Backward-angle Exclusive pi0 Production above the Resonance Region
W.B. Li, G.M. Huber, J.R. Stevens, K. Semenov-Tian-Shansky, L. Szymanowski, B. Pire, M. Amaryan, D. Androic, K. Aniol, D. Armstrong, T. Averett, C. Ayerbe Gayoso, W. Boeglin, M. Boer, A. Camsonne, J. Chen, S. Covrig Dusa, W. Deconinck, M. Defurne, F. Delcarro, M. Diefenthaler, S. Diehl, M. Elaasar, C. Fanelli, S. Fegan, E. Fuchey, D. Gaskell, O. Hansen, F. Hauenstein, D. Higinbotham, A. Hiller Blin, A. Hurley, C. Hyde, K. Joo, M. Junaid, N. Kalantarians, S. Kay, M. Khachatryan, P. King, V. Kumar, D. Lersch, L. Lorenti, P. Markowitz, M. McCaughan, A. Mkrtchyan, H. Mkrtchyan, G. Niculescu, I. Niculescu, Z. Papandreou, R. Paremuzyan, K. Park, D. Paudyal, J. Roche, A. Rodas, B. Sawatzky, A. Schertz, G. Smith, I. Strakovsky, V. Tadevosyan, A. Usman, H. Voskanyan, C. Yero
aa r X i v : . [ nu c l - e x ] A ug A Jefferson Lab PAC 48 Experiment Proposal
Backward-angle Exclusive π Production above the Resonance Region
Wenliang Li (Spokesperson and contact person), ∗ Justin Stevens (Spokesperson), DavidArmstrong, Todd Averett, Andrew Hurley, Lydia Lorenti, Arkaitz Rodas, and Amy Schertz
College of William and Mary, Williamsburg, VA, USA
Garth Huber (Spokesperson), Muhammad Junaid, Stephen Kay,Vijay Kumar, Zisis Papandreou, Dilli Paudyal, and Ali Usman
University of Regina, Regina, SK Canada
Kirill Semenov-Tian-Shansky
National Research Centre Kurchatov Institute: PetersburgNuclear Physics Institute, RU-188300 Gatchina, Russia andSaint Petersburg National Research Academic University of theRussian Academy of Sciences, RU-194021 St. Petersburg, Russia
Bernard Pire
CPHT, CNRS, ´Ecole Polytechnique, IP Paris, 91128-Palaiseau, France
Lech Szymanowski
National Centre for Nuclear Research (NCBJ), 02-093 Warsaw, Poland
Alexandre Camsonne, Jian-Ping Chen, Silviu Covrig Dusa, Filippo Delcarro,Markus Diefenthaler, Dave Gaskell, Ole Hansen, Doug Higinbotham,Astrid Hiller Blin, Mike McCaughan, Brad Sawatzky, and Greg Smith
Jefferson Lab, Newport News, Virginia, USA
Arthur Mkrtchyan, Vardan Tadevosyan, Hakob Voskanyan, and Hamlet Mkrtchyan
A. Alikhanyan National Science Laboratory (Yerevan Physics Institute), Yereven, Armenia
Stefan Diehl, Eric Fuchey, and Kyungseon Joo
University of Connecticut, Mansfield, Connecticut, USA
Werner Boeglin, Mariana Khachatryan, Pete E. Markowitz, and Carlos Yero
Florida International University, Miami, Florida, USA
Moskov Amaryan, Florian Hauenstein, and Charles Hyde
Old Dominion University, Norfolk, VA, USA
Gabriel Niculescu and Ioana Niculescu
James Madison University, Harrisonburg, Virginia, USA
Paul King and Julie Roche
Ohio University, Athens, Ohio, USA
Darko Androi´c
University of Zagreb, Zagreb , Croatia
Konrad Aniol
California State University, Los Angeles, California, USA
Marie Boer
University of New Hampshire, Durham, New Hampshire, USA andVirginia Polytechnic Institute and State University, Blacksburg, Virginia, USA
Wouter Deconinck
University of Manitoba, Winnipeg, Manitoba, Canada
Maxime Defurne
CEA, Universit´e Paris-Saclay, Gif-sur-Yvette, France
Mostafa Elaasar
Southern University at New Orleans, New Orleans, Louisiana, USA
Cristiano Fanelli
Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
Stuart Fegan
University of York, Heslington, York, UK
Carlos Ayerbe Gayoso
Mississippi State University, Starkville, MS, USA
Narbe Kalantarians
Virginia Union University, Richmond, VA, USA
Daniel Lersch
Florida State University, Tallahassee, Florida, USA
Rafayel Paremuzyan
University of New Hampshire, Durham, New Hampshire, USA
Kijun Park
Hampton University Proton Therapy Institute, Hampton, Virginia, USA
Igor Strakovsky
The George Washington University, Washington, DC, USA (Dated: August 26, 2020)The proposed measurement is a dedicated study of the exclusive electroproduction pro-cess, H ( e, e ′ p ) π , in the backward-angle regime ( u -channel process) above the resonanceregion. Here, the produced π is emitted 180 degrees opposite to the virtual-photon momen-tum (at large momentum transfer). This study also aims to apply the well-known Rosenbluthseparation technique that provides the model-independent (L/T) differential cross-section atthe never explored u -channel kinematics region ( − t = − t max , − u = − u min ).Currently, the “soft-hard transition” in u -channel meson production remains an interest-ing and unexplored subject. The available theoretical frameworks offer competing inter-pretations for the observed backward-angle cross section peaks. In a ”soft” hadronic Reggeexchange description, the backward meson production comes from the interference betweennucleon exchange and the meson produced via re-scattering within the nucleon. Whereasin the “hard” GPD-like backward collinear factorization regime, the scattering amplitudefactorizes into a hard subprocess amplitude and baryon to meson transition distribution am-plitudes (TDAs), otherwise known as super skewed parton distributions (SuperSPDs). BothTDAs and SPDs are universal non-perturbative objects of nucleon structure accessible only through backward-angle kinematics.The separated cross sections: σ T , σ L and ( σ T / σ L ) ratio at Q = , provide a directtest of two predictions from the TDA model: σ T ∝ /Q and the σ T ≫ σ L in u -channelkinematics. The magnitude and u -dependence of the separated cross sections also provide adirect connection to the re-scattering Regge picture. The extracted interaction radius (from u -dependence) at different Q can be used to study the soft-hard transition in the u -channelkinematics. The acquisition of these data will be an important step forward in validating theexistence of a backward factorization scheme (TDA and SuperSPD) of the nucleon structurefunction and establishing its applicable kinematic range. Contents
I. Introduction II. Summary of Backward-angle Physics from JLab 6 GeV π Electroproduction at Hall A and C 8B. High − t charged π Electroproduction at Hall B 10C. BSA on High- t charged π Electroproduction at Hall B 11D. Backward ω Electroproduction at Hall C 14
III. Theoretical Context for Backward-angle π Electroproduction π N TDAs 212. Two Predictions from TDA Collinear Factorization 24C. Complementary Objective: the Hadronic Approach 24
IV. Studying TDA through VCS and DEMP from JLab 12 GeV to EIC π Electroproduction 29
V. Experiment Kinematics and Configuration ∗ E-mail: [email protected]
A. L/T/LT/TT Separation 30B. Choice of Kinematics 34C. “Soft-hard” Transition through u -channel Phenomenology Study 36D. Beam Spin Asymmetry Measurement 38E. W Scaling Correction and Q = 2 GeV , W = 3 GeV Setting 39F. Singles Rate Estimation 40G. Kinematic Checks and Normalization with Elastic Scattering 431. Single Arm Elastic Checks 432. Elastic Coincidence Checks 44H. Particle Identification 451. Critical Hardware Replacement 48I. Physics Background Contribution 48J. Non-Physics Background 50K. Systematic Errors 51L. Projected Error Bars, Rates and Time Estimation 53
VI. Closing Remarks A. Monte Carlo model of Deep Exclusive π Production in u -channel B. Monte Carlo model of γ Production in u -channel C. Differences and Connections to Other Approved π Measurements D. Further Details on u Channel Workshop in September 2020 u Channel Workshop Agenda and Speakers List 62
References I. INTRODUCTION
In this proposal, we present a unique opportunity to access deep exclusive meson produc-tion (DEMP) in the backward-angle ( u -channel kinematics) regime. The primary experimentalobservable involves exclusive π electroproduction: H ( e, e ′ p ) π , with a kinematic coverage of < Q < . GeV at fixed x B = 0 . and W > GeV. Since the π is produced al-most at 180 ◦ opposite to the direction of the virtual-photon momentum (corresponding to ex-treme backward angles), the Mandelstam variable for crossed four-momentum transfer squared is u ′ = u − u min ≈ GeV . At selected Q settings, the full L/T/LT/TT cross section separationwill be performed. Due to its unusual kinematics, the backward-angle meson production reactionis often referred to as a “knocking a proton out of a proton process”, as shown in Fig. 1. e e ′ qqqqqp pπ γ ∗ qqqqq FIG. 1: Cartoon demonstration of a “knocking a proton out of a proton process” above theresonance region ( √ s = W > GeV) [5]. In this case, a backward π is produced nearly at rest.The proposed measurement uses the standard Hall C equipment, polarized electron beam up to70 µ A (at standard accelerator gradient settings at the time of running) and standard unpolarizedliquid hydrogen (LH ) target. Since the produced π are not directly detected, the missing massreconstruction method will be applied. This technique permits access to a unique backward-anglekinematics region which was previously unexplored. The L/T separation technique used here isidentical to the ones used successfully by many previous Hall A and C experiments during the6 GeV era of CEBAF, an example being the pion form factor experiment [1, 2].The most important goals of the proposed measurement are to:1. Determine if exclusive π electroproduction has a significant backward-angle peak, as itwas demonstrated recently in exclusive ω electroproduction [3], where backward angle datafrom Hall C were combined with forward-angle data from CLAS. Here, we have chosenkinematics compatible with E12-13-010 [4] (forward-angle) and CLAS 12 (forward andwide angle) to facilitate a complete coverage in − t for the π production at certain W and Q settings. A complete − t evolution would reveal a forward-angle peak (at t min ), a wideangle plateau ( − t ≈ − u ) and a backward-angle peak (at t max ).2. A phenomenology study of extracting the u -dependence for the separated cross sectionswould be the good handle to determine transverse size of interaction, which can be used tostudy the transition from a “soft” Regge-exchange type picture (transverse size of interactionis of order of the hadronic size) to the “hard” QCD regime (transverse size of interaction ≪ hadronic size). See further detail in Sec. V C.3. Assuming the backward-angle peak is present, as expected, the next important objectiveof the proposed measurement is to demonstrate the (model independent) dominance of thetransverse cross section ( σ T ) over the longitudinal: σ T > σ L , at < Q < GeV abovethe resonance region ( W > GeV).4. The last objective is to measure the Q -dependence of the σ T cross section at fixed x B =0 . .The outcome of the measured result is a critical step towards finding the applicable factoriza-tion region in the backward-angle ( u -channel) kinematic regime. These scientific motivations arefurther elaborated in Sec. III B. Additional to the main objectives, there are two potential oppor-tunistic studies which will come for free with the planned measurements:1. As part of the physics background to π , backward-angle Virtual Compton Scattering (VCS)above the resonance region has generated high community interest. An exploratory effortwith proposed data into this challenging measurement will gain important experimental in-sights which may lead to a dedicated study of this interesting process. See more detail inSec V I.2. By default, CEBAF offers polarized electron beam, this provides an opportunity to studythe Beam Spin Asymmetry (BSA) with the proposed data. See detail in Sec. V D.It is also worth mentioning that the proposed π measurement was submitted to PAC 46 asa letter of intent with reference number LOI-12-18-005. In the final PAC report, the committeemembers acknowledged the uniqueness of the proposed study and the fact that Jefferson Lab is thebest venue to carry out such study. In addition, the feedback by experimental experts regarded themeasurements as technically straightforward. It is also important to remember that the u -channelmeson production reaction is not a new concept among past experiments at JLab, Sec. II providesa brief summary of these experimental efforts. In addition, the difference between the proposed π measurement and other approved Hall C π experiments is addressed in Sec. C.We also would like to emphasize that the proposed π measurement is not an isolated measure-ment; it marks the beginning of a comprehensive plan to study the backward-angle ( u -channel)factorization scheme of nucleon structure. This plan involves JLab 12 GeV measurements, collab-orative efforts with the P ANDA experiment and the future EIC. See Sec. IV for full details.
II. SUMMARY OF BACKWARD-ANGLE PHYSICS FROM JLAB 6 GEV
At Jefferson Lab, direct or indirect measurements of exclusive meson electroproduction at largescattering angles are not a new concept. Here, indirect measurement implies the usage of themissing mass reconstruction technique. During the 6 GeV era, there have been a few examplesof such studies. In this section, we present a short overview of some of the important pioneeringstudies of backward-angle physics.
A. Backward VCS and π Electroproduction at Hall A and C
Since the early stage of JLab (1993), backward angle H ( e, e ′ p ) γ and H ( e, e ′ p ) π measure-ments were attempted by a dedicated Hall A experiment E93-050 [6–8] in the nucleon resonanceregion. E93-050 used the 4 GeV electron beam colliding with a liquid hydrogen target, wherea pair of High Resolution Spectrometers (HRSs) were used to detect the scattered electron andproton in coincidence. The forward-going proton was detected in parallel kinematics and the ‘re-coil’ π , γ was emitted at backward angle at low momentum. The missing mass reconstructiontechnique was used to reconstruct the final state γ as well as π events. An example of the recon-structed missing mass squared distribution from E93-050 is shown in Fig. 2.The physics objective was to access the Compton photon scattered at backward angles in thenucleon resonance region ( S and D ), whereas the π was detected as the dominant background.Thanks to the good particle momentum resolution of the HRSs, separating the γ and π peaks wasa relatively easy task (as shown in Fig, 2).E93-050 (and later E00-110) obtained a great deal of information about VCS, even when theFIG. 2: Squared missing mass M X for an experimental setting W = 1 . GeV is shown in plot (a).The zoomed distribution around γ peak is shown in (b). These plots were published in Ref. [9].BH amplitude was larger than VCS (DVCS-BH interfere is at the amplitude level). Also, ( e, e ′ p ) γ and ( e, e ′ p ) π cross sections were published in Ref. [7, 9].In 2008, Laveissiere, et al., published the first measurement of the backward-angle VCS crosssection with the data from E93-050 [9]. This experiment was performed at Q = 1 GeV in thenucleon resonance region from threshold to W = 1 . GeV. Despite the differences in physicsmotivations, Experiment E93-050 provides important insight to this proposal. The relative heightand width of γ and π peaks from this measurement are useful benchmarks for estimating crosssections and determining the mass resolution requirements.In the 12 GeV era, the backward-angle VCS program is further explored by E12-15-001 at HallC [10]. The measurement aims to extract the two scalar Generalized Polarizabilities of the protonin the range of Q = 0 . to . GeV , near the ∆(1232) resonance region. Most interestingly, theequipment configuration (including 10 cm target cell) and missing mass reconstruction techniqueused by E12-15-001 are identical to the one used in this proposal. The partial completion ofE12-15-001 (in 2019) is a great validation to the experimental methodology.0 B. High − t charged π Electroproduction at Hall B
The CLAS detector, in comparison to the Halls A and C spectrometers, offers the great advan-tage of a wide angular acceptance. Since the cross section for a given electroproduction reactionfalls exponentially as a function of − t (a larger − t value corresponds to a wider scattering angle),it is difficult to determine the detector efficiency for wide scattering angles. After years of carefulstudy, K. Park et al. published results for exclusive π + electroproduction, H ( e, e ′ π + ) n , near thebackward angle above the resonance region [11]. The Q coverage is . < Q < . GeV , at W ∼ − u = . The publication of this result was an important step for u -channelphysics. Evidence of Q -scaling (particularly for Q > GeV ) was observed, and it is consis-tent with the prediction of the GPD-like Transition Distribution Amplitude (TDA) factorizationscheme at a much lower Q range than originally expected. This is demonstrated by the closeagreement between the blue TDA band and the unseparated σ U in Fig. 3. -40-200204060801.5 2 2.5 3 3.5 4Q (GeV ) s U , s LT , s TT ( nb / s r ) FIG. 3: The structure functions σ u , σ TT and σ LT as a function of Q . The bands refer to modelcalculations of σ u in the TDA description with different nucleon DA models; dark blue band:COZ [13] N DA model, light blue band: KS [14], black band: BLW NNLO [15]. This plot waspublished in Ref. [11].1
C. BSA on High- t charged π Electroproduction at Hall B
Recently [12], the CLAS collaboration reported the results of the analysis of hard exclusivesingle pion ( π + ) electroproduction with CEBAF at GeV aiming on the study of the beam-spinasymmetry of the reaction above the resonance region.The beam spin asymmetry for the reaction is defined as
BSA (cid:0) t, φ, x B , Q (cid:1) = dσ + − dσ − dσ + + dσ − = A sin φLU sin φ A cos φUU cos φ + A cos 2 φUU cos 2 φ , (1)where σ ± is the differential cross section for each beam helicity state ( ± ). For the positive/negativehelicity the spin is parallel/anti-parallel to the beam direction. The subscripts ij represent the lon-gitudinal ( L ) or unpolarized ( U ) state of the beam and the target, respectively. φ is the azimuthalangle between the electron scattering plane and the hadronic reaction plane, on which the dif-ferential cross sections depend. The extraction of the beam spin asymmetry (1) provides accessto the A sin φLU moment. It probes the interference between the amplitudes for longitudinal ( L ) andtransverse ( T ) virtual-photon polarizations and is proportional to the polarized structure function σ LT ′ (the symbol ′ signifies the structure function is the backward-angle): A sin φLU = p ε (1 − ε ) σ LT ′ σ T + εσ L , (2)where ε is the polarization parameter of the virtual- photon. These measurements were performedwith nearly full coverage from forward to backward angles in the center-of-mass pion scatteringangle. As shown in Fig. 4, the kinematic region for the extraction of A sin φLU was extended up to − t = 6 . , which is close to the maximal accessible − t value for given kinematical setup.The presented data provides important constraints for the development of a factorized reactionmechanism describing the complete kinematic regime, including the near-forward regime, with apossible collinear factorized description in terms of GPDs and pion DAs, the intermediate kine-matical regime and the near-backward kinematical regime, with the eventual collinear factorizeddescription in terms of πN TDAs and nucleon DAs.In particular, the sign of A sin φLU in near-forward kinematics (GPD region) is clearly positive(Fig. 4). However, a sign change of A sin φLU has been observed around θ CM = 90 ◦ , bringing theBSA clearly negative in the backward hemisphere, and quite small in near-backward kinematics.2This suggests a completely distinct reaction mechanism in the backward regime and hints at theleading twist dominance in the small ( − u ) domain for Q < GeV , which is a central feature ofthe u -channel TDA factorization mechanism.The data presented in Ref. [12] provide important constraints for the development of reactionmechanisms that describe the complete kinematic regime including GPDs and TDAs, as well asthe intermediate regime. Fig. 5 shows A sin φLU as function of Q (top) and x B (bottom) for pionsgoing in the near-forward (left) and near-backward (right) kinematics.FIG. 4: A sin φLU as a function of − t . The data are binned in − t and integrated over the complete Q distribution ranging from GeV to . GeV and x B ranging from . to . . The shaded arearepresents the systematic uncertainty (see detailed discussion in Ref. [12]).Here, we focus on the impact of these measurements in the near-backward kinematical regime,where a description in terms of πN TDAs and nucleon DAs might be applied. Assuming thecollinear factorized description of the single pion electroproduction in the near-backward kine-matics regime in terms of πN TDAs and nucleon DAs, the cross section σ LT turns to be a sublead-ing twist- effect. Therefore, the expression for the BSA involves the twist- nucleon DAs andnucleon-to-pion TDAs. • For the leading twist transverse amplitude in terms of πN TDAs, H tw=3 i , and nucleon DAs, φ tw=3 i , we employ the notation h H tw=3 i φ tw=3 j i . • To describe the next-to-leading twist longitudinal amplitude we need to introduce πN TDAs, H tw=4 i , and nucleon DAs, φ tw=4 i [16, 17].3FIG. 5: A sin φLU as function of Q (top) and x B (bottom) for pions going in the forward (left) andbackward (right) regions. The shaded area represents the systematic uncertainty (see details inRef. [12]).Then, to twist- accuracy the appropriate amplitude can be written as h H tw=4 i φ tw=3 j i + h H tw=3 i φ tw =4 j i . Therefore, the cross section σ LT ′ within the TDA framework can be written as: σ LT (cid:12)(cid:12) Backwardregime ≈ Im (cid:2) h H tw=3 i φ tw =3 j i (cid:0) h H tw =4 i φ tw =3 j i + h H tw =3 i φ tw =4 j i (cid:1) ∗ (cid:3) . (3)A complete theoretical study of this twist- longitudinal amplitude is not yet available, but isanticipated to be quite similar to the analysis done in Ref. [18] for the calculation of the Paulinucleon form factor. From Fig. 5 (top right), one could see the size and the sign flip in the Q behavior of BSA fits (despite large error bars) in the backward angle, when compared to theforward-angle counterpart in Fig. 5 (top-left), this is similar to the prediction by the twist countingrules of collinear TDA/DA factorization mechanism in the near-backward regime. These findings4are further elaborated in Ref. [12].A dedicated higher precision BSA measurement in a larger range of Q will be enabled with theupgraded GeV CEBAF accelerator at JLab. This definitely would boost the theoretical studies,needed to provide the still lacking quantitative estimates of the effect.
D. Backward ω Electroproduction at Hall C
The recently published results from Hall C [3, 19] demonstrated that the missing mass recon-struction technique, in combination with operating the Hall C high precision spectrometers incoincidence mode, can be used to extract the backward-angle ω cross section reliably through theexclusive reaction H ( e, e ′ p ) ω , while performing a full L/T separation. The experiment has central Q values of 1.60 and 2.45 GeV , at W = 2 . GeV. There was significant coverage in φ and ǫ ,which allowed separation of σ T,L,LT,T T . The data set has a unique u coverage near − u ∼ , whichcorresponds to − t > GeV .The extracted cross sections (red crosses) show evidence of a backward-angle peak for ω exclu-sive electroproduction; angular distributions at Q = 1 . and 2.35 GeV are shown in Fig. 6. Theforward-angle ( t -channel) peak from the CLAS-6 data [20] is also shown. Previously, the the ap-pearance of both forward and backward-angle peaks was only observed in meson photoproductiondata [21, 22]. Furthermore, the Regge model description of Laget [23], involving re-scattering,provides a natural description of both the magnitude and slope of the observed backward-anglepeak (discussed further in Sec. III C). The investigation whether such a backward-angle peak alsoexists in π electroproduction, and whether it persists over a wide Q range, is the first goal of thisproposal.The extracted dσ L /dt and dσ T /dt from the Hall C p ( e, e ′ p ) ω data are shown versus − u inFig. 7. The data are compared to the TDA model prediction [25]. At Q = , the TDApredictions are within the same order of magnitude as the data, whereas at Q = , theTDA model overpredicts the data by a factor of ∼
10. This is very similar to the behavior shownfor the CLAS data in Fig. 3. Together, the datasets suggest that the backward-angle collinearfactorization (TDA model) regime may begin to apply around Q ≈ . The data proposedhere would go a long way to confirm or reject whether this interpretation is correct.The most important finding from the backward-angle ω analysis was the demonstration of σ T dominance over σ L at Q = 2 . GeV , see Fig. 8(b) for the σ L /σ T cross section ratio as function5 - - -
10 1 ] b / G e V m [ d t s d JML18JML04CLASFpi-2 (Scaled)TDA COZTDA KS = 1.75 GeV Q = 2.48 GeV, W , w p+ fi +p * g ] [GeV -t = 2.35 GeV Q =2.47 GeV, W , w p+ fi +p * g FIG. 6: Total differential cross section, dσ u /dt versus − t for W =2.48 GeV, Q =1.75 (left) GeVand W =2.47 GeV, Q =2.35 GeV (right). The black dots are published CLAS results [20]. Thered crosses are reconstructed σ u using σ T and σ L from Hall C (scaled to same kinematics) [3, 19],the systematic error bands are shown in blue. The magenta and blue dashed lines represents theprediction of the hadronic Regge-based model, without [23], and with ρ − N and ρ − ∆ unitaryrescattering (Regge) cuts [24]. This plot was published in Ref. [3].of Q . Note that this was predicted by the TDA framework. As the JLab 12 GeV experiments canreach higher Q values, the TDA formalism must be carefully studied and tested in more mesonchannels. An example of further study is this proposed π meson measurement. III. THEORETICAL CONTEXT FOR BACKWARD-ANGLE π ELECTROPRODUCTION
In the framework of the e - p scattering representation, the exclusive π electroproduction H ( e, e ′ p ) π reaction can be written as e ( k ) + p ( p ) → e ′ ( k ′ ) + π ( p π ) + p ′ ( p ) . (4)If the virtual-photon is considered as the projectile, then γ ∗ ( q ) + p ( p ) → π ( p π ) + p ′ ( p ) . (5)6 - -
10 110 ] b / G e V m [ L s , T s =1.6 GeV Q ] [GeV -u =1.60) (Q T s =1.60) (Q L s =2.45) (Q T s =2.45) (Q L s TDA COZTDA KS =2.45 GeV Q FIG. 7: Separated differential cross section σ T versus − u for Q = 1 . GeV (left) and Q = 2 . GeV (right). The blue dashed and red solid and lines represent the TDA calculation[25] using the COZ [13] and KS [14] nucleon DA models, respectively. The green bands indicatecorrelated systematic uncertainties for σ T . The separated cross sections shown in this figure aredetermined at the Q and W values at individual − u bin, therefore cant not be used to determinethe u dependence directly. A scaling procedure is required when making the comparison at anominal set of Q and W values, such as in Fig. 6. This plot was published in Ref. [3, 19].Here, p and p ′ are the proton before and after the interaction; e and e ′ are the electron before andafter the interaction; γ ∗ is the space-like virtual-photon. The associated four-momentum for eachparticle is given inside the bracket. For this reaction, the Mandelstam variables are defined as s = ( p + q ) ; u = ( p π − p ) ; t = ( p − p ) . (6)In the case of the forward-angle ( t -channel) meson production process, the π is produced in thesame direction as the virtual-photon momentum q (known as the q -vector), and − t → t min (i.e.parallel kinematics). Correspondingly, the backward-angle ( u -channel) process produces π in theopposite direction as the q -vector, and − u → u min (anti-parallel kinematics).In the different kinematic regions, backward meson production can be explained using differentnucleon structure models. When the process is within the resonance region ( W < GeV), the u -channel process can be described using the nucleon fragmentation model which has a mild Q dependence [5]; when above the resonance region ( W > GeV) a more complicated parton7 ] [GeV Q ] b / G e V m [ T s , L s L s T s ] [GeV Q R a t i o T s L s T s / L s FIG. 8: (a) σ T and σ L as function of Q at u ′ = 0 GeV ; (b) σ L /σ T ratio as function of Q .Thisplot was published in Ref. [3, 19].based model is required to describe the Q n dependence. The latter is the research interest of thisproposal.Within the 6 GeV JLab kinematics coverage: W > GeV, Q < GeV , x B = 0 . , there aretwo independent models capable of describing the existing backward angle data. The first is a QCDGPD-like model known as the TDA [26] (also Skewed Distribution Amplitude in the pioneeringwork of Ref. [27]), which offers direct description of the individual partons within the nucleon;the other model, a hadronic Regge-based model known as the JML model [23, 24], that exploresmeson-nucleon dynamics of hadron production reactions. In this section, we introduce how abackward-angle π is produced according to both models and describe the benefits of studyingthem. L/T-separated cross sections can be calculated in both models and the leading twist TDAspredict σ L ∼ [28]. A. GPDs and Skewed Parton Distributions (SPDs)
Generalized parton distributions (GPDs) are a modern description of the complex internal struc-ture of the nucleon, which provides access to the correlations between the transverse position andlongitudinal momentum distribution of the partons in the nucleon. In addition, GPDs give accessto the orbital momentum contribution of partons to the spin of the nucleon [29, 30].Currently, there is no known direct experimental access to the information encoded inGPDs [31]. The prime experimental channels for studying the GPDs are through the DVCS and8DEMP processes [29]. Both processes rely on the collinear factorization (CF) scheme [32, 33]. Anexample DEMP reaction, γ ∗ p → pπ , is shown in Fig. 9(a). In order to access the forward-angleGPD collinear factorization regime ( γ ∗ p → pπ interaction), the kinematic variable requirementsare as follows: sufficiently high Q , large s , fixed x B and t ∼ [25, 31]. Here, the definition of“sufficiently high Q ” is process-dependent terminology. Based on the existing DIS data [34–36],the GPD physics has shown that the range of “sufficiently high Q ” lies between 1 and 5 GeV ;this is sometimes referred to as “early scaling” [37, 38].Under the collinear factorization regime, a parton is emitted from the nucleon GPDs ( N GPDs)and interacts with the incoming virtual-photon, then returns to the N GPDs after the interac-tion [31]. Studies [39, 40] have shown that perturbative calculation methods can be used to calcu-late the CF process (top oval in Fig. 9 (a)) and extract GPDs through factorization, while preservingthe universal description of the hadronic structure in terms of QCD principles. One limitation inthe GPD description of Fig. 9 (a) requires t ∼ t min , namely, the process defaults a fast-mesonand slow nucleon final state. Processes such as the one in this proposal could not be correctlyaccounted for by such a description.In a 2002 paper [27], M. Strikman and others presented an innovative approach to resolve thisissue. In it, they discuss the specific scenario when three valence quarks collapse to a small sizecolor singlet configuration in a nucleon, or of valence quark and antiquark in a meson. As a result,a fast proton and a slow meson are created. Such a setup implies the manifestation of a “cluster”structure within the initial state that co-existed with three valence quarks. See a visualization ofsuch a process in Fig. 1.Application of this knowledge to the context of Reaction 5: In the QCD description, the hardexclusive processes one needs to use generalized (skewed) parton distributions. In the case ofdescribing the N → N transitions and non-diagonal transitions like N → Λ , ∆ , the first typeof distributions are known as generalized parton distributions (GPDs), while in the case of non-diagonal transitions (latter case) the used term is skewed Parton Distributions (SPDs). Underthe case of extreme skewness, ξ → , (for appropriate quantum numbers of the current) onewould use super-SPDs to describe nucleon distribution amplitude [27]. Note that this proposed π measurement fulfills the super-SPD kinematics.Although the above stated qualitative prediction was not made regarding π electroproduction,one could still examine the predicted / (1 − t/m ) (where m ∼ GeV ) cross section dependenceusing the proposed data.9 B. Meson-Nucleon Transition Distribution Amplitude γ ∗ p pπtN GPD π D A Q CFs = W γ ∗ p pπuπN TDA N D A CFQ s = W FIG. 9: (a) shows the π electroproduction interaction ( γ ∗ p → pπ ) diagram under the(forward-angle) GPD collinear factorization regime (large Q , large s , fixed x B , fixed t ∼ ). N GPD is the quark nucleon GPD (note that there are also gluon GPD that is not shown). π DAstands for the vector meson distribution amplitude. The CF corresponds to the calculable hardprocess amplitude. (b) shows the (backward-angle) TDA collinear factorization regime (large Q ,large s , fixed x B , u ∼ ) for γ ∗ p → pπ . The πN TDA is the transition distribution amplitudefrom a nucleon to a vector meson. These plots were created based on the original ones publishedin Ref. [41].A few years after this pioneering work, B. Pire, L. Szymanowski, J.P Lansberg and K.Semenov-Tian- Shansky rediscovered and developed the QCD formalism appropriate to describethe backward electroproduction of photons or mesons. In their transition distribution amplitude(TDA) formalism, they called baryon-to-meson transition distribution amplitude ( π N TDA) thebackward analog of GPDs. TDAs describe the underlying physics mechanism of how the targetproton transitions into a π meson in the final state, shown as the gray oval in Fig. 9(b). One fun-damental difference between GPDs and TDAs is that the TDAs require three parton exchangesbetween πN TDA and CF.Relevant to this discussion is the definition of skewness. For forward-angle kinematics, in theregime where the handbag mechanism and GPD description may apply, the skewness is defined inthe usual manner, ξ t = p +1 − p +2 p +1 + p +2 , (7)where p +1 , p +2 refer to the light-cone plus components of the initial and final proton momenta inEqn. 5, calculated in the CM frame [42]. The subscript t has been added to indicate that this0skewness definition is typically used for forward-angle kinematics, where − t → − t min . In thisregime, ξ t is related to Bjorken- x , and is approximated by ξ t = x/ (2 − x ) , up to correctionsof order t/Q < [43]. This relation is an accurate estimate of ξ t to the few percent level forforward-angle electroproduction.In backward-angle kinematics, where − t → − t max and − u → − u min , also − t/Q > .The skewness is defined with respect to u -channel momentum transfer in the TDA (TransitionDistribution Amplitude) formalism [28], ξ u = p +1 − p + π p +1 + p + π . (8)The GPDs depend on x , ξ t and t , whereas the TDAs depend on x , ξ u and u . The π productionprocess through GPDs in the forward-angle ( t -channel) and through TDAs in the backward-angle( u -channel) are schematically shown in Figs. 9(a) and (b), respectively. In terms of the formalism,TDAs are similar to the GPDs, except they depend on three quark momentum fractions x i (with x + x + x = 2 ξ u .The backward-angle TDA collinear factorization scheme has similar requirements: x is fixed,the u -momentum transfer is required to be small compared to Q and s ; u ≡ ∆ , which impliesthat Q and s need to be sufficiently large. Recall an optimistic estimate of early scaling for GPDphysics occurs between < Q < GeV (although the σ T T for π forward electroproduction isquite far from expectation). The case for the backward processes was open before the pioneeringstudies from JLab 6 GeV [3, 11, 19]. The backward π + and ω production results have shownindications of TDA Q -scaling at Q ≪ GeV . Furthermore, the parameter ∆ = p π − p is considered to encode new valuable complementary information on the hadronic 3-dimensionalwave functions, whose detailed physical meaning still awaits clarification [25].Beyond the JLab 12 GeV program, backward π production will be studied by the P ANDA ex-periment at FAIR [44]. This experimental channel can be accessed through observables including p + p → γ ∗ + π and p + p → J/ψ + π . Note that this backward π production involves the sameTDAs as in the electroproduction case. They will serve as very strong tests of the universality ofTDAs in different processes [28].1
1. Further Detail on the π N TDAs
At leading twist-3, the parameterization of the Fourier transform of the πN transition matrix el-ement of the three-local light cone quark operator b O ρτχ ( λ n, λ n, λ n ) [45] can be written as [46] F h π α ( p π ) | b O ρτχ ( λ n, λ n, λ n ) | N ι ( p ) i = 4( P · n ) Z " Y j =1 dλ j π e i P k =1 x k λ k ( P · n ) h π α ( p π ) | b O ρτχ ( λ n, λ n, λ n ) | N ι ( p ) i = δ ( x + x + x − ξ u ) X s.f. ( f a ) αβγι s ρτ,χ H πNs.f. ( x , x , x , φ, ∆ ; µ F ) (9)where F represents the Fourier transform; P = p + p π is the average u -channel momentum, and ∆ = p π − p is the u -channel momentum transfer, recall ∆ ≡ u . The spin-flavor ( s.f. ) sum overall independent flavor structure ( f a ) αβγι and Dirac structure s ρτ,χ relevant at the leading twist; ι ( a ) is the nucleon (pion) isotopic index. The invariant transition amplitudes, H πNs.f. , which are oftenreferred to as the leading twist πN TDAs, are functions of the light-cone momentum fraction x i ( i = 1 , , , the skewness variable ξ u , the u -channel momentum-transfer squared ∆ , and thefactorization scale µ F [41]. The full extended expression of H πNs.f. can be found in Ref. [46]. Notethat the cross section depends upon the squared modulus of defined amplitude in Eqn. 9.In a simplified notation, H πN ( x, ξ u , ∆ ) can be written in terms of invariant amplitudes V πN , , A πN , , T πN , , , [41, 46], H πNs.f. = { V πN , , A πN , , T πN , , , } . (10)Each invariant amplitude V πN , , A πN , , T πN , , , is also a function of x i , ξ u and ∆ . It is important tonote that not all of the πN TDA invariant amplitudes are independent [41], and their relations aredocumented in Ref. [41].Similar to early attempts in the GPD case [47], the most straightforward solution to determinea reasonable ∆ dependence is to perform a factorized form of ∆ dependence for quadrupledistributions. Thus, the πN factorized form of ∆ dependence can be written as [41]: H πN ( x, ξ u , ∆ ) = H πN ( x i , ξ u ) × G (∆ ) , (11)where G (∆ ) is the πN transition form factor of the three local quarks. Note that the determination2of the ∆ dependence and extraction of the G (∆ ) form factor will be a distant goal for backward-angle physics. x B d Σ T (cid:144) d W Π @ nb (cid:144) s r D Γ * p ® Π p CZ v.s. COZ v.s. KS v.s GS input nucleon DA
FIG. 10: d σ T /d Ω π for backward γ ∗ p → pπ as a function of x B for πN TDAs at Q = 10 GeV , u = − . CZ (solid line) [48], COZ (dotted line) [13], KS (dashed line) [14] and GS(dash-dotted line) [49] nucleon DAs were used as input. This plot was published in Ref. [41]FIG. 11: π p TDAs V π p , A π p and T π p , computed as functions of quark-diquark coordinates, inthe limit ξ u → . CZ N DAs are used as numerical input. These plots were published inRef. [41].In the ξ u = 1 limit, the π p TDAs: V π p , A π p , T π p can be simplified to the following combi-3nation of nucleon DAs [46]: V π p ( x , x , x , ξ u = 1) = − × V P (cid:16) x , x , x (cid:17) (12) A π p ( x , x , x , ξ u = 1) = − × A P (cid:16) x , x , x (cid:17) (13) T π p ( x , x , x , ξ u = 1) = 32 × T P (cid:16) x , x , x (cid:17) . (14)A variety of nucleon ( N ) DAs, such as Chernyak-Zhitnitsky (CZ) [48], Chernyak-Ogloblin-Zhitnitsky (COZ) [13], King and Sachrajda (KS) [14] and Gari and Stefanis (GS) [49] can beused as numerical input for V P , A P and T P , see their graphical representation in Fig. 11. A TDAcalculation for π production cross section versus x B is shown in Fig. 10, where all four N DAsare used.The N DA model is an important part of the TDA model prediction, and depending on thechoice of the N DAs, the predicted experimental observables can change significantly. Therefore,improvements to the TDA parameterized formalism will rely on an accurate nucleon spectral dis-tribution by the N DA models. In the same time, as more data are collected during JLab 12 GeV,a refined TDA model will help to discriminate between different N DAs. This healthy iterativeprocess can help improve our knowledge of proton structure [28].According to the TDA framework, the leading order (LO) backward angle γ + p → π + p unpolarized cross section can be written as [28, 41] d σ T d Ω π = |C | Q Λ( s, m , M )128 π s ( s − M ) 1 + ξξ ( |I| − ∆ T M |I ′ | ) . (15) Λ( s, m , M ) is the Mandelstam function [41], where m corresponds to the meson mass and M isthe nucleon mass. In the backward-angle kinematics, ∆ T = (1 − ξ ) (cid:16) ∆ − ξ (cid:16) M ξ − m − ξ (cid:17)(cid:17) ξ . (16)The coefficients I and I ′ are defined as [28] I = Z X α =1 T α + X α =8 T α ! , I ′ = Z X α =1 T ′ α + X α =8 T ′ α ! , (17)4where the coefficients T α and T ′ α ( α = 1 , ..., are functions of x i , y j , ξ u and ∆ . Here, x i and y j represent the momentum fractions for the initial and final state quarks. Also recall ∆ ≡ u .Each of the components of T α and T ′ represent one of the 21 diagrams contributing to the hard-scattering amplitudes (note that the last seven diagrams are the duplicates the of the first sevendiagrams).Furthermore, T α ( α = 1 , ..., can be written in terms of V pπ , A pπ , T pπ , T pπ and N DA( V p , A p , T p ); T ′ α ( α = 1 , ..., can be written in terms of V pπ , A pπ , T pπ , T pπ and N DA [28].This work has genuinely established the connection between the TDAs amplitudes to the crosssection observables.
2. Two Predictions from TDA Collinear Factorization
The TDA collinear factorization has made two specific qualitative predictions regarding back-ward meson electroproduction, which can be verified experimentally [25, 37, 41, 50]: • The dominance of the transverse polarization of the virtual-photon results in the suppressionof the σ L cross section by a least ( /Q ): σ L /σ T < /Q , • The characteristic /Q -scaling behavior of the transverse cross section for fixed x B (or atfixed ξ u ), following the quark counting rules.The goal of the proposed π measurement is to challenge these predictions. In addition, the − u dependence of the separated experimental cross section will provide insight for the extraction ofthe πN transition form factor G (∆ ) (from Eqn. 11), C. Complementary Objective: the Hadronic Approach
The development of Regge-trajectory-based models has created a useful linkage betweenphysics kinematic quantities and experimental observables. Experimental observables in the JLabphysics regime are often parameterized in terms of W , x B , Q and t . By varying a particularparameter while fixing others, one can perform high precision studies to investigate the isolateddependence of the varied parameter for a given interaction.In the Regge models, the exchange of high-spin, high-mass particles is normally taken intoaccount by replacing the pole-like Feynman propagator of a single particle (i.e. t − M with the5 pπ + , π − ρp γ ∗ π ∆ , ∆ ++ π, ρ pρ + , ρ − πp γ ∗ π ∆ , ∆ ++ π, ρ FIG. 12: Examples of meson exchange diagrams which contribute to forward-angle π production. The left plot is an example of charged π rescattering [53]; the right plot is an examplevector meson contribution. These plots were created based on the original ones published inRef. [24]. π ρp γ ∗ pω FIG. 13: Example of a possible meson exchange diagram which contributes to backward-angle π production. This plot was created based on the original one published in Ref. [24].Regge (trajectory) propagator). Meanwhile, the exchange process involves a series of particles ofthe same quantum number (following the same Regge trajectory α ( t ) ), instead of single particleexchange [51, 52]. In the forward-angle π electroproduction study [53], J. M. Laget linked theelastic π cross section to the scattering channels of ωp , ρ + n , ρ − ∆ ++ , diagrams shown in Fig. 12.This treatment significantly improved the predictive power of the hadronic Regge-based modeland led to a good agreement with the data [53].Recently [24], J. M. Laget indicated that the hadronic Regge-based model is capable of de-scribing the data trend of the backward ω cross section (shown in Figs. 6,14), at Q = 1 . and . GeV . The preliminary conclusion from this study was that the nucleon pole contribution(baryon exchange) alone is not enough to account for the measured cross section [53]. Backward ω production requires ρ , ρ + n , ρ ∆ scattering channels, in addition to the nucleon pole ampli-tude [24]. Note, this approach is very similar to that used for the π forward-angle study. Forreference purpose, a possible u -channel baryon trajectory exchange diagram for π production is6 -2 -1 p( g * , w )p Q = 0 GeV W = 2.48 GeVSLAC Q = 0.84 GeV W = 2.30 GeVDESY10 -3 -2 -1 -t (GeV ) d s / d t ( m b / G e V ) Q = 1.75 GeV W = 2.476 GeVJLab 0 2 4 6Q = 2.35 GeV W = 2.472 GeVJLab
FIG. 14: The cross section evolution for exclusive ω meson production as a function of − t . Notethat the bottom panels show ω electroproduction data from CLAS (dots) and Hall C (circles), the Q = 1 . GeV on the left and Q = 2 . GeV on the right. The dashed red curves are thepredictions of the basic model when a constant cutoff mass is used in the meson electric andmagnetic form factors [23]. The full red curves are the predictions when a t -dependent cutoffmass was used [23]. The black dashed line is the prediction of the nucleon degenerated pole only.The full line curves take into account the interference between nucleon exchange and the ω produced via nucleon exchange re-scattering on the nucleon. Plot provided by J. M. Lagetthrough private communication [24].shown in Fig. 13, and this diagram is based on the knowledge of forward-angle π production(shown in Fig. 12). Currently, a publication is in preparation which will contain more findings ofthe backward ω cross section using the hadronic Regge-based model [24].7TABLE I: Status table showing the progress of TDA validation in Jefferson Lab 12 GeV. (cid:13) : thisproposal; △ : in the early planning stage; X : parasitic data may be available to perform study; XX : confirmed by existing data. σ T > σ L /Q Scaling π (cid:13) (cid:13) π + XX π − K K ± η X X ρω XX X η ′ X X φ X X
VCS △ △
Due to a lack of systematic studies, currently available backward-angle physics data above theresonance region (most of them are summarized in Sec. II) have limited coverage in terms of W , Q and t (or u ) and therefore cannot support a full phenomenological study. However, the u -channel Regge-exchange study is still a useful tool to verify the key knowledge gained from theforward-angle physics program, i.e. to map out the full − t evolution and extract the backward-angle slope for a given meson production process, such as the example shown in Fig. 6. Note thatthe chosen kinematic setting in the proposal is made based on the existing and proposed forward-angle π measurements [4, 55], i.e. Q = 2 . , 3.0 and . GeV at fixed x B = IV. STUDYING TDA THROUGH VCS AND DEMP FROM JLAB 12 GEV TO EIC
With caution in mind, the /Q scaling behavior observed in the π + production (Sec. II B)and the separated cross section ratio have shown an indication that σ T ≫ σ L for the exclusive ω electroproduction channel (Sec: II D), which could be considered as initial evidence needed todemonstrate the validity of the TDA factorization approach in backward-angle kinematics.These initial successes of the TDA framework raise important and urgent questions:1. Would the dominance of σ T be observed in other u -channel exclusive meson productionchannels (e.g. π , ρ , η , η ′ and u -channel φ ), and VCS?2. What is the applicability region (in Q and W ) for a perturbative QCD description in the8 - - Q50 - ) ( f b / G e V / dq s d PANDAJLab 12 GeVEIC u-Channel Production p , s = 10 GeV FIG. 15: Projected Q evolution of u -channel π electroproduction measurement combiningcoverage from projected data from Panda (blue circle), JLab 12 GeV from this proposal (magentasquare) and future EIC (red circle). Note that the projected PANDA measurements is given in dσ/dq at u ′ ∼ GeV with the assumption of σ T ≫ σ L . The cross sections from this proposal(JLab 12 GeV) and EIC will be provided in dσ/dt , which can be easily converted into dσ/dq . Inaddition, one also would have to apply a W and x B correction before forming the combined plot.backward-angle region?These important questions need to be answered by 12 GeV measurements. At the current stage,we are in a process of establishing a coherent and comprehensive program (with significant the-ory insights) that will prioritize the backward-angle observable and lay down a path to continuestudying u -channel physics in the future EIC. Note that an important part of this program is tofurther support the development of the TDA. It is our pleasure to inform the PAC that the 2020JSA postdoc prize was awarded to one of the authors (Wenliang Li) of this proposal to develop thedescribed u -channel physics program.Tentatively, we envision a program to study TDAs systematically that consists of three stages: Stage 0:
Continuing to demonstrate the existence of u -channel signals and studying the “soft-hard” transition with increasing Q . Stage 1:
Measuring the /Q n scaling trend of σ T and attempting to extract σ T ≫ σ L for allsingle-meson production channels and VCS.9 Stage 2:
Extraction of TDAs by probing the single and double spin asymmetries for backward-angle meson productions. This step implies the determination of the ∆ dependence and the πN transition form factor G (∆ ) defined in Eqn. 11.Stage 0 has been underway for some time, but as shown in Table I there are still unexploredsingle-meson channels. We are in the early period of the Stage 1, which includes this proposaland will likely continue through the entire 12 GeV era. Only after most of the tasks in Table I arecompleted with corresponding measurements can we provide answers to the important questionsraised earlier. Special Role of π Electroproduction
In comparison to the u -channel ω or η electroproduction processes, the reconstructed missingmass distribution for π has little physics background underneath its narrow peak. This signifi-cantly reduces the complication associated with the background removal during the analysis. Inaddition, π production has been a popular candidate for theoretical studies [23, 28]. All thesefeatures make it a prime choice to initiate backward-angle studies in the JLab 12 GeV era. In ad-dition, backward π production has received significant interest beyond the JLab physics programand will be studied by the P ANDA experiment at FAIR [44] through the complementary process p + p → γ ∗ + π . See Fig. 15 for a Q evolution of the π cross section after combining pro-jected data coverage from P ANDA, JLab 12 GeV (this proposal) and future EIC measurements.This combined − < Q < GeV range would offer a unique opportunity to challenge theuniversality of the TDA.In a recent publication, L. Szymanowski, B. Pire and K. Semonov-Tian-Shansky laid out thepath for TDA from JLab 12 GeV to the EIC [56]. Through private communication, experts unan-imously agreed the electroproduction of π is an ideal candidate to initiate a 12 GeV to EIC tran-sition study. In parallel to this proposal, the feasibility study of probing the u -channel π processat kinematics shown in Fig. 15, has begun. The preliminary result of the study will be includedin the upcoming EIC Yellow Report (YR) as one of the benchmark observables. Recently, oneof the authors from this proposal (Wenliang Li) was awarded with the EIC fellowship, which willsignificantly accelerate the completion of the π feasibility study and corresponding section of theYR.In a broader scope, u -channel electroproduction is only one aspect of probing nucleon structure0through backward-angle observables. The diversified experimental programs and equipment fromJLab 12 GeV offer other exciting u -channel physics opportunities, such as the backward-anglevector meson production and hyperon production at GlueX. The authors of this proposal are ex-cited to inform the PAC that the first backward-angle physics focused workshop is taking place atJLab in September, 2020. One major objective is to offer a platform to connect scattered exper-imental and theoretical efforts together, thus, potentially forming small backward-angle physicsworking groups. See Appx. D for the full objectives of the workshop, topics of discussion andparticipants list.Additionally, the inclusion of u -channel exclusive reactions in the July 14 mini-workshop on“Physics Opportunities for Large Angle Production with CLAS” [54] indicates the growing inter-est in the physics opportunities available in this regime. V. EXPERIMENT KINEMATICS AND CONFIGURATION
The exclusive backward-angle π electroproduction measurement is proposed to use the stan-dard Hall C equipment: SHMS and HMS in coincidence mode, the standard-gradient electronbeam and the liquid hydrogen (LH ) target. For most of settings, SHMS will be used to detect theforward going (fast) proton and HMS will be used to detect the scattered electron. The π eventswill be selected by using the missing mass reconstruction technique. A schematic diagram of theexperimental configuration for the H ( e, e ′ p ) π is shown in Fig. 16. A. L/T/LT/TT Separation
In the one-photon-exchange approximation, the H ( e, e ′ p ) X cross section of the π and othermeson production interactions ( X = π ± , ρ , ω , φ , 2 π , η and η ′ ) can be written as the contractionof a lepton tensor L µν and a hadron tensor W µν : d σd Ω e ′ dE e ′ d Ω p dE p = | p p | E p α Q E e ′ E e L µν W µν , (18)where the L µν can be calculated exactly in QED, and the explicit structure of the W µν is yet to bedetermined. Since the final states are over constrained (either detected or can be reconstructed),as in the case of the H ( e, e ′ p ) π reaction, the cross section can be reduced further to a five-fold1FIG. 16: Experimental configuration for H ( e ′ , ep ) π with the standard Hall C equipment. SHMSand HMS are located on the left and right side of the beam line, respectively.differential form: d σdE ′ d Ω e ′ d Ω ∗ p = Γ v d σd Ω ∗ p , (19)where the asterisks denote quantities in the center-of-mass frame of the virtual-photon-nucleonsystem; Γ V is the virtual-photon flux factor: Γ v = α π E e ′ E e q L Q − ǫ ) , where α is the fine structure constant, the factor q L = ( W − m p ) / (2 M p ) is the equivalent real-photon energy, which is the laboratory energy a real photon would need to produce a system withinvariant mass W ; and ǫ is the polarization of the virtual-photon which is defined as ǫ = (cid:18) | q | Q tan θ e (cid:19) − . d σd Ω ∗ ω = d σdt dφ · dtd cos θ ∗ , (20)where dtd cos θ ∗ = 2 | p ∗ || q ∗ | is the Jacobian factor, and p ∗ and q ∗ are the three momentum of the proton and the virtual-photonin the CM frame.The general form of two-fold differential cross section can be expressed in terms of the structurefunctions as: π d σdt dφ = dσ T dt + ǫ dσ L dt + p ǫ (1 + ǫ ) dσ LT dt cos φ + ǫ dσ TT dt cos 2 φ . (21)FIG. 17: The scattering and reaction planes for π electroproduction: H ( e, e ′ p ) π . The scatteringplane is shown in blue and the reaction plane is shown in orange. Note that the forward-goingproton after the interaction is labelled p ; γ ν represents the exchanged virtual-photon and itsdirection defines the q -vector; φ p ( φ p = φ π + 180 ◦ ) is defined as the angle between the scatteringand reaction planes (the azimuthal angle around the q -vector); θ p and θ π denote the scatteringangles of the p and π with respect to the q -vector, respectively. The definition of the Lorentzinvariant variables such as W , Q , t and u are also shown.A schematic diagram of the exclusive π electroproduction reaction, H ( e, e ′ p ) π , giving thedefinition of the kinematic variables in Eqn. 21 is shown in Fig. 17. The three-momentum vectorsof the incoming and the scattered electrons are denoted as ~p e and ~p e ′ , respectively. Together theydefine the scattering plane, which is shown as a blue box. The corresponding four momentaare p e and p ′ e . The electron scattering angle in the lab frame is labelled as θ e . The transferred3four-momentum vector q ( ν, ~q ) is defined as (p e − p e ′ ). The three-momentum vectors of the recoilproton target ( ~p p ) and produced π ( ~p π ) define the reaction plane, is shown as the orange box. Theazimuthal angle between the scattering plane and the reaction plane is denoted by the recoil protonangle φ p . From the perspective of standing at the Hall C beam entrance and looking downstream ofthe spectrometer, the forward going proton angle φ p = 0 points to horizontal left of the q -vector,and it follows a counterclockwise rotation. The lab frame scattering angles between ~p p (or ~p π )and ~q are labeled θ p (or θ π ). Unless otherwise specified, the symbols θ and φ without subscriptare equivalent to θ p and φ p , since the recoil protons will be detected during the experiment. Theparallel and antiparallel kinematics are unique circumstances, and occur at θ = 0 ◦ and θ = 180 ◦ ,respectively.The Rosenbluth separation, also known as the longitudinal/transverse (L/T) separation, is aunique method of isolating the longitudinal component of the differential cross section from thetransverse component. The method requires at least two separate measurements with differentexperimental configurations, such as the spectrometer angles and electron beam energy, whilefixing the Lorentz invariant kinematic parameters such as x B and Q . The only physical parameterthat is different between the two measurements is ǫ = (cid:16) | ~q | Q tan θ (cid:17) − , which is directlydependent upon the incoming electron beam energy ( E e ) and the scattering angle of the outgoingelectron.Even though the SHMS setting at θ = 0 (or θ pq = 0 for clarity) is centered with respect to the q -vector, corresponding to the parallel scenario for the proton (anti-parallel for π ), the spectrometeracceptance of the SHMS (proton arm) is not wide enough to provide uniform coverage in φ (blackevents in Fig. 18). A complete φ coverage over a full u range is critical for the extraction ofthe interference terms (LT and TT) during the L/T separation procedure. To ensure an optimal φ coverage, additional measurements are required at the θ = ± ◦ SHMS angles (blue and redevents). Constrained by the minimum SHMS angle from the beam line of θ SHMS = 5 . ◦ , the lower ǫ measurement is only possible at two angles at some Q . However, this can be compensated bythe full φ coverage at the higher ǫ measurement and the simulated distribution, thus determiningthe interference components (LT and TT) of the differential cross section.The last step of the L/T separation is to fit the experimental cross section versus φ for a given u bin. The lower and higher epsilon data will be fitted simultaneously using Eqn. 21 to ensuresuccessful extraction of the σ T , L , LT , TT . The common offset between and difference between thelower and higher ǫ data set give raise to the σ T and σ L ; whereas the φ dependence signifies the σ LT σ TT contribution. B. Choice of Kinematics
The H ( e, e ′ p ) π experimental yield will be measured at Q = 2 . , . , . , . and . GeV ,at common Bjorken x B = 0 . . We intend to perform L/T/LT/TT separations for all except the Q = 6 . GeV setting. One additional L/T separation study at Q = 2 GeV , W = 3 GeVwill provide W scaling information needed to achieve the projected experimental objective. See asummary table that includes relevant kinematics variables and spectrometer settings in Table II.FIG. 18: u ′ - φ polar distributions at Q = 2 GeV and ǫ = 0 . . ( − u + 0 . is plotted as the radialvariable and φ as the angular variable. The blue points represent data at θ pq = +3 ◦ , black pointsrepresent data at θ pq = 0 ◦ , and red data points represent data at θ pq = − ◦ . The center of the plotrepresents u = +0 . GeV and the outer circle is at u = − . GeV .Using the backward H ( e, e ′ p ) π and H ( e, e ′ p ) γ physics models (see Appx. A for a detaileddescription), the estimated event rates and times for collecting the required event samples are5TABLE II: Proposed kinematics for the H ( e, e ′ p ) π measurement. Note that the W and Q arethe same for the H ( e, e ′ pγ ) reaction. For most of the settings, HMS will detect the scatteredelectron and the SHMS will detect the recoiled proton. For Q = 2 GeV (indicated by ∗ ), theSHMS will detect the electron and the HMS will detect the proton because the scattered electronmomentum and angle at high ǫ are too high and too far forward for the HMS. For all settings, u ′ = 0 GeV . Note that at Q = 3 . , and 4.0 GeV , E12-13-010 will provide the x B =0.36 L/Tseparated cross section at t ′ ∼ [4], while Q = 2 . GeV x B = Q W x B E Beam ǫ θ
HMS P HMS θ SHMS P SHMS θ pq − t (GeV ) (GeV) (GeV) (deg) (GeV/c) (deg) (GeV/c) (deg) (GeV )2.0 2.11 0.36 4.4 ∗ ∗ ∗ ∗ − ∗ − ∗ − + ∗ ∗ ∗ ∗ ∗ − ∗ − ∗ − + ∗ − − − − − − + − − − + − − − + − − − + − − − + − − − − − − + − − φ coverage, each ( Q , ǫ ) point requiresthree proton spectrometer (SHMS in most cases) angle settings: left ( θ pq = − ◦ ), center ( θ pq = 0 ◦ )and right ( θ pq = +3 ◦ ) with respect to the q -vector, as shown in Fig. 18. W versus Q distributions (the ‘diamond’ distributions) for all settings are shown in Fig. 20.The L/T separated cross sections are planned at Q = 2 . , . , . and . GeV . These measure-ments will provide the − u dependence for σ L and σ T at nearly constant Q and W , in addition tothe behavior of σ L / σ T ratio as function of Q . Note there are two ǫ measurements at each Q set-ting: the red diamonds indicate low ǫ measurements and black for the high ǫ measurements. The Q = 6 . GeV setting is chosen to test the Q scaling nature of the unseparated cross section,but only one ǫ setting is available due to limitations on the accessible spectrometer angles.These proposed measurements will provide the following insights into the Q dependence ofthe TDA formalism: L/T Separation at Q =2 GeV The experimental insights from Figs. 3, 7 and 8 reveal TDA’sability of capturing the general trend of cross section and σ L /σ T ratio behaviour as a function of Q . However, it is interesting to note that, at Q ∼ , the TDA completely mis-predictedthe cross section and the T/L ratio for π + and ω production. Based on these observations, we would6draw a tentative conclusion: in the region of . < Q < . GeV , the nucleon wavefunctionundergoes a transition; Q =2.5 GeV is a boundary point where the TDA factorization starts tobecome valid. Also, we expect the σ T begins to become dominant (due to the fast drop of σ L inthis transition region), and the predicted σ T /σ L ratio ( R ) in the range of < R < . L/T Separation at Q =3, 4, 5 GeV The L/T separated cross sections at these kinematicpoints yield the core data of the proposed experiment. In this kinematics region, if the TDAcollinear factorization hypothesis is valid, we expect to clearly observe: σ T > σ L . Based onour parameterization of the Defurne, et al. Hall A data [55], our parameterization of a GPDcalculation by Goloskokov & Kroll [57], and the observed forward-backward peak ratios in ω electroproduction [3], we estimate ratios of: R > at Q = 3 GeV and R ∼ at Q = 5 GeV .However, the expected extremely low σ L contribution to the cross section ( σ L ∼ ), will make theaccurate determination of σ L impossible within proposed running time. For these extremely largevalues of R , our goal is to set a lower bound on R , as accurate very large ratio values within theproposed running time are not feasible. Cross Section at Q =6.25 GeV As it will be shown in the time estimation in Sec. V L,the measurement at Q = 6 . GeV requires less time than Q = 5 . , since no L/T separationis intended due to the spectrometer angle limitations. At this kinematics point, we assume thecomplete domination of σ T over σ L and σ L ∼ (based on the results from the lower Q points).In this case, the measured cross section would only consist of contributions from σ T and theinterference term σ T T . With the cross section model extracted from the Q = 2 , , and GeV ,we will be able to complete the projected σ T ∝ /Q test from 2 to 6.2 GeV . This spread ofcoverage q ( ∆ q ∼ GeV ) is similar to the spread of q coverage of the P ANDA TDA study.
C. “Soft-hard” Transition through u -channel Phenomenology Study As described in the introduction, one of the main objectives of the proposed measurementis to better understand the mechanism of a “soft-hard transition” in u -channel physics, whichremains an important open question. To accomplish the stated objective, we propose to utilize thephenomenology tools developed at HERA, by examining the t (or u in this case) dependence of thecross section at different Q settings, then comparing the extracted transverse size of interaction(using the fitted slope) to the hadronic size. For a “soft” Regge-exchange picture, the transversesize of the interaction is on the order of the hadronic size; for a “hard” QCD regime, the transverse7 . .
157 0 .
219 0 .
278 0 .
435 0 .
475 0 .
433 0 .
324 0 . =0 pq q o =-3.0 pq q =+3.0 pq q ) -u + 0.5 (GeV - -
10 1 T s = 2.00 GeV Q = 3.00 GeV Q = 4.00 GeV Q = 5.00 GeV Q = 6.25 GeV Q FIG. 19: (a) shows the simulated − u distributions at Q = 4 GeV , ǫ = 0 . generated by theHall C Monte Carlo (SIMC). Distributions from all three angle settings: 0 ◦ (black), -3 ◦ (red) and+3 ◦ (cyan) are overlapped. The spectrometer acceptance and the diamond (Fig. 20) cuts areapplied. (b) demonstrates − u coverage for σ T as a function of − u . In the presence ofbackward-angle peaks, the total (or separated) cross sections are expected to fall as − u increases(as described by Eqn. 8). Here, in order the maintain the continuity of the u dependence, an offsetof +0 . GeV to the u coverage is introduced for all settings. The σ T is in arbitrary unit. Notethat the purpose of the plot is to demonstrate the − u coverage and falling behaviour of crosssections in all Q settings. The shown slope values do not represent any accurate prediction.size of interaction ≪ hadronic size [58]. Note that the proposed measurement is the first attemptto apply such a methodology in the u -channel kinematics.In the presence of a backward-angle peak, the events ( θ pq = 0 ◦ , ± ◦ ) will be binned in five u bins, as shown in Fig. 19(a). All u bins will have equal statistics. Also see the projected − t coverage for all Q settings in Fig. 19(b).The standard formula with the exponential − t dependence is replaced by the − u dependenceto address the rising cross section in the u-channel kinematics, given by dσ L,T du = A · e − b ·| u | , (22)where A and b are free fitted parameters. The parameter, b , in the above equation can be rigorouslylinked to the transverse size of the γ ∗ p interaction region, as given below, r int = p | b | ~ c, (23)where ~ c = 0 . GeV · fm. In some terminology, r int is also referred to as the interaction radius.The same approach for extracting r int was successfully applied to extract the − t dependence in8the forward-wide angle π + exclusive electroproduction in Hall C (using the 6 GeV data) [59].At the same time, one must point out the two foreseeable challenges when performing thedescribed phenomenological study: • The proposed measurement only offers ∼ . GeV coverage in − u , which provides asmaller lever arm than other similar studies in meson production channels. See − u coverageat different Q settings in Fig. 19(b). • The expected suppression and large uncertainty in σ L will make the extraction of its − u dependence less conclusive in that case. See Table II for reference.With these in mind, we are optimistic that the proposed measurement will determine the − u de-pendence of σ T at the proposed Q settings where L/T separation separation data is available. The Q dependence of the measured slopes, and corresponding size of interaction region, will providenew insights into the “soft-hard” transition in this unique kinematic regime D. Beam Spin Asymmetry Measurement
As mention in the introduction section, the π BSA will come for free with the planned data ofthis proposal. The study is similar to the one descried in Sec. II C.The A sin φLU will be obtained at fixed x B = 0 . and Q = 2 , , and GeV . At each setting,BSA will be obtained in two separate methodologies:1. Reconstruct A sin φLU based on the separated cross sections: σ T , σ L , and σ LT using Eqn. 2.2. Utilize the readily available electron beam polarization and extract A sin φLU directly from thedata as described by Eqn. 1.Here, σ LT and σ LT ′ are identical in terms of helicity amplitude product terms. The only differencebetween the two is that sigma LT is a real part of the products and σ LT p rime is an imaginary part ofthe products. The determinations of A sin φLU with σ LT and σ LT ′ would provide information sensitiveto the phase between the two.The interference ( σ LT and σ T T ) contributions at the parallel kinematics: u min < u < . GeV ,are expected to be very small (in this proposal), which makes the accurate extraction of the BSA adifficult task (with significant uncertainty comparable to − t ∼ . GeV in Fig. 4). However, we9will be able to offer two separate verification (as stated above) to validate the phenomenologicalobservation from CLAS: sign change of BSA in u -channel kinematics like in Fig. 4 and 5. In orderto accomplish this we would request to utilize the existing beam polarimeter at Hall C.We aim to obtain the beam polarization to a precision of dP/P = 2 - %. This will require semi-regular polarization measurements with the existing Moller polarimeter. These measurements willbe made after every beam pass change and each measurement lasts one shift. E. W Scaling Correction and Q = 2 GeV , W = 3 GeV Setting
In the ideal case, one would select kinematics regions where W and x B are fixed among mea-surements at different Q settings. However, the narrow spectrometer acceptance and correlatednature of Q - W (within the coincidence acceptance) make measurements at a fixed W and x B animpossible task, as demonstrated in Fig. 20.FIG. 20: W vs Q diamonds for the Q = 2 . , . , . , . and . GeV settings. The blackdiamonds are for the higher ǫ settings and the red diamonds are for the lower ǫ settings. Thediamonds for the W scaling setting are shown separately (in blue and red). Note that there is onlyone ǫ setting for Q = 6 . GeV . The overlap between the black and red diamond is critical forthe L/T separation at each setting. The boundary of the low ǫ (red) data coverage will become acut for the high ǫ data.When performing a study of L/T-separated cross sections versus u or t , one must use a scalingprocedure to correct for the small W dependence in the measured cross section, namely, scaling0different W values to a common W norm . The standard W scaling formalism is as follows: W − M p W norm − M p , (24)where M p is the proton mass. This procedure is our best estimate for the W correction based on the t -channel meson production data. The necessary data to validate this relationship for backward-angle meson production have never been acquired, so this relationship was assumed to apply inthe backward ω case ( ∆ W ∼ . GeV) [3, 19].In this proposal, keeping x B fixed at 0.36 means that ∆ W ∼ GeV as Q is varied from 2to 5 GeV . This raises an important question, could Eqn. 24 correct the W dependence? The W dependence across the diamond for a single L/T separation is not that large (Fig. 20). However,it will play a role when comparing data-sets from the different Q with each other. An inaccu-rate W dependence measurement would increase the uncertainty, or even bias the result, whendetermining the exponent factor ( n ) of the σ T ∝ /Q n scaling test.To verify the W scaling procedure with experimental data, we propose performing a u -channelL/T separated cross section measurement at Q = 2 . GeV , W = 3 GeV and x B = 0 . . Alongwith planned separated cross section at Q = 2 . GeV , W = 2 . , one can get a clean W correction for σ L and σ T independently. F. Singles Rate Estimation
All singles rate estimates assume the use of a 70 µ A beam on a 8 cm LH cryogenic target andthe detection efficiencies listed in Table VI.For the purpose of calculating online random coincidence rates, the hadron arm trigger ratewas taken as equal to the raw trigger rate, i.e. no distinguishing between pions, kaons and protonsin the hadron trigger. Assuming the online ELREAL trigger is set up to suppress the combinedtrigger contribution from π − and K − by 5:1, the electron arm trigger rate was taken to be electronsplus ( π − + K − + ¯ p ) / . The electron arm rate is calculated as E Rate = e − + ( π − + K − + ¯ p ) / , and hadron arm rate as: H Rate = e + + π + + K − + p . β versus HMS-SHMS coincidence time distribution from the recent 12GeV Kaon-LT experiment (E12-09-011). No acceptance or PID cuts are applied. The 4 nsindividual beam bunch spacing from the CEBAF accelerator is easily resolved. Within eachbunch, a doublet is observed, with the left darker band due primarily to p ( e, e ′ π + ) n and p ( e, e ′ K + )Λ coincidences, and the right fainter band due to p ( e, e ′ p ) X coincidences, the offsetbeing due to the longer time of flight of protons in the hadron arm.Here, we also assume the trigger contribution from e + is small compared to other particles, there-fore is neglected. The random coincidence rate is then given by (E Rate) · (H Rate) · ∆ t , where the coincidence resolving time was taken to be ∆ t = 70 ns.Since the roles of SHMS and HMS differ, the singles rate estimations for the Q = and W -scaling settings are listed in Table III; the singles rate estimation for Q = is in Table IV. In all cases, the resulting online real + random rates are well below the expectedcapability of the HMS+SHMS data acquisition system. The real coincidence rate is based on ourestimated SIMC yield without any cuts applied.It is important to note that the random and real coincidence rates listed in Table III and Table IVare for the online DAQ rates, which only take into account the π − rejection ratio on the electronarm (assuming 70 ns resolving time). Fig. 21 shows an example of SHMS β vs. the HMS-SHMS2coincidence time spectrum from a recent 12 GeV Hall C experiment. It shows we typically resolvethe prompt coincidence peak to less than 2 ns, so a cut on this automatically would reduce therandom contributions by a factor of ∼
35 (i.e. 70 ns/2 ns). Random coincidence subtraction andparticle ID cuts will reduce this further, to a few percent of the reals physics events in the offlineanalysis.TABLE III: Calculated singles rates for Q =
3, 4, 5 and 6.25 GeV and the W -scaling settings.In these settings, the SHMS is on the positive and the HMS is on the negative polarity. Note thatthese calculated random and real rates are for the online DAQ only, and do not correspond to theactual event and background offline analysis rates.SHMS HMS ǫ π + K + p e π − K − ¯ p Random Coin. Real Coin.(kHz) (kHz) (kHz) (kHz) (kHz) (kHz) (kHz) (Hz) (Hz) Q = , W = 3 GeV, x B = 0 . Q = , W = 2 . GeV, x B = 0 . Q = , W = 2 . GeV, x B = 0 . Q = , W = 3 . GeV, x B = 0 . Q = , W = 3 . GeV, x B = 0 . Q = . In these settings, the HMS is on the positiveand the SHMS is on the negative polarity.HMS SHMS ǫ π + K p e π − K − ¯ p Random Coin. Real Coin.(kHz) (kHz) (kHz) (kHz) (kHz) (kHz) (kHz) (Hz) (Hz) Q = , W = 2 . GeV, x B = 0 . G. Kinematic Checks and Normalization with Elastic Scattering
The elastic H ( e, e ′ ) p and H ( e, e ′ p ) measurements are extremely useful tests for determiningthe systematic uncertainties in single arm and coincidence measurements. The fixed position ofthe elastic peak ( W for the single arm case and along missing energy/missing momentum inthe coincidence) allows one to verify the spectrometer central angle momentum and determinepotential offsets. In addition, the well known e - p elastic cross section provides verification of thenormalization and establishes acceptance boundaries.The use of elastic scattering for calibration has been performed extensively and has been astandard procedure in Hall C measurements in both the 6 GeV and 12 GeV eras. Due to the strictsystematic requirements of the L/T separation procedure, we will perform additional checks atkinematics close to the planned measurements. In this section, we briefly discuss our kinematicschoices and beam time requirements for the elastic checks.
1. Single Arm Elastic Checks
As described above, the single arm elastic scan measures the hydrogen elastic peak. There arethree unknown parameters to be determined from these measurements: beam energy, spectrometercentral angle, and spectrometer central momentum. A series of high quality measurements willprovide precise constraints to these parameters.For the HMS, we can rely on the rigid connection to the target station pivot to ensure thatvariations in the pointing angle are relatively small when rotating the spectrometer to variousangles. Because of this excellent pointing reproducibility, one can assume that, by and large,any offset to the spectrometer central angle is a fixed value, with minimal variation (on the orderof 0.2 mrad) as the spectrometer is rotated. Similarly, one can assume that the deviation of thespectrometer central momentum is a fixed value due to the very linear response of the HMS dipole.Hence, by measuring the position of the reconstructed proton mass peak ( W = M p ) over a rangeof angles and at several beam energies, one has several constraints on the spectrometer kinematicoffsets. In certain Q measurement settings, the HMS spectrometer momentum is considerablyhigher than during the standard 6 GeV operation. We will take additional optics data at P HMS =5 . GeV/c (our highest HMS momentum setting) to ensure that any saturation corrections thatmay be needed are properly understood and applied.4A similar study will be carried out with the SHMS. In this case, the smaller angles andhigher energies accessible provide a very large lever-arm for constraining the central scatteringangle and momentum. The elastic peak position is expected to shift from ∼ − MeV/mrad to ∼ − MeV/mrad as one rotates from 5.5 degrees to 18 degrees. At fixed beam energy, the de-pendence on the central spectrometer momentum is relatively flat, but by making measurementsat several beam energies, one can extract the angle and momentum offsets.A study of this nature can be carried out in concert with data taking during the π measurement.At each setting, one need acquire 10,000 elastic events each. Note that this is not an issue since all π measurements require much longer running times.
2. Elastic Coincidence Checks
TABLE V: Elastic coincidence H ( e, e ′ p ) kinematics. Settings indicated by ∗ will have HMSdetecting proton and SHMS detecting electron; Setting indicated by + will have HMS detectingelectron and SHMS detecting proton. Assumed 70 µ A beam current. 10000 events whichcorresponds to 1% statistical error. E beam Q θ ′ e p ′ e θ p p p Coincidence Rate Time(GeV) (GeV ) (deg) (GeV) (deg) (GeV) (Hz) (Hours)4.4 + + + + + +
371 14.4 ∗ ∗ ∗ ∗ ∗ ∗
251 16.6 + + + + + +
30 16.6 + + + + + +
170 16.6 ∗ ∗ ∗ ∗ ∗ ∗
323 16.6 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
167 1In addition to constraining the kinematic offsets as described above, the elastic data taken willenable us to check the normalization of the single arm and coincidence acceptance. In particular,examining the elastic yield across the spectrometer momentum acceptance has provided rigorouschecks of our knowledge of the spectrometer response. In the H( e, e ′ p ) reaction, the scatteredelectron is detected in one arm and recoil proton in another, and these signals are in a coincidencemode. The missing energy: E m and three components of missing momentum: p ⊥ m , p k m , p oopm .Ideally, one would choose a coincidence elastic data point at each beam energy used by theexperiment. It is also of benefit to choose the kinematics to sample a similar angle and momentum5range used in the experiment. Table V shows the selected H( e, e ′ p ) kinematics. One would like touse the elastic coincidence data to totally overlap the kinematics used in the experiment, however,this is not possible for all settings. We chose these kinematics in strong favor of SHMS, due toits shorter operation history. The electron and proton arm assignments are described in detail inthe Table V caption. In order to utilize the available spectrometer angle and momentum ranges,while minimizing the run time (run at lowest possible Q value), the spectrometer magnets are atopposite polarities to the corresponding π measurement. This is based on the assumption thatreversing the magnet polarity will not affect the spectrometer response. Note that there are twoelastic settings at E beam = 6 . GeV, Q = 3 . GeV , where each of the SHMS and HMS willperform both the e and p arm role. This will serve as a good check to see any potential offsets anddiscrepancies in the spectrometer response when the polarity reversed. Additionally, one can alsocheck for potential effects from hadrons (protons in this case) punching through the collimator andinvestigate hadron absorption effects in the detector stack.The expectation for the length of the elastic runs is to collect 10,000 coincidence events at70 µA of beam current. Due to magnet polarity reversal and other experimental overhead, theminimum time listed for each setting in Table V is 1 hr. H. Particle Identification htempEntries 6435Mean 9.522RMS 6.832 haero_su-5 0 5 10 15 20 25 30050100150200250300350400450 htempEntries 6435Mean 9.522RMS 6.832 haero_su {abs(hsytar) <= 1.75 && abs(hsdelta) <= 8.0 && abs(hsxptar) <= 0.080 && abs(hsyptar) <= 0.035 && abs(ssdelta) <= 15. && abs(ssxfp) <= 20. && ssytar < 1.5 && abs(ssxptar) <= 0.04 && abs(ssyptar) <= 0.065 && hcer_npe <= 2.0 && ssshtrk >= 0.70 && scer_npe >= 0.50 && (Pm**2-(0.938272-Em)**2) > 0} p p FIG. 22: (a) HMS aerogel Cherenkov cuts that discriminate protons from pions at P HMS = 2 . GeV/c; (b) HMS proton loss percentage due to scattering in the HMS as function of centralmomentum. Red dots are for the ω data; where black dots are for the elastic scattering data. Bothresults are from 6 GeV studies in the F π -2 experiment.For the most π settings, the SHMS will detect the forward going protons and HMS will detect6TABLE VI: Anticipated HMS and SHMS detection efficiency based on past operationexperience.HMS Tracking 0.95SHMS Tracking 0.95HMS Aerogel for proton PID 0.95SHMS Aerogel for proton PID 0.95HMS proton scattering loss 0.95SHMS proton scattering loss 0.95HMS:5.9 msr Acceptance for δ = − to +10% δ = − to +20% π and K events.For the Q = 2 GeV settings, the SHMS will be used as the electron arm and it requiresthe installation of the Noble Gas Cherenkov detector to reject π − events. A variety of gases withdifferent refractive indices are available to perform e / π separation at different particle momenta.We propose at P SHMS = − . GeV/c, the NGC is filled and circulated with mixture of Argon &Nitrogen gas at 1 atm; and at P SHMS = − . GeV/c, the NGC is filled and circulated with Neongas at 1 atm. Note that at P SHMS ∼ − . GeV/c, the β vs. coincidence beam bunch structure willbe capable of providing a clean e/π separation.In the SHMS proton detection case, the Noble Gas Cherenkov detector will remain installeddespite higher proton multiple scattering probability. The reason is to benefit the e - p elastic scat-tering measurement, where in some settings, the SHMS will detect the scattered electron in theelastic runs. At P SHMS =
3, 5, 6.5 and 9.5 GeV/c, the NGC will be filled and circulated withArgon-Nitrogen, Hydrogen, Neon and Helium gas respectively. The Heavy Gas Cherenkov (HGC)detector will be filled with C F at 1 atm. Under normal operation, the HGC gas pressure is re-duced at momenta higher than 7 GeV/c to ensure good K / π separation. However, this step is notnecessary for the proton identification (to save time). The primary methodology for the protonidentification during the π measurement involves: 1) Examining the β vs. coincidence time dis-tribution; 2) Placing a threshold cut on the Aerogel Cherenkov Detector (ACD). These points arefurther elaborated below.At 8 GeV/c SHMS momentum, the coincidence time information provided by the RF reference7and hodoscope triggers (from both spectrometers) cannot provide clean π / K separation. However,proton coincidence triggers arrive significantly later ( ∼
10 ns) than those of π / K , therefore a cleanproton separation is expected in the β vs. cointime distribution for all kinematics settings (lessseparation at the highest momentum setting).As complementary to examining the coincidence timing, the method of placing ACD cuts toexclude events beyond the applied threshold further cleans the proton event sample. An exampleof ACD distribution from the 6 GeV analysis is shown in Fig. 22(a). For the π measurement, wepropose to utilize the SHMS aerogel tray with n = 1 . throughout the run. This corresponds toa Cherenkov threshold momentum of 3.315 GeV/c for K and 6.307 GeV/c for proton. The ACDthreshold cut will be applied at . photoelectrons. It is worth noting that the HGC (at 1 atm) hasa threshold momentum of 9 GeV/c for kaon and ( P th ≪
11 GeV/c) for proton, and it can be usedto help
K/P separation at the high SHMS momentum settings ( P SHMS = 9 . GeV/c). Note that π will generate HGC and ACD signals in all measurement settings.In the HMS detecting proton case ( P SHMS = 3 . GeV/c), we propose the HMS aerogel traywith n = 1 . throughout the run. This corresponds to a Cherenkov threshold momentum of2.0 GeV/c for K and 3.8 GeV/c for proton. The proton PID efficiency has been studied during the6 GeV operation (see efficiency in Table VI) at a similar momentum setting, the ACD PID (negli-gible π contamination) and tracking efficiency of 95% were determined with the elastic data. Weare confident that the combined β vs coincidence information and cuts from the Cherenkov pack-age provide clean separation of proton events from pion and kaon events at the highest momentumsetting.The dominant reaction for the recoil protons inside of the spectrometers is inelastic scattering(mainly pion production), elastic and (quasi) elastic scattering (with heavier elements than hydro-gen). In the case of pion production and (quasi) elastic scattering, a secondary pion, proton orneutron is emitted along the path of the recoil proton momentum, and therefore has a probabilityto generate a valid trigger. The pp and pn total cross sections are dependent on the proton momen-tum, and are estimated to be 43 mb at 3.5 GeV/c, where the elastic cross section is 1/3 of the totalcross section. Extensive studies during the 6 GeV era, as an example, revealed the proton loss isaround for the HMS in the momentum range of 2 to 4 GeV/c. For the SHMS, we assume sim-ilar 5% loss (based on similar material in radiation length to the HMS) for all momentum range,and detailed studies will be performed with elastic data at the π measurement momentum range.8
1. Critical Hardware Replacement
As described in the previous section, the HMS ACD plays an important role in the proton PIDwhen used as the proton arm at Q = 2 GeV/c , W = 2 . GeV. Based on recent operationexperience, the performance of the HMS ACD PMT tubes were identified as degraded. We wouldlike to urge for the refurbishment or replacement of the damaged HMS ACD PMTs before thisproposed measurement, so that one could achieve the stated systematic uncertainties in Table VII.
I. Physics Background Contribution
In comparison to backward-angle ω electroproduction [3, 19], π production has much lessphysics background from other mesons (such as η and ρ ). A contributing physics backgroundunder the coincidence missing mass peak comes from the VCS process, whose missing mass peakis near m x = 0 GeV. An improved u -channel VCS model (based on the results from E93-050and forward-angle DVCS model) and standard Bethe-Heitler (BH) formulation are used, see theirimplementation in Fig. 23). The simulated m x distributions for the backward π and γ at Q = are shown in Fig.24. The black distributions are for the π events and the magentadistributions are for single γ events, both distributions are normalized to 1 µ C of beam charge.The π : γ production ratio is ∼ m x cut of 90 MeV should exclude most of thesingle γ events. After events are binned in the u and φ , the shape and width of the m x peak willchange slightly due to differences in the kinematics coverage ( Q and W ). Given the reconstructedresolution, the standard missing mass cut will not completely separate the two event distributions.Monte Carlo simulation will be needed to estimate the single γ contamination for backgroundsubtraction purposes and place sensible cuts. This contamination is expected to be much less than1%. Furthermore, the single γ physics background is unlikely to come from the Bethe-Heitler(BH) contribution. In Fig. 23, the Virtual Compton Scattering (VCS) model used to estimate the γ background is plotted on the same − t axis as the classic BH formalism, and in the backwardkinematics regime ( − t reaches the maximum value), the BH contribution is suppressed by a factorof 1000 compared to VCS.The lowest possible limit for the two pion production phase-space is likely to start playing arole at M m ∼ MeV, which corresponds to M m = 0 . GeV , it will continue into the η and9FIG. 23: t evolution of the VCS model and BH at Q =
2, 3, 4 and 5.5 GeV (red lines). The BHcontribution is plotted in black. The objective of these plots is to demonstrate the general trend ofthe projected VCS in relation to the classic BH contribution at t = t min and t = t max . The mostimportant observation here is: BH dominates at small − t , and is suppressed at extremely large − t . The discontinuation (in the middle − t range) is due to the fact that the backward-angle VCSmodel is parameterized separately from the forward-angle model. We have not focused onparameterizing the model that handles the forward-backward transition region, since this is faroutside the kinematics of this proposed measurement. For further details, see Appx. B. ρ mass region. We will place a cut at M m = 0 . GeV to reject potential 2 π contamination dueto resolution effects. We expect the background contribution from 2 π production (after cuts) to bevery small and it will be included as part of the analysis simulation.0 - - GeV Mm - - - - - - -
10 110 C oun t s / h r p DVCS+BH2
Missing mass (a) Q =2.0 GeV, ǫ = - - GeV Mm - - - - - - -
10 110 C oun t s / h r p DVCS+BH2
Missing mass (b) Q =2.0 GeV, ǫ = - - GeV Mm - - - - - - -
10 110 C oun t s / h r p DVCS+BH2
Missing mass (c) Q =3.0 GeV, ǫ = - - GeV Mm - - - - - - -
10 110 C oun t s / h r p DVCS+BH2
Missing mass (d) Q =3.0 GeV, ǫ = - - GeV Mm - - - - - - -
10 110 C oun t s / h r p DVCS+BH2
Missing mass (e) Q =4.0 GeV, ǫ = - - GeV Mm - - - - - - -
10 110 C oun t s / h r p DVCS+BH2
Missing mass (f) Q =4.0 GeV, ǫ = FIG. 24: Simulated missing-mass-squared (M m ) distribution of H ( e, e ′ p ) X process at Q =
2, 3and 4 GeV. The π distributions are in black; the backward VCS is in magenta. A cut at M m = will be applied to minimize the VCS background. Simulation will be used toexamine/subtract the small level of VCS contamination directly underneath the π peak. J. Non-Physics Background
Once a combination of online hardware and offline software cuts had determined that there isa coincidence between an electron in the HMS and a proton in the SHMS, there remain severalbackgrounds of the incoherent non-physics variety: random coincidences and events from the endcaps of the target cell. The online coincidence rates are presented and described in Sec. V F, and1the online electronic coincidence resolving window will be roughly 70 ns.Offline, our excellent coincidence time resolution enables us to reduce the relevant resolvingtime to 2 ns with negligible inefficiency. This is the first level of suppression of random coinci-dences. A cut on the missing mass variable reduces the final random coincidence contaminationto the few percent level. The missing (or undetected residual) mass is reconstructed from the finalelectron and detected hadron 4-momenta: M = ( P e − P e ′ + P tar − P h ) . (25)The missing mass cut does a lot more than random coincidence reduction. To the extent thatparticle identification is flawless, real coincidences with larger inelasticity than p ( e, e ′ p ) π arecompletely removed. Finally, the model dependence of the experimentally determined cross sec-tions due to radiative effects are reduced as well.Both spectrometers will detect the aluminum target end windows in all configurations, so win-dow background subtractions are necessary. Because the aluminum windows are each 4 mil thick,the ratio of protons in the windows to protons in the liquid hydrogen is about 10%. However,based on the previous operational experience such as the F π -2 experiment, the surviving windowbackground: p ( e, e ′ π + ) n and p ( e, e ′ p ) ω , after cuts was found to be only 1% [19, 60]. The HallC “empty” target consists of two 40 mil thick aluminum windows separated by 8 cm, which cantolerate up to 30 µ A beam current. Thus, our “empty” data come in 3 times = (40 mil × µ A)/(4mil × µ A) faster than window events on the real target. The empty target measurement over-head will be about 10% of total data taking.
K. Systematic Errors
For all of the measurements proposed here, we have chosen the target length to be 10 cm. Thisis longer than the 6 cm used in L/T separation experiments in Hall C during the 6 GeV era. Usinga longer target is possible because of the larger SHMS y -target acceptance compared to the ShortOrbit Spectrometer (SOS). Even at very small angles, the extended target presents no problem forSHMS to project the background events from the target cell wall. The HMS y -target acceptancecould potentially be problematic since it will be used at rather large angles (up to ∼
37 degrees).We anticipate minimal extra uncertainty due to the use of the longer target.2TABLE VII: Estimated systematic uncertainties for the proposed π measurement. Thesystematic uncertainties in each column are added quadratically to obtain the total systematicuncertainty shown in the last row. The systematic uncertainty from the F π -2- ω analysis [19] isalso listed for comparison purpose. The π has smaller systematic uncertainties since it does notrequire multiple background fitting and subtraction from other neighboring mesons (such as the ρ underneath the ωpeak ) .Correction Uncorrelated ǫ Uncorrelated Correlated(Pt-to-Pt) u Correlated (scale)(%) (%) (%)SHMS+HMS Tracking 0.6 1.2SHMS+HMS Triggers 0.1SHMS/HMS Detectors 0.2Target Thickness 0.2 0.8CPU Live Time 0.2Electronic Live Time 0.2Coincidence Blocking 0.2Beam charge 0.5 0.5PID 0.2Acceptance 0.6 0.6 1.0Proton Interaction 1.0Radiative Corrections 0.3 1.5Kinematics Offset 0.4 1.0Model Dependence 0.7 π Total 1.0 1.4 2.5F π -2- ω Total 2.9 1.9 2.7TABLE VIII: Statistical error projection for each Q - ǫ setting. These estimates are based on thecross section model (presented in Table IX); and the assumed cross section ratio ( σ T /σ L ) basedon the previous u -channel ω analysis [19]. Q W ǫ ǫ ∆ ǫ δσ δσ δσ L δσ T σ T /σ L δ ( σ T /σ L ) (GeV ) (GeV) (%) (%) (%) (%) (%)2.0 2.11 0.52 0.94 0.42 3.0 2.5 25 12 2 202.0 3.00 0.32 0.79 0.47 5.6 5.5 46 12 2 253.0 2.49 0.54 0.86 0.32 3.0 2.6 70 12 5 204.0 2.83 0.55 0.73 0.18 3.5 3.1 800 20 30 405.0 3.13 0.27 0.55 0.28 8.6 7.8 8300 20 200 506.25 3.46 - 0.36 - - 8.5 - - - -The resulting anticipated systematic uncertainties are listed in Table VII. Our estimates arebased on the proven experience with the HMS+SOS during the 6 GeV era and HMS+SHMS3operation since the 12 GeV commissioning. L. Projected Error Bars, Rates and Time Estimation
The unseparated cross sections at low and high ǫ values: ǫ and ǫ , can be expressed in termsof the separated cross sections σ L and σ T , σ = σ T + ǫ σ L = σ T (1 + ǫ R ) (26) σ = σ T + ǫ σ L = σ T (1 + ǫ R ) (27)where σ and σ represent the unseparated cross sections at ǫ and ǫ , respectively; R is thetransverse-longitudinal (T-L) ratio defined as R = σ T σ L . (28)Through substitution and manipulation of equations above, σ T and σ L can be expressed interms of σ and σ : σ L = σ − σ ǫ − ǫ (29) σ T = σ ǫ − σ ǫ ǫ − ǫ (30)By differentiating σ L and σ T , the percentage errors can be expressed as, δσ T σ T (%) = 1 ǫ − ǫ s ǫ (cid:18) δσ σ (cid:19) (cid:16) ǫ R (cid:17) + ǫ (cid:18) δσ σ (cid:19) (cid:16) ǫ R (cid:17) (31) δσ L σ L (%) = 1 ǫ − ǫ s ǫ (cid:18) δσ σ (cid:19) ( R + ǫ ) + ǫ (cid:18) δσ σ (cid:19) ( R + ǫ ) (32)where δσ and δσ are the total statistical uncertainties of σ and σ , respectively. The errormagnification factor is / ( ǫ − ǫ ) .The determination of the running time at each Q - ǫ setting is dictated by the observable, shownin Figs. 25 (a) and 26, for testing the TDA hypothesis. The logic behind our decisions is as follows:1. For the σ T scaling shown in Fig. 25 (a), our goal is to demonstrate that σ T ( u ′ = 0) ∝ / ( Q ) n and with an uncertainty for n = 4 ± . , assuming the / ( Q ) scaling hypothesis.2. In order to demonstrate the dominance of σ T ≫ σ L separated cross section, i.e. the ratio R = σ T /σ L , the uncertainty δ ( σ L /σ T ) should be kept less than < .3. For a fixed Q , the statistical uncertainty balance between different epsilon settings, δσ and δσ ,after the acceptance and diamond cuts must be maintained. Note that all dataare divided in five different u bins, where δσ and δσ are the average value between theuncertainty at the lowest − u and the five-bin-average for low and high ǫ , respectively.The total time required for each Q - ǫ setting is listed in Table IX, this time will be sharedequally by the hadron arm angle settings. Times have been increased by 10% to account for datataking from the aluminum “dummy” target, needed to subtract contributions from the target cellwalls. The time required to complete the π measurement is 706 (PAC) hours. ) (GeV Q -
10 110 T s T s (cid:181) T s (cid:181) T s ) (GeV Q L s FIG. 25: σ T as function of Q . This is the figure of merit that demonstrates the second TDApostulation: σ T ∝ /Q . Our estimate of σ L as function of Q . VI. CLOSING REMARKS
In a short summary, the proposed exclusive electroproduction H ( e, e ′ p ) π measurement is astandard Hall C L/T experiment which utilizes the HMS+SHMS configuration, unpolarized LH µ A beam current is assumed. Note thatthe estimated time presented in the table includes ( π +Heep) dummy target running time (10%).Heep time is scaled up be additional 1 hours per setting inelastic optics study (verificationpurpose) on a carbon target. Four E beam polarization measurements are planned, onemeasurement at each beam energy. All planed measurements assume the standard acceleratorgradient at the time of running (here we assume 2.2 GeV/pass). Q W ǫ E
Beam [Pass] Physics Rate Background Rate PAC Time PAC Time(GeV ) (GeV) (GeV) (per Hour) (per Hour) (Hours) (Days)2.0 2.11 0.52 4.4 [2] 140 0.01 33 1.40.94 10.9 [5] 500 0.05 10 0.42.0 3.00 0.32 6.6 [3] 14 < < < < < < < < H ( e, e ′ p )
28 1.2 E Beam change 52 2.2Optics study 4 0.2 E Beam
Polar. 32 1.3Total Time 706 29.4target and standard accelerator gradient (at the time of running). The total run time is 706 PAChours (29.4 PAC days). Technically, the experiment is “straightforward”.We aim to perform L/T separation over a Q range of 2-5 GeV and two additional settings: Q =2 GeV ( W = 3 GeV) and Q =6.25 GeV to check the W and Q scaling.1. At each measurement setting, data are binned in five − u bins, with the lowest valued − u bin corresponding to − t = − t max . The observed − t dependence will reveal the anticipated u -channel π peak, which will facilitate the comparison to the forward-angle exclusive datafrom E12-13-010.2. Once the backward-angle peaks are confirmed, the separated cross sections: σ T and σ L at Q =
2, 3, 4 and 5 GeV will be extracted. The transverse size of interaction that is6 ) (GeV Q R a t i o L s T s FIG. 26: σ T /σ L ratio as function of Q . This is the figure of merit that demonstrates the firstTDA postulation: σ T ≪ σ L or σ T /σ L < /Q .determined from the u -dependence of the separated cross sections at each setting, whichprovides a good phenomenological handle to study the “soft-hard” transition in u -channelphysics.3. The anticipated σ T /σ L ratios, will test the TDA predicted dominance of σ T over σ L . Quan-titatively, the criteria for σ T ≫ σ L : σ T /σ L increases as a function of Q and reaches σ T /σ L > at Q = 5 GeV.4. The σ T ∝ /Q n scaling test for < Q < . GeV , especially, at Q = 6 . GeV isdependent on the σ T ≫ σ L from the above item. The scaling result will further validate theTDA factorization scheme and we expect to extract the exponent factor to an accuracy of δn = ± . ,With these results, one would conclude the backward-angle ( u -channel) factorization scheme of-fered by TDA is valid for the exclusive π electroprodcution channel.In our opinion, the proposed measurement is an example of how we can utilize the existingexperimental apparatus and refined techniques, but slightly tweak our perspective to explore alargely unknown kinematics territory. After this proposal, we anticipate a wave of u -channel7meson electroproduction and photoproduction measurements to emerge ( η , ρ , ω , φ , even J/ ψ ) andperhaps u -channel VCS.8 Appendix A: Monte Carlo model of Deep Exclusive π Production in u -channel The Monte Carlo studies needed for this proposal require a reaction model for an experimen-tally unexplored region of kinematics. This appendix describes the model and the constraints used.The differential cross section for exclusive π production from the nucleon can be written as d σdE ′ d Ω e ′ d Ω π = Γ V d σd Ω π . (A1)The virtual-photon flux factor Γ V is defined as Γ v = α π E ′ E KQ − ǫ , (A2)where α is the fine structure constant, K is the energy of real photon equal to the photon energyrequired to create a system with invariant mass equal to W and ǫ is the polarization of the virtual-photon. K = ( W − M p ) / (2 M p ) (A3) ǫ = (cid:18) | q | Q tan θ e (cid:19) − , (A4)where θ e is the scattering angle of scattered electron.The two-fold differential cross section d σd Ω π in the lab frame can be expressed in terms of theinvariant cross section in center of mass frame of the photon and nucleon, d σd Ω π = J d σdtdφ , (A5)where J is the Jacobian of transformation of coordinates from lab Ω π to t and φ (CM).In the one-photon exchange approximation, the unpolarized nucleon cross section for n ( e, e ′ π − ) p can be expressed in four terms. Two terms correspond to the polarization states ofthe virtual-photon (L and T) and two states correspond to the interference of polarization states(LT and TT), dσ UU = ǫ dσ L dt + dσ T dt + p ǫ ( ǫ + 1) dσ LT dt cos φ + ǫ dσ TT dt cos 2 φ, (A6)where φ is the angle between lepton plane and hadron plane (Fig. 17). The first two terms of9Eqn. A6 correspond to the polarization states of the virtual-photon (L and T) and last two termscorrespond to the interference of polarization states (LT and TT).The following data and calculations were used as constraints on the parameterizations used inthis model: • From Hall A,
L/T /LT /T T separated experimental data of exclusive electroproduction of π on H are available at x B = Q values ranging from 1.5 to 2GeV [55]. Of these three, we use only the data set at Q =1.75 GeV , as it spans the widest t -range, . < − t < . GeV [55]. • A GPD-based handbag-approach calculation by Goloskokov and Kroll [57] for the E12-13-010 proposal [4] at x B = Q =3.0, 4.0, 5.5 GeV [57].Since both of these data and calculations are for forward-angle kinematics, we used the fol-lowing prescription to obtain a crude model for the unique backward-angle kinematics proposedhere. • The t -dependence of the T/LT/TT structure functions at each Q were fitted with functionsof the form a + b/ ( − t ) , which gave good fits over the range − t min < − t < . GeV with aminimum of fit parameters. σ L displayed very little t -dependence over the region for whichthere were data, so it was simply taken as a small constant value with t (about 1 nb/GeV ,but with magnitude dropping as Q increases). • Since the electroproduction data in Fig. 6 display a forward to backward-angle peak ratio ofabout 10:1, we estimate the magnitude of the backward angle cross sections by switchingthe u -slope for t -slope in the above equations, and divide by ten. • Linear interpolation was performed between the parameterized values at fixed Q =1.75, 3.0,4.0, 5.5 GeV to obtain the L/T/LT/TT cross sections for the exact Q needed for each eventin the SIMC Hall C Monte Carlo simulation. • After the parameterization of σ L,T,LT,T T for − u and Q , we assume the same W dependenceas used in [2] for exclusive π + electroproduction at similar x B , which is ( W − M ) − where M is the proton mass.Clearly, this model can only be described as a ‘best guess’ of the actual DEMP π cross sec-tions in this unexplored regime. It is anticipated that some parasitic π backward angle data will0be acquired in the Hall C DEMP experiments, E12-09-011, E12-19-006, which can be used toimprove the crude model used here. Appendix B: Monte Carlo model of γ Production in u -channel The backward-angle VCS+BH model is based on the C++ code by C. Munoz Camacho and H.Moutarde (CEA-Saclay, IRFU/SPhN), which itself is derived from a Mathematica package writtenby P. Guichon and M. Vanderhaeghen.The Bethe-Heitler (BH) formulas are exact and valid over the full kinematic range of the ex-periment. As shown in Fig. 23, the BH process does not exhibit a backward-angle peak, and hasminimal contribution in the large − t probed by this experiment.For the Virtual Compton Scattering (VCS) process, the code makes use of a file of ComptonForm Factors which were computed by Guichon and Vanderhaeghen in a grid over the range: . In this section, we attempt to connect the proposed π measurement with other approved π measurements. It is interesting that the combined kinematics coverage from different experimentsoffer a complete − t evolution at certain kinematics settings. Also note that these measurementsutilize direct detection method (of the decayed photons) to reconstruct the π events, in contrast tothe missing mass technique utilized in this proposal.There were two approved π production oriented measurements at Hall C, see their brief de-scription as follows: E12-13-010 and E12-06-114 Hall C E12-13-010 proposed by the Neutral Particle Spectrometer(NPS) Collaboration in PAC 41, is a dedicated measurement that will map out the DVCSand exclusive π electroproduction cross sections at various bins of x B , W and Q . Its1main physics objective involves studying GPDs and establish the applicable range of theforward-angle factorization scheme. At the same kinematic settings as this proposal, x B =0 . , Q = 3 and GeV , the NPS will offer L/T separated cross sections at low − t , inparticular, − t = − t min or − t ′ = 0 GeV . Note that in this kinematics region − u = − u max .This is sometimes referred to as the “low − t region”. Hall A E12-06-114 experiment hasthe same physics goal and similar kinematics coverage as E12-13-010, which will offercomplementary data set. The full L/T separation for all settings depends on the completionof both programs. E12-14-005 is a companion program to the Wide Angle Compton scattering experiment (WACS),and it focuses on the detection of π photoproduction at a range of large scattering angles.In the range of s : . < s < . GeV , the NPS will be used to detect π produced in ◦ to ◦ in CM angle. In this region, u ≈ t . This is sometimes referred to as the “high − t ≈ − u transition region”. Large Angle π Measurement at CLAS 12 With its large acceptance, the default CLAS 12 de-tector package is capable of extracting π cross sections up to a large angle ( < θ < ◦ in the lab frame) at the same kinematics setting as the proposed measurement. These re-sults would serve as an even better complementary data set which allows one to study withhigh precision the hard → soft → hard structure transition, which corresponds to forward-angle-peak → wide-angle-plateau → backward-angle-peak in the measured cross section.Interestingly, the measurement of BSA and cross section ( φ ) modulation sensitive to σ LT has already generated interesting physics insight, see Sec. II C.As described in the main section of the proposal, this proposal π has unique − t = − t max or − u = − u min kinematics (a low u region). No other physics measurement acquires such kine-matics at JLab. The combined − t dependence (two mentioned π measurements along with thisproposal) will yield a complete − t evolution (after appropriate kinematics correction). Such ex-ample is shown in Fig. 6: the forward-peak, the wide-angle-plateau and the backward-peak willbe projected on the same − t axis. These combined features will significantly reduce the Regge-model uncertainty in any description of proposed data, specially at the overlapped Q = 3 and GeV settings.2 Appendix D: Further Details on u Channel Workshop in September 2020 The first backward-angle ( u -channel) physics workshop is set to take place in Jefferson Lab inSeptember 21-22, 2020. The scientific program is expected to comprise roughly 15 talks. If anin-person meeting is not possible, the workshop will be held virtually.The objectives of the workshop are as follows: • Offer a platform to connect scattered experiment and theory efforts together, thus, poten-tially forming small backward-angle physics working groups. • Generate discussions on the implications the backward-angle physics and probe the physicscase for a systematic backward-angle physics research program. • Inspire future backward-angle physics data mining or dedicated studies, including the JLab12 GeV program, and rmP ANDA/FAIR. • Discuss the feasibility of including backward-angle physics in the EIC scientific program.JLab event page: u Channel Workshop Agenda and Speakers List Tentative list of discussion topics and speakers are listed below. A up-to-date agenda can alsobe found on the JLab indico page: . • Meson-Nucleon Transition Distribution Amplitude, Lech Szymanowski (NCNR, Poland). • Studying Vector Meson Electroproduction at Large Momentum Transfer, J-M Laget (Jeffer-son Lab, USA). (Not confirmed) • Regge Phenomenology through u -channel Processes , Christian Weiss (Jefferson Lab,USA). • Backward Charged π + Electroproduction from CLAS 6, Kijun Park (Hampton UniversityProton Therapy Institute, USA). • Backward Exclusive ω Electroproduction from JLab 6 GeV Hall C, Garth Huber (Universityof Regina, Canada).3 • Backward Meson Electroproduction from JLab 12 GeV Hall C Kaon LT Experiment,Stephen Kay (University of Regina, Canada). • Backward Meson Electroproduction Opportunities at EIC, Wenliang (Bill) Li (William andMary, USA). • Backward Opportunities at JLab 12 GeV Hall A with BigBite and Super BigBite Spectrom-eters, Carlos Ayerbe (Mississippi State, USA). • Large Angle π Production at CLAS 12, Stefan Diehl (University of Connecticut, USA). • Studying TDA with pp → e + e − π at the P ANDA Experiment, Stefan Diehl (University ofConnecticut, USA). • Wide Compton Scattering at Hall C, Bogdan Wojtsekhowski (Jefferson Lab, USA) • ω Photoproduction off Proton Target at Backward Angles, B.-G. 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