From U(1) to E8: soft theorems in supergravity amplitudes

Abstract

It is known that for N = 8 supergravity, the double-soft-scalar limit of a n-point amplitude is given by a sum of local SU(8) rotations acting on a (n−2)-point amplitude. For N < 8 supergravity theories, complication arises due to the presence of a U(1) in the U(N ) isotropy group, which introduces a soft-graviton singularity that obscures the action of the duality symmetry. In this paper, we introduce an anti-symmetrised extraction procedure that exposes the full duality group. We illustrate this procedure for tree-level amplitudes in 4 ≤ N < 8 supergravity in four dimensions, as well as N = 16 supergravity in three dimensions. In three dimensions, as all bosonic degrees of freedom transform under the E 8 duality group, supersymmetry ensures that the amplitude vanishes in the single-soft limit of all particle species, in contrast to its higher dimensional siblings. Using recursive formulas and generalized unitarity cuts in three dimensions, we demonstrate the action of the duality group for any tree-level and one-loop amplitudes. Finally we discuss the implications of the duality symmetry on possible counter terms for this theory. As a preliminary application, we show that the vanishing of single-soft limits of arbitrary component fields in three-dimensional supergravity rules out the direct dimensional reduction of D 8 R 4 as a valid counter term.

Double soft theorems in gauge and string theories

“…We find that the double soft behavior is governed by R-symmetry generators acting on a lower-point amplitude, resembling the results of supergravity theories found in [54,56,57], although the vacuum structure of SYM is quite different from that of supergravity theories. Finally, we consider double-soft scalars in the open superstring theory.…”

“…For example, Arkani-Hamed et al [54] have shown that the double soft limit of scalars in N = 8 supergravity exhibit the expected E 7(7) symmetry of the scalar moduli space, in a manner analogous to the classic soft-pion theorem of [52,55]. This result was recently extended to the four-dimensional supergravity theories with N < 8 supersymmetry, and the N = 16 supergravity in three dimensions in [56]. Furthermore, supergravity amplitudes in both four and three dimensions with two soft fermions were studied in [57], and new soft theorems were proposed.…”

“…Recently, this result was extended to more general supersymmetric gravity theories [56], including 4 ≤ N < 8 supergravity theories in four dimensions as well as N = 16 supergravity in three dimensions. Soft-scalar theorems have been very useful in determining the UV counter terms in supergravity theories [56,67,68].…”

“…Soft-scalar theorems have been very useful in determining the UV counter terms in supergravity theories [56,67,68]. It is known that for supersymmetric gauge theories (in particular N = 4 SYM), a generic vacuum has mostly massive particles, and the massless S-matrix only exists at the origin of moduli space.…”

Soft theorems from anomalous symmetries

“…(2.8), stems from the original three-point vertices in the supergravity action as well as those that arise from the anomalous effective action. For the former, the contribution is zero since it can be derived from the tree-level soft theorems [32], which must vanish due to the scalar's role as parameterising the degenerate vacua. For the later, the corresponding three point vertex is R 2 φ and R 2φ , which would generate a term that is of order q 2 in the numerator, and thus, in the q → 0 limit, suppresses the 1/p i · q propagator leading to a vanish result.…”