Chiral symmetry restoration and scalar-pseudoscalar partners in QCD

Abstract

We describe Scalar-Pseudoscalar partner degeneration at the QCD chiral transition in terms of the dominant low-energy physical states for the light quark sector. First, we obtain within modelindependent one-loop Chiral Perturbation Theory (ChPT) that the QCD pseudoscalar susceptibility is proportional to the quark condensate at low T . Next, we show that this chiral-restoring behaviour for χP is compatible with recent lattice results for screening masses and gives rise to degeneration between the scalar and pseudoscalar susceptibilities (χS, χP ) around the transition point, consistently with an O(4)-like current restoration pattern. This scenario is clearly confirmed by lattice data when we compare χS(T ) with the quark condensate, expected to scale as χP (T ). Finally, we show that saturating χS with the σ/f0(500) broad resonance observed in pion scattering and including its finite temperature dependence, allows to describe the peak structure of χS(T ) in lattice data and the associated critical temperature. This is carried out within a unitarized ChPT scheme which generates the resonant state dynamically and is also consistent with partner degeneration.

Chiral symmetry restoration for the large-Npion gas

“…Finally, we also compare our results with the analysis performed in [14] and [16], from which one can define a scalar susceptibility saturated by the thermal f 0 ð500Þ state, with a mass corresponding to that scalar resonance defined as M 2 is the position of the pole in the second Riemann sheet of the scattering partial wave with isospin and angular momentum I ¼ J ¼ 0 calculated also at finite temperature within the large-N limit in [16] and with the IAM in [14] around the chiral limit. In order for such saturated susceptibility to comply properly with the expected low-T behavior given by the leading order ChPT in the chiral limit, we define it as…”

“…Critical exponents for different universality classes and the present analysis. We include for reference also the result obtained from saturating the susceptibility with the thermal f 0 ð500Þ pole as discussed in [14,16] and in the main text.…”

“…It provides also a novel understanding of the role of the σ=f 0 ð500Þ I ¼ J ¼ 0 thermal pole (a broad resonant state) in chiral symmetry restoration. Thus, the scalar susceptibility saturated with this σ-like state within the so-called inverse amplitude method (IAM) unitarization develops a maximum near T c [14] compatible with lattice data and chiral partners in the scalar-pseudoscalar sector are understood through degeneration of correlators and susceptibilities. The role of the f 0 ð500Þ state for chiral restoration could become more complicated if its possible tetraquark component is also considered at finite temperature [15].…”

“…Therefore, mass corrections might not be reliable for physical masses. A related complication introduced by the large-N resummation is that, as shown above, from z 2 onwards the effective vertex gives rise actually to M 2g ðG 1 =F 2 Þ, withg given in (14), which may introduce spurious divergences near T c when keeping a finite mass M. For these reasons, we stick here to the strict chiral limit M → 0 þ since it ensures that those two dimensionless scales can be treated as perturbatively equivalent. Thus, we keep only the M 0 term in (17) for the condensate and the M −1 term for the scalar susceptibility.…”

“…n from the modified effective vertex (14), counting all the ways to take in each effective vertex the two pion lines shared with the adjacent vertices, and where we have taken into account that there is a tadpole function G 1 attached to every vertex in the ring diagrams; (ii) by 2 n for the two possible ways to take those shared lines in every link; (iii) by ð1=n!Þ from the series expansion of the interaction Lagrangian; (iv) by ðn − 1Þ!=2, which are the topologically different ways of sorting n points in a circle; and (v) by N nþ1 from the loops. Finally we obtain…”