Demonstration of a novel technique to measure two-photon exchange effects in elastic e ± p scattering
M. Moteabbed, M. Niroula, B.A. Raue, L.B. Weinstein, D. Adikaram, J. Arrington, W.K. Brooks, J. Lachniet, Dipak Rimal, M. Ungaro, K.P. Adhikari, M. Aghasyan, M.J. Amaryan, S. Anefalos Pereira, H. Avakian, J. Ball, N.A. Baltzell, M. Battaglieri, V. Batourine, I. Bedlinskiy, R. P. Bennett, A.S. Biselli, J. Bono, S. Boiarinov, W.J. Briscoe, V.D. Burkert, D.S. Carman, A. Celentano, S. Chandavar, P.L. Cole, P. Collins, M. Contalbrigo, O. Cortes, V. Crede, A. D'Angelo, N. Dashyan, R. De Vita, E. De Sanctis, A. Deur, C. Djalali, D. Doughty, R. Dupre, H. Egiyan, L. El Fassi, P. Eugenio, G. Fedotov, S. Fegan, R. Fersch, J.A. Fleming, N. Gevorgyan, G.P. Gilfoyle, K.L. Giovanetti, F.X. Girod, J.T. Goetz, W. Gohn, E. Golovatch, R.W. Gothe, K.A. Griffioen, M. Guidal, N. Guler, L. Guo, K. Hafidi, H. Hakobyan, C. Hanretty, N. Harrison, D. Heddle, K. Hicks, D. Ho, M. Holtrop, C.E. Hyde, Y. Ilieva, D.G. Ireland, B.S. Ishkhanov, E.L. Isupov, H.S. Jo, K. Joo, D. Keller, M. Khandaker, A. Kim, F.J. Klein, S. Koirala, A. Kubarovsky, V. Kubarovsky, S.E. Kuhn, S.V. Kuleshov, S. Lewis, H.Y. Lu, M. MacCormick, I .J .D. MacGregor, D. Martinez, M. Mayer, B. McKinnon, T. Mineeva, M. Mirazita, V. Mokeev, R.A. Montgomery, K. Moriya, H. Moutarde, E. Munevar, C. Munoz Camacho, et al. (50 additional authors not shown)
aa r X i v : . [ nu c l - e x ] J u l Demonstration of a novel technique to measure two-photon exchange effects in elastic e ± p scattering M. Moteabbed, M. Niroula, B.A. Raue, L.B. Weinstein, D. Adikaram, J. Arrington, W.K. Brooks, J.Lachniet, Dipak Rimal, M. Ungaro,
5, 6
A. Afanasev, K.P. Adhikari, M. Aghasyan, M.J. Amaryan, S.Anefalos Pereira, H. Avakian, J. Ball, N.A. Baltzell,
3, 35
M. Battaglieri, V. Batourine, I. Bedlinskiy, R.P. Bennett, A.S. Biselli, J. Bono, S. Boiarinov, W.J. Briscoe, V.D. Burkert, D.S. Carman, A. Celentano, S. Chandavar, P.L. Cole,
18, 6
P. Collins,
11, 7
M. Contalbrigo, O. Cortes, V. Crede, A. D’Angelo,
22, 33
N. Dashyan, R. De Vita, E. De Sanctis, A. Deur, C. Djalali, D. Doughty,
13, 6
R. Dupre, H. Egiyan,
6, 28
L. El Fassi, P. Eugenio, G. Fedotov,
35, 34
S. Fegan, R. Fersch, ∗ J.A. Fleming, N. Gevorgyan, G.P. Gilfoyle, K.L. Giovanetti, F.X. Girod,
6, 12
J.T. Goetz, W. Gohn, E. Golovatch, R.W. Gothe, K.A. Griffioen, M. Guidal, N. Guler, † L. Guo,
1, 6
K. Hafidi, H. Hakobyan,
4, 39
C. Hanretty,
37, 16
N. Harrison, D. Heddle,
13, 6
K. Hicks, D. Ho, M. Holtrop, C.E. Hyde, Y. Ilieva,
35, 17
D.G. Ireland, B.S. Ishkhanov, E.L. Isupov, H.S. Jo, K. Joo, D. Keller, M. Khandaker, A. Kim, F.J. Klein, S. Koirala, A. Kubarovsky,
V. Kubarovsky,
6, 31
S.E. Kuhn, S.V. Kuleshov,
4, 24
S. Lewis, H.Y. Lu,
10, 35
M. MacCormick, I .J .D. MacGregor, D. Martinez, M. Mayer, B. McKinnon, T. Mineeva, M. Mirazita, V. Mokeev,
6, 34
R.A. Montgomery, K. Moriya, ‡ H. Moutarde, E. Munevar,
6, 17
C. Munoz Camacho, P. Nadel-Turonski,
6, 17
R. Nasseripour,
25, 35
S. Niccolai, G. Niculescu, I. Niculescu, M. Osipenko, A.I. Ostrovidov, L.L. Pappalardo, R. Paremuzyan, § K. Park,
6, 26
S. Park, E. Phelps, J.J. Phillips, S. Pisano, O. Pogorelko, S. Pozdniakov, J.W. Price, S. Procureur, D. Protopopescu, A.J.R. Puckett, M. Ripani, G. Rosner, P. Rossi, F. Sabati´e, M.S. Saini, C. Salgado, D. Schott, R.A. Schumacher, E. Seder, H. Seraydaryan, Y.G. Sharabian, E.S. Smith, G.D. Smith, D.I. Sober, D. Sokhan,
36, 14
S. Stepanyan, S. Strauch, W. Tang, C.E. Taylor, Ye Tian, S. Tkachenko,
37, 2
H. Voskanyan, E. Voutier, N.K. Walford, M.H. Wood,
9, 35
N. Zachariou, L. Zana, J. Zhang,
6, 2
Z.W. Zhao,
37, 35 and I. Zonta ¶ (The CLAS Collaboration) Florida International University, Miami, Florida 33199, USA Old Dominion University, Norfolk, Virginia 23529, USA Argonne National Laboratory, Argonne, Illinois 60439, USA Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V Valpara´ıso, Chile University of Connecticut, Storrs, Connecticut 06269, USA Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA Arizona State University, Tempe, Arizona 85287, USA California State University, Dominguez Hills, Carson, California 90747, USA Canisius College, Buffalo, New York 14208, USA Carnegie Mellon University, Pittsburgh Pennsylvania 15213, USA Catholic University of America, Washington, D.C. 20064, USA CEA, Centre de Saclay, Irfu/Service de Physique Nucl´eaire, 91191 Gif-sur-Yvette, France Christopher Newport University, Newport News, Virginia 23606, USA Edinburgh University, Edinburgh EH9 3JZ, United Kingdom Fairfield University, Fairfield, Connecticut 06824, USA Florida State University, Tallahassee, Florida 32306, USA The George Washington University, Washington, DC 20052, USA Idaho State University, Pocatello, Idaho 83209, USA INFN, Sezione di Ferrara, 44100 Ferrara, Italy INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy INFN, Sezione di Genova, 16146 Genova, Italy INFN, Sezione di Roma Tor Vergata, 00133 Rome, Italy Institut de Physique Nucl´eaire ORSAY, Orsay, France Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia James Madison University, Harrisonburg, Virginia 22807, USA Kyungpook National University, Daegu 702-701, Republic of Korea LPSC, Universite Joseph Fourier, CNRS/IN2P3, INPG, Grenoble, France University of New Hampshire, Durham, New Hampshire 03824, USA Norfolk State University, Norfolk, Virginia 23504, USA Ohio University, Athens, Ohio 45701, USA Rensselaer Polytechnic Institute, Troy, New York 12180, USA University of Richmond, Richmond, Virginia 23173, USA Universita’ di Roma Tor Vergata, 00133 Rome Italy Skobeltsyn Nuclear Physics Institute, 119899 Moscow, Russia University of South Carolina, Columbia, South Carolina 29208, USA University of Glasgow, Glasgow G12 8QQ, United Kingdom University of Virginia, Charlottesville, Virginia 22901, USA College of William and Mary, Williamsburg, Virginia 23187, USA Yerevan Physics Institute, 375036 Yerevan, Armenia (Dated: September 4, 2018)
Background:
The discrepancy between proton electromagnetic form factors extracted using unpolarized and polarized scat-tering data is believed to be a consequence of two-photon exchange (TPE) effects. However, the calculations of TPEcorrections have significant model dependence, and there is limited direct experimental evidence for such corrections.
Purpose:
The TPE contributions depend on the sign of the lepton charge in e ± p scattering, but the luminosities of secondarypositron beams limited past measurement at large scattering angle where the TPE effects are believe to be most significant.We present the results of a new experimental technique for making direct e ± p comparisons, which has the potential tomake precise measurements over a broad range in Q and scattering angles. Methods:
We use the Jefferson Lab electron beam and the Hall B photon tagger to generate a clean but untagged photonbeam. The photon beam impinges on a converter foil to generate a mixed beam of electrons, positrons, and photons. Achicane is used to separate and recombine the electron and positron beams while the photon beam is stopped by a photonblocker. This provides a combined electron and positron beam, with energies from 0.5 to 3.2 GeV, which impinges on aliquid hydrogen target. The large acceptance CLAS detector is used to identify and reconstruct elastic scattering events,determining both the initial lepton energy and the sign of the scattered lepton.
Results:
The data were collected in two days with a primary electron beam energy of only 3.3 GeV, limiting the data fromthis run to smaller values of Q and scattering angle. Nonetheless, this measurement yields a data sample for e ± p with statistics comparable to those of the best previous measurements. We have shown that we can cleanly identifyelastic scattering events and correct for the difference in acceptance for electron and positron scattering. Because weran with only one polarity for the chicane, we are unable to study the difference between the incoming electron andpositron beams. This systematic effect leads to the largest uncertainty in the final ratio of positron to electron scattering: R = 1 . ± . ± .
05 for h Q i = 0 .
206 GeV and 0 . ≤ ǫ ≤ . Conclusions:
We have demonstrated that the tertiary e ± beam generated using this novel technique provides the opportunityfor dramatically improved comparisons of e ± p scattering, covering a significant range in both Q and scattering angle.Combining data with different chicane polarities will allow for detailed studies of the difference between the incoming e + and e − beams. PACS numbers: 14.20.Dh,13.40.Gp,13.60.Fz
I. INTRODUCTION
Electron scattering is one of the most powerful toolsavailable for measurements involving the quark structureof nucleons and nuclei. The dominant one-photon ex-change (OPE) mechanism is well understood, and therelatively weak electromagnetic coupling means that thescattering uniformly probes the matter within even adense nucleus. This weak coupling also implies smallhigher-order corrections to the cross section related totwo-photon exchange (TPE), which are suppressed byan additional power of the fine structure constant α ≈ / ∗ Current address: Christopher Newport University, NewportNews, Virginia 23606 † Current address: Los Alamos National Laboratory, Los Alamos,NM 87544 USA ‡ Current address: Indiana University, Bloomington, IN 47405 § Current address: Institut de Physique Nucl´eaire ORSAY, Orsay,France ¶ Current address: Universita’ di Roma Tor Vergata, 00133 RomeItaly elastic electromagnetic form factors of the proton [1–3].There is renewed interest in two-photon exchangecontributions due to new polarization measurements ofthe proton electromagnetic form factors, G E ( Q ) and G M ( Q ). High- Q measurements using recoil polariza-tion techniques to extract the ratio µ p G E /G M [4–6] indi-cated a significant discrepancy [7] with extractions basedon the Rosenbluth separation technique [8–11]. This ledto the concern that TPE corrections to the cross sectionmay be more important than previously thought [12–14], with implications for not only the form factors, butalso other precision measurements using electron scatter-ing [15–20].Theoretical investigations suggest that the TPE contri-butions may be sufficient to resolve the discrepancy [21–23]. For the most part, the calculations indicate that theTPE effects are small, but have a significant angle depen-dence, increasing in magnitude at larger scattering an-gles. Because the charge form factor, G E ( Q ), is relatedto the angular dependence of the elastic cross section, theimpact of small TPE corrections can be very large if theangular dependence associated with G E becomes small,e.g. at large Q values. Thus, the charge form factor is aspecial case with exceptional sensitivity to TPE correc-tions. Nonetheless, other high-precision measurementsmay still need to evaluate these small contributions, andthere are essentially no direct measurements of TPE thatcan be used to validate these calculations.There is clear evidence for TPE contributions in otherprocesses and other observables [24–28], but little directevidence for TPE contributions in the unpolarized elas-tic electron–proton cross section. The cleanest and mostdirect way to study TPE contributions to the cross sec-tion is through the comparison of electron and positronscattering [29, 30]. The interference between the single-photon exchange and the TPE diagrams yields the largestTPE contribution to the cross section, and its sign de-pends on the sign of the lepton charge. Most other ra-diative corrections are identical for electron and positronscattering, with the only other charge-dependent contri-bution being from the interference between lepton andproton bremsstrahlung, which is relatively small at low Q where the proton momentum is small.The main difficulty in measuring TPE contributionsin fixed-target experiments is that the low luminositiesof the secondary positron beams have historically lim-ited measurements to regions where the cross section islarge: low Q and/or very forward angle scattering. TheTPE contributions needed to explain the form factor dis-crepancy are relatively small, and become important atlarger Q and scattering angles. Thus, a significant in-crease in the luminosity is required to make meaningfulmeasurements in the kinematic region of interest.We present here the results from an experiment thatused a novel technique to make a simultaneous measure-ment of positron–proton and electron–proton elastic scat-tering. While the data from this brief run are limitedto low Q and small scattering angles, the experimentprovides statistics comparable to the best previous mea-surements. It also demonstrates the possibility to covera large range of Q and scattering angles with the preci-sion and accuracy necessary to determine whether TPEcorrections can explain the observed form factor discrep-ancy. Such data can also constrain calculations of thecorrections at low-to-moderate Q values, allowing vali-dation of the calculations that may be needed to evaluatepotential TPE impacts beyond elastic scattering. II. TWO-PHOTON EXCHANGE
Figure 1 shows the Born contribution and higher or-der QED corrections to lepton–proton elastic scattering.The TPE contribution (diagrams (e) and (f)) is diffi-cult to calculate because the intermediate hadronic statemust be integrated over all baryonic resonance and con-tinuum states that can be excited by the virtual pho-ton. Therefore, TPE is typically neglected in calculat-ing radiative corrections [31–33], with the exception ofthe contribution needed to cancel infrared divergences inbremsstrahlung terms.A direct measurement of the TPE correction can beachieved experimentally in the ratio of the positron-
FIG. 1: Feynman diagrams for the elastic lepton–proton scat-tering, including the 1st-order QED radiative corrections. Di-agram (a) shows the electron vertex renormalization term,(b) shows the photon propagator renormalization term, (c)and (d) show the electron bremsstrahlung terms, (g) showsthe proton vertex renormalization term, (h) shows the protonbremsstrahlung term, and (e) and (f) show the two-photonexchange terms, where the intermediate state can be an un-excited proton, a baryon resonance or a continuum of hadrons. proton to electron-proton elastic cross sections. Neglect-ing bremsstrahlung terms, the Born term and first or-der corrections from Fig. 1 yield a total amplitude for ep → ep scattering of A ep → ep = q e q p [ A γ + q e A e.vertex + q p A p.vertex + q e A loop + q e q p A γ ] , (1)where q e and q p are the lepton and proton charges andthe amplitudes A γ , A e.vertex , A p.vertex , A e.loop and A γ respectively describe one-photon exchange, electron andproton vertex corrections [Figs. 1(a) and 1(g)], loop cor-rections [1(b)], and two-photon exchange [1(e) and 1(f)].Squaring the above amplitude and keeping only the cor-rections up to order α , we have | A ep → ep | ≈ e [ A γ + 2 e A γ ℜ ( A loop + vertex )+2 q e q p A γ ℜ ( A γ ) ] , (2)where we have simplified the expression by replacing q e and q p with e and taken A loop + vertex to be the sum ofthe 1st order corrections where the lepton (and proton)charges appear in even powers, and thus are identicalfor electron and positron scattering. Note that because A γ is real and large compared to the other terms inEq. 1, the contribution from the imaginary part of A γ has a negligible contribution to the squared amplitude,and it is common to include only the real part of the TPEamplitude.Experientally, one cannot always separate true elasticscattering from events with a radiated photon in the fi-nal state. A cut on the missing energy or the invariantmass is often used to exclude events with a high-energyphoton in the final state from the case with low energyreactions. The interference between electron and protonbremsstrahlung yields another contribution that changessign with the lepton charge, yielding a final cross sectionthat is proportional to | A ep → ep | = e { A γ + 2 e C even +2 q e q p [ A γ ℜ ( A γ ) + ℜ ( A ∗ e.br. A p.br. )] } , (3)where C even is the sum of the charge-even part of theradiative contributions, including both the loop and ver-tex diagrams and the charge-even contributions from theelectron and proton bremsstrahlung diagrams [Figs. 1(c),1(d), and 1(h)]. There is no interference between theBorn term and the bremsstrahlung terms because theyhave different final states. The only portion of thebremsstrahlung term that is not charge-even is the inter-ference between A e.br. and A p.br. , the electron and protonbremsstrahlung terms.The total charge-even radiative correction factor isthen: σ = σ Born (1 + δ even ) ,δ even = 2 e C even /A γ . (4)Two terms contribute to the charge asymmetry in elastic e ± p scattering: the interference between the Born andtwo-photon exchange diagrams and the interference be-tween electron and proton bremsstrahlung. Both of theseterms have infrared divergent contributions, but thesedivergences cancel in the sum of the two contributions,making the QED description of the e ± p -scattering self-consistent. This interference effect for the standard kine-matics of elastic e ± p -scattering experiments is dominatedby soft-photon emission and results in a factorizable cor-rection already included in the standard calculations ofradiative corrections [31–33].The ratio of e ± p scattering cross sections can thus bewritten as follows: R = σ ( e + p ) σ ( e − p ) ≈ δ even − δ γ − δ e.p.br. δ even + δ γ + δ e.p.br. ≈ − δ γ + δ e.p.br. ) / (1 + δ even ) , (5)where δ even is the total charge-even radiative correctionfactor and δ γ and δ e.p.br. are the fractional TPE andlepton–proton interference contributions. Note that thesign of δ γ and δ e.p.br. are chosen by convention such thatthey appear as additive corrections for electron scatter-ing. However, the sign of these corrections is determinedfrom the evaluation of the full expression given in Eq. 3.Typically, a correction is applied to account for the effect δ e.p.br to isolate the TPE contribution: R γ ≈ − δ γ / (1 + δ even ) . (6) Where R is the measured e + /e − ratio and R γ is theratio after applying corrections for the e – p interferenceterm. The quantity R γ corresponds to the quantity thatis typically quoted by such measurements, although thenotation is not always consistent.Note that most previous extractions neglect thecharge-even contributions, assuming that R = 1 − δ γ − δ e.p.br. and R γ = R + 2 δ e.p.br. = 1 − δ γ . Because thefactor δ even is typically small (20–30%) and negative, thismeans that assuming δ γ = (1 − R γ ) / δ γ = 0, this rescal-ing of the TPE contribution has minimal effect. Moresignificant is the fact that δ even is neglected when apply-ing the correction for the e – p interference term. Becausethis correction is always a reduction in the ratio, typically1–5%, this yields a systematic underestimate of R γ upto ∼ Q polarization trans-fer measurements [4–6] of the proton form factor ratio G pE /G pM and Rosenbluth separation extractions [9–11]utilizing unpolarized scattering. Rosenbluth extractionsgenerally showed that both G pE and G pM approximatelyfollow the dipole form, G D = (1 + Q / (0 .
71 GeV )) − ,so that the ratio G pE /G pM is constant, while polarizationmeasurements showed the ratio decreasing linearly with Q .One possible explanation of this discrepancy is aTPE contribution to the cross section. Explaining thedifference between these techniques requires an angle-dependent cross-section correction of 5–8% at large Q [46–50]. However, this assumes that the cross sectionchange fully resolves the discrepancy. The form factor ra-tio discrepancy does not provide significant cross sectionconstraints at low Q .Calculations of the TPE corrections were revisited [12–14] in the wake of the form-factor discrepancy and initialcalculations of the TPE correction brought the Rosen-bluth results into near agreement with the polarizationresults. While low Q calculations generally agree [51–56], all of the available calculations have significant modeldependence at large Q . While the hadronic calcula-tions of Blunden, Melnitchouk and Tjon [13, 51] includeonly the proton intermediate state, they fully reconcilethe cross section and polarization measurements up to Q ≈ and resolve most of the discrepancy athigher Q [21]. The effect of an intermediate ∆ contri-bution in diagrams (e) and (f) of Fig. 1 has been esti-mated and has a much smaller contribution, although itmay have a more significant contribution to the polariza-tion measurements [55, 57]. Calculations using a gener-alized parton distribution formalism [14, 16], dispersionrelations [54], and a QCD factorization approach [58, 59]also yield TPE contributions that can resolve a large partof the discrepancy. Details of these calculations and theissues involved can be found in recent reviews [22, 23].Some limits for TPE contributions can be set basedon existing cross section and polarization measurements,combined with the known properties of the OPE andTPE contributions. In the Born approximation, the re-duced cross section depends linearly on ǫ = (1 + 2(1 + τ ) tan ( θ/ − , where τ = Q / M p . Corrections be-yond single-photon exchange can yield nonlinearities inthe reduced cross section, but existing data show that thecorrections are nearly linear [60–62]. A recent measure-ment of the ǫ dependence of the polarization transfer [63]also sets limits on TPE corrections, but the precisionis not sufficient to rule out the available calculations asonly the ǫ dependence, and not the overall size of theextracted form factor, can be constrained. In addition,even if there is no contribution to the polarization trans-fer data, there can still be a significant impact on thecross section [64, 65].It is clear that a direct confirmation of the presenceof TPE corrections is needed, as well as the data neces-sary to validate calculations required for measurementsthat may be sensitive to TPE effects. Since δ γ is ex-pected to be on the order of a few percent, one needs tomeasure R to within an uncertainty of ∼ Q values or smallangle, where TPE contributions are not expected to belarger than 1%.There are several recent attempts to improve on pre-vious TPE measurements comparing e ± p and µ ± p scat-tering. Two of these are straightforward experimentally,utilizing electron and positron beams from the VEPP-3storage ring [66–68] or the DORIS ring at DESY [69, 70].The storage rings allow for good control of systematics,though the available luminosities limit the measurementsto be done at lower beam energies and thus lower Q val-ues and also limit the statistical precision of the dataat small ǫ where TPE contributions are believed to belarge. The MUSE Collaboration [71] has proposed tocompare e ± p and µ ± p scattering at very low Q . This ismotivated by the “proton radius puzzle”; the differencebetween proton radius extractions involving muonic hy-drogen [72] and those involving electron–proton interac-tions [73–75]. The MUSE experiment will compare elec-tron and muon scattering to look for indications of lepton non-universality, but will also examine TPE corrections,which are important in the radius extraction from elec-tron scattering data [51, 76–78].We have taken a very different approach to compar-ing e + p and e − p scattering. Rather than alternatingbetween mono-energetic e + and e − beams, we gener-ate a combined beam of positrons and electrons coveringa range of energies and use the large-acceptance CLASspectrometer in Hall B of Jefferson Lab to detect boththe scattered lepton and struck proton. The kinematicsfor elastic scattering are overconstrained in such a mea-surement, allowing us to reconstruct the initial leptonenergy, as well as ensuring that the scattering was elas-tic. This allows a simultaneous measurement of electronand positron scattering, while also covering a wide rangein ǫ and Q . As such, the full experiment utilizing thesetup described here [79] is the only TPE measurementthat will extract the ǫ dependence of the TPE correc-tions at fixed Q , such that they can be directly appliedto Rosenbluth separations of the form factors. III. EXPERIMENTAL DETAILS
This section and the next describe the novel tech-nique used to create a mixed electron-positron beam,the methods for extracting elastic scattering events usingthis beam, and the initial measurement of the positron-electron elastic scattering ratio over a narrow kinematicrange.The experiment took place at the Thomas JeffersonNational Accelerator Facility (Jefferson Lab) and usedthe CEBAF Large Acceptance Spectrometer (CLAS) [80]in Hall B to detect scattered particles. CLAS (see Fig. 2)is a nearly 4 π detector. The magnetic field is provided bysix superconducting coils that produce an approximatelytoroidal field in the azimuthal direction around the beamaxis. The regions between the six magnet cryostats areinstrumented with identical detector packages called sec-tors. Each sector consists of three regions of drift cham-bers (R1, R2, and R3) to determine the trajectories ofcharged particles [81], Cherenkov Counters (CC) for elec-tron identification [82], scintillation counters for Timeof Flight (TOF) information [83], and ElectromagneticCalorimeters (EC) for electron identification and neu-tral particle detection [84]. The R2 drift chambers arein the region of the magnetic field and provide trackingthat is then used to determine particle momenta with δp/p ∼ .
6% This experiment did not use the CC andused the EC only in the trigger.We produced a simultaneous mixed beam of electronsand positrons by using the primary electron beam to pro-duce photons and then using the photon beam to produce e + e − pairs (see Fig. 3). A 40–80 nA 3.3 GeV electronbeam struck a 4 . × − radiation-length (RL) gold ra-diator to produce a bremsstrahlung photon beam. Theelectrons were diverted by the Hall B tagger magnet [85]into the standard underground beam dump. The photon FIG. 2: Three dimensional view of CLAS showing the beam-line, drift chambers (R1, R2, and R3), the Cherenkov Counter(CC), the Time of Flight system (TOF) and the Electromag-netic Calorimeter (EC). In this view, the beam enters thepicture from the upper left corner. flux was about 10 greater than previous Hall B photonfluxes, requiring substantial additional shielding aroundthe beam dump.The photon beam passed through a 12.7 mm diame-ter nickel collimator and then struck a 5 . × − RLgold converter to produce e + /e − pairs. The combinedphoton-lepton beam then entered a three-dipole chicaneto horizontally separate the electron, positron and pho-ton beams. The photon beam was stopped by a tungstenblock in the middle of the second dipole. The leptonbeams were recombined into a single beam by the thirddipole, which then proceeded to a liquid hydrogen targetat the center of CLAS. Fig. 3 shows the layout of thebeamline and Table I lists the relevant parameters.The TPE chicane consisted of three dipole magnets.The first and third dipoles were the so-called “ItalianDipoles” and the second dipole was the pair spectrome-ter magnet (PS). The Italian Dipoles were operated witha magnetic field of B ≈ ± . B ≈ ∓ .
38 T and was about1 m long. The oppositely charged leptons were separatedhorizontally and recombined by the chicane. The photonbeam was absorbed by a 4-cm wide and 35-cm long tung-sten photon blocker positioned with its upstream face atthe entrance aperture of the PS magnet.The momentum acceptance of the chicane is deter-mined by the width of the photon blocker and theapertures of the PS. The width of the photon blocker
Primary Beam 40 ≤ I ≤
80 nA E = 3 . . × − RLPhoton Collimator 12.7 mm IDlength = 30 cmConverter (gold) 5 . × − RLItalian Dipole B ≈ . L ≈ . B ≈ .
38 T L ≈ × ± ( ± ± . ×
16 scintillating fibers read outby a multichannel PMT and was located approximately15 cm upstream of the entrance to CLAS. The fibers were1 mm × e + e − /primaryelectronbeam photonbeam positronselectrons3−dipole Chicanephoton blocker Targetbeamcombinedto tagger dumpTagger magnetradiatorconverter CLASLead wall Collimator 2BPMCollimator 1Concrete wall FIG. 3: (Color online) Beamline sketch for the CLAS TPE experiment. Shielding elements around the chicane and taggerare not shown. The chicane bends the electron and positron trajectories in the horizontal plane, not the vertical plane. Theelectron and positron directions are selected by the chicane polarity. Drawing is not to scale.
This experiment was primarily an engineering test runto determine the feasibility of using this mixed electron-positron beam line to definitively measure the ratio R = σ ( e + p ) /σ ( e − p ) to resolve the proton form factor discrep-ancy. Prior to the data taking phase of the test run wevaried the experimental parameters to optimize the lep-ton beam luminosity. The beam luminosity was limitedby requiring that the CLAS drift chamber occupancy ineach sector and each region all be less than 3%. We var-ied the incident beam current, the radiator thickness, thephoton collimator diameter, the converter thickness, thechicane magnet currents, and the first lepton collimatordiameter. We also greatly improved the shielding. Thevalues listed in Table I show the final optimized values.For example, in order to make sure that both leptonbeams had the same centroid position, we varied thecurrent in the Italian Dipole magnets while keeping thePS dipole current fixed. We blocked one lepton beamand measured the position of the centroid of the otherbeam as a function of the Italian Dipole current. Wethen blocked the other beam and repeated the measure-ment for the first beam. Fig. 4 shows the results. Wefit straight lines to the linear parts of the beam positionvs. magnet current data. The intersection of the two lin-ear fits indicates the Italian Dipole magnet current thatoptimized the centering of the two lepton beam spots.This optimized centering is approximately 5 mm off theexpected beam center indicating a likely misalignment ofthe BPM.The reconstructed beam energy distribution for elasticscattering events detected in CLAS is shown in Fig. 5.The shape of this distribution is a convolution of the in-cident beam energy distribution, the elastic scatteringcross section and the CLAS acceptance. The maximumflux is at low energies, consistent with the bremsstrahlungcross section. This feature limits the measurement here to low Q and high ǫ. We estimate the integrated beamcurrent of each beam to be on the order of 1 pA. Thewidth of the beam varied as a function of energy froman RMS of 1.6 cm for beam energies in the range 0 . ≤ E beam ≤ . . ≤ E beam ≤ . FIG. 4: (Color online) Position of the individual lepton beamsas a function of the current in the first and third dipoles.Data points are measured beam centroid positions at the fiberdetector and the lines are fits to points 2–10.
FIG. 5: Reconstructed beam energy (electrons and positronscombined) at the target for elastic events detected in CLAS.Each event is weighted by one over the elastic cross sectionto recover the initial beam energy distribution. particles) and a hit in a TOF counter in the oppositesector.The magnetic fields of the CLAS torus magnet andthe beamline chicane can be reversed periodically toreduce lepton charge-dependent experimental asymme-tries. However, during this run only the CLAS torusfield was reversed. By using simultaneous mixedelectron-positron beams we eliminated the effect of time-dependent detector efficiencies. By taking data with bothchicane polarities, we would eliminate, within uncertain-ties, any flux-dependent differences between the left andright beams.The data were taken as part of a test run to verifythe feasibility of the experiment. The test run was suf-ficiently successful that we took about 1.5 days of ex-perimental data after commissioning the e + e − beamline.These data allowed us to test our data analysis tech-niques and to measure the ratio R at low Q and high ǫ as shown in Fig. 6. IV. DATA ANALYSIS
This analysis confronted us with a number of issues un-common to other CLAS experiments. The primary prob-lems were (a) determining the energy of the incident lep-ton, (b) making the analysis lepton-charge-independent,and (c) identifying the lepton and proton without usingthe Cherenkov Counter because its efficiency depends onwhether the lepton is in-bending or out-bending. Oursolution to this was to require the detection of both theproton and the lepton in each event, to exploit the re-stricted kinematics of elastic-scattering to identify elasticevents, and to match the detector acceptances for the twotypes of events (electron-proton and positron-proton). Adescription of the important analysis procedures is given
FIG. 6: (Color online) The experimental acceptance in mo-mentum transfer and virtual photon polarization shown fornegative torus polarity and both lepton charges. The redboxes show the binning for the data presented here. below.
A. Elastic Event Identification
We first selected events with only two detected chargedparticles in opposite sectors and where their charges wereeither positive/negative or positive/positive.For positive/positive events, we used information fromthe TOF counters to determine which particle was theproton and which was the positron. We initially identi-fied the positron by requiring that β = v/c > .
9, notingthat at these kinematics protons have β < .
9. With thisloose PID cut in place we then verified the PID assign-ment by following the rest of the elastic event identifi-cation chain. If the event did not subsequently satisfythe elastic scattering cuts listed below, we swapped theidentities of the two positive particles and checked tosee if the event then satisfied the elastic cuts. None ofthese swapped ++ events passed the elastic cuts, indi-cating that the β > . Bad paddle removal.
As CLAS has aged, someof the TOF detector photomultiplier tubes (PMT)have deteriorated and have low gain leading to verypoor efficiency. Events with particles that hit oneor more of these detectors were removed from theanalysis.2. Z -vertex. The particle origin along the beamline( z -vertex) was reconstructed as part of the trajec-tory measurement. A cut was placed on z -vertexto ensure that events came from the LH target.3. Distance of closest approach between leptonand proton candidates.
This is defined as thedistance between the two tracks (lepton and pro-ton) at their closest point. A cut was placed onthis distance to ensure the two tracks came fromthe same interaction.4.
Fiducial cuts.
Fiducial cuts in angle and momen-tum were used to select the region of CLAS withuniform acceptance for both lepton polarities, thusmatching the acceptances for electron and positron.5.
Azimuthal opening angle (co-planarity).
Since there are only two particles in the final state,these events must be co-planar. Fig. 7 shows theazimuthal-angle difference between events beforeand after all other cuts. [deg] f D
160 170 180 190 200 c oun t s · positive polarityp - e [deg] f D
160 170 180 190 200 c oun t s · positive polarityp + e [deg] f D
160 170 180 190 200 c oun t s · negative polarityp - e [deg] f D
160 170 180 190 200 c oun t s · negative polarityp + e FIG. 7: Angle between lepton and proton (∆ φ ) distributionsfor event type and torus polarity as indicated. The solid his-togram is the data with only the opposite sector cut. Thedotted histogram is after all other cuts. The dashed linesshow the ∆ φ cut. Transverse momentum.
Conservation of mo-mentum requires the total transverse momentum, p t (with respect to the incident beam) of the final-state elastic scattering products to be zero. SeeFig. 8. [GeV/c] t p-0.4 -0.2 0 0.2 0.4 c oun t s · positive polarityp - e [GeV/c] t p-0.4 -0.2 0 0.2 0.4 c oun t s · positive polarityp + e [GeV/c] t p-0.4 -0.2 0 0.2 0.4 c oun t s · negative polarityp - e [GeV/c] t p-0.4 -0.2 0 0.2 0.4 c oun t s · negative polarityp + e FIG. 8: Reconstructed transverse momentum distributionsfor event type and torus polarity as indicated. The solid his-togram is the data with only the opposite sector cut. Thedotted histogram is after all other cuts. The dashed linesshow the p t cut. Beam energy difference.
Because we measuredthe 3-momenta for both particles in the final state,our kinematics are over constrained. This allows usto reconstruct the unknown energy of the incidentlepton in two different ways. Eq. 7 calculates theincident energy using the scattered lepton and pro-ton angles, whereas Eq. 8 calculates this from thetotal momentum along the z -direction. E anglesbeam = m p (cid:18) cot θ e θ p − (cid:19) (7) E mombeam = p e cos θ e + p p cos θ p (8)For elastic scattering events, these two quantitiesshould be equal. We cut on ∆ E beam = E anglesbeam − E mombeam (see Fig. 9). The 7 to 22 MeV shift in thecentroids from zero is due to particle energy loss inthe target, which reduces the value of E mombeam . Notethat we used E anglesbeam in all calculations that dependon beam energy (e.g., Q , W , ǫ ).8. Momentum polar angle.
We cut on the po-lar angle of the final-state total-momentum ( ~P f = ~p l ′ + ~p p ′ ) with respect to the z -axis. Devia-tions from zero may be due to inelastic events,mis-reconstructed scattered particles, or multiple-scattered final particles. To discard these back-ground events, we required θ P f < ◦ (see Fig. 10).This cut is largely redundant to the transverse mo-mentum cut.The cuts for items 3, 5, 6, 7 and 8 were determinedby fitting a Gaussian to the peak of the combined distri-bution for that variable, including both event types ( e + p [GeV] beam E D -1 -0.5 0 0.5 1 c oun t s · positive polarityp - e [GeV] beam E D -1 -0.5 0 0.5 1 c oun t s · positive polarityp + e [GeV] beam E D -1 -0.5 0 0.5 1 c oun t s · negative polarityp - e [GeV] beam E D -1 -0.5 0 0.5 1 c oun t s · negative polarityp + e FIG. 9: ∆ E beam for event type and torus polarity as indi-cated. The solid histogram is the data with only the oppositesector cut. The dotted histogram is after all other cuts. Thedashed lines show the ∆ E beam cut. [deg] beam q c oun t s · positive polarityp - e [deg] beam q c oun t s · positive polarityp + e [deg] beam q c oun t s · negative polarityp - e [deg] beam q c oun t s · negative polarityp + e FIG. 10: Reconstructed polar angle of the beam for eventtype and torus polarity as indicated. The solid histogram isthe data with only the opposite sector cut and the dottedhistogram is after all other cuts. The dashed lines show the θ P f cut. and e − p ) and both torus polarities and setting the cut to ± σ . The widths of these distributions did not dependsignificantly on either torus polarity or event type.The cleanliness of the final data sample after these cutswere applied is shown in Fig. 11, which shows the in-variant mass of the virtual photon plus target proton, W = p m p + 2 m p ν − Q , distribution for one of our binsin ǫ . The peak is at the proton mass and shows virtually no hint of non-elastic background. Using side bands oneither side of the peak we estimate the background tobe in the range of 0.3 to 0.4%. Since the background isequal to within uncertainties for both lepton species theeffect on R is negligible. W [GeV] 0.6 0.8 1 1.2 1.4 1.6 c oun t s positive polarityp - e W [GeV] 0.6 0.8 1 1.2 1.4 1.6 c oun t s positive polarityp + e W [GeV] 0.6 0.8 1 1.2 1.4 1.6 c oun t s negative polarityp - e W [GeV] 0.6 0.8 1 1.2 1.4 1.6 c oun t s negative polarityp + e FIG. 11: Top two panels show the W distributions (in GeV)for positive torus polarity electron (left) and positron (right)events for 0 . ≤ ǫ ≤ .
840 and h Q i = 0 .
206 GeV withall cuts applied. The bottom two panels show the same fornegative torus polarity events. The distribution of elastic events in Q vs. ǫ after allcuts is shown in Fig. 6. The boxes in the figure show thebins used for this analysis. The final results cover a single Q bin (0 . ≤ Q ≤ .
400 GeV with h Q i = 0 . ) and seven bins in ǫ (0 . ≤ ǫ ≤ . ǫ bin. B. Acceptance Matching and Corrections
In order to calculate the ratio R = σ ( e + p ) /σ ( e − p ),we must ensure that the detector acceptance does notdepend on lepton charge. We first calculate R by calcu-lating the ratio of e + p to e − p events for a given torus po-larity. In this ratio, the proton acceptance cancels. How-ever, the CLAS acceptances for electrons and positronsfor a given kinematic bin differ because one bends awayfrom the beamline while the other bends toward thebeamline in the CLAS magnetic field.We have accounted for acceptance differences in threesteps. First, we match acceptances by using the fidu-cial cuts to select regions in ( p, θ, φ ) space where CLASis almost 100% efficient in detecting both electrons andpositrons. Second, we correct for differences due todead detectors using a “swimming” algorithm to checkwhether an e + p would have been detected had it been an e − p event (and vice versa).1For example, if an e + p event is detected and passes allthe elastic cuts, then the swimming algorithm generatesa conjugate lepton, in this case an e − , with the samemomentum ~p as the e + and calculates (“swims”) its tra-jectory through the CLAS detector system and magneticfield. If the conjugate lepton falls within the CLAS ac-ceptance, then the original event is kept. If the conju-gate lepton falls outside of the CLAS acceptance (eitheroutside fiducial cuts or hits a dead paddle), the event isdiscarded.In the third step, any remaining acceptance differencescan be removed by measuring R for both torus polar-ities and constructing a double ratio. The number ofdetected elastic events for a given torus polarity ( t = ± )and a given lepton charge ( l = ± ) should be proportionalto the cross section times the unknown torus-polarity-related and lepton-charge-related detector efficiency andacceptance function f lt : N lt ∝ σ ( e l p ) f lt . Thus, for one torus polarity, the simultaneously measuredratio R t will be R t = N + t N − t = σ ( e + p ) f + t σ ( e − p ) f − t . Taking the square-root of the product of the single-polarity ratios we get R = p R + R − = s N ++ N − + · N + − N −− = s σ ( e + p ) f ++ σ ( e − p ) f − + · σ ( e + p ) f + − σ ( e − p ) f −− = σ ( e + p ) σ ( e − p ) , (9)where by charge symmetry, one expects f ++ = f −− and f + − = f − + . The unknown lepton acceptance functionsare expected to cancel in the double ratio. The protonacceptance cancels out independently in the single ratios R + and R − .We checked the quality of the corrections describedabove by comparing it to two other methods. Thefirst approach is to apply an acceptance correction tothe e + and e − data separately based on a full MonteCarlo (MC) study using GSIM, the GEANT-based CLASMonte Carlo simulation that included all dead detectors.The second approach was to calculate the double ratiowith no acceptance corrections at all since, in princi-ple, all acceptances cancel out in the double-ratio. Wefound the differences among the three values of R to besmaller than their statistical uncertainties. We used thedifference between our swimming results and our MC-corrected results to estimate the dead-detector-relatedsystematic uncertainty. C. Systematic Uncertainties
The four major categories of systematic uncertaintiesin this analysis are:1.
Luminosity differences between electronsand positrons.
In this test run we could not in-dependently measure the lepton beam luminositiesand we did not have the time to take data for bothpolarities of the beam-line chicane magnetic field.Therefore we determined the relative luminosityuncertainty by a detailed GEANT4-based MonteCarlo study of the beam line that included allknown lepton interactions. The MC study showedthat the relative flux difference between positronsand electrons on the target was less than 1% for anideal beamline. Based on survey results, the align-ment of beam line elements was within 1 mm. TheMC beam-line simulation showed that, for a singlechicane polarity, a 1-mm change in the relative po-sition of the collimators and magnets leads to a 5%change in the electron-positron luminosity ratio.2.
Effects of elastic event ID cuts.
This was stud-ied by varying the widths of these cuts (from thenominal 4- σ cut to a 3- σ cut or removing the cutentirely). The differences in the final double-ratioresults between the nominal and the varied cuts re-sult in an estimated absolute uncertainty in R of0.0040.3. Effects of fiducial cuts.
We also varied the cutsthat define the good region of CLAS, again compar-ing the nominal double-ratio results to those withthe varied fiducial cuts. The estimated absoluteuncertainty in R is 0.0011.4. Acceptance (dead detector) corrections.
Aspreviously mentioned, this was done by compar-ing our nominal double-ratio results (using “swim-ming”) to results using a MC correction. The es-timated absolute uncertainty in R is 0.0071 and isthe largest of our point-to-point systematic uncer-tainties.The 5% luminosity-related uncertainty is a scale typeuncertainty affecting all points in the same way. Theother three items represent uncorrelated point-to-pointuncertainties, which added in quadrature give an overalluncertainty of 0.0083 in R . V. RESULTS
Our final results are presented in Table II and Fig. 12.Fig. 13 and Table II show the results for R γ (Eq. 6) af-ter correcting the measured ratio R for the lepton-protonbremsstrahlung interference [31]. The corrections reducethe measured ratio by 0.0049 at ǫ = 0 .
830 and decreasegradually to 0.0034 at ǫ = 0 . W used in the correc-tion calculation and the uncertainty in the term δ even .This uncertainty is far smaller than our other systematicuncertainties and can be ignored. The average of ourresults, with the point-to-point systematic uncertaintycombined in quadrature with the statistical uncertainty,is 1.027 ± . ± .
05, with the last uncertainty being dueto the luminosity uncertainty.
FIG. 12: (Color online) Measured ratio R for acceptancematched data at h Q i = 0 .
206 GeV before radiative correc-tions. Vertical error bars are statistical only and horizontalerror bars show the range of the bin. The red shaded band in-dicates the point-to-point systematic uncertainty (1 σ ) of thepresent data. The 5% luminosity-related systematic uncer-tainty is not shown. These measurements cover a very narrow range in ǫ . R γ is not expected to vary over this narrow range of ǫ ,especially at this low momentum transfer. For example,see the BMT calculation [51] shown in Fig. 13. There-fore, the variation of these data should be consistent withits uncertainties. The standard deviation of the sevendata points is 0.01, which is consistent with and slightlysmaller than the statistical plus point-to-point uncertain-ties.We compare our results with the world’s data at a sim-ilar value of Q as a function of ǫ in Fig. 13. There areseven previous data points in this range of Q . Our dataare compatible with these points, although with signifi-cantly smaller statistical uncertainties. However, the 5%systematic uncertainty due to the luminosity prevents usfrom extracting any significant conclusions about the sizeof the TPE effect. VI. CONCLUSION AND FUTURE PROSPECTS
We have presented a new technique for producing amixed electron-positron beam using bremsstrahlung toproduce a secondary photon beam from the primary elec-tron beam and then pair-production to produce a ter-tiary electron-positron beam from the photon beam. We h Q i (GeV ) h ǫ i R R γ δR stat δR sys δR lum h Q i and h ǫ i show the average momentum transfer and photon po-larization for that bin respectively, R and R γ show the mea-sured value of R = σ ( e + p ) /σ ( e − p ) before and after radia-tive corrections respectively, δR stat , δR sys and δR lum showthe statistical uncertainty, the point-to-point systematic un-certainty and the luminosity-related systematic uncertaintyrespectively.FIG. 13: (Color online) Ratio R γ overlaid on the world data.Black filled squares are from this experiment at h Q i = 0 . and have had radiative corrections applied, blue filledcircles are previous world data at similar Q , and green hollowpoints the rest of the previous world data with Q < [29]. The red densely shaded band indicates the point-to-point systematic uncertainty (1 σ ) and the black shaded bandrepresents the scale-type systematic uncertainty (due to rel-ative luminosity) on the present data. The red dashed curveis the BMT calculation [51] at Q = 0 . . aimed this beam at a liquid hydrogen target in the centerof the CLAS spectrometer. We have presented analysistechniques to cleanly extract elastic-scattering electron-proton and positron-proton events and to minimize thecharge-dependent experimental asymmetries.We then used these techniques to extract R = σ ( e + p ) /σ ( e − p ), the ratio of positron-proton to electron-proton elastic scattering cross sections over a limitedkinematic range at large ǫ and small Q . The extractedratio is consistent with the world’s data. This ratio R isdirectly related to the magnitude of the Two Photon Ex-change contribution to electron-proton elastic scattering.During late 2010 and early 2011 we conducted the fullCLAS TPE experiment using an incident beam energy3of 5.5 GeV and significantly greater luminosity. This ex-periment covered a much larger kinematic range, up to Q = 2 GeV and ǫ values down to about 0.3. We ex-pect similar systematic uncertainties related to data andfiducial cuts and dead detector corrections as were de-termined for the results presented here. We expect toreduce the systematic uncertainty for positron/electronluminosity differences to ∼
1% for the full run by formingan additional double ratio of results for the two differentchicane polarities. We also utilized a beam-profile mon-itor at the downstream end of CLAS to verify that theratio of positron luminosity for a given chicane polarityto electron luminosity in the opposite chicane polaritywas flat to within ∼
1% as a function of lepton energy.Analysis of these data is underway and we expect finalresults soon.Two other experiments are measuring R to determinethe TPE effect using electron and positron beams at in-ternal storage rings. The Novosibirsk group [66–68] mea-sured R at six different kinematic points and in 2012,the OLYMPUS Collaboration [69] took data at a singlelepton beam energy for Q < . . These exper-iments have very different systematic uncertainties andkinematic coverages from the CLAS experiment.These experiments will provide information that is vi-tal to our understanding of the electron-scattering pro-cess as well as our understanding of the proton struc-ture. We have heard the common statement that “theelectromagnetic probe is well understood.” However, thediscrepancy between Rosenbluth and polarization mea-surements of the form-factor ratio indicates otherwise.Indeed, if we don’t understand elastic electron scatter-ing at high precision or when higher order contributions become significant, then similar measurements will bein doubt. There are important implications for many ofthe nuclear physics quantities being studied ranging fromhigh-precision quasi-elastic experiments to strangenessand parity violation experiments. Acknowledgments
We acknowledge the efforts of the staff of the Accel-erator and Physics Divisions at Jefferson Lab that madethis experiment possible. We are especially grateful tothe Hall B staff members who tirelessly reconfigured thebeamline and stacked (and restacked) shielding blocks.Thanks also to Dave Kashy who made the crucial sugges-tion of narrowing the post-chicane collimator. This workwas supported by the U.S. Department of Energy andNational Science Foundation, the Israel Science Founda-tion, the US-Israeli Bi-National Science Foundation, theChilean Comisi´on Nacional de Investigaci´on Cient´ıficay Tecnol´ogica (CONICYT) grants FB0821, ACT-119,1120953, 11121448, and 791100017, the French CentreNational de la Recherche Scientifique and Commissariata l’Energie Atomique, the French-American Cultural Ex-change (FACE), the Italian Istituto Nazionale di FisicaNucleare, the National Research Foundation of Korea,and the United Kingdom’s Science and Technology Fa-cilities Council (STFC). The Jefferson Science Associates(JSA) operates the Thomas Jefferson National Accelera-tor Facility for the United States Department of Energyunder contract DE-AC05-84ER40150. [1] J. Arrington, C. D. Roberts, and J. M. Zanotti, J. Phys.
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