N-Delta(1232) axial form factors from weak pion production

0.14 MB·English

by  E. Hernandez

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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