^{1}

^{1}

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^{3}.

We elucidate chirality production under parity breaking constant electromagnetic fields, with which we clarify qualitative differences in and out of equilibrium. For a strong magnetic field the pair production from the Schwinger mechanism increments the chirality. The pair production rate is exponentially suppressed with mass according to the Schwinger formula, while the mass dependence of chirality production in the axial Ward identity appears in the pseudoscalar term only. We demonstrate that, in a real-time formulation with in and out states, the axial Ward identity with an in-in expectation value leads to a chirality production rate consistent with the Schwinger formula, while the axial anomaly with an in-out expectation value is canceled by the pseudoscalar condensate for any mass. We illuminate that such an in- and out-state formulation clarifies subtleties in the chiral magnetic effect in and out of equilibrium, and we discuss further applications to real-time condensates.

Chirality is a topical keyword for anomalous phenomena in physics and related subjects. In the context of high-energy physics in which the fermion mass is often neglected, the chirality and the helicity are identifiable, which has also motivated a modern redefinition of chirality in chemistry

The most notable feature of chirality in relativistic fermionic systems is the realization of the quantum anomaly. Since relativistic fermionic dispersion relations are realized in not only 2D but also 3D materials, as in the Weyl and Dirac semimetals

In all ideas to access the chiral anomaly, the generation of finite chirality imbalance is indispensable. The simplest optical setup is, as discussed in Ref.

Even though the parallel electromagnetic fields are simple to treat, there are still some controversies especially on different manifestations of the chiral anomaly in and out of equilibrium. In this Letter we clarify these controversies by addressing the following two closely related problems, namely: (i) The effect of fermion mass

Answering these questions will naturally lead us to a clear picture of chiral dynamics. Moreover, we will see that our present considerations have many applications to be studied in the future.

We choose constant and parallel electric

Schematic picture of the pair production in a reduced (

A pair of

The right-hand side

To resolve this puzzle, the crucial observation is that the vacua at

Fortunately, for the constant

For the standard propagator,

After some calculations we find the integration kernel with parallel

Let us first consider,

Interestingly, using our method, we can directly evaluate

Now, we have clarified that

The situation is even more transparent if we performed the Euclidean Monte Carlo calculation on the lattice. As discussed in Ref.

We note that

One important extension along these lines of

It is a straightforward exercise to compute the vector current associated with the chiral magnetic effect. In Ref.

Here, interestingly, a simple calculation yields

Such a statement about equilibrium chiral magnetic effect itself is not quite new, see Ref.

Now, let us turn to another problem, that is, a scalar condensate (which is commonly called the chiral condensate in QCD). Since we already discussed the pseudoscalar condensates,

From Eq.

In view of the fact that

From the point of view of the spontaneous symmetry breaking, Eq.

We have clarified important differences associated with the in and out states in the presence of electric field

The authors thank Stefan Flörchinger, Xu-Guang Huang, Niklas Mueller, and Naoto Tanji for useful comments and discussions. K. F. is grateful for a warm hospitality at the Fudan University where he stays as a Fudan University Fellow. This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. 18H01211. S. P. was supported by the JSPS postdoctoral fellowship for foreign researchers.

To justify the imaginary rotation with singularities from