We present a general systematic formalism for describing dynamics of fluctuations in an arbitrary relativistic hydrodynamic flow, including their feedback (known as long-time hydrodynamic tails). The fluctuations are described by two-point equal-time correlation functions. We introduce a definition of equal time in a situation where the local rest frame is determined by the local flow velocity, and a method of taking derivatives and Wigner transforms of such equal-time correlation functions, which we call confluent. We find that the equations for confluent Wigner functions not only resemble kinetic equations, but that the kinetic equation for phonons propagating on an arbitrary background nontrivially matches the equations for Wigner functions, including relativistic inertial and Coriolis forces due to acceleration and vorticity of the flow. We also describe the procedure of renormalization of short-distance singularities which eliminates cutoff dependence, allowing efficient numerical implementation of these equations.
To describe dynamics of bulk and fluctuations near the QCD critical point we develop general relativistic fluctuation formalism for a fluid carrying baryon charge. Feedback of fluctuations modifies hydrodynamic coefficients including bulk viscosity and conductivity and introduces nonlocal and noninstantaneous terms in constitutive equations. We perform necessary ultraviolet (short-distance) renormalization to obtain cutoffindependent deterministic equations suitable for numerical implementation. We use the equations to calculate the universal nonanalytic small-frequency dependence of transport coefficients due to fluctuations (long-time tails). Focusing on the critical mode we show how this general formalism matches existing Hydro+ description of fluctuations near the QCD critical point and nontrivially extends it inside and outside of the critical region.
We determine the scaling properties of the Yang-Lee edge singularity as described by a onecomponent scalar field theory with imaginary cubic coupling, using the nonperturbative functional renormalization group in 3 ≤ d ≤ 6 Euclidean dimensions. We find very good agreement with high-temperature series data in d = 3 dimensions and compare our results to recent estimates of critical exponents obtained with the four-loop = 6−d expansion and the conformal bootstrap. The relevance of operator insertions at the corresponding fixed point of the RG β functions is discussed and we estimate the error associated with O(∂ 4 ) truncations of the scale-dependent effective action.
We address a number of outstanding questions associated with the analytic properties of the universal equation of state of the φ 4 theory, which describes the critical behavior of the Ising model and ubiquitous critical points of the liquid-gas type. We focus on the relation between spinodal points that limit the domain of metastability for temperatures below the critical temperature, i.e., T < T c , and Lee-Yang edge singularities that restrict the domain of analyticity around the point of zero magnetic field H for T > T c . The extended analyticity conjecture (due to Fonseca and Zamolodchikov) posits that, for T < T c , the Lee-Yang edge singularities are the closest singularities to the real H axis. This has interesting implications, in particular, that the spinodal singularities must lie off the real H axis for d < 4, in contrast to the commonly known result of the mean-field approximation. We find that the parametric representation of the Ising equation of state obtained in the ε = 4 − d expansion, as well as the equation of state of the O(N )-symmetric φ 4 theory at large N , are both nontrivially consistent with the conjecture. We analyze the reason for the difficulty of addressing this issue using the ε expansion. It is related to the long-standing paradox associated with the fact that the vicinity of the Lee-Yang edge singularity is described by Fisher's φ 3 theory, which remains nonperturbative even for d → 4, where the equation of state of the φ 4 theory is expected to approach the mean-field result. We resolve this paradox by deriving the Ginzburg criterion that determines the size of the region around the Lee-Yang edge singularity where mean-field theory no longer applies. 1 arXiv:1707.06447v2 [hep-th]
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