a Xi :1108.5202 1 [nucl- h] 25 Aug 2011
Rela i is ic Models o Quasi-Elas ic Neu ino-Nucleus
Sca e ing
M.B. Ba ba o∗, J.E. Ama o†, J.A. Caballe o∗∗, T.W. Donnelly‡and J.M. Udías§
∗Uni e si à di To ino and INFN, Sezione di To ino, 10125 To ino, ITALY
†Uni e sidad de G anada, 18071 G anada, SPAIN
∗∗ Uni e sidad de Se illa, 41080 Se illa, SPAIN
‡CTP, LNS and Depa men o Physics, MIT, Camb idge, MA 02139, USA
§Uni e sidad Complu ense de Mad id, 28040 Mad id, SPAIN
Abs ac . Two ela i is ic app oaches o cha ged-cu en quasielas ic neu ino-nucleus sca e ing a e illus a ed and com-
pa ed: one is phenomenological and based on he supe scaling beha io o elec on sca e ing da a and he o he elies on
he mic oscopic desc ip ion o nuclea dynamics in ela i is ic mean ield heo y. The ole o meson exchange cu en s in he
wo-pa icle wo-hole sec o is explo ed. The p edic ions o he models o di e en ial and o al c oss sec ions a e p esen ed
and compa ed wi h he MiniBooNE da a.
Keywo ds: Neu ino in e ac ions; Rela i is ic mean ield heo y; Meson exhcange cu en s; Supe scaling; Quasielas ic sca e ing.
PACS: 25.30.P , 13.15.+g, 24.10.J [P esen ed by M.B. Ba ba o a PANIC 2011, MIT, Camb idge, MA. July 2011]
The ecen MiniBooNE da a on muon neu ino cha ged-cu en quasielas ic (CCQE) sca e ing [1] ha e aised
an impo an deba e on he ole played by bo h nuclea and nucleonic ing edien s en e ing in he desc ip ion o he
eac ion. Unexpec edly, he c oss sec ion u ns ou o be subs an ially unde es ima ed by he Rela i is ic Fe mi Gas
(RFG) p edic ion, unless an unusually la ge ad hoc alue o he axial mass MA≃1.35 GeV/c2(as compa ed o he
s anda d alue MA≃1 GeV/c2) is employed in he dipole pa ame iza ion o he nucleon axial o m ac o . F om
compa ison wi h elec on sca e ing da a he RFG model is known, howe e , o be oo c ude o accoun o he nuclea
dynamics: he e o e his esul should be aken mo e as an indica ion o incomple eness o he heo e ical desc ip ion
o he nuclea many-body p oblem a he han as a ue indica ion o a la ge axial mass.
A he le el o he impulse app oxima ion (IA), a numbe o much mo e sophis ica ed desc ip ions o he nuclea
dynamics o he han he RFG also unde p edic he measu ed CCQE c oss sec ion (see, e.g., Re . [2] o a ull
lis o e e ences, ha we omi he e o loss o space). Possible explana ions o his puzzle ha e been p oposed in
he li e a u e, based ei he on mul inucleon knockou o on pa icula ea men s o inal s a e in e ac ions h ough
phenomenological op ical po en ials, indica ing ha con ibu ions beyond he simple IA play an impo an ole in QE
neu ino eac ions.
He e we summa ize he esul s o Re s. [3, 4, 5], whe e he p edic ions o he ollowing wo ela i is ic models we e
compa ed wi h each o he and wi h he MiniBooNE da a:
1. he Supe Scaling App oach (SuSA) including 2p2h Meson Exchange Cu en s (MEC);
2. he Rela i is ic Mean Field model (RMF).
Bo h models, al hough being a mo e ealis ic han he RFG, sha e wi h i he impo an p ope y o ea ing exac ly
he ela i is ic aspec s o he p oblem: hese canno be neglec ed o he kinema ics o MiniBooNE, whe e he neu ino
ene gy eaches alues as high as 3 GeV.
The “SuSA” app oach [3] is based on he assumed uni e sali y o he scaling unc ion o elec omagne ic and
weak in e ac ions. Analyses o inclusi e (e,e′)da a ha e demons a ed ha a ene gy ans e s below he QE peak
supe scaling is ul illed a he well: his means ha he educed c oss sec ion is la gely independen o he momen um
ans e (I-kind scaling) and nuclea a ge (II-kind scaling), when ep esen ed as a unc ion o he app op ia e scaling
a iable. F om hese analyses a phenomenological scaling unc ion, d ama ically di e en in size and shape om he
RFG pa abola, has been ex ac ed om he longi udinal QE elec on sca e ing esponse and used o p edic neu ino-
nucleus c oss sec ions by mul iplying i by he co esponding elemen a y weak c oss sec ions. The model ep oduces
by cons uc ion he longi udinal elec on sca e ing esponse a all kinema ics and o all nuclei. I s limi a ions come
om he assump ions on which he app oach is based, namely: 1) he equali y o he longi udinal and ans e se
SuSA+MEC
SuSA
RMF
0.8<cos θ < 0.9
Tµ(GeV)
d2σ/d cos θ/dTµ(10−39 cm2/GeV)
21.510.50
25
20
15
10
5
0
0.6<cos θ < 0.7
Tµ(GeV)
d2σ/d cos θ/dTµ(10−39 cm2/GeV)
1.210.80.60.40.20
20
15
10
5
0
0.4<cos θ < 0.5
Tµ(GeV)
d2σ/d cos θ/dTµ(10−39 cm2/GeV)
1.210.80.60.40.20
16
14
12
10
8
6
4
2
0
FIGURE 1. Flux-in eg a ed
νµ
-12C CCQE double di e en ial c oss sec ion pe a ge nucleon e alua ed in he SuSA model wi h
and wi hou inclusion o 2p2h MEC and in he RMF model and displayed e sus he muon kine ic ene gy T
µ
o a ious bins o he
muon sca e ing angle cos
θ
. He e and in he ollowing igu es he da a a e om MiniBooNE [1].
SuSA+MEC
SuSA
RMF
0.2< Tµ<0.3 GeV
cos θ
d2σ/d cos θ/dTµ(10−39 cm2/GeV)
0.80.60.40.20-0.2-0.4-0.6-0.8-1
14
12
10
8
6
4
2
0
0.4< Tµ<0.5 GeV
cos θ
d2σ/d cos θ/dTµ(10−39 cm2/GeV)
0.80.60.40.20-0.2-0.4
20
15
10
5
0
0.6< Tµ<0.7 GeV
cos θ
d2σ/d cos θ/dTµ(10−39 cm2/GeV)
0.90.80.70.60.50.40.30.20.1
30
25
20
15
10
5
0
FIGURE 2. Same as Fig. 1, bu now displayed e sus he sca e ing angle cos
θ
o a ious bins o T
µ
.
scaling unc ions (0-kind scaling), a p ope y iola ed by he L/T sepa a ed da a, which show a ans e se scaling
unc ion ypically la ge han he longi udinal one; 2) he equali y o he scaling unc ions in di e en isospin channels
(III-kind scaling), which allows o use he longi udinal elec on sca e ing da a (ha ing bo h isoscala and iso ec o
componen s) o p edic he pu ely iso ec o CC neu ino c oss sec ion.
The esul s o he SuSA model o he double di e en ial, single di e en ial and o al CCQE neu ino c oss sec ions
a e shown in Figs. 1-3, whe e hey appea o all below he da a o mos o he angle and ene gy bins. No e ha we do
no compa e wi h he mos o wa d angles (0.9<cos
θ
<1) since o such kinema ics oughly 1/2 o he c oss sec ion
has been p o ed [4] o a ise om e y low exci a ion ene gies (<50 MeV), whe e he c oss sec ion is domina ed by
collec i e exci a ions and any app oach based on IA is bound o ail.
To go beyond he SuSA app oach one mus ake in o accoun supe scaling iola ions, which occu mainly in he
ans e se channel a ene gies abo e he QE peak and a e associa ed o non-impulsi e e ec s, like inelas ic sca e ing
and meson-exchange cu en s. The la e a e wo-body cu en s ca ied by a i ual meson exchanged be ween wo
bound nucleons and can exci e bo h 1p1h and 2p2h s a es. In he 1p1h sec o , s udies o elec omagne ic (e,e′)p ocess
ha e shown ha he MEC, when combined wi h he co esponding co ela ions, which a e needed o p ese e gauge
in a iance, gi e a small con ibu ion o he QE c oss sec ion and can be neglec ed in i s app oxima ion. On he o he
hand in he 2p2h sec o he MEC a e known o gi e a signi ican posi i e con ibu ion o he (e,e′)c oss sec ion a
high ene gy ans e s, leading o a pa ial illing o he “dip” be ween he QE and ∆- esonance peaks. This egion is
ele an o he MiniBooNE expe imen , whe e “QE” e en s (namely wi h no eal pions in he inal s a e) can in ol e
ans e ed ene gies a beyond he QE peak, due o he la ge ene gy ange spanned by he neu ino lux.
In he esul s p esen ed in Figs. 1-3 we ha e used a ully ela i is ic model, de eloped o use in elec on sca e ing
s udies, whe e all he MEC many-body diag ams con aining wo pionic lines ha con ibu e o he elec omagne ic
2p2h ans e se esponse a e aken in o accoun . In o de o apply he model o neu ino sca e ing, we obse e ha
in lowes o de he 2p2h sec o is no di ec ly eachable o he axial- ec o ma ix elemen s. Hence a his o de he
MEC a ec only he ans e se pola ec o esponse. As shown in Figs. 1-3, he inclusion o 2p2h MEC in he SuSA
app oach yields la ge c oss sec ions and acco dingly be e ag eemen wi h he da a, bu heo y s ill lies below he da a
a la ge angles whe e he c oss sec ions a e smalle . I should be no ed, howe e , ha he p esen app oach s ill lacks
RMF
SuSA+MEC
SuSA
RFG
Tµ(GeV)
dσ/dTµ(10−39 cm2/GeV)
21.510.50
16
14
12
10
8
6
4
2
0
RMF
SuSA+MEC
SuSA
RFG
cos θ
dσ/d cos θ(10−39 cm2)
10.50-0.5-1
25
20
15
10
5
0
RMF
SuSA+MEC
SuSA
RFG
Eν(GeV)
σ(10−39 cm2)
21.510.50
16
14
12
10
8
6
4
2
0
FIGURE 3. Flux-a e aged
νµ
-12CCCQE c oss sec ion in eg a ed o e he sca e ing angle and displayed e sus he muon kine ic
ene gy (le panel), in eg a ed o e he muon kine ic ene gy and displayed e sus sca e ing angle (cen e panel), in eg a ed o e
he muon kine ic ene gy and sca e ing angle and displayed e sus he un olded neu ino ene gy ( igh panel). Beyond he models
desc ibed in he ex , he RFG esul is also shown o compa ison.
he con ibu ions om he co ela ion diag ams associa ed wi h he MEC which a e equi ed by gauge in a iance;
hese migh imp o e he ag eemen wi h he da a, as sugges ed by ecen esul s o inclusi e elec on sca e ing [6].
Be o e d awing de ini i econclusions on he anomalousaxial mass, i is impo an o explo e al e na i e app oaches
ha ha e been shown o be success ul in desc ibing inclusi e QE (e,e′)p ocesses. This is he case o he RMF
model, whe e a ully ela i is ic desc ip ion (kinema ics and dynamics) o he p ocess is inco po a ed, and inal s a e
in e ac ions a e aken in o accoun by using he same ela i is ic scala and ec o ene gy-independen po en ials
conside ed in he desc ip ion o he ini ial bound s a es. The RMF model applied o inclusi e QE (e,e′)p ocesses
has been shown o desc ibe he scaling beha iou and, in con as wi h mos o he nuclea models, o gi e ise o
a supe scaling unc ion wi h a signi ican asymme y, in comple e acco d wi h da a. Mo eo e , con a y o SuSA,
whe e scaling o he ze o h kind is assumed, he RMF model p o ides longi udinal and ans e se scaling unc ions
which di e by ypically 20%, he T one being la ge . When applied o he desc ip ion o CCQE neu ino-nucleus
c oss sec ions, he 0-kind scaling iola ion in oduced by he RMF app oach, as well as he di e en isospin cha ac e
shown by he elec omagne ic and weak nucleon o m ac o s, can lead o signi ican disc epancies be ween he esul s
p o ided by SuSA and RMF app oaches. This is illus a ed in Figs. 1, 2 and 3, whe e he di e ences be ween he
SuSA and RMF p edic ions a e especially isible in he double di e en ial c oss sec ions (Figs. 1 and 2), which a e
be e desc ibed by he RMF model, and end ins ead o be washed ou by he in eg a ion (Fig. 3).
Summa izing, we ha e applied wo ela i is ic models, SuSA and RMF, bo h able o desc ibe wi h good accu acy he
longi udinal (e,e′)da a, o CCQE neu ino sca e ing, inding ha bo h unde es ima e he MiniBooNE c oss sec ions:
al hough he RMF does be e han SuSA in ep oducing he shape o he double di e en ial c oss sec ions, he wo
app oaches p o ide almos iden ical esul s o he single-di e en ial and o al c oss sec ions. Al hough ou scope
he e is no o ex ac a alue o he axial mass o he nucleon, bu a he o unde s and which nuclea e ec s a e
e ec i ely accoun ed o by a la ge axial cu o pa ame e , le us men ion ha a bes i o he RMF and SuSA esul s
o he MiniBooNE expe imen al c oss sec ion gi es an e ec i e axial mass Me
A≃1.5 GeV/c2and alues in he ange
1.35 <Me
A<1.65 GeV/c2yield esul s compa ible wi h he MiniBooNE da a wi hin he expe imen al e o s.
The inclusion o 2p2h MEC con ibu ions in he SuSA app oach inc eases bo h he di e en ial and he in eg a ed
c oss sec ions and hus seems o imp o e he ag eemen wi h he da a, sugges ing ha he da a can be explained
wi hou he need o a la ge nucleon axial mass. Howe e , in he p esen scheme, mo e e ined calcula ions aking ca e
o co ela ion cu en s and MEC e ec s in he axial- ec o channel should be pe o med be o e inal conclusions can
be d awn. We e e he eade o Re s. [3, 4, 5] o u he de ails and esul s.
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