On RG potentials in Yang-Mills theories

5 March 1998

Ž .Physics Letters B 421 1998 217–222

On RG potentials in Yang-Mills theories

J.I. Latorre a,1, C.A. Lutken b,2¨a Departament d’Estructura i Constituents de la Materia, Facultat de Fısica, UniÕersitat de Barcelona, Diagonal 647,` ´

E-08028 Barcelona, Spainb Department of Physics, UniÕersity of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway

Received 24 November 1997Editor: L. Alvarez-Gaume

Abstract

Ž .We construct an RG potential for Ns2 supersymmetric SU 2 Yang-Mills theory, and extract a positive definite metricby comparing its gradient with the recently discovered beta-function for this system, thus proving that the RG flow isgradient in this four-dimensional field theory. We also discuss how this flow might change after supersymmetry breaking,provided the quantum symmetry group does not, emphasizing the non-trivial problem of asymptotic matching ofautomorphic functions to perturbation theory. q 1998 Elsevier Science B.V.

Our understanding of the non-perturbative aspectsof non-abelian gauge theories is at this point in timevery limited. Most of our knowledge of the stronglycoupled domain comes from exploiting approximateglobal continuous symmetries, whose origin are notwell understood and may be coincidental, to con-struct low-energy effective field theories encodingthe dynamics of the effectively massless modeswhose existence is guaranteed by Goldstone’s theo-rem. Recently it has been realized that certain theo-ries may possess exact global discrete quantum sym-metries which are so constraining that the theory canessentially be solved exactly, even in the non-per-turbative domain. Such symmetries may underlie the

1 e-mail: [email protected] e-mail: [email protected]

remarkable phase- and fixed point structure of thew xsubtle quantum Hall system 1 , and their discovery

w x2 in Ns2 supersymmetric YM theories has led toa deep understanding of these systems, recently cul-minating in the construction of an exact representa-

Ž . w xtion of the contravariant physical beta-function 3 .It is the purpose of this letter to further explore theconsequences of such symmetries and their relation-ship to supersymmetry in non-abelian gauge theories.

The main idea to be discussed here, originallyw xdevised for the quantum Hall system 4 , is that the

combined constraints of a quantum symmetry groupŽ .contained in the modular group and matching toperturbation theory in the asymptotically flat domainŽ .weak coupling may be sufficient to completelyharness the beta-function. In particular, automorphyof the contravariant beta-function is incompatiblewith holomorphy on the entire fundamental domain,and the way in which the analytical structure must be

0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0370-2693 97 01588-8

( )J.I. Latorre, C.A. LutkenrPhysics Letters B 421 1998 217–222¨218

relaxed is indicated by the asymptotic behaviour ofthe beta-function:

dz b b1 2zb s sb q q q . . . q q-expansion ,Ž .0 2dt y y

1Ž .

where t is a scale parameter and for YM theory2 Ž .z s x q iy s ur2p q 4p irg , and q z s

Ž .exp 2p iz .If the theory is Ns2 supersymmetric, a strong

non-renormalization theorem implies that there existsa renormalization scheme in which only one loop

w xcontributes, so that all b except b vanish 5 .i 0

Hence the beta-function is asymptotically holomor-phic, and analyticity is only violated by a simplepole at zs0, which is the unique point on theboundary of the fundamental domain where the ef-fective field theory breaks down. This is as close to

Ž .holomorphic as a sub- modular beta-function canget, and its form is then completely fixed by thediscrete quantum symmetry and asymptotic matching

Žat z™ i` to perturbation theory and a single-instan-.ton calculation .

3 w xThis idea was very recently 7 applied to theŽ .contravariant beta-function for SU 2 and Ns2 in-

w xvariant YM theory 3 , thus simplifying its derivationenormously. We first extend this line of reasoning tothe Ns2 covariant beta-function, which by compar-ison with the contravariant one allows us to extract apositive definite metric with reasonable physicalproperties. Since the covariant beta-function is gradi-ent, this proves that the RG flow in this system is

w xgradient 8 .If the theory is not supersymmetric, b does not1

vanish in any scheme and we see that the analyticstructure changes dramatically. It seems unlikely thatsuch a function could still be automorphic, but aremarkable ‘‘quasi-holomorphic’’ exception exists,and this can be used to build a candidate beta-func-tion with similar uniqueness properties to the super-symmetric case. Thus, it is not inconsistent to dis-cuss the properties of non-supersymmetric beta-func-tions automorphic under quantum symmetries. If suchsymmetries do exist they may or may not be sur-

3 w xSimilar ideas have been discussed in Ref. 6 in the context ofnon-linear sigma models.

vivors of supersymmetry breaking, but it seems plau-sible that it is the interplay of supersymmetry, quan-tum duality and holomorphy which must be eluci-dated if the spectacular success of Ns2 theories isto be extended to non-supersymmetric models.

A simple yet precise picture of the RG propertiesŽ .of Ns2 SU 2 YM has recently emerged from the

explicit construction of an exact non-perturbativew xbeta-function 3 :

1y4 f z i 1 1Ž .zb z s sy qŽ . X 4 4ž /f z pŽ . u z u zŽ . Ž .3 4

2Ž .Ž 2 .4where fsy u u ru is invariant under the quan-3 4 2

Ž .tum symmetry group G 2 of this system, andTŽ .u is2,3,4 are the conventional elliptic theta-func-i

4 4 4 w xtions, related by u su yu 9 . This infinite2 3 4

non-abelian discrete quantum symmetry group is thelargest sub-group of the modular group which con-

Ž .tains translations by one. This is why it is G 2T

which appears both in YM and in the quantum Hallsystem. Translations by one 4 T : z™zq1 is a sym-metry because the theta parameter is periodic undershifts by 2p , and guarantees that all functions auto-morphic under this group have q-expansions with

Ž .qsexp 2p iz . The other generator of the group canbe conveniently chosen as ST 2S.

Very recently a simple and illuminating derivationw xof this beta-function was given by Ritz 7 , which

relies on the observation that the contravariant beta-function transforms like a weight wsy2 automor-phic function 5:

dg y2z z zb g z s b z s czqd b z 3Ž . Ž . Ž . Ž . Ž .Ž .dzŽ .for any ggG 2 acting holomorphically by frac-T

tional linear transformations on the modular parame-Ž . Ž . Ž . w xter z: g z s azqb r czqd . As observed in 4 ,

4 w xWe follow the conventions of Rankin 9 as closely as possi-ble, but adopt the standard physics notation for the generators of

Ž .the modular group SL 2,Z . Thus, translations are denoted by TŽ .rather than U , and simple duality z™y1r z is denoted by SŽ .rather than V .

5 Mathematical textbooks like to reserve the words ‘‘function’’and ‘‘form’’ for automorphic objects with special analyticityproperties, but their usage is inconsistent and confusing, so we donot qualify our language in this way. We shall therefore have tobe explicit about the number of poles possessed by our functions.

( )J.I. Latorre, C.A. LutkenrPhysics Letters B 421 1998 217–222¨ 219

together with asymptotic matching to the perturba-tively evaluated beta-function, this is a powerfulconstraint on the possible forms of this function. Inthe Ns2 case, where the beta-function is meromor-phic, it does in fact uniquely fix the beta-function tothe one given above, up to a single constant whichwill be discussed shortly.

We now adapt the argument to the covariantbeta-function, which transforms as a weight wsq2

Ž w x .function. The first observation Rankin 9 , p.111 isŽ Ž .that for groups of genus zero which includes G 1

Ž . Ž .. ŽsSL 2,Z and G 2 there exists an invariant wsT. Ž0 meromorphic function with a single pole and a

.single zero in the fundamental domain, which wehave called f above. Any other weight zero function

Ž .with total order q of zeros or poles is given by aŽ . Ž .rational function P f rQ f , where the degree in f

of both the polynomials P and Q do not exceed q.Applying this theorem to the invariant function gsb z f X, and using the asymptotic behaviour of f X and

z Žthe Ns2 beta-function, it follows that b s c q1. Xc f rf , where the conventional choice of scheme2

Ž .corresponds to c sy4 from perturbation theory2Ž .and c s1 one-instanton computation .1

Similarly, assuming that there exists an asymptot-ically flat metric, the same argument applied to theinvariant function hsb rf X givesz

f X

b s sE F withz z1y4 f

1 2Fsy log 1y4 f . 4Ž .

4

A contour plot of the RG potential is given in Fig. 1,where the horizontal axis is xsRe zsur2p andthe vertical axis is ys Im zs4prg 2.

A similar flow was very recently displayed inw xRef. 10 . It is remarkable that this RG potential for

Ns2 YM is so simply related to the unique univa-Ž .lent holomorphic function f in G 2 . From thisT

construction we can further extract a metric2Xf

G sb b s sE E Kz z z z z z1y4 f

12with KscFq F 5Ž .

2

which behaves correctly under the action of the Liew xderivative 4 , and is flat, finite and positive definite

Fig. 1. The phase and RG flow diagram of Ns2 supersymmetricŽ .SU 2 Yang-Mills theory, obtained from the RG potential F

Ž .exhibited in Eq. 4 . Flow lines are downward and displayed onlyin the fundamental domain, while equipotential lines are shownfor u between yp and p .

except at the fixed points. The Kahler potential K¨can have any linear term since E b s0. At zs i` itz z

is finite, as behooves a metric adapted to a basis ofnoninteracting operators in the perturbative domain.Furthermore, it only carries mixed components, asexpected from the OPE of YM in the perturbative

²Ž .Ž ) .:domain, since FF F F s0. At the IR fixedŽ .point, zs 1q i r2, it has a coordinate singularity,

which is to be expected if it is adapted to theperturbative basis of weakly interacting operators. At

Ž .strong coupling we expect a different ‘‘dual’’ basisof weakly interacting composite operators to be moreconvenient, so it is natural for the metric adapted tothe perturbative domain to have a coordinate singu-larity in the infrared. The curvature is zero, except at

Ž .zs0, where the metric is ill defined not positive .This is also to be expected since this is where theeffective field theory breaks down due to the pres-ence of new massless modes. The Seiberg-Wittenmetric is quite different. It also has an essentialsingularity at zs0, but it has a coordinate singular-


ity in the perturbative domain, and is well behaved atthe IR fixed point, indicating that it is better adaptedto describing strong coupling.

These issues are not of any great relevance for us,since we have now achieved our goal: by exhibitinga potential function and a positive definite metric, wehave shown that the RG flow is gradient. Moreprecisely, we have just specified the elements in thefollowing chain of equations:

yb iE Fsyb ib syb ib jG F0 , 6Ž .i i i j

where the inequality is obtained because G is posi-i j

tive definite. It is worth noting that this is the firstexplicit construction of a non-perturbative functionof this kind for a ds4 field theory. The muchharder question of whether or not these functionshave physical interpretations as the C-function, whichcounts effectively massless degrees of freedom atfixed points, and the Zamolodchikov metric, whichis given by the correlators of relevant and marginaloperators, is logically independent. All efforts ad-dressing the irreversibility of RG flows are ulti-mately motivated by such physical considerations.

w xDifferent approaches 11 have recently convergedŽ .on a solid candidate for the c-number central charge

of the conformal field theories appearing at non-triv-ial fixed points. It is the coefficient of the Euler

Ždensity in the trace anomaly in curved space labeledw x w x.as a in 12 and b in 13 , and this candidate hasb

been shown, empirically, to be positive and to de-crease along RG flows in a large number of non-triv-

w xial examples 14,12 . Further work has related thesign of b to a new form of the weak energyb

w xpositivity theorem 13 .Although we at present are unable to relate F to a

distinct physical observable, this is not required inorder to prove that the RG flow is gradient. We arealso not able to relate our metric G to the OPE ofz z

the operators in this theory. So, while it has manygood properties, being non-singular and automorphicas well as having correct asymptotics and Lie trans-port, we do not know how to relate it to the basiccorrelators of the theory.

Let us now turn to the issue of scheme depen-w xdence 15 alluded to above. As we have seen, the

Ž .G 2 -symmetry reduces the mathematical freedomT

in the construction of the b z-function to finding theappropriate values for c and c . It was argued in1 2

w xRef. 7 that c controls the asymptotic behaviour of2z Ž .b , and therefore is fixed by the one-loop perturba-

tive coefficient b sy2 irp of the beta-function,0

while c was fixed by a one-instanton computation.1

It is obvious that c is scheme independent. Let us2

argue that c does not bring any scheme dependence1

either. A change of scheme corresponds to a finiteredefinition of coupling constants which, in turn,leads to a different value of the RG invariant:

Ž .g myH y1Lsme . 7Ž .b

In our case we find:yc 2

Lsc qc f1 2ž /m

yc 3c 69c2 2 2 2s q c q y q q . . . .12 ž /32 64256q

8Ž .

Although c appears as a constant on the rhs of the1

above equation and looks suspiciously similar to thelhs, it cannot be absorbed by a finite redefinition ofthe coupling z. Furthermore, by plotting Fs

2y1c log c qc f for different values of c it is2 1 2 1

immediately evident that this function yields flowswith different topologies, moving the IR attractor allover the z-plane. Therefore, rather than parametriz-ing different schemes, c labels different universality1

Žclasses: c s1 corresponds to Ns2 YM see Fig.1.1 , c s0 leads to flows of the type discussed below1Ž .see Fig. 2 , while other values of c give exotic1

flows with no obvious interpretation.In short, the contravariant beta-function for Ns2Ž .SU 2 YM is completely fixed by asymptotic match-

ing to a one-loop and a one-instanton computation.The scheme is selected in such a way that thebeta-function takes its simplest form with respect tomodular transformations. Coupling constantreparametrizations will produce a change of schemewhere the entanglement of the perturbative and non-perturbative parts of the beta-function can be verycomplicated. A deeper understanding of scheme de-pendence and the way in which reparametrizations ofthe moduli space can be used to simplify the descrip-tion of a C-theorem is needed.

Finally we consider the possibility that non-super-symmetric gauge theories are also constrained by a

( )J.I. Latorre, C.A. LutkenrPhysics Letters B 421 1998 217–222¨ 221

quantum duality. The simplest origin of such a quan-tum symmetry G would be if the Ns0 theory is a

Ž .broken version of Ns2, and GsG 2 is suffi-T

ciently robust to survive supersymmetry breakingŽ w x.see the relevant instance proposed in 16 , but wedo not need to commit ourselves to this scenario.There are several curious properties of ordinary YMgauge theory that may be construed as hinting atsuch a structure. The natural parametrization of YMmoduli space is obtained by writing the action in

Ž . Ž .terms of self-dual F and anti-self-dual F fieldq y2 2configurations: LszF qzF , where z is the sameq y

complexified coupling constant that appears in theNs2 case. Instanton corrections to the beta-function

w xhave been argued 17 to be periodic in theta andexponentially suppressed in perturbation theory, inprecisely such a way that they are holomorphic in

Ž 2 2 . Ž .this parameter: qsexp iuy8p rg sexp 2p iz ,as required by the q-expansion of automorphic func-tions. Note also that the u-parameter is ill-defined at

Ž .the fixed point zs i` of the sub- modular group, asrequired by perturbation theory. Furthermore, werecall that the original introduction of the modulargroup into physics was precisely in an attempt tounderstand the oblique confinement scenario advo-

w xcated by t’Hooft 18 . Unless this remarkable conflu-ence of convenient results is mere coincidence, it isnot unnatural to conjecture that they are orchestratedby a hidden quantum symmetry contained in themodular group. Since the vacuum is invariant undertranslations of the theta-parameter by 2p , T is oneof the generators and it follows that if G is not too

Ž .small then GsG 2 .T

We now formulate a precise conjecture whichimplies a unique form for the RG potential, andtherefore the phase- and RG flow diagram in thenon-supersymmetric case. We exploit the remarkable

Ž .mathematical fact that for G 2 there exist twoT

basic functions that transform properly with weightw xwsq2 9,4 :

TE z sEF z , z s1q q-expansionŽ . Ž . Ž .2 E

1TH z , z sEF z , z s q q-expansionŽ . Ž . Ž .2 H

p y

9Ž .

where, curiously, the potentials are built only fromŽ .the lowest weight cusp forms ‘‘instanton forms’’

Fig. 2. The phase and RG flow diagram of the potential F . FlowE

lines are downward and displayed only in the fundamental do-main, while equipotential lines are shown for u between yp andp .

Ž . Ž . Ž .for G 2 and G 1 sSL 2,Z , and ys Im z. TheyT

can be conveniently written as:i

< <F z , z s log f ,Ž .Ep

i4 8 8< <F z , z s log y u u . 10Ž . Ž .H 3 42p

Notice that while both these functions are fullyT TŽ . Ž .automorphic, and E z is holomorphic, H z, z is2 2

not. It is the unlikely existence of the latter, whichwe call a ‘‘quasi-holomorphic’’ or ‘‘Hecke’’ func-tion, which allows us to formulate our conjecture. Acontourplot of F is shown in Fig. 2, and the plotE

for F is very similar.H

Our conjecture for the non-supersymmetric casecan now be stated succinctly. If the quantum symme-

Ž .try group is G 2 , and the holomorphic structure isT

violated only in the benign way exhibited above inthe beta-function and in the Hecke form, by admit-ting only non-analytical terms of the type 1ry in the

Žasymptotic domain ‘t Hooft scheme, where only b0.and b are different from zero , then the candidate1

covariant beta-function is a linear combination of ET2

and H T where one of the constants can be fixed by2


comparison with perturbation theory, provided thecontravariant beta-function is constructed using anasymptotically flat metric. Thus, the final form of theconjecture is that Fsb F qbF , where b con-0 E H

spires with the sub-leading term in the metric to giveb etc. The phase- and RG flow diagram implied by1

this potential is evident from Fig. 2, showing an IRfixed point at zs0, except when usp in whichcase u is not renormalized and the IR fixed point is acritical point at a sg 2r4ps2. So this potentials

has the desirable property of having only one phasein which the theta-parameter renormalizes to zero inthe IR domain, unless theta is exactly p . This sug-gests that the strong CP problem may be automati-cally resolved within YM theory by quantum duality,since it forces theta to flow towards zero at strongcoupling, thus eliminating the need for axions orother ad-hoc constructions. The running of thetheta-parameter would stop at the mass gap, yieldinga finite but very small value of the physical theta-parameter. The existence of the critical point is inour context inescapable, and it may be possible totest this if the region with theta near p can be

Ž .probed in lattice simulations of SU 2 Yang-Mills.

We are grateful to J. Ambjorn, C. Burgess, D. Z.Freedman, P. E. Haagensen and F. Quevedo foruseful discussions and constructive comments on themanuscript. Financial support from CICYT, contractAEN95-0590, and from CIRIT, contract GRQ93-1047, and the Norwegian Science Council are ac-knowledged.

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On RG potentials in Yang-Mills theories