Hard-thermal-loop Perturbation Theory Research Articles

Estimation of the diffusion coefficient of heavy quarks in light of Gribov-Zwanziger action

The heavy quark momentum diffusion coefficient (κ) is one of the most essential ingredients for the Langevin description of heavy quark dynamics. In the temperature regime relevant to the heavy ion collision phenomenology, a substantial difference exists between the lattice estimations of κ and the corresponding leading order (LO) result from the hard thermal loop (HTL) perturbation theory. Moreover, the indication of poor convergence in the next-to-leading order (NLO) perturbative analysis has motivated the development of several approaches to incorporate the non-perturbative effects in the heavy quark phenomenology. In this work, we estimate the heavy quark diffusion coefficient based on the Gribov-Zwanziger prescription. In this framework, the gluon propagator depends on the temperature-dependent Gribov mass parameter, which has been obtained self-consistently from the one-loop gap equation. Incorporating this modified gluon propagator in the analysis, we find a reasonable agreement with the existing lattice estimations of κ within the model uncertainties.

Scheme dependence of two-loop hard thermal loop perturbation theory resummed SYM4,4 thermodynamics

The resummed thermodynamics of ${\cal N}=4$ supersymmetric Yang-Mills theory in four space-time dimensions ($\text{SYM}_{4,4}$) has been calculated previously to two loop order within hard thermal loop perturbation theory (HTLpt) using the canonical dimensional regularization (DRG) scheme. Herein, we revisit this calculation using the regularization by dimensional reduction (RDR) scheme. Since the RDR scheme manifestly preserves supersymmetry it is the preferred scheme, however, it is important to assess if and by how much the resummed perturbative results depend on the regularization scheme used. Comparing predictions for the scaled entropy obtained using the DRG and RDR schemes we find that for $\lambda \lesssim 6$ they are numerically very similar. We then compare the results obtained in both schemes with the strict perturbative result, which is accurate up to order $\lambda^2$, and a generalized Pad\'{e} approximant constructed from the known large-$N_c$ weak- and strong-coupling expansions. Comparing the strict perturbative expansion of the two-loop HTLpt result with the perturbative expansion to order $\lambda^2$, we find that both the DRG and RDR HTLpt calculations result in the same scheme-independent predictions for the coefficients at order $\lambda$, $\lambda^{3/2}$, and $\lambda^2 \log\lambda$, however, at order $\lambda^2$ there is a residual regularization scheme dependence.

QCD pressure: Renormalization group optimized perturbation theory confronts lattice

The quark contribution to the QCD pressure, ${P}_{q}$, is evaluated up to next-to-leading order (NLO) within the renormalization group optimized perturbation theory (RGOPT) resummation approach. To evaluate the complete QCD pressure we simply add the perturbative NLO contribution from massless gluons to the resummed ${P}_{q}$. Despite this unsophisticated approximation our results for $P={P}_{q}+{P}_{g}$ at the central scale $M\ensuremath{\sim}2\ensuremath{\pi}T$ show a remarkable agreement with lattice predictions for $0.25\ensuremath{\lesssim}T\ensuremath{\lesssim}1\text{ }\text{ }\mathrm{GeV}$. We also show that by being imbued with RG properties, the RGOPT produces a drastic reduction of the embarrassing remnant scale dependence that plagues both standard thermal perturbative QCD and hard thermal loop perturbation theory applications.

Renormalization group improved pressure for hot and dense quark matter

We apply the renormalization group optimized perturbation theory (RGOPT) to evaluate the quark contribution to the QCD pressure at finite temperatures and baryonic densities, at next-to-leading order (NLO). Our results are compared to NLO and state-of-the-art higher orders of standard perturbative QCD (pQCD) and hard thermal loop perturbation theory (HTLpt). The RGOPT resummation provides a nonperturbative approximation, exhibiting a drastically better remnant renormalization scale dependence than pQCD, thanks to built-in renormalization group invariance consistency. At NLO, upon simply adding to the RGOPT-resummed quark contributions the purely perturbative NLO glue contribution, our results show a remarkable agreement with ab initio lattice simulation data for temperatures $0.25 \lesssim T \lesssim 1 \, {\rm GeV}$, with a remnant scale dependence drastically reduced as compared to HTLpt.

Pressure of a weakly magnetized hot and dense deconfined QCD matter in one-loop hard-thermal-loop perturbation theory

We consider our recently obtained general structure of two point (self-energy and propagator) functions of quarks and gluons in a nontrivial background like a heat bath and an external magnetic field. Based on this, here we have computed free energy and pressure of quarks and gluons for a magnetized hot and dense deconfined QCD matter in weak field approximation. For heat bath we have used hard thermal loop perturbation theory (HTLpt) in presence of finite chemical potential. For weak field approximations we have obtained the pressure of QCD matter, both with and without the high temperature expansion. The results with high $T$ expansions are completely analytic and gauge independent but depends on the renormalization scale in addition to the temperature, chemical potential and the external magnetic field. We also discuss the modification of QCD Debye mass of such matter for an arbitrary magnetic field. Analytic expressions for Debye mass are also obtained for both strong and weak field approximation. It is found to exhibit some interesting features depending upon the three different scales, i.e, the quark mass, temperature and the strength of the magnetic field. The various divergences appearing in the quark and gluon free energies are regulated through appropriate counter terms. In weak field approximation, the low temperature behavior of the pressure is found to strongly depend on the magnetic field than that at high temperature. We also discuss the specific problem with one-loop HTLpt associated with the over-counting of certain orders in coupling.

Nonperturbative finite-temperature Yang-Mills theory

We present nonperturbative correlation functions in Landau-gauge Yang-Mills theory at finite temperature. The results are obtained from the functional renormalisation group within a self-consistent approximation scheme. In particular, we compute the magnetic and electric components of the gluon propagator, and the three- and four-gluon vertices. We also show the ghost propagator and the ghost-gluon vertex at finite temperature. Our results for the propagators are confronted with lattice simulations and our Debye mass is compared to hard thermal loop perturbation theory.

Three-loop hard-thermal-loop perturbation theory thermodynamics at finite temperature and finite baryonic and isospin chemical potential

In a previous paper (JHEP {\bf 05} (2014) 27), we calculated the three-loop thermodynamic potential of QCD at finite temperature $T$ and quark chemical potentials $\mu_q$ using the hard-thermal-loop perturbation theory (HTLpt) reorganization of finite temperature and density QCD. The result allows us to study the thermodynamics of QCD at finite temperature and isospin chemical potential $\mu_I$. We calculate the pressure, energy density, and entropy density, the trace anomaly, and the speed of sound at zero and nonzero $\mu_I$. The second, fourth, and sixth-order isospin susceptibilities are calculated at zero $\mu_I$. Our results can be directly compared to lattice QCD without Taylor expansions around $\mu_q=0$ since QCD has no sign problem at finite isospin chemical potential.

Renormalization group optimized perturbation theory at finite temperatures

A recently developed variant of the so-called optimized perturbation theory (OPT), making it perturbatively consistent with renormalization group (RG) properties, RGOPT, was shown to drastically improve its convergence for zero temperature theories. Here the RGOPT adapted to finite temperature is illustrated with a detailed evaluation of the two-loop pressure for the thermal scalar $ \lambda\phi^4$ field theory. We show that already at the simple one-loop level this quantity is exactly scale-invariant by construction and turns out to qualitatively reproduce, with a rather simple procedure, results from more sophisticated resummation methods at two-loop order, such as the two-particle irreducible approach typically. This lowest order also reproduces the exact large-$N$ results of the $O(N)$ model. Although very close in spirit, our RGOPT method and corresponding results differ drastically from similar variational approaches, such as the screened perturbation theory or its QCD-version, the (resummed) hard thermal loop perturbation theory. The latter approaches exhibit a sensibly degrading scale dependence at higher orders, which we identify as a consequence of missing RG invariance. In contrast RGOPT gives a considerably reduced scale dependence at two-loop level, even for relatively large coupling values $\sqrt{\lambda/24}\sim {\cal O}(1)$, making results much more stable as compared with standard perturbation theory, with expected similar properties for thermal QCD.

QCD thermodynamics at finite temperature and finite chemical potential

We calculate thermodynamic functions viz. pressure, change in pressure due to chemical potential, quark number density of hot and dense nuclear matter at finite temperature and chemical potential using three-loop Hard Thermal Loop perturbation theory.

Three loop HTL perturbation theory at finite temperature and chemical potential

In this proceedings contribution we present a recent three-loop hard-thermal-loop perturbation theory (HTLpt) calculation of the thermodynamic potential for a finite temperature and chemical potential system of quarks and gluons. We compare the resulting pressure, trace anomaly, and diagonal/off-diagonal quark susceptibilities with lattice data. We show that there is good agreement between the three-loop HTLpt analytic result and available lattice data.

Three-loop HTLpt thermodynamics at finite temperature and chemical potential

We calculate the three-loop thermodynamic potential of QCD at finite temperature and chemical potential(s) using the hard-thermal-loop perturbation theory (HTLpt) reorganization of finite temperature and density QCD. The resulting analytic thermodynamic potential allows us to compute the pressure, energy density, and entropy density of the quark-gluon plasma. Using these we calculate the trace anomaly, speed of sound, and second-, fourth-, and sixth-order quark number susceptibilities. For all observables considered we find good agreement between our three-loop HTLpt calculations and available lattice data for temperatures above approximately 300 MeV.

Hard-Thermal-Loop QCD thermodynamics and quark number susceptibility

The weak-coupling expansion of the QCD pressure is known up to the order g 6 log g . However, at experimentally relevant temperatures, the corresponding series is poorly convergent. In this proceedings, we discuss at which extent the gauge-invariant resummation scheme, Hard-Thermal-Loop perturbation theory (HTLpt), improves the apparent convergence. We first present HTLpt results for QCD thermodynamic functions up to three-loop order at vanishing chemical potential. Then, we report a preliminary HTLpt result of one-loop quark number susceptibility, probing the finite density equation of state. Our results are consistent with lattice data down to 2 − 3 T c , reinforcing the weakly-coupled quasiparticle picture in the intermediate coupling regime.

Equation of state of hot and dense QCD: resummed perturbation theory confronts lattice data

We perform a detailed analysis of the predictions of resummed perturbation theory for the pressure and the second-, fourth-, and sixth-order diagonal quark number susceptibilities in a hot and dense quark-gluon plasma. First, we present an exact one-loop calculation of the equation of state within hard-thermal-loop perturbation theory (HTLpt) and compare it to a previous one-loop HTLpt calculation that employed an expansion in the ratios of thermal masses and the temperature. We find that this expansion converges reasonably fast. We then perform a resummation of the existing four-loop weak coupling expression for the pressure, motivated by dimensional reduction. Finally, we compare the exact one-loop HTLpt and resummed dimensional reduction results with state-of-the-art lattice calculations and a recent mass-expanded three-loop HTLpt calculation.

Quark number susceptibilities from two-loop hard thermal loop perturbation theory

We use the recently obtained two-loop hard thermal loop perturbation theory thermodynamics functions of a plasma of quarks and gluons to compute the diagonal second- and fourth-order quark number susceptibilities. The two-loop hard thermal loop perturbation theory thermodynamic functions used are reliable in the limit that the ratio of the quark chemical potential to temperature is small. Using this result, we are able to obtain (semi-)analytic expressions for the quark number susceptibilities at leading- and next-to-leading-order in hard thermal loop perturbation theory. We compare the hard thermal loop perturbation theory results with perturbative quantum chromodynamics calculations, a Polyakov-loop Nambu-Jona-Lasinio model calculation, and lattice quantum chromodynamics results.

Two-loop hard thermal loop pressure at finite temperature and chemical potential

We calculate the two-loop pressure of a plasma of quarks and gluons at finite temperature and chemical potential using the hard thermal loop perturbation theory reorganization of finite temperature/density quantum chromodynamics. The computation utilizes a high temperature expansion through fourth order in the ratio of the chemical potential to temperature. This allows us to reliably access the region of high temperature and small chemical potential. We compare our final result for the leading-and next-to-leading-order hard thermal loop perturbation theory pressure at finite temperature and chemical potential with perturbative quantum chromodynamics calculations and available lattice quantum chromodynamics results.

Hard-thermal-loop QCD trace anomaly

Abstract In this proceedings I summarize results of QCD trace anomaly from recent three-loop hard-thermal-loop perturbation theory (HTLpt) calculations. I focus on the trace anomaly scaled by T 2 for pure-glue and N f = 3 QCD. The comparison to available lattice data suggests that for pure-glue QCD agreement between HTLpt results and lattice data for the trace anomaly begins at temperatures above 8 T c while when including quarks (N f = 3) agreement begins already at temperatures above 2 T c. The results in both cases indicate that at very high temperatures the T 2-scaled trace anomaly increases with temperature in accordance with the predictions of HTLpt.

A Brief Overview of Hard-Thermal-Loop Perturbation Theory

The poor convergence of quantum field theory at finite temperature has been one of the main obstacles in the practical applications of thermal QCD for decades. Here we briefly review the progress of hard-thermal-loop perturbation theory (HTLpt) in reorganizing the perturbative expansion in order to improve the convergence. The quantum mechanical anharmonic oscillator is used as a simple example to show the breakdown of weak-coupling expansion, and variational perturbation theory is introduced as an effective resummation scheme for divergent weak-coupling expansions. We discuss HTLpt thermodynamic calculations for QED, pure-glue QCD, and QCD with Nf = 3 up to three-loop order. The results suggest that HTLpt provides a systematic framework that can be used to calculate both static and dynamic quantities for temperatures relevant at LHC.

NNLO hard-thermal-loop trace anomaly for QCD

Recent results of a next-to-next-to-leading order hard-thermal-loop perturbation theory (HTLpt) calculation of the QCD thermodynamics are revisited. The focus is on the trace anomaly scaled by T2 for pure-glue and Nf = 3 QCD. The comparison to available lattice data suggests that for pure-glue QCD agreement between HTLpt and lattice data for the trace anomaly begins at temperatures above 8 Tc while when including quarks (Nf = 3) agreement begins already at temperatures above 2 Tc. The results in both cases indicate that at very high temperatures the T2-scaled trace anomaly increases with temperature in accordance with the predictions of HTLpt.

QCD trace anomaly

In this brief report we compare the predictions of a recent next-to-next-to-leading order hard-thermal-loop perturbation theory (HTLpt) calculation of the QCD trace anomaly to available lattice data. We focus on the trace anomaly scaled by ${T}^{2}$ in two cases: ${N}_{f}=0$ and ${N}_{f}=3$. When using the canonical value of $\ensuremath{\mu}=2\ensuremath{\pi}T$ for the renormalization scale, we find that for Yang-Mills theory (${N}_{f}=0$) agreement between HTLpt and lattice data for the ${T}^{2}$-scaled trace anomaly begins at temperatures on the order of $8{T}_{c}$, while when including quarks (${N}_{f}=3$) agreement begins already at temperatures above $2{T}_{c}$. In both cases we find that at very high temperatures the ${T}^{2}$-scaled trace anomaly increases with temperature in accordance with the predictions of HTLpt.

Conserved density fluctuation and temporal correlation function in hard thermal loop perturbation theory

Considering recently developed Hard Thermal Loop perturbation theory that takes into account the effect of the variation of the external field through the fluctuations of a conserved quantity we calculate the temporal component of the Euclidian correlation function in the vector channel. The results are found to be in good agreement with the very recent results obtained within the quenched approximation of QCD and small values of the quark mass ($\sim 0.1T$) on improved lattices of size $128^3\times N_\tau$ at ($N_\tau=40, \ T=1.2T_C$), ($N_\tau=48, \ T=1.45T_C$), and ($N_\tau=16, \ T=2.98T_C$), where $N_\tau$ is the temporal extent of the lattice. This suggests that the results from lattice QCD and Hard Thermal Loop perturbation theory are in close proximity for a quantity associated with the conserved density fluctuation.