,?U2SG Я* ITEP -76 INSTITUTE OF THEORETICAL AND EXPERIMENTAL PHYSICS V.A.Novikov, M.A.Shifman, A.I.Vainshtein, M.B.Voloshin, V.I.Zakharov SUPER SYMMETRY TRANSFORMATIONS OF INSTANTONS MOSCOW 1983 УДК 530.145:538.3 И-Г6 Abstract Instantons in the simplest supereymmetric Yang-Mills theory are considered. We introduce boeonic and feraionic collective coordinates and study now they change under the eupersymmetry transformations. The tnstanton measure is shown to be explicitly invariant under the transformations. We discuss the relation between quantum anomalies and the functional form of the instanton aeasure. Имститут т*орг»ческоЯ а ••сперяи*ыт«льной фвзшм, 1883 1. Introduction la this paper we consider instazxtons is supersyamstrlc gluodyaamics. Heedless to say that non-perturbative aspect» — о and the BFST instantoos represent the most faaous non-pertur- batlve fluctuation — may play the key role la the theory. (Рог as illuminating introduction to the subject and recent deve- lopment see ref. 1.J Below we investigate technical iaauea of instant on calculus specific for supersyaaetrlc theories. As vell-Jcnown, instantone of small sise induce effective oultifermlon interaction [2J, Suspicion exists I?'4 that this interaction breaks the symmetry between feraione and bosons expected in the supersymmetric Xang-Vills theory. Jor definite- ness concentrate on pure gluodyoaaica with SU(2) as the gauge group. (A nice review and an exhaustive list of references is given in I J.) In the «ess-Zuaino (superJgange the i«gr« takes the form (1) 1,2,3) is the giuon field, Q^ is the glaon field strength tensor, 4 is the coupling constant, Л is the color triplet of Hajorana spinora, OL is the covariant derivative, and (...) stand for the gauge-fixing term and the ghosts. The lagrangian (1) is invariant, up to a total derivative, under the aupersyaastry tranvforBations: where £ is the transformation paraaeter (actually gauge fixing breaks the supersyametry, out this ie irrelevant to aoet of the paper* for further discussion see sect. 5). The instanton-induced effective lagrangian is determined by feraion zero modes L2J. In oar case it looks as follows LAJ 7 (3) V*' where O is the instanton size, and it is assumed that the ex- ternal momenta are much less than О ~1. The la&rangian (3) apparently violates the symmetry under the transformations (г), and this is just the starting point of our discussion. We will argue that it is the approximation of the effec- tive lagrangian , or fixing the instanton size, that is incon- sistent with the supersymmetry. Notice that naively supersya- metry is perfectly consistent with choosing some particular instanton size since the supersymmetry transformations do not involve dilatations. This simple argument is false, however. The reason is that the classical la£ran&ian (1) posesses additional invariance — it is invariant under conforaal and •uperconforaal transformations. To be^in with, consider the instanton measure It/ as a function of the following collec- tive coordinates: 3 Here x is the instanton position, О is its sise, and eC and A are the Grassmannian coordinates corresponding to super- symmetric and superconformal transformations, respectively. Eq. (4) represents the result of explicit calculation of the measure in the lowest order in the coupling constant. We will aryie that these collective coordinates natural- ly split into two pairs (x0, <* ) and ( £ ,f> ) . The first pair (x , ct ) is the instanton center coordinate, et playing the role of the supersymmetric partner of xQ . Notice that x transforms under the supersymmetry transformation as xchiral • ^ *s aDalosotts to 0 , and there is no analogue of 0, The second pair ( P ,"5 ) is, so to say, a superfield pa- rameter labelling the instanton solution. The central point is that supersymmetry induces simulta- neous changes in © and A . On the other hand, the effective lagrangian (3) is obtained by integration over all fermion zero modes while О is fixed. This is inconsistent with super- symmetry. The supersymmetry restores itself either after integration over О , or, alternatively, if some particular о is chosen, one should necessarily fix ft . 2. lostanton Preliminaries Let us recall first the basic facts on instantons and supersymmetry. BP5T instantons L Jare the classical solutions of the Yang-Mills equations (more exactly, duality equations). In particular, where fofcV ere *Ье ** Hooft symbols. Below we shall sometimes omit the subscript cl for brevity. The supersymmetry transformations, when applied to the bo eonic solution (5), generate nothing else but the feroion zero •odes: where •< is the вр1вот parameter of the transformation, Indeed, 7l8B satisfies the equation 3\3>\%= 0 if (<^* )cl Howerer, not all zero nodes are constructed in this way. Two remaining modes (euperconforaal modes) are given by л л with the independent transformation parameter (X - ЭС» } Jb . One caa readily check again that A,, satisfy the Dirac ©qjiation, As was first explained by Zumlno >-7j this pattern of the aero modes is a manifestation of the superconfozoal symmetry of the classical equations of motion. Likewise, the existence of the parameter P le a reflection of the ixnrariance under change of the «c«l*.tej So far we have used the Ma;) or ana notations. Actually, the instanto.a field Qp+ ie self-dual. It is more appropriate, there- fore, to proceed to chlral notations. Then the equation for the 5 inetanton field is u"*^= 0, and only transformations witb the right-handed parameters o^ generate non-trivial variations, ^55 ~* 4 £ • Similarly» 1A the case of th» superconforaal transformations only the left-handed parameters &% are rele- тал*, Я5С "» Q ** OCpiJb " . Below we shall use as a rule the chlral representation. 1 few foxoalas, howerer, «ill be given in the Xajorana representation. *e hope that this will cause no confusion. Two aore reaarks on inatantone and superayaaetzy axe in order sow. first, instantons are solutions in the Xaclidean space-tiae. On the other hand, it is not straightforward to hare a theory of a Majorana or leyl spimr in the JBuclidea& space. Рог this reason one starts soaetiaes L »*J with the S - 2 saper- syaaetric lajag-Vills theory which can be foraolated directly in terns of the. Dirac fields, and there are no probleas with con- tixuation to the Euclidean space. All calcolatipns becoae aore involved, however. In particular, nobody succeeded so far in de- riving the analogue of the lagrangian (3) in this case because averaging over the instanton orientation 1A the color «pace Ъе- coaes auch aore tedious. On the other hand, probleas with conti- ination of the S>1 theory to the Euclidean space do not seea to be fundaaental. An explicit treatment ot the issue can be found in ref. 4 «here the effective lagrangian (5) has been derived. Secondly, one might think that the supersyaaetxy breaking figuring in eq. (?) is a manifestation of spontaneous supersya- aetry breaking induced by the instant one. This is not so — we have convinced ourselves that the instantone do not bring about б the spontaneous breaking of the supersymmetry in the case considered L?J. 5. Supereymnetric Instanton Measure Since supersymmetry is a true symmetry of the action, in- cluding quantum corrections, the instanton measure must be in- variant under the supersymmetry transformations as well. In this section we find transformations of the collective coordina tes that are induced by supersymmetry transformations and show that the measure (4) is indeed invariant. As was mentioned in sect. 2 there exist both fermionic and bosonic solutions of the classical equations cf motion. To keep this symmetry between fermions and bosons explicit at each step let us Introduce a vector superfield V(x, О , & ,& ) composed of the classical solutions. We define V(x, P , Ы , is the following way where Q and S are the generators of supersymmetric and super- confonnal transformations, respectively, P is the generator of translations, x t 0 , et and A are the instanton parameters mentioned above, V(x =O, P )*Wb denotes an "initial" super- field containing purely bosonic solution (5) with no fermion components. Hotice that QoC and S 5" commute. The spinor generators act on V in the standard way, for instance, u$4] [ $^ yMSiJ* о How, the sapersyametry transformation, ot the ia*tafl£o& field •(s^.-J , о* ,j| ) ie gemitid by ваф(-1в<1) or eap(-i5« ) action of the right-banded generator 1* trivial and re&ces to a shift of o< . Indeed, The standard inataston measure is obviously inrariant under this shift The effect of the left-handed generator Q Is less trivial. To find the corresponding transformation of the collective coordi- nates «e «rite i" ехр(1Рж0) ехр(-1Л <k - IS*) % (12) where 0 is an operator, to the first order in 6 reducing to The spinor generators 4 and S, together with bosonic generator», font the euperconforaal algebra which is a generalization of the 15-diaeasional confornal algebra of space—tiae (see ref. 8, 8 ••ct. Z.Z). The algebra fixes the set of cosumtation relations, ie reproduce here thoae (anti) cosmutator3 L6J which will be needed below (H) 1Я Her* M, D, fl denote the following generators: Ж - Lorentz rotations, D - dilatations, ft - chiral rotations of spinors. Moreover, M acting en Vi__.t ia equivalent to a color rota- tion and can be disregarded as far as we average over the instan- v ton color orientation» П does not act on in3t et all. Using eq. (14 we readily find commutators figuring in eq. (15). Notice that triple and further commutators Сdenoted by dots in eg. (13)) vanish identically. The result for О can be written as follows: Bow we are finally able to answer the question bow the col- lect lve coordinates ere transformed. Recalling that exp(-lQi) acting on Viaat reduces to unity we represent eq.