Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction

It is shown in the theory of usual electron–ion plasma that weak turbulence of the plasma can be of two types (see, for example, [1]). Weak turbulence of the first type is caused by scattering of waves by plasma particles. Weak turbulence of the second type is due to decay, fusion, and scattering of waves off one another, which occur without energy exchange between particles and waves. In a number of publications [2, 3, 4, 5, 6, 7], the kinetic equations for the simplest collective excitations (Langmuir plasmons) of the electron–ion plasma, which describe elastic scattering of plasmons off one another, were constructed and analyzed in detail.
At present, a certain interest is shown in the construction of kinetic description of the new fundamental state of matter (quark–gluon plasma that consists of asymptotically free quarks, antiquarks, and gluons; see, for example, review [8]), which is probably formed during ultrarelativistic heavy ion collisions. It is shown that in the high-temperature limit, quark–gluon plasma is successfully described by the effective perturbation theory by Braaten and Pisarski [9] reformulated in the terms of the Blaizot-Iancu kinetic equations [10]. A gluon plasma (here, we will disregard for simplicity the presence of quarks and antiquarks) can be represented as a combination of two subsystems, viz., the subsystem of hard thermal gluons and the subsystem of soft plasma excitations, which exchange energy with each other. In a high-temperature gluon plasma, as well as in the usual electron–ion plasma, two types of collective plasma excitations exist, viz., transverse-polarized and longitudinal-polarized excitations (plasmons). In the absence of external chromomagnetic and chromoelectric fields, the color matrix of the number density of collective gluon excitations is diagonal; therefore, these excitations should be treated as colorless.
In [11], a kinetic description of the nonlinear interaction of colorless and color plasmons in the hard thermal loop approximation [9, 10] was developed. This approach is based on calculation of some effective currents generating these processes. Using these currents, the matrix elements of nonlinear interaction of an arbitrary (even) number of colorless plasmons [9, 10] are determined. In this study, we propose an alternative method for kinetic description of the nonlinear plasmon dynamics, which is based on the classical Hamiltonian formalism for systems with distributed parameters and which has been systematically developed by Zakharov [5, 6, 7] and Gitman and Tyutin[12]. In our case, this approach is based on the fact that equations describing a collisionless high-temperature plasma in the hard thermal loop approximation have the Hamiltonian structure that has been determined in papers by Nair [13, 14], Blaizot и Iancu [15]. This enables us to develop (at least for weakly excited states; see Conclusions) an independent approach to the derivation of the kinetic equation for soft longitudinally polarized gluonic plasma excitations. In the Hamiltonian approach, the matrix elements of the plasmon–plasmon interaction are obtained using special canonical transformations simplifying the plasmon interaction Hamiltonian.
This article has the following structure. In Section 2, we derive the fourth-order effective Hamilton operator $\widetilde{\hat{H}}_{4}$ describing elastic scattering of two colorless plasmons off each other. In Section 3, we introduce plasmon distribution function $N^{l}_{\bf k}$ and analyze the fourth- and sixth-order correlation functions in plasmon creation and annihilation operators $\hat{c}^{{\dagger}\ \!\!a}_{{\bf k}}$ and $\hat{c}^{\phantom{\dagger}\!a}_{{\bf k}}$. Section 4 is devoted to the derivation of the Boltzmann-type kinetic equation for soft gluon excitations with allowance for the nonlinear Landau damping effect for plasmons. Sections 5 and 6 are connected with the determination of the explicit form of three - and four-plasmon vertex functions using the hard thermal loop approximation and the approximation of the effective gluon propagator at the plasmon pole. In concluding Section 7, we outline possible ways for generalization of the Hamiltonian description to the case of a strongly excited gluon plasma.
In Appendix, we give all basic expressions for the effective gluon vertex functions and gluon propagator in the high-temperature approximation of hard thermal loops.

Let us consider the application of the general Zakharov theory to a specific system (high-temperature gluon plasma) in the semi-classical approximation. The gauge field potentials describing the gluon field in the system are $N_{c}\times N_{c}$ matrices in the color space and are defined in terms of $A_{\mu}(x)=A_{\mu}^{a}(x)\,t^{a}$ with $N^{2}_{c}-1$ Hermitian generators $t^{a}$ of the color $SU(N_{c})$ group in the fundamental representation¹¹1 The color index $a$ runs through values $1,2,\,\ldots\,,N^{2}_{c}-1$, while vector index $\mu$ runs through values $0,1,2,3$. Everywhere in this article, we imply summation over repeated indices and use the system of units with $\hbar=c=1$.. Field strength tensor $F_{\mu\nu}(x)=F^{a}_{\mu\nu}(x)\,t^{a}$, where

$$F^{a}_{\mu\nu}(x)=\partial^{\phantom{a}}_{\mu}A_{\nu}^{a}-\partial^{\phantom{a}}_{\nu}A_{\mu}^{a}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c}$$

obeys the Yang–Mills equation in the $A_{0}$ - gauge:

$$\partial_{\mu}F^{\mu\nu}(x)-ig\hskip 0.85355pt[A_{\mu}(x),F^{\mu\nu}(x)]-{\xi}_{0}^{-1}n_{\mu}n^{\nu}A_{\nu}(x)=-j^{\nu}(x),$$

where $\xi_{0}$ is the gauge parameter in the given gauge. We will henceforth identify the four-vector $n_{\mu}$ with global four-velocity $u_{\mu}$ of the plasma. Color current $j^{\nu}$ is defined conventionally:

$$j^{\nu}(x)=g\hskip 0.56917ptt^{a}\!\!\int\!d^{4}p\,p^{\nu\,}{\rm Tr}\,(T^{a}f_{g}(x,p)).$$

Here, $x=(t,{\bf x})$ is the space–time variable of the initial dynamical system and $(T^{a})^{\ \!\!b\ \!\!c}\equiv-if^{\ \!\!a\ \!\!b\ \!\!c}$ is the color matrix in the adjoint representation. Gluon distribution function $f_{g}=f_{g}(x,p)$ is an $(N_{c}^{2}-1)\times(N_{c}^{2}-1)$ Hermitian matrix in the colour space.
It is known that there exist two types of physical boson soft (transverse- and longitudinal-polarized) fields in an equilibrium hot quark–gluon plasma [8]. For simplicity, we confine our analysis only to processes involving longitudinally polarized plasma excitations, which are known as plasmons. These excitations are a purely collective effect of the medium, which has no analogs in the conventional quantum field theory. Let us consider the longitudinal part of the gauge field potential in the form of expansion

$$\hat{A}^{a}_{\mu}(x)=\int\!\frac{d{\bf k}}{(2\pi)^{3}}\!\left(\frac{Z^{l}({\bf k})}{2\omega^{l}_{{\bf k}}}\right)^{\!\!1/2}\!\!\left\{\epsilon^{\ \!l}_{\mu}\ \hat{a}^{\phantom{{\dagger}}\!\!a}_{{\bf k}}\ \!e^{-i\hskip 0.56917pt{\bf k}\cdot{\bf x}}+\epsilon^{*l}_{\mu}\ \hat{a}^{{\dagger}\ \!\!a}_{{\bf k}}\ \!e^{i\hskip 0.56917pt{\bf k}\cdot{\bf x}}\right\},\quad k_{0}=\omega^{l}_{{\bf k}},$$

$(2.1)$

where $\epsilon^{\ \!l}_{\mu}=\epsilon^{\ \!l}_{\mu}({\bf k})$ is the polarization vector of a longitudinal plasmon; its explicit form depends on the choice of the gauge (in particular, in the $A_{0}$ - gauge, this vector is defined by expression (5.6)). Factor $Z^{l}({\bf k})$ is the residue of the effective gluon propagator at the plasmon pole. Coefficients $\hat{a}^{\phantom{{\dagger}}\!\!a}_{{\bf k}}$ and $\hat{a}^{{\dagger}\ \!\!a}_{{\bf k}}$ will be treated as quasiparticle creation and annihilation operators for plasmons obeying the commutation relations for Bose operators:

$$\Bigl{[}\hat{a}^{\phantom{{\dagger}}\!a}_{{\bf k}},\,\hat{a}^{\phantom{{\dagger}}\!\!b}_{{\bf k}^{\prime}}\Bigr{]}=\Bigl{[}\hat{a}^{{\dagger}\ \!\!a}_{{\bf k}},\,\hat{a}^{{\dagger}\ \!\!b}_{{\bf k}^{\prime}}\Bigr{]}=0\ ,\ \ \ \Bigl{[}\hat{a}^{\phantom{{\dagger}}\!\!a}_{{\bf k}},\,\hat{a}^{{\dagger}\ \!\!b}_{{\bf k}^{\prime}}\Bigr{]}=\delta^{\phantom{{\dagger}}\!\!ab}\ \!\!(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}-{\bf k}^{\prime}).$$

$(2.2)$

Multiplasmon states are obtained by the multiple action of operator $\hat{a}^{{\dagger}\ \!\!a}_{{\bf k}}$ on vacuum state $|\ \!0\rangle$, which obeys the following condition:

$$\hat{a}^{a}_{{\bf k}}|\ \!0\rangle=0.$$

Therefore, we refer as vacuum to the ground unexcited state of the system (i.e., the state without elementary collective excitations). In operators $\hat{a}^{\phantom{{\dagger}}\!a}_{{\bf k}}$ and $\hat{a}^{{\dagger}\ \!\!a}_{{\bf k}}$, only matrix elements corresponding to a change in the number of plasmons by unity differ from zero.
Let us write the quantum-mechanical analogue of the Hamilton equation, namely, the Heisenberg equation for operator $\hat{a}^{a}_{{\bf k}}$:

$$\frac{\partial\hskip 0.56917pt\hat{a}^{a}_{{\bf k}}}{\partial t}=i\hskip 0.85355pt\Bigl{[}\widehat{H},\hat{a}^{a}_{{\bf k}}\ \!\Bigr{]}.$$

$(2.3)$

Here, $\widehat{H}$ is the Hamiltonian of the plasmon system, which is a sum $\widehat{H}=\widehat{H}_{0}+\widehat{H}_{int}$, where

$$\widehat{H}_{0}=\!\int\!\frac{d{\bf k}}{(2\pi)^{3}}\ \omega^{l}_{{\bf k}}\ \!\hat{a}^{{\dagger}\ \!\!a}_{{\bf k}}\ \!\hat{a}^{\phantom{{\dagger}}\!a}_{{\bf k}}\$$

$(2.4)$

is the Hamiltonian of noninteracting plasmons and $\widehat{H}_{int}$ is interaction Hamiltonian. The dispersion relation $\omega^{\ \!l}_{{\bf k}}$ for plasmons satisfies the following dispersion equation [16]:

$${\rm Re}\ \!\varepsilon^{l}(\omega,{\bf k})=0\ \!,$$

$(2.5)$

where

$$\varepsilon^{l}(\omega,{\bf k})=1+\frac{3\hskip 0.85355pt\omega^{2}_{pl}}{{\bf k}^{\ \!2}}\left[1-F\Biggl{(}\frac{\omega}{|{\bf k}|^{2}}\Biggr{)}\right],\quad F(x)=\frac{x}{2}\left[\ln\left|\frac{1+x}{1-x}\right|-i\pi\hskip 0.85355pt\theta(1-|x|)\right]$$

is the longitudinal permittivity, $\omega^{2}_{pl}=g^{2}N_{c}T^{2}/9$, $T$ is the temperature of the system, and $g$ is the strong interaction constant. In the small amplitude approximation, the interaction Hamiltonian can be written as a formal integral-power series in $\hat{a}^{a}_{{\bf k}}$ and $\hat{a}^{{\dagger}a}_{{\bf k}}$:

$$\widehat{H}_{int}=\widehat{H}_{3}+\widehat{H}_{4}+\,\ldots\,\,,$$

where the third- and fourth-order interaction Hamiltonians have the following structure:

$$\widehat{H}_{3}=\int\frac{d{\bf k}\,d{\bf k}_{1}\,d{\bf k}_{2}}{(2\pi)^{9}}\left\{V^{\ a\ a_{1}\ a_{2}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2}}\ \hat{a}^{{\dagger}\ \!a}_{{\bf k}}\ \hat{a}^{\phantom{{\dagger}}\!\!a_{1}}_{{\bf k}_{1}}\ \hat{a}^{\phantom{{\dagger}}\!\!a_{2}}_{{\bf k}_{2}}+V^{*a\ a_{1}\ a_{2}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2}}\ \hat{a}^{{\dagger}\ \!a_{1}}_{{\bf k}_{1}}\ \hat{a}^{{\dagger}\ \!a_{2}}_{{\bf k}_{2}}\ \hat{a}^{\phantom{{\dagger}}\!\!a}_{{\bf k}}\right\}$$

$(2.6)$

$$\times(2\pi)^{3}\delta({\bf k}-{\bf k}_{1}-{\bf k}_{2})$$

$$+\ \frac{1}{3}\int\frac{d{\bf k}\,d{\bf k}_{1}\,d{\bf k}_{2}}{(2\pi)^{9}}\left\{U^{\ a\ a_{1}\ a_{2}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2}}\ \hat{a}^{a}_{{\bf k}}\ \hat{a}^{a_{1}}_{{\bf k}_{1}}\ \hat{a}^{a_{2}}_{{\bf k}_{2}}+U^{*a\ a_{1}\ a_{2}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2}}\ \hat{a}^{{\dagger}\ \!a}_{{\bf k}}\ \hat{a}^{{\dagger}\ \!a_{1}}_{{\bf k}_{1}}\ \hat{a}^{{\dagger}\ \!a_{2}}_{{\bf k}_{2}}\right\}$$

$$\times(2\pi)^{3}\delta({\bf k}+{\bf k}_{1}+{\bf k}_{2}),$$

$$\widehat{H}_{4}=\frac{1}{2}\int\frac{d{\bf k}\,d{\bf k}_{1}\,d{\bf k}_{2}\,d{\bf k}_{3}}{(2\pi)^{12}}\ T^{\ a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ \hat{a}^{{\dagger}\ \!a}_{{\bf k}}\ \hat{a}^{{\dagger}\ \!a_{1}}_{{\bf k}_{1}}\ \hat{a}^{\phantom{{\dagger}}\!\!a_{2}}_{{\bf k}_{2}}\ \hat{a}^{\phantom{{\dagger}}\!\!a_{3}}_{{\bf k}_{3}}\ (2\pi)^{3}\delta({\bf k}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3})$$

$(2.7)$

and so on. Symbol ‘‘$*\,$’’ indicates complex conjugation. In expression (2.7), we retained only the ‘‘essential’’ contribution in Zakharov’s terminology because the resonance conditions

$$\left\{\begin{array}[]{l}{\bf k}+{\bf k}_{1}+{\bf k}_{2}+{\bf k}_{3}=0\\[5.0pt] \omega^{l}_{{\bf k}}+\omega^{l}_{{\bf k}_{1}}+\omega^{l}_{{\bf k}_{2}}+\omega^{l}_{{\bf k}_{3}}=0\ ,\end{array}\right.\ \ \ \left\{\begin{array}[]{l}{\bf k}={\bf k}_{1}+{\bf k}_{2}+{\bf k}_{3}\\[5.0pt] \omega^{l}_{{\bf k}}=\omega^{l}_{{\bf k}_{1}}+\omega^{l}_{{\bf k}_{2}}+\omega^{l}_{{\bf k}_{3}}\end{array}\right.$$

have no solutions for the plasmon spectrum defined by dispersion equation (2.5).
It should be noted that such a representation of the interaction Hamiltonian in the form of formal infinite power series in the creation and annihilation operators was considered in the monograph by Schwarz [17] based on the quantum field theory for scalar fields.

Hamiltonian (2.12) describes elastic scattering of color plasmons by one another (i.e., $2\rightarrow 2$ process). The equations of motion for $\hat{c}^{\phantom{\dagger}\!a}_{{\bf k}}$ and $\hat{c}^{\dagger\ \!\!b}_{{\bf k}}$ are determined in this case by the corresponding Heisenberg equations:

$$\frac{\partial\hat{c}^{\ \!a}_{{\bf k}}}{\partial t}=i\Bigl{[}\widetilde{\widehat{H}}_{0}+\widetilde{\widehat{H}}_{4},\,\hat{c}^{a}_{{\bf k}}\ \!\Bigr{]}=-i\hskip 0.85355pt\omega^{\ \!l}_{{\bf k}}\ \!\hat{c}^{\ \!a}_{{\bf k}}\$$

$(3.1)$

$$-\;i\!\int\frac{d{\bf k}_{1}\,d{\bf k}_{2}\,d{\bf k}_{3}}{(2\pi)^{9}}\ \widetilde{T}^{\ a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ \hat{c}^{{\dagger}\ \!\!a_{1}}_{{\bf k}_{1}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{2}}_{{\bf k}_{2}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{3}}_{{\bf k}_{3}}\ (2\pi)^{3}\hskip 0.85355pt\delta({\bf k}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3})$$

and

$$\frac{\partial\hat{c}^{{\dagger}\ \!\!b}_{{\bf k}}}{\partial t}=i\Bigl{[}\widetilde{\widehat{H}}_{0}+\widetilde{\widehat{H}}_{4},\,\hat{c}^{{\dagger}\ \!\!b}_{\bf k}\ \!\Bigr{]}=i\hskip 0.85355pt\omega^{\ \!l}_{{\bf k}}\ \hat{c}^{{\dagger}\ \!\!b}_{{\bf k}}\$$

$(3.2)$

$$+\;i\!\int\frac{d{\bf k}_{1}\,d{\bf k}_{2}\,d{\bf k}_{3}}{(2\pi)^{9}}\ \widetilde{T}^{\,*\ \!b\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ \hat{c}^{\phantom{{\dagger}}\!\!a_{1}}_{{\bf k}_{1}}\ \!\hat{c}^{{\dagger}\ \!\!a_{2}}_{{\bf k}_{2}}\ \!\hat{c}^{{\dagger}\ \!\!a_{3}}_{{\bf k}_{3}}\ (2\pi)^{3}\hskip 0.85355pt\delta({\bf k}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3}).$$

These exact equations in the absence of an external color field in the system enable us to determine the kinetic equation for the number density $N^{ab\ \!l}_{{\bf k}\!\!}\!\equiv\delta^{ab}N^{l}_{{\bf k}}$ of colorless plasmons.
If the set of waves for a low level of nonlinearity of (2.14) has random phases, this set can be described statistically by introducing correlation function

$$\langle\,\hat{c}^{{\dagger}\ \!\!a}_{{\bf k}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!b}_{{\bf k}^{\prime}}\rangle=\delta^{ab}(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}-{\bf k}^{\prime})\hskip 0.85355ptN^{l}_{{\bf k}}.$$

$(3.3)$

It should be emphasized that the introduction of distribution function $N^{l}_{{\bf k}}\equiv N^{l}({\bf k},{\bf x},t)$ of quasiparticles (plasmons), which depends on plasmon momentum $\hbar\hskip 1.13791pt{\bf k}$ as well as on coordinate ${\bf x}$ and time $t$, makes sense only in the case when the number of plasmons varies slowly in space and time. This means that the variation of the function over distances on the order of wavelength $\lambda=2\pi/k$ and in time intervals on the order of wave period $T=2\pi/\omega^{l}_{\bf k}$ must be much smaller than function $N^{l}_{\bf k}$ itself.
Proceeding from Heisenberg equations (3.1) and (3.2), we can determine the kinetic equation for plasmon number density $N^{l}_{{\bf k}}$. For this purpose, we multiply Eqs. (3.1) and (3.2) by $\hat{c}^{{\dagger}\ \!\!b}_{{\bf k}^{\prime}}$ and $\hat{c}^{\phantom{\dagger}\!a}_{{\bf k}}$, respectively:

$$\frac{\partial\hskip 0.56917pt\hat{c}^{\ \!a}_{\bf k}}{\partial\hskip 0.56917ptt}\ \hat{c}^{{\dagger}\ \!\!b}_{{\bf k}^{\prime}}=-\,i\hskip 0.85355pt\omega^{\ \!l}_{{\bf k}}\ \!\hat{c}^{\phantom{\dagger}\!a}_{{\bf k}}\ \!\hat{c}^{{\dagger}\ \!\!b}_{{\bf k}^{\prime}}$$

$$-\;i\!\int\frac{d{\bf k}_{1}\,d{\bf k}_{2}\,d{\bf k}_{3}}{(2\pi)^{9}}\ \widetilde{T}^{\ a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ \hat{c}^{{\dagger}\ \!\!b}_{{\bf k}}\ \hat{c}^{{\dagger}\ \!\!a_{1}}_{{\bf k}_{1}}\ \!\!\hat{c}^{\phantom{{\dagger}}\!\!a_{2}}_{{\bf k}_{2}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{3}}_{{\bf k}_{3}}\ (2\pi)^{3}\hskip 0.85355pt\delta({\bf k}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3}),$$

$$\hat{c}^{\phantom{\dagger}\!\!a}_{{\bf k}}\ \!\frac{\partial\hskip 0.56917pt\hat{c}^{{\dagger}\ \!\!b}_{{\bf k}^{\prime}}}{\partial\hskip 0.56917ptt}=i\hskip 0.85355pt\omega^{\ \!l}_{{\bf k}}\ \!\hat{c}^{\phantom{\dagger}\!\!a}_{{\bf k}}\ \!\hat{c}^{{\dagger}\ \!\!b}_{{\bf k}^{\prime}}$$

$$+\;i\!\int\frac{d{\bf k}_{1}\,d{\bf k}_{2}\,d{\bf k}_{3}}{(2\pi)^{9}}\ \widetilde{T}^{\,*\!\ b\ a_{1}\ a_{2}\ a_{3}}_{{\bf k}^{\prime},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ \hat{c}^{\phantom{\dagger}\!a}_{{\bf k}}\ \!\hat{c}^{\ \!a_{1}}_{{\bf k}_{1}}\ \!\hat{c}^{{\dagger}\ \!\!a_{2}}_{{\bf k}_{2}}\ \!\hat{c}^{{\dagger}\ \!\!a_{3}}_{{\bf k}_{3}}\ (2\pi)^{3}\hskip 0.85355pt\delta({\bf k}^{\prime}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3}).$$

Summing these two equations and averaging them, we obtain

$$\delta^{ab}(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}-{\bf k}\!\ ^{\prime})\frac{\partial N^{l}_{{\bf k}}}{\partial t}=$$

$(3.4)$

$$=-\,i\!\int\frac{d{\bf k}_{1}\,d{\bf k}_{2}\,d{\bf k}_{3}}{(2\pi)^{9}}\ \biggl{\{}\widetilde{T}^{\ a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ I^{\ b\ a_{1}\ a_{2}\ a_{3}}_{{\bf k}^{\prime},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ \!(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3})$$

$$-\ \widetilde{T}^{\,*\!\ b\ a_{1}\ a_{2}\ a_{3}}_{{\bf k}^{\prime},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ I^{\ a_{2}\ a_{3}\ a\ a_{1}}_{{\bf k}_{2},\ {\bf k}_{3},\ {\bf k},\ {\bf k}_{1}}\ \!(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}^{\prime}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3})\biggr{\}},$$

where

$$I^{\ a\;a_{1}\;a_{2}\;a_{3}}_{{\bf k},\,{\bf k}_{1},\,{\bf k}_{2},\,{\bf k}_{3}}=\langle\ \!\hat{c}^{{\dagger}\ \!\!a}_{{\bf k}}\ \!\hat{c}^{{\dagger}\ \!\!a_{1}}_{{\bf k}_{1}}\ \!\hat{c}^{\phantom{{\dagger}}\!a_{2}}_{{\bf k}_{2}}\ \!\hat{c}^{\phantom{{\dagger}}\!a_{3}}_{{\bf k}_{3}}\ \!\rangle$$

is the four-point correlation function. Differentiating the correlation function $I^{\ a\;a_{1}\;a_{2}\;a_{3}}_{{\bf k},\,{\bf k}_{1},\,{\bf k}_{2},\,{\bf k}_{3}}$ with respect to $t$ and considering Eqs. (3.1) and (3.2), we obtain the following equation, the right-hand side of which contains sixth-order correlation functions in operators $\hat{c}^{{\dagger}\ \!\!a}_{{\bf k}}$ and $\hat{c}^{\phantom{\dagger}\!a}_{{\bf k}}$:

$$\frac{\partial I^{\ a\;a_{1}\;a_{2}\;a_{3}}_{{\bf k},\,{\bf k}_{1},\,{\bf k}_{2},\,{\bf k}_{3}}}{\partial t}=i\bigl{[}\ \!\omega^{l}_{{\bf k}}+\omega^{l}_{{\bf k}_{1}}-\omega^{l}_{{\bf k}_{2}}-\omega^{l}_{{\bf k}_{3}}\bigr{]}\ \!I^{\ a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ +$$

$(3.5)$

$$+\,i\!\int\frac{d{\bf k}^{\prime}_{1}\,d{\bf k}^{\prime}_{2}\,d{\bf k}^{\prime}_{3}}{(2\pi)^{9}}\ \widetilde{T}^{\,*\!\ \!a\ a^{\prime}_{1}\ a^{\prime}_{2}\ a^{\prime}_{3}}_{{\bf k},\ {\bf k}^{\prime}_{1},\ {\bf k}^{\prime}_{2},\ {\bf k}^{\prime}_{3}}\ \!\langle\ \!\hat{c}^{\ \!a^{\prime}_{1}}_{{\bf k}^{\prime}_{1}}\ \!\hat{c}^{{\dagger}\ \!\!a^{\prime}_{2}}_{{\bf k}^{\prime}_{2}}\ \!\hat{c}^{{\dagger}\ \!\!a^{\prime}_{3}}_{{\bf k}^{\prime}_{3}}\ \!\hat{c}^{{\dagger}\ \!\!a_{1}}_{{\bf k}_{1}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{2}}_{{\bf k}_{2}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{3}}_{{\bf k}_{3}}\ \!\rangle(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}+{\bf k}^{\prime}_{1}-{\bf k}^{\prime}_{2}-{\bf k}^{\prime}_{3})$$

$$+\,i\!\int\frac{d{\bf k}^{\prime}_{1}\,d{\bf k}^{\prime}_{2}\,d{\bf k}^{\prime}_{3}}{(2\pi)^{9}}\ \widetilde{T}^{\,*\!\ \!a_{1}\ a^{\prime}_{1}\ a^{\prime}_{2}\ a^{\prime}_{3}}_{{\bf k}_{1},\ {\bf k}^{\prime}_{1},\ {\bf k}^{\prime}_{2},\ {\bf k}^{\prime}_{3}}\ \!\langle\ \!\hat{c}^{{\dagger}\ \!a}_{{\bf k}}\ \!\hat{c}^{a^{\prime}_{1}}_{{\bf k}^{\prime}_{1}}\ \!\hat{c}^{{\dagger}\ \!\!a^{\prime}_{2}}_{{\bf k}^{\prime}_{2}}\ \!\hat{c}^{{\dagger}\ \!\!a^{\prime}_{3}}_{{\bf k}^{\prime}_{3}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{2}}_{{\bf k}_{2}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{3}}_{{\bf k}_{3}}\ \!\rangle(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}_{1}+{\bf k}^{\prime}_{1}-{\bf k}^{\prime}_{2}-{\bf k}^{\prime}_{3})$$

$$-\,i\!\int\frac{d{\bf k}^{\prime}_{1}\,d{\bf k}^{\prime}_{2}\,d{\bf k}^{\prime}_{3}}{(2\pi)^{9}}\ \widetilde{T}^{\ \!a_{2}\ a^{\prime}_{1}\ a^{\prime}_{2}\ a^{\prime}_{3}}_{{\bf k}_{2},\ {\bf k}^{\prime}_{1},\ {\bf k}^{\prime}_{2},\ {\bf k}^{\prime}_{3}}\ \!\langle\ \!\hat{c}^{{\dagger}\ \!\!a}_{{\bf k}}\ \!\hat{c}^{{\dagger}\ \!\!a_{1}}_{{\bf k}_{1}}\ \!\hat{c}^{{\dagger}\ \!\!a^{\prime}_{1}}_{{\bf k}^{\prime}_{1}}\ \!\hat{c}^{\ \!a^{\prime}_{2}}_{{\bf k}^{\prime}_{2}}\ \!\hat{c}^{\ \!a^{\prime}_{3}}_{{\bf k}^{\prime}_{3}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{3}}_{{\bf k}_{3}}\ \!\rangle(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}_{2}+{\bf k}^{\prime}_{1}-{\bf k}^{\prime}_{2}-{\bf k}^{\prime}_{3})$$

$$-\,i\!\int\frac{d{\bf k}^{\prime}_{1}\,d{\bf k}^{\prime}_{2}\,d{\bf k}^{\prime}_{3}}{(2\pi)^{9}}\ \widetilde{T}^{\ \!a_{3}\ a^{\prime}_{1}\ a^{\prime}_{2}\ a^{\prime}_{3}}_{{\bf k}_{3},\ {\bf k}^{\prime}_{1},\ {\bf k}^{\prime}_{2},\ {\bf k}^{\prime}_{3}}\ \!\langle\ \!\hat{c}^{{\dagger}\ \!\!a}_{{\bf k}}\ \!\hat{c}^{{\dagger}\ \!\!a_{1}}_{{\bf k}_{1}}\ \!\hat{c}^{\ \!a_{2}}_{{\bf k}_{2}}\ \!\hat{c}^{{\dagger}\ \!\!a^{\prime}_{1}}_{{\bf k}^{\prime}_{1}}\ \!\hat{c}^{\ \!a^{\prime}_{2}}_{{\bf k}^{\prime}_{2}}\ \!\hat{c}^{\ \!a^{\prime}_{3}}_{{\bf k}^{\prime}_{3}}\ \!\rangle(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}_{3}+{\bf k}^{\prime}_{1}-{\bf k}^{\prime}_{2}-{\bf k}^{\prime}_{3}).$$

We close this set of equations for correlation functions by expressing the formulas for sixth-order correlation functions in terms of pair correlation functions. For example, the first sixth-order correlation function on the right-hand side of Eq. (3.5) has the following structure:

$$\langle\ \!\hat{c}^{\ \!\!a^{\prime}_{1}}_{{\bf k}^{\prime}_{1}}\ \!\hat{c}^{{\dagger}\ \!\!a^{\prime}_{2}}_{{\bf k}^{\prime}_{2}}\ \!\hat{c}^{{\dagger}\ \!\!a^{\prime}_{3}}_{{\bf k}^{\prime}_{3}}\ \!\hat{c}^{{\dagger}\ \!a_{1}}_{{\bf k}_{1}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{2}}_{{\bf k}_{2}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{3}}_{{\bf k}_{3}}\ \!\rangle=$$

$(3.6)$

$$=3\hskip 0.85355pt(2\pi)^{9}\Bigl{\{}\delta^{\ \!\!a^{\phantom{\prime}}_{3}a^{\prime}_{3}}\delta^{\ \!\!a^{\phantom{\prime}}_{2}a^{\prime}_{2}}\delta^{\ \!\!a^{\phantom{\prime}}_{1}a^{\prime}_{1}}\ \!\delta({\bf k}^{\prime}_{3}-{\bf k}^{\phantom{\prime}}_{3})\delta({\bf k}^{\prime}_{2}-{\bf k}^{\phantom{\prime}}_{2})\delta({\bf k}^{\prime}_{1}-{\bf k}^{\phantom{\prime}}_{1})\ \!N^{l}_{{\bf k}_{3}}N^{l}_{{\bf k}_{2}}N^{l}_{{\bf k}_{1}}$$

$$+\ \delta^{\ \!\!a^{\phantom{\prime}}_{3}a^{\prime}_{3}}\delta^{\ \!\!a_{1}a_{2}}\delta^{\ \!\!a^{\prime}_{2}a^{\prime}_{1}}\ \!\delta({\bf k}^{\prime}_{3}-{\bf k}^{\phantom{\prime}}_{3})\delta({\bf k}_{1}-{\bf k}_{2})\delta({\bf k}^{\prime}_{2}-{\bf k}^{\prime}_{1})\ \!N^{l}_{{\bf k}_{3}}N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}^{\prime}_{1}}$$

$$+\ \delta^{\ \!\!a^{\prime}_{3}a^{\prime}_{1}}\delta^{\ \!\!a_{1}a_{3}}\delta^{\ \!\!a^{\prime}_{2}a^{\phantom{\prime}}_{2}}\ \!\delta({\bf k}^{\prime}_{3}-{\bf k}^{\prime}_{1})\delta({\bf k}_{1}-{\bf k}_{3})\delta({\bf k}^{\prime}_{2}-{\bf k}^{\phantom{\prime}}_{2})\ \!N^{l}_{{\bf k}^{\prime}_{1}}N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}_{2}}$$

$$+\ \delta^{\ \!\!a^{\prime}_{3}a^{\prime}_{1}}\delta^{\ \!\!a_{1}a_{2}}\delta^{\ \!\!a^{\prime}_{2}a^{\phantom{\prime}}_{3}}\ \!\delta({\bf k}^{\prime}_{3}-{\bf k}^{\prime}_{1})\delta({\bf k}_{1}-{\bf k}_{2})\delta({\bf k}^{\prime}_{2}-{\bf k}^{\phantom{\prime}}_{3})\ \!N^{l}_{{\bf k}^{\prime}_{1}}N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}_{3}}$$

$$+\ \delta^{\ \!\!a^{\prime}_{3}a^{\phantom{\prime}}_{2}}\delta^{\ \!\!a_{1}a_{3}}\delta^{\ \!\!a^{\prime}_{2}a^{\prime}_{1}}\ \!\delta({\bf k}^{\prime}_{3}-{\bf k}^{\phantom{\prime}}_{2})\delta({\bf k}_{1}-{\bf k}_{3})\delta({\bf k}^{\prime}_{2}-{\bf k}^{\prime}_{1})\ \!N^{l}_{{\bf k}_{2}}N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}^{\prime}_{1}}$$

$$\hskip 12.80365pt+\ \delta^{\ \!\!a^{\prime}_{3}a^{\phantom{\prime}}_{2}}\delta^{\ \!\!a^{\prime}_{1}a^{\phantom{\prime}}_{1}}\delta^{\ \!\!a^{\prime}_{2}a^{\phantom{\prime}}_{3}}\ \!\delta({\bf k}^{\prime}_{3}-{\bf k}^{\phantom{\prime}}_{2})\delta({\bf k}^{\prime}_{1}-{\bf k}^{\phantom{\prime}}_{1})\delta({\bf k}^{\prime}_{2}-{\bf k}^{\phantom{\prime}}_{3})\ \!N^{l}_{{\bf k}_{2}}N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}_{3}}\Bigr{\}}.$$

In this expression, only the first and last terms make the required contribution to the required kinetic equation. Substituting these terms into the first integral on the right-hand side of Eq. (3.5) and performing summation over color indices $a^{\prime}_{1},\,a^{\prime}_{2}$ and $a^{\prime}_{3}$ and integration over momenta ${\bf k}^{\prime}_{1},\,{\bf k}^{\prime}_{2}$ и ${\bf k}^{\prime}_{3}$, we obtain

$$3\hskip 0.85355pti\biggl{\{}\widetilde{T}^{\,*\ \!a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ \!N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}_{2}}N^{l}_{{\bf k}_{3}}+\widetilde{T}^{*\ \!a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ \!N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}_{2}}N^{l}_{{\bf k}_{3}}\biggr{\}}\ \!(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3})$$

$$=6\hskip 0.85355pti\ \!\widetilde{T}^{\,*\ \!a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ \!N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}_{2}}N^{l}_{{\bf k}_{3}}\ \!(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3}).$$

$(3.7)$

By way of example, let us consider the explicit form of the contribution that generates the second term in the decomposition of correlation function (3.6):

$$N^{l}_{\bf k}N^{l}_{{\bf k}_{1}}\delta({\bf k}_{2}-{\bf k}_{1})\ \!\delta({\bf k}_{3}-{\bf k})\ \!\delta^{a_{1}a_{2}}\ \!i\!\int\!\widetilde{T}^{\,*\ \!a\ b\ b\ a_{3}}_{{\bf k},\ {\bf k}^{\prime},\ {\bf k}^{\prime},\ {\bf k}}\ \!N^{l}_{{\bf k}^{\prime}}\ \!d{\bf k}^{\prime}.$$

Comparing the last two expressions, we see that they have quite different structures.
Let us now consider the second six-point correlation function in expression (3.5). In this correlator, we write in explicit form only “regular” terms:

$$\langle\ \!\hat{c}^{{\dagger}\ \!\!a}_{{\bf k}}\ \!\hat{c}^{a^{\prime}_{1}}_{{\bf k}^{\prime}_{1}}\ \!\hat{c}^{{\dagger}\ \!\!a^{\prime}_{2}}_{{\bf k}^{\prime}_{2}}\ \!\hat{c}^{{\dagger}\ \!\!a^{\prime}_{3}}_{{\bf k}^{\prime}_{3}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{2}}_{{\bf k}_{2}}\ \!\hat{c}^{\phantom{{\dagger}}\!\!a_{3}}_{{\bf k}_{3}}\ \!\rangle=$$

$$=3\hskip 0.85355pt(2\pi)^{9}\Bigl{\{}\delta^{\ \!\!a^{\phantom{\prime}}\!a^{\prime}_{1}}\delta^{\ \!\!a^{\phantom{\prime}}_{2}a^{\prime}_{2}}\delta^{\ \!\!a^{\phantom{\prime}}_{3}a^{\prime}_{3}}\ \!\delta({\bf k}^{\prime}_{1}-{\bf k}^{\phantom{\prime}})\delta({\bf k}^{\prime}_{2}-{\bf k}^{\phantom{\prime}}_{2})\delta({\bf k}^{\prime}_{3}-{\bf k}^{\phantom{\prime}}_{3})\ \!N^{l}_{{\bf k}}N^{l}_{{\bf k}_{2}}N^{l}_{{\bf k}_{3}}+\ (2\rightleftarrows 3)\ +\ \ldots\,\Bigr{\}}.$$

Substituting this expression into the second integral in (3.5), we obtain the following expression analogous to (3.7):

$$6\hskip 0.85355pti\ \!\widetilde{T}^{\,*\ \!a_{1}\ a\ a_{2}\ a_{3}}_{{\bf k}_{1},\ {\bf k},\ {\bf k}_{2},\ {\bf k}_{3}}\,N^{l}_{{\bf k}}N^{l}_{{\bf k}_{2}}N^{l}_{{\bf k}_{3}}\ \!(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3}).$$

Similar arguments for the third and fourth correlators in expression (3.5) give the remaining two contributions:

$$-6\hskip 0.85355pti\ \!\widetilde{T}^{\ \!a_{2}\ a_{3}\ a\ a_{1}}_{{\bf k}_{2},\ {\bf k}_{3},\ {\bf k},\ {\bf k}_{1}}\,N^{l}_{{\bf k}}N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}_{3}}\ \!(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3})$$

and

$$-6\hskip 0.85355pti\ \!\widetilde{T}^{\ \!a_{3}\ a_{2}\ a\ a_{1}}_{{\bf k}_{3},\ {\bf k}_{2},\ {\bf k},\ {\bf k}_{1}}\,N^{l}_{{\bf k}}N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}_{2}}\ \!(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3}).$$

Considering the symmetry relations for the scattering amplitude

$$\widetilde{T}^{\ \!a_{2}\ a_{3}\ a\ a_{1}}_{{\bf k}_{2},\ {\bf k}_{3},\ {\bf k},\ {\bf k}_{1}}=\widetilde{T}^{*\ \!a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}},\quad\widetilde{T}^{\ \!a_{3}\ a_{2}\ a\ a_{1}}_{{\bf k}_{3},\ {\bf k}_{2},\ {\bf k},\ {\bf k}_{1}}=\widetilde{T}^{*\ \!a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}},$$

we obtain the following equation for the fourth-order correlation function, instead of (3.5):

$$\frac{\partial I^{\ a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}}{\partial t}=i\bigl{[}\ \!\omega^{l}_{{\bf k}}+\omega^{l}_{{\bf k}_{1}}-\omega^{l}_{{\bf k}_{2}}-\omega^{l}_{{\bf k}_{3}}\bigr{]}\ \!I^{\ a\;a_{1}\;a_{2}\;a_{3}}_{{\bf k},\,{\bf k}_{1},\,{\bf k}_{2},\,{\bf k}_{3}}$$

$(3.8)$

$$+\;6\hskip 1.13791pti\ \!\widetilde{T}^{\,*\ \!a\ a_{1}\ a_{2}\ a_{3}}_{{\bf k},\ {\bf k}_{1},\ {\bf k}_{2},\ {\bf k}_{3}}\ \!\Bigl{(}N^{l}_{\bf k}N^{l}_{{\bf k}_{2}}N^{l}_{{\bf k}_{3}}+N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}_{2}}N^{l}_{{\bf k}_{3}}-N^{l}_{{\bf k}}N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}_{3}}-N^{l}_{{\bf k}}N^{l}_{{\bf k}_{1}}N^{l}_{{\bf k}_{2}}\Bigr{)}$$

$$\times\ \!(2\pi)^{3}\hskip 0.85355pt\delta({\bf k}+{\bf k}_{1}-{\bf k}_{2}-{\bf k}_{3}).$$