SUSY on the Lattice I

SUSY on the lattice I. Montvay a aDeutsches Elektronen Synchrotron DESY, Notkestr. 85, D-22603 Hamburg, Germany

The motivation and perspectives of numerical simulations of supersymmetric Yang-Mills theories are reviewed.

1. INTRODUCTION combining the gauge coupling g with the Θ- parameter. The first line of eq. (1) is in superfield Supersymmetric quantum field theories, re- notation, in terms of the spinorial field strength viewed for instance in refs. [1]-[3], have peculiar superfield W (x, θ, θ¯)α which depends on the four- properties. They represent an interesting special coordinate x and the anticommuting Weyl-spinor subset of quantum field theories. Because of their variables θα, θ¯α˙ (α, α˙ =1,2). After performing highly symmetric nature, they are best suited for the grassmanian integration on θ, one obtains the analytical studies, which sometimes lead to exact second form of the SYM action in eq. (1) in terms solutions [4]. of the component fields: the field strength tensor ˜ 1 1.1. Supersymmetric Yang-Mills theory Fµν , its dual Fµν ≡ 2 µνρσ Fρσ, the Majorana ¯ Since local gauge symmetries play a very im- fermionfieldintheadjointrepresentationλ, λ portant rôle in Nature, there is a particular inter- and the auxiliary field D.Ofcourse,Dµdenotes µ 1,2,3 est in supersymmetric gauge theories. The sim- gauge covariant derivative and σ ≡{1,σ }. plest examples are supersymmetric Yang-Mills (For details of the formalism see refs. [1]-[3].) (SYM) theories which are supersymmetric ex- Since the auxiliary field without kinetic term tensions of pure gauge theories. Depending on D can be trivially integrated out, SSY M is noth- the number N of supersymmetry generator pairs ing else but a Yang-Mills theory with a massless Majorana fermion in the adjoint representation. Qiα, Q¯iα˙ ,(i=1,2,...,N), one speaks about N = 1 “simple” supersymmetry or N ≥ 2 “ex- Hence the Yang-Mills theory of such a field, called tended” supersymmetry. Interesting examples of “gaugino” or, in the context of strong interac- SYM theories have, for instance, N =1or2su- tions, “gluino”, is automatically supersymmetric. persymmetry. Introducing a non-zero gaugino mass mg˜ breaks The SYM action with N = 1 supersymmetry supersymmetry “softly”. Such a mass term is: can be written as follows: α α˙ mg˜(λ λα + λ¯ λ¯α˙ )=mg˜(ΨΨ)¯ . (3) 1 S = Im τ d4xd2θTr(W α W ) SY M 4π α Here in the first form the Majorana-Weyl compo- Z 1 1 nents λ, λ¯ are used, in the second form the Dirac- = d4 x Tr − F F µν g 2 2 µν Majorana field Ψ. Z µ ¯ ¯ µ 2 SSY M in eq. (1) defines the simplest, N =1 − iλσ (Dµλ)+i(Dµλ)¯σ λ + D SYM theory. It is reasonable to start the non- Θ 4 µν perturbative lattice studies by this. An impor- + d x Tr Fµν F˜ . (1) 16π2 tant feature of this theory is its global U(1) chi- Z λ h i ral symmetry which coincides with the so called Here τ is a complex coupling parameter R-symmetry generated by the transformations Θ 4πi τ ≡ + (2) 0 iϕ ¯0 −iϕ ¯ 2π g2 θα = e θα , θα˙ = e θα˙ . (4) 2

This is equivalent to which is equal to the difference of the number of bosonic minus fermionic states with zero en- 0 iϕ ¯0 −iϕ ¯ 0 −iϕγ5 λα = e λα , λα˙ = e λα˙ , Ψ = e Ψ . (5) ergy. It is supposed not to change with the pa- rameters of the theory. For SYM theory we have The U(1)λ-symmetry is anomalous: for the cor- w = N , therefore there is no spontaneous responding axial current Jµ ≡ Ψ¯ γµγ5Ψ, in case of SY M c supersymmetry breaking. (The N ground states SU(Nc) gauge group with coupling g,wehave c discussed above correspond to this Witten index.) 2 µ Ncg µνρσ r r ∂ Jµ = F F . (6) 32π2 µν ρσ 1.2. Seiberg-Witten solution The SYM theory with N = 2 extended super- However, the anomaly leaves a Z2Nc subgroup of U(1)λ unbroken. This can be seen, for instance, symmetry is a highly constrained theory which by noting that the transformations has, however, more structure than the relatively simple N = 1 case discussed above. In particular, iϕγ iϕγ Ψ → e− 5 Ψ , Ψ¯ → Ψ¯ e− 5 (7) besides the N = 1 “vector superfield” containing the gauge boson and gaugino (A ,λ), it also in- are equivalent to µ volves an N = 1 “chiral superfield” (φ, λ0)inthe −2iϕγ5 mg˜ → mg˜e , ΘSY M → ΘSY M − 2Ncϕ,(8) adjoint representation which consists of the complex scalar φ and the Majorana fermion λ0.The where ΘSY M is the Θ-parameter of gauge dynam- Majorana pair (λ, λ0) can be combined to a Dirac- ics. Since ΘSY M is periodic with period 2π,for fermion ψ and then the vector-like (non-chiral) mg˜=0theU(1)λ symmetry is unbroken if nature of this theory can be made explicit. kπ The main new feature of N = 2 SYM is that ϕ = ϕk ≡ , (k =0,1,...,2Nc −1) . (9) it contains also scalar fields, hence there is the Nc possibility of Higgs mechanism. Let us here For this statement it is essential that the topolog- consider only an SU(2) gauge group, when the ical charge is integer. Higgs mechanism implies the symmetry break- The discrete global chiral symmetry Z2Nc is ex- ing SU(2) → U(1). The complex expectation pected to be spontaneously broken by the non- value of the scalar field hφi parametrizes the mod- zero gaugino condensate hλλi6=0toZ2 defined uli space of zero-energy degenerate vacua. The by {ϕ0,ϕNc} (λ →−λis a rotation). Instan- degeneracy is a consequence of extended super- ton calculations (for a review and references see symmetry. Due to the Higgs mechanism the [5]) give at ΘSY M =0inNc degenerate vacua “charged” gauge bosons become heavy and the (k =0,...,Nc −1) low-energy effective theory is N =2SYMwith

2 2πik/Nc U(1) gauge group. g 4 e 3 hλλi0 = Λ (10) As it has been shown by Seiberg and Wit- 2 1/Nc SY M 32π [(Nc − 1)!(3Nc − 1)] ten [4], extended supersymmetry and asymptotic

Here ΛSY M denotes the Λ-parameter of the gauge freedom can be used to determine exactly the low- coupling. The transformation with ΘSY M is energy effective action in the Higgs phase, if the vacuum expectation value is large. It turns out iΘSY M /Nc hλλiΘSY M = e hλλi0 . (11) that, besides the singularity which corresponds In general, an important and interesting ques- to asymptotic freedom at infinity, at strong cou- tion is whether supersymmetry can be broken plings there are two singularities of the effective spontaneously or not. In pure SYM theory, with- action which describe light monopoles and dyons, out additional matter supermultiplets, this is not respectively. expected to occur. An argument for this is given The unprecedented achievement of an exact so- by the non-zero value of the Witten index [6] lution for a non-trivial four-dimensional quantum field theory induced a considerable activ- F w ≡ Tr (−1) = nboson − nfermion , (12) ity among theoreticians. Recently there is a new 3 way of considering non-perturbative problems in The doubling of the eigenvalues of Q˜ and/or Q quantum field theory. It is an interesting question implies that for fermions in the adjoint represen- how much the more conventional approach based tation zero eigenvalues always occur in pairs, as in on lattice regularization can contribute to these the continuum [8]. (This is different for fermions new insights. in the fundamental representation: see for instance [9].) 2. LATTICE FORMULATION Of course, the lattice action in eq. (13) is not unique. Another possibility is based on five- The simplest supersymmetric gauge theory is dimensional domain walls [10]. In this approach N =1SU(2) SYM. Massive gluinos (Majorana one knows the value of the bare fermion mass fermions in the adjoint representation) break su- where the supersymmetric continuum limit is persymmetry softly. As discussed in section 1.1, best approached and one may have advantages the supersymmetric point is at mg˜ =0. from the point of view of the speed of symme- 2.1. Actions try restoration, but one has to pay with the pro- The Curci-Veneziano action of N =1SYM[7] liferation of (auxiliary) fermion flavours. There is based on Wilson fermions. The effective gauge are also speculations about direct dimensional re- action obtained after integrating out the gluino duction on the lattice from d =10ord=6di- field is given by mensions down to the physically interesting case d = 4, started by ref. [11] and continued by 1 1 ref. [12]. However, up to now it has not yet SCV = β 1 − Tr Upl − log det Q[U] ,(13) 2 2 been demonstrated that this approach does really pl X work. where the fermion matrix is 2.2. Algorithms Qyv,xu = δyxδvu −K δy,x+ˆµ(1+γµ)Vvu,xµ(14) As one can see in the lattice action in eq. (13), µ=± Majorana fermions effectively mean a flavour X number N = 1 . This can be achieved by the and the gauge link in the adjoint representation f 2 hybrid classical dynamics algorithm [13], which is is defined as applicable to any number of flavours. Since it is a 1 † finite step size algorithm, an extrapolation to zero Vrs,xµ = Tr (UxµτrUxµτs) . (15) 2 step size is needed. This approach has been im- The Majorana nature of the gaugino is rep- plemented and successfully tested for the Curci- 1 Veneziano action (13) by Donini and Guagnelli in resented in eq. (13) by the factor 2 in front of log det Q. There is no problem with the sign of ref. [14]. det Q becausewehavedetQ=detQ˜,where Another possibility to simulate non-even numbers of flavours is based on the multi-bosonic algo- ˜ ˜† Q≡γ5Q=Q (16) rithm [15]. A two-step variant using a noisy correction step [16] has been developed in ref. [17]. is the hermitean fermion matrix. From the rela- This algorithm is based, for general N ,onthe tions f approximation CQC−1 = QT ,BQB˜−1 = Q˜T , (17) 1 det(Q†Q)Nf /2 , (19) ' † det Pn(Q Q) with the charge conjugation matrix C and B ≡ where the polynomial Pn satisfies Cγ5, it follows that every eigenvalue of Q and Q˜ −Nf/2 is (at least) doubly degenerate. Therefore, with lim Pn(x)=x (20) n→∞ the (real) eigenvalues λ˜i of Q˜,wehave within a suitably chosen positive interval x ∈ ˜ ˜2 [, λ], which covers the spectrum of Q†Q in typi- det Q = λi ≥ 0. (18) i cal gauge configurations. In the two-step variant Y 4

one uses a ﬁrst polynomial P¯n¯ for a crude approximation and realizes a ﬁne correction by another polynomial Pn, where now

−Nf/2 lim P¯n¯ (x)Pn(x)=x for x ∈ [, λ] . (21) n→∞ The fermion determinant is approximated as 1 det(Q†Q)Nf /2 .(22) ' † † det P¯n¯ (Q Q)detPn(Q Q) The advantage of this two-step variant is its smaller storage requirement and shorter autocor- relations, because the order of the first polyno- mialn ¯ can be kept small. (In praxis, for instance, n¯ ≤ 10−12 for lattice sizes ≤ 83 ·16.) As a consequence of the smallness ofn ¯, this remains a large step size algorithm, also in the limit of good pre- cision. In order to achieve an exact algorithm in this two-step variant of the multi-bosonic algorithm, Figure 1. The deviation norm δ2 of the polyno- in principle, an extrapolation to n →∞is mial approximations of x−1/4 as function of the needed, together with measurement-corrections order for different values of λ/.Theasterisks for eigenvalues outside [, λ]. In praxis, however, show the /λ → 0 limit. (Ref. [19]) (¯n, n)and[, λ] can be chosen such that these kinds of corrections are smaller than the statisti- lowest masses is shown in figure 2. Notation con- cal errors. ventions are as follows:π ˜ andσ ˜ denote, respec- Random matrix models suggest that the fluc- tively, the pion-like and sigma-like states made tuations of the minimal eigenvalue are inversely out of gluinos.χ ˜ is a Majorana fermion state proportional to the lattice volume. (For refer- made out of gluons and gluinos. ences and a recent summary see ref. [18].) This is advantageous for the choice of the interval [, λ]. 3. FIRST RESULTS AND OUTLOOK For the calculation of the necessary polynomials procedures written in Maple are available [19]. The goal of the DESY-Münster-Athens collab- The quality of the polynomial approximations is oration [24] is to study non-perturbative proper- illustrated by figure 1. ties of SYM theories by Monte Carlo simulations. Besides the above “hermitean” version “com- The computations are performed on the APE- plex” (non-hermitean) variants of the multi-boso- Quadrics at DESY-Zeuthen and on the CRAY- nic algorithms [20] can also be exploited, prefer- T3E at HLRZ-Jülich by using the two-step multi- ably also in two-step variants. In order to im- bosonic fermion algorithm for gluinos discussed in prove performance, preconditioning according to section 2.2. ref. [21] turned out to be very useful. 3.1. Chiral symmetry breaking 2.3. Quenched computations An important first step in the numerical inves- Quenched calculations without the effect of dy- tigations is to find the critical hopping parame- namical gluinos can be useful for roughly localiz- ter Kcr corresponding to zero gluino mass. As ing the critical region in the hopping parameter K it is discussed in section 1.1, the supersymmetric and for testing the methods to determine masses. point is at zero gluino mass. This tells how the Such calculations have been performed recently supersymmetric continuum limit β →∞has to in refs. [22] and [23]. An example of the obtained be performed in the (β,K)-plane. The basic ex- 5

Figure 2. Masses ofπ, ˜ σ,˜ χ˜ and 2ρ as a function of 1/K (the mass ofπ ˜ is squared). The extrapo- Figure 3. Expected phase structure in the (β,K)- plane. The dashed-dotted line is a ﬁrst order lation at Kcr is also shown (curves). (Ref. [23]) phase transition at zero gluino mass. pectation is that at K = Kcr, for gauge group SU(Nc), there are Nc degenerate ground states rameter K>Kcr we expect to be at ΘSY M = π. with diﬀerent values of the gluino condensate (see Since the Monte Carlo simulations are usually eq. (10)). This corresponds to the expected spon- done at non-zero gluino mass mg˜ 6=0,thebest taneous global chiral symmetry breaking pattern procedure is to derive analytically the dependence

Z2Nc → Z2 [6],[25]. of different quantities on mg˜ [27] and compare The coexistence of several different vacua im- them directly with numerical data. In case of the plies a first order phase transition at K = Kcr, gluino condensate the best possibility is to study at least in the continuum limit. At finite lattice the difference ∆hλλi between the values of hλλi spacing in the (β,K)-plane this might be repre- in different vacua. The advantage is that the ad- sented just by cross-over lines. Nevertheless, the ditive ultraviolet divergences, due to the breaking simplest possibility is a first order phase transi- of supersymmetry at finite lattice spacing, cancel. tion already for finite β. For instance, in case of SU(2) gauge group the phase structure may be 3.2. Bound state spectrum as shown by figure 3. An alternative phase struc- The low energy effective actions predict that at ture can be motivated by the modified Veneziano- the supersymmetric point the low lying states are Yankielowicz effective low energy action proposed organized in massive supermultiplets. The sim- recently by Kovner and Shifman [26]. They sug- plest possibility is to have a lowest supermultiplet gest the existence of an additional massless phase generated by a superfield made out of gluinos [25]. with no chiral symmetry breaking. On the lat- On the lattice such states areπ ˜,˜σandχ ˜ discussed tice this could lead to a more complicated struc- already in section 2.3. Other obvious candidates ture, for instance, because the symmetric phase are glueball states observed in pure Yang-Mills could be stable or metastable in some intermedi- theory, which might also belong to another super- ate range of the hopping parameter. multiplet. Of course, the notion of constituents An interesting point is the dependence of the is in general not expected to work perfectly in a phase structure on the gauge group. For example, strongly interacting theory. Therefore one has to in case of SU(3) there are at least three degen- be open minded towards non-trivial mixings. erate vacua and beyond the critical hopping pa- One of the main goals of the DESY-Münster- 6

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