Measurement of Compton Scattering at MAMI for the Extraction of the Electric and Magnetic Polarizabilities of the Proton

A precise measurement of the differential cross sections dσ=dΩ and the linearly polarized photon beam asymmetry Σ 3 for Compton scattering on the proton below pion threshold has been performed with a tagged photon beam and almost 4π detector at the Mainz Microtron. The incident photons were produced by the recently upgraded Glasgow-Mainz photon tagging facility and impinged on a cryogenic liquid hydrogen target, with the scattered photons detected in the Crystal Ball/TAPS setup. Using the highest statistics Compton scattering data ever measured on the proton along with two effective field theories (both covariant baryon and heavy-baryon) and one fixed-t dispersion relation model, constraining the fits with the Baldin sum rule, we have obtained the proton electric and magnetic polarizabilities with unprecedented precision: α E1 ¼ 10.99 AE 0.16 AE 0.47 AE 0.17 Introduction.-The study of hadron structure in terms of quantum chromodynamics and the underlying quarks and gluons is a major focus of modern physics. Because of the nature of confinement and the complex internal dynamics involved, however, QCD calculations of hadron properties have proved challenging. The recent proton radius "puzzle" [1,2] and the many measurements and theoretical developments it has spurred, have emphasized that, while the proton is one of the basic building blocks of matter and the most familiar of all hadrons, we still do not fully understand its properties and structure. Advances in effective field theories [3][4][5][6], dispersion relation analyses [7][8][9][10], and lattice QCD [11] have added impetus to obtain more accurate measurement of hadron structure observables, such as polarizabilities and charge radii.

An object's polarizabilities characterize its internal response to applied external electric ( E) and magnetic ( H) fields; they are fundamental properties such as mass and the charge and at the microscopic level, they can be accessed via Compton scattering. In the expansion of the effective Hamiltonian in incident photon energy ω γ , the electric (α E1 ) and magnetic (β M1 ) polarizabilities enter at Oðω 2 γ Þ [12]:

while the four spin polarizabilities (γ EiMj ) are included at Oðω 3 γ Þ:

where σ are the proton's Pauli spin matrices,

are partial derivatives with respect to time and space, respectively. Considerable experimental effort has been expended over the last half century to obtain the scalar polarizabilities of the proton [13][14][15][16], and recent measurements have resulted in extractions of the proton's heretofore unknown individual spin polarizabilities [17,18]. Work has also been carried out on the neutron's scalar polarizabilities, but due to the absence of a freeneutron target their unambiguous extraction has proved more challenging [19][20][21][22].

Polarizabilities are of interest, not only in the study of hadron structure where they can provide input to the QCD puzzle, but also in other fields including precision atomic physics and astrophysics. They yield an appreciable correction to the proton charge radius via the Lamb shift and hyperfine structure [23][24][25][26] and influence neutron star properties [27]. Moreover, a precise determination of the proton scalar polarizabilities is integral to the extraction of the proton spin polarizabilities, as the latter appear at higher order in the expansion of the Compton scattering Hamiltonian [12,[28][29][30][31].

Furthermore, based solely on improved chiral effective field theory analyses and attempts to curate a statistically consistent database [3][4][5][6], the Particle Data Group have recently adjusted their values of the proton scalar polarizabilities [32] without using new experimental data. This clearly demonstrates the necessity to obtain a new Compton scattering dataset with high statistical accuracy and low systematic errors in order to constrain the extraction of the scalar polarizabilities.

The measurement reported here represents the highest statistics Compton scattering data ever obtained on the proton-roughly one million Compton events below pion photoproduction threshold-resulting in an improvement of a factor of approximately 5 over the world's previous best measurement [16]. It builds on our recently reported pilot experiment where the photon beam asymmetry below pion threshold was obtained for the very first time [33].

Experimental setup and data handling.-The experimental data were obtained in two beamtimes in 2018 using tagged photons at the MAMI electron microtron facility [34,35]. The electron beam, with an energy of 883 MeV, impinged on a 10 μm thin diamond radiator, producing a linearly polarized photon beam via coherent Bremsstrahlung [36], with a degree of polarization up to 78%. The recoiling electrons from the Bremsstrahlung process were momentum analyzed using the Glasgow photon tagging spectrometer [37]. Only photons with energies in the range ω γ ¼ 85-140 MeV were considered in this analysis. The resulting photon beam passed a 3-mmdiameter lead collimator and was incident on a 10-cm-long liquid hydrogen target. The final state particles were detected using the same Crystal Ball/TAPS detector system as in the pilot experiment [33] with a nearly complete solid angle coverage. Additional information on the apparatus used for these measurements can be found in Refs. [38,39].

The experiment was performed with two different orthogonal orientations of the photon polarization vector (formed by the momentum of the incoming photon and its electric field vector) and to minimize the systematic uncertainty the polarization vector was flipped approximately every two hours. The degree of linear polarization, that depends on the photon energy and crystal orientation, was directly extracted from experimental data following the procedure described in Ref. [40].

The Compton scattering events, γp → γp, were selected by requiring exactly one photon in the Crystal Ball (for the incident photon energies of interest the recoil protons are typically not energetic enough to reach the detector apparatus). The photon is identified as a cluster in the calorimeter without any associated hit in the charged particle identification detector nor in any of the two multiwire proportional chambers. Because of the significant beam-related electromagnetic background in the forward region, only events with outgoing photon scattering angle θ γ 0 > 30°were considered. Moreover, due to the relatively high photon flux, a time coincidence within 3 ns was required between the scattered photon and the hits in the tagger focal plane. To remove the random coincidences in the selected time window, a side-band subtraction was also performed by selecting a background sample on each side of the prompt peak. Furthermore, the background generated from the nonhydrogenic components of the target (Kapton cell, insulating material, etc.) was sampled during dedicated empty-target runs, and subtracted from the full-target sample.

Figure 1 shows a sample of the missing mass distribution calculated as

where k ¼ ðω γ ; kÞ and k 0 ¼ ðω γ 0 ; k 0 Þ are the incoming and scattered photon four-momenta, respectively, and m p is the target proton mass at rest. For a Compton scattering event from a proton in the target cell we expect M miss to be in agreement with the proton mass. The distribution is plotted together with the one obtained from a Monte Carlo simulation of the full experimental apparatus, based on the GEANT4 package [41]. The good agreement between the data and the simulated distribution indicates a low background contamination in the final sample, and to remove any remaining background a cut on missing mass is applied (Fig. 1), which was optimized using the Monte Carlo simulation.

From the final dataset, the unpolarized differential cross section was extracted for five approximately 10 MeV-wide photon energy bins spanning the range ω γ ¼ 86 to 140 MeV and twelve 10°-wide polar angular bins from θ γ 0 ¼ 30°to 150°, for a total of 60 points. A sample of the obtained results is plotted in Fig. 2(a) as the statisticalerror-weighted average between the two beamtimes. The error bars are statistical only with the systematic uncertainty given by gray bars. They include both correlated and point-to-point uncorrelated errors. The former comes from three independent sources that give in total a systematic uncertainty of 3%: target density (1%), photon flux normalization (2%), and analysis cuts and Monte Carlo simulation (2%). The latter comes from the remaining background contamination in the final sample and it was estimated from the small yield evident to the right of the proton peak before the missing mass cut (Fig. 1). This background yield was found to be energy-and polar-angledependent and ranges from 10% at low beam energy and scattering angle to 0.2% at high energy and central angular bins. The consistency of the two beamtimes was checked looking at the distribution of the normalized residuals of the two cross sections calculated using the two different periods. The results were found to be in perfect agreement for all energy bins, but the lowest one (ω γ ¼ 86.3-98.2 MeV) in which one of the two beamtimes was about 5% higher than the other one. To account for this, a 3% systematic uncertainty was included in the uncorrelated systematic uncertainties of this lowest energy bin. The numerical values are tabulated and plotted in the Supplemental Material [42]. Our results in Fig. 2(a) are compared with the previously published data from the TAPS Collaboration [16]. The Born term, describing the proton as a pointlike particle without internal structure aside from the anomalous magnetic moment, is shown I, obtained within DR [7,8,10], BχPT [3], and HBχPT [43] frameworks, respectively. All the measured points are reported and plotted in the Supplemental Material [42].

along with the fit results reported in Table I, obtained within different theoretical frameworks: dispersion relation (DR) [7,8,10], baryon chiral perturbation theory (BχPT) [3], and heavy baryon chiral perturbation theory (HBχPT) [43]. The main difference between the two χPT variants is whether the nucleon is treated relativistically (BχPT) or an amplitude expansion in powers of 1=M N is performed (HBχPT) (see Ref. [44] for a comprehensive comparison). The three theories fit our data equally well, as confirmed also from the χ 2 values in Table I. The experimental data show divergence from a calculation based on only the Born term, showing the sensitivity of the data to the proton internal structure, which at this energy is mainly, but not exclusively, described by the scalar polarizabilities. The improvement in the statistical quality of the new data compared to the previous TAPS measurement is clear from Fig. 2(a). The new measurement also provides a wider angular coverage and smaller systematic errors. The combination of the linearly polarized photon beam with the unpolarized LH 2 target results in azimuthal dependence to the Compton scattering cross section, allowing for the determination of the single polarization observable Σ 3 defined as [12]

where dσ kð⊥Þ is the polarized cross section obtained with one of the two orthogonal orientations of the photon polarization vector, usually named "parallel" and "perpendicular."

The beam asymmetry was extracted in the same energy and angular range as the unpolarized cross section, using a procedure similar to the one in Ref. [33]. Because of the larger statistical uncertainties, to achieve adequate statistics the current data were binned in three photon energy regions, for a total of 36 new points. A sample of our results is shown in Fig. 2(b). The error bars shown are statistical only. The systematic uncertainties are depicted as gray bars. The main contribution to the systematic uncertainty comes from the procedure to extract the linear polarization degree from the data, and an upper limit to this is estimated to be 5%, uniformly distributed. An additional uncorrelated point-to-point contribution coming from the background contamination was also estimated, using a method similar to that employed for the unpolarized cross section. The numerical values are tabulated and plotted in the Supplemental Material [42]. Our asymmetry results in Fig. 2(b) are plotted together with the fit results obtained using the same theoretical frameworks as for the differential cross sections shown in Fig. 2(a). Also in this case, the three models can fit our results equally well. However, in all three photon energy bins the fit results are distinct from the leading Born term, indicating that the asymmetry is sensitive to the proton structure constants.

A more comprehensive description of this work can be found in Ref. [45].

Results and discussion.-A fit to extract the proton scalar polarizabilities from all of the new data was performed using three different models: fixed-t DRs [7,8,10], BχPT [3], and HBχPT [43]. The unpolarized cross section and the beam asymmetry were given as input to the fitter as two independent datasets. The uncorrelated point-to-point systematic errors were added in quadrature to the statistical ones. The correlated systematic uncertainties were included in the fit as common normalization factors, one for each dataset, and treated as additional fit parameters. Their deviations from the expected value of 1 were also accounted for in the χ 2 function to be minimized [46]. The minimization was performed by using the MINUIT minimization routine [47]. In order to emphasize the sensitivity of the new data to α E1 and β M1 as much as possible, the spin polarizabilities were kept fixed to the most recent experimental values [18]. Moreover, to minimize the statistical uncertainty, the well-known Baldin Sum rule was included as an additional data point to be fitted at α E1 þ β M1 ¼ 13.8 AE 0.4 (in the usual units) [16].

The fit results are summarized in Table I. The errors quoted in the central values of α E1 and β M1 are statistical, systematic, and from the spin polarizabilities, respectively. The first one was obtained by performing the fit without the normalization factors. The second error is given by how much the errors on the parameters changed by the inclusion of the systematic errors. The third error is given by the variation in the best value of α E1 and β M1 when the spin TABLE I. Scalar polarizabilities extracted by fitting the new unpolarized cross section and beam asymmetry data using HDPV DR [7,8,10], BχPT [3], and HBχPT [43] code. The errors are statistical, systematic, and from the spin polarizabilities, respectively. The spin polarizabilities were fixed to the last experimental values available [18]. s σ and s Σ are the normalization factors for the unpolarized cross section and the beam asymmetry, respectively. The scalar polarizability values are given in units of 10 -4 fm 3 . polarizabilities are not fixed, but rather free to vary within their experimental errors. The small systematic error for the latter term indicates the new dataset has only a limited dependency on the spin polarizabilities, and thus making it well suited for a precise study of the two scalar terms. The extractions of the scalar polarizabilities reported in Table I-in particular of β M1 -exhibit a moderate model dependence. To provide a best estimate of the central values for the two parameters, the results from the three theories were combined using weighted average, taking the quadratic sum of the statistical and systematic uncertainties as weights. For each error the largest contributions among the different theories was assigned. Additionally, the largest of the differences between each theory and the average was used to estimate an additional error due to the model dependence for both α E1 and β M1 . The best values for the extraction of the scalar polarizabilities from the new data using the Baldin sum rule constraint are α E1 ¼ 10.99 AE 0.16 AE 0.47 AE 0.17 AE 0.34;