We discuss non-renormalization properties of some composite operators in N=4
supersymmetric Yang-Mills theory.Comment: Contribution to the proceedings of the conference SUSY01 held in
  Dubna, June 2001; Latex, 4

Arutyunov, G.

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We discuss non-renormalization properties of some composite operators in N=4 supersymmetric Yang-Mills theory

Arutyunov, Gleb

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On Exceptional Non-renormalization Properties of N=4 SYM

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arXiv:hep-th/0108168 v1   22 Aug 2001AEI-2001-107ON EXCEPTIONAL NON-RENORMALIZATIONPROPERTIES OF N = 4 SYM4G. ArutyunovMax-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut,Am Mühlenberg 1, D-14476 Golm, GermanyEmail:agleb@aei-potsdam.mpg.deAbstractWe discuss non-renormalization properties of some composite operatorsin N = 4 supersymmetric Yang-Mills theory.Recently considerable attention was attracted to the N = 4 super Yang-Mills theory basically due to the prominent role it plays among the modelsrealizing the holographic AdS/CFT duality [1]. In the superconformal phasethe dynamics of the gauge theory is encoded in the correlation functions ofthe composite gauge invariant operators, which might exhibit in general anon-trivial behavior under the RG flow. In particular it is of great interestto determine the 4-point correlation functions (both holographic and weakcoupling) of the N = 4 supercurrent (stress-tensor) multiplet L and its OPE;the latter contains an information about many other composite operatorspresent in the theory.A superconformal primary operator generating L is a scalar OI of di-mension 2 transforming in the irrep 20 of the R-symmetry group SU(4),I = 1, . . . , 20. Presently both the holographic [2] and the weak coupling [3]4-point correlators of OI and their OPE studies [4, 5, 6] are available.1 Sur-prisingly composite operators2 with vanishing anomalous dimensions werefound [4] though naively unitarity allows the latter to appear in quantuminteracting theory.This note is based on the paper [6] and reviews a statement that the OPEof two primary operators from the multiplet L can contain superconformal1For studies of other correlation functions from stress-tensor multiplet see e.g. [7].2They saturate the bound of the so-called series A) of unitary irreps of SU(2, 2|4) andtransform non-trivially under R-symmetry [8].1primary operators with a non-vanishing anomalous dimension only in thesinglet of SU(4).It was found non-perturbatively [9] that the “quantum” part of the four-point function of OI comprising all possible quantum corrections to thefree-field result is given by a single function F (v, u) of conformal cross-ratios, which we choose to be v =x212x234x214x223and u = 1 −x213x224x214x223. Under SU(4)the product of two OI decomposes as 20×20 = 1+20+105+84+15+175 .The “quantum” part of the four-point function of the operators OI projectedon different irreps is〈O(x1)O(x2)O(x3)O(x4)〉i =1x412x434Pi(v, u)vF (v, u)(1 − u)2, (1)where Pi(v, u) are certain polynomials [4, 5]. Every irrep i of SU(4) in theOPE of two OI represents a contribution from an infinite tower of operatorsOi∆,l, where ∆ is the conformal dimension of the operator, l is its Lorentzspin. The corresponding contribution to the four-point function can thenbe represented as an expansion of the type〈O(x1)O(x2)O(x3)O(x4)〉i =∑∆,lai∆,lH∆,l(x1,2,3,4) . (2)Here H∆,l(x1,2,3,4) denotes the (canonically normalized) Conformal PartialWave Amplitude (CPWA) for the exchange of an operator Oi∆,l and ai∆,l is anormalization constant. We treat the CPWA as a double series of the typeH∆,l =1x412x434vh2∞∑n,m=0c∆,lnmvnum , (3)where the dimension ∆ was split into a canonical part ∆0 and an anomalouspart h: ∆ = ∆0 + h. Assigning the grading parameter T = 2n + m to themonomial vnum one can show that the monomials in (3) with the lowestvalue of T have T = ∆0, where ∆0 is the canonical (free-field) dimension ofthe corresponding operator.Comparing (1) and (2) one finds, within every fractional power vh2 , thefollowing compatibility conditionsPi∑∆,laj∆,lH∆,l(x1,2,3,4) = Pj∑∆,lai∆,lH∆,l(x1,2,3,4) (4)which hold for all pairs. Here the sums are taken over operators which havethe same h. Thus, eqs. (4) imply non-trivial relations between the CPWAs2of primary operators belonging to the same supersymmetry multiplet(s)with anomalous dimension h. Only one of these primary operators is thesuperconformal primary operator, i.e., it generates under supersymmetrythe whole multiplet, while the others are its descendents.Now we see that a superconformal primary operator appears only in thesinglet of SU(4). Indeed, let us choose in (4) the irrep j to be the singlet.The polynomial P1 is distinguished from the other Pi’s by the presence ofa constant term. Suppose that a superconformal primary operator with acanonical dimension ∆0 contributes to the OPE and transforms in someirrep i which is not a singlet. Due to the constant in P1, the lowest-ordermonomials on the r.h.s. of (4) would have T = 2n+m = ∆0. Clearly, all theother Pi’s always raise the T -grading by at least unity. The lowest dimensionoperator with canonical dimension ∆′0 in the singlet would have the lowestterms with at least T = ∆0 − 1 (or lower) to saturate (4). Hence, ∆′0 isalways lower then ∆0, and therefore the corresponding operator cannot bea supersymmetry descendent of an operator in the irrep i. This shows thatanomalous superconformal primary operators are occure in the singlet of theR-symmetry group.References[1] O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri, Y. Oz, Phys. Rept.323, 183 (2000), hep-th/9905111.[2] G.E. Arutyunov and S.A. Frolov, Nucl. Phys. B 544, 576 (1999),hep-th/9806216; Phys. Rev. D 61, 064009 (2000), hep-th/9907085;Nucl. Phys. B 579, 117 (2000), hep-th/9912210; Phys. Rev. D 62,064016 (2000), hep-th/0002170; Phys. Rev. D 60, 026004 (1999), hep-th/9901121.[3] B. Eden, P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Nucl.Phys. B 557, 355 (1999), hep-th/9811172; Phys. Lett. B 466, 20 (1999),hep-th/9906051; B. Eden, C. Schubert and E. Sokatchev, Phys. Lett. B482, 309 (2000), hep-th/0003096; M. Bianchi, S. Kovacs, G. Rossi andY. S. Stanev, JHEP 9908, 020 (1999), hep-th/9906188; Nucl. Phys.B 584, 216 (2000), hep-th/0003203; JHEP 0105, 042 (2001), hep-th/0104016.3[4] G. Arutyunov, S. Frolov and A.C. Petkou, Nucl. Phys. B 586,547 (2000), hep-th/0005182; Nucl. Phys. B 602, 238 (2001), hep-th/0010137.[5] G. Arutyunov, B. Eden, A.C. Petkou and E. Sokatchev, Exceptionalnon-renormalization properties and OPE analysis of chiral four-pointfunctions in N = 4 SYM(4), hep-th/0103230.[6] G. Arutyunov, B. Eden and E. Sokatchev, On non-renormalization andOPE in superconformal field theories, hep-th/0105254.[7] E. D’Hoker, S. D. Mathur, A. Matusis and L. Rastelli, Nucl. Phys. B589, 38 (2000); E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusisand L. Rastelli, Nucl. Phys. B 562, 353 (1999).[8] S. Ferrara and A. Zaffaroni, Superconformal field theories, multipletshortening, and the AdS5/SCFT4 correspondence, hep-th/9908163.[9] B. Eden, A. Petkou, C. Schubert and E. Sokatchev, Partial non-renormalization of the stress-tensor four-point function in N = 4 SYMand AdS/CFT, hep-th/0009106.4

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