Soft-photon radiative corrections to the $e^{-}p\to e^{-}p{l}^{-}{l}^{+}$ process

We calculate the leading-order QED radiative corrections to the process $e^{-}p\to e^{-}p{l}^{-}{l}^{+}$ in the soft-photon approximation, in two different energy regimes which are of relevance to extract nucleon structure information. In the low-energy region, this process is studied to better constrain the hadronic corrections to precision muonic hydrogen spectroscopy. In the high-energy region, the beam-spin asymmetry for double-virtual Compton scattering allows us to directly access the generalized parton distributions. We find that the soft-photon radiative corrections have a large impact on the cross sections and are therefore of paramount importance to extract the nucleon structure information from this process. For the forward-backward asymmetry, the radiative corrections are found to affect the asymmetry only around or below the 1% level, whereas the beam-spin asymmetry is not affected at all in the soft-photon approximation, which makes them gold-plated observables to extract nucleon structure information in both the low- and high-energy regimes.

Figure 1

Tree level QED diagrams contributing to the $e^{-}p\to e^{-}p{l}^{-}{l}^{+}$ process. We distinguish between the spacelike (left) and the timelike (right) Bethe-Heitler processes. The crossed diagrams, for which in the spacelike process the order of the vertices on the produced dilepton line are interchanged, and for which in the timelike process the order of the vertices on the electron beam line are interchanged, are not shown.

Figure 2

Tree level diagrams for the Compton scattering. The blob represents the (elastic and inelastic) interaction of the virtual photon with the nucleon.

Figure 3

Planes defining the scattering angles which characterize the $e^{-}p\to e^{-}p{l}^{-}{l}^{+}$ process. The angles $\mathrm{\Phi }$ and ${\varphi }_l^{*}$ are defined with respect to the blue plane, which is the scattering plane of the virtual photons with four-momenta $q$ and $q^{\prime }$.

Figure 4

Diagram representing the double-virtual Compton process.

Figure 5

Born contribution (upper panel) and $s$-channel $\mathrm{\Delta }$-pole contribution (lower panel) to the Compton amplitude. While for the Born contribution only the sum of $s$- and $u$-channel diagrams is gauge invariant, the $s$-channel $\mathrm{\Delta }$-pole contribution is gauge invariant by itself.

Figure 6

The ${\gamma }^{*}N\mathrm{\Delta }$ vertex ${\mathrm{\Gamma }}_{\gamma N\mathrm{\Delta }}^{\beta \mu }$ (left diagram) and its adjoint, ${\tilde{\mathrm{\Gamma }}}_{\gamma N\mathrm{\Delta }}^{\alpha \nu }$ (right diagram).

Figure 7

Handbag diagrams for the double deeply virtual Compton amplitude. The single (composite) lines represent quarks (nucleons), respectively. The blobs represent the GPDs.

Figure 8

Virtual corrections of class (a) (left) and class (b) (right) to the spacelike BH process with two virtual photons. One also has to consider the corresponding counterterm diagrams. The crossed diagrams with $l_{-}$ and $l_{+}$ interchanged yield the same correction.

Figure 9

Left panel: one-loop vertex diagram. Right panel: vertex counterterm.

Figure 10

Left panel: fermion self-energy at one-loop order. Right panel: counterterm for fermion self energy.

Figure 11

Virtual photon corrections of class (c) to the spacelike BH proces. The crossed diagrams with $l_{-}$ and $l_{+}$ interchanged yield the same correction.

Figure 12

Virtual corrections of class (a) (left) and class (b) (right) to the timelike BH process. One also has to consider the corresponding counterterm diagrams. For the correction, the crossed diagrams with $\mathrm{\Delta }$ and $q^{\prime }$ interchanged yield the same result.

Figure 13

Contributing diagrams from class (c) for the timelike Bethe-Heitler process. The crossed diagrams with $\mathrm{\Delta }$ and $q^{\prime }$ interchanged yield the same result.

Figure 14

Virtual corrections of class (a) (top left), class (b) (top right), and class (c) (lower two rows) to the dVCS process. One also has to consider the corresponding counterterm diagrams. The crossed diagrams with $q$ and $q^{\prime }$ interchanged yield the same result.

Figure 15

Kinematic quantities entering the exchange dVCS amplitude for the $e^{-}p\to e^{-}p{e}^{-}{e}^{+}$ process in the $\mathrm{\Delta }(1232)$ region. In the upper panel, we show the c.m. energy $W_{ex}$, as function of ${\theta }_e^{*}$, compared with the value $W=1.25\mathrm{GeV}$ of the direct process. In the lower panel, we compare the ${\theta }_e^{*}$ dependence of both photon virtualities in the exchange dVCS amplitude with their constant values for the direct dVCS amplitude.

Figure 16

${\theta }_l^{*}$ dependence of the $e^{-}p\to e^{-}p{l}^{-}{l}^{+}$ cross section (upper panels), forward-backward asymmetry (middle panels), and beam-spin asymmetry (lower panels) for $e^{-}{e}^{+}$ production (left panels) and ${\mu }^{-}{\mu }^{+}$ production (right panels) in the $\mathrm{\Delta }(1232)$ region, for $\mathrm{\Phi }=90°$. The curves show the predictions for BH and $\mathrm{BH}+\mathrm{dVCS}$ for two models showing the sensitivity to the low-energy constant $b_{3,0}$. The black solid curves show the effect of the radiative corrections for the hadronic model of the green dashed-dotted curves (these curves exactly coincide in the lower panels).

Figure 17

${\theta }_e^{*}$ dependence of the $e^{-}p\to e^{-}p{e}^{-}{e}^{+}$ cross section (upper panels) and forward-backward asymmetry (lower panels) in the $\mathrm{\Delta }(1232)$ region for $\mathrm{\Phi }=30°$ (left panels) and $\mathrm{\Phi }=45°$ (right panels). Curve conventions as in Fig. 16.

Figure 18

Radiative corrections for the $e^{-}p\to e^{-}p{e}^{-}{e}^{+}$ process in the $\mathrm{\Delta }(1232)$ region in the limit $Q^2\to 0$ for the comparable kinematic setup as was studied before for the $\gamma p\to e^{-}{e}^{+}p$ process in Ref. [40]. The corrections are for soft-photon cutoff energy of $\mathrm{\Delta }{E}_s=0.01\mathrm{GeV}$.

Figure 19

Upper panels: ${\theta }_l^{*}$ dependence of the radiative corrections to the $e^{-}p\to e^{-}p{l}^{-}{l}^{+}$ cross section in the $\mathrm{\Delta }(1232)$ region for $e^{-}{e}^{+}$ production (left) and ${\mu }^{-}{\mu }^{+}$ production (right), for the different classes of radiative corrections for $\mathrm{\Delta }{E}_s=0.01\mathrm{GeV}$. Lower panels: ${\theta }_l^{*}$ dependence of the total radiative correction for different values of $\mathrm{\Delta }{E}_s$ for both $e^{-}{e}^{+}$ production (left) and ${\mu }^{-}{\mu }^{+}$ production (right).

Figure 20

${\theta }_l^{*}$ dependence of the $e^{-}p\to e^{-}p{l}^{-}{l}^{+}$ cross section in the DDVCS regime for $e^{-}{e}^{+}$ production (left panels) and ${\mu }^{-}{\mu }^{+}$ production (right panels), for different values of the dilepton invariant mass $s_{ll}$. The curves show the predictions for BH and $\mathrm{BH}+\mathrm{dVCS}$ for two models showing the sensitivity to the D term in the GPD parametrization. The black solid curves show the effect of the radiative corrections for the hadronic model of the green dashed-dotted curves.

Figure 21

Upper panel: ${\theta }_l^{*}$ dependence on the soft-photon radiative corrections to the $e^{-}p\to e^{-}p{l}^{-}{l}^{+}$ cross section in the DDVCS regime for different values of $s_{ll}$. Upper (lower) panel is for $e^{-}{e}^{+}$ (${\mu }^{-}{\mu }^{+}$) production, respectively. The corrections are for a soft-photon cutoff energy of $\mathrm{\Delta }{E}_s=0.05\mathrm{GeV}$.

Figure 22

Upper panels: ${\theta }_l^{*}$ dependence of the radiative corrections to the $e^{-}p\to e^{-}p{l}^{-}{l}^{+}$ cross section in the DDVCS regime for $e^{-}{e}^{+}$ production (left) and ${\mu }^{-}{\mu }^{+}$ production (right) for the different classes of radiative corrections for $\mathrm{\Delta }{E}_s=0.05\mathrm{GeV}$. Lower panels: ${\theta }_l^{*}$ dependence of the total radiative correction for different values of $\mathrm{\Delta }{E}_s$ for both $e^{-}{e}^{+}$ production (left) and ${\mu }^{-}{\mu }^{+}$ production (right).

Figure 23

Radiative correction for the $e^{-}p\to e^{-}p{e}^{-}{e}^{+}$ process in the TCS limit $Q^2\to 0$ in a kinematic setup comparable to Ref. [40].

Figure 24

${\theta }_l^{*}$ dependence of the $e^{-}p\to e^{-}p{l}^{-}{l}^{+}$ beam-spin asymmetry $A_{\odot }$ (upper panels) and forward-backward asymmetry $A_{FB}$ (lower panels) in the DDVCS regime, for $e^{-}{e}^{+}$ production (left) and ${\mu }^{-}{\mu }^{+}$ production (right). Curve conventions as in Fig. 20.

Figure 25

Upper panel: photon virtualities entering the dVCS amplitude in the exchange diagrams of the $e^{-}p\to e^{-}p{e}^{-}{e}^{+}$ process for the kinematics of Fig. 24. Lower panel: scaling variables entering the GPDs for the exchange diagrams.

Figure 26

Lab momenta (upper panel) and Lab scattering angles (lower panel) of the dilepton pair as function of dilepton rest frame angle ${\theta }_l^{*}$ for the kinematics of Fig. 24.