Beyond cusp anomalous dimension from integrability

Journal of High Energy Physics, 2010

We compute the anomalous dimension of a length-five operator at five-loop order in the SU(2) sector of N = 4 SYM theory in the planar limit. This is critical wrapping order at five loops. The result is obtained perturbatively by means of N = 1 superspace techniques. Our result from perturbation theory confirms explicitly the formula conjectured in arXiv:0901.4864 for the fiveloop anomalous dimension of twist-three operators. We also explicitly obtain the same result by employing the recently proposed Y-system.

2011

We are considering the semi-classical string soliton solution of Gubser, Klebanov and Polyakov which represents highly excited states on the leading Regge trajectory, with large spin in AdS 5. A prescription relates this soliton solution with the corresponding field theory operators with many covariant derivatives, whose anomalous scaling dimension grows logarithmically with the space-time spin. We develop an iteration procedure which, in principle, allows to derive all terms in the large spin expansion of the anomalous scaling dimension of twist two operators at strong coupling. We explicitly derive the dependence of anomalous dimension on spin for all leading and next-to-leading orders. Our string theory results are consistent with the conjectured "reciprocity" relation, which has been verified to hold in perturbation theory up to five loops in N = 4 SYM. We also derive a duality relation between long and short strings.

Journal of Physics A: Mathematical and Theoretical, 2011

We are considering the semi-classical string soliton solution of Gubser, Klebanov and Polyakov which represents highly excited states on the leading Regge trajectory, with large spin in AdS 5 . A prescription relates this soliton solution with the corresponding field theory operators with many covariant derivatives, whose anomalous scaling dimension grows logarithmically with the space-time spin. We develop an iteration procedure which, in principle, allows to derive all terms in the large spin expansion of the anomalous scaling dimension of twist two operators at strong coupling. We explicitly derive the dependence of anomalous dimension on spin for all leading and next-to-leading orders. Our string theory results are consistent with the conjectured "reciprocity" relation, which has been verified to hold in perturbation theory up to five loops in N = 4 SYM. We also derive a duality relation between long and short strings.

2006

As the Hubbard energy at half filling is believed to reproduce at strong coupling (part of) the all loop expansion of the dimensions in the SU(2) sector of the planar N=4 SYM, we compute an exact non-perturbative expression for it. For this aim, we use the effective and well-known idea in 2D statistical field theory to convert the Bethe Ansatz equations into two coupled non-linear integral equations (NLIEs). We focus our attention on the highest anomalous dimension for fixed bare dimension or length, L, analysing the many advantages of this method for extracting exact behaviours varying the length and the 't Hooft coupling, λ. For instance, we will show that the large L (asymptotic) expansion is exactly reproduced by its analogue in the BDS Bethe Ansatz, though the exact expression clearly differs from the BDS one (by non-analytic terms). Performing the limits on L and λ in different orders is also under strict control. Eventually, the precision of numerical integration of the N...

2020

In the present paper in the Section 1, we have described some equations concerning the cusp anomalous dimension in the planar limit of N = 4 super Yang-Mills from a Thermodynamic Bethe Antsaz (TBA) system, the Luscher correction at strong coupling and the strong coupling expansion of the function F. In the Section 2, we have described some equations concerning a two-parameter family of Wilson loop operators in N = 4 supersymmetric Yang-Mills theory which interpolates smoothly between the 1/2 BPS line or circle, principally some equations concerning the one-loop determinants. In the Section 3, we have described some results and equations of the mathematician Ramanujan concerning some definite integrals and an infinite product and some equations concerning the development of derivatives of order n (n positive integer) of various trigonometric functions and divergent series. Thence, we have described some mathematical connections between some equations concerning this Section and the S...

2009

We compute the anomalous dimension of a length-five operator at five-loop order in the SU(2) sector of N=4 SYM theory in the planar limit. This is critical wrapping order at five loops. The result is obtained perturbatively by means of N=1 superspace techniques. Our result from perturbation theory confirms explicitly the formula conjectured in arXiv:0901.4864 for the five-loop anomalous dimension

2006

Some non-perturbative considerations on finite length operators. Abstract As the Hubbard energy at half filling is believed to reproduce at strong coupling (part of) the all loop expansion of the dimensions in the SU (2) sector of the planar N = 4 SYM, we compute an exact non-perturbative expression for it. For this aim, we use the effective and well-known idea in 2D statistical field theory to convert the Bethe Ansatz equations into two coupled non-linear integral equations (NLIEs). We focus our attention on the highest anomalous dimension for fixed bare dimension or length, L, analysing the many advantages of this method for extracting exact behaviours varying the length and the 't Hooft coupling, λ. For instance, we will show that the large L (asymptotic) expansion is exactly reproduced by its analogue in the BDS Bethe Ansatz, though the exact expression clearly differs from the BDS one (by non-analytic terms). Performing the limits on L and λ in different orders is also unde...

Journal of Statistical Mechanics: Theory and Experiment, 2007

As the Hubbard energy at half filling is believed to reproduce at strong coupling (part of) the all loop expansion of the dimensions in the SU (2) sector of the planar N = 4 SYM, we compute an exact non-perturbative expression for it. For this aim, we use the effective and well-known idea in 2D statistical field theory to convert the Bethe Ansatz equations into two coupled non-linear integral equations (NLIEs). We focus our attention on the highest anomalous dimension for fixed bare dimension or length, L, analysing the many advantages of this method for extracting exact behaviours varying the length and the 't Hooft coupling, λ. For instance, we will show that the large L (asymptotic) expansion is exactly reproduced by its analogue in the BDS Bethe Ansatz, though the exact expression clearly differs from the BDS one (by non-analytic terms). Performing the limits on L and λ in different orders is also under strict control. Eventually, the precision of numerical integration of the NLIEs is as much impressive as in other easierlooking theories.

Journal of High Energy Physics, 2012

We derive an analytic formula at three loops for the cusp anomalous dimension Γ cusp (φ) in N = 4 super Yang-Mills. This is done by exploiting the relation of the latter to the Regge limit of massive amplitudes. We comment on the corresponding three loops quark anti-quark potential. Our result also determines a considerable part of the threeloop cusp anomalous dimension in QCD. Finally, we consider a limit in which only ladder diagrams contribute to physical observables. In that limit, a precise agreement with strong coupling is observed.

Journal of High Energy Physics, 2011

Twist operators in the closed sl´¾µ sector of planar AE SYM are characterized by their spin. The explicit dependence of anomalous dimensions on this important parameter is a source of interesting information. Wrapping corrections are a non trivial part of the calculation and are under control in the framework of thermodynamical Bethe Ansatz valid for the full theory and thoroughly checked in that sector. The extension to more general twist operators beyond sl´¾µ has been recently accomplished for the so-called 3-gluon operators that are a special case of the generalized twist operators introduced by Freyhult, Rej and Zieme. Such operators are dual to spinning strings configurations with two spins Ë ½ , Ë ¾ in Ë and charge in Ë. We compute the expansion of the weak-coupling leading order wrapping correction in the gauge theory limit dual to large Ë ½ and fixed Ë ¾. We present a simple algorithm for the calculation and provide explicit results illustrating the general structure of the expansion.