Supersymmetry on the lattice and the status of the Super-Yang-Mills simulations
aa r X i v : . [ h e p - l a t ] N ov Supersymmetry on the lattice and the status of theSuper-Yang-Mills simulations
Georg Bergner ∗ Theoretisch-Physikalisches Institut, Universität Münster, D-48149 Münster, GermanyE-mail:
Supersymmetry (SUSY) and supersymmetric field theories are an interesting topic for numericallattice simulations. Similar to the chiral symmetry there is also no local realization of (interact-ing) supersymmetry on the lattice. I briefly review the basic reasons for the breaking of super-symmetry. One attempt to solve the problem uses a Ginsparg-Wilson relation for supersymmetry.However, apart from the free theory a solution of this relation has so far not been found. For su-persymmetric Yang-Mills (SYM) theory a fine-tuning of the bare gluino mass is enough to arriveat a supersymmetric continuum limit. The last part of this work contains a short status report ofrecent SYM simulations.
The XXVIII International Symposium on Lattice Field Theory, Lattice2010June 14-19, 2010Villasimius, Italy ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ attice SUSY and the SYM simulations
Georg Bergner
1. Introduction
Supersymmetric lattice simulations have been the subject of several recent investigations. Thereason for this can be seen in the growing interest for possible extensions of the standard model.A complete understanding of supersymmetric theories, as they are used in these extensions, needsnon-perturbative methods. Supersymmetry transforms fermionic and bosonic particles into eachother. As a consequence it predicts a pairing of bosonic and fermionic states and has nontrivialcommutation relations with the Poincare-symmetry of space-time.The simplest examples for supersymmetric field theories are Wess-Zumino modells. Theycan be seen as the matter sector of the supersymmetric extensions of the standard model. Thecontinuum action of such a model has the following form S = Z d x (cid:20) ¶ m f¶ m f + | W ′ ( f ) | + y ( / ¶ + W ′′ ( f ) P + + W ′′ ( f ) P − ) y (cid:21) . (1.1)It consists of bosonic and fermionic kinetic terms and a Yukawa-type interaction term. The su-persymmetry transforms the fields y and f into each other. Due the nontrivial interplay with thePoincaré symmetry the transformations contain derivatives of the fields. The variation of the actionin the continuum is d S = − e Z dt (cid:2) W ′ ( j )( ¶ t y ) + y W ′′ ( j ) ¶ t j (cid:3) (1) = − e Z dt ¶ t (cid:2) y W ′ ( j ) (cid:3) (2) = . (1.2)On the lattice the equality (2) can be easily satisfied using periodic or open boundary conditions.The violation of the first equality (1) is, however, unavoidable for a local lattice theory. It is due tothe breaking of the Leibniz rule on the lattice. This fact can be stated in terms of a No-Go theoremfor local lattice supersymmetry. A version of such a No-Go theorem was presented in [1]. Asshown in [2], even stronger conditions for a local lattice action must be fullfilled. The findings in[2], that include also the results of a simulation with intact supersymmetry on the lattice, will besummarized in the next section. A No-Go theorem for a local realization of a supersymmetry seemsto be similar to the Nielson-Ninomiya theorem [3] for chiral symmetry. In this case the solutionwas found in terms of a modified symmetry relation on the lattice. The Section 3 is therefore abriefly comment on the current investigations of this approach. The last Section 4, consists of areport of the simulations of a supersymmetric gauge theory. In this case the unavoidable breakingof supersymmetry can be controlled via fine tuning.
2. No-Go theorem for local lattice supersymmetry
A simple form of a No-Go theorem for the Leibniz rule, and therefore for the invariance of thelattice action can be found in terms of the following statement.
Lemma 1 (Simple form of the No-Go theorem)
For all lattice derivative operators (cid:209) nm the Leibniz-rule is violated, (cid:229) m (cid:209) nm ( f m g m ) − f n (cid:229) m (cid:209) nm g m − g n (cid:229) m (cid:209) nm f m = . (2.1)Note that this is true also for a nonlocal lattice derivative. It is therefore a stronger restriction thenin [1]. In this case the on-shell version of the N = attice SUSY and the SYM simulations Georg Bergner
Already in the early days of lattice supersymmetry a generalization of the discretization wasproposed in [4], that circumvents this simple form of the No-Go theorem using a modification ofthe product rule on the lattice. It leads, e. g., to the following relplacement for a product of threecontinuum fields on the lattice Z dx f ( x ) → (cid:229) i , j , k C i jk f i f j f k . (2.2)However, the lattice action in [4] includes a nonlocal C i jk . The nonlocality of C i jk is indeed un-avoidable to circumvent the simple form of the No-Go theorem. Furthermore, C i jk projects oneof the interacting fields to its zero momentum part and can hence be considered only as a trivialsolution. To exclude such kind of trivial solutions one has to require an additional condition fora possible lattice action. The interaction should not be restricted to a only a discrete number ofmodes.A realization that fullfills this condition has been proposed in [5]. In contrast to the previouslymentioned approach it involves not only a nonlocal interaction term but also a nonlocal SLAC typederivative. This seems to be a severe violation of locality, but for a supersymmetric lattice theoryone can find no better solution. This can be stated in terms of the following No-Go statement.
Lemma 2 (No-Go theorem)
In order to get a nontrivial interacting supersymmetric lattice theoryone needs a nonlocal derivative operator and a nonlocal interaction term.
Besides the violation of the Leibniz rule a second source of the supersymmetry breaking on thelattice must be mentioned. Due to the doubling problem in the fermionic sector a Wilson massterm must be added to the fermion action. If not consistently added also to the bosonic potentialthis leads to a supersymmetry breaking. Note that the doubling problem can also be avoided witha nonlocal lattice action.According to the No-Go theorem it is hence possible to realize a fully supersymmetric theoryon the lattice, if one accepts a nonlocality of the action. A local continuum limit of such a theorycan be proven to all orders of perturbation theory in Wess-Zumino models up to three dimensions[7]. To perform numerical simulations one has to seek for an efficient realization of the nonlocalinteraction term on the lattice. This can be done by performing the complete simulation in Fourierspace. Equivalently the nonlocal interaction term can be realized using Fourier transformations anda local product on a larger lattice (cf. [2] for details). Figure 1 shows the result for the supersym-metric Ward-Identities, that verify the intact supersymmetry on the lattice.
3. Solutions similar to the Ginsparg-Wilson relation
Considering the presented No-Go theorem, the problem of realizing supersymmetry on thelattice seems to be much similar to the one for chiral symmetry on the lattice. The correspondingNo-Go theorem for local realizations of chiral symmetry is the Nielsen-Ninomiya theorem. In caseof this symmetry a solution for a lattice realization was found in terms of the Ginsparg-Wilson re-lation, a modified symmetry relation on the lattice [9]. The basic idea behind the Ginsparg-Wilsonrelation is a blocking transformation (corresponding to a renormalization group step) from the con-tinuum to the lattice. In this way not only the action is mapped onto a (perfect) lattice action. For a definition of this derivative see [6]. attice SUSY and the SYM simulations Georg Bergner -0.006-0.004-0.00200.0020.0040.006 0 2 4 6 8 10 12 14 R ( ) n , R ( ) n lattice point n The Ward-idenities of the full supersymmetric model ( m = 10 , g = 800 , N = 15 )SLAC unimproved Ward-Identity 1SLAC improved Ward-Identity 1Ward-Identity 1Ward-Identity 2 Figure 1:
The Ward-identities measured in supersymmetric quantum mechanics using a model with com-pletely realized supersymmetry on the lattice. A nonlocal SLAC derivative and a nonlocal interaction termwere used in the action. The figure shows for comparison also the Ward-identities of the unimproved andimproved SLAC model introduced in [8]. These models contain only the nonlocal SLAC derivative but nononlocal interaction term.
Also the symmetry transformations get a modification due to the symmetry breaking induced bythe regulator. In case of chiral symmetry the relevant part of the action is only quadratic and asolution for the modified symmetry transformation can be found easily. In contrast, the interestingcase of supersymmetry includes non-quadratic interaction terms. A modified symmetry relationcan, nevertheless, be found, as presented in [10]. It is, however, hard to find an action, that isinvariant under it. The form of the modified symmetry leads generically to non-polynomial ac-tions and involves possible nonlocal terms. This might not be unexpected since a renormalizationgroup transformation generically generates operators of higher order in the fields and interactionsbetween distant lattice sites. In case of the Ginsparg-Wilson relation a solution for a lattice actioninvariant under the modified chiral symmetry was found, e. g., in terms of the overlap operator. Forsupersymmetry this problem remains unresolved.A different way to ensure chiral symmetry on the lattice uses fine-tuning of the parametersin the action. The number of parameters that must be fine tuned depends on the mixing of thesupersymmetry breaking terms with operators of equal or lower dimension. Based on similar argu-ments one can find that no fine tuning is needed for the Poincaré symmetry and only a fine tuningof the mass parameter for the chiral symmetry [11]. A fine tuning of all necessary parameters fora Wess-Zumino model (in more than one dimension) seems currently impossible. However, fora supersymmetric Yang-Mills theory it was found that only one parameter is needed for the finetuning [12].
4. Supersymmetric Yang-Mills theory
The field content of the supersymmetric ( N =
1) Yang-Mills theory contains, besides the4 attice SUSY and the SYM simulations
Georg Bergner usual bosonic gauge fields (field strength F mn ), the Majorana fermion l in the adjoint representa-tion. Adding a gluino mass term the Lagrangian of the continuum theory has the following form L = Tr (cid:20) − F mn F mn + i l / D l − m g ll (cid:21) . (4.1)Note that supersymmetry is only established at m g =
0. Low energy effective actions of the theoryhave been constructed in [14, 15].The lattice action proposed by Veneziano-Curci in seems, in comparison to the previouslymentioned attempts, to be a rather “brute force” approach. It constits of the usual gauge action(gauge group SU( N C )) and the common fermionic action with a Wilson term, S L = b (cid:229) P (cid:18) − N c ´ U P (cid:19) + (cid:229) xy l x ( D w ( m g )) xy l y . (4.2)It breaks the chiral symmetry and the supersymmetry of the model. However, a detailed analysisshows that both symmetries can be recovered with a fine-tuning of the bare gluino mass m g . Thechiral limit of the theory corresponds to the supersymmetric limit.We applied this approach to perform numerical simulations using a PHMC algorithm. For thegauge action (gauge group SU(2)) an additional tree-level Symanzik improvement was used andstout smeared links for the fermion action. We have considered lattice sizes of 16 ×
32, 24 × ×
64 lattice points. If one sets the Sommer scale of the theory to the usual QCD value( r = . .
09 fm corresponding to a lattice volume of L ≈ . − . It was confirmed thatthe supersymmetric Ward identities would lead to the same extrapolated point. Although we haveused anti-periodic boundary conditions for the fermions in time direction, the finite volume effects,including the supersymmetry breaking by the boundary conditions, seem to be under control.The proposed low energy effective action contains operators for adjoint mesons (gluino-balls),glueballs, and, since the fermions are in an adjoint representation, compound operators formedfrom gluon and gluino fields (gluino-glue-balls). It is a nontrivial task to measure the correspond-ing correlations on the lattice. All mesonic states contain disconnected contributions. We havemeasured them with a stochastic estimator technique including, where necessary, a separate deter-mination of the contributions form the lowest eigenvalues. For the glueballs we applied variationalsmearing methods, and the gluino-glue-ball correlation was measured with a combination of Jacobiand APE smearing.The presence of Majorana fermions in the theory leads to additional difficulties. Instead ofthe determinant the fermionic path integral corresponds to a Pfaffian of D w . Up to a sign thePfaffian is the square root of the determinant. Hence one gets a sign problem in the theory, althoughthe determinant stays always positive. The signs are represented in our approach by positive andnegative reweighting factors.The masses of the particles obtained from the correlation functions are shown in figure 2. The U P denotes the usual plaquette and D w the Dirac matrix including the gauge field, the Wilson term, and the massterm of m g . The mass of the connected part of the lg l correlation vanishes at that point. It corresponds to the adjoint versionof a pion mass ( m a − p ) in a partially quenched framework The renormalized masses obtained form the chiral and the supersymmetric Ward identities vanish at the samepoint. Up to O ( a ) effects the breaking of the corresponding symmetry is determined by these masses. attice SUSY and the SYM simulations Georg Bergner (r m p ) r M L = 1.54 fmL = 2.3 fmL = 2.11 fm (stout) a- h ’Gluino-glueGlueball 0 ++ a-f Spectrum of N=1 SU(2) Super-Yang-Mills theory on the lattice
Figure 2:
The masses of the particles obtained in our simulation of supersymmetric (SU(2)) gauge theoryas a function of the adjoint pion mass. The blue symbols correspond to the values extrapolated to the chirallimit. All masses are given in units of the Sommer scale r . gluino-glue-ball operator Tr c [ F sl ] should have an overlap with the lightest fermionic state ofthe theory. If supersymmetry is not broken, it should be paired with bosonic particles of the samemass. We have obtained masses for the gluino-ball a - f (operator ll ) and a - h ′ (operator lg l ) aswell as the 0 ++ glueball. However, all of these particles have a much smaller mass than the gluino,when extrapolated to the chiral limit. This is in contradiction with the theoretical predictions sinceit indicates a breaking of supersymmetry.In our recent studies we have increased the value of b to approach a smaller lattice spacing(around 0 .
06 fm). The first results of these simulations indicate that the gap between the masses ofthe fermionic and bosonic masses is reduced towards the continuum limit. However, it is still tooearly to draw a conclusion from these preliminary data.
5. Conclusions
Supersymmetry on the lattice remains an interesting subject of theoretical investigations. It isby now well understood that this symmetry can be realized only in a nonlocal theory on the lattice.It has been shown in [16] that a nonlocal lattice gauge theory leads to a nonlocal continuum limit.However, in case of the Wess-Zumino modells a full supersymmetric theory yields the correct localcontinuum limit at least in lower dimensions.In a local lattice theory it is necessary to control the breaking of supersymmetry. The mostelegant way to control the breaking would be a Ginsparg-Wilson relation for supersymmetry. Al-though it is possible to establish such an relation, it is hard to find a solution in the interacting case.Perhaps an approximation can lead to a useful lattice realization [17]. F mn ( x ) is replaced by the clover plaquette on the lattice, and s mn is the commutator of two gamma matrices. attice SUSY and the SYM simulations Georg Bergner
For the supersymmetric counterpart of a pure gauge theory the supersymmetry breaking canbe controlled with the same fine tuning as needed for the chiral symmetry. Only a single parameteris used to arrive at a supersymmetric continuum limit. This theoretical prediction is verified bythe supersymmetric Ward-Identities. The investigation of the mass spectrum of the theory stilldemands further efforts. The first results at one lattice spacing show a gap between the bosonicand fermionic masses. This gap may, however, be due to the lattice artifacts and vanish in thecontinuum limit.
Acknowledgments
The results of the simulations of supersymmetric Yang-Mills theory have been obtained incollaboration with K. Demmouche, F. Farcioni, G. Münster, I. Montvay and J. Wuilloud. I thankG. Münster for corrections and useful remarks.
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