Abstract
We clarify the structure of the four-dimensional low-energy effective action that encodes the conformal and $U(1)$ R-symmetry anomalies in an $\mathcal{N}=1$ supersymmetric field theory. The action depends on the dilaton, $\tau$, associated with broken conformal symmetry, and the Goldstone mode, $\beta$, of the broken $U(1)$ R-symmetry. We present the action for general curved spacetime and background gauge field up to and including all possible four-derivative terms. The result, constructed from basic principles, extends and clarifies the structure found by Schwimmer and Theisen in arXiv:1011.0696 using superfield methods. We show that the Goldstone mode $\beta$ does not interfere with the proof of the four-dimensional $a$-theorem based on $2 \to 2$ dilaton scattering. In fact, supersymmetry Ward identities ensure that a proof of the $a$-theorem can also be based on $2 \to 2$ Goldstone mode scattering when the low-energy theory preserves $\mathcal{N}=1$ supersymmetry. We find that even without supersymmetry, a Goldstone mode for any broken global $U(1)$ symmetry cannot interfere with the proof of the four-dimensional $a$-theorem.
Highlights
Single-axion exchanges, which would be the only option in the low-energy effective action
The action depends on the dilaton, τ, associated with broken conformal symmetry, and the Goldstone mode, β, of the broken U(1) R-symmetry
We present the action for general curved spacetime and background gauge field up to and including all possible fourderivative terms
Summary
Scattering amplitudes in supersymmetric theories obey supersymmetry Ward identities [13, 14]. Assuming the vacuum is supersymmetric, i.e. Q|0 = Q†|0 = 0, we can derive supersymmetry Ward identities for the amplitudes. The free-field commutators (2.1) can be used here because the supercharges are acting on the asymptotic states This is the statement that at any loop-order, the on-shell four-scalar amplitude A4(ζ ζ ζ ζ) must vanish (where we mean the particles created by the field ζ). Dot in p1| and use the antisymmetry of the angle bracket to eliminate the first term on the right hand side in (2.5) For generic momenta, this leads to the statement that A4( ζ ζ ζ ζ ) = 0. If we expand around another vacuum, we generate mass-terms and we are only interested in the case of massless particles If the resulting amplitudes are compatible with (2.7)–(2.8), the operator has a supersymmetric extension at the level of four fields (though not necessarily beyond that order)
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