Continuum extrapolation of Ward identities in N=1 supersymmetric SU(3) Yang-Mills theory
Sajid Ali, Georg Bergner, Henning Gerber, Istvan Montvay, Gernot Münster, Stefano Piemonte, Philipp Scior
MMS-TP-20-17
Continuum extrapolation of Wardidentities in N = supersymmetricSU(3) Yang-Mills theory Sajid Ali ∗ , Georg Bergner † , Henning Gerber ‡ , IstvanMontvay §4 , Gernot Münster ∗ , Stefano Piemonte ¶ , and PhilippScior k University of Münster, Institute for Theoretical Physics,Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany Government College University Lahore, Department of Physics,Lahore 54000, Pakistan University of Jena, Institute for Theoretical Physics,Max-Wien-Platz 1, D-07743 Jena, Germany Deutsches Elektronen-Synchrotron DESY, Notkestr. 85,D-22607 Hamburg, Germany University of Regensburg, Institute for Theoretical Physics,Universitätsstr. 31, D-93040 Regensburg, Germany Universität Bielefeld, Fakultät für Physik, Universitätsstr. 25,D-33615 Bielefeld, Germany14th May 2020 ∗ {sajid.ali,munsteg}@uni-muenster.de † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] k [email protected] a r X i v : . [ h e p - l a t ] J un bstract: In N = 1 supersymmetric Yang-Mills theory, regularised on aspace-time lattice, in addition to the breaking by the gluino mass term,supersymmetry is broken explicitly by the lattice regulator. In addition tothe parameter tuning in the theory, the supersymmetric Ward identities canbe used as a tool to investigate lattice artefacts as well as to check whethersupersymmetry can be recovered in the chiral and continuum limits. In thispaper we present the numerical results of an analysis of the supersymmetricWard identities for our available gauge ensembles at different values of theinverse gauge coupling β and of the hopping parameter κ . The results clearlyindicate that the lattice artefacts vanish in the continuum limit, confirmingthe restoration of supersymmetry. Supersymmetry (SUSY) is an elegant idea which relates fermions and bosons, whose spindiffers by 1/2, through supercharges [1]. SUSY provides dark matter candidates, arisingfrom the lightest supersymmetric particles [2]. In addition to that, supersymmetricextensions of the Standard Model would resolve the hierarchy problem [3]. N = 1supersymmetric 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 describesthe strong interactions between gluons and gluinos, the superpartners of the gluons.Gluinos are Majorana particles that transform under the adjoint representation of thegauge group. The on-shell Lagrangian of N = 1 SYM theory, which consists of thegluon fields A aµ ( x ) and the gluino fields λ a ( x ), where a = 1 , . . . , N c −
1, can be writtenin Minkowski space as L SYM = − F aµν F a,µν + i2 ¯ λ a γ µ ( D µ λ ) a − m ˜ g λ a λ a , (1)where the first term, containing the field strength tensor F aµν , is the gauge part, and D µ in the second term is the covariant derivative in the adjoint representation of the gaugegroup SU( N c ), N c being the number of colors. The last part of the above Lagrangian isa gluino mass term which breaks SUSY softly for m ˜ g = 0, which means that it does notaffect the renormalisation properties of the theory and that the spectrum of the theorydepends on the gluino mass in a continuous way. The physical spectrum of this theory isexpected to consist of bound states of gluons and gluinos, arranged in mass degeneratesupermultiplets if SUSY is not broken [5, 6].In order to perform Monte-Carlo simulations of the theory, we discretise the Euclideanaction and put it onto a four-dimensional hypercubic lattice. We use the Curci-Venezianoversion [7] of the lattice action S = S g + S f , where the gauge part S g is defined by theusual plaquette action S g = − βN c X p Re [tr ( U p )] , (2)2ith the inverse gauge coupling given by β = 2 N c /g , and the fermionic part S f = 12 X x ¯ λ ax λ ax − κ X µ =1 h ¯ λ ax +ˆ µ V ab,xµ (1 + γ µ ) λ bx + ¯ λ ax V Tab,xµ (1 − γ µ ) λ bx +ˆ µ i (3)implements the gluinos as Wilson fermions. Here the adjoint link variables are definedby V ab,xµ = 2 tr ( U † xµ T a U xµ T b ), where T a are the generators of the gauge group, and thehopping parameter κ is related to the bare gluino mass m ˜ g by κ = 1 / (2 m ˜ g + 8). Inorder to approach the limit of vanishing gluino mass, the hopping parameter has to betuned 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 thetheory with gauge group SU(2) and SU(3) non-perturbatively from first principles usingMonte Carlo techniques [4, 9, 10, 11], and obtained mass degenerate supermultiplets [12]. In classical physics, Noether’s theorem provides a relation between symmetries and con-servation laws. In the case of quantum field theories, symmetries are translated to Wardidentities, representing quantum versions of Noether’s theorem. In N = 1 supersym-metric Yang-Mills theory a gluino mass term breaks SUSY softly. The soft breakingeffects vanish in the chiral limit, a limit where theory is characterised by massless glui-nos. 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 supersym-metry is broken explicitly due to the introduction of the discretisation of space-timelattice as a regulator of the theory. SUSY Ward identities can be used to check whethersupersymmetry is restored in the continuum limit.In the Euclidean continuum, on-shell supersymmetry transformations of the gaugeand gluino fields are given by δA aµ = − λ a γ µ ε , δλ a = − σ µν F aµν ε , (4)where the transformation parameter ε is an anticommuting Majorana spinor. Fromthe variation of the action under a supersymmetry transformation with a space-time-dependent parameter ε ( x ) one derives the SUSY Ward identities. For any suitable gaugeinvariant local operator Q ( y ), they read h ∂ µ S µ ( x ) Q ( y ) i = m ˜ g h χ ( x ) Q ( y ) i − * δQ ( y ) δ ¯ (cid:15) ( x ) + , (5)where S µ ( x ) = ( S αµ ( x )) is the supercurrent of spin 3/2, and the term m ˜ g h χ ( x ) Q ( y ) i isdue to the gluino mass in the action of the theory. In the continuum the supercurrent3 µ ( x ) and the operator χ ( x ) are given by S µ ( x ) = − g tr [ F νρ ( x ) σ νρ γ µ λ ( x )] , (6) χ ( x ) = + 2 i g tr [ F µν ( x ) σ µν λ ( x )] . (7)The last term of Eq. (5) is a contact term, which contributes only if x = y , and it canbe avoided if Q ( y ) is not localised at x . Therefore the contact term is ignored in thefollowing 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 byan additional term h X S ( x ) Q ( y ) i . The explicit form of this term is known, but neednot 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 perturbationtheory, nevertheless, the contribution of this term is finite in the continuum limit dueto divergences proportional to 1/ a that multiply the factor a . This plays a role for therenormalisation of the supercurrent and of the gluino mass [7, 13]. In the renormalisationof SUSY Ward identities, operators of dimensions ≤ / T µ , mixing withthe supercurrent, appears, corresponding to an operator of dimension 9 /
2. Consequently,on the lattice the following Ward identities are obtained Z S h∇ µ S µ ( x ) Q ( y ) i + Z T h∇ µ T µ ( x ) Q ( y ) i = m S h χ ( x ) Q ( y ) i + O ( a ) , (8)where Z S and Z T are renormalisation coefficients. The subtracted gluino mass is definedas m S = m ˜ g − ¯ m , where ¯ m is the mass subtraction coming from the operators ofdimension 7 /
2. The mixing current is defined as T µ ( x ) = 2 i g tr [ F µν ( x ) γ ν λ ( x )] . (9)Regarding the local insertion operator Q ( y ), our choice is the spinor Q ( y ) = χ (sp) ( y ),with χ (sp) ( y ) = X i 3) andgiven numerical data x iα , the probability distribution P ∼ exp( − L ) of the quantitiesˆ x iα , subject to the constraints (13), has its maximum at a point where L = L min , with L min = 12 X i,α,j,β ( A α x iα )( D − ) ij ( A β x jβ ) , (14)5here D ij = X α,β A α A β ( h x iα x jβ i − h x iα ih x jβ i ) . (15)Next, the desired coefficients A α have to be found such that L min as a function of A and A is minimised. This cannot be solved analytically, and we find A α numericallysuch that the global minimum of L min ( A , A ) is reached; for details see Ref. [15]. Inparticular, owing to A = − am S Z − S this provides us with the subtracted gluino mass m S up to the renormalisation factor. To estimate the statistical uncertainties we employthe standard Jackknife procedure. All terms in the Ward identity (8), including the O ( a ) term h X S ( x ) Q ( y ) i , are correl-ation functions of gauge invariant operators. In the corresponding Eqs. (11) they arecorrelation functions of operators localised on time slices or pairs of adjacent time slicesat distance t . As for any gauge invariant correlation function of this type, they decayexponentially in t , with a decay rate given by the mass gap of the theory. For verysmall t the contributions of higher masses will affect the impact of the O ( a ) term on theWard identities. Therefore we expect that the value of the obtained gluino mass willdepend on the minimal time slice distance t min . This effect should become negligible atsufficiently large t min . On the other hand, if t min is chosen too large, noise in the datawill dominate. The behaviour that can be observed in Fig. 1 is compatible with theseexpectations. An adequate choice of t min is therefore important for the quality of theresults. We cope with this in two ways.In order to avoid perturbing effects at too small t min and a poor signal-to-noise ratioat too large t min , 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 subtractedgluino mass begins. The results are presented in Tab. 1. β = 5 . β = 5 . β = 5 . β = 5 . β = 5 . V = 12 × V = 16 × V = 16 × V = 16 × V = 24 × κ t min κ t min κ t min κ t min κ t min t min for all available gauge ensembles, chosen such that a plateauis formed. 6 . . . . . a m S Z − S t min κ = 0.1645GLS . . . . . 25 2 4 6 8 10 12 a m S Z − S t min κ = 0.1650GLS . . . . . a m S Z − S t min κ = 0.1655GLS . . . . . 15 2 4 6 8 10 12 a m S Z − S t min κ = 0.1660GLS Figure 1: The subtracted gluino mass am S Z − S as a function of t min calculated with theGLS Method at β = 5 . 6. At small values of t min the subtracted gluino mass isaffected by contact terms and by O ( a ) terms. Data from t min = 2 and t min = 3are shown, but do not enter our final analysis.In the second approach, we consider that our simulations of the theory are done atdifferent values of the lattice spacing a , which leads to different O ( a ) terms in the Wardidentities. A fixed value of t min in lattice units would mean a lower limit on the time-slice distances in physical units, that is on the cutoff-scale and shrinks to zero in thecontinuum limit. Instead it would be more appropriate to consider t min at constantphysical distance for all gauge ensembles. This is done in the following way.At the coarsest lattice spacing, at inverse gauge coupling β , the value of t min isselected according to the plateau criterion explained above. For finer lattice spacingsat inverse gauge couplings β i the corresponding t min are then obtained by scaling witha physical scale. In order to determine the physical scale we use the mass m g ˜ g of thegluino-glue particle and the Wilson flow parameter w . Correspondingly, t min is scaled7ccording to t min,β i = t min,β m g ˜ g,β m g ˜ g,β i , (16)or t min,β i = t min,β w ,β i w ,β , (17)where β = 5 . β = 5 . β = 5 . 5, and β = 5 . 6. The resulting t min is rounded to thenearest integer value. The values obtained by this method are collected in Tab. 2. Inmost points they are equal or almost equal to those in Tab. 1. β t min from m g ˜ g t min from w t min at fixed physical temporal distance from scaling with thegluino-glue mass m g ˜ g and with the Wilson flow parameter w . The chiral limit is defined by the vanishing of the subtracted gluino mass. Its measuredvalues can therefore be employed for the tuning of the hopping parameter κ to thechiral 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 SYMtheory, that can be defined in partially quenched chiral perturbation theory [17]. In thenumerical simulations its correlation function can be computed as the connected pieceof the correlation function of the a- η particle. Similar to the Gell-Mann-Oakes-Rennerrelation of QCD [5], in the continuum limit there is a linear relation between the adjointpion mass squared and the gluino mass: m π ∝ m ˜ g .The numerical results for the subtracted gluino mass from the Ward identities andthe adjoint pion mass squared in lattice units are shown for β = 5 . am S Z − S ), and it is expected to vanish in the continuum limit. The values of theremnant gluino mass, obtained by taking an average of the values calculated using theprocedures explained above, are presented in Tab. 3.8 . 00 3 . 01 3 . 02 3 . 03 3 . / κ . . . . . . . a m β = 5 . am S Z − S ( am a − π ) (a) The subtracted gluino mass am S Z − S and theadjoint pion mass squared ( am a- π ) as a functionof 1 / (2 κ ), and the corresponding extrapolationstowards the chiral limit ( κ c ). − . 05 0 . 00 0 . 05 0 . 10 0 . 15 0 . 20 0 . 25 0 . am a − π ) . . . . . . a m S Z − S β = 5 . (b) The subtracted gluino mass am S Z − S asa function of the adjoint pion mass squared( am a- π ) in order to obtain the remnant gluinomass ∆( am S Z − S ). Figure 2: Chiral limit and determination of the remnant gluino mass at β = 5 . 6. Allquantities are in lattice units. β am S Z − S ) 0.0334(48) 0.019(12) 0.0099(88) 0.0103(33)Table 3: The values of the remnant gluino mass ∆( am S Z − S ) obtained at four differentvalues of the inverse gauge coupling. 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, argumentsbased on partially quenched chiral perturbation theory suggest that the remnant gluinomass is of order a at m π = 0 [13]. In order to investigate this relation, the remnantgluino mass has to be expressed in physical units. Our choice for the scale is the Wilsonflow parameter w , which is defined through the gradient flow [10]. We use its valuesextrapolated to the chiral limit, w ,χ . Similarly the lattice spacing is represented by a/w ,χ . Our numerical results for the remnant gluino mass as a function of the latticespacing and its extrapolation towards the continuum limit are shown in Fig. 3. The datapoints in Fig. 3(a) show the results from separate chiral extrapolations for each latticespacing and the corresponding extrapolation to the continuum limit. The extrapolationto the continuum and the error of this extrapolation are obtained by means of parametricbootstrap with linear fits. On the other hand, Fig. 3(b) is obtained by means of asimultaneous fit of the dependence on the hopping parameter and the lattice spacing [18].9 . 00 0 . 05 0 . 10 0 . 15 0 . 20 0 . a/w ,χ )2 − . − . . . . . . ∆ ( w , χ m S Z − S ) (a) The remnant gluino mass from separate extra-polations to the chiral limit where m π is zero, andthe extrapolation to the continuum limit. . 00 0 . 05 0 . 10 0 . 15 0 . 20 0 . a/w ,χ ) . . . . . ∆ ( w , χ m S Z − S ) (b) The remnant gluino mass from a simultan-eous chiral and continuum extrapolation. Byconstruction, in this method the data points co-incide with the error band. Figure 3: The remnant gluino mass ∆( w m S Z − S ) in physical units w as a function ofthe lattice spacing squared, and its linear extrapolation towards the continuumlimit.The remnant gluino mass in the continuum limit is compatible with zero within onestandard-deviation, confirming the preliminary results present in Ref. [15] with only twodata points. Lattice artefacts vanish in the continuum limit as expected, and supersym-metry is recovered in the chiral and continuum limits, in agreement with our findingsfrom the mass spectrum [12]. In this paper we have presented numerical results of an analysis of SUSY Ward identit-ies 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 controlledby suitable choices of time-slice distances. Ensembles of gauge configurations at fourdifferent values of the lattice spacing and various hopping parameters have been ana-lysed, 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 intwo alternative ways, on the one hand by extrapolating to the chiral limit at each latticespacing separately and then to the continuum limit, and on the other hand by means of asimultaneous extrapolation to the chiral and continuum limit. With both extrapolationsthe lattice artefacts in the subtracted gluino mass appear to scale to zero as of order a in agreement with the theoretical expectations. Our findings support the validity ofSUSY Ward identities and the restoration of supersymmetry in the continuum limit.10 cknowledgments References [1] J. Wess and J. Bagger, Supersymmetry and Supergravity , Princeton UniversityPress, 1992.[2] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 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