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In quantum chromodynamics, axial charge is known to be nonconserved due to chiral anomaly and nonvanishing quark mass. In this paper, we explore the role of quark mass in axial charge fluctuation and dissipation. We present two separate calculations of the axial charge correlator, which describe dynamics of axial charge. The first is free quarks at a finite temperature. We find that axial charge can be generated through effective quantum fluctuations in free theory. However, the fluctuation does not follow a random walk behavior. Because of the presence of an axial symmetry breaking mass term, the axial charge also does not settle asymptotically to the thermodynamic limit given by susceptibility. The second calculation is in a weakly coupled quark gluon plasma. We find in the hard thermal loop (HTL) approximation, the quark-gluon interaction leads to random walk growth of the axial charge, but dissipation is not visible. We estimate the relaxation timescale for the axial charge, finding it lies beyond the HTL regime.

The chiral anomaly is one of the most intriguing discoveries in quantum field theory. Over the past ten years, its manifestations in macroscopic phenomena such as the chiral magnetic effect and chiral vortical effect have triggered significant interest across different communities

One of the well-known generation mechanisms of axial charge is through a topological fluctuation of the gluon field

This paper aims at providing a unified description of the two effects in the same setting. To be specific, we study dynamics of total axial charge in QCD in the weakly coupled regime, where perturbative calculation is possible. The results apply equally well to QED. For pedagogical reasons, we begin with axial charge dynamics in free quark theory in Sec.

For simplicity, we first study axial charge dynamics in free theory. This allows us to set the stage and gain insights into the dynamics. We first note that classical fermions satisfying the Dirac equation do not have a net axial charge, even though axial symmetry is broken by fermion mass. In order to generate the net axial charge, we need the quantum fluctuation to push fermions off shell. The quantity characterizing axial charge dynamics is the following Wightman correlator:

It is instructive to compare

Normalized susceptibility versus

The fluctuation of

In

Now we turn to the evaluation of the contribution from

The two seemingly odd features are, in fact, related: Although quarks are free at the Lagrangian level, Fermi-Dirac statistics obeyed by quarks in equilibrium provides effective interaction; thus quantum fluctuation is present. Furthermore, this also gives a quantitative explanation of the negative sign in the regularized fluctuations. The effect of Fermi-Dirac statistics becomes prominent as we lower the temperature. In the vacuum case, quantum fluctuation is maximal, and thus the vacuum fluctuation is larger than any finite temperature fluctuation, giving rise to negative regularized fluctuation. Equation

Contributions from intrinsic fluctuation

To summarize, we find the fluctuation of

After the warm-up, we move on to the calculation in weakly coupled quark gluon plasma (QGP). We expect the same structure of the Wightman correlator as

At one loop order, the

Three leading one loop diagrams contributing to

In the spirit of HTL, we will drop any contributions at

Our calculation heavily relies on Ward identities. We note the

We proceed by evaluating the tadpole diagram

It is instructive to look at the result in the massless limit first. Setting

Now we move on to a massive case. We note that the first two terms of

For the purpose of demonstrating late time dynamics of

To evaluate

We thank Pengfei Zhuang for pointing this out for us. See also related work

Note that in the weakly coupled case, only the generation of the axial charge is obtained, and the damping effect is not visible. The reason can be understood by making an estimate of the damping timescale. Using the fluctuation-dissipation theorem, the damping timescale due to the quark mass is given by

Let us compare the fluctuation of the axial charge in free theory and weakly coupled QGP. First of all, the fluctuations in both cases contain two contributions: one is proportional to susceptibility, originating from the charge exchange with a heat bath; the other contribution is intrinsic to the breaking of axial symmetry. Focusing on the contribution from the breaking of axial symmetry, we find that unlike the susceptibility term, quantum fluctuation is needed to give a nonvanishing contribution. In the case of free theory, the quantum fluctuation is provided by effective Pauli repulsion. This also explains the counterintuitive result we find: the fluctuation maximizes at zero temperature. It also implies that it is misleading to interpret this contribution as a correction to susceptibility. In the case of weakly coupled QGP, the fluctuation is given by quark-gluon interaction, which is enhanced by the presence of the thermal medium. The frequency dependence of the Wightman correlator of the two cases is given by the following:

We are grateful to Kenji Fukushima, Lianyi He, Anping Huang, Rob Pisarski, and especially Pengfei Zhuang for insightful discussions. S. L. thanks Central China Normal University, Beihang University, Tsinghua University, Peking University, Institute of High Energy Physics for hospitality where part of this work is done. This work is in part supported by the Ministry of Science and Technology of China (MSTC) under the “973” Project No. 2015CB856904(4) (D. H.), by NSFC under Grants No. 11375070, No. 11735007, No. 11521064 (D. H.), by One Thousand Talent Program for Young Scholars (S. L.), and by NSFC under Grants No. 11675274 and No. 11735007 (S. L.).