Continuum extrapolation of Ward identities in $\mathbf{\mathcal{N}=1}$ supersymmetric SU(3) Yang-Mills theory

In $\mathcal{N}=1$ supersymmetric Yang-Mills theory, regularised on a space-time lattice, in addition to the breaking by the gluino mass term, supersymmetry is broken explicitly by the lattice regulator. In addition to the parameter tuning in the theory, the supersymmetric Ward identities can be used as a tool to investigate lattice artefacts as well as to check whether supersymmetry can be recovered in the chiral and continuum limits. In this paper we present the numerical results of an analysis of the supersymmetric Ward identities for our available gauge ensembles at different values of the inverse gauge coupling $\beta$ and of the hopping parameter $\kappa$. The results clearly indicate that the lattice artefacts vanish in the continuum limit, confirming the restoration of supersymmetry.


Introduction
Supersymmetry (SUSY) is an elegant idea which relates fermions and bosons, whose spin differs by 1/2, through supercharges [1]. SUSY provides dark matter candidates, arising from the lightest supersymmetric particles [2]. In addition to that, supersymmetric extensions of the Standard Model would resolve the hierarchy problem [3]. N = 1 supersymmetric Yang-Mills (SYM) theory, which is being considered in this article, provides an extension of the pure gluonic part of the Standard Model [4]. It describes the strong interactions between gluons and gluinos, the superpartners of the gluons. Gluinos are Majorana particles that transform under the adjoint representation of the gauge group. The on-shell Lagrangian of N = 1 SYM theory, which consists of the gluon fields A a µ (x) and the gluino fields λ a (x), where a = 1, . . . , N 2 c − 1, can be written in Minkowski space as where the first term, containing the field strength tensor F a µν , is the gauge part, and D µ in the second term is the covariant derivative in the adjoint representation of the gauge group SU(N c ), N c being the number of colors. The last part of the above Lagrangian is a gluino mass term which breaks SUSY softly for mg = 0, which means that it does not affect the renormalisation properties of the theory and that the spectrum of the theory depends on the gluino mass in a continuous way. The physical spectrum of this theory is expected to consist of bound states of gluons and gluinos, arranged in mass degenerate supermultiplets if SUSY is not broken [5,6].
In order to perform Monte-Carlo simulations of the theory, we discretise the Euclidean action and put it onto a four-dimensional hypercubic lattice. We use the Curci-Veneziano version [7] of the lattice action S = S g + S f , where the gauge part S g is defined by the usual plaquette action with the inverse gauge coupling given by β = 2N c /g 2 , and the fermionic part implements the gluinos as Wilson fermions. Here the adjoint link variables are defined by V ab,xµ = 2 tr (U † xµ T a U xµ T b ), where T a are the generators of the gauge group, and the hopping parameter κ is related to the bare gluino mass mg by κ = 1/(2mg + 8). In order to approach the limit of vanishing gluino mass, the hopping parameter has to be tuned properly. In our numerical investigations the fermionic part is additionally O(a) improved by adding the clover term −(c sw /4)λ(x)σ µν F µν λ(x) [8].
In our previous investigations we have determined the low-lying mass spectrum of the theory with gauge group SU(2) and SU(3) non-perturbatively from first principles using Monte Carlo techniques [4,9,10,11], and obtained mass degenerate supermultiplets [12].

SUSY Ward identities
In classical physics, Noether's theorem provides a relation between symmetries and conservation laws. In the case of quantum field theories, symmetries are translated to Ward identities, representing quantum versions of Noether's theorem. In N = 1 supersymmetric Yang-Mills theory a gluino mass term breaks SUSY softly. The soft breaking effects vanish in the chiral limit, a limit where theory is characterised by massless gluinos. In order to analyse this breaking of supersymmetry and to identify the chiral limit, we employ the Ward identities for supersymmetry. Moreover, on the lattice supersymmetry is broken explicitly due to the introduction of the discretisation of space-time lattice as a regulator of the theory. SUSY Ward identities can be used to check whether supersymmetry is restored in the continuum limit.
For any suitable gauge invariant local operator Q(y), the SUSY Ward identities read where S µ (x) = (S α µ (x)) is the supercurrent of spin 3/2, and the term mg χ(x)Q(y) is due to the gluino mass in the action of the theory. In the continuum the supercurrent S µ (x) and the operator χ(x) are given by The last term of Eq. (4) is a contact term, which contributes only if x = y, and it can be avoided if Q(y) is not localised at x. Therefore the contact term is ignored in the following discussions.
The four-dimensional space-time lattice breaks SUSY explicitly. As a consequence, the lattice versions of the Ward identities differ from their continuum counter parts by an additional term X S (x)Q(y) . The explicit form of this term is known, but need not be displayed here. At tree level this term is proportional to the lattice spacing a and vanishes in the limit of zero lattice spacing. At higher orders in perturbation theory, nevertheless, the contribution of this term is finite in the continuum limit due to divergences proportional to 1/a that multiply the factor a. This plays a role for the renormalisation of the supercurrent and of the gluino mass [7,13]. In the renormalisation of SUSY Ward identities, operators of dimensions ≤ 11/2 have to be taken into account. They lead to a modification of the gluino mass, and in addition a current T µ , mixing with the supercurrent, appears, corresponding to an operator of dimension 9/2. Consequently, on the lattice the following Ward identities are obtained where Z S and Z T are renormalisation coefficients. The subtracted gluino mass is defined as m S = mg −m, wherem is the mass subtraction coming from the operators of dimension 7/2. The mixing current is defined as Regarding the local insertion operator Q(y), our choice is Q(y) = χ (sp) (y), with where the indices i, j ∈ {1, 2, 3}. The reason behind this choice is that it gives the best signal [13].

Numerical analysis of SUSY Ward identities
We have analysed the SUSY Ward identities numerically, employing the configurations produced in our project on N = 1 supersymmetric Yang-Mills theory with gauge group SU(3). Numerically it is convenient to use integrated Ward identities where integration or sum is performed over all three spatial coordinates. The resulting identities will then hold for every time-slice distance t. We consider the data starting from t = t min to t = t max where t min ≥ 3 to avoid contamination from contact terms, and t max is the half of the time extent of the lattice because of (anti-)symmetrisation. Each term in Eq. (7) is a 4×4 matrix in spin-space and can be expanded in the basis of 16 Dirac matrices, i. e. {1, γ 5 , γ µ , γ µ γ 5 , iσ µν }. It can be shown, with the help of discrete symmetries, that only the following two contributions are non-zero [13]: where and traces over repeated Dirac indices are implied. O(a) terms that vanish in the continuum limit are not written explicitly in these equations. Introducing a double index i = (b, t), running over 2T values, where T is the time extent of the lattice, and In these equations thex iα = x iα are the expectation values of random variables x iα , which themselves are considered to be the results of a finite Markov chain. We compute the estimators x iα for the correlation functionsx iα numerically using high performance facilities, and solve Eq. (12) for A α by means of minimal chi-squared methods. Two methods, namely the so-called Local Method and Global Method, have been used in the past by our collaboration [4,13]. These methods, however, do not take properly into account correlations between the different quantities appearing in Eq. (12). For this purpose we have developed a new method based on a generalised least squares fit, the so-called GLS Method [14], based on the maximum likelihood. For fixed A α (α = 1, 2, 3) and given numerical data x iα , the probability distribution P ∼ exp(−L) of the quantitieŝ x iα , subject to the constraints (12), has its maximum at a point where L = L min , with where Next, the desired coefficients A α have to be found such that L min as a function of A 2 and A 3 is minimised. This cannot be solved analytically, and we find A α numerically such that the global minimum of L min (A 2 , A 3 ) is reached; for details see Ref. [15]. To estimate the statistical uncertainties we employ the standard Jackknife procedure.

Discretisation effects
Some of the O(a) terms of Eq. (7) come from the correlation function X S (x)Q(y) and they depend on the time slice distance t. They are expected to decay exponentially with t min . This is compatible with the behaviour that can be observed in Fig. 1. We cope with this in two ways.
As some O(a) terms in the Ward identities decay with t min , their effects become negligible at sufficiently large t min . On the other hand, if t min is chosen too large, noise in the data will dominate, see Fig. 1. Therefore, for each hopping parameter and inverse gauge coupling, the value of t min is selected by finding an optimal starting point where a plateau in the subtracted gluino mass begins. The results are presented in Tab. 1.
In the second approach, we consider the values of the subtracted gluino mass am S Z −1 S at constant physical distance in the t direction for all gauge ensembles. In order to determine the physical scale we use the mass m gg of the gluino-glue particle and the Wilson flow parameter w 0 . We select the maximal possible value of t min for the coarsest lattice  spacing and find the corresponding t min for finer lattice spacings by scaling according to where β 0 = 5.4, β 1 = 5.45, β 2 = 5.5, and β 3 = 5.6. The resulting t min is rounded to the nearest integer value, see Tab. 2.  Table 2: The values of t min at fixed physical temporal distance from scaling with the gluino-glue mass m gg and from the Wilson flow parameter w 0 .

Adjoint pion and remnant gluino mass
The chiral limit is defined by the vanishing of the subtracted gluino mass. Its measured values can therefore be employed for the tuning of the hopping parameter κ to the chiral limit. On the other hand, we can also use the vanishing of the adjoint pion mass m a-π for the tuning [16]. The adjoint pion a-π is an unphysical particle in the SYM theory, that can be defined in partially quenched chiral perturbation theory [17]. In the numerical simulations its correlation function can be computed as the connected piece of the correlation function of the a-η particle. Similar to the Gell-Mann-Oakes-Renner relation of QCD [5], in the continuum limit there is a linear relation between the adjoint pion mass squared and the gluino mass: m 2 a-π ∝ mg. The numerical results for the subtracted gluino mass from the Ward identities and the adjoint pion mass squared in lattice units are shown for β = 5.6 in Fig. 2 together with their extrapolations towards the chiral limit.  In the continuum the subtracted gluino mass and the adjoint pion mass should vanish at the same point. On the lattice, however, this is not the case due to lattice artefacts. As an estimate for this discrepancy we determine the value of the subtracted gluino mass at vanishing adjoint pion mass. This quantity is called the remnant gluino mass ∆(am S Z −1 S ), and it is expected to vanish in the continuum limit. The values of the remnant gluino mass, obtained by taking an average of the values calculated using the procedures explained above, are presented in Tab. 3.

Continuum limit
The remnant gluino mass is a lattice artefact and should vanish in the continuum limit a → 0. It is therefore a quantity to check on whether supersymmetry is recovered or not.
Concerning the dependence of the remnant gluino mass on the lattice spacing, arguments based on partially quenched chiral perturbation theory suggest that the remnant gluino mass is of order a 2 at m 2 a-π = 0 [13]. In order to investigate this relation, the remnant gluino mass has to be expressed in physical units. Our choice for the scale is the Wilson flow parameter w 0 , which is defined through the gradient flow [10]. We use its values extrapolated to the chiral limit, w 0,χ . Similarly the lattice spacing is represented by a/w 0,χ . Our numerical results for the remnant gluino mass as a function of the lattice spacing and its extrapolation towards the continuum limit are shown in Fig. 3. The data points in Fig. 3(a) show the results from separate chiral extrapolations for each lattice spacing and the corresponding extrapolation to the continuum limit. The extrapolation to the continuum and the error of this extrapolation are obtained by means of parametric bootstrap with linear fits. On the other hand, Fig. 3(b) is obtained by means of a simultaneous fit of the dependence on the hopping parameter and the lattice spacing [18].  The remnant gluino mass in the continuum limit is compatible with zero within one standard-deviation, confirming the preliminary results present in Ref. [15] with only two data points. Lattice artefacts vanish in the continuum limit as expected, and supersymmetry is recovered in the chiral and continuum limits, in agreement with our findings from the mass spectrum [12].

Conclusion
In this paper we have presented numerical results of an analysis of SUSY Ward identities in N = 1 supersymmetric Yang-Mills theory on the lattice with gauge group SU(3). Contact terms and O(a) lattice artefacts in the Ward identities have been controlled by suitable choices of time-slice distances. Ensembles of gauge configurations at four different values of the lattice spacing and various hopping parameters have been analysed, allowing us for the first time to perform an extrapolation to the continuum limit, where the lattice artefacts vanish. The remnant gluino mass has been extrapolated in two alternative ways, on the one hand by extrapolating to the chiral limit at each lattice spacing separately and then to the continuum limit, and on the other hand by means of a simultaneous extrapolation to the chiral and continuum limit. With both extrapolations the lattice artefacts in the subtracted gluino mass appear to scale to zero as of order a 2 in agreement with the theoretical expectations. Our findings support the validity of SUSY Ward identities and the restoration of supersymmetry in the continuum limit.