Spin polarizabilities of the proton by measurement of Compton double-polarization observables

The Compton double-polarization observable $\Sigma_{2z}$ has been measured for the first time in the $\Delta(1232)$ resonance region using a circularly polarized photon beam incident on a longitudinally polarized target at the Mainz Microtron. This paper reports these results, together with the model-dependent extraction of four proton spin polarizabilities from fits to additional asymmetry data using dispersion relation and chiral perturbation theory calculations, with the former resulting in: $\gamma_{E1E1} = -3.18 \pm 0.52$, $\gamma_{M1M1} = 2.98 \pm 0.43$, $\gamma_{E1M2} = -0.44 \pm 0.67$ and $\gamma_{M1E2} = 1.58 \pm 0.43$, in units of $10^{-4}~\mathrm{fm}^{4}$.

The Compton double-polarization observable Σ2z has been measured for the first time in the ∆(1232) resonance region using a circularly polarized photon beam incident on a longitudinally polarized target at the Mainz Microtron. This paper reports these results, together with the modeldependent extraction of four proton spin polarizabilities from fits to additional asymmetry data using dispersion relation and chiral perturbation theory calculations, with the former resulting in: γE1E1 = −3.18 ± 0.52, γM1M1 = 2.98 ± 0.43, γE1M2 = −0.44 ± 0.67 and γM1E2 = 1.58 ± 0.43, in units of 10 −4 fm 4 .
PACS numbers: 25.20.Lj,13.40.-f, 13.60.Fz, 13.88.+e The electromagnetic interaction of a photon with a nucleon can be studied through Compton scattering experiments. It is best described using an effective Hamiltonian expanded in terms of the incident photon energy. Structure observables of these composite systems are experimentally accessible by elastically scattering real photons from the nucleon in Real Compton Scattering (RCS).
Over decades, RCS has been established as a benchmark for understanding the ground-state properties of the nucleon, such as the magnetic moment. However, the leading order properties that are sensitive to the internal quark dynamics of the nucleon are still poorly understood experimentally. This paper uses RCS in the ∆(1232) resonance region as a probe to understand some arXiv:1909.02032v2 [nucl-ex] 17 Sep 2019 internal structure observables of a nucleon, the nucleon polarizabilities. These are fundamental properties that describe how its internal structure deforms under an applied electromagnetic field [1,2].
The electromagnetic field of the photon undergoes transitions of certain definite multipolarities while attempting to deform the nucleon. The effective Hamiltonian at second order in incident photon energy, E γ , depends on the electric and magnetic scalar polarizabilities, α E1 and β M 1 , and at third order depends on the spin polarizabilities (SPs).
The third-order effective Hamiltonian term in the spindependent interaction is where˙ E,˙ H, E ij and H ij are the partial derivatives with respect to time and space defined as˙ , and γ E1E1 , γ M 1M 1 , γ M 1E2 and γ E1M 2 are the four SPs. The physics behind these leading-order SPs involves the excitation of the spin-1 2 target nucleon to some intermediate state (∆ or N ) via an electric or magnetic (E1 or M 1) dipole transition and a successive de-excitation back to a spin-1 2 nucleon final state via an electric or magnetic dipole (E1 or M 1) or quadrupole (E2 or M 2) transition. These internal structure constants are manifestations of the spin structure of the nucleon, which parameterize the "stiffness" of the nucleon's spin against the electromagnetically induced deformations relative to the spin axis.
Measurements of two linear combinations of these four SPs-the forward spin polarizability, γ 0 [3,4], and the backward spin polarizability, γ π [5]-have been reported for the proton by several experiments. An individual extraction of these proton SPs was recently published via measurement of the double-polarization Compton asymmetry-Σ 2x -using a transversely polarized target at the Mainz Microtron (MAMI) [6] in conjunction with the γ 0 and γ π results and measurement of the beampolarization Compton asymmetry Σ 3 performed at the LEGS facility [7]. This paper describes an improvement to the extraction of these proton SPs from the measurement of the double-polarization asymmetry Σ 2z using a longitudinally polarized target at MAMI. Σ 2z is defined as where N R ±z and N L ±z are the normalized yield for righthanded and left-handed helicity states of the beam with the target polarized in the ±z direction, and P γ circ and P t z are the degrees of the photon beam circular polarization and target polarization, respectively. The experiment was performed in the A2 hall at MAMI [8,9], a facility composed of a cascade of three Race Track Microtrons that can provide both unpolarized and longitudinally polarized electron beams with energies up to 1.6 GeV [8]. The longitudinally polarized electron beam was produced by irradiating a strained GaAsP III-V semiconductor with circularly polarized laser light [10]. A standard Mott polarimeter [11], installed near the MAMI accelerator cascade, was used for polarization measurements. The average beam polarization was 86.8 ± 0.1% [12]. A 180 • polarization flip was provided by reversing the helicity of the laser light with a Pockels cell at a rate of approximately 1 Hz, thereby reducing the systematic uncertainty resulting from a fixed beam polarization. For this measurement, a 450 MeV polarized electron beam passed through an alloy radiator of cobalt and iron, producing circularly polarized Bremsstrahlung photons. The energy of the radiated photon was determined by detecting the Bremsstrahlung electrons in the tagged photon spectrometer [13], and only photons in the energy range E γ = 265 − 305 MeV were used for this analysis. The photon beam was passed through a 2.5-mm-diameter lead collimator, resulting in a beam spot size of 9 mm on the longitudinally polarized Frozen Spin Target (FST) [14] located in the center of the Crystal Ball spectrometer (CB) [15]. The FST used dynamic nuclear polarization, and its polarization was measured with a nuclear magnetic resonance coil; both are described in detail in Ref. [14]. Polarization of up to 80% and relaxation times of nearly 1000 hours were achieved [13,16], and the direction of proton polarization was reversed approximately once per week to further remove systematic effects. Polarization measurements were completed at the start and end of each data taking period for different polarization orientations. Corrections to the target polarization due to ice buildup on the NMR coil [17] were determined with π 0 asymmetries as well as comparisons of unpolarized and polarized total inclusive and π 0 cross sections. To reflect inconsistencies between these methods, a liberal systematic error of 10% for the target polarization was utilized.
Data were collected during two beamtimes in 2014 and 2015 using the nearly 4π CB-TAPS detector system [18]: the CB as a central calorimeter, and TAPS as a forward calorimeter. The CB consists of 672 optically isolated NaI(Tl) crystals with a truncated triangular pyramid shape arranged in two hemispheres. It covers about 94% of 4π steradians and an angular range of 21 • ≤ θ ≤ 159 • [7]. TAPS consists of 366 hexagonal BaF 2 crystals and two inner rings totaling 72 PbWO 4 crystals and covers an angular range of 2 • ≤ θ ≤ 20 • [19]. Charged particles were identified using energy deposition in the particle identification detector and tracked by a pair of multi-wire proportional chambers or TAPS-veto detectors and their corresponding calorimetric detector. Although the CB-TAPS system covers the angular range of 2 − 159 • , there are regions near the entrance and exit through the detectors that are less efficient. These regions are: (i) the forward hole in the TAPS detector, 2 − 6 • , and (ii) the backward hole in the CB, 150 − 159 • . Fiducial cuts were applied to remove all the data from these angular regions of reduced detection efficiency.
The Compton scattering channel, γp → γp, has a simple final state, but it is very important to correctly identify background from competing reactions because its cross section is only about 1% of the cross section for the dominant π 0 photoproduction process. In addition, under certain conditions, π 0 photoproduction can mimic the Compton scattering signature if one of the photons escapes the detector, or if the electromagnetic showers from the two photons overlap due to finite angular resolution. The Compton channel was identified by selecting a single neutral particle and a single charged track, each with deposited energies above 40 MeV within a 20 ns wide timing coincidence window of an event in the photon tagger. In order to remove uncorrelated events between CB-TAPS and the photon tagger, the random background was sampled in timing windows on each side of the prompt peak: one 450 ns wide on the right and one 460 ns wide on the left. The random sample was normalized by the relative window widths and subtracted from the prompt timing signal.
To eliminate competing backgrounds from coherent and incoherent Compton scattering and π 0 photoproduction off of non-hydrogen nuclei in the FST from the windows and shells of the cryostat material (mainly 3 He/ 4 He, 12 C and 16 O), separate data were taken by inserting a carbon foam target with density 0.55 g/cm 3 into the same cryostat and the normalized yield was subtracted. To remove background from π 0 photoproduction off of the proton, the coincidence of a recoil charged particle in addition to the neutral hit was required, as mentioned above. However, since protons suffer a significant amount of energy loss when they travel from the event vertex through the target material, a 3 He/ 4 He refrigeration bath, various cryostat shells and a longitudinal holding coil on their way to a detector crystal, the analysis was limited to an incident photon energy range of E γ = 265 − 305 MeV. Further details on the background cuts, subtractions, and normalization factors can be found in Ref. [12].
To identify events of interest, four-momentum conservation was used to constrain the observed reaction kinematics. As the background varies significantly across both energy and angle, their dependencies were studied. The tagged photon energy bins below γp → π 0 π 0 p threshold were divided into five θ bins, and were analyzed separately. The opening angle (Ω OA ), defined as the angle between detected proton, p recoil , and where the proton was expected, p miss = p γi − p γ f , cos(Ω OA ) = pmiss. p recoil |p miss |×| p recoil | , was used for a two-body reaction selection. The Monte Carlo simulated opening angle results show a sharp peak around 5 • , which is in good agreement with the data. The large background, as seen in Fig. 1, is mainly due to the π 0 photoproduction process from the proton. This can be suppressed by applying a 10 • opening angle cut, as indicated by the green vertical line. The Compton coplanarity angle, defined as the dif-FIG. 2. Coplanarity distribution for simulated Compton scattering events (magenta), and simulated π 0 events that were analyzed as if they were a Compton photon (red), compared with the carbon-subtracted data (blue) at Eγ = 285 − 305 MeV and over all Compton angles (ΩOA cut from Fig. 1 is applied). ference in the azimuthal angles of a scattered photon and a recoil proton, ∆φ = |φ γf − φ p |, was used to suppress additional background. A cut on the fixed coplanarity angle, ∆φ = 180 ± 15 • , as indicated by the two vertical green lines in Fig. 2, was applied to the reconstructed events. Fig. 3 shows the missing mass (M miss ) distribution for events with a single neutral and a single charged track, where missing mass is defined as where (E γi , p γi c) and (E γ f , p γ f c) are the four vectors of the incident and scattered photon, respectively, and m p is the proton mass. The carbon-subtracted M miss spectrum using the corrected carbon target scaling factors [12] is shown in Fig. 3. The corrected scaling factors were determined based on the ratio of live-time corrected tagger scalers, the ratio of incoherent nucleon π 0 yield (to account for additional nuclear effects) and the simulation of coherent π 0 production on 4 He. Simulation of Compton scattering shows good agreement between data in the region from 900 − 940 MeV/c 2 but there is some inconsistency in the region above 980 MeV/c 2 .
To investigate this, simulated π 0 events were analyzed as if they were from the Compton reaction and were added together (according to their known cross section at given energy and angle) with simulated Compton scattering events to create an expected distribution. From these spectra, it is clear that there is good agreement of the data with the expected distribution up to M miss ≈ 980 MeV/c 2 . It is observed that π 0 photoproduction is the major source of background, which has a larger impact above a M miss of approximately 940 MeV/c 2 and hence it is necessary to set a clear upper M miss limit that coincides with the turn-on point of π 0 photoproduction. It is also clear from the simulations that there is very little or no π 0 background below a M miss of approximately 938 MeV/c 2 . Though M miss spectra can be integrated up to the most conservative limit 938 MeV/c 2 (proton mass), it is desirable to have as many Compton events as possible to minimize statistical error and maximize the physics impact of results. As a result, some steps were taken. First, the lower M miss limit was fixed at 900 MeV/c 2 for each energy and angle bin. Second, a safe conventional M miss range, 900 − 938 MeV/c 2 , and the resulting asymmetry for this region was taken as a reference. Finally, the asymmetry was allowed to vary a maximum of 5% by moving the M miss upper limit to higher values compared to the reference. This ±5% is based on the systematic uncertainties from the choice of carbon target length and the ratio of π 0 photoproduction background to Compton scattering determined from simulation. The resulting final M miss upper limits are between 940 − 948 MeV/c 2 , and further details on this work can be found in Ref. [12].
The Σ 2z asymmetries for E γ = 265 − 285 MeV and E γ = 285 − 305 MeV, obtained by combining the results from the two beamtimes via their weighted average, are shown in Fig. 4 along with determinations at 0 • through dispersive sum rules [20,21]. The systematic errors from the three different sources: target polarization (10%), beam polarization (2.7%), and carbon subtraction (3 − 6%), were added in quadrature and their average between the 2014 and 2015 beamtimes for each Compton angle is shown as a separate block above the horizontal axis. To study the sensitivity of the Σ 2z results on the SPs, a fixed-t dispersion relation code (HDPV) [2,22,23] was used to generate predicted asymmetries at fixed lab energies for various values of the scalar and spin polarizabilities. Predictions within Baryon Chiral Perturbation Theory (BχPT) [24] and Heavy Baryon Chiral Perturbation Theory [25,26] were also available, but are not shown here to preserve readability. The code used nominal values for the scalar polarizabilities of: α E1 + β M 1 = 13.8 ± 0.4 (Baldin sum rule) [27] and α E1 − β M 1 = 8.7 ± 0.7 (in units of 10 −4 fm 3 ) [28], and for the SPs of: γ 0 = −0.929 ± 0.105 [20,21] and γ π = 8 ± 1.8 (in units of 10 −4 fm 4 ) [5]. It should be noted that the value for α E1 −β M 1 was chosen as the current PDG numbers [28], despite the debate regarding them [29,30], as the focus of this study is on the spin polarizabilities. It should also be noted that this value for γ π does not include the π 0 -pole component, set as −46.7×10 −4 fm 4 [30] in all of these studies.
Though γ 0 and γ π can form a basis of the SPs with γ E1E1 and γ M 1M 1 , they can alternatively form an orthogonal basis with [26]. In Fig. 4, γ E− was fixed at −3.5 × 10 −4 fm 4 and γ M − was set at −0.5, 1.5, or 3.5 in the same units. The various bands represent the different values for γ M − , while the spread of each band is a result of allowing γ 0 , γ π , α E1 + β M 1 and α E1 − β M 1 , to vary by their experimental errors. It is clear from Fig. 4 that the Σ 2z data in this energy range indicate a sensitivity to γ M − of approximately ±2 in the standard units. Alternatively, γ M − can be fixed at 1.5 × 10 −4 fm 4 and γ E− set at −5.5, −3.5, or −1.5 in the same units. Unlike the previous case, Σ 2z in this energy range showed a weak sensitivity to γ E− .
A global analysis of Σ 2z data from this measurement, along with the published Σ 2x and Σ LEGS  [20,21]. The curves are from the HDPV dispersion theory calculation of Pasquini, et al., [2,23], where γE− [26] is fixed at −3.5 × 10 −4 fm 4 and γM− [26] is set at −0.5, 1.5, or 3.5 × 10 −4 fm 4 , in the green, red, or blue bands, respectively. The width of each band represents the other parameters, γ0, γπ, αE1 + βM1 and αE1 − βM1 varying within their experimental errors. The error bars shown are point-to-point statistical plus random systematic errors added in quadrature. The correlated systematic uncertainties are shown as a separate block above the horizontal axis for each Compton angle.
the model dependence and extract the SPs. This was done by fitting the asymmetry data using the HDPV calculation [2,22,23] and a BχPT calculation [24]. The extracted SPs determined using each model are summarized in Table I. The fit with HDPV results in γ E− = −2.74 × 10 −4 fm 4 and γ M − = 1.4, in the same units, similar to the values used for the theoretical bands in Fig. 4. The values from the two models are fairly consistent, and the best estimate of a central value is given by the weighted average in the last column of Table I, with the error conservatively taken as the larger of the HDPV and BχPT errors. In summary, model dependent extractions of the SPs from a combined data fit of double-and single-polarized Compton scattering asymmetry results in the ∆(1232) resonance region are presented. These extracted SPs are also in good agreement with dispersion relation [2,22,23], Baryon Chiral Perturbation Theory [31], Heavy Baryon Chiral Perturbation Theory [25,26], K-matrix theory [32], and chiral Lagrangian [33] predictions. Although the uncertainties in the SPs are significantly improved compared to previously reported results [6], forthcoming Σ 3 results from MAMI experiments [34] are expected to provide further improvements in the determination of these fundamental nuclear structure terms.