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

Global analysis o pa i y- iola ing asymme y da a o elas ic elec on sca e ing R. González-Jiménez,1J. A. Caballe o,1and T. W. Donnelly2 1Depa amen o de Física A ómica, Molecula y Nuclea , Uni e sidad de Se illa, 41080 Se illa, Spain 2Cen e o Theo e ical Physics, Labo a o y o Nuclea Science and Depa men o Physics, Massachuse s Ins i u e o Technology, Camb idge, Massachuse s 02139, USA (Recei ed 20 Ma ch 2014; published 1 Augus 2014) We pe o m a s a is ical analysis o he ull se o pa i y- iola ing asymme y da a o elas ic elec on sca e ing including he mos ecen high p ecision measu emen om Q-weak. Gi en he basis o he p esen analysis, ou es ima es appea o a o nonze o ec o s angeness, speci ically, posi i e (nega i e) alues o he elec ic (magne ic) s ange o m ac o s. We also p o ide an accu a e es ima e o he axial- ec o nucleon o m ac o a ze o momen um ans e , Gep Að0Þ. Ou s udy shows Gep Að0Þ o be impo an ly educed wi h espec o he cu en ly accep ed alue. We also ind ou analysis o da a o be compa ible wi h he S anda d Model alues o he weak cha ges o he p o on and neu on. DOI: 10.1103/PhysRe D.90.033002 PACS numbe s: 12.15.-y, 14.20.Dh, 14.65.B , 25.30.B O e he yea s pa i y- iola ing (PV) elec on sca e ing has p o ided a g ea deal o p ecise in o ma ion on he s uc u e o he nucleon. A a ie y o expe imen s unning on di e en a ge s, om hyd ogen o hea ie sys ems wi h emphasis on deu e ium and helium, ha e added s ong cons ain s on he elec oweak o m ac o s. Mo eo e , he high p ecision eached by he mos ecen expe imen s [1,2] will se e as a es o he S anda d Model (SM) p o iding a signi ican cons ain on non- pe u ba i e QCD e ec s. The Q-weak Collabo a ion has ecen ly de e mined he weak cha ge o he p o on co esponding o he analysis o app oxima ely 4% o he da a collec ed in he expe imen [2]. The main objec i e o he Q-weak expe imen is o p o ide a alue o sin2θWwi h a 0.3% p ecision, ha is, he weak cha ge o he p o on o 4%. This ex emely small unce ain y will p o ide a signi ican es o he SM. In [2] a global i o da a aken o hyd ogen, deu e ium and helium a ge s up o jQ2j¼0.6ðGeV=cÞ2was also done p o iding some es ima es o he weak neu al cu en (WNC) couplings. These esul s a e compa ible wi h he ones ob ained in p e ious analyses pe o med by Young and collabo a o s [3,4]. Howe e , he use o da a only up o jQ2j¼0.6ðGeV=cÞ2[in he case o [3] only da a o jQ2j<0.3ðGeV=cÞ2we e conside ed], in addi ion o pa icula assump ions o he Q2expansion o he o m ac o s, ha e con inced us o he necessi y o a new and mo e comple e analysis o he p ocess. Mo eo e , when such high le els o p ecision a e he goal, mixing da a o elas ic sca e ing on he p o on and 4He wi h hose co esponding o he quasielas ic (QE) p ocess, make i di icul o disen angle e ec s due o he nucleon s uc u e om o he s di ec ly linked o inal s a e in e ac ions, o - shell e ec s, ew-body nuclea s uc u e, e c. Acco dingly, in his wo k we es ic ou sel es o elas ic elec on sca e ing p ocesses, and make use o all a ailable da a in he li e a u e wi h no es ic ion on he Q2 ange conside ed. The PV asymme y (APV) in he case o elas ic elec on- p o on (ep) sca e ing may be w i en as ollows [5]: APV ep ¼A0 2G½aAðεGp E~ Gp EþτGp M~ Gp MÞ −aVffiffiffiffiffiffiffiffiffiffiffiffi 1−ε2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τð1þτÞ pGp MGep A;ð1Þ whe e τ≡jQ2j=ð4M2Þand G≡εðGp EÞ2þτðGp MÞ2wi h M he nucleon mass and Gp E;M he elec omagne ic (EM) o m ac o s o he p o on. The e m A0de e mines he scale o he PVasymme y and is gi en by A0¼GFjQ2j=ð2ffiffiffi 2 pπαÞ wi h GF he Fe mi coupling and α he ine s uc u e cons an . Finally, aV¼−1þ4sin2θWand aA¼−1a e he ec o and axial- ec o WNC elec on couplings, and we ha e in oduced he kinema ical ac o ε¼½1þ 2τð1þτÞ an2θe=2−1which depends on he sca e ing angle θe. Assuming cha ge symme y, he WNC o m ac o s can be w i en as ollows: ~ Gp E;MðQ2Þ¼ξp VGp E;M þξn VGn E;M þξð0Þ VGs E;M Gep AðQ2Þ¼ξT¼1 AG3 AþξT¼0 AG8 Aþξð0Þ AGs A; whe e Gs E;M a e he elec ic (E) and magne ic (M) s ange o m ac o s and G3;8;s Aa e he iso ec o (3), isoscala (8) and s ange (s) con ibu ions o he axial- ec o o m ac o o he p o on (Gep A). The ξcoe icien s ep esen he WNC e ec i e coupling cons an s ha a e gi en in e ms o he weak mixing angle (θW) and adia i e co ec ions (see [5,6] o explici exp essions). In his wo k we also conside he da a ob ained by he HAPPEX Collabo a ion on elas ic e-4He sca e ing. In his PHYSICAL REVIEW D 90, 033002 (2014) 1550-7998=2014=90(3)=033002(5) 033002-1 © 2014 Ame ican Physical Socie y case he PVasymme y in ol es he a io o he wo nuclea monopole o m ac o s: he EM and he WNC ones. Hence, nuclea e ec s also play a ole in he desc ip ion o he p ocess— o discussions o how such e ec s a ise, see [7–9]. I can be shown ha he use o one-body ope a o s (leading-o de app oxima ion) and he assump ion ha isospin mixing can be igno ed o he 4He g ound s a e yield o he a io o nuclea o m ac o s simply a a io o single-nucleon o m ac o s. The la e poin was i s discussed in [7] and hen e isi ed in a la e s udy [10] (see, especially, Fig. 2in ha e e ence). Wi hin he con ex o he app oaches discussed he e, he e ec s o isospin mixing in 4He we e ypically ound o be mo e han an o de o magni ude smalle han he e ec s om s ange- ness con en in he nucleon, in con as o he expec a ions o hea ie nuclei [8,9]. This issue is no comple ely closed, howe e , as o he app oaches ind somewha la ge e ec s om isospin mixing [11]. In u u e expe imen al s udies, p e e ably a low momen um ans e s, using a a ie y o N¼Znuclei i should be possible o add ess his p oblem mo e de ini i ely; o his s udy we con inue o make he abo e assump ions, in which case he PVelas ic asymme y can be w i en in he o m [6] APV eHe ¼− A0 2ðξp Vþξn VÞþ2ξð0Þ VGs E Gp EþGn E:ð2Þ Thus, hese da a a e also sensi i e o he WNC nucleon s uc u e, and hey gi e in o ma ion ha complemen s wha is ob ained h ough elas ic ep sca e ing. In a ecen s udy [5] we showed ha a jQ2j¼ 1ðGeV=cÞ2( he cu en limi o asymme y da a) he dispe sion in APV due o he use o di e en p esc ip ions o he EM o m ac o s (some o hem accoun ing o wo-pho on exchange con ibu ions) was ∼3% in he e y o wa d limi (θe¼5°) ge ing much smalle o la ge angles and lowe jQ2j. The impac o hese unce ain ies a he pa icula Q-weak kinema ical condi ions has been explo ed in de ail in [5]. The e o e, in he p esen wo k we neglec he unce ain ies associa ed wi h he pa icula desc ip ion o he EM nucleon o m ac o s. On he con a y, APV is highly sensi i e o he elec ic and magne ic s ange o m ac o s Gs E;MðQ2Þand o he axial- ec o one Gep AðQ2Þ. In he la e adia i e co ec ions can in oduce e y s ong e ec s. In his wo k we pe o m a global i o he i e pa ame e s: ξp V,ξn V,ρs,μs( hese wo linked o he elec ic and magne ic s ange o m ac o s [6]) and he axial- ec o o m ac o a ze o momen um ans e : Gep A≡Gep Að0Þ. Ou p edic ions, based on a s a is ical analysis o he ull se o PV asymme y da a o elas ic ep sca e ing (SAMPLE [12], HAPPEX [13–16],PVA4 [17–19],G0[20,21] and Q-weak [2]) as well as he wo PV elas ic e-4He da a (HAPPEX [15,22]), a e compa ed wi h all p e ious analyses p esen ed in he li e a u e. Assuming he adia i e co ec ions (RC) o be mildly dependen on he ene gy, namely, on he pa icula kinema ics selec ed, he esul s o ou analysis could se e as a es o hei s eng h. Thus he dispe sion ound in he alues o ξp Vand ξn Vwi h espec o he co esponding ee- le el esul s (see [6] o de ails) could p o ide in o ma ion on Rp Vand Rn V. Simila ly, he de ia ion in Gep Að0Þwould cons ain he RC con ibu ions in he axial cu en as well as possible e ec s linked o he anapole momen [23–25]. Recen ly a la ge amoun o heo e ical wo k has been ca ied ou ega ding he ene gy-dependen γZ-box co - ec ion [26–29]. Ob iously, he esul s o any global analysis o da a a e a ec ed by he co ec ion associa ed wi h he γZbox. Howe e , a p esen his co ec ion can only be applied o a limi ed se o da a: e y o wa d sca e ing angles. Hence in his wo k we do no in oduce such co ec ions, so ha all da a a e ea ed consis en ly in he global analysis. Ou p esen analysis o he weak cha ges o he p o on and neu on, wi h hei unce ain ies linked o he pa ame e s ha ha e been es ima ed, is consis en wi h he s udy epo ed by he Q-weak Collabo a ion [2], whe e he γZ-box ene gy-dependen co ec ion was applied. Howe e , some cau ion should be d awn conce ning his ou come because o he p esen limi ed analysis o Q-weak da a (only 4% o da a ha e been conside ed). Once he analysis is ex ended o he 100% o he Q-weak da a, and aking in o accoun ha ξp Vis basically no cons ained by any o he da a, he γZ-box co ec ion o he Q-weak da a will become c ucial. Be o e en e ing in o a de ailed discussion o he esul s, a gene al commen on he i p ocedu e should be made. Speci ically, he cu en ly accep ed alue o he axial- ec o o m ac o a Q2¼0, namely, Gep Að0Þ¼−1.04 0.44 (see [30]), has been included as an addi ional expe imen al cons ain in he global i . Al hough a signi ican con i- bu ion o Gep Acomes om G3 A≡gA¼1.2695 ha is well de e mined om Gamow-Telle β-decay measu emen s, adia i e co ec ions can in oduce an impo an unce ain y in Gep AðQ2Þ(see discussion in [5]). Hence he e is s ill oom in he global i p ocedu e o a ia ion o he speci ic alue o Gep AðQ2Þ. In summa y, ou analysis akes in o accoun all 31 expe imen al da a o he PVasymme y a ailable in he li e a u e plus he es ic ion applied o Gep Að0Þ. The Q2dependence in he ec o s ange and axial- ec o o m ac o s has been aken in hei usual o m [5,6], namely, dipole (dipole imes τ) o Gs M(Gs E) wi h a ec o mass MV¼0.84 GeV, and dipole shape o he axial o m ac o wi h he axial mass MA¼1.03 GeV. The GKex p esc ip ion [31–33] has been used o he EM o m ac o s. The i p ocedu e consis s in minimizing he χ2 unc ion: χ2¼½Aexp −A heT½V−1½Aexp −A he;ð3Þ whe e Aexp con ains all da a and A he akes ca e o he co esponding heo e ical p edic ions ha depend on he i e pa ame e s conside ed: ξp V;ξn V;ρs;μs;G ep A. The e m V ep esen s he co a iance e o ma ix de ined as GONZÁLEZ-JIMÉNEZ, CABALLERO, AND DONNELLY PHYSICAL REVIEW D 90, 033002 (2014) 033002-2 Vij ¼ðσunco iÞ2δij þσco iσco jð4Þ wi h σunco iand σco i he unco ela ed and co ela ed unce - ain ies o he i h measu emen , espec i ely. In his wo k only co ela ed e o s epo ed by G0 Collabo a ion a e conside ed. The esul s o he χ2 i a e summa ized in Table Iwhich con ains he alues o he i e ee pa ame e s as well as hei 1σe o s (χ<χ2 MIN þ1). These alues a e in good ag eemen wi h he SM weak cha ges o he p o on and neu on [34]:ξp V¼0.0710 0.0007 and ξn V¼−0.9890  0.0007. On he con a y, Gep Aex ac ed om he i , Gep A¼−0.62 0.41, is signi ican ly lowe han he alue Gep A¼−1.04 0.44 gi en in [30]. Howe e , no e ha bo h esul s a e a ec ed by la ge e o s ha make he wo p edic ions o e lap. These esul s may e lec impo an e ec s coming om highe -o de con ibu ions ( adia i e co ec ions) as well as om al e na i e desc ip ions o he Q2dependence. The use o he s anda d dipole jQ2j dependence o he axial o m ac o in oduces a sys em- a ical unce ain y in he i ha p opaga es in o he speci ic alue o Gep Aa jQ2j¼0. O he unc ional dependences [5] as well as he use o a di e en alue o he axial mass MA≈1.35 GeV=c as ound by he MiniBooNE Collabo a ion [35] may lead o sligh ly di e en esul s. Finally, in spi e o he signi ican e o s associa ed wi h he s ange o m ac o s, ou global i a o s a la ge-posi i e s ange cha ge adius ρs(elec ic s angeness), whe eas he s ange magne ic momen μs ends o be nega i e al hough s ill being compa ible wi h ze o con ibu ion. The co ela ion coe icien s be ween pai s o he i e pa ame e s conside ed in he χ2 i a e gi en in Table II. No ice he ex emely small co ela ion o he pai ðξp V;G ep AÞ. This esul suppo s he idea ha a de e mina ion o he weak cha ge o he p o on is no a ec ed by he la ge unce ain y in Gep A. The co ela ion alues be ween ξp Vand he h ee emaining pa ame e s, ξn V,ρsand μs, al hough ela i ely small, should be aken in o accoun ca e ully i a high p ecise de e mina ion o he p o on weak cha ge is desi ed. To conclude, he es o pa ame e s: ξn V,Gep A,ρs and μs, a e e y s ongly co ela ed. In Fig. 1we show all PV asymme y da a o elas ic elec on sca e ing compa ed wi h ou heo e ical p edic- ions (ze o line). The inne e o ba s ep esen he expe imen al e o s while he ou e ones (in ed) include he heo e ical unce ain y p o ided by he global i co esponding o he 1σcon idence le el. The s ong co ela ion be ween some o he pa ame e s has been aken in o accoun in o de o de e mine he heo e ical e o . In Fig. 2we show he 95% con idence con ou s ob ained in ou analysis ex apola ed o jQ2j¼1ðGeV=cÞ2. The esul s o ou p esen i ( ed ellipse) a e compa ed wi h hose om ou p e ious wo k [5] (blue cu e). The signi ican di e ence in he a eas spanned by he wo ellipses is due o he di e en app oaches conside ed in he wo cases. In [5]a global analysis o he ull se o PV TABLE I. Values o he ee pa ame e s om he global i . The educed χ2 alue 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. Co ela ion coe icien s be ween he ee pa ame e s o he i . 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 - A heo) / A0 HAPPEX 4He Qweak HAPPEXa SAMPLE HAPPEXb G0- o wa d G0-backwa d HAPPEXIII PVA4-145ª PVA4-35.5ª HAPPEX99 -0.1 -0.05 0 0.05 0.1 FIG. 1 (colo online). Full se o PV asymme y expe imen al da a o elas ic elec on sca e ing. The da a a e no malized by sub ac ing ou heo e ical asymme y and di iding by A0. -1 -0.5 00.5 11.5 GM s -0.1 -0.05 0 0.05 0.1 GE s This wo k (95%) Young e al. (95%) Liu e al. (95%) Phys. Rep. (95%) Leinwebe e al. Doi e al. |Q2|=0.1 (GeV/c)2 FIG. 2 (colo online). 95% con idence le el cons ain ellipses in he plane Gs E-Gs Ma jQ2j¼0.1ðGeV=cÞ2: ed ↔ his wo k, g een ↔[3],b own↔[30], blue ↔[5]. Black and o ange c osses a e he heo e ical p edic ions p esen ed in [36,37] and [38], espec i ely. GLOBAL ANALYSIS OF PARITY-VIOLATING ASYMMETRY …PHYSICAL REVIEW D 90, 033002 (2014) 033002-3 asymme y da a, including only ep sca e ing, was pe - o med (28 da a) aking only ρsand μsas ee pa ame e s (see [5] o de ails). On he con a y, he ed ellipse co esponds o he p edic ions ob ained by simul aneously i ing i e pa ame e s. This la ge numbe o ee pa am- e e s explains he signi ican inc ease in he a ea. No e ha he p esen analysis seems o a o posi i e (nega i e) alues o ρs(μs). Howe e , he case o ze o s angeness, i.e., ρs¼μs¼0, is s ill inside he 95% con idence le el in ou p esen analysis ( ed cu e), and only sligh ly ou side he egion ob ained om ou p e ious s udy (blue). We also compa e ou p edic ions wi h he s a is ical analyses o da a pe o med by Liu e al. [30] (b own ellipse) and Young e al. [3] (g een ellipse). Also shown o e e ence he heo e ical p edic ions p o ided by Leinwebe e al. [36,37] (black c oss) and Doi e al. [38] (o ange c oss). The ellipses co esponding o he wo ks [3,30] a e he esul s o χ2 i s o he PV asymme y da a o elec on sca e ing on helium, deu e ium and hyd ogen in he icini y o jQ2j≈0.1ðGeV=cÞ2. In pa icula , in he wo k o Liu e al. [30] a o al o 10 da a in he ange 0.091 ≤ jQ2j≤0.136 ðGeV=cÞ2we e employed wi h only wo ee pa ame e s: he elec ic and magne ic s ange o m ac o s, Gs EðjQ2j¼0.1ðGeV=cÞ2Þ,Gs MðjQ2j¼0.1ðGeV=cÞ2Þ.On he con a y, in [3] a o al o 19 da a in he ange 0.038 ≤ jQ2j≤0.299 ðGeV=cÞ2we e used. In his case, ou ee pa ame e s we e conside ed in he i : he wo ec o s ange o m ac o s and he axial- ec o o m ac o o he p o on and neu on a ze o momen um ans e ed, Gep Að0Þand Gen Að0Þ. In Fig. 3we p esen he 95% con idence le el ellipse in he (μs;G ep A) plane. Resul s ha e been ex apola ed o he kinema ical si ua ion jQ2j¼0.1ðGeV=cÞ2. Ou p edic ion ( ed ellipse) is compa ed wi h he one o Young e al. [3] (g een ellipse). As shown, ou analysis imp o es signi i- can ly he p e ious p edic ion (impo an educ ion in he a ea o he ellipse). To conclude, we apply ou analysis o he de e mina ion o he WNC e ec i e couplings. Thus, we build he 95% con idence le el con ou in he plane ðC1u−C1d; C1uþC1dÞ. This is shown in Fig. 4by he ed ellipse. Ou p edic ion is also compa ed wi h p e ious analyses gi en in he li e a u e: blue ellipse [5], g een illed ellipse [4], and b own ellipse [2]. The yellow one is he esul o combining he cons ain s coming om ou p esen analysis ( ed ellipse) wi h hose om A omic PV in Cesium (APV-Cs) expe imen s (magen a ho izon al band, see [34,39,40] o de ails). Likewise, he cyan ellipse is he esul o he combined analysis p esen ed in [2]. This combined analysis allows us o ge a mo e accu a e de e mina ion o he WNC qua k coupling cons an s: C1u¼−0.186 0.006,C1d¼0.338 0.006 wi h a co - ela ion coe icien −0.966. Simila ly, he weak cha ges o he p o on and neu on esul : ξp V¼0.070 0.013 and ξn V¼−0.978 0.012 wi h a co ela ion coe icien −0.716. Ou esul s a e in excellen ag eemen wi h he S anda d Model p edic ions [34]:ξp V¼0.0710 0.0007, ξn V¼−0.9890 0.0007 [C1u¼−0.1885 0.0002,C1d¼ 0.3415 0.0002]. Summa izing, we ha e p esen ed a comple e s a is ical analysis o all PV asymme y da a a ailable o elas ic sca e ing [41] on he p o on and on 4He. The χ2 es is based on he simul aneous i o i e ee pa ame e s ha a e shown o be s ongly co ela ed in mos o he cases. This esul may indica e ha some cau ion should be exe cised when conside ing p e ious analyses ha a e based on a educed numbe o pa ame e s. Ou new s udy seems o a o ec o s angeness di e en om ze o. This is consis en wi h ou p e ious indings [5]. Mo eo e , a s iking esul in ou analysis is he unexpec edly lowe alue o Gep A. Howe e , he axial- ec o o m ac o is -3 -2 -1 0 1 2 3 GA ep -1.5 -1 -0.5 0 0.5 1 1.5 GM s Young e al. (PVES, 95%) This wo k (95%) |Q2|=0.1 (GeV/c)2 FIG. 3 (colo online). 95% con idence le el cons ain ellipse in he plane Gs M-Gep Aa jQ2j¼0.1ðGeV=cÞ2: ed ↔ his wo k, g een ↔[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 e al. (PVES, 1σ) Qweak Collab. (PVES, 95%) Phys. Rep. (PVep, 95.45%) This wo k (95%) APV Cs + This wo k APV Cs + Qweak Collab. S anda d Model FIG. 4 (colo online). Con idence le el ellipses om di e en wo ks: ed ↔ his wo k, blue ↔[5], g een illed ↔[4], b own ↔ [2]. The magen a ho izon al band ep esen s he cons ain om 133Cs APV esul [39] (ex ac ed om [2]). The yellow (cyan) ellipse is he esul o combining he analysis o his wo k (Q-weak Collab a ion [2]) and he 133Cs APV esul . The S anda d Model p edic ion is also ep esen ed as e e ence (black c oss). GONZÁLEZ-JIMÉNEZ, CABALLERO, AND DONNELLY PHYSICAL REVIEW D 90, 033002 (2014) 033002-4 known o be highly sensi i e o adia i e co ec ions ha could explain his esul . In any case, mo e s udies a e needed and ou p edic ion could simply be conside ed as a signal o possible al e na i e desc ip ions o he Q2 dependence o Gep A. Finally, ou esul s a e in acco dance wi h he weak cha ges o p o on and neu on p o ided by he S anda d Model. This wo k was pa ially suppo ed by DGI (Spain): FIS2011-28738-C02-01, by he Jun a de Andalucía (FQM-160) and he Spanish Consolide -Ingenio 2000 p og ammed CPAN, and in pa (T. W. D.) by U.S. Depa men o Ene gy unde coope a i e ag eemen DE- FC02-94ER40818. R. G. J. acknowledges inancial help om VPPI-US (Uni e sidad de Se illa). [1] The Hall A Collabo a ion, JLab, h p://hallaweb.jlab.o g/ expe imen /PVDIS/, 2007, ~ e−2HPVDIS a CEBAF 6 GeV. [2] D. And oic e al. (Q-weak Collabo a ion), Phys. Re . Le . 111, 141803 (2013). [3] R. D. Young, J. Roche, R. D. Ca lini, and A. W. Thomas, Phys. Re . Le . 97, 102002 (2006). [4] R. D. Young, R. D. Ca lini, A. W. Thomas, and J. Roche, Phys. Re . Le . 99, 122003 (2007). [5] R. 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