Quantum field renormalization group in the theory of stochastic Langmuir turbulence

Quantum field theory renormalization group is used for analysing the Langmuir stochastic turbulence of plasma described by the Zakharov equations [1] with random noises. Existence of the dissipative scaling critical regime is proved, for which all the critical exponents are nontrivial and are calculated in the framework of the $4-\varepsilon$ expansion up to $\varepsilon^2$. An explicit expression is obtained for the scaling asymptotics of the longitudinal dielectric permeability $\varepsilon_\parallel(\omega,k)$ in the neighbourhood of a “critical point” $\varepsilon_\parallel(\omega_e,0)=0$ ($\omega_e$ is the Langmuir frequency). The expression implies, in particular, that the usual dispersion law of the Langmuir waves $\omega-\omega_l\sim k^2$ is substitutted for small $k$ by the law $\omega-\omega_l\sim k^{2-\gamma_a}$ in which the exponent $\gamma_a$ is known up to $\varepsilon^2$.