Species Doublers as Super Multiplets in Lattice Supersymmetry: Exact Supersymmetry with Interactions for D=1 N=2
Alessandro D'Adda, Alessandra Feo, Issaku Kanamori, Noboru Kawamoto, Jun Saito
aa r X i v : . [ h e p - l a t ] J un DFTT 05/2010EPHOU 10-002June, 2010
Species Doublers as Super Multiplets in Lattice Supersymmetry:Exact Supersymmetry with Interactions for D = 1 N = 2 Alessandro D’Adda a ∗ , Alessandra Feo a † , Issaku Kanamori a ‡ , Noboru Kawamoto b § and Jun Saito b ¶ . a INFN Sezione di Torino, andDipartimento di Fisica Teorica, Universita di TorinoI-10125 Torino, Italy b Department of Physics, Hokkaido UniversitySapporo, 060-0810 Japan
Abstract
We propose a new lattice superfield formalism in momentum representation which ac-commodates species doublers of the lattice fermions and their bosonic counterparts as supermultiplets. We explicitly show that one dimensional N = 2 model with interactions hasexact Lie algebraic supersymmetry on the lattice for all super charges.In coordinate representation the finite difference operator is made to satisfy Leibnitz ruleby introducing a non local product, the “star” product, and the exact lattice supersymmetryis realized. The standard momentum conservation is replaced on the lattice by the conser-vation of the sine of the momentum, which plays a crucial role in the formulation. Halflattice spacing structure is essential for the one dimensional model and the lattice supersym-metry transformation can be identified as a half lattice spacing translation combined withalternating sign structure. Invariance under finite translations and locality in the continuumlimit are explicitly investigated and shown to be recovered. Supersymmetric Ward identitiesare shown to be satisfied at one loop level. Lie algebraic lattice supersymmetry algebra ofthis model suggests a close connection with Hopf algebraic exactness of the link approachformulation of lattice supersymmetry. PACS codes: 11.15.Ha, 11.30.Pb, 11.10.Kk.Keywords: lattice supersymmetry, lattice field theory. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] Introduction
If we regularize massless fermions naively on a lattice, it is unavoidable that species doublersappear. Since massless particles cannot be put in the rest frame by means of a Lorentz transfor-mation, helicity or chirality cannot be changed with a momentum change, while species doublersin different momentum region may have different helicity. Therefore species doublers have to beconsidered as different particles [1]. However, species doublers of chiral fermions on a lattice areusually considered as unwanted particles, the so called doubling problem, although the enlargeddegree of freedom (d.o.f.) is customarily identified as a flavor (taste) d.o.f..The equivalence of the above naive fermion formulation and the staggered fermion formu-lation can be shown by a spin diagonalization procedure [2] and the staggered fermion can betransformed into the Kogut-Susskind type fermion formulation [3] by considering double sizelattice structure, where the flavor d.o.f. was identified [4]. This double size structure makesit possible to have a correspondence with differential forms and then the equivalence of thestaggered fermion and Dirac-K¨ahler fermion on the lattice can be proved by introducing anoncommutativity between differential forms and fields [5]. Therefore all these lattice fermionformulations are exactly equivalent.In the link approach of lattice supersymmetry [6], the super charges are expanded on the basisof Dirac matrices by the Dirac-K¨ahler twisting procedure [7]. The corresponding d.o.f. of thesuper charges are then exactly the same as those of fermionic species doublers and the geometriccorrespondence between the particles as species doublers and super multiplets is expected fromthe equivalence of the naive fermion and Dirac-K¨ahler fermion formulations on the lattice. Inother words the species doublers are necessary fields to construct the super multiplets of extendedsupersymmetry: N = 2 in two dimensions and N=4 in four dimensions. The flavor (taste)d.o.f. of chiral fermions are thus expected to be identified as extended supersymmetry d.o.f..In this paper we explicitly show how the species doublers for both fermions and bosons can beidentified as super multiplets of extended supersymmetry for the simplest model of N = 2 in onedimension. We propose to introduce lattice counter parts of bosonic and fermionic “superfields”where species doublers are accommodated.The one dimensional N = 2 model was proposed as a supersymmetric quantum mechanicsby Witten [8] and the lattice version has been investigated by several authors [9, 10] as thesimplest model to clarify the fundamental problems of lattice supersymmetry. It was shownthat a species doubler of the Wilson fermion term, having a mass proportional to the inverselattice constant, breaks supersymmetry and a bosonic counter term is needed to remove theunwanted contribution [10]. Numerical evaluation of boson and fermion masses show that theyapproach the same value in the continuum limit, suggesting the recovery of supersymmetry, onlywhen the counter term is introduced [9, 11]. In this model the species doubler interferes withsupersymmetry and its influence has to be removed by the bosonic counter term. It has alsobeen recognized that only one exact supersymmetry of the type Q = 0, which can be identifiedas the scalar part of a twisted supersymmetry for this supersymmetric quantum mechanics, isrealized when interaction terms are included [11]. This system also provides a nice arena for anumerical method for detecting spontaneous supersymmetry breaking [12].One dimensional N = 2 lattice supersymmetric model of this paper is constructed in parallelto the continuum superspace formulation [13, 14, 15] and is slightly different from the super-symmetric quantum mechanics model. The lattice model we propose in this paper is exactlysupersymmetric for two supersymmetry charges even with interaction terms and the counterterm is not necessary to fulfill Ward-Takahshi identity since the bosonic and fermionic species2oublers are identified as physical particles in supermultiplets. In the momentum representationthis model has, however, a lattice counter part of trigonometric momentum conservation, whichwas first proposed by Dondi and Nicolai [16] in the very first paper of lattice supersymmetry.In the coordinate space we introduce a new star product which makes the lattice differenceoperator satisfy Leibniz rule and then the exact lattice supersymmetry is realized. The modelhas mildly nonlocal interactions which approach local interactions in the continuum limit.There is a long history of attempts to realize exact sypersymmetry on a lattice. See [11, 17]for some references. However exact lattice supersymmetry with interactions for full extendedsupersymmetry has never been realized except for the nilpotent super charge. This difficultyis essentially related to the lattice chiral fermion problem and to the breakdown of the Leibnizrule for the lattice difference operator. Instead of formulating an exact supersymmetry for localinteractions in the coordinate space, there has been several attempts that approach the problemfrom momentum representation point of view [18, 19, 20]. This may be related to the followingclaim: if one tries to include the difference operator in a supersymmetry algebra, one cannotavoid introducing nonlocal interactions [21]. The momentum representation can take care thenonlocal nature of the formulation. In this paper we first establish a formulation of exact latticesupersymmetry in the momentum representation. Then we reformulate the momentum spaceversion into the coordinate space by introducing a new non local “star” product.In the link approach for lattice supersymmetry [6] the claim that exact lattice supersymmetryhad been realized for all super charges was questioned by several authors [14, 22]. In factit was stressed that all the extended supersymmetry are broken when the shift parameter ofthe scalar super charge is non zero a = 0 [23] while it exactly coincides with the orbifoldconstruction of lattice supersymmetry when a = 0 [24, 25, 26]. It was however shown lateron that lattice supersymmetry can be formulated consistently within the framework of a Hopfalgebra, accounting for the breaking of the Leibniz rule for the difference operator, and of themild noncommutativity between fields carrying a shift. Thus exact lattice supersymmetry holdswithin the Hopf algebraic symmetry [27].One of the important aims of the present paper is to clarify the fundamental nature ofthe link approach within a simple one-dimensional model. Since we find a new exact latticesupersymmetry formulation, it would be interesting to compare the algebraic structure of thelink approach with this new formulation. We point out the interesting possibility, supported byseveral arguments, that the star product formulation of current model and the link approachformulation are equivalent.This paper is organized as follows: We explain the basic ideas of the formulation in section 2.We then briefly explain the continuum version of the model which we investigate in this paperin section 3. Then it will be explained in section 4 how the species doublers can be naturallyaccommodated into supersymmetry transformations together with trigonometric momentumconservation. In section 5 an exact supersymmetry invariant action with interaction terms inmomentum representation on the infinite lattice will be proposed. In section 6 the recoveryof the translational invariance of this model in the continuum limit is investigated. It will beconfirmed that supersymmetric Ward-Takahashi identities are satisfied. In section 7 we proposea new star product which makes lattice difference operator satisfy Leibniz rule and exact latticesupersymmetry be realized in the coordinate space. A close connection with the link approachwill be discussed. We then summarize the result of this paper and discuss remaining problems inthe final section. In the appendix 1-loop radiative corrections of propagators in Ward-Takahashiidentity are summarized. 3 Basic ideas
One of the most distinctive features of the so called link approach to lattice supersymmetry isthe introduction in place of the usual hyper cubic lattice of extended lattices that account forthe underlying supersymmetry algebra. The idea is the following: on the lattice infinitesimaltranslations are replaced by finite displacements, or shifts, represented typically for an hypercubic lattice by orthogonal vectors ~n µ of length equal to the lattice spacing a . The vectors ~n µ generate the whole lattice and each point of the lattice can be reached from a given pointby means of a finite number of such displacements. It follows that a translationally invariantfield configuration, or vacuum, is a constant field configuration on the lattice. In the linkapproach a shift ~a A is associated also to each supersymmetry charge Q A , in the same way as ~n µ is associated to the generator P µ of translations . The shifts ~a A are not arbitrary, but theyare constrained by the supersymmetry algebra: consistency with the algebra requires that theconstraint ~a A + ~a B = ± ~n µ must be satisfied for each non-vanishing anticommutator { Q A , Q B } = P µ of the superalgebra. Only a limited number of supersymmetry algebras, in particular the N = 2 SUSY algebra in 2 dimensions and the N = 4 SUSY algebra in 4 dimensions, arecompatible with these constraints. The extended lattices introduced in the link approach aregenerated by the displacements ~a A and ~n µ and hence contain, with respect to the standardhyper cubic lattices, new links of the type ( ~x, ~x + ~a A ) which we will call “fermionic” and newpoints. The key remark now is that not all points of the extended lattices can be reached froma given one simply by translations. An extended lattice will consist in general of more copiesof the hyper cubic lattice connected by “fermionic” links, and translations will only make usmove within each copy. So for a field configuration to be invariant under translations it is notnecessary to be constant over the whole extended lattice but only separately over each copy of thehyper cubic lattice. In other words the number of field configurations which are invariant undertranslations in an extended lattice is equal to the number of hyper cubic sublattices contained init. Consider as an example the N = 2 superalgebra in 2 dimensions. This superalgebra containsfour supersymmetry charges Q A besides the generators of translations in two dimensions. In itsdiscrete lattice version, described in detail in ref [28], four shifts ~a A are associated to the foursupesymmetry charges, constrained by the non-vanishing anticommutators of the superalgebra,as discussed above. The constraints do not determine ~a A completely: one of them is arbitrary.However if we further require the resulting extended lattice to be invariant under π rotations,then the solution is unique and given by ~a A = ( ± a , ± a , (2.1)where a denotes the lattice spacing. The vectors (2.1), together with the shifts associated totranslations, namely ~n = ( a,
0) and ~n = (0 , a ), generate the extended lattice of the N = 2, D = 2 SUSY algebra. This is shown in fig. (1). The points ~x of the lattice have coordinates: ~x = ( na , ma , (2.2)where n + m is even ( n and m both even or both odd).The extended lattice is then made by two copies of the cubic lattice, in fact by the originallattice ( n and m even) and its dual ( n and m odd) connected by the “fermionic” links ( ~x, ~x + ~a A ).It is clear that there are two independent field configurations on the extended lattice that areinvariant under translation, namely:Φ ( ~x ) = c , Φ ( ~x ) = ( − n c , (2.3)4 Figure 1: Φ ( ~x ) takes the same value c in both • and × points, while Φ ( ~x ) takes the value + c in the • points and − c in the × points. Adjoining • and × points are joined by the fermioniclinks which are not shown in the figure.with c and c constant.It is interesting to note that this phase remind us of the phase of staggered fermion [2] Thisis relevant because we expect the degrees of freedom of the theory in the continuum limit tobe associated to small fluctuations around translationally invariant vacua, so that fluctuationsaround the two configurations of eq. (2.3) will correspond to two distinct degrees of freedomin the continuum limit. Hence one degree of freedom on the extended lattice will absorb twodegrees of freedom of the continuum theory. This fact, namely that the extended lattice impliesa correspondingly reduced number of independent fields, was not fully appreciated in the originalformulation of the link approach and is one of the key points of the present paper. It is clearthat since the different copies of the hyper cubic lattice in an extended lattice are generatedby the extra links associated to supersymmetry charges it is natural to expect the differentcontinuum degrees of freedom associated to a single lattice degree of freedom to be part of asupersymmetric multiplet. It is instructive to look at the previous example from the point ofview of the momentum space representation, which will play a crucial role in what follows. Thefirst Brillouin zone associated to the lattice (2.2) is the square defined by − πa ≤ p + p ≤ πa and − πa ≤ p − p ≤ πa . That means that in the momentum space fields will be periodic withperiod πa in the variables p + p and p − p . The translationally invariant field configurations(2.7) correspond respectively to the center of the square, namely ~p = 0, and to the four vertices(which are all equivalent due to the periodicity), namely ~p = ( ± πa ,
0) or ~p = (0 , ± πa ). Thelatter is exaclty the species doubler. A solution of the fermion doubling problem becomes nowpossible. Fermion doubling originates from the fact that the fermionic kinetic term has a simplezero at ~p = 0 and hence has to vanish somewhere else in the Brillouin zone due to the latticeperiodicity. Within the extended lattice scheme if the second zero occurs in correspondence of5he other translationally invariant vacuum the would be doubler can be interpreted as a physicalfield, in fact as a supersymmetric partner of the original one at ~p = 0.We have used the example of the extended lattice of the N = 2, D = 2 supersymmetry toillustrate the ideas that we are going to develop in this paper. However the explicit examplethat we are going to study is a simpler one, N = 2 supersymmetric model in one dimension( D = 1). Before going into that we are going to consider an even simpler example, that hasbeen considered in the present context in ref. [29]. This is a one dimensional model with an N = 1 supersymmetry. It is described in terms of a superfield:Φ( x, θ ) = ϕ ( x ) + iθψ ( x ) , (2.4)with a supersymmetry charge given by: Q = ∂∂θ + iθ ∂∂x , (2.5)and Q = i ∂∂x . (2.6)On the lattice derivatives are replaced by finite shifts of length a , hence consistency with thealgebra (2.6) requires that the supercharge Q is associated to a shift a . The extended lattice isthen a one dimensional lattice with spacing a , which can be thought of as the superposition of twolattices with spacing a each invariant under translations and separated by a an a shift associatedto the SUSY charge. Again there are two field configurations invariant under translations,namely: Φ ( x ) = c , Φ ( x ) = ( − xa c (2.7)with x = na . According to the previous discussion fluctuations around Φ ( x ) and Φ ( x ) willdescribe in the continuum limit two distinct degrees of freedom. However we have just twodegrees of freedom in this model, one bosonic and one fermionic, so the natural thing is toassociate the bosonic degree of freedom to fluctuations around Φ ( x ) and the fermionic one tofluctuations around Φ ( x ).To understand the origin of the hyper lattice structure of half lattice step and the alternatingsign states, let us look at the algebraic relation of the superfield and the supercharge from amatrix point of view [15]. We now introduce the following matrix form of the super coordinateand its derivative as: θ = (cid:18) (cid:19) , ∂∂θ = (cid:18) (cid:19) , (2.8)which satisfy the following anticommutation relation: { ∂∂θ , θ } = 1 . (2.9)We may consider this matrix structure as an internal structure of the space time coordinate.With respect to this internal structure the boson ϕ is considered as a field which commutes with θ and ∂∂θ and the fermion ψ as a field which anticommutes with them. The component fields ofboson and fermion with respect to this internal structure has then the following form: ϕ ( x ) = (cid:18) ϕ ( x ) 00 ϕ ( x ) (cid:19) , (2.10)6 ( x ) = (cid:18) ψ ( x ) 00 − ψ ( x ) (cid:19) . (2.11)In the matrix formulation of fields the coordinate dependence can be introduced by diagonalentries of a big matrix as direct product to the internal matrix structure [15]. From the degreesof freedom point of view for N=1 model in one dimension two fields of boson and fermion on thesame lattice site have this internal matrix structure (2.10) and (2.11). If we consider the N = 2model in one dimension, which we will consider later in this paper, there are four independentfields on one site we may then consider that the four fields of internal matrix structure may beidentified with the four independent d.o.f. of fields.We now ask a question: “How do we interpret this internal space time structure on thelattice ?” A natural identification is an introduction of a half lattice step structure to make acorrespondence with two translational invariant states in (2.7). We can then identify a constantfield of ϕ ( x ) in (2.10) as Φ ( x ) in (2.7) and a constant field of ψ ( x ) in (2.11) as Φ ( x ) in (2.7).One can then write a lattice “superfield” corresponding to (2.4) asΦ( x ) = ϕ ( x ) + 12 ( − xa ψ ( x ) , (2.12)where we have introduced a factor for later convenience and taken away the factor i sincethe second term is not a product of two Grassmann numbers to keep hermiticity. Since θ and ∂∂θ are not Hermitian by them self in this matrix representation we have to take care thehermiticity separately. It is crucial to recognize at this stage that the super coordinate structureand fermionic nature of ψ can be accommodated by the alternating sign factor of half latticespacing if this simple lattice representation works as a superfield. We now introduce a matrixform of a fermionic super parameter by α = (cid:18) α − α (cid:19) , (2.13)where α is a Grassmann odd parameter. This parameter can then be expressed as α ( − xa inaccordance with the representation of the lattice superfield in (2.12).We now propose lattice supersymmetry transformations as a finite difference over a halflattice spacing a : δ Φ( x ) = a − α ( − xa (cid:16) Φ( x + a − Φ( x ) (cid:17) . (2.14)In terms of the component fields the supersymmetry transformations (2.14) are: δϕ ( x ) = − α (cid:20) ψ ( x + a ) + ψ ( x ) (cid:21) −−−→ a → − αψ ( x ) , (2.15) δψ ( x ) = 2 a − α (cid:20) ϕ ( x + a ) − ϕ ( x ) (cid:21) −−−→ a → α ∂ϕ ( x ) ∂x . (2.16)It is surprising that the half lattice translation together with alternating sign structure (staggeredphase) for the lattice superfields generates a correct lattice supersymmetry transformation. Weconsider that this observation is a key of our formulation.Although supersymmetry transformations (2.15) and (2.16) have the correct structure, theyviolate hermiticity: a factor i is missing at the l.h.s. of (2.15). In order to restore the hermiticityof the supersymmetry transformations symmetric finite differences must be used, introducing a7hift of a of the fermionic fields sites with respect to the bosonic ones. Hence, instead of writingthe superfield on the lattice as in (2.12) we shall introduce Φ( x ), with x = n a , defined by:Φ( x ) = ( ϕ ( x ) for x = na/ , a / e iπxa ψ ( x ) for x = (2 n + 1) a/ . (2.17)Again the supersymmetry transformations can be written in terms of Φ( x ): δ Φ( x ) = αa − / e iπxa [Φ( x + a/ − Φ( x − a/ . (2.18)By separating Φ( x ) into its component fields according to (2.17) we find: δϕ ( x ) = iα (cid:20) ψ ( x + a ψ ( x − a (cid:21) −−−→ a → iαψ ( x ) , (2.19) δψ ( x ) = 2 a − α (cid:20) ϕ ( x + a − ϕ ( x − a (cid:21) −−−→ a → α ∂ϕ ( x ) ∂x , (2.20)where x is an even multiple of a/ a . For instance we have for ϕ ( x ) (the same applies to ψ ( x )): δ β δ α ϕ ( x ) − δ α δ β ϕ ( x ) = 2 iαβa [ ϕ ( x + a/ − ϕ ( x − a/ . (2.21)It is instructive to look at the supersymmetry transformations given above from the point ofview of of the momentum space representation. Let us consider first the Fourier transform ofthe component fields ψ ( x ) and ϕ ( x ), and denote them by ˜ ψ ( p ) and ˜ ϕ ( p ) respectively. The latticespacing being a/
2, the Brillouin zone extends over a πa interval with the two vacua Φ ( x ) andΦ ( x ) corresponding respectively to p = 0 and p = πa . The periodicity conditions are:˜ ϕ ( p + 4 πa ) = ˜ ϕ ( p ) , ˜ ψ ( p + 4 πa ) = − ˜ ψ ( p ) , (2.22)where the minus sign in the case of ˜ ψ is due to the a/ δ ˜ ϕ ( p ) = i cos ap α ˜ ψ ( p ) , (2.23) δ ˜ ψ ( p ) = − i a sin ap α ˜ ϕ ( p ) . (2.24)Eqs. (2.23) and (2.24) are consistent with both the periodicity conditions (2.22) and with thereality conditions expressed in momentum space by: ˜ ϕ ( p ) † = ˜ ϕ ( − p ) and ˜ ψ ( p ) † = ˜ ψ ( − p ). A moreextensive analysis of the D = 1, N = 1 model, including the lattice action, can be found in [29].The point we want to emphasize here is the following: in order to derive the supersymmetrytransformations (2.19) and (2.20), or equivalently their momentum space representation (2.23)and (2.24), we started from a bosonic field ϕ ( x ) and a fermionic one ψ ( x ) interpreted respectivelyas fluctuations around Φ ( x ) (i.e. p = 0) and Φ ( x ) (i.e p = πa ). Either these two fields representon the lattice a single degree of freedom whose statistic changes from bosonic to fermionic as themomentum moves from zero to πa ( and we don’t know how to implement that consistently) oreach field has a doubler with the same statistic in correspondence of the other vacuum. In this8ase the system will contain two bosonic and two fermionic degrees of freedom in the continuum,for which there is no room in the D = 1, N = 1 supersymmetry. Furthermore the action ofthis model is fermionic and thus the vacuum is not well defined. We will show in the followingsection that it actually provides a consistent formulation of the D = 1, N = 2 supersymmetry ,whose algebra contains a bosonic field ϕ and a bosonic auxiliary field D , described by the latticebosonic field at p = 0 and p = πa respectively and, in the fermionic sector, two fields ψ and ψ described on the lattice by the fluctuations of a single field around the two vacua. We will showin the following sections that the N = 2 supersymmetry can be explicitly formulated on thelattice (one of the transformations is essentially already given in (2.23) and (2.24) ), an invariantaction can be constructed (including the mass and interaction term) and the continuum limittaken keeping exact supersymmetry at all stages. The doubling problem does not arise sincethe would be doublers are physical degrees of freedom in the same supermultiplet as the originalfield. We briefly summarize the continuum version of one dimensional N = 2 supersymmetric modelwith two supersymmetry charges Q and Q [13], whose matrix version on the lattice wasdiscussed in [15]. Its supersymmetry algebra is given by: Q = Q = P t , { Q , Q } = 0 , (3.1)[ P t , Q ] =[ P t , Q ] = 0 , (3.2)where P t is the generator of translations in the one-dimensional space-time coordinate t : P t = − i ∂∂t . (3.3)A superspace representation of the algebra may be given in terms of two Grassmann odd, realcoordinates θ and θ , namely: Q = ∂∂θ − iθ ∂∂t , Q = ∂∂θ − iθ ∂∂t . (3.4)The field content of the theory is described by a hermitian superfield Φ( t, θ , θ ):Φ( t, θ , θ ) = ϕ ( t ) + iθ ψ ( t ) + iθ ψ ( t ) + iθ θ D ( t ) , (3.5)where ψ and ψ are Majorana fermions. The supersymmetry transformations of the superfieldΦ are given by: δ j Φ = [ η j Q j , Φ] j = 1 , , (3.6)where η i are the Grassmann odd parameters of the transformation. In terms of the componentfields eq. (3.6) reads: δ j ϕ = iη j ψ j , (3.7) δ j ψ k = δ j,k η j ∂ t ϕ + ǫ jk η j D, (3.8) δ j D = iǫ jk η j ∂ t ψ k . (3.9) Unlike reference [15] we use here a Lorentzian metric. The euclidean formulation of [15] can be obtainedwith a Wick rotation t → − ix .
9t is important to note that since the supersymmetry transformations are defined in (3.6) ascommutators, supersymmetry transformations of superfields products obey Leibniz rule: δ i (Φ Φ ) = ( δ i Φ )Φ + Φ ( δ i Φ ) . (3.10)In order to write a supersymmetric action we need to introduce the super derivatives, definedas D j = ∂∂θ j + iθ j ∂∂t , (3.11)which anticommute with the supersymmetry charges Q j and satisfy the algebra: D j = i ∂∂t , { D , D } = 0 . (3.12)The supersymmetric action can then be defined in terms of the superfield Φ as: Z dtdθ dθ (cid:20) D Φ D Φ + iV (Φ) (cid:21) , (3.13)where V (Φ) is a superpotential which may includes any powers of superfields together withcoupling constants. By integrating over θ and θ in (3.13) one can obtain the action written interms of the component fields. If we take the super potential in the following form: V (Φ) = 12 m Φ + 14 g Φ , (3.14)we obtain the following action: S = Z dt { (cid:2) − ( ∂ t ϕ ) − D + iψ ∂ t ψ + iψ ∂ t ψ (cid:3) − m ( iψ ψ + Dϕ ) − g (3 iϕ ψ ψ + Dϕ ) } . (3.15)As we can see, the general interaction terms in Φ n are of the forms of ϕ n − ψ ψ and Dϕ n − .It is convenient for later use to write the SUSY transformations in the Lorentzian metricand in the momentum representation. They are given by: δ ϕ ( p ) = iη ψ ( p ) , δ ϕ ( p ) = iη ψ ( p ) ,δ ψ ( p ) = − iη pϕ ( p ) , δ ψ ( p ) = − η D ( p ) ,δ ψ ( p ) = η D ( p ) , δ ψ ( p ) = − iη pϕ ( p ) ,δ D ( p ) = η pψ ( p ) , δ D ( p ) = − η pψ ( p ) . (3.16)Notice that δ is obtained from δ with the discrete symmetry: ψ → ψ an ψ → − ψ . Finallywe write the action in the momentum representation: S = Z dp n (cid:2) − p ϕ ( − p ) ϕ ( p ) − D ( − p ) D ( p ) + iψ ( − p ) pψ ( p ) + iψ ( − p ) pψ ( p ) (cid:3) − m ( iψ ( − ) ψ ( p ) + D ( − p ) ϕ ( p )) o − g Z dp dp dp dp (3 iϕ ( p ) ϕ ( p ) ψ ( p ) ψ ( p ) + D ( p ) ϕ ( p ) ϕ ( p ) ϕ ( p )) δ ( p + p + p + p ) . (3.17)10 Φ Φ Φ Φ ΦΨ Ψ Ψ Ψ Ψ a a/2 aa/40 3a/4 ...... Figure 2: One dimensional lattice with the bosonic field Φ located on the points multiple of a ,and the fermionic field Ψ on points shifted by a . According to the discussion of section 2 the formulation of the D = 1, N = 2 supersymmetricmodel on a lattice with spacing a should involve on the lattice two fields, one bosonic andone fermionic. In fact the lattice consists of two sub lattices with lattice spacing a invariantunder translations and hence each degree of freedom on the lattice corresponds to two degreesof freedom in the continuum limit. Let us denote the bosonic “superfield” by Φ( x ) with x = na ,and the fermionic “superfield” by Ψ( x ) with x = na + a . The shift of a in the fermionic superfieldwith respect to the bosonic one has been introduced to have symmetric finite differences in thesupersymmetry transformations and implement hermiticity in a natural way as discussed insection 2. A picture of the lattice is given in fig. 2.Let us proceed now to define the supersymmetry transformations on the lattice. Thereare two supercharges in the N = 2 model, whose algebra was given in (3.1). One of thesupersymmetry transformation, which we shall denote by δ , was already formulated on thelattice in the context of the N = 1 model and can be written as: δ Φ( x ) = iα (cid:20) Ψ( x + a x − a (cid:21) x = na , (4.1) δ Ψ( x ) = 2 α (cid:20) Φ( x + a − Φ( x − a (cid:21) x = na a . (4.2)We assume here that Φ( x ) and Ψ( x ) are dimensionless, so that no dependence on the latticespacing a appears at the r.h.s. of (4.1) and (4.2). Of course a rescaling of the fields with powersof a will be needed to make contact with the fields of the continuum theory. Let us introducenow the superfields in the momentum space defined as the Fourier transform of Φ( x ) and Ψ( x ) :Φ( p ) = 12 X x = na e ipx Φ( x ) , Ψ( p ) = 12 X x = na + a e ipx Ψ( x ) . (4.3)The corresponding inverse transformations are:Φ( x ) = a Z πa dp π Φ( p ) e − ipx , Ψ( x ) = a Z πa dp π Ψ( p ) e − ipx . (4.4)From(4.3) it is clear that Φ( p ) and Ψ( p ) satisfy the following periodicity conditions:Φ( p + 4 πa ) = Φ( p ) , Ψ( p + 4 πa ) = − Ψ( p ) . (4.5) For simplicity we shall denote fields in momentum and coordinate representation with the same symbols. Ar-guments x , y , z will always refer to coordinate representations, arguments p , q , r to the momentum representation.
11n momentum representation the supersymmetry transformations (4.1) and (4.2) read: δ Φ( p ) = i cos ap α Ψ( p ) , (4.6) δ Ψ( p ) = − i sin ap α Φ( p ) . (4.7)The commutator of two supersymmetry transformations δ with parameters α and β defines aninfinitesimal translation on the lattice. From (4.6) and (4.7) one finds: δ β δ α F ( p ) − δ α δ β F ( p ) = 4 sin ap αβF ( p ) , (4.8)where F ( p ) stands for either Φ( p ) or Ψ( p ). In coordinate space this is completely equivalent to(2.21), so that an infinitesimal translation of parameter λ on the lattice is defined by; F ( x ) → F ( x ) + λ F ( x + a ) − F ( x − a ) a , (4.9)which clearly reduces to F ( x ) → F ( x + λ ) in the continuum limit. Translations defined in (4.9)are however conceptually different from the discrete translations on the lattice that would bedefined as: F ( x ) → F ( x + a ) = F ( x ) + a F ( x + a ) − F ( x ) a . (4.10)The difference between the two definitions is even more apparent in the momentum represen-tation, where (4.10) is simply F ( p ) → e iap F ( p ) and applied to a product of fields leads to thestandard form of momentum conservation. Invariance under (4.9) instead leads to a non localconservation law where p is replaced by sin ap , namely , for a product of fields of momenta p , p ,..., p n : sin ap ap · · · + sin ap n . (4.11)This conservation law on the lattice was first pointed out by Dondi and Nicolai [16]. Theimplications of this conservation law, in particular with respect to the validity of the Leibnizrule and the relation of the present approach to the link approach will be discussed in section7. In the continuum limit ( ap i ≪
1) (4.11) reduces to the standard momentum conservationlaw and locality is restored. The conservation law (4.11) is not affected if any momentum p i init is replaced by πa − p i due to the invariance of the sine. In view of last section’s discussionthe interpretation is clear: in the continuum limit ( ap ≪ F ( p ) and F ( πa − p ) representfluctuations of momentum p respectively around the vacuum of momentum zero and πa on thelattice. So the symmetry p → πa − p amounts to exchanging the two vacua keeping the physicalmomentum unchanged and will play an important role in supersymmetry transformations.We now want to match the superfields Φ( p ) and Ψ( p ) appearing in the supersymmetrytransformations (4.6) and (4.7) with the component fields of the N = 2 D = 1 supersymmetrydescribed in the previous section. Working in the momentum representation we shall associate ϕ and D with the fluctuations of Φ respectively around 0 and πa and similarly ψ and ψ withthe fluctuations of Ψ. More specifically we assume the following correspondence:Φ( p ) = a − ϕ ( p ) , (4.12)Ψ( p ) = a − ψ ( p ) , (4.13)Ψ( 2 πa − p ) = ia − ψ ( p ) , (4.14)Φ( 2 πa − p ) = − a − D ( p ) , (4.15)12here p is restricted in (4.12-4.15) to the interval ( − πa , πa ), which is also the range of definition ofthe component fields (although with a possible discontinuity at p = ± πa ) which corresponds to alattice of spacing a . A rescaling of the fields with powers of a has been also introduced in (4.12-4.15) to account for the dimensionality of the component fields in momentum representation(remember that Φ and Ψ were defined to be dimensionless). A similar rescaling will be assumedfor the supersymmetry parameter α : α = a η. (4.16)By inserting (4.12-4.15) and (4.16) into the supersymmetry transformations (4.6) and (4.7) weobtain for the component fields the following transformations: δ ϕ ( p ) = i cos ap ηψ ( p ) −→ ap ≪ iηψ ( p ) , (4.17) δ ψ ( p ) = − i a sin ap ηϕ ( p ) −→ ap ≪ − ipηϕ ( p ) , (4.18) δ ψ ( p ) = cos ap ηD ( p ) −→ ap ≪ ηD ( p ) , , (4.19) δ D ( p ) = 4 a sin ap ηψ ( p ) −→ ap ≪ pηψ ( p ) . (4.20)In the continuum limit ap ≪ p appearing as the argument of the component fields in (4.12-4.15) andin (4.17-4.20) is restricted to the interval ( − πa , πa ). So we could introduce a lattice of spacing a and coordinates ˜ x = na and define the component fields ϕ (˜ x ), etc. on such lattice by taking theFourier transform on the πa interval of ϕ ( p ), etc. and finally write the supersymmetry trans-formations (4.17-4.20) of the component fields in the ˜ x coordinate representation. However thetrigonometric functions at the r.h.s. of (4.17-4.20) are not periodic of period πa , and as a resultthe supersymmetry transformation are non-local in the ˜ x coordinate representation, implyingthat the natural representation for the supersymmetry is on the lattice with a spacing.We have now to identify the second supersymmetry transformation δ . In the continuum δ is obtained from δ by replacing everywhere ψ ( p ) with ψ ( p ), and ψ ( p ) with − ψ ( p ). It is easyto see from (4.12-4.15) that this corresponds on the lattice to the replacement:Ψ( p ) −→ − i Ψ( 2 πa − p ) . (4.21)By performing this replacement on the supersymmetry transformations (4.6) and (4.7) oneobtains the expression for δ : δ Φ( p ) = cos ap α Ψ( 2 πa − p ) , (4.22) δ Ψ( 2 πa − p ) = 4 sin ap α Φ( p ) . (4.23)The supersymmetry transformation δ , defined by (4.22) and (4.23), satisfies together with δ an N = 2 supersymmetry algebra. It is easy to check in fact that the commutator of two δ transformations gives an infinitesimal translation ( namely eq. (4.8) holds also for δ ) and thatthe commutator of a δ and δ transformation vanishes, namely: δ β δ α F ( p ) − δ α δ β F ( p ) = 0 . (4.24)13n terms of the component fields, and in the continuum limit, the explicit expression for the δ transformation can be obtained from (4.22) and (4.23) by using eq.s (4.12-4.15): δ ϕ ( p ) = i cos ap ηψ ( p ) −→ ap ≪ iηψ ( p ) , (4.25) δ ψ ( p ) = − i a sin ap ηϕ ( p ) −→ ap ≪ − ipηϕ ( p ) , (4.26) δ ψ ( p ) = − cos ap ηD ( p ) −→ ap ≪ − ηD ( p ) , (4.27) δ D ( p ) = − a sin ap ηψ ( p ) −→ ap ≪ − pηψ ( p ) . (4.28)As for δ in the limit ap ≪ Q in the continuum theory.The coordinate representation of δ can be obtained directly from (4.25-4.28) by Fouriertransform, or from (4.1-4.2) by performing the following substitution:Ψ( x ) −→ ( − n Ψ( − x ) x = na − a , (4.29)which is the same as (4.21) in the coordinate representation. Either way the result is: δ Φ( x ) = iα − n (cid:20) Ψ( − x + a − Ψ( − x − a (cid:21) x = na , (4.30) δ Ψ( x ) = 2 α ( − n (cid:20) Φ( − x + a − Φ( − x − a (cid:21) x = na a . (4.31)It is clear from (4.30) and (4.31) that the supersymmetry transformation δ is local in thecoordinate representation only modulo the reflection x → − x . This was already implicit in thecorrespondence (4.12-4.15) between the lattice fields and the ones of the continuum theory. Infact it is clear from (4.12-4.15) that while for instance ϕ ( x ) is associated to the fluctuations ofΦ( x ) around the constant configuration ( p = 0), the fluctuations of Φ( x ) around the constantconfiguration with alternating sign ( p = πa ) correspond in the continuum to D ( − x ). Forfermions this parity change leads to a physical meaning. Since ψ ( p ) ↔ ψ ( x ) is defined as aspecies doubler of ψ ( p ) ↔ ψ ( x ), the chirality of ψ is the opposite of ψ . However by thechange of p → πa − p equivalently x → − x , chirality of ψ and ψ are adjusted to be the same.Thus this bi local nature in the coordinate space may be transfered to a local interpretation. In order to construct a lattice action invariant under the two supersymmetry transformations δ and δ defined in the previous section we consider first the invariance under translations, whichfollows from supersymmetry, and it is expressed by the sine conservation law given in (4.11).Any vertex of a supersymmetric invariant theory will have to include a delta function enforc-ing the conservation law (4.11). Unlike the standard momentum conservation this conservationlaw does not lead to a local action in coordinate space, and in fact it makes it impossible to writethe action in coordinate space without using transcendental function (more specifically Besselfunctions). For this reason we shall first formulate the action in the momentum representation.We provide a full treatment of coordinate prescription later in section 7. There is however an14mportant exception to this, namely the case n = 2, that is the kinetic term and the mass term.In fact for n = 2 the conservation law (4.11) has two solutions: p + p = 0 (mod 4 πa ) (5.1)and p − p = 2 πa (mod 4 πa ) (5.2)and the delta function of momentum conservation does not need in this case to include any sinefunction. The two point term in the action with the momentum conservation (5.1) describes,as we shall see, the kinetic term and is local when expressed in the coordinate representation.The mass term will be expressed instead by a term with the conservation law (5.2), involvingin coordinate space a coupling between fields in x and − x , in agreement with the discussion atthe end of the previous section. Let us introduce now a supersymmetric action on the lattice.All terms of this action have the same structure, which for an n -point term is the following: S ( n ) = g ( n )0 a n n ! Z πa − πa dp π · · · dp n π πδ n X i =1 sin ap i ! × G ( p , p , · · · , p n ) h ap p )Φ( p ) · · · Φ( p n )+ (5.3)+ n −
14 sin a ( p − p )4 Ψ( p )Ψ( p )Φ( p ) · · · Φ( p n ) i . A direct check shows that S ( n ) is invariant under the supersymmetry transformation δ definedin (4.1,4.2) as well as under the replacement (4.21), which in turn implies the invariance under δ provided the otherwise arbitrary function G satisfies the following properties: i ) it is symmetricunder permutations of the momenta p i , ii ) it is periodic with period πa in all the momenta and iii ) it is invariant when an even number of p i is replaced by πa − p i . An example of functionsatisfying the above requirements is: C ( p ) = n Y i =1 cos ap i . (5.4)For all interaction terms ( n >
2) we will take G ( p ) = C ( p ). In fact, thanks to the cosine factorsthe function C ( p ) vanishes if any of the momenta p i is equal to ± πa , so a factor C ( p ) is neededto cancel the singularities arising at p i = ± πa from the integration volume as a consequenceof the delta function. Later in section seven we find out another natural reason why thisfactor (5.4) comes out. All terms in (5.3) are periodic with period πa in the momenta, andthe momentum integration is over a whole period: the specific choice here (from − πa to πa )is for future convenience. The integrand can be made explicitly symmetric with respect topermutations of the momenta, so the factor 2 sin ap in front of the bosonic part can be replacedby 2 sin ap → − n n X cos ap i . (5.5)It should be noticed also that, although we kept the dependence on the lattice spacing a explicit,this could be completely absorbed in the definition of the momenta, by introducing a-dimensionalmomenta ˜ p i = ap i . 15 .1 Kinetic term and Mass term The case n = 2 is special because only in that case the sine conservation law splits into the twoseparate conservation laws (5.1) and (5.2) which are linear in the momenta. The delta functionin (5.3) can then be replaced by the sum, with arbitrary coefficients, of the delta functionsenforcing (5.1) and (5.2). This amounts ( for n = 2 only) to perform in (5.3) the followingreplacement: a g (2)0 G ( p , p ) δ (sin ap ap −→ δ ( p + p ) + m δ ( p − p − πa ) , (5.6)where m is a free parameter. The delta functions at the r.h.s. of (5.6) do not give rise toany singularity at p i = ± πa so no factor C ( p ) is required in this case We are going to showhere that the first delta in (5.6) generates the supersymmetric kinetic term, the second deltathe supersymmetric mass term. By inserting the r.h.s. of (5.6) into (5.3) and performing onemomentum integration we obtain: S (2) = S kin + S mass , (5.7)with S kin = 4 a Z πa − πa dp π (cid:20) ap − p )Φ( p ) −
14 sin ap − p )Ψ( p ) (cid:21) , (5.8)and S mass = 4 am Z πa − πa dp π (cid:20) Φ( p + 2 πa )Φ( p ) + 14 Ψ( p + 2 πa )Ψ( p ) (cid:21) . (5.9)One could regard the kinetic term (5.8) as the kinetic term of a lattice theory with lattice spacing a ′ = a and just one bosonic and one fermionic degree of freedom. The invariance under the δ supersymmetry transformations (4.6,4.7) would be described as an N = 1 supersymmetry.However, as shown by (5.8), the fermion would have a doubler at p = πa ′ , as expected. Ourinterpretation is different. Since supersymmetry transformations (4.2) are related to shifts of a ,we consider a as the fundamental lattice spacing and the a spacing as the signal that we aredescribing with the same lattice field two distinct degrees of freedom in the continuum: hencethe fermion and its doubler are interpreted as partners in an N = 2 supersymmetry generatedby δ and δ , the latter given by (4.22,4.23). In order to separate the degrees of freedom let ussplit the integration region in (5.8) and (5.9) into ( − πa , πa ) and ( πa , πa ). In the first interval weuse the correspondence (4.12,4.13), in the second the correspondence (4.14,4.15) and find: S kin = Z πa − πa dp π h a (1 − cos ap ϕ ( − p ) ϕ ( p ) + 14 (1 + cos ap D ( − p ) D ( p ) −− a sin ap ψ ( − p ) ψ ( p ) − a sin ap ψ ( − p ) ψ ( p ) i . (5.10)Similarly for the mass term we get: S mass = 2 m Z πa − πa dp π [ − ϕ ( − p ) D ( p ) − iψ ( − p ) ψ ( p )] , (5.11) Of course it would be possible to treat the n = 2 case in the same way as the interaction terms, keeping thesine delta function with the factor C ( p ) in front. However this would fix the relative coefficient of the mass termand of the kinetic term. The smoothness of the continuum limit would then be spoiled since, as we shall see, m need to scale with a for such limit to be smooth. m is now the physical mass: m = m a . Thanks to the rescaling all fields in (5.10) and(5.11) have the correct canonical dimension, and the continuum limit is smooth. The componentfields ϕ ( p ), D ( p ), ψ ( p ) and ψ ( p ) are defined for p in the interval ( − πa , πa ). This is the Brillouinzone corresponding to a lattice of spacing a , so we could define a lattice with coordinates ˜ x = na and the component fields on it as the Fourier transforms of the momentum space components.For instance we could define: ϕ (˜ x ) = Z πa − πa dp π ϕ ( p ) e − ip ˜ x . (5.12)However the action written in the coordinate ˜ x space is non-local, since the finite differenceoperators appearing in (5.10) are periodic with period πa and not πa that would be needed fora local expression on a lattice with spacing a .Instead it is possible, using (4.3), to write (5.8) in the coordinate space x with lattice spacing a : S kin = X x = n a (cid:20) Φ( x ) (cid:16) x ) − Φ( x + a − Φ( x − a (cid:17) + i x + a x − a (cid:21) . (5.13)In the bosonic part of the action the equivalent of a second finite difference appears. In fact ifwe define the finite difference on the lattice of spacing a as ∂ ± F ( x ) = F ( x ± a − F ( x ) , (5.14)the kinetic term can be rewritten as: S kin = X x = n a (cid:20) Φ( x ) ∂ − ∂ + Φ( x ) + i x + a ∂ − Ψ( x + a (cid:21) . (5.15)The coordinate representation for the mass term (5.9) reveals some new features, namely acoupling between fields in x and − x . In fact, by using again (4.3), one finds: S mass = m X x = n a ( − xa (cid:20) Φ( − x )Φ( x ) + i − x − a x + a (cid:21) . (5.16)The bi local structure of (5.16) shows that the extended lattice with spacing a has not a straight-forward relation to the coordinate space in the continuum limit. This is related to the fact thatwhile the fluctuations of Φ( x ) (resp. Ψ( x )) around a constant field configuration are associatedto the component field ϕ ( x ) (resp. ψ ( x )), its fluctuations around ( − xa are associated to D ( − x ) (resp. ψ ( − x )). In other words the way the two bosonic (resp. fermionic) componentsof the superfield are embedded in a single bosonic (resp. fermionic) field on the extended latticeis non trivial and exhibits a bi local structure. Although the extended lattice is not a discreterepresentation of superspace (bosonic and fermionic fields have to be introduced separately onit) it carries some information about the superspace structure and as such it does not simplymap onto the coordinate space in the continuum limit.From (5.10) and (5.11)one can then easily derive the free propagators. For this purpose it isconvenient to introduce the following notations:ˆ p = 2 a sin ap , (5.17)17nd c (ˆ p ) = r − a ˆ p . (5.18)Moreover, for each component field f ( p ) we define f ′ (ˆ p ) = f ( p (ˆ p )) . (5.19)With these notations the two point bosonic correlation function can be written as: h ϕ ′ (ˆ p ) ϕ ′ ( − ˆ p ) i = c (ˆ p )ˆ p − m
12 (1 + c (ˆ p )) , (5.20) h D ′ (ˆ p ) D ′ ( − ˆ p ) i = c (ˆ p )ˆ p − m a (1 − c (ˆ p )) , (5.21) h ϕ ′ (ˆ p ) D ′ ( − ˆ p ) i = c (ˆ p )ˆ p − m m, (5.22)whereas for the fermionic ones we have: h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = − c (ˆ p )ˆ p − m a ˆ p, (5.23) h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = −h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = c (ˆ p )ˆ p − m im. (5.24) The interaction terms are obtained from the general invariant expression (5.3) with n >
2. Weshall choose the arbitrary function G to be equal to the function C ( p ) defined in (5.4). Thisis needed to cancel the divergences occurring in the integration volume at p i = ± πa due to thedelta function. With this choice the n -point interaction term reads: S ( n ) = g ( n )0 a n n ! Z πa − πa dp π · · · dp n π πδ n X i =1 sin ap i ! × n Y i =1 cos ap i ! h ap p )Φ( p ) · · · Φ( p n )+ (5.25)+ n −
14 sin a ( p − p )4 Ψ( p )Ψ( p )Φ( p ) · · · Φ( p n ) i . Unlike the case of S (2) for n ≥ p i ,hence the coordinate representation for S ( n ) cannot be given in terms of elementary functionsand it is non-local.When expressed in terms of the component fields (5.25) contains many terms, as each fieldin (5.25) can correspond to different components of the superfield depending on the value ofits momentum. These terms however have different powers of the lattice spacing a accordingto the rescaling properties given in (4.12–4.15). We have therefore to select the terms that areleading in the continuum limit. In the bosonic sector D scales with an extra power of a withrespect to ϕ , so that the leading term seems to be obtained by replacing Φ( p i ) with a − ϕ ( p i )and restricting p i between − πa and πa . This term, however, is not the leading term, becauseof a factor with the momentum labelled p in the bosonic part of the action. The sin factor18ultiplying Φ( p ) is of order a at p ≃ p ≃ πa , so that the latter vacuumbecomes dominant and Φ( p ) should be identified with D . That is, the leading term in thebosonic sector is Dϕ n − . In the fermionic part of the action Ψ( p ) scales in the same way at p = 0 and p = πa , but p − p has to be πa and not zero to avoid an extra factor a comingfrom the sin factor. So Ψ( p ) and Ψ( p ) must correspond one to ψ and one to ψ . By carefullycounting the powers of a , one finds that in order to have a finite and non vanishing continuumlimit for the leading term of (5.25) the physical coupling constant g ( n ) must be defined as: g ( n ) = a − n g n (5.26)With this normalization the leading term in the interaction term becomes: S ( n ) = g ( n ) n ! Z πa − πa dp π · · · dp n π πδ a n X i =1 sin ap i ! × n Y i =1 cos ap i ! h − cos ap D ( p ) ϕ ( p ) · · · ϕ ( p n )+ (5.27)+ ( n −
1) cos a ( p + p )4 ψ ( p ) ψ ( p ) ϕ ( p ) · · · ϕ ( p n ) i + O ( a ) , where the terms of order a or higher are included in O ( a ). The leading term corresponds to theusual Φ n interaction: S ( n ) i = Z dxd θ Φ n . (5.28)The O ( a ) terms are of two types: some contain higher powers of D ( p ), namely terms that donot appear in the continuum in any superfield action for dimensional reasons, but are needed onthe lattice for exact supersymmetry. Then there are terms where all momenta are fluctuatingaround the p = 0 vacuum and have the same structure as the kinetic term. They correspond inthe continuum to derivative interactions given in terms of the superfield Φ by S ( n ) k = Z dxd θ Φ n − D Φ D Φ . (5.29)These derivative interaction terms are sub leading (of order a ) with the choice of the function G ( p ) given above, namely G ( p ) = C ( p ). However with a different choice of G ( p ) they can bethe leading terms in the continuum limit. For instance, if we choose G ( p ) = a C ( p )(1 + C ( p )),kinetic-like terms in (5.3) with 0, or 2 momenta around the vacuum at πa would be of order1 in the continuum limit which would contain derivative interactions of the form (5.29). It israther surprising that the same action on the lattice, namely the one given in (5.3), can giveorigin to terms which are entirely different when written in the superfield formalism. This seemto indicate that some deeper understanding of supersymmetry may be achieved by the presentapproach. One of the key features of the present approach is that momentum conservation is replaced bythe sin conservation law (4.11). This means that invariance under finite translations is violated The reason is that C ( p ) is 1 when an even number of momenta are equal to πa and the remaining are zero,it is − πa is odd.
19n the lattice. It is then crucial that translational invariance is recovered in the continuum limit.This is not obvious and it requires the analysis of the UV properties of the theory when thecontinuum limit is taken. Recovery of translational invariance is the subject of the first partof this section. In the second part we check explicitly at one loop level that supersymmetricWard-Takahashi identities are preserved in the continuum limit.
As a preliminary step towards a proof that invariance under finite translations is recovered inthe continuum limit, we proceed to analyze such a limit in the ultraviolet region.The lattice theory described in the previous section in terms of the fields Φ and Ψ is freeof ultraviolet divergences. In fact everything in that theory can be written in terms of thedimensionless momentum variables ˜ p i = ap i , which are angular variables with periodicity 4 π .Momentum integrations reduce to integrations over trigonometric functions of ˜ p i , and ultravioletdivergences never appear. All correlation functions of Φs and Ψs integrations are therefore finite.This however is not enough to ensure that the continuum limit is smooth and that ultravioletdivergences do not appear in the limiting process. The continuum limit in fact involves arescaling of fields with powers of a , which is singular in the a → a → p i are in the neighborhood of one of the vacua, namelyat ˜ p i = 0 or ˜ p i = 2 π . The limit being a singular one, the ultraviolet behavior has to be checkedexplicitly.Let us consider then the action written in terms of the rescaled component fields. Thestructure of its interaction terms (neglecting momentum integrations and delta functions) is thefollowing: S ( n )B ∼ g ( n ) a ( aD ) k ϕ n − k , (6.1) S ( n )F ∼ g ( n ) a aψ i ψ j ( aD ) k ϕ n − k − . (6.2)Therefore it is convenient to introduce for the internal lines in loop integrations the rescaledfields D int = aD and ψ i, int = a ψ i . In this way the effective coupling in the perturbativeexpansion is g ( n ) /a and each vertex contributes with a factor a in the ultraviolet region. Nextwe consider the UV behavior (including momentum integration in the variable ˆ p = a sin ap ) ofthe propagators: h ϕϕ i ∼ h D int D int i ∼ Z /a d ˆ p p − m ∼ a, (6.3) h ϕD int i ∼ Z /a d ˆ p am ˆ p − m ∼ a , (6.4) h ψ , int ψ , int i ∼ h ψ , int ψ , int i ∼ Z /a d ˆ p a ˆ p ˆ p − m ∼ a, (6.5) h ψ , int ψ , int i ∼ Z /a d ˆ p am ˆ p − m ∼ a . (6.6)The diagonal propagators contribute with a factor a in the UV region, off-diagonal ones with afactor a . Considering now an amputated diagram with V vertices and I internal lines of which20 off have an off-diagonal propagator, its UV contribution will be:(the total UV contribution) ∼ a I + I off a V − a − V ∼ a I + I off − , (6.7)where the extra factor a V − comes from the delta functions which reduces the number of themomentum integrations. This calculation shows that the superficial degree of divergence is I + I off −
1, which can give a logarithmic divergence only for I = 1, I off = 0. However ifwe consider that a factor a − V comes from the vertices, the contribution of the momentumintegration alone is given by: (the integration only) ∼ a I + I off a V − , (6.8)that is momentum integrations are always convergent in the UV.How about the IR divergences? All the propagators are convergent in the IR. All the in-teractions are finite as well. Note that φ n term in the n -point interaction has a factor of 1 /a but it is compensated by (1 − cos ap ) in the IR. Therefore momentum intergrations are alwaysconvergent in the IR as well.We are now in position to prove the main result, namely that translational invariance isrecovered in the continuum limit. Since the conserved quantity is not the momentum itself p but sin ap finite translational invariance is explicitly broken at a finite lattice spacing. Indeed,if we denote the component fields by φ A = ( ϕ, D, ψ , ψ ), the sine conservation law implies thatcorrelation functions are invariant under the transformation: φ A ( p ) → exp( il a sin ap φ A ( p ) l : a finite length (6.9)whereas invariance under finite translation would require the invariance under the transformation φ A ( p ) → exp( ilp ) φ A ( p ) . (6.10)To prove that invariance under finite translations is recovered we need to prove that in thecontinuum limit (6.9) and (6.10) are equivalent. For an n -point correlation function of φ A ,transformation (6.9) is equivalent to h φ A ( p ) φ A ( p ) . . . φ A n ( p n ) i → exp n X i =1 la sin ap ! h φ A ( p ) φ A ( p ) . . . φ A n ( p n ) i≃ − i a l n X i =1 p i ! exp il n X i =1 p i ! h φ A ( p ) φ A ( p ) . . . φ A n ( p n ) i . (6.11)where in the last step higher order terms in the expansion of sin ap have been neglected since ap → a in thecontinuum limit if we assume lp i to be of order 1 so that this term can be neglected as long as nodivergence ( of order at least a ) arises in the correlation function h φ A ( p ) φ A ( p ) . . . φ A n ( p n ) i .As shown in the first part of this section this is not the case, so we can conclude that invarianceunder finite translations is recovered in the continuum limit.21 .2 Ward-Takahashi identities Invariance under supersymmetry transformations is exact at the finite lattice spacing and itis not spoiled by radiative corrections, which are all finite in the lattice theory. Since thecontinuum limit is smooth, we expect that exact supersymmetry is preserved also in this limit.This can be confirmed explicitly, by checking that the corresponding Ward-Takahashi identities(WTi) are satisfied. We shall consider here the case of a four points interaction and checkthat the supersymmetric Ward-Takahashi identities are satisfied at 1-loop level. For the 2-pointcorrelation function, there are 3 independent WTis:cos ap h ψ ( p ) ψ ( − p ) i + 4 a sin ap h ϕ ( − p ) ϕ ( p ) i = 0 , (6.12)4 a sin ap h ψ ( p ) ψ ( − p ) i + cos ap h D ( − p ) D ( p ) i = 0 , (6.13) i h ψ ( p ) ψ ( − p ) + h ϕ ( p ) D ( − p ) i = 0 . (6.14)There are also 2 more identities obtained from the above by replacing ψ → ψ , ψ → − ψ , butthey are automatically satisfied if (6.12) and (6.13) are satisfied.At the tree level, it is easy to see that the WTi (6.12)–(6.14) are satified using propagators(5.20)–(5.24).The 1-loop radiative corrections to these propagators are calculated in Appendix A. Theresult is: h ϕ ′ (ˆ p ) ϕ ′ ( − ˆ p ) i = h ϕ ′ (ˆ p ) ϕ ′ ( − ˆ p ) i tree F (ˆ p ) , (6.15) h D ′ (ˆ p ) D ′ ( − ˆ p ) i = h D ′ (ˆ p ) D ′ ( − ˆ p ) i tree F (ˆ p ) , (6.16) h D ′ (ˆ p ) ϕ ′ ( − ˆ p ) i = h D ′ (ˆ p ) ϕ ′ ( − ˆ p ) i tree F (ˆ p ) , (6.17)where F (ˆ p ) = − ig a D (ˆ p ) Z /a − /a dk π D ( k ) 2(1 + am ) (cid:2) (1 + am ) − c (ˆ p ) (cid:3) , (6.18) F (ˆ p ) = − ig a D (ˆ p ) Z /a − /a dk π D ( k ) 2(1 + am ) am (cid:2) (1 + am ) − (1 + 2 am ) c (ˆ p ) (cid:3) (6.19)with D (ˆ p ) = c (ˆ p )( a ˆ p − a m ) . (6.20)So 1-loop radiative corrections to the diagonal (resp. off-diagonal) propagators are given by thefunction F (ˆ p ) (resp. F (ˆ p )). The same thing happen in the case of fermionic propagators: h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i tree F (ˆ p ) , (6.21) h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i tree F (ˆ p ) . (6.22)It follows that since the WT identities are satisfied at the tree level they are also satisfied at1-loop level. Do not confuse with the auxiliary field. Leibniz rule and new star product in coordinate spaceand the link approach
Since we have established a new exactly supersymmetric lattice model, it is instructive to com-pare the algebra of this model with that of link approach. We first find out the algebraic structureof the model. The momentum representation of supersymmetry transformations (4.6,4.7) and(4.22,4.23) can be related to the supercharges Q and Q as δ = α √ aQ ,δ = α √ aQ , (7.1)where the lattice constant dependence is introduced to recover the dimensionality of super-charges. We find the following algebraic relation: Q = Q = 2 a sin ap , { Q , Q } = 0 . (7.2)The coordinate representation of the super charges Q and Q for the supersymmetry trans-formations (4.1 ,4.2) and (4.30 ,4.31) can be defined exactly same as (7.1), then we find thefollowing supersymmetry algebra: Q = Q = i ˆ ∂, { Q , Q } = 0 , (7.3)where the symmetric difference operator ˆ ∂ is defined as:ˆ ∂F ( x ) ≡ ( ∂ + − ∂ − ) a F ( x ) = F ( x + a ) − F ( x − a ) a , (7.4)with ∂ ± given in (5.14). We find the following algebraic correspondence between the momentumrepresentation of derivative operator and the coordinate counterpart of difference operator:2 a sin ap ←→ i ˆ ∂. (7.5)This lattice version of supersymmetry algebra coincides with the continuum algebra of (3.1)in the continuum limit ap →
0. As we have stressed in section 2, the lattice counter part ofmomentum operator generates the lattice constant a step translation of fields although the basiclattice structure of this model has half lattice nature.We have pointed out that the supersymmetry transformation is essentially the half latticespacing translation of lattice superfields with an alternating sign factor as we can see in (2.18).The operation of the lattice half step translation needs special care since ∂ ± ( − xa = ( − xa ∂ ± . (7.6)On the other hand the translation generator ˆ ∂ commutes with supersymmetry generators: δ ˆ ∂ = ˆ ∂δ , δ ˆ ∂ = ˆ ∂δ , (7.7)which are equivalent to [ Q , ˆ ∂ ] = [ Q ˆ ∂ ] = 0 . (7.8)23his leads to the continuum algebra (3.2) in the continuum limit. The full lattice constantspacing differential operator is the translation generator, which is consistent with (7.3). Wehave thus confirmed that the lattice supersymmetry algebra of this model leads to the continuumalgebra in the continuum limit. We can, however, recognize that lattice supersymmetry algebrahas the same form with the continuum super algebra even with a finite lattice constant.We have shown already that the lattice version of this model have exact supersymmetryat least in the momentum representation even with the interaction terms. The coordinateformulation of the model should have exact supersymmetry as well since it should in principlebe equivalent with the formulation of momentum representation. On the other hand the latticesupersymmetry algebra of this model includes symmetric difference operator as in (7.3). It isa well known fact that the symmetric difference operator (7.4) does not satisfy Leibniz rule forthe product of fields:ˆ ∂ ( F ( x ) G ( x )) = 1 a (cid:16) F ( x + a G ( x + a − F ( x − a G ( x − a (cid:17) = ˆ ∂F ( x ) G ( x + a F ( x − a ∂G ( x )= ˆ ∂F ( x ) G ( x − a F ( x + a ∂G ( x ) . (7.9)Here comes a crucial question: “How can the lattice supersymmetry algebra be consistent since the difference operator does notsatisfy Leibniz rule while the super charges satisfy Leibniz rule ?” In the link approach of the lattice supersymmetry formulation this problem was avoided byintroducing shifting nature for the super charges: Q ( F ( x ) G ( x )) = Q F ( x ) G ( x + a F ( x − a Q G ( x ) ,Q ( F ( x ) G ( x )) = Q F ( x ) G ( x − a F ( x + a Q G ( x ) , (7.10)where F and G are assumed to be bosonic lattice superfields. In the case of fermionic superfieldsit works same if the anticommuting Grassmann nature is taken into account. We can confirmthat the lattice supersymmetry algebra (7.3) is consistently fulfilled. There is, however, orderingambiguity for the product of fields: Q ( F ( x ) G ( x )) = Q F ( x ) G ( x + a F ( x − a Q G ( x )= Q ( G ( x ) F ( x )) = Q G ( x ) F ( x + a G ( x − a Q F ( x ) , (7.11)since F ( x ) G ( x ) = G ( x ) F ( x ). We obtain a similar relation for Q . Since the right hand sides of(7.11) are different, this discrepancy was criticized as an inconsistency of the formulation [14, 22].It was, however, recognized that if there is a mild noncommutativity between fields having ashifting nature: Q F ( x ) G ( x + a G ( x − a Q F ( x ) ,Q G ( x ) F ( x + a F ( x − a Q G ( x ) , (7.12)where F and G are shiftless while Q F and Q G carry a shift of − a , then there is no incon-sistency. This algebraic consistency was carefully investigated and it was discovered that these24lgebraic relations (7.9),(7.10) and (7.12) are consistently treated by Hopf algebraic symme-try [27]. Thus we may claim that the model has exact Hopf algebraic lattice supersymmetry.The exact invariance of the momentum representation of the action (5.3) under the super-symmetry transformations (4.6, 4.7) and (4.22, 4.23) suggests that there should be coordinatecounterpart which reflect this exact invariance including interactions. In the proof of the super-symmetry invariance, Leibniz rule is used for the operation of super charges Q j . It then leadsto the following relation: Q j ( F ( x ) ∗ G ( x )) = ( Q j F ( x )) ∗ G ( x ) + F ( x ) ∗ ( Q j G ( x )) , (7.13)or equivalently ˆ ∂ ( F ( x ) ∗ G ( x )) = ( ˆ ∂F ( x )) ∗ G ( x ) + F ( x ) ∗ ( ˆ ∂G ( x )) , (7.14)where we have tentatively introduced a new type of ∗ product which satisfies Leibniz rule even forthe difference operator Q j = ˆ ∂ which is Euclidean version of (7.3) and normally satisfies shiftedLeibniz rule (7.9) for the normal product of fields. However in the proof of the supersymmetryinvariance of the kinetic terms and the mass terms in the coordinate representation Leibniz rulehas been used for the normal products and thus the relations (7.13) and (7.14) should hold forthe normal products, at least for the product of two fields, which seems to be inconsistent withthe shifted Leibniz rule of symmetric difference operator (7.9). This is rephrasing the puzzle ofthe current problem.Assuming that the Leibniz rule for the difference operator works for the normal product, wefind the following difficulty. That is, supose we had defined ˆ ∂ operation on a product of fieldsasˆ ∂ ( F ( x ) G ( x ) H ( x ) · · · ) ? = ( ˆ ∂F ( x )) G ( x ) H ( x ) · · · + F ( x )( ˆ ∂G ( x )) H ( x ) · · · + F ( x ) G ( x )( ˆ ∂H ( x )) · · · . (7.15)This new definition does not necessarily lead to a cancellation of surface terms for the product ofsuperfields and thus supersymmetry cannot be kept exactly. Up to the product of two superfields,the surface terms cancel using the r.h.s of (7.15) X x ˆ ∂F ( x ) = 0 , X x ˆ ∂ ( F ( x ) G ( x )) = 0 . (7.16)However, in general the surface terms for a product of more than three superfields do not cancel: X x ˆ ∂ ( F ( x ) G ( x ) H ( x ) · · · ) = 0 . (7.17)We must find a formulation of a new ∗ product which satisfies the ∗ product version of (7.17)where the nonequality changes to equality: X x ˆ ∂ ( F ( x ) ∗ G ( x ) ∗ H ( x ) · · · ) = 0 . (7.18)We have recognized in the previous sections that non locality plays an important role in thepresent formulation. We have also recognized that non locality stems from the sine momentumconservation (4.11) which, in turn, arises from the necessity on the lattice to have completeperiodicity in the momentum and to have the species doublers on the same footing as theoriginal fields. With ordinary momentum conservation the product of a field F of momentum25 and a field G of momentum p is a composite field Φ = F · G of momentum p = p + p ,namely the momentum is the additive quantity under product:Φ( p ) ≡ ( F · G )( p ) = a π Z dp dp F ( p ) G ( p ) δ ( p − p − p ) . (7.19)In coordinate space this amounts to the ordinary local product:Φ( x ) ≡ ( F · G )( x ) = F ( x ) G ( x ) . (7.20)On the lattice momentum conservation is replaced by the lattice (sine) momentum conservation(4.11), which means that ˆ p = a sin ap is the additive quantity when taking the product of twofields. In other words the product of a field F of momentum p and a field G of momentum p is a composite field Φ = F ∗ G of momentum p with sin ap = sin ap + sin ap . This amounts tochanging the definition of the “dot” product to that of a “star” product defined in momentumspace as : Φ( p ) ≡ ( F ∗ G )( p ) = a π Z d ˆ p d ˆ p F ( p ) G ( p ) δ (ˆ p − ˆ p − ˆ p ) (7.21)As we shall see this product is not anymore local in coordinate space but satisfies the Leibniz rulewith respect to the symmetric difference operator ˆ ∂ . This is easily checked in the momentumrepresentation. In fact, according to (7.5) acting with ˆ ∂ corresponds in momentum space tomultiplication by ˆ p = a sin ap , so that from (7.21) we get:ˆ p Φ( p ) = a π Z d ˆ p d ˆ p [ˆ p F ( p ) G ( p ) + F ( p ) ˆ p G ( p )] δ (ˆ p − ˆ p − ˆ p ) . (7.22)Explicit form of the coordinate representation of the star product is given by( F ∗ G )( x ) = F ( x ) ∗ G ( x ) = a Z d ˆ p π e − ipx ( F ∗ G )( p )= Z π − π d ˜ p cos ˜ p e − ipx Z π − π d ˜ p π d ˜ p π cos ˜ p cos ˜ p Z ∞−∞ dτ π e iτ (sin ˜ p − sin ˜ p − sin ˜ p ) × X y,z e i ( m ˜ p + l ˜ p ) F ( y ) G ( z )= Z ∞−∞ dτ J n ± ( τ ) X m,l J m ± ( τ ) J l ± ( τ ) F ( y ) G ( z ) , (7.23)where ˜ p = ap , and x = na , y = ma , z = la should be understood and where the integrationvariable is not p but ˆ p .The lattice delta function is parametrized by τδ (cid:18) a sin ˜ p i (cid:19) = a π Z ∞−∞ dτ e iτ sin ˜ p i . (7.24) With this definition the star product is periodic in p with period πa . So while it is suitable for bosons, inorder to apply it to fermions we have to redefine the fermion fields and make them periodic. This can be doneby defining Ψ p ( p ) = e − ipa/ Ψ( p ) and use Ψ p in the definition of the “star” product. Ψ p ( p ) satisfies the realitycondition Ψ † p ( p ) = Ψ p ( − p ) and can be used in the action and in the SUSY transformations instead of Ψ( p ). Themain difference is that with the use of Ψ p fermions are like the bosons at the sites x = n a in the coordinaterepresentation and the SUSY transformations are expressed in terms of right (or left) finite differences of spacing a instead of the symmetric one as in (4.1,4.2). n ( τ ) is a Bessel function defined as J n ( τ ) = 12 π Z π + αα e i ( nθ − τ sin θ ) dθ, (7.25)and we use the following notation: J n ± ( τ ) = 12 ( J n +1 ( τ ) + J n − ( τ )) . (7.26)It is obvious that the star product is commutative: F ( x ) ∗ G ( x ) = G ( x ) ∗ F ( x ) . (7.27)We can now check how the difference operator acts on the star product of two lattice superfieldsand find that the difference operator action on a star product indeed satisfies Leibniz rule: i ˆ ∂ ( F ( x ) ∗ G ( x )) = a Z d ˆ p π i ˆ ∂ x e − ipx ( F ∗ G )( p )= a Z d ˆ p e − ipx X y,z Z d ˆ p π d ˆ p π e ip y + ip z × (cid:16) ( i ˆ ∂ y F ( y )) G ( z ) + F ( y ) ( i ˆ ∂ z G ( z )) (cid:17) δ (ˆ p − ˆ p − ˆ p )= ( i ˆ ∂F ( x )) ∗ G ( x ) + F ( x ) ∗ ( i ˆ ∂G ( x )) . (7.28)Eq. (7.28) already answer the question posed at the beginning of this section: the Leibniz rulefor the symmetric finite difference operator ˆ ∂ is recovered by the redefinition of the product offields, in agreement with the sine momentum conservation on the lattice.Similar to the case for star product of two fields we can show that the difference operatoracting on the star product of three fields satisfies the Leibniz rule and the surface terms of astar product for more than 3 lattice superfields vanishes: X x i ˆ ∂ ( F ( x ) ∗ G ( x ) ∗ H ( x )) = X x (cid:16) ( i ˆ ∂F ( x )) ∗ G ( x ) ∗ H ( x ) + F ( x ) ∗ ( i ˆ ∂G ( x )) ∗ H ( x )+ F ( x ) ∗ G ( x ) ∗ ( i ˆ ∂H ( x )) (cid:17) = X x i ˆ ∂ Z d ˆ p e − ipx ( F ∗ G ∗ H )( p )= Z dp cos ˜ p δ ( p ) sin ˜ p ( F ∗ G ∗ H )( p )= 0 , (7.29)where we have used the following relation: X x e − ipx = 4 πa δ ( p ) . (7.30)We are considering that our lattice coordinate space has infinite extension and thus the latticeis discrete but the momentum can be continuous. The relation (7.29) works exactly similar toa star product of more than 3 fields. Thus this star product has the desired property of (7.18).27fter defining the new star product we may look at the kinetic and mass terms. The S (2) without g (2)0 but with the regularization factor of (5.4) is given by S (2) = 2 a Z πa − πa dp π dp π πδ X i =1 sin ap i ! cos ap ap × h (1 − cos ap p )Φ( p ) + 12 sin a ( p − p )4 Ψ( p )Ψ( p ) i = X x h Φ( x ) ∗ (cid:16) x ) − Φ( x + a − Φ( x − a (cid:17) + i x + 3 a ∗ Ψ( x + a i . (7.31)It is interesting to recognize that the coordinate representation of the action with star producthas almost the same form of the kinetic term of the local action, S kin in (5.13), where the starproduct is just replaced by the normal product. The arguments of the fermionic lattice superfieldin S kin is shifted with a from that of (7.31). This is due to the loss of lattice translationalinvariance in the star product formulation while in the local expression the lattice translationalinvariance is recovered and thus a half lattice shift is equivalent in the action. This equivalentform correspondence between the star product action and the kinetic term of the local actionleads to a recognizability that the star product action is invariant under the supersymmetrytransformations by Q and Q acting on the star product of fields by keeping Leibniz rule sinceit is invariant for the local action. This correspondence in return to a result that the differenceoperator acting on local fields should satisfies Leibniz rule since the difference operator actingon the star product fields satisfies Leibniz rule. This solves the puzzle of the problem posed inthis section.This S (2) action in the star products form, however, is equivalent to a sum of both thekinetic terms and the mass terms with fixed coefficient, which include product of local fields. Inderiving the local action of the S kin in (5.13) and S mass in (5.16) the regularization factor forlattice momentum conservation (5.4) has not been included. In the star product formulation oflattice field theory the regularization factor is automatically involved since the lattice momentumitself ˆ p = a sin ap is the integration variable.Similarly to S (2) we can now derive the coordinate representation of the general interactionaction S ( n ) . We first note the following relation:2 πδ n X j =1 sin ap j = a X x Z d (sin ˜ p ) e − ipx δ sin ˜ p − n X j =1 sin ˜ p j . (7.32)Then the general interaction action (5.25) can be given by the following form: S ( n ) = 4 g ( n )0 n !(2 π ) n X x Z d ˜ p cos ˜ pe − ipx Z n Y j =1 ( d ˜ p j cos ˜ p j ) Z dτ e iτ (sin ˜ p − P nj =1 sin ˜ p j ) X y ,y , ··· y n e i ( P nj =1 ˜ p j m j ) h(cid:18) −
12 ( e i ˜ p + e − i ˜ p ) (cid:19) Φ( y )Φ( y ) · · · Φ( y n ) − ( n − i e i ˜ p − e i ˜ p )Ψ( y + a y + a y ) · · · Φ( y n ) i = 4 n ! g ( n )0 X x h(cid:16) x ) − Φ( x + a − Φ( x − a (cid:17) ∗ Φ( x ) n − ( x )+ ( n − i x + 3 a ∗ Ψ( x + a ∗ Φ n − ( x ) i , (7.33)28here the relations: x = na , ˜ p j = p j a , y j = m j a should be understood and J n ± ( τ ) is given in(7.26). Φ n ( x ) is a star product of n bosonic superfields. We have defined the star product of n fields as: F ( x + b ) ∗ F ( x + b ) ∗ · · · ∗ F n ( x + b n ) = Z ∞−∞ dτ J n ± ( τ ) X m , ··· ,m n n Y j =1 J m j ± ( τ ) F j ( y j + b j ) . (7.34)The non local nature of the star product should disappear in the continuum limit. This ishowever non trivial due to the p → πa − p symmetry of the sin ap function and the existence oftwo translationally invariant vacua at p = 0 and p = πa . It was shown by Dondi and Nicolai [16]that in the continuum limit namely at fixed x with a → J xa ( τ ) → δ ( τ − xa ) . (7.35)However in the present context the continuum limit picks up also the configuration at p = πa and the previous result has to be replaced by: J xa ( τ ) → δ ( τ − xa ) + ( − xa δ ( τ + 2 xa ) . (7.36)Thus locality is recovered in the continuum limit, but with an extra coupling of fields in thepoints x and − x accompanied with the alternating sign factor ( − xa . Such remaining nonlocality disappears when the lattice field Φ and Ψ are reinterpreted in terms of component fieldsas discussed in the previous section.We have now found a consistent definition of supersymmetry algebra in the coordinate spaceas well by the star product which assures the Leibniz rule operation of difference operator. Thesupercharge operation for star product of fields satisfies Leibniz rule and has the following form: Q j ( F ( x ) ∗ G ( x )) = Q j F ( x ) ∗ G ( x ) + ( − | F | F ( x ) ∗ Q j G ( x ) . (7.37)Thus the operation of supersymmetry charges and translation generators on lattice superfieldsare consistently defined as an algebra both in momentum and coordinate representation.We finally comment on an interesting possibility: “The formulation of the present model with lattice momentum conservation, equivalently the starproduct formulation of lattice theory, and that of link approach are equivalent.” This is based on the following observation: the algebraic correspondence of ˆ ∂ and Q j , respec-tively, (7.14) and (7.37) with respect to (7.9) and (7.10) are exactly parallel and algebraicallyboth of frameworks have one to one correspondence if the mild noncommutativity of (7.12) isintroduced in the link approach. The current formulation of algebra is Lie algebraic latticesupersymmetry with a new star product in the coordinate representation while the link ap-proach is Hopf algebraic lattice supersymmetry. The nonlocality in the star product and thenoncommutativity in the link approach are corresponding. We have proposed a new lattice supersymmetry formulation which ensures an exact Lie algebraicsupersymmetry invariance on the lattice for all super charges even with interactions. We have29ntroduced bosonic and fermionic lattice superfields which accommodate species doublers asbosonic and fermionic particle fields of super multiplets. This lattice superfield formulationis, however, not a naive extension of continuum superfield formulation in the sense that thereappear higher order irrelevant terms, including species doubler particle fields, which do notappear in the continuum formulation because of dimensional reason. These irrelevant termsare, however, crucial to assure the exact lattice supersymmetry invariance. We consider thatthis lattice superfield formulation is fundamental for a regularization of supersymmetry on thelattice.As the simplest model we have explicitly investigated N = 2 model in one dimension. Themodel includes interaction terms and the exact lattice supersymmetry invariance of the actionfor two supersymmetry charges are shown explicitly. In the momentum representation of theformulation the standard momentum conservation is replaced by the lattice counterpart of mo-mentum conservation: the sine momentum conservation. The basic lattice structure of this onedimensional model is half lattice spacing structure and the lattice supersymmetry transforma-tion is essentially a half lattice spacing translation. The super coordinate structure and themomentum representation of species doubler fields is hidden implicitly in the alternating signstructure of a half lattice spacing in the coordinate space. This sign change with a half latticespacing shift is a typical phenomenon of lattice regularization and crucially related to the bothlattice supersymmetry and the chiral fermion regularization. The sign change is a typical of lat-tice regularization and can never appear in the continuum regularization and thus is consideredto be fundamental for the lattice supersymmetry.Since we introduce the lattice (sine) momentum conservation: the translational invariance isbroken. We have investigated this problem and shown explicitly how the translational invariancerecovers in the continuum limit. The Ward-Takahashi identity is investigated for a model withΦ interaction term and it is confirmed that the identity is satisfied in one loop level and issatisfied in all orders since this model is shown to be super renormalizable. In the previousinvestigation a fermionic species doubler contribution from Wilson term, having inverse latticeconstant mass, was responsible to break the identity and the bosonic counter term was necessaryto cancel this unwanted term [10, 11]. In our model the species doubler contribution of fermionis identified as a physical contribution as a super multiplet and this fermionic contribution canbe compensated by the bosonic counterpart of species doubler.Since the symmetric difference operator does not satisfy Leibniz rule, it was very natural toask how the supersymmetry algebra be consistent in the coordinate space since super chargessatisfy Leibniz rule. In the link approach this problem was avoided by introducing shift naturefor super charges. In the current formulation this puzzle is beautifully solved by introducinga new star product of lattice superfields: The difference operator satisfies Leibniz rule on thestar products of lattice super fields. Then Lie algebraic exact supersymmetry is realized in thiscoordinate formulation of star product lattice field theory. This formulation provides a newwell defined regularization scheme of fermions and bosons without species doubling problem offermions. One may say otherwise, the regularization of fermions on the lattice inevitably leadsto supersymmetric lattice fields theories. It is recognized that the Lie algebraic structure of lattice supersymmetry and the algebraof link approach are totally one to one corresponding if mild noncommutativity is introducedto the link approach. This suggests an interesting possibility that the Lie algebraic latticesupersymmetry and the Hopf algebraic supersymmetry of link approach are equivalent.
Thenonlocality in the star product and the noncommutativity in the link approach are corresponding.The confirmation of this interesting possibility will be left for future investigation.30ince we have established a new lattice supersymmetry formulation which has exact super-symmetry on the lattice, it would be important to extend the formulation into higher dimensionsand to the models with gauge fields.
Acknowledgments
This work was supported in part by Japanese Ministry of Education, Science, Sports andCulture under the grant number 50169778 and also by Insituto Nazionale di Fisica Nucleare(INFN) research funds. I. Kanamori is financially supported by Nishina memorial foundation.
AppendixA Calculation of the 1-loop contribution to 2-point functions
Let us consider the four points interaction term. To write it down it is convenient to introducethe following combination: φ ± (ˆ p ) ≡ a (cid:16) ϕ ′ (ˆ p ) ± a D ′ (ˆ p ) (cid:17) . (A.1)The 4 point interaction terms become S (4)B = 163 ga Z /a − /a d ˆ p π d ˆ p π d ˆ p π d ˆ p π πδ (ˆ p + ˆ p + ˆ p + ˆ p ) × [ φ + (ˆ p ) φ + (ˆ p ) φ + (ˆ p ) φ + (ˆ p ) − c (ˆ p ) φ − (ˆ p ) φ + (ˆ p ) φ + (ˆ p ) φ + (ˆ p )] , (A.2) S (4)F = 8 ga Z /a − /a d ˆ p π d ˆ p π d ˆ p π d ˆ p π πδ (ˆ p + ˆ p + ˆ p + ˆ p ) × h i cos a ( p + p )4 ψ ′ (ˆ p ) ψ ′ (ˆ p ) φ + (ˆ p ) φ + (ˆ p ) (A.3)+ 12 sin a ( p − p )4 ψ ′ (ˆ p ) ψ ′ (ˆ p ) φ + (ˆ p ) φ + (ˆ p ) (A.4)+ 12 sin a ( p − p )4 ψ ′ (ˆ p ) ψ ′ (ˆ p ) φ + (ˆ p ) φ + (ˆ p ) i . (A.5)For the S (4)F , the momenta insides the trigonometric function is p instead of ˆ p .The propagators are h φ + (ˆ p ) φ + ( − ˆ p ) i = 1 + amD (ˆ p ) (A.6) h φ − (ˆ p ) φ − ( − ˆ p ) i = 1 − amD (ˆ p ) (A.7) h φ + (ˆ p ) φ − ( − ˆ p ) i = c (ˆ p ) D (ˆ p ) (A.8) h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = − a ˆ pD (ˆ p ) (A.9) h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = −h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = a imD (ˆ p ) , (A.10)31here 1 D (ˆ p ) = c (ˆ p ) a ˆ p − a m . (A.11)Since the propagators are not diagonal, we first calculate diagrams without contracting withthe external fields. They are φ + (ˆ p ) (cid:1) φ + ( − ˆ p ) = φ + (ˆ p ) φ + ( − ˆ p )( − ig )8 a Z /a − /a dk π D ( k ) 4(1 + am ) , (A.12) φ + (ˆ p ) (cid:1) φ − ( − ˆ p ) = φ + (ˆ p ) φ − ( − ˆ p )( − ig )8 a Z /a − /a dk π D ( k ) [ − am ) c (ˆ p )] , (A.13) φ − (ˆ p ) (cid:1) φ − ( − ˆ p ) = 0 (A.14) ψ ′ (ˆ p ) (cid:1) ψ ′ ( − ˆ p ) = ψ ′ (ˆ p ) ψ ′ ( − ˆ p )( − ig )8 a Z /a − /a dk π D ( k ) a ˆ p am ) , (A.15) ψ ′ (ˆ p ) (cid:1) ψ ′ ( − ˆ p ) = ψ ′ (ˆ p ) ψ ′ ( − ˆ p )( − ig )8 a Z /a − /a dk π D ( k ) a ˆ p am ) , (A.16) ψ ′ (ˆ p ) (cid:1) ψ ′ ( − ˆ p ) = ψ ′ (ˆ p ) ψ ′ ( − ˆ p )( − ig )8 a Z /a − /a dk π D ( k ) i (1 + am ) . (A.17)After contracting with the external lines, we obtain the 2-point functions for φ ± as h φ + (ˆ p ) φ + ( − ˆ p ) i = − ig a D (ˆ p ) Z /a − /a dk π D ( k ) 4(1 + am ) (cid:2) (1 + am ) − (1 + am ) c (ˆ p ) (cid:3) , (A.18) h φ − (ˆ p ) φ − ( − ˆ p ) i = − ig a D (ˆ p ) Z /a − /a dk π D ( k ) 4(1 + am ) amc (ˆ p ) , (A.19) h φ + (ˆ p ) φ − ( − ˆ p ) i = − ig a D (ˆ p ) Z /a − /a dk π D ( k ) 2(1 + am ) c (ˆ p ) (cid:2) (1 + am ) − c (ˆ p ) (cid:3) . (A.20)In terms of the scalar ϕ and the auxiliary field D these become: h ϕ ′ (ˆ p ) ϕ ′ ( − ˆ p ) i = h ϕ ′ (ˆ p ) ϕ ′ ( − ˆ p ) i tree F (ˆ p ) , (A.21) h D ′ (ˆ p ) D ′ ( − ˆ p ) i = h D ′ (ˆ p ) D ′ ( − ˆ p ) i tree F (ˆ p ) , (A.22) h D ′ (ˆ p ) ϕ ′ ( − ˆ p ) i = h D ′ (ˆ p ) ϕ ′ ( − ˆ p ) i tree F (ˆ p ) , (A.23)where F (ˆ p ) = − iga D (ˆ p ) Z /a − /a dk π D ( k ) 2(1 + am ) (cid:2) (1 + am ) − c (ˆ p ) (cid:3) , (A.24) F (ˆ p ) = − iga D (ˆ p ) Z /a − /a dk π D ( k ) 2(1 + am ) am (cid:2) (1 + am ) − (1 + 2 am ) c (ˆ p ) (cid:3) . (A.25)32or the fermionic 2-point functions, we obtain: h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i tree F (ˆ p ) , (A.26) h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i = h ψ ′ (ˆ p ) ψ ′ ( − ˆ p ) i tree F (ˆ p ) . (A.27) References [1] H.B.Nielsen and M.Ninomiya, Nucl.Phys.
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