On the exactness of soft theorems

TL;DR

This work analyzes the exactness of subleading soft theorems for massless S-matrices by distinguishing type I (vertex-structure dependent) from type II (enhanced broken space-time symmetry) origins. Using current algebra Ward identities, it derives single- and double-soft theorems for spontaneously broken conformal and translational symmetries, and demonstrates their protection by symmetry against quantum and string-theoretic UV completions. The authors compute one-loop UV divergences for DBI, conformal DBI, and Akulov-Volkov theories and show that type II soft theorems survive these corrections, while type I theorems can be broken by higher-derivative counterterms; they extend the analysis to alpha' corrections in open strings, confirming that the massless S-matrix of string theory encodes D-brane information through these soft theorems. Overall, the results establish soft theorems derived from nonlinearly realized space-time symmetries as robust constraints on EFTs and their UV completions, with open-string amplitudes offering a nontrivial check of these ideas in a UV-complete setting.

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 alpha' expansion of both super and bosonic open strings amplitudes, and verify the validity of the translation symmetry breaking soft-theorems up to O(alpha'^6). Thus the massless S-matrix of string theory "knows" about the presence of D-branes.

Figures (3)

• Figure 1: The Feynman diagrams contribute to four and six-point amplitudes in DBI at one-loop order. One should also sum over all other independent permutations.
• Figure 2: The Feynman diagrams contributing to five and six-point amplitudes in conformal DBI at one loop of order $s^4$ at $D=4$ and $s^5$ at $D=6$, where the four-point vertex is identical to that of DBI, and five- and six-point vertices are $\phi (\partial \phi \cdot \partial \phi )^2$ and $\phi^2 (\partial \phi \cdot \partial \phi )^2$. One should also sum over all other independent permutations.
• Figure 3: The Feynman diagrams contributing to four- and six-point fermionic amplitudes in K-S action at one loop. One should also sum over all other independent permutations.