Doubling down on proton structure: what double parton scattering can and cannot tell us about partonic separation

Abstract

Double parton scattering provides a unique probe of proton structure and is commonly characterized by the effective cross section ${\sigma }_{\mathrm{eff}}$. We examine, under the factorized ansatz for the double-parton-scattering cross section and without assuming any specific parametric form for the transverse profile, what the effective cross section can reveal about the transverse separation, $r$, between partons in the proton. We derive a sharp lower bound on the transverse partonic separation, $⟨r^2⟩\ge 4{\sigma }_{\mathrm{eff}}/(9\pi )$, and demonstrate through explicit counterexamples that previously posited upper bounds do not hold. Our results establish, within the effective-cross-section parametrization of double parton scattering framework, that ${\sigma }_{\mathrm{eff}}$ can provide a sharp lower bound on partonic separation but cannot provide a general upper bound.