Abstract
The Ward identities involving the currents associated to the spontaneously broken scale and special conformal transformations are derived and used to determine, through linear order in the two soft-dilaton momenta, the double-soft behavior of scattering amplitudes involving two soft dilatons and any number of other particles. It turns out that the double-soft behavior is equivalent to performing two single-soft limits one after the other. We confirm the new double-soft theorem perturbatively at tree-level in a D-dimensional conformal field theory model, as well as nonperturbatively by using the “gravity dual” of mathcal{N}=4 super Yang-Mills on the Coulomb branch; i.e. the Dirac-Born-Infeld action on AdS5 × S5.
Highlights
The two kinds of NG bosons differ in their soft behavior: in the case of a spontanously broken internal symmetry, amplitudes involving the NG bosons vanish when the momentum of one of the NG bosons goes to zero.1 A famous example is the non-linear σmodel (NLSM) describing the low-energy behaviour of SU(n)×SU(n) theory spontaneously broken to the vectorial subgroup SU(n)
In the following subsections we study the Ward identity implications for the case of spontaneously broken symmetries, and in particular the specific cases of theories with broken dilatation and special conformal transformations
In this paper we have studied the Ward identities of spontaneously broken scale and special conformal invariance, and from them derived the consequences for scattering amplitudes describing the interaction between the dilaton and other spinless particles
Summary
We start by briefly reviewing aspects of conformal symmetry and its representations in field theory. The group transforms space-time coordinates and fields as follows xμ → x μ = xμ + MN fMμ N (x) Φ(x) → Φ (x) = Φ(x) + i MN ΓMN (x)Φ(x). Of dilatation, and Kμ are the generators of special conformal transformation. The currents associated to the conformal generators can be constructed by varying the conformal invariant action as in eq (2.1) assuming that the infinitesimal parameters MN are arbitrary functions of x. In this way, for the conformal group, one can get: δS = dDx MN (x)(∂μJMμ N ) = dDx MN (x)∂μ (fMν N Tμν ).
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