Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD

Abstract

We investigate the Dirac eigenvalue spectrum (𝜌(𝜆, 𝑚 𝑙 )) to study the microscopic origin of axial anomaly in high temperature phase of QCD. We propose novel relations between the derivatives (𝜕 𝑛 𝜌(𝜆, 𝑚 𝑙 )/𝜕𝑚 𝑛 𝑙 ) of the Dirac eigenvalue spectrum with respect to the quark mass (𝑚 𝑙 ) and the (𝑛+1)-point correlations among the eigenvalues (𝜆) of the massless Dirac operator. Based on these relations, we present lattice QCD results for 𝜕 𝑛 𝜌(𝜆, 𝑚 𝑙 )/𝜕𝑚 𝑛 𝑙 (𝑛 = 1, 2, 3) with 𝑚 𝑙 corresponding to pion masses 𝑚 𝜋 = 160 − 55 MeV, and at a temperature of about 1.6 times the chiral phase transition temperature. Calculations were carried out using (2+1)-flavors of highly improved staggered quarks and the tree-level Symanzik gauge action with the physical strange quark mass, three lattice spacings 𝑎 = 0.12, 0.08, 0.06 fm, and lattices having aspect ratios 4 − 9. We find that 𝜌(𝜆 → 0, 𝑚 𝑙 ) develops a peaked structure. This peaked structure, which arises due to non-Poisson correlations within the infrared part of the Dirac eigenvalue spectrum, becomes sharper as 𝑎 → 0, and its amplitude is proportional to 𝑚 2 𝑙 . After continuum and chiral extrapolations, we find that the axial anomaly remains manifested in two-point correlation functions of scalar and pseudo-scalar mesons in the chiral limit. We demonstrate that the behavior of 𝜌(𝜆 → 0, 𝑚 𝑙 ) is responsible for it.

Gliding Down the QCD Transition Line, from Nf = 2 till the Onset of Conformality

“…By use of a subtracted condensate and related susceptibilities, as well as finite volume scaling, Refs. [13,17] find a satisfactory O(4) scaling up to nearly physical pion mass, with T c = 132 +3 −6 MeV. A recent FRG study [57] confirms these findings, but with a slightly larger T c = 142 MeV in the chiral limit.…”

“…Beyond two flavours, the issue of the anomaly becomes more subtle: the definition of a proper order parameter for axial symmetry is entangled with different susceptibilities associated with different flavours [16]. Some studies indicate restoration above T c [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], others find hints of a near-coincidence of the two transitions [5,29]. Our recent study [32], which will be reviewed in detail in Section 4, attempts at quantifying the limit of the scaling window and finds compatibility with 3D O(4), thus implicitly suggesting a separation between the two transitions.…”

“…All these approaches are prone to suffer from the contamination of regular terms, especially when the pseudo-critical temperature T s c associated with the particular observable s under consideration has a strong dependence on the breaking field, i.e., on the pion mass (see also Refs. [17,32]). These considerations suggest an alternative order parameter [32], see also [48,49], free from linear contributions:…”

“…One final comment concerns the U(1) A symmetry: given its prominent role, it is natural to resort to its analysis to try to shed more light on the symmetry pattern. However, again, the problem remains open: the current understanding is that it seems to be effectively restored above T c [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], but there is no consensus on the restoration temperature. For instance, Ref.…”

The role of strangeness in chiral and $$U(1)_A$$ restoration

“…However, there is currently no agreement as to whether this symmetry is effectively restored close to the critical temperature. While several phenomenological [13][14][15][16][17] and lattice [18][19][20][21][22] analyses for N f = 2 light flavors of mass m l = m u = m d support the idea that the U (1) A symmetry can be effectively restored at the chiral transition in the chiral limit, lattice results for N f = 2 + 1 (i.e., including the strange quark flavor with mass m s m l ) suggest that the anomalous U (1) A symmetry is still broken in the chiral crossover region [23][24][25][26]. This phenomenon has also implications for the hadron spectrum [11,14,27] as well as phenomenological effects driven by the associated reduction of the anomalous η mass and the topological susceptibility in a thermal environment [28][29][30][31].…”