# Anomalous Chiral Symmetry Breaking above the QCD Phase Transition

###### Abstract

We study the anomalous breaking of symmetry just above the QCD phase transition for zero and two flavors of quarks, using a staggered fermion, lattice discretization. The properties of the QCD phase transition are expected to depend on the degree of symmetry breaking in the transition region. For the physical case of two flavors, we carry out extensive simulations on a lattice, measuring a difference in susceptibilities which is sensitive to symmetry and which avoids many of the staggered fermion discretization difficulties. The results suggest that anomalous effects are at or below the 15% level.

###### pacs:

11.15.H, 11.30.R, 12.38, 12.38.G, 12.38.M^{†}

^{†}preprint: CU-TP-902

The breaking of classical chiral symmetry by quantum effects is a theoretical phenomenon of considerable physical importance which has direct implications on the order of the finite temperature, QCD phase transition. If we consider only the two light and quarks, anomalous symmetry breaking reduces the flavor symmetry from to . It is only this reduced symmetry that is consistent with second-order, critical behavior[4]. Although present lattice simulations suggest that the QCD phase transition is indeed second-order for two flavors, it is an important consistency check to establish the required anomalous symmetry breaking directly. In addition, the size of these symmetry breaking effects will determine the width in temperature and quark mass of the region showing universal critical behavior.

The plasma phase of QCD is a particularly good place to study the axial anomaly. For , the dynamical breaking of chiral symmetry obscures the effects of the axial anomaly. For , chiral symmetry is restored and thermal Greens’ functions are explicitly symmetric under transformations. We can then look directly for anomalous symmetry breaking by comparing Greens functions that are related by the anomalous symmetry.

Such breaking of global symmetry, associated with a zero-momentum Ward identity, is especially interesting, since at zero momentum the anomalous term in the chiral Ward identity becomes times the topological charge , a quantity which vanishes to all orders in conventional perturbation theory. The mass in QCD and the non-conservation of baryon number in the standard model are other examples in which such non-perturbative anomalous effects should occur[5].

This sort of anomalous symmetry breaking can be understood from two perspectives: i) As the physical remnant of an ultraviolet ambiguity in the theory. Here modifications made to regulate the divergences present in the continuum gauge theory, necessarily introduce explicit chiral symmetry breaking whose effects remain visible, even on energy scales small compared to those of the regulator. ii) As resulting from an infrared singularity which permits chiral asymmetry to survive the chiral limit of vanishing fermion mass. While chiral symmetry breaking effects might naively be expected to be proportional to the explicit quark mass, , the presence of infrared singular, behavior can allow such chirally asymmetric effects to remain even in the limit. In a semiclassical calculation, such behavior appears for gauge backgrounds with fermion zero modes. These two views are related by the Atiyah-Singer index theorem[6].

Most lattice calculations[7] explore the first approach, studying explicit chiral symmetry breaking inherent in the lattice regulation which should reduce to anomalous effects as the continuum limit is taken[8]. In this paper we explore the second approach, searching for anomalous asymmetries that arise from infrared singularities in the limit of small quark mass. For lattice calculations these two approaches are not equivalent since the Atiyah-Singer theorem applies only in the limit of an infinite number of degrees of freedom.

It is important to pursue both methods. The first approach may over-estimate anomalous effects, confusing them with simple lattice artifacts which vanish in the continuum limit. The second approach may miss anomalous effects since infrared singularities are often softened by lattice effects, e.g. the zero-mode-shift of Smit and Vink[9], and become apparent only as .

The question of anomalous symmetry breaking above has now been studied by a number of groups. For a general review see the article of Laermann[10]. Preliminary versions of our results can be found in Ref. [11], while an alternative calculation, also taking approach ii), can be found in Bernard, et al.[12]. Finally, Kogut, et al.[13] use a combination of methods examining signals for anomalous symmetry breaking of both types i) and ii) above.

In this paper, we study both zero- and two-flavor QCD, just above . For , anomalous effects are expected in the chiral condensate, , while for we must examine a more infra-red singular, -noninvariant quantity, here a difference of iso-vector susceptibilities which we refer to as where

(1) |

for space-time volume and flavor generator . The scalar susceptibility is defined similarly, by omitting the internal factors.

We adopt the staggered fermion, lattice discretization. This is the approach used most successfully to date in finite temperature, lattice QCD studies. While the quantity can be calculated directly using this formalism, the more physically interesting can not. Direct definitions of and using staggered fermions will necessarily introduce ambiguities, with potentially large lattice artifacts obscuring the anomalous effects of interest.

Here, we take an indirect approach, expressing , in the continuum, as a spectral integral whose singular behavior as gives rise to anomalous symmetry breaking. We then demonstrate that this spectral integral can be directly evaluated using staggered lattice fermions and use this result to provide a lattice calculation of .

Consider the spectral representations:

(2a) | |||||

(2b) |

Here is the average density of Dirac eigenvalues . The first formula is due to Banks and Casher[14] and the second is derived in a similar fashion. In Eq. 2a we distinguish the fermion mass that appears in the fermion line attached to and , , from that entering through the fermion determinant, . The factors of or in the numerators of Eq. 2 reflect the chiral symmetry breaking character of and . However, an anomalous, small-mass limit can result if the integral over is sufficiently singular for small .

Now let us investigate what might be expected for these quantities in continuum QCD. For , the small mass limit of and in the continuum theory can be analyzed for both the case of very small volume and in the limit of infinite volume. For finite volume, the Dirac spectrum will be discrete for each gauge configuration in the path integral. The only non-zero contributions to either or as will come from gauge configurations with at least one exact Dirac zero mode. In very small volumes, these zero modes can be predicted semiclassically and give the anomalous, small-mass behaviors: , for and constant, for .

The case where first is more interesting and can be analyzed using the methods of Leutwyler and Smilga[15]. Above , there are no massless modes so the free energy should be proportional to the volume and analytic in the fermion mass:

(3) |

where is the complex fermion mass matrix and the usual theta parameter. Defining the topological susceptibility as , one easily derives for and I, a multiple of the identity.

We can similarly obtain expressions for and :

(4a) | |||||

(4b) |

where in Eq. 4a we have divided by to define as coming from a single fermion specie while in Eq. 4b we have used a complex .

If we make the possibly reasonable assumption that the quenched value of can be obtained as the limit of Eq. 4a, then we can combine Eq. 4 with the formula for to obtain:

(5) |

The last relation is of particular interest, implying that above the quantity provides an alternative measure of the topological susceptibility. As is shown below, can be easily determined using lattice methods, without the normal difficulties of defining topological winding on a discrete lattice.

We will now compare these continuum expectations with lattice calculations. Because of the remnant chiral symmetry of staggered fermions, Eq. 2a is also valid on the lattice, allowing us to relate and , where is the single component, staggered fermion field. Viewing as a function of and , we can express as a function of and then use this continuum result to define on the lattice:

(6) |

where these two terms correspond precisely to the terms in the difference . In the remainder of this paper we quote values of , and normalized according to Eq. 6 where is normalized to behave as in the large mass limit.

First consider . In Fig. 1 we show for two distinct phases distinguished by the complex phase of the Wilson line, , computed at , just above . (Recall the Wilson line, , is the volume average of the trace of the ordered product of link variables along a line in the time direction.) For the case where is real, we see the power law for both and volumes suggesting this power law description holds in the infinite volume limit. We see no sign of the anomalous behavior in expected in the continuum.

For the case of complex , we see an unexpected spontaneous breaking of chiral symmetry above , with approaching a constant as decreases. The eventual decrease in for very small is the normal finite-volume behavior expected with spontaneous symmetry breaking, with for both volumes.

Next we examine and the more physical case of two flavors, at , just above (recall for and =0.01), on a lattice for five different values of the dynamical quark mass. The results are summarized in Table 1 and plotted in Fig.2. This figure shows the chiral condensate, approaching zero linearly as is expected for . Likewise, shows the expected regular, constant behavior as . However, rather than showing the anomalous behavior, , expected from Eq. 5, Fig.2 suggests a nearly linear as goes to zero.

Four fitted curves are also shown in Fig. 2. The two linear fits to and have a of 2.2 and 2.7 respectively. Both of these fits are constrained to vanish at . If that constraint is dropped for the fit, the intercept moves upward slightly to 0.15(5) and the falls to 0.34. A fit to the expected form is worse, with a of 3.4.

While these results are consistent with those reported by Bernard et al.[12], our conclusions are different. That calculation examines a smaller lattice spacing than considered here but with larger statistical errors. Their analysis adopts the quadratic small-mass dependence for . However, such quadratic behavior is only guaranteed theoretically in the unphysical limit where vanishes at fixed lattice spacing and needs to be established numerically for the case of interest.

From Fig. 2 one observes that for quark masses in the range , our results are consistent with an unusual but non-anomalous, linear behavior . At our smallest mass, 0.005, is significantly higher than such a linear extrapolation, suggesting that anomalous effects may be emerging. However, such effects are clearly quite small and occur for quark masses that are below those used in present studies of QCD thermodynamics, suggesting little connection between this anomalous behavior and the observed second-order QCD phase transition.

In order to describe the physical size of a possible non-zero value of , we must address the potential cut-off dependence of the quantities being discussed. While a thorough analysis of this question lies beyond the scope of the present paper[16], there are two issues that are important to recognize. First, the intercept of requires the same -dependent, multiplicative renormalization as the inverse square of the quark mass , as is suggested by Eq. 5. We will ignore such a factor for our present rough estimate, since this factor should be of order 1 for current lattice spacings.

Since is the difference of and , it is natural to compare to either of these quantities. However, both quantities contain a piece when evaluated in physical units. Thus, we choose to compare to a quantity we will call , obtained as the difference between evaluated at and evaluated at , just below the transition:

(7) |

Such a subtraction removes the unwanted, quadratically divergent term at tree level and leaves an expression finite up to an (1), -dependent multiplicative factor and a much smaller term suppressed by the factor . We find .

Within the expected critical region, the light modes (, ) should be much less massive than the non-universal degrees of freedom suggesting , not the observed here. Thus, our results suggest that critical behavior should not be seen in thermodynamics at least for .

In conclusion, we have numerically studied anomalous symmetry breaking by examining quantities whose anomalous behavior comes directly from infrared effects. Given the relatively coarse lattice spacing in our simulations 1 Fermi, our failure to find such effects above the 15% level is far from conclusive evidence that such effects are suppressed in Nature[17]. However, this represents a first step in a systematic lattice calculation of such phenomena and must be followed by more demanding calculations on finer lattices and calculations using fermion formulations with improved chiral properties.

We thank Edward Shuryak and Andre Smilga for helpful discussions and Yubing Luo for assistance.

## References

- [*] Present address, Department of Physics, Duke University, PO Box 90305, Durham, NC 27708.
- [‡] Present address, CTP-LNS, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139.
- [†] Present address, MS-B285, T-8 Group, Theory Division, Los Alamos National Lab, Los Alamos, NM 87545.
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- [13] See also J. B. Kogut, J.-F. Lagaë and D. K. Sinclair, Nucl. Phys. B (Proc. Suppl.) 63 (1997) 433 and hep-lat-9801020.
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- [16] S. Chandrasekharan, et al., in preparation.
- [17] For example, a recent numerical study of in smooth gauge field backgrounds suggests that staggered fermion lattice artifacts will sufficiently spread the continuum zero eigenvalues to obscure anomalous effects at least on lattices with . See A. Kaehler, in preparation.

run length | ||||
---|---|---|---|---|

0.005 | 4464 | 0.02256(21) | 3.945(43) | 0.559(37) |

0.01 | 2550 | 0.04374(50) | 3.369(68) | 0.932(56) |

0.015 | 2600 | 0.06517(82) | 3.010(52) | 1.322(62) |

0.02 | 2992 | 0.0896(10) | 2.697(34) | 1.722(62) |

0.025 | 3072 | 0.1141(38) | 2.38(11) | 2.24(12) |