Justification of the complex Langevin method with the gauge cooling procedure

Recently there has been remarkable progress in the complex Langevin method, which aims at solving the complex action problem by complexifying the dynamical variables in the original path integral. In particular, a new technique called the gauge cooling was 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 a rigorous justification of the complex Langevin method including the gauge cooling procedure. We first show that the gauge cooling can be formulated as an extra term in the complex Langevin equation involving a gauge transformation parameter, which is chosen appropriately as a function of the configuration before cooling. The probability distribution of the complexified dynamical variables is modified by this extra term. However, this modification is shown not to affect the Fokker-Planck equation for the corresponding complex weight as far as observables are restricted to gauge invariant ones. Thus we demonstrate explicitly that the gauge cooling can be used as a viable technique to satisfy the convergence conditions for the complex Langevin method. We also discuss the"gauge cooling"in 0-dimensional systems such as vector models or matrix models.


Introduction
Monte Carlo calculation has been playing an important role in nonperturbative studies of quantum field theories. However, the usefulness becomes quite limited when the action S becomes complex because the integrand e −S in the path integral can no longer be regarded as the Boltzmann weight. This occurs in many interesting cases such as QCD at finite density or with a theta term, gauge theories with a Chern-Simons term, chiral gauge theories, and so on. It also occurs in supersymmetric gauge theories and matrix models relevant to nonperturbative studies of superstring theory.
Amongst various approaches to this complex action problem, the one based on the complex Langevin equation has been attracting a lot of attention recently. The original idea was proposed by Parisi [1] and Klauder [2] in 1983, and since then it was applied to various systems with complex actions. A salient feature of the method is that it works beautifully in some fairly nontrivial cases, but it fails completely in the other cases. For a long time, theoretical understanding of this feature was missing, and that led to the gradual decline of interest in this approach. However, in 2011 one of the problems of the method in the case it fails has been clearly identified [3,4]. The authors first derived the key relation between the complex Langevin process and the Fokker-Planck equation for the complex weight. Then it was found that the integration by parts used in the derivation may not be justified unless the probability distribution of the complexified dynamical variables is suppressed strongly enough when they take large values.
The gauge cooling has been proposed to cure this problem in the case of gauge theories [5]. It has been applied to finite density QCD in the heavy dense limit and shown to work in the whole parameter regime in that limit [6]. More recently it has been applied to finite density QCD without taking the heavy dense limit, and it is shown to work at least in the deconfined phase [7]. These are already quite remarkable since the cases that have been studied include a parameter region, which would be hardly accessible by other methods such as reweighting. On the other hand, it is also realized in a solvable gauge theory with a complex coupling constant that there exists some parameter regime in which the gauge cooling cannot cure completely the insufficient fall-off of the probability distribution [8].
In fact there is another problem anticipated to occur when one applies the complex Langevin method (CLM) to QCD with light quarks at low temperature. This was realized in ref. [9] by applying the CLM to the Random Matrix Theory for finite density QCD. It turned out that a naive implementation of the method fails as the quark mass is decreased (See ref. [10], however.). The reason for this failure was speculated to have something to do with the logarithmic singularity in the action due to the fermion determinant [9,10,11]. On the other hand, it was also pointed out [12] that the problem occurs due to a singular drift term, which breaks the requirement of holomorphy in the derivation of the key relation between the complex Langevin process and the Fokker-Planck equation for the complex weight [3,4]. Recently two of the authors (J.N. and S.S.) [13] have argued that it is actually the integration by parts used in the derivation that is invalidated by the singular drift term. According to this understanding, the problem can be avoided if the probability distribution is suppressed strongly enough near the singularity. In a separate paper [14] we show that this can also be achieved by the gauge cooling if one chooses appropriately the quantity that should be reduced by the cooling procedure.
While intuitive arguments for justification of the gauge cooling are given in the literature (See, for instance, section 5 of ref. [15]), a rigorous justification has been missing. Actually there is even some suspicion in the community that the procedure may not be fully justified. Some of the concerns we have encountered in private communications are: 1) the gauge cooling uses a complexified gauge symmetry, which is not a symmetry of the original system. 2) the noise term in the complex Langevin equation is invariant under the original gauge transformation but not under the complexified gauge transformation. 3) the quantity one tries to reduce by the complexified gauge transformation is not holomorphic, which may spoil the justification of the CLM. In view of this situation, here we provide a rigorous justification of the CLM including the gauge cooling procedure. We first show that the gauge cooling can be formulated as an extra term in the complex Langevin equation involving a gauge transformation parameter, which is chosen appropriately as a function of the configuration before cooling. The probability distribution of the complexified dynamical variables is modified by this extra term. However, this modification is shown not to change the Fokker-Planck equation for the corresponding complex weight as long as the observables are restricted to gauge-invariant ones. Thus we conclude that the gauge cooling can be used to realize the properties of the probability distribution that are required for its relation to the complex weight without affecting the Fokker-Planck equation.
We also discuss the "gauge cooling" in 0-dimensional systems such as vector models or matrix models, which is simpler than 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 [16,17].
The rest of this paper is organized as follows. In section 2 we briefly review the Langevin method starting from the well-established case of real action, and discuss the conditions for correct convergence in the case of complex action. In section 3 we discuss the "gauge cooling" in 0-dimensional systems and provide its justification. In section 4 we discuss the application of the CLM to lattice gauge theory. In section 5 we present an explicit justification of the gauge cooling in lattice gauge theory. Section 6 is devoted to a summary and discussions.

Brief review of the Langevin method
In this section we briefly review the Langevin method. (As a comprehensive review on this subject, we recommend ref. [18].) Here we consider a system of n real variables x k (k = 1, · · · , n) given by the partition function where the action S(x) is a function of x = (x 1 , · · · , x n ). We start with the well-established case of real action, which is known also under the name of stochastic quantization. Then we discuss the case of complex action focusing on the conditions for correct convergence.

the case of real action
When the action S(x) is real, we can use the ordinary Langevin method to study this system [19]. Introducing a fictitious time t, we consider the t-evolution governed by the Langevin equationẋ where η k (t) are probabilistic variables obeying the probability distribution e − 1 4 dt η k (t) 2 . The first term and the second term on the right-hand side of the Langevin equation (2.2) are commonly called the drift term and the noise term, respectively, for historical reasons.
The probability distribution of x (η) (t) can be defined as where the expectation value · · · η is defined by (2.4) Using this notation, one obtains, for instance, One can actually show that P (x, t) satisfies the Fokker-Planck (FP) equation (See section 2.2 for derivation.) which has a time-independent solution Under quite general conditions [18], one can show that the eigenvalues of the differential operator acting on P on the right-hand side of (2.6) are strictly negative except for the zero eigenvalue corresponding to (2.7). This implies that the probability distribution P (x, t) approaches (2.7) exponentially. One can therefore obtain a vacuum expectation value (VEV) with respect to the partition function (2.1) as (2.8) In the last step, the statistical average over η is replaced by the time average assuming the ergodicity of the stochastic process as is done in usual Monte Carlo methods.

the discretized Langevin equation
When one tries to solve the Langevin equation (2.2) numerically, one has to discretize the fictitious time t and solve, for instance, 4 x (η) where the probabilistic variables η k (t) obey the probability distribution e − 1 4 ǫ t η k (t) 2 . Let us rescale them asη k = √ ǫη k so that they obey the probability distribution e − 1 4 tη k (t) 2 and hence, in particular, (2.10) With this normalization, the discretized Langevin equation (2.9) becomes Below we omit the tilde on η to simplify the notation. With this discretized version, we can derive the FP equation (2.6) in a more elementary manner than in the continuum [18]. Let us consider a test function f (x) and its expectation value at a fictitious time t. The t-evolution of this quantity is given by There are more sophisticated ways for discretization that can be used to reduce the systematic errors due to the discretization. See ref. [20] and references therein.

the case of complex action
Let us apply the same method to the case in which the action S is a complex-valued function of the real variables x k (k = 1, · · · , n). In that case, however, the first term on the right-hand side of the Langevin equation (2.2) becomes complex, which means that x (η) k (t) becomes complex even if one starts from a real configuration x (η) k (0) ∈ R. Let us therefore complexify the variables 5 as x k → z k = x k + iy k , and solve the complex Langevin equatioṅ where the action S is now considered as a function of the complex variables z k (k = 1, · · · , n) by analytic continuation. It is important for the method that the action S(z) thus obtained is a holomorphic function of z k . The probabilistic variables η k (t) in (2.17) are, in general, complex and obey the probability distribution e where the expectation value · · · η is defined by (2.20) With this notation, we have, for instance, In what follows we assume for a reason that becomes clear later. For practical purposes, one should actually use N R = 1, N I = 0, corresponding to real η k (t) with the distribution (2.4), to reduce the excursion in the imaginary directions [3,4], which spoils the validity of the method as we review below.
Repeating the analysis given in section 2.2, one can easily show that P (x, y; t) satisfies the FP-like equation In fact, for observables O(x) that admit holomorphic extension to O(x + iy), one can show under certain conditions that there exists a complex function ρ(x; t), which satisfies and obeys the differential equation The convergence to this solution in the t → ∞ limit requires that all the eigenvalues of the operator acting on ρ on the right-hand side of (2.25) should have strictly negative real part except for the zero eigenvalue corresponding to (2.26). While this is not guaranteed in general unlike in the real action case, one can argue that the convergence to (2.26) should occur if the relation (2.24) holds and the solution to the FP-like equation (2.23) converges to some function uniquely. Suppose the operator acting on ρ has an eigenvalue with a positive real part. Then the overall magnitude of ρ increases exponentially with t, and (2.24) cannot be satisfied. Also, suppose the operator acting on ρ has an eigenvalue with a vanishing real part other than the zero eigenvalue corresponding to (2.26). Then the asymptotic behavior of ρ depends on the initial condition, and (2.24) cannot be satisfied with P having a unique asymptotic behavior. To the best of our knowledge, this argument has been given for the first time in ref. [13] with explicit examples. Thus, provided that the relation (2.24) holds and the solution to the FP-like equation (2.23) converges to some function uniquely, we can calculate the VEV with respect to the partition function (2.1) as ( 2.27) In what follows, we review the derivation 6 of the key relation (2.24) given in refs. [3,4]. At t = 0, we can choose where ρ(x; 0) ≥ 0 so that (2.24) holds trivially. In order to prove the relation (2.24) at arbitrary t > 0, we are going to show that each side of (2.24) can be rewritten as In eq. (2.29), we have introduced the time-dependent observables O(z; t) defined by solving with the initial condition Let us recall that we are considering holomorphic observables O(z). One can actually show that the time-evolved observables O(z; t) remain to be holomorphic when S(z) is a holomorphic function [3]. The observables O(x; t) that appear in (2.30) are obtained by setting y = 0 in O(x + iy; t), and they satisfy the differential equation Since the right-hand sides of (2.29) and ( which interpolates each side of (2.29) with 0 ≤ τ ≤ t. Taking the derivative with respect to τ , we get where L ⊤ denotes the operator acting on P on the right-hand side of (2.23). The operator L is then defined as an operator satisfying Lf, g = f, L ⊤ g , where f, g ≡ f (x, y)g(x, y)dxdy, assuming that f and g are functions that allow integration by parts. The explicit form of the operator L can be obtained as An important observation here is that when L acts on a holomorphic function f (z) of z k , it can be replaced byL since which interpolates each side of (2.30) for 0 ≤ τ ≤ t. Taking the derivative with respect to τ , we get where we have used (2.34) and (2.25). Here the integration on the right-hand side involves the real directions x k only, so we can perform integration by parts without any problem due to the effects of the action, which make ρ(x; t) well localized. Thus (2.41) vanishes, and (2.30) follows.
On the other hand, the integration by parts that one needs to use to show that (2.37) vanishes involves the imaginary directions y k . It can therefore be justified only if the probability distribution P (x, y; t) has a sharp fall-off in the imaginary directions [3,4].
Recently it has been pointed out that the integration by parts can be invalidated also when the drift term includes a singularity [13]. This issue is relevant, in particular, to complex action systems involving fermions such as finite density QCD since the fermion determinant gives rise to a singular drift term. The CLM still works if the probability distribution P (x, y; t) is suppressed strongly enough near the singularity.

"Gauge cooling" in 0-dimensional systems
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 [5] in gauge theories. In ref. [14] we propose that it can be applied also to cure the second problem, and demonstrate that it does in the Random Matrix Theory for finite density QCD. In this section we consider the "gauge cooling" in 0-dimensional systems such as the Random Matrix Theory and provide an explicit justification. Apart from pedagogical purposes, we consider that it is useful, in particular, in matrix models relevant to superstring theory [16,17]. Generalization to the lattice gauge theory is straightforward and it is given in sections 4 and 5.
Let us consider a system of N real variables with a symmetry under where g is a representation matrix of a Lie group. An infinitesimal transformation is denoted as Here λ is an element of the Lie algebra, which can be expanded as in terms of the generators t a of the Lie group under consideration with real coefficients λ a ∈ R . Upon complexifying the variables x k → z k = x k + iy k , the symmetry of the action and the observables naturally enhances from (3.1) to where g is an element of the Lie group that can be obtained by complexifying the original Lie group. In particular, an infinitesimal transformation of the complexified symmetry is given by Here λ is an element of the Lie algebra for the complexified Lie group, which can be expanded as (3.3) but now with complex coefficients λ a ∈ C.
As a simple example, let us consider an O(N) vector model with σ ∈ C, which is invariant under (3.1) with g ∈ O(N). An infinitesimal transformation is given by (3.2), where λ jk is a purely imaginary antisymmetric N × N matrix. Upon complexification x k → z k = x k + iy k , the action becomes which is invariant under (3.4) with g ∈ O(N, C), namely with g being an N × N complex matrix satisfying g ⊤ g = gg ⊤ = 1. (The symbol g ⊤ here represents the transpose of the matrix g.) An infinitesimal transformation is given by (3.5), where λ jk is a complex antisymmetric N × N matrix. The discretized version of the complex Langevin equation (2.17) can be written as analogously to (2.11). The probabilistic variables obey the probability distribution e The gauge cooling [5] is a procedure of making a complexified symmetry transformation (3.4) between the Langevin steps. Thus it amounts to modifying the complex Langevin equation ( where g is an element of the complexified Lie group chosen appropriately as a function of the configuration before cooling. The basic idea is to determine g in such a way that the modified Langevin process (3.10) does not suffer from the problem of the original Langevin process (3.8). Clearly this idea will have a chance to work only if the degrees of freedom in the symmetry transformation have at least the same order of magnitude as those of the dynamical system itself. Gauge theories are one such example, but 0-dimensional models such as vector models and matrix models would be equally good as demonstrated explicitly in the Random Matrix Theory [14]. For instance, if the excursions in the imaginary directions are problematic in studying the model (3.6) by the CLM, one can introduce the norm 7 which measures the distance from the real region, and determine the transformation g in (3.10) in such a way that the norm N for z (η) (t) is reduced by the transformation. In ref. [14] we propose to combine this norm with another norm to cure also the problem caused by a singular drift term. Typically the norm one tries to reduce is invariant under transformations in the original Lie group but not under transformations in the complexified Lie group as in the case of (3.11). The main issue we address below is whether the modification of the Langevin process by the "gauge cooling" spoils the equivalence to the path integral reviewed in section 2.3.
Note that the gauge cooling is a completely deterministic procedure. In particular, the transformation g in (3.10) is determined only by the configuration z (η) (t) before cooling. Therefore, for our purpose, it is convenient to rewrite (3.10) into the form which involves z (η) (t) only. 8 Let us assume that g is given by where λ(x, y) is an element of the Lie algebra of O (N, C). For instance, one may use which can be obtained by calculating the gradient of the norm (3.11) with respect to the O(N, C) transformation. The real positive function α(x, y) can be chosen to optimize the reduction of the norm. (In practice, one can simply determine g by using the steepest descent method with respect to the norm (3.11) [5].) Note that λ(x, y) is not a holomorphic function of z k in general as in (3.14). Using (3.13) in eq. (3.12) and taking the ǫ → 0 limit, we obtain the continuum complex Langevin equation for z (η) (t) aṡ z (η) where the effect of the gauge cooling is represented by the last term on the right-hand side. Then we can easily find that the FP-like equation (2.23) that P (x, y; t) satisfies is modified by the gauge cooling as This modifies the differential operator L in eq. (2.38) into Acting this modified L on a holomorphic function f (z), we obtain

Application of the CLM to lattice gauge theory
In this section we discuss the application of the CLM to lattice gauge theory, which is defined by the partition function where the action S is a complex-valued function of the configuration U = {U nµ } composed of link variables U nµ ∈ SU(3), and the integration measure dU nµ represents the Haar measure for the SU(3) group. The only complication compared with the case discussed in the previous sections comes from the fact that the dynamical variables take values on a group manifold. The Langevin equation in such a case with a real action is discussed intensively in refs. [29,30,31,32,33]. Using this formulation, we can easily generalize our discussions to the case of lattice gauge theory. When the action S is complex, the drift term in the Langevin equation makes the link variables evolve into SL(3, C) matrices (i.e., 3 × 3 general complex matrices with the determinant one) even if one starts from a configuration of SU(3) matrices. Let us therefore complexify the link variables as U nµ ∈ SL(3, C), and solve the complex Langevin equatioṅ where the action S(U) is now considered as a holomorphic function of the complexified configuration U nµ , and t a are the generators of the SU(3) group normalized by tr (t a t b ) = δ ab .
The probabilistic variables η anµ (t) are defined similarly to (3.9). The derivative operator D anµ is defined as 9 Here f (U) are functions on the complexified group manifold, which are not necessarily holomorphic, and x and y in eqs. (4.4) and (4.5) are real parameters. Note that for a holomorphic function f (U), we haveD anµ f (U) = 0, wherē and hence Then we define the probability distribution where the delta function is defined by for any function f (U). The integration measure that appears on the left-hand side represents the Haar measure for the SL(3, C) group normalized appropriately. One can show that the probability distribution P (U; t) obeys the FP-like equation (See Appendix A for derivation.) Here we have defined the derivative operator D anµ , which acts on a function f (U) of the unitary gauge configuration as . (4.13) Note that the FP equation (4.12) has a time-independent solution (4.14) As we argued in section 2.3, the convergence to (4.14) should occur if the relation (4.11) holds and the FP-like equation (4.10) converges to some function uniquely. In that case, we can calculate the VEV with respect to the partition function (4.1) as Let us briefly discuss how one can derive the relation (4.11). At t = 0, we choose with ρ(U; 0) ≥ 0 so that (4.11) holds trivially. In order to prove the relation (4.11) at arbitrary t > 0, we are going to show that each side of (4.11) can be rewritten as Let us recall that we are considering holomorphic observables O(U). One can actually show that the time-evolved observables O(U; t) remain to be holomorphic when S(U) is a holomorphic function [3]. The observables O(U; t) that appears in (4.18) are obtained by setting U = U in O(U; t), and they satisfy the differential equation In order to show (4.17), we introduce the function which interpolates each side of (4.17) with 0 ≤ τ ≤ t. Taking the derivative with respect to τ , we get where L ⊤ denotes the operator acting on P on the right-hand side of (4.10). The operator L is then defined as an operator satisfying Lf, g = f, L ⊤ g , where f, g ≡ f (U)g(U)dU, assuming that f and g are functions that allow integration by parts. The explicit form of the operator L can be obtained as where we have used (4.7) and (2.22). This implies thatL in the first term of (4.25) can be replaced by L, and hence (4.25) vanishes if one can perform integration by parts. In that case, F (t, τ ) is independent of τ , and (4.17) follows. A similar argument can be used to show (4.18). We define which interpolates each side of (4.18) for 0 ≤ τ ≤ t. Taking the derivative with respect to τ , we get where we have used (4.22) and (4.12). Here the integration on the right-hand side involves the real directions only, which are compact in the present case, so we can perform integration by parts without any problem to show that (4.29) vanishes. Thus G(t, τ ) is independent of τ , and (4.18) follows.
On the other hand, the integration by parts in (4.25) is justified only if the probability distribution P (U; t) has a sharp fall-off in the noncompact imaginary directions. The gauge cooling [5] was originally proposed to solve this problem. As we mentioned in section 2.3, the integration by parts can be invalidated also when the drift term includes a singularity [13]. This problem is anticipated to occur when one applies the CLM to finite density QCD at low temperature with light quarks. The CLM still works if the probability distribution P (U; t) is suppressed strongly enough near the singularity. We consider that the gauge cooling is useful also in solving this problem as in the case of Random Matrix Theory [14].

Gauge cooling in lattice gauge theory
The lattice gauge theory is invariant under the SU(3) gauge transformation. For instance, the plaquette action where g n ∈ SU(3). An infinitesimal transformation is denoted as Here λ n is an element of the Lie algebra, which can be expanded as (λ n ) jk = a λ na (t a ) jk (5.4) in terms of the generators t a of SU(3) with real coefficients λ na ∈ R . When one complexifies the variables U nµ → U nµ ∈ SL(3, C), the symmetry of the action and the observables naturally enhances to the SL(3, C) gauge symmetry that can be obtained by complexifying the original Lie group. For instance, the plaquette action (5.1) becomes which is invariant under with g n ∈ SL(3, C). An infinitesimal transformation is given by Here λ is an element of the Lie algebra for the complexified Lie group, which can be expanded as (5.4) but now with complex coefficients λ na ∈ C.
The discretized version of the complex Langevin equation (4.2) can be written as The gauge cooling [5] modifies the complex Langevin equation (5.8) into where g n is an element of the complexified Lie group. The basic idea is to determine g n in such a way that the modified Langevin process (5.9) does not suffer from the problem of the original Langevin process (5.8).
For instance, if the excursions in the imaginary directions are problematic, one can introduce a positive semi-definite quantity [34] (We call it the "norm" in this paper.) which measures the distance from the unitary region, and determine the transformation g n in (5.9) in such a way that the norm N for U (η) (t) is reduced by the transformation.
Typically the norm one tries to reduce is invariant under transformations in the original Lie group but not under transformations in the complexified Lie group as in the case of (5.10). Below we demonstrate that the modification of the Langevin process by the gauge cooling does not spoil the equivalence to the path integral reviewed in the previous section.
Note that the gauge cooling is a completely deterministic procedure. In particular, the transformation g n in (5.9) is determined only by the configuration U (η) (t) before cooling. Therefore, for our purpose, it is more convenient to rewrite (5.9) into the form which involves U (η) (t) only. 10 Let us assume that g n is given as where λ n (U) is an element of the Lie algebra of SL (N, C). For instance, one may use which can be obtained by calculating the gradient of the norm (5.10) with respect to the SL(3, C) gauge transformation of the configuration U. The real positive function α(U), which is not necessarily holomorphic, can be chosen to optimize the reduction of the norm.
(In practice, one can simply determine g n by using the steepest descent method with respect to the norm (5.10) [5].) Note that λ n (U) is not a holomorphic function of U nµ in general as in (5.13). Using (5.12) in eq. (5.11) and taking the ǫ → 0 limit, we obtain the continuum complex Langevin equation for U (η) (t) aṡ where the effect of the gauge cooling is represented by the last term on the right-hand side. Then we can easily find that the FP-like equation (4.10) that P (U; t) satisfies is modified by the gauge cooling as Therefore, theL in the first term of (4.25) can be replaced by the modified L. Hence (4.25) vanishes if one can perform integration by parts for the modified L and P . In that case, the crucial identity (4.11) holds for the modified P with the same ρ. Thus we have shown explicitly that the gauge cooling provides a possibility to improve the property of the probability distribution P (U; t) so that (4.11) holds, without affecting the FP equation (4.12) for ρ(U; t).

Summary and discussions
In this paper we have provided a rigorous 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 P associated with the complex Langevin process and the complex weight ρ associated with the original path integral problem. This relation holds if and only if the probability distribution P satisfies the following two properties. One is that it has a rapid fall-off in the imaginary directions in the complexified configuration space [3,4]. The other is that the distribution is such that complexified configurations which make the drift term singular are strongly suppressed [13]. Since the gauge cooling modifies the probability distribution P , one may hope to make it satisfy the above two properties by appropriately choosing the complexified symmetry transformation to be used in the cooling procedure. What we have shown in this paper is that the modification of the probability distribution P due to the gauge cooling does not alter the FP equation that the complex weight ρ obeys if the relation between P and ρ holds at all. For a long time it has been thought that the convergence of the FP equation for the complex weight ρ is not guaranteed unlike in the real action case. However, once the relation between the probability distribution P and the complex weight ρ is established, one may argue [13] that the unique convergence of the probability distribution P already implies the unique convergence of ρ to the desired complex weight e −S . Therefore, if one can satisfy the above two properties of the probability distribution P by using the gauge cooling appropriately, the CLM is guaranteed to give the correct results.
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 the gauge cooling with a new type of norm [14] look very promising.