Inelastic Neutrino-Nucleus Interactions within the Spectral Function Formalism

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by  Erica Vagnoni

Inelastic Neutrino-Nucleus Interactions within the Spectral Function Formalism
Erica Vagnoni,1,∗ Omar Benhar,2,3,† and Davide Meloni1,‡
1INFN and Dipartimento di Matematica e Fisica, Universita` Roma Tre, I-00146 Roma, Italy
2INFN and Dipartimento di Fisica, “Sapienza” Universita` di Roma, I-00185 Roma, Italy
3Center for Neutrino Physics, Virginia Tech, Blacksburg, Virginia 24061, USA
(Dated: January 9, 2017)
Wereporttheresultsofastudyofneutrino-carboninteractionsatbeamenergiesrangingbetween
few hundreds MeV and few tens of GeV, carried out within the framework of the impulse approx-
imation using a realistic spectral function. The contributions of quasi elastic scattering, resonance
production and deep inelastic scattering—consistently obtained, for first time, from a model based
on a realistic description of the nuclear ground state—are compared and analyzed.
PACSnumbers: 13.15.+g,25.30.Pt,24.10.Cn
7
1 Over the past decade, the broad effort aimed at im- isthat,atlargemomentumtransfer,nuclearinteractions
0 provingtheoversimplifieddescriptionofneutrino-nucleus reduce to the incoherent sum of elementary processes in-
2
interactions based on the Relativistic Fermi Gas Model volving individual nucleons. As a consequence, nuclear

n (RFGM), has led to the development of a number of and weak interaction dynamics are decoupled, and—to
a more advanced approaches, capable of providing a fairly the extent to which the corresponding neutrino-nucleon
J
accurate description of part of the available data [1– cross section can be measured using hydrogen and deu-

6 13]. Most existing studies are restricted to the charged- terium targets—the formalism based on the IA can be

current quasi-elastic (CCQE) sector, which makes the usedtodescribeneutrino-nucleusscatteringinanychan-
]
h dominant contribution to the neutrino-carbon cross sec- nels.
t tionsmeasuredbytheMiniBooNECollaborationusinga In this Letter, we report the results of the first
-
l neutrino flux of mean energy (cid:104)E (cid:105) ∼ 800 MeV [14, 15]. comprehensive study of the neutrino-carbon cross sec-
c ν
However, theinterpretationofthesignalsrelevanttoon- tion—including CCQE interactions, resonance produc-
u
n goingandfutureexperimentsathigherneutrinoenergies, tionandDIS—carriedoutwithintheIAusingarealistic
[ such as MINERνA [16], NOνA [17] and DUNE [18] re- spectral function.

quires accurate predictions of the nuclear cross sections The differential cross section of the process
1
in inelastic channels. For example, at E (cid:46) 2 GeV, cor-
v ν ν +12C→µ−+X , (1)
8 respondingtothepeakenergyoftheNOνAoscillatedνe µ
1 events,thetotalcrosssectionisexpectedtoreceivecom- in which a neutrino of four-momentum k =(E ,k) scat-
ν
7 parablecontributionsfromCCQE,resonanceproduction
ters off a carbon nucleus producing a muon of four-
1 and deep inelastic scattering (DIS) processes [17]. momentum k(cid:48) = (E ,k(cid:48)), with the nuclear final state
0 µ
Theoretical calculations of the neutrino-nucleus cross
being undetected, can be written in the form
.
1 section involve three main elements—the target initial
0 and final states and the nuclear weak current—whose d2σ G2 V2 |k(cid:48)|
7 consistent description in the broad kinematical region dΩ dE = 1F6πu2d |k| LµνWAµν , (2)
1 µ µ
correspondingtoneutrinoenergiesbetweenfewhundreds
v: MeV and few GeV poses severe difficulties. The initial where Ωµ is the solid angle specified by the direction
i statecanbesafelymodeledwithinthenonrelativisticap- of the vector k(cid:48), GF is the Fermi constant and Vud is
X
the element of the Cabibbo-Kobayashi-Maskawa (CKM)
proximation,independentofkinematics,whereasatlarge
ar momentum transfer q=k−k(cid:48), the same approximation matrix coupling u and d quarks.
The tensor L is completely determined by lepton
cannot be used to describe either the nuclear final state, µν
kinematics, whereas the nuclear response to weak inter-
comprisingatleastoneparticlecarryingmomentum∼q,
actions is described by the tensor
orthenuclearcurrentoperator,whichdependsexplicitly
on momentum transfer. Wµν =(cid:88)(cid:104)0|Jµ†|X(cid:105)(cid:104)X|Jν|0(cid:105)δ(4)(p +q−p ), (3)
The impulse approximation (IA)—a detailed deriva- A A A 0 X
tion of which can be found in Refs. [1, 19]—provides a X
conceptual framework ideally suited to circumvent the where|0(cid:105)and|X(cid:105)denotethetargetgroundstateandthe
above problem. The main tenet underlying this scheme hadronic final state, carrying four momenta p and p ,
0 X
respectively, Jµ is the nuclear weak current and the sum
A
is extended to all hadronic final states.
The formalism of IA is based on the factorization
∗ vagnoni@fis.uniroma3.it ansatz, which amounts to replacing [1, 19]
† [email protected]
‡ meloni@fis.uniroma3.it |X(cid:105)−→|x,p(cid:105)⊗|R,pR(cid:105) , (4)
2
where |x,p(cid:105) is the hadronic state produced at the elec- scattering, charged pion electro-production off nucleons
tromagneticvertexwithmomentump, while |R,p (cid:105) de- and muon capture on the proton [25, 26].
R
scribes the recoiling nucleus, carrying momentum p .
R The results reported in this Letter have been obtained
It follows that Eq.(2) reduces to the simple and trans-
using the state-of-the-art parametrization of the vector
parent form
form factors of Ref. [23], and the dipole parametrization
d2σIA =(cid:90) d3kdEP(k,E) d2σνN , (5) of the axial-vector form factor with MA =1.03 GeV.
dΩ dE dΩ dE Conceptually, the generalization to describe resonance
µ µ µ µ
production, driven by elementary processes such as
where the elementary νN cross section—written in
terms of five structure functions W —describes the in-
i
teraction between the incoming neutrino and a mov- ν +p→µ−+∆++ →µ−+p+π+ , (9)
µ
ing bound nucleon, while the nuclear spectral function
P(k,E)—trivially related to the imaginary part of the
two-point Green’s function [20, 21]—yields the probabil- where ∆++ denotes the P (1232) nucleon resonance,
33
ityofremovinganucleonofmomentumkfromthetarget onlyrequiresminorchanges[2]. Inthiscase,theνN cross
groundstate,leavingtheresidualnucleuswithexcitation section involves the matrix elements of the weak current
energy E. describingthenucleon-resonancetransitions. Asaconse-
Equation(5)clearlyillustratesthepotentialofthefor- quence, the structure functions—which can still be writ-
malism based on the factorization ansatz of Eq. (4). Be- tenintermsofphenomenologicalvectorandaxial-vector
cause the spectral function is an intrinsic property of form factors—depend on both Q2 and W2, the squared
the target ground state, it can be obtained from non invariant mass of the state |x,p(cid:105). In addition, the en-
relativistic nuclear many-body theory, and employed to ergy conserving δ-function is replaced by a Breit-Wigner
carry out calculations of the nuclear cross section in any factor, accounting for the finite width of the resonance.
channels,providedthecorrespondingν-nucleoncrosssec-
Besides the prominent P (1232) state, providing the
tion in vacuum is known. The elementary cross section 33
largest contribution to the cross section, we have taken
can be treated using the relativistic formalism without
into account the three isospin 1/2 states—D (1520),
any problems, nuclear medium effects being taken into 13
P (1440), and S (1535)—comprised in the so-called
account through the replacement [19] 11 11
second resonance region. The numerical results have
been obtained using the parametrization of the struc-
ω =E −E →ω =ω+M −E −E , (6)
ν µ (cid:101) A p R
ture functions described in Refs [27–29]. Within this
whereM isthetargetmass,whileE andE denotethe approach, the vector form factors are constrained by
A p R
energies of the hadronic state produced at the neutrino electroproduction data, while the axial couplings are ex-
interaction vertex and of the recoiling nucleus, respec- tracted from the measured resonance decay rates, ex-
tively. Equation (6) allows to account for the fact that, ploiting the Partially Conserved Axial Current (PCAC)
even though the weak interaction involves an individual hypothesis.
nucleon, a fraction of the energy transfer in the scatter-
From the observational point of view, Deep Inelastic
ing process goes into excitation energy of the spectator
Scattering (DIS) is associated with hadronic final states
system.
comprising more than one pion.
In the CCQE channel, characterized by the absence of
pions in the final state, the relevant elementary interac- In principle, the three nucleon structure functions de-
tion process is termining the νN cross section in the DIS regime—W1,
W andW —maybeobtainedcombiningmeasuredneu-
2 3
νµ+n→µ−+p , (7) trino and antineutrino scattering cross sections. How-
ever, as the available structure functions have been ex-
and the nucleon structure functions—involving a δ-
tracted from nuclear cross sections (see, e.g., Ref. [30]),
functionconstrainingthemassofthehadronicfinalstate
their use in ab initio theoretical studies, aimed at identi-
to be equal to the proton mass, m —can be written in
p fying nuclear effects, entails obvious conceptual difficul-
termsofthenucleonvectorandaxial-vectorformfactors.
ties.
The former have been accurately measured in electron-
proton and electron-deuteron experiments [22, 23], while An alternative approach, allowing to obtain the struc-
the latter is usually written in the dipole form turefunctionsdescribingDISonisolatednucleons,canbe
developedwithintheconceptualframeworkofthequark-
F (Q2)=g (cid:0)1+Q2/M2(cid:1)−2 , (8) parton model, exploiting the large database of accurate
A A A
DIS data collected using charged lepton beams and hy-
with Q2 = −(k −k(cid:48))2. The axial-vector coupling con- drogenanddeuterontargets(see,e.g.,Ref.[31]). Within
stant, g = − 1.2761+14, is known from neutron β- this scheme, the function FνN = ωW , where W is the
A −17 2 2 2
decay [24], while the value of the axial mass is deter- structure function of an isoscalar nucleon, can be simply
mined from elastic neutrino- and antineutrino-nucleon relatedtothecorrespondingstructurefunctionextracted
3
from electron scattering data, FeN through1
2
18
FνN(Q2,x)= FeN(Q2,x) , (10)
2 5 2
where x is the Bjorken scaling variable. In addition, the
relation
xFνN(Q2,x)=x [ u (Q2,x)+d (Q2,x) ] , (11)
3 v v
where FνN = ωW and u and d denote the valence
3 3 v v
quark distributions, implies
xFνN(Q2,x)=FeN(Q2,x) (12)
3 2
−2x [u(Q2,x)+d(Q2,x)] .
Using Eqs. (10)-(12) and the Callan-Gross relation [31],
linking FνN = mW to FνN, one can readily obtain all
1 1 2
the relevant weak structure functions from the existing
parametrizations of the measured electromagnetic struc-
ture function and of the antiquark distributions u and d
(see, e.g., Ref .[32]). Alternatively, the quark and anti-
quark distributions can be also used to obtain the struc-
ture function FeN from FIG. 1. Double-differential cross section of the scattering
2 process ν +12C → µ− +X at fixed muon emission angle
µ
5 θ =30deg,andbeamenergiesE =1GeV(A)and1.5GeV
FeN(Q2,x)=x [ u(Q2,x)+u(Q2,x) µ ν
2 18 (B), displayed as a function of ω = Eν −Eµ. The dashed,
dot-dash and dotted lines correspond to CCQE scattering,
+d(Q2,x)+d(Q2,x) ] . (13)
resonance production and DIS, respectively. The sum of the
three contributions is represented by the full line.
In this work, we have used Eqs.(10)-(13) and the par-
ton distributions of Ref. [32], which are available for
Q2 ≥ Q2 = 0.8 GeV2. At lower values of Q2,
min and inelastic sectors [36, 37]. Figure 1 shows the results
wehaveassumedthepartondistributionstobethesame
oftheextensionoftheseanalysestothecaseofneutrino-
as at Q2 =Q2 .
min carbon interactions. The calculations have been carried
Note that the above procedure rests on the tenet, un-
out using the spectral function of Ref. [37] and setting
derlying the IA scheme, that the elementary neutrino-
the muon emission angle to θ = 30 deg. Comparison
nucleon interaction is not affected by the presence of the µ
between panels (A) and (B), corresponding to E = 1
nuclear medium, the effects of which are accounted for ν
and 1.5 GeV, respectively, illustrates how the relative
withthesubstitutionofEq.(6). Whilethisassumptionis
weightofthedifferentreactionmechanismschangeswith
stronglysupportedbyelectron-nucleusscatteringdatain
increasing neutrino energy.
thequasielasticchannel,showingnoevidenceofmedium
modificationsofthenucleonvectorformfactors,ithasto The Q2-distributions, obtained from the double-
be mentioned that analyses of neutrino DIS data are of- differential cross section of Fig. 2 by integrating over
ten carried out within a conceptually different approach, cosθµ, are displayed in Fig. 2. At both Eν = 1 and 1.5
allowing for medium modifications of either the nucleon GeV, the full dσ/dQ2, corresponding to the solid line,
structurefunctions[33,34],orofthepartondistributions exhibits a pronounced maximum at Q2 (cid:46)0.2 GeV2.
entering their definitions [35]. Finally, integration over Q2 yields the total cross sec-
Theresultsofcalculationsoftheelectron-nucleuscross tion, σ, whose behavior as a function of the neutrino en-
sectionshaveprovidedampleevidencethattheapproach ergy Eν is illustrated in Fig. 3. Panels (B) and (A) show
based on IA and the spectral function formalism, involv- σ and the ratio σ/Eν, respectively, as well as the con-
ingnoadjustableparameters,iscapabletodeliveraquan- tributions corresponding to the CCQE, resonance pro-
titative description of the double-differential electron- duction, and DIS channels. It is apparent that, while at
nucleus cross sections—measured at fixed beam energy Eν (cid:46)0.8GeVCCQEinteractionsdominate,theinelastic
and electron scattering angle—in both the qualsielastic cross section rapidly increases with energy. At Eν ≈1.3
GeV, the contributions arising from the three reaction
channels turn out to be about the same.
For comparison, in panel (B) we also report, as di-
1 For the sake of simplicity, here, and in what follows, we will amonds, the ν -carbon total cross section measured by
µ
ignorethecontributionsofsandcquarks. the NOMAD collaboration [38]. It turns out that, while
4
FIG.3. Totalcrosssectionofthereactionν +12C →µ−+X
FIG.2. Q2-distributionoftheprocessν +12C →µ−+X at µ
µ as a function of neutrino energy. The dashed, dot-dash and
fixedneutrinoenergyE =1Gev(A)and1.5GeV(B).The
ν dottedlinesofpanel(A)representthecontributionsofCCQE,
meaning of the lines is the same as in Fig. 1.
resonanceproductionandDISprocesses. Panel(B)showsthe
E -dependenceoftheratioσ/E . Themeaningofthedashed,
ν ν
dot-dash and dotted lines is the same as in panel (A). The
the energy-dependence of the data at E (cid:38) 10 GeV is
ν full line corresponds to the sum of the three contributions.
well reproduced by our prediction of the DIS contribu- Diamonds and squares represent the data of Refs.[38, 39],
tion, represented by the dotted line, the results of the respectively.
full calculation, corresponding to the solid line, sizably
exceed the measured cross section. In view of the fact
thattheCCQEcrosssectionobtainedfromtheNOMAD Theoreticalstudiesoftheelectron-carboncrosssection
data of Ref. [39], shown by the open squares, turns out provideconvincingevidencethatMECcontributionscan
to be in close agreement with the results of our calcula- be consistently included in the spectral function formal-
tions, this discrepancy is likely to be ascribed to double ism, through a generalization of the factorization ansatz
counting between resonance production and DIS contri- [41,42],whileFSIcorrectionsinthequasielasticchannel
butions, which are very hard to identify in a truly model are understood at quantitative level [43, 44]. Note, how-
independent fashion. ever,thatFSIdo notaffecttheCCQEtotalcrosssection
In conclusion, we have carried out a calculation based shown in Fig. 3.
on the IA and the spectral function formalism, in which Theemergingpicturesuggeststhattheapproachbased
the contributions of CCQE processes, resonance produc- on spectral functions strongly constrained by both inclu-
tion and DIS are taken into account, for the first time, sive and exclusive electron-nucleus scattering data, such
in a fully consistent fashion. The present implementa- asthosederivedinRef.[37],hasthepotentialtodescribe
tion of the factorization scheme does not take into ac- both elastic and inelastic neutrino-nucleus interactions
count the occurrence of processes involving more than at the level of accuracy required to face the outstanding
onenucleon—suchasthoseinwhichtheneutrinocouples challenges of neutrino physics.
to nuclear Meson-Exchange-Currents (MEC)—as well as
final state interactions (FSI) between the nucleon par-
ticipating in the weak interaction process and the spec-
ACKNOWLEDGMENTS
tator particles. The inclusion of MEC contributions
to the neutrino-nucleus cross section is believed to be
needed to explain the flux-integrated double-differential The authors are grateful to Artur M. Ankowski and
cross section measured by the MiniBooNE collabora- Camillo Mariani for countless illuminating discussions.
tion [7, 10, 13], while the understanding of FSI is re- The work of E.V. and D.M. was supported by INFN
quired, e.g., to determine the nuclear transparency to through grant WSIP. The work of O.B. was supported
the hadrons produced at the interactions vertex [40]. by INFN through grant MANYBODY.
5
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