Article

# Operator mixing N = 4 SYM: The Konishi anomaly revisited

Dipartimento di Fisica, Università di Roma “Tor Vergata”, INFN, Sezione di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy
(Impact Factor: 3.93). 01/2005; 722(1):119-148. DOI: 10.1016/j.nuclphysb.2005.06.005
Source: arXiv

ABSTRACT

In the context of the superconformal N=4 SYM theory the Konishi anomaly can be viewed as the descendant K10 of the Konishi multiplet in the 10 of SU(4), carrying the anomalous dimension of the multiplet. Another descendant O10 with the same quantum numbers, but this time without anomalous dimension, is obtained from the protected half-BPS operator O20′ (the stress-tensor multiplet). Both K10 and O10 are renormalized mixtures of the same two bare operators, one trilinear (coming from the superpotential), the other bilinear (the so-called “quantum Konishi anomaly”). Only the operator K10 is allowed to appear in the right-hand side of the Konishi anomaly equation, the protected one O10 does not match the conformal properties of the left-hand side. Thus, in a superconformal renormalization scheme the separation into “classical” and “quantum” anomaly terms is not possible, and the question whether the Konishi anomaly is one-loop exact is out of context. The same treatment applies to the operators of the BMN family, for which no analogy with the traditional axial anomaly exists. We illustrate our abstract analysis of this mixing problem by an explicit calculation of the mixing matrix at level g4 (“two loops”) in the supersymmetric dimensional reduction scheme.

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• "Further, Z F also receives nonvanishing finite corrections at higher loops. Our work fully confirms the general considerations about the singularities of the Z-factors put forward in [9] and extends the leading order perturbative analysis presented there. The descendant operator K has the anomalous dimension of the Konishi operator as required by supersymmetry. "
##### Article: Operator mixing in N=4 SYM: The Konishi anomaly re-re-visited
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ABSTRACT: The supersymmetry transformation relating the Konishi operator to its lowest descendant in the 10 of SU(4) is not manifest in the N=1 formulation of the theory but rather uses an equation of motion. On the classical level one finds one operator, the unintegrated chiral superpotential. In the quantum theory this term receives an admixture by a second operator, the Yang–Mills part of the Lagrangian. It has long been debated whether this “anomalous” contribution is affected by higher loop corrections. We present a first principles calculation at the second non-trivial order in perturbation theory using supersymmetric dimensional reduction as a regulator and renormalisation by Z-factors. Singular higher loop corrections to the renormalisation factor of the Yang–Mills term are required if the conformal properties of two-point functions are to be met. These singularities take the form determined in preceding work on rather general grounds. Moreover, we also find non-vanishing finite terms.The core part of the problem is the evaluation of a four-loop two-point correlator which is accomplished by the Laporta algorithm. Apart from several examples of the T1 topology with two lines of non-integer dimension we need the first few orders in the ϵ expansion of three master integrals. The approach is self-contained in that all the necessary information can be derived from the power counting finiteness of some integrals.
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• "The difficulty lies in the fact that the classical on-shell supersymmetry transformations, if used for instance in schemes related to dimensional regularisation like in ours, do not yield the additional term when applied to K 1 ; hence the term " anomaly " . It has to be fixed by independent means, i.e. by finding conformal eigenstates, or in other words by diagonalising the mixing matrix [13] [10] [15]. In [13] [10] the obstacle was circumvented by going one step higher in the multiplet: The operators B and F both transform into the same supersymmetry descendent Y = Tr([Φ 1 , Φ 2 ][Φ 1 , Φ 2 ]) (16) so that K 10 has a descendent K 84 = a(g 2 )Y. "
##### Article: Boxing with Konishi
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ABSTRACT: The spin chain formulation of the operator spectrum of the N=4 super Yang-Mills theory is haunted by the problem of ``wrapping'', i.e. the inapplicability of the formalism for short spin chain length at high loop-order. The first instance of wrapping concerns the fourth anomalous dimension of the Konishi operator. While we do not obtain this number yet, we lay out an operational scheme for its calculation. The approach passes through a five- and six-loop sector. We show that all but one of the Feynman integrals from this sector are related to five master graphs which ought to be calculable by the method of partial integration. The remaining supergraph is argued to be vanishing or finite; a numerical treatment should be possible. The number of numerator terms remains small even if a further four-loop sector is included. There is no need for infrared rearrangements.
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##### Article: More on "La Grande Bouffe": Towards higher spin symmetry breaking in AdS
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ABSTRACT: We discuss higher spin gauge symmetry breaking in AdS space from a holographic prespective. Indeed, the AdS/CFT correspondence implies that N=4 SYM theory at vanishing coupling constant is dual to a theory in AdS which exhibits higher spin gauge symmetry enhancement. When the SYM coupling is non-zero, the current conservation condition becomes anomalous, and correspondingly the local higher spin symmetry in the bulk gets spontaneously broken. In agreement with previous results and holographic expectations, we find that the Goldstone mode responsible for the symmetry breaking in AdS has a non-vanishing mass even in the limit in which the gauge symmetry is restored. Moreover, we show that the mass of the Goldstone mode is exactly the one predicted by the correspondence. Finally, we obtain the precise form of the higher spin supercurrents in the SYM side. Comment: LaTeX file, JHEP3 class, 24 pages. Corrections to formulae in section 5.1 and reference added
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