Rosenbluth separation of the $\pi^0$ Electroproduction Cross Section off the Neutron

TL;DR

This study presents the first separation of longitudinal and transverse components of the $c ext{pi}^0$ electroproduction cross section off the neutron, providing insights into nucleon structure and quark contributions.

Contribution

It introduces the first longitudinal/transverse separation of the $c ext{pi}^0$ electroproduction cross section off the neutron, revealing the dominance of transversely polarized photon responses.

Findings

Cross sections align with theoretical model predictions.

Transversely polarized photon response dominates in neutron $c ext{pi}^0$ production.

Flavor decomposition of quark contributions achieved.

Abstract

We report the first longitudinal/transverse separation of the deeply virtual exclusive $\pi^0$ electroproduction cross section off the neutron and coherent deuteron. The corresponding four structure functions $d\sigma_L/dt$, $d\sigma_T/dt$, $d\sigma_{LT}/dt$ and $d\sigma_{TT}/dt$ are extracted as a function of the momentum transfer to the recoil system at $Q^2$=1.75 GeV$^2$ and $x_B$=0.36. The $ed \to ed\pi^0$ cross sections are found compatible with the small values expected from theoretical models. The $en \to en\pi^0$ cross sections show a dominance from the response to transversely polarized photons, and are in good agreement with calculations based on the transversity GPDs of the nucleon. By combining these results with previous measurements of $\pi^0$ electroproduction off the proton, we present a flavor decomposition of the $u$ and $d$ quark contributions to the cross section.

Equations

$$Q^2$$

$$x_A$$

$$W^2$$

$$y$$

$$t$$

$$t^{\prime }$$

$$\frac{d^{4}\sigma}{dQ^{2}dx_{A}dtd\phi}=\frac{1}{2\pi}\frac{d^{2}\Gamma_{A}}{dQ^{2}dx_{A}}\Big{[}\frac{d\sigma_{T}}{dt}+\epsilon\frac{d\sigma_{L}}{dt}\\ \sqrt{2\epsilon(1+\epsilon)}\frac{d\sigma_{TL}}{dt}\cos\phi+\epsilon\frac{d\sigma_{TT}}{dt}\cos 2\phi\Big{]}\;,$$

$$\frac{d^2{\Gamma }_A}{d{Q}^{2}d{x}_A}$$

$$ϵ$$

$$\left|m_{\gamma \gamma }-m_{\pi }\right|$$

$$M_X^{\prime 2}=M_X^2+C(m_{\gamma \gamma }-m_{\pi })$$

$$D(e,e^{\prime }{\pi }^0)X=d(e,e^{\prime }{\pi }^0)d+n(e,e^{\prime }{\pi }^0)n+p(e,e^{\prime }{\pi }^0)p.$$

$${\chi }^2=\sum _{i=1}^{3600}{\left(\frac{N_i^{exp}-N_i^{sim}}{{\delta }_i^{exp}}\right)}^2,$$

$$\frac{d{\sigma }_T}{dt}$$

$$\frac{d{\sigma }_{TT}}{dt}$$

$${\left|⟨H_T^{p,n}⟩\right|}^2$$

Full text

Understanding the internal three-dimensional structure of nucleons in terms of quarks and gluons is a major challenge of modern hadronic physics. Two complementary approaches have been used in the past in order to achieve this goal. On the one hand, nucleon form factors (FFs) measured in elastic electron scattering provide information on the transverse charge and current distributions inside the nucleon Hofstadter and McAllister (1955). On the other hand, parton distribution functions (PDFs) measured in Deeply Inelastic Scattering (DIS) characterize the longitudinal momentum distribution of the underlying quarks and gluons Friedman et al. (1991). Twenty years ago, FFs and PDFs were unified within the formalism of Generalized Parton Distributions (GPDs) Mueller et al. (1994); Ji (1997); Radyushkin (1997). GPDs are universal functions encoding a wealth of information about the nucleon internal structure such as the correlation between the transverse position of quarks and gluons (partons) and their longitudinal momenta Burkardt (2003). GPDs also provide access to the contribution of quark and gluon orbital angular momenta to the nucleon spin Ji (1997). Eight GPDs for each quark flavor $q$ describe nucleon structure at leading order in $1/Q$ (twist-2). They correspond to each combination of nucleon and parton helicities. The four chiral-even GPDs ($H^{q}$, $E^{q}$, $\widetilde{H}^{q}$ and $\widetilde{E}^{q}$) conserve the helicity of the parton whereas the four chiral-odd, or transversity GPDs ($H_{T}^{q}$, $E_{T}^{q}$, $\widetilde{H}_{T}^{q}$ and $\widetilde{E}_{T}^{q}$), flip the parton helicity Diehl (2003); Hoodbhoy and Ji (1998).

GPDs parametrize the structure of the target independently of the reaction. Chiral-even GPDs can be accessed experimentally via hard exclusive processes such as deeply virtual Compton scattering (DVCS) and deeply virtual meson electroproduction (DVMP) in the Bjorken limit $Q^{2}\to\infty$ and $t/Q^{2}\ll 1$ at fixed $x_{B}$. Recent results on DVCS show the validity of this limit at values of $Q^{2}$ as low as 1.5 GeV2 C. Muñoz Camacho et al. (2006); Defurne et al. (2015); Jo et al. (2015). In the case of DVMP, the longitudinal scattering amplitude factorizes into a hard perturbative contribution and a soft convolution of the nucleon GPDs and the meson distribution amplitude (DA). The transverse virtual photo-production amplitude is proven to be suppressed by a factor of $1/Q^{2}$ at sufficiently high values of $Q^{2}$ Collins et al. (1997). In the case of $\pi^{0}$ electroproduction, it was suggested in Ahmad et al. (2009); Goloskokov and Kroll (2011) that a large contribution to the transverse amplitude could arise from the convolution of the transversity GPDs of the nucleon with a twist-3 quark-helicity flip pion DA. Model calculations including the transversity GPDs have successfully described recent $\pi^{0}$ electroproduction data on a proton target, measured at Jefferson Lab (JLab) Fuchey et al. (2011); Bedlinskiy et al. (2012, 2014); Defurne et al. (2016). Measurements of $\pi^{0}$ electroproduction on the neutron are extremely interesting as they provide the exciting possibility to separate the individual contributions of the $u$ and $d$ quarks to the cross sections, when combined with measurements from a proton target at the same kinematics.

The differential cross section of deeply virtual $\pi^{0}$ production is given by Drechsel and Tiator (1992):

[TABLE]

where $\phi$ is the angle between the hadronic and leptonic planes following the Trento Convention Bacchetta et al. (2004). The virtual photon flux factor $d^{2}\Gamma_{A}$ and photon polarization $\epsilon$ are defined by:

[TABLE]

Fig. 1 shows the lowest order Feynman diagram of the reaction and includes definitions of the kinematic variables. The $\phi$ dependence in Eq. (1) allows the extraction of the interference terms $d\sigma_{TL}/dt$ and $d\sigma_{TT}/dt$ while measurements of the total cross section at two incident beam energies and fixed $Q^{2}$ and $x_{B}$ separate $d\sigma_{T}/dt$ and $d\sigma_{L}/dt$.

In JLab Hall A experiment E08-025, we measured the D$(e,e^{\prime}\pi^{0})X$ reaction, with the primary goal of extracting the $n(e,e\pi^{0})n$ cross section in the quasi-free approximation. We perform a Rosenbluth separation, based on data taken with incident beam energies $E=4.455$ and 5.550 GeV. A 15-cm-long liquid deuterium (LD2) target was used as a quasi-free neutron target. The quasi-free $\pi^{0}$ electroproduction events off the proton are subtracted using the data from experiment E07-007 Defurne et al. (2016). These two experiments ran concurrently with liquid hydrogen (LH2) and LD2 targets interchanged daily to minimize systematic uncertainties. Scattered electrons were detected in the left High Resolution Spectrometer (HRS) of Hall A Alcorn et al. (2004), which determined accurately the electron scattering kinematics centered at $x_{B}=0.36$ and $Q^{2}=1.75\text{ GeV}^{2}$. The two photons from the $\pi^{0}$ decay were detected in an electromagnetic calorimeter composed of a $13\times 16$ array of $3\times 3\times 18.6\text{ cm}^{3}$ PbF2 crystals, resulting in a $[0,2\pi]$ coverage in $\phi$ and $[0,0.25]$ GeV2 range in $t^{\prime}=t_{min}-t$. A 0.6 ns $\pi^{0}$-electron coincidence time resolution was achieved by means of a 1 GHz flash ADC system in each calorimeter channel. The calibration of the calorimeter was performed with elastic H($e$,$e_{\text{Calo}}^{\prime}p_{\text{HRS}}$) data from dedicated runs in which the scattered electrons were detected in the calorimeter, with energy predetermined by the kinematics of the elastic recoil proton in the HRS. The calorimeter calibration was monitored continuously * a postiori* by tracking the 2-photon invariant mass $m_{\gamma\gamma}=\sqrt{(q_{1}+q_{2})^{2}}$ and the $ep\to e\pi^{0}X$ missing mass squared $M_{X}^{2}=(q+p-q_{1}-q_{2})^{2}$. Exclusive $\pi^{0}$ electroproduction events are selected for each $(t^{\prime},\phi)$ bin by applying a bidimensional cut:

[TABLE]

where $\sigma_{m_{\gamma\gamma}}$ is the resolution of the reconstructed $\pi^{0}$ invariant mass, and the empirical factor $C=13$ GeV takes into account the natural correlation between the invariant mass and missing mass originating from energy fluctuations in the calorimeter. Fig. 2 shows the corrected missing mass squared $M_{X}^{\prime 2}$ obtained at $E$=4.455 GeV for LH2 and LD2 data sets where $M_{X}^{2}$ is calculated with a target corresponding to a nucleon at rest. Accidentals were subtracted from these spectra and the LH2 data were normalized to the same integrated luminosity as the LD2 data.

The average momentum transfer to the target $\langle|\vec{\Delta}|\rangle=\langle|\vec{q}-\vec{q^{\prime}}|\rangle$ in the kinematics of this experiment is much larger than the average $np$ relative momentum in the deuteron wavefunction $\langle|\vec{p_{F}}|\rangle$ . Below the threshold for the production of a second pion, the impulse approximation is expected to accurately describe the exclusive D$(e,e^{\prime}\pi^{0})X$ yield, with $X=np\oplus d$. Thus we write the cross section as the sum of the coherent elastic channel $d(e,e^{\prime}\pi^{0})d$ and two incoherent quasi-elastic contributions:

[TABLE]

We subtract the $p(e,e^{\prime}\pi^{0})p$ yield from the deuterium data by normalizing our H$(e,e^{\prime}\pi^{0})X$ data to the luminosity of the LD2 data. The Fermi-momentum $\vec{p_{F}}$ of bound protons inside the deuteron is statistically added to the LH2 data following the distribution given in Lacombe et al. (1980) since this effect is intrinsically present in the $M_{X}^{\prime 2}$ spectrum of the LD2 data. The result of the subtraction of the H$(e,e^{\prime}\pi^{0})X$ data from the D$(e,e^{\prime}\pi^{0})X$ yield is shown in Fig. 2. The $d(e,e^{\prime}\pi^{0})d$ and $n(e,e^{\prime}\pi^{0})n$ channels are in-principle kinematically separated by $\Delta M_{X}^{\prime 2}=t(1-M/M_{d})\approx t/2$ where $M_{d}$ is the deuteron mass. This kinematic shift is exploited in the procedure described below to separate the contributions of the quasi-free neutron and coherent deuteron channels in the total $\pi^{0}$ electroproduction cross section.

Fig. 2 illustrates that the exclusive $\pi^{0}$ electroproduction events are primarily localized below the production threshold for a second pion: $M_{X}^{\prime 2}<(M+m_{\pi})^{2}\approx 1.15\text{ GeV}^{2}$. However, we apply a nominal cut of $M_{X}^{\prime 2}<0.95\text{ GeV}^{2}$ to minimize any contamination of inclusive events that might arise from resolution effects. The resulting events below this $M_{X}^{\prime 2}$ cut are divided into $12\times 2\times 5\times 30$ bins in $\phi$, $E$ , $t^{\prime}$ and $M_{X}^{\prime 2}$ respectively. The first two variables allow the independent extraction of the four structure functions of the $\pi^{0}$ electroproduction cross section while the binning in $M_{X}^{\prime 2}$ enables the separation of the $d(e,e^{\prime}\pi^{0})d$ and $n(e,e^{\prime}\pi^{0})n$ contributions.

A Monte-Carlo simulation of the experimental setup is based on the Geant4 toolkit Agostinelli and others (GEANT4 collaboration). It includes both external and real internal radiative effects based on calculations described in  Vanderhaeghen et al. (2000). The virtual internal effects are applied as a global correction factor to the extracted cross sections. The HRS acceptance is modeled by an R-function Rvachev (2001) defining correlated multi-dimensional boundaries. Only the overlapping ($Q^{2}$,$x_{B}$) phase-space between the two beam energy settings is considered. The calorimeter energy resolution in the $p(e,e^{\prime}\pi^{0})p$ simulation is smeared to match the $M_{X}^{\prime 2}$ distribution in each ($E$, $t^{\prime}$, $\phi$) bin of the LH2 data. These bin-by-bin resolution smearing factors are also applied to the $n(e,e^{\prime}\pi^{0})n$ and $d(e,e^{\prime}\pi^{0})d$ simulated data. The Fermi-smearing described above is also applied to the simulated $n(e,e^{\prime}\pi^{0})n$ yields. The systematic uncertainty of this smearing procedure as well the asymmetric systematic uncertainty originated from the inclusive yield under the $M_{X}^{\prime 2}$ cut are evaluated by varying the cut applied around its nominal value. They are found to be bin-dependent and were added quadratically to the $3.1\%$ normalization uncertainty listed in Defurne et al. (2016).

For each $t^{\prime}$ we fit the simulated yield to the experimental distributions of the $\phi$- and $M_{X}^{\prime 2}<\text{cut}$-bins. To wit, we minimize the $\chi^{2}$:

[TABLE]

where $N^{exp}_{i}$ ($N^{sim}_{i}$) is the number of experimental (simulated) events in bin $i$ and $\delta^{exp}_{i}$ is the corresponding uncertainty. The kinematic factors appearing in Eq. (1) are convoluted with the experimental acceptance and resolution in the computation of $N^{sim}_{i}$. The eight cross-section structure functions $d\sigma_{\Lambda}^{n,d}(t^{\prime})/dt$ ($\Lambda=T,L,LT,TT$) which define $N^{sim}_{i}$ are the free parameters of the fit for each $t^{\prime}$ bin.

Fig. 3 shows the measured $\phi$-dependent photo-absorption cross section for both beam energies and for the lowest $t^{\prime}$ bin. The $d^{2}\sigma^{n}/{dtd\phi}$ cross section is almost independent of the beam energy indicating a dominance of the transverse response. The ${d^{2}\sigma^{d}}/{dtd\phi}$ cross section is found negligible within uncertainties for all $\phi$ bins. The fit to the $M_{X}^{\prime 2}$-distribution is shown in Fig. 2 which also illustrates that the LD2–LH2 yield is dominated by the neutron contribution in the exclusive region. In Fig. 4, we display $\phi$-independent cross section $d\sigma_{T}+\epsilon d\sigma_{L}$ for the two beam energies, separated into the fitted quasi-free neutron and coherent deuteron channels. The highest $t^{\prime}$ bin is used in the analysis to treat bin migration effects and is not shown herein. The figure again shows the clear separation of the neutron signal. The coherent deuteron cross sections are found to be very small and compatible with theoretical calculations based on chiral-even deuteron GPDs, which predict cross-section values smaller than 1 nb/GeV2 in similar kinematics Cano and Pire (2004).

Fig. 5 shows the four extracted structure functions for the neutron and the deuteron as functions of $t^{\prime}$. The neutron cross sections are dominated by $d\sigma_{T}^{n}/dt$ and $d\sigma_{TT}^{n}/dt$, while the terms involving a longitudinal response are compatible with zero within uncertainties and are in good agreement with previous results off a proton target at the same kinematics Defurne et al. (2016). The neutron measurements are compared to a calculation based on both quark helicity-conserving GPDs and quark helicity-flip (transversity) GPDs Goloskokov and Kroll (2011), and show good agreement for all structure functions, with a slight overestimation of $|d\sigma_{TT}^{n}/dt|$. The experimental $d\sigma_{L}^{n}/dt$ term is also compatible with the VGG model Vanderhaeghen et al. (1999) based on chiral-even GPDs, which predicts $d\sigma_{L}^{n}/dt<4\text{ nb/GeV}^{2}$ for all $t^{\prime}$ bins.

Together with previous measurements of $d\sigma_{T}/dt$ and $d\sigma_{TT}/dt$ on the proton Defurne et al. (2016) and extensive unseparated measurements before Fuchey et al. (2011); Bedlinskiy et al. (2012, 2014), these new results provide strong support to the exciting idea that transversity GPDs can be accessed via neutral pion electroproduction in the high $Q^{2}$ regime.

Within the modified factorization approach of Goloskokov and Kroll (2011), $d\sigma_{T}/dt$ and $d\sigma_{TT}/dt$ are functions of $\langle H_{T}\rangle$ and $\langle\bar{E}_{T}\rangle$, which are convolutions of the elementary $\gamma^{*}q\to q^{\prime}\pi^{0}$ amplitude with the transversity GPDs $H_{T}$ and $\bar{E}_{T}=2\widetilde{H}_{T}+E_{T}$:

[TABLE]

In these equations $\Lambda(Q^{2},x_{B})$ is a phase space factor Bedlinskiy et al. (2014) and $\xi\simeq x_{B}/(2-x_{B})$ is the skewness variable. For a proton and a neutron target, the quark-flavor structures of $\left|\left<H_{T}\right>\right|^{2}$ (neglecting strange quarks) are:

[TABLE]

with similar equations for $\left|\left<\bar{E}_{T}\right>\right|^{2}$. The different flavor weights of the proton and neutron targets allow us to separately determine $\left|\left<H_{T}^{u}\right>\right|$ and $\left|\left<H_{T}^{d}\right>\right|$ (similarly $\left|\left<\bar{E}_{T}^{u}\right>\right|$ and $\left|\left<\bar{E}_{T}^{d}\right>\right|$) by combining the data we report herein and $\pi^{0}$ electroproduction cross sections on the proton measured at the same kinematics as in Defurne et al. (2016). The unknown relative phase between the $u$ and $d$ convolutions is treated as a systematic uncertainty in the separation. The flavor-separated results assuming no relative phase between the $u$ and $d$ convolutions are presented in Fig. 6, with the bands indicating their variation when the phase takes all possible values between 0 and $\pi$. This phase could be resolved with exclusive $p(\gamma^{\ast},\eta p)$ data in the same kinematics. Fig. 6 shows that the magnitudes of the $u$-quark convolutions are larger than the $d$-quark convolutions for all $t$ bins. The results in Fig. 6 also demonstrate that the $u$-quark nucleon helicity non-flip term $\left|\left<\bar{E}_{T}^{u}\right>\right|$, is larger than the nucleon helicity flip term $\left|\left<H_{T}^{u}\right>\right|$. The comparison to the Goloskokov-Kroll model Goloskokov and Kroll (2011) shows good agreement for $\left|\left<H_{T}\right>\right|$ for both quark flavors but an underestimation for $\left|\left<\bar{E}_{T}^{u}\right>\right|$. The GPD $H_{T}$ parametrization is constrained in the forward limit by the transversity parton distributions. However, no similar experimental constraint is available for $\bar{E}_{T}$. The constraints on $\bar{E}_{T}$ are mainly taken from lattice QCD calculations Gockeler et al. (2007).

In conclusion, we have separated the four unpolarized structure functions of $\pi^{0}$ electroproduction off the neutron at $Q^{2}$=1.75 GeV2 and $x_{B}$=0.36 in the $t^{\prime}$ range $[0,0.2]$ GeV2. Similar measurements are obtained for coherent $\pi^{0}$ electroproduction off the deuteron at $x_{d}$=0.18. The latter are found to be very small and according to theoretical expectations. Neutron results show a dominance of the transverse response confirming the transversity GPD approach for the description of this process. By combining neutron and proton results, we have performed the first flavor decomposition of the $u$ and $d$ quark contributions to the cross section.

We thank P. Kroll, S. Goloskokov, M. Guidal, M. Vanderhaeghen and B. Pire for valuable information about their work and providing the results of their models. We acknowledge essential work of the JLab accelerator staff and the Hall A technical staff. This work was supported by the Department of Energy (DOE), the National Science Foundation, the French Centre National de la Recherche Scientifique, the Agence Nationale de la Recherche, the Commissariat à l’énergie atomique et aux énergies alternatives and P2IO Laboratory of Excellence. Jefferson Science Associates, LLC, operates Jefferson Lab for the U.S. DOE under U.S. DOE contract DE-AC05-060R23177.