Abstract
In the non-supersymmetric gamma_i-deformed N=4 SYM theory, the scaling dimensions of the operators tr[Z^L] composed of L scalar fields Z receive finite-size wrapping and prewrapping corrections in the 't Hooft limit. In this paper, we calculate these scaling dimensions to leading wrapping order directly from Feynman diagrams. For L>=3, the result is proportional to the maximally transcendental `cake' integral. It matches with an earlier result obtained from the integrability-based Luescher corrections, TBA and Y-system equations. At L=2, where the integrability-based equations yield infinity, we find a finite rational result. This result is renormalization-scheme dependent due to the non-vanishing beta-function of an induced quartic scalar double-trace coupling, on which we have reported earlier. This explicitly shows that conformal invariance is broken - even in the 't Hooft limit.
Highlights
In the non-supersymmetric γi-deformed N = 4 SYM theory, the scaling dimensions of the operators tr[ZL] composed of L scalar fields Z receive finite-size wrapping and prewrapping corrections in the ’t Hooft limit
I.e. in the absence of finite-size effects, the dilatation operator can be obtained directly from its undeformed counterpart via a relation2 between planar singletrace Feynman diagrams of elementary interactions: in the deformed theory such a diagram is given by its undeformed counterpart multiplied by a phase factor which is determined from the order and (q1, q2, q3)-charge of the external fields alone
A simple test of the claimed integrability, beyond the asymptotic regime, can be performed by analyzing the spectrum of composite operators that are protected in the N = 4 SYM theory but acquire anomalous dimensions in the β- and γi-deformation
Summary
We analyze the diagrams which contribute to the renormalization of the composite operators (1.5) at any loop order K. As mentioned in the introduction, a planar single-trace diagram of elementary interactions in the γi- and β-deformation is given by its counterpart in the undeformed parent theory times a phase factor which is determined from the order and (q1, q2, q3)-charge of the external fields alone. This relation is based on the adaption of Filk’s theorem for spacetime-noncommutative field theories [9], and in the formulation of [38] it reads. At K ≥ L−1 loops, diagrams containing connected subdiagrams with double-trace structure tr[(φi)L] tr[(φi)L] can contribute They are associated with finite-size effects, i.e with the prewrapping and wrapping corrections at K ≥ L − 1 and K ≥ L loops, respectively. All deformation-dependent wrapping diagrams must be contained in the first class
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