Abstract
We present a complete one-loop renormalization of the Special Galileon Smatrix. Especially, we give a complete list of the higher derivative operators which are necessary for the one-loop on-shell renormalization and prove the invariance of the oneloop on-shell effective action with respect to the Special Galileon symmetry. This enables us to enlarge the validity of the enhanced O(p3) soft behavior of the scattering amplitudes to the one-loop level. As an illustration, we discuss explicitly the four-point and five-point one-loop scattering amplitudes and comment on some conjectures appearing in the existing literature.
Highlights
By the requirement of yielding a second-order equation of motion and by invariance with respect to the polynomial shift symmetry of the first order, δφ = a + b · x
As we have proved in the previous section, the effective action Se1ff−loop [φcl] is invariant19 with respect to the hidden Special Galileon symmetry
We have studied the issue of the one-loop renormalization of the Special Galileon S-matrix
Summary
For the reader’s convenience and in order to fix our notation, let us first briefly review the interrelation of the S-matrix and the on-shell effective action (for original papers, see [33,34,35]). I∆F (x − y) is the Feynman propagator, Sint [φ] is the interaction part of the action with UV counterterms included and the functional differential operator exp δ δφ. Provided we are interested only in the on-shell scattering amplitudes up to one loop order, we need to know the one-loop on-shell effective action Γ1−loop [φcl], i.e. the oneloop effective action with the argument satisfying the classical equation of motion. This well known fact will be crucial for our following calculations
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