Measurement of the Generalized Polarizabilities of the Proton at Intermediate Q 2
H.Fonvieille, J. Beričič, L. Correa, M. Benali, P. Achenbach, C. Ayerbe Gayoso, J.C. Bernauer, A. Blomberg, R. Böhm, D. Bosnar, L. Debenjak, A. Denig, M.O. Distler, E.J. Downie, A. Esser, I. Friščić, S. Kegel, Y. Kohl, M. Makek, H. Merkel, D.G. Middleton, M. Mihovilovič, U. Müller, L. Nungesser, M. Paolone, J. Pochodzalla, S. Sánchez Majos, B.S. Schlimme, M. Schoth, F. Schulz, C. Sfienti, S. Širca, N. Sparveris, S. Štajner, M. Thiel, A. Tyukin, A. Weber, M. Weinriefer
aa r X i v : . [ nu c l - e x ] A ug Measurement of the Generalized Polarizabilities of the Proton at Intermediate Q H. Fonvieille, ∗ J. Beriˇciˇc, L. Correa,
1, 3
M. Benali, P. Achenbach, C. Ayerbe Gayoso, † J.C. Bernauer,
4, 5
A. Blomberg, R. B¨ohm, D. Bosnar, L. Debenjak, A. Denig, M.O. Distler, E.J. Downie, A. Esser, I. Friˇsˇci´c, S. Kegel, Y. Kohl, M. Makek, H. Merkel, D.G. Middleton, M. Mihoviloviˇc,
U. M¨uller, L. Nungesser, M. Paolone, J. Pochodzalla, S. S´anchez Majos, B.S. Schlimme, M. Schoth, F. Schulz, C. Sfienti, S. ˇSirca,
10, 2
N. Sparveris, S. ˇStajner, M. Thiel, A. Tyukin, A. Weber, and M. Weinriefer (A1 Collaboration) Universit´e Clermont Auvergne, CNRS/IN2P3, LPC, F-63000 Clermont-Ferrand, France Joˇzef Stefan Institute, SI-1000 Ljubljana, Slovenia Institut f¨ur Kernphysik, Johannes Gutenberg-Universit¨at Mainz, D-55099 Mainz, Germany Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794-3800, USA RIKEN BNL Research Center, Upton, NY 11973-5000, USA Temple University, Philadelphia, PA 19122, USA Department of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia Institute for Nuclear Studies, Department of Physics, The George Washington University, Washington DC 20052, USA Laboratory for Nuclear Science, Massachussetts Institute of Technology, Cambridge, MA 02139, USA Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia (Dated: August 26, 2020)
Background:
Generalized polarizabilities (GPs) are important observables to describe the nucleon structure,and measurements of these observables are still scarce.
Purpose:
This paper presents details of a virtual Compton scattering (VCS) experiment, performed at the A1setup at the Mainz Microtron by studying the ep → epγ reaction. The article focuses on selected aspects of theanalysis. Method:
The experiment extracted the P LL − P TT /ǫ and P LT structure functions, as well as the electric andmagnetic GPs of the proton, at three new values of the four-momentum transfer squared Q : 0.10, 0.20 and 0.45GeV . Results:
We emphasize the importance of the calibration of experimental parameters. The behavior of themeasured ep → epγ cross section is presented and compared to the theory. A detailed investigation of thepolarizability fits reveals part of their complexity, in connection with the higher-order terms of the low-energyexpansion. Conclusions:
The presented aspects are elements which contribute to minimize the systematic uncertainties andimprove the precision of the physics results.
I. INTRODUCTION
Nucleon polarizabilities are fundamental observableswhich describe how the charge, magnetization and spindensities in the nucleon are deformed when an exter-nal quasi-static electromagnetic field is applied. Theycan be accessed through the Compton scattering pro-cess γN → N γ , and owe their small magnitude [1] tothe strong binding force of quantum chromodynamics.Polarizabilities extend to finite momentum transfer, byreplacing the incoming real photon with a space-like vir-tual one ( γ ∗ ), of virtuality Q . This leads to the con-cept of generalized polarizabilities (GPs) [2], i.e., Q -dependent observables describing the spatial distributionof the polarization density in the composite system. Nu-cleon GPs are accessed in the virtual Compton scattering(VCS) process γ ∗ N → N γ , via the eN → eN γ reaction. ∗ [email protected] † Now at College of William and Mary, Williamsburg, VA 23185,USA
The associated theoretical framework was first estab-lished in Ref. [3]. Further developments [4] led to six in-dependent GPs at lowest order: two scalar ones, the elec-tric GP α E ( Q ) and the magnetic GP β M ( Q ), plusfour spin GPs. These observables have a well-definedcontinuity to the polarizabilities in real Compton scat-tering (RCS) at Q = 0.The low-energy regime is defined by small values of thetotal energy W in the γ ∗ N center-of-mass (c.m.), typ-ically below the pion production threshold, or slightlyabove it. In this regime, the photon electroproduc-tion cross section is dominated by the so-called Bethe-Heitler(BH)+Born cross section, dσ BH+Born , that con-tains no polarizability effect and is entirely calculablein quantum electrodynamics. The effect of the GPs con-sists of a small deviation of the experimental eN → eN γ cross section from dσ BH+Born . The electric and magneticGPs of the proton have been measured by several exper-iments, at various four-momentum transfers in the Q range from 0.06 to 1.76 GeV [5–16]. GPs are extractedfrom ep → epγ cross sections by fitting methods basedeither on the low-energy theorem [3] (low-energy expan-sion or “LEX fit”) or the dispersion relation model forCS [17, 18] (“DR fit”). A more complete presentationcan be found in the recent review [19].Our VCS experiment has been conducted at the MainzMicrotron (MAMI) at various times from 2011 to 2015,to perform new measurements of the electric and mag-netic GPs of the proton in the intermediate Q range.The results have been published in Ref. [16], in termsof GPs and structure functions. The experiment wasperformed essentially below the pion production thresh-old, and GPs were extracted from the measurement ofabsolute ep → epγ cross sections, using the two fittingmethods cited above. The aim of the present paper is togive complementary accounts of this experiment. Aftera brief review of the instrumental configuration (Sec. II),details of the analysis are provided, with a focus oncalibration aspects (Sec. III), photon electroproductioncross sections (Sec. IV) and polarizability fits (Sec. V).Cross-section data are available electronically as sup-plemental material to this article [20] and at arXiv.orgin the source files. II. THE EXPERIMENT
The experiment uses the unpolarized MAMI electronbeam and the A1 setup with a 5 cm long liquid hydro-gen target and the two high-resolution, small solid-anglemagnetic spectrometers A and B in coincidence. Werefer to Ref. [21] for a detailed description of the appa-ratus. The detector package comprises a set of verti-cal drift chambers and scintillators in each arm, plus aCherenkov detector in the electron arm. The beam ofintensity 5-15 µ A is rastered on the target by 1-2 mm inboth transverse directions. The instantaneous luminos-ity of the experiment reaches (0.6-1.8) × cm − /s.The detected particles are the scattered electron andthe outgoing proton of the ep → epγ reaction. Theevent reconstruction yields the particles’ four-momentaat the vertex, denoted by k ′ and p ′ for the final electronand final proton, respectively. The four-momentum ofthe missing particle (the outgoing photon), denoted by q ′ , can then be reconstructed as q ′ = k + p − k ′ − p ′ ,where k and p are the four-momenta of the target pro-ton and the incoming electron, respectively. The missingmass squared, noted M X = ( q ′ ) , exhibits a clear peakcorresponding to a single undetected photon, the so-called “VCS events” (cf. Fig. 3). The four-momentumof the virtual photon is q = k − k ′ , with Q ≡ − q .The experiment studies VCS at three yet unexploredvalues of Q : 0.10, 0.20 and 0.45 GeV . The aim istwofold: to cover a rather large Q range, while sur-rounding the point at Q = 0 .
33 GeV where previ-ous measurements exist and are intriguing. An impor-tant variable for the design is the modulus of the three-momentum of the outgoing photon in the ( γ ∗ p ) c.m., de-noted by q ′ c . m . . Two other main kinematical variables arethe polar and azimuthal angles of the outgoing photonwith respect to the virtual photon in the c.m., denoted TABLE I. The main kinematical settings, in terms of beamenergy E beam , spectrometer central momenta P A and P B ,spectrometer angles relative to the beamline, θ A and θ B , andthe out-of-plane angle of spectrometer B ( OOP B ). The scat-tered electron is detected in spectrometer B (resp. A) at Q = 0.10 and 0.20 GeV (resp. 0.45 GeV ). A few com-plementary settings are also used as slight variants of theseones.Setting E beam P A θ A P B θ B OOP B name (MeV) (MeV/ c ) ( ◦ ) (MeV/ c ) ( ◦ ) ( ◦ ) Q = 0 .
10 GeV INP 872 425 53.1 700 22.9 0OOP 872 343 52.6 693 21.9 9.0LOW 872 365 58.0 745 22.4 0 Q = 0 .
20 GeV INP 1002 580 51.5 766 30.4 0OOP 1002 486 51.0 766 29.2 8.5LOW 905 462 52.2 723 32.5 0 Q = 0 .
45 GeV INP 1034 650 51.2 634 32.7 0OOP 1034 647 51.0 750 39.2 8.0LOW 938 645 52.3 713 40.5 0 by θ c . m . and φ c . m . respectively.The low-energy theorem [3] is valid only below thepion production threshold, corresponding to W = m N + m π and q ′ c . m . = 126 MeV/ c . Given the fact that the ef-fect of the GPs in the cross section increases with q ′ c . m . ,different energy regions are defined, according to theirincreasing sensitivity to the GPs: “low- q ′ c . m . ” ( q ′ c . m . < c ) and “high- q ′ c . m . ” ( q ′ c . m . >
50 MeV/ c ). At each Q , three kinematical settings are chosen, each one witha different goal: i ) a high- q ′ c . m . , out-of-plane setting(“OOP”) with large sensitivity to the electric GP, ii )a high- q ′ c . m . , in-plane setting (“INP”) with mixed sensi-tivity to the electric and magnetic GPs, and iii ) a low- q ′ c . m . setting (“LOW”) with no sensitivity to the GPsbut useful for normalization. These settings are listedin Table I. Note that the OOP B angle of 8-9 ◦ in thelaboratory frame allows one to reach φ c . m . = 90 ◦ in thec.m. At each Q , the experiment is performed at a sin-gle value of the virtual photon polarization parameter ǫ .The settings are designed to maximize this parameter,since large values of ǫ enhance the GP effect in the crosssection.High statistics are achieved in the experiment, withabout 900k, 1100k and 300k VCS events recorded at Q = 0.10, 0.20 and 0.45 GeV , respectively. About onethird of the statistics corresponds to low- q ′ c . m . ( q ′ c . m . <
50 MeV/ c ) and is used for absolute normalization (cf.Sec. IV). The remaining two thirds of events correspondto higher q ′ c . m . and are used in the polarizability fits. Themotivation for such high statistics is driven by consider-ations on the GP effect at backward θ c . m . angles. Thisangular region is important because of its high sensitiv-2ty to the magnetic GP, via the structure function P LT .However, in this region the GP effect exhibits very rapidvariations (cf. Fig. 9) and the LEX fit may not be ap-plicable everywhere (cf. Sec. V). To be able to includethis region selectively in the fit, one needs a fine 2D-binning in (cos θ c . m . , φ c . m . ), with reasonable statistics ineach bin. III. DATA ANALYSIS
The experiment is quite demanding in terms of accu-racy of the measured ep → epγ cross section. Indeedthe effect of the GPs in the cross section is very small,ranging from a few percent to at most 15%. The qual-ity of the event reconstruction, the calibration of ex-perimental parameters and the reliability of the simula-tion are key factors to minimize the systematic error andachieve competitive uncertainties of the physics results.A few-percent systematic error on the cross section in-duces non-negligible biases in the polarizability fits. Wetherefore aim at a precision of 1% on the knowledge ofthe solid angle, a goal that can be reached thanks tothe excellent performances of the MAMI beam and theA1 setup. Sections III A to III D describe the steps to-wards this goal. Sections III E and III F summarize theanalysis cuts and the corrections to the event rate, whileSec. III G recalls a few features of the simulation. A. Event reconstruction
The event reconstruction is carried out by the A1COLA software. In each spectrometer, vertical driftchambers provide a track of the detected particle in thefocal plane, characterized by two transverse coordinates( x fp , y fp ) and two projected angles ( θ fp , φ fp ). This trackis transformed into variables of the particle at the targetby using the spectrometer optics, described by the op-tical transfer matrix. One obtains four variables at thevertex: the relative momentum δ = ( P − P ref ) /P ref where P ref is the reference momentum, the projected verticaland horizontal angles, θ and φ , respectively, as well asthe transverse horizontal coordinate y in the spectrom-eter frame. Some of this information is then coupledbetween the two spectrometers, to build more elaboratevariables in the laboratory frame, such as the missingmass squared M X . The longitudinal coordinate of thevertex, Z vertex , is obtained by intersecting the beam di-rection with the direction of the particle going into spec-trometer B. This spectrometer is chosen for the vertexreconstruction, since its point-to-point focusing proper-ties provide the optimal resolution in the y coordinate. Z vertex depends therefore directly on y B ) . The trans-verse coordinates of the vertex, horizontal ( Y vertex ) andvertical ( X vertex ), are obtained solely from the beam po-sition, and are formally equal to the instantaneous valuesof the beam transverse positions Y beam and X beam , re- spectively, corrected for the raster pattern. The time ofcoincidence between the two detected particles is formedby using the TDC information of the scintillators in eachspectrometer. Three other variables coupling the twospectrometers: q ′ c . m . , cos θ c . m . and φ c . m . , are constructedfor defining the 3D cross-section bins. B. Experimental calibration
An important step of the analysis is the calibration ofexperimental parameters. After the raw calibration ofdetectors (documented, e.g., in Ref. [22]), a second levelof calibration involves additional items, such as: opticaltransfer matrix elements, various offsets in momenta, an-gles and positions, and a specific parameter describingthe cryogenic deposit on the walls of the target cell (cf.Sec. III D). A major tool for judging the overall qual-ity of the calibration is the missing mass squared M X .It is sensitive to almost all parameters, but as a singlevariable it does not permit to adjust them all. Thus,different studies were developed off-line in order to fixall the experimental parameters. They are described inSects. III C and III D. C. Optical studies
A first study concerns the optical transfer matricesof the spectrometers. The work of Ref. [21] has es-tablished that, for spectrometer B, a single set of op-tical coefficients can be used for dipole magnetic fieldsup to 1.2 T, i.e., a reference momentum of 600 MeV/ c .Ref. [21] also reports that, for spectrometer A, no field-dependent effects are seen up to 600 MeV/ c , a valueat which first indications of field saturation effects be-come visible. Above 600 MeV/ c , the optical properties ofthe spectrometers may change increasingly due to mag-netic saturation. In our experiment, spectrometer mag-nets are operated in the saturation region all the timefor spectrometer B and about one third of the time forspectrometer A (cf. Table I). Calibration data takenduring the experiment allow one to make some improve-ments with respect to the available spectrometer opticsat high fields, namely for spectrometer B. This optimiza-tion work is outlined below.Data taken with a stack of thin foils regularly spacedalong the beam axis are used to optimize the optics in y B ) at several central momenta between 635 and 765MeV/ c . The y B ) variable is of special importance sinceit determines the longitudinal coordinate of the interac-tion point, Z vertex , on which one of the main analysiscuts is applied (see Fig. 4 and Sec. III E). Data with asieve-slit collimator are taken to control the optics in the( θ , φ ) ( B ) angles.For the relative momentum δ ( B ) , a few lowest-orderoptical coefficients can be partially adjusted on our( e, e ′ p ) coincidence data. The method is based on op-3 E miss (MeV) c oun t s -5000500 -2 -1 0 1 2 3 4 5 6 7 E miss (MeV) x f p ( B ) ( mm ) -5000500 -2 -1 0 1 2 3 4 5 6 7 E miss (MeV) x f p ( B ) ( mm ) FIG. 1. (Color online) Top plot: nuclear levels in the reaction C ( e, e ′ p ) X where the missing energy E miss (corrected forkinematical broadening) represents the excitation energy ofthe B nucleus. The peak FWHM is 0.30 MeV. Middle plot:the nuclear peaks versus the focal plane coordinate x fp( B ) , fora well-adjusted first-order coefficient ( δ | x ) of spectrometerB. Bottom plot: the same thing for ( δ | x ) of spectrometer Bdecreased by 1%. timizing the width of the narrow peaks correspondingto nuclear levels in the missing energy spectrum. Suchpeaks originate from processes of the type A ( e, e ′ p ) A -1 ( ∗ ) and are observed in various calibration runs using a car-bon target ( A = 12). They are also seen in “VCS runs”when the q ′ c . m . variable, which actually corresponds tothe missing energy, is small enough. In this last case, thenuclear ( e, e ′ p ) events take place at the extreme ends ofthe cryotarget, where the beam crosses the walls of thecell and the cryogenic deposit. An example of nuclearpeaks observed with a carbon target is given in Fig. 1.The figure also illustrates the high sensitivity one canreach in the adjustment of the main first-order element( δ | x ) (see Eq. (1) for definition) with such events. Sincethis method uses both spectrometers at the same time,it relies on the good knowledge of the δ -optics of onespectrometer, in order to tune the δ -optics of the otherspectrometer.Based on the above adjustments, dedicated transfermatrices for spectrometer B have been devised and usedat each central momentum setting. The optical trans-port is expressed by a polynomial expansion of the focalplane variables, given by the following set of equations (we adopt notations similar to Ref. [21]): δ = ( δ | x ) x fp + ( δ | θ ) θ fp + ... ,θ = ( θ | x ) x fp + ( θ | θ ) θ fp + ... ,φ = ( φ | y ) y fp + ( φ | φ ) φ fp + ... ,y = ( y | y ) y fp + ( y | φ ) φ fp + ... . (1)Here, only the eight first-order (and dominant) termshave been explicitly written out, and the dots indicatethe series of higher-order terms, which are proportionalto x i fp θ j fp y k fp φ l fp . Figure 2 shows the eight first-order el-ements of the spectrometer B transfer matrices used inthe experiment, as a function of the central momentum P B . Although not deduced from a dedicated calibra-tion campaign, and therefore not very accurate, theygive an idea of the magnitude of the saturation effects inthis spectrometer. Overall, the observed variations aresmooth versus P B . The main terms: ( δ | x ) , ( θ | θ ) , ( y | y )and ( φ | φ ), are only slightly affected by saturation effects,showing at most a 2.5% relative change in the displayedmomentum range. For instance, the element ( y | y ), whichessentially gives the scale of the y B ) reconstruction, isfound to vary only by ≈
1% in the saturation region.However, ignoring this change would induce an error ofup to 1% on the scale of the target length, and hencea systematic error of similar size on the measured crosssection. Other first-order terms in Fig. 2, such as ( δ | θ )or ( y | φ ), show larger relative variations, but their con-tribution is comparatively small.For spectrometer A, the same optimization work hasnot been done, since available optics in the saturation re-gion (at P A = 645 MeV/ c ) give essentially satisfactoryresults, in terms of sieve-slit reconstruction, M X widthor nuclear peaks width. We just note that hints of sat-uration are observed for a central momentum P A as lowas 580 MeV/ c . D. Offsets and other calibration parameters
Many parameters are continuously monitored on-line in order to ensure stable data taking conditions.While the AQUA program performs data acquisition,the MEZZO software performs the slow control of basi-cally every instrumental device in the A1 Hall: magnets,detectors, cryotarget, beam delivery, etc., and most ofthese items are known with high precision in real time.Table II gives a list of the parameters that have an im-pact on either the particle reconstruction, the missingmass squared M X , or the acceptance as calculated bythe simulation. Some of these items do not need ad-justment since they are measured with high precision: ≈ − relative for the beam energy E beam , ≈ < − relative for thecentral momentum P A . The other items of Table II po-tentially need to be adjusted, essentially by off-line re-calibrations. The corresponding methods, listed in Ta-ble II, are outlined below.4 ABLE II. Various parameters having a direct impact on the reconstructed missing mass squared and/or the simulated accep-tance. Items are listed in the first column. The second column indicates the existence of a real-time measuring device, or theorigin of the offset. The third column specifies the potential need for an off-line adjustment. How often the latter should bedone is indicated in the fourth column. The different adjustment methods are numbered in the last column.Type of offset or Source of information or need to Time basis Methodcalibration constant measuring device adjust?Beam energy E beam measured by MAMI noSpectrometer angles relative to the beam on-line readout noSpectrometer A central momentum P A measured by NMR probe noTransverse beam position, horizontal Y beam punctual screenshots yes per run IOffset in horizontal angles φ A ) and φ B ) related to spectrom. optics yes once for all IOffset in horiz. coordinates y A ) and y B ) related to spectrom. optics yes once for all ICryotarget longitudinal centering Z target pre-experiment surveys yes per cooldown ITransverse beam position, vertical X beam punctual screenshots yes per run IIOffset in vertical angles θ A ) and θ B ) related to spectrom. optics yes once for all IIICryogenic deposit on target walls e frost none yes per run IVSpectrometer B central momentum P B measured by Hall probe yes per field setting IV P B (MeV/c) P B (MeV/c) -0.595-0.59-0.585-0.58 500 600 700 800 ( d| x) ( d|Q ) ( Q| x) ( Q|Q ) (y | y) (y |F ) ( F| y) ( F|F ) FIG. 2. (Color online) The eight first-order elements (cf.Eq. (1)) of spectrometer B optics as a function of the cen-tral momentum, used in the VCS analysis. The experimentcovers the region P B ∈ [634-770] MeV/ c . The starred pointindicates the non-saturated value at P B = 495 MeV/ c . Theunits in ordinate combine cm, mrad and percent. Method I focuses on a set of variables pertaining tothe horizontal plane, and treats them altogether. Forconvenience, time-independent offsets are introduced forthe y A ) and y B ) coordinates, and for the φ A ) and φ B ) angles of the reconstructed particles. The longi-tudinal position of the center of the cryotarget alongthe beamline, Z target , is known only to a limited preci-sion. Indeed, the target may slightly move when going from warm to cold state, with a degree of reproducibilitythat is unknown. We therefore consider one adjustablevalue of Z target for each new establishment of the coldstate. The beam position on the target is not contin-uously monitored during the experiment, but only in-spected visually at discrete times, by inserting a scin-tillating Al O screen. The Y beam parameter (averagedover the raster) is thus re-determined for each run.A global fit of these different parameters is realized,based on several constraints on reconstructed variables: i ) the target center, Z target , must be the same when seenby both spectrometers A and B, and must be constantover given periods of time; ii ) the Z -position of the thincarbon target used in calibration runs must be as closeas possible to zero, to agree with precise pre-experimentsurveys; iii ) the edges of the entrance collimators mustdisplay a left-right symmetry in their positioning. In-deed, each collimator is centered by construction on thespectrometer’s optical axis. The variable allowing thistest is the reconstructed impact coordinate at the col-limator plane: Y colli = y + D tan φ , where D is thetarget-to-collimator distance.As a result of this global optimization, performed onthe entire data set, the center of the cryotarget is foundto be shifted upstream along the beamline, by 1.4 mmto 3.3 mm depending on the data taking period. Thisknowledge serves as an input to the simulation. Thebeam horizontal position is found to be very stable intime, with excursions smaller than ± φ B ) , and an extra-offset in y B ) in therange (0.5-0.9) mm. These very small values testify tothe remarkable stability of the spectrometer’s mechani-cal alignment during out-of-plane motions.5ethod II allows to adjust the vertical beam position X beam in-between the daily visual inspections. It usesthe fact that variations in X beam induce visible shiftsin the sharp edges of the θ A ) distribution (the verticalangle of the particle), due to the very small target-to-collimator distance (0.56 m) in spectrometer A. Fittingthe centroid of the θ A ) spectrum for each run providesan efficient follow-up of the X beam variations with time.Observed excursions with respect to the nominal set-point do not exceed ± M X photon peak, i.e., to center it on its nominalposition and minimize its width. This peak width is rep-resentative of the resolution achieved by the apparatus.As already mentioned, the M X variable is kinematicallysensitive to all particles’ momenta and angles, and to thethickness of the cryogenic deposit on the target walls. Awrong value of these parameters causes distortions of the M X peak, which in turn allow for diagnostics on someglobal offsets.Method III focuses on possible global offsets attachedto the vertical angles θ A ) and θ B ) of the reconstructedparticles. The M X optimization does not constrain bothparameters, but only a linear combination of them, of thetype ( P A sin θ A ) + P B sin θ B ) ). The main finding isthat the adjustment hints at a small but noticeable ver-tical misalignment with respect to an ideal setup. Anoffset is needed that de-centers the distribution of ei-ther the θ A ) angle or the θ B ) angle. In the absenceof further identification of its origin, this misbalance isentirely attributed to the θ A ) angle, de-centering itsdistribution by about 3.2 mr for the settings at Q =0.10 and 0.20 GeV , and 0.6 mr for the settings at Q =0.45 GeV . In the simulation, this departure from anideal setup is reproduced by shifting the entrance colli-mator of spectrometer A by about 1.8 mm downwardsfor the settings at Q = 0.10 and 0.20 GeV , and 0.3mm downwards for the settings at Q = 0.45 GeV .Method IV determines the last two unknown param-eters. The first one is related to the cryogenic depositaround the target cell, due to residual nitrogen, oxygenand water vapor present in the scattering chamber. Thisdeposit varies with time in an unpredictable way, and af-fects the acceptance through particle energy losses. Thisextra-material is modeled in the analysis codes by a uni-form layer over the cell, leading to one single adjustableitem: the layer thickness, e frost , in g.cm − . The sec-ond parameter is the value of the central momentum inspectrometer B, P B . It is measured with a rather lim-ited accuracy (a few per mil or more) by a Hall probe,and needs to be more finely determined at each new fieldsetting.The key variables to optimize these two parametersare the position and the width of the M X photon peak.Contrarily to the previous methods based solely on ex-perimental data, here the simulation is also used. Onecan then exploit the two most-sensitive features: the high sensitivity of the peak position to P B in the experi-ment, and the high sensitivity of the peak width to e frost in the simulation. We note in passing that, apart fromthe cryogenic deposit, all other sources contributing tothe resolution in the simulation (cf. Sec. III G) are wellconstrained by other means.This two-fold optimization leads to a unique solutionin terms of P B and the cryogenic deposit h e frost i averagedover the setting. As a last step, e frost is finely tunedrun-per-run in the experimental sample, by requiring theposition of the M X photon peak to be stable in time.Overall, the thickness of the cryogenic deposit is foundto vary in the range (0 - 0.1) g.cm − throughout thewhole data taking. The adjusted values of P B departfrom the Hall probe readings by ≈ a few per mil, which isconsistent with the expected accuracy of the measuringdevice.The resulting M X distributions of Fig. 3 show thegood level of agreement obtained between the experi-ment and the simulation. Depending on the setting, thephoton peak is centered on values ranging from 20 to100 MeV and the optimized width is in the range (300-1300) MeV (FWHM). On average, the simulation andthe experiment agree to ≈ ±
10 MeV on the peak cen-tering, and to ≈ ±
20 MeV on the peak width. Thisgood agreement is also verified locally in the VCS phasespace.As a conclusion to this section, a good calibration ofall the mentioned parameters is important to get the cor-rect experimental event rate, as well as a faithful simu-lation. The accuracy reached by the above methods isestimated to be below ± X beam and Y beam ) and on y offsets, ± θ , φ ) angles, ± c on P B and ± − on h e frost i . Dedicated simulation studiesshow that, for each parameter varying within its quotedprecision, the corresponding uncertainty, or systematicerror on the integrated solid angle is in most cases wellbelow 1% relative. The most crucial case is the knowl-edge of h e frost i for the settings at Q = 0 .
10 GeV . Inthese kinematics, the outgoing protons have the lowestmomenta (kinetic energies of 70-90 MeV) and the simu-lated acceptance is very sensitive to the proton’s energyloss through the layer of cryogenic deposit. This param-eter has to be known to better than ± − inorder to control the solid angle to ± y A ) and y B ) offsets are replaced by asingle offset in Z target (known to better than ± e frost parameter (sector 8)when Q increases. The results differ from one Q toanother; dominant calibration uncertainties come from e frost at Q = 0 .
10 GeV , Z target at Q = 0 .
20 GeV and6 M x2 (MeV ) c oun t s M x2 (MeV ) M x2 (MeV ) M x2 (MeV ) M x2 (MeV ) M x2 (MeV ) M x2 (MeV ) M x2 (MeV ) M x2 (MeV ) LOW INP OOPLOW INP OOPLOW INP OOP (a)(b)(c)
FIG. 3. (Color online) The experimental (solid red) and simulated (dotted blue) distributions of the missing mass squared, foreach type of setting. Plots in columns refers to LOW, INP and OOP settings, while rows (a), (b), (c) refer to Q = 0 . , .
20 and0 .
45 GeV , respectively. All the analysis cuts are applied. Both the experimental and simulated distributions are normalized tothe same luminosity (i.e., there is no free adjustment). The lower and upper cuts in M X (vertical dashed green lines) correspondto − σ , where σ is the r.m.s. of the photon peak. X beam at Q = 0 .
45 GeV . E. Analysis cuts
The VCS sample is obtained from the experimentaldata by selecting the true coincidences via a timingcut, and essentially applying two main analysis cuts, in Z vertex and M X .The coincidence time spectrum exhibits a narrowpeak, over a wide plateau formed by random events. TheFWHM of the peak is in the range (0.8-1.7) ns. The truecoincidences are kept in a window of ± Z vertex is obvious from Fig. 4,which compares the experiment and the simulation atthe same level of cuts. While both event rates agree wellin the central part of the target cell, they disagree at theextreme ends. For most settings, this region of the tar-get shows an excess of experimental events relative to the simulation, due to ( e, e ′ p ) reactions on nuclei, not con-sidered in the simulation. In one case (setting “LOW” at Q = 0.20 GeV ), a loss of experimental events, insteadof an excess, is seen at the downstream end of the tar-get. It may come from particles absorbed in the magnetsfor the events most close to elastic ep → ep kinematics.The cut (dashed vertical lines in Fig. 4) selects the cen-tral part of the Z vertex spectrum, reducing the usabletarget cell length to about 3 cm.As the second main cut, events are required to be inthe photon peak of the missing mass squared spectrum.The wide selection window around the peak center (cf.Fig. 3) allows one to include a large fraction of the radia-tive tail that develops on the positive- M X side. Theseradiative events are well reproduced by the simulation.The cut in Z vertex is the only one that eliminates alarge fraction of VCS events. The cut in M X just re-moves the distant part of the radiative tail. We nowmention a few auxiliary cuts, which remove even smallerfractions of good events. Firstly, events are excludedwhen they are reconstructed far out of the nominal ac-ceptance, either in the ( θ , φ ) angles, or in the impactpoint at the collimator, or in the relative momentum δ .The selected window for δ is ( − − Z vertex (mm) c oun t s Z vertex (mm) Z vertex (mm) Z vertex (mm) Z vertex (mm) Z vertex (mm) Z vertex (mm) Z vertex (mm) Z vertex (mm) LOW INP OOPLOW INP OOPLOW INP OOP (a)(b)(c)
FIG. 4. (Color online) The experimental (solid red) and simulated (dotted blue) distributions of the longitudinal vertexcoordinate Z vertex , for each type of setting, with the same nomenclature for the plots as in the previous figure. The selectedevents are true coincidences within the M X cut. Both the experimental and simulated distributions are normalized to the sameluminosity. The useful part of the spectrum is the central region, delimited by the two vertical dashed green lines. settings a 2D-cut in the ( M X , q ′ c . m . ) plane is designed toeliminate the few events at the most negative values of M X , which are seen in the experiment but not in thesimulation. These events may come from ep → ep elas-tic scattering followed by particle rescattering inside thespectrometers.After having applied all the cuts, one obtains a “pureVCS” experimental sample, very clean, as seen fromFig. 3. In particular, there is no need for particle-identification (PID) cuts. This can be checked by testingthe response of the PID detectors, i.e., the Cherenkovdetector in the electron arm and the scintillators in theproton arm. At this stage of the analysis, there is ex-tremely small trace, if any, of π − in the distribution ofthe Cherenkov signal, or π + in the distribution of scin-tillator ADC signals. F. Event rate corrections and luminosity
The rate of experimental events, obtained after allcuts and the subtraction of random coincidences, is cor-rected for data acquisition deadtime. Since the scintilla-tors are trigger elements, the event rate is also correctedfor scintillator inefficiency. The latter is mapped in the( x, y ) coordinates in the scintillator planes, and found to be negligible almost everywhere, except in some lo-calized regions at the overlap of the scintillator paddles.The efficiency of the vertical drift chambers is consid-ered to be 100% in all cases. At this stage, one obtainsthe number of experimental events N exp in each of the3D cross-section bins. The precise measurement of theexperimental luminosity L exp relies on two inputs: thebeam current, given by a fluxgate magnetometer, andthe liquid hydrogen density, determined from pressureand temperature sensors. The continuous monitoring ofthese target parameters, together with the beam raster-ing, ensure a very stable liquid hydrogen density. G. Simulation
The acceptance, or solid angle ∆Ω that is needed todetermine the ep → epγ cross section, is too complexto be calculated by simple means. It requires the useof a simulation, as complete and faithful to the exper-iment as possible. We only summarize here the mainfeatures of the calculation of this acceptance, noted here-after ∆Ω sim . A more detailed description can be foundin Ref. [23]. The simulation only deals with ep → epγ events in the hydrogen volume of the cell, and doesnot consider any physical background or secondary pro-8 ABLE III. Results of the normalization test at each Q , us-ing the data of the “LOW” settings (and their variants) at q ′ c . m . = 37 . c . The fitted value of the normalizationfactor is given in the third column, together with its statisti-cal uncertainty obtained at ( χ +1) (non-reduced χ ). Thereduced χ of the fit and the number of degrees of freedomare given in the fourth and fifth columns, respectively. Thetest uses the proton form factors parametrization of Ref. [24]for calculating dσ BH+Born . Q Setting fitted F norm χ n.d.f.0.10 GeV LOW (I) 0.9856 ± LOW (II) 1.0092 ± LOW (III) 0.9704 ± LOW (I) 0.9894 ± LOW (II) 0.9885 ± LOW 1.0173 ± cesses. ∆Ω sim is an “effective” and not purely geomet-rical solid angle, in the sense that all resolution effectsare taken into account. The simulation includes the ra-diative effects which generate the tail in missing masssquared, and the effect of the cryogenic deposit aroundthe target cell. Other sources of resolution consist inmultiple Coulomb scattering, energy losses and strag-gling in the known materials, tracking errors in the focalplane and reconstruction errors at the target level. Thedescription of the apparatus is based on the nominalcharacteristics (cf. Ref. [21]). Namely, the acceptanceof the spectrometers is defined solely by the geometri-cal aperture of their entrance collimator, plus the nomi-nal momentum acceptance. The simulation incorporatesfurthermore the results of the calibration described inSec. III, using setting-averaged parameter values. Asimulated sample is obtained for each kinematical set-ting separately, together with its associated luminosity L sim . The simulated events are weighed by the realisticBH+Born cross section. Analysis cuts are then appliedto the simulated sample in a way similar to the experi-ment. IV. CROSS SECTIONS AND NORMALIZATION
The ep → epγ absolute cross section is the five-fold quantity d σ exp / ( dE ′ e d Ω ′ e d cos θ c . m . dφ c . m . ), denotedhereafter by dσ exp . dE ′ e and d Ω ′ e are the differential en-ergy and solid angle of the scattered electron in the lab-oratory frame, while ( d cos θ c . m . dφ c . m . ) is the differentialsolid angle of the emitted photon in the c.m. At eachof the three Q , dσ exp is determined at fixed q c . m . andfixed ǫ , in a three-dimensional binning in the variables( q ′ c . m . , cos θ c . m . , φ c . m . ). One obtains dσ exp ( i ) in each bin i as (cf. Ref. [23]): dσ exp ( i ) = N exp ( i ) L exp · (cid:20) L sim N sim ( i ) · dσ BH+Born ( i ) (cid:21) , (2)where N exp ( i ) is the number of experimental events inthe bin, and N sim ( i ) the weighed sum of simulated eventsin this bin. The cross section dσ BH+Born ( i ) is evaluatedat the center of each bin, and the bracket represents theinverse of the five-fold solid angle ∆Ω sim .The chosen bin size is small: 25 MeV/ c in q ′ c . m . , 0.05in cos θ c . m . and 10 ◦ in φ c . m . , allowing one to follow therapidly varying effect of the GPs in this 3D phase space.As a result, many cross-section points are generated, ofthe order of a thousand at each Q . Our measured cross-section data are provided as supplemental material tothis article [20].As explained in Ref. [16], the final normalization ofthe experiment is based on the very low- q ′ c . m . data, here q ′ c . m . = 37 . c . The method uses the fact that,at these low final photon energies, the measured crosssection must coincide with the theoretical one, composedof the BH+Born cross section plus a very small GP effect( < dσ BH+Born is entirely calculable when one makesa choice for the electric and magnetic form factors ofthe proton, G pE ( Q ) and G pM ( Q ). Here and in all thefollowing, the form-factor parametrization of Ref. [24]is used. The comparison of the experimental and thetheoretical cross sections at low q ′ c . m . is then realized bya χ -minimization, in which the fitted parameter is theglobal normalization factor F norm to apply to dσ exp . Asshown in Table III, we obtain in all cases a very goodfit (reduced χ of ≈
1, for about 400 to 900 data pointsinvolved) and a normalization factor F norm very closeto 1.00, within ≈ G pE ( Q ) and G pM ( Q ), thenormalization factors of Table III may change. However,the physics results of the experiment, i.e., the fitted GPsand structure functions, remain essentially unchanged,as long as the same form factor choice is used for thenormalization of dσ exp and for the polarizability fits (seeRef. [19] for more details).The next four figures show selected examples of ourcross-section data. Figure 5 displays the low- q ′ c . m . crosssection obtained at Q = 0 .
45 GeV . As expected, nopolarizability effect is observed here, and the measure-ment matches well the BH+Born cross section. Figures 6and 7 display the high- q ′ c . m . data obtained at Q = 0.10and 0.20 GeV , respectively. On these figures one candiscern in some angular regions the small departure from dσ BH+Born due to the GPs (the dashed green curves in-clude the GP effect). Figure 7 shows the quality of thesymmetry of the cross section relative to φ c . m . = 0 ◦ , aproperty that is required theoretically for an unpolarizedexperiment. Our final cross-section data [20] are subse-quently symmetrized in φ c . m . . An overview of the ex-perimental coverage in the (cos θ c . m . , φ c . m . ) phase space9 d s ( pb / ( M e V s r )) f cm = 95 o f cm = 105 o f cm = 115 o f cm = 125 o f cm = 135 o f cm = 145 o f cm = 155 o f cm = 165 o -1 0 1 cos q cm f cm = 175 o -1 0 1 Q = 0.45 GeV , q' cm = 37.5 MeV/c FIG. 5. (Color online) An example of the measured crosssection at Q = 0 .
45 GeV and q ′ c . m . = 37 . c . Thesolid (red) curve is the BH+Born calculation. Error bars arestatistical only. is given in Fig. 8, for the three q ′ c . m . -bins considered inthe LEX fit. Each plot of this figure receives contri-butions from several kinematical settings, which are insome cases visible as isolated angular regions. Althoughmost of the events are below the pion production thresh-old, the acceptance extends slightly beyond this limit.Namely, a small subset of cross-section values is obtainedfor the q ′ c . m . -bins [125-150] MeV/ c and [150-175] MeV/ c and will be considered in the DR fit. V. EXTRACTION OF THE GENERALIZEDPOLARIZABILITIES
We refer to Ref. [19] for the detailed aspects of theformalism of VCS at low energy and methodologies forextracting the GPs from data. This section recalls theingredients of the two fits using cross-section measure-ments below the pion production threshold: the LEXand DR fits. We further develop on an estimator of thehigher-order terms of the low-energy expansion, whichis used to make a detailed presentation of the fit results.Statistical and systematic errors are also discussed.
A. Theoretical tools
The LEX fit is based on the low-energy theorem [3], amodel-independent approach which expresses the ep → d s ( pb / ( M e V s r )) f cm = 25 o f cm = 35 o f cm = 45 o f cm = 55 o f cm = 65 o f cm = 75 o f cm = 85 o f cm = 95 o -1 0 1 cos q cm f cm = 105 o Q = 0.10 GeV , q' cm = 112.5 MeV/c FIG. 6. (Color online) An example of the measured crosssection at Q = 0 .
10 GeV and q ′ c . m . = 112 . c . Thesolid (red) curve is the BH+Born calculation and the dashed(green) curve includes in addition a first-order GP effect fromthe LEX. Error bars are statistical only. -1 d s ( pb / ( M e V s r )) cos q cm = -0.975 -1 cos q cm = -0.925 cos q cm = -0.875cos q cm = -0.825 cos q cm = -0.775 cos q cm = -0.725cos q cm = -0.675 cos q cm = -0.625 -180 0 180 f cm (deg) cos q cm = -0.575 Q = 0.20 GeV , q' cm = 112.5 MeV/c FIG. 7. (Color online) An example of the measured crosssection at Q = 0 .
20 GeV and q ′ c . m . = 112 . c (be-fore symmetrization in φ c . m . ). The solid (red) curve is theBH+Born calculation and the dashed (green) curve includesin addition a first-order GP effect from the LEX. Error barsare statistical only. f c m ( d e g ) cos q cm -1 0 1 (a)(b)(c) q' cm = 62.5 q' cm = 87.5 q' cm = 112.5MeV/c MeV/c MeV/c FIG. 8. (Color online) A view of the VCS phase space cov-ered by the experiment, in the (cos θ c . m . , φ c . m . ) plane. Plotsare made for three q ′ c . m . -bins, of central values 62.5, 87.5 and112.5 MeV/ c , from left to right. Rows (a), (b), (c) refer to Q = 0 . , .
20 and 0 .
45 GeV , respectively. The content ofeach filled bin is the measured cross section, with the cor-responding scale (one per Q ) given at the right side of thefigure, in pb/(MeV sr ). -1800180 f c m ( d e g ) . -1800180 -1 0 1 -0.21-0.15-0.09-0.030.030.090.150.210.270.330.39 cos q cm -1 0 1 LEX DR (DR-LEX)
FIG. 9. The GP effect in the 2D-plane (cos θ c . m . , φ c . m . ) atfixed Q = 0.20 GeV , q ′ c . m . = 110 MeV/ c and ǫ = 0.9.Left: the GP effect from the LEX, defined as ( dσ LEX − dσ BH+Born ) /dσ BH+Born . Center: the GP effect from DR, de-fined as ( dσ DR − dσ BH+Born ) /dσ BH+Born . Right: their dif-ference, also equal to ( dσ DR − dσ LEX ) /dσ BH+Born . The cal-culation uses P LL − P TT /ǫ = 15 . − and P LT = − . − as input values. epγ cross section as: dσ = dσ BH+Born + (Φ q ′ c . m . ) Ψ + O ( q ′ . m . ) , Ψ = V ( P LL − P T T /ǫ ) + V P LT , (3)where Φ q ′ c . m . , V , V are known kinematical factors. Thethree VCS response functions are the structure functions P LL ∝ α E ( Q ), P LT ∝ ( β M ( Q ) + spin GPs), and P T T ∝ spin GPs (see [25] for details). The dσ BH+Born -1800180 f c m ( d e g ) . -1800180 -1 0 1 cos q cm -1 0 1 K=0.01 K=0.025 K=0.04
FIG. 10. Example of bin selection in the (cos θ c . m . , φ c . m . )plane, based on Eqs. (4) and (5) (see text). Kinematics cor-respond to Q = 0 .
20 GeV , ǫ = 0 .
90 and q ′ c . m . = 112 . c . The plots show from left to right three increasingvalues of the cut threshold K , from 1% to 4%. Bins filledin black correspond to the condition O ( q ′ . m . ) DR ≤ K . Thecalculation of O ( q ′ . m . ) DR uses P LL − P TT /ǫ = 15 . − and P LT = − . − as input values. cross section contains no polarizability effect and repre-sents typically 90% or more of the cross section below thepion production threshold. Ψ is the first-order polariz-ability term, and the quantity [ dσ BH+Born + (Φ q ′ c . m . ) Ψ ]will be denoted hereafter by dσ LEX . The higher-orderterms O ( q ′ . m . ) are unknown and supposed to be small.They are neglected in the standard LEX fit, whichtherefore uses Eq. (3) in its truncated form withoutthe O ( q ′ . m . ) term. A linear χ -minimization compares dσ exp with dσ LEX and yields the two structure functions P LL − P T T /ǫ and P LT , at a given value of Q and ǫ .The electric and magnetic GPs are obtained only indi-rectly by this approach; an input from a model (herethe DR model) is needed to subtract the spin-GP partof the fitted structure functions. The LEX fit is per-formed for q ′ c . m . -bins below the pion threshold, in ourcase including the three bins [50-75], [75-100] and [100-125] MeV/ c . The lowest q ′ c . m . -bin [25-50] MeV/ c servesessentially to fix the normalization and does not bringfurther constraint to the polarizability fit.The DR fit is based on the dispersion relations modelfor VCS [17, 18], which has a wide range of applica-bility in energy, up to the ∆ resonance region. In theDR formalism, the electric and magnetic GPs have anunconstrained part, which can be fitted to the experi-ment. α E ( Q ) and β M ( Q ) then become the two freeparameters of the adjustment. dσ exp is compared withthe model cross section, dσ DR , calculated for all possiblevalues of the free parameters, and α E ( Q ) and β M ( Q )are fitted by a numerical χ -minimization. The structurefunctions P LL − P T T /ǫ and P LT are obtained from thescalar GPs in a straightforward way, by adding the con-tribution of the spin GPs, which is entirely fixed in theDR model. The DR fit uses the same q ′ c . m . -bins as theLEX fit, with the optional inclusion of bins at higher q ′ c . m . , above the pion production threshold.11 . Higher-order estimator The LEX and DR fits are a priori very different, inthe sense that dσ LEX ignores the higher-order terms O ( q ′ . m . ), while dσ DR includes by construction all or-ders in q ′ c . m . . When these two fits are performed onthe same data set, the appropriate comparison betweentheir results is at the level of the structure functions P LL − P T T /ǫ and P LT , since these are the only directoutputs of the LEX fit. If both types of results agree, it isa strong indication that the higher-order terms O ( q ′ . m . )of the LEX are indeed negligible. Among the variousVCS experiments performed [5, 6, 8, 10, 16], some findan agreement between the two types of fits, while othersfind a significant disagreement (see [19] for more details).As a general statement, not much is known yet aboutthese higher-order terms of the q ′ c . m . -expansion and theirimpact on the polarizability fits. In the present experi-ment, we have studied this question more systematically,using a novel method which is described in the remainderof this section.Among its many advantages, the DR model can beutilized to provide an estimate of the higher-order termsof the LEX expansion. One just needs to calculate boththeoretical cross sections, dσ LEX and dσ DR , using thesame input values of structure functions P LL − P T T /ǫ and P LT . Since dσ DR includes all orders in q ′ c . m . , the dif-ference ( dσ DR − dσ LEX ) is a measure of the higher-orderterms O ( q ′ . m . ) of Eq. (3), as given by the DR model.Accordingly, we build the following dimensionless esti-mator: O ( q ′ . m . ) DR = dσ DR − dσ LEX dσ BH+Born (4)at each point in the VCS phase space. Figure 9 showsan example of the GP effect calculated from the LEX,from the DR model, and their difference.This (model-dependent) estimator has been used firstin the design of the experiment [26], to define kinematicswhere O ( q ′ . m . ) DR is expected to be small. It is furtheremployed in the analysis phase, to study the behaviorof the LEX fit under varying conditions. More precisely,we perform the LEX fit of Eq. (3) in its truncated form,including a varying number of experimental bins, corre-sponding to gradually increased values of the O ( q ′ . m . ) DR estimator. This is realized by setting the condition |O ( q ′ . m . ) DR | ≤ K , (5)and letting the threshold K vary. An example of theaccepted bins is given in Fig. 10. In principle, this “cur-sor” for higher-order terms is not relevant for the DR fit,since the DR calculation is a priori valid in the wholeVCS phase space. We have nevertheless performed thesame study versus K for the DR fit as well.The K parameter acts as a threshold for bin exclusion,or “bin masking”. A very tight cut, e.g., K = 0.005,eliminates many bins in the ( q ′ c . m . , cos θ c . m . , φ c . m . ) phasespace, mainly at high q ′ c . m . . In these conditions, The LEX and DR fits should give very similar results, since dσ exp is compared to two model calculations, dσ LEX and dσ DR , that almost do not differ. As the cut thresholdloosens, e.g., to K = 0.02 or 0.03, more bins are in-cluded, larger differences between the two model calcu-lations are allowed, and the LEX and DR fits may yieldmore different results. At the largest value of the cut,e.g., K = 0.18 at Q = 0.20 GeV , all bins below thepion production threshold are included, and the LEXand DR fits become fully independent. This configura-tion is the one of the published LEX fits of all previousexperiments [5, 8, 10, 14]. C. Fit results
Results of our fine scan in K are shown in Figs. 11and 12 for the LEX and DR fits at each Q . At verysmall values of K , the two types of fits give very simi-lar results, as expected. When K increases, the two fitstend to deviate, more or less quickly, indicating the effectof the higher-order terms O ( q ′ . m . ) that are neglected inthe LEX fit. The divergence between the two types offits versus K is maximal for Q = 0 .
10 GeV , and de-creases when Q increases. At Q = 0 .
45 GeV , the twofits show no difference, suggesting that the higher-orderterms, as given by the DR model, are very small.Another clear feature of Figs. 11 and 12 is the betterstability of the DR fit versus K relative to the LEX fit, inmost cases. This demonstrates the good ability of DRsto evaluate the higher-order terms in q ′ c . m . and to modelthe ep → epγ cross section over a large phase space. Onenotices a few localized exceptions to the stability of theDR fit versus K , for which possible origins can be in-voked. At very small K (Fig. 12, plots (b) and (c) for K ≤ . q ′ c . m . bins, and possible biases may arise. At the other end ofthe K “cursor” (Fig. 11, starred point in plot (a), andFig. 12, starred points in plots (a) and (b)), the addedcross-section data above the pion production thresholdcorrespond to acceptance edges, where experimental sys-tematics may be larger.We now discuss how to choose the optimal value of K ,for the LEX fit with bin exclusion. For Eq. (3) to be validin its truncated form, the higher-order terms should besmall relative to the overall magnitude of the first-orderGP effect, i.e., the Ψ term. One is then led to choosesmall K values, typically K optimal < K optimal ≃ . ± .
5% on the measured cross sections(cf. Sec. V E). Lastly, as mentioned above, the stabilityplateau for the DR fit in Figs. 11 and 12 does not alwaysstart at the smallest value of K but sometimes at K ≥ .
02. Based on the above arguments, K optimal = 0 . K P LL - P TT / e ( G e V - ) K K (a)(b)(c) FIG. 11. (Color online) The behavior of the LEX fit (redfilled circles) and the DR fit (blue open circles) as a functionof the cut threshold K (see text) for the structure function P LL − P TT /ǫ . Plots (a), (b), (c) refer to Q = 0 . , .
20 and0 .
45 GeV , respectively. In each plot, the rightmost filled(red) and open (blue) circles correspond to the inclusion ofall data points in the q ′ c . m . -range (50-125) MeV/ c (i.e., the K -cut is inactive). The cyan starred points are placed atarbitrarily abscissa and refer to the DR fit with the inclusionof the q ′ c . m . -bin [125-150] MeV/ c (plot (a)) and additionallythe q ′ c . m . -bin [150-175] MeV/ c (plots (b) and (c)). Error barsare statistical. The supplementary (green) error bar at K =0 .
025 represents the total systematic error, for our final choiceof fit results. is finally chosen, and considered as providing the mostreliable LEX fit. This point is represented in Figs. 11and 12 with the attached total systematic error (thicksolid green error bar).In practice, the computation of O ( q ′ . m . ) DR dependson input values for the structure functions, therefore thewhole procedure (bin masking + polarizability fit) needsa few iterations. Figures 11 and 12 are produced at thelast iteration step. The results of both LEX and DRfits, obtained without bin masking and with bin maskingat K = 0 . χ between 1.1 and 1.3, for ≈
400 to 1000 degrees offreedom.We consider the results with bin masking (at K =0 . -12-10-8-6-40 0.05 0.1 0.15 0.2 0.25 K P LT ( G e V - ) -12-10-8-6-40 0.05 0.1 0.15 0.2 0.25-8-6-40 0.05 0.1 0.15 0.2 0.25 K -8-6-40 0.05 0.1 0.15 0.2 0.25-2020 0.01 0.02 0.03 0.04 0.05 0.06 K -2020 0.01 0.02 0.03 0.04 0.05 0.06 (a)(b)(c) FIG. 12. (Color online) The behavior of the fits for the struc-ture function P LT . Conventions are the same as in Fig. 11. a deeper complexity of the polarizability fits, of whichsome aspects have been explored and presented here. D. Statistical errors
Statistical errors on the physics observables are pro-vided for each fit by the minimization itself, in whicheach term contributing to the χ is weighed by the sta-tistical error on the measured cross section. The contourat ( χ + 1) (non-reduced χ ) is used, corresponding toa confidence level of 70% on each parameter separately.Error correlations between the two fitted parameters aresmall in all cases. E. Systematic errors
The dominant errors are the systematic ones. Thenormalization method based on the low- q ′ c . m . data (cf.Sec. IV) helps to reduce them substantially, in the sensethat all the global normalization uncertainties commonto all settings, related for instance to the experimen-tal luminosity or radiative corrections, are absorbed inthe F norm factor. However, residual normalization dif-ferences may still exist from setting to setting. They aretaken into account in a simplified way by consideringan overall, intrinsic error of ± F norm . Anotheruncertainty comes from the calibration of experimentalparameters and the solid angle calculation. Here again,13 ABLE IV. Results of the LEX and DR fits, obtained with bin masking at K = 0 .
025 (see text). The q ′ c . m . -bins cover the range(50,125) MeV/ c . The first error is statistical. The second one is the total systematic error, whose sign indicates the correlationto the ( ± ) sign of the overall normalization change. In the LEX part of the table, the GPs are obtained only indirectly, bysubtracting from the structure functions the spin-GP contribution calculated by the DR model. Q ǫ P LL − P TT /ǫ P LT α E ( Q ) β M ( Q ) reduced χ (GeV ) (GeV − ) (GeV − ) (10 − fm ) (10 − fm ) / n.d.f.LEX fit0.10 0.91 33.15 ± ∓ − ± ∓ ± ∓ ± ± ± ∓ − ± ∓ ± ∓ ± ± ± ∓ − ± ∓ ± ∓ ± ± ± ∓ − ± ∓ ± ∓ ± ± ± ∓ − ± ∓ ± ∓ ± ± ± ∓ − ± ∓ ± ∓ ± ± the problem is simplified by considering the error glob-ally, instead of possible point-to-point error correlations.The resulting uncertainty is estimated to be ±
1% on thecross section, relying on the work exposed in Sec. III D.Lastly, another ± q ′ c . m . , cos θ c . m . , φ c . m . ) phase space, residual depen-dence of the physics results on the proton form factorchoice, or versus the cut threshold K , etc.Figure 13 displays the systematic error budget at each Q , with the detailed contribution of each calibration pa-rameter (corresponding to the nine colored sectors), ascoming from simulation studies mentioned in the con-cluding part of Sec. III D. Summed quadratically, theeleven sources of error of Fig. 13 yield a total system-atic error of ± dσ exp changed globally by ± F norm factor.This “one-shot” method for obtaining the final system-atic error is quick and efficient, but in some cases itis not realistic enough. We have tested the validity ofthis method by comparing it to more traditional means,such as performing various analyses with different cali-brations, cut conditions, etc., and measuring the corre-sponding spread of the fitted results. On the one hand,the “quick method” works well at Q = 0 .
20 GeV , asshown explicitly in Ref. [27], and is further assumed towork satisfactorily at Q = 0 .
10 GeV , due to highlysimilar ( q ′ c . m . , cos θ c . m . , φ c . m . ) kinematics. On the otherhand, this quick method works only partly at Q = 0 . , giving in particular an excessively small system-atic error on P LT (of ± − for the LEX fit).The more traditional test of multiple analyses gives anerror about ten times larger ( ± − ), which is Q = 0.10 GeV Q = 0.20 GeV Q = 0.45 GeV beam F F target beam Q Q frost B
10 = intrinsic F norm uncertainty 11 = auxiliary sources
FIG. 13. (Color online) The detailed contributions to the to-tal systematic error, for each Q . The pie chart represents therelative weights w i of each source of error ( i = 1 , ..., i = 1 , ..., i = 10 corresponds to the intrinsic uncertainty of F norm and i = 11 corresponds to the other auxiliary sources of error(see text, Sec. V E). Due to our method of calculation, thesecharts apply equally well to the cross section, the structurefunctions and the GPs. Note that the total systematic er-ror δ tot is given by the quadratic sum ( P i =1 δ i ) / , so thateach partial error δ i is given by δ tot w i ( P i =1 w i ) − / (with P i =1 w i = 1). clearly more realistic, and chosen as the final value. Be-sides, both methods give a similar systematic error on14 LL − P T T /ǫ at Q = 0 .
45 GeV . Although such dis-parities in the behavior of systematic errors are not fullytraced, they could originate in differences of angular cov-erage in (cos θ c . m . , φ c . m . ) versus Q , which induce differ-ences in the weighing factors V and V of the low-energytheorem (cf. Eq. (3)). VI. PHYSICS RESULTS AND CONCLUSIONS P LL - P TT / e ( G e V - ) This ExperimentPreviousExperiments [ ] -15-10-50 0 0.2 0.4 0.6 0.8 Q (GeV ) P LT ( G e V - ) -15-10-50 0 0.2 0.4 0.6 0.8 DR model [ ] Cov.BChPT [ ] FIG. 14. (Color online) The structure functions P LL − P TT /ǫ and P LT of the proton (see text for details). Filled (magenta)circles and filled (red) squares at Q = 0.10, 0.20 and 0.45GeV are from this experiment. Open circles and squaresare from previous experiments at MIT-Bates [10] ( Q = 0.06GeV ), MAMI [5, 6, 14] ( Q = 0.33 GeV ) and JLab [8]( Q = 0.92 GeV ). Open and filled circles correspond to DRanalyses, while open and filled squares refer to LEX analyses.The triangular (cyan) point in the upper plot is from the re-cent measurement of the electric GP at Q = 0 .
20 GeV [15],converted to P LL − P TT /ǫ using the DR model. The RCSpoint ( ⋄ ) is from Ref. [1]. The dashed curve is obtained usingthe DR model [17] with dipole mass parameters Λ α = Λ β =0.7 GeV. The solid curve with its error band (shaded area)is from covariant BChPT [28]. Some data points are slightlyshifted in abscissa for visibility. The inner and outer errorbars are statistical and total, respectively. Our final results are shown in Figs. 14 and 15, in-cluding the world data in terms of structure functionsand scalar GPs of the proton. These results havebeen discussed in Ref. [16] and in a broader context in a E ( - f m ) This ExperimentPreviousExperiments [ ] Q (GeV ) b M ( - f m ) DR model [ ] Cov.BChPT [ ] FIG. 15. (Color online) The electric and magnetic GPs of theproton (top and bottom plots, respectively). The notationsand conventions are the same as in Fig. 14.
Ref. [19], so we just summarize here the main findings.The present measurements provide important new in-sights into the Q -behavior of the VCS observables un-der study. A consistent and smooth behavior starts toemerge in the whole Q range from 0 to 1 GeV , withthe exception of the existing data at Q = 0 .
33 GeV [5, 6, 14]. The tension or lack of smoothness at thisvalue of Q , observed especially for the P LL − P T T /ǫ structure function and the electric GP, remains presentlyunexplained and would require new investigations. A re-cently performed VCS experiment at Jefferson Lab [29]is expected to shed light on this anomaly, by measuringthe electric and magnetic GPs in the Q range from 0.3to 0.7 GeV . At Q = 0 .
20 GeV , results from the twomost recent and independent experiments are shown forthe electric GP and the P LL − P T T /ǫ structure function:the present measurement (filled circles and squares in thefigures) and the one of Ref. [15] (cyan triangular point).These two results show a rather good compatibility, al-though they involve different c.m. energy regimes: belowthe pion production threshold (our experiment) and the∆ resonance region [15].The DR model does not give a prediction of the elec-tric and magnetic GPs. However, it uses a convenientparametrization of their Q -dependence, that allows toprovide predictions for VCS observables. This is real-ized by assuming a single dipole behavior for the un-15onstrained part of the scalar GPs [17, 18]. Namely,with dipole mass parameter values Λ α = Λ β = 0 . Q -behavior suggested by the world data. Thelow- Q data for the magnetic GP and the P LT structurefunction show also good agreement with the recent co-variant BChPT calculation of Ref. [28] (solid curve in thefigures), despite the large theoretical uncertainty. Ourexperiment provides for the first time a precise measure-ment of β M ( Q ) at very low Q (0.10 GeV ), stronglyconstraining the way the two large components, diamag-netic and paramagnetic, nearly cancel in this polarizabil-ity.In conclusion, a new, high-statistics VCS experimentperformed at MAMI has yielded precise measurementsof the proton electric and magnetic GPs at three yetunexplored values of Q . Although measurements oflow-energy VCS observables are still rather scarce, theygradually improve in precision, as experiments are betterdesigned and GP extraction methods become more ma-ture. Examples along these lines have been given in thisarticle. We have demonstrated how one can minimizesystematic errors, by performing a careful experimentalcalibration and using the normalization constraint pro-vided by low- q ′ c . m . data. We have also shown how one can deepen the study of the polarizability fits themselves, inrelation with the higher-order terms of the low-energyexpansion. Nucleon GPs are valuable observables whichbring specific constraints to models of nucleon structure.Improving their knowledge is a long-term challenge thatwill require inventive strategies for new measurements.The DR model, with its unique advantages and evolu-tive capabilities, serves as a precious and reliable toolfor designing and analyzing VCS experiments, and willhelp in pursuing further developments in the field. ACKNOWLEDGMENTS
We wish to thank our theoretician colleagues Bar-bara Pasquini, Marc Vanderhaeghen, Vladimir Pasca-lutsa and Vadim Lensky for their support, and VadimLensky for providing the results of the covariant BChPTcalculation. We gratefully acknowledge the MAMI-C ac-celerator group for the excellent beam quality. This workwas supported by the Deutsche Forschungsgemeinschaftwith the Collaborative Research Center 1044, the Fed-eral State of Rhineland-Palatinate and the FrenchCNRS/IN2P3. Some of the authors would like to ac-knowledge the support by the Croatian Science Founda-tion under the project 8570. [1] M. Tanabashi et al. (Particle Data Group),Phys. Rev.
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