N-∆(1232) axial form factors from weak pion production E. Hern´andez,1 J. Nieves,2 M. Valverde,3 and M.J. Vicente Vacas4 1Departamento de F´ısica Fundamental e IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain 2Instituto de F´ısica Corpuscular (IFIC), Centro Mixto CSIC-Universidad de Valencia, Institutos de Investigacio´n de Paterna, Aptd. 22085, E-46071 Valencia, Spain 3Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan 4Departamento de F´ısica Te´orica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigacio´n de Paterna, Aptd. 22085, E-46071 Valencia, Spain The N∆ axial form factors are determined from neutrino induced pion production ANL & BNL databyusingastateofthearttheoreticalmodel,whichaccountsbothforbackgroundmechanisms anddeuteroneffects. WefindviolationsoftheoffdiagonalGoldberger-Treimanrelationatthelevel 0 of 2σ which might havean impact in background calculations for T2K andMiniBooNE low energy 1 0 neutrinooscillation precision experiments. 2 PACSnumbers: 25.30.Pt,13.15.+g n a J The ∆(1232)resonanceis the lightestbaryonicexcita- less well suited for the extraction of the N∆ axial form 5 tion of the nucleon. In addition, it couples very strongly factors. 2 to the lightest meson, the pion, and to the photon. As Besides the original experimental publications, there ] a consequence, the ∆(1232) is of the utmost importance are many studies of the ANL and/or the BNL data in h in the description of a wide range of hadronic and nu- theliterature[18–23]withdifferentadvantagesandshort- p clear phenomenology going from low and intermediate - comings. Some of those studies are discussed below. In p energy processes [1, 2] to the GZK cut-off of the cos- this letter, we analyze the ANL and BNL data incorpo- e mic ray flux [3, 4]. On the other hand, despite its h ratingthedeuteroneffects,withaproperconsiderationof large width, it is well separated from other resonances [ statistical and systematical uncertainties and taking ad- whatfacilitates itsexperimentalinvestigation. Inpartic- vantage of severalrecentdevelopments: improvedvector 1 ular, the electromagnetic nucleon to ∆(1232) excitation form factors and a new model for weak pion production v processes, induced by electrons and photons, have been 6 off the nucleon that includes background terms. extensively studied at many experimental facilities like 1 AconvenientparameterizationoftheW+n→∆+ ver- 4 LEGS, BATES, ELSA, MAMI, and J-LAB. For a recent tex is given in terms of eight q2 (momentum transfer 4 review see Ref. [5], where also many of the recent the- 1. oretical advances in the understanding of the resonance square)dependentform-factors: fourvectorandfourax- ial (CA ) ones. We follow the conventions and no- 0 have been addressed. 3,4,5,6 0 tation of Ref. [21]. Vector form factors have been deter- 1 There has also been a great theoretical interest in the mined from the analysis ofphoto and electro-production : axial nucleon ∆ transition form factors. Recently, they data. Here, we use the parameterization of Lalakulich v i have been studied using quark models [6], Light Cone et al. [24], as done in Ref. [21]. Among the axial form X QCDSum Rules [7], Lattice QCD[8]andChiralPertur- factorsthe mostimportantcontributioncomesfromCA. 5 r bation Theory (χPT) [9, 10]. These form factors are of The form factor CA, which contribution to the differen- a 6 topicalimportanceinthebackgroundanalysisofsomeof tial cross section vanishes for massless leptons, can be theneutrinooscillationexperiments(e.g.[11]). However, related to CA thanks to the partial conservation of the 5 their experimental knowledge is less than satisfactory. axial current (CA(q2) = CA(q2) M2 , with m and Although the feasibility of their extraction in parity- 6 5 m2π−q2 π M the pion and nucleon masses, respectively). Since violatingelectronscatteringhasbeenconsidered[12],the there are no other theoretical constraints for CA (q2), best available information comes from old bubble cham- 3,4,5 they have to be fitted to data. Most analysis, includ- ber neutrino scatteringexperiments at ANL [13, 14] and ing the ANL and BNL ones, adopt Adler’s model [25] BNL [15, 16]. These experiments measured pion pro- where1 CA(q2) = 0 and CA(q2) = −CA(q2)/4. For ductionindeuteriumatrelativelylowenergieswherethe 3 4 5 CA severalq2 parameterizationshavebeen used[19,22], dominantcontributionisgivenbythe∆pole(∆P)mech- 5 though given the limited range of statistically signifi- anism: weak excitationofthe ∆(1232)resonanceandits cant q2 values accessible in the ANL and BNL data, it subsequent decay into Nπ. Only very recently, π0 pro- duction cross sections have been measured at low neu- trino energies and with good statistics [17]. However, the target was mineral oil what implies large and diffi- 1 Setting C3A to zero seems to be consistent with SU(6) symme- cult to disentangle nuclear effects. Thus, these data are try[26]andrecentlatticeQCDresults[27]. 2 should be sufficient to consider for it a dipole depen- A second re-analysis [23] brings in the discussion two dence, CA(q2) = C5A(0) , where one would expect interesting points. First that both ANL and BNL data 5 (1−q2/MA2∆)2 were measured in deuterium, and second, the uncertain- M ∼0.85−1GeV,toguaranteeanaxialtransitionra- A∆ ties in the neutrino flux normalization. Deuteron struc- dius2 R in the range of 0.7−0.8 fm, and CA(0)∼1.2, A 5 ture effects in the νd → µ−∆++n reaction, sometimes which is the prediction of the off-diagonal Goldberger- Treiman relation (GTR), CA(0)= 2f f∗ =1.2, with ignored, were estimated from the results of Ref. [18] to ∗ 5 q3 πmπ produce a reduction of the cross section from 5–10%. In the πN∆ coupling f = 2.2 fixed to the ∆ width and what respects to the ANL and BNL flux uncertainties, fπ ∼93 MeV, the pion decay constant. the procedure followedin[23] is notrobustfromthe sta- There is no constraint from χPT and lattice calcula- tistical point of view, since it ignores the correlations of tionsarestillnotconclusiveaboutthesizeofpossiblevio- these systematic errors4. Nevertheless, this latter work lationsoftheGTR.Forinstance,thoughvaluesforC5A(0) constitutes a clear step forward, and from a combined as low as 0.9 can be inferred in the chiral limit from the best fit to the ANL & BNL data, the authors of [23] resultsofRef.[27],theyalsopredictCA(0)/ 2f f∗ find CA(0) = 1.19±0.08 in agreement with the GTR 5 (cid:16)q3 πmπ(cid:17) 5 to be greater than one. estimate. C5A(q2) ASSUMING ∆P DOMINANCE CAHXIIRAALLFNOORNM-RFAESCOTNOARNSTINBCALCUKDGINRGOUTNHDE. Traditionally, Adler’s model and the GTR have been Alltheabove-mentioneddeterminationsofCA(q2)suf- assumed, being the M axial mass adjusted in such 5 A∆ ferfromaserioustheoreticallimitation. Thoughthe∆P a way that the ∆P contribution alone would lead to a mechanism dominates the neutrino pion production re- reasonable description of the shape of the BNL q2 dif- ferential ν p→µ−pπ+ cross section (see e.g. Ref. [19]). action, specially in the ∆++ channel, there exist sizable µ non-resonant contributions of special relevance for low These fits also describe reasonably well the q2 depen- neutrino energies (below 1 GeV) of interest in T2K and dence of the ANL data and the BNL total cross section MiniBooNE experiments. These background terms are but overestimate the size of the ANL data by 20% near totally fixed by the pattern of spontaneous chiral sym- the maximum [20]. Thus, ANL data might favor CA(0) 5 metry breaking of QCD, and are given in terms of the values smaller than the GTR prediction. nucleonandpionmasses,the axialchargeof the nucleon Recently, two re-analysis have been carried out trying and the pion decay constant. When background terms to make compatible the GTR prediction for CA(0) and 5 are considered, the tension between ANL data and the ANL data. In Ref. [22], CA(0) is kept to its GTR value 5 GTR prediction for CA(0) substantially increases. In- andthreeadditionalparameters,thatcontroltheCA(q2) 5 5 deed, the fit carried out in [21] to the ANL data finds fall off, are fitted to the ANL data. In fact CA(q2 ∼ 0) 5 a value for CA(0) as low as 0.87±0.08 with a reason- is not so relevant due to phase space, and what is actu- 5 able axialtransitionradiusof0.75±0.06fm, anda large ally important is the CA(q2) value in the region around 5 Gaussian correlation coefficient (r = 0.85), as expected −q2 ∼ 0.1GeV2. Although ANL data are well repro- from the above discussion of the results of Ref. [22]. duced, we findthe outcomein [22]to be unphysical,be- Here, we follow the approach of Ref. [21], but imple- causeitprovidesaquitepronouncedq2−dependencethat menting four major improvements: i) we include in the givesrisetoatoolargeaxialtransitionradius3 ofaround fit the BNL total ν p → µ−pπ+ cross section measure- 1.4 fm. Moreover, neither the fitted parameter statisti- µ ments of Ref. [15]. Since there is no cut in the outgoing cal errors, nor the corresponding correlation coefficients are calculated in [22]. Undoubtedly, the fit carried out there shouldbe quite unstable,fromthe statisticalpoint ofview,becauseofthedifficultyofdeterminingthreepa- 4 There exist some other aspects that might require further in- rameters given the limited range of q2 values covered in vestigation. Forinstance,additionalparameterspANLandpBNL the ANL dataset. Furthermore,the consistencyofthese areintroducedin[23](seeχ2functioninEq.(37))toaccountfor results with the BNL data has not been tested. the flux uncertainties. At very low q2 values, dσ/dq2 is totally dominated by C5A. If we had infinitely precise statistical mea- surements,thefitcarriedoutin[23]wouldprovideaveryprecise determinationoftheratioC5A(0)/√p,butnotoftheformfactor C5A(0). However,insuchsituation,oneexpectstoextractC5A(0), 23 FIutritshderefidneetadilfsroamndCp5Aos(sqi2b)le/Cre5Ap(e0r)cu=ss1io+nsqi2nRn2Aeu/t6r+inoOi(nqd4u)c.edco- tflhuoxugnhorwmiathlizaantiounn.ceBrteasiindteys,dtohmeinfiattetdo tbhyetBhaNtLofdtahtae nuesuestrtinhoe herentpionproductioncalculationsarediscussedin[28]. There, totalcross-sectiondata,forwhichthehadronicinvariantmassis ANLdata fits of the type proposed in[22],but including chiral unconstrained,andtheneutrinoenergyvariesintherange0.5–3 non-resonantcontributionsarealsoperformed,findingthatthen GeV. Above 1 GeV, heavier resonances than the ∆(1232), and theaxialtransitionradiusbecomesevenlarger,about2.5fm. notconsideredin[23],shouldplayarole[24]. 3 pion-nucleon invariantmass in the BNL data, and in or- 13%,whiledeuteroneffectsincreaseitbyabout5%,con- dertoavoidheavierresonancesfromplayingasignificant sistentlywiththeresultsof[21]and[18,23],respectively. role, we have just included the three lowest neutrino en- Third, the fitted data are quite insensitive to CA (0), as 3,4 ergies: 0.65, 0.9 and 1.1 GeV. We do not use the BNL fit V–VII results show. This is easily understood, taking measurement of the q2−differential cross section, since for simplicity the massless lepton limit. In that case it lacks an absolute normalization. ii) we take into ac- dσ count deuteron effects in our theoretical calculation, iii) ∝ [CA(0)]2+q2a(q2) (2) dq2 5 wetreatthe uncertaintiesinthe ANL andBNL neutrino (cid:8) (cid:9) flux normalizations as fully correlatedsystematic errors, and CA (0) start contributing to a(q2), i.e. to O(q2), 3,4 improving thus the treatment adopted in Ref. [23], and which also gets contributions from vector form factors finally iv) in some fits, we relax the Adler’s model con- and terms proportional to dCA/dq2 . This also ex- straints, by setting C3A,4(q2) = C3A,4(0)(C5A(q2)/C5A(0)), plains the large statistical cor5relatio(cid:12)(cid:12)nq2s=d0isplayed in fits and explore the possibility of extracting some direct in- V–VII.Moreover,dCA /dq2 appearsatorderO(q4), formation on C3A,4(0). which has prevented3u,4s to fi(cid:12)(cid:12)tqt2i=n0g the q2−shape of these Let us consider first the neutrino–deuteron reaction νd→µ−pπ+nmeasuredinANLandBNL.Owingtothe form factors. Fourth, fit IV is probably the most robust from the statistical point of view. In Fig. 1, we display inclusionofbackgroundterms,theformalismofRef.[18], fit IV results for the ANL and BNL νd→µ−pπ+n cross where the pπ+ pair was replaced by a ∆++, cannot be sections. LookingatthecentralvaluesofCA(0),wecon- used to account for deuteron corrections, and we must 5 cludethattheviolationoftheoff-diagonalGTRisabout work with four particles in the final state. Neglecting 15% smaller than that suggested in Ref. [21], though it the D−wave deuteron component and considering the is definitely greaterthanthat claimedin [23],mostly be- neutron as a mere spectator, we find for the differential causeinthislatterworkbackgroundtermswerenotcon- cross section on deuteron sidered. However,GTR andfit IV CA(0) values differ in 5 dσ M dσ lessthantwosigmas,andthediscrepancyisevensmaller = d3p |Ψ (p~ )|2 (1) dq2dW(cid:12)d Z d d d Ep,ddq2dW(cid:12)p−offshell if Adler’s constraints are removed. These new results (cid:12) (cid:12) (cid:12) (cid:12) arequite relevantforthe neutrinoinduced coherentpion where E = m − M2+~p 2, with m the deuteron productionprocessinnucleiwhichismuchmoreforward p,d d d d mass, is the energypof the off-shell proton inside the peaked than the incoherent reaction. For instance, we deuteron which has four-momentum pµ = (E ,p~ ). W expect the results in Ref. [30], based in the determina- p,d d is the final pπ+ invariant mass. The differential cross tionofCA(0)ofRef.[21],tounderestimatecrosssections 5 sectiondσ/dq2dW is computedusingthe model by at least 30%. (cid:12)p−offshell By using a state of the art theoretical model, we have of Ref. [21]. Final(cid:12)ly, Ψ is the S−wave Paris potential (cid:12) d determined the N∆ axial form factors from statistically deuteron wave function [29] normalized to 1. improvedfitstothecombinedANL&BNLdata. Thein- Inwhatrespectstotheneutrinofluxnormalizationun- clusion of chiral background terms significantly modifies certainties,we consider them as sources of20%and 10% the form factors. We have found violations of the GTR systematic errors for the ANL and BNL experiments re- at the level of 2σ, when the usual Adler’s constraints spectively(seediscussionin [23]). Wehaveassumedthat areadopted. This will influence backgroundcalculations the ANL andBNL input data haveindependent statisti- for T2K and MiniBooNE low energy neutrino precision calerrors(σ )andfully-correlatedsystematicerrors(ǫ ), i i oscillation experiments. but no correlations linking the ANL and BNL sets. We end up with a 12×12 covariance matrix, C, with two diagonal blocks. The first 9 ×9 block is for the ANL flux averaged q2−differential νd → µ−pπ+n cross sec- M.Valverdeacknowledgesthe JapaneseSociety forthe Promotion of Science (JSPS) for a Postdoctoral Fellow- tion data (with a 1.4 GeV cut in W), while the second ship. Research supported by DGI contracts FIS2008- 3×3block is for the BNL totalcrosssectionsmentioned 01143, FIS2006-03438, FPA2007-65748 and CSD2007- above. Both blocks have the form C = σ2δ +ǫ ǫ . ij i ij i j 00042,JCyLcontractsSA016A07andGR12,Generalitat The χ2 function is constructed by using the inverse of ValencianacontractPROMETEO/2009/0090andbyEU the covariance matrix. HadronPhysics2 contract 227431. Results from severalfits are compiled in Table I, from where we draw several conclusions. First, by comparing ∗ fit II with Ref. [21], we deduce that the consideration ofBNLdataandflux uncertaintiesincreasesthevalueof C5A(0) by about 9%, while strongly reduces the statisti- [1] G. E. Brown and W.Weise, Phys.Rept.22 (1975) 279. cal correlations between C5A(0) and MA∆. Second, the [2] G. Cattapan and L. S. Ferreira, Phys. Rept. 362 (2002) inclusion of background terms reduces CA(0) by about 303. 5 4 TABLE I: Results from different fits to the ANL and BNL data. Deuteron effects are included in all cases except for the two first fits (marked with ∗). The non-resonant chiral background contributions are not included in fits I and III. In the C3A,4 columns, Ad indicates that Adler’s constraints (C3A = 0, C4A = −C5A/4) are imposed. Finally, rij are Gaussian correlation coefficients between parameters i and j. For C5A(q2) a dipole form has been used. C5A(0) MA∆/GeV C3A(0) C4A(0) r12 r13 r14 r23 r24 r34 χ2/dof I∗ (only ∆P) 1.08±0.10 0.92±0.06 Ad Ad −0.06 0.36 II∗ 0.95±0.11 0.92±0.08 Ad Ad −0.08 0.49 III (only ∆P) 1.13±0.10 0.93±0.06 Ad Ad −0.06 0.32 IV 1.00±0.11 0.93±0.07 Ad Ad −0.08 0.42 V 1.08±0.14 0.91±0.10 −1.0±1.4 Ad −0.48 −0.61 0.81 0.40 VI 1.08±0.14 0.86±0.15 Ad −1.0±1.3 −0.57 −0.66 0.93 0.40 VII 1.07±0.15 1.0±0.3 1±4 −2±4 −0.62 −0.45 0.30 0.89 −0.77 −0.97 0.44 1 ) 0.8 − + −2eV 0.8 ν d → µ−p π+n νµ d → µ p π n 2 G µ 2m ) 0.6 m 0.6 c −382 (10c 0.4 −38σ ( 10 0.4 dq 0.2 0.2 ANL σ/ ANL BNL d 0 0 0 0.2 0.4 0.6 0.8 1 0.25 0.5 0.75 1 1.25 2 2 -q (GeV ) E (GeV) FIG. 1: Comparison of the ANL dσ/dq2 differential (left panel) and ANL & BNL total (right panel) cross section data with fit IV theoreticalresults. Theoretical68%confidencelevelbandsarealsodisplayed. Datainbothplotsincludeasystematicerror(20%forANL and10%forBNLdata) addedinquadraturetothestatistical ones. Intheleftpanel,bothdataandresultsincludeacutW <1.4GeV. 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