# Analysis of Ward identities in supersymmetric Yang–Mills theory

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## Abstract

In numerical investigations of supersymmetric Yang–Mills theory on a lattice, the supersymmetric Ward identities are valuable for finding the critical value of the hopping parameter and for examining the size of supersymmetry breaking by the lattice discretisation. In this article we present an improved method for the numerical analysis of supersymmetric Ward identities, which takes into account the correlations between the various observables involved. We present the first complete analysis of supersymmetric Ward identities in \(\mathcal {N}=1\) supersymmetric Yang–Mills theory with gauge group SU(3). The results indicate that lattice artefacts scale to zero as \(O(a^2)\) towards the continuum limit in agreement with theoretical expectations.

## 1 Introduction

Ward identities are the key instruments for studying symmetries in quantum field theory. They represent the quantum counterparts to Noether’s theorem, expressing the realisation of a classical symmetry at the quantum level in terms of relations between Green’s functions. They also allow to characterise sources of explicit symmetry breaking. In the case of theories that are regularised non-perturbatively by means of a space-time lattice, Ward identities are a useful tool for the investigation of lattice artefacts, which are related to the breaking of symmetries. In lattice QCD, for example, chiral Ward identities in the form of the PCAC relation are being used to quantify the breaking of chiral symmetry by the lattice discretisation, and thereby to control the approach to the continuum limit [1].

For supersymmetric (SUSY) theories the corresponding relations are the supersymmetric Ward identities. In the context of numerical investigations of supersymmetric Yang–Mills theory on a lattice, SUSY Ward identities are being employed for a twofold purpose [2]. First, in numerical simulations using Wilson fermions a gluino mass is introduced, which breaks supersymmetry softly. With the help of SUSY Ward identities the parameters of the model can be tuned such that an extrapolation to vanishing gluino mass is possible. Second, the discretisation on a lattice generically breaks supersymmetry [3], leading to lattice artefacts of order *a* in the lattice spacing. By means of SUSY Ward identities it can be checked if lattice artefacts are small enough for an extrapolation to the continuum limit.

Our collaboration has employed SUSY Ward identities in previous investigations of \(\mathcal {N}=1\) supersymmetric Yang–Mills theory with gauge group SU(2); for recent result see [4]. In the analysis of SUSY Ward identities, following the methods introduced in [2], the correlations between the various quantities entering the calculation are, however, not being taken into account. Therefore, for our present studies with gauge group SU(3) we developed a method, based on a generalised least squares fit, that incorporates these correlations. In this article we describe the method and present the results of the first complete analysis of SUSY Ward identities for supersymmetric Yang–Mills theory with gauge group SU(3).

## 2 Supersymmetric Ward identities on the lattice

*C*, thus being their own antiparticles. Gluinos transform under the adjoint representation of the gauge group, so that the gauge covariant derivative is given by \((\mathcal {D}_{\mu } \lambda )^{a} = \partial _{\mu } \lambda ^{a} + g\,f_{abc} A^{b}_{\mu } \lambda ^{c}\). In the Euclidean continuum the (on-shell) Lagrangian of the theory, where auxiliary fields have been integrated out, is

*Q*(

*y*) is any suitable insertion operator, and the last term represents a contact term given by the SUSY variation of

*Q*(

*y*), which vanishes if

*Q*(

*y*) is localised at space-time points different from

*x*.

*p*, and

*O*(

*a*) improved by addition of the clover term with the one-loop coefficient specific for this model [9].

*O*(

*a*) terms. We choose the local transcriptions of the continuum forms,

The supersymmetric continuum limit is obtained at vanishing gluino mass \(m_S\). The value of the critical hopping parameter \(\kappa _c\), where \(m_S\) is zero, has to be determined numerically. With suitable choices of *Q*(*y*), this can be achieved with the lattice SUSY Ward identity. The expectation values appearing in Eq. (8) can be evaluated in the Monte Carlo calculations. This allows to obtain the coefficient \(m_S / Z_S\), which in turn enables us to locate the point \(m_S = 0\). An alternative tuning is obtained from the signals of a restored chiral symmetry, see below. It is expected that both are consistent up to lattice artefacts. The investigation of the SUSY Ward identities allows to confirm this scenario and to estimate the relevant lattice artefacts.

## 3 Numerical analysis of SUSY Ward identities

*O*(

*a*) terms are omitted, and

The six different correlators \(\hat{x}_{b,t,\alpha }\) are estimated numerically in our Monte Carlo simulations for gauge group SU(3). The usual estimators for these expectation values are the numerical averages of the corresponding observables over the Monte Carlo run. Let us call these averages \(x_{b,t,\alpha }\). They are random variables with expectation values \(\hat{x}_{b,t,\alpha } \equiv \langle x_{b,t,\alpha } \rangle \). It should be noted that only data at \(t \ge 3\) are being considered in order to avoid contamination by contact terms.

*t*the two equations (13) could be solved for

*t*together, however, we have an overdetermined set of equations for these two coefficients. The aim is to find solutions for

*A*and

*B*numerically such that with the measured values \(x_{b,t,\alpha }\) the equations are satisfied approximately in an optimal way. In previous studies for gauge group SU(2) the coefficients

*A*and

*B*have been calculated by means of a minimal chi-squared method, as proposed in [2]. The correlators \(x_{b,t,\alpha }\) are, however, statistically correlated amongst each other, in particular for nearby values of

*t*, and these correlations have not been taken into account.

In order to improve on this point, we have developed a method, which takes all correlations fully into account, so that more reliable results and error estimates can be obtained. The approach is based on the method of generalised least squares [10].

Results for \(a m_S Z_S^{-1}\) from the previous method and from the generalised least squares (GLS) method for our ensembles at \(\beta = 5.5\)

\(\kappa \) | 0.1637 | 0.1649 | 0.1667 | 0.1673 | 0.1678 | 0.1680 | 0.1683 |
---|---|---|---|---|---|---|---|

Previous | 0.489(26) | 0.343(7) | 0.176(4) | 0.123(3) | 0.081(3) | 0.057(4) | 0.025(4) |

GLS | 0.494(42) | 0.348(8) | 0.178(4) | 0.123(3) | 0.081(2) | 0.056(5) | 0.024(6) |

- 1.
For given \(x_{i\alpha }\), consider \(A_{\alpha }\) to be fixed and determine \(\hat{x}_{i\alpha }\) such that

*P*is maximal under the constraint \(\sum _{\alpha } A_{\alpha } \, \hat{x}_{i\alpha } = 0\). The value \(P_{\text {max}}(A_\alpha )\) at maximum depends on \(A_\alpha \). - 2.
Find \(A_\alpha \) such that \(P_{\text {max}}(A_\alpha )\) is maximal.

*L*with the help of Lagrange multipliers gives

Now the minimum of \(L_{\text {min}}(A_{\alpha })\) as a function of the parameters \(A_2\) and \(A_3\) (\(A_1=1\)) has to be found. Because \(D_{ij}\) depends on the \(A_{\alpha }\), it is not possible to do this analytically, and we determine the global minimum numerically, thus obtaining \(A_2\) and \(A_3\). To get the statistical errors we re-sample the data and apply the jackknife method, repeating the whole procedure for each jackknife sample. In this way we arrive at our final result for \(B = a m_S Z_S^{-1}\).

## 4 Results for SU(3) SYM

For SYM theory with gauge group SU(3) we have applied the method to our current simulation ensembles obtained with *O*(*a*) improved clover fermion action [11] at different inverse gauge couplings \(\beta \) and hopping parameters \(\kappa \). At two lattice spacings, corresponding to \(\beta = 5.4\) and 5.5, the available statistics has allowed to obtain reliable results for the Ward identities. From the results for the gluino mass parameter \(a m_S Z_S^{-1}\) the value of \(\kappa _c\), where \(m_S\) vanishes, can be estimated.

Comparing the results for \(a m_S Z_S^{-1}\) with those from the earlier method, which does not properly take the correlations into account, we find that the values are compatible within errors, but this time we have a precise and reliable estimate of the errors. As examples, the results of both methods for \(\beta = 5.5\) are shown in Table 1.

An alternative way to estimate \(\kappa _c\) in the Monte Carlo calculations employs the mass of the adjoint pion \(\text {a--}\pi \), see e. g. [12]. The \(\text {a--}\pi \) is an unphysical particle in SYM theory. However, by arguments based on the OZI-approximation [13], and in the framework of partially quenched chiral perturbation theory [14], the squared mass \(m_{\text {a--}\pi }^2\) is expected to vanish linearly with the gluino mass close to the chiral limit.

In Fig. 1 we show \(a m_S Z_S^{-1}\) and \((a m_{\text {a--}\pi })^2\) as a function of \(1/(2\kappa )\) for our two values of \(\beta \). Both quantities depend linearly on \(\kappa ^{-1}\) within errors, as expected, and yield independent estimates of the value of \(\kappa _c\).

*a*in the dependence of the squared pion mass on the quark mass:

*a*this is by far not the case. Having only two points available, one has to be cautious drawing conclusions, but the result clearly indicates that the remnant gluino mass \(\Delta m_S\) vanishes proportional to \(a^2\) in the continuum limit.

## 5 Conclusions

We have presented a method for the numerical analysis of SUSY Ward identities in supersymmetric Yang–Mills theory on a lattice, which employs the expectation values of the relevant operators on a range of time slices. The statistical correlations between all observables are taken into account by means of a generalised least squares procedure. Applied to SUSY Yang–Mills theory with gauge group SU(3), the value of the hopping parameter, where the renormalised gluino mass vanishes, can be estimated, and is in rough agreement with the estimation using the adjoint pion mass. The difference between the estimates appears to vanish in the continuum limit. Our results represent the first continuum extrapolation of SUSY Ward identities. The scaling of lattice artefacts as of \(O(a^2)\) is in agreement with theoretical expectations.

## Notes

### Acknowledgements

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 JUQUEEN and JURECA at Jülich Supercomputing Centre (JSC) and SuperMUC at Leibniz Supercomputing Centre (LRZ). Further computing time has been provided on the compute cluster PALMA of the University of Münster. This work is supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group “GRK 2149: Strong and Weak Interactions – from Hadrons to Dark Matter”. G.B. acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) Grant No. BE 5942/2-1.

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