Modeling Backward-Angle ($u$-channel) Virtual Compton Scattering at an Electron-Ion Collider

Backward ($u$-channel) Compton scattering (CS) occurs when a photon scatters backwards from a proton, with a large momentum transfer between the two as shown in Fig. LABEL:fig:COMFrame. This is in stark contrast to the more common $t$-channel process which dominates the CS cross section. In $t$-channel (forward/small-angle) Compton scattering, the momentum transfer between the photon and proton is small, as is the scattering angle, e.g. $\theta\approx 0$, so $|t|\approx 0$. $u$-channel CS has a near-maximal momentum transfer $|t|$, and small $|u|$ with $\theta\approx 180^{\circ}$.

If the initial-state photon is a virtual photon, Compton scattering is referred to as virtual Compton scattering (VCS) or deeply-virtual Compton scattering (DVCS) depending on the photon’s invariant mass squared ($-Q^{2}$). DVCS is considered a “golden channel” of the future U.S. Electron-Ion Collider because $t$-channel DVCS provides access to proton Generalized Parton Distributions (GPDs) at non-zero skewedness [1]. Much attention has been paid to $t$-channel Compton scattering due to its dominance of the total cross section and straightforward interpretation in terms of GPDs.

Forward DVCS is used for proton tomography [2] because the transverse component of the Mandelstam $t$ is conjugate to the distribution of partons in the transverse plane [3], i.e. as a function of impact parameter. The majority of DVCS measurements to-date have been collected at low $|t|$ primarily from experiments at Jefferson Lab [4, 2] and DESY [5, 6]. These low-$t$ measurements map the proton at large impact parameters, but there is little to constrain parton distributions at small radii. For this reason, the EIC White Paper [7] stresses the need for DVCS measurements up to large momentum transfers.

Recent theoretical work on baryon-to-photon/meson transition distribution amplitudes (TDAs) proposes a $u$-channel factorization scheme similar to the $t$-channel factorization with an impact-parameter interpretation of backward amplitudes [8]. In this view, the TDAs encode information about the transverse distribution of di-quark and tri-quark clusters within the proton. Furthermore, backward DVCS and other $u$-channel processes may play a role in baryon stopping in heavy-ion collisions, in which nucleons undergo large momentum transfers and are detected near midrapidity [9].

VCS analyses at the EIC should attempt to measure the magnitude of the $u$-channel contribution to the VCS cross section, and how it scales with $Q^{2}$, $W$, and $t$. After transforming the cross section from transverse momentum to impact-parameter space, these measurements may allow the EIC to map those partonic constituents that contribute to reactions involving baryon-number transfer.

Moreover, without knowing if a backward VCS peak exists, the magnitude of $u$-channel contributions to forward DVCS cross section measurements at the EIC are unknown. Toward low (threshold) $\gamma^{*}p$ collision energies, the difference between the $|t|$ values corresponding to forward and backward scattering becomes small. As a result, it is difficult to isolate the $t$-channel contribution at threshold, as the $u$-channel mechanism may contribute an unknown amount. Therefore it may benefit DVCS studies to understand the magnitude of the contribution that $u$-channel exchange adds to the cross section.

In this paper, we examine the prospects of measuring backward VCS at the EIC, in the face of a large background from backward $\pi^{0}$ production. In Sec. II, we define kinematic variables and provide background information on backward Compton scattering. Section III presents a model of backward VCS developed from existing data. Section IV discusses backgrounds to the backward VCS signal, including $u$-channel $\pi^{0}$ production. Details about the simulations that were developed are provided in Sec. V. Section VI discusses the prospects for detecting these simulated events at the EIC, given current detector expectations. This section also demonstrates how the background may be reduced to a few percent of the backward VCS signal.

The kinematic variables used to describe DVCS and electroproduction processes are labeled in Fig. LABEL:fig:Mandelstam. $Q^{2}$ is the negative square of the four-momentum of the virtual photon, $Q^{2}=-q^{2}$, and is quantifiable from the electron’s final energy and scattering angle $\theta_{e^{\prime}}$:

where $E_{e}$ and $E_{e^{\prime}}$ are the energies of the initial and final-state electron, respectively. The backward DVCS cross section is expected to drop quickly with increasing $Q^{2}$. Therefore we often refer to this process as $u$-channel VCS rather than $u$-channel DVCS in order to not overstate the virtuality that may be expected of $u$-channel Compton scattering. The center-of-mass energy of the $p\gamma^{*}$ system is $W=\sqrt{s_{p\gamma^{*}}}=\sqrt{(p+q)^{2}}$, and is measurable through the momenta of the outgoing proton and photon. The Mandelstam $t=(p-p^{\prime})^{2}$ is the square difference in four-momenta of the initial and final-state proton, and $u=(p-k)^{2}$ is the square difference in four-momenta of the beam proton and final-state photon.

Two additional variables are often used to describe VCS in the $\gamma^{*}p$ center-of-mass frame. The scattering angle, $\theta$, is shown in Fig. LABEL:fig:COMFrame. The momentum transfer involved in the scattering process is described equivalently in terms of $\theta$, $u$, or $t$. In this paper, we parameterize cross sections in terms of $u$ and $t$. Much of the literature quantifies measurements in terms of $\theta$ [14, 15], so it is often necessary to convert between the $\theta$ and Mandelstam parameterizations. $\phi$ is the azimuthal rotation of the final-state $\gamma p^{\prime}$ plane with respect to the electron scattering plane. The $\phi$-dependence of the VCS cross section is related to orbital angular momentum contributions to proton GPDs [16].

We can construct the $ep\rightarrow e^{\prime}p^{\prime}\gamma$ cross section using these quantities:

where $\Gamma(Q^{2},W)$ is the virtual photon flux [17]. Given a $\gamma^{*}p$ system with a defined $Q^{2}$ and $W$, the probability of the photon scattering with the proton to produce a final state with some $t$ and $\phi$ is thus proportional to $\frac{d^{2}\sigma}{d\phi dt}(Q^{2},W,\phi,t)$. It is this reduced cross section for $\gamma^{*}p\rightarrow p^{\prime}\gamma$ that is of primary interest to future EIC analyses. The form of this cross section is the subject of Sec. III.

It is often more convenient to discuss the cross section in terms of $u$ rather than the Mandelstam $t$, because the backward-production cross section as a function of $u$ behaves similarly to the forward-production cross section as a function of $t$. For this reason, we will also refer to the similar cross section:

The Mandelstam $u$ is related to the scattering angle $\theta$ via:

where $G=m_{p}^{2}+Q^{2}+W^{2}$. Equation 4 can be rearranged to give $u$ in terms of the scattering angle:

Equation 5 is used to compare our models with differential cross-section measurements at fixed scattering angles in Sec. III. The most positive $u$ value is $u_{0}(Q^{2},W)=u(Q^{2},W,\cos\theta=-1)$, corresponding to $180^{\circ}$ backward Compton scattering.

A general relation from two-to-two particle scattering is useful here as well. For particles 1 and 2 scattering to produce particles 3 and 4, the Mandelstam relations give $s+t+u=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}$. For VCS:

This equation relates the $t$-dependence of the cross section to the $u$-dependence. At fixed $W$ and $Q^{2}$ the exponential rise of the cross section toward the most negative $t$ values in Fig. 2 can be translated into an exponential rise in the cross section toward the most positive possible $u$ values. Taken together, Equations 5 and 6 relate the scattering angle to $t$.

As discussed in Sec. II, VCS may be described by the kinematic variables: $Q^{2}$, $W$, $u$ (or $t$), and $\phi$. The event-plane rotation in azimuth does not affect the feasibility of detecting VCS events, so for this paper, all cross sections and rates are integrated over $\phi$.

DVCS off of protons is often modeled with a differential cross section of the form:

Exclusive vector-meson production can be modeled by the same functional form, in agreement with data at forward angles. This dependence is a somewhat simplified picture as was demonstrated by the H1 Collaboration, which measured the $B$ parameter and showed it to vary slowly with $Q^{2}$ [18]. The DVCS event-generator MILOU allows users to express $B$ as a linear function of $\ln Q^{2}$ [19], and eSTARlight expresses cross sections for vector-meson production on protons in terms of a dipole form factor [20] that reduces to Eq. 7 as $t\rightarrow 0$.

Despite their differences, these parameterizations all include a sharp peak at $t\sim 0$ and a vanishing cross section as $|t|$ rises. However, early photoproduction data found a breakdown of these parameterizations at very large $|t|$ [21, 22]. Instead of an ever-decreasing cross section with increasing $|t|$, an enhancement was observed at the maximal $|t|$ values. This is interpreted as coming from contributions from baryon (Reggeon) exchange trajectories. Recent measurements, seen in Fig. 2, extend $u$-channel $\omega$ electroproduction data to high $Q^{2}$ [10]. However, a peak in the VCS cross section at maximal $|t|$ has not yet been observed, likely due to the challenges of detecting backward VCS in fixed-target experiments. Experiments have been proposed at Jefferson Lab to establish the existence of this peak [23].

We exploit expected similarities between the $t$ and $u$-channel exchanges and model the backward cross section with the form:

which describes an exponentially falling cross section as $u$ deviates from its most positive possible value $u_{0}$. There exists little data on VCS at backward angles to constrain the value of $D$, often called the “slope parameter.” We can estimate $D$ using data from backward production of $\omega$ mesons.

Daresbury Laboratory’s NINA 5 GeV electron synchrotron [21] and Jefferson Lab Hall C [10] have both measured the $u$-channel peak in backward $\omega$ production. The NINA measurements correspond to photoproduction ($Q^{2}$=0) and the Hall C measurements are for electroproduction at $Q^{2}=1.75$ GeV² and 2.45 GeV². The Hall C data had fewer bins in $u$, measuring the cross section at three $u$ values for each set of kinematics, compared to around twenty measurements in $u$ from NINA for each configuration. The Hall C and NINA data are at similar values of $u$ and $W$, and differ primarily in their $Q^{2}$ and the number of measurements in $u$. The NINA measurements show a steep exponential drop-off near $|u|\sim 0$ and a slower drop-off at large $|u|$, with a dip in between. The Hall C measurement does not show a dip in the cross section, and is well-described by the simple exponential of Eq. 8. The slope parameter obtained from the Hall C data is $D=2.4\pm 1.8$ GeV^-2 [10] at $W=2.5$ GeV and $0.03<-u<0.28$ GeV². The slope parameter does not depend on $Q^{2}$ over the measured range. The steep $u$-exponential portion of the NINA photoproduction cross section was fit for 4.7 GeV photons ($W=3.1$ GeV), resulting in a slope parameter of $D=21.8\pm 1.2$ GeV^-2, valid over $0.01<-u<0.12$ GeV².

It is not clear what causes the difference between the Hall C and NINA slope parameters. Photoproduction of $\omega$ mesons might have a very different behavior than electroproduction at low $|u|$. Another possibility is that a steep slope and dip structure are integrated over in the Hall C data or that these features are not present at the lower $W$ measured by Hall C. Additional electroproduction measurements with fine binning in $u$ are needed for a decisive comparison. In the absence of backward VCS measurements, we use these very different slopes to develop two alternate models of backward VCS (referred to as models 1 and 2) that, taken together, provide us with a sense for the range of possible cross sections and kinematics.

Model 1 uses the $D=2.4$ GeV^-2 value measured by Hall C. Model 2 uses the $D=21.8$ GeV^-2 value from the NINA data. In a vector-meson-dominance framework, it is not unreasonable to assume that $u$-channel VCS would behave similarly to $u$-channel $\omega$ production. We use these two slope-parameter values for our backward VCS models in lieu of more representative data. In order to minimize uncertainties on the cross section caused by these two different slope parameters, we describe below how both cross section models are scaled to the limited backward VCS data that is available.

We next model the $W$-dependence of the cross section. Backward-production cross sections scale with a negative power of the center-of-mass energy $W$, representing the scaling behavior of Reggeon exchange trajectories [24, 25]:

There is currently no data on the $W$-dependence of the backward VCS cross section at both fixed $u-u_{0}$ and fixed $Q^{2}$. A reasonable starting point is $d\sigma/du(W)\sim(W^{2}-m_{p}^{2})^{-2}$ [14, 26], which has also been used to model backward meson production [10]. This is similar to the $(W^{2}-m_{p}^{2})^{-2.7}$ dependence previously used in backward vector-meson simulations [9].

We employ a squared nucleon dipole form factor for the explicit $Q^{2}$-dependence. This goes as $\sim(Q^{2}+\Lambda^{2})^{-4}$ for some constant $\Lambda$. Backward VCS data were collected at a constant scattering angle at $W$=1.53 GeV in the resonance region, where hadronic resonances give structure to the cross section in $W$. A fit to this data using the dipole form-factor scaling found $\Lambda^{2}=2.77\ \text{GeV}^{2}$ [14]. This is the best measurement of the $Q^{2}$ scaling of backward VCS, but it should be noted that above the resonance region the cross section may scale differently. For example, in backward $\omega$ production at $W=2.21$ GeV, the cross section goes as $\sim$$Q^{-1.08}$ for transversely-polarized photons and as $\sim$$Q^{-10.22}$ for longitudinally-polarized photons [27]. Our models use the $(Q^{2}+2.77\ \text{GeV}^{2})^{-4}$ scaling, which is representative of the data that is most relevant here. Combining this scaling with the $W$ scaling, the backward VCS cross-section model is:

where $\alpha=2$, $\Lambda^{2}=2.77$ GeV², and $A$ is a normalization factor. The Jefferson Lab Hall A Collaboration’s VCS data [14] at fixed angle ($\cos\theta\ =\ -0.975$), and $Q^{2}$ = 1 GeV² are used to anchor the cross-section amplitude. To limit effects of nucleon resonances that decay to $\gamma p$, the model amplitudes were fit to cross section measurements at the eleven highest $W$ values, from 1.77 to 1.97 GeV. Statistical and systematic uncertainties on the data points were combined in quadrature.

The fit finds an amplitude of $A=32$ $\mu$b/GeV² for model 1 and $A=65$ $\mu$b/GeV² for model 2, each with a 7$\%$ uncertainty from the normalization fit. The two models are compared in Fig. 3 along with the VCS data. These measurements were taken at constant scattering angles, $\theta$. For the comparison to data, both $u$ and $u_{0}$ were calculated at each $W$ value, so the differential cross sections shown in Fig. 3 do not have the simple $\sim(W^{2}-m_{p}^{2})^{-2}$ dependence as might be expected. The parameters for both models and the background model discussed in Sec. IV are summarized in Tab. 1. The table also summarizes the kinematic ranges over which the data informing each parameter were taken.

With the amplitudes fixed, each model was then used to predict the differential cross section as a function of $W$ for wide-angle real Compton scattering (RCS), and compared with data [28] at $\cos\theta=-0.62$. Models 1 and 2 both demonstrate plausible scaling behavior when compared to the backward-angle VCS data. For the RCS data at the wide backward angle, model 1 performs significantly better, although it still overshoots the data. Although the RCS data with $\cos\theta=-0.62$ corresponds to backward scattering, it is not close to the backward peak, and may not be well-described by the simple exponential $u$-dependence employed at the most backward angles. It is not surprising then that neither model matches the data, but the near-agreement of model 1 is a good reason to move forward with this model. We therefore use model 1 ($D=2.4$ GeV^-2, $A=32$ $\mu$b/GeV²) in the simulations described in Sec. V.

Backgrounds in VCS measurements are dominated by the Bethe-Heitler process and $\pi^{0}$ production. However the Bethe-Heitler cross section, which peaks in the forward region, does not have a complementary backward peak. Thus there is almost no Bethe-Heitler contribution to the background for backward Compton scattering [8, 29]. The $\pi^{0}\rightarrow\gamma\gamma$ process does have a peak in the backward limit [22] and is expected to be the primary background to $u$-channel VCS.

In Sec. VI.3 we discuss the separation of backward $\pi^{0}$ production from backward Compton events. The model used to generate these $u$-channel $\pi^{0}$ events is similar to the backward VCS model. Unlike Compton scattering, a backward peak in the cross section has already been observed for $\pi^{0}$ production in fixed-target experiments at the Stanford Linear Accelerator [22]. At photon energies between 6 and 18 GeV, the cross section was found to scale as $d\sigma/du\sim W^{-6.0\pm 0.4}$. We performed a fit to the data assuming a scaling of the form $d\sigma/du\sim(W^{2}-m_{p}^{2})^{-\alpha}$, which described the data well with $\alpha=2.8\pm 0.1$.

Similar to backward VCS, the backward $\pi^{0}$ production cross section may be expected to scale with $Q^{2}$ according to a squared nucleon dipole form factor: $\sim(Q^{2}+\Lambda^{2})^{-4}$. Jefferson Lab’s Hall A Collaboration measured this $Q^{2}$-dependence of backward $\pi^{0}$ production [30] which, at $W=2$ GeV, was found to be consistent with the $\Lambda^{2}=2.77~{}\rm{GeV}^{2}$ measured in backward VCS [14]. We therefore use the same $Q^{2}$ dependence in our model of the $u$-channel $\pi^{0}$ cross section. Taken together, these scalings lead to the following model for backward $\pi^{0}$ production:

In order to extract the $A$ and $D$ parameters, we performed fits to the SLAC backward $\pi^{0}$ production data [22], accounting for the $W$ and $Q^{2}$ scalings. The fits were done in the region $-0.34$$<$$u$$<$0.0 GeV², over which the cross section is nicely described by the exponential behavior in $u$. The amplitude $A$ and slope $D$ were found to be $1.26\pm 0.07$ mb/GeV² and $4.2\pm 0.4$ GeV^-2 respectively. A comparison of the model to the photoproduction data is shown in Fig. 4.

Backward VCS and $\pi^{0}$-production simulations were performed using the eSTARlight Monte Carlo event generator [20], which was modified by the authors to include these processes with the kinematics described in Sections III and IV. eSTARlight was developed for modeling exclusive meson production in $ep$ and $eA$ collisions, and thus already includes much of the framework needed for simulating backward Compton scattering and $\pi^{0}$ production.

The code first generates a virtual photon spectrum using a lookup table, representing the photon flux $\Gamma(k,Q^{2})=d^{2}N_{\gamma^{*}}/dkdQ^{2}$ [20]. Here $k$ refers to the energy of the virtual photon. For each $k$ and $Q^{2}$ selected, the center-of-mass energy, $W$, is calculated. The generated photon is then compared against the cross section models for backward VCS or backward $\pi^{0}$ production, $d\sigma/du(Q^{2},W)$, and is accepted or rejected by Monte Carlo sampling. After the virtual photon kinematics have been selected, $u_{0}$ is calculated and the Mandelstam $u$ of the process is then selected according to random sampling from the $d\sigma/du(u)\sim\exp{(-D|u-u_{\text{0}}|)}$ distribution. Figure 6 shows several example exponential differential cross sections using model 1 for a given $W$ and $Q^{2}$. The $u_{0}$ for each set of kinematics, as calculated using Eq. 4, is shown as a vertical dashed line corresponding to each cross section.

The event is treated in the center-of-mass frame as in Fig. LABEL:fig:COMFrame. For a given value of $u$, the polar angle $\theta$ of the outgoing photon with respect to the initial-state axis is given by Eq. 4. With $\theta$ chosen, the next step is to generate a value of the azimuthal rotation $\phi$, which is uniformly sampled. For $\pi^{0}$ production, the $\pi^{0}$ is then decayed isotropically via $\pi^{0}\rightarrow\gamma\gamma$. Finally the event is rotated and boosted back to the laboratory frame, and the final-state particles are written to file.

The models considered in Sec. III include a divergence as $W\rightarrow m_{p}$, which is consistent with expectations. Compton scattering does not have a lower threshold on the energy of produced photons, so there is no restriction on $W$ approaching the proton mass. However, it becomes increasingly difficult to discriminate forward and backward scattering for Compton scattering at low $W$. From Eq. 5, the difference between $u_{\text{f}}$ for forward scattering at $\cos\theta=1$ and $u_{\text{b}}$ representing backward scattering at $\cos\theta=-1$ vanishes at threshold. We therefore place a lower-limit on $W$ at 2 GeV, which also limits contributions from the resonance region.

We consider three standard EIC collision energies: 5 GeV electrons on 41 GeV protons (5$\times$41 GeV), 10 GeV electrons on 100 GeV protons (10$\times$100 GeV), and 18 GeV electrons on 275 GeV protons (18$\times$275 GeV). These three energy combinations will allow us to study how predicted acceptances for the planned ePIC detector will compare to the phase space of final-state particles from backward VCS and $\pi^{0}$ production.

The $u$-channel contribution to the virtual Compton scattering cross section is an opportunity to measure and interpret new physics at the future EIC. Measurement of a backward peak in the VCS cross section will help clarify the mechanism by which $u$-channel production proceeds, and may shed light on which processes contribute to baryon stopping. The cross section can also be interpreted in the baryon-to-photon TDA formalism as a description of the partonic makeup of the proton in the transverse plane.

We have developed models of both $u$-channel Compton scattering and its dominant background, $u$-channel $\pi^{0}$ production. The Monte Carlo generator eSTARlight was used to simulate $u$-channel VCS and $\pi^{0}$ production at the EIC. Event rates and the distribution of final-state particles within projected detector acceptances were predicted. As the ePIC design takes shape in the coming years, more work will be needed in order to understand the effect of bremsstrahlung electrons on low-$Q^{2}$ event reconstruction.

In backward VCS simulations, the scattered proton lands within the acceptance of the central trackers, the B0 trackers, and the Roman pots. Protons from high-$Q^{2}$ events will be detected primarily in the B0 and central trackers. The Compton photon rarely ends up in the central detectors ($\eta<3.5$), and requires calorimetry in the far-forward region. The inclusion of electromagnetic calorimetry in the B0 magnet system should improve the $u$-channel VCS acceptance by a factor of two at $10\times 100$ GeV and a factor of ten at $5\times 41$ GeV. The additional calorimeter would also aid in the reduction of the $u$-channel $\pi^{0}$ background.

In $ep$ collisions at 18$\times$275 GeV, backward VCS photons land primarily within the projected ZDC. If a $u$-channel peak has a large enough cross section ($\sim$1/10th the $\pi^{0}$ peak) then the $\pi^{0}$ background will be reducible to a few-percent level with appropriate exclusivity cuts. The feasibility of measuring backward VCS depends largely on the acceptance of the ZDC, and will be aided by the inclusion of electromagnetic calorimeters in the B0 magnet system. Assuming that the cross-sections are similar to our model predictions, $u$-channel virtual Compton scattering should be measurable at the EIC.

We acknowledge useful conversations with Bill Li, Feng Yuan, and Alex Jentsch. Aaron Stanek participated in an early phase of this work. This work is supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract numbers DE-AC02-05CH11231, by the US National Science Foundation under Grant No. PHY-2209614, and by the University of California Office of the President Multicampus Research Programs under grant number M21PR3502.