Unambiguously Testing Positivity at Lepton Colliders

Abstract

The diphoton channel at lepton colliders, $e^+e^- (μ^+μ^-) \to γγ$, has a remarkable feature that the leading new physics contribution comes only from dimension-eight operators. This contribution is subject to a set of positivity bounds, derived from the fundamental principles of Quantum Field Theory, such as unitarity, locality, analyticity and Lorentz invariance. These positivity bounds are thus applicable to the most direct observable -- the diphoton cross section. This unique feature provides a clear, robust, and unambiguous test of these principles. We estimate the capability of various future lepton colliders in probing the dimension-eight operators and testing the positivity bounds in this channel. We show that positivity bounds can lift certain flat directions among the effective operators and significantly change the perspectives of a global analysis. We also discuss the positivity bounds of the $Zγ/ZZ$ processes which are related to the $γγ$ ones, but are more complicated due to the massive $Z$ boson.

Figures (6)

• Figure 1: $\Delta \chi^2=1$ contours for CEPC/FCC-ee 240 GeV and ILC 250 GeV. The green shaded region is allowed by the positivity bounds.
• Figure 2: The reach on the scale of the dim-8 operators $\Lambda_8 (\equiv v/a^{\frac{1}{4}})$ as a function of the center-of-mass energy $\sqrt{s}$ from the measurement of the $e^+e^- \to \gamma \gamma$ (or $\mu^+\mu^- \to \gamma \gamma$) process. The band covers $1-5\,{\rm ab}^{-1}$ and various beam polarization scenarios. The circle represents the best reach of each collider scenario. The LEP2 Schael:2013ita reach is shown assuming a SM central value. For linear colliders, the triangle (square) shows the reach for $a_L$ ($a_R$) in a simultaneous fit of both parameters. Note that the luminosities of FCC $Z$-pole (150$\,{\rm ab}^{-1}$), LEP2 (3$\,{\rm fb}^{-1}$) and muon collider 30 TeV (90$\,{\rm ab}^{-1}$) are all very different from the 1-5$\,{\rm ab}^{-1}$ of the band.
• Figure 3: Feynman diagrams for $e^- \gamma \to e^- \gamma$ in the SM.
• Figure 4: Momenta of the forward scattering process before and after the collision.
• Figure 5: The differential cross section $d\sigma/d|\!\cos\theta|$ for the SM and the d8 contribution (as in Eq. (4) in the main text) for $\sqrt{s}=240\,$GeV, unpolarized beams. For the d8 contribution, an arbitrary benchmark of $a_L+a_R=2$ is chosen.