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

TL;DR

This work derives and interprets one-loop positivity bounds in a Scalar-QED EFT from forward dispersion relations for the elastic processes $\\gamma\\phi \\to \\gamma\\phi$ and $\\gamma\\gamma \\to \\gamma\\gamma$. Under the assumptions of a weakly coupled UV with heavy states of spin ≤1, the leading dim-8 operators arise at one loop, and gauge invariance plus IR subtleties shape the loop contributions and their subtraction, with $s_0 \ \\ll \\Lambda^2$ to maintain EFT validity. For $\\gamma\\phi \\to \\gamma\\phi$, the tree-level bound $c^{(1)}_{F^2D^2\\phi^2} \\le 0$ is not guaranteed, but the one-loop $\\beta$-function satisfies $\\beta(c^{(1)}_{F^2D^2\\phi^2}) \\ge 0$, tending to restore the bound in the IR if the operator is UV-generated at one loop. In contrast, for $\\gamma\\gamma \\to \\gamma\\gamma$, the tree-level positivity bounds $c^{(1)}_{F^4} \\ge 0$ and $c^{(2)}_{F^4} \\ge 0$ remain robust at one loop, with the corresponding $\\beta$-functions shown to be model-dependent and not universal at this order. These conclusions are corroborated by two explicit UV models with heavy scalars, and the results highlight the importance of including all contributions up to the specified loop order when applying positivity bounds to gauge theories and EFT extensions of the Standard Model.

Abstract

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 ($γφ\to γφ$) and the photon-photon scattering ($γγ\to γγ$), 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 $\leq1$, 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 $γφ\to γφ$, 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 $γγ\to γγ$, 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.

Figures (6)

• Figure 1: Contours in the complex $s$-plane. The blue line represents the branch cut on the real axis. The contours $\gamma$ around $s=\pm is_0$ are deformed to the contours $\Gamma$, where the big semi-circles of $\Gamma$ is at $|s| \to \infty$.
• Figure 2: If the $\gamma\phi \to \gamma\phi$ amplitude contain a $1/t$ pole, it would correspond to a massless $t$-channel propagator (represented by the solid line). In the $t\to 0$ limit the massless propagator goes on-shell and the amplitude factorizes into two sub-amplitudes (cut by the red dashed line). The shadowed circle represents a generic vertex that include both tree-level and loop diagrams.
• Figure 3: An illustration of some typical contributions to the one-loop forward elastic amplitude of $\gamma^+\phi \to \gamma^+\phi$ in the UV theory. Only the left halves of the one-loop diagrams are drawn, which correspond to the diagrams for the tree-level $\sigma(\gamma^+ \phi \to {\rm 2~particles} )$. The thick red line represents a generic heavy particle. The heavy particle only contributes where at least one of the final state particles is heavy. $\sigma(\gamma^+ \phi \to \gamma \phi)$ receives no contribution from the heavy particle at the tree level since it is not allowed by the structure of the gauge interaction. Note that we assume the UV theory is weakly coupled and heavy particles have spin $\leq 1$. Not all diagrams are shown here.
• Figure 4: The corresponding Feynman diagrams for \ref{['eq: sigma log t']} and \ref{['eq: sigma log m']}. The tree-level contribution comes from $c_{F^{2}D^{2}\phi^{2}}^{(1)}$. The one-loop diagrams are proportional to the combination $2 c_{D^4\phi^4}^{(1)} + c_{D^4\phi^4}^{(2)}$. Additional one-loop diagrams which contain vanishing scaleless integrals (in the massless case) are not included here.
• Figure 5: The positivity bound of $c_{F^4}^{(1)}$ and $c_{F^4}^{(2)}$. The red lines stand for $c_{F^4}^{(1)}=0$ and $c_{F^4}^{(2)}=0$. The green region bounded by $c_{F^4}^{(1)}\geq 0$ and $c_{F^4}^{(2)}\geq 0$ already satisfies the more general set of bounds given by \ref{['eq:cf4posgen']}.