rm v} \approx (\mathbf{u}\_{\rm s} / 2\pi) (10 / 263 \text{ MeV}) (\mathbf{m}\_{\rm p} / \mathbf{m}\_{\rm v})^{\odot} (\Sigma\_{\rm i} (\mathbf{q}\_{\rm i})^{4}) \tag{9}
$$

Yet more formally, we finally obtain:

$$
\Gamma\_{\rm v} \approx (\mathbf{u}\_{\rm s} / 2\pi)(10,042)(2\,\mathrm{m}\_{\rm e})(\mathbf{m}\_{\rm p} / \mathbf{m}\_{\rm v})^{3}(\Sigma\_{\rm i}(\mathbf{q}\_{\rm i})^{4})\tag{10}
$$

#### **3. The Feynman Diagrams of the GEM**

As per F. Close, the GEM treats the virtual photon and the gluon as, essentially, two aspects of the same entity, which we will call "the four-momentum propagator" designated as "ζ". Thus, as stated in Section 2 above, the ratio of the partial width associated with a given decaying pair of quarks comprising a given vector meson associated with electron-positron decay to the hadronic width of same is simply (/s), where, again, "" represents the fine structure constant = (1/137.036). Hence, the general form for the partial width of a vector meson undergoing e+e decay would be given by

$$
\Gamma\_{\rm v-ee} \approx (\mathrm{a}/2\pi)(10,042)(2\,\mathrm{m}\_{\odot})(\mathrm{m}\_{\oplus}/\mathrm{m}\_{\mathrm{v}})^{3}(\Sigma\_{\mathrm{i}}(\mathrm{q}\_{\mathrm{i}})^{4})\tag{11}
$$

A relevant Feynman Diagram will make the various aspects of the GEM easier to picture, so let us look now to Fig. 1 below, which represents the Feynman Diagram (FD) associated the formation and decay of vector meson "X" in its simplest possible form.

Fig. 1. Basic Feynman Diagram for Conventional Vector Meson Formation/Decay via the GEM

In Figure 1 ζ1 represents, in part, a virtual photon created at the e+e- annihilation vertex, coupling at said vertex represented as ; then, in Close's terms, the virtual photon couples to a gluon with coupling strength "1", which then couples to the xx\* … a given quark – antiquark pair, also with coupling strength "1". In our notation ζ1 simply represents a fourmomentum propagator, created at the e+e- vertex and absorbed (as a gluon) at the xx\* node. The details of the absorption of ζ1 are contained in the integrated absorption cross-section as exhibited in the Introduction, and lVl2, proportional to qx4, describes the formation of the spin one resonance. From there ζ2 (a gluon) is emitted, resulting in coupling to hadrons (h; h\*), the coupling at the latter vertex of magnitude s . The calculation of the width of the xx\* state then, given the stated mechanism of a spin-flip of one of the "x quarks" due to a spin – 232 Measurements in Quantum Mechanics

As per F. Close, the GEM treats the virtual photon and the gluon as, essentially, two aspects of the same entity, which we will call "the four-momentum propagator" designated as "ζ". Thus, as stated in Section 2 above, the ratio of the partial width associated with a given decaying pair of quarks comprising a given vector meson associated with electron-positron decay to the hadronic width of same is simply (/s), where, again, "" represents the fine structure constant = (1/137.036). Hence, the general form for the partial width of a vector

A relevant Feynman Diagram will make the various aspects of the GEM easier to picture, so let us look now to Fig. 1 below, which represents the Feynman Diagram (FD) associated the

Fig. 1. Basic Feynman Diagram for Conventional Vector Meson Formation/Decay via the

In Figure 1 ζ1 represents, in part, a virtual photon created at the e+e- annihilation vertex, coupling at said vertex represented as ; then, in Close's terms, the virtual photon couples to a gluon with coupling strength "1", which then couples to the xx\* … a given quark – antiquark pair, also with coupling strength "1". In our notation ζ1 simply represents a fourmomentum propagator, created at the e+e- vertex and absorbed (as a gluon) at the xx\* node. The details of the absorption of ζ1 are contained in the integrated absorption cross-section as exhibited in the Introduction, and lVl2, proportional to qx4, describes the formation of the spin one resonance. From there ζ2 (a gluon) is emitted, resulting in coupling to hadrons (h; h\*), the coupling at the latter vertex of magnitude s . The calculation of the width of the xx\* state then, given the stated mechanism of a spin-flip of one of the "x quarks" due to a spin –

decay would be given by

formation and decay of vector meson "X" in its simplest possible form.

Expressing Eq. 8 in more conventional form, we have:

Yet more formally, we finally obtain:

meson undergoing e+e-

GEM

**3. The Feynman Diagrams of the GEM** 

Γ<sup>v</sup> ≈ (1633 Mev) (mρ/mv)3(Σi(qi)4) s (8)

Γ<sup>v</sup> ≈ (s /2*π*)(10,263 Mev)(mρ/mv)3(Σi(qi)4) (9)

Γ<sup>v</sup> ≈ (s /2*π*)(10,042)(2me)(mρ/mv)3(Σi(qi)4) (10)

Γv-ee ≈ (/2*π*)(10,042)(2me)(mρ/mv)3(Σi(qi)4) (11)

spin interaction proportional to qx2, proceeds straight along the dictates of standard QED, except for the replacement of by s at the hh\* vertex.

For comparison, immediately below we present the FD associated with the same X meson, assumed to exist in the realm of asymptotic freedom, decaying into an electron-positron pair.

Fig. 2. Basic Feynman Diagram for Conventional Vector Meson Formation & Decay into an Electron/Positron Pair via the GEM

The only fundamental difference between Figure 1 and Figure 2 is that in Figure 2 ζ2 starts out as a gluon and ends up as a virtual photon at the right hand vertex, at which point the coupling, of course, is now . Hence, all in the width calculation associated with Figure 1 is the same in Figure 2, except that s in Eq. 10 is replaced by . Of note, too, and we shall return to the point made here, Figure 2 represents rigorously a straight-forward calculation in QED, again, given the stated mechanism for the formation of the resonance state. However, it is also important to note that Figure 2 applies only to vector mesons existing in the realm of "asymptotic freedom", i.e., to the J(3097), the *Y*(1S), and "Toponium", or the "T" meson.

Immediately below we will view the detailed FDs required by the GEM to describe the widths of the ρ, the , the K\*(892) … a very interesting case, as the K\*(892) is not conventionally thought of as a vector meson per se, though it is of the spin one variety … the J(3097), the *Y*(1S), and the "T".

#### **The ρ-meson**

Although the width of the ρ (and the ) as determined by the GEM is guaranteed to be a match to experiment by construction, the ρ is a good place to start with the elucidation of the application of the GEM to the various spin one mesons because of the simplicity involved. Let us begin by viewing Figure 3 below … the FD associated with the formation and decay of the ρ meson.

In Figure 3 ζ1 represents a virtual photon created at the e+e- vertex which transmutes to a gluon, which, in turn, is absorbed by the [ququ\* + qdqd\*] combination; ζ2 represents the emitted gluon, which converts to pion pairs. The application of Eq. 10 results in the following for the hadronic width of the ρ:

$$
\Gamma\_{\rho} \approx (\mathbf{a}\_{\circ} / 2\pi)(10.042)(2\mathbf{m}\_{\circ})(\Sigma\_{\mathrm{i}}(\mathbf{q}\_{\mathrm{j}})^{4}) \approx (\mathbf{a}\_{\circ} / 2\pi)(10.042)(2\mathbf{m}\_{\circ})(17/81)\tag{12a}
$$

where

$$
\alpha\_s = 1.2[\ln(776/50)]^{\cdot 1} = 0.4376\tag{12b}
$$

The Gluon Emission Model for Vector Meson Decay 235

Again applying Eq. 11 to the kaon branch of the , we obtain for its e+e- partial width the

a figure7 still way too high as compared to experiment, but here about 52 times so, indicating that the ζ2 gluon to virtual photon transmutation coupling has risen to 0.0194.

The situation regarding the K\*(892) is highly interesting. Close had developed the GEM in the 1970s to describe two distinct processes: (1) the production of pion pairs associated with the ρ resonance and (2) the production of kaon pairs associated with the resonance. In a sense, then, the GEM was first envisioned to be "route specific", i.e., the spin-flip process involving up and down quarks, which resonates at the ρ mass, was thought of as "the pion route" in thinking of the decay of quark – anti-quark structures, while the spin-flip process involving the strange quark, which resonates at the mass, was thought of as the corresponding "kaon route". At that time no one had thought of applying the GEM to the K\*(892), because, although energetically possible, the K\*(892) did not exhibit "a pion route" in its decay; rather, the K\*(892) decays almost exclusively into various {π, K} combinations, with equal probability of occurrence among the various allowed decay products. Such circumstance led to the invention of the "isospin" quantum number, a half integer value for which signifying a forbidden decay route that is energetically possible. However, since the spin associated with the K\*(892) is one, it is quite feasible that the GEM, appropriately mitigated to fit the situation pertaining to the K\*(892)'s isospin, may be applied to the K\*(892) resonance. In fact, the GEM has been applied to the K\*(892) quite successfully4, 8.

Since pions and kaons are the decay products of the K\*(892), with the various types of pions combining with correspondingly allowed various types of kaons and all types showing up with equal probability, it is reasonable to assume that the K\*(892) … for purposes of discussion here considered as a composite entity of mass, 894 Mev, i.e., no distinction as to charged mode versus neutral mode being made … comprises a linear combination of {uu\*, dd\*, and ss\*} in equal measure. Symbolically, we may represent the K\*(892), therefore, as

Now, the associated value of (Σi(qi)4) would be (18/81), but the "pion route" does not occur, though it is energetically possible. So, segmenting the decay in terms of "routes", the {π, K} route, whose (Σi(qi)4) = (18/81) does occur, whereas the "pion route", whose (Σi(qi)4) = (17/81) does not occur. The allowed route is thus favored over the forbidden route by the factor (18/17), therefore. Hence, we postulate that the isospin quantum number = (1/2) assigned to the K\*(892) signifies that of the energetically possible routes available to the K\*(892) resonance, (18/35) of them manifests in the decay process (the {π, K} route), whereas (17/35) of them fails to materialize (the pion route). We thus multiply the right hand side of Eq. 10 by (18/35) to obtain the width of the K\*(892). First, let us view the associated FD:

ΓK\* ≈ (18/35)(s /2*π*)(10,042)(2me)(m**ρ**/mv)3(Σi(qi)4)

K\*(892) = (1/√3)[uu\* + dd\* + ss\*] (14)

The reasoning leading to the proper mitigation is as follows:

The GEM yields for the width of the K\*(892) the following:

Γ-K-ee ≈ ( /2*π*)(10,042)(2me)(776/1019)3(1/81 ) ≈ 0.0650 Mev

following:

**The K\*(892)** 

Γ-K ≈ 3.55 Mev (13c)

Fig. 3. Basic Feynman Diagram for Formation and Decay of the ρ meson via the GEM Hence,

$$
\Gamma\_{\rho} \approx 150 \text{ MeV} \tag{12c}
$$

Though adaptation of Figure 2 and Eq. 11 do not formally apply, as asymptotic freedom does not apply to the ρ, we note that in the event that it were to apply, we would obtain for the electron/positron partial width, Γρ-ee , the following:

Γρ-ee ≈ ( /2*π*)(10,042)(2me)(17/81) ≈ 2.50 Mev

a figure7 about 355 times too high, indicating that the transmutation coupling of the ζ2 gluon to its virtual photon identity is only 0.0028, as opposed to 1 in the asymptotically free energy regime.

#### **The -meson**

Application of the GEM to the kaon branch of the meson (K) follows similar lines as to the ρ. The FD associated with the formation and decay of the kaon branch of the follows:

Fig. 4. Basic Feynman Diagram for Formation and Decay of the Kaon Branch of the Meson via the GEM

For the hadronic width of the kaon branch of the we obtain:

Γ-K ≈ (s /2*π*)(10,042)(2me)(mρ/mv)3(Σi(qi)4) ≈ (s /2*π*)(10,042)(2me)(776/1019)3(1/81) (13a) where

$$
\alpha\_s = 1.2[\ln(1019/50)]^{\cdot\_1} = 0.3981\tag{13b}
$$

Hence,

$$
\Gamma\_{\text{\\$} \text{-K}} \approx \text{3.55 MeV} \tag{13c}
$$

Again applying Eq. 11 to the kaon branch of the , we obtain for its e+e- partial width the following:

$$
\Gamma\_{\text{\\$-K\text{-}ee}} \approx (\text{a} \,/2\pi)(10.042)(2\text{m}\_{\text{e}})(776/1019)^{\text{3}}(1/81) \approx 0.0650 \text{ MeV}
$$

a figure7 still way too high as compared to experiment, but here about 52 times so, indicating that the ζ2 gluon to virtual photon transmutation coupling has risen to 0.0194.

#### **The K\*(892)**

234 Measurements in Quantum Mechanics

Fig. 3. Basic Feynman Diagram for Formation and Decay of the ρ meson via the GEM

the electron/positron partial width, Γρ-ee , the following:

For the hadronic width of the kaon branch of the we obtain:

Though adaptation of Figure 2 and Eq. 11 do not formally apply, as asymptotic freedom does not apply to the ρ, we note that in the event that it were to apply, we would obtain for

Γρ-ee ≈ ( /2*π*)(10,042)(2me)(17/81) ≈ 2.50 Mev a figure7 about 355 times too high, indicating that the transmutation coupling of the ζ2 gluon to its virtual photon identity is only 0.0028, as opposed to 1 in the asymptotically free energy

Application of the GEM to the kaon branch of the meson (K) follows similar lines as to the ρ. The FD associated with the formation and decay of the kaon branch of the follows:

Fig. 4. Basic Feynman Diagram for Formation and Decay of the Kaon Branch of the Meson

Γ-K ≈ (s /2*π*)(10,042)(2me)(mρ/mv)3(Σi(qi)4) ≈ (s /2*π*)(10,042)(2me)(776/1019)3(1/81) (13a)

s = 1.2[ln(1019/50)]-1 = 0.3981 (13b)

Γρ ≈ 150 Mev (12c)

Hence,

regime.

**The -meson** 

via the GEM

where

Hence,

The situation regarding the K\*(892) is highly interesting. Close had developed the GEM in the 1970s to describe two distinct processes: (1) the production of pion pairs associated with the ρ resonance and (2) the production of kaon pairs associated with the resonance. In a sense, then, the GEM was first envisioned to be "route specific", i.e., the spin-flip process involving up and down quarks, which resonates at the ρ mass, was thought of as "the pion route" in thinking of the decay of quark – anti-quark structures, while the spin-flip process involving the strange quark, which resonates at the mass, was thought of as the corresponding "kaon route". At that time no one had thought of applying the GEM to the K\*(892), because, although energetically possible, the K\*(892) did not exhibit "a pion route" in its decay; rather, the K\*(892) decays almost exclusively into various {π, K} combinations, with equal probability of occurrence among the various allowed decay products. Such circumstance led to the invention of the "isospin" quantum number, a half integer value for which signifying a forbidden decay route that is energetically possible. However, since the spin associated with the K\*(892) is one, it is quite feasible that the GEM, appropriately mitigated to fit the situation pertaining to the K\*(892)'s isospin, may be applied to the K\*(892) resonance. In fact, the GEM has been applied to the K\*(892) quite successfully4, 8. The reasoning leading to the proper mitigation is as follows:

Since pions and kaons are the decay products of the K\*(892), with the various types of pions combining with correspondingly allowed various types of kaons and all types showing up with equal probability, it is reasonable to assume that the K\*(892) … for purposes of discussion here considered as a composite entity of mass, 894 Mev, i.e., no distinction as to charged mode versus neutral mode being made … comprises a linear combination of {uu\*, dd\*, and ss\*} in equal measure. Symbolically, we may represent the K\*(892), therefore, as

$$\mathbf{K^\*(892) = (1/\sqrt{3})[\mathbf{u}\mathbf{u^\*} + \mathbf{d}\mathbf{d^\*} + \mathbf{ss^\*}]} \tag{14}$$

Now, the associated value of (Σi(qi)4) would be (18/81), but the "pion route" does not occur, though it is energetically possible. So, segmenting the decay in terms of "routes", the {π, K} route, whose (Σi(qi)4) = (18/81) does occur, whereas the "pion route", whose (Σi(qi)4) = (17/81) does not occur. The allowed route is thus favored over the forbidden route by the factor (18/17), therefore. Hence, we postulate that the isospin quantum number = (1/2) assigned to the K\*(892) signifies that of the energetically possible routes available to the K\*(892) resonance, (18/35) of them manifests in the decay process (the {π, K} route), whereas (17/35) of them fails to materialize (the pion route). We thus multiply the right hand side of Eq. 10 by (18/35) to obtain the width of the K\*(892). First, let us view the associated FD: The GEM yields for the width of the K\*(892) the following:

$$\Gamma\_{\mathbf{K^{\*}}} \approx (18/35)(\mathbf{a\_{\*}}/2\pi)(10,042)(2\mathbf{m\_{e}})(\mathbf{m\_{p}}/\mathbf{m\_{v}})^{3}(\Sigma\_{i}(\mathbf{q\_{i}})^{4})$$

The Gluon Emission Model for Vector Meson Decay 237

point-like" transition to ss\*, described by a form factor, f < 1, which, in turn, decays into both hadrons and leptons as per Eq. 10 and Eq. 11, respectively, while (1/9)th of the original cc\* structure remains to decay into leptons exclusively. We may picture the complete details of the J formation and decay via the following two arrays of FDs, the first such array descriptive of what we may now call "the first order approximation" to the width of the J,

the second such array descriptive of what we call "the second order approximation".

Fig. 6a. Feynman Diagram Array Characterizing the Formation and Decay of the J(3097) in

In Figure 6a above "l" represents a leptonic decay product, ζ2a represents the gluon involved in a point-like transition from cc\* to ss\*, and all other "ζ" designates should be understood from previous discussion. Transforming the schematic representation of Figure 6a into the calculation of the full (hadronic plus leptonic) width of the J in first order approximation, denoted as ΓJ-full-1, proceeds as follows (the factors of "2" in Eq. 16a, immediately in front of the factors "( /2*π*)" take into account muon pair production in accord with "e-μ

ΓJ-full-1 ≈ (8/9){(s /2*π*)(10,042)(2me)(mρ/mJ)3(qs)4 + 2( /2*π*)(10,042)(2me)(mρ/mJ)3(qs)4}

First Order Approximation via the GEM

universality"):

#### Fig. 5. Basic Feynman Diagram for Formation and Decay of the K\*(892) via the GEM

$$^{\approx} (18/35) (\text{u}\_{\text{s}}/2\pi) (10,042) (2\text{m}\_{\text{e}}) (776/894) ^{\circ} (18/81) \tag{15a}$$

where

$$
\alpha\_{\rm s} = 1.2 \left[ \ln(894/50) \right] \cdot 1 = 0.4161 \tag{15b}
$$

Hence,

$$
\Gamma\_{\text{K}^\*} \approx 50.80 \,\text{MeV} \tag{15c}
$$

The average of the widths associated with the charged and neutral modes of the K\*(892) is stated9 as ΓK\*(PDG) = 50.75 Mev. Hence, the GEM as applied to the K\*(892) provides for fabulous agreement with experiment. Moreover, the GEM demonstrates quite clearly that the K\*(892) is not a "strange meson" in the usual sense, i.e., it is seen not as a us\*, su\*, ds\*, or sd\* structure at all; rather it is seen, similar to the theoretical structures of the ρ and the , as comprising a linear combination of more than one type of quark – anti-quark pair, its specific nature expressed via Eq. 14.

#### **The J(3097)**

Application of the GEM in accord with Figure 1, with x = c, seems reasonably straightforward, but it turns out to be problematic. However, when one sees that the hadronic width of the J(3097), designated as simply the "J" henceforth, given by the application of Eq. 10 in accord with Figure 1 with x = c, is roughly sixteen times too large, as compared to experimental results, coupled with the fact that the hadronic width of the *Y*(1S) given by the application of Eq. 10 in accord with Figure 1 with x = b is roughly sixteen times too small, as compared with experimental results, it becomes obvious as to what physically must transpire as regards both the J and the *Y*(1S). Restricting the discussion to the J for the time being, in what we call "the zeroth order approximation", the basic cc\* structure of the J must make a point-like transition to an ss\* structure of equal mass, whereupon one of the s quarks undergoes a spin flip to form the associated resonance10, the point-like transition from cc\* to ss\* instantaneous, thus having no influence on the J's width. Indeed, the resonance does not even form until an s (or s\*) quark undergoes a spin-flip. That the cc\* to ss\* transition is necessary is quite understandable: The J is not massive enough for it to be able to decay into hadrons via emission of two c quarks; hence, it must transition to a quark pair of lesser bare mass each. The simplest possible assumption is that the cc\* transitions to the quark pair type characterized by the next smallest mass, viz., the s type. Nothing prevents the cc\* structure from decaying into leptons (e+e and μ+μ-), however. It is found10 in fact, that in order for both the hadronic width of the J and the leptonic width of the J as determined via the GEM to match the results of experiment, (8/9)ths of the cc\* structure must undergo a slightly "un236 Measurements in Quantum Mechanics

Fig. 5. Basic Feynman Diagram for Formation and Decay of the K\*(892) via the GEM

The average of the widths associated with the charged and neutral modes of the K\*(892) is stated9 as ΓK\*(PDG) = 50.75 Mev. Hence, the GEM as applied to the K\*(892) provides for fabulous agreement with experiment. Moreover, the GEM demonstrates quite clearly that the K\*(892) is not a "strange meson" in the usual sense, i.e., it is seen not as a us\*, su\*, ds\*, or sd\* structure at all; rather it is seen, similar to the theoretical structures of the ρ and the , as comprising a linear combination of more than one type of quark – anti-quark pair, its

Application of the GEM in accord with Figure 1, with x = c, seems reasonably straightforward, but it turns out to be problematic. However, when one sees that the hadronic width of the J(3097), designated as simply the "J" henceforth, given by the application of Eq. 10 in accord with Figure 1 with x = c, is roughly sixteen times too large, as compared to experimental results, coupled with the fact that the hadronic width of the *Y*(1S) given by the application of Eq. 10 in accord with Figure 1 with x = b is roughly sixteen times too small, as compared with experimental results, it becomes obvious as to what physically must transpire as regards both the J and the *Y*(1S). Restricting the discussion to the J for the time being, in what we call "the zeroth order approximation", the basic cc\* structure of the J must make a point-like transition to an ss\* structure of equal mass, whereupon one of the s quarks undergoes a spin flip to form the associated resonance10, the point-like transition from cc\* to ss\* instantaneous, thus having no influence on the J's width. Indeed, the resonance does not even form until an s (or s\*) quark undergoes a spin-flip. That the cc\* to ss\* transition is necessary is quite understandable: The J is not massive enough for it to be able to decay into hadrons via emission of two c quarks; hence, it must transition to a quark pair of lesser bare mass each. The simplest possible assumption is that the cc\* transitions to the quark pair type characterized by the next smallest mass, viz., the s type. Nothing prevents the cc\* structure

both the hadronic width of the J and the leptonic width of the J as determined via the GEM to match the results of experiment, (8/9)ths of the cc\* structure must undergo a slightly "un-

where

Hence,

**The J(3097)** 

specific nature expressed via Eq. 14.

from decaying into leptons (e+e-

≈ (18/35)(s /2*π*)(10,042)(2me)(776/894)3(18/81) (15a)

s = 1.2[ln(894/50)]-1 = 0.4161 (15b)

and μ+μ-), however. It is found10 in fact, that in order for

ΓK\* ≈ 50.80 Mev (15c)

point-like" transition to ss\*, described by a form factor, f < 1, which, in turn, decays into both hadrons and leptons as per Eq. 10 and Eq. 11, respectively, while (1/9)th of the original cc\* structure remains to decay into leptons exclusively. We may picture the complete details of the J formation and decay via the following two arrays of FDs, the first such array descriptive of what we may now call "the first order approximation" to the width of the J, the second such array descriptive of what we call "the second order approximation".

Fig. 6a. Feynman Diagram Array Characterizing the Formation and Decay of the J(3097) in First Order Approximation via the GEM

In Figure 6a above "l" represents a leptonic decay product, ζ2a represents the gluon involved in a point-like transition from cc\* to ss\*, and all other "ζ" designates should be understood from previous discussion. Transforming the schematic representation of Figure 6a into the calculation of the full (hadronic plus leptonic) width of the J in first order approximation, denoted as ΓJ-full-1, proceeds as follows (the factors of "2" in Eq. 16a, immediately in front of the factors "( /2*π*)" take into account muon pair production in accord with "e-μ universality"):

$$\Gamma\_{\text{I-full-1}} \approx (8/9) \langle (\mathbf{a}\_{\text{s}}/2\pi)(10.042)(2\text{m}\_{\text{e}}) (\mathbf{m}\_{\text{p}}/\text{m})^{\natural} (\mathbf{q}\_{\text{f}})^{\natural} + 2 \langle \mathbf{a}\_{\text{s}}/2\pi \rangle (10.042)(2\text{m}\_{\text{e}}) (\mathbf{m}\_{\text{p}}/\text{m})^{\natural} (\mathbf{q}\_{\text{f}})^{\natural}$$

The Gluon Emission Model for Vector Meson Decay 239

Fig. 6b. Feynman Diagram Array Characterizing the Formation and Decay of the J(3097) in

The full width of the J under second order approximation is thus nearly an exact match to experiment (93.4 Kev via the GEM vs. 93.2 Kev from PDG (2009)). The hadronic width of the J is unchanged from first to second approximation; so, it remains a match with experiment (82.0 Kev via the GEM vs. 81.7 Kev from PDG (2009)). As well, the leptonic width of the J via the GEM (11.4 Kev) is now only 2.7% higher than that reported by the PDG ((11.1 ± 0.2)

ΓJ-full-2 ≈ (8/9)[92.2491 Kev + 4.6298 Kev + (1/9){74.0769 Kev}] ≈ 93.43 Kev (16e)

Second Order Approximation via the GEM

Therefore,

Kev).

$$+ \quad \text{(1/9)}\\ \text{[2(a/2\pi)(10,042)(2\text{m}\_e)(\text{m}\_p/\text{m}\_l)](\text{q}\_c)^4] \tag{16a}$$

Thus,

$$\begin{array}{l} \Gamma\_{\text{f-full-1}} \approx (8/9)\{ (a\_\text{s} / 2\pi)(10,042)(2\text{m}\_\text{e})(776/3097)^3(1/81) + 10(10)(10)(10)^2(10)^2(10)^3(10)^1(10)^1(10)^2(10)^3(10)^1(10)^2(10)^3(10)^1(10)^2(10)^3(10)^1(10)^2(10)^3(10)^1(10)^2(10)^3(10)^3(10)^2(10)^3(10)^1(10)^2(10)^3(10)^3(10)^2(10)^3(10)^4) \} \\ \qquad \qquad 2(\text{a} \; / 2\pi)(10,042)(2\text{m}\_\text{e})(776/3097)^3(16/81)) \} \end{array}$$

The value of the strong coupling parameter at the J mass is given by

$$
\alpha\_s = 1.2[\ln(3097/50)]^{\cdot 1} = 0.2908 \tag{16b}
$$

Therefore,

$$
\Gamma\_{\text{l-full-1}} \approx (8/9)[92.2491 \text{ Kev} + 4.6298 \text{ Kev}] + (1/9)[74.0769 \text{ Kev}] \approx 94.35 \text{ Kev} \tag{16c}
$$

The value for ΓJ-full-1 obtained via the first approximation of the GEM is a match to experiment, as according to PDG (2009), the full width of the J via experiment is (93.2 ± 2.1) Kev . As well, the hadronic width alone via the first approximation of the GEM is a match to experiment (82.00 Kev via the GEM vs. (81.7 ± 0.5) Kev via experiment (PDG (2009)); the leptonic width via the first approximation of the GEM is 12.35 Kev, which is about 11% more than that reported by the PDG currently (11.10 ± 0.16) Kev (PDG (2009)).

The first approximation assumes that (8/9)ths of the original cc\* state undergo a point-like transition to an excited ss\* state, leaving (1/9)th of the original cc\* state to decay into leptons. A point-like transition is instantaneous, so it has no effect on the width of the original construction (i.e., the J). In terms of a *form factor*, f, a point-like transition is consistent with f = 1. As it is difficult to see how *any fraction* of the original cc\* state could "know" to make an instantaneous transition, leaving a remnant to do other things, we believe a second order approximation is in order. Our reasoning is simply that, logically, we feel that there simply *must* be some type of communication between the cc\* and ss\* states *before* the cc\* to ss\* transition takes place in order for the proper remnant to consistently remain to decay into leptons. Hence, we reason that f < 1 describes the cc\* to ss\* transition. Statistically, f = (1 – qs 2) = (8/9) is necessary to describe the hadronic width of the J. Since f is not appreciably different than 1, the leptonic width of the J, relative to the first order approximation, will be mitigated slightly. The second order FD for the J follows:

In Figure 6b f = (8/9) multiplies the entire array. Denoting the full width of the J in second order approximation by ΓJ-full-2, we find in accord with Figure 6b:

$$\begin{array}{l} \Gamma\_{\text{f-full-2}} \approx (8/9)[(\alpha\_s/2\pi)(10,042)(2\text{m}\_e)(\text{m}\_\wp/\text{m})^3(\text{q})^4 + 2(\alpha\_s/2\pi)(10,042)(2\text{m}\_e)(\text{m}\_\wp/\text{m}\_\wp)^3(\text{q})^4] \\ \quad / 2\pi)(10,042)(2\text{m}\_e)(\text{m}\_\wp/\text{m}\_\wp)(\text{m}\_\wp/\text{m}\_\wp/\text{m}\_\wp)^3(\text{q})^4] \end{array} \tag{16d}$$

Thus,

$$\begin{array}{c} \Gamma\_{\text{f-full-2}} \approx (8/9) [(\text{a}\_{\text{s}} / 2\text{r})(10,042)(2\text{m}\_{\text{e}})(776/3097)^{3}(1/81) + \\ 2(\text{a} / 2\text{r})(10,042)(2\text{m}\_{\text{e}})(776/3097)^{3}(1/81) + \\ (1/9) [2(\text{a} / 2\text{r})(10,042)(2\text{m}\_{\text{e}})(776/3097)^{3}(16/81)] \end{array}$$

Again, the value of the strong coupling parameter at the J mass is given by Eq. 16b, viz.,

$$\alpha\_\* = 1.2[\ln(3097/50)]^{\cdot 1} = 0.2908$$

238 Measurements in Quantum Mechanics

ΓJ-full-1 ≈ (8/9){(s /2*π*)(10,042)(2me)(776/3097)3(1/81) + 2( /2*π*)(10,042)(2me)(776/3097)3(1/81)} + (1/9){2(/2*π*)(10,042)(2me)(776/3097)3(16/81)}

ΓJ-full-1 ≈ (8/9){92.2491 Kev + 4.6298 Kev} + (1/9){74.0769 Kev} ≈ 94.35 Kev (16c)

The value for ΓJ-full-1 obtained via the first approximation of the GEM is a match to experiment, as according to PDG (2009), the full width of the J via experiment is (93.2 ± 2.1) Kev . As well, the hadronic width alone via the first approximation of the GEM is a match to experiment (82.00 Kev via the GEM vs. (81.7 ± 0.5) Kev via experiment (PDG (2009)); the leptonic width via the first approximation of the GEM is 12.35 Kev, which is about 11%

The first approximation assumes that (8/9)ths of the original cc\* state undergo a point-like transition to an excited ss\* state, leaving (1/9)th of the original cc\* state to decay into leptons. A point-like transition is instantaneous, so it has no effect on the width of the original construction (i.e., the J). In terms of a *form factor*, f, a point-like transition is consistent with f = 1. As it is difficult to see how *any fraction* of the original cc\* state could "know" to make an instantaneous transition, leaving a remnant to do other things, we believe a second order approximation is in order. Our reasoning is simply that, logically, we feel that there simply *must* be some type of communication between the cc\* and ss\* states *before* the cc\* to ss\* transition takes place in order for the proper remnant to consistently remain to decay into leptons. Hence, we reason that f < 1 describes the cc\* to ss\* transition. Statistically, f = (1 –

2) = (8/9) is necessary to describe the hadronic width of the J. Since f is not appreciably different than 1, the leptonic width of the J, relative to the first order approximation, will be

In Figure 6b f = (8/9) multiplies the entire array. Denoting the full width of the J in second

ΓJ-full-2 ≈ (8/9)[(s /2*π*)(10,042)(2me)(776/3097)3(1/81) + 2( /2*π*)(10,042)(2me)(776/3097)3(1/81) + (1/9){2(/2*π*)(10,042)(2me)(776/3097)3(16/81)}] Again, the value of the strong coupling parameter at the J mass is given by Eq. 16b, viz.,

s = 1.2[ln(3097/50)]-1 = 0.2908

/2*π*)(10,042)(2me)(mρ/mJ)3(qs)4

ΓJ-full-2 ≈ (8/9)[(s /2*π*)(10,042)(2me)(mρ/mJ)3(qs)4 + 2(

+ (1/9){2( /2*π*)(10,042)(2me)(mρ/mJ)3(qc)4}] (16d)

more than that reported by the PDG currently (11.10 ± 0.16) Kev (PDG (2009)).

mitigated slightly. The second order FD for the J follows:

order approximation by ΓJ-full-2, we find in accord with Figure 6b:

The value of the strong coupling parameter at the J mass is given by

Thus,

Therefore,

qs

Thus,

+ (1/9){2( /2*π*)(10,042)(2me)(mρ/mJ)3(qc)4} (16a)

s = 1.2[ln(3097/50)]-1 = 0.2908 (16b)

Fig. 6b. Feynman Diagram Array Characterizing the Formation and Decay of the J(3097) in Second Order Approximation via the GEM

Therefore,

$$
\Gamma\_{\text{f-full-2}} \approx (8/9) [92.2491 \text{ Kev} + 4.6298 \text{ Kev} + (1/9)[74.0769 \text{ Kev}]] \approx 93.43 \text{ Kev} \tag{16e}
$$

The full width of the J under second order approximation is thus nearly an exact match to experiment (93.4 Kev via the GEM vs. 93.2 Kev from PDG (2009)). The hadronic width of the J is unchanged from first to second approximation; so, it remains a match with experiment (82.0 Kev via the GEM vs. 81.7 Kev from PDG (2009)). As well, the leptonic width of the J via the GEM (11.4 Kev) is now only 2.7% higher than that reported by the PDG ((11.1 ± 0.2) Kev).

The Gluon Emission Model for Vector Meson Decay 241

Specific to the e+e- partial width (Γ*Y*-ee), the GEM obviously determines Γ*<sup>Y</sup>*-ee ≈ 1.30 Kev, while the PDG in the above-mentioned source (p. 119) states Γ*Y*-ee(PDG1) ≈ 1.34 Kev directly, but indirectly, in terms of its stated fractional branching ratio on p.119, a different value is inferred, viz., Γ*Y*-ee(PDG2) ≈ 1.29 Kev. From the latter we infer that according to the PDG (2008), the experimentally determined value for the e+e- partial width of the *Y*(1S) is given

Herein (i.e., the match between Eq. 18d and Eq. 18e) lies the source of a paradox that the hadronic width as given by the GEM (i.e., ~ 41 Kev) should be correct, though it is so seriously discrepant with that reported by the PDG (i.e., 50 Kev). The paradox unfolds as follows: In order to obtain the constant "B" in the general expression for s, once Λ was determined, the assumption was made that, since the *Y*(1S) exists well into the realm of

In Section 2 we inserted Γ*Y*-ee(PDG) = 1.31 Kev for the e+e- partial width, and for the hadronic partial width, we inserted the *GEM-theoretical width*, i.e., Γ*Y-*<sup>H</sup> ≈ 41 Kev. We then obtained

In turn, as "B" is a multiplier on the right hand sides of all width calculations via the GEM theory, and as all width calculations, as seen above, represent nearly exact matches with experiment in all cases except as to the hadronic width of the *Y*(1S), it is difficult to fathom

After a good number of years of pondering, it turns out that there is, actually, a very simple, and at the same time a very plausible solution to the paradox mentioned above, viz., we postulate an additional route for *Y*(1S) decay into hadrons, a route assumed not to have a high probability of occurrence for the J or the other vector mesons of mass less than that of the J. As the basis for the existence of the additional route available to the *Y*(1S), we point to the fact that there is roughly three times the energy spectrum available to the *Y*(1S) in its decay (9460 Mev worth) as compared to the next lightest vector meson, i.e., the J (3097 Mev worth). With three times the energy spectrum (as compared to the J) available to the *Y*(1S), we think it plausible that decays resulting in hadrons as products may be allowed to take place through the *bifurcation* of the gluon emitted from the resonance state (or more simply stated: via emission of two gluons), rather than what has heretofore been assumed in accord

the source of the disparity between Γ*Y-*H = 41 Kev and Γ*Y-*H(PDG) = 50 Kev.

s = B[ln(9460/50]-1 = (41/1.31)

which represents a match to the PDG's report from the same 2008 Meson Table of

by

a match to that of the GEM, i.e.,

/s = (e+e-

from which we solved for "B" to obtain, B = 1.2

asymptotic freedom,

the general relation,

as associated with the *Y*(1S).

Γ*<sup>Y</sup>*-L ≈ 3.90 Kev (18b)

Γ*Y*-L(PDG) = (4.03 ± 0.14) Kev (18c)

Γ*Y*-ee(PDG) = (1.31 ± 0.03) Kev (18d)

partial width)/(hadronic partial width)

Γ*<sup>Y</sup>*-ee ≈ 1.30 Kev (18e)

#### **The** *Y***(1S)**

Analogous to the J, the *Y*(1S), originally a bb\* construction, must transition to a cc\* excited state of the same mass as that of the bb\* state in order to decay into hadrons. Unlike the J, however, there is no reason to suspect that leptons emerge from the bb\* state. Hence, we assume that all types of *Y*(1S) decays ensue from the cc\* excited state. Corroborative evidence abounds in support of such assumption, as we shall see, so let us proceed with the viewing of the two FDs which depict the hadronic decay of the *Y*(1S) and the leptonic decay of the *Y*(1S), respectively:

Fig. 7a. Basic Feynman Diagram for *Y*(1S) Formation and Decay into Hadrons via the GEM

Fig. 7b. Basic Feynman Diagram for *Y*(1S) Formation and Decay into Leptons via the GEM

From Eq. 10 the hadronic width of the *Y*(1S), denoted by Γ*Y*-H, via the GEM theoretical structure is given by:

$$\Gamma\_{\rm Y\cdot H} \approx (\mathfrak{a}\_{\rm s}/2\pi)(10.042)(2\mathfrak{m}\_{\rm e})(\mathfrak{m}\_{\rm f}/\mathfrak{m})^{3}(\mathfrak{q}\_{\rm f})^{4} \approx (\mathfrak{a}\_{\rm s}/2\pi)(10.042)(2\mathfrak{m}\_{\rm e})(776/9460)^{3}(16/81) \quad \text{(17a)}$$
 where

$$
\alpha\_s = 1.2[\ln(9460/50)]^{\cdot 1} = 0.2289 \tag{17b}
$$

Hence,

$$
\Gamma\_{\text{Y-H}} \approx 40.76 \,\text{Kev} \tag{17c}
$$

The PDG in the 2008 Meson Table (PDG (2008), p.119) reports the corresponding figure as

$$
\Gamma\_{Y\text{-H}} \text{(PDG)} = 49.99 \text{ Kev} \tag{17d}
$$

a figure 23% higher than the GEM-theoretical result.

However, if we look at the leptonic width of the *Y*(1S), denoted by Γ*Y-*L , as derived via the GEM, we find from Eq. 11 (the right hand side of same multiplied by "3" to take into account muon and tauon pairs in accord with "e-μ-τ universality") that

$$\Gamma\_{\rm Y\perp} \approx \Im(\mathbf{a}/2\pi)(10.042)(2\text{m}\_{\uparrow})(\mathbf{m}\_{\uparrow}/\mathbf{m})^{\natural}(\mathbf{q}\_{\uparrow})^{\natural} \approx \Im(\mathbf{a}/2\pi)(10.042)(2\text{m}\_{\uparrow})(776/9460)^{\natural}(16/81) \tag{18a}$$

Hence,

240 Measurements in Quantum Mechanics

Analogous to the J, the *Y*(1S), originally a bb\* construction, must transition to a cc\* excited state of the same mass as that of the bb\* state in order to decay into hadrons. Unlike the J, however, there is no reason to suspect that leptons emerge from the bb\* state. Hence, we assume that all types of *Y*(1S) decays ensue from the cc\* excited state. Corroborative evidence abounds in support of such assumption, as we shall see, so let us proceed with the viewing of the two FDs which depict the hadronic decay of the *Y*(1S) and the leptonic decay

Fig. 7a. Basic Feynman Diagram for *Y*(1S) Formation and Decay into Hadrons via the GEM

Fig. 7b. Basic Feynman Diagram for *Y*(1S) Formation and Decay into Leptons via the GEM From Eq. 10 the hadronic width of the *Y*(1S), denoted by Γ*Y*-H, via the GEM theoretical

Γ*Y-*<sup>H</sup> ≈ (s /2*π*)(10,042)(2me)(mρ/mY)3(qc)4 ≈ (s /2*π*)(10,042)(2me)(776/9460)3(16/81) (17a)

The PDG in the 2008 Meson Table (PDG (2008), p.119) reports the corresponding figure as

However, if we look at the leptonic width of the *Y*(1S), denoted by Γ*Y-*L , as derived via the GEM, we find from Eq. 11 (the right hand side of same multiplied by "3" to take into

Γ*<sup>Y</sup>*-L ≈ 3(/2*π*)(10,042)(2me)(mρ/m*Y*)3(qc)4 ≈ 3(/2*π*)(10,042)(2me)(776/9460)3(16/81) (18a)

a figure 23% higher than the GEM-theoretical result.

account muon and tauon pairs in accord with "e-μ-τ universality") that

s = 1.2[ln(9460/50)]-1 = 0.2289 (17b)

Γ*Y-*<sup>H</sup> ≈ 40.76 Kev (17c)

Γ*Y-*H(PDG) = 49.99 Kev (17d)

**The** *Y***(1S)** 

of the *Y*(1S), respectively:

structure is given by:

where

Hence,

Hence,

$$
\Gamma\_{\rm Y\therefore L} \approx 3.90 \,\text{Kev} \tag{18b}
$$

which represents a match to the PDG's report from the same 2008 Meson Table of

$$
\Gamma\_{Y\text{-L}}\text{(PDG)} = (4.03 \pm 0.14)\,\text{Kev} \tag{18c}
$$

Specific to the e+e- partial width (Γ*Y*-ee), the GEM obviously determines Γ*<sup>Y</sup>*-ee ≈ 1.30 Kev, while the PDG in the above-mentioned source (p. 119) states Γ*Y*-ee(PDG1) ≈ 1.34 Kev directly, but indirectly, in terms of its stated fractional branching ratio on p.119, a different value is inferred, viz., Γ*Y*-ee(PDG2) ≈ 1.29 Kev. From the latter we infer that according to the PDG (2008), the experimentally determined value for the e+e- partial width of the *Y*(1S) is given by

$$
\Gamma\_{Y\text{-ee}}(\text{PDG}) = (1.31 \pm 0.03) \,\text{Kev} \tag{18d}
$$

a match to that of the GEM, i.e.,

$$
\Gamma\_{\text{Y-ee}} \approx \textbf{1.30} \,\text{Kev} \tag{18e}
$$

Herein (i.e., the match between Eq. 18d and Eq. 18e) lies the source of a paradox that the hadronic width as given by the GEM (i.e., ~ 41 Kev) should be correct, though it is so seriously discrepant with that reported by the PDG (i.e., 50 Kev). The paradox unfolds as follows: In order to obtain the constant "B" in the general expression for s, once Λ was determined, the assumption was made that, since the *Y*(1S) exists well into the realm of asymptotic freedom,

> /s = (e+e partial width)/(hadronic partial width)

as associated with the *Y*(1S).

In Section 2 we inserted Γ*Y*-ee(PDG) = 1.31 Kev for the e+e partial width, and for the hadronic partial width, we inserted the *GEM-theoretical width*, i.e., Γ*Y-*<sup>H</sup> ≈ 41 Kev. We then obtained the general relation,

$$\alpha\_{\sf s} = \sf B[\ln(9460/50)^\square = \alpha(41/1.31)]$$

from which we solved for "B" to obtain, B = 1.2

In turn, as "B" is a multiplier on the right hand sides of all width calculations via the GEM theory, and as all width calculations, as seen above, represent nearly exact matches with experiment in all cases except as to the hadronic width of the *Y*(1S), it is difficult to fathom the source of the disparity between Γ*Y-*H = 41 Kev and Γ*Y-*H(PDG) = 50 Kev.

After a good number of years of pondering, it turns out that there is, actually, a very simple, and at the same time a very plausible solution to the paradox mentioned above, viz., we postulate an additional route for *Y*(1S) decay into hadrons, a route assumed not to have a high probability of occurrence for the J or the other vector mesons of mass less than that of the J. As the basis for the existence of the additional route available to the *Y*(1S), we point to the fact that there is roughly three times the energy spectrum available to the *Y*(1S) in its decay (9460 Mev worth) as compared to the next lightest vector meson, i.e., the J (3097 Mev worth). With three times the energy spectrum (as compared to the J) available to the *Y*(1S), we think it plausible that decays resulting in hadrons as products may be allowed to take place through the *bifurcation* of the gluon emitted from the resonance state (or more simply stated: via emission of two gluons), rather than what has heretofore been assumed in accord

The Gluon Emission Model for Vector Meson Decay 243

BGRHD is a viable one. In fact, if we postulate that in addition to the BGRHD there is a companion route for leptons, i.e., a bifurcated gluon route for lepton decay (BGRLD), whose FD is identical to that of Figure 9, except that on the far right hand side of the diagram, each

respectively, where "i" and "j" denote lepton types and i = j is allowed, (3.90 Kev / 137.036)

To address the T-meson, thought to be a tt\* (where "t" represents the top quark) state of mass approximately 340000 Mev, but never "discovered" to date, is quite speculative on our part, but we think it important to do so because the GEM provides a perfectly logical reason as to why the T has yet to be "found", i.e., unequivocally shown to exist by experiment. Said reason is just the opposite of the prevailing view as to the "invisibility" of the T, which is: "the T doesn't last long enough for it to be found." In a sense such is true; after all, the bb\* of the *Y*(1S) transitions instantaneously to a cc\* state according to the GEM, but the mass of the original bb\* state is preserved in the resulting cc\* state, thus allowing for the "finding" of a resonance at the *Y*(1S) mass. Assuming the T to act in like manner to the *Y*(1S), the following

Fig. 9. Basic Feynman Diagram for T Formation and Decay into Hadrons via the GEM

Γ<sup>T</sup> ≈ (s /2*π*)(10,042)(2me)(mρ/mT)3(qb)4

(In Eq. 23b the constant "1.2" in the expression for s becomes11 "0.90" beyond 100000 Mev,

≈ (s /2*π*)(10,042)(2me)(776/340000)3(1/81) (23a)

s = 0.90[ln(340000/50)]-1 = 0.1020 (23b)

Γ<sup>T</sup> ≈ 0.024 ev (23c)

Γ*Y-*full(GEM 2010)

Γ*Y-*full(GEM 2010) → 54.02 Kev (22)

+, li - , lj

+, and lj - ",

"s" is replaced by "" and "h, h', h'', and h''' " are replaced by "li

= 0.03 Kev would be added to

i.e., the realization of an exact match to experiment.

**4. Speculations based upon the GEM** 

FD would apply as regards hadron production:

The hadronic width of the T, from Eq. 4 would be:

above, thus bringing

**The T-Meson** 

where

Hence,

and in Eq. 23a qb = -1/3.)

with Figure 7 a, in which a single gluon, ζ3a , converts to hadrons to mark the final stage of the decay process. Specifically, we propose that, in addition to the route as described immediately above, a route exists in which ζ3a bifurcates into two gluons, each of which then converts to hadrons. The FD associated with the proposed additional route is seen immediately below.

Fig. 8. Basic Feynman Diagram for Postulated Additional Route for *Y*(1S) Formation and Decay into Hadrons (h, h', h'', and h''') via the GEM

The additional route, which we denote as the "bifurcated gluon route for hadron decay" (BGRHD), effectively adds s times Γ*Y-*H, or (0.2289)(40.76 Kev) = 9.33 Kev to the GEMtheoretical width of the *Y*(1S). The reformulated situation regarding the *Y*(1S) may be summarized, therefore, as follows:

Denoting the partial width due to the BGRHD as Γ*Y-*BGH, we have

$$
\Gamma\_{\text{Y-BGH}} = \text{9.33 Kev} \tag{19}
$$

From above we have

Γ*Y-*H = 40.76 Kev

Also from above we have

Γ*Y-*L = 3.90 Kev

The net hadronic width of the *Y*(1S) as per the GEM would now be given by

Γ*Y-*(H+BGH)(GEM 2010) = 50.09 Kev (20)

which now represents a nearly perfect match to

Γ*Y-*H(PDG) = 49.99 Kev

In addition the full width of the *Y*(1S) as per the GEM would now be given by

$$
\Gamma\_{Y\text{-full}}(\text{GEM 2010}) = 53.99 \text{ Kev} \tag{21}
$$

which also represents a nearly perfect match to Γ*Y-*full (PDG) = (54.02 ± 1.25) Kev

With the addition of the BGRHD the calculation of "B" in the expression for s is uncompromised, while at the same time the major discrepancy between the hadronic width of the *Y*(1S) as determined via the GEM versus via the methods engaged by the PDG is completely removed. For that reason we believe the postulate as to the addition of the BGRHD is a viable one. In fact, if we postulate that in addition to the BGRHD there is a companion route for leptons, i.e., a bifurcated gluon route for lepton decay (BGRLD), whose FD is identical to that of Figure 9, except that on the far right hand side of the diagram, each "s" is replaced by "" and "h, h', h'', and h''' " are replaced by "li +, li - , lj +, and lj - ", respectively, where "i" and "j" denote lepton types and i = j is allowed, (3.90 Kev / 137.036) = 0.03 Kev would be added to

$$\Gamma\_{Y\text{-full}}(\text{GEM 2010})$$

above, thus bringing

242 Measurements in Quantum Mechanics

with Figure 7 a, in which a single gluon, ζ3a , converts to hadrons to mark the final stage of the decay process. Specifically, we propose that, in addition to the route as described immediately above, a route exists in which ζ3a bifurcates into two gluons, each of which then converts to hadrons. The FD associated with the proposed additional route is seen

Fig. 8. Basic Feynman Diagram for Postulated Additional Route for *Y*(1S) Formation and

The additional route, which we denote as the "bifurcated gluon route for hadron decay" (BGRHD), effectively adds s times Γ*Y-*H, or (0.2289)(40.76 Kev) = 9.33 Kev to the GEMtheoretical width of the *Y*(1S). The reformulated situation regarding the *Y*(1S) may be

Γ*Y-*H = 40.76 Kev

Γ*Y-*L = 3.90 Kev

Γ*Y-*H(PDG) = 49.99 Kev

With the addition of the BGRHD the calculation of "B" in the expression for s is uncompromised, while at the same time the major discrepancy between the hadronic width of the *Y*(1S) as determined via the GEM versus via the methods engaged by the PDG is completely removed. For that reason we believe the postulate as to the addition of the

Γ*Y-*BGH = 9.33 Kev (19)

Γ*Y-*(H+BGH)(GEM 2010) = 50.09 Kev (20)

Γ*Y-*full(GEM 2010) = 53.99 Kev (21)

Decay into Hadrons (h, h', h'', and h''') via the GEM

which now represents a nearly perfect match to

Denoting the partial width due to the BGRHD as Γ*Y-*BGH, we have

The net hadronic width of the *Y*(1S) as per the GEM would now be given by

In addition the full width of the *Y*(1S) as per the GEM would now be given by

which also represents a nearly perfect match to Γ*Y-*full (PDG) = (54.02 ± 1.25) Kev

summarized, therefore, as follows:

From above we have

Also from above we have

immediately below.

$$
\Gamma\_{\text{Y-full}}(\text{GEM 2010}) \to \text{54.02 Kev} \tag{22}
$$

i.e., the realization of an exact match to experiment.

#### **4. Speculations based upon the GEM**

#### **The T-Meson**

To address the T-meson, thought to be a tt\* (where "t" represents the top quark) state of mass approximately 340000 Mev, but never "discovered" to date, is quite speculative on our part, but we think it important to do so because the GEM provides a perfectly logical reason as to why the T has yet to be "found", i.e., unequivocally shown to exist by experiment. Said reason is just the opposite of the prevailing view as to the "invisibility" of the T, which is: "the T doesn't last long enough for it to be found." In a sense such is true; after all, the bb\* of the *Y*(1S) transitions instantaneously to a cc\* state according to the GEM, but the mass of the original bb\* state is preserved in the resulting cc\* state, thus allowing for the "finding" of a resonance at the *Y*(1S) mass. Assuming the T to act in like manner to the *Y*(1S), the following FD would apply as regards hadron production:

Fig. 9. Basic Feynman Diagram for T Formation and Decay into Hadrons via the GEM

The hadronic width of the T, from Eq. 4 would be:

$$\begin{array}{l} \Gamma\_{\mathrm{T}} \approx (\mathrm{a\_{s}} \,/\, 2\pi) \text{(10.042)} \text{(2m\_{e})} \text{(m\_{p}/m\_{r})} \mathrm{^{3}} \mathrm{(q\_{b})}^{4} \\ \approx (\mathrm{a\_{s}} \,/\, 2\pi) \text{(10.042)} \text{(2m\_{e})} \text{(776/340000)} \text{(1/81)} \end{array} \tag{23a}$$

where

$$
\alpha\_s = 0.90 [\ln(340000/50)]^{\frac{1}{1}} = 0.1020\tag{23b}
$$

(In Eq. 23b the constant "1.2" in the expression for s becomes11 "0.90" beyond 100000 Mev, and in Eq. 23a qb = -1/3.) Hence,

$$
\Gamma\_{\rm T} \approx 0.024 \text{ ev} \tag{23c}
$$

The Gluon Emission Model for Vector Meson Decay 245

**Meson Mass (Mev) Γee(GEM) Γee(PDG) # of Colors** 

operative or an even mix of two colors and one color taking part in the decay), n = 4 associated with the *Y*(5S) (two colors operative), and n = 2 associated with the *Y*(6S) (one color operative). At the present juncture, perhaps the best guess as how to handle the results associated with the *Y*(3S) and *Y*(4S) would be to speculate that the GEM description should be due to a nearly even mix of two and three contributing colors in one case, and a nearly even mix of one and two contributing colors in the other case, rather than making the claim that the GEM has shown that the quark colors become fragmented beyond 10 000 Mev, especially when one notes that all results associated with integer color contributions in Chart 2 represent statistical matches with experiment. Note, as well, that the progressive color disengagement behavior with increasing mass is preserved if only integer color

One basic reality demonstrated by the GEM is the prime role that the electromagnetic interaction plays in vector meson formation and decay. The spin-flip responsible for all spin one mesons takes place via the electromagnetic interaction. The transitioning of the fourmomentum from cc\* states to ss\* in the ψ(NS) decays, as evidenced via the mathematical form of the form factor, f1, is seen to take place via the electromagnetic interaction, as is the case regarding the analogous transition from bb\* to cc\* in the *Y*(NS) decays. A second reality demonstrated by the GEM represents a radical departure from current assumptions about the structures of vector mesons in general … but especially about the structure of the K\* …

where Qi represents a quark of flavor, "i", Qi\* represents the associated anti-quark, and "n" represents the number of flavors operative at the energy scale of the relevant meson's rest energy. Thus, for example, the neutral K\* does *not* comprise a ds\* or an sd\*, and the charged K\* does *not* comprise a us\* or an su\*. Rather, there is a general "K\* construction" given by Eq. 14, the decay of which features a "favored energy" of 892 Mev resulting in a net charge of ± 1 amongst its decay products … *and* another "favored energy" at 896 Mev resulting in no net charge amongst its decay products. In the literature similar considerations apply to the various D\* and B\* states13. In other words the GEM illustrates that vector mesons are not actually "unstable particles" which form at collision sites, but rather are manifestations of a "quark sea" as part of the construction of what we call "the vacuum" … much analogous to Dirac's idea of the "electron sea" of old. Just as the electrons in the Dirac Sea were thought to be excited via the electromagnetic interaction, we see from the above that one may surely think of the formation of vector mesons as an electromagnetic excitation of relevant quarks in a "quark sea" … where a given quark is promoted to a positive energy state by a virtual gluon, which, unlike in electron/positron production, where the electron flies away from its

χv = (1/√n)[Σi=1n (QiQi\*)] (24)

**Y(2S) 10023 0.624 0.612 ± 0.11 3 Y(3S) 10355 0.471 0.443 ± 0.008 2½ Y(4S) 10579 0.266 0.272 ± 0.029 1½ Y(5S) 10860 0.33 0.31 ± 0.07 2 Y(6S) 11019 0.157 0.13 ± 0.03 1** 

Chart 2. Color Participation in Lepton Production in the Y-Series

contributions are considered.

**Vector Mesons as Vacuum Excitations** 

viz., that all vector mesons are represented by

**Operative** 

Thus, we see that, contrary to the "convenient explanation" as to why the T has not so far been observed, the T lives for a *very long time* (about 6 ps)! It's just that its width to mass ratio makes it impossible right now for the experimental apparatus to pick up such a narrow signal amongst the "noise" inherent in the energy background needed to produce the T.

#### **Color-by-Color Disengagement from Lepton Production**

We must go a bit beyond the scope of the material thus far presented in order to discuss the phenomenon of color-by-color disengagement from lepton production, as we must now make reference to the excited states of the Ψ-series and *Y*-series mesons, designated respectively as Ψ(NS) and *Y*(NS), where N > 1 designates an excited state. It turns out that as regards lepton decay, partial widths of the above objects, when N = 2, the associated form factors (fi) which provide for excellent agreement with experiment are as follows12:

For the Ψ(2S) f1 = 1 – qs 2 = 8/9 (as was found above for the J = Ψ(1S)), and for the *Y*(2S) f2 = 1 – qc 2 = 5/9 (analogous to the Ψ(2S)). We will designate the form factor for the *Y*(1S) (as per above) as f3 = 1. What is highly interesting, as it turns out, is that in terms of the above form factors, the leptonic partial widths of many of the Ψ(NS) and *Y*(NS) states are excellently described by the GEM if the latitude exists to multiply the resulting GEM-width formulas involving the appropriate form factors by "(n/6)", where "n" is an integer, the interpretation being that as N increases, the number of quark colors participating in lepton decay decreases … effectively by "half colors" at a time. We illustrate the quark color disengagement phenomenon by reproducing the results from the reference12 associated with footnote # 12 below:


Chart 1. Color Participation in Lepton Production in the Ψ-Series

In Chart 1 all electron/positron partial widths are expressed in Kev. Note that a near-match with experiment occurs for the J(3097), as per Section C above, assuming three colors are operative in electron/positron decay (n = 6). A near-match with experiment results for the Ψ(2S) assuming two colors are operative in said decay (n = 4), and statistical matches with experiment are evident if we assume only one color participates in electron/positron decay (n = 2) as associated with the Ψ(3S), Ψ(4S), and Ψ(5S). As to the Ψ(NS) objects, then, the GEM provides an excellent match to experiment (from the PDG's 2009 "Meson Table") if we assume that sequentially progressive disengagement … color-by-color … of quark colors manifests in lepton decay.

The situation is similar to, but slightly different than the above, as regards the *Y*(NS) series, as seen in Chart 2 below.

Again, all partial widths are expressed in Kev, and, again, we see excellent agreement with experiment if we assume n = 6 associated with the *Y*(2S) (three colors operative), n = 5 associated with the *Y*(3S) (either "2½ colors" operative or an even mix of three colors and two colors taking part in the decay), n = 3 associated with the *Y*(4S) (either "1½ colors"

244 Measurements in Quantum Mechanics

Thus, we see that, contrary to the "convenient explanation" as to why the T has not so far been observed, the T lives for a *very long time* (about 6 ps)! It's just that its width to mass ratio makes it impossible right now for the experimental apparatus to pick up such a narrow signal amongst the "noise" inherent in the energy background needed to produce the T.

We must go a bit beyond the scope of the material thus far presented in order to discuss the phenomenon of color-by-color disengagement from lepton production, as we must now make reference to the excited states of the Ψ-series and *Y*-series mesons, designated respectively as Ψ(NS) and *Y*(NS), where N > 1 designates an excited state. It turns out that as regards lepton decay, partial widths of the above objects, when N = 2, the associated form

2 = 5/9 (analogous to the Ψ(2S)). We will designate the form factor for the *Y*(1S) (as per above) as f3 = 1. What is highly interesting, as it turns out, is that in terms of the above form factors, the leptonic partial widths of many of the Ψ(NS) and *Y*(NS) states are excellently described by the GEM if the latitude exists to multiply the resulting GEM-width formulas involving the appropriate form factors by "(n/6)", where "n" is an integer, the interpretation being that as N increases, the number of quark colors participating in lepton decay decreases … effectively by "half colors" at a time. We illustrate the quark color disengagement phenomenon by reproducing the results from the reference12 associated with

**Meson Mass (Mev) Γee(GEM) Γee(PDG) # of Colors** 

**Ψ(1S) 3097 5.72 5.55 ± 0.16 3 Ψ(2S) 3686 2.26 2.36 ± 0.04 2 Ψ(3S) 4039 0.86 0.86 ± 0.07 1 Ψ(4S) 4153 0.79 0.83 ± 0.07 1 Ψ(5S) 4421 0.65 0.58 ± 0.07 1** 

In Chart 1 all electron/positron partial widths are expressed in Kev. Note that a near-match with experiment occurs for the J(3097), as per Section C above, assuming three colors are operative in electron/positron decay (n = 6). A near-match with experiment results for the Ψ(2S) assuming two colors are operative in said decay (n = 4), and statistical matches with experiment are evident if we assume only one color participates in electron/positron decay (n = 2) as associated with the Ψ(3S), Ψ(4S), and Ψ(5S). As to the Ψ(NS) objects, then, the GEM provides an excellent match to experiment (from the PDG's 2009 "Meson Table") if we assume that sequentially progressive disengagement … color-by-color … of quark colors

The situation is similar to, but slightly different than the above, as regards the *Y*(NS) series,

Again, all partial widths are expressed in Kev, and, again, we see excellent agreement with experiment if we assume n = 6 associated with the *Y*(2S) (three colors operative), n = 5 associated with the *Y*(3S) (either "2½ colors" operative or an even mix of three colors and two colors taking part in the decay), n = 3 associated with the *Y*(4S) (either "1½ colors"

Chart 1. Color Participation in Lepton Production in the Ψ-Series

2 = 8/9 (as was found above for the J = Ψ(1S)), and for the *Y*(2S) f2 =

**Operative** 

factors (fi) which provide for excellent agreement with experiment are as follows12:

**Color-by-Color Disengagement from Lepton Production** 

For the Ψ(2S) f1 = 1 – qs

footnote # 12 below:

manifests in lepton decay.

as seen in Chart 2 below.

1 – qc


Chart 2. Color Participation in Lepton Production in the Y-Series

operative or an even mix of two colors and one color taking part in the decay), n = 4 associated with the *Y*(5S) (two colors operative), and n = 2 associated with the *Y*(6S) (one color operative). At the present juncture, perhaps the best guess as how to handle the results associated with the *Y*(3S) and *Y*(4S) would be to speculate that the GEM description should be due to a nearly even mix of two and three contributing colors in one case, and a nearly even mix of one and two contributing colors in the other case, rather than making the claim that the GEM has shown that the quark colors become fragmented beyond 10 000 Mev, especially when one notes that all results associated with integer color contributions in Chart 2 represent statistical matches with experiment. Note, as well, that the progressive color disengagement behavior with increasing mass is preserved if only integer color contributions are considered.

#### **Vector Mesons as Vacuum Excitations**

One basic reality demonstrated by the GEM is the prime role that the electromagnetic interaction plays in vector meson formation and decay. The spin-flip responsible for all spin one mesons takes place via the electromagnetic interaction. The transitioning of the fourmomentum from cc\* states to ss\* in the ψ(NS) decays, as evidenced via the mathematical form of the form factor, f1, is seen to take place via the electromagnetic interaction, as is the case regarding the analogous transition from bb\* to cc\* in the *Y*(NS) decays. A second reality demonstrated by the GEM represents a radical departure from current assumptions about the structures of vector mesons in general … but especially about the structure of the K\* … viz., that all vector mesons are represented by

$$\chi\_{\mathbf{v}^\mathbf} = (\mathbf{1}/\sqrt{\mathbf{n}})[\Sigma\_{\mathbf{i}-\mathbf{1}^\mathbf{n}}(\mathbf{Q}\mathbf{Q}\mathbf{i}^\mathbf{\*})] \tag{24}$$

where Qi represents a quark of flavor, "i", Qi\* represents the associated anti-quark, and "n" represents the number of flavors operative at the energy scale of the relevant meson's rest energy. Thus, for example, the neutral K\* does *not* comprise a ds\* or an sd\*, and the charged K\* does *not* comprise a us\* or an su\*. Rather, there is a general "K\* construction" given by Eq. 14, the decay of which features a "favored energy" of 892 Mev resulting in a net charge of ± 1 amongst its decay products … *and* another "favored energy" at 896 Mev resulting in no net charge amongst its decay products. In the literature similar considerations apply to the various D\* and B\* states13. In other words the GEM illustrates that vector mesons are not actually "unstable particles" which form at collision sites, but rather are manifestations of a "quark sea" as part of the construction of what we call "the vacuum" … much analogous to Dirac's idea of the "electron sea" of old. Just as the electrons in the Dirac Sea were thought to be excited via the electromagnetic interaction, we see from the above that one may surely think of the formation of vector mesons as an electromagnetic excitation of relevant quarks in a "quark sea" … where a given quark is promoted to a positive energy state by a virtual gluon, which, unlike in electron/positron production, where the electron flies away from its

The Gluon Emission Model for Vector Meson Decay 247

towards any reasonable modifications that might remove the above-mentioned disparity, with the satisfactory result being the assumption of an additional decay route for the *Y*(1S),

Unlike the theoretical structures prevalent in the literature that one encounters as to determining the width of the vector mesons, the GEM theory is about as simple as it gets: One fundamental process is posited for the formation and decay of any spin one meson, i.e., a quark spin-flip; the gluon absorption cross-section for said process is then integrated over energy, and from there, the Feynman Diagram resulting in hadron or lepton pairs is then

Form factors associated with the ψ(1S) and *Y*(2S), calculated directly from relevant

of the strange quark and qc = 2/3 represents the charge of the charm quark. The form factors represent the fraction of the originally produced quark/anti-quark (QQ\*) state which makes a transition to a QQ\* state of the next lowest mass … ss\* in the case of the ψ-series mesons and cc\* in the case of the *Y*-series mesons … and thus figure prominently into the calculation of the hadronic and leptonic widths of a given meson via the constructs of the Gluon Emission Model. We have seen that f1 = (8/9) is representative of all ψ-states, if, and only if, it is assumed that one quark color (in the case of the ψ(2S)) or two quark colors (in all other cases) become disengaged from lepton production. A similar set of circumstances is observed as to the *Y*-series mesons, such illustrating that all three quark colors are functional in lepton production in *Y*(2S) decay, fewer than three functional in *Y*(3S) and *Y*(4S) decay, with likely only one color functioning in *Y*(5S) and *Y*(6S) decay. For each meson series, then, lepton decay is characterized by the phenomenon of sequential disengagement

Finally, we have seen that the GEM suggests, contrary to a rigid interpretation of the Standard Model, in which vector mesons are treated as unstable particles, that vector mesons are quite realizable as electromagnetic vacuum excitations of a constituent "quark sea", analogous to the "Dirac Sea" of electrons of old. Specifically, the GEM construct yields agreement with experiment only if it is assumed that vector mesons are represented as linear combinations of quark spin-flip excitation possibilities. The K\*(892) is a case in point, but, further, there appears to be no hope for reliable width calculations of any ψ-series mesons if such elements of said series are represented solely as cc\* objects. A like statement,

[2] R. H. Dalitz (1977), "Glossary for New Particles and New Quantum Numbers",

[6] M. Schmelling (1996), arxiv.org/abs/hep-ex/9701002, "Status of the Strong Coupling

[3] S. Gasiorowicz and J. L. Rosher, *American Journal of Physics* 49, pp. 954 & ff. (1981)).

*Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences*,

Constant", *Plenary Talk given at the XXVIII International Conference on High Energy* 

of quark color from lepton production as a function of increasing mass.

of course, holds for the elements of the *Y*-series as immutably bb\* "particles".

[1] F. E. Close (1979), *An Introduction to Quarks and Partons*, Academic Press.

Vol. 355 (1683), pp. 601 & ff.

[4] Brian Greene (2004), *The Fabric of the Cosmos,* Knopf*.*  [5] E. Merzbacher (1970), *Quantum Mechanics*, Wiley.

*Physics*, Warsaw, Poland, July, 1996.

2) = (5/9) in the case of the *Y*-series mesons, where qs = -1/3 represents the charge

2) = (8/9) in the case of the ψ-series mesons and

i.e., the bifurcated gluon route.

experimental data, are given by f1 = (1-qs

calculated.

f2 = (1-qc

**6. References** 

"vacuum hole", produces the circumstance that a tightly bound quark/anti-quark pair becomes virtually extant, in which vicinity the spin-flip of one of the quarks occurs, thus producing what we call "a vector meson". Whether one thinks of the vacuum in accord with Brian Greene as "the fabric of the cosmos", or, more conventionally, in accord with the pioneers of Quantum Electrodynamics (QED), as "a sea of leptons" (i.e., electrons, muons, tauons, and six flavors of quarks), the vacuum is certainly a "something" as opposed to "nothing at all". If one treats vector mesons as vacuum excitations, i.e., "an excitation of the sea of leptons" and follows the rules laid down by the pioneers of QED (i.e., the calculation of the relevant Feynman Diagrams) with the one exception of replacing by s (it derived solely via the GEM) in the hadronic partial width calculations, one acquires fine agreement with experiment. Hence, we believe, vector mesons thought of as vacuum excitations makes perfect sense.

#### **5. Summary**

The Gluon Emission Model has been shown to serve very nicely as a basis for calculations of not only the widths of the ρ meson, the meson, the K\*(892), the J meson, and the *Y* meson, but also for the determination of the strong coupling parameter, s , over the entire range of energy over which the above objects exist. We have seen that the GEM has built into its framework two precepts of prime importance for the carrying out of the above types of calculations: (1) the specification of a quark spin-flip matrix element as the central determinant of a vector meson resonance and (2) the virtual photon and the gluon as two aspects of the same entity, viz., the four-momentum propagator. The prime significance of (1) is that the square of the quark spin-flip matrix elements in vector meson width calculations are proportional to qi 4, where qi represents the magnitude of the charge of quark type "i". The significance of (2) is that the virtual photon and the gluon essentially obtain their identities from what the vertices of origin and termination are in the relevant Feynman Diagram. The ramifications of (1) are that, as (2/3)4 is 16 times (1/3)4, it is quite easy to determine that the cc\* (charm – anti-charm) structure of the J(3097) must transmute to an ss\* (strange – anti-strange) in nearly a point-like manner, such that it is the ss\* structure that undergoes the spin-flip at the J(3097) resonance. Similarly, the *Y*(1S) must transmute in point-like manner from its original bb\* (bottom – anti-bottom) structure to a cc\* structure before decaying. The ramification of (2) is that the leptonic width to hadronic width ratio associated with the same basic decaying structure must be in the ratio of to s. We saw that the GEM predicts the hadronic width of the *Y*(1S) to be ~ 41 Kev, assuming that the *Y*(1S), as lower energy mesons do, decays solely via the emission of a single gluon, whereas the figure for same as stated in the 2008 Meson Table from the Particle Data Group (PDG) is ~ 50 Kev. The discrepancy noted above (23%) is seen to be extremely important, because, if we were to assume that the GEM was in error by such amount, it turns out that all other GEM calculations, currently essentially exactly on the mark as to the ρ, the , the K\*(892), the J, and s , would have to be rendered as 23% too large by bringing the GEM's determination of the *Y*(1S) in line with the PDG's determination of same through adjustment of the GEM's determination of s . Hence, in order to make the GEM as currently constructed fit the PDG as to the hadronic width of the Y(1S), all other GEM calculations would be discrepant by the same amount, i.e., 23%, at each diverse point of the energy spectrum where the GEM has been successfully applied. Clearly, then, what needed to be addressed are the details in the GEM's determination of the width of the Y(1S), with an eye 246 Measurements in Quantum Mechanics

"vacuum hole", produces the circumstance that a tightly bound quark/anti-quark pair becomes virtually extant, in which vicinity the spin-flip of one of the quarks occurs, thus producing what we call "a vector meson". Whether one thinks of the vacuum in accord with Brian Greene as "the fabric of the cosmos", or, more conventionally, in accord with the pioneers of Quantum Electrodynamics (QED), as "a sea of leptons" (i.e., electrons, muons, tauons, and six flavors of quarks), the vacuum is certainly a "something" as opposed to "nothing at all". If one treats vector mesons as vacuum excitations, i.e., "an excitation of the sea of leptons" and follows the rules laid down by the pioneers of QED (i.e., the calculation of the relevant Feynman Diagrams) with the one exception of replacing by s (it derived solely via the GEM) in the hadronic partial width calculations, one acquires fine agreement with experiment. Hence, we believe, vector mesons thought of as vacuum excitations makes

The Gluon Emission Model has been shown to serve very nicely as a basis for calculations of not only the widths of the ρ meson, the meson, the K\*(892), the J meson, and the *Y* meson, but also for the determination of the strong coupling parameter, s , over the entire range of energy over which the above objects exist. We have seen that the GEM has built into its framework two precepts of prime importance for the carrying out of the above types of calculations: (1) the specification of a quark spin-flip matrix element as the central determinant of a vector meson resonance and (2) the virtual photon and the gluon as two aspects of the same entity, viz., the four-momentum propagator. The prime significance of (1) is that the square of the quark spin-flip matrix elements in vector meson width

quark type "i". The significance of (2) is that the virtual photon and the gluon essentially obtain their identities from what the vertices of origin and termination are in the relevant Feynman Diagram. The ramifications of (1) are that, as (2/3)4 is 16 times (1/3)4, it is quite easy to determine that the cc\* (charm – anti-charm) structure of the J(3097) must transmute to an ss\* (strange – anti-strange) in nearly a point-like manner, such that it is the ss\* structure that undergoes the spin-flip at the J(3097) resonance. Similarly, the *Y*(1S) must transmute in point-like manner from its original bb\* (bottom – anti-bottom) structure to a cc\* structure before decaying. The ramification of (2) is that the leptonic width to hadronic width ratio associated with the same basic decaying structure must be in the ratio of to s. We saw that the GEM predicts the hadronic width of the *Y*(1S) to be ~ 41 Kev, assuming that the *Y*(1S), as lower energy mesons do, decays solely via the emission of a single gluon, whereas the figure for same as stated in the 2008 Meson Table from the Particle Data Group (PDG) is ~ 50 Kev. The discrepancy noted above (23%) is seen to be extremely important, because, if we were to assume that the GEM was in error by such amount, it turns out that all other GEM calculations, currently essentially exactly on the mark as to the ρ, the , the K\*(892), the J, and s , would have to be rendered as 23% too large by bringing the GEM's determination of the *Y*(1S) in line with the PDG's determination of same through adjustment of the GEM's determination of s . Hence, in order to make the GEM as currently constructed fit the PDG as to the hadronic width of the Y(1S), all other GEM calculations would be discrepant by the same amount, i.e., 23%, at each diverse point of the energy spectrum where the GEM has been successfully applied. Clearly, then, what needed to be addressed are the details in the GEM's determination of the width of the Y(1S), with an eye

4, where qi represents the magnitude of the charge of

perfect sense.

**5. Summary** 

calculations are proportional to qi

towards any reasonable modifications that might remove the above-mentioned disparity, with the satisfactory result being the assumption of an additional decay route for the *Y*(1S), i.e., the bifurcated gluon route.

Unlike the theoretical structures prevalent in the literature that one encounters as to determining the width of the vector mesons, the GEM theory is about as simple as it gets: One fundamental process is posited for the formation and decay of any spin one meson, i.e., a quark spin-flip; the gluon absorption cross-section for said process is then integrated over energy, and from there, the Feynman Diagram resulting in hadron or lepton pairs is then calculated.

Form factors associated with the ψ(1S) and *Y*(2S), calculated directly from relevant experimental data, are given by f1 = (1-qs 2) = (8/9) in the case of the ψ-series mesons and f2 = (1-qc 2) = (5/9) in the case of the *Y*-series mesons, where qs = -1/3 represents the charge of the strange quark and qc = 2/3 represents the charge of the charm quark. The form factors represent the fraction of the originally produced quark/anti-quark (QQ\*) state which makes a transition to a QQ\* state of the next lowest mass … ss\* in the case of the ψ-series mesons and cc\* in the case of the *Y*-series mesons … and thus figure prominently into the calculation of the hadronic and leptonic widths of a given meson via the constructs of the Gluon Emission Model. We have seen that f1 = (8/9) is representative of all ψ-states, if, and only if, it is assumed that one quark color (in the case of the ψ(2S)) or two quark colors (in all other cases) become disengaged from lepton production. A similar set of circumstances is observed as to the *Y*-series mesons, such illustrating that all three quark colors are functional in lepton production in *Y*(2S) decay, fewer than three functional in *Y*(3S) and *Y*(4S) decay, with likely only one color functioning in *Y*(5S) and *Y*(6S) decay. For each meson series, then, lepton decay is characterized by the phenomenon of sequential disengagement of quark color from lepton production as a function of increasing mass.

Finally, we have seen that the GEM suggests, contrary to a rigid interpretation of the Standard Model, in which vector mesons are treated as unstable particles, that vector mesons are quite realizable as electromagnetic vacuum excitations of a constituent "quark sea", analogous to the "Dirac Sea" of electrons of old. Specifically, the GEM construct yields agreement with experiment only if it is assumed that vector mesons are represented as linear combinations of quark spin-flip excitation possibilities. The K\*(892) is a case in point, but, further, there appears to be no hope for reliable width calculations of any ψ-series mesons if such elements of said series are represented solely as cc\* objects. A like statement, of course, holds for the elements of the *Y*-series as immutably bb\* "particles".

#### **6. References**


**10** 

*China* 

Xian-Fang Yue *Jining University* 

**Vector Correlations in Collision of Atom and Diatomic Molecule** 

The use of crossed molecule beam (CMB) and polarized laser techniques has allowed the precisely measurements of the vector correlations in atom and diatomic molecule collision reactions. There are a variety of experimental studies aiming at exploring the vector correlations underlying a collision reaction.[1-15] To name a few are those of Zare and coworkers, [1-3] those of Herschbach and co-workers [4-5], those of Fano and co-workers [6], as well as those of Han and co-workers [7-9]. These experimental investigations impelled the development of theoretical methods to simulate the measured profiles and to explore the unmeasured ones. Generally, theoretical explorations on vector correlations of collision reactions fall into two categories: one is the reagent *k*-*j, j*-*k*-*k'* vector correlations, and the other is the product *k*-*j', k*-*k'*-*j'* vector correlations. Where *k*/*k'* and *j*/*j'* denotes the relative velocity of reagent/product and reagent/product rotational angular momentum in the centre of mass (CM) frame. As reported in many previous papers, both of the quantum scattering and the quasi-classical trajectory (QCT) calculations can be used to investigate the vector correlations. In this chapter, we report the product *k*-*j', k*-*k'*-*j'* vector correlations of

the N(2D) + H2(v, j) → NH(v', j') + H collision reactions with the QCT method.

experiments with mass spectrometric detection technique.

The N(2D) + H2 → NH + H collision reaction plays an important role in the chemistry of nitrogen containing fuels and of nitrogen in the atmosphere [16]. It has attracted many investigations from not only experimental viewpoints [17-25], but also theoretical viewpoints [26-37]. Experimentally, Suzuki et al. [17] measured the rate constants for this reaction by employing a pulse radiolysis-resonance absorption technique at temperatures between 213 and 300 K. They found that the temperature dependence of the rate constants exhibits an Arrhenius behavior. Umemoto and coworkers [18-20] measured the vibrational and rotational state distributions of the nascent NH and ND molecules formed in the N(2D) + H2 and N(2D) + D2 reactions. In their experiments, the N(2D) atoms were generated by two-photon dissociation of NO, while the nascent NH and ND molecules were detected by laser-induced uorescence (LIF) technique. They found that the nascent vibrational distributions have NH(v=1)/NH(v=0) = 0.8±0.1 and ND(v=1)/ND(v=0) = 1.0±0.1, and that the rotational populations of these vibrational states are broad and hot. More recently, Casavecchia et al. [21-25] carried out a series of experimental and theoretical studies on the N(2D) + H2 and N(2D) + D2 reactions. They have measured the angular and velocity distributions of the nascent NH and ND products in the title reactions under CMB

**1. Introduction** 

