Positivity bounds in scalar-QED EFT at one-loop level

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Abstract
A bstract: 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 ( γϕ → γϕ ) and the photon-photon scattering ( γγ → γγ ), 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 ≤ 1, 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 γϕ → γϕ , while the tree-level bound does not necessarily hold, the one-loop β -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 γγ → γγ , 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 β -functions that we obtain should be considered as an accidental feature at one loop, rather than a fundamental property of the theory.

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