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In this work, we study the nonperturbative renormalization of the supercurrent operator in

Supersymmetric (SUSY) theories have long been considered a promising completion of the Standard Model with appealing properties like natural explanations for dark matter. Unbroken supersymmetry implies that particles arrange themselves into multiplets with the same number of fermionic and bosonic degrees of freedom. The supersymmetric partners of the Standard Model particles have not been observed in experiments so far. An unknown breaking mechanism is therefore required in a realistic extension of the Standard Model. In order to understand possible breaking scenarios, it is essential to investigate the nonperturbative regime of SUSY theories.

Another important motivation for nonperturbative investigations of SUSY theories are theoretical conjectures about the confinement mechanism and relations to gauge/gravity duality. These have their foundations in the enhanced symmetries of SUSY gauge theories and it would be interesting to extend and relate them to QCD or Yang-Mills theory. This requires more general insights into the nonperturbative regime of SUSY theories.

Numerical lattice simulations would be an ideal general nonperturbative first principles tool to investigate SUSY gauge theories. However, it is unavoidable to break SUSY in any nontrivial theory on the lattice. In general, fine-tuning is required to restore supersymmetry in the continuum limit, which can be guided by signals provided by the SUSY Ward identities. The analysis of SUSY Ward identities requires the renormalization of the supercurrent, which mixes due to broken supersymmetry with other operators of the same or lower dimension. In our current studies, we investigate how to determine the mixing in a perturbative and nonperturbative way. The first step is a study of

In Ref.

This approach, despite being successful for SYM, has limitations when applied to more general SUSY gauge theories. The number of tuning parameters and mixing coefficients is significantly larger in this case. Therefore it is essential to find alternative ways to determine the renormalization of the supercurrent and reduce the number of parameters that need to be determined from the WI. In this work, we explore an alternative way of renormalizing the supercurrent on the lattice, using a gauge-invariant renormalization scheme (GIRS).

GIRS has been employed in recent studies of operator mixing regarding the renormalization of the traceless gluon and quark energy-momentum tensor operators in QCD

The paper is organized as follows: Sec.

In this section, we introduce the setup of our calculation. We provide details on the action, the operators and the Green’s functions that we calculate in this work.

In our study, we consider

The definitions of the covariant derivatives are as follows:

In standard notation, the supercurrent

For ease of notation, we leave out the one free Dirac index in all operators appearing in the text. Similarly, we drop the two Dirac indices from all the Green’s functions appearing in the sequel.

The supercurrent is the conserved quantity associated with SUSY. When SUSY is broken (as is the case in both dimensional and lattice regularizations),

Class G: Gauge-invariant operators.

Class A: BRST variations of some operator.

Class B: Operators which vanish by the equations of motion.

Class C: Any other operators which share the same global symmetries, but do not belong to the above classes; these can at most have finite mixing with

To simplify the mixing problem, we implement a renormalization scheme in which only gauge-invariant Green’s functions are considered; thus a nonperturbative implementation of such a scheme avoids gauge fixing altogether. In particular, by extending the X-space scheme

In the case of multiplicatively renormalizable operators

The case at hand does not require Green’s functions containing products of more than two operators, whose evaluation is more demanding, both perturbatively and nonperturbatively. Moreover, the gauge-noninvariant operators of classes A–C cannot contribute to such Green’s functions and they need not be considered any further. As a consequence, the set of mixing operators in GIRS is greatly reduced and includes only gauge-invariant operators, which are accessible by lattice simulations.

There is only one gauge-invariant operator

The determination of the four elements of the mixing matrix requires four conditions. A maximum of three conditions can be imposed by considering expectation values between the two mixing operators:

A fourth condition can be obtained by considering two-point Green’s functions involving products of

Given Eq.

In this section we investigate the mixing problem in the continuum by regularizing the theory in

First, we present our results for the

One-loop and two-loop Feynman diagrams contributing to the tree-level and one-loop two-point Green’s functions of Eqs.

As is standard practice, the pole terms (

Next, we define appropriate renormalization conditions in GIRS, which must be applicable in both continuum and lattice. There is,

On the lattice, integration over time slices is replaced by summation, which is expected to reduce statistical errors in the nonperturbative data of the numerical simulations. The explicit form of the corresponding conditions on the lattice can be found in the next section.

Alternative definitions of the GIRS conditions may involve higher moments of the Green’s functions, e.g.,

While in

Integration over spatial components of

Equations

The final step in our perturbative calculation is the evaluation of the conversion matrix

Terms of the form

The GIRS renormalization conditions [Eqs.

In the following we apply this approach on a first set of test ensembles selected from the ones presented in

Details of the ensembles of gauge configurations can be found in earlier publications

The Dirac-Wilson operator breaks supersymmetry and chiral symmetry, but the critical point

Value of the

The parameters of these ensembles have been checked extensively in previous investigations. Finite size effects and the Pfaffian sign are under control. The lattice spacing is small enough to induce only a rather small supersymmetry breaking, while effects like topological freezing become relevant at even smaller lattice spacings

The supercurrent operators

It is worth noting that the definition of the gluino-glue operator, Eq.

The spatial projectors

This is opposite to the behavior of Eqs.

Correlators

For our choice of spatial projectors and after applying a summation over

The results on the right-hand side of Eqs.

The GIRS renormalization conditions, Eqs.

As already anticipated, all

Combining the nonperturbative result for the GIRS mixing matrix

The numerical values obtained for the ratio

Note that the perturbative estimates are positive, while the nonperturbative determination leads to a negative value. We have checked that the numerically determined correlators have the same sign as their perturbative counterparts. Consequently the effect cannot be due to some different overall sign factors. Nonperturbative effects should be relevant and our parameter range is most likely far outside the perturbative regime where the one-loop computations can be reliable. In order to provide a further illustration for the importance of higher perturbative or nonperturbative corrections, we derived an alternative determination of

In order to reduce higher loop corrections one would ideally like to simulate at smaller couplings close to the perturbative regime, however the cost of such simulations can increase rather fast. As an exploratory and alternative approach we have done measurements on smeared configurations to investigate how much the noise is reduced and whether the results get closer to the perturbative predictions. We applied six levels of stout smearing with smearing parameter

For improving the estimates, one may employ a tree-level correction by subtracting tree-level discretization effects from the correlators. This can be useful especially for small values of

Ratios between lattice (all orders in

In this paper, we have presented a concrete prescription to renormalize nonperturbatively the supercurrent operator in

We have employed GIRS on two lattices of SU(2) SYM with three different values of critical mass parameter, on a smeared ensemble, and on an ensemble based on the gauge group SU(3). We have presented nonperturbative results for the mixing matrix. In parallel, we have performed a one-loop perturbative calculation of the mixing matrix in both GIRS and

One means of improving the nonperturbative estimates is further elimination of the discretization errors. For small values of

A second improvement is the elimination of truncation effects coming from the conversion matrix. For large values of

From the numerical point of view, more statistics, especially for larger values of

We thank G. Münster for helpful discussions and comments. G. Bergner and I. Soler acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) Grants No. 432299911 and No. 431842497. This work was co-funded by the European Regional Development Fund and the Republic of Cyprus through the Research and Innovation Foundation (Projects No. EXCELLENCE/0918/0066 and No. EXCELLENCE/0421/0025). M. Costa also acknowledges partial support from the Cyprus University of Technology under the “POST-DOCTORAL” programme. G. Spanoudes acknowledges financial support from H2020 project PRACE-6IP (Grant Agreement No. 823767). The authors gratefully acknowledge the Gauss Centre for Supercomputing e. V.