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Register a new userTurbulent spectra in real-time gauge field evolution
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Authors
Sebastian Scheffler
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We investigate ultraviolet fixed points in the real-time evolution of non-Abelian gauge fields. Classical-statistical lattice simulations reveal equal-time correlation functions with a spectral index 3/2. Analytical understanding of this result is achieved by employing a 2PI- loop expansion for the quantum theory.
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We have written a {it Mathematica} program that calculates the integrand corresponding to any amplitude in the closed-time-path formulation of real time statistical field theory. The program is designed so that it can be used by someone with no previous experience with {it Mathematica}. It performs the contractions over the tensor indices that appear in real time statistical field theory and gives the result in the 1-2, Keldysh or RA basis. We have used the program to calculate the ward identity for the QED 3-point function, the QED 4-point function for two photons and two fermions, and the QED 5-point function for three photons and two fermions. In real time statistical field theory, there are seven 3-point functions, 15 4-point functions and 31 5-point functions. We produce a table that gives the results for all of these functions. In addition, we give a simple general expression for the KMS conditions between $n$-point green functions and vertex functions, in both the Keldysh and RA bases
Real-time classical SU($N$) gauge theories at non-zero temperature contain linear divergences. We introduce counterterms for these divergences in the equations of motion in the continuum and on the lattice. These counterterms can be given in terms of auxiliary fields that satisfy local equations of motion. We present a lattice model with 6+1D auxiliary fields that for IR-sensitive quantities yields cut-off independent results to leading order in the coupling. Also an approximation with 5+1D auxiliary fields is discussed.
Real-time evolution of replicas of classical field is proposed as an approximate simulator of real-time quantum field dynamics at finite temperatures. We consider $N$ classical field configurations dubbed as replicas which interact with each other via the $tau$-derivative terms and evolve with the classical equation of motion. The partition function of replicas is found to be proportional to that of quantum field in the imaginary time formalism. As the replica index $tau$ can be regarded as the imaginary time index, the replica evolution is technically the same as the molecular dynamics part of the hybrid Monte-Carlo sampling and the replica configurations should reproduce the correct quantum equilibrium distribution after the long-time evolution. At the same time, evolution of the replica-index average of field variables is described by the classical equation of motion when the fluctuations are small. In order to examine the real-time propagation properties of replicas, we first discuss replica evolution in quantum mechanics. Statistical averages of observables are precisely obtained by the initial condition average of replica evolution, and the time evolution of the unequal-time correlation function, $langle x(t) x(t)rangle$, in a harmonic oscillator is also described well by the replica evolution in the range $T/omega > 0.5$. Next, we examine the statistical and dynamical properties of the $phi^4$ theory in the 4+1 dimensional spacetime, which contains three spatial, one replica index or the imaginary time, and one real-time. We note that the Rayleigh-Jeans divergence can be removed in replica evolution with $N geq 2$ when the mass counterterm is taken into account. We also find that the thermal mass obtained from the unequal-time correlation function at zero momentum grows as a function of the coupling as in the perturbative estimate in the small coupling region.
We compute the suppression and elliptic flow of bottomonium using real-time solutions to the Schr{o}dinger equation with a realistic in-medium complex-valued potential. To model the initial production, we assume that, in the limit of heavy quark masses, the wave-function can be described by a lattice-smeared (Gaussian) Dirac delta wave-function. The resulting final-state quantum-mechanical overlaps provide the survival probability of all bottomonium eigenstates. Our results are in good agreement with available data for $R_{AA}$ as a function of $N_{rm part}$ and $p_T$ collected at $sqrt{s_{rm NN}} =$ 5.02 TeV. In the case of $v_2$ for the various states, we find that the path-length dependence of $Upsilon(1s)$ suppression results in quite small $v_2$ for $Upsilon(1s)$. Our prediction for the integrated elliptic flow for $Upsilon(1s)$ in the $10{-}90$% centrality class is $v_2[Upsilon(1s)] = 0.0026 pm 0.0007$. We additionally find that, due to their increased suppression, excited bottomonium states have a larger elliptic flow and we make predictions for $v_2[Upsilon(2s)]$ and $v_2[Upsilon(3s)]$ as a function of centrality and transverse momentum. Similar to prior studies, we find that it is possible for bottomonium states to have negative $v_2$ at low transverse momentum.
We derive an effective classical theory for real-time SU($N$) gauge theories at high temperature. By separating off and integrating out quantum fluctuations we obtain a 3D classical path integral over the initial fields and conjugate momenta. The leading hard mode contribution is incorporated in the equations of motion for the classical fields. This yields the gauge invariant hard thermal loop (HTL) effective equation of motion. No gauge-variant terms are generated as in treatments with an intermediate momentum cut-off. Quantum corrections to classical correlation functions can be calculated perturbatively. The 4D renormalizability of the theory ensures that the 4D counterterms are sufficient to render the theory finite. The HTL contributions of the quantum fluctuations provide the counterterms for the linear divergences in the classical theory.
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