# Nonperturbative renormalization of the supercurrent in $\cal{N}$ = 1
Supersymmetric Yang-Mills Theory

G. Bergner<EMAIL_ADDRESS>University of Jena, Institute for
Theoretical Physics,
Max-Wien-Platz 1, D-07743 Jena, Germany
M. Costa<EMAIL_ADDRESS>Department of Physics, University of Cyprus,
P.O. Box 20537, 1678 Nicosia, Cyprus
Department of Chemical Engineering, Cyprus University of Technology,
30 Archbishop Kyprianou Str., 3036, Limassol, Cyprus
H. Panagopoulos<EMAIL_ADDRESS>Department of Physics, University of Cyprus,
P.O. Box 20537, 1678 Nicosia, Cyprus
S. Piemonte<EMAIL_ADDRESS>University of Regensburg, Institute for
Theoretical Physics,
Universitätsstr. 31, D-93040 Regensburg, Germany
I. Soler Calero<EMAIL_ADDRESS>University of Jena, Institute
for Theoretical Physics,
Max-Wien-Platz 1, D-07743 Jena, Germany
G. Spanoudes
<EMAIL_ADDRESS>Computation-based Science and Technology Research
Center,
The Cyprus Institute, 20 Constantinou Kavafi Str., 2121 Nicosia, Cyprus

###### Abstract

In this work, we study the nonperturbative renormalization of the supercurrent
operator in $\mathcal{N}=1$ Supersymmetric Yang-Mills (SYM) theory, using a
gauge-invariant renormalization scheme (GIRS). The proposed prescription
addresses successfully the unwanted mixing of the supercurrent with other
operators of equal or lower dimension, which respect the same global
symmetries. This mixing is introduced by the unavoidable breaking of
supersymmetry on the lattice. In GIRS all gauge-noninvariant operators, which
mix with the supercurrent, are excluded from the renormalization procedure.
The one remaining mixing operator is accessible by numerical simulations. We
present results for the renormalization of the supercurrent using a GIRS
scheme. We also compute at one-loop order the conversion matrix which relates
the nonperturbative renormalization factors in GIRS to the reference scheme
${\overline{\rm MS}}$.

##  I Introduction

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
(SM) 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 non-trivial 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
$\mathcal{N}=1$ supersymmetric Yang-Mills theory (SYM).

In Ref. Curci and Veneziano (1987) Curci and Veneziano have shown that only a
tuning of the gluino mass is required to restore supersymmetry in the
continuum limit with Wilson fermions. There is even a very basic connection
between chiral symmetry and supersymmetry in SYM, as shown in Suzuki (2012).
The theory has been successfully simulated using this approach and the results
show consistency with supersymmetry Bergner  _et al._ (2016); Ali  _et al._
(2019), where the low energy spectrum was obtained in the continuum limit. The
tuning has been done based on chiral symmetry, confirming a consistency with
supersymmetric Ward identities (WI) afterwards Farchioni  _et al._ (2002); Ali
_et al._ (2018, 2020). On the lattice the renormalized WI get corrections due
to mixing of the supercurrent with all possible operators of the same or lower
dimension that respect all unbroken symmetries on the lattice and share the
same quantum numbers. Mixing also accounts for an additive renormalization of
the fermion mass Curci and Veneziano (1987). The fermion mass renormalization
provides a signal for the gluino mass tuning. The mixing of the supercurrent
appears as an additional undetermined parameter. In the standard approach,
mixing and mass renormalization have to be determined simultaneously from the
same set of numerically determined WI. In previous investigations, the mixing
has also been determined to one loop order in perturbation theory Taniguchi
(2000).

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 trace-less gluon and quark energy-momentum tensor
operators in QCD Costa  _et al._ (2021a) and the gluino-glue operator in SYM
theory Costa  _et al._ (2021b). In the spirit of X-space scheme Gimenez  _et
al._ (2004); Chetyrkin and Maier (2011); Cichy  _et al._ (2012); Tomii and
Christ (2019), GIRS involves Green’s functions of two or more gauge-invariant
operators in coordinate space. The main advantage of this scheme is that all
gauge-noninvariant operators appearing in the set of mixing operators are
automatically excluded from the renormalization procedure since their Green’s
functions are zero. In order to set appropriate conditions in GIRS, which are
applicable in both continuum and lattice, we first perform a perturbative
calculation of certain Green’s functions involving all the gauge-invariant
mixing operators in a continuum regularization. Our final goal is to
renormalize the supercurrent in the ${\overline{\rm MS}}$ scheme, which is the
typical scheme used in the analysis of experimental data. Since
${\overline{\rm MS}}$ is defined perturbatively in dimensional regularization
(DR), we employ DR and we extract the one-loop conversion matrix between GIRS
and ${\overline{\rm MS}}$. The renormalization factors and mixing coefficients
of the supercurrent in ${\overline{\rm MS}}$ on the lattice will be extracted
at the end by combining our results for the perturbative conversion matrix
with the nonperturbative GIRS mixing matrix on the lattice.

The paper is organized as follows: Sec. II contains the formulation of our
calculation including the $\mathcal{N}=1$ SYM action, the set of operators
under study, as well as an introduction to the GIRS renormalization
prescription and to the Green’s functions calculated in this work. In Sec.
III, we present our perturbative calculation in dimensional regularization and
we provide one-loop results for the Green’s functions, the renormalization
factors/mixing coefficients, and the conversion matrix between the GIRS and
${\overline{\rm MS}}$ schemes. A description of the lattice setup along with
the nonperturbative results for the renormalization of the supercurrent is
presented in Sec. IV. Finally, we conclude in Sec. V with a summary and a
discussion of our results and possible future extensions of our work.

##  II GIRS in supersymmetric Yang-Mills

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.

###  II.1 The action of supersymmetric Yang-Mills theory

In our study, we consider $\mathcal{N}=1$ supersymmetric Yang-Mills (SYM)
theory with gauge group SU($N_{c}$) in the continuum and on the lattice. By
applying the Wess-Zumino gauge Wess and Bagger (1992) and by eliminating
auxiliary fields of the theory, we end up with the following gauge-fixed
continuum action in Euclidean space

$\mathcal{S}_{\rm SYM}=\int
d^{4}x\left[\frac{1}{4}u_{\mu\nu}^{\alpha}(x)u_{\mu\nu}^{\alpha}(x)+\frac{1}{2}\bar{\lambda}^{\alpha}(x)\gamma_{\mu}\mathcal{D}_{\mu}\lambda^{\alpha}(x)-\frac{1}{2\xi}(\partial_{\mu}u_{\mu}^{\alpha}(x))^{2}-\bar{c}^{\alpha}(x)\partial_{\mu}D_{\mu}^{\alpha\beta}c^{\beta}(x)\right],$
(1)

where $u_{\mu}^{\alpha}(x)$ ($\lambda^{\alpha}(x)$, $c^{\alpha}(x)$) is the
gluon (gluino, ghost) field, $u_{\mu\nu}^{\alpha}$ is the gluon field-strength
tensor and $\xi$ is the gauge parameter [$\xi=1(0)$ corresponds to Feynman
(Landau) gauge]. The gluino field is a Majorana fermion, and thus, $\lambda$
and $\bar{\lambda}$ are related through a charge conjugation transformation.
Since supersymmetry is broken in all known regularizations (including
dimensional and lattice regularizations) at intermediate steps, we choose the
gauge-fixing and ghost terms arising from the Faddeev-Popov procedure to be
the same as in the nonsupersymmetric case. Due to the gauge fixing, the total
action is no longer gauge-invariant but it is Becchi-Rouet-Stora-Tyutin (BRST)
invariant. On the lattice, we employ a tree-level Symanzik improved gauge
action Symanzik (1983) and Wilson/clover fermions Sheikholeslami and Wohlert
(1985) for the gluino fields; the action reads

$\displaystyle{\cal S}^{L}_{\rm SYM}=a^{4}\sum_{x}$ $\displaystyle\Bigg{\\{}$
$\displaystyle\frac{2}{g^{2}}\left[\frac{5}{3}\sum_{\rm plaq.}{\rm Re}\ {\rm
tr}_{c}(1-U_{\rm plaq.})-\frac{1}{12}\sum_{\rm rect.}{\rm Re}\ {\rm
tr}_{c}(1-U_{\rm rect.})\right]$ (2) $\displaystyle+$
$\displaystyle\sum_{\mu}\left[{\rm
tr}_{c}(\bar{\lambda}\gamma_{\mu}D_{\mu}\lambda)-\frac{ar}{2}{\rm
tr}_{c}(\bar{\lambda}D^{2}\lambda)\right]-\sum_{\mu,\nu}\left(\frac{c_{\rm
SW}\
a}{4}\bar{\lambda}^{\alpha}\sigma_{\mu\nu}\hat{F}_{\mu\nu}^{\alpha\beta}\lambda^{\beta}\right)+m_{0}{\rm
tr}_{c}(\bar{\lambda}\lambda)\Bigg{\\}},$

where $r\ (c_{\rm SW})$ is the Wilson (clover) parameter,
$\sigma_{\mu\nu}=\frac{1}{2}[\gamma_{\mu},\gamma_{\nu}]$, $U_{\rm
plaq.}(U_{\rm rect.})$ denotes $1{\times}1$ ($2{\times}1$) rectangular Wilson
loops, $\hat{F}_{\mu\nu}^{\alpha\beta}$ is the clover definition of the field
strength tensor in the adjoint representation and $m_{0}$ is the Lagrangian
mass. The explicit definitions of $U_{\rm plaq.}$, $U_{\rm rect.}$,
$\hat{F}_{\mu\nu}^{\alpha\beta}$ in terms of link variables $U_{\mu}(x)$, and
the covariant derivatives are standard; they can be found in, e.g. Ref. Costa
_et al._ (2021c). Note that gauge-fixing and ghost terms must be added in the
above action, as well as a “measure” term coming from the change of
integration variables $U_{\mu}\to u_{\mu}$. These terms are chosen to be equal
to the corresponding terms in the nonsupersymmetric case (see, e.g. Costa and
Panagopoulos (2017)).

###  II.2 Supercurrent and mixing operators

In standard notation, the supercurrent de Wit and Freedman (1975) is defined
as111For 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.:

$S_{\mu}(x)\equiv-\sigma_{\nu\rho}\gamma_{\mu}{\rm
tr}_{c}(\,u_{\nu\,\rho}(x)\lambda(x)),\qquad\sigma_{\nu\rho}=\frac{1}{2}[\gamma_{\nu},\gamma_{\rho}].$
(3)

The supercurrent is the conserved quantity associated with SUSY. When SUSY is
broken (as is the case in both dimensional and lattice regularizations),
$S_{\mu}(x)$ is no longer conserved and must be renormalized. $S_{\mu}(x)$ can
mix with other operators of the same or lower dimensionality which respect the
same global symmetries. On general grounds Joglekar and Lee (1976), such
operators can be separated into four classes:

  * •

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
$S_{\mu}(x)$.

Mixing with gauge-noninvariant (class A-C) operators results from the
introduction of a gauge-fixing term in the action. A list of all possible
candidate operators which can mix with $S_{\mu}(x)$ can be found in Refs.
Bergner  _et al._ (2022a, b).

###  II.3 Supercurrent renormalization in GIRS

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 Gimenez  _et al._ (2004); Chetyrkin and Maier
(2011); Cichy  _et al._ (2012); Tomii and Christ (2019), we consider Green’s
functions of products of gauge-invariant operators (at different spacetime
points, in a way as to avoid potential contact singularities), e.g.,

$\langle S_{\mu}(x)\ S_{\nu}(y)\rangle,\qquad(x\neq y).$ (4)

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 $T_{\mu}(x)$ (see Ref. Ali  _et
al._ (2018) and references therein), which can mix with the supercurrent:

$T_{\mu}(x)\equiv 2\,\gamma_{\nu}{\rm tr}_{c}(\,u_{\mu\,\nu}(x)\lambda(x)).$
(5)

Thus, we construct a $2\times 2$ mixing matrix, which relates the bare (B) to
the renormalized (R) operators:

$\displaystyle\begin{pmatrix}{S}^{R}_{\mu}(x)&\\\\[8.61108pt]
{T}^{R}_{\mu}(x)\end{pmatrix}=\begin{pmatrix}Z_{SS}^{B,R}&Z_{ST}^{B,R}\\\\[8.61108pt]
Z_{TS}^{B,R}&Z_{TT}^{B,R}\end{pmatrix}\begin{pmatrix}{S}^{B}_{\mu}(x)&\\\\[8.61108pt]
{T}^{B}_{\mu}(x)\end{pmatrix}.$ (6)

[The index $B$ refers to dimensional (${\rm DR}$) or lattice (${\rm L}$)
regularization, and $R={\overline{\rm MS}},{\rm GIRS}$ to the renormalization
scheme. For simplicity, the renormalization functions $Z^{B,R}$ will be often
denoted merely as $Z$.]

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:

$\displaystyle G^{SS}_{\mu\nu}(x,y)$ $\displaystyle\equiv$
$\displaystyle\langle S_{\mu}(x)\ \overline{S}_{\nu}(y)\rangle,$ (7)
$\displaystyle G^{TT}_{\mu\nu}(x,y)$ $\displaystyle\equiv$
$\displaystyle\langle T_{\mu}(x)\ \overline{T}_{\nu}(y)\rangle,$ (8)
$\displaystyle G^{ST}_{\mu\nu}(x,y)$ $\displaystyle\equiv$
$\displaystyle\langle S_{\mu}(x)\ \overline{T}_{\nu}(y)\rangle,$ (9)
$\displaystyle G^{TS}_{\mu\nu}(x,y)$ $\displaystyle\equiv$
$\displaystyle\langle T_{\mu}(x)\ \overline{S}_{\nu}(y)\rangle,$ (10)

where

$\displaystyle\overline{S}_{\mu}(x)$ $\displaystyle\equiv$
$\displaystyle\,{\rm
tr}_{c}(\,\bar{\lambda}(x)u_{\nu\,\rho}(x))\gamma_{\mu}\sigma_{\nu\rho},$ (11)
$\displaystyle\overline{T}_{\mu}(x)$ $\displaystyle\equiv$ $\displaystyle
2\,{\rm tr}_{c}(\,\bar{\lambda}(x)u_{\mu\,\nu}(x))\gamma_{\nu}.$ (12)

For convenience, we choose to express the “bar” operators in terms of
$\bar{\lambda}$ rather than $\lambda$. The bar operators
$\overline{A}=\overline{S}_{\mu},\overline{T}_{\mu}$ are related to the
original operators $A=S_{\mu},T_{\mu}$ through charge conjugation
transformations, as follows:

$\overline{A}\equiv A_{C}^{T}\ C^{T},\qquad A=C\overline{A}_{C}^{T},$ (13)

where $A_{C}$ is the operator $A$ with its fields replaced by their charge
conjugates, and $C$ is the charge conjugation matrix satisfying
$C\gamma_{\mu}C^{-1}=-\gamma_{\mu}^{T}$. The transformations of fields under
charge conjugation are given below:

$\displaystyle u_{\mu}$ $\displaystyle\rightarrow$ $\displaystyle-
u_{\mu}^{T},$ (14) $\displaystyle\lambda$ $\displaystyle\rightarrow$
$\displaystyle C\overline{\lambda}^{T},$ (15)
$\displaystyle\overline{\lambda}$ $\displaystyle\rightarrow$
$\displaystyle-\lambda^{T}C^{-1}.$ (16)

A fourth condition can be obtained by considering two-point Green’s functions
involving products of $S_{\mu}(x)$ or $T_{\mu}(x)$ with the Gluino-Glue
operator $\mathcal{O}(x)$, which is the only other gauge-invariant operator of
equal or lower dimension:

$\displaystyle G^{S{\cal O}}_{\mu}(x,y)$ $\displaystyle\equiv$
$\displaystyle\langle S_{\mu}(x)\ \overline{{\cal O}}(y)\rangle,$ (17)
$\displaystyle G^{T{\cal O}}_{\mu}(x,y)$ $\displaystyle\equiv$
$\displaystyle\langle T_{\mu}(x)\ \overline{{\cal O}}(y)\rangle,$ (18)
$\displaystyle G^{{\cal O}S}_{\mu}(x,y)$ $\displaystyle\equiv$
$\displaystyle\langle{\cal O}(x)\ \overline{S}_{\mu}(y)\rangle,$ (19)
$\displaystyle G^{{\cal O}T}_{\mu}(x,y)$ $\displaystyle\equiv$
$\displaystyle\langle{\cal O}(x)\ \overline{T}_{\mu}(y)\rangle,$ (20)

where

$\displaystyle{\cal O}(x)$ $\displaystyle\equiv$
$\displaystyle\sigma_{\mu\nu}\,{\rm{tr}}_{c}(\,u_{\mu\nu}(x)\lambda(x)),$ (21)
$\displaystyle\overline{{\cal O}}(x)$ $\displaystyle\equiv$
$\displaystyle\,{\rm{tr}}_{c}(\,\bar{\lambda}(x)u_{\mu\nu}(x))\sigma_{\mu\nu}.$
(22)

$\overline{{\cal O}}(x)$ satisfies Eq. (13). The operator ${\cal O}(x)$ is
multiplicatively renormalizable in GIRS Costa  _et al._ (2021b) and its
renormalization factor is obtained by considering the following Green’s
function:

$G^{{\cal O}{\cal O}}(x,y)\equiv\langle{\cal O}(x)\ \overline{{\cal
O}}(y)\rangle.$ (23)

Given Eq. (13), the above Green’s functions are related among themselves
through charge conjugation transformations, as follows:

$\left\langle A(x){\overline{B}}(y)\right\rangle=C{\left\langle
B(y){\overline{A}}(x)\right\rangle}^{T}C^{-1},$ (24)

where $A,B=S_{\mu},T_{\mu},{\cal O}$. Note that for determining the mixing
matrix, we may not need to implement all choices of Lorentz/Dirac indices in a
nonperturbative evaluation of the above Green’s functions; the exact choice of
the renormalization conditions will dictate which components of the Green’s
functions are needed. However, in order to determine a consistent and solvable
set of nonperturbative renormalization conditions, we need to calculate all
Green’s functions perturbatively in dimensional regularization. Also, since
there is no unique way of selecting solvable conditions in GIRS, a
perturbative calculation of all the above Green’s functions will be useful for
determining conversion factors from all possible variants of GIRS to the
${\overline{\rm MS}}$ scheme.

##  III Perturbative calculation in dimensional regularization

In this Section we investigate the mixing problem in the continuum by
regularizing the theory in $d\equiv 4-2\varepsilon$ dimensions. We apply both
${\overline{\rm MS}}$ and ${\rm GIRS}$ and we extract the conversion matrix,
$C^{{\rm GIRS},{\overline{\rm MS}}}$, between the two schemes to one loop. The
conversion matrix along with the lattice mixing matrix in ${\rm GIRS}$,
$Z^{{\rm L},{\rm GIRS}}$, (computed nonperturbatively) allow us to determine
the lattice mixing matrix in the ${\overline{\rm MS}}$ scheme, $Z^{{\rm
L},{\overline{\rm MS}}}$, through the following relation:

$\displaystyle\begin{pmatrix}Z_{SS}^{{\rm L},{\overline{\rm MS}}}&Z_{ST}^{{\rm
L},{\overline{\rm MS}}}\\\\[8.61108pt] Z_{TS}^{{\rm L},{\overline{\rm
MS}}}&Z_{TT}^{{\rm L},{\overline{\rm
MS}}}\end{pmatrix}=\begin{pmatrix}C_{SS}^{{\rm GIRS},{\overline{\rm
MS}}}&C_{ST}^{{\rm GIRS},{\overline{\rm MS}}}\\\\[8.61108pt] C_{TS}^{{\rm
GIRS},{\overline{\rm MS}}}&C_{TT}^{{\rm GIRS},{\overline{\rm
MS}}}\end{pmatrix}\begin{pmatrix}Z_{SS}^{{\rm L},{\rm GIRS}}&Z_{ST}^{{\rm
L},{\rm GIRS}}\\\\[8.61108pt] Z_{TS}^{{\rm L},{\rm GIRS}}&Z_{TT}^{{\rm L},{\rm
GIRS}}\end{pmatrix}.$ (25)

###  III.1 One-loop calculation in the ${\overline{\rm MS}}$ scheme

First, we present our results for the ${\overline{\rm MS}}$-renormalized
Green’s functions. Due to the $2\times 2$ mixing, the renormalized Green’s
functions are linear combinations of bare Green’s functions, which are given
below in matrix form:

$\displaystyle\begin{pmatrix}G^{SS,\,R}_{\mu\nu}&G^{ST,\,R}_{\mu\nu}\\\\[8.61108pt]
G^{TS,\,R}_{\mu\nu}&G^{TT,\,R}_{\mu\nu}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\quad\ \ \ \begin{pmatrix}Z_{SS}&Z_{ST}\\\\[8.61108pt]
Z_{TS}&Z_{TT}\end{pmatrix}\begin{pmatrix}G^{SS,\,B}_{\mu\nu}&G^{ST,\,B}_{\mu\nu}\\\\[8.61108pt]
G^{TS,\,B}_{\mu\nu}&G^{TT,\,B}_{\mu\nu}\end{pmatrix}\begin{pmatrix}Z_{SS}&Z_{TS}\\\\[8.61108pt]
Z_{ST}&Z_{TT}\end{pmatrix},$ (26)
$\displaystyle\begin{pmatrix}G^{S\mathcal{O},\,R}_{\mu}\\\\[8.61108pt]
G^{T\mathcal{O},\,R}_{\mu}\end{pmatrix}$ $\displaystyle=$ $\displaystyle
Z_{\mathcal{O}}\ \begin{pmatrix}Z_{SS}&Z_{ST}\\\\[8.61108pt]
Z_{TS}&Z_{TT}\end{pmatrix}\begin{pmatrix}G^{S\mathcal{O},\,B}_{\mu}\\\\[8.61108pt]
G^{T\mathcal{O},\,B}_{\mu}\end{pmatrix},$ (27)
$\displaystyle\begin{pmatrix}G^{\mathcal{O}S,\,R}_{\mu}\\\\[8.61108pt]
G^{\mathcal{O}T,\,R}_{\mu}\end{pmatrix}$ $\displaystyle=$ $\displaystyle
Z_{\mathcal{O}}\ \begin{pmatrix}Z_{SS}&Z_{ST}\\\\[8.61108pt]
Z_{TS}&Z_{TT}\end{pmatrix}\begin{pmatrix}G^{\mathcal{O}S,\,B}_{\mu}\\\\[8.61108pt]
G^{\mathcal{O}T,\,B}_{\mu}\end{pmatrix},$ (28) $\displaystyle
G^{\mathcal{O}\mathcal{O},\,R}$ $\displaystyle=$
$\displaystyle(Z_{\mathcal{O}})^{2}\ \ G^{\mathcal{O}\mathcal{O},\,B}.$ (29)

The evaluation of the corresponding bare Green’s functions at tree-level and
to one-loop order in dimensional regularization (DR) involve, respectively,
the one-loop and two-loop Feynman diagrams of Fig. 1.

Figure 1: One-loop and two-loop Feynman diagrams contributing to the tree-
level and one-loop two-point Green’s functions of Eqs. (7 – 10, 17 – 20, 23).
A wavy (solid, dashed) line represents gluons (gluinos, ghosts). The two
crosses denote the insertions of operators $S_{\mu},T_{\nu},\mathcal{O}$
appearing in the definition of each two-point function.

As is standard practice, the pole terms ($1/\varepsilon^{n}$,
$n\in\mathbb{Z}^{+}$) are removed by defining the ${\overline{\rm MS}}$ mixing
matrix elements to have only negative integer powers of $\varepsilon$, i.e.,
$Z_{ij}^{{\rm DR},{\overline{\rm
MS}}}=\delta_{ij}+g^{2}(z_{ij}/\varepsilon)+\mathcal{O}(g^{4})$, where
$i,j=S,T$ and $Z_{\mathcal{O}}^{{\rm DR},{\overline{\rm
MS}}}=1+g^{2}(z_{\mathcal{O}}/\varepsilon)+\mathcal{O}(g^{4})$. Our results
for $Z_{ij}^{{\rm DR},{\overline{\rm MS}}}$, $Z_{\mathcal{O}}^{{\rm
DR},{\overline{\rm MS}}}$ read:

$\displaystyle Z_{SS}^{{\rm DR},{\overline{\rm MS}}}$ $\displaystyle=$
$\displaystyle 1+\mathcal{O}(g^{4}),$ (30) $\displaystyle Z_{ST}^{{\rm
DR},{\overline{\rm MS}}}$ $\displaystyle=$ $\displaystyle\mathcal{O}(g^{4}),$
(31) $\displaystyle Z_{TS}^{{\rm DR},{\overline{\rm MS}}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{16\,\pi^{2}}\;\frac{3N_{c}}{2\varepsilon}+\mathcal{O}(g^{4}),$
(32) $\displaystyle Z_{TT}^{{\rm DR},{\overline{\rm MS}}}$ $\displaystyle=$
$\displaystyle
1-\frac{g^{2}}{16\,\pi^{2}}\;\frac{3N_{c}}{\varepsilon}+\mathcal{O}(g^{4}),$
(33) $\displaystyle Z_{\mathcal{O}}^{{\rm DR},{\overline{\rm MS}}}$
$\displaystyle=$ $\displaystyle
1-\frac{g^{2}}{16\,\pi^{2}}\;\frac{3N_{c}}{\varepsilon}+\mathcal{O}(g^{4}),$
(34)

which agree with our recent one-loop calculations of Refs. Costa  _et al._
(2021c); Bergner  _et al._ (2022b), where gauge-noninvariant Green’s functions
of all mixing operators with external elementary fields are considered. The
one-loop expressions for the ${\overline{\rm MS}}$-renormalized Green’s
functions are given below:

$\displaystyle G^{SS,\,{\overline{\rm MS}}}_{\mu\nu}(x,y)$ $\displaystyle=$
$\displaystyle-\frac{2(N_{c}^{2}-1)}{3\pi^{4}{(z^{2})}^{4}}\,(3-5\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c})\,(4s_{\mu\nu}^{[3]}+s_{\mu\nu}^{[4]}),$ (35)
$\displaystyle G^{TT,\,{\overline{\rm MS}}}_{\mu\nu}(x,y)$ $\displaystyle=$
$\displaystyle-\frac{(N_{c}^{2}-1)}{6\pi^{4}{(z^{2})}^{4}}\Big{[}(3-5\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c})\,(4s_{\mu\nu}^{[3]}-2s_{\mu\nu}^{[4]})$ (36)
$\displaystyle\qquad\qquad\qquad+9\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c}\Big{(}2s_{\mu\nu}^{[1]}+2s_{\mu\nu}^{[2]}-3\,(3+4\gamma_{E}+2\ln(\bar{\mu}^{2}z^{2}/4))\,s_{\mu\nu}^{[4]}\Big{)}\Big{]},$
$\displaystyle G^{ST,\,{\overline{\rm MS}}}_{\mu\nu}(x,y)$ $\displaystyle=$
$\displaystyle-\frac{(N_{c}^{2}-1)}{3\pi^{4}{(z^{2})}^{4}}\Big{[}(3-5\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c})(4s_{\mu\nu}^{[3]}+s_{\mu\nu}^{[4]})+18\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c}(s_{\mu\nu}^{[2]}-s_{\mu\nu}^{[4]})\Big{]},$ (37)
$\displaystyle G^{TS,\,{\overline{\rm MS}}}_{\mu\nu}(x,y)$ $\displaystyle=$
$\displaystyle-\frac{(N_{c}^{2}-1)}{3\pi^{4}{(z^{2})}^{4}}\Big{[}(3-5\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c})(4s_{\mu\nu}^{[3]}+s_{\mu\nu}^{[4]})+18\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c}(s_{\mu\nu}^{[1]}-s_{\mu\nu}^{[4]})\Big{]},$ (38)
$\displaystyle G^{S{\cal O},\,{\overline{\rm MS}}}_{\mu}(x,y)$
$\displaystyle=$
$\displaystyle-\frac{(N_{c}^{2}-1)}{\pi^{4}{(z^{2})}^{4}}12\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c}s_{\mu}^{[6]},$ (39) $\displaystyle G^{{\cal
O}S,\,{\overline{\rm MS}}}_{\mu}(x,y)$ $\displaystyle=$
$\displaystyle-\frac{(N_{c}^{2}-1)}{\pi^{4}{(z^{2})}^{4}}12\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c}s_{\mu}^{[6]},$ (40) $\displaystyle G^{T{\cal
O},\,{\overline{\rm MS}}}_{\mu}(x,y)$ $\displaystyle=$
$\displaystyle-\frac{(N_{c}^{2}-1)}{\pi^{4}{(z^{2})}^{4}}\Big{[}\Big{(}3+2\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c}(8+18\gamma_{E}+9\ln(\bar{\mu}^{2}z^{2}/4))\Big{)}(\phantom{+}s_{\mu}^{[5]}+s_{\mu}^{[6]})-6\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c}s_{\mu}^{[5]}\Big{]},$ (41) $\displaystyle G^{{\cal
O}T,\,{\overline{\rm MS}}}_{\mu}(x,y)$ $\displaystyle=$
$\displaystyle-\frac{(N_{c}^{2}-1)}{\pi^{4}{(z^{2})}^{4}}\Big{[}\Big{(}3+2\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c}(8+18\gamma_{E}+9\ln(\bar{\mu}^{2}z^{2}/4))\Big{)}(-s_{\mu}^{[5]}+s_{\mu}^{[6]})+6\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c}s_{\mu}^{[5]}\Big{]},$ (42) $\displaystyle G^{{\cal
O}{\cal O},\,{\overline{\rm MS}}}(x,y)$ $\displaystyle=$
$\displaystyle-\frac{2(N_{c}^{2}-1)}{\pi^{4}{(z^{2})}^{4}}\Big{[}3+2\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c}(5+18\gamma_{E}+9\ln(\bar{\mu}^{2}z^{2}/4))\Big{]}\not{z},$
(43)

where $z\equiv x-y$,

$\displaystyle s_{\mu\nu}^{[1]}(z)$ $\displaystyle\equiv$
$\displaystyle\gamma_{\mu}z_{\nu},\qquad
s_{\mu\nu}^{[2]}(z)\equiv\gamma_{\nu}z_{\mu},\qquad
s_{\mu\nu}^{[3]}(z)\equiv(\delta_{\mu\nu}-2\frac{z_{\mu}z_{\nu}}{z^{2}})\not{z},$
(44) $\displaystyle s_{\mu\nu}^{[4]}(z)$ $\displaystyle\equiv$
$\displaystyle\gamma_{\mu}\not{z}\gamma_{\nu},\quad\ \,s_{\mu}^{[5]}(z)\equiv
z_{\mu}\mathbf{1},\qquad\ \,s_{\mu}^{[6]}(z)\equiv\sigma_{\mu\rho}z_{\rho}.$
(45)

If a gluino mass were present the above Green’s functions would also contain
the structures of Eqs. (44 – 45) multiplied by an extra $\not{z}$.

###  III.2 Renormalization conditions in GIRS

Next, we define appropriate renormalization conditions in GIRS, which must be
applicable in both continuum and lattice. There is, a priori, wide flexibility
in determining conditions in GIRS. In our study we consider the following set
of conditions, in which we integrate over the spatial components of
$z=x-y=(\vec{z},t)$ :

$\displaystyle\int d^{3}\vec{z}\ {\rm Tr}\left[G^{SS,\,{\rm
GIRS}}_{\mu\nu}(x,y)\ P_{\nu\mu}\right]$ $\displaystyle=$ $\displaystyle\int
d^{3}\vec{z}\ {\rm Tr}\left[G^{SS,\,{\rm tree}}_{\mu\nu}(x,y)\
P_{\nu\mu}\right],$ (46) $\displaystyle\int d^{3}\vec{z}\ {\rm
Tr}\left[G^{TT,\,{\rm GIRS}}_{\mu\nu}(x,y)\ P_{\nu\mu}\right]$
$\displaystyle=$ $\displaystyle\int d^{3}\vec{z}\ {\rm Tr}\left[G^{TT,\,{\rm
tree}}_{\mu\nu}(x,y)\ P_{\nu\mu}\right],$ (47) $\displaystyle\int
d^{3}\vec{z}\ {\rm Tr}\left[G^{ST,\,{\rm GIRS}}_{\mu\nu}(x,y)\
P_{\nu\mu}\right]$ $\displaystyle=$ $\displaystyle\int d^{3}\vec{z}\ {\rm
Tr}\left[G^{ST,\,{\rm tree}}_{\mu\nu}(x,y)\ P_{\nu\mu}\right],$ (48)
$\displaystyle\int d^{3}\vec{z}\ {\rm Tr}\left[G^{S{\cal O},\,{\rm
GIRS}}_{\mu}(x,y)\ P_{\mu}\ \right]$ $\displaystyle=$ $\displaystyle\int
d^{3}\vec{z}\ {\rm Tr}\left[G^{S{\cal O},\,{\rm tree}}_{\mu}(x,y)\ P_{\mu}\
\right].$ (49)

where

$P_{\nu\mu}=\gamma_{\nu}\gamma_{4}\gamma_{\mu},\qquad
P_{\mu}=\gamma_{4}\gamma_{\mu},$ (50)

and the repeated indices $\mu,\nu$ are not summed over. The tree-level values
in the r.h.s. of Eqs. (46 – 49) are given below:

$\displaystyle\int d^{3}\vec{z}\ {\rm Tr}\left[G^{SS,\,{\rm
tree}}_{\mu\nu}(x,y)\ P_{\nu\mu}\right]$ $\displaystyle=$
$\displaystyle-\frac{(N_{c}^{2}-1)\ t}{\pi^{2}|t|^{5}}(1-\delta_{\mu
4}-\delta_{\nu 4}-3\,\delta_{\mu\nu}+4\,\delta_{\mu 4}\,\delta_{\nu 4}),$ (51)
$\displaystyle\int d^{3}\vec{z}\ {\rm Tr}\left[G^{TT,\,{\rm
tree}}_{\mu\nu}(x,y)\ P_{\nu\mu}\right]$ $\displaystyle=$
$\displaystyle\phantom{+}\frac{(N_{c}^{2}-1)\
t}{4\pi^{2}|t|^{5}}(2+\delta_{\mu 4}+\delta_{\nu
4}+3\,\delta_{\mu\nu}-4\,\delta_{\mu 4}\,\delta_{\nu 4}),$ (52)
$\displaystyle\int d^{3}\vec{z}\ {\rm Tr}\left[G^{ST,\,{\rm
tree}}_{\mu\nu}(x,y)\ P_{\nu\mu}\right]$ $\displaystyle=$
$\displaystyle-\frac{(N_{c}^{2}-1)\ t}{2\pi^{2}|t|^{5}}(1-\delta_{\mu
4}-\delta_{\nu 4}-3\,\delta_{\mu\nu}+4\,\delta_{\mu 4}\,\delta_{\nu 4}),$ (53)
$\displaystyle\int d^{3}\vec{z}\ {\rm Tr}\left[G^{S\mathcal{O},\,{\rm
tree}}_{\mu}(x,y)\ P_{\mu}\right]\ $ $\displaystyle=$
$\displaystyle\phantom{+}0.$ (54)

On the lattice, integration over timeslices is replaced by summation, which is
expected to reduce statistical errors in the nonpertubative 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., $\int d^{3}\vec{z}\ z_{\mu}z_{\nu}{\rm
Tr}\left[G^{A\,B}_{\mu\nu}P_{\nu\mu}\right],(A,B=S,T)$. Such choices can
affect statistical error either positively, since some contributions from
positive and negative directions will now add up, rather than cancel, or
negatively, since contributions from larger values of $z$, which are expected
to be more noisy, will now appear multiplied by extra powers of $z$.

While in ${\overline{\rm MS}}$ the renormalization factors for each operator
are independent of Lorentz components, in GIRS different choices of the
Lorentz components (spatial or temporal) for each operator can, in principle,
give different renormalization factors. Thus, in order to determine a
consistent set of conditions in GIRS, we need to use the same components
(spatial or temporal) for each operator in all conditions. Also, due to the
integration over the spatial components of $z$, the possible choices of
Lorentz indices in each operator which give a solution to the system of
conditions are further limited. By writing down the conditions of Eqs. (46 –
49) in terms of bare Green’s functions222Integration over spatial components
of $z$ is understood in each trace.:

$\displaystyle Z_{SS}^{2}\ {\rm
Tr}\left[G^{SS}_{\mu\nu}P_{\nu\mu}\right]+Z_{SS}\ Z_{ST}\ ({\rm
Tr}\left[G^{ST}_{\mu\nu}P_{\nu\mu}\right]+{\rm
Tr}\left[G^{TS}_{\mu\nu}P_{\nu\mu}\right])+Z_{ST}^{2}\ {\rm
Tr}\left[G^{TT}_{\mu\nu}P_{\nu\mu}\right]={\rm Tr}\left[G^{SS,{\rm
tree}}_{\mu\nu}P_{\nu\mu}\right],$ (55) $\displaystyle Z_{TS}^{2}\ {\rm
Tr}\left[G^{SS}_{\mu\nu}P_{\nu\mu}\right]+Z_{TS}\ Z_{TT}\ ({\rm
Tr}\left[G^{ST}_{\mu\nu}P_{\nu\mu}\right]+{\rm
Tr}\left[G^{TS}_{\mu\nu}P_{\nu\mu}\right])+Z_{TT}^{2}\ {\rm
Tr}\left[G^{TT}_{\mu\nu}P_{\nu\mu}\right]={\rm Tr}\left[G^{TT,{\rm
tree}}_{\mu\nu}P_{\nu\mu}\right],$ (56) $\displaystyle Z_{SS}\ \left(Z_{TS}\
{\rm Tr}\left[G^{SS}_{\mu\nu}P_{\nu\mu}\right]+Z_{TT}\ {\rm
Tr}\left[G^{ST}_{\mu\nu}P_{\nu\mu}\right]\right)+Z_{ST}\ \left(Z_{TS}\ {\rm
Tr}\left[G^{TS}_{\mu\nu}P_{\nu\mu}\right]+Z_{TT}\ {\rm
Tr}\left[G^{TT}_{\mu\nu}P_{\nu\mu}\right]\right)=$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{\rm
Tr}\left[G^{ST,{\rm tree}}_{\mu\nu}P_{\nu\mu}\right],$ (57) $\displaystyle
Z_{O}\left(Z_{SS}\ {\rm Tr}\left[G^{SO}_{\mu}P_{\mu}\right]+Z_{ST}\ {\rm
Tr}\left[G^{TO}_{\mu}P_{\mu}\right]\right)={\rm Tr}\left[G^{SO,{\rm
tree}}_{\mu}P_{\mu}\right]=0,$ (58)

we conclude that indices $\mu$ and $\nu$ must be both spatial; the choice of a
temporal component in Eq. (55) gives a vanishing tree-level value (see Eq.
(51)), which in combination with Eq. (58) results in either vanishing or
indeterminate values for $Z_{SS}$ and $Z_{ST}$. There is no restriction in
choosing which spatial components $\mu$ and $\nu$ will be in each condition
(same or different). However, some choices may be preferable from the
simulation point of view. In particular, we set: $\mu=\nu=i$ in Eq. (55),
$\mu=\nu=j$ in Eq. (56), $\mu=\nu=k$ in Eq. (57) and $\mu=\ell$ in Eq. (58),
where $i,j,k,\ell$ can be equal or different among themselves (of course, at
least two of the four indices will be equal). These choices of indices give
larger signal-to-noise ratios, as is observed in our nonperturbative study
described in the next section. Also, averages over $i,j,k,\ell$ can be
employed for improving the signal.

Eq. (55) and Eq. (58) lead to a second degree equation from which $Z_{SS}$
(and subsequently $Z_{ST}$) can be determined. Following that, Eq. (56) and
Eq. (57) lead to another second degree equation for $Z_{TT}$ (and also
$Z_{TS}$). We thus obtain:

$\displaystyle Z_{SS}$ $\displaystyle=$ $\displaystyle\sqrt{2c\ {\\{{\rm
Tr}\left[G^{SS}_{ii}P_{ii}\right]-({\rm Tr}\left[G^{ST}_{ii}P_{ii}\right]+{\rm
Tr}\left[G^{TS}_{ii}P_{ii}\right])R_{1}+{\rm
Tr}\left[G^{TT}_{ii}P_{ii}\right]R_{1}^{2}\\}}^{-1}},$ (59) $\displaystyle
Z_{ST}$ $\displaystyle=$ $\displaystyle-Z_{SS}R_{1},$ (60) $\displaystyle
Z_{TT}$ $\displaystyle=$ $\displaystyle R_{4}+\sqrt{R_{4}^{2}+R_{5}},$ (61)
$\displaystyle Z_{TS}$ $\displaystyle=$ $\displaystyle
Z_{SS}^{-1}R_{2}-Z_{TT}R_{3},$ (62)

where

$\displaystyle c$ $\displaystyle\equiv$
$\displaystyle\frac{(N_{c}^{2}-1)t}{\pi^{2}|t|^{5}},$ (63) $\displaystyle
R_{1}$ $\displaystyle\equiv$ $\displaystyle\frac{{\rm
Tr}\left[G^{SO}_{\ell}P_{\ell}\right]}{{\rm
Tr}\left[G^{TO}_{\ell}P_{\ell}\right]},$ (64) $\displaystyle R_{2}$
$\displaystyle\equiv$ $\displaystyle\frac{c}{{\rm
Tr}\left[G^{SS}_{kk}P_{kk}\right]-{\rm
Tr}\left[G^{TS}_{kk}P_{kk}\right]R_{1}},$ (65) $\displaystyle R_{3}$
$\displaystyle\equiv$ $\displaystyle\frac{{\rm
Tr}\left[G^{ST}_{kk}P_{kk}\right]-{\rm
Tr}\left[G^{TT}_{kk}P_{kk}\right]R_{1}}{{\rm
Tr}\left[G^{SS}_{kk}P_{kk}\right]-{\rm
Tr}\left[G^{TS}_{kk}P_{kk}\right]R_{1}},$ (66) $\displaystyle R_{4}$
$\displaystyle\equiv$ $\displaystyle\frac{R_{2}\ \\{2{\rm
Tr}\left[G^{SS}_{jj}P_{jj}\right]R_{3}-({\rm
Tr}\left[G^{ST}_{jj}P_{jj}\right]+{\rm
Tr}\left[G^{TS}_{jj}P_{jj}\right])\\}}{2Z_{SS}\ \\{{\rm
Tr}\left[G^{SS}_{jj}P_{jj}\right]R_{3}^{2}-({\rm
Tr}\left[G^{ST}_{jj}P_{jj}\right]+{\rm
Tr}\left[G^{TS}_{jj}P_{jj}\right])R_{3}+{\rm
Tr}\left[G^{TT}_{jj}P_{jj}\right]\\}},$ (67) $\displaystyle R_{5}$
$\displaystyle\equiv$ $\displaystyle\frac{(5/4)\ c-{\rm
Tr}\left[G^{SS}_{jj}P_{jj}\right]Z_{SS}^{-2}\ R_{2}^{2}}{{\rm
Tr}\left[G^{SS}_{jj}P_{jj}\right]R_{3}^{2}-({\rm
Tr}\left[G^{ST}_{jj}P_{jj}\right]+{\rm
Tr}\left[G^{TS}_{jj}P_{jj}\right])R_{3}+{\rm
Tr}\left[G^{TT}_{jj}P_{jj}\right]}.$ (68)

The choice of plus/minus signs for the square roots appearing in the solutions
of the second degree equations (Eqs. (55–58)) is dictated by the requirement
that the Z matrix be equal to the unit matrix at tree level. Because of the
zero tree-level value in the r.h.s. of Eq. (58), the renormalization factor
$Z_{O}$ is eliminated and thus does not enter the solution.

###  III.3 Results for the conversion matrix

The final step in our perturbative calculation is the evaluation of the
conversion matrix $C^{{\rm GIRS},{\overline{\rm MS}}}$ between ${\rm GIRS}$
and ${\overline{\rm MS}}$ scheme. Given that the conversion factors between
two renormalization schemes are regularization-independent, we calculate
$C^{{\rm GIRS},{\overline{\rm MS}}}$ by using results for the mixing matrix in
dimensional regularization, instead of lattice regularization, through the
following relation:

$\displaystyle\begin{pmatrix}C_{SS}^{{\rm GIRS},\overline{\rm
MS}}&C_{ST}^{{\rm GIRS},\overline{\rm MS}}\\\ \\\ C_{TS}^{{\rm
GIRS},\overline{\rm MS}}&C_{TT}^{{\rm GIRS},\overline{\rm MS}}\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}Z_{SS}^{{\rm L},\overline{\rm
MS}}&Z_{ST}^{{\rm L},\overline{\rm MS}}\\\ \\\ Z_{TS}^{{\rm L},\overline{\rm
MS}}&Z_{TT}^{{\rm L},\overline{\rm
MS}}\end{pmatrix}\cdot\begin{pmatrix}Z_{SS}^{{\rm L},{\rm GIRS}}&Z_{ST}^{{\rm
L},{\rm GIRS}}\\\ \\\ Z_{TS}^{{\rm L},{\rm GIRS}}&Z_{TT}^{{\rm L},{\rm
GIRS}}\end{pmatrix}^{-1}$ (69) $\displaystyle=$
$\displaystyle\begin{pmatrix}Z_{SS}^{{\rm DR},\overline{\rm MS}}&Z_{ST}^{{\rm
DR},\overline{\rm MS}}\\\ \\\ Z_{TS}^{{\rm DR},\overline{\rm MS}}&Z_{TT}^{{\rm
DR},\overline{\rm MS}}\end{pmatrix}\cdot\begin{pmatrix}Z_{SS}^{{\rm DR},{\rm
GIRS}}&Z_{ST}^{{\rm DR},{\rm GIRS}}\\\ \\\ Z_{TS}^{{\rm DR},{\rm
GIRS}}&Z_{TT}^{{\rm DR},{\rm GIRS}}\end{pmatrix}^{-1}.$

By combining our one-loop results for the mixing matrix in ${\rm GIRS}$ (Eqs.
(59 – 62)) and ${\overline{\rm MS}}$ (Eqs. (30 – 33)), we extract the one-loop
conversion matrix elements 333Terms of the form $\ln(\bar{\mu}^{2}t^{2})$ have
arisen as a result of integrating $\ln(\bar{\mu}^{2}z^{2}/4)/(z^{2})^{4}$ (cf.
Eqs. (35–43)) over spatial z-components.:

$\displaystyle C_{SS}^{{\rm GIRS},\overline{\rm MS}}$ $\displaystyle=$
$\displaystyle 1-\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}\frac{17N_{c}}{6}+\mathcal{O}(g^{4}_{{\overline{\rm MS}}}),$
(70) $\displaystyle C_{ST}^{{\rm GIRS},\overline{\rm MS}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}4N_{c}+\mathcal{O}(g^{4}_{{\overline{\rm MS}}}),$ (71)
$\displaystyle C_{TS}^{{\rm GIRS},\overline{\rm MS}}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}\frac{3N_{c}}{2}\left(\frac{2}{3}+2\gamma_{E}+\ln(\bar{\mu}^{2}t^{2})\right)+\mathcal{O}(g^{4}_{{\overline{\rm
MS}}}),$ (72) $\displaystyle C_{TT}^{{\rm GIRS},\overline{\rm MS}}$
$\displaystyle=$ $\displaystyle 1+\frac{g^{2}_{{\overline{\rm
MS}}}}{16\pi^{2}}N_{c}\left(\frac{7}{6}+6\gamma_{E}+3\ln(\bar{\mu}^{2}t^{2})\right)+\mathcal{O}(g^{4}_{{\overline{\rm
MS}}}).$ (73)

The results in Eqs. (70–73) are independent of the choices of spatial indices
($i,j,k,\ell$) in the GIRS renormalization conditions.

##  IV Nonperturbative calculation

The GIRS renormalization conditions (Eqs. (55–58)), involving only Green’s
functions between gauge-invariant operators, provide a scheme that can be used
for a nonperturbative determination of the supercurrent renormalization
factors, Eq. (6), on the lattice. The conversion matrix (Eqs. (70–73)) allows
to translate the results from GIRS to the ${\overline{\rm MS}}$ scheme.

In the following we apply this approach on a first set of test ensembles
selected from the ones presented in Bergner  _et al._ (2016); Ali  _et al._
(2019). For comparison, we consider the two gauge groups SU($2$) and SU($3$)
and two different lattice actions. The simulations of SU($2$) SYM have been
done with a tree level Symanzik improved gauge action and unimproved Wilson
fermions ($c_{\rm SW}=0$, $r=1$). One level of stout smearing is used for the
gauge links in the Dirac-Wilson operator in the SU(2) case. The SU($3$)
simulations use an unimproved gauge action and clover improved Wilson fermions
where $c_{\rm SW}$ determined by one-loop perturbation theory Musberg  _et
al._ (2013). The lattice action has been introduced in Eq. (2). The lattice
Wilson operator is represented in terms of the hopping parameter $\kappa\equiv
1/(2m_{0}+8)$

$D_{W}=1-\kappa\big{[}(1-\gamma_{\mu})(V_{\mu}(x))\delta_{x+\mu,y}+(1+\gamma_{\mu})(V^{\dagger}{}_{\mu}(x-\mu))\delta_{x-\mu,y}\big{]},$
(74)

where the links $V_{\mu}(x)$ in the adjoint representation are given by
$V_{\mu}^{ab}=2\,{\rm tr_{c}}[U^{\dagger}_{\mu}(x)T^{a}U_{\mu}(x)T^{b}]$. The
clover term is included in a similar way.

Details of the ensembles of gauge configurations can be found in earlier
publications Bergner  _et al._ (2013, 2016). For SU($2$) SYM we selected
ensembles of two different lattice sizes, one with $V\equiv L^{3}\times
T=24^{3}\times 48$ and a larger one with $V=32^{3}\times 64$. Anti-periodic
boundary conditions have been implemented in the temporal direction and the
gauge coupling constant $\beta=2N_{c}/g^{2}$ is $\beta=1.75$ in both cases. In
order to investigate possible effects of the finite gluino mass, we considered
$\kappa=0.14920$ and $\kappa=0.14925$ for the small lattice, while for the
larger one we have selected $\kappa=0.1494$. In case of SU($3$) SYM we
selected only a single test ensemble with $\beta=5.6$, $\kappa=0.1655$,
$c_{\rm SW}=1.587$, and $V=24^{3}\times 48$.

The Dirac-Wilson operator breaks supersymmetry and chiral symmetry, but the
critical point $\kappa_{c}(\beta)$ can be extrapolated according to signals
for a vanishing renormalized gluino mass. Supersymmetry and chiral symmetries
are recovered in the continuum limit at this point. As explained in our
previous studies, the adjoint pion mass provides a signal for chiral symmetry
breaking. In the selected ensembles, the pion mass and lattice artefacts are
already quite small and nearly degenerate supersymmetry multiplets are
observed. For reference, we provide the parameters in Table 1.

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 Bergner  _et al._ (2016).

###  IV.1 Correlators

The supercurrent operators $S_{\mu},T_{\mu}$ and the gluino-glue operator
$\cal{O}$ are generated from clover plaquettes
$\hat{F}_{\mu\nu}^{\alpha\beta}(x,t)$ and gluino fields. Their correlators,
after integrating out the gluino fields and omitting Lorentz and color indices
for clarity, take the following form

$\langle A(t)\overline{B}(0)\rangle\equiv\sum_{\vec{x},\vec{y}}\langle
A(\vec{x},t)\overline{B}(\vec{y},0)\rangle=\sum_{\vec{x},\vec{y}}\langle{\rm
Tr}[\Gamma\hat{F}(\vec{x},t)D^{-1}(\vec{x},t|\vec{y},0)\hat{F}(\vec{y},0)\Gamma^{\prime}]\rangle,$
(75)

where $\Gamma,\Gamma^{\prime}$ are combinations of gamma matrices coming from
the respective operators $A,B=S_{\mu},T_{\mu},{\cal O}$ and the inverse of the
Dirac operator $D^{-1}(\vec{x},t|\vec{y},0)$ propagates a gluino from the
point $x$ to $y$ .To estimate these correlators, we introduced wall sources
while the sinks were summed over the spatial positions.

It is worth noting that the definition of the gluino-glue operator, Eq. (22),
includes both spatial and temporal directions for the gauge links. This
symmetric choice of the operator $\mathcal{O}$ is in contrast to earlier
works; in particular it is different from the insertion operator chosen in the
WI analysis Farchioni  _et al._ (2002); Ali  _et al._ (2018). The symmetric
choice is, however, more natural in GIRS as only operators that transform as
fully covariant tensors mix with the supercurrent.

The spatial projectors $P_{i}=\gamma_{4}\gamma_{i}$ and
$P_{ij}=\gamma_{i}\gamma_{4}\gamma_{j}$ with $i=1,2,3$ are the most suitable
to use numerically, as they lead to a solvable set of GIRS conditions (Eqs.
(55–58)) and a better signal-to-noise ratio. We plot the results for two of
the correlators in Fig. 2 to show the quality of the numerical signal. In
order to alleviate the noise as much as possible, we use lattice symmetries to
average different components of the correlators. Taking
$\langle\mathcal{O}(t)S_{i}(0)P_{i}\rangle$ as an example (no sum over $i$),
we can average all the different spatial directions $\mu,\nu=x,y,z$

$\langle\mathcal{O}(t)\overline{S}_{i}(0)P_{i}\rangle\rightarrow\frac{\langle\mathcal{O}(t)\overline{S}_{x}(0)P_{x}\rangle+\langle\mathcal{O}(t)\overline{S}_{y}(0)P_{y}\rangle+\langle\mathcal{O}(t)\overline{S}_{z}(0)P_{z}\rangle}{3}.$
(76)

Secondly, because all the correlators are even under time reversal444This is
opposite to the behavior of Eqs. (35-43), due to the anti-periodic boundary
conditions. one can also perform the average

$\langle\mathcal{O}(t)S_{i}(0)P_{i}\rangle\rightarrow\frac{\langle\mathcal{O}(t)\overline{S}_{i}(0)P_{i}\rangle+\langle\mathcal{O}(N_{t}-t)\overline{S}_{i}(0)P_{i}\rangle}{2}.$
(77)

Finally due to charge conjugation Eq (24) the following relation between
Green’s functions holds

$\text{Tr}\Big{[}\langle\mathcal{O}(t)\overline{S}_{i}(0)\rangle
P_{i}\Big{]}=\text{Tr}\Big{[}C\langle
S_{i}(0)\overline{\mathcal{O}}(t)\rangle^{T}C^{-1}P_{i}\Big{]}=\text{Tr}\Big{[}\langle
S_{i}(t)\overline{\mathcal{O}}(0)\rangle P_{i}\Big{]},$ (78)

so that one can also average the reverse ordered operators,

$\text{Tr}\Big{[}\langle\mathcal{O}(t)\overline{S}_{i}(0)\rangle
P_{i}\Big{]}\rightarrow\frac{\text{Tr}\Big{[}\langle\mathcal{O}(t)\overline{S}_{i}(0)\rangle
P_{i}\Big{]}+\text{Tr}\Big{[}\langle S_{i}(t)\overline{\mathcal{O}}(0)\rangle
P_{i}\Big{]}}{2}.$ (79)

The same arguments hold for the rest of the Green’s functions. The signal can
improve substantially after the averaging procedure. Due to possible
autocorrelation we have only considered every 8th configuration of the
ensemble.

t$\text{Tr}\langle\mathcal{O}(t)S(0)P\rangle$

t$\text{Tr}\langle\mathcal{O}(t)T(0)P\rangle$

Figure 2: Correlators $\text{Tr}\langle\mathcal{O}(t)S_{i}(0)P_{i}\rangle$ and
$\text{Tr}\langle\mathcal{O}(t)T_{i}(0)P_{i}\rangle$ computed numerically on
the ensemble with $\kappa=0.14925$ and $\beta=1.75$ of the $V=24^{3}\times 48$
lattice.

###  IV.2 Renormalization coefficients

For our choice of spatial projectors and after applying a summation over
$\vec{x}$, the GIRS conditions on the lattice take the following form

$\displaystyle\frac{1}{3L^{3}}\sum_{\vec{x},\vec{y}}\sum_{i}\ {\rm
Tr}\left[G^{SS,\,{\rm GIRS}}_{ii}((\vec{x},t),(\vec{y},0))\
\gamma_{i}\gamma_{4}\gamma_{i}\right]$ $\displaystyle=$
$\displaystyle\frac{2(N_{c}^{2}-1)t}{\pi^{2}{|t|}^{5}},$ (80)
$\displaystyle\frac{1}{3L^{3}}\sum_{\vec{x},\vec{y}}\sum_{i}\ {\rm
Tr}\left[G^{TT,\,{\rm GIRS}}_{ii}((\vec{x},t),(\vec{y},0))\
\gamma_{i}\gamma_{4}\gamma_{i}\right]$ $\displaystyle=$
$\displaystyle\frac{5(N_{c}^{2}-1)t}{4\pi^{2}{|t|}^{5}},$ (81)
$\displaystyle\frac{1}{3L^{3}}\sum_{\vec{x},\vec{y}}\sum_{i}\ {\rm
Tr}\left[G^{ST,\,{\rm GIRS}}_{ii}((\vec{x},t),(\vec{y},0))\
\gamma_{i}\gamma_{4}\gamma_{i}\right]$ $\displaystyle=$
$\displaystyle\frac{(N_{c}^{2}-1)t}{\pi^{2}{|t|}^{5}},$ (82)
$\displaystyle\frac{1}{3L^{3}}\sum_{\vec{x},\vec{y}}\sum_{i}\ {\rm
Tr}\left[G^{S{\cal O},\,{\rm GIRS}}_{i}((\vec{x},t),(\vec{y},0))\
\gamma_{4}\gamma_{i}\right]$ $\displaystyle=$ $\displaystyle 0.$ (83)

The renormalized Green’s functions appearing in Eqs. (80 – 83) are related to
the bare Green’s functions extracted from lattice simulations via Eqs. (55 –
58), where the Z factors stand for $Z^{{\rm L},{\rm GIRS}}$.

The GIRS renormalization conditions, Eqs. (55 – 58), are enough to solve for
all the renormalization factors of the supercurrents using the Green’s
functions computed on the lattice. The results for $Z_{ST}/Z_{SS}$ after
Jackknife analysis are shown on the left side of Fig. 3. Due to gauge links
extending in the temporal direction, short range effects persist to larger
time separation $t$. On the other hand for large $t$ intervals $t>9$ the noise
starts to become dominant and so the relevant signal is contained in only a
rather small $t$ window (filled dots).

As already anticipated, all Z factors in GIRS can be translated to the
${\overline{\rm MS}}$ scheme by applying the conversion factors Eqs. (70–73).
All ${\overline{\rm MS}}$ renormalization functions should turn out to be
independent of the GIRS renormalization scale $t$ and a plateau-like behaviour
of the Z factors in the ${\overline{\rm MS}}$ scheme is expected. For
instance, the ratio $Z_{ST}^{{\rm L},{\overline{\rm MS}}}/Z_{SS}^{{\rm
L},{\overline{\rm MS}}}$ is extracted by combining the nonperturbative result
for the GIRS mixing matrix $Z^{{\rm L},{\rm GIRS}}$ with the perturbative
result for the conversion matrix $C^{{\rm GIRS},{\overline{\rm MS}}}$, through
the following relation:

$\frac{Z_{ST}^{{\rm L},{\overline{\rm MS}}}}{Z_{SS}^{{\rm L},{\overline{\rm
MS}}}}=\frac{C_{SS}^{{\rm GIRS},{\overline{\rm MS}}}Z_{ST}^{{\rm L},{\rm
GIRS}}+C_{ST}^{{\rm GIRS},{\overline{\rm MS}}}Z_{TT}^{{\rm L},{\rm
GIRS}}}{C_{SS}^{{\rm GIRS},{\overline{\rm MS}}}Z_{SS}^{{\rm L},{\rm
GIRS}}+C_{ST}^{{\rm GIRS},{\overline{\rm MS}}}Z_{TS}^{{\rm L},{\rm GIRS}}}.$
(84)

where we substituted the coupling constant $g^{2}_{{\overline{\rm MS}}}$ on
the conversion factors Eqs. (70–73) by the bare coupling constant of the
lattice action, $\beta=2N_{c}/g^{2}$, since the difference would be
$\mathcal{O}(g^{4}_{{\overline{\rm MS}}})$. For comparison of the results, the
lattice spacing can be estimated using the QCD Sommer scale value
$r_{0}=0.5\text{ fm}$ which leads to a lattice spacing of $0.0554(11)\text{
fm}$ for SU($2$) and $0.0532(8)\text{ fm}$ for SU($3$) SYM.

The numerical values obtained for the ratio $Z_{ST}^{{\rm L},{\overline{\rm
MS}}}/Z_{SS}^{{\rm L},{\overline{\rm MS}}}$ are presented on the right side of
Fig. 3. The final values are determined by a constant fit in the plateau-like
interval marked by filled dots. The results are collected in Table 1.
$Z_{ST}^{{\rm L},{\overline{\rm MS}}}/Z_{SS}^{{\rm L},{\overline{\rm MS}}}$ is
much smaller than the perturbative predictions. Our perturbative estimate for
SU($2$) SYM is $Z_{ST}^{{\rm L},{\overline{\rm MS}}}/Z_{SS}^{{\rm
L},{\overline{\rm MS}}}=0.100809$, but compared to the nonperturbative
simulations it does not include stout smearing. The value for SU($3$) SYM
without clover improvement is $Z_{ST}^{{\rm L},{\overline{\rm
MS}}}/Z_{SS}^{{\rm L},{\overline{\rm MS}}}=0.0656238$. In a consistent
truncation at a given order in the coupling constant, one would use the tree
level clover coefficient ($c_{\rm SW}=1.0$) and obtain $Z_{ST}^{{\rm
L},{\overline{\rm MS}}}/Z_{SS}^{{\rm L},{\overline{\rm MS}}}=0.0508682$; with
the one-loop improved value used in the actual simulations ($c_{\rm
SW}=1.587$), one obtains $Z_{ST}^{{\rm L},{\overline{\rm MS}}}/Z_{SS}^{{\rm
L},{\overline{\rm MS}}}=0.0373759$. The one-loop expressions for $Z_{ST}^{{\rm
L},{\overline{\rm MS}}}$ and $Z_{SS}^{{\rm L},{\overline{\rm MS}}}$ are
explicitly shown in Ref. Bergner  _et al._ (2022b), where the number of
colors, $N_{c}$, the coupling constant $g$, and the clover parameter, $c_{\rm
SW}$ are left unspecified.

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 $Z_{ST}^{{\rm L},{\overline{\rm MS}}}/Z_{SS}^{{\rm
L},{\overline{\rm MS}}}$ using the substitution on the conversion matrix
$C(g^{2})^{{\rm GIRS},{\overline{\rm MS}}}\rightarrow C^{-1}(-g^{2})^{{\rm
GIRS},{\overline{\rm MS}}}$. At one-loop order this amounts to taking the
inverse twice and so the result should be left invariant. Due to higher order
corrections, this equivalence is, however, significantly violated as shown in
Fig. 4.

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 6 levels of stout smearing with smearing
parameter $\rho=0.15$ and adjusted the mass parameter such that the pion mass
in lattice units stays approximately the same. The resulting ratio
$Z_{ST}/Z_{SS}$ is presented in Fig. 5 both in GIRS and in the ${\overline{\rm
MS}}$ scheme. We can observe that the noise is drastically decreased and a
plateau-like intermediate $t$ range becomes much more visible. In GIRS scheme
we observe a clear $t$ dependence which is erased in the ${\overline{\rm MS}}$
scheme by the conversion factors up to a certain $t$, where higher order
corrections of the conversion matrix seem to become relevant. We note that the
fitted value has now moved away even further from the perturbative result;
however, a faithful comparison would require performing perturbative
computation with smeared gauge links, which we have not done so far.

GIRS schemet$Z_{ST}/Z_{SS}$

${\overline{\rm MS}}$ schemet

(a) $V=24^{3}\times 48$ lattice with $\kappa=0.14925$ (blue dots) and
$\kappa=0.14920$ (orange squares). The points on the $\kappa=0.14920$ ensemble
are shifted in $t$ by $+0.33$ for visibility.

GIRS schemet$Z_{ST}/Z_{SS}$

${\overline{\rm MS}}$ schemet

(b) $V=32^{3}\times 64$ lattice with $\kappa=0.1494$.

GIRS schemet$Z_{ST}/Z_{SS}$

${\overline{\rm MS}}$ schemet

(c) SU(3) ensemble on a $V=24^{3}\times 48$ lattice with $\kappa=0.1655$ and
$\beta=5.6$.

Figure 3: $Z_{ST}/Z_{TT}$ in the GIRS (left) and in the ${\overline{\rm MS}}$
scheme (right) as a function of time for the first time slices. The filled
dots represent the regime where the effects of contact terms and noise are
minor. The colour bands correspond to the linear extrapolation using the
plateau interval. ${\overline{\rm MS}}$ schemet$Z_{ST}/Z_{SS}$ Figure 4:
$Z_{ST}^{\overline{\rm MS}}/Z_{SS}^{\overline{\rm MS}}$ computed with
$C(g^{2})^{{\rm GIRS},{\overline{\rm MS}}}$ (blue dots) and
$C^{-1}(-g^{2})^{{\rm GIRS},{\overline{\rm MS}}}$ (orange squares). The
difference between the two computations $\Delta(Z_{ST}^{\overline{\rm
MS}}/Z_{SS}^{\overline{\rm MS}})=0.0277$ is due to higher loop corrections.

GIRS schemet$Z_{ST}/Z_{SS}$

${\overline{\rm MS}}$ schemet

Figure 5:  $V=24^{3}\times 48$ lattice at $\kappa=0.14920$ with smearing. The fitted value corresponds to $Z_{ST}/Z_{SS}=-0.0848(15)$. A more pronounced plateau is observed in the ${\overline{\rm MS}}$ data, as expected. Gauge group | $\beta$ | $\kappa$ | $w_{0,\chi}m_{\pi}$ | Scheme | L | T | $Z_{SS}$ | $Z_{ST}$ | $Z_{TS}$ | $Z_{TT}$ | $Z_{ST}/Z_{SS}$
---|---|---|---|---|---|---|---|---|---|---|---
SU($2$) | 1.75 | 0.14920 | 0.6915(62) | ${\overline{\rm MS}}$ | 24 | 48 |  0.745(20) | -0.0362(48) | 0.1759(12) | 0.3518(30) | -0.0499(55)
SU($2$) | 1.75 | 0.14925 | 0.6467(92) | ${\overline{\rm MS}}$ | 24 | 48 | 0.783(24) | -0.0312(26) | 0.201(16) | 0.3612(67) | -0.0404(38)
SU($2$) | 1.75 | 0.14940 | 0.5471(80) | ${\overline{\rm MS}}$ | 48 | 64 | 0.877(25) | -0.0417(38) | 0.275(20) | 0.275(20) | -0.0477(44)
SU($3$) | 5.6 | 0.16550 | 1.204(27) | ${\overline{\rm MS}}$ | 24 | 48 | 1.127(43) | -0.0703(56) | 0.323(20) | 0.4302(25) | -0.0640(55)
Table 1: Value of the Z renormalization factors of the supercurrents obtained
nonperturbatively on the GIRS and then translated to the ${\overline{\rm MS}}$
scheme by using the conversion matrix $C^{{\rm GIRS},{\overline{\rm MS}}}$.
The errors presented here are purely statistical, resulting from a Jacknife
analysis. For reference, the parameters of the different ensembles are also
provided. Each simulation has been done on an $L^{3}\times T$ lattice and the
adjoint pion mass is provided in units of the gradient flow scale $w_{0,\chi}$
extrapolated to the chiral limit ($w_{0,\chi}/a=3.411(18)$ for SU(2) SYM and
$w_{0,\chi}/a=3.485(71)$ for SU(3) SYM).

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 $t$, where large cut-off effects are
present. An equivalent procedure (up to higher-order effects) is to replace
the continuum tree-level values in the r.h.s. of Eqs. (80 – 83) with the
corresponding lattice tree-level values, which include artifacts to all orders
in the lattice spacing $a$. We examine the effect of this tree-level
correction by calculating the ratio between the lattice and continuum tree-
level values: $R^{AB}(t)\equiv{\rm Tr}[{(G_{ii}^{AB}(t))}^{\rm tree\
lat.}\gamma_{i}\gamma_{4}\gamma_{i}]/{\rm Tr}[{(G_{ii}^{AB}(t))}^{\rm tree\
cont.}\gamma_{i}\gamma_{4}\gamma_{i}]$, where $A,B=S,T$. In Fig. 6, we show
the ratios $R^{SS}(t),R^{ST}(t)$ and $R^{TT}(t)$. We observe that the most
important differences between $\mathcal{O}(a^{0})$ and all orders in $a$
corrections of tree-level values regard very small values of $t$, for which
results in $Z_{ST}/Z_{SS}$ are far from a plateau; thus these differences do
not significantly affect the plateau values.

Figure 6: Ratios between lattice (all orders in $a$) and continuum tree-level
values: $R^{AB}(t)\equiv{\rm Tr}[{(G_{ii}^{AB}(t))}^{\rm tree\
lat.}\gamma_{i}\gamma_{4}\gamma_{i}]/{\rm Tr}[{(G_{ii}^{AB}(t))}^{\rm tree\
cont.}\gamma_{i}\gamma_{4}\gamma_{i}]$, for $AB=SS,ST,TT$. For better
visibility, $R^{ST}(t)$ and $R^{TT}(t)$ are shifted in $t$ by +0.25 and +0.50,
respectively.

##  V Discussion

In this paper, we have presented a concrete prescription to renormalize
nonperturbatively the supercurrent operator in $\mathcal{N}=1$ Supersymmetric
Yang-Mills (SYM) theory on the lattice, using a gauge-invariant scheme (GIRS).
The employed scheme addresses the mixing between the supercurrent and all
gauge-invariant operators which respect the same global symmetries and share
the same quantum numbers. An advantageous feature of GIRS is that whole
classes of gauge-noninvariant operators, which, in principle, can and will mix
with the supercurrent, have zero contributions to the gauge-invariant Green’s
functions of GIRS. In this respect, the set of mixing operators that one needs
to study within GIRS is greatly reduced. This makes GIRS a good alternative
for renormalizing operators in the presence of mixing, compared to more
standard methods, such as employing Ward Identities on the lattice.

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 ${\overline{\rm MS}}$
schemes in order to deduce the conversion matrix between the two schemes. At
the end, we have combined the nonperturbative data with the one-loop
conversion matrix and we have extracted the ${\overline{\rm MS}}$ mixing
matrix. Our result for the ratio $Z_{ST}^{{\rm L},{\overline{\rm
MS}}}/Z_{SS}^{{\rm L},{\overline{\rm MS}}}$ is different from our previous
perturbative computation at one loop Bergner  _et al._ (2022b). However, the
parameter regime for our first test is most likely far outside the
perturbative regime.

One means of improving the nonperturbative estimates is further elimination of
the discretization errors. For small values of $t$, large cutoff effects are
present in our calculation. In order to address this issue, a higher-loop
perturbative evaluation of lattice artifacts, to all orders in the lattice
spacing, for different values of $t$, could be performed. As has been observed
in other contexts Capitani  _et al._ (2001); Constantinou  _et al._ (2009);
Gockeler  _et al._ (2010); Constantinou  _et al._ (2013); Alexandrou  _et al._
(2017); Constantinou and Panagopoulos (2022), subtraction of the unwanted
contributions of the finite lattice spacing from the nonperturbative estimates
can lead to a more rapid convergence to the continuum limit. Such a procedure
has been successfully employed to Green’s functions of operators in momentum
space up to one loop. However, in a coordinate-space scheme, such as GIRS, a
one-loop computation involves two-loop Feynman diagrams, which make the
application of this subtraction in GIRS more difficult.

A second improvement is the elimination of truncation effects coming from the
conversion matrix. For large values of $t$, the one-loop conversion matrix is
not enough to guarantee convergence, since strong higher-loop logarithmic
divergences in $t$ are present. A natural continuation of the present work is
the two-loop calculation of the conversion matrix.

From the numerical point of view, more statistics, especially for larger
values of $t$, can enlarge the window of time that can be used for reaching
the plateau in the renormalization factors. Finally simulating closer to a
perturbative regime would be a very useful way to see if one can match the
perturbative with the nonperturbative result.

###### Acknowledgements.

We thank G. Münster for helpful discussions and comments. G. Bergner and I.
Soler acknowledge financial support from the Deutsche Forschungsgemeinschaft
(DFG) Grant No. BE 5942/3-1 and 5942/4-1. This work was co-funded by the
European Regional Development Fund and the Republic of Cyprus through the
Research and Innovation Foundation (Projects: EXCELLENCE/0918/0066 and
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 ID: 823767). The authors gratefully acknowledge the Gauss Centre for
Supercomputing e. V. (www.gauss-centre.eu) for funding this project by
providing computing time on the GCS Supercomputer SuperMUC-NG at Leibniz
Supercomputing Centre (www.lrz.de). Further computing time has been provided
on the compute cluster PALMA of the University of Münster.

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