Subsubleading soft theorems of gravitons and dilatons in the bosonic string

TL;DR

Extends soft-theorem analysis to the bosonic string at subsubleading order for soft gravitons and dilatons, revealing string corrections for gravitons but none for dilatons. The authors derive a gauge-invariant soft operator that reproduces the full amplitude through subsubleading order by leveraging gauge invariance and the three-point massless-string vertex, including alpha' corrections from the three-point function. The graviton soft theorem acquires string corrections, while the dilaton soft theorem remains field-theory-like, with its finite piece corresponding to the momentum-space generator of special conformal transformations. These results clarify the string-theoretic origin of soft theorems and suggest extensions to heterotic/superstrings and Kalb-Ramond states, linking soft behavior to conformal symmetry structures in momentum space.

Abstract

Starting from the amplitude with an arbitrary number of massless closed states of the bosonic string, we compute the soft limit when one of the states becomes soft to subsubleading order in the soft momentum expansion, and we show that when the soft state is a graviton or a dilaton, the full string amplitude can be expressed as a soft theorem through subsubleading order. It turns out that there are string corrections to the field theoretical limit in the case of a soft graviton, while for a soft dilaton the string corrections vanish. We then show that the new soft theorems, including the string corrections, can be simply obtained from the exchange diagrams where the soft state is attached to the other external states through the three-point string vertex of three massless states. In the soft-limit, the propagator of the exchanged state is divergent, and at tree-level these are the only divergent contributions to the full amplitude. However, they do not form a gauge invariant subset and must be supplemented with extra non-singular terms. The requirement of gauge invariance then fixes the complete amplitude through subsubleading order in the soft expansion, reproducing exactly what one gets from the explicit calculation in string theory. From this it is seen that the string corrections at subsubleading order arise as a consequence of the three-point amplitude having string corrections in the bosonic string. When specialized to a soft dilaton, it remarkably turns out that the string corrections vanish and that the non-singular piece of the subsubleading term of the dilaton soft theorem is the generator of space-time special conformal transformation.

Figures (2)

• Figure 1: Soft dilaton (dashed line) scattering on other massless states (solid lines), and the only three types of exchange diagrams appearing in field theory, involving i) another external dilaton and an internal graviton (double-wavy line), ii) an external graviton and an internal dilaton, iii) an external and an internal Kalb-Ramond field (wavy line).
• Figure 2: Decomposition of the $n+1$-point massless closed string amplitude, $M_{n+1}$, into a factorizing part involving the three-point amplitude $M_3$, where the soft state with momentum $q$ directly interacts with the $i$th state, exchanging a third massless closed state with momentum $k_i + q$ with the $n$-point amplitude $M_n$, and the reminding part of the amplitude, $N_{n+1}$, which excludes factorization through the former channel.