Stringy sums and corrections to the quantum string Bethe ansatz

TL;DR

This work critically tests zeta-function regularization for summing one-loop quantum corrections to spinning strings in AdS5×S5. By comparing naive zeta-regularized sums to exact evaluations across folded-string, toy-model, and circular-string (su(2)) cases, the authors show that ζ-regularization misses both perturbative odd-power terms and non-perturbative exponential corrections in the effective coupling. They employ multiple complementary methods—asymptotic/gamma integral representations, Bessel-function representations, and generalized zeta functions—to demonstrate the complete correction structure and identify non-analytic contributions that are not encoded in the quantum string Bethe equations. The findings explain observed discrepancies at large winding and suggest that non-perturbative, non-analytic effects must be incorporated (potentially via S-matrix or Bethe-ansatz refinements) to achieve accurate quantum-string descriptions. Overall, the paper argues for going beyond zeta-regularization when matching semiclassical string results with Bethe-ansatz based quantum strings, emphasizing the importance of exponential and non-perturbative terms in the spectrum.

Abstract

We analyze the effects of zeta-function regularization on the evaluation of quantum corrections to spinning strings. Previously, this method was applied in the sl(2) subsector and yielded agreement to third order in perturbation theory with the quantum string Bethe ansatz. In this note we discuss related sums and compare zeta-function regularization against exact evaluation of the sums, thereby showing that the zeta-function regularized expression misses out perturbative as well as non-perturbative terms. In particular, this may imply corrections to the proposed quantum string Bethe equations. This also explains the previously observed discrepancy between the semi-classical string and the quantum string Bethe ansatz in the regime of large winding number.