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Hamiltonian Formalism for Bose Excitations in a Plasma with a Non-Abelian Interaction

  • STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS
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Abstract

We have developed the Hamiltonian theory for collective longitudinally polarized colorless excitations (plasmons) in a high-temperature gluon plasma using the general formalism for constructing the wave theory in nonlinear media with dispersion, which was developed by V.E. Zakharov. In this approach, we have explicitly obtained a special canonical transformation that makes it possible to simplify the Hamiltonian of interaction of soft gluon excitations and, hence, to derive a new effective Hamiltonian. The approach developed here is used for constructing a Boltzmann-type kinetic equation describing elastic scattering of collective longitudinally polarized excitations in a gluon plasma as well as the effect of the so-called nonlinear Landau damping. We have performed detailed comparison of the effective amplitude of the plasmon–plasmon interaction, which is determined using the classical Hamilton theory, with the corresponding matrix element calculated in the framework of high-temperature quantum chromodynamics; this has enabled us to determine applicability limits for the purely classical approach described in this study.

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Notes

  1. Color index a runs through values 1, 2, …, \(N_{c}^{2}\) – 1, while vector index μ 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.

  2. Variational derivatives with respect to operators \(\hat {c}_{{\mathbf{k}}}^{a}\) and \(\hat {c}_{{\mathbf{k}}}^{{\dag a}}\) should be treated as the limits of the corresponding functional derivatives with respect to the classical additions \(\varphi _{{\mathbf{k}}}^{a}\) and \(\varphi _{{\mathbf{k}}}^{{*\,a}}\) to quantum operators \(\hat {c}_{{\mathbf{k}}}^{a}\) and \(\hat {c}_{{\mathbf{k}}}^{{\dag a}}\) [18]: \(\hat {c}_{{\mathbf{k}}}^{a} \to \hat {c}_{{\mathbf{k}}}^{a} + \varphi _{{\mathbf{k}}}^{a}\), \(\hat {c}_{{\mathbf{k}}}^{{\dag a}} \to \hat {c}_{{\mathbf{k}}}^{{\dag a}} + \varphi _{{\mathbf{k}}}^{{*\,a}}\).

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Funding

The research of D.M.G. and Yu.A.M. was supported by the Program for Improving Competitiveness of National Research Tomsk State University among the Leading World Scientific and Educational Centers. The work of D.M.G. was also supported in part by the Russian Foundation for Basic Research (project no. 18-02-00149), San Paolo Research Foundation (FAPESP, project no. 2016/03319-6), and the National Science Council (CNPq).

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Correspondence to Yu. A. Markov, M. A. Markova or D. M. Gitman.

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Translated by N. Wadhwa

EFFECTIVE VERTICES AND GLUON PROPAGATOR

EFFECTIVE VERTICES AND GLUON PROPAGATOR

In this Appendix, we consider the explicit form of vertex functions and gluon propagator in the high-temperature hard thermal loop (HTL) approximation [8, 9].

Effective three-gluon vertex

$$\text{*}{{\Gamma }^{{\mu \nu \rho }}}(k,{{k}_{1}},{{k}_{2}}) \equiv {{\Gamma }^{{\mu \nu \rho }}}(k,{{k}_{1}},{{k}_{2}}) + \delta {{\Gamma }^{{\mu \nu \rho }}}(k,{{k}_{1}},{{k}_{2}})$$
(A.1)

is the sum of bare three-gluon vertex

$$\begin{gathered} {{\Gamma }^{{\mu \nu \rho }}}(k,{{k}_{1}},{{k}_{2}}) \\ = {{g}^{{\mu \nu }}}{{(k - {{k}_{1}})}^{\rho }} + {{g}^{{\nu \rho }}}{{({{k}_{1}} - {{k}_{2}})}^{\mu }} + {{g}^{{\mu \rho }}}{{({{k}_{2}} - k)}^{\nu }} \\ \end{gathered} $$
(A.2)

and the corresponding HTL correction

$$\begin{gathered} \delta {{\Gamma }^{{\mu \nu \rho }}}(k,{{k}_{1}},{{k}_{2}}) = 3\omega _{{{\text{pl}}}}^{2}\int {\frac{{d\Omega }}{{4\pi }}\frac{{{{{v}}^{\mu }}{{{v}}^{\nu }}{{{v}}^{\rho }}}}{{{v} \cdot k + i\epsilon }}} \\ \times \left( {\frac{{{{\omega }_{2}}}}{{{v} \cdot {{k}_{2}} - i\epsilon }} - \frac{{{{\omega }_{1}}}}{{{v} \cdot {{k}_{1}} - i\epsilon }}} \right), \\ \end{gathered} $$
(A.3)

where \({{{v}}^{\mu }}\) = (1, v), k + k1 + k2 = 0, and dΩ is a differential solid angle. We consider below useful properties of the three-gluon HTL resummed vertex function for complex conjugation and permutation of momenta:

$$\begin{gathered} (\text{*}{{\Gamma }_{{\mu {{\mu }_{1}}{{\mu }_{2}}}}}( - {{k}_{1}}\, - \,{{k}_{2}},{{k}_{1}},{{k}_{2}})){\text{*}} \\ = \, - {\kern 1pt} {\text{*}}{{\Gamma }_{{\mu {{\mu }_{1}}{{\mu }_{2}}}}}({{k}_{1}}\, + \,{{k}_{2}}, - {{k}_{1}}, - {{k}_{2}}) \\ = {\text{*}}{{\Gamma }_{{\mu {{\mu }_{1}}{{\mu }_{2}}}}}({{k}_{1}} + {{k}_{2}}, - {{k}_{2}}, - {{k}_{1}}). \\ \end{gathered} $$
(A.4)

Further, effective four-gluon vertex

$$\begin{gathered} \text{*}{{\Gamma }^{{\mu \nu \lambda \sigma }}}(k,{{k}_{1}},{{k}_{2}},{{k}_{3}}) \\ \equiv {{\Gamma }^{{\mu \nu \lambda \sigma }}}(k,{{k}_{1}},{{k}_{2}},{{k}_{3}}) + \delta {{\Gamma }^{{\mu \nu \lambda \sigma }}}(k,{{k}_{1}},{{k}_{2}},{{k}_{3}}) \\ \end{gathered} $$
(A.5)

is the sum of bare four-gluon vertex

$${{\Gamma }^{{\mu \nu \lambda \sigma }}} = 2{{g}^{{\mu \nu }}}{{g}^{{\lambda \sigma }}} - {{g}^{{\mu \sigma }}}{{g}^{{\nu \lambda }}} - {{g}^{{\mu \lambda }}}{{g}^{{\sigma \nu }}}$$
(A.6)

and the corresponding HTL correction

$$\begin{gathered} \delta {{\Gamma }^{{\mu \nu \lambda \sigma }}}(k,{{k}_{1}},{{k}_{2}},{{k}_{3}}) = 3\omega _{{{\text{pl}}}}^{2}\int {\frac{{d\Omega }}{{4\pi }}\frac{{{{{v}}^{\mu }}{{{v}}^{\nu }}{{{v}}^{\lambda }}{{{v}}^{\sigma }}}}{{{v} \cdot k + i\epsilon }}} \\ \times \left[ {\frac{1}{{{v} \cdot (k + {{k}_{1}}) + i\epsilon }}\left( {\frac{{{{\omega }_{2}}}}{{{v} \cdot {{k}_{2}} - i\epsilon }} - \frac{{{{\omega }_{3}}}}{{{v} \cdot {{k}_{3}} - i\epsilon }}} \right)} \right. \\ \left. { - \frac{1}{{{v} \cdot (k + {{k}_{3}}) + i\epsilon }}\left( {\frac{{{{\omega }_{1}}}}{{{v} \cdot {{k}_{1}} - i\epsilon }} - \frac{{{{\omega }_{2}}}}{{{v} \cdot {{k}_{2}} - i\epsilon }}} \right)} \right]. \\ \end{gathered} $$
(A.7)

Finally, expression

$$\begin{gathered} \text{*}{{{\tilde {\mathcal{D}}}}_{{\mu \nu }}}(k) = - {{P}_{{\mu \nu }}}(k)\,{\text{*}}{{\Delta }^{t}}(k) - {{{\tilde {Q}}}_{{\mu \nu }}}(k)\,{\text{*}}{{\Delta }^{l}}(k) \\ \, - {{\xi }_{0}}\frac{{{{k}^{2}}}}{{{{{(k \cdot u)}}^{2}}}}{{D}_{{\mu \nu }}}(k) \\ \end{gathered} $$
(A.8)

is a gluon (retarded) propagator in the A0 gauge, which is modified by effects of the medium. Here, “scalar” transverse and longitudinal propagators have form

$$\text{*}{{\Delta }^{t}}(k) = \frac{1}{{{{k}^{2}} - {{\Pi }^{t}}(k)}},\quad {\text{*}}{{\Delta }^{l}}(k) = \frac{1}{{{{k}^{2}} - {{\Pi }^{l}}(k)}},$$
(A.9)

where

$${{\Pi }^{t}}(k) = \frac{1}{2}{{\Pi }^{{\mu \nu }}}(k){{P}_{{\mu \nu }}}(k),\quad {{\Pi }^{l}}(k) = {{\Pi }^{{\mu \nu }}}(k){{\tilde {Q}}_{{\mu \nu }}}(k).$$

Polarization tensor Πμν(k) in the HTL approximation has form

$${{\Pi }^{{\mu \nu }}}(k) = 3\omega _{{pl}}^{2}\left( {{{u}^{\mu }}{{u}^{\nu }} - \omega \int {\frac{{d\Omega }}{{4\pi }}\frac{{{{{v}}^{\mu }}{{{v}}^{\nu }}}}{{{v} \cdot k + i\epsilon }}} } \right),$$

and the longitudinal and transverse projectors are defined by expressions

$$\begin{gathered} {{{\tilde {Q}}}_{{\mu \nu }}}(k) = \frac{{{{{\tilde {u}}}_{\mu }}(k){{{\tilde {u}}}_{\nu }}(k)}}{{{{{\bar {u}}}^{2}}(k)}}, \\ {{P}_{{\mu \nu }}}(k) = {{g}_{{\mu \nu }}} - {{u}_{\mu }}{{u}_{\nu }} - {{{\tilde {Q}}}_{{\mu \nu }}}(k)\frac{{{{{(k \cdot u)}}^{2}}}}{{{{k}^{2}}}}, \\ \end{gathered} $$
(A.10)

respectively, where Lorentz-covariant four-vector \({{\tilde {u}}_{\mu }}\)(k) is defined by formula (5.5).

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Markov, Y.A., Markova, M.A., Markov, N.Y. et al. Hamiltonian Formalism for Bose Excitations in a Plasma with a Non-Abelian Interaction. J. Exp. Theor. Phys. 130, 274–286 (2020). https://doi.org/10.1134/S1063776120010082

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