The AdS(n) x S(n) x T(10-2n) BMN string at two loops

TL;DR

This work performs the first full two-loop BMN analysis for strings in $AdS_n\times S^n\times T^{10-2n}$, systematically regularizing divergences and computing two-loop dispersion relations for both massive and massless worldsheet modes. Massive sectors across backgrounds agree with symmetry-derived dispersions, with no $h$-corrections, while massless modes in $AdS_3\times S^3\times T^4$ display a significant discrepancy in the two-loop coefficient relative to symmetry expectations. The massless S-matrix is computed up to one loop, aligning with the massless limit of the Hernández-Lopez phase after IR subtraction, and a two-loop forward scattering amplitude vanishes on shell, suggesting no finite two-loop phase in the massless sector. The results underscore the consistency of integrability-based predictions for massive modes and reveal subtle issues in the massless sector that warrant further investigation, including potential quantum corrections to central charges and extensions to more intricate backgrounds.

Abstract

We calculate the two-loop correction to the dispersion relation for worldsheet modes of the BMN string in AdS(n) x S(n) x T(10-2n) for n=2,3,5. For the massive modes the result agrees with the exact dispersion relation derived from symmetry considerations with no correction to the interpolating function h. For the massless modes in AdS(3) x S(3) x T(4) however our result does not match what one expects from the corresponding symmetry based analysis. We also derive the S-matrix for massless modes up to the one-loop order. The scattering phase is given by the massless limit of the Hernandez-Lopez phase. In addition we compute a certain massless S-matrix element at two loops and show that it vanishes suggesting that the two-loop phase in the massless sector is zero.

Figures (6)

• Figure 1: Three out of the six wineglass diagrams. The other three are obtained by flipping the position of the bubble loop.
• Figure 2: Double bubble diagrams
• Figure 3: Another double-bubble type diagram. Some trivial permutations are not written out.
• Figure 4: Bubble and a tadpole with four- and six-vertices.
• Figure 5: Another bubble-tadpole type diagram. We have suppressed the three additional diagrams with the tadpole attached to the second vertex.