Lepton Universality Test in the Photoproduction of $e^{-}{e}^{+}$ Versus ${\mu }^{-}{\mu }^{+}$ Pairs on a Proton Target

In view of the significantly different proton charge radius extracted from muonic hydrogen Lamb shift measurements as compared to electronic hydrogen spectroscopy or electron-scattering experiments, we study in this Letter the photoproduction of a lepton pair on a proton target in the limit of very small momentum transfer as a way to provide a test of the lepton universality when extracting the proton charge form factor. By detecting the recoiling proton in the $\gamma p\to l^{-}{l}^{+}p$ reaction, we show that a measurement of a ratio of $e^{-}{e}^{+}+{\mu }^{-}{\mu }^{+}$ over $e^{-}{e}^{+}$ cross sections with an absolute precision of $7\times 10^{-4}$ would allow for a test to distinguish, at the $3\sigma$ level, between the two different proton charge radii currently extracted from muonic and electronic observables.

Figure 1

Bethe-Heitler (BH) mechanism to the $\gamma p\to l^{-}{l}^{+}p$ process, where the four-momenta of the external particles are $k$ for the photon, $p(p^{\prime })$ for initial (final) protons, and $l_{-}$, $l_{+}$ for the lepton pair.

Figure 2

The upper panel shows a comparison of the (lepton pair)-invariant mass dependence of the $\gamma p\to e^{-}{e}^{+}p$ process (upper three curves) versus the $\gamma p\to {\mu }^{-}{\mu }^{+}p$ process (lower three curves) at $E_{\gamma }=0.5\mathrm{GeV}$, and for three values of the momentum transfer $t$ as indicated. The lower panel shows the corresponding kinematic relation between the lepton-pair-invariant mass and the recoiling proton lab angle.

Figure 3

Ratio $R_{\mu /e}$ of the $\gamma p\to (e^{-}{e}^{+}+{\mu }^{-}{\mu }^{+})p$ versus $\gamma p\to (e^{-}{e}^{+})p$ differential cross sections, according to Eq. (10). The lower (blue) curve corresponds with the lepton universality result, for which $G_{Ep}^{\mu }=G_{Ep}^e$. The upper (red) curve assumes a violation of lepton universality, for which $G_{Ep}^{\mu }/G_{Ep}^e=1.01$. The blue bands denote $1\sigma$, $3\sigma$, and $5\sigma$ bands corresponding with a $\sigma$ of $7\times 10^{-4}$ around the blue curve. The difference between the blue curve and the dotted curve is an estimate of the physical background due to the contribution of the timelike Compton process.

Figure 4

Linear photon asymmetry $A_{lU}$ of the $\gamma p\to (l^{-}{l}^{+})p$ process. The dashed (blue) curve corresponds with $e^{-}{e}^{+}$ production, and the dashed-dotted (red) curve corresponds with ${\mu }^{-}{\mu }^{+}$ production. The solid (black) curve is the asymmetry corresponding with the sum of the $e^{-}{e}^{+}+{\mu }^{-}{\mu }^{+}$ channels according to Eq. (12).