Soft theorem for the graviton, dilaton and the Kalb-Ramond field in the bosonic string

TL;DR

The authors derive universal soft-theorem behavior for bosonic closed-string amplitudes involving one soft massless state (graviton, dilaton, or Kalb-Ramond) and various external states, including tachyons and massless strings. By constructing a common tensor $M_{\mu\nu}$ and saturating it with the appropriate polarization, they reproduce the known soft-graviton results from gauge invariance and obtain new, gauge-consistent soft limits for the dilaton and Kalb-Ramond field; notably, the dilaton soft factor does not acquire extra ${\eta}_{\mu\nu}$ terms in these contexts. Extending the analysis to amplitudes with one soft and $n$ massless closed strings, they formulate a soft operator ${\hat S}$ that yields the correct leading and subleading terms for all three massless states, tying the results to BCJ/KLT perspectives. The work highlights a unified, string-theoretic mechanism for soft behavior and points to future extensions to open strings, superstrings, and loop-level amplitudes.

Abstract

We study the behavior of the scattering amplitudes of the bosonic string involving a soft massless state (graviton, dilaton and Kalb-Ramond antisymmetric tensor) and closed string tachyons or other closed string massless states. For a soft graviton we confirm the results, obtained in Ref. [24] using just gauge invariance, up to terms of ${\cal O}(q^1)$ for external tachyons and up to terms of ${\cal O} (q^0)$ for external massless closed string states. Furthermore, we also derive the behavior of the scattering amplitude when a dilaton or a Kalb-Ramond field becomes soft. These results are new and cannot, to our knowledge, be derived by using gauge invariance. It turns out, in the cases examined, that the soft amplitude for a dilaton or for an antisymmetric tensor is obtained by saturating the tensor, $M_{μν}$, derived from gauge invariance for gravitons, with their respective polarization tensors. Thus extra terms that could have appeared in $M_{μν}$ in the case of a soft dilaton, in fact do not appear.