Supersymmetric QCD

VOLUME 78, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 3 FEBRUARY 1997

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Charge Fractionization in N 5 2 Supersymmetric QCD

Frank Ferrari *Laboratoire de Physique Théorique de l’École Normale Supérieure,†

24 rue Lhomond, 75231 Paris Cedex 05(Received 11 October 1996)

It is shown that the physical “quark number” charges which appear in the central chargethe supersymmetry algebra ofN 2 supersymmetric QCD can take irrational values and depennontrivially on the Higgs expectation value. This gives a physical interpretation of the constant shwhich the “electric” and “magnetic” variablesaD and a undergo when encircling a singularity, andshow that duality in this model is truly an electric-magnetic-quark number duality. Also included iscomputation of the monodromy matrices directly in the microscopic theory. [S0031-9007(96)02244

PACS numbers: 11.30.Pb, 11.15.Kc, 11.30.Er

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During the last couple of years, huge progress hbeen made in the understanding of four-dimensionalN 2 supersymmetric gauge theories, following the worof Seiberg and Witten [1,2], where the low energ(Wilsonian) effective action was computed exactly up ttwo derivatives and four fermions terms for the gauggroup SU(2). The mathematical structure that emergthere was then generalized to obtain very plausibsolutions for general gauge groupsG (see [3] and [4], andreferences therein). All these theories have a complmanifold of inequivalent vacua (moduli space). Thidegeneracy comes from flat directions in the scalpotential which cannot be lifted quantum mechanicaldue to tight constraints imposed byN 2 supersymmetry[5]. The moduli space has a Coulomb branch where tgauge group, of rankr, is spontaneously broken downto Us1dr . Many interesting physical phenomena occualong the Coulomb branch, including the appearanof massless solitonic states at strong coupling [1,2a subtle realization of electric-magnetic duality at thlevel of the low energy physics [1,2], the existence onontrivial conformal field theory in four dimensions [6],and discontinuities in the spectrum of stable BPS stat[7]. These “exactly soluble” theories are likely to playan outstanding role in our understanding of more realisgauge theories as QCD, like the Ising model did focritical phenomena.

In this Letter is explained how to compute from firsprinciples some Abelian charges appearing in the centchargeZ of the supersymmetry algebra. Then, knowinZ, one can compute the massm of any state lying in asmall representation of the supersymmetry algebra (eigdimensional for a CPT conjugate multiplet):

m p

2 jZj . (1)

Such states are called BPS states. All the elementexcitations (quarks,W bosons, etc.), as well as the knownsolitonic states (monopoles and dyons), are BPS staand their masses are thus given by (1). We will alsobtain an interesting physical interpretation of curiou

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monodromy properties, and we will be able to compute thmonodromy matrices directly in the microscopic theoryThis study will finally solve some puzzles about the BPSmass formula and the renormalization group flow.

(I) Presentation of the problem.—Along the Coulombbranch, the central chargeZ of the N 2 supersymmet-ric QCD contains, in addition to the electric and magnetic chargesQe and Qm, the “quark number” chargesSf . These charges correspond to the invariance of the Lgrangian under the transformationsQf ° eiaQf , Q̃f °e2iaQ̃f , whereQf and Q̃f are theN 1 chiral super-fields making up a matterN 2 hypermultiplet. Classi-cally, Z reads

Zcl 2a

µ1g

Qe 1ig

Qm

∂1

1p

2

NfXf1

mfSf , (2)

where Nf is the number of flavors,mf the bare massof the hypermultipletsQf , Q̃fd, and g the gauge cou-pling constant. For the gauge group SU(2), on whichwill focus for conciseness, the Dirac quantization condition can be writtenQm 4pnmyg, where nm is aninteger also called the “magnetic charge.” In [2], anexact quantum formula forZ was proposed:Z ane 1

aDnm 11

p2

Pf mfSf . Herea is the Higgs expectation

value, kfl as3, and aD 12 ≠aF sad is the dual vari-

able which can be expressed in terms of the prepotetial F governing the low energy effective action. Thisformula for Z is a straightforward generalization of thecorresponding formula for the pure gauge theory derivein [1] using electric-magnetic duality arguments. How-ever, we will see that interpretingSf in this formula asbeing the physical quark number is not free from contradictions. Problems also arise when considering that thterm ane 1 aDnm stems from corrections to the physicalelectric and magnetic charges alone.

To understand the origin of these difficulties, let usset for a moment the bare masses to zero. In this casthe quantum formulaZ ane 1 aDnm is not obtainedfrom (2) by simply replacingg by the running coupling

© 1997 The American Physical Society 795


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constant, even at one loop. One also has to take inaccount that, because CP invariance is spontaneoubroken by Ima fi 0, the physical electric charge can pickup terms in addition togney2. The simplest example ofthis phenomenon was first studied by Witten in [8]. Inthe theories with zero bare masses, all CP violation cabe absorbed in au angle by performing a chiral Us1dR

transformation. The formula of [8] can then be readilyadapted to our case and yieldsQe sgy2d hne 2 fs4 2

Nfdnmargagyp. Note that theSf charges are not affectedby a u term, since the latter appears in the Lagrangianfront of FF̃, which does not transform underSf . Thus,when mf 0, Sf is expected to have the value one cacompute in the CP conserving theories. For instancSf 6

12 whennm 1 [9].

However, when the bare masses are nonzero, wwould expectSf to depend on the Higgs expectationvalue, or alternatively on the gauge invariant coordinaon the Coulomb branch,u ktrf2l. This is strictlyanalogous to the phenomenon first discovered in [10where irrational fermion number values were found insome CP violating field theories. At first sight, thisseems bizarre. AsSf are real numbers, a nontrivialudependence would seem to violate holomorphy. Oncould then be tempted to forget about CP breaking anargue that, because of supersymmetry, theSf chargesmust be constant and equal to the values one compuin CP conserving theories. But one then faces anothemore subtle, difficulty. Suppose one is studying threnormalization group flow, say, from theNf 1 to theNf 0 theory, and that, in particular, one is trying todeduce the spectrum of stable BPS states of theNf 0theory from the one of theNf 1 theory. What shouldoccur is that some states of theNf 1 theory, becominginfinitely massive in the process, disappear from thspectrum of theNf 0 theory, and that other states,remaining of finite mass, finally constitute the stable BPstates of theNf 0 theory. Since the work in [7],we know that the spectrum of theNf 0 theory isindeed strictly included in the spectrum of theNf 1theory. Limiting the discussion to the weak-couplingspectra, which can be described semiclassically, all thmonopoles of oddne disappear when one goes fromNf 1 to Nf 0, while the monopoles of evenne formthe solitonic spectrum of the pure gauge theory. Buthis is incompatible with the previous formula forZ. IfS 6

12 for the monopoles, their massesm

p2 jZj will

diverge whatever their electric chargene, sinceaD andamust flow towards the solution of the pure gauge theorunder the action of the renormalization group. [Notethat the ambiguity coming from the nonzero residues othe Seiberg-Witten differential form can only account fointeger jumps inS in the caseNf 1, and this is notdirectly related to our problem.] So one must definitivelygive up the idea that the constantsSf appearing inZ could

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be the physical charges. I will rename these constantssf

and reserve the symbolSf for the physical charges. Thesf may be zero even for monopolesnm 1. The exactquantum formula for the central charge is then

Z ane 1 aDnm 11

p2

NfXf1

mfsf . (3)

In the remainder of this Letter I will explain where thephysical chargesSf hide and how to compute the numbersf . We will also find a physical explanation of the curiousmonodromy propertiesaD and a have in the massivetheories [they pick up constants in addition to the standaSL(2,Z) transformations] and a new method to computthe monodromy matrices, directly in the microscopitheory.

(II) Computation of the physical charges.—In this sec-tion, I outline the computation of the physical chargesSf

and of the electric charge, focusing on the contributioof the fermions. This yields the most interesting resultA detailed discussion of possible additional contributionthe generalization to any gauge group, as well as the dcussion of other physical aspects related to CP invarianin our theories shall be published elsewhere.

To study the semiclassical contributions to theSf chargeof the Dirac fermionsxf belonging to the hypermultipletQf , Q̃f , we need to quantize the Dirac field aroundnontrivial monopole background characterized by a fixemagnetic chargenm. The Dirac equation is

ip

2 gmDmxf sM 1 ig5Ndxf

11

p2

fsRemf dxf 1 isImmfdg5xf g ,

(4)

where M Ref and N Imf. Usually one focuseson the (complex) zero modes of this equation, whosnumberksnmd is given by an index theorem of Callias[11]. Each zero mode carries one unit ofSf charge,which shows that, if2Sf,msnmd is the minimal valueSf can reach in the monopole sectornm, the set ofallowed values ofSf will be h2Sf,msnmd, 2Sf,msnmd 1

1, . . . , 2Sf,msnmd 1 ksnmdj. If (4) were CP invariant, onewould deduce thatSf,msnmd ksnmdy2, becauseSf is oddunder CP. However, in our case we cannot forget about tmassive modes of (4). When CP is violated, the densof states having positive energy differs from the density ostates having negative energy. This means that the Diroperator under consideration has a nonzero Atiyah-PatoSingerh invariant, which formally readsh

PEn.0 1 2P

En,0 1, wheren labels the energy levels. [Note that theoperator is defined on an open space here, thus wedealing with a generalizedh invariant.] It is not difficultto relate the spectral asymmetry quantified byh to theSf

charge, carefully taking into account the fact thatSf must


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be odd under CP. One finds

Sf,msnmd 12

h . (5)

Fortunately, the computation of theh invariants of variousDirac operators and the application to charge fractioniztion have been extensively studied in the literature. Theworks were motivated not only for purely field theoreticreasons, but also because of their important phenomelogical applications in the physics of linearly conjugatepolymers (see, e.g., [12]). In particular, in [13], a vergeneral method, applicable to our Dirac operator, was dveloped. See also [14], and [15] for a review. The resuof the computation is

Sf,msnmd 2nm

2parg

a 1 mfyp

2

mfyp

2 2 a. (6)

The same type of technique can be applied to evaluatephysical electric charge, which picks up terms in additioto the standard Witten effect [8] term. One can, foinstance, use the Gauss law, following [16], and find foour theory

2g

Qe ne 24p

nmarga 1nm

2p

NfXf1

argsm2f 2 2a2d .

(7)

(III) Physical Analysis.—Let us discuss the physicalmeaning of the formulas found in the previous sectioNote that we have a singular point whena 6mfy

p2,

which corresponds to a quark becoming massless. Itvery instructive to study the monodromy properties oSf around this singularity. Since all the nontrivial,u-dependent parts ofSf are included in2Sf,m, encirclingthe singularity ata mfy

p2 yields Sfsud ° Sfsud 2

nm. This is reminiscent of the shiftsf ° sf 1 nm theconstantsf undergoes [2]. [Though constants, thesf dotransform nontrivially when encircling a singularity, in thesame sense that the constantsne and nm are mixed bythe monodromy matrix.] However, the sign differencbetween the transformations ofSf and sf is crucial. Itdefinitively proves thatsf cannot be identified withSf .Moreover, it shows thatSfsud andaDsud pick up the sameterm under the monodromy. This simply means thatSf sudis already included inaDsud (at weak coupling) and isresponsible for the curious constant shiftaD was knownto undergo since the work in [2]. In the strong couplinregion, sincea andaD are intimately related due to the nonAbelian monodromies, theSf charges will also contributeto a. Note, however, that the distinction betweenQe, Qm,andSf is very unclear in the strong coupling region, anthat the natural quantities to use area andaD.

The fact that the variablesa and aD do pick upcontributions from the electric, magnetic,and Sf chargesis very interesting from the physical point of view. Thismeans that duality in the theories with nonzero bamasses is really an electric-magnetic-Sf duality. Thisphenomenon is likely to be quite general when Abelia

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charges appear in the central charge in addition toelectric and magnetic charges.

Now, it should be clear that the variablessf appearingin the formula (3) are just constant parts of theSf

charges not already included inaD and a. In particular,one can computesf in the weak-coupling region bystudying the asymptotics ofaD when u and mf arelarge compared to the dynamically generated scale oftheory, then extracting the terms contributing toSf fromthis asymptotics, and choosingsf in order to match withthe formulaSf 2Sf,msnf d 1 p. Herep is an integerbetween 0 andksnf d; see section II.

Let us close this section computing the SL(2,Z) mono-dromy matrixM corresponding to a singularity due toquark becoming massless. This can easily be done us(7), which shows thatQe contribute toaD through theterm 2s4ypda arga 1 s1y2pda

Pargsm2

f 2 2a2d; see(2). Thus, when no bare masses coincide,

M

µ10

11

∂, (8)

since a, being a good local coordinate aroundu m2f ,

obviously does not transform. This result agrees wthe standard computation from the low energy effectitheory. When some bare masses coincide, we obtainmonodromy matrix corresponding to several hypermulplets becoming massless at the same time. Then, pforming SL(2,Z) transformations, it is possible to deducthe most general monodromy matrix corresponding to anumber ofsnm, ned states becoming massless.

In conclusion, we gained interesting physical insightthe meaning of duality inN 2 supersymmetric QCDby using semiclassical methods. This was possible sinwhen bare masses are much larger than the dynamicgenerated scale of the theory, some singularities canpresent at weak coupling. We found that in this regimthe contribution of the physical electric chargeQe to aD

can account for the SL(2,Z) monodromy matrices associated with the weak-coupling singularities. This providea new way to derive these matrices directly in the micrscopic theory. More importantly, we saw that the physicSf charges contribute toaD and can account for the constant shift this variable undergoes. At strong coupling, tthree Abelian charges appearing in the central chargethe supersymmetry algebra will intimately mix togetheproving an example of an electric-magnetic-quark numbduality.

I wish to thank Adel Bilal, Eugène Cremmer, and JeaLoup Gervais for useful discussions and encouragemeas well as Tom Wynter who kindly read the manuscript

Note added.—After the first appearance of this workon the hep-th archive (9609101), a preprint appear[17] where the scenario described in section I for threnormalization group flow is shown to occur. For thNf 1 theory, the authors of [17] were able to findwithin a string theory framework, thats 0 for the

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monopole (1,0) ands 21 for the dyon (1,1). Thiswould contradict the semiclassical computation ifs werethe physical charge, but is in perfect agreement with thdiscussion of section I.

*Electronic address: [email protected]†Unité Propre de Recherche 701 du CNRS, associée

l’École Normale Supérieure et á l’Université Paris Sud.[1] N. Seiberg and E. Witten, Nucl. Phys.B426, 19 (1994);

B430, 485(E) (1994).[2] N. Seiberg and E. Witten, Nucl. Phys.B431, 484 (1994).[3] A. Klemm, W. Lerche, S. Yankielowicz, and S. Theisen

Phys. Lett. B344, 169 (1995); A. Klemm, W. Lerche, andS. Theisen, Int. J. Mod. Phys. A11, 1929 (1996); P. C.Argyres and A. Faraggi, Phys. Rev. Lett.73, 3931 (1995).

[4] M. R. Abolhasani, M. Alishahiha, and A. M. GhezelbashNucl. Phys.B480, 279 (1996).

[5] N. Seiberg, Phys. Rev. D49, 6857 (1994).[6] P. C. Argyres and M. R. Douglas, Nucl. Phys.B448, 93

(1995); P. C. Argyres, M. R. Plesser, N. Seiberg, anE. Witten, Nucl. Phys.B461, 71 (1996).

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[7] F. Ferrari and A. Bilal, Nucl. Phys.B469, 387 (1996);A. Bilal and F. Ferrari, Nucl. Phys.B480, 589 (1996).

[8] E. Witten, Phys. Lett. B86, 283 (1979).[9] R. Jackiw and C. Rebbi, Phys. Rev. D13, 3398 (1976).

[10] J. Goldstone and F. Wilczek, Phys. Rev. Lett.47, 986(1981).

[11] C. Callias, Commun. Math. Phys.62, 213 (1978); R. Bottand R. Seeley, Commun. Math. Phys.62, 235 (1978).

[12] W. P. Su, inHandbook of Conducting Polymers,edited byT. Skotheim (Marcel-Dekker, New York, 1984).

[13] A. J. Niemi and G. W. Semenoff, Nucl. Phys.B269, 131(1986).

[14] M. B. Paranjape and G. W. Semenoff, Phys. Lett. B132,369 (1983); A. J. Niemi and G. W. Semenoff, Phys. ReD 30, 809 (1984).

[15] A. J. Niemi and G. W. Semenoff, Phys. Rep.135, 99(1986).

[16] A. J. Niemi, M. B. Paranjape, and G. W. Semenoff, PhyRev. Lett.53, 515 (1984).

[17] A. Brandhuber and S. Stieberger, Report No. CERTHy96-263, NEIP-96y7, hep-thy9610053 (unpublished).

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Supersymmetric QCD