Curvature of the pseudocritical line in QCD
Candidate: Kevin Zambello
Supervisor: Prof. Massimo D’Elia
1
Abstract
The phase diagram of QCD on the µB − T plane, where T is the temperature and µB is the baryochemical
potential, is a subject of great interest due to its relevance to various fields of physics, like heavy-ion collision experiments, the study of the early Universe and the physics of neutron stars. The boundary between the confined phase and the deconfined phase in the small µB region is known as pseudocritical line and can be
parametrized as Tc(µB) = Tc0 1 − kB µB T0 c 2 + O(µ4B) ! .
The determination of the pseudocritical line is a difficult task, because at µB 6= 0 numerical simulations
are hindered by the sign problem. However, many methods have been introduced to tackle this issue, like reweighting, Taylor expansion and analytic continuation. Recently, many numerical investigations have been carried out to determine the curvature coefficient kB of the pseudocritical line. Estimates obtained by Taylor
expansion are generally lower than those obtained by analytic continuation, but, since the transition is a crossover, care is needed when comparing results obtained by studying different observables. Because refs. [5] and [1] adopted the same discretization and used the same observable, the chiral condensate, to locate the transition, they are natural candidates for a direct comparison. Ref. [5] found kB = 0.0066(20) by Taylor
expansion, while ref. [1] found kB = 0.0132(18) by analytic continuation. The estimate by Taylor expansion
is significantly lower than the estimate by analytic continuation despite the fact that the same observable and discretization have been used in the simulations. This discrepancy shows that further investigation could be useful in order to study possible sources of systematic uncertainty.
In this work, the curvature of the pseudocritical line has been studied through numerical simulations per-formed using the tree-level Symanzik gauge action and the stout-smeared staggered fermion action; this is the same discretization adopted in refs. [1, 5]. The location of the phase transition has been determined from the inflection point of the chiral condensate using two renormalization prescriptions, ¯ψψr1 and ¯ψψr2, respectively
adopted by refs. [1] and [5]. The curvature coefficient has been calculated by Taylor expansion, using two definitions for the pseudocritical temperature: k1 has been computed by defining Tc(µB) under the hypothesis
adopted in ref. [5] of constant h ¯ψψri at Tc (hence defining Tc0 as the inflection point of h ¯ψψri(T, µB = 0) and
Tc(µB) by the relation h ¯ψψri(Tc(µB), µB) = h ¯ψψri(Tc0, 0)), while k2 has been computed, like in ref. [1], by
defining Tc(µB) as the actual inflection point of h ¯ψψri. This set-up allows to give an independent estimate for
kB, investigate the systematics and make a proper comparison with the results reported by refs. [1, 5].
No statistically significant effect due to the renormalization prescription has been observed and finite size effects are found to be negligible. The values obtained for k2on the 163× 6 and 243× 6 lattices are compatible
with those reported by ref. [1]. They are generally higher than the values obtained in this work for k1, but the
difference is within statistical errors. Current statistics is not enough to determine a reliable estimate for k2on
finer lattices and to perform an extrapolation to the continuum. The continuum extrapolation for k1, quoting
the intermediate value kB = 0.0129(40) between k1cont( ¯ψψr1) and kcont1 ( ¯ψψr2), is in agreement with previous
determinations by refs. [1, 3–5], as shown in fig. 1. Specifically the quoted estimate is found to be compatible with the determinations reported by refs. [1] and [5], but more in agreement with the former. In fig. 2, the pseudocritical line resulting from the curvature coefficient kB = 0.0129(40) and the currently accepted value
for the critical temperature Tc≈ 155(5) [2, 6] is compared with the phenomenological analyses of experimental
data from heavy-ion collisions by refs. [7, 8]. The determination of the pseudocritical line is compatible with the experimental data points, though one must stress that the pseudocritical temperature Tc resulting from
lattice calculations is here compared with the chemical freeze-out temperature TF Z, which is expected to be
lower than Tc, and more precise experimental data might not agree in the region µB ≈ 0.
0 0.005 0.01 0.015 0.02 0.025 kB This work arXiv:1102.1356 [hep-lat] arXiv:1011.3130 [hep-lat] arXiv:1410.5758 [hep-lat] arXiv:1410.5758 [hep-lat] arXiv:1403.0821 [hep-lat]
Figure 1: Curvature coefficient kB: comparison with previous determinations. From the left to the right: estimate by this work,
ref. [5], ref. [4], ref. [1], ref. [1] and ref. [3]. The red, magenta, blue and cyan colors indicate that the estimate has been obtained respectively by Taylor expansion + chiral condensate, Taylor expansion + chiral susceptibility, analytic continuation + chiral condensate and analytic continuation + chiral susceptibility.
0 20 40 60 80 100 120 140 160 180 0 100 200 300 400 500 600 700 800 Tc (µB ) (MeV) µB (MeV) arXiv:0511094 [hep-ph] arXiv:1212.2431 [nucl-th]
Figure 2: Comparison of the pseudocritical line with the chemical freeze-out curve. Red data points taken from ref. [7], blue data points taken from [8].
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