Abstract
In theories of Einstein gravity coupled with a dilaton and a two-form, a soft theorem for the two-form, known as the Kalb-Ramond B-field, has so far been missing. In this work we fill the gap, and in turn formulate a unified soft theorem valid for gravitons, dilatons and B-fields in any tree-level scattering amplitude involving the three massless states. The new soft theorem is fixed by means of on-shell gauge invariance and enters at the subleading order of the graviton’s soft theorem. In contrast to the subsubleading soft behavior of gravitons and dilatons, we show that the soft behavior of B-fields at this order cannot be fully fixed by gauge invariance. Nevertheless, we show that it is possible to establish a gauge invariant decomposition of the amplitudes to any order in the soft expansion. We check explicitly the new soft theorem in the bosonic string and in Type II superstring theories, and furthermore demonstrate that, at the next order in the soft expansion, totally gauge invariant terms appear in both string theories which cannot be factorized into a soft theorem.
Highlights
The B-field appears in gravity as a double-copy Yang-Mills theory
We check explicitly the new soft theorem in the bosonic string and in Type II superstring theories, and demonstrate that, at the order in the soft expansion, totally gauge invariant terms appear in both string theories which cannot be factorized into a soft theorem
We have shown using gauge invariance that the soft behavior of the antisymmetric B-field is fixed at the order q0 in the soft momentum in amplitudes involving gravitons, dilatons and other B-fields
Summary
We derive the soft theorem for the antisymmetric tensor Bμν in an amplitude with only massless particles, i.e. Kalb-Ramond fields, gravitons and dilatons. In the case of a soft antisymmetric tensor scattering on other massless states, gauge invariance fixes the amplitude only through the order q0. The term of order q contains a totally antisymmetric tensor that cannot be fixed by gauge invariance. It is convenient for later use to introduce a new tensor Aρμν for the leading order expression of Aρμν, in the following way This is possible since the operator in the squared bracket is just another totally antisymmetric tensor. This expression can be written more compactly, by defining holomorphic and antiholomorphic total angular momentum operator, as follows: Jiμν = Lμi ν + Siμν , Jiμν = Lμi ν + Siμν , Lμi ν = i (kiμ∂iν − kiν ∂iμ) , These operators especially turn useful, when considering the action on superstring amplitude. The conclusion about the field theory limit of Aρμν remains open
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