Spontaneous supersymmetry breaking in two dimensional lattice super QCD
PPreprint typeset in JHEP style - HYPER VERSION
Spontaneous supersymmetry breaking in twodimensional lattice super QCD
Simon Catterall and Aarti Veernala
Department of Physics, Syracuse University, Syracuse, NY13244, USA
Abstract:
We report on a non-perturbative study of two dimensional N = (2 ,
2) super QCD.Our lattice formulation retains a single exact supersymmetry at non-zero lattice spacing, andcontains N f fermions in the fundamental representation of a U ( N c ) gauge group. The latticeaction we employ contains an additional Fayet-Iliopoulos term which is also invariant underthe exact lattice supersymmetry. This work constitutes the first numerical study of thistheory which serves as a toy model for understanding some of the issues that are expectedto arise in four dimensional super QCD. We present evidence that the exact supersymmetrybreaks spontaneously when N f < N c in agreement with theoretical expectations. Keywords:
Supersymmetry, SQCD, lattice, quiver. a r X i v : . [ h e p - l a t ] S e p ontents
1. Introduction 12. The starting point: twisted Q = 8 SYM in three dimensions 23. Two dimensional quivers from three dimensional lattice Yang-Mills 44. Vacuum Structure and SUSY Breaking Scenarios 75. Numerical Results 96. Conclusions 12A. Calculating the Bosonic Action 14
1. Introduction
In recent years a new approach to the problem of putting supersymmetric theories on thelattice has been developed based on discretization of a topologically twisted version of thecontinuum theory [1, 2, 3, 4, 5, 6]. Initially the focus was on lattice actions that targetpure super Yang-Mills theories in the continuum limit, in particular N = 4 super Yang-Mills[12, 13, 14, 15, 16]. For alternative approaches to numerical studies of N = 4 Yang-Mills seerefs. [17, 18, 19, 20]. However in [21] [22] these formulations were extended to the case oftheories incorporating fermions transforming in the fundamental representation of the gaugegroup and hence targeting super QCD. The starting point for these later lattice constructionsis a continuum quiver theory containing fields that transform as bifundamentals under aproduct gauge group U ( N c ) × U ( N f ). After discretization these bifundamental fields connecttwo separate lattices and, in the limit that the U ( N f ) gauge coupling is sent to zero, yield asuper QCD theory with a global U ( N f ) flavor symmetry. This construction is described indetail in section 3. The lattice action we have employed in this work includes an additionalFayet-Illopoulos term which, while invariant under the exact lattice supersymmetry, generatesa potential for the scalar fields. It is straightforward to show that this yields a non-zerovacuum expectation value for the auxiliary field (D term supersymmetry breaking) if N f < N c .In section 4. we show the results from numerical simulations of this theory which support this The same lattice theories can be obtained using orbifold methods and indeed supersymmetric latticeactions for Yang-Mills theories were first constructed using this technique [7, 8, 9, 10] and the connectionbetween twisting and orbifold methods forged in [11] – 1 –onclusion; we measure a non-zero vacuum energy and show that a light state - the Goldstino-appears in the spectrum of the theory if N f < N c . In contrast we show that vacuum energyis zero and this state is absent from the spectrum when N f > N c which is consistent with theprediction that the theory does not spontaneously break supersymmetry in that case.
2. The starting point: twisted Q = 8 SYM in three dimensions
We start from the continuum eight supercharge ( Q = 8) theory in three dimensions which iswritten in terms of twisted fields which are completely antisymmetric tensors in spacetimeunder the twisted SO(3) group. The original two Dirac fermions reappear in the twistedtheory as the components of a K¨ahler-Dirac field Ψ = ( η, ψ a , χ ab , θ abc ) where the indices a, b, c = 1 . . .
3. The bosonic sector of the twisted theory comprises a complexified gaugefield A a = A a + iB a containing the original gauge field A a and an additional vector field B a . This additional field contains the three scalars expected of the eight supercharge theorywhich, being vectors under the R symmetry, transform as a vector field after twisting. Thecorresponding action S = S exact + S closed where S exact = 1 g Q Λ = 1 g Q (cid:90) d x Tr (cid:20) χ ab ( x ) F ab ( x ) + η ( x ) (cid:2) D a , D a (cid:3) + 12 η ( x ) d ( x ) (cid:21) , (2.1) S closed = − g (cid:90) d x Tr (cid:2) θ abc ( x ) D [ c χ ab ] ( x ) (cid:3) . (2.2)Here all fields are in the adjoint representation of a U ( N ) gauge group X = (cid:80) N a =1 X a T a and we adopt an antihermitian basis for the generators T a . D a and D a are the continuumcovariant derivatives defined in terms of the complexified gauge fields as D a = ∂ a + A a and D a = ∂ a + A a . The action of the scalar supersymmetry on the fields is given by QA a = ψ a QA a = 0 Q ψ a = 0 Q χ ab = −F ab Q η = d Q θ abc = 0 (2.3)Notice that we have included an auxiliary field d ( x ) that allows the algebra to be off-shellnilpotent Q = 0. This feature then guarantees that S exact is supersymmetric. The equationof motion for this auxiliary field is then d ( x ) = (cid:2) D a , D a (cid:3) (2.4)– 2 –he Q -invariance of S closed follows from the Bianchi identity (cid:15) abc D c F ab = 0 . (2.5)To discretize this theory we place all fields on the links of a lattice. This 3d lattice consists ofthe usual hypercubic vectors plus additional face and body links. In detail these assignmentsare continuum field lattice link A a ( x ) x → x + ˆ a A a ( x ) x + ˆ a → xψ a ( x ) x → x + ˆ aχ ab x + ˆ a + ˆ b → xη ( x ) x → xd ( x ) x → xθ abc x → x + ˆ a + ˆ b + ˆ c The lattice gauge field will be denoted U µ ( x ) in the following discussion. For the scalarfields d ( x ), η ( x ) the link degenerates to a single site. Notice that the orientation of a givenfermion link field is determined by the even/odd character of its corresponding continuumantisymmetric form. The link character of a field determines its transformation propertiesunder lattice gauge transformations eg. U a ( x ) → G ( x ) U a ( x ) G † ( x + ˆ a ). To complete the con-struction of the lattice action it is necessary to replace continuum covariant derivatives byappropriate gauged lattice difference operators. The necessary prescription was described in[4], [21], [23]. It is essentially determined by the simultaneous requirements that the latticedifference agree with the continuum derivative as the lattice spacing is sent to zero and thatit yields expressions that transform as the appropriate link field under lattice gauge transfor-mations. The lattice difference operators acting on a field f ( ± ) a , where ( ± ) corresponding tothe orientation of the field , are given by: D (+) a f (+) b ,b ,...,b n ( x ) = U a ( x ) f (+) b ,b ,...,b n ( x + ˆ a ) − f (+) b ,b ,...,b n ( x ) U a ( x + ˆ b ) (2.6) D (+) a f ( − ) b ,b ,...,b n ( x ) = U a ( x + ˆ b ) f ( − ) b ,b ,...,b n ( x + ˆ a ) − f ( − ) b ,b ,...,b n ( x ) U a ( x ) (2.7) D (+) a f (+) b ,b ,...,b n ( x ) = f (+) b ,b ,...,b n ( x + ˆ a ) U a ( x + ˆ b ) − U a ( x ) f (+) b ,b ,...,b n ( x ) (2.8) D (+) a f ( − ) b ,b ,...,b n ( x ) = f ( − ) b ,b ,...,b n ( x + ˆ a ) U a ( x ) − U a ( x + ˆ b ) f ( − ) b ,b ,...,b n ( x ) (2.9) D ( − ) a f ( ± ) b ,b ,...,b n ( x ) = D ( ± ) f ( ± ) b ,b ,...,b n ( x − ˆ a ) (2.10) D ( − ) a f ( ± ) b ,b ,...,b n ( x ) = = D ( ± ) f ( ± ) b ,b ,...,b n ( x − ˆ a ) , (2.11) Note that it is also possible to write the 3d action completely in terms of an Q -exact form without a Q -closed term by employing an additional auxiliary field B abc Note that ψ a ( x ) and θ abc (x) originate from lattice site x and are, thus, positively oriented. χ ab ( x ), however,terminates at lattice site x and this therefore assigned a negative orientation. – 3 –here ˆ b = (cid:80) ni =1 ˆ b i in equations (2.6) to (2.9). For example the continuum derivative D a ψ b becomes D (+) a ψ b ( x ) = U a ( x ) ψ b ( x + ˆ a ) − ψ b ( x ) U a ( x + ˆ b ) (2.12)This prescription yields a set of link paths which, when contracted with the link field χ ab ( x ),yields a closed loop whose trace is gauge invariant:Tr (cid:104) χ ab ( x ) (cid:16) U a ( x ) ψ b ( x + ˆ a ) − ψ b ( x ) U a ( x + ˆ b ) (cid:17)(cid:105) (2.13)It has the correct naive continuum limit provided that (in some suitable gauge) we can expand U a ( x ) = I N + A a ( x ). The field strength on the lattice, F ab ( x ), is defined using the forwarddifference operator as: F ab ( x ) = D (+) a U b ( x ) . (2.14)In lattice QCD the unit matrix arising in this expansion is automatic since the link fields taketheir values in the group. However the constraints of exact lattice supersymmetry require thatthe lattice gauge fields take their values, like the fermions, in the algebra. In this case the unitmatrix can then be interpreted as arising from giving a vev to the trace mode of the originalscalar fields B a . This feature is required by lattice supersymmetry but is only possible becausewe are working with a complexified U ( N ) theory - another indication of the tight connectionbetween twisting and exact supersymmetry. It also implies that the path integral definingthe quantum theory will use a flat measure rather than the usual Haar measure employedin conventional lattice gauge theory. Such a prescription would usually break lattice gaugeinvariance but again complexification comes to the rescue since the Jacobian resulting froma gauge transformation of the D U measure cancels against an equivalent one coming from D U .We now show how to use this three dimensional lattice model to construct a two dimen-sional quiver theory while maintaining the exact lattice supersymmetry.
3. Two dimensional quivers from three dimensional lattice Yang-Mills
Consider a lattice whose extent in the 3-direction comprises just two 2d slices. Furthermore weshall assume free boundary conditions in the 3-direction so that these two slices are connectedby just a single set of links in the 3-direction - those running from x = 0 to x = 1. Ignoringfor the moment any fields that live on these latter links it is clear that the gauge group canbe chosen independently on these two slices. We choose a group U ( N c ) for the slice at x = 0and U ( N f ) at x = 1 and will henceforth refer to them as the N c and N f lattices. Denotingdirections on the 2d slices by Greek indices µ, ν = 1 , N c : Ψ( x ) = ( η, ψ µ , χ µν ) , U µ = I N c + A µ , d (3.1) N f : ˆΨ( x ) = (cid:16) ˆ η, ˆ ψ µ , ˆ χ µν (cid:17) , ˆ U µ = I N f + ˆ A , µ , ˆ d (3.2)– 4 –n these expressions x ( x ) denotes the coordinates on the N c ( N f ) lattice and 1 N c ( N f ) denotethe N c ( N f ) × N c ( N f ) unit matrix respectively. Now consider fields that live on the linksbetween the N c and N f lattice. These must necessarily transform as bi-fundamentals under U ( N c ) × U ( N f ). We have, N c × N f : Ψ bi-fund ( x, x ) = ( ψ , χ µ , θ µν ) = ( λ, λ µ , λ µν ) , φ (3.3)The second equality in the above equation is a mere change of variables and corresponds tolabeling fields according to their two dimensional character. The complete field content ofthis model is summarized in the table below: N c -lattice Bi-fundamental fields N f -lattice x ( x, x ) , ( x, x ) x A µ ( x ) φ ( x, x ) ˆ A µ ( x ) η ( x ) λ ( x, x ) ˆ η ( x ) ψ µ ( x ) λ µ ( x + µ, x ) ˆ ψ µ ( x ) χ µν ( x ) λ µν ( x, x + µ + ν ) ˆ χ µν ( x )Defining G(x) as a group element belonging to U ( N c ) and H(x) to U ( N f ) the lattice gaugetransformations for the bi-fundamental fields are as follows: φ ( x ) → G ( x ) φ ( x ) H † ( x ) λ ( x ) → G ( x ) λ ( x ) H † ( x ) λ µ ( x ) → H ( x + µ ) λ µ ( x ) G † ( x ) λ µν ( x ) → G ( x ) λ µν ( x ) H † ( x + µ + ν ) (3.4)It is crucial to note that this generalization of the original lattice super Yang-Mills theory toa quiver model is completely consistent with both the quiver gauge symmetries and the exactsupersymmetry. For example the 3d term given in eqn. 2.13 yields a bi-fundamental term ofthe form Tr (cid:104) λ µ ( x ) (cid:16) U µ ( x ) λ ( x + µ ) − λ ( x ) ˆ U µ ( x ) (cid:17)(cid:105) (3.5)which is invariant under the the generalized gauge transformations given in eqn. 3.4. Thus,the above construction lends us a consistent lattice quiver gauge theory containing bothadjoint and bi-fundamental fields transforming under a product U ( N c ) × U ( N f ) gauge group.Consider now setting the U ( N f ) gauge coupling to zero. This sets ˆ U µ = I N f up to gaugetransformations and it is then consistent to set all other fields on the N f lattice to zero. Theoriginal U ( N f ) gauge symmetry now becomes a global U ( N f ) flavor symmetry which acts ona set of complex scalar fields φ transforming in the fundamental representation of the gaugegroup and their fermionic superpartners ( λ, λ µ , λ µν ). The situation is depicted in figure 1. At– 5 –his point we have the freedom to add to the action one further supersymmetric and gaugeinvariant term - namely r (cid:80) x Tr d ( x ) = r Q (cid:80) x Tr η . This is a Fayet-Iliopoulos term. Itspresence changes the equation of motion for the auxiliary field d ( x ) = D ( − ) µ U µ ( x ) + φ ( x ) φ ( x ) − rI N c (3.6)with I N c a N c × N c unit matrix. The SUSY transformations for the remaining adjoint andfundamental fields are: Adjoint Fields Fundamental fields QA µ = ψ µ Q φ = λ QA µ = 0 Q φ = 0 Q ψ µ = 0 Q λ = 0 Q χ µν = −F µν Q λ µ = −D µ φ Q η = d Q λ µν = 0After integration over d the Fayet-Iliopoulos term yields a scalar potential term which will playa crucial role in determining whether the system can undergo spontaneous supersymmetry ⇥ µ , ⇥ µ ( , µ , µ ) ⇥ U µ , U µ , ( , ⇤ µ , ⇥ µ ) ⇤ Frozen (Non-dynamical) ( b i ) F und a m e n t a l M a tt e r ( b i ) F und a m e n t a l M a tt e r U ( N c ) SYM Adjoint Model U ( N F ) SYM Adjoint Model Figure 1:
3d quiver model – 6 –reaking. The final action may be written as S adj = κ (cid:88) x Tr (cid:20) −F µν ( x ) F µν ( x ) −
12 ( D ( − ) µ U µ ) − η ( x ) D ( − ) µ ψ µ ( x ) − χ µν ( x ) D (+)[ µ ψ ν ] ( x ) (cid:21) , (3.7) S fund = κ (cid:88) x Tr (cid:20) −D (+) µ φ ( x ) D (+) µ φ ( x ) − (cid:104)(cid:0) φ ( x ) φ ( x ) − rI (cid:1) (cid:105) + (cid:104) D ( − ) µ U µ ( x ) (cid:105) (cid:0) φ ( x ) φ ( x ) − rI (cid:1)(cid:21) − (cid:104) η ( x ) λ ( x ) φ ( x ) + (cid:110) λ µ ( x ) D (+) µ λ ( x ) − λ µ ( x ) ψ µ ( x ) φ ( x + µ ) (cid:111) − (cid:110) λ µν ( x ) D (+) µ λ ν ( x ) − λ µν ( x ) φ ( x + µ + ν ) χ µν ( x ) (cid:111)(cid:105) , (3.8)In practice we have also included the following soft SUSY breaking mass term, S soft , in theadjoint action, S adj in equation (3.7): S soft = µ (cid:20) N c Tr (cid:0) U µ U µ (cid:1) − (cid:21) . (3.9)Such a term is necessary to create a potential for the trace mode of the twisted scalar fieldsas we have discussed earlier. In principle we should extrapolate µ → µ . In practice we observethat these soft breaking effects are rather small.Finally, the lattice coupling κ appearing above is given by: κ = N c LT λA . (3.10)Here, λ = g N c is the dimensionful ‘t Hooft coupling, L and T are the numbers of points ineach direction of the 2d lattice and A is a continuum area - the importance of interactionsin the theory being controlled by the dimensionless combination λA . When we later discussour numerical results we refer to this dimensionless combination as simply λ .
4. Vacuum Structure and SUSY Breaking Scenarios
Let us return to the equation of motion for the auxiliary field d ( x ). If we sum the trace ofthis expression over all lattice sites and take its expectation value we find (cid:104) (cid:88) x Tr d ( x ) (cid:105) = (cid:104) (cid:88) x Tr (cid:0) φ ( x ) φ ( x ) − rI N c (cid:1) (cid:105) (4.1)Since the lefthand side of this expression is the expectation value of the Q -variation of someoperator the question of whether supersymmetry breaks spontaneously or not is determined– 7 –y whether the righthand side is non-zero. Indeed after we integrate over the auxiliary field d we find a scalar potential of the form S Dterm = N f (cid:88) x,f =1 κ (cid:16) φ f ( x ) φ f ( x ) − rI N c (cid:17) , (4.2)Consider the case where N f < N c . Using SU ( N c ) transformations one can diagonalize the N c × N c matrix φφ . In general it will have N f non-zero real, positive eigenvalues and N c − N f zero eigenvalues. This immediately implies that there is no configuration of the fields φ wherethe potential is zero. Indeed the minimum of the potential will have energy r ( N c − N f ) andcorresponds to a situation where N f scalars develop vacuum expectation values breaking thegauge group to U ( N c − N f ). The situation when N f ≥ N c is qualitatively different; now therank of φφ is at least N c and a zero energy vacuum configuration is possible. In such a sit-uation N c scalars pick up vacuum expectation values and the gauge symmetry is completelybroken.For the case when N f < N c where Q -supersymmetry is expected to break we would ex-pect the spectrum of the theory to contain a massless fermion - the goldstino [24]. To seehow this works in the twisted theory consider the vacuum energy (cid:104) | H | (cid:105) (cid:54) = 0 , (4.3)which is equivalent to < {Q , O} > (cid:54) = 0 for some operator O . In the two dimensional twistedtheory the relevant part of the supersymmetry algebra is {Q , Q µ } = P µ [25] so that eqn. 4.3is equivalent to (cid:104) | {Q , Q } | (cid:105) (cid:54) = 0 , (4.4)Note that the equation above involves both the scalar Q and the 1-form supercharge Q µ .Corresponding to these supercharges are a set of supercurrents, J and J µ whose form canbe derived in the usual manner by varying the continuum twisted action under infinitesimalspacetime dependent susy transformations. This yields gauge invariant supercurrents on thelattice of the following form J ( x ) = (cid:88) µ (cid:2) ψ µ ( x ) U µ ( x ) (cid:3) d ( x ) + ..., (4.5) J ( x ) = η ( x ) d ( x ) + ..., (4.6)and using the equations of motion, the auxiliary field d(x) can be replaced by d ( x ) = (cid:88) µ =1 , (cid:2) D µ , D µ (cid:3) + (cid:2) φ ( x ) φ ( x ) − rI N c (cid:3) (4.7)We therefore expect a possible Goldstino signal to manifest itself in the contribution of a lightstate to the two-point function: C ( t ) = (cid:104) |O ( x ) O (cid:48) ( y ) | (cid:105) , (4.8)– 8 –here ‘t’ corresponds to ( x − y ) and a suitable set of lattice interpolating operators aregiven by: O ( x ) = Tr (cid:34)(cid:88) µ ψ µ ( x ) U µ ( x ) (cid:0) φ ( x ) φ ( x ) − rI N c (cid:1)(cid:35) . (4.9)and O (cid:48) ( y ) = Tr (cid:2) η ( y ) (cid:0) φ ( y ) φ ( y ) − rI N c (cid:1)(cid:3) . (4.10)
5. Numerical Results
We employ a RHMC algorithm to simulate our system having first replaced all the twistedfermions in our model by corresponding pseudofermions - see for example [26] [27]. Thesimulations are performed by imposing anti-periodic (thermal) boundary conditions on thefermions along one of the two space-time directions. This is done to avoid running into thefermion zero modes resulting from the scalar component of the twisted fermion, η . As dis-cussed in [28] [29] this has the added benefit of ameliorating the sign problem for these latticetheories. This breaks supersymmetry explicitly by a term that vanishes as the lattice volumeis increased.In this section, we contrast results from simulations with N f = 2, N c = 3 correspondingto the predicted susy breaking scenario with results from simulations with N f = 3, N c = 2- the susy preserving case. We ran our simulations for three different values of the ‘t Hooftcoupling, λ = 0 . , . λ . The results presented in this section correspond to λ = 1 .
0. The FI parameter,r, is a free parameter and is set to 1.0 for the rest of the discussion.As a first check, we compared the expectation value of the bosonic action with the theo-retical value obtained using a supersymmetric Ward identity < κS boson > = (cid:18) N c + N c N f (cid:19) V. (5.1)In appendix A. we show how to compute this value. Figure 2 shows a plot of the bosonicaction for various values of the soft SUSY breaking coupling µ . In principle we should takethe limit µ → µ dependence is in factrather weak. We have normalised the data to its value obtained by assuming supersymmetryis unbroken. The red points at the bottom of the figure denote the SUSY preserving case andit can be observed that they agree with the theoretical prediction. This is to be contrastedwith the case when N f < N c denoted by the blue points which shows a large deviation fromeqn. 5.1 and is the first sign that supersymmetry is spontaneously broken in this case.The spatial Polyakov lines shown in figure 3 also show a distinct difference between the N f < N c and N f > N c cases. The red lines where | P | ≈ c = 3; N f = 2 N c + N c N f V Soft SUSY breaking mass, µ ⇥ = 1 . Figure 2:
Normalized bosonic action vs soft breaking coupling µ for λ = 1 . case and are consistent with a deconfined or fully Higgsed phase. Indeed the Polyakov line is atopological operator and in a susy preserving phase should be coupling constant independentconsistent with what is seen. The blue line in the lower half of the plot corresponds to smallervalues which is qualitatively consistent with the predicted partial Higgsing of the gauge fieldin the phase where supersymmetry is spontaneously broken.One of clearest signals of supersymmetry breaking can be obtained if one considers the equa-tion of motion for the auxiliary field eqn. 4.2. We expect the susy preserving case to obey1 N c Tr (cid:2) φ ( x ) φ ( x ) (cid:3) = 1 . (5.2)The red points, corresponding to ( N f > N c ) are consistent with this over a wide range of µ . We attribute the small residual devaition as µ → Q susy breaking into the system. The simulations with N f < N c (blue points) however show a clear signal for spontaneous supersymmetry breakingwith the value of this quantity deviating dramatically from its supersymmetric value even as µ → C ( t ) = (cid:88) x,y < O (cid:48) ( y, t ) O ( x, > (5.3)– 10 – × λ = 1 . Soft SUSY breaking mass, µ Figure 3:
Spatial Polyakov line vs µ for λ = 1 . × λ = 1 . Soft SUSY breaking mass, µ N c Tr φφ Figure 4: N c Tr φφ vs µ for a ’t Hooft coupling of λ = 1 . where O (cid:48) ( y, t ) and O(x,0) are fermionic operators given by: O ( x,
0) = ψ µ ( x, U µ ( x, (cid:2) φ ( x, φ ( x, − rI N c (cid:3) (5.4) O (cid:48) ( y, t ) = η ( y, t ) (cid:2) φ ( y, t ) φ ( y, t ) − rI N c (cid:3) . (5.5)– 11 – = 1 . µ = 0 . Figure 5:
Correlation function C(t) for λ = 1 . µ = 0 . Since it is computationally very cumbersome to evaluate the above correlation function forevery lattice site x at the source we instead evaluate the correlator for every lattice site y fora few randomly chosen source points x. In figure 5 we show the logarithm of this correlatoras a function of temporal distance for a range of spatial lattice size, L = 6 , ,
12 and 14.The anti-periodic boundary condition is applied along the temporal direction correspondingto T=16 and for both N f > N c and N f < N c . The approximate linearity of these curves isconsistent with the correlator being dominated by a single state in both cases. However when N f > N c the amplitude of this correlator is strongly suppressed relative to the case where N f < N c . Furthermore the effective mass extracted from fits to this latter correlator (figure6) falls as the spatial lattice size (L) increases, consistent with a vanishing mass in the largevolume limit. The lines in figure 6 show fits to 1 /L - the smallest mass consistent with theboundary conditions - the dashed green line is a fit constrained to go through the origin whilethe dotted red line allows the intercept to float. This is just what we would expect of a wouldbe Goldstino arising from spontaneous breaking of the exact Q -symmetry.
6. Conclusions
In this paper, we have reported on a numerical study of super QCD in two dimensions. Themodel in question possesses N = (2 ,
2) supersymmetry in the continuum limit while ourlattice formulation preserves a single exact supercharge for non zero lattice spacing. It isexpected that the single supersymmetry will be sufficient to ensure that full supersymmetryis regained without fine tuning in the continuum limit. This constitutes the first lattice study– 12 – = 1 . µ = 0 . L Figure 6:
Goldstino mass derived from fits M eff vs inverse transverse lattice size, L − of a supersymmetric theory containing fields which transform in both the fundamental andadjoint representations of the gauge group. Our lattice action also contains a Q -exact Fayet-Iliopoulos term which yields a potential for the scalar fields. The lattice theory possessesseveral exact symmetries; U ( N c ) gauge invariance, Q -supersymmetry and a global U ( N f )flavor symmetry.It is expected that the system will spontaneously break supersymmetry if N f < N c . Thearguments that lead to this conclusion depend on the inclusion of the Fayet-Iliopoulos term.Such a term is rather natural in our lattice model since the formulation requires U ( N c ) gaugesymmetry. Notice, though, that the free energy of the lattice model does not naively dependon the coupling r as long as it is positive since the Fayet-Iliopoulos term is Q -exact. Ournumerical work is fully consistent with this picture; we have examined several supersymmetricWard identities which clearly distinguish between the N f < N c and N f > N c situations andwe have observed a would be Goldstino state in the former case.There are many directions for future work; inclusion of anti-fundamentals fields is straight-forward since it merely corresponds to including the bifundamental fields truncated from the N f -lattice. Observations of phase transitions in such models as the parameters are varied canthen potentially probe sigma models based on Calabi-Yau hypersurfaces [30]. It is possiblethat the SU ( N ) theories could be studied by deforming the moduli space of the lattice theory In contrast for r < N f /N c . Thus one expectsa phase transition in the N f ≥ N c theory at r = 0. – 13 –sing ideas similar to those presented in [31]. This would allow direct contact to be made tothe continuum calculations of Hori and Tong [32]. Finally the lattice constructions discussedin this paper generalize [33] to three dimensional quiver theories leaving open the possibilityof studying 3D super QCD using lattice simulations. A. Calculating the Bosonic Action
Consider the partition function Z = (cid:90) DXe − κ ( Q Λ+ S c ) (A.1)where DX denotes the measure over all boson and fermion fields and S c the Q -closed term.We start by rescaling the field θ abc → κθ abc to remove the coupling κ from in front of the Q -closed term. This yields Z = κ N c N f V (cid:90) DXe − κ Q Λ − S c = κ N c N f V Z (cid:48) (A.2)with V the two dimensional volume. Notice that N c N f is the number of fermions at each siteresulting from the 3d θ field. Differentiating with respect to κ gives − ∂ ln Z∂κ = − N c N f Vκ − ∂ ln Z (cid:48) ∂κ (A.3)The last term in the righthand side being Q -exact would yield zero in the original theorycontaining a d -field. However in the action we simulate this field is integrated out yieldinginstead a contribution of − κ N c V Putting these pieces together we find κ < S b > + κ < S f > = − N c N f V − N c V (A.4)The expectation value of the fermionic action can be gotten by scaling arguments since thefermions occur only quadratically in the action yielding κ < S f > = − V (cid:0) N c + N c N f (cid:1) (A.5)Collecting terms yields the final result quoted previously κ < S b > = V (cid:18) N c + N c N f (cid:19) (A.6) Acknowledgments
SMC is supported in part by DOE grant DE-SC0009998. SMC and AV would like to thankDavid Tong and David Schaich for useful discussions. The simulations were carried out usingUSQCD resources at Fermilab. – 14 – eferences [1] F. Sugino,
SuperYang-Mills theories on the two-dimensional lattice with exact supersymmetry , JHEP (2004) 067, [ hep-lat/0401017 ].[2] S. Catterall,
A Geometrical approach to N=2 super Yang-Mills theory on the two dimensionallattice , JHEP (2004) 006, [ hep-lat/0410052 ].[3] S. Catterall,
Lattice formulation of N=4 super Yang-Mills theory , JHEP (2005) 027,[ hep-lat/0503036 ].[4] S. Catterall, D. B. Kaplan, and M. Unsal,
Exact lattice supersymmetry , Phys.Rept. (2009)71–130, [ arXiv:0903.4881 ].[5] A. D’Adda, I. Kanamori, N. Kawamoto, and K. Nagata,
Exact extended supersymmetry on alattice: Twisted N=2 super Yang-Mills in two dimensions , Phys.Lett.
B633 (2006) 645–652,[ hep-lat/0507029 ].[6] P. H. Damgaard and S. Matsuura,
Geometry of Orbifolded Supersymmetric Lattice GaugeTheories , Phys.Lett.
B661 (2008) 52–56, [ arXiv:0801.2936 ].[7] A. G. Cohen, D. B. Kaplan, E. Katz, and M. Unsal,
Supersymmetry on a Euclidean space-timelattice. 1. A Target theory with four supercharges , JHEP (2003) 024, [ hep-lat/0302017 ].[8] A. G. Cohen, D. B. Kaplan, E. Katz, and M. Unsal,
Supersymmetry on a Euclidean space-timelattice. 2. Target theories with eight supercharges , JHEP (2003) 031, [ hep-lat/0307012 ].[9] D. B. Kaplan and M. Unsal,
A Euclidean lattice construction of supersymmetric Yang-Millstheories with sixteen supercharges , JHEP (2005) 042, [ hep-lat/0503039 ].[10] P. H. Damgaard and S. Matsuura,
Classification of supersymmetric lattice gauge theories byorbifolding , JHEP (2007) 051, [ arXiv:0704.2696 ].[11] M. Unsal,
Twisted supersymmetric gauge theories and orbifold lattices , JHEP (2006) 089,[ hep-th/0603046 ].[12] S. Catterall, D. Schaich, P. H. Damgaard, T. DeGrand, and J. Giedt,
N=4 Supersymmetry on aSpace-Time Lattice , Phys.Rev.
D90 (2014), no. 6 065013, [ arXiv:1405.0644 ].[13] S. Catterall and J. Giedt,
Real space renormalization group for twisted lattice N =4 superYang-Mills , JHEP (2014) 050, [ arXiv:1408.7067 ].[14] S. Catterall, P. H. Damgaard, T. Degrand, R. Galvez, and D. Mehta,
Phase Structure of LatticeN=4 Super Yang-Mills , JHEP (2012) 072, [ arXiv:1209.5285 ].[15] S. Catterall, E. Dzienkowski, J. Giedt, A. Joseph, and R. Wells,
Perturbative renormalization oflattice N=4 super Yang-Mills theory , JHEP (2011) 074, [ arXiv:1102.1725 ].[16] S. Catterall, J. Giedt, and A. Joseph,
Twisted supersymmetries in lattice N = 4 superYang-Mills theory , JHEP (2013) 166, [ arXiv:1306.3891 ].[17] M. Hanada, Y. Hyakutake, G. Ishiki, and J. Nishimura,
Holographic description of quantumblack hole on a computer , arXiv:1311.5607 .[18] M. Hanada, S. Matsuura, and F. Sugino, Two-dimensional lattice for four-dimensional N = 4 supersymmetric Yang-Mills , Prog. Theor. Phys. (2011) 597–611, [ arXiv:1004.5513 ]. – 15 –
19] M. Honda, G. Ishiki, J. Nishimura, and A. Tsuchiya,
Testing the AdS/CFT correspondence byMonte Carlo calculation of BPS and non-BPS Wilson loops in 4d N=4 super-Yang-Mills theory , PoS
LATTICE2011 (2011) 244, [ arXiv:1112.4274 ].[20] G. Ishiki, S.-W. Kim, J. Nishimura, and A. Tsuchiya,
Testing a novel large-N reduction for N=4super Yang-Mills theory on R x S**3 , JHEP (2009) 029, [ arXiv:0907.1488 ].[21] S. Matsuura,
Two-dimensional N=(2,2) Supersymmetric Lattice Gauge Theory with MatterFields in the Fundamental Representation , JHEP (2008) 127, [ arXiv:0805.4491 ].[22] F. Sugino,
Lattice Formulation of Two-Dimensional N=(2,2) SQCD with Exact Supersymmetry , Nucl.Phys.
B808 (2009) 292–325, [ arXiv:0807.2683 ].[23] S. Catterall,
From Twisted Supersymmetry to Orbifold Lattices , JHEP (2008) 048,[ arXiv:0712.2532 ].[24] E. Witten,
Dynamical Breaking of Supersymmetry , Nucl.Phys.
B188 (1981) 513.[25] S. Catterall,
Simulations of N=2 super Yang-Mills theory in two dimensions , JHEP (2006) 032, [ hep-lat/0602004 ].[26] S. Catterall and A. Joseph,
An Object oriented code for simulating supersymmetric Yang-Millstheories , Comput.Phys.Commun. (2012) 1336–1353, [ arXiv:1108.1503 ].[27] D. Schaich and T. DeGrand,
Parallel software for lattice N = 4 supersymmetric YangMillstheory , Comput.Phys.Commun. (2015) 200–212, [ arXiv:1410.6971 ].[28] S. Catterall, R. Galvez, A. Joseph, and D. Mehta,
On the sign problem in 2D lattice superYang-Mills , JHEP (2012) 108, [ arXiv:1112.3588 ].[29] S. Catterall, J. Giedt, D. Schaich, P. H. Damgaard, and T. DeGrand,
Results from latticesimulations of N=4 supersymmetric Yang–Mills , PoS
LATTICE2014 (2014) 267,[ arXiv:1411.0166 ].[30] E. Witten,
Phases of N=2 theories in two-dimensions , Nucl.Phys.
B403 (1993) 159–222,[ hep-th/9301042 ].[31] S. Catterall and D. Schaich,
Deforming the moduli space of lattice super yang-mills, inpreparation , .[32] K. Hori and D. Tong,
Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N=(2,2)Theories , JHEP (2007) 079, [ hep-th/0609032 ].[33] A. Joseph,
Lattice formulation of three-dimensional N=4 gauge theory with fundamental matterfields , JHEP (2013) 046, [ arXiv:1307.3281 ].].