^{3}

We establish a concrete correspondence between a gradient flow and the renormalization group flow for a generic scalar field theory. We use the exact renormalization group formalism with a particular choice of the cutoff function.

The gradient flow, introduced in Refs. [

The gradient flow has been used for scale setting and for the definition of a topological charge [

The similarity between the gradient flow and the renormalization group flow was already pointed out at the beginning [

In the gradient flow we introduce a diffusion time

Based on the correspondence in Eq. (

We organize the paper as follows. In

We use the following shorthand notation for the momentum integrals:

We give a brief overview of the exact renormalization group. There are many reviews available on the subject (see Ref. [

Let

We have introduced a constant anomalous dimension

In this paper we choose

Given a bare action

The dependence on the reference momentum

As it is, the ERG differential equation of Eq. (

Defining

With an appropriate choice of

To derive a gradient flow for the scalar field, we need to rewrite Eq. (

We introduce the generating functionals for

By substituting Eq. (

The extra quadratic term in the

The result in Eq. (

The connected parts of the higher-point functions are simply related by the same change of normalization as

To understand the above results better, let us introduce a

Using the explicit form in Eq. (

In coordinate space, this gives

Integrating Eqs. (

Using Eqs. (

Similarly, we obtain

Suppose the bare theory

We then obtain

The large-

We next consider a bare action

Let us outline the construction of the renormalized trajectory, following Sect. 12 of Ref. [

Given a bare action

We can take the dimensionless squared mass

This means that the solution

We assume that the fixed point

This implies that

We can then define a renormalized trajectory by the limit

For the limit to exist, we must find

For an explanation that such a limit exists, we refer the reader to standard references such as Sect. 12 of Ref. [

Hence, from Eq. (

ERG flows

Now that we have constructed a renormalized trajectory

The squared mass of

We then define a renormalized Wilson action with cutoff

Since

Eq. (

Since

This implies that the correlation functions are related by

Note that the diffused field only needs the standard wave function renormalization in the continuum limit

Before closing this section, we would like to relate the correlation functions in the continuum limit to those obtained by the Wilson action

The field of the Wilson action corresponds to a diffused field of the continuum limit, and we use the factor

The correlation functions with double brackets are the continuum limit defined at renormalization scale

This explains the powers of

In the previous section we obtained the relation in Eqs. (

By construction, see Eqs. (

Hence, integrating over the momenta, we obtain

Let us introduce the dimensionless analogs of Eqs. (

These satisfy the scaling laws

Correspondingly, the correlation functions in coordinate space, defined by

Thus, we obtain

The functions

Since the mass scale of

Hence, for such large

The coefficient functions satisfy the RG equations:

For

We thus obtain the large-

Using Eqs. (

This is the analog of the small-

In this paper we have considered the gradient flow of a real scalar field obeying the simple diffusion equation without potential terms. We have then shown that the correlation functions of diffused fields match with those of elementary fields of a Wilson action that has a finite momentum cutoff. We have only discussed formalism, and we plan to provide concrete examples of the correspondence in a future publication.

Obviously we have scratched only the tip of an iceberg. In theories such as gauge theories and non-linear sigma models, the fields are continuous but live naturally in a compact space, and the diffusion equations that respect the geometry of the compact space should be and have been introduced [

The work of H. Suzuki is supported in part by Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research Grant Number JP16H03982.

Open Access funding: SCOAP