Evaluation of theoretical uncertainties in parity-violating electron scattering from nucleons and nuclei

arXiv:1408.3511v1 [nucl-th] 15 Aug 2014 Evaluation of theoretical uncertainties in parity-violating electron scattering from nucleons and nuclei O. Moreno and T. W. Donnelly Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA R. Gonz´alez-Jim´enez and J. A. Caballero Departamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Sevilla, 41080 Sevilla, Spain Abstract Parity-violating polarized electron scattering from nucleons and nuclei provides an excellent tool to extract valuable information on nuclear and nucleon structure, as well as to determine Standard Model couplings and higher-order radiative corrections. As measurements become more precise, theoretical models should improve accordingly in order to exploit the experimental data fully in extracting meaningful information. At the same time, it is crucial that theoretical evaluations come with realistic estimations of the corresponding theoretical uncertainties to establish that the precision reached in the measurements is not compromised. Here we consider isospin mixing and the charge distribution in nuclei, and strangeness content of the nucleon together with its axial form factor. PACS numbers: 12.15.-y; 12.15.Lk; 12.15.Mm; 24.80.+y; 25.30.Bf; 21.60.Jz 1 I. INTRODUCTION Parity-violating (PV) elastic scattering of longitudinally polarized electrons by nucleonic or nuclear targets is an excellent tool to determine the electroweak coupling constants, to test Standard Model features such as the mixing parameters or the role of higher-order radiative corrections, or to explore the neutron distribution in the case of nuclear targets [1]. We begin by briefly summarizing some of the basic ideas behind such studies. Parity violation originates in the interference between the vector electromagnetic (EM) current term and the axial weak neutral current (WNC) term of the electron-hadron interaction, these showing opposite behavior under a parity transformation. The degree of violation is usually cast in terms of the PV asymmetry, involving the cross sections of incoming electrons longitudinally polarized parallel (σ+) and antiparallel (σ−) to their momentum: AP V =dσ+−dσ− dσ++dσ−.(1) By considering a single exchanged boson for each interaction and neglecting the distortion of the electron wave function due to the target Coulomb field (plane-wave Born approximation (PWBA) — see below), the PV asymmetry is proportional to the ratio of parity-violating to parity-conserving responses: A=A0WP V /WP C, with a scale given by A0=GF|Q2|/(2√2πα)≈1.8·10−4|Q2|, where GFand αare the Fermi (weak) and the fine-structure (electromagnetic) coupling constants and the only kinematic dependence is on the four-momentum transfer |Q2|, given here in (GeV/c)2. In this report we consider two specific cases of elastic PV electron scattering, viz., from protons (ep) and from N=Z, spin-0, nominally isospin-0 nuclei (eA), in the latter case focusing on 12C as a practical example. In the case of elastic ep scattering the PV asymmetry can be written as AP V ep =A0 aAεGp Ee Gp E+τGp Me Gp M−aV√1−ε2qτ(1 + τ)Gp MGep A 2(ε(Gp E)2+τ(Gp M)2),(2) where aVand aArepresent the vector and axial WNC electron couplings and Gp E,M are the electric and magnetic EM form factors of the proton; we have introduced the kinematical factor ε= (1 + 2τ(1 + τ) tan2(θe/2))−1which depends on the scattering angle θe, and τ≡ |Q2|/(4M2) with Mthe nucleon mass. The WNC form factors can be written as follows (charge symmetry is assumed): e Gp E,M (Q2) = ξp VGp E,M +ξn VGn E,M +ξ(0) VGs E,M (3) 2 Gep A(Q2) = ξT=1 AG3 A+ξT=0 AG8 A+ξ(0) AGs A,(4) where Gs E,M,A are the strange form factors and G3,8 Athe isovector triplet (3) and isoscalar octet (8) contributions to the axial 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 [2] for details). Parity-violating scattering from the proton is free from nuclear structure ambiguities and can therefore provide detailed information on the WNC nucleon form factors. Our interest in Sec. II concerns the uncertainties associated with the pure EM properties as well as with the axial and vector WNC form factors through the electric and magnetic strangeness contributions and the effective weak couplings. It is clearly important to assess these theoretical uncertainties before drawing definite conclusions concerning higher-order electroweak contributions to the asymmetry. The eA PV asymmetry is again proportional to the ratio of parity-violating to parityconserving nuclear responses, with the same scale A0as for the ep case. Under further assumptions, namely elastic scattering from N=Znuclear targets with pure isospin T= 0 and angular momentum Jπ= 0+ground states and absence of nucleon strangeness content, the parity-violating and parity-conserving responses are such that the nuclear structure effects cancel out (see [3] and references therein) and one gets a reference PV asymmetry: AP V ref ≡ −2A0aAsin2θW.(5) The extent to which some of the above-mentioned conditions are not actually fulfilled needs to be modeled by theory and extracted from the measured PV asymmetry, so that the remaining effects can be pinned down or the values of the constants accurately determined. This procedure introduces theoretical uncertainties in the analysis due to the spread of results when, for instance, different microscopic nuclear structure models are used, or when different values of the parameters within a model are considered. To extract new information on electroweak couplings or higher-order interaction effects from eA PV scattering the uncertainty of the PV observables must lie at the few per mil level, typically around 0.3% [4] (both experimental and theoretical uncertainties). On the experimental side, interest has been shown recently in developing relatively low energy, high luminosity polarized electron beams for PV experiments with the hope of reaching the required precision of [4] — proposals such as the MESA accelerator at Mainz [5] or an 3 upgraded FEL facility at Jefferson Lab [6] go in this direction. On the theoretical side, uncertainties related to the modeling of different nuclear effects are the focus of Sec. III. Having as a goal the basic objectives summarized above, and following the spirit of this JPG Focus Issue, we discuss how one has attempted to evaluate the theoretical uncertainties in extracting specific information on hadronic form factors, electroweak corrections beyond tree level and nuclear many-body effects such as isospin mixing. We begin in Sec. II with ep PV scattering, followed in Sec. III with elastic eA scattering from 12C and concluding in Sec. IV with a brief summary. II. PARITY-VIOLATING ELECTRON SCATTERING FROM THE PROTON In recent work [7, 8] a systematic analysis of the effects on the PV asymmetry introduced by the various ingredients involved has been presented. Specifically, a variety of prescriptions for the EM form factors, some of them accounting for two-photon exchange effects and other higher-order corrections, was considered. The radiative corrections in the axial form factor and their effects on AP V ep were studied in detail, providing results linked to the specific functional dependence with Q2(value of the axial mass) and to its value at Q2= 0. Finally, the sensitivity shown by the PV asymmetry with ss(strangeness) content in the nucleon was discussed for both electric and magnetic channels. A comparison of our theoretical predictions for AP V ep with available data is also provided. Let us begin with a few words on one type of uncertainty which is not really a theoretical one, but which affects the extraction of the other ingredients discussed below, viz. uncertainties introduced by the lack of precise enough knowledge of the EM form factors themselves. As seen in Eq. (2), the PV asymmetry involved an interference of EM with WNC form factors and thus depends on these both via the numerator and denominator in that equation. In principle this source of uncertainty can be controlled by PC electron scattering from the nucleon; however, at present, while knowledge of the EM form factors is very good, nevertheless some uncertainty is inevitable in the PV asymmetry from this source. To summarize, in [7] it was found that the typical present uncertainty in the asymmetry amounts to ∼2-3% for θe= 5◦, but only ∼0.7% for θe= 170◦for |Q2|= 1 (GeV/c)2. However, the experimental situation is not fully settled and with specific data sets the deviation from the average is larger: ∼5% (∼4.3%) for θe= 5◦(θe= 170◦) for |Q2|= 0.8 (GeV/c)2. 4 These issues are discussed more completely in [7], although in the present study we do not pursue this problem, since our main focus is on evaluations of theoretical uncertainties, with which we now continue. A basic ingredient in the study of PV electron scattering is the WNC axial form factor and how its description may modify the results for AP V ep . It is well known that, contrary to neutrino reactions, radiative corrections play an important role in the description of Gep A for PV electron scattering. On the other hand, strangeness effects in Gep Aare generally very small. Thus in the discussion that follows we focus on the effects in AP V ep associated with the present knowledge of Gep A(Q2= 0) and the “possible” contributions of radiative corrections. In this work we assume the radiative corrections to be constant. Recently some authors [9] have studied the possible effects associated with energy-dependence in the γZ-box correction. However, at present all calculations have been performed only at very specific kinematics: forward scattering angles and low energy. Although different studies conclude results that differ by some amount, all calculations lead to uncertainties that are much smaller than the ones corresponding to the parameter range variation considered in this work. Here we follow the analysis presented in [7] and show results corresponding to the currently accepted value of the axial form factor at Q2= 0, namely, Gep A(0) = −1.04 ±0.44 (see [10]). Notice that the value at tree-level, −gA=−1.27, is included within the above range. To analyze how this large variation in Gep A(0) modifies the PV asymmetry, we compare in Fig. 1 our predictions for the two extreme values of Gep A(0) with different data (see [7, 8] for references to the data). The shaded areas represent the uncertainty linked to the particular description of the functional dependence of the axial form factor with the transferred four-momentum Q2. We assume the usual dipole form with two values of the axial mass: the standard one MA= 1.032 GeV/c2(lower limit in the bands) and MA= 1.35 GeV/c2(upper limit). The latter is connected with the recent analysis of quasielastic neutrino reactions performed by the MiniBooNE Collaboration [11, 12]. The study of other functional dependences, such as a monopole shape, has been considered in [7]. Note that, whereas G0 at lower |Q2|and the SAMPLE data seem to be consistent with the smaller (in absolute value) Gep A(red band), the reverse occurs for G0 at higher |Q2|and PVA4 data (green band). Also, the global uncertainty associated with radiative corrections for 110◦is ∼12% at |Q2|= 0.75 (GeV/c)2 and ∼9% at |Q2|= 0.25 (GeV/c)2. Similar results are found for 145◦. Concerning the role of MAin the Q2dependence of the form factor, its global impact depends on the specific value 5 selected for Gep A(0). For 110◦(left panel) the uncertainty shown by the green (red) band is ∼3.5% (∼1.8%) at |Q2|= 0.25 (GeV/c)2and ∼6.85% (∼3.15%) at |Q2|= 0.75 (GeV/c)2. Similar comments apply to 145◦. 0.2 0.4 0.6 0.8 |Q2| (GeV/c)2 1 2 3 4 5 6 |APV| (10-5) GA ep(0) = -1.48 GA ep(0) = -0.60 00.05 0.1 0.15 0.2 0.25 |Q2| (GeV/c)2 0.5 1 1.5 2 PVA4 SAMPLE 0.2 0.4 0.6 0.8 |Q2| (GeV/c)2 1 2 3 4 5 6 |APV| (10-5) G0 θe = 110º θe = 145º FIG. 1. Dependence of the PV asymmetry on the nucleon axial form factor including radiative corrections (see text for details). The upper (lower) extreme line of each band corresponds to MA= 1.35 GeV/c2(MA= 1.03 GeV/c2). The magnetic strangeness has been fixed to zero. An important objective in studies of PV electron scattering concerns the role of strange quarks in the electric and magnetic sectors. At backward angles the electric channel gives a negligible contribution and hence AP V ep can give detailed information on the strangeness content in the magnetic form factors. However, notice that in this situation Gep A(see discussion above) can also play a significant role. At forward angles, where the contribution in the asymmetry from Gep Ais smaller, both the magnetic and electric channels contribute, and accordingly a combined analysis of the two kinematical situations can shed light on the strangeness content in the nucleon. In what follows we present a detailed study comparing our theoretical predictions with all available data up to date, and we provide an estimate of the uncertainty in AP V ep due to the strange quark contribution. The Q2- dependence for the strange form factors are taken in their usual form, namely, dipole for Gs M(Q2) and dipole times τfor Gs E(Q2). The amount of strangeness content is given through the parameters µsand ρscorresponding to the magnetic and electric channels, respectively. Results are shown in Fig. 2 (backward angles) and Fig. 3 (forward angles). The prescription GKex for the EM form factors has been used (see [7]) with the dipole-shape description for the standard axial form factor. At backward angles the PV asymmetry depends mainly on the magnetic strangeness content through µs. Hence, we show in Fig. 2 the band (shaded 6 0.2 0.4 0.6 0.8 |Q2| (GeV/c)2 1 2 3 4 5 6 |APV| (10-5) µs = -0.3 µs = 0.0 µs = 0.3 0.2 0.4 0.6 0.8 |Q2| (GeV/c)2 1 2 3 4 5 6 |APV| (10-5) G0 00.05 0.1 0.15 0.2 0.25 |Q2| (GeV/c)2 0.5 1 1.5 2 PVA4 SAMPLE θe = 110º θe = 145º FIG. 2. Dependence of the PV asymmetry on the vector magnetic strangeness (µs) at backward angles. The value MA= 1.03 GeV has been employed. 0 0.2 0.4 0.6 0.8 1 |Q2| (GeV/c)2 10-7 10-6 10-5 |APV| G0 HAPPEXa HAPPEXb HAPPEX99 HAPPEXIII Qweak 00.05 0.1 0.15 0.2 0.25 |Q2| (GeV/c)2 1 2 3 4 5 6 |APV| (10-6) µs = -0.3 µs = 0.0 µs = 0.3 00.05 0.1 0.15 0.2 0.25 |Q2| (GeV/c)2 1 2 3 4 5 6 |APV| (10-6) PVA4 θe< 21º θe = 35.5º 0 0.2 0.4 0.6 0.8 1 |Q2| (GeV/c)2 10-7 10-6 10-5 |APV| G0 HAPPEXa HAPPEXb HAPPEX99 HAPPEXIII Qweak 00.05 0.1 0.15 0.2 0.25 |Q2| (GeV/c)2 1 2 3 4 5 6 |APV| (10-6) ρs = -1.0 ρs = 0.5 ρs = 2.0 00.05 0.1 0.15 0.2 0.25 |Q2| (GeV/c)2 1 2 3 4 5 6 |APV| (10-6) PVA4 θe < 21º θe = 35.5º FIG. 3. (Top panels) Dependence of the PV asymmetry at forward angles on magnetic strangeness. The electric strangeness has been fixed to ρs= 0.5. (Bottom panels) Dependence of the PV asymmetry at forward angles on electric strangeness. The magnetic strangeness has been fixed to µs= 0.0. 7 region) spanned by µs-values in the range [−0.3,0.3], which is consistent with previous analyses [7]. As observed, the global theoretical uncertainty is large enough to cover the four data analyzed. However, whereas G0 at low |Q2|and SAMPLE seem to be consistent with positive µs, SAMPLE and G0 at higher |Q2|fit the region of negative µsbetter. Only the case of magnetic strangeness close to zero seems to be the case where theory and data agree the best. In summary, the uncertainty in the PV asymmetry associated with the variation assumed in µsis of the order of 24–25%. This result applies to the two scattering angles as well as to low and high |Q2|-values. The case of forward kinematics is analyzed in Fig. 3, where the two panels on the top show the sensitivity of AP V ep with respect to the magnetic strange parameter µswith ρs (strangeness in the electric channel) fixed. Likewise, the two bottom panels refer to the theoretical uncertainty introduced by the electric strangeness (ρs) for µsfixed. In both cases the range of variation selected for µs[−0.3,0.3] and ρs[−1.0,2.0] is consistent with previous work [7]. Moreover, the region spanned by the above selection includes all data and its width (global uncertainty) is slightly larger for the specific ρs-variation considered. In the case of θe<21◦(left panels) the width in the band is ∼30% (top panel: magnetic strangeness) and ∼53% (bottom panel: electric strangeness) at |Q2|= 0.25 (GeV/c)2. Similar results are found for |Q2|= 0.8 (GeV/c)2. For θe∼35.5◦(right panels) the dispersion in the asymmetry is ∼26% (top panel →µs) and ∼39% (bottom panel →ρs) at |Q2|= 0.15 (GeV/c)2. In the particular case of the Qweak experiment [13], i.e., θe≈8◦and |Q2|= 0.025 (GeV/c)2, the sensitivity of the PV asymmetry with strangeness results: ∼12% (∼20%) for µs(ρs). Finally, let us note that data agree better with theoretical results for positive µsin the case in which ρsis fixed to 0.5 (top panels). Although not shown, such agreement also emerges for negative values of the magnetic parameter, µs=−0.3, if the electric strangeness (ρs-value) is moved to more positive values (region spanned by ρs∈[1.25 ,2.5]). This clearly indicates that strangeness in the electric and magnetic sectors are strongly correlated. A similar comment also applies to the axial form factor and its correlation with Gs E,M (see discussion below). To complete this study we present the results obtained from a statistical analysis of the full set of PV asymmetry data for elastic ep scattering including the most recent high precision measurement from Qweak [13]. The fit procedure is based on the work presented in [8] where, in addition to Gep A(0), µsand ρs, the proton and neutron WNC 8 effective coupling constants were also included in the analysis. The results obtained for the five free parameters with 1σ-errors are given in [8], also showing a strong correlation in most of the cases. As already mentioned, forward angle asymmetry data provide information on the electric and magnetic strangeness while backward angle data are mainly sensitive to the magnetic and axial contributions. Therefore, the strong correlation between the electric (ρs) and magnetic (µs) strangeness parameters comes basically from data taken at forward kinematics (HAPPEX and G0 experiments). In fact, the Jefferson Lab experimentalists often show their results as a specific combination of electric and magnetic strangeness due to this correlation. Likewise, the strong correlation that exists between µsand Gep Ais mainly linked to backward scattering data (SAMPLE, PVA4, and G0 experiments). The statistical analysis presented in [7] combines experimental data at forward and backward kinematics. This explains the large correlation coefficient (0.711) found also for ρsand Gep A(0) (see [7, 8] for details). It is worth commenting on the role played by the axial mass connected with the specific functional Q2-dependence of Gep A. This topic was discussed above, showing that the role of MAis only significant at backward kinematics and for large enough values of |Q2|. Thus, given the current experimental situation with only very few data at backward angles (specifically, the G0 experiment at |Q2| ≈ 0.63 (GeV/c)2), the statistical analysis does not seem to be appropriate in order to get clear information on the value of MA. This justifies the use of the standard value MA= 1.036 GeV/c2in the global analysis presented in [8]. In Fig. 4 we show the 95% confidence level contour ellipses in the [µs, ρs] (right) and [µs, Gep A(0)] (left) planes (red curves). Notice that the values spanned by the ellipses for the three parameters are consistent with the range considered above. Furthermore, the shape of the ellipses indicates the strong correlation existing between the axial form factor and the strangeness content in the nucleon. The blue curve in the right hand panel corresponds to the results obtained in [7] where the fit procedure was performed taking only ρsand µsas free parameters. This explains the much smaller region spanned by the blue curve compared with the red one (based on five independent parameters). Finally, although not shown here, the PV asymmetry also shows significant sensitivity to the proton and neutron WNC effective coupling constants. These couplings, that incorporate contributions from higher-order processes (γZ-box diagram, γZ-mixing diagram, etc.), were also analyzed in [8] providing constraints on their values which can be combined with atomic 9 less subject to modeling uncertainties than the latter, since, clearly, in general it is hard to deal with nuclear structure issues when very high precision is demanded. However, the case of interest here is a very special one: the main focus for eA scattering has been placed on elastic scattering from N=Z, nominally isospin-0 nuclei. There, having only a single monopole form factor, the reference PV asymmetry is independent of any nuclear structure uncertainties, and it is only when one proceeds beyond the na¨ıve starting point that these enter. Specifically, when one takes into account that the nuclear ground states of such nuclei are not at some level eigenstates of isospin and therefore that small isovector contributions must be included, then some relatively weak dependence on structure issues arises. Also, when possible strangeness contributions in the nucleons in the nucleus are considered, there are new dependences to take into account. In contrast, the nucleon with its spin-1/2 and isospin-1/2 nature, while more fundamental, nevertheless has a more complicated dependence on the underlying hadronic structure via the various form factors that enter (nine, in fact), none of which is known as well as one might like. Indeed, we have not even considered that it also is likely not an eigenstate of isospin at some level and, furthermore, that both the nucleon and nuclei are not perfect parity eigenstates, which introduces other structure issues — presumably these are small, and accordingly we have neglected them in this brief analysis. Finally, the ‘doability’ (figure-of-merit) for eA PV forward-angle elastic scattering is considerably larger than it is for ep scattering (see [2], Fig. 3.10, and [3]), which means that parts per mil determinations of the asymmetry in the former case can be contemplated. ACKNOWLEDGMENTS This research was supported by a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Programme, by MINECO (Spain) under Research Grants Nos. FIS2011-23565 (O. Moreno) and FIS2011-28738-C02-01 (R. Gonz´alez-Jim´enez and J.A. Caballero), by the Junta de Andaluc´ıa (FQM-160) and the Spanish Consolider- Ingenio 2000 programmed CPAN. R.G.J. also acknowledges financial support from VPPI-US (Universidad de Sevilla). Also supported in part (T. W. Donnelly) by the US Department 16 of Energy under cooperative agreement DE-FC02-94ER40818. [1] T. W. Donnelly, J. Dubach and I. Sick, Nucl. Phys. A 503, 1989, pp. 589. [2] M. J. Musolf et al.,Phys. Rep. 239, 1994, pp. 1. [3] O. Moreno and T. W. Donnelly, Phys. Rev. C 828, 2014, pp. 015501. [4] T. W. Donnelly and O. Moreno, AIP Conf. Proc. 1563, 2013, pp. 82. [5] K. Aulenbacher, Hyperfine Interact. 200, 2011, pp. 3. [6] G. R. Neil et al.,Phys. Rev. Lett. 84, 2000, pp- 662; D. 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