Abstract
We study effective actions for simultaneous breaking of space-time and internal symmetries. Novel features arise due to the mixing of Goldstone modes under the broken symmetries which, in contrast to the usual Adler's zero, leads to non-vanishing soft limits. Such scenarios are common for spontaneously broken SCFT's. We explicitly test these soft theorems for $\mathcal{N}=4$ sYM in the Coulomb branch both perturbatively and non-perturbatively. We explore the soft constraints systematically utilizing recursion relations. In the pure dilaton sector of a general CFT, we show that all amplitudes up to order $s^{n} \sim \partial^{2n}$ are completely determined in terms of the $k$-point amplitudes at order $s^k$ with $k \leq n$. Terms with at most one derivative acting on each dilaton insertion are completely fixed and coincide with those appearing in the conformal DBI, i.e. DBI in AdS. With maximal supersymmetry, the effective actions are further constrained, leading to new non-renormalization theorems. In particular, the effective action is fixed up to eight derivatives in terms of just one unknown four-point coefficient and one more coefficient for ten-derivative terms. Finally, we also study the interplay between scale and conformal invariance in this context.
Highlights
The question can be framed as follows: “to what extent does the sub-leading soft theorem, due to broken conformal boost symmetry, follow from the leading behaviour stemming from broken dilation symmetry?” First of all, we find that any five-point amplitude constrained by the leading soft theorem automatically satisfy the sub-leading soft theorem
We initiate the systematic study of constraints on effective actions due to soft theorems of spontaneously broken symmetries where multiple GB modes are mixed under the broken symmetry
Using the one-loop and one-instanton effective action for N = 4 super Yang-Mills (SYM) in the Coulomb branch, we demonstrated the validity of the dilaton soft theorems as well as that of the newly derived R-symmetry pion soft theorems, both perturbatively and non-perturbatively
Summary
Soft behaviour of amplitudes with massless particles are often dictated by Ward identities of the underlying symmetries. If δφ does produce a particle in the spectrum the r.h.s. is non-zero, and is given by the sum of Fourier transformed amplitude with the i-th field transformed under the generator of the broken generator If the dilaton is identified with one of the scalars that transforms non-trivially under the broken Rsymmetry generator, following the above discussion the soft limit of the R-symmetry GB is non-vanishing. In N = 4 SYM, the scalars form a 6 of SO(6), any one of the scalars taking a vev (say φ6) breaks R-symmetry down to SO(5), with 5 GB’s associated with the broken rotation generators R6I with I = 1, · · · , 5.
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