Extracting transition generalized parton distributions from hard exclusive pion-nucleon scattering

With a high energy charged $\pi$ beam at J-PARC [1] and AMBER at CERN [2], hard exclusive processes on a nucleon target ($N$) can provide unique opportunities to extract transition generalized parton distributions (GPDs), $\mathcal{F}_{NN^{\prime}}(x,\xi,t)$, which could offer fundamental nonperturbative transition information from a hadron $N$ to $N^{\prime}$. If $N$ and $N^{\prime}$ are nucleons, the transition GPDs can be related to nucleon GPDs through isospin symmetry [3, 4]. Nucleon GPDs are nonperturbative hadron correlation functions which incorporate the concepts of parton distribution functions, distribution amplitudes (DAs), and form factors in their various aspects (for reviews, see [5, 6, 7, 8]). The Fourier transform with respect to $t$ at $\xi=0$ maps out the parton density distribution $f(x,\bm{b}_{T})$ at a position $\bm{b}_{T}$ in the transverse plane perpendicular to the direction of the colliding hadron and in slices of different values of longitudinal parton momentum fraction $x$. The various $x$ moments, on the other hand, provide important information on the hadron’s emergent properties such as mass [9, 10, 11, 12], spin [13], and internal pressure and shear force distributions [14, 15, 16].

Experimental extraction of GPDs of a hadron $h$ of momentum $p$ can be achieved by measurements of QCD factorizable single diffractive hard exclusive processes (SDHEPs) [17] in its collisions with a beam particle $B(p_{2})$, $h(p)+B(p_{2})\to h^{\prime}(p^{\prime})+C(q_{1})+D(q_{2})$, where $h^{\prime}(p^{\prime})$ represents a diffractively produced hadron along with two (or more) exclusively produced particles $C(q_{1})$ and $D(q_{2})$ with a large transverse momentum $q_{T}\sim q_{1T}\sim q_{2T}$ that defines the hard scale of the collision. Such processes can be thought of as occurring in two stages,

where the diffractive subprocess is connected to the hard scattering through a virtual state $A^{*}$ of momentum $\Delta=p-p^{\prime}$, which has a low invariant mass $t=\Delta^{2}$ but high energy, $\Delta^{+}=2\xi P^{+}$, where $P=(p+p^{\prime})/2$, and propagates for a long lifetime before entering the short-time hard interaction. The two-stage picture provides a clear view of factorization, which necessarily requires $q_{T}\gg\sqrt{|t|}$. Then the diffractive subprocess is factorized to a universal GPD function, separated from the hard interaction. The latter, on the other hand, serves as a probe for the GPD to map out partonic information of the transition from $h(p)$ to $h^{\prime}(p^{\prime})$.

Depending on the beam particle $B(p_{2})$ and specific hard scattering subprocesses, the SDHEPs can probe various types of GPDs [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. With an electron beam, we have well studied electroproduction processes, such as deeply virtual Compton scattering (DVCS) [18, 19] and deeply virtual meson production (DVMP) [20, 21]. With a photon beam, we have timelike Compton scattering (TCS) [22]. The pion (and also kaon) beams at J-PARC and AMBER at CERN allow “mesoproduction” processes to probe transition GPDs, which can be expressed in terms of flavor diagonal GPDs and are only sensitive to the valence parts of the latter. Two processes of this type have been identified: the Drell-Yan (DY) dilepton production process [38, 39, 40],

and the diphoton production process,

which was first introduced in Ref. [41]. See also Refs. [42, 43, 44] for discussions of pion beam use to probe transition distribution amplitudes.

For exclusive hard scattering like that in Eqs. (2) and (3), QCD factorization requires at least two active partons from each identified hadron in the scattering amplitude due to the color singlet nature of the identified hadron. The hadron correlation functions associated with the factorization of exclusive processes are generally harder to extract from experimental data than their inclusive counterparts, especially for their dependence on the relative momentum fraction of the two active partons, e.g., the $x$ dependence of GPDs. This is because at the amplitude level, $x$ is fully integrated from $-1$ to $1$ in the factorization formulas. Kinematically, owing to its exclusive nature, the invariant mass of the produced $C(q_{1})$ and $D(q_{2})$ of the $2\to 3$ SDHEPs in Eq. (1) is uniquely determined by the observed $p^{\prime}$. It is the distribution of the transverse momentum $q_{T}$ of $C(q_{1})$ [or $D(q_{2})$] in Eq. (1b) that could provide additional details of the state $A^{*}(\Delta)$ and the $x$ dependence of the GPDs if the $x$ integration is entangled with the observed $q_{T}$ distribution. However, such entanglement is absent from most processes, including the DVCS, DVMP, TCS, and DY dilepton production, where the $q_{T}$ dependence simply factors out of the $x$ dependence in the hard scattering, at least to the leading order, leaving only the GPD moment [see Eq. (15)] to be measured. Without considering the relatively weak QCD evolution effects [45], extracting GPDs merely from such moments gives infinite families of solutions quantified by the so-called shadow GPDs [46].

Fortunately, we could have processes [41, 37] whose hard coefficients do not factorize in the above sense, allowing us to have more direct access to the hadron’s partonic structure or the $x$ dependence of the GPDs. As demonstrated in Ref. [37], such new processes can greatly reduce the ambiguity caused by shadow GPDs due to the entanglement between the measured hard scale and $x$ dependence of the GPDs.

In this paper, we explore the role of SDHEPs with a charged $\pi$ beam for extracting transition GPDs while focusing on the diphoton production process in Eq. (3), where the two observed photons do not come from the decay of a single virtual particle, unlike the DY dilepton process in Eq. (2). We will evaluate both the production rate and the target spin asymmetry, then explore how they can help distinguish the shadow GPDs from the real GPDs, complementing the DY dilepton process at the same facilities. Our study could be extended further by replacing the two photons with two back-to-back high transverse momentum mesons, for example.

While the $\xi$ and $t$ dependence of GPDs directly corresponds to the diffractive kinematics which can be measured in experiments, the $x$ dependence is only indirectly accessed through convolutions with hard coefficients like those in Eqs. (4) and (6). It is only the hard scale $q_{T}$ (or, equivalently, $\theta$) that can help probe the hadron’s partonic structure via the $x$ dependence of the GPDs.

One can, however, immediately notice from Eq. (32) that the $\theta$ dependence of the leading-order hard coefficient for the DY dilepton process completely factorizes from the $x$ and $z$ dependence. Its amplitude simply reduces to the moments of the GPDs,

with $\mathcal{P}$ referring to principle-value integration, with a completely predictable $\sin\theta$ distribution decoupled from the $x$ dependence of the GPDs. Here $\widetilde{F}^{\pm}(x,\xi,t)=\widetilde{F}(x,\xi,t)\pm\widetilde{F}(-x,\xi,t)$ is the charge-conjugation-even or -odd GPD component [see Eq. (29)] and we used the same moment notations as in Eq. (27).

This simple factorization property also applies to many other processes, such as DVCS, DVMP, and TCS, which yield only moment-type sensitivity at leading order. As first demonstrated in Ref. [46], there are infinite families of analytic functions, called shadow GPDs, that give zero to the moments in Eq. (15) and cannot be distinguished from the “real GPD” by those processes alone.

In contrast, for the diphoton process, the hard coefficients in Eq. (22) contain terms in which the $\theta$ dependence is entangled with the $x$ (and $z$) dependence, resulting in a special convolution integral for the GPDs,

with a new pole for the $x$ integration,

The special integral appears in the two helicity amplitudes $\mathcal{M}_{-}$ and $\widetilde{\mathcal{M}}_{-}$ in Eq. (25), with $\mathcal{F}=\widetilde{F}$ and $F$, respectively, and the “$-$” subscripts referring to the two photons having aligned spins. The new pole in Eq. (17) scans through $[\xi,\infty)\cup(-\infty,-\xi]$ as $z$ goes from $0$ to $1$ and reflects the GPD $x$ dependence in the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) region. In comparison, the DY dilepton and other similar processes provide sensitivity only on the ridges $x=\pm\xi$.

The exact form of $x$ sensitivity by Eq. (16) is further complicated by the integration with the DA and depends on the specific form of the latter. As a simple demonstration in this paper, we take $D^{[u\bar{d}]}_{\pi^{+}}(z)=-D^{[d\bar{u}]}_{\pi^{-}}(z)=(if_{\pi}/2)\phi(z)$, with $f_{\pi}=0.13~{}{\rm GeV}$ being the pion decay constant and $\phi(z)=6z(1-z)$ the asymptotic form. Then for the two amplitudes $\mathcal{M}_{-}$ and $\widetilde{\mathcal{M}}_{-}$ in Eqs. (25b) and (25d), we define

where $a=+1$ for $\mathcal{F}=F$ and $-1$ for $\mathcal{F}=\widetilde{F}$, and the kernel $K^{a}$ is easily obtained with the asymptotic pion DA,

where $c\equiv\cos\theta$ and

As shown in Fig. 2, the kernel $K^{a}$ exhibits a strong logarithmic behavior at $x\to\pm\xi$ and becomes flatter as $|x|\gg|\xi|$ or $|x|\ll|\xi|$.

Clearly, the diphoton process does have the capability of distinguishing between two GPD functions that differ only by some shadow GPD by yielding different $\theta$ (or $q_{T}$) distributions, while the DY dilepton process does not. To demonstrate this, we construct the transition GPD models by adding two shadow GPDs, $S_{i}(x,\xi)$ [$\widetilde{S}_{i}(x,\xi)$] with $i=1$ or $2$, on $H^{[u\bar{d}]}_{pn}$ and $E^{[u\bar{d}]}_{pn}$ ($\widetilde{H}^{[u\bar{d}]}_{pn}$ and $\widetilde{E}^{[u\bar{d}]}_{pn}$) in the Goloskokov-Kroll (GK) model [69, 70, 71, 72], as in Ref. [37]. The specific forms of the shadow GPDs used in this paper are collected in Appendix C. The GPDs $H$ and $E$ also allow a $D$-term that lives only in the Efremov-Radyushkin-Brodsky-Lepage (ERBL) region, for which we may similarly construct a shadow $D$-term. The resultant GPDs, denoted as $(H_{i},E_{i},\widetilde{H}_{i},\widetilde{E}_{i})$, with $i=0$ referring to the GK model, are shown in Figs. 3 and 4, where one can notice the strong pion-pole enhancement in the ERBL region of the GPD $\widetilde{E}$.

Evaluating the integral $\bar{I}^{\pm}$ for these GPD models gives different $\cos\theta$ distributions, as shown in Figs. 5 and 6. $\bar{I}^{+}$ ($\bar{I}^{-}$) evaluated on $H^{[u\bar{d}]+}_{pn}$ or $E^{[u\bar{d}]+}_{pn}$ ($\widetilde{H}^{[u\bar{d}]+}_{pn}$ or $\widetilde{E}^{[u\bar{d}]+}_{pn}$) gives an odd (even) distribution of $\cos\theta$, so it is shown only for the positive $\cos\theta$. This property also strongly suppresses the value of $\bar{I}^{+}$ in the central rapidity region, so the value of $\bar{I}^{-}$ is generally larger than $\bar{I}^{+}$. We note that the shadow GPDs give substantial contributions to the special integrals, altering both the magnitudes and the shapes of the GK model results. The real and imaginary parts are both important in $\bar{I}^{+}$, whereas the real part dominates in $\bar{I}^{-}$, which agrees with the kernel behavior in Fig. 2.

Charged pion beams can be accessed in fixed target experiments at J-PARC [1] and AMBER at CERN [2] facilities as secondary beams, with wide energy coverage. It is not the purpose of this paper to perform a realistic experimental simulation; rather, we explore how the diphoton process can complement the DY dilepton one to help disentangle transition GPDs from shadow GPDs. At a benchmark study, we choose the two pion beam energies, $E_{\pi}=20$ and $100~{}{\rm GeV}$, for J-PARC and AMBER, respectively, and examine only the nucleon transitions.

Evaluating the unpolarized cross sections in Eq. (9) and the SSAs in Eq. (12) for the GPD models in Figs. 3 and 4 yields different distributions of $q_{T}$, as shown in Figs. 7 and 8 at $(t,\xi)$ $=$ $(-0.2~{}{\rm GeV}^{2},0.2)$ for the $p\pi^{-}$ and $n\pi^{+}$ collisions at $E_{\pi}=20$ and $100~{}{\rm GeV}$, respectively. We require $q_{T}\geq 1~{}{\rm GeV}$ to constrain the power correction from $\sqrt{|t|}/q_{T}$. The $q_{T}$ peaks of cross sections on the right are due to Jacobian effects. In each choice of the GPD models, we vary $H^{[u\bar{d}]}_{pn}$ or $\widetilde{H}^{[u\bar{d}]}_{pn}$ one at a time and fix $E^{[u\bar{d}]}_{pn}$ and $\widetilde{E}^{[u\bar{d}]}_{pn}$ to be of the GK model; varying the latter makes minimal differences due to the kinematic suppression factors in Eq. (10). We note that varying the GPDs yields not only different magnitudes for both production rates and SSAs but also different shapes in the $q_{T}$ distributions. Thus, in experimental analyses with binned $q_{T}$ observables, each bin offers an independent constraint on the GPDs, in contrast to the DY dilepton process, where the $q_{T}$ shape is universal (at leading order), as shown in Fig. 9.

The AMBER energy gives much wider $q_{T}$ coverage, but much smaller (differential) production rates, and leads to similar sensitivity in terms of the rate distribution ratios. In all cases, we notice that the shadow $D$-term added to $H^{[u\bar{d}]}_{pn}$ does not make as much an impact as it does for the photoproduction process in Ref. [37], as a result of the special pole $x_{p}$ in Eq. (17) lying only in the DGLAP region.

The difference between the $p\pi^{-}$ and $n\pi^{+}$ collisions is caused by the isospin-breaking effect of QED interactions, namely, the $(e_{1}^{2}-e_{2}^{2})$ terms in Eq. (25). Numerically, the diagonal values $\mathcal{F}(\xi,\xi,t)$ of transition GPDs $\mathcal{F}$ dominate over the their moments $\mathcal{F}_{0}(\xi,t)$ for a wide range of $\xi$ at a given $t$, although sea quark components do not contribute to transition GPDs. The $(e_{1}^{2}-e_{2}^{2})$ terms are therefore non-negligible and cause significant differences in the imaginary parts of the amplitudes in Eq. (25) between the $p\pi^{-}$ and $n\pi^{+}$ collisions. This induces the differences of cross sections and SSAs between the two channels.

The GPDs $H^{[u\bar{d}]}_{pn}$ and $E^{[u\bar{d}]}_{pn}$ make much smaller contributions to the cross sections and SSAs than $\widetilde{H}^{[u\bar{d}]}_{pn}$ and $\widetilde{E}^{[u\bar{d}]}_{pn}$ because (1) when making the flavor transition GPDs, there is a cancellation (enhancement) between $u$ and $d$ quark GPDs for the former (latter) in the GK model employed here; (2) aside from the $(e_{1}^{2}-e_{2}^{2})$ terms in Eq. (25), it is the $C$-even GPD combinations that contribute to the amplitudes, which gives a further cancellation (enhancement) for the former (latter) [see Eq. (29)]; and (3) most importantly, the pion-pole term in $\widetilde{E}^{[u\bar{d}]}_{pn}$ severely leverages the contributions from $\widetilde{H}^{[u\bar{d}]}_{pn}$, overshadowing the effects from $H^{[u\bar{d}]}_{pn}$ and $E^{[u\bar{d}]}_{pn}$. Hence, the $q_{T}$ distributions are more sensitive to the GPD $\widetilde{H}^{[u\bar{d}]}_{pn}$ than $H^{[u\bar{d}]}_{pn}$ for both the cross sections and SSAs. However, one would still be able to probe the GPD $H^{[u\bar{d}]}_{pn}$ near $|x|\sim\xi$, which is why the shadow GPDs $S_{i}(x,\xi)$ for $H^{[u\bar{d}]}_{pn}$ have been constructed to be much more dramatic in this region than the $\widetilde{S}_{i}(x,\xi)$ for $\widetilde{H}^{[u\bar{d}]}_{pn}$. If we remove or decrease the pion-pole contribution from $\widetilde{E}^{[u\bar{d}]}_{pn}$, the sensitivity to $H^{[u\bar{d}]}_{pn}$ would become more manifest.

Despite the kinematic suppression in Eq. (12), we get significant SSAs, especially for the $\pi^{+}n$ channel, which is dominated by the large pion-pole enhancement in the GPD $\widetilde{E}^{[u\bar{d}]}_{pn}$. Hence, the SSAs can also serve as useful probes of GPDs, particularly of $\widetilde{H}^{[u\bar{d}]}_{pn}$, since it is the one that interferes with $\widetilde{E}^{[u\bar{d}]}_{pn}$ in Eq. (12), although it has one extra suppression factor of $\xi$, as shown in Figs. 7 and 8. For the same reason, varying the GPDs $E^{[u\bar{d}]}_{pn}$ and $\widetilde{E}^{[u\bar{d}]}_{pn}$ by the same shadow GPDs does not yield appreciable effects. Nevertheless, by simply scaling them with some factor, the SSAs are proportionally changed. Measuring the SSA gives sensitive probes to both the magnitudes and the signs of the GPDs $E^{[u\bar{d}]}_{pn}$ and $\widetilde{E}^{[u\bar{d}]}_{pn}$.

In comparison, for the DY dilepton process, the shadow GPDs are constructed to give null impacts. The above GPD models thus give the same prediction for both the unpolarized cross section in Eq. (11) and the SSA in Eq. (13), shown in Fig. 9 for the J-PARC and AMBER energies, respectively. At leading power, the virtual photon decaying into the lepton pair is purely longitudinally polarized, giving a $\sin^{2}\theta$ angular distribution [Eq. (33)], so the cross section is finite at small $q_{T}$. This feature contrasts with the diphoton process, whose production rate is enhanced in the forward region, with a much larger overall rate as well. Interestingly, the DY dilepton process produces a large SSA. It does not depend on $q_{T}$, and the energy dependence cancels in the ratio of Eq. (13). This measurement could set a useful constraint on the overall magnitude of GPD $\widetilde{E}^{[u\bar{d}]}_{pn}$, especially on the pion-pole peak in the GK model, but has no capability of probing its $x$ dependence.

While GPDs encode valuable information on hadrons’ emergent properties and tomographic structures, it is a challenging task to fully extract such information out of experiments, especially the $x$ dependence. In this paper, we compared two single diffractive hard exclusive mesoproduction processes, the DY dilepton production and diphoton production, which can be studied at hadron experiment facilities like J-PARC and AMBER at CERN using secondary pion beams. At leading power, the scattering amplitudes of both processes can be factorized into transition GPDs and provide constraints on the nucleon GPDs through isospin symmetry, complementary to the electroproduction and photoproduction processes.

We showed that both the unpolarized production rate and the target spin asymmetry can help probe the transition GPDs and clearly demonstrated that, while the DY dilepton process only constrains the polarized GPDs through simple moments, the diphoton process can provide enhanced $x$ sensitivity from transverse momentum ($q_{T}$) distributions of the final-state particles that are entangled with the $x$ dependence of the GPDs. This enhanced $x$ sensitivity shares the same physics origin as that appearing in photoproduction of photon-meson pair studied in Ref. [37] but complements it by being more sensitive to the DGLAP region near $x=\pm\xi$. Both J-PARC and AMBER at CERN give similar sensitivity; the former produces larger rates, but the latter covers a wider kinematical range.

Given the fact that we need to solve an inverse problem to extract GPDs from experimental data, we would not expect one or two simple processes to be sufficient to fully map out the $x$ dependence of the GPDs. We therefore advocate that global analyses should be performed by combining data from various processes that are sensitive to the universal GPDs, especially those with enhanced $x$ sensitivity. Only in this way would it be possible to obtain the full tomographic images of the hadrons.

We thank W.-C. Chang, P. Kroll, J.-C. Peng, B. Pire, M. G. Santiago, M. Strikman, and N. Tomida for the helpful discussions and communications. This work is supported in part by U.S. Department of Energy (DOE) Contract No. DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, operates Jefferson Lab. The work of Z.Y. at MSU is partially supported by the U.S. National Science Foundation under Grant No. PHY-2013791 and the fund from the Wu-Ki Tung endowed chair in particle physics.