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 mathcal{O}left({alpha}^{prime 6}right) . Thus the massless S-matrix of string theory “knows” about the presence of D-branes.
Highlights
In Adler’s zero [2]
We study sub-leading soft theorems that arise from the enhanced broken symmetries
This occurs for space-time symmetries, where some generators of the broken symmetries are derivatively related
Summary
We review the derivation of soft theorems from current algebra, where we use the currents associated with the broken symmetries to excite the Goldstone boson. Where ai label the distinct generators and jV b1 is a current of the unbroken invariant subgroup and does not produce physical states in the spectrum, i.e. jVμi|0 = 0 For cases that it does produce physical states, the single-soft limit no-longer is zero. Similar to the single-soft discussion, the resulting double-soft limit depends on the nature of the broken symmetry. The first is similar to the single-soft limit, where one considers the variation of remaining fields under the broken symmetries. As with the single soft discussion, employing jKμ ν instead of jDμ will lead to sub-leading soft theorems. We will derive the double soft theorems for susy and conformal susy breaking in appendix A and appendix B
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