On the exactness of soft theorems

Abstract

Soft behaviours of S-matrix for massless theories reflect the underlying symmetry principle that enforces its masslessness. As an expansion in soft momenta, sub-leading soft theorems can arise either due to (I) unique structure of the fundamental vertex or (II) presence of enhanced broken-symmetries. While the former is expected to be modified by infrared or ultraviolet divergences, the latter should remain exact to all orders in perturbation theory. Using current algebra, we clarify such distinction for spontaneously broken (super) Poincaré and (super) conformal symmetry. We compute the UV divergences of DBI, conformal DBI, and A-V theory to verify the exactness of type (II) soft theorems, while type (I) are shown to be broken and the soft-modifying higher-dimensional operators are identified. As further evidence for the exactness of type (II) soft theorems, we consider the α ′ expansion of both super and bosonic open strings amplitudes, and verify the validity of the translation symmetry breaking soft-theorems up to O(α ′ 6 ). Thus the massless S-matrix of string theory "knows" about the presence of D-branes.

Double-soft behavior of the dilaton of spontaneously broken conformal invariance

“…(4.11). This confirms our statement that the current algebra derivation can be performed by either working with the broken, partially conserved current (our approach), or by defining new conserved currents and working with those (as done in [52] As explained in the main text, the soft limit k 6 → 0 of A 6 reproduces the correct soft theorem, since A ∂ 6 6 and A pole 6 both vanish in this limit. At subleading order they do not vanish, but should instead cancel each other, since they cannot contribute to the soft theorem due to the coupling being λ 4 …”

“…Shortly after our work, another preprint appeared [52], in which the soft theorems here presented were also derived by similar methods. Additionally, the authors of ref.…”

“…The general formulas presented here are compatible also with this case and therefore one can straightforwardly from our formulas derive the soft theorems presented in ref. [52]. A small technical difference in our approach with respect to theirs lies in the comment we gave from eq.…”

“…Additionally, the authors of ref. [52] also consider the case of spontaneously broken translational invariance, and realize cleverly the connection to string theories restricted on D-branes. The general formulas presented here are compatible also with this case and therefore one can straightforwardly from our formulas derive the soft theorems presented in ref.…”

Spontaneously broken spacetime symmetries and the role of inessential Goldstones

“…[7][8][9][10], and the study of soft limits of general scattering amplitudes, e.g. [11][12][13][14][15]. See also [16] for a discussion on spontaneous breaking of spacetime symmetries in condensed matter systems, [17,18] for a discussion on the coset construction for superfluids etc and [19][20][21][22] for more examples related to cosmology and gravity.…”