Chiral Symmetry Breaking and Cooling in Lattice QCD

TL;DR

This paper investigates how cooling affects chiral symmetry breaking in quenched lattice QCD, revealing persistent symmetry breaking linked to instantons even after extensive cooling, though the condensate diminishes.

Contribution

It provides the first detailed analysis of the evolution of chiral symmetry breaking under cooling in lattice QCD, highlighting the role of instantons.

Findings

Chiral condensate remains non-zero after over 100 cooling steps.

Chiral symmetry breaking is associated with instantons.

Cooling reduces the magnitude of the chiral condensate.

Abstract

Chiral symmetry breaking is calculated as a function of cooling in quenched lattice QCD. A non-zero signal is found for the chiral condensate beyond one hundred cooling steps, suggesting that there is chiral symmetry breaking associated with instantons. Quantitatively, the chiral condensate in cooled gauge field configurations is small compared to the value without cooling.

Tables

Table 1. TABLE I.: Chiral order parameter values calculated for different values of the quark mass along with the value extrapolated to zero mass.

		ma
$\beta$	Cooling step	0.2	0.15	0.1	0.05	Extrapolated value
5.7	0	0.889(2)	0.788(2)	0.660(2)	0.494(3)	0.30(1)
	20	0.3659(3)	0.2835(5)	0.1986(7)	0.115(2)	0.028(4)
	40	0.3613(5)	0.2783(5)	0.1930(8)	0.110(2)	0.023(5)
	60	0.3600(4)	0.2768(6)	0.1913(9)	0.109(2)	0.020(4)
	80	0.3593(5)	0.2761(6)	0.1907(9)	0.107(2)	0.020(4)
	100	0.3592(5)	0.2757(6)	0.1899(9)	0.107(2)	0.020(5)
5.9	0	0.7775(7)	0.6589(8)	0.5104(9)	0.316(1)	0.091(4)
	20	0.3595(2)	0.2760(2)	0.1892(3)	0.1009(5)	0.009(2)
	40	0.3583(2)	0.2745(2)	0.1874(3)	0.099(6)	0.007(2)
	60	0.3576(2)	0.2738(2)	0.1866(3)	0.098(6)	0.005(2)
	80	0.3576(2)	0.2736(2)	0.1862(3)	0.098(6)	0.005(2)
	100	0.3573(2)	0.2732(2)	0.1859(3)	0.097(6)	0.005(2)

Equations

$$S_f$$

$$\equiv$$

$$⟨\overline{\chi }\chi ⟩=\frac{1}{V}⟨\text{mbox}Tr{\mathcal{M}}^{-1}(\{U\})⟩,$$

$$⟨\overline{\chi }\chi ⟩(m)=⟨\overline{\chi }\chi {⟩}_0+⟨\overline{\chi }\chi {⟩}_{1}m+⟨\overline{\chi }\chi {⟩}_2{m}^2.$$

$$q_t(x)=\frac{1}{32{\pi }^2}\sum _{\mu \nu \rho \sigma }{ϵ}_{\mu \nu \rho \sigma }\text{mbox}Tr\{U_{\mu \nu }(x)U_{\rho \sigma }(x)\},$$

Full text

††preprint:

*to appear in Phys. Rev. D *TRI-PP-95-58

Recently Chu et al [1] presented evidence from a quenched SU(3) lattice gauge theory that instantons play a dominant role in determining hadronic correlation functions. Motivated by this work, a calculation was done in quenched SU(2) lattice gauge theory exploring the effect of cooling [2]. In Ref. [2] it was found that the spin-dependent potential and the chiral symmetry breaking order parameter disappear very quickly upon cooling. In a subsequent nonquenched SU(3) calculation by Edwards et al [3], it was found that cooling has a large effect on the hadron spectrum, tending, for example, to diminish spin-slittings. This supports the conclusions of Ref. [2]. However, Edwards et al [3] did not see the rapid disappearance of the chiral order parameter found in the quenched SU(2) calculation. For this reason we have decided to revisit the problem of chiral symmetry breaking and cooling. A calculation was done in quenched SU(3) lattice gauge theory to see if the different chiral symmetry breaking pattern observed in Ref. [2] and in Edwards et al [3] is due to the use of SU(2) versus SU(3) color or due to the quenched versus nonquenched nature of the simulation.

In this paper we present results for the chiral order parameter calculated in a set of quenched SU(3)-color gauge configurations to which local cooling [4, 5] has been applied. A persistent non-vanishing signal for chiral symmetry breaking was found even for cooling of a hundred sweeps or more. This differs from what was observed in the earlier quenched SU(2) simulation [2] but is consistent with the finding of Edwards et al [3] in their nonquenched SU(3) calculation. Quantitatively, after cooling, the chiral order parameter is small (in lattice units) compared to the uncooled value.

The calculations were done using standard methods of Ref. [2] carried over to SU(3). The fermion action is

[TABLE]

where $\overline{\chi},\chi$ are single-component fermion fields, $\eta_{\mu}(x)$ is the staggered fermion phase  [6], $m$ is the mass (in units of the inverse of the lattice spacing $a$) and the $U^{\prime}s$ are gauge field links. Antiperiodic boundary conditions were used for the staggered fermion fields in all directions.

The chiral symmetry order parameter is calculated from the inverse of the fermion matrix ${\cal M}$ of Eq. (2)

[TABLE]

where V is the lattice volume and the angle brackets indicate a gauge field configuration average. The random source method  [7] was used to calculate $\mbox{Tr}{\cal M}^{-1}(\{U\})$. Twelve Gaussian random sources were used for each gauge field configuration.

Calculations were done on a $10^{4}$ lattice at $\beta=5.7$ and on a $12^{4}$ at $\beta=5.9$. The SU(3) pseudo-heatbath method was used for updating the quenched gauge field configurations. After 4000 sweeps to thermalize, configurations were analyzed every 300 sweeps. A total of 30 configurations at $\beta=5.7$ and 20 configurations at $\beta=5.9$ were included in the analysis. Each configuration was cooled using a local cooling algorithm [4, 5] and $\langle\overline{\chi}\chi\rangle$ was calculated along the way.

The results are tabulated in Table I. The results for $\langle\overline{\chi}\chi\rangle$ as a function of quark mass at no cooling and after 20 cooling sweeps are shown in Fig. 1 and Fig. 2 for $\beta=5.7$ and $\beta=5.9$ respectively. The value of $\langle\overline{\chi}\chi\rangle$ obtained in a free field theory is also plotted in the figures.

Following Bowler et al [8], the values at zero quark mass (extrapolated values) were obtained by fitting to the function

[TABLE]

The resulting chiral condensate values $\langle\overline{\chi}\chi\rangle_{0}$ are plotted in Fig. 3 versus the number of cooling sweeps. A clear nonzero signal for chiral symmetry breaking is obtained confirming the conclusion of Edwards et al [3]. This is in clear contrast to the results of Ref. [2] (see Figs. 9-11) where the chiral order parameter was found to disappear very quickly with cooling. It is fair to say, however, that the value of the chiral order parameter found in our calculation, although nonzero after cooling, is small compared (in lattice units) to the uncooled value.

The behavior of topological properties under cooling is well known [9]. Here we give a brief description to put our calculation into context. The simple transcription of $F_{\mu\nu}(x)\tilde{F}_{\mu\nu}(x)$ to the lattice [10] is used for the topological charge density

[TABLE]

where $U_{\mu\nu}$, $U_{\rho\sigma}$ are plaquettes of gauge field links. Fig. 4 shows the expectation values of topological charge squared $Q^{2}$ ($Q=\sum_{x}q_{t}(x)$) and a measure of the topological activity, $A_{t}=\sum_{x}|q_{t}(x)|$. Note that without cooling the topological activity $A_{t}$ is very large due to short distance fluctuations in $q_{t}(x)$. The numbers are too large to be plotted in Fig. 4(b) and are $45.02\pm 0.05$ at $\beta=5.7$ ($10^{4}$ lattice) and $87.6\pm 0.1$ at $\beta=5.9$ ($12^{4}$ lattice). In the first ten or so cooling steps, short distance fluctuations disappear very quickly, the topological activity drops rapidly and $\langle Q^{2}\rangle$ takes a value which changes only slowly with more cooling. In the region up to about $60$ cooling steps $\langle A_{t}\rangle^{2}$ is larger than $\langle Q^{2}\rangle$, indicating that one does not yet have gauge field configurations consisting of single isolated instantons. Rather, the vacuum in this region of cooling is sometimes characterized in terms of “overlapping” or “interacting” instantons [1, 9]. Beyond about $60$ cooling steps $\langle A_{t}\rangle^{2}<\langle Q^{2}\rangle$ as one would expect for a sample of instanton configurations with differing topological charge. The rate at which the topological activity changes becomes very small as instantons, which are not absolutely stable on the lattice, slowly shrink and fade away. By $100$ cooling steps one expects to be in the domain of a vacuum dominated by isolated instantons [1, 2, 9]. Our finding that the value of the chiral order parameter is non-zero and essentially constant beyond $60$ cooling steps, we would say, gives an indication that, in an SU(3) gauge theory, instantons produce chiral symmetry breaking, although it would not suggest that instantons dominate chiral symmetry breaking.

At present we have no explanation for the different behavior observed in the SU(2) and SU(3) calculations. One possibility is that the SU(2) calculation, which was done at $\beta=2.4$, is just deeper in the weak coupling region than the SU(3) calculations done here at $\beta=5.7$ and $5.9$. Indeed, comparing the SU(3) calculations at the two $\beta$ values may suggest a diminished persistence of chiral symmetry breaking at $\beta=5.9$ but this conclusion can not be made definitively due to the smallness of the chiral condensate (and concomitant large uncertainty) at the larger $\beta$ value. It may also be that there is still some more fundamental difference between the SU(2) and SU(3) theories which has yet to be found.

It is a pleasure to thank H.D. Trottier for helpful discussions. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.