Description
Understanding the implication of positivity bounds on loop-generated dim-8 operator coefficients is a nontrivial task, as these bounds only strictly hold when all the contributions are included in the dispersion relation up to a certain loop order in the UV theory.
As a step towards more realistic gauge theories such as the Standard Model, in this paper we study the positivity bounds in the Scalar QED Effective Field Theory (EFT) from the scalar-photon scattering and the photon-photon scattering, derived from the dispersion relation of the full one-loop EFT amplitudes. Assuming the UV theory is weakly coupled and all heavy particles have spin $\leq1$, the leading dim-8 interaction for both amplitudes are generated at the one-loop level in the UV theory.
Gauge invariance imposes strong constraints on the loop structures, while potential IR divergences also require careful treatments. Our findings reveal that, for Photon-scalar scattering, while the tree-level bound does not necessarily hold, the one-loop $\beta$-function of the corresponding coefficient always tends to restore the tree-level bound in the IR, unless its actual loop order in the UV theory is further suppressed.
For Photon-photon scattering, on the other hand, the tree-level positivity bound is still robust at the one-loop level in the UV theory. These findings are verified in two example UV models with a heavy scalar extension.
Importantly, the bounds on the $\beta$-functions that we obtain should be considered as an accidental feature at one loop, rather than a fundamental property of the theory.
