Leading order corrections to the Bethe-Heitler process in the $\gamma p\to l^{+}{l}^{-}p$ reaction

The ratio of dilepton production cross sections on a proton, using the $\gamma p\to l^{+}{l}^{-}p$ process, above and below the dimuon production threshold allows one to extract the effective lepton-proton interaction, which is required to be identical for electrons and muons if lepton universality is exact. To test for a scenario of broken universality at the percent level, of the size that could explain the different proton charge radii extracted from electron scattering and from muonic hydrogen spectroscopy, we evaluate all one-loop QED corrections to this process, including the full lepton mass dependencies. We furthermore show that two photon–exchange processes with both photons attached to the proton line vanish after averaging over dilepton angles and estimate the relatively small radiation off the proton. We compare the full one-loop calculation with a soft-photon approximation of the same order and present estimates for a planned experiment.

Figure 1

The Bethe-Heitler process at tree level.

Figure 2

Ratio of the cross sections for $\gamma p\to (e^{+}{e}^{-}+{\mu }^{+}{\mu }^{-})p$ vs $\gamma p\to (e^{+}{e}^{-})p$. The blue band corresponds to a $3\lambda$ band around the lepton universality result, with standard deviation $\lambda =7\times {10}^{-4}$.

Figure 3

Generic one-loop leptonic diagram with its crossed counterpart.

Figure 4

Lepton self-energy diagram with $k^{\prime 2}=s^{\prime }$.

Figure 5

Lepton self-energy contributing to the Bethe-Heitler process.

Figure 6

Half–off shell vertex with $k^2=m^2$, $k^{\prime }$ off shell, and $q=k^{\prime }-k$.

Figure 7

Vertex diagrams contributing to the Bethe-Heitler process.

Figure 8

Vacuum polarization diagram. The fermion loop can be either electrons or muons.

Figure 9

Vacuum polarization diagram contributing to the Bethe-Heitler process. In the ratio of electron- vs muon-pair production cross sections, this contribution drops out.

Figure 10

Lepton box diagrams contributing to the Bethe-Heitler process.

Figure 11

Diagrams with real photon emission from the lepton lines of the Bethe-Heitler process. In the soft-photon limit, the diagram with the photon attached to the internal (off-shell) fermion line does not contribute.

Figure 12

One-loop radiative corrections to the $\gamma p\to l^{+}{l}^{-}p$ process. Upper diagrams: box graphs with protons; lower right diagram: the proton vertex correction; and two lower left diagrams: soft bremsstrahlung from the proton line. The graphs with an interchange of final leptons are not shown.

Figure 13

Comparison of first order QED corrections to the cross section, including soft-photon bremsstrahlung with $\mathrm{\Delta }{E}_s=0.01\mathrm{GeV}$ (solid lines), with the calculation in the soft-photon approximation (dashed lines). The vertical dashed red line indicates the muon-pair production threshold at $s_{ll}\approx 0.045{\mathrm{GeV}}^2$.

Figure 14

Effect of the one-loop radiative correction on the ratio $(d\sigma (e^{+}{e}^{-})+d\sigma ({\mu }^{+}{\mu }^{-}))/d\sigma (e^{+}{e}^{-})$, comparing with the soft-photon result of Ref. [35].

Figure 15

Ratio of cross sections between electron- and muon-pair production at tree level (dotted blue curve) with account of full one-loop QED corrections estimated using $\mathrm{\Delta }{E}_s=0.01\mathrm{GeV}$ (green curve) with corresponding $3\lambda$ bands, with standard deviation $\lambda =7\times {10}^{-4}$. The dashed red curve denotes the scenario in which lepton universality is broken with $G_E^{\mu }/G_E^e=1.01$, including the one-loop radiative corrections.

Figure 16

First order radiative correction factor $\delta$, due to radiative corrections on the proton side. The vertical dashed red line indicates the muon-pair production threshold at $s_{ll}\approx 0.045{\mathrm{GeV}}^2$.

Figure 17

First order radiative correction factor $\delta$ on the proton side as a function of $t$, with $\mathrm{\Delta }{E}_s=0.01\mathrm{GeV}$.