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Register a new userThe energy-momentum tensor in lattice QCD and the Equation of State
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We present a new theoretical and practical strategy to renormalize non-perturbatively the energy-momentum tensor in lattice QCD based on the framework of shifted boundary conditions. As a preparatory step for the fully non-perturbative calculation, we apply the strategy at 1-loop order in perturbation theory determining the renormalization constants of both the gluonic and the fermionic components of the energy-momentum tensor. Using shifted boundary conditions, the entropy density of QCD is directly related to the expectation value of the space-time components of the renormalized energy-momentum tensor. We then discuss its practical implementation by numerical simulations of QCD with 3 flavours of Wilson quarks for temperatures between 2.5 GeV and 80 GeV.
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We report results on the proton mass decomposition and also on related quark and glue momentum fractions. The results are based on overlap valence fermions on four ensembles of $N_f = 2+1$ DWF configurations with three lattice spacings and three volumes, and several pion masses including the physical pion mass. With fully non-perturbative renormalization (and universal normalization on both quark and gluon), we find that the quark energy and glue field energy contribute 33(4)(4)% and 37(5)(4)% respectively in the $overline{MS}$ scheme at $mu = 2$ GeV. A quarter of the trace anomaly gives a 23(1)(1)% contribution to the proton mass based on the sum rule, given 9(2)(1)% contribution from the $u, d,$ and $s$ quark scalar condensates. The $u,d,s$ and glue momentum fractions in the $overline{MS}$ scheme are in good agreement with global analyses at $mu = 2$ GeV.
The energy density and the pressure of SU(3) gauge theory at finite temperature are studied by direct lattice measurements of the renormalized energy-momentum tensor obtained by the gradient flow. Numerical analyses are carried out with $beta=6.287$--$7.500$ corresponding to the lattice spacing $a= 0.013$--$0.061,mathrm{fm}$. The spatial (temporal) sizes are chosen to be $N_s= 64$, $96$, $128$ ($N_{tau}=12$, $16$, $20$, $22$, $24$) with the aspect ratio, $5.33 le N_s/N_{tau} le 8$. Double extrapolation, $arightarrow 0$ (the continuum limit) followed by $trightarrow 0$ (the zero flow-time limit), is taken using the numerical data. Above the critical temperature, the thermodynamic quantities are obtained with a few percent precision including statistical and systematic errors. The results are in good agreement with previous high-precision data obtained by using the integral method.
We employ a new strategy for a non perturbative determination of the renormalized energy momentum tensor. The strategy is based on the definition of suitable lattice Ward identities probed by observables computed along the gradient flow. The new set of identities exhibits many interesting qualities, arising from the UV finiteness of flowed composite operators. In this paper we show how this method can be used to non perturbatively renormalize the energy momentum tensor for a SU(3) Yang-Mills theory, and report our numerical results.
We present results for the equation of state upto previously unreachable, high temperatures. Since the temperature range is quite large, a comparison with perturbation theory can be done directly.
We calculated the QCD equation of state using Taylor expansions that include contributions from up to sixth order in the baryon, strangeness and electric charge chemical potentials. Calculations have been performed with the Highly Improved Staggered Quark action in the temperature range $Tin [135~{rm MeV}, 330~{rm MeV}]$ using up to four different sets of lattice cut-offs corresponding to lattices of size $N_sigma^3times N_tau$ with aspect ratio $N_sigma/N_tau=4$ and $N_tau =6-16$. The strange quark mass is tuned to its physical value and we use two strange to light quark mass ratios $m_s/m_l=20$ and $27$, which in the continuum limit correspond to a pion mass of about $160$ MeV and $140$ MeV espectively. Sixth-order results for Taylor expansion coefficients are used to estimate truncation errors of the fourth-order expansion. We show that truncation errors are small for baryon chemical potentials less then twice the temperature ($mu_Ble 2T$). The fourth-order equation of state thus is suitable for the modeling of dense matter created in heavy ion collisions with center-of-mass energies down to $sqrt{s_{NN}}sim 12$ GeV. We provide a parametrization of basic thermodynamic quantities that can be readily used in hydrodynamic simulation codes. The results on up to sixth order expansion coefficients of bulk thermodynamics are used for the calculation of lines of constant pressure, energy and entropy densities in the $T$-$mu_B$ plane and are compared with the crossover line for the QCD chiral transition as well as with experimental results on freeze-out parameters in heavy ion collisions. These coefficients also provide estimates for the location of a possible critical point. We argue that results on sixth order expansion coefficients disfavor the existence of a critical point in the QCD phase diagram for $mu_B/Tle 2$ and $T/T_c(mu_B=0) > 0.9$.
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