Global analysis of parity-violating asymmetry data for elastic electron scattering

Global analysis of parity-violating asymmetry data for elastic electron scattering R. González-Jiménez,1J. A. Caballero,1and T. W. Donnelly2 1Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, 41080 Sevilla, Spain 2Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 20 March 2014; published 1 August 2014) We perform a statistical analysis of the full set of parity-violating asymmetry data for elastic electron scattering including the most recent high precision measurement from Q-weak. Given the basis of the present analysis, our estimates appear to favor nonzero vector strangeness, specifically, positive (negative) values for the electric (magnetic) strange form factors. We also provide an accurate estimate of the axialvector nucleon form factor at zero momentum transfer, Gep Að0Þ. Our study shows Gep Að0Þto be importantly reduced with respect to the currently accepted value. We also find our analysis of data to be compatible with the Standard Model values for the weak charges of the proton and neutron. DOI: 10.1103/PhysRevD.90.033002 PACS numbers: 12.15.-y, 14.20.Dh, 14.65.Bt, 25.30.Bf Over the years parity-violating (PV) electron scattering has provided a great deal of precise information on the structure of the nucleon. A variety of experiments running on different targets, from hydrogen to heavier systems with emphasis on deuterium and helium, have added strong constraints on the electroweak form factors. Moreover, the high precision reached by the most recent experiments [1,2] will serve as a test of the Standard Model (SM) providing a significant constraint on nonperturbative QCD effects. The Q-weak Collaboration has recently determined the weak charge of the proton corresponding to the analysis of approximately 4% of the data collected in the experiment [2]. The main objective of the Q-weak experiment is to provide a value of sin2θWwith a 0.3% precision, that is, the weak charge of the proton to 4%. This extremely small uncertainty will provide a significant test of the SM. In [2] a global fit of data taken for hydrogen, deuterium and helium targets up to jQ2j¼0.6ðGeV=cÞ2was also done providing some estimates for the weak neutral current (WNC) couplings. These results are compatible with the ones obtained in previous analyses performed by Young and collaborators [3,4]. However, the use of data only up to jQ2j¼0.6ðGeV=cÞ2[in the case of [3] only data for jQ2j<0.3ðGeV=cÞ2were considered], in addition to particular assumptions for the Q2expansion of the form factors, have convinced us of the necessity of a new and more complete analysis of the process. Moreover, when such high levels of precision are the goal, mixing data for elastic scattering on the proton and 4He with those corresponding to the quasielastic (QE) process, make it difficult to disentangle effects due to the nucleon structure from others directly linked to final state interactions, offshell effects, few-body nuclear structure, etc. Accordingly, in this work we restrict ourselves to elastic electron scattering processes, and make use of all available data in the literature with no restriction on the Q2range considered. The PV asymmetry (APV) in the case of elastic electronproton (ep) scattering may be written as follows [5]: APV ep ¼A0 2G½aAðεGp E~ Gp EþτGp M~ Gp MÞ −aVffiffiffiffiffiffiffiffiffiffiffiffi 1−ε2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τð1þτÞ pGp MGep A;ð1Þ where τ≡jQ2j=ð4M2Þand G≡εðGp EÞ2þτðGp MÞ2with M the nucleon mass and Gp E;M the electromagnetic (EM) form factors of the proton. The term A0determines the scale of the PVasymmetry and is given by A0¼GFjQ2j=ð2ffiffiffi 2 pπαÞ with GFthe Fermi coupling and αthe fine structure constant. Finally, aV¼−1þ4sin2θWand aA¼−1are the vector and axial-vector WNC electron couplings, and we have introduced the kinematical factor ε¼½1þ 2τð1þτÞtan2θe=2−1which depends on the scattering angle θe. Assuming charge symmetry, the WNC form factors can be written as follows: ~ Gp E;MðQ2Þ¼ξp VGp E;M þξn VGn E;M þξð0Þ VGs E;M Gep AðQ2Þ¼ξT¼1 AG3 AþξT¼0 AG8 Aþξð0Þ AGs A; where Gs E;M are the electric (E) and magnetic (M) strange form factors and G3;8;s Aare the isovector (3), isoscalar (8) and strange (s) contributions to the axial-vector form factor of the proton (Gep A). The ξcoefficients represent the WNC effective coupling constants that are given in terms of the weak mixing angle (θW) and radiative corrections (see [5,6] for explicit expressions). In this work we also consider the data obtained by the HAPPEX Collaboration on elastic e-4He scattering. In this PHYSICAL REVIEW D 90, 033002 (2014) 1550-7998=2014=90(3)=033002(5) 033002-1 © 2014 American Physical Society case the PVasymmetry involves the ratio of the two nuclear monopole form factors: the EM and the WNC ones. Hence, nuclear effects also play a role in the description of the process—for discussions of how such effects arise, see [7–9]. It can be shown that the use of one-body operators (leading-order approximation) and the assumption that isospin mixing can be ignored for the 4He ground state yield for the ratio of nuclear form factors simply a ratio of single-nucleon form factors. The latter point was first discussed in [7] and then revisited in a later study [10] (see, especially, Fig. 2in that reference). Within the context of the approaches discussed there, the effects of isospin mixing in 4He were typically found to be more than an order of magnitude smaller than the effects from strangeness content in the nucleon, in contrast to the expectations for heavier nuclei [8,9]. This issue is not completely closed, however, as other approaches find somewhat larger effects from isospin mixing [11]. In future experimental studies, preferably at low momentum transfers, using a variety of N¼Znuclei it should be possible to address this problem more definitively; for this study we continue to make the above assumptions, in which case the PVelastic asymmetry can be written in the form [6] APV eHe ¼− A0 2ðξp Vþξn VÞþ2ξð0Þ VGs E Gp EþGn E:ð2Þ Thus, these data are also sensitive to the WNC nucleon structure, and they give information that complements what is obtained through elastic ep scattering. In a recent study [5] we showed that at jQ2j¼ 1ðGeV=cÞ2(the current limit for asymmetry data) the dispersion in APV due to the use of different prescriptions for the EM form factors (some of them accounting for two-photon exchange contributions) was ∼3% in the very forward limit (θe¼5°) getting much smaller for larger angles and lower jQ2j. The impact of these uncertainties at the particular Q-weak kinematical conditions has been explored in detail in [5]. Therefore, in the present work we neglect the uncertainties associated with the particular description of the EM nucleon form factors. On the contrary, APV is highly sensitive to the electric and magnetic strange form factors Gs E;MðQ2Þand to the axialvector one Gep AðQ2Þ. In the latter radiative corrections can introduce very strong effects. In this work we perform a global fit of the five parameters: ξp V,ξn V,ρs,μs(these two linked to the electric and magnetic strange form factors [6]) and the axial-vector form factor at zero momentum transfer: Gep A≡Gep Að0Þ. Our predictions, based on a statistical analysis of the full set of PV asymmetry data for elastic ep scattering (SAMPLE [12], HAPPEX [13–16],PVA4 [17–19],G0[20,21] and Q-weak [2]) as well as the two PV elastic e-4He data (HAPPEX [15,22]), are compared with all previous analyses presented in the literature. Assuming the radiative corrections (RC) to be mildly dependent on the energy, namely, on the particular kinematics selected, the results of our analysis could serve as a test of their strength. Thus the dispersion found in the values of ξp Vand ξn Vwith respect to the corresponding treelevel results (see [6] for details) could provide information on Rp Vand Rn V. Similarly, the deviation in Gep Að0Þwould constrain the RC contributions in the axial current as well as possible effects linked to the anapole moment [23–25]. Recently a large amount of theoretical work has been carried out regarding the energy-dependent γZ-box correction [26–29]. Obviously, the results of any global analysis of data are affected by the correction associated with the γZbox. However, at present this correction can only be applied to a limited set of data: very forward scattering angles. Hence in this work we do not introduce such corrections, so that all data are treated consistently in the global analysis. Our present analysis for the weak charges of the proton and neutron, with their uncertainties linked to the parameters that have been estimated, is consistent with the study reported by the Q-weak Collaboration [2], where the γZ-box energy-dependent correction was applied. However, some caution should be drawn concerning this outcome because of the present limited analysis of Q-weak data (only 4% of data have been considered). Once the analysis is extended to the 100% of the Q-weak data, and taking into account that ξp Vis basically not constrained by any other data, the γZ-box correction to the Q-weak data will become crucial. Before entering into a detailed discussion of the results, a general comment on the fit procedure should be made. Specifically, the currently accepted value of the axial-vector form factor at Q2¼0, namely, Gep Að0Þ¼−1.04 0.44 (see [30]), has been included as an additional experimental constraint in the global fit. Although a significant contribution to Gep Acomes from G3 A≡gA¼1.2695 that is well determined from Gamow-Teller β-decay measurements, radiative corrections can introduce an important uncertainty in Gep AðQ2Þ(see discussion in [5]). Hence there is still room in the global fit procedure for variation of the specific value of Gep AðQ2Þ. In summary, our analysis takes into account all 31 experimental data for the PVasymmetry available in the literature plus the restriction applied to Gep Að0Þ. The Q2dependence in the vector strange and axialvector form factors has been taken in their usual form [5,6], namely, dipole (dipole times τ) for Gs M(Gs E) with a vector mass MV¼0.84 GeV, and dipole shape for the axial form factor with the axial mass MA¼1.03 GeV. The GKex prescription [31–33] has been used for the EM form factors. The fit procedure consists in minimizing the χ2function: χ2¼½Aexp −AtheT½V−1½Aexp −Athe;ð3Þ where Aexp contains all data and Athe takes care of the corresponding theoretical predictions that depend on the five parameters considered: ξp V;ξn V;ρs;μs;G ep A. The term V represents the covariance error matrix defined as GONZÁLEZ-JIMÉNEZ, CABALLERO, AND DONNELLY PHYSICAL REVIEW D 90, 033002 (2014) 033002-2 Vij ¼ðσuncor iÞ2δij þσcor iσcor jð4Þ with σuncor iand σcor ithe uncorrelated and correlated uncertainties of the ith measurement, respectively. In this work only correlated errors reported by G0 Collaboration are considered. The results of the χ2fit are summarized in Table Iwhich contains the values for the five free parameters as well as their 1σerrors (χ<χ2 MIN þ1). These values are in good agreement with the SM weak charges of the proton and neutron [34]:ξp V¼0.0710 0.0007 and ξn V¼−0.9890  0.0007. On the contrary, Gep Aextracted from the fit, Gep A¼−0.62 0.41, is significantly lower than the value Gep A¼−1.04 0.44 given in [30]. However, note that both results are affected by large errors that make the two predictions overlap. These results may reflect important effects coming from higher-order contributions (radiative corrections) as well as from alternative descriptions of the Q2dependence. The use of the standard dipole jQ2j dependence for the axial form factor introduces a systematical uncertainty in the fit that propagates into the specific value of Gep Aat jQ2j¼0. Other functional dependences [5] as well as the use of a different value of the axial mass MA≈1.35 GeV=c as found by the MiniBooNE Collaboration [35] may lead to slightly different results. Finally, in spite of the significant errors associated with the strange form factors, our global fit favors a large-positive strange charge radius ρs(electric strangeness), whereas the strange magnetic moment μstends to be negative although still being compatible with zero contribution. The correlation coefficients between pairs of the five parameters considered in the χ2fit are given in Table II. Notice the extremely small correlation for the pair ðξp V;G ep AÞ. This result supports the idea that a determination of the weak charge of the proton is not affected by the large uncertainty in Gep A. The correlation values between ξp Vand the three remaining parameters, ξn V,ρsand μs, although relatively small, should be taken into account carefully if a high precise determination of the proton weak charge is desired. To conclude, the rest of parameters: ξn V,Gep A,ρs and μs, are very strongly correlated. In Fig. 1we show all PV asymmetry data for elastic electron scattering compared with our theoretical predictions (zero line). The inner error bars represent the experimental errors while the outer ones (in red) include the theoretical uncertainty provided by the global fit corresponding to the 1σconfidence level. The strong correlation between some of the parameters has been taken into account in order to determine the theoretical error. In Fig. 2we show the 95% confidence contours obtained in our analysis extrapolated to jQ2j¼1ðGeV=cÞ2. The results of our present fit (red ellipse) are compared with those from our previous work [5] (blue curve). The significant difference in the areas spanned by the two ellipses is due to the different approaches considered in the two cases. In [5]a global analysis of the full set of PV TABLE I. Values of the free parameters from the global fit. The reduced χ2value is χ2 MIN=27 ¼1.30. ξp Vξn VGep A 0.070 0.013 −0.946 0.044 −0.62 0.41 ρsμs 0.92 0.58 −0.26 0.26 TABLE II. Correlation coefficients between the free parameters of the fit. ξn VGep Aρsμs ξp V−0.191 0.0469 0.262 0.162 ξn V0.392 0.552 −0.775 Gep A0.711 −0.749 ρs−0.870 0.1 1 |Q2| (GeV/c)2 (Aexp - Atheo) / A0 HAPPEX 4He Qweak HAPPEXa SAMPLE HAPPEXb G0-forward G0-backward HAPPEXIII PVA4-145ª PVA4-35.5ª HAPPEX99 -0.1 -0.05 0 0.05 0.1 FIG. 1 (color online). Full set of PV asymmetry experimental data for elastic electron scattering. The data are normalized by subtracting our theoretical asymmetry and dividing by A0. -1 -0.5 00.5 11.5 GM s -0.1 -0.05 0 0.05 0.1 GE s This work (95%) Young et al. (95%) Liu et al. (95%) Phys. Rep. (95%) Leinweber et al. Doi et al. |Q2|=0.1 (GeV/c)2 FIG. 2 (color online). 95% confidence level constraint ellipses in the plane Gs E-Gs Mat jQ2j¼0.1ðGeV=cÞ2: red ↔this work, green ↔[3],brown↔[30], blue ↔[5]. Black and orange crosses are the theoretical predictions presented in [36,37] and [38], respectively. GLOBAL ANALYSIS OF PARITY-VIOLATING ASYMMETRY …PHYSICAL REVIEW D 90, 033002 (2014) 033002-3 asymmetry data, including only ep scattering, was performed (28 data) taking only ρsand μsas free parameters (see [5] for details). On the contrary, the red ellipse corresponds to the predictions obtained by simultaneously fitting five parameters. This larger number of free parameters explains the significant increase in the area. Note that the present analysis seems to favor positive (negative) values of ρs(μs). However, the case of zero strangeness, i.e., ρs¼μs¼0, is still inside the 95% confidence level in our present analysis (red curve), and only slightly outside the region obtained from our previous study (blue). We also compare our predictions with the statistical analyses of data performed by Liu et al. [30] (brown ellipse) and Young et al. [3] (green ellipse). Also shown for reference the theoretical predictions provided by Leinweber et al. [36,37] (black cross) and Doi et al. [38] (orange cross). The ellipses corresponding to the works [3,30] are the results of χ2fits of the PV asymmetry data for electron scattering on helium, deuterium and hydrogen in the vicinity of jQ2j≈0.1ðGeV=cÞ2. In particular, in the work of Liu et al. [30] a total of 10 data in the range 0.091 ≤ jQ2j≤0.136 ðGeV=cÞ2were employed with only two free parameters: the electric and magnetic strange form factors, Gs EðjQ2j¼0.1ðGeV=cÞ2Þ,Gs MðjQ2j¼0.1ðGeV=cÞ2Þ.On the contrary, in [3] a total of 19 data in the range 0.038 ≤ jQ2j≤0.299 ðGeV=cÞ2were used. In this case, four free parameters were considered in the fit: the two vector strange form factors and the axial-vector form factor of the proton and neutron at zero momentum transferred, Gep Að0Þand Gen Að0Þ. In Fig. 3we present the 95% confidence level ellipse in the (μs;G ep A) plane. Results have been extrapolated to the kinematical situation jQ2j¼0.1ðGeV=cÞ2. Our prediction (red ellipse) is compared with the one of Young et al. [3] (green ellipse). As shown, our analysis improves significantly the previous prediction (important reduction in the area of the ellipse). To conclude, we apply our analysis to the determination of the WNC effective couplings. Thus, we build the 95% confidence level contour in the plane ðC1u−C1d; C1uþC1dÞ. This is shown in Fig. 4by the red ellipse. Our prediction is also compared with previous analyses given in the literature: blue ellipse [5], green filled ellipse [4], and brown ellipse [2]. The yellow one is the result of combining the constraints coming from our present analysis (red ellipse) with those from Atomic PV in Cesium (APV-Cs) experiments (magenta horizontal band, see [34,39,40] for details). Likewise, the cyan ellipse is the result of the combined analysis presented in [2]. This combined analysis allows us to get a more accurate determination of the WNC quark coupling constants: C1u¼−0.186 0.006,C1d¼0.338 0.006 with a correlation coefficient −0.966. Similarly, the weak charges of the proton and neutron result: ξp V¼0.070 0.013 and ξn V¼−0.978 0.012 with a correlation coefficient −0.716. Our results are in excellent agreement with the Standard Model predictions [34]:ξp V¼0.0710 0.0007, ξn V¼−0.9890 0.0007 [C1u¼−0.1885 0.0002,C1d¼ 0.3415 0.0002]. Summarizing, we have presented a complete statistical analysis of all PV asymmetry data available for elastic scattering [41] on the proton and on 4He. The χ2test is based on the simultaneous fit of five free parameters that are shown to be strongly correlated in most of the cases. This result may indicate that some caution should be exercised when considering previous analyses that are based on a reduced number of parameters. Our new study seems to favor vector strangeness different from zero. This is consistent with our previous findings [5]. Moreover, a striking result in our analysis is the unexpectedly lower value for Gep A. However, the axial-vector form factor is -3 -2 -1 0 1 2 3 GA ep -1.5 -1 -0.5 0 0.5 1 1.5 GM s Young et al. (PVES, 95%) This work (95%) |Q2|=0.1 (GeV/c)2 FIG. 3 (color online). 95% confidence level constraint ellipse in the plane Gs M-Gep Aat jQ2j¼0.1ðGeV=cÞ2: red ↔this work, green ↔[3]. -0.6 -0.55 -0.5 -0.45 -0.4 C1u - C1d 0.12 0.14 0.16 0.18 C1u + C1d APV Cs Young et al. (PVES, 1σ) Qweak Collab. (PVES, 95%) Phys. Rep. (PVep, 95.45%) This work (95%) APV Cs + This work APV Cs + Qweak Collab. Standard Model FIG. 4 (color online). Confidence level ellipses from different works: red ↔this work, blue ↔[5], green filled ↔[4], brown ↔ [2]. The magenta horizontal band represents the constraint from 133Cs APV result [39] (extracted from [2]). The yellow (cyan) ellipse is the result of combining the analysis of this work (Q-weak Collabration [2]) and the 133Cs APV result. The Standard Model prediction is also represented as reference (black cross). 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