On a new type of divergence for spiky Wilson loops and related entanglement entropies
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Springer
Received: February 12, 2018 Accepted: March 15, 2018 Published: March 21, 2018
Harald Dorn Institut f¨ ur Physik und IRIS Adlershof, Humboldt-Universit¨ at zu Berlin, Zum Großen Windkanal 6, D-12489 Berlin, Germany
E-mail: [email protected] Abstract: We study the divergences of Wilson loops for a contour with a cusp of zero opening angle, combined with a nonzero discontinuity of its curvature. The analysis is performed in lowest order, both for weak and strong coupling. Such a spike contributes a leading divergent term proportional to the inverse of the square root of the cutoff times the jump of the curvature. As nextleading term appears a logarithmic one in the supersymmetric case, but it is absent in QCD. The strong coupling result, obtained from minimal surfaces in AdS via holography, can be used also for applications to entanglement entropy in (2+1)-dimensional CFT’s. Keywords: Wilson, ’t Hooft and Polyakov loops, AdS-CFT Correspondence, Renormalization Regularization and Renormalons ArXiv ePrint: 1801.10367
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP03(2018)124
JHEP03(2018)124
On a new type of divergence for spiky Wilson loops and related entanglement entropies
Contents 1
2 Lowest order perturbation theory
2
3 Holographic evaluation for strong coupling
5
4 Conclusions
8
A Integral for lowest order perturbation theory
9
B Integral for holographic evaluation
11
C Subtleties due to touching of two spikes
12
1
Introduction
Wilson loops for smooth contours in non-supersymmetric gauge theories, besides a linear divergence proportional to the length, do not require any further renormalisation beyond that of the coupling constant [1, 2]. For the local supersymmetric generalisation in N = 4 super Yang-Mills [3, 4] the situation is even more comfortable: the coupling is not renormalised, and the linear divergence cancels between the gauge boson and scalar contribution [5]. For contours with cusps or self-intersections each of these singular points generates renormalisation Z-factors [1, 6]. The corresponding cusp anomalous dimension depends on the cusp angle1 ϑ and the coupling constant and has been calculated in the 80-ies up to perturbative two loop level [1, 7, 8]. Later on it turned out to be related to various other physical situations, and with the advent of AdS-CFT holography it became one of the most studied quantities from both the weak as well as the strong coupling side. It is now available up to three loops both for N = 4 SYM and QCD [9, 10]. For strong coupling one has the leading and next leading contribution [5, 11]. Of special interest is also the limit of large imaginary angle [8]. It plays a crucial role for scattering amplitudes and the dimensions of large spin operators, and using techniques of integrability even an interpolation between weak and strong coupling has been found [12, 13]. The minimal string surfaces, one has to consider for the strong coupling evaluation, play still another prominent role. They carry all t
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