?U2sg Я* Itep -76

,?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 external 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 effective lagrangian , or fixing the instanton size, that is inconsistent with the supersymmetry. Notice that naively supersyametry 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 collective coordinates: 3 Here x is the instanton position, О is its sise, and eC and A are the Grassmannian coordinates corresponding to supersymmetric 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 parameter labelling the instanton solution. The central point is that supersymmetry induces simultaneous 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 supersymmetry. 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;) oran a notations. Actually, the instanto.a field Qp+ ie self-dual. It is more appropriate, therefore, 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 invariant 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" superfield 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 coordinates «e «rite

i" ехр(1Рж0) ехр(-1Л

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. (12) in the following identical fora

The last exponential rescales the instaston solution The corresponding effect is equivalent to the following transformation of J> i J/*j»C**^ff*>). (17.) Moreover, occurence of the first exponential meant redefinition of x0, XQ^-» x -Zi*GL<£* Tbe measure (4) ie trivially inva - riant under the shift of x . Transformations ot 9i and 5" stemming from eq. (1S) are

One can easily check that the instanton measure (4) is invariant under the changes of the collective coordinates (17).

Actually, the products d X94 «C and Р4РД в are invariant by themselves* 10 We pause here to draw toe reader'в attention to the fact that there le as absolute parallel between the standard chiral realization of the supergroup in the (x, 6 & ) superspace and

the transformation law of tx0,oO. Indeed, supertransformatione corresponding *° parameters €• and € , respectively, look as follows

(ch в chiral, ach • antichlral^. Compare these expressions witb

(a) et t

Thus, (-xfl) plays the role of xcairai , while * plays the role ! of e.

Similar exercise in commutation relations can be readily made with the superconformal transformations. This allows one to answer the question bow the instanton collective coordinates are transformed under the action of £ p and S £ • The measure (4) ie trivially Invariant under the S£ transformation. Indeed, the only change of the coordinates is a shift of

On the contrary, an infinitesimal € p transformation results in a more complicated law .

Apart from eq. (14) we use here the following commutation relations LSJ : jy *Jr = -.г *•*, £ (4-#

€) + ft

Now the inatanton measure is not invariant under these chan-

ges:

(20) This noninvariance is a ma:tLfestatioii of breaking of auper- conformal symmetry by qnantvaa corrections.

4< Instantons and Anomalies

Thus, we have shown that the standard instanton measure reapects 3upers;nnirietry. The only unusual thing about inatantons is that the aupersyiarretry is realized in a rather peculiar way, аз a combination of dilatationc and (super) conforms! tranaforraationo none of which is a true symmetry of the quantum theory because ot anproalies. Izi this section we discuss in more detail the relation betiveen variations of the inr,tenton measure and the quantum pjiomalies.

As mentioned in sect. 2 the super-Yang-Hills action admits dilatations and superconfornal transformations. Quan- tum corrections do, not respect these symmetries, however — taking them into account yields nonvanishing divergencies of the corresponding currents. The anomalies are fixed to all orders in the-couplin£ constant by renormalizaoility and supersymmetry of the theory: 12

whore ^y is the energy-momentum tensor, jL* Is the con- served veetor-spinor current (the corresponding charge wm* denoted above Ъу Q), C^- J^^eX^^ » and 6^) ie the Gell-Mann-Lovr function,

We use in this section the Majorana notations. The instanton measure (4) is actually a normalized

pre-oxpone-ntial factor in front of exp(-Scl) » ехр(-2Л*/в/ао). It emerges when one performes functional integration - 3 ('S+

wb.era the integration runs over quantum fluctuations around the classical solution. Recall that the solution family (8) introduces both bosonic and lermionlc classical fields, see eqs. (5)-(7). Perform now a change of scale X -» X.(i-fY) (and the corresponding rescaling of fields ) in the original functional integral (22). Because of the conformal anomaly the action is changed under this transformation as follows

*»

In order to find the variation of the integral (22) we proceed to euclidean space and substitute , to the leading order in

ei& , the operator GMV by its classical part (QjuY/e£ * 13 Talcing into account that

we get that the right-hand side of eq. (22) is multiplied by 1 + вУ. The instanton measure (4) must acquire this factor under the scale transformation of the collective coordinates. This, of course, happens df

Moreover, one could fix the powtr of О in the instanton measure U, in this way, without performing explicit calculation of the instanton determinant. Let us discuss now superconformal transformation with parameter £ . Dae to the superconformal anomaly (21) the action is also changed:

Proceeding to euclidean space and subotituting eqs. (5), (б) instead of the operators C*»y , Л we get the variation of the integral I (22) in the form i

It is a pleasure to notice that variation (20) of the instanton measure just reproduces this anomaly. 14 6. Ward Identities So far *• bar* discussed the instanton манит* using only group-theoretical arguaeats. It is reasonable to ask «bather tba standard instanton calculus reproduces the results obtained. In otter words, one can check the lard identities. The Ward identities relevant to inatantons дата been die- cue aed previously L8J , and an apparent violation of the Ward identities baa bees found. Mo satisfactory explanation to this observation has been given, however, end *e feel it worthwhile to consider the question anew. Our conclusion is that the violation of the Ward identities disappears after integration over 0 . This is just another forsulation of the result obtained in sect. 3. . First let us review briefly some points of ref. 8. We start with the following aatrlx element

and calculate its divergence

1*"

+ other coaoutator tars». : Here !o) and \o'> are"vacuua*statee with a unit difference of topo- lo^ical charge, X| is the supercurrent introduced in eq. (21), aad I . »](%\ are *xteri>ai bo sonic and fernionic sources, respectively. Pour fersionic sources, three J- . and J^ , are needed ' because of the instanton zero oodes. for the aaae reason matrix 15 elements of the commutator terms omitted in eq. (29) actually vanish. Haively 9u J-^ = 0, and substituting this equality in eq. (29) we get the desired Ward identity. Actually it should be corrected for explicit supersynuaetry breaking broubbt in by the

procedure of £aufoe fixing and — quite unexpectedly — for one more effect to be discussed below. Eq. (29) takes especially simple form if we concentrate on the iVard identities for the supercharge and, to this end, integrate over d^x. Barring the pos- sibility of spontaneous symmetry breaking the result of the integration should vanish (we have checked that the surface term does vanish, so that there is no spontaneous breaking L8J)- 3?hus, one expects that the contact term in the ri^ht-hand side of eq. (SS'j also vanishes. This is indeed the case,

One can check the fact by usin^, simple 'dimensional arguments and some additional results which will be published in a separate paper. The latter will also toive the details of toe corresponding scenario. It is instructive to confront the above derivation v.ith sn

explicit evaluation of Мм, in a biven instanton field. Common wisdom says that introduction of such a field does not affect the Ward identity since the boson field does not vary under the supersymmetry transformation if there is no external feraion field, as we assume now. Moreover, M; is readily calculable (fig. 1): 16

(30)

+ permutations of fermion zero modes ( ^Js fa,, ^ici 2 )•

iiere ns.b±tZ. » ^SCI,Z denote four independent solutions of the Birac equation in the instanton field, *"Bf is the gluon Green function built from non-zero gluon modes. Dilierentiating (30) with respect to x yields [ j

where .ve have used the equations S> Я =O (38) -1 » , ]

and the bosonic zero modes are denoted by «^ . The О (х -• y) term in eq. C33) just reproduces the contact tens in the general relation (29).. The зшп over the bosonic zero modes produces an extra piece, however, iwhich corresponds to nothing else but

nonconservation of the supercurrent, UU*JM£ 0, as was first noticed in ref. [ej.

To investigate further the effect of the new term let us integrate ЛцМщОУег & x. As seen from eq. ^31) the result de- pends on the overlap of the bosonic and fermionic zero modes. The latter look very similar, and the overlap integral turns out vo be either zero or unity (in a proper normalization) depen- ding on the modes considered. For example, the translational modes are proportional to the eiluon fi.ld stren&cn tensor*. 17

Since the supersyiametrie zero modes are also proportional to the integration over & x is readily pei-formed witn the help of the following relation

(35)

On the other hand, the overlap of ? Ip' with the supercomor- mal zero modes,/\$c > vanishes because the product is odd in

In this way one can convince oneself that the extra term in the Ward identity corresponds to substitution of the variation 6v^ i of the classical vector field in the presence of the classical fermion field «Ag£. in the interaction term *|„ fa. '- П

It seems amis ing that in this rather complicated say v,e have come back to consideration of the classical fields. Compa- ring the situation with the results of sect. 3 we find that we

have reproduced the variations of xQ and p but not of S . However, the latter is readily recovered as well. The point is that we have arrived at the shifted bosonic field and unshifted fermionic field 18 Redefinition of ft is needed if we wish to preserve the para- metrization of tne instanton solution, see eq. (8). On the other hand, v,e have already learned that the variation of the classical field corresponding to OJ^, Op ,oS" , eqs. (17), 1з completely consistent with supersymmetry. Of course, to recover the supersyjnmetry it is absolutely necessary to integrate over all collective coordinates. To sunuarize, the consideration of the Ward identities de- monstrates once more that fixing О violates supersymmetry and that integrating over О restores it completely. Still, theve is one puzzle left, fiow can 01:0 understand ^X^ ° in the presence of the instanton field if «ц «L= 0 is an operator equality? The explanation seeias to be as follows. Introduce a small mass term to the quantum vector field. This is useful since the gluon Green function in the instanton field is not defined otherwise due to zero modes. Then the divf;r;i,ence of the supercurrent is proportional to this mass, f*' ' If we consider now the gluon propat,ation and tend ш -^ О the corresponding effects will be negligible о everywhere except fcr the zero modes. In the latter case the m factor just cancels out and we are left with some ^Ju^ 0» Therefore,yi чм г О in the presence of any external field which posesses zero modes. The phenomenon is due to infrared regu- larization. Moreover, thi3 supercurrent nonconservation is not manifested in the standard Ward identities. However, it is harmless iu the sense that it vanishes after integration over the collective coordinates. 19 в. ConclnaiasB We have shown thst the standard instanton calculus is perfectly consistent wit* aapexsymmetry provided that simultaneous integration over all collective coordinates is performed. Al- though we have discussed «sly the simplest example with the SU(2) gauge group, the assertion i* of more general nature and is valid for any augMi IJW—»T1C l&ng-Mills theory. The notion of an instantoa-iauuced effective lagrangian with fixed 0 is contradictory — it remolts in a conflict with supersymaetry * . We would like to emphasise ta» most general aspect» in any super- symmetrie theory with, сотГаи—Л invariance at the classical level fixing 0 wool* -wfmicte superiijmmulij **K There is no analogue of this phenmmemes. in ordinary QpDf л'пЛ this ml^ht indicate that dynamics; inherent to supersymmetric theories Is quite specific.

The authors ax* £tm£eful to M.Vysotsky and B.Wittea. for useful discussions.

*' The necessity of мидшиИйи, for J> variations first by four of OK gpmwttmaly PJ. Bowevvr, at tkat tlav Itite idea has not Ъеежпшшш&ар to the logical «aA axi *&• ешь- г» с шаюп ox rsXs |№Ц оспшшпк xrosi mcmtc w% xlma now. **) Тле opposite Is аХта> 1m—r. If the cXaeaieal coafianial ance is absent, опк ОШт complete a£reem*JBC *it*i ry for any (gives. Ш „ «ее aat.EJ, see*, в. 20

<*>

. 1 Reference* 1. l.ffltttn, Duel. Phye. Bjfifc (1961) 815 ». e^'t Eooft, Pbye. Нет., Ц* (1976) 3438 J. Ь.» .Abbot, M.T.Grieadni, H.J.3chnitBer, Pays. R«T. £>]£ (1977) 8998, 58O£ 4. A.I.Vainsbtein, T.I.Zakbaror, Pie'«a Zhetf, i£ (19в£> see 5. P.Fayet, S.lerreira, Phye. Reports, .ЗД (1977) 849 в. J.BelaTin, A.Polj»kOT, A.Schwar», Xu. Tyupkin, Phye, bett., 89B (1975) 85 7. B.Zunino, Phye. Lett., Д9Л (1977) 369 в. T.Hovlkor, H.Shlfaan, A.Vainehtein, V.Zakbaror, Ineterrtone in Supersyaaetric Theorize, Preprint ITEP-93, Moscow 1988, fuel. Phys. B, in press.

В. А. Нотисов i др. Преобревоваввя оупвронмивтрии для инотантонов Работа поотупила в ОНТИ 13.05.83 / Подетоано к пвчап I8.05.83_ «TI27I8 • Офоетн.пвч. 7ол.-печ.л.1,25. У9.и*гя.л.0,9. Тнрвх 290»m. { Saxas 76 Индекс 3S24 Цена 13 коп. I II. Ill Р II • Р • I, 11 I. ... • ... • • '•••!• ••! -Ill I..I. I II. |j I I I | I »| I Отпечатано в ИТЭФ, If7259, Мооква, БЛере19шс|нокая, 25 ИНДЕКС 3624

М.,ПРЕПРИНТ ИТЭФ, 1983, № 76, с.1-21