Supersymmetry, chiral symmetry and the generalized BRS transformation in lattice formulations of 4D $\mathcal{N}=1$ SYM

In the context of the lattice regularization of the four-dimensional $\mathcal{N}=1$ supersymmetric Yang--Mills theory (4D $\mathcal{N}=1$ SYM), we formulate a generalized BRS transformation that treats the gauge, supersymmetry (SUSY), translation and axial U(1) ($U(1)_A$) transformations in a unified way. A resultant Slavnov--Taylor identity or the Zinn-Justin equation gives rise to a strong constraint on the quantum continuum limit of symmetry breaking terms with the lattice regularization. By analyzing the implications of the constraint on operator-mixing coefficients in the SUSY and the $U(1)_A$ Ward-Takahashi (WT) identities, we prove to all orders of perturbation theory in the continuum limit that, (i) the chiral symmetric limit implies the supersymmetric limit and, (ii) a three-fermion operator that might potentially give rise to an exotic breaking of the SUSY WT identity does not emerge. In previous literature, only a naive or incomplete treatment on these points can be found. Our results provide a solid theoretical basis for lattice formulations of the 4D $\mathcal{N}=1$ SYM.

based on the lattice regularization was pointed out [1], 1 there has been considerable research based on the proposed scenario of the SUSY restoration. Classified by the method for simulating the gaugino or gluino, a fermionic superpartner of the gauge boson, Refs. [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] use the Wilson fermion [23] (see Ref. [24] for a very readable review), Refs. [25,26,27,28] employ the domain-wall fermion [29,30] and, most recently, Ref. [31] uses the overlap fermion [32,33]. Some related works are Refs. [34,35,36,37,38]. 2 The central issue in the scenario [1] is the restoration of SUSY and the axial U(1) symmetry, U(1) A , in the quantum continuum limit; the lattice regularization generally breaks these symmetries that define 4D N = 1 SYM. The basic physical picture of Ref. [1] is very simple (see also Ref. [2]): In terms of the effective field theory, a unique relevant operator that breaks SUSY is the gluino mass term. If one can have a U(1) A symmetry (either by the finetuning of the bare gluino mass [40,41,42] or by use of the Ginsparg-Wilson relation [43]; see below), the U(1) A would forbid the gluino mass term and one would end up with a supersymmetric theory in the continuum limit. In this way, SUSY might emerge as an accidental symmetry accompanied by the chiral symmetry. The situation is a little bit more complicated because the U(1) A suffers from the axial anomaly. Still, one could use a remaining Z 2Nc ⊂ U(1) A symmetry 3 to forbid the gluino mass term. Thus one expects that, in the continuum limit, the restoration of SUSY and that of the chiral symmetry occur simultaneously.
The best characterization of a symmetry property of a quantum field theory is given by the WT identity. Although the basic physical picture of the above scenario is simple, to show the validity of the scenario in terms of WT identities is not so simple. 4 For example, the analysis of Ref. [1] considers a lattice "identity", such as (Eq. (14) of Ref. [1] with Ω = A ρ (y)λ(z), in our 1 See also Ref. [2] for an earlier consideration. 2 Ref. [39] is discussing a possible implication of the lattice BRS invariance (in the non-perturbative level) in supersymmetric gauge theories. 3 Throughout this paper, we assume that the gauge group is SU (N c ). 4 For a lattice formulation [44,45] of the two-dimensional N = (2, 2) SYM, the restoration of the SUSY WT identity without fine-tuning can be argued perturbatively [46] and has been confirmed non-perturbatively [47].

notation)
where S µ (x) is the supercurrent on the lattice, X S (x) is an explicit SUSY breaking term in the lattice action, A a ν (y) is the gauge potential andψ b (z) is the gluino field. δ ξ in the last term is the SUSY transformation on the lattice. This relation, however, does not hold even in the tree level approximation of the perturbation theory. 5 The point is that the inserted operator A a ν (y)ψ b (z) is neither gauge invariant nor covariant thus the contribution of the gaugefixing term (and the ghost action for higher orders) cannot be neglected. Moreover, as found in Ref. [37] through an explicit one-loop calculation, X S (x) mixes even with gauge non-invariant operators, a situation that was not presumed in Ref. [1]. In these aspects, the WT identity associated with SUSY is much more complicated than that for the chiral symmetry [41,42] as noted sometime ago [15]. It is certainly unlikely that those gauge noninvariant elements affect a physical question. Nevertheless, it is not quite obvious whether or not these complications modify the conclusion of Ref. [1] that both the U(1) A WT identity (with the axial anomaly) and the SUSY WT identity are restored by a single fine-tuning of the bare gluino mass. Since the axial U(1) current and the supercurrent belong to a single SUSY multiplet (at least in the classical theory) [48], it is quite conceivable that one can relate the U(1) A WT identity and the SUSY WT identity by considering an algebra involving both SUSY and the R-symmetry, U(1) A . It is not a priori clear how to carry out this program, however, especially with the lattice regularization that generally breaks both symmetries explicitly.
The original motivation of the present work was to obtain a transparent understanding on the above issue. In the course of the analysis, however, we encountered another difficulty that has not been noted in previous literature. That is that, generally, one might have a mixing of X S (x) with a gauge invariant three-fermion operator. Since this three-fermion operator has the same mass-dimension as X S (x) and same transformation properties as X S (x) under lattice symmetries, a dimensional counting and a simple symmetry argument alone cannot exclude the possibility of such a mixing. But then, one would have an exotic breaking of the SUSY WT identity even in the continuum limit (and even with the fine-tuning of the gluino mass elucidated above). See Eq. (4.20). According to the supposed scenario of the SUSY restoration, we should not have such an exotic breaking.
Considering recent increases in computing power that would enable a true realization of 4D N = 1 SYM and a fundamental role that the SUSY WT identity should play there, it must be important to clarify the above issues; this is the objective of the present paper. We will find that the adoption of the generalized BRS transformation [49,50,51] in the lattice framework, that treats the gauge, SUSY, translation and U(1) A transformations in a unified way, does the required job.
This paper is organized as follows. In Sec. 2, we summarize basic assumptions we make on the lattice formulation. They are physically natural and very mild; actually all the lattice formulations of 4D N = 1 SYM considered so far are covered by our analysis. Sections. 3 and 4 are mostly a review of well-known facts concerning the operator mixing in the U(1) A WT identity [41,42] and the SUSY WT identity [1,37,15]; however, we tried not to make any ad hoc assumption on the possible operator-mixing structure. Eqs. (4.22) and (4.23) summarize our main assertions in this paper. After these rather long preparations, in Sec. 5, we introduce the generalized BRS transformation on lattice variables. There, some complications inherent in the present supersymmetric system in the Wess-Zumino (WZ) gauge are explained. After identifying O(a) breaking terms attributed to the lattice regularization, we obtain a Wess-Zumino (WZ)-like consistency condition [52] by using the algebraic property of the generalized BRS transformation. In Sec. 6, finally, combining the consistency condition and the general structure of the operator mixing explored in Secs. 3 and 4, we prove the main assertions, Eqs. (4.22) and (4.23). Sec. 7 concludes the paper. Appendices contain some useful information required in the main text.

Basic lattice framework
Our arguments below are quite general and rather independent of a particular lattice formulation we adopt. 6 Our lattice action consists of the sum of a gauge boson (gluon) action S gluon , the kinetic term of the gluino S gluino and the mass term of the gluino, S (0) mass . A minimal requirement for the lattice actions is that they have the correct classical continuum limit such that , ] is the covariant derivative with respect to the adjoint representation, and where M is the bare gluino mass. The gluino field ψ(x) on the Euclidean lattice (and in the Euclidean continuum theory as well) is subject to the constraintψ written. The gauge potentials A µ (x) are defined from the link variables by where g is the bare gauge coupling constant. All gamma matrices are hermitian and obey {γ µ , γ ν } = 2δ µν ; γ 5 ≡ −γ 0 γ 1 γ 2 γ 3 and σ µν ≡ [γ µ , γ ν ]/2. The charge conjugation matrix C satisfies, C −1 γ µ C = −γ T µ , C −1 σ µν C = −σ T µν , C −1 γ 5 C = γ T 5 and C T = −C. ǫ µνρσ denotes the totally antisymmetric tensor and ǫ 0123 = −1. The generators of the gauge group SU (N c ), T a , are normalized as tr(T a T b ) = (1/2)δ ab . Φ a (x) denotes the gauge component of a generic field Φ(x), Φ(x) ≡ Φ a (x)T a . Throughout this article, the symbol tr denotes the trace over gauge indices. The forward and backward difference operators respectively are defined by The symbol of the functional derivative with respect to a lattice variable implies for Φ a (x), for example.
to express the degrees of freedom of the Majorana fermion in the Minkowski space. Consequently, the lattice Dirac operator should satisfy (C −1 D) T = −C −1 D. We further assume that the lattice actions are invariant under the hypercubic transformations, the parity transformation P, 8) and the time reversal transformation T , The simplest lattice formulation which fulfills the above requirements is the sum of the Wilson plaquette action, (2.10) and the Wilson fermion action, where lattice covariant differences in the adjoint representation are defined by In what follows, we often consider correlation functions containing elementary fields that are not gauge invariant. To define such correlation functions, we have to introduce the gauge-fixing term and the Faddeev-Popov (FP) ghost and anti-ghost fields. We thus define wherec(x) and B(x) are the anti-ghost and auxiliary fields, respectively and α is the gauge parameter. The nilpotent BRS transformation s 0 for lattice variables is defined by (see, for example, Ref. [53]), and ∆ X is the adjoint action, 3. U (1) A WT identity on the lattice

Derivation of the U(1) A WT identity
Let us begin our discussion with a WT identity associated with the U(1) A transformation. We define a localized version of the U(1) A transformation on lattice variables by and δ θ = 0 on other variables. Then, from a variation of the lattice action (note that S (0) GF+FP (2.13) does not contain the gluino field), we obtain the divergence of the axial-vector current j 5µ (x) whose classical continuum limit is given by 7 the pseudo-scalar density P (x), which arises from the variation of the gluino mass term S (0) mass (2.11), and X A (x), a U(1) A symmetry breaking term associated with the lattice regularization (e.g., the Wilson term, for the case of Eq. (2.11)). In Eq. (3.2), the separation between −∂ * µ j 5µ (x) and X A (x) is not unique and there remains O(a) ambiguity in the definition of X A (x) even with requirement (3.3). We partially fix this ambiguity by requiring that the breaking X A (x) is a gaugeinvariant local combination of lattice fields that behaves in an identical way as P (x) (3.4) under lattice discrete transformations. In particular, we assume that Then by considering the variation of the lattice action (3.2) in the functional integral, we find the identity where O(y, z, . . . ) is any multi-local operator. This is an identity that exactly holds on the lattice.

X A (x) in the continuum limit
We next consider, on the basis of the perturbation theory, how the U(1) A breaking term X A (x) in Eq. (3.6) behaves in the quantum continuum limit. Since X A (x) is a dimension 4 operator that is proportional to the lattice spacing a (because it arises from lattice artifacts), we set where O 5 (x) is a dimension 5 operator. Then, from dimension counting and the covariance under lattice symmetries (3.5), subtractions which are required to define a renormalized composite operator O R 5 (x) in the continuum limit are given by [41] (see also Ref. [42]), where the explicit form of power-subtraction operators is written only for BRS invariant ones. 8 As indicated in Eq. We have used the continuum theory language in the above expression (P (x) is the classical continuum limit of P (x) (3.4)), because the present considerations are meaningful only in the continuum limit.
Eqs. (3.7) and (3.8) show that and the lattice identity (3.6) then reads Now, for simplicity, we assume temporarily that the inserted operator O(y, z, . . . ) is gauge (or BRS) invariant and renormalizable without subtracting powerdivergent terms. That is, where all renormalization constants are at most logarithmically divergent. We further assume that all the points x, y, z, . . . , in Eq. (3.10) differ from each other. In this situation, the contact term (3.10f) vanishes. The contribution of the dimension 5 operators (3.10e) also vanishes in the continuum limit because of the overall factor of a (when all the points x, y, z, . . . , differ, no O(1/a) ultraviolet divergence arises that can cancel the overall factor a).
There is no mixing of X A (x) with BRS non-invariant operators because the inserted operator O(y, z, . . . ) is BRS invariant. Therefore, in this assumed situation, we have (3.12) From this (anomalous) conservation law of the axial U(1) current, one might infer that for an infinitesimal lattice spacing the U(1) A symmetry broken by the lattice regularization is restored by tuning the bare mass parameter M such that We call this a chiral symmetric limit [40,41,1]. Incidentally, as for the domain-wall fermion [29,30] and for the overlap fermion [32,33], if the lattice Dirac operator D satisfies the GW rela- . Suppose that M = 0. Then, in the correlation function (3.6), contractions of gluino fields in this O 5 (x) and gluino fields in the inserted operator O(y, z, . . . ) produce a term being proportional to δ x,y δ x,z , . . . , which does not contribute to the divergence of the operator O 5 (x). On the other hand, the contraction of gluino fields within O 5 (x) produces the combination where tr D is the trace over Dirac indices. The modified chiral matrix in this expression is defined by [54,55] From the GW relation, one has the identity z,cγ 5 ac (x, z)γ 5 cb (z, y) = δ ab δ x,y and, from this [54], where δ denotes an arbitrary infinitesimal local variation of the gauge field. This relation implies that the vertices that result from the composite operator (3.14) identically vanish when the momentum being conjugate to the position x vanishes. In possible subtractions (3.8), this property is shared by operators ∂ µ5µ (x) and ǫ µνρσ tr [F µν (x)F ρσ (x)], but not byP (x). This shows that Z P = 0 when M = 0. 9 That is, when D satisfies the GW relation, we generally have 10 Z P ∝ aM.

Derivation of the SUSY WT identity
A lattice WT identity associated with the SUSY transformation can also be derived in a similar way as the above U(1) A WT identity. The localized version of the SUSY transformation in the continuum is given by Corresponding to this, we may adopt the following localized lattice SUSY transformation, where P µν (x) is the clover plaquette, defined by and This definition of δ ξ [37] is advantageous because P µν (x) possesses the same transformation properties as the continuum F µν (x) under P and T . That is, and We define that the ghost, anti-ghost and auxiliary fields as singlets under the above (localized) SUSY transformation δ ξ . Then one sees that the BRS transformation s 0 (2.14) and δ ξ anti-commute with each other, and as the consequence, the SUSY variation of the gauge fixing and ghost actions is BRS exact: The explicit form of δ ξ A µ (x), a transformation of the gauge potential induced by lattice SUSY transformation (4.2), is Now, we consider the variation of the lattice action (2.17) under Eq. (4.2). As the general structure of the variation (noting Eq. (4.7)), we have Here, χ(x) is a gauge invariant fermionic field, which arises from the variation of the gluino mass term S (0) mass , In Eq. (4.9), we defined a lattice supercurrent S µ (x), whose classical continuum limit is However, in Eq. (4.9), the separation between −∂ * µ S µ (x) and X S (x), a SUSY breaking associated with the lattice regularization, is not unique even under condition (4.11); there remains an O(a) ambiguity in the definition of X S (x). We thus require that the breaking X S (x) is a gauge-invariant local combination of lattice fields which behaves in an identical way as χ(x) (4.10) under lattice discrete transformations. In particular, we require that Then, for any multi-local operator O(y, z, . . . ), we have an exact identity on the lattice,

X S (x) in the continuum limit
We next investigate, within perturbation theory, how the SUSY breaking term X S (x) in Eq. (4.13) behaves in the continuum limit. X S (x) is a dimension 9/2 operator that is proportional to the lattice spacing a (because it results from the lattice regularization). Thus we set (4.14) where O 11/2 (x) is a dimension 11/2 operator. To define a renormalized finite operator O R 11/2 (x) in the continuum limit, we generally need the subtraction of operators of mass-dimension 9/2 or less with power-diverging coefficients. Those operators must, by our assumption, behave identically to χ(x) (4.10) under lattice discrete transformations (as Eq. (4.12)). A gauge (or BRS) invariant class of such operators, without containing the ghost, anti-ghost and auxiliary fields, are enumerated in Appendix B of Ref. [15]. Utilizing this result, we have 11 In the above expression, the last line represents possible mixing with other (renormalized) dimension 11/2 operators. From this and Eq. (4.14), we have and thus, combined with Eq. (4.13), where we have expressed operator (4.17e) in terms of the variations of the total action of the continuum theory,S that is cubic in the fermion field. In Refs. [1,15], on the other hand, one does not encounter such a mixing with a three-fermion operator. In Ref. [1], this operator was not noticed at all, while in Ref. [15], it seems that only the case of the gauge group SU (2) is considered for which this three-fermion operator identically vanishes O S (x) ≡ 0. The fact is that, as we analyse in detail in Appendix A, O S does not generally vanish, and since O S (x) and χ(x) transform in a completely identical manner under lattice symmetries, 13 we cannot exclude O S (x) from the operator renormalization by a simple symmetry argument alone. For simplicity, as we did in Eq. (3.12), let us temporarily assume that the inserted operator O(y, z, . . . ) in Eq. (4.18) is BRS invariant and renormalizable without subtracting power-divergent terms as Eq. (3.11); we assume also that all the points x, y, z, . . . , differ. Then, by a similar argument used to obtain Eq. (3.12), we have in the continuum limit where we neglected contribution (4.18e), because any BRS invariant operator with the dimension 9/2 or less that contains c(x),c(x) or B(x) is always BRS exact; the proof is given in Appendix B.
After the redefinition of the supercurrent, Z SSµ +Z TTµ ∝S new µ , 14 Eq. (4.20) takes the form of the conservation law of a supercurrent. Then the breaking 13 Since they rotate in opposite angles under U (1) A , if the U (1) A symmetry were exactly preserved, one could get rid of the possibility of O S (x). The lattice regularization inevitably breaks the U (1) A , however, to reproduce the axial anomaly.
of SUSY due to the lattice regularization is characterized by the combination M − (1/a)Z χ . One may remove this unphysical SUSY breaking by tuning the bare mass parameter M so that this combination vanishes: We call this a supersymmetric limit [1] (a non-perturbative method to impose this condition has been proposed in Ref. [36]). It is considered that the chiral symmetric limit (3.13) coincides with this supersymmetric limit and thus condition (3.13) defines a unique supersymmetric theory [1]. For this, we must have Eq. (4.20) shows that, however, the conservation law of the supercurrent suffers from an exotic breaking if Z 3F = 0. If this breaking cannot be removed by a local counter term, this would imply an exotic SUSY anomaly which certainly what we do not expect in the present system. Even if this anomaly could be removed by a local counter term, its presence implies that, as Eq. (4.20) shows, tuning the mass parameter (4.21) alone will not lead to a supersymmetric theory, contradicting with what is believed. Thus we have to ensure that Z 3F = 0. (4.23) As noted above, however, it is not possible to conclude Z 3F = 0 by a simple symmetry argument alone. It appears that one needs some powerful machinery such as the WZ consistency condition [52]. The rest of this paper will be entirely devoted to the construction of the required consistency condition and its application to the proof of Eqs. (4.22) and (4.23).

General framework
We want to formulate a certain WZ consistency condition that constrains the quantum continuum limit of symmetry breaking terms attributed to the lattice regularization. Because of several reasons, unfortunately, this is a somewhat complicated task.
First, the algebraic structure of symmetry transformations is rather involved even in the continuum target theory. In 4D N = 1 SYM in the WZ gauge, the algebra of SUSY transformations contains field-dependent gauge transformations. Moreover, the algebra closes only under the equation of motion of the (massless) gluino (i.e., on-shell closure). Thus, to construct a BRS-like nilpotent operation that contains SUSY (which will be a building block in the WZ consistency condition), one has to treat the SUSY, translation and gauge transformations on an equal footing. For this, we adopt the generalized BRS transformation developed in the continuum 4D N = 1 SYM in the WZ gauge. See Refs. [49,50,51] and references therein. The onshell closure can also be incorporated into this framework by introducing a term that is quadratic in the "anti-field" of the gluino, K ψ (see, for example, Refs. [70,50]).
Second, we have a bare-mass term of the gluino that is inevitable for the tuning in lattice formulations; this term however explicitly breaks SUSY and the U(1) A symmetry. If one wants to constrain the structure of radiative corrections by using the Slavnov-Taylor (ST) identity or the Zinn-Justin equation, those relations should not contain tree-level symmetry-breakings such as the gluino mass term. To circumvent this point, we introduce generalized BRS doublet fields (spurions) (u V , v V ) and (u A , v A ) which make the mass term "formally" BRS invariant [71,51].
Finally, of course, SUSY and the infinitesimal translation transformations are not properly realized on lattice variables and, as the consequence, the nilpotency of the generalized BRS transformation on some of lattice variables is broken by O(a). Then we have to carefully separate those O(a) breakings from the main part of the ST identity or the Zinn-Justin equation.
We thus define a generalized BRS transformation s in our lattice system by where ∂ S µ denotes the symmetric difference operator In the above expressions, the gauge-ghost c(x), anti-ghostc(x) and the auxiliary field B(x) are common to the conventional lattice BRS transformation s 0 (2.14), while ξ, t µ and θ are newly-introduced constant ghosts associated with the SUSY, translation and U(1) A transformations, respectively. 15 These constant ghosts possess opposite Grassmann parity to the original transformation parameters; thus, ξ is Grassmann-even, and t µ and θ are Grassmann-odd. The constant Grassmann-even ghost ξ is subject to the Majorana constraint, 16ξ = ξ T (−C −1 ).

(5.4)
If there were only the gauge-ghost c(x) and the U(1) A ghost θ in Eq. (5.1), we would simply have s 2 = 0 on all lattice variables, because algebras of the gauge and of the U(1) A close on lattice variables. However, since the SUSY algebra (that contains the infinitesimal translation) is not properly realized on lattice variables, the nilpotency of s is broken by lattice artifacts. That is, on lattice variables, we have We define the gauge-fixing term and the ghost-anti-ghost action with respect to the generalized BRS transformation s by Note that this S GF+FP reduces to our previous one S This is not the end of the story, however. To incorporate the mass term of the gluino that explicitly breaks SUSY and U(1) A , we must further introduce s-doublet fields (u V , v V ) and (u A , v A ) following the procedure of Refs. [71,51]: where u V (x) and u A (x) are Grassmann-odd, while v V (x) and v A (x) are Grassmann-even. On these variables, we have an exact nilpotency of s: Using these fields, we define a "generalized" mass term by where∂ S µ is defined by the Leibniz rule, ∂ S µ (X 1 . . . X n ) ≡ (∂ S µ X 1 ) . . . X n + · · · + X 1 . . . X n−1 (∂ S µ X n ). (5.10) Again, S mass reduces to our original mass term S We also introduce source terms for lattice dynamical variables ) and, following the standard procedure (see, for example, Ref. [72]), source terms associated with the generalized BRS transformation s, .) For the massive theory, it turns out that we need one more term, The last term of Eq. (5.12), which is quadratic in the source K ψ (x), and S ∆ are required to make the following ST identity or the Zinn-Justin equation hold off-shell up to O(a) lattice artifacts. Thus, our generalized total lattice action is S tot ≡ S gluon + S gluino + S GF+FP + S mass + S source1 + S source2 + S ∆ . (5.14) Using this, we define the generating functional for the connected diagram W , 15) where dµ denotes the integration measure for dynamical variables, U(x, µ), ψ(x), c(x),c(x) and B(x). An important point to recognize here is that those , are all non-dynamical, as the argument of the above W indicates. One can always set those external variables zero and then the system reduces to our original lattice theory in Sec. 2. In this way, the dynamics of the original system can always be reproduced; yet, those new variables are quite useful to organize the underlying symmetry structure.
For P and T transformation properties of new variables, we can set so that s in Eqs. (5.1) and (5.7) preserves transformation properties under P and T ; for example, one sees that [ξγ µ ψ] L (x) transforms in an identical way as A µ (x). Then, it is easy to find appropriate P and T transformations of source fields J and K such that the total action S tot (5.14) is invariant under P and T . Now, a crucial property of the above total action S tot , that can be obtained by a careful calculation using relations in Appendix C, is In deriving Eq. (5.18), we parametrized the breaking of the super, U(1) A and translation symmetries in our original lattice action as This X S is thus identical to our previous definition in Eq. (4.9) and X A is also identical to X A in Eq. (3.2); this correspondence will be the key to our analysis below. Since the continuum action without the gluon mass term possesses the super, U(1) A and translation symmetries, all these breakings, Explicit forms of the last three combinations in Eq. (5.18) are given bȳ It is obvious that these three combinations are of O(a), especially in view of Eq. (5.5). Another crucial property of S tot to note is

Generalized ST identity or Zinn-Justin equation and an associated WZ consistency condition
Now, the combination in the left-hand side of relation (5.18) prompts us to consider the following infinitesimal change of variables for the functional integral (5.15). Letting ε be a Grassmann-odd parameter, and v A (x), except the gluino field ψ(x), for which, Although the last term depends on ψ(x) itself through the definition of K ′ ψ (x) (5.19), the Jacobian J associated with this infinitesimal change of variable is unity, with the lattice regularization (note Eq. (C.1)). From the above change of variables, using Eq. (5.18), we have the identity As usual, we define the 1PI effective action Γ by the Legendre transform of W only with respect to the source J: Here and in what follows, we use the same symbols for the integration variables and the expectation values for notational simplicity; for example, A a µ (x) in the argument of the effective action Γ actually means A a µ (x) J,K,ξ,t,θ,u,v . Then noting relations such as, (the last one follows from Eq. (5.24)) etc., we can express identity (5.28) in terms of the effective action Γ . This gives rise to the following identity (the ST identity or the Zinn-Justin equation in the present context): 33) where the combination S(F ) is defined for an arbitrary functional F by, . (5.34) Corresponding to this combination, we introduce an operation D(F ) by . As one can verify straightforwardly, this relation follows solely from the nilpotency s 2 = 0 on the variables,c(x), and v A (x); recall Eqs. (5.5d) and (5.8). Although the nilpotency of s on the variables, A µ (x), ψ(x) and c(x), is broken by the lattice regularization, the nilpotency on these variables is not necessary to derive Eq. (5.36); this is a crucial observation. Since Eq. (5.36) holds for arbitrary F , it holds of course for Γ . Then, combined with Eq. (5.33), Eq. (5.36) provides a strong constraint on the possible form of the breaking terms, the right-hand side of Eq. (5.33). That is, we have We now expand the effective action Γ in the powers of the loop-counting parameter as Γ = Γ 0 + Γ 1 + Γ 2 + · · · , where Γ 0 is the tree level action, (5.38) In the continuum limit, the expectation value in Eq.
This is the WZ consistency condition that we were seeking.

Simplified consistency condition
To simplify the analysis, we consider the consistency condition (5.39) for a special configuration of expectation values and external variables. First, we assume that expectation values, A µ (x), ψ(x) and c(x), satisfy the equations of motion of the total action S tot . This removes the functional derivative with respect to K from the expression for D(S tot ) (see Eq. (5.35)) and D(S tot ) simply becomes the s transformation. We further set We can now relate Eqs. (3.9) and (4.17) to Eq. (5.39). Since we assumed in Eq. (6.2) that all K vanish, expectation values in Eq. (5.39) are computed without the source term for the composite operators, S source2 (5.12); we have only the source term for the elementary fields, S source1 (5.11). In the language of Eqs. In this way, in the continuum limit, consistency condition (5.39) is simplified to and r ≡ 0 on other variables. Note that the s 2 ψ a -term in Eq. (6.3) is quadratic in ξ, i.e., O(ξ 2 ).

6.2.
O(θ 1 , ξ 0 ) term Relation (6.3) must hold order by order in θ and ξ. Let us first consider the O(θ 1 , ξ 0 ) term. This gives When θ = ξ = 0, r is nothing but the conventional gauge BRS transformation in the continuum theory. Thus, Eq. (6.5) simply tells that X A (x) must be (gauge) BRS invariant up to a total divergence. 17 This is trivially true for Eqs. (3.9a)-(3.9c), but it constrains the possible structure of the BRS non-invariant piece, Eq. (3.9d).
Similarly, we have from the O(θ 0 , ξ 1 ) term of Eq. (6.3), As an application of this relation, let us ask whether a BRS non-invariant combination can appear in Eq. (4.17g) or not. For this operator, we have which is proportional to the totally-symmetric tensor Then the question is that whether Eq. (6.8) can be made into a totaldivergence by adding the r-transformation of some operator other than minus Eq. (6.7). We see that this is impossible as follows. Note first that except the term ∂ µ c(x) in rA µ (x) of Eq. (6.4), c(x) in the r-operation appears in the adjoint action. Since the adjoint action is proportional to the antisymmetric structure constant, the adjoint action with the ghost c(x) cannot produce the totally-symmetric tensor d abc . The only way to form d abc is to use the ∂ µ c(x) in rA µ (x). Thus the generic operator to be added to Eq. (6.8) must have the following structure, 18 For the sum of this and Eq. (6.8) to become a total divergence, terms propor- should cancel out (because it cannot be a total divergence). This requires C µνρσ = δ µρ δ νσ . However, then the first term in Eq. (6.10) is just the r-transformation of minus Eq. (6.7). This shows that Eq. (6.7) cannot appear in the BRS non-invariant piece in X S (x) (4.17g); this fact has an important implication below.
First, since the O(θ) term in Eq. (6.4) does not change the gauge index structure and Eq. (6.13) is proportional to δ ab = 2 tr(T a T b ), the possible contribution from Eq. (4.17g) to Eq. (6.13) comes from . (6.14) However, for (the integral of) this to coincide with Eq. (6.13) up to a total divergence, we have to have G µνρσ = cδ µρ δ νσ , because the combination . (6.15) However, this cannot contribute to Eq. (6.13), because rA µ (x) (6.4) produces ξγ µ ψ(x), but this combination is linearly independent ofξγ 5 σ µν ψ(x) (the coefficient of Eq. (6.13)). 19 By this way, we see that Eqs. (3.9d) and (4.17g) do not contribute to combination (6.13). Besides these, in the present case, we have to take into consideration also the possible modification of Eqs. (3.9) and (4.17) by the presence of newlyintroduced external variables. From Eqs. (6.4), we see that if there exist mixings such that then these produce combination (6.13) under r. These possibilities are excluded, however, because in our new total action S tot , M, v V (x) and v A (x) appear always in the particular combination, In this way, we establishes relation (6.12). In Eq. (6.12), the integer n ≥ 1 was the lowest-order of the loop expansion in which Z O( n ) χ = 0 and Z O( n ) P = 0. However, once Eq. (6.12) is fulfilled, then the consistency condition (6.11) applies to the next leading order O( n+1 ). Thus, repeating the above argument, we have Then, if we substitute Eqs. (4.17a)-(4.17e) in the first term, we have = 0, (6.19) as the coefficient of the combination 20 (6.20) In fact, it is easy to see without any calculation that among Eqs. (4.17a)-(4.17e) only the three-fermion operator (4.17d) contributes to Eq. (6.20): Eq. (4.17d) is proportional to the totally-symmetric tensor (6.9), while others are proportional to δ ab = 2 tr(T a T b ).
Again, it is easy to see that the terms containing c(x),c(x) or B(x), Eq. (4.17f), do not contribute to Eq. (6.20).
Concerning the possible contribution of the BRS non-invariant terms (4.17g) to Eq. (6.20), the unique possibility is .

(6.22)
This possibility is again excluded, because u V (x) should appear only in the combination M + u V (x). In this way, we establish Eq. (6.19), and repeating the argument for higher n, we have That is, we establish Eq. (4.23).

Conclusion
In this paper, in the context of the lattice regularization of 4D N = 1 SYM, we formulated a generalized BRS transformation that treats the gauge, SUSY, translation and U(1) A transformations in a unified way. On the basis of this (almost-nilpotent) transformation on lattice variables, we obtained a generalized WZ consistency condition for the symmetry breaking effects in the lattice formulation. Utilizing this powerful machinery, we then proved that Z χ = (1/2)Z P which implies the coincidence of the chiral symmetric limit (3.13) and the supersymmetric limit (4.21), and that Z 3F = 0 which implies that there is no exotic breaking of the SUSY WT identity by the three-fermion operator (4.19). Our these results provide a solid theoretical basis for lattice formulations of 4D N = 1 SYM. It is interesting to investigate further consequences of the consistency condition (5.39) or (6.3).
In the continuum theory, the generalized BRS symmetry has been formulated also for 4D N = 4 SYM [74] and for 4D N = 2 SYM [75,76]; in the former, such a framework is crucially important because no off-shell multiplet is known. Adopting this framework to the lattice formulation might be useful to systematically classify the necessary fine-tuning in such formulations. 21 as a higher-order quantity in the lattice spacing. What we have proven in Eq. (6.17) is thus the equality Z χ (g, 0) = 1 2 Z P (g, 0), (7.2) to all orders in the power series of g.
On the other hand, in actual numerical simulations with a small but fixed lattice spacing a, the chiral limit would be specified by the tuning, Similarly, the supersymmetric limit would be specified by, M → M SUSY cr (g), where [36] aM SUSY cr (g) − Z χ (g, aM SUSY cr (g)) = 0. (7.4) Then the question is whether from Eq. (7.2) one can draw any conclusion concerning the relation between M chiral cr (g) and M SUSY cr (g) which are defined by Eqs. (7.3) and (7.4), respectively. The answer is affirmative as follows: The point is that the mixing coefficients Z P and Z χ are of O(g 2 ), thus for a fixed aM, Z P (g, aM) and Z χ (g, aM) become arbitrarily small as g → 0. This shows that, irrespective of how Z P and Z χ depend on aM, we have as we approach to the continuum limit. Then Eq. (7.2) implies that M chiral cr (g) and M SUSY cr (g) coincide in the continuum limit.

Appendix A. Three-fermion spinorial operators
In this appendix, we explore the relations between various gauge-invariant three-fermion operators and show that the operator O S (4.19) generally does not vanish.