NNuclear Physics A 00 (2018) 1–4
NuclearPhysics A
The QCD Equation of State to O ( µ B ) from Lattice QCD Prasad Hegde (for the BNL-Bielefeld-CCNU Collaboration) Key Laboratory of Quark and Lepton Physics (MOE) & Institute of Particle Physics, Central China Normal University, Wuhan 430079, China.
Abstract
We present first results from a first-principles calculation of the QCD equation of state to O ( µ B ), where µ B is the baryon chemicalpotential. We find that second-order corrections are su ffi cient for a large part of the freeze-out temperature and baryon chemicalpotential range achieved by the RHIC beam energy scan. Nevertheless, higher-order corrections are necessary to extend the validityof the equation of state down to beam energies s / NN ∼
20 GeV.
Keywords:
Heavy-Ion Collisions, Lattice QCD, Equation of State, Beam Energy Scan, Quark Number Susceptibilities.
1. Introduction
Lattice results for the Equation of State (EoS) have long been used to model the hydrodynamic evolution of thethermal matter that is created in heavy-ion collisions. Although state-of-the-art results exist for the EoS at µ B = µ B . Specifically,values µ B ≈ µ B = µ B > ffi cients of the expansion bear a straightforward interpretation as either the cumulants of the various conservedcharge distributions (diagonals), or as the correlations between them (o ff -diagonals). Because of this, they can beused to probe deconfinement [7] and they can also be measured in experiments via the moments of di ff erent hadronmultiplicity distributions [8].The basic thermodynamic quantity is the pressure p , which at µ B > pT = ∞ (cid:88) i , j , k = χ i jk i ! j ! k ! (cid:18) µ B T (cid:19) i (cid:18) µ Q T (cid:19) j (cid:18) µ S T (cid:19) k −→ ∞ (cid:88) n = c n (cid:18) µ B T (cid:19) n . (1)With three flavors of quarks, one has three chemical potentials. A change of basis allows us to express these interms of conserved charge chemical potentials: baryon number, electric charge and strangeness: ( µ B , µ Q , µ S ). Eq. (1) Email: [email protected]. a r X i v : . [ h e p - l a t ] A ug . Hegde / Nuclear Physics A 00 (2018) 1–4 is a completely general expression. However, by specializing to the case of heavy-ion collisions and taking intoaccount the constraints coming from the initial conditions , we can express µ Q and µ S in terms of µ B . This makesour expansion e ff ectively one-dimensional i.e. in the variable µ B . The c n coe ffi cients, as well as µ Q and µ S , can allbe expressed in terms of the χ i jk ; thus it su ffi ces to calculate all the χ i jk upto a certain order. Current perturbativecalculations are applicable only for temperatures T (cid:38)
350 MeV or so [9, 10]. Thus the calculation of susceptibilitiesaround and just above the crossover region is a non-perturbative problem requiring the use of lattice techniques.
2. Results S = 0, n Q = 0.4n B O( µ B2 ) T[MeV] SBN τ = 86HRG 0.00000.00050.00100.00150.00200.0025 140 160 180 200 220 240 260 280n S = 0, n Q = 0.4n B O( µ B4 ) T[MeV]SBN τ = 86HRG -0.0003-0.00010.00010.00030.0005 140 150 160 170 180 190 200 210 220n S = 0, n Q = 0.4n B O( µ B6 ) SBT[MeV]N τ = 86HRG Figure 1. (From left to right) Lattice results for the coe ffi cients c , c and c with initial conditions appropriate to Pb-Pb collisions ( n S = N p = . N p + N n )) for N τ = T c = Fig. 1 shows our preliminary results for the Taylor coe ffi cients c , c and c . These were calculated with staggeredfermions using the state-of-the-art HISQ action [12]. We computed all the susceptibilities upto sixth order at twolattice spacings, namely a = / N τ T with N τ = (cid:46) T (cid:46)
330 MeV. Sincewe had only two values of the lattice spacing, we did not attempt to perform a continuum extrapolation. Neverthelesswe found that cuto ff e ff ects were under control, especially for c but also for c as well . Our dominant errors infact were statistical, as Fig. 1 shows, especially for c and to an extent for c . Even taking the large errors on c into consideration, we still found that | c | < c (cid:28) c . As a result we were able to extrapolate to O ( µ B ) for variousobservables for fairly large values of µ B / T . to O( µ B2 ) n S = 0, n Q = 0.4n B N τ = 8 T[MeV] µ B /T = 02.03.0 0.51.01.52.02.5 145 155 165 175 185 195 205p/T to O( µ B4 ) n S = 0, n Q = 0.4n B N τ = 8 T[MeV] µ B /T = 02.03.0 0.51.01.52.02.5 145 155 165 175 185 195 205p/T to O( µ B4 ) n S = 0, n Q = 0.4n B N τ = 6 T[MeV] µ B /T = 02.03.0 Figure 2. p / T calculated for N τ = O ( µ B ) (left) and O ( µ B ) (center), and to O ( µ B ) for N τ = T ≤ T c are the corresponding HRG results. The zeroth-order results are taken from Ref. [1]. We show our results for the pressure, energy density and entropy density in Figs. 2 and 3. Wherever possible, wehave shown results for both N τ = ff e ff ects are small for all the observables shown here.The energy and entropy densities are obtained from the pressure from ε T = ∞ (cid:88) n = (cid:18) µ B T (cid:19) n (cid:40) T dc n dT + c n (cid:41) and sT = ∞ (cid:88) n = (cid:18) µ B T (cid:19) n (cid:40) T dc n dT + (4 − n ) c n (cid:41) . (2) The initial conditions are: zero net strangeness ( n S = N p / ( N p + N n ) = r ). Using these, µ Q and µ S may be determined upto any given order in µ B [13]. In fact, the electric charge sector does su ff er from cuto ff e ff ects at these spacings [14]. However, since we are expanding with respect to µ B ,the contribution of this sector is suppressed. . Hegde / Nuclear Physics A 00 (2018) 1–4 ε /T : 2 nd (open) vs. 4 th order (filled)n S = 0, n Q = 0.4n B N τ = 6 µ /T = 02.0 3.0 2 4 6 8 10 12 14 150 160 170 180 190 200 210 220T[MeV] ε /T : 2 nd (open) vs. 4 th order (filled)n S = 0, n Q = 0.4n B N τ = 8 µ /T = 02.0 3.0 2 4 6 8 10 12 14 150 160 170 180 190 200 210 220T[MeV]s/T : 2 nd (open) vs. 4 th order (filled)n S = 0, n Q = 0.4n B N τ = 8 µ /T = 02.0 3.0 Figure 3. Comparision between second and fourth orders for the energy density ε normalized to T for N τ = N τ = N τ = It is readily seen that the main correction for these values of µ B / T comes from second-order susceptibilities. Howeverfourth-order corrections do contribute, especially for temperatures in the important crossover region and lower. Inthe case of the energy and entropy densities (Fig. 3), the contribution of the derivative term in Eq. (2) also becomessignificant as the coe ffi cient c rises more rapidly than c in the crossover region. The fourth-order contribution willalso be larger for larger values of µ B / T ; however for these values, it is likely that the sixth-order contribution cannotbe neglected any more.
3. Observables on the Freezeout Curve [GeV] n B [fm -3 ]N τ = 8n S = 0, n Q = 0.4n B O( µ B )O( µ B3 )O( µ B5 ) NN1/2 ε [GeV/fm ] n S = 0, n Q = 0.4n B N τ = 8 O(µ B0 )O(µ B2 )O(µ B4 ) Figure 4. (Left) The baryon density at freezeout when calculated upto leading ( O ( µ B )), next-to-leading ( O ( µ B )) and next-to-next-to-leading ( O ( µ B ))order. (Right) Energy density at freeze-out. In the BES at RHIC, as the center-of-mass energy s / NN is decreased, the chemical potential at freeze-out µ fB increases steadily while the freeze-out temperature T f changes by only about 10-15%. While at present the EoS isknown upto O ( µ B ) [15], as one goes to lower energies, fourth-order and possibly even higher-order corrections mayhave to be taken into account. The determination of freeze-out parameters T f and µ fB is ongoing in RHIC and LHCexperiments. In particular, the estimate for the freeze-out temperature has decreased recently [16]. In our currentpreliminary analysis we nonetheless use the well-known parametrization of the freeze-out curve by Cleymans etal. [3, 17] to point out some basic features of our Taylor expansion on the freeze-out curve, such as the value of s / NN at which higher-order corrections start to become important.As we saw in the previous section, in both the pressure and the energy density, the combination of zeroth andsecond-order terms accounted for practically the entire contribution. By contrast, the baryon number density receivesits leading contribution from O ( µ B ) susceptibilities and its first corrections from the O ( µ B ) ones. This makes it a goodobservable to study the impact of higher-order corrections.Fig. 4 (left) plots the baryon number density, in units of fm − , on the freezeout curve as a function of the beamenergy. We see that the leading-order description is a good one down to s / NN ∼
30 GeV, at which point the leading and3 . Hegde / Nuclear Physics A 00 (2018) 1–4 next-to-leading order results start to di ff er. Going to still lower energies, we find that similarly, the O ( µ B ) and O ( µ B )terms seem to start to di ff er below s / NN ∼
15 GeV.A second way in which higher-order corrections might be important is seen from the energy density plot in thesame figure. Freeze-out is believed to happen when the energy density drops to a certain, constant value. We can checkthis hypothesis by calculating the energy density on the freeze-out curve. The zeroth-order result, which only takesthe temperature dependence into account, remains roughly constant down to s / NN ∼
50 GeV, below which it seems todip. When O ( µ B ) and O ( µ B ) terms are included, the constant region is extended to slightly lower energies s / NN ∼ ε ∼ . / fm , is veryclose to the value in the crossover region and at the “softest point” of the µ B =
4. Conclusions
The QCD equation of state is a necessary input in hydrodynamic models of heavy-ion collisions, and calculating itfrom first principles has been one of the major programs in lattice QCD. With experiments to probe nuclear matter atfinite density either running (BES) or planned (BES-II and FAIR), the interest has shifted to equations of state at µ B >
0. In this work, we presented first results from an ongoing calculation by the BNL-Bielefeld-CCNU collaborationto calculate the EoS to fourth order in the baryon chemical potential. We expect that this will eventually yield anequation of state that is valid down to beam energies of s / NN ∼
20 GeV and lower.The author is partially supported by grant QLPL2014P01 of the Ministry of Education, China. The numericalcalculations described here have been performed at JLab and at Indiana University in the United States and at BielefeldUniversity and Paderborn University in Germany. We acknowledge the support of Nvidia through the CUDA ResearchCenter at Bielefeld University.
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