Table Of ContentThe Relativistic Green’s function model and charged-current inclusive
neutrino-nucleus scattering at T2K kinematics
Andrea Meucci and Carlotta Giusti
Dipartimento di Fisica, Università degli Studi di Pavia and
INFN, Sezione di Pavia, via A. Bassi 6, I-27100 Pavia, Italy
(Dated: January 15, 2015)
We compare the results of the relativistic Green’s function model with the experimental data
of the charged-current inclusive differential neutrino-nucleus cross sections published by the T2K
Collaboration. The model, which is able to describe both MINERνA and MiniBooNE charged-
current quasielastic scattering data, underpredictsthe inclusive T2K cross sections.
PACSnumbers: 25.30.Pt;13.15.+g;24.10.Jv
5
Keywords: Neutrinoscattering; Neutrino-induced reactions;Relativisticmodels
1
0
2
I. INTRODUCTION the charged-current(CC) [24, 32–36] and in the neutral-
n current (NC) [37–39] sector. Although different, the two
a
situations present many similar aspects and the exten-
J In recent years many Collaborations have presented
sion to neutrino scattering of the electron scattering for-
4 very interesting results of neutrino oscillations that aim
malism is straightforward. In the QE kinematic region,
1 at a precise determination of mass-squaredsplitting and
where the nuclear response to an electroweak probe is
] mixing angles in νµ disappearance and νe appearance dominated by single-nucleon scattering with direct one-
h measurements. Sincemodernexperiments,suchasMini-
nucleonemission,a reliabledescriptionofthe FSI effects
t BooNE, SciBooNE, ArgoNeuT, MINERνA, and T2K,
- between the ejected nucleon and the residual nucleus is
l are performed with detectors made of medium-heavy
c very important for the comparison with data. In the
nuclear targets, e.g, Carbon, Oxygen, Argon, or Iron,
u RGF model FSI are described in the inclusive scattering
a clear understanding of neutrino-nucleus interactions,
n consistently with the exclusive scattering by the same
[ whereallnucleareffectsarewellundercontrol,isrequired
complex optical potential (OP) and the components of
for a proper analysis of experimental data. The recent
1 the nuclear response are obtained in terms of matrix el-
progress, the questions and challenges in the physics of
v ements of the same type as the distorted wave impulse
3 neutrino cross sections are reviewed in [1–3]. approximationonesofthe exclusive(e,e′p) process,but
1 The first measurements of the charged-current involve eigenfunctions of the OP and of its Hermitian
2 quasielastic (CCQE) flux-averaged double-differential conjugate,wherethe opposite signofthe imaginarypart
3 12
0 νµ(ν¯µ) cross section on C in the few GeV region by gives in one case an absorption and in the other case a
. the MiniBooNE Collaboration [4, 5] have raised exten- gain of strength. In the exclusive scattering, where only
1 sive discussions that effects beyond the impulse approx- onechannelisconsidered,theimaginarypartgivesanab-
0
imation (IA) may play a significant role in this energy sorptionthataccountsforthefluxlosttootherchannels.
5
domain [6–17]. Multinucleon mechanisms, 2p-2h excita- In the inclusive scattering,where all elastic and inelastic
1
: tionsandmeson-exchangecurrents(MEC),andalsoRPA channels are included, the imaginary part redistributes
v correctionsappearessentialtodescribethedata. Models the flux in all the channels and in the sum over all the
i
X basedontheIA,whichmakeuseeitherofarealisticspec- channels the total flux is conserved. More details on the
tralfunctionobtainedwithinanonrelativisticframework model can be found in our previous papers [25–28, 32].
r
a [18,19]orofarelativisticIA(RIA)[20–23],generallyun- In other approaches based on the RIA, FSI are in-
derestimate the MiniBooNE CCQE cross sections. Only cluded in the emitted nucleon state with real poten-
the relativistic Green’s function (RGF) model is able to tials,eitherretainingonlythe realpartoftherelativistic
give a good description of the data [24]. Although un- energy-dependent complex OP, or using distorted waves
dermanyaspects basedonthe RIA,the RGFmodelcan obtained with the same relativistic energy-independent
recover contributions of final-state channels that are not mean-field potential considered in describing the initial
included in other models based on the RIA. These con- nucleon state (RMF) [23, 40, 41]. In the relativistic
tributionsarerecoveredbytheimaginarypartoftherel- plane-waveimpulseapproximation(RPWIA)FSIarene-
ativistic opticalpotential that is used in the RGF model glected.
to describe final-state interactions (FSI).
The results of these different descriptions of FSI have
TheRGFmodelwasoriginallydevelopedwithinanon- beencomparedin[30]forthe inclusiveQEelectronscat-
relativistic [25, 26] and then a relativistic [27, 28] frame- tering, in [33] for the CCQE neutrino scattering, and
work to describe FSI in the inclusive quasielastic (QE) in [24, 38] with the CCQE and NC elastic MiniBooNE
electron scattering. The model was successfully tested data. Both RGF and RMF models can describe success-
against electron scattering data [25, 26, 29–31] and it fully electron scattering data and their related scaling
waslaterextendedtoneutrino-nucleusscattering,bothin functions. Bothmodelsareabletoprovideasatisfactory
2
descriptionoftheCCQEMINERνAdata[36,41]. Inthe
case of the MiniBooNE CCQE data, both models repro-
20
duce the shape of the experimental cross sections, but ]
2m 18
onlythe RGFgivescrosssectionsofthesamemagnitude 9c νµ CCQE
as the experimental ones without the need to increase 30 16
the world average value of the axial mass MA [24, 34]. [1 14
The largerRGF crosssections are due to the translation
σ 12
to the inclusive strength of the overall effect of inelastic
channels, including rescattering and some multinucleon 10
contributions, that are recovered in the model by the 8 RGF EDAI
imaginary part of the relativistic OP and that are not 6 RGF DEM
included in the RMF and in other models based on the RPWIA
4
IA. MiniBooNE
The optical potential is a powerfultool to recoverand 2 T2K
includeimportantcontributions. Theavailabilityofphe- 0
0.5 1 1.5 2 2.5
nomenological relativistic OP’s, obtained through a fit Eν [GeV]
of elastic proton-nucleus scattering data, is essential to
make RGF calculations feasible, but the use of a phe-
nomenological OP does not allow us to disentangle the Figure 1. Total CCQE νµ-12C cross sections per target
roleofaspecificinelasticchannelandcanthereforeintro- neutron versus the neutrino energy. The experimental data
duce uncertainties and ambiguities in the interpretation are from MiniBooNE [4] and T2K [43].
of the RGF results. The imaginary part can recover,
to some extent, contributions beyond direct one-nucleon
emission,suchas,forinstance, rescatteringofthe outgo- whether the effects of the inelastic channels recoveredin
ing nucleon and some multinucleon processes,which can theRGFmodelbytherelativisticOPgiveordonotgive
beincludedinCCQEmeasurements,buttheRGFmodel enough strength to reproduce these data.
isbasedontheuseofaone-bodynuclearcurrentanddoes I
not containMEC mechanisms that in other models have
been found to be significant. On the other hand, the
imaginary part of the OP can include pion-absorption II. RESULTS
and pion-emission processes, that should have already
been subtractedin the MiniBooNE analysis. It has been
In all the calculations presented in this work we have
written in [3] that the good agreement of the RGF re-
adopted the standard value for the nucleon axial mass
sults with the MiniBooNE data “should be interpreted 2
M = 1.03 GeV/c . The bound nucleon states are
A
with care” and that “it would be very interesting to con-
taken as self-consistent Dirac-Hartree solutions derived
front the RGF results with the fully CC-inclusive data”
within a relativistic mean field approach using a La-
[3]. The comparison with the fully CC-inclusive data,
grangian containing σ, ω, and ρ mesons [46–50]. We
which include also pion production, is the motivation of
have used two different parametrizations for the rela-
the present paper. 12
tivistic OP of C that is adopted in our RGF cal-
TheT2Kcollaborationhasrecentlypublished[42]new culations: the Energy-Dependent and A-Independent
results on the CC-inclusive double differential cross sec- EDAI (where the E represents the energy and the A
12
tion on C, which includes also pion production, and of the atomic number) OP of [51], and the more recent
the CCQE cross section [43]. The T2K νµ energy range Democratic (DEM) phenomenological OP of [52]. The
is the same as for MiniBooNE, the beam peaks at ∼600 EDAI OP is a single-nucleus parametrization, which is
MeV,similartothatofMiniBooNE,butitissignificantly constructed to better reproduce the elastic proton-12C
narrower and receives almost neglible contributions for phenomenology, whereas the DEM parametrization is a
energies larger than 1 GeV. In view of these differences, global parametrization, which depends on the atomic
the analysis of T2K data is another useful and indepen- number A and is obtained through a fit to more than
dent test for a theoretical description. 200datasetsofelasticproton-nucleusscatteringdataon
InthispaperwecomparetheresultsoftheRGFmodel a wide range of nuclei and that is not limited to doubly
with the CC-inclusive ν and ν T2K cross sections on closed shell nuclei. In comparison with electron scatter-
µ e
12C [42–45]. As a firststep, we compareour results with ing data, the DEM parametrization produces in general
the CCQE ν −12C crosssections measuredby the Mini- good results for doubly magic nuclei and less good but
µ
BooNE and T2K collaborations, for which there is con- stillacceptableresults fornucleiwitha number ofnucle-
sistency betweenthe twoexperiments within the current ons far from the magic numbers [31, 53].
12
statistical and systematic uncertainties [43]. Then, we In Fig. 1 we show our calculated CCQE ν - C cross
µ
consider the flux-averagedCC-inclusive ν and ν differ- sections per target neutron as a function of the neutrino
µ e
entialcrosssectionsfromT2Kwiththeaimtoinvestigate energycomparedwiththe MiniBooNE[4]andT2Kdata
3
] 10×10-42 ] 14×10-42
eV ν T2K RGF EDAI eV
M 9 RGF DEM M
2/cm 8 RPWIA 2/cm 12
) [ 7 T2K data ) [ 10 0.84≤cosϑµ≤0.90
µ µ
ϑs 6 0.00≤cosϑµ≤0.84 ϑs
o o 8
c c
dPµ 5 dPµ
d d 6
( 4 (
/ /
σ σ
2d 3 2d 4
2
2
1
0 0
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
Pµ [GeV/c] Pµ [GeV/c]
] ×10-42 ] 10×10-42
V 12 V
e e
M M 9
/ /
2cm 10 0.90≤cosϑµ≤0.94 2cm 8
[ [
) ) 7
ϑµ 8 ϑµ
s s 6
o o
c c
dPµ 6 dPµ 5
(d (d 4 0.94≤cosϑµ≤1
/ /
σ2d 4 σ2d 3
2
2
1
0 0
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1
Pµ [GeV/c] Pµ [GeV/c]
Figure 2. Flux-averaged CC-inclusive double differential νµ-12C cross sections per target nucleon as a function of the muon
momentum. The data are from T2K [42].
[43]. Althoughtheresultsofthesetwoindependentmea- in general results somewhat lower than the RPWIA one
surements can be consistently compared in the entire and therefore lower than the data.
rangeofenergies,withtheonlyexceptionoftheT2Kda- In Fig. 2 we present the CC-inclusive double differen-
tum in the energy bin 1−1.5 GeV, we observe that the tial ν -12C cross section d2σ/(dP dcosϑ ) as a function
µ µ µ
average magnitude of the MiniBooNE dataset is larger oftheoutgoingmuonmomentumtransferP forfourdif-
µ
than that of the T2K one. The differences between the ferent bins in the scattering angle. The calculated cross
RGF-EDAIandRGF-DEMresultsaresizable. Thesedif- sections are flux-averagedover the T2K ν flux [54] and
µ
ferences are due to the different imaginary parts of the compared with the experimental data of [42].
twoOPs,particularlyfortheenergiesconsideredinkine- The RPWIA results in Fig. 2 are approximately 50%
matics with the lowest scattering angles and the largest lowerthanthedata. AlsotheRGFresultsunderestimate
kinetic energies of the muon [24]. The RGF-EDAI cross thedata. BothRGF-EDAIandRGF-DEMcrosssections
sectionislargerthantheRGF-DEMone,inbetteragree- are generally lower than the data, although within the
ment with the MiniBooNE data and in agreement with error bars for low values of P and large angular bins.
µ
bothMiniBooNEandT2Kcrosssectionswithintheerror A satisfactory agreement with the data is obtained with
bars in the entire energy range of the data. The RGF- the model of [55], which includes np-nh excitations and
DEM cross section underpredicts the MiniBooNE data single-pionproduction. IntheRGFmodeltheimaginary
at low Eν and it is in better agreement with the T2K partoftheOPcanincludetheexcitationofmultinucleon
data. The RPWIA cross section, which is also shown in channels. We cannot exclude that it can contain some
the figure for a comparison, is similar to the RGF-DEM contributionduetopionemission,wecannotdisentangle
one. We note that other models based on the IA give and evaluate the relevance of this contribution, but in
4
of the total cross section arises from excitation energies
0.1×10-36 below ∼ 50 MeV. If we consider that in the RGF calcu-
n] T2K νe data lationscollectiveeffectsareneglected,itisnotsurprising
o0.09
e RGF EDAI that our results are significantly lower than the data in
l
uc0.08 RGF DEM the forward angular bin.
n
2/cm0.07 RPWIA 12CIndFiffiger.en3tiaanldcrFosigs.se4cttihonescaarlceudlaistpeldayCedC-ainscalufusinvcetiνoen-
ϑ [0.06 of the electron momentum and scattering angle, respec-
os0.05 tively,andcomparedwiththeT2Kdataof[45]. Forsake
c
/d0.04 p > 0.550 GeV/c of simplicity the calculations have been performed only
σ
2d0.03 cosϑ > 0.72 for the reduced phase-space (momentum > 550 MeV/c
and cosϑ > 0.72) and not for the full phase-space of
0.02 T2K. The T2K ν beam peaks at ∼ 500 MeV and, in
e
0.01 contrast to the νµ one, extends to energies larger than 1
GeV. Thus the CC ν cross section at T2K may receive
e
0.75 0.8 0.85 0.9 0.95 1
cosϑ alsohigher-ordercontributionslike two-pionproduction.
Also in this case our calculations are significantly lower
thanthedata. TheRPWIAandthetwoRGFresultsare
Figure 3. Flux-averaged CC-inclusive νe-12C differential very similar in Fig. 3, where the three curves practically
cross section per nucleon as a function of cosθ. Only elec- overlap,while in Fig. 4 the differences are small but vis-
trons corresponding to p>0.550 GeV/c and cosθ>0.72 are ible. The fact that collective effects are not included in
considered [45]. The T2K data can befound at [44]. our model is one of the reasons of the large underesti-
mation of the experimental data at smaller angle in Fig.
3.
×10-39
4
} T2K νe data
n]3.5 RGF EDAI III. CONCLUSIONS
o
e RGF DEM
l
uc 3 RPWIA In this paper we have comparedthe predictions of the
n
) 2.5 RGF model with the CCQE and CC-inclusive νµ and νe
c
/ scattering T2K data. The RGF model is able to give
V
e 2 a satisfactory description of inclusive QE electron scat-
G
(
[ tering cross sections and of the CCQE MiniBooNE and
/1.5
2m MINERνA data, both for νµ and ν¯µ scattering, without
c
{ 1 theneedtoincreasethestandardvalueoftheaxialmass.
/dp0.5 p > 0.550 GeV/c The RGF results are usually larger than the results of
σ2 cosϑ > 0.72 othermodelsbasedontheIA.IntheRGFmodelFSIare
d
0 described using a complex energy-dependent relativistic
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
p [GeV/c] OP whose imaginary part includes the overall effect of
the inelastic channels, which give different contributions
at different energies, and makes the RGF results sensi-
Figure 4. Flux-averaged CC-inclusive νe-12C differential tivetothekinematicconditionsofthecalculations. With
cross section per target nucleon as a function of the elec- a complex OP the model can include all the available
tron momentum. Only electrons corresponding to p > 0.550 final-statechannelsandnotonlydirectone-nucleonemis-
GeV/c and cosθ > 0.72 are considered [45]. The T2K data sionprocesses. Theimportantroleofcontributionsother
can befound at [44]. than direct one-nucleon emission has been confirmed by
different and independent models in the case of CCQE
MiniBooNE cross sections, but the same conclusion is
any case the results in Fig. 2 indicate that this is not doubtful in the case of MINERνA data.
enough to reproduce CC-inclusive data. TheRGFmodelcanincludecontributionsoffinal-state
Wenotethatforthemostforwardangularbin(0.94≤ channels like, e.g., rescattering processes of the nucleon
cosϑ ≤ 1.00) the RGF results are significantly smaller in its way out of the nucleus, non-nucleonic ∆ excita-
µ
than the RPWIA ones. In this small bin the trans- tions, which may arise during nucleon propagation,with
verse and charge-isovector contributions are suppressed or without real-pion production, and also some multin-
and the longitudinal response gives the main contribu- ucleon processes. These contributions are not incorpo-
tion to the cross section. In addition, it has been shown rated explicitely in the model with a microscopic cal-
in[56]thatmodels basedonquasi-freescatteringcannot culation: they can be recovered, to some extent, at a
describe properly this kinematic situation where ∼ 1/2 phenomenologicallevelbytheimaginarypartofthephe-
5
nomenologicalOP which is adopted in the RGF calcula- disentanglethepion-productioncontributionthatcanbe
tion. TheuseofaphenomenologicalOPdoesnotallowus included in the phenomenologicalOP, this is not enough
to disentangleandevaluatethe roleofa specific reaction to reproduce the CC-inclusive T2K data. If we consider
process. Different available parametrizations of the phe- thattheRGFmodelwasdevelopedtodescribeFSIinthe
nomenologicalrelativisticOPcanintroduceuncertainties inclusive QE scattering and that it is able to give a rea-
in the predictions of the model. The determination of a sonably good agreement with QE electron and CCQE
theoretical OP, which fulfills the dispersion relations in neutrino-scattering data, the present comparison with
the whole energy regionof interest, would be very useful T2K data can be interpreted as an indication that the
to reduce the theoretical uncertainties. pion-productionchannel gives only a minor contribution
In the RGF model the nuclear response is written in to the RGF results.
terms of the single-particle optical-model Green’s func- The full and explicit inclusion of multinucleon chan-
tion. This result is obtainedretainingonly the one-body nels is required to successfully reproduce CC-inclusive
part of the nuclear current. The inclusion of two-body data. Other models, which explicitly include multinu-
MECwouldrequireanextendedmodelbasedonthetwo- cleon emission channels, obtain a satisfactory agreement
particle Green’s function, whose evaluation represents a with the T2K data [55].
very hard task.
The imaginary part of the OP can include pion-
productionprocessesthat shouldhavealreadybeen sub-
ACKNOWLEDGMENTS
tracted in the analysis of CCQE data. In this paper we
have shown that the fully CC-inclusive T2K cross sec-
tions, which include pion production, are clearly under- WethankMarcoMartiniforinterestingandusefuldis-
estimated by the RGF calculations. Even if we cannot cussions.
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