Breaking up the Proton: An Affair with Dark Forces

Deep inelastic scattering of $e^{\pm }$ off protons is sensitive to contributions from “dark photon” exchange. Using HERA data fit to HERA’s parton distribution functions (PDFs), we obtain the model-independent bound $\epsilon \lesssim 0.02$ on the kinetic mixing between hypercharge and the dark photon for dark photon masses $\lesssim 10\mathrm{GeV}$. This slightly improves on the bound obtained from electroweak precision observables. For higher masses, the limit weakens monotonically; $\epsilon \lesssim 1$ for a dark photon mass of 5 TeV. Utilizing PDF sum rules, we demonstrate that the effects of the dark photon cannot be (trivially) absorbed into refit PDFs and, in fact, lead to non–Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (Bjorken $x_B$-independent) scaling violations that could provide a smoking gun in data. The proposed $e^{\pm }p$ collider operating at $\sqrt{s}=1.3\mathrm{TeV}$ (Large Hadron Electron Collider) is anticipated to accumulate ${10}^3$ times the luminosity of HERA, providing substantial improvements in probing the effects of a dark photon: sensitivity to $\epsilon$ well below that probed by electroweak precision data is possible throughout virtually the entire dark photon mass range, as well as being able to probe to much higher dark photon masses, up to 100 TeV.

Figure 1

DIS of $e^{\pm }$ on the proton, mediated by the SM photon and $Z$, and a dark photon arising from kinetic mixing with an Abelian hidden sector. Measurements at HERA and Large Hadron Electron Collider (LHeC) probe the mixing parameter and $A_D$ mass relying on no assumptions about the production and decay properties of $A_D$.

Figure 2

DIS neutral-current reduced cross sections for a representative $x_B={10}^{-2}$, as estimated in Eq. (4) using herapdf2.0 lo. The blue band covers $2\sigma$ uncertainties in the SM cross section, obtained by using PDF uncertainties and summing in quadrature the terms in Eq. (4). In this regime, where $Q\ll m_Z$, the effects of $Z$ are negligible and $A_D$ behaves like a massive photon constructively interfering with $\gamma$, leading to SM cross sections upscaled by $[1+{\epsilon }^2{Q}^2/(Q^2+m_{A_D}^2){]}^2$. Importantly, the cross sections at all $x_B$ experience a shift at the same $|Q|\simeq m_{A_D}$ that is a non-DGLAP scaling violation, providing a prominent feature of the contribution from $A_D$. Already the $\epsilon =0.1$ curve can be visually distinguished from the SM, and when combined with the plethora of measurements at other $x_B$ values, it is clear that DIS at HERA can probe the considerably smaller values of $\epsilon \simeq 0.02$, as we show later.

Figure 3

95% C.L. limits from DIS measurements at HERA and future sensitivities at LHeC; similar sensitivities are expected at FCC-eh. For comparison are shown other decay-agnostic limits from EWPO measurements and the muon $g-2$, and, in yellow regions, decay-mode-dependent limits from collider searches. Also shown are hypothetical limits obtained by rescaling SM DIS cross sections by a factor of $[1+{\epsilon }^2{Q}^2/(Q^2+m_{A_D}^2){]}^2$, amounting to accounting only for interference between $A_D$ and $B$ exchange. In the gray-shaded region, there is no physical value of $m_{A_D}$ in the neighborhood of $m_Z=91.1876\mathrm{GeV}$ due to repulsion of eigenmasses. The change in slope of the HERA and LHeC sensitivity curves at large $\epsilon \gtrsim 0.7$ and $m_{A_D}\gtrsim 1\mathrm{TeV}$ occurs due to a factor of $1/\sqrt{1-{\epsilon }^2/\cos {\theta }_W^2}$ enhancement in the $A_D$-fermion coupling. See text for further details.