Elastic Positron-Proton Scattering at Low Q^2
Tyler J. Hague, Dipangkar Dutta, Douglas W. Higinbotham, Xinzhan Bai, Haiyan Gao, Ashot Gasparian, Kondo Gnanvo, Vladimir Khachatryan, Mahbub Khandaker, Nilanga Liyanage, Eugene Pasyuk, Chao Peng, Weizhi Xiong, Jingyi Zhou
EEur. Phys. J. A manuscript No. (will be inserted by the editor)
Elastic Positron-Proton Scattering at Low Q Tyler J. Hague , Dipangkar Dutta , Douglas W. Higinbotham ,Xinzhan Bai , Haiyan Gao , Ashot Gasparian , Kondo Gnanvo ,Vladimir Khachatryan , Mahbub Khandaker , Nilanga Liyanage ,Eugene Pasyuk , Chao Peng , Weizhi Xiong , Jingyi Zhou North Carolina A&T State University, Greensboro, NC 27411 Jefferson Lab, Newport News, VA 23601 Department of Physics and Astronomy, Mississippi State University, Starkville, MS 39762, USA Department of Physics, Duke University, Durham, NC 27708, USA Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA Department of Physics, University of Virginia, Charlottessville, VA 22904, USA Energy Systems, Davis, CA 95616, USA Physics Division, Argonne National Laboratory, Lemont, IL 60439, USA Department of Physics, Syracuse University, Syracuse, NY 13244, USAReceived: date / Accepted: date
Abstract
Systematic differences in the the proton’scharge radius, as determined by ordinary atoms andmuonic atoms, have caused a resurgence of interest inelastic lepton scattering measurements. The proton’scharge radius, defined as the slope of the charge formfactor at Q =0, does not depend on the probe. Anydifference in the apparent size of the proton, when de-termined from ordinary versus muonic hydrogen, couldpoint to new physics or need for the higher order cor-rections. While recent measurements seem to now bein agreement, there is to date no high precision elas-tic scattering data with both electrons and positrons.A high precision proton radius measurement could beperformed in Hall B at Jefferson Lab with a positronbeam and the calorimeter based setup of the PRad ex-periment. This measurement could also be extendedto deuterons where a similar discrepancy has been ob-served between the muonic and electronic determina-tion of deuteron charge radius. A new, high precisionmeasurement with positrons, when viewed alongsideelectron scattering measurements and the forthcomingMUSE muon scattering measurement, could help pro-vide new insights into the origins of the proton radiuspuzzle, and also provide new experimental constraintson radiative correction calculations. Keywords
Elastic Scattering · Proton Radius · Positrons a e-mail: [email protected] Elastic lepton scattering at low four-momentum trans-fer can be used to determine the charge and magneticradii of a nucleus. For the special case of a lepton scat-tering from a spin-1/2 nucleus, such as the proton or He, the cross section for the scattering process can bewritten as σ = σ Mott × (cid:34) G E (cid:0) Q (cid:1) + τ G M (cid:0) Q (cid:1) τ + 2 τ G M (cid:0) Q (cid:1) tan (cid:18) θ (cid:19)(cid:35) ,(1)where G E ( Q ) and G M ( Q ), are the charge and mag-netic form factors, Q is the four momentum transfersquared, and m is the mass of the nucleus, and τ = Q m .By making measurements with multiple energies butover the same Q ranges, the form factors G E ( Q ) and G M ( Q ) can be determined. In the limit of Q = 0,these form factors can be used to extract the chargeradius, r E , and magnetic radius, r M , of the proton: r pE = (cid:32) − G pE ( Q )d Q (cid:12)(cid:12)(cid:12)(cid:12) Q =0 (cid:33) / ,r pM = (cid:32) − µ p d G pM ( Q )d Q (cid:12)(cid:12)(cid:12)(cid:12) Q =0 (cid:33) / (2)where µ p is the magnetic moment of the proton. It is im-portant to note, that this definition of the proton’s ra-dius is consistent with the definition used by the atomic a r X i v : . [ nu c l - e x ] F e b and muonic lamb shift measurements [1]. Experimen-tally, the data cannot extend to exactly Q = 0, thusvarious methods of extrapolation are employed. To min-imize the model dependence of these extrapolation, itis desirable for experiments to measure at Q as lowas achievable and over a sufficiently large Q range.To best understand this effect, the experiment ideallywould be performed with both electrons and positrons.In 2010, Lamb shift measurements in muonic hydro-gen ( µ H) [2,3] with their unprecedented < r pE that was a combined eight-standarddeviations smaller than the average value from all pre-vious experiments. This discrepancy triggered the “pro-ton radius puzzle” [4,5]. The puzzle prompted new scat-tering experiments [6,7,8] and numerous reanalyses ofexisting electron scattering data [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23].The most recent electron scattering [24] and atomichydrogen spectroscopy [25] results seem to be in agree-ment with the µ H results [26]. Nonetheless, the new re-sults do not rule out one of the original explanations forthe proton radius puzzle [4], a fundamental differencebetween electrons and muons that violates lepton uni-versality. Previous experiments, performed in the ’70sand ’80s, showed that lepton universality holds at the10% level [27,28]. While this is often theoretically ac-cepted to be true in the standard model, there has yetto be any experimental validation. The MUSE exper-iment [29,30], which has begun running at PSI, maybe able to determine if universality holds, and thus ifthe proton radius puzzle is truly solved. However, itis highly desirable to verify the results from MUSEwith high precision measurements with electrons andpositrons.The PRad experiment [24] has credibly demonstratedthe advantages of the calorimetric method in e − p scat-tering experiments to measure r pE with high accuracy.An upgraded experiment (PRad-II), which will reducethe overall experimental uncertainties by a factor of 3.8compared to PRad has recently been proposed. ThePRad setup could also be used with a positron beamto measure r pE with high precision and thereby helpverify lepton universality in the electron sector withsub-percent precision. In addition, it would allow us tovalidate the radiative correction calculations for elec-tron scattering that account for internal and externalBremsstrahlung suffered by the incident and scatteredelectrons and contributions from two-photon exchange(TPE) processes. The original PRad experiment was designed to use amagnetic-spectrometer-free, calorimeter based method [24].The innovative design of the PRad experiment enabledthree major improvements over previous e − p exper-iments: (i) The large angular acceptance (0 . ◦ − . ◦ )of the hybrid calorimeter (HyCal) allowed for a large Q coverage spanning two orders of magnitude (2 . × − − × − ) (GeV / c) , in the low Q range. The sin-gle fixed location of HyCal eliminated the multitude ofnormalization parameters that have affected magneticspectrometer based experiments, where the spectrome-ter must be physically moved to many different anglesto cover the desired range in Q . In addition, the PRadexperiment reached extreme forward scattering anglesdown to 0 . ◦ achieving the lowest Q (2 . × − (GeV / c) ) in e − p experiments, an order of magni-tude lower than previously achieved. Reaching a lower Q range is critically important since r pE is extractedas the slope of the measured G pE ( Q ) at Q = 0.(ii) The extracted e − p cross sections were normalizedto the well known quantum electrodynamics process - e − e − → e − e − Møller scattering from the atomic elec-trons - which was measured simultaneously with the e − p within the same detector acceptance. This leadsto a significant reduction in the systematic uncertain-ties of measuring the e − p cross sections. (iii) Thebackground generated from the target windows, one ofthe dominant sources of systematic uncertainty for allprevious e − p experiments, is highly suppressed in thePRad experiment.The PRad experimental apparatus consisted of thefollowing four main elements (see Fig. 1): (i) a 4-cm-long, windowless, cryo-cooled hydrogen (H ) gas flowtarget with a density of 2 × atoms/cm . It elim-inated the beam background from the target windowsand was the first such target used in non storage-ring e − p experiments; (ii) the high resolution, large accep-tance HyCal electromagnetic calorimeter [32,33]. Thecomplete azimuthal coverage of HyCal for the forwardscattering angles allowed simultaneous detection of thepair of electrons from e − e scattering, for the firsttime in these types of measurements; (iii) a single planeof coordinate detectors made of two high resolution X − Y gas electron multipliers (GEM) located in frontof HyCal; and (iv) a two-section vacuum chamber span-ning the 5.5 m distance from the target to the detectors.The PRad experiment was the first electron scat-tering experiment to utilize a new technique with com-pletely different systematics compared to all previousmagnetic-spectrometer-based e − p experiments. Thefirst generation PRad experiment was able to determine Fig. 1
A schematic layout of the PRad experimental setup in Hall B at Jefferson Lab, with the electron beam incident fromthe left. The key beam line elements are shown along with the windowless hydrogen gas target, the two-segment vacuumchamber, and the two detector systems, GEM and HyCal.
Fig. 2
The proposed experimental setup for PRad-II. [31] the proton radius to ± . stat ± . syst fm [34,24].The PRad experiment has convincingly demonstratedthe validity and advantage of the new calorimetric tech-nique, but further improvements are possible. The second generation experiment – PRad-II, whichwill reduce the overall experimental uncertainties by afactor of 3.8 compared to PRad, has been approved bythe JLab 2020 Program Advisory Committee (PAC)with an A rating. PRad-II will be the first lepton scat-tering experiment to reach the Q range of 10 − GeV allowing a more accurate and robust extraction of theproton radius. This new experiments will push the pre-cision of the proton radius extraction to 0.003 fm, allow- ing it to address possible systematic difference between e − p and the µ H experiments.
Fig. 3
A schematic of the cylindrical recoil detector consist-ing of 20 silicon strip detector modules, held inside the targetcell.
Additionally, a proposal for a high precision elas-tic e − d scattering cross section measurement (DRad) atvery low scattering angles, θ e = 0 . ◦ − . ◦ ( Q = 2 × − to 5 × − (GeV / c) ), using the PRad-II experimentalsetup has also been submitted to the 2020 PAC. Thisexperiment has one major modification to the PRad-IIsetup. To ensure the elasticity of the e − d scatteringprocess a low energy Si-based cylindrical recoil detectorwill be included within the windowless gas flow targetcell (see Fig. 3). As in the PRad experiment, to con-trol the systematic uncertainties associated with mea-suring the absolute e − d cross section, a well knownQED process, e − e Møller scattering, will be simulta-neously measured in this experiment. The DRad exper-iment will provide a new measurement of the deuteronradius with a precision of 0.4%. [35] Q [fm ] G E Pseudo DataRational Function
Fig. 4
Shown are the expected precision of e − p elastic scat-tering in Hall B using the PRad experimental setup. Data ofthis quality, would allow the proton radius to be extractedusing a low order rational function and would achieve a pre-cision approximately ± Jefferson Lab, with a positron beam, would be idealfor performing a high precision follow-up experimentto MUSE and the PRad family of experiments. Thesetup used for the PRad-II experiment in Hall B couldbe reused to measure the cross sections and extractthe proton radius, thereby verifying whether the pro-ton radius is identical when measured with electronsand positrons.Positrons, being the same mass as electrons, followthe same scattering kinematics as electron-proton scat-tering. Thus, the success of the PRad experiment and Q [fm ] G E Pseudo DataRational Function
Fig. 5
Log scale version of Fig. 4 to highlight the low Q data. expected results of PRad-II and DRad serve as proof-of-concept for the proposed experiment.The proposed experiment would use the same setupas the PRad-II and DRad experiments, as shown inFig. 2. By using this improved setup, a positron scat-tering measurement of the proton radius would be ableto reach a minimum Q in the range of 10 − GeV .This setup also provides ready-made solutions to sev-eral systematic issues that existed in the original PRadexperiment.The PRad-II setup improves on the PRad setup bythe use of two novel spacerless GEM detectors and alarger angle scintillator detector. This setup pushes Hy-Cal back by 40 cm to make room for the second GEMdetector. The spacerless GEM technology will greatlyreduce the inefficiencies that particularly affected thevery forward scattering angles. Using two GEM detec-tors will also allow them to use each other to determinetheir efficiencies and avoid using the lower position res-olution of HyCal for calibration. A helium bag will spanthe 40 cm distance between these detectors. This spac-ing will allow for accurate target- z resolution (wherethe interaction occurred along the length of the target),which will help to mitigate beamline backgrounds.As scattering angle decreases, the energy carried byelectrons in both e − p and e − e scattering approach avalue that is indistinguishable within the resolution ofHyCal. This places a limit on the lowest usable scatter-ing angles and, consequentially, lowest Q measurableas shown in Fig. 6. A solution to this is to use double-arm Møller scattering. At low scattering angles, how-ever, the higher angle Møller electron is outside of theacceptance of the main PRad setup. The PRad-II ex-periment will add a scintillator detector 25 cm fromthe target in a cross shape. This addition will allow forthe detection of the high scattering angle Møller elec-trons. Kinematic selection with these high scattering Fig. 6
The reconstructed energy vs angle for e − p and e − e events for the electron beam energy of 2.2 GeV. The red andblack lines indicate the event selection for e − p and e − e ,respectively. The angles ≤ . ◦ are covered by the PbWO crystals and the rest by the Pb-glass part of HyCal. [31] angle electrons will allow for clear discernment between e − p and e − e scattering. The PRad-type experiments have and will measure e − e Møller scattering (see Fig. 7), a well known QEDprocess, simultaneously with the e − p cross sectionmeasurements. Being exactly calculable in the next-to-leading order makes the e − e process an ideal candidatefor monitoring the luminosity and controlling system-atic uncertainties of the experiments. As an analog tothis, the proposed positron scattering experiment woulduse e + e − → e + e − Bhabha scattering (see Fig. 8) as aluminosity monitor.The PRad-II detector setup is designed so that itcould measure both e − − p and e − − e − scattering si-multaneously. The Bhabha scattering process followsidentical kinematics to Møller scattering. Thus, a suc-cessful simultaneous measurement of e − − p ( e − − d )and e − − e − scattering in the PRad-II and DRad exper-iments will serve to prove the success of a simultaneous e + − p ( e + − d ) and e + − e − measurement. By measur-ing an exactly calculable process at the same time asthe elastic cross section, uncertainties associated withoverall normalization are kept to a minimum. The analysis of this data is a two-step process: the e + − p elastic cross section extraction and the proton radius e − e − e − e − e − e − e − e − Fig. 7
Feynman diagrams of the leading order Møller scat-tering e + e + e − e − e + e − e + e − Fig. 8
Feynman diagrams of the leading order Bhabha scat-tering extraction. The cross section extraction requires a real-istic simulation of both e + − p elastic scattering and of e + − e − Bhabha scattering. The PRad experiment useda bin-by-bin Møller technique, which is analogous toa bin-by-bin Bhabha technique. This technique allowsfor the cancellation of the inefficiencies of the GEM, butintroduces a Q -dependent uncertainty to the measure-ment. With the improved efficiencies of the spacerlessGEM detectors, an integrated Bhabha technique couldbe used to normalize the e + − p elastic cross section.This involves using the Bhabha scattering measurementfrom the entire detector to normalize each e + − p crosssection bin. This technique does not allow for the can-cellation of the GEM efficiencies, but would make any Q -dependent uncertainties into overall normalizationuncertainties. By implementing the integrated Bhabhatechnique, the e + − p elastic cross section would be cal-culated as: (cid:18) d σ d Ω (cid:19) Born e + p ( θ i ) = N e + p ( θ i ) ε sim e + p N sim e + p ( θ i ) ε e + p × (cid:20) N sim e + e − (PbO ) ε e + e − N e + e − (PbO ) ε sim e + e − (cid:21) (cid:18) d σ d Ω (cid:19) sim,Born e + p ( θ i ) ,(3) where N are the counts of the respective processes fromdata and simulation without radiative corrections and ε are the efficiencies and acceptances of the processesfrom data and simulation. The e + − e − associated valuesare integrated over the PbWO region of HyCal to forman overall normalization applied to all bins.After the cross section has been calculated, the elec-tric form factor, G pE is extracted from the Rosenbluthformula (Eq. 1) by assuming a model for G pM . This formfactor must then be fit in order to determine the slopeas Q →
0. As this is an extrapolation, great care mustbe taken to ensure that the functional form used yieldsan unbiased extraction of r pE . A study has shown thatthe Rational (1 ,
1) function, f (cid:0) Q (cid:1) = nG E (cid:0) Q (cid:1) = n p Q p Q , (4)yields consistent results with minimal uncertainties [34].This study generated pseudo-data from various modelsof G pE and fit the pseudo-data with several proposedfunctional forms. The goodness of each fit was then de-termined by the Root Mean Square Error, RMSE = (cid:112) bias + σ . Once the data has been fit, Eq. 4 can becombined with Eq. 2 to yield a charge radius of r E = (cid:112) p − p ). (5)The deuteron is a spin-1 nucleus, which means thatan equation for the cross section different from Eq. 1 isnecessary. The cross section for e − d elastic scatteringis given by σ = σ Mott × (cid:20) A d (cid:0) Q (cid:1) + B d (cid:0) Q (cid:1) tan (cid:18) θ (cid:19)(cid:21) , (6)where A d and B d are structure functions that are de-fined in terms of three form factors (as opposed to twoin the spin-1/2 case): the deuterons charge ( G dC ), mag-netic dipole ( G dM ), and electric quadrupole ( G dQ ) formfactors. The structure functions and form factors arerelated by A d (cid:0) Q (cid:1) = (cid:0) G dC (cid:0) Q (cid:1)(cid:1) + 23 τ (cid:0) G dM (cid:0) Q (cid:1)(cid:1) +89 τ (cid:0) G dQ (cid:0) Q (cid:1)(cid:1) , B d (cid:0) Q (cid:1) = 43 τ (1 + τ ) (cid:0) G dM (cid:0) Q (cid:1)(cid:1) . (7)As in the proton case, this cross section would be nor-malized to Bhabha scattering that is measured simul-taneously. The e + − d cross section would be extractedfrom the data using an analog to Eq. 3 where the e + − p terms are replaced by e + − d terms. The deuteron charge radius is then given by r dE = (cid:32) − G dC ( Q )d Q (cid:12)(cid:12)(cid:12)(cid:12) Q =0 (cid:33) / . (8)As this is calculated in the Q → Q as possible. To extract G dC from the measuredcross section, data-driven models of G dM and G dQ wouldbe used. The models that would be used can be exploredin Appendix A of Ref. [35]. A thorough analysis of radiative corrections is neces-sary to minimize systematic uncertainties on the finalmeasurement. The PRad experiment has estimated theradiative correction related systematic uncertainty onthe extracted r pE to be δr pE = 0 . e − p and e − e Møller scatterings. These areobtained within a covariant formalism and beyond theultrarelativistic approximation [36], as well as using themethod of Ref. [37] for evaluation of the contributioncoming from higher order corrections. The estimatedsystematic uncertainties for both e − p and e − e scat-terings are correlated and Q -dependent, where the Q -dependence is much larger for the corrections from theMøller process.The PRad-II experiment will reduce the overall ex-perimental uncertainties by a factor of 3.8 compared toPRad [31]. In order to succeed in this goal, it is nec-essary to perform radiative correction calculations tobeyond the next-to-leading order and beyond the ul-trarelativistic limit. This step is crititcal in order tosufficiently reduce the systematic uncertainty on r pE as-sociated with radiative corrections. Plans have alreadybeen put forward and detailed by PRad’s theory col-leagues to perform improved calculations at the next-to-next leading order level for elastic e − p and Møllerscatterings beyond ultrarelativistic limit for the PRad-II kinematics (including the contributions of the two-photon exchange processes). This work will employ newcalculation methods which they will develop in collabo-ration with a PSI-based group that is performing inde-pendent calculations. Similar radiative correction cal-culations will also be performed by PRad’s theory col-leagues for e − d scattering pertaining to the DRad ex-periment.Based on PRad’s results, it is clear that the system-atic uncertainties associated with radiative corrections stemming from e + − p ( e + − d ) and e + − e − Bhabha scat-tering processes would also require thorough and metic-ulous studies. In this case, it is also critical to calculatethem at and beyond the next-to-leading order and be-yond the ultrarelativistic limit for these scattering pro-cesses. We believe that this goal may be accomplishedafter our theory colleagues successfully obtain the cor-responding results for PRad-II. The methods that aredeveloped for calculations of the higher order radiativecorrections in elastic e − p and Møller scatterings should,in principle, be applicable to calculations for e + − p andBhabha scatterings. Other approaches are also possiblefor achieving this goal, such as using the method de-veloped in Ref. [38] for numerical calculations of thesecond-order leptonic radiative corrections for lepton-proton scattering. We can additionally use the meth-ods developed in Refs. [39,40,41,42] for calculating thetwo-loop corrections to Bhabha scattering in the casethat these methods are applicable to the kinematics ofthe PRad/PRad-II setup. The impact of this measurement is largely dependent onthe findings of the PRad-II, DRad, and MUSE exper-iments. If the proton radius puzzle is not solved, thenlepton universality will still be in question. This datawould then provide a measure of the extent to whichlepton measurements differ with electrons, positrons,and muons.On the other hand, it may be found that the protonradius puzzle is solved and that lepton universality stillholds. If that is the case, then this data would be anideal measure of positron radiative corrections, specif-ically internal and external Bremsstrahlung as well astwo-photon exchange processes. Precise knowledge ofthe proton radius would allow for use of that as a fixedparameter for determining the correct radiative correc-tions to be applied.
Using the PRad setup in Hall B would allow for an ex-tremely precise comparison of the proton and deuteronradii as extracted from positrons and electrons along-side world muon data. While currently the initial protonradius puzzle seems to be solved, there is still a hint at adifference between muonic and atomic results which canonly be resolved with higher precision experiments. Inaddition, even if the proton radius puzzle is solved, ourunderstanding of radiative corrections and two-photon exchange processes can be improved by studying thedifferences between electrons and positrons.
Acknowledgements
This work is supported in part by theU.S. Department of Energy, Office of Science, Office of Nu-clear Physics under contract DE-FG02-03ER41231 and DE-AC05-060R23177. This work is supported in part by NSFGrant NSF PHY-1812421.
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