Renormalization group and diffusion equation

We study relationship between renormalization group and diffusion equation. We consider the exact renormalization group equation for a scalar field that includes an arbitrary cutoff function and an arbitrary quadratic seed action. As a generalization of the result obtained by Sonoda and Suzuki, we find that the correlation functions of diffused fields with respect to the bare action agree with those of bare fields with respect to the effective action, where the diffused field obeys a generalized diffusion equation determined by the cutoff function and the seed action and agrees with the bare field at the initial time.


Introduction
It is recognized that the diffusion equation has to do with renormalization group, since diffusion can be regarded as a continuum analog of block-spin transformation or coarsegraining. Indeed, the solution to the diffusion equation in d dimensions with an initial condition ϕ(0, x) = φ(x) is given by where K(x, x ′ , τ ) is the heat kernel Here ϕ(τ, x) can be viewed as a coarse-grained field obtained by smearing φ in a ball centered at x with the radius √ τ . On the other hand, the gradient flow equation in gauge theories, which is regarded as a generalized diffusion equation respecting gauge symmetry, has been recently used to renormalize composite operators and so on [1][2][3].
Thus it is natural to expect that there is relationship between (generalized) diffusion equations or gradient flow equations and renormalization group equations. Indeed, the relationship has been studied in [4][5][6][7][8][9][10][11]. In particular, it was shown in [11] that the correlation functions of the diffused fields (1.2) with respect to the bare action agree with those of bare fields with respect to the effective action, where the effective action obeys the exact renormalization group (ERG) equation with a cutoff function and a seed action. This implies that correlation functions of the composite operators consisting of the diffused fields (1.2) are finite so that the diffused fields can be used to renormalize the composite operators.
In this paper, we study generalization of the result in [11]. We consider the ERG equation for a scalar field with an arbitrary cutoff function and an arbitrary quadratic seed action.
We show that the correlation functions of diffused fields with respect to the bare action agree with those of bare fields with respect to the effective action obeying the above ERG equation, where the diffused field obeys a generalized diffusion equation determined by the cutoff function and the seed action and agrees with the bare field at the initial τ .
We also perform the ǫ expansion using the derivative expansion as a a check of validity of the ERG equation. We reproduce the well-known scaling dimensions of operators around the Wilson-Fisher fixed point.
Throughout this paper except section 5, we work in the momentum space, and introduce a notation Note that the Fourier transforms of (1.1), (1.2) and (1.3) are and respectively.
This paper is organized as follows. In section 2, we give our statement on relationship between the ERG equation and a generalized diffusion equation. In section 3, we prove our statement using functional differential equations for generating functionals of the correlation functions. In section 4, we provide another proof of our statement by solving the ERG equation using a functional integration kernel. In section 5, we perform the ǫ expansion using the derivative expansion. We reproduce the well-known scaling dimensions of operators around the Wilson-Fisher fixed point. Section 6 is devoted to conclusion and discussion. We clarify the reason why we need to restrict ourselves to seed actions that are quadratic in φ.

Relation between renormalization group and diffusion equation
The ERG equation is a functional differential equation that describes nonperturbatively how the effective action S Λ at the energy scale Λ changes when Λ is decreased to Λ − dΛ. The one for a scalar field in d dimensions is specified by a cutoff functionĊ Λ (p 2 ) and a seed action S Λ , and takes the form [12][13][14][15] (2.1) C Λ is quasi-local and incorporates UV regularization, while the seed actionŜ Λ is a functional of φ that has derivative expansion. Here we consider an arbitrary cutoff functionĊ(p 2 ) and an arbitrary quadratic seed action that satisfy the above conditions. The coarse graining procedure is fixed by the cutoff function and the seed action. We parametrize the seed action asŜ where χ Λ is an arbitrary regular function. The initial condition for (2.1) is given at a bare cutoff scale Λ 0 , and S Λ 0 = S Λ=Λ 0 is a bare action.
We define the vev with respect to the effective action S Λ as We show the following relation on the correlation functions where c stands for the connected part, and (2.6) ϕ(τ, p) is a solution to the differential equation with the initial condition r(Λ, p 2 ) obeys a differential equation and satisfies an initial condition r(Λ 0 , p 2 ) = 0 . For later convenience, we solve (2.10) and (2.8). The solution to (2.10) with the initial condition (2.11) is given by 14) The solution to (2.8) with the initial condition (2.9) is given by We will give a proof of (2.5) in the next section. Before closing this section, we give an example ofĊ and χ Λ (p 2 ): and Setting ζ = η in (2.16) yields an ERG equation studied in [11,16]. We rescale ϕ as (2.20) We see from (2.7) that ϕ ′ satisfies (1.5) and is given by (1.6). (2.20) is nothing but the relation obtained in [11].

Proof of the relation (2.5)
In this section, we give a proof of the relation (2.5). For this purpose, we define two functionals of J(p): U is the generating functional for the connected correlation functions of φ with respect to S Λ , while V ′ is the one for the connected correlation functions of ϕ with respect to S Λ 0 . U and V agree at Λ = Λ 0 because ϕ and r(Λ, p 2 ) satisfy (2.9) and (2.11), respectively: In the following, we will show that U and V satisfy the same functional differential equation, which is the first order in the Λ derivative. First, we calculate as follows: .
Here we have used (2.1) in the second equality and performed partial path-integrations in the third equality. In the fourth equality, In the fifth equality, we have used the explicit form ofŜ Λ (2.2).
as follows: Here we have used ( Thus, we have completed the proof of (2.5).

Functional integration kernel
In this section, we show (2.5) by solving (2.1) in terms of a functional integration kernel, which is a functional analog of the heat kernel (1.3). We seek for a functional integra-tion kernel K[φ, φ ′ , Λ, Λ 0 ] that is a solution to the ERG equation (2.1) satisfying an initial where ∆ is the delta functional. It is easy to show that K is given by where A Λ,Λ 0 and B Λ,Λ 0 are defined in (2.13) and (2.14), respectively, and satisfy the initial Then, (2.1) is solved in terms of K as

ǫ expansion
In this section, as a check of validity of the ERG equation (2.1), we perform the ǫ expansion in d = 4 − ǫ dimensions using the derivative expansion. We restrict ourselves to the case in whichĊ Λ (p 2 ) and χ Λ (p 2 ) are given by (2.16) and (2.18), respectively. We will see that the scaling dimensions of operators around the Wilson-Fisher fixed point are reproduced for arbitrary η and ζ. The ǫ expansion using the derivative expansion was studied for the Polchinski equation [18], which can be viewed as an ERG equation, in [17] and for another specific ERG equation in [9]. Here we perform procedure done in [9].
We set Λ 0 = ∞ so that (2.6) reduces to τ = 1/Λ 2 . We expandĊ Λ in terms of τ as 1 Then, we rewrite (2.1) in the coordinate space as In the following, we make the derivative expansion keeping up to two derivatives. The effective action is expanded as By using a formula we obtain from (5.2) On the other hand, we obtain from (5.3) Comparing (5.5) and (5.6) gives rise to We normalize the kinetic term in (5.3) as W τ (φ) = 1, so that Then, (5.7) and (5.8) reduce to We make a change of variable φ → ϕ such that W τ +δτ = 1: Then, the Jacobian for the change of variable is where we have reguralized the delta function δ (d) (x − y) by aK(x, y, τ ) with a being a constant.
We further make quantities dimensionless as Then, from (5.7), (5.13) and (5.14), we obtain We expand V τ in terms of φ as (5.16) By substituting (5.16) into (5.15), we obtain We look for the Wilson-Fisher fixed point by assuming that We set τ ∂ τ v * n = 0 in (5.17)-(5.19), and expand the RHS of (5.17) up to the first order in ǫ and those of (5.18) and (5.19) up to the second order in ǫ. Then, we obtain equations determining the fixed point as follows: We solve these equations as We take a such that A = 0. Note that We regard these as a linear transformation for δv n . Then, the eigenvalue equation reads Here we have ignored (5.37) and dropped the term proportional to δv 12 in (5.36). By substituting (5.27) and (5.28) into this equation, we obtain The solutions to (5.39) are

Conclusion and discussion
In this paper, we studied relationship between renormalization group and diffusion equation.
We considered the ERG equation for a scalar field that includes an arbitrary cutoff function and an arbitrary quadratic seed action. As a generalization of the result in ref. [11], we found that the correlation functions of diffused fields with respect to the bare action agree with those of bare fields with respect to the effective action, where the diffused field obeys a generalized diffusion equation determined by the cutoff function and the seed action and agrees with the bare field at the initial time. This result is reasonable in that diffusion is associated with coarse graining and that the coarse graining procedure is fixed by the cutoff function and the seed action. We performed the ǫ expansion using the derivative expansion as a check of validity of the ERG equation. We reproduced the well-known scaling dimensions of operators around the Wilson-Fisher fixed point.
We make a comment on a case in which the seed action includes terms that are higher than the second order in φ. In this case, U satisfies a functional differential equation given by the fourth equality in (3.7), while V satisfies a functional differential equation given by the third equality in (3.9). The latter equation includes terms that mix δV /δJ with δR/δJ, and those terms do not exist in the former equation. This implies that the relation (2.5) does not hold for this case even if r is modified. We need to generalize (2.5) in some way.