The beta-function in N=2 supersymmetric Yang-Mills theory

20 August 1998

Ž .Physics Letters B 434 1998 54–60

The beta-function in Ns2 supersymmetric Yang-Mills theory

Adam Ritz 1

Theoretical Physics Group, Blackett Laboratory, Imperial College, Prince Consort Rd., London, SW7 2BZ, UK

Received 31 October 1997; revised 6 May 1998Editor: P.V. Landshoff

Abstract

The constraints of Ns2 supersymmetry, in combination with several other quite general assumptions, have recentlybeen used to show that Ns2 supersymmetric Yang-Mills theory has a low energy quantum parameter space symmetry

Ž .characterised by the discrete group G 2 . We show that if one also assumes the commutativity of renormalization groupU

flow with the action of this group on the complexified coupling constant t , then this is sufficient to determine thenon-perturbative b-function, given knowledge of its weak coupling behaviour. The result coincides with the outcome ofdirect calculations from the Seiberg-Witten solution. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 02.20.Rt; 11.10.Hi; 11.30.Pb

1. Introduction

In recent years there has been remarkable progressin understanding the non-perturbative dynamics ofsupersymmetric gauge theories. The powerful holo-morphy constraints imposed by supersymmetryŽ .SUSY have allowed results, calculable at weakcoupling, to be analytically continued to the strongcoupling regime along quantum-mechanically flat

Ždirections in the space of inequivalent vacua see e.g.w x.1 . In particular, the constraints of Ns2 SUSYhave allowed an exact solution for the holomorphic

1 E-mail: [email protected].

two-derivative contribution to the Wilsonian effec-tive action to be determined by Seiberg and Wittenw x Ž .2 for Ns2 supersymmetric Yang-Mills SYMtheory and SQCD. More recently, it has been arguedw x w x3,4 that a number of the conjectures made in 2 , inparticular those of physical importance such as elec-tric-magnetic duality, are strictly unnecessary in or-der to obtain a unique solution. Indeed the existenceof an underlying discrete parameter-space symmetrygroup of the full quantum theory, by which we meantransformations the couplings of the theory whichleave the vacuum state and full mass spectrum in-variant, has been determined uniquely from severalrather general requirements including unbroken Ns

w x2 supersymmetry 4 .Quite generally, the existence of such parameter

space symmetries is of great utility, as they act onŽ .the same space as renormalization group RG trans-

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 00748-5

( )A. RitzrPhysics Letters B 434 1998 54–60 55

formations of the theory, and thus place restrictionson the structure of RG flow. In particular, whendiscrete symmetries associated with a subgroup of

Ž .the modular group SL 2,Z hold at all scales in aquantum theory, then the b-function must satisfycertain modular transformation properties. When the

Ž .symmetry group is large enough, for example SL 2,Zitself, the constraint on the b-function is generallystrong enough to force it to vanish, rendering thetheory scale invariant. However, in cases where thesymmetry group is smaller, the RG flow may still benontrivial albeit highly constrained. In such cases,study of the required modular transformation proper-ties of b often provides nonperturbative informationabout the RG flow. Such arguments have been usedto highly constrain the structure of the RG b-func-

w xtions in statistical systems 5 , and also the non-linearw xsigma model 6 .

In this letter, we re-analyse renormalization groupflow in Ns2 SYM from this perspective of parame-ter space symmetries. A particular nonperturbativedefinition of the b-function, associated with the flowof the couplings along the moduli space, arises natu-rally in this context. This b-function has been ob-tained previously from the Seiberg–Witten solution

w xby Minahan and Nemeschansky 7 , and also Bonelliw xand Matone 8 . Here we show that this result fol-

lows from the following conditions, along withknowledge of its weak-coupling behaviour. Theseconditions, although not in their most basic form, aregiven by:1. The maximal parameter-space symmetry group of

the low energy effective theory, when acting onŽ . 2the complexified coupling t , is given by G 2 ,U

Ž . Ž .the index-3 subgroup of PSL 2,Z sSL 2,Z rZ .2

2. This equivalence group commutes, in an appro-priate sense to be defined below, with the flow oft induced by the renormalization group.

In order to justify condition 1, we also require thew xconditions shown in 4 to be sufficient to ensure the

uniqueness of this symmetry group. We shall discussthese additional assumptions shortly, although wenote that no assumption about the existence of a dual

2 w xThroughout we shall use the notation of Rankin 9 forŽ . Ž .subgroups of SL 2,Z and their generators. Note that G 2 isU

Ž .also commonly denoted G 2 .0

magnetic description at strong coupling will be re-quired.

The second condition above is crucial, and is thestatement of compatibility between the action of thesymmetry group and the renormalization group flow.Once satisfied, it implies that if the parameter spacesymmetry holds at one scale, then it will be pre-served by the RG flow, and continue to hold at lowerscales.

In Section 2, after briefly reviewing some generalfeatures of the effective theory, we recall the condi-

w xtions imposed in 4 which were used to ensurecondition 1. Although it is not clear at this stage ifall these assumptions are strictly necessary, certainholomorphy conditions on the coupling, followingfrom unbroken Ns2 SUSY, and the allowed singu-larity structure structure over the moduli space ofvacua, will be important in the following analysis. InSection 3 we discuss the precise characterization ofthe renormalization group flow to be studied, and

Ž .give a definition of b t . We then illustrate howŽ .condition 2 allows determination of b t where all

unknown parameters may be fixed purely via knowl-edge of the behaviour in the semi-classical limit.

2. The maximal equivalence group

Unbroken supersymmetry ensures that the poten-tial for the scalar component f of the Ns2 multi-plet vanishes, even when quantum corrections areincluded, when f takes values in the Cartan subal-gebra of the gauge group, which we shall take to be

Ž . ² :SU 2 . Classically, for non-zero a' f the Higgsmechanism spontaneously breaks the gauge symme-

Ž . ² :try to U 1 , with the arbitrariness in f , or more² 2:precisely a gauge invariant parameter such as Trf ,

leading to a moduli space of vacua MM. It is aconsequence of Ns2 supersymmetry that the struc-ture of the low energy Wilsonian effective action for

Ž .the light U 1 multiplet may be represented in termsw xof an Ns2 superfield AAsfqucq PPP as 10

14 2 2w xG AA s Im d xd u d u FF AA q PPP ,Ž .HW 1 24p

1Ž .

( )A. RitzrPhysics Letters B 434 1998 54–6056

up to non-holomorphic higher derivative terms, whereFF is the holomorphic prepotential. In Ns1 super-space this action takes the form

w xG A ,WW a

14s Im d xH

4p

14 2 a= d u K A , A q d u t A W WŽ . Ž .H H a2

q PPP , 2Ž .where A and W are Ns1 chiral and vector super-a

XŽ .fields, K'FF A A is the Kahler potential, and¨

E 2FFt A ' . 3Ž . Ž .2E A

Ž .Since t A is the coefficient of the kinetic term, itsimaginary part must be positive. Its real part alsoplays a role similar to the theta parameter in themicroscopic theory, and thus it is natural to definethe corresponding effective parameters in the manner

2p 4p it u ' q , 4Ž . Ž .2u u g uŽ . Ž .eff eff

where u is a gauge invariant parameter labelling thew xmoduli space. Careful analysis in 11 has shown that

one may match these effective parameters to theŽ .underlying microscopic SU 2 parameters at small

scales via an appropriate identification of the L scaleof the effective theory with the renormalization-scheme-dependent dynamically generated micro-scopic scale.

Although we shall need the relationship only inthe weak-coupling region, it has been shown quite

w x 3generally 12 that for the pure Yang-Mills case u² 2: ² 2:may be identified as us Trf . Thus Trf is a

good global coordinate on the moduli space, as wasw xconjectured in 2 . The point is that a, when defined

as the expectation value of the scalar component ofthe Ns2 superfield AA, only corresponds to a usefulparametrisation in the semi-classical domain.

An important insight is that the moduli spacecoordinatized by u, may also be parametrized in

3 The situation is more complex for certain models with higherw xmatter field content 13 .

terms of t which plays the role of a convenientuniformizing parameter. This requires knowledge of

Ž .the multivaluedness of the relation tst u . While,for unitarity tgH, the upper-half complex plane, itis clear from the presence of an effective theta

Ž .parameter in 4 that there exist equivalence relationsbetween various values of t . In other words themapping between t and u is not one-to-one. In

Ž .particular, one may identify t¨U t stq1, cor-responding to a rotation of the effective u-angle by

w x2p . Following 4 , we define the maximal equiÕa-lence group G as the group of all transformations of

Ž .t which leave the vacuum, u t gMM, and the fullmass spectrum, invariant. Since g 2 must be posi-eff

Ž .tive, this ensures that G;SL 2,R . In order to con-strain G further, we note that Ns2 SUSY impliesthat the mass of charged particles is BPS saturated

'w x < <14 , i.e. Ms 2 Z , where Z is the SUSY centralw xcharge. It has been argued in 2,15,16 that quantum

mechanically this relation has the form

' < <Ms 2 q aqq a , 5Ž .e m D

XŽ .where a 'FF a , and q and q are integer val-D e m

ued charges. The invariance of this spectrum, and theŽ .fact that t satisfies tsda rda as follows from 3 ,D

Ž .then ensures that G is a subgroup of PSL 2,Z , andthus the physical moduli space may be representedas the fundamental t-domain 4 DsHrG.

Having presented evidence above for the nontrivi-Ž .ality of G;PSL 2,Z , we now recall the analysis of

w x4 which argued that the following assumptionsconstitute a set of sufficient conditions to determineG uniquely:1. t takes all values in H.2. There are a finite number of singular points in MM.3. The BPS mass M is single-valued on MM.4. The mass of the lightest charged field is finite

except in the perturbative region.With these assumptions, it was shown that G is

Ž . Ž . Ž .given by G 2 ;G 1 sSL 2,Z , and thus theseU

requirements also serve as sufficient conditions toensure the validity of our initial assumption that the

Ž .symmetry group of the quantum theory is G 2 .U

Conditions 2 and 4 deserve further comment. In

4 Note that a more general definition is necessary when onew xadds matter hypermultiplets 17,16 .


particular, condition 2 ensures that functions, ormore generally sections, defined over the modulispace, will be meromorphic. Since singular points inthe moduli space will be mapped to the vertices ofthe fundamental domain of G, this ensures that thereare no singularities in the interior of this fundamentaldomain. Meanwhile, condition 4 is the expectedbehaviour for a theory with only one asymptoticallyfree regime, as we observe in the present case.

3. Determination of the beta-function

The above definition of the equivalence group, asacting at a fixed point of the moduli space, isimportant in determining the precise characterisationof RG flow we shall use. In particular, in order toconsistently apply the second assumption of Section1, we need to consider the infinitesimal flow at afixed point of the moduli space. Since the effectivetheory has been constructed via recourse only to thegeneral constraints imposed by supersymmetry, thereis no explicit cutoff scale, and the only dimensionful

Ž .parameters of the theory are u or a and L. In orderto allow compatibility with the action of the equiva-

w xlence group it is then appropriate, following 7,18,8 ,to define the b-function for the evolution of t as

Et ub t 'L sy2 , 6Ž . Ž .X

EL u Lu

where uX denotes E urEt . The latter relation, noted inw x8 , follows from the fact that, since t is dimension-

Ž 2 .less, bsb urL and the above definition is physi-cally equivalent to considering the flow induced bymotion over the moduli space, i.e. changing u withL fixed. The latter description is perhaps more phys-ically relevant, and closer in spirit to a more standard

w x a <weak-coupling definition such as 10 , b 'aE t ,La

where the renormalization scale is chosen equal tothe vev. The choice of u as the scale in the nonper-

Ž .turbative definition 6 is motivated by the fact thatu is a more appropriate coordinate for the modulispace at strong coupling. However, while these defi-nitions may differ in the strong coupling region oneexpects that they should be equivalent, at least up toan overall constant, in the perturbative region a4L.This may be verified by considering the appropriate

w xdifferentials of a, u, and t 8 , from which oneobtains the following relation,

Ea absb L lna y1 ™yb q PPP . 7Ž .ž /EL u

The final limit is taken in the perturbative region,where the relation simplifies as expected since in this

'case a; 2u q PPP which is independent of L.Thus perturbatively the definitions agree up to asign, while non-perturbatively there is a discrepancydue to the fact that a is not a good global coordinateon the moduli space.

At weak coupling, we may match this b-functionfor the effective coupling to the running of themicroscopic coupling. The 1-loop contribution is

5 Ž . w xthen given by b t sy2 irp 19 . Note thatpertw xthere are no higher loop contributions 20,21,10 , but

w xinstantons 21 lead to additional nonperturbativeeffects as first discussed for Ns2 theories by Seiberg

w x Ž .in 10 . It is these contributions to b t that we shalldetermine below.

Ž .It is important to note that b t inherits certainŽ .analyticity properties from those of t u . The holo-

morphy of t follows from its definition as thederivative of FF and the existence of unbroken su-persymmetry. Furthermore, as was noted in the pre-vious section, since possible singular points in themoduli space are mapped to the vertices of the

Ž .fundamental domain D, b t must be regular in theinterior of this domain.

We now concentrate on the global definition of b

Ž .given in Eq. 6 and make use of the second assump-tion of Section 1 in order to obtain an explicitexpression. Recall that the existence of a nontrivial

Ž .equivalence group G 2 implies that the physicalU

moduli space MM reduces to the corresponding funda-Ž .mental domain DsHrG 2 . The action of a gen-U

Žeral element of this equivalence group, gPts atq. Ž .b r ctqd , where adybcs1, a,b,c,dgZ, may

be conveniently represented in terms of the genera-2 w xtors U and VU V 9 ,

t2U :t™tq1 VU V :t™ . 8Ž .

1y2t

The assumed validity of this symmetry at all scales

5 w xNote that there is a relative minus sign compared to 19,10Ž .due to the definition of the b-function in 6 .


leads to the condition that the RG flow commutewith this action of the equivalence group. From therestrictions on its allowed analytic structure, this

Ž . w ximplies that b t for tgH, satisfies 5

d gPtŽ . y2b gPt s b t s ctqd b t ,Ž . Ž . Ž . Ž .

dt

9Ž .

Ž . Ž .where ggG 2 . That is to say, b t transforms asUŽ .a modular form of G 2 of weight y2. Note thatU

the fact that b is not invariant under such transfor-mations is due to its definition as a contravariantvector field on the space of couplings.

Positive weight modular forms may be obtainedconstructively in terms of generalised Eisenstein se-ries. However, for negative weight forms no suchconstruction exists. Nevertheless, we may use a gen-

w xeral theorem, valid for all even weight forms 9 ,which states that any weight y2 modular form of a

Ž .discrete group G;G 1 , may be represented in termsof a univalent automorphic function f of G , via

1 P fŽ .b t s , 10Ž . Ž .Xf Q fŽ .

where f X denotes E frEt , and P and Q are polyno-Ž .mials in f. In the present case GsG 2 , and aU

convenient automorphic function is given by fsŽ . 4 4 8 Ž < .F t syu u ru , where u su 0 t , for is3 4 2 i i

2,3,4, are the theta constants.We could now proceed to place constraints on the

polynomials P and Q from the known weak-cou-pling asymptotics. However, an alternative approach

Ž .is to recall that the definition of the b-function 6 ,when expressed in terms of usurL2, has precisely˜

Ž .the form 10 , with u playing the role of f. Further-˜more, since u is clearly automorphic from the defini-˜tion of the equivalence group, and univalent due toits parametrisation of the moduli space, we may

Ž .equally well identify fsu t .˜Ž .Obtaining an explicit expression for 10 thenŽ .reduces to determining the relation usu t for the˜ ˜

automorphic function u. Using only the required˜Ž .transformation properties under G 2 , and theU

known weak coupling perturbative asymptotics forŽ 2 .t u;a r2 arising from the 1-loop b-function, this

w xrelationship was obtained by Nahm 17 . These arethe assumptions of the present paper, and thus wemay use this result which, in corrected form and with

w xa convenient normalization of the scale L, reads 17

2us1y4F . 11Ž .2ž /L

This simple relationship between F and u2 is˜essentially demanded by their required transforma-

Ž .tion properties under G 2 . To gain a little moreU

insight into this expression, we may also recover theŽ .functional relation, F u , as follows. Since u and F˜ ˜

Ž .are both univalent and automorphic under G 2 ,U

they may be functionally related by a polynomial ofw xdegree determined by their singularity structure 9 .

Importantly, since the singularity of u is at the˜weak-coupling vertex, ts i`, of D, we may extractthe order of the pole from weak coupling asymp-

2Ž .totics. Introducing an elliptic modulus k t su 4ru 4 which has a zero at ts i`, and using the2 3

Ž 2 .weak coupling relation for t u;a r2 , obtained byintegrating the 1-loop b-function, we find that u™

2ky2 at the weak coupling vertex. Similarly, usingthe explicit representation for F in terms of complete

w x y4elliptic integrals 22 , we find F™yk . Thus Fhas a pole of order 2 at the pole of u, and conse-˜

w xquently we may write 9

2u uFsc qc qc . 12Ž .1 2 32 2ž / ž /L L

The univalence of F and u implies that c s0,˜ 2

while the perturbative asymptotics, arising from theŽ . w x1-loop b-function, b t sy2 irp 19 , impliespert

c sy1r4. From the earlier discussion, we now1

expect that instanton contributions to b are deter-mined by the value of c .3

Since F and u are univalent, we may fix c by a˜ 3w xchoice of the zero. In the analysis of 17 discussed

Ž .above, it was pointed out that the zero of u t˜Ž . Žshould lie at the orbifold vertex of G 2 , ts iyU

. Ž .1 r2. This fixes c s1r4 and 12 then reduces to3Ž .11 .

Ž .More generally, the zero of u t may not be fixed˜by the group structure and thus, from a calculationalpoint of view, it is helpful to consider an expansion


of this result near the weak coupling vertex, withoutfirst fixing c ,3

4c y4F3bs XF

2 i1 4;y 1q 32c y3 k q PPP . 13Ž . Ž .Ž .332

p

Ž 4.The O k correction may be associated with aw x1-instanton contribution 10 . This may be seen by

2 Ž .noting that k ;2exp ipt q PPP in this limit andtherefore, setting the u-angle to zero for clarity, wehave

2 i y8p 21

b;y 1q 32c y3 exp q PPP ,Ž .38 2ž /ž /p g uŽ .14Ž .

which exhibits the standard 1-instanton exponentialfactor.

As a consequence, we may also fix the constantc from knowledge of the 1-instanton correction3

which is calculable at weak coupling. In effect, whenrestricted to the 1-instanton level, the constant c has3

implicitly been fixed by the choice of perturbativerenormalization scheme. In order to see this moreclearly we note that in the weak coupling limit wecan identify L with the perturbative renormalizationgroup invariant scale, given by L 4 s

2 Ž Ž .. 4 4 2u exp 2 ipt u . Thus we have k ;4L ru q PPPŽ .and, from the structure of 13 , we observe that a

change of renormalization scheme corresponding toa change in L may be compensated by a change inc at the 1-instanton level. The 1-instanton term is3

therefore scheme dependent and a choice of schemefixes the coefficient of the associated exponentialfactor, and thus the value of c , unambiguously.3

Therefore, the final constant may be fixed viaknowledge of the 1-instanton contribution at weakcoupling, calculable by saddle point methods in ascheme such as Pauli-Villars. Such a scheme was

w xshown in 11 to be equivalent to the implicit schemew xused in 2 . However, a direct instanton calculation

w x11 gives the first power correction to the perturba-tive result at a fixed value of a, rather than u.

< <Nevertheless, while b and b differ at this order,u a2 4 Ž 2 .the relationship, usa r2qL r 4a , is again cal-

culable at weak coupling to the required 1-instanton

6 ² 2:order using the fact that us Trf . Thus, con-verting from u to k, the 1-instanton induced powercorrection to b takes the form

2 i5 4b sy 1q k q PPP , 15Ž .Ž .weak coupling 32

p

where the dots represent higher order instanton con-Ž . Ž .tributions. Comparing 13 with 15 leads to the

identification c s1r4, consistent with our earlier3

conclusion.The final result for the b-function is then given

by

1y4F i 1 1b t s sy q . 16Ž . Ž .X 4 4ž /F p u u3 4

After accounting for the alternative u-function nota-tion used, one may readily verify that this resultcoincides with that obtained by Minahan and

w xNemeschansky 7 from the elliptic curve of theSeiberg–Witten solution. This expression may alsobe shown to coincide with the result obtained by

w xBonelli and Matone 8 from the Picard–Fuchs equa-tion for the vevs a and a of the chiral superfield.D

However, this requires use of an alternative choice ofboundary conditions in this equation in order to beconsistent with the choice k 2 su 4ru 4. The singular-2 3

ity structure of this b-function has been discussedw x w xpreviously in 7 and 8 . We recall here that the

Ž .fixed points, located at ts iy1 r2, and the equiv-Ž .alent points under G 2 , correspond to us0, whereU

the full gauge symmetry is classically restored. ThisŽ 2 .result is not unexpected recalling that bsb urL .

Finally we note that the b-function is singular at thevertices of the fundamental domain which lie on thereal axis, i.e. tgZ in general. The gauge couplingdiverges at these points, which are associated with abreakdown of the effective theory due to the pres-ence of extra massless monopoles and dyons.

Ž .Finally, we note that one may expand 16 tohigher order allowing comparison of the 2-instantoncoefficient with semi-classical calculations. Expand-

Ž . Ž 8.ing 16 up to O k , which includes all 2-instanton

6 w xOur normalisation of L differs from that of 11 by a factor of22'2 . i.e. L s2 L , which fixes the renormalization scheme.DR


effects plus partial corrections due to three instan-tons, we obtain

2 i5 5 12294 6 8b;y 1q k q k q k q PPP . 17Ž .Ž .32 32 8192

p

In order to compare this result with saddle-pointcalculations, which are obtained at fixed a, ratherthan u, it is necessary to carefully convert the 2-in-

w x Ž .stanton results 23 for FF or t to functions of u orŽ w x.k see e.g. 24 . Once one does this, and accounts

w xfor the different normalisation of L in 23 , onereadily verifies the explicit 2-instanton coefficientobtained by evaluation of the induced vertex.

Acknowledgements

I would like to thank D.Z. Freedman, J.I. Latorre,D.F. Litim, A.A. Tseytlin, and A.I. Vainshtein, forhelpful comments and correspondence, and the fi-nancial support of the Commonwealth ScholarshipCommission and the British Council is also grate-fully acknowledged.

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The beta-function in N=2 supersymmetric Yang-Mills theory