Beam-Normal Single Spin Asymmetry in Elastic Electron Scattering off 28 Si and 90 Zr
A.Esser, M.Thiel, P.Achenbach, K.Aulenbacher, S.Aulenbacher, S.Baunack, D.Bosnar, S.Caiazza, M.Christmann, M.Dehn, M.O.Distler, L.Doria, P.Eckert, M.Gorchtein, P.Guelker, P.Herrmann, M.Hoek, S.Kegel, P.Klag, H.-J.Kreidel, M.Littich, S.Lunkenheimer, F.E.Maas, M.Makek, H.Merkel, M.Mihovilovivc, J.Mueller, U.Mueller, J.Pochodzalla, B.S.Schlimme, R.Spreckels, V.Tioukine, C.Sfienti
BBeam-Normal Single Spin Asymmetry in Elastic Electron Scattering o ff Si and Zr A. Esser a , M. Thiel a , P. Achenbach a , K. Aulenbacher a , S. Aulenbacher a , S. Baunack a , D. Bosnar b , S. Caiazza a , M. Christmann a ,M. Dehn a , M. O. Distler a , L. Doria a , P. Eckert a , M. Gorchtein a , P. G¨ulker a , P. Herrmann a , M. Hoek a , S. Kegel a , P. Klag a ,H.-J. Kreidel a , M. Littich a , S. Lunkenheimer a , F. E. Maas a , M. Makek b , H. Merkel a , M. Mihoviloviˇc a,c , J. M¨uller a , U. M¨uller a ,J. Pochodzalla a , B. S. Schlimme a , R. Spreckels a , V. Tioukine a , C. Sfienti a a Institut f¨ur Kernphysik, Johannes Gutenberg-Universit¨at Mainz, D-55099 Mainz, Germany b Department of Physics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia c Joˇzef Stefan Institute, SI-1000 Ljubljana, Slovenia
Abstract
We report on a new measurement of the beam-normal single spin asymmetry A n in the elastic scattering of 570 MeV transverselypolarized electrons o ff Si and Zr at Q = .
04 GeV / c . The studied kinematics allow for a comprehensive comparison withformer results on C. No significant mass dependence of the beam-normal single spin asymmetry is observed in the mass regimefrom C to Zr.
Keywords: transverse asymmetry, elastic scattering, polarized beam, multi-photon exchange
PACS: [2010] 25.30.Bf, 27.30. + t, 27.60. + j, 29.30.Aj
1. Beam-Normal Single Spin Asymmetry
Parity violation in weak interactions is a well establishedexperimental technique in atomic, particle and nuclear physics.Over the past 30 years, precision experiments have allowed toprobe hadron [1, 2, 3, 4, 5, 6] and nuclear structure [7] and newproposals have recently been put forward which will consider-ably improve our understanding of the electroweak interactionand will allow us to explore physics beyond standard model[8, 9, 10].The interpretation of these future measurements requirestheoretical predictions with uncertainties below those of the ex-periments. To that end it is mandatory to go beyond the one-photon exchange approximation and include higher-order cor-rections (such as γ Z - [11], γ W -, [12] or γγ -box graphs [13]) inthe calculations.The measurement of observables sensitive to two-photonexchange processes is essential to benchmark such higher-ordercalculations.For this purpose the beam-normal single spin asymmetry(the so-called transverse asymmetry) A n in polarized electron-nucleus scattering is an ideal candidate. Since A n is a parityconserving asymmetry, arising from the interference of one-and two (or more)-photon exchange amplitudes, it gives directaccess to the imaginary part of the two-photon exchange pro-cess. A n can be observed when the polarization vector (cid:126) P e of theelectrons is aligned parallel or antiparallel to the normal vectorˆ n = ( (cid:126) k × (cid:126) k (cid:48) ) / | (cid:126) k × (cid:126) k (cid:48) | of the scattering plane, where (cid:126) k ( (cid:126) k (cid:48) ) are thethree-momenta of the incident (scattered) electrons. The mea-sured beam-normal single spin asymmetry in the two-photon Email address: [email protected] (A. Esser) approximation can be expressed as A n = σ ↑ − σ ↓ σ ↑ + σ ↓ = (cid:16) M ∗ γ · M γγ (cid:17)(cid:12)(cid:12)(cid:12) M γ (cid:12)(cid:12)(cid:12) , (1)where σ ↑ ( σ ↓ ) denotes the cross section for electrons with spinparallel (antiparallel) to the normal vector ˆ n . In Eq. 1, Im( M ∗ γ ·M γγ ) denotes the imaginary part of the one- and two-photonexchange amplitudes M γ and M γγ [14], respectively. The mea-sured asymmetry is related to A n by A exp = A n (cid:126) P e · ˆ n . (2)The transverse asymmetry roughly scales as m e E α em , with m e theelectron mass, E the beam energy, and α em the electromagneticcoupling constant [15]. Asymmetries as small as 10 − to 10 − are therefore expected for beam energies of several hundredMeV. This makes the experiments particularly challenging, asstatistical and systematic errors in the measurement need to bekept well below 10 − .The theoretical treatment of A n is nontrivial as well, sincethe absorptive part of the two-photon exchange amplitude has tobe related to the sum of all possible physical (on-mass-shell) in-termediate states. While several approaches are available to cal-culate the transverse asymmetry for the reaction p ( e , e (cid:48) ) p [15,16, 17, 18], only two di ff erent calculations, exploiting di ff er-ent ansatzes, allow for extension to nuclei with Z ≥
2. Cooperand Horowitz [19] are numerically solving the Dirac equationto calculate Coulomb distortion e ff ects. To do so, they assumethat only the ground state contributes, especially with increas-ing Z . In contrast, Gorchtein and Horowitz [20] include a fullrange of intermediate states (elastic and inelastic), but limit Preprint submitted to Physics Letters B May 1, 2020 a r X i v : . [ nu c l - e x ] A p r heir calculation to the very low four-momentum transfer re-gion ( m e c (cid:28) Q (cid:28) E / c ). In this model, the asymmetry can bewritten as: A n ∼ C log (cid:32) Q m c (cid:33) F Compton ( Q ) F ch ( Q ) , (3)with C being the energy-weighted integral over the total pho-toabsorption cross section. It can be model-independently ob-tained from the optical theorem and it depends on mass number A and charge number Z of the target nucleus. The last termin Eq. 3, the ratio of Compton to charge form factor, allows themodel to be generalized to nuclear targets. Up to now, measure-ments of the Compton slope parameter are available only for Hand He targets (see [20] and references therein). These datasuggest an approximate independence of F Compton ( Q ) / F ch ( Q )from the target nucleus. Moreover, in the low momentum trans-fer region the Q dependence of A n is dominated by the loga-rithmic term.
2. Previous Studies
So far, the transverse asymmetry at forward angles ( θ < ◦ )has been measured at the Thomas Je ff erson National Accelera-tor Facility (JLab) for H, He, C, and
Pb [21]. Althoughthe data span the entire nuclear chart, a systematic interpretationin terms of Q , Z , and E dependence is hindered by the di ff er-ent kinematics of each measurement. A comparison to avail-able theoretical calculations [17, 18, 20] shows a good agree-ment for light nuclei, with the corresponding asymmetry beingdominated by inelastic contributions. At the same time, a strik-ing disagreement in the case of Pb was observed: this mayindicate the inadequacy of the two-photon exchange (TPE) ap-proximation in [20] given that the expansion parameter of theperturbation theory is not small ( Z α ∼ Q dependence of A n for carbon has beenmeasured in the range between 0 .
02 GeV / c and 0 .
05 GeV / c .The obtained results show reasonable agreement with the exist-ing theoretical calculation [22]. The deviations from the the-oretical description have been related to the assumption of thedominance of the log( Q / m c ) term and the independence of F Compton ( Q ) / F ch ( Q ) from the target nucleus. The result em-phasizes that the Q behavior of the asymmetry cannot be treat-ed independently of the target nucleus. Even larger discrepan-cies could be expected for heavier nuclei.Therefore a new experiment has been performed with thesame setup and within the same four-momentum transfer rangewith the aim of investigating heavier target materials such as Si and Zr.
3. New Measurements
These experiments were carried out at the Mainz MicrotronMAMI [23] using the spectrometer setup of the A1 Collabora- tion [24], a well established facility for high resolution spec-troscopy in electron scattering experiments. To allow for com-parison with previous results, the data were taken in the samekinematics as reported in [22]. Minor adjustments due to thedi ff erent target materials led to slightly di ff erent spectrometerangles and Q values as given in Table 1. In order to study thetransverse asymmetry A n , the A1 setup was slightly modifiedby inserting additional fused-silica Cherenkov detectors in thefocal plane of the two high-resolution spectrometers A and B .Corresponding to the di ff erent focal plane geometries of spec-trometers A and B , the size of the fused-silica bars were chosento be (300 × ×
10) mm and (100 × ×
10) mm , respec-tively. The fused-silica detectors were orientated at 45 ◦ withrespect to the direction of the elastically scattered electrons inthe spectrometer. The produced Cherenkov light was collectedby photomultiplier tubes (PMTs) with fused-silica windows.In the MAMI beam source, the primary electrons were pro-duced by illuminating a strained GaAs / GaAsP super lattice pho-tocathode with circularly polarized laser light [25, 26]. In orderto measure the transverse asymmetry with the described spec-trometer setup, the polarization vector of the emitted – longitu-dinally polarized – electrons had to be aligned vertically in or-der to be perpendicular to the scattering plane. In this two-stepprocess, the longitudinal spin is first rotated to transverse orien-tation in the horizontal plane using a Wien filter [27]. Secondly,the polarization vector is rotated to the vertical orientation usinga pair of solenoids. The polarization was verified to be solelyvertical to within 1% using a Mott polarimeter [28] locateddownstream of the 3.5 MeV injector linac and a Møller po-larimeter [29] close to the interaction point in the spectrometerhall. Details on this procedure can be found in [30]. Measure-ments with both polarimeters determined the absolute degree ofpolarization. During each experimental campaign, the degreeof polarization was monitored by frequent measurements withthe Mott polarimeter. The full range of variation of the absolutedegree of polarization amongst the di ff erent measurements wasbetween 78.2 % and 83.6 %.The polarized electron beam had an energy of 570 MeVand was impinging on a 1.17 g / cm (1 .
11 g / cm ) Si ( Zr)target with an intensity of 20 µ A. Both targets needed to becooled during the measurement to avoid variation in their den-sities due to melting. For this purpose a custom-made cool-ing frame was constructed. The targets with an active area of10 mm ×
10 mm each were attached to a copper support struc-ture, which was mounted on an outer aluminum frame. In thisouter frame a mixture of water and ethanol was circulated at astabilized temperature T circ = . ◦ C. To spread the heat load ofthe point-like beam spot, the electron beam was rastered overan area of 4 mm × ∆ E ≈ . Si Excitation Spectrum (all events)Covered by Detector Acceptance C o un t s / ( . M e V ) ( ) Zr Excitation Spectrum (all events)Covered by Detector Acceptance
Figure 1: The excitation energy spectra of Si (top panel) and Zr (bottompanel) show the acceptance of the spectrometer without (black line) and with(filled areas) a cut on the Cherenkov detector. By changing the magnetic field ofthe spectrometer the elastic peak was aligned with the position of the Cherenkovdetector. current of I ≈
20 nA was used. In this mode, the events wereprocessed individually by a conventional data acquisition sys-tem measuring timing and charge of the PMT pulses in parallelwith the other detectors in the spectrometers. The accurate po-sition information obtained from a set of drift chambers allowedto match the position of the elastic line of the scattered electronsto the Cherenkov detectors by tuning the magnetic field of thehigh-resolution spectrometers. The resulting detector coverageis illustrated in Fig. 1.In the integrating mode of data taking used for the asymme-try measurements, the beam current was raised to 20 µ A. Thecurrent produced by each detector PMT was integrated overmains-synchronized 20 ms long periods (so-called polarization-state windows). These windows were arranged in a random se-quence of quadruples with the orientation of the electron beampolarization being either ↑↓↓↑ or ↓↑↑↓ . The polarization statewas reversed by setting the high voltage of a fast Pockels cellin the optical system of the polarized electron source. A 80 µ stime window between the polarization-state windows allowedfor the high voltage of the Pockels cell to be changed. The inte-grated PMT signal for each polarization-state window was thendigitized and recorded.In order to identify and reduce polarity correlated instru-mental asymmetries several methods have been applied to re-verse the sign of the measured asymmetry. Besides revers-ing the polarization vector orientation between the measuringgates, the di ff erential electrical signal switching the polarity atthe beam source was reversed every five minutes. Additionally,a half-wave plate in the optical system at the beam source [31] was used to reverse the beam polarity on a time scale of 24hours.Fluctuations of beam parameters such as current ( I ), energy( E ), horizontal and vertical position ( x and y ) and horizontaland vertical slope ( x (cid:48) and y (cid:48) ) are partly correlated to the re-versal of the polarization vector orientation. This can intro-duce instrumental asymmetries. Therefore it is of utmost im-portance to constantly control these beam parameters. Theyhave been measured by a set of monitors, PIMO (Phase and In-tensity MOnitor), ENMO (ENergy MOnitor), and XYMO (XYMOnitor) which were used in a dedicated stabilization systemto minimize polarity correlated beam fluctuations (see Fig. 2)[31, 32].In parallel, the output signals of the monitors were acquiredin the same way as the detector signals, to correct for instru-mental asymmetries in the o ffl ine analysis. C o un t s ( n o r m a li z e d ) Beam Current Asymmetry (ppm) unstabilizedstabilized 0 100 200 300 400 500 600 700 800 900 1000 -2000 -1000 0 1000 2000 C o un t s ( n o r m a li z e d ) Energy Fluctuation (eV) unstabilizedstabilized 0 1000 2000 3000 4000 5000 6000-50 -40 -30 -20 -10 0 10 20 30 40 50 C o un t s ( n o r m a li z e d ) Vertical Position Fluctuation ( µ m) unstabilizedstabilized 0 1000 2000 3000 4000 5000 6000 7000-50 -40 -30 -20 -10 0 10 20 30 40 50 C o un t s ( n o r m a li z e d ) Vertical Slope Fluctuation ( µ rad) unstabilizedstabilized Figure 2: Comparison between the beam parameters observed in a run withbeam stabilization o ff (black) and with beam stabilization on (red). able 1: Measured beam-normal single spin asymmetries for each spectrometer and kinematical setting with the corresponding statistical and systematic uncertaintycontributions in units of parts per million (ppm). Target Si ZrSpectrometer A B A BScattering angle 23.51 ◦ ◦ ◦ ◦ Q (GeV / c ) 0.038 0.036 0.042 0.042 A n (ppm) − − − − Total systematic error + + + + − − − −
4. Data Analysis
As a first step, all acquired values were corrected for fluctu-ations in the integration gate length. Secondly, after the detectorsignals were o ff set-corrected, the raw asymmetry could be cal-culated: A raw = N ↑ e − N ↓ e N ↑ e + N ↓ e , (4)where N ↑ ( ↓ )e is the corrected detector signal. Assuming a lin-ear behaviour of detectors and data acquisition, N ↑ ( ↓ )e is propor-tional to the number of elastically scattered electrons for eachpolarization state. To determine the experimental asymmetry A exp = A raw − c A I − c ∆ x − c ∆ y −− c ∆ x (cid:48) − c ∆ y (cid:48) − c ∆ E (5)the raw asymmetry needs to be corrected for instrumental asym-metries. Therefore the physical parameters of the beam have tobe extracted from the beam monitor data and the correction fac-tors c i ( i = , ...,
6) need to be determined. Due to the beam sta-bilization system, the helicity-correlated changes of the beamparameters were small, but not negligible.For the calibration of the beam monitors, dedicated runswere performed. The PIMO signal together with the PMT gainwas automatically calibrated in special runs, which have beenperformed approximately every three hours. For these specialruns the beam current was ramped up in steps of 0 . µ A from17 . µ A to 22 . µ A, covering the nominal beam current setting.The integrated PMT signal was calibrated against the beam cur-rent allowing for the extraction of an individual o ff set for everyPMT. This procedure also allowed to constantly check the lin-earity of the PMT responses and to monitor any gain variations.A precise calibration of the PIMO was also essential for the cal-ibration of XYMOs and ENMO, since their signals scale withthe beam current.For the XYMO calibrations, the beam was slowly rasteredover a wire target with known wire positions. For the ENMOcalibration an electronic, polarity-correlated signal correspond-ing to a defined energy variation was superimposed on the rawenergy signal. Both, XYMOs and ENMO were calibrated onceper experimental campaign.The correction factors c i were obtained by an iterative opti-mization procedure. The data were first analyzed with a given set of correction parameters. The remaining asymmetry in thedetector signal was then linearly fitted against the di ff erenceof each of the beam parameters. Finally, the correction factorwas modified to counteract this e ff ect. An exception to this isthe correction factor c for the beam current asymmetry, whichwas set to 1.For 0.01 % of all events, the correction of the detector signalwas not possible. In these cases, either the gate-length signalwas significantly smaller than the one chosen for the experi-ment or the determined correction was larger than the physi-cally possible value. These events have been excluded from theanalysis.
5. Results
The results obtained for A n together with their uncertain-ties are shown in Table 1. Large beam fluctuations and shortrunning time a ff ects the Zr result, which exhibits larger sta-tistical and systematic errors compared to both the Si mea-surement and our former C result [22]. The systematic errorsconsist of a set of contributions arising from di ff erent sources.The contributions introduced by fluctuations of position, angle,and energy of the beam were determined by varying the correc-tion factors by ±
25 %, and calculating the maximum change inthe resulting asymmetry. The same procedure was applied tothe PMT signal o ff set allowing for a variation of up to ±
100 %.In a similar way, the contribution from cuts with unphysicallylarge corrections was determined, by varying the cut threshold.The current and gate-length asymmetry was measured for everyevent. Therefore, the remaining statistical error contributed tothe systematic uncertainty. The contribution of possible nonlin-earities in the asymmetry correction was estimated by exclud-ing 0.1 % of the events with the largest absolute correction foreach term in Eq. 5, respectively. The absolute values of the re-sulting changes in the asymmetry were then added up. A smalldi ff erence in the number of events between the two di ff erentstates of the half-wave plate and a slight variation in the mea-sured asymmetry of both states also contributed to the system-atic error.Further uncertainties in the measurement on Si arose fromadditional beam fluctuations during the XYMO calibration run.The resulting transverse asymmetries for Si and Zr includ-ing our recent result for C [22] are shown in Fig. 3 togetherwith an extension of the theoretical calculation from Refs. [20,42] to Si and Zr. The error bands assigned to the theoreticalpredictions are computed by varying the Compton slope param-eter by 10 % and 20 %. For identical kinematics, the theoreticalcalculation depends only on the mass to charge ratio of the nu-cleus. Thus the same asymmetry is expected for both C and Si. -30-25-20-15-10-5 0 C Si Zr B e a m - N o r m a l S i n g l e S p i n A s y mm e t r y ( pp m ) M. Gorchtein et al.Spectrometer ASpectrometer BSystematic Uncertainty
Figure 3: (color online). Extracted transverse asymmetries A n for C (fromRef. [22]), Si and Zr. The error bars mark the statistical uncertainty andthe boxes show the systematic error. The theoretical calculation for E b = .
570 GeV and Q = .
04 GeV / c of Ref. [20] (black line) is shown for com-parison. The given bands indicate the theoretical error for uncertainties of theCompton slope parameter of 10 % (light grey) and 20 % (dark grey). Within the estimated theoretical uncertainty due to the un-known Compton slope parameter of 20 %, the measurementsare in agreement with the theoretical prediction.A dramatic disagreement, as it was obtained for
Pb [21],has not been observed for Zr. Though our result is a ff ected bya large statistical uncertainty, its value is not compatible withzero, unlike for the Pb measurement. While the mean valueof the asymmetry for the zirconium target slightly deviates fromthe values for the lighter nuclei, the experimental statistical er-rors and the theoretical uncertainty on the Compton slope pa-rameter do not allow for a quantitative statement concerning aclear dependence of A n on the nuclear charge. The discrepancywith the theoretical prediction seems to be roughly independenton the target nucleus.The experimental results for the beam-normal single spinasymmetry on Si and Zr presented in this work contributesignificantly to the study of this observable across the nuclearchart from hydrogen through lead. Our results are in agree-ment with all previous measurements on light and intermediatenuclei confirming that the theoretical model of Ref. [20] cor-rectly grasps the relevant physics. Several explanations for thedisagreement with the
Pb result [21] are conceivable.The coe ffi cient C - Eq. 3 - in front of the logarithmicallyenhanced term could be suppressed for Pb. However, thiscoe ffi cient is fixed by the total photoabsorption cross section,which to a good approximation is known to scale with the massof the nucleus [33] in the relevant energy range. Contributionsfrom nuclear excitations are suppressed as E Nucl / E beam , with E Nucl a characteristic scale of nuclear excitations (of the orderof several MeV).A possible underestimation of the systematic uncertainty ofthe theoretical calculation could also explain the observed dis-agreement. This uncertainty arises from two sources: the termthat is not enhanced by the large logarithm was assigned a con-servative 100 % uncertainty; the Compton form factor has theexponential form, and the respective slope parameter was al-lowed to vary by 10 % - 20 %. Given the agreement of themodel and the data for all nuclei up to Zr, an abrupt change inat least one of these terms is needed to reconcile the calculationwith
Pb.Eventually the two-photon approximation used in [20], whileappropriate for light and intermediate mass nuclei might be in-adequate for heavy nuclei. However, the reasonable agreementof the theory with the Zr data (see Fig. 3), as well as a prelim-inary result of new calculations accounting for Coulomb distor-tion e ff ects (thus summing corrections ∼ Z α to all orders) [34]seem to disprove this explanation.A new experimental program on Compton scattering at MAMIwill permit to reduce the uncertainty of the Compton parame-ter for intermediate mass nuclei. In addition measurements of A n with a C target at di ff erent beam energies will allow tobenchmark the energy dependence of the beam-normal singlespin asymmetry in the theoretical treatment. Furthermore a newmeasurement of A n for Pb by the PREX-II experiment [35]might provide additional clues to the solution of the current ten-sion.
Acknowledgments
We acknowledge the MAMI accelerator group and all theworkshop sta ff members for outstanding support. This workwas supported by the PRISMA + (Precision Physics, Funda-mental Interactions and Structure of Matter) Cluster of Excel-lence, the Deutsche Forschungsgemeinschaft through the Col-laborative Research Center 1044, and the Federal State of Rhine-land-Palatinate. The work of M. Gorchtein was supported bythe German-Mexican research collaboration grant No. 278017(CONACyT) and No. SP 778 /
4- 1 (DFG), and by the EU Hori-zon 2020 research and innovation programme, project STRONG-2020, grant agreement No. 824093.
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