^{3}

In K. Hieda, A. Kasai, H. Makino, and H. Suzuki, Prog. Theor. Exp. Phys.

In quantum field theory, lattice regularization enables nonperturbative computation from first principles but it breaks preferred symmetries in the target theory (such as the chiral and spacetime symmetries) quite often. For this reason, nonperturbative computation in supersymmetric field theory from first principles, especially on the basis of the lattice [

For the supersymmetric continuum limit, the lattice parameters should be tuned so that the Ward–Takahashi (WT) relations associated with the supersymmetry (SUSY) hold up to the finite lattice spacing effect. More specifically, one imposes the conservation law of the supercurrent—the Noether current associated with SUSY. See Refs. [

Having the above problem in mind, in Ref. [

This type of construction of the Noether current by the gradient flow was first considered for the energy–momentum tensor—the Noether current associated with the translational invariance [^{1}

In this paper, as a natural extension of the study of Ref. [^{2}

Since our analysis in this paper is rather lengthy, we summarize the basic line of reasoning and the main results in this section.

Our strategy is basically identical to that of Ref. [^{3}

In ^{4}

Contribution of each diagram in

Diagram | |
---|---|

By substituting the small time expansion obtained in

In this way, we have the supercurrent

These are our main results in this paper.^{5}

In

The Euclidean action of the ^{6}

In this expression,

The easiest way to derive these formulas is dimensional reduction from the 6D

To apply the perturbation theory, we also introduce the gauge-fixing and ghost–anti-ghost terms by

To derive SUSY WT relations, we consider the SUSY transformations in Eqs. (

The above supercurrents would be regarded as “canonical” ones. From the perspective of the conformal or scale symmetry of the classical theory, Eq. (

It can be seen that, noting that

Through the following analyses, however, we find that

Here, the added term in Eq. (

The effect of the insertion of such a term in a correlation function can be deduced by the infinitesimal change of variable

Some calculation shows that these finite supercurrents enjoy extremely simple forms:

At

The gauge-fixing and ghost–anti-ghost terms also break SUSY. We define the ghost and anti-ghost fields as SUSY singlets. Then

We note that

Because of Eq. (

In what follows, we consider SUSY WT relations following from the identities

In these identities, the parameters of the SUSY transformation,

From Eq. (

These are

Before going into the problem of renormalization, we analyze the effect of

One-loop Feynman diagrams containing

We see that the effect of

(Consideration of the correlation function

Interestingly, this counterterm breaks the global axial

In what follows, we assume the presence of the counterterm

In this section, we will work out the renormalization in the one-loop order. We set

The present 4D

For parameters and elementary fields, to the one-loop order, the renormalization is accomplished by

For some gauge-covariant composite operators that appear on the right-hand sides of SUSY WT relations, after some calculation, we find

From the renormalization factors in Eqs. (

Our present calculation of diagrams in

One can confirm that Eq. (^{7}

Combining the result of Ref. [

From Eqs. (

We thus observe that, to the one-loop order, the combination

We now rewrite the SUSY WT relations of Eqs. (

Then, the left-hand side of Eq. (

On the other hand, the right-hand side of Eq. (

We will be sloppy about the indices of

Starting from Eq. (

Also, Eq. (

Thus, in all the above WT relations, we have observed that the combination

We have observed that the combination in Eq. (

Now, in on-mass-shell correlation functions, equations of motion identically hold. Under tree-level equations of motion, Eq. (

Moreover, since

This is the current appearing in Eq. (

The bottom line of the above very lengthy one-loop analysis is that, in on-mass-shell correlation functions that contain gauge-invariant operators only, the combination in Eq. (

We now express these currents by fields defined by the gradient flow.

Our flow equations for the gauge field and the fermion field are standard ones [^{8}

The fields

For the scalar field, we adopt

This is also not the gradient flow in the narrow sense. In this case, for the renormalizability of the flowed scalar field, it is important not to include other terms of the equation of motion (such as the term arising from the Yukawa coupling) on the right-hand side of the flow equations. We refer the reader to Ref. [

A remarkable feature of the “gradient” flow is that any composite operator of flowed fields for ^{9}

Although the wave function renormalization of flowed fields renders all composite operators finite, the wave function renormalization factors themselves depend on the regularization, and this is not satisfactory from the perspective of a universal representation of composite operators. To avoid this point, we introduce the following ringed fields, following Ref. [

The wave function renormalization factor is canceled out in the ringed fields, and any composite operator of the ringed fields becomes finite without an explicit wave function renormalization. The expectation value

Similarly, for the scalar fields, we introduce [

For the following calculations, we need to compute the expectation value

In total,

In this subsection we present the computation of the small flow time expansion [

The calculations of the small flow time expansion are presented in Refs. [^{10}

Now, for diagrams

For

From diagrams B01, B04, B06, B08, B10, and B12, we have

Using the relation

On the other hand, by applying the parity transformations of Eqs. (

Finally, for

It is easy to invert the above relations and obtain expressions for composite operators of the unflowed fields in terms of composite operators of flowed fields to the one-loop order. For example, Eq. (

Similar inversions can be made for other relations.

We substitute the relations in the small flow time expansion presented in the last subsection into the expression of the supercurrent, Eq. (

The conjugate supercurrent

Some remarks are in order: (1) The composite operators in Eqs. (^{11}

By the standard argument, this implies that we can set the renormalization scale

Here, we have used the definition of the beta function

We would like to thank Hiroki Makino for discussions. This work was supported by Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research Grant Numbers JP16J02259 (A. K.), JP18J20935 (O. M.), and JP16H03982 (H. S.).

Open Access funding: SCOAP

^{1}A similar construction has also been considered for fermion bilinear operators [

^{2}This is a natural extension in the sense that we need the asymptotic freedom for the construction; in the 4D

^{3}This is the first explicit demonstration to our knowledge.

^{4}Flow equations in supersymmetric theories that are alternative to our choice are given in Refs. [

^{5}Our notational convention is summarized in

^{6}Here, we assume the spacetime dimension is

^{7}In principle, we should include the counterterm of Eq. (

^{8}The term that is proportional to the “gauge-fixing parameter”

^{9}It is shown that the parameter

^{10}If only the “topology” of the diagram is concerned, there also exist other diagrams which are not included in

^{11}Note that we can set