Abstract
We consider the tree-level scattering amplitudes in the NS-NS (Neveu-Schwarz) massless sector of closed superstrings in the case where one external state becomes soft. We compute the amplitudes generically for any number of dimensions and any number and kind of the massless closed states through the subsubleading order in the soft expansion. We show that, when the soft state is a graviton or a dilaton, the full result can be expressed as a soft theorem factorizing the amplitude in a soft and a hard part. This behavior is similar to what has previously been observed in field theory and in the bosonic string. Differently from the bosonic string, the supersymmetric soft theorem for the graviton has no string corrections at subsubleading order. The dilaton soft theorem, on the other hand, is found to be universally free of string corrections in any string theory.
Highlights
JHEP12(2016)020 factorizing soft operator proportional to the string slope α′ appears at subsubleading order, when the soft state is a graviton, while the dilaton soft theorem remains equal to the field theory result of ref
By including the α′ corrections in the three-point amplitude for massless states, this was shown to again follow from gauge invariance of the scattering amplitudes. While this shows that the graviton soft theorem at subsubleading order is not universal, but depends on higher-order operators in its effective action, it is intriguing to think that the soft behavior of the dilaton is universal in any theory through subsubleading order, signalling some underlying hidden symmetry. (For recent discussions on existing relations between broken symmetries of Lagrangians and soft theorems, see refs. [28,29,30,31,32,33,34,35].) as shown in ref. [28], the dilaton soft theorem does bare striking resemblance to the soft theorem of the Nambu-Goldstone boson of spontaneously broken conformal symmetry, which is universal through subsubleading order
The calculation is an extension of the one done in ref. [2] for the bosonic string and despite the much more complicate expressions of the amplitudes it requires exactly the same ingredients and techniques developed for the bosonic theory
Summary
We review the closed superstring amplitude and rewrite it in a convenient form, when one particle is soft, which allows us to directly express the results using the calculations already done in the bosonic string in ref. [1, 2]. Overall integration measures dθiθi and dθiθi for i = 1, 2 Since this choice could have been made for any two of the n states, we will not explicitly impose these zero conditions in the expressions that follow. The n+1 point amplitude, with the help of the correlation functions written in eq (2.4) and after having integrated over the variables θ and θ, reduces to an expression which can be factorized at the integrand level as follows: Mn+1 = Mn ∗ S ,. Ss and Ss are the complex conjugates of each other and they provide the contributions from the additional supersymmetric states The second part gives the supplement of the additional superstring states and reads
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