Measurement of the proton spin structure at long distances
X. Zheng, A. Deur, H. Kang, S.E. Kuhn, M. Ripani, J. Zhang, K.P. Adhikari, S. Adhikari, M.J. Amaryan, H. Atac, H. Avakian, L. Barion, M. Battaglieri, I. Bedlinskiy, F. Benmokhtar, A. Bianconi, A.S. Biselli, S. Boiarinov, M. Bondi, F. Bossu, P. Bosted, W.J. Briscoe, J. Brock, W.K. Brooks, D. Bulumulla, V.D. Burkert, C. Carlin, D.S. Carman, J.C. Carvajal, A. Celentano, P. Chatagnon, T. Chetry, J.-P. Chen, S. Choi, G. Ciullo, L. Clark, P.L. Cole, M. Contalbrigo, V. Crede, A. D'Angelo, N. Dashyan, R. De Vita, M. Defurne, S. Diehl, C. Djalali, V.A. Drozdov, R. Dupre, M. Ehrhart, A. El Alaoui, L. Elouadrhiri, P. Eugenio, G. Fedotov, S. Fegan, R. Fersch, A. Filippi, T.A. Forest, Y. Ghandilyan, G.P. Gilfoyle, K.L. Giovanetti, F.-X. Girod, D.I. Glazier, R.W. Gothe, K.A. Griffioen, M. Guidal, N. Guler, L. Guo, K. Hafidi, H. Hakobyan, M. Hattawy, T.B. Hayward, D. Heddle, K. Hicks, A. Hobart, T. Holmstrom, M. Holtrop, Y. Ilieva, D.G. Ireland, E.L. Isupov, H.S. Jo, K. Joo, S. Joosten, C.D. Keith, D. Keller, A. Khanal, M. Khandaker, C.W. Kim, W. Kim, F.J. Klein, A. Kripko, V. Kubarovsky, L. Lanza, M. Leali, P. Lenisa, K. livingston, E. Long, I.J.D. MacGregor, N. Markov, L. Marsicano, V. Mascagna, B. McKinnon, et al. (51 additional authors not shown)
MMeasurement of the proton spin structure at long distances
X. Zheng, A. Deur,
2, 1, ∗ H. Kang, S.E. Kuhn, M. Ripani, J. Zhang, K.P. Adhikari,
4, 2, 6, † S. Adhikari, M.J. Amaryan, H. Atac, H. Avakian, L. Barion, M. Battaglieri,
2, 5
I. Bedlinskiy, F. Benmokhtar, A. Bianconi,
12, 13
A.S. Biselli, S. Boiarinov, M. Bond`ı, F. Boss`u, P. Bosted, W.J. Briscoe, J. Brock, W.K. Brooks,
18, 2
D. Bulumulla, V.D. Burkert, C. Carlin, D.S. Carman, J.C. Carvajal, A. Celentano, P. Chatagnon, T. Chetry, J.-P. Chen, S. Choi, G. Ciullo,
9, 20
L. Clark, P.L. Cole,
22, 23
M. Contalbrigo, V. Crede, A. D’Angelo,
25, 26
N. Dashyan, R. De Vita, M. Defurne, S. Diehl,
28, 29
C. Djalali,
30, 31
V.A. Drozdov, R. Dupre, M. Ehrhart, A. El Alaoui, L. Elouadrhiri, P. Eugenio, G. Fedotov, S. Fegan, R. Fersch,
35, 16
A. Filippi, T.A. Forest, Y. Ghandilyan, G.P. Gilfoyle, K.L. Giovanetti, F.-X. Girod,
2, 29
D.I. Glazier, R.W. Gothe, K.A. Griffioen, M. Guidal, N. Guler, L. Guo,
7, 2
K. Hafidi, H. Hakobyan,
18, 27
M. Hattawy, T.B. Hayward, D. Heddle,
35, 2
K. Hicks, A. Hobart, T. Holmstrom, M. Holtrop, Y. Ilieva,
31, 17
D.G. Ireland, E.L. Isupov, H.S. Jo,
40, 19
K. Joo, S. Joosten, C.D. Keith, D. Keller, A. Khanal, M. Khandaker, ‡ C.W. Kim, W. Kim, F.J. Klein, A. Kripko, V. Kubarovsky,
2, 43
L. Lanza, M. Leali,
12, 13
P. Lenisa,
9, 20
K. livingston, E. Long, I.J.D. MacGregor, N. Markov, L. Marsicano, V. Mascagna,
44, 13
B. McKinnon, D.G. Meekins, T. Mineeva, M. Mirazita, V. Mokeev,
2, 32
C. Mullen, P. Nadel-Turonski,
2, 17
K. Neupane, S. Niccolai, M. Osipenko, A.I. Ostrovidov, M. Paolone, L. Pappalardo,
9, 20
K. Park,
40, 2
E. Pasyuk, W. Phelps, S.K. Phillips, O. Pogorelko, J. Poudel, Y. Prok,
4, 1
B.A. Raue,
7, 2
J. Ritman, A. Rizzo,
25, 26
G. Rosner, P. Rossi,
2, 45
J. Rowley, F. Sabati´e, C. Salgado, A. Schmidt, R.A. Schumacher, M.L. Seely, Y.G. Sharabian, U. Shrestha, S. ˇSirca, K. Slifer,
1, 39
N. Sparveris, S. Stepanyan, I.I. Strakovsky, S. Strauch, V. Sulkosky, N. Tyler, M. Ungaro,
2, 43
L. Venturelli,
12, 13
H. Voskanyan, E. Voutier, D.P. Watts, X. Wei, L.B. Weinstein, M.H. Wood,
49, 31
B. Yale, N. Zachariou, and Z.W. Zhao
4, 31 (The Jefferson Lab CLAS Collaboration) University of Virginia, Charlottesville, Virginia 22904, USA Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA Seoul National University, Seoul 08826, Korea Old Dominion University, Norfolk, Virginia 23529, USA INFN, Sezione di Genova, 16146 Genova, Italy Mississippi State University, Mississippi State, MS 39762, USA Florida International University, Miami, Florida 33199, USA Temple University, Philadelphia, PA 19122, USA INFN, Sezione di Ferrara, 44100 Ferrara, Italy National Research Centre Kurchatov Institute - ITEP, Moscow, 117259, Russia Duquesne University, 600 Forbes Avenue, Pittsburgh, PA 15282, USA Universit`a degli Studi di Brescia, 25123 Brescia, Italy INFN, Sezione di Pavia, 27100 Pavia, Italy Fairfield University, Fairfield, Connecticut 06824, USA IRFU, CEA, Universit´e Paris-Saclay, F-91191 Gif-sur-Yvette, France College of William and Mary, Williamsburg, Virginia 23187, USA The George Washington University, Washington, DC 20052, USA Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V Valpara´ıso, Chile Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France Universit`a di Ferrara , 44121 Ferrara, Italy University of Glasgow, Glasgow G12 8QQ, United Kingdom Lamar University, 4400 MLK Blvd, PO Box 10009, Beaumont, Texas 77710, USA Idaho State University, Pocatello, Idaho 83209, USA Florida State University, Tallahassee, Florida 32306, USA INFN, Sezione di Roma Tor Vergata, 00133 Rome, Italy Universit`a di Roma Tor Vergata, 00133 Rome Italy Yerevan Physics Institute, 375036 Yerevan, Armenia II Physikalisches Institut der Universitaet Giessen, 35392 Giessen, Germany University of Connecticut, Storrs, Connecticut 06269, USA Ohio University, Athens, Ohio 45701, USA University of South Carolina, Columbia, South Carolina 29208, USA Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119234 Moscow, Russia Argonne National Laboratory, Argonne, Illinois 60439, USA University of York, York YO10 5DD, United Kingdom Christopher Newport University, Newport News, Virginia 23606, USA INFN, Sezione di Torino, 10125 Torino, Italy University of Richmond, Richmond, Virginia 23173, USA a r X i v : . [ nu c l - e x ] F e b James Madison University, Harrisonburg, Virginia 22807, USA University of New Hampshire, Durham, New Hampshire 03824, USA Kyungpook National University, Daegu 41566, Republic of Korea Norfolk State University, Norfolk, Virginia 23504, USA Catholic University of America, Washington, D.C. 20064, USA Rensselaer Polytechnic Institute, Troy, New York 12180-3590 Universit`a degli Studi dell’Insubria, 22100 Como, Italy INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy Institute fur Kernphysik (Juelich), Juelich, Germany Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA University of Ljubljana, Slovenia Joˇzef Stefan Institute, Ljubljana, Slovenia Canisius College, Buffalo, NY 14208, USA
Measuring the spin structure of nucleons (protons and neutrons) extensively tests our understanding of hownucleons arise from quarks and gluons, the fundamental building blocks of nuclear matter. The nucleon spinstructure is typically probed in scattering experiments using polarized beams and polarized nucleon targets, andthe results are compared with predictions from Quantum Chromodynamics directly or with effective theories thatdescribe the strong nuclear force. Here we report on new proton spin structure measurements with significantlybetter precision and improved coverage than previous data at low momentum transfer squared between . and . GeV . This kinematic range provides unique tests of effective field theory predictions. Our resultsshow that a complete description of the nucleon spin remains elusive. They call for further theoretical worksthat include the more fundamental lattice gauge method. Finally, our data agree with the Gerasimov-Drell-Hearnsum rule, a fundamental prediction of quantum field theory. Understanding how hadronic matter arises from its funda-mental constituents, quarks and gluons, is central to the studyof nuclear and particle physics. Although the strong interac-tion is described by Quantum Chromodynamics (QCD), it re-mains the least understood force in the Standard Model. Thedifficulty arises because the QCD coupling constant α s be-comes large at long distances [1], making traditional pertur-bative expansions in powers of α s infeasible. Consequently,complex phenomena like quark confinement are hard to un-derstand quantitatively. The most fundamental approach tocalculate QCD non-perturbatively is lattice gauge theory [2].A second approach is provided by Effective Field Theories(EFT), which maintain rigorous, traceable connections to theunderlying fundamental theory. A popular approach is chi-ral effective field theory ( χ EFT) [3, 4], which is constructedfrom hadronic degrees of freedom and incorporates the sym-metries of QCD, including its approximate chiral symmetry.By making use of a perturbative expansion in small parame-ters, χ EFT predicts experimental observables from a limitedset of phenomenological inputs. Although generally success-ful, χ EFT has been challenged by experimental data that de-pend explicitly on spin degrees of freedom [5, 6]. This is notunprecedented: other theoretical predictions had been thoughtto be robust until confronted with spin observables, includingparity symmetry [7], the Ellis-Jaffe spin sum rule [8], the nu-cleon spin asymmetry A [9], and lattice QCD calculationsof the nucleon axial charge [10]. Therefore, fully understand-ing QCD and nuclear matter requires an extensive set of spinobservables.We report on the measurements performed using a polar-ized electron beam to probe a polarized proton at the Thomas ∗ email: [email protected] † Now at Hampton University, Hampton, Virginia 23669, USA ‡ Now at Idaho State University, Pocatello, Idaho 83209, USA
Jefferson National Accelerator Facility (Jefferson Lab), inVirginia, USA. We measured spin-dependent cross sectionsin the nucleon resonance region at very low Q , i.e. atlong distances. Here, Q is the square of the 4-momentumtransferred from the electron to the proton and representsthe inverse of the distance scale probed by the scattering.Polarized electrons with energies of 3.0, 2.3, 2.0, 1.3 and1.1 GeV, produced by Jefferson Lab’s Continuous ElectronBeam Accelerator Facility (CEBAF), were scattered from apolarized proton target [11, 12]. The beam polarization ( P b )was measured to be 85% with a total uncertainty of 2% us-ing a Møller polarimeter [13]. The target contained gran-ules of NH that were dynamically polarized [11] at 1K ina 5 T magnetic field. The target polarization ( P t ) varied from75% to 90%, as monitored by nuclear magnetic resonancepolarimetry. As described below and in the Methods sec-tion, the product P b P t was measured to a relative precisionof (2 − . The scattered electrons were identified using theCEBAF Large Acceptance Spectrometer (CLAS) [13], whichwas equipped with a multi-layer drift chamber detector forcharged particle tracking, a scintillator hodoscope for parti-cle time-of-flight measurement, an electromagnetic calorime-ter and a Cherenkov Counter for discriminating scatteredelectrons from other background particles. The CherenkovCounter in one of the six sectors of CLAS was modifiedspecifically for this experiment to detect electron scatteringat angles as low as ◦ . Only this sector was used to collect theinclusive electron scattering data reported here.The dominant scattering process is the one-photon ex-change, in which the incident electron exchanges a single vir-tual photon with the nucleon of mass M , see Fig. 1. The4-momentum transferred from the electron to the nucleon is q µ = k µ − k (cid:48) µ = ( ν, q ), in which k µ and k (cid:48) µ are the4-momenta of the incident and the scattered electrons, re-spectively, and ν is the energy transfer. In the following,we describe this process using the Lorentz-invariant vari- k =(E, )k’ =(E’, ) q =( , ) P =(M, ) electronscatteredelectronincident exchanged photon µ µ µ µ targetnucleon ν k’ 0k q FIG. 1. The one-photon exchange process of polarized electron scat-tering off a polarized nucleon. The 4-momenta of the incident andthe scattered electrons are k µ = ( E, k ) and k (cid:48) µ = ( E (cid:48) , k (cid:48) ) , respec-tively. The spin direction of the incident electron is indicated by thearrows ↑↓ . The nucleon, if at rest, has P = ( M, ) and its spin isindicated by the outlined arrow ⇑ . ables Q = − q , and the Bjorken scaling variable x ≡− q / (2 P · q ) or the invariant mass of the photon-nucleon sys-tem W ≡ (cid:112) ( P + q ) = (cid:112) P + (1 /x − Q . The inclu-sive electron scattering cross section can be written as a lin-ear combination of structure functions, of which F ( x, Q ) and F ( x, Q ) represent the spin-independent part of thecross section, and the spin structure functions g ( x, Q ) and g ( x, Q ) describe its dependence on the beam and target spinpolarization. These structure functions encode the internalstructure of the target. Alternatively, one can describe thespin-dependent part of the nucleon response in terms of virtualphoto-absorption asymmetries A = [ g − ( Q /ν ) g ] /F and A = ( (cid:112) Q /ν )( g + g ) /F [14]. The polarized crosssection difference ∆ σ ≡ σ ↓⇑ − σ ↑⇑ , with ↑↓ representingthe beam helicity state and ⇑⇓ the target spin orientation, islargely proportional to g (or equivalently A F ) with a smallcontribution from A F .The proton spin structure function g and the product A F were extracted from the difference in the measured yield, N ,of scattered electrons from a longitudinally polarized targetbetween opposite beam helicity states: N ↓⇑ Q ↓ b − N ↑⇑ Q ↑ b = ∆ σ ( W, Q ) L P b P t a ( W, Q ) , (1)where Q b is the time-integrated beam current, L is the arealdensity of polarized protons in the target, and a ( W, Q ) ac-counts for the detector acceptance and efficiency. The product L P b P t was measured directly using elastic scattering on theproton and a ( W, Q ) was determined using a Monte Carlosimulation of the experiment; see the Methods section for de-tails. Examples of our g results on the proton are shown inFig. 2. Our results extend the measured Q range down tobelow the pion mass squared ( m π ), three times smaller thanprevious data [14–22], which makes it possible to rigorouslytest χ EFT calculations for spin-dependent observables.In our study, we utilize sum rules that relate integralsof structure functions to amplitudes calculable by latticeQCD [23, 24] or χ EFT, or to known static properties of the tar-get. One such relation is the Gerasimov-Drell-Hearn (GDH)sum rule [25, 26] for real photon absorption ( Q = 0 ): (cid:90) ∞ ν ∆ σ ( ν ) dνν = − π αM κ , (2) W (GeV) p g Q =(0.011,0.016)GeV g W(GeV) −0.5 1.9 Q =(0.131,0.156) GeV FIG. 2. Results on g of the proton ( solid circles) vs invariant mass W for the lowest ( . ≤ Q ≤ . GeV ) bin and an interme-diate ( . ≤ Q ≤ . GeV ) bin, compared to a parameteri-zation of previous world data ( dotted curve) [14]. The error bars arestatistical. The solid and the vertically-hatched bands show the ex-perimental and the parameterization uncertainties, respectively. Re-sults from a previous experiment carried out in Jefferson Lab’s HallB [14] are shown when available (crosses). with κ the anomalous magnetic moment of the target particle, ν the inelastic threshold and α the fine-structure constant.Theoretical arguments indicate that the divergence of the /ν factor is compensated by the fast decrease of ∆ σ with ν . Thisis supported by experiments which have verified the GDH sumrule for the proton within about 7% accuracy [27, 28]. Thereexist several prescriptions that generalize the GDH sum ruleto electron scattering in terms of moments of spin structurefunctions integrated over x (which is equal to Q / M ν in thelaboratory frame). One often-used generalization is [29]: Γ ( Q ) ≡ (cid:90) x g ( x, Q ) dx = Q M I ( Q ) , (3)where x = Q / ( W thr − M + Q ) corresponds to the elec-troproduction threshold W thr = M + m π = 1.073 GeV. Equa-tion (3) defines the integral I , which is related to the firstpolarized doubly-virtual Compton scattering (VVCS) ampli-tude that is calculable in the ν → limit with lattice QCDor χ EFT [3, 4, 30–39]. The other prevailing generalization ofthe GDH integral is [40]: I ( Q ) = 2 M Q (cid:90) x (cid:2) A ( x, Q ) F ( x, Q ) (cid:3) dx, (4)which can be calculated from both the first and the secondspin-dependent VVCS amplitudes in the ν → limit. The I ( Q ) thus obtained can be extrapolated to Q = 0 to testthe original GDH prediction I (0) = κ / . In this work, wepresent results on both generalizations.To form the spin structure integrals in Eqs. (3 & 4), themeasured values of g or A F were used whenever availablefrom our experiment up to a maximum x corresponding to W = 1 . GeV, which was chosen to limit the backgroundfrom the elastic radiative tail (see Methods section) and downto a minimum x determined by the beam energy and the ac-ceptance of CLAS. Contributions from regions at low x (downto x = 10 − ) and at high x from W thr to W = 1 . GeV wereevaluated using a parameterization of previous data [14].Results on Γ ( Q ) and I ( Q ) are shown in Figs. 3 and 4. Bernard et al.GDH slopeBurkert et al.Soffer et al.ParameterizationThis work (full integral)0.1 Fersch et al. (full integral)0.080.060.040.020.0−0.02−0.04 1.00.20.160.120.080.040 This work (measured range only)Alarcon et al.
Q (GeV ) Γ FIG. 3. Results on Γ ( Q ) for the proton. Integrals over the exper-imentally covered x range are shown as open circles. Full integralsare shown as solid circles. The inner and the outer error bars (some-times too small to be seen) are for statistical and total uncertainties,respectively. Results from a previous experiment [14] are shownas solid triangles. The solid and the vertically-hatched bands showthe experimental and the parameterization uncertainties, respectively.Also shown are the latest χ EFT predictions by Bernard et al. [36]( diagonally hatched band) and Alarc´on et al. [37] ( cross-hatchedband), phenomenological models by Burkert et al. [41] (solid curve)and Soffer et al. [42] ( dashed curve), as well as our spin structurefunction parameterization [14] (dotted curve). The dash-dotted lineis the slope predicted by the GDH sum rule as Q → . To quantify the degree of agreement between our data and therecent χ EFT predictions [36, 37], we computed the χ per de-gree of freedom between these predictions and our results. Wefind that the predictions in [36] agree with our results only atthe lowest few Q points, up to Q = 0 . . GeV for Γ ( I ), if we require a χ < . On the other hand, thepredictions in [37] agree with our data over their full range,with χ < up to Q = 0 . GeV . The phenomeno-logical models [41, 42] agree well with our results for all Q values. The new results on Γ ( Q ) generally agree with a pre-vious experiment [14] in the overlapping Q region. However,there exist visible differences between our results and the spinstructure function parameterization [14], indicating that it canbe improved with our new data. Extrapolating our results on I ( Q ) to Q = 0 yields I exp (0) = − . ± . (5)assuming the Q -dependence of I predicted by Alarc´on etal. [37] within their quoted theoretical uncertainty (see detailsin the Methods section). This result is in good agreement withthe GDH sum rule prediction I GDH = − κ / − . for the proton and with the experimental photoproduction re-sult − . ± . ± . [27, 28]. Our resultsprovide, for the first time, a test of the GDH sum independent Q (GeV ) I GDH sum rule
FIG. 4. Results on I for the proton, with symbols the same as inFig. 3. The GDH value is shown by the arrow at I GDH = − . .The experimental photoproduction result [27, 28] is shown by thesolid square. from exclusive photoproduction [27, 28].Predictions from χ EFT for I ( Q ) and Γ ( Q ) are con-strained at Q = 0 by the GDH sum rule. No such con-straint is available for γ ( Q ) , the generalized longitudinalspin polarizability, related by a sum rule to the integral of A F [40, 43]: γ ( Q ) = 16 αM Q (cid:90) x x A ( x, Q ) F ( x, Q ) dx. (6)This endows γ ( Q ) with additional resolving power to testthe several theoretical predictions available. Furthermore, the x weighting in Eq. (6) suppresses the low- x contribution.This is beneficial since the low- x region is inaccessible ex-perimentally and must be estimated using models, which in-troduces model uncertainty. The two integrals I and γ havedifferent systematic uncertainties and therefore provide com-plementary tests of theoretical predictions.Our results for γ ( Q ) are shown in Fig. 5. Neither of thenew χ EFT calculations describes the full data set well: Thecalculation from Ref. [36] agrees in magnitude (but not inslope) with our lowest Q results up to Q ≈ . GeV ,while the calculation from Ref. [37] describes the shape ofthe data only marginally below that Q value. Together withthe photoproduction data point [27, 28, 44], our data indi-cate a strong change in Q slope towards a value consistentwith that predicted in Ref. [36] at very low Q . Classically, γ represents the distortion of the proton spin structure in re-sponse to the interference between various transverse electricand magnetic field components of the virtual photon shown inFig. 1. In a hadronic picture, γ is principally due to the dif-ference between the contribution from the pion cloud of theproton (positive) and the excitation of the ∆ (negative) [37].The data thus indicate that the ∆ contribution dominates atthe photon point and becomes even more important for small- Q virtual photons. This may be pictured intuitively from theextended size of the pion cloud whose contribution is quicklysuppressed with increasing Q . However, at higher Q , theslope turns over since the polarizability is a global feature ofthe proton which must vanish as Q → ∞ , as seen from the /Q factor in Eq. (6). Bernard et al.ParameterizationThis work (measured range only)This work (full integral)Fersch et al. (full integral)1.00−0.5 0.20.160.120.080.040 1.00.5−3.5−4.0−3.0−2.5−2.0−1.5−1.0 Alarcon et al.Ahrens et al. (real photon) γ x Q (GeV ) (f m ) FIG. 5. Results on γ ( Q ) for the proton, with symbols the same asin Fig. 3. The photoproduction data point [27, 28, 44] is shown asthe solid square. Although the upper bound of the validity domain of χ EFTis not known, the kinematic coverage of our data is well withinits expected range between m π ≈ . GeV and the chi-ral symmetry breaking scale, Λ χ ≈ GeV . The actual va-lidity range depends on the orders of the expansion parame-ter m π / Λ χ at which the calculations are done, the expansionmethod, and the observable. One reason for the limited suc-cess of χ EFT in describing our results may be coming fromthe difficulty to fully account for the ∆ resonance, the proton’sfirst excited state. In fact, early χ EFT calculations [30–32]did not explicitly include the ∆ excitation, which slows downthe convergence of the χ EFT perturbation series, or they in-cluded it phenomenologically [33–35]. This was thought tobe the reason why many of the early nucleon spin structurefunction data [15–22] disagreed with calculations [30–35].This disagreement prompted refined χ EFT calculations [36–39] and a new experimental program at Jefferson Lab op-timized to cover the χ EFT domain [45, 46], including themeasurement reported here. The latest calculations [36–39]both include the ∆ but differ in their expansion method to ac-count for the π - ∆ corrections. Ref. [36] treats the nucleon- ∆ mass difference δM as a small parameter of the same order as m π . Refs. [37–39] use δM as an intermediate scale such that δM/ Λ χ ≈ m π /δM is used as the expansion parameter to ac-count for the ∆ . In addition, the calculations [37–39] includeempirical form factors in the relevant couplings to approxi- mate the expected impact of high-order contributions. Theymake γ vanish at large Q , as observed, in contrast to calcu-lation [36] which purely contains terms computed with χ EFTand has no free-parameter that can be adjusted. For γ , whicharises at third order in the π -N loops there are large cancel-lations between π -N loops and the ∆ contribution. There-fore, the calculations are very sensitive to the expansion andrenormalization scheme, and the order of the expansion. Thisis why γ is especially well-suited to test χ EFT. Finally, theintegrals Γ and I contain Born terms in addition to the po-larizability contributions calculated in χ EFT. These terms areconstrained by the GDH sum rule at Q = 0 . Refs [37–39]assume that their Q -dependence follows the correspondingproton form factors. This Q -dependence leads to the differ-ence with Ref. [36] and the agreement with our data.In summary, the proton polarized structure functions g and A F and their integrals Γ , I and γ have been measured inthe very low Q region, down to . GeV . Our resultson I ( Q ) extrapolated to Q = 0 agree well with the origi-nal GDH sum rule. For Q > , they provide precise testsof predictions from χ EFT, the leading effective theory for thestrong interaction. These tests use for the first time the protonspin degrees of freedom in the Q region where χ EFT shouldbe the most applicable. Although it is essential to understandthe fundamental forces of nature from first principles, such de-scriptions are often impossible and one must use effective the-ories based on the new degrees of freedom that emerge fromcomplexity [47]. Our data show that it remains difficult for χ EFT to precisely describe all observables in which spin de-grees of freedom are explicit. They provide strong incentivefor future improvements of calculations using χ EFT, the lead-ing approach to the effective theory emerging directly fromQCD, and for extending the more fundamental lattice QCDcalculations to the spin-dependent structure of the nucleon.
Methods
We measured the spin difference yields on the l.h.s. ofEq. (1) and solved that equation for ∆ σ ( W, Q ) , from whichwe extracted g and A F as functions of W and Q . We re-lied on the standard CLAS G EANT -3 Monte Carlo simulationpackage to fully simulate the spin-dependent yields, includ-ing all radiative effects and detector responses. The efficiencyof the modified Cherenkov Counter was determined by com-paring data taken with only the Electromagnetic Calorimeterin the trigger to those taken with the standard trigger that re-quires a coincidence between both detectors. The ratio of thelatter to the former gave the Cherenkov efficiency. We se-lected only detector regions of well-understood acceptance inboth the data and the simulation. This process fully deter-mined the function a ( W, Q ) in Eq. (1). The same Eq. (1)was also used to extract the product L P b P t by comparing themeasured yield difference (l.h.s. of Eq. (1)), integrated overthe elastic peak region 0.85 GeV < W < ∆ σ ( W = M, Q ) which canbe calculated from the known electromagnetic form factorsof the proton [48]. The polarized cross section ∆ σ ( W, Q ) in the simulation was calculated using an event generator forinclusive electron scattering [49] with up-to-date models ofstructure functions and asymmetries, including near-final datafrom JLab experiment E08-027. We extracted our results on g and A F by varying our input parameterization for thesequantities and finding the required values to make our simu-lation for the polarized yield agree with data. Corrections forhigher-order quantum electromagnetic effects (radiative cor-rections) were applied in the simulation, of which one effectis the high-energy tail from elastic scattering (elastic radiativetail).We propagated the uncertainties on the polarized yields tothe final values for g and A F . Systematic uncertaintieswere studied by changing model parameters, or other inputs,and re-running the simulation. The overall uncertainty on thenormalization factor L P b P t for each beam energy varied from2% to 5%, dominated by the statistics of the measured elas-tic peak and , to a lesser extent, the accuracy of the protonelastic form factors [48] that enter into ∆ σ ( W = M, Q ) andhence into our determination of that factor. Smaller contribu-tions, all less than 1%, came from π − and e + e − backgrounds,and scattering off the slightly polarized N in the target. Thereconstruction of W has an uncertainty of less than 2 MeV,which was studied by shifting the simulated W spectrum andrepeating the extraction. Uncertainties due to trigger and par-ticle reconstruction and identification inefficiencies, as well asparameterizations for the structure functions, F , and A , ,were studied by varying them in the simulation. Uncertain-ties in the radiative corrections were estimated by varying theamount of material the electron passed through in the simu-lation, and by adjusting the elastic radiative tail within rea-sonable limits. In all, the total experimental uncertainty isdominated by statistics.To extrapolate our results on I ( Q ) to Q = 0 , we fit ourdata with a form obeying the Q -dependence of the Alarc´on etal. χ EFT calculation [37]. We chose this calculation becauseits Q -dependence agrees well with our data over a wide Q range. We found the intercept of our fit with the Q = 0 axis to be I exp (0) = − . ± . ± . ± . ± . , with χ = 2 . deter-mined with the “uncor” uncertainty. Here, “uncor” and “cor”refer to the experiment point-to-point uncorrelated and cor-related uncertainties, respectively; “range” refers to the un-certainty due to the Q range ( Q ≤ . GeV ) used forthe fit. The last contribution, “ form”, is the uncertainty onthe Q -dependence used for the fit. It is calculated from theuncertainty band given by the χ EFT calculation [37]. Sincethe various uncertainties are largely independent, they areadded quadratically, giving a total uncertainty of ± . . Thisis about twice smaller than that from photoproduction mea-surements of I (0) because the Q → extrapolation un-certainty calculated using [37] is negligible and because in-clusive electroproduction automatically sums over all reac-tion channels, thereby removing uncertainties associated withthe detection of final states needed in photoproduction. Onthe other hand, the extrapolation uncertainty is calculatedfrom [37], which disagrees with [36]. This indicates thatthe uncertainty bands provided in the calculations may notreflect the full theoretical uncertainties. Extrapolating usingthe Q -dependence from [36] yields I exp (0) = − . ± . (uncor) ± . (cor) ± . . (range) ± . (form), with χ = 2 . determined with the “uncor” uncertainty.The “uncor” value here is larger because the fit is limitedto very few data points ( Q ≤ . GeV ). This resultdiffers notably from our main result, as expected from thevery different slope of [36]. This discrepancy exemplifiesthe importance of testing and improving χ EFT calculations,since well-controlled predictions would make electroproduc-tion data very competitive for verifying the GDH sum rule andother real photon observables.
Data availability
Experimental data that support the find-ings of this study will be posted on the CLAS database,https://clasweb.jlab.org/physicsdb/, or are available from X.Zheng upon request.
Code availability
The computer codes that support the plotswithin this paper and the findings of this study are availablefrom X. Zheng upon request.
Author contributions
The members of the Jefferson LabCLAS Collaboration constructed and operated the experimen-tal equipment used in this experiment. A large number of col-laboration members participated in the data collection. Thefollowing authors provided various contributions to the exper-iment design and commissioning, data processing, data anal-ysis and Monte Carlo simulations: M. Battaglieri, R. De Vita,V. Drozdov, L. El Fassi, H. Kang, K. Kovacs, E. Long, M.Osipenko, S. Phillips, K. Slifer. The authors who performedthe final data analysis and Monte Carlo simulations were A.Deur, S.E. Kuhn, M. Ripani, J. Zhang, and X. Zheng.The manuscript was reviewed by the entire CLAS collabo-ration before publication, and all authors approved the finalversion of the manuscript.
Competing interests
The authors declare no competing inter-ests.
Additional information
Supplementary information are available online that includesall numerical results reported here.Correspondence and requests for materials should be ad-dressed to A. Deur.Reprints and permissions information is available at ... (to beupdated upon publication)
ACKNOWLEDGMENTS
We thank the personnel of Jefferson Lab for their effortsthat resulted in the successful completion of the experiment.We are grateful to U.-G. Meißner and V. Pascalutsa for usefuldiscussions on the theoretical χ EFT calculations. This workwas supported by the U.S. Department of Energy (DOE), theU.S. National Science Foundation, the U.S. Jeffress MemorialTrust; the United Kingdom Science and Technology FacilitiesCouncil (STFC), the Italian Istituto Nazionale di Fisica Nu-cleare; the French Institut National de Physique Nucl´eaire etde Physique des Particules, the French Centre National de laRecherche Scientifique; and the National Research Founda-tion of Korea. This material is based upon work supported bythe U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177.
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