Dimensional Reduction, Hard Thermal Loops and the Renormalization Group

We study the realization of dimensional reduction and the validity of the hard thermal loop expansion for lambda phi^4 theory at finite temperature, using an environmentally friendly finite-temperature renormalization group with a fiducial temperature as flow parameter. The one-loop renormalization group allows for a consistent description of the system at low and high temperatures, and in particular of the phase transition. The main results are that dimensional reduction applies, apart from a range of temperatures around the phase transition, at high temperatures (compared to the zero temperature mass) only for sufficiently small coupling constants, while the HTL expansion is valid below (and rather far from) the phase transition, and, again, at high temperatures only in the case of sufficiently small coupling constants. We emphasize that close to the critical temperature, physics is completely dominated by thermal fluctuations that are not resummed in the hard thermal loop approach and where universal quantities are independent of the parameters of the fundamental four-dimensional theory.Comment: 20 pages, 13 eps figures, uses epsfig and pstrick

The pressure of deconfined QCD for all temperatures and quark chemical potentials

We present a new method for the evaluation of the perturbative expansion of the QCD pressure which is valid at all values of the temperature and quark chemical potentials in the deconfined phase and which we work out up to and including order g^4 accuracy. Our calculation is manifestly four-dimensional and purely diagrammatic -- and thus independent of any effective theory descriptions of high temperature or high density QCD. In various limits, we recover the known results of dimensional reduction and the HDL and HTL resummation schemes, as well as the equation of state of zero-temperature quark matter, thereby verifying their respective validity. To demonstrate the overlap of the various regimes, we furthermore show how the predictions of dimensional reduction and HDL resummed perturbation theory agree in the regime T~\sqrt{g}*mu. At parametrically smaller temperatures T~g*mu, we find that the dimensional reduction result agrees well with those of the nonstatic resummations down to the remarkably low value T~0.2 m_D, where m_D is the Debye mass at T=0. Beyond this, we see that only the latter methods connect smoothly to the T=0 result of Freedman and McLerran, to which the leading small-T corrections are given by the so-called non-Fermi-liquid terms, first obtained through HDL resummations. Finally, we outline the extension of our method to the next order, where it would include terms for the low-temperature entropy and specific heats that are unknown at present.Comment: 45 pages, 21 figures; v2: minor corrections and clarifications, references added; v3: Fig 16 added, version accepted for publication in PR

Non-perturbative Debye mass in finite T QCD

Employing a non-perturbative gauge invariant definition of the Debye screening mass m_D in the effective field theory approach to finite T QCD, we use 3d lattice simulations to determine the leading O(g^2) and to estimate the next-to-leading O(g^3) corrections to m_D in the high temperature region. The O(g^2) correction is large and modifies qualitatively the standard power-counting hierarchy picture of correlation lengths in high temperature QCD.Comment: 4 pages, Late

Two-color QCD via dimensional reduction

We study the thermodynamics of two-color QCD at high temperature and/or density using a dimensionally reduced superrenormalizable effective theory, formulated in terms of a coarse grained Wilson line. In the absence of quarks, the theory is required to respect the Z(2) center symmetry, while the effects of quarks of arbitrary masses and chemical potentials are introduced via soft Z(2) breaking operators. Perturbative matching of the effective theory parameters to the full theory is carried out explicitly, and it is argued how the new theory can be used to explore the phase diagram of two-color QCD.Comment: 17 pages, 1 eps figure, jheppub style; v2: minor update, references added, published versio

On the Infrared Behavior of the Pressure in Thermal Field Theories

We study non-perturbatively, via the Schwinger-Dyson equations, the leading infrared behavior of the pressure in the ladder approximation. This problem is discussed firstly in the context of a thermal scalar field theory, and the analysis is then extended to the Yang-Mills theory at high temperatures. Using the Feynman gauge, we find a system of two coupled integral equations for the gluon and ghost self-energies, which is solved analytically. The solutions of these equations show that the contributions to the pressure, when calculated in the ladder approximation, are finite in the infrared domain.Comment: 20 pages plus 4 figures available by request, IFUSP/P-100

One-Loop QCD Corrections to the Thermal Wilson Line Model

We calculate the time independent four-point function in high temperature (T) QCD and obtain the leading momentum dependent terms. Furthermore, we relate these derivative interactions to derivative terms in a recently proposed finite T effective action based on the SU(3) Wilson Line and its trace, the Polyakov Loop. By this procedure we thus obtain a perturbative matching at finite T between QCD and the effective model. In particular, we calculate the leading perturbative QCD-correction to the kinetic term for the Polyakov Loop.Comment: Minor changes, one reference adde

Pressure to order g8∗log(g)g^8*log(g) in ϕ4\phi^4-theory at weak coupling

We calculate the pressure of massless $\phi^4$-theory to order $g^8\log(g)$ at weak coupling. The contributions to the pressure arise from the hard momentum scale of order $T$ and the soft momentum scale of order $gT$. Effective field theory methods and dimensional reduction are used to separate the contributions from the two momentum scales: The hard contribution can be calculated as a power series in $g^2$ using naive perturbation theory with bare propagators. The soft contribution can be calculated using an effective theory in three dimensions, whose coefficients are power series in $g^2$. This contribution is a power series in $g$ starting at order $g^3$. The calculation of the hard part to order $g^6$ involves a complicated four-loop sum-integral that was recently calculated by Gynther, Laine, Schr\"oder, Torrero, and Vuorinen. The calculation of the soft part requires calculating the mass parameter in the effective theory to order $g^6$ and the evaluation of five-loop vacuum diagrams in three dimensions. This gives the free energy correct up to order $g^7$. The coefficients of the effective theory satisfy a set of renormalization group equations that can be used to sum up leading and subleading logarithms of $T/gT$. We use the solutions to these equations to obtain a result for the free energy which is correct to order $g^8\log(g)$. Finally, we investigate the convergence of the perturbative series.Comment: 29 pages and 12 figs. New version: we have pushed the calculations to g^8*log(g) using the renormalization group to sum up log(g) from higher orders. Published in JHE

Entropy from AdS(3)/CFT(2)

We parametrize the (2+1)-dimensional AdS space and the BTZ black hole with Fefferman-Graham coordinates starting from the AdS boundary. We consider various boundary metrics: Rindler, static de Sitter and FRW. In each case, we compute the holographic stress-energy tensor of the dual CFT and confirm that it has the correct form, including the effects of the conformal anomaly. We find that the Fefferman-Graham parametrization also spans a second copy of the AdS space, including a second boundary. For the boundary metrics we consider, the Fefferman-Graham coordinates do not cover the whole AdS space. We propose that the length of the line delimiting the excluded region at a given time can be identified with the entropy of the dual CFT on a background determined by the boundary metric. For Rindler and de Sitter backgrounds our proposal reproduces the expected entropy. For a FRW background it produces a generalization of the Cardy formula that takes into account the vacuum energy related to the expansion.Comment: major revision with several clarifications and corrections, 22 page

A fresh look on three-loop sum-integrals

In order to prepare the ground for evaluating classes of three-loop sum-integrals that are presently needed for thermodynamic observables, we take a fresh and systematic look on the few known cases, and review their evaluation in a unified way using coherent notation. We do this for three important cases of massless bosonic three-loop vacuum sum-integrals that have been frequently used in the literature, and aim for a streamlined exposition as compared to the original evaluations. In passing, we speculate on options for generalization of the computational techniques that have been employed.Comment: 19 page

Quark-Gluon Plasma as a Condensate of Z(3) Wilson Lines

Effective theories for the thermal Wilson line are constructed in an SU(N) gauge theory at nonzero temperature. I propose that the order of the deconfining phase transition for Z(N) Wilson lines is governed by the behavior of SU(N) Wilson lines. In a mean field theory, the free energy in the deconfined phase is controlled by the condensate for Z(N) Wilson lines. Numerical simulations on the lattice, and the mean field theory for Z(3) Wilson lines, suggest that about any finite temperature transition in QCD, the dominant correlation length increases by a large, uniform factor, of order five.Comment: 5 pages, LaTe