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Recently, there has been remarkable progress in the complex Langevin method, which aims to solve the complex action problem by complexifying the dynamical variables in the original path integral. In particular, a new technique, called gauge cooling, has been introduced and the full QCD simulation at finite density has been made possible in the high-temperature (deconfined) phase or with heavy quarks. Here we provide an explicit justification of the complex Langevin method including the gauge cooling procedure. We first show that the gauge cooling can be formulated in the form of a modified complex Langevin equation involving a complexified gauge transformation, which is chosen appropriately as a function of the configuration before cooling. The probability distribution of the complexified dynamical variables is modified accordingly. However, this modification is shown

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Monte Carlo calculation plays an important role in nonperturbative studies of quantum field theories. However, its usefulness becomes quite limited when the action

Amongst various approaches to this complex action problem, that based on the complex Langevin equation has recently been attracting a lot of attention. The original idea was proposed by Parisi [

Gauge cooling has been proposed to cure this problem in the case of gauge theories [

In fact, there is another problem that is anticipated to occur when one applies the complex Langevin method (CLM) to QCD with light quarks at low temperature. This was realized in Ref. [

While intuitive arguments for justification of the gauge cooling are given in the literature (see, for instance, Sect. 5 of Ref. [

We also discuss “gauge cooling” in 0D systems such as vector models or matrix models, which is simpler than that in lattice gauge theory. Apart from pedagogical purposes, we consider that it is useful, for instance, in studying the matrix models relevant to superstring theory [

The rest of this paper is organized as follows. In Sect.

In this section, we briefly review the Langevin method. (For a comprehensive review on this subject, we recommend Ref. [

When the action

The probability distribution of

Under quite general conditions [

When one tries to solve the Langevin equation (^{1} ,

With this discretized version, we can derive the FP equation (

Let us apply the same method to the case in which the action ^{2} as

Repeating the analysis given in Sect.

In what follows, we review the derivation^{3} of the key relation (

In order to show (

A similar argument can be used to show (

On the other hand, the integration by parts that one needs to use to show that (

Recently, it has been pointed out that the integration by parts can also be invalidated when the drift term includes a singularity [

At the end of the previous section, we discussed two possible problems, which can make the CLM give wrong results. The gauge cooling was originally proposed to cure the first problem [

Let us consider a system of

As a simple example, let us consider an O(

The discretized version of the complex Langevin equation (

For instance, if the excursions in the imaginary directions are problematic in studying the model (^{4}

Note that gauge cooling is a completely deterministic procedure. In particular, the transformation ^{5} .

In this section, we discuss the justification of gauge cooling, assuming for simplicity that the asymptotic behavior of

Using (

where

In practical applications, the asymptotic behavior (

First let us derive the discretized FP-like equation for

Note that the effect of the gauge cooling comes only through

Let us then define the function

In this section, we discuss the application of the CLM to lattice gauge theory, which is defined by the partition function

When the action ^{6}

Then we define the probability distribution

Let us briefly discuss how one can derive the relation (

In order to show (

A similar argument can be used to show (

On the other hand, the integration by parts in (

In this section, we discuss gauge cooling in lattice gauge theory and provide the justification of the CLM including gauge cooling. The argument is a straightforward generalization of that given in Sect.

The lattice gauge theory is invariant under the

When one complexifies the variables

The discretized version of the complex Langevin equation (

For instance, if the excursions in the imaginary directions are problematic, one can introduce a positive semidefinite quantity [

Note that gauge cooling is a completely deterministic procedure. In particular, the transformation ^{7} .

In what follows, we assume for simplicity that the asymptotic behavior of

Using (

In practical applications, the asymptotic behavior (

First let us derive the discretized FP-like equation for

Let us then define the function

In this paper, we have provided an explicit justification of the CLM with the gauge cooling procedure. As we have reviewed in detail, the CLM relies crucially on the relation between the probability distribution

For a long time, it has been thought that the convergence of the FP equation for the complex weight

While the gauge cooling certainly enlarges the range of applicability of the CLM, it remains to be seen how powerful it is in studying various interesting systems with complex actions. In this regard, our results for the random matrix theory using gauge cooling with a new type of norm (K. Nagata et al., manuscript in preparation) look very promising.

Open Access funding: ^{3}

The authors would like to thank J. Bloch, K. Fukushima, and D. Sexty for valuable discussions. We are also grateful to E. Seiler for correspondence on the first version of this paper. K.N. was supported by JSPS Grants-in-Aid for Scientific Research (Kakenhi) Grants No. 00586901, MEXT SPIRE, and JICFuS. The work of J.N. was supported in part by a Grant-in-Aid for Scientific Research (No. 23244057) from the Japan Society for the Promotion of Science.