Assessing the theory-data tension in neutrino-induced charged pion production the eect of nal-state nucleon distortion A. Nikolakopoulos1R. Gonz alez-Jim enez2N. Jachowicz3and J. M. Ud as2

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Assessing the theory-data tension in neutrino-induced charged pion production: the

eﬀect of ﬁnal-state nucleon distortion

A. Nikolakopoulos,1, ∗R. Gonz´alez-Jim´enez,2N. Jachowicz,3and J. M. Ud´ıas2

1Theoretical Physics Department, Fermilab, Batavia IL 60510, USA

2Grupo de F´ısica Nuclear, Departamento de Estructura de la Materia,

F´ısica T´ermica y Electr´onica, Facultad de Ciencias F´ısicas,

Universidad Complutense de Madrid and IPARCOS, CEI Moncloa, Madrid 28040, Spain

3Department of Physics and Astronomy, Ghent University, B-9000 Gent, Belgium

Background: Pion production on nuclei constitutes a signiﬁcant part of the total cross section in experiments

involving few-GeV neutrinos. Combined analyses of data on deuterium and heavier nuclei points to tensions be-

tween the bubble chamber data and the data of the MINERνA experiment, which are often ascribed to unspeciﬁed

nuclear eﬀects.

Purpose: In experimental analysis use is made of approximate treatments of nuclear dynamics, usually in a

Fermi gas approach with classical treatments of the reaction mechanism, and ﬁts are often performed by simply

rescaling cross sections. To understand the origin of these tensions, check the validity of approximations, and

to further advance the description of neutrino pion production on nuclei, a microscopic quantum mechanical

framework is needed to compute nuclear matrix elements.

Method: We use the local approximation to the relativistic distorted wave impulse approximation (RDWIA) to

calculate nuclear matrix elements. We include the distortion of wave functions of the ﬁnal-state nucleon in a real

energy-dependent potential. We compare results with and without distortion. To perform this comparison under

conditions relevant to neutrino experiments, we compute cross sections for the MINERνA and T2K charged pion

production datasets.

Results: The inclusion of nucleon distortion leads to a reduction of the cross section up to 10%, but to no

signiﬁcant change in shape of the ﬂux-averaged cross sections. Results with and without distortion compare

favorably to experimental data, with the exception of the low-Q2MINERνAπ+data. We point out that hydrogen

target data from BEBC is also overpredicted at low-Q2, and the data-model discrepancy is similar in shape and

magnitude as what is found in comparison to MINERνA data.

Conclusions: Including nucleon distortion alone cannot explain the overprediction of low-Q2cross sections mea-

sured by MINERνA. The similar overprediction of BEBC data on hydrogen means that it is impossible to ascribe

this discrepancy solely to a nuclear eﬀect. Axial form factors might not be constrained in a satisfactory way by

the ANL/BNL data alone. Axial couplings and their Q2dependence should ideally be derived from more precise

data on hydrogen and deuterium. Nuclear matrix elements should be tested with e.g. electron scattering data

for which nucleon level physics is better constrained.

I. INTRODUCTION

In neutrino experiments such as DUNE and NOνA in-

elastic interactions with the nucleon constitutes a large

part of the total event rate [1,2]. Experiments such as

T2K, MiniBooNE, MicroBooNE and the short-baseline

program at Fermilab are more sensitive to quasi-elastic

scattering and meson-exchange currents. However these

experiments are still sensitive to inelastic processes,

mostly single pion production (SPP) in the ∆ region [3–

6]. Experiments often adopt a signal deﬁnition in which

events where a pion is present are rejected, but this proce-

dure does not not fully remove inelastic contributions [7–

9]. The inelastic contribution to a 0πsignal is often

labeled ‘pion-absorption’, but may also consist of pions

below detector thresholds, or non-pionic decays of res-

onances. Comparisons to electron scattering data show

that this region is problematic in the commonly used GE-

NIE event generator, due to double counting of resonance

∗anikolak@fnal.gov

and DIS descriptions [9,10].

Over the past couple of years measurements of cross

sections for neutrino-induced pion production on nu-

clei have been performed. Notably the MINERνA

experiment has reported cross sections for both neu-

trino and antineutrino production of both neutral and

charged pions on carbon [11–17]. Analysis of the dif-

ferent MINERνA datasets in Ref. [18] highlights that

there are tensions between the diﬀerent datasets and with

the ANL/BNL hydrogen and deuterium bubble cham-

ber data. A notable ad-hoc modiﬁcation introduced to

improve model-data agreement is a large suppression of

the cross section at low four-momentum transfer Q2, al-

though the amount of suppression needed varies with the

dataset. A similar modiﬁcation was used by the NOνA

collaboration in ﬁtting their measurements of the inclu-

sive cross section [1]. These modiﬁcations are applied to

the interactions on nuclei only, motivated by the analogy

to quasielastic interactions. In that case it is indeed es-

tablished that inclusion of Pauli blocking, the distortion

of the ﬁnal state, long and short-range correlations yield

smaller cross sections at low-Q2compared to equivalent

calculations that omit these eﬀects [19,20]. It is impor-

arXiv:2210.12144v1 [nucl-th] 21 Oct 2022

tant to actually compute the eﬀect of these mechanisms,

based on established microscopic approaches, which can

be validated with diﬀerent interaction mechanisms. Em-

pirical ﬁts based on a limited amount of data, the inter-

pretation of which might be highly model dependent, is

unsatisfactory and can bias analyses.

This becomes especially important as the amplitudes

for neutrino SPP on nucleons, that serve as input to nu-

clear models, are poorly constrained [21–23]. While theo-

retical approaches for electroweak amplitudes may diﬀer

in sophistication, models for electroweak SPP amplitudes

that span the delta region and beyond rely on the analysis

of precise data of diﬀerential cross sections, which is avail-

able only for electromagnetic pion production. For the

neutrino-induced processes such high-quality data does

not exist, and one has to rely on total and ﬂux-averaged

cross sections obtained in deuterium and hydrogen bub-

ble chambers [24–26]. As such it is hard, if not impossi-

ble, to identify whether model-data discrepancies in cur-

rent ﬂux-averaged neutrino cross sections are due to any

speciﬁc mechanism, be it nucleus or nucleon speciﬁc.

The relativistic distorted wave impulse approximation

(RDWIA) provides the framework of choice to compute

nuclear eﬀects in direct pion production reactions, where

a pion is produced through a single nucleon operator.

The RDWIA and non-relativistic DWIA, have proven to

be successful tools for interpreting and describing nu-

cleon knockout in electromagnetic interactions [27–29]

and photon and electron-induced pion production [30–

32]. The relativistic formulation of the RDWIA has the

advantage that the full Dirac structure of the single nu-

cleon operators is retained. One does not need a non-

relativistic reduction, which is usually obtained by cut-

ting oﬀ the expansion of a current between free-nucleon

spinors in orders of p/MN[33,34]. Additionally, rela-

tivistic models for the nucleus based on density functional

theory such as the relativistic mean ﬁeld (RMF) [35,36],

although they are phenomenological, provide an excellent

description of many nuclear phenomena with relatively

few free parameters [35,37,38].

Matrix elements for SPP in the RDWIA require as

input a single-nucleon operator, wavefunctions for the

initial and ﬁnal-state nucleon and pion. We currently

cannot provide all these ingredients, notably a potential

for the distorted pion wavefunctions which is suitable for

the experimental signature in neutrino experiments is not

available. In this work we isolate and tackle speciﬁcally

the eﬀect of distortion of the ﬁnal-state nucleon. The

outgoing nucleon wavefunctions are obtained with the

energy-dependent RMF (EDRMF) potential introduced

in Ref. [19]. The main appeal of this treatment is that

initial and ﬁnal state potentials are identical for low nu-

cleon energies, this leads naturally to Pauli blocking [39],

and the conservation of the Dirac current [40]. This

consistency combined with the energy dependence leads

to an excellent description of (e, e0) data from small to

large momentum transfers in the quasielastic region [41].

We use the common approximation where the asymp-

k’

FIG. 1. kinematics of single pion production on the nucleus.

totic value of nucleon momentum is used in evaluating

the single-nucleon operator [30,31]. We perform direct

comparisons of the RDWIA with the equivalent calcula-

tion in the relativistic plane wave impulse approximation

(RPWIA) in order to asses the eﬀect and importance of

nucleon distortion.

This paper is structured as follows: In Section II we

describe the RDWIA formalism, ﬁrst the general expres-

sion of the cross section and kinematics in the RMF are

discussed. In Sec. II B we start from the general form

of the nuclear current in RDWIA and discuss the ap-

proximations made that lead to the local RDWIA and

the RPWIA expressions. Finally in Sec. II C we discuss

the single-nucleon operator used in our calculations, and

provide a comparison of the isovector contribution with

more advanced analyses of electron scattering data. The

comparison of RPWIA and RDWIA results with each

other and with experimental data is shown in Sec. III.

Finally we brieﬂy illustrate the uncertain status of the

delta coupling and present our conclusions in Sec. IV.

II. SINGLE PION PRODUCTION ON THE

NUCLEUS

We consider the process of single pion production oﬀ

a nucleus Athrough a charged current interaction where

a single gauge boson with four-momentum

qµ=kµ−k0µ,(1)

is exchanged between the lepton vertex and the hadron

system. As usual we denote the squared four momentum

transfer Q2=−q2as positive. We describe a ”direct”

reaction in which the pion is produced oﬀ a single nucleon

which is excited to the continuum. The kinematics of

nucleon and pion are shown in Fig. 1, the full process

satisﬁes the conservation of the four-momentum

qµ+Pµ

A=kµ

π+kµ

N+Pµ

B,(2)

here, PAis the initial nucleus, and PBrepresents the

undetected residual hadronic system. The cross section

for the charged current process can be written as

d9σ(E)

dE0dΩdEπdΩπdENdΩN

=G2

Fcos2θc

2(2π)8

MNkπkNMB

δ(ω+MA−Eπ−EN−EB)Lµν Hµν ,(3)

as Q2is negligible compared to the squared mass of the

exchanged W-boson. The lepton tensor, when the initial

lepton mass is neglected, is given by

Lµν =kµk0ν+kνk0µ−gµν kαk0α−ihµναβ kαk0

β(4)

with hthe initial leptons helicity, i.e. −1 and +1 for neu-

trino and antineutrino reactions respectively. The depen-

dence on the nuclear and hadron dynamics is captured in

the values EBwhich the residual system may take and in

the hadron tensor Hµν . These are respectively described

in the following subsections.

A. Kinematics

For the following discussion we assume that the four

vectors of the initial lepton kµand the nucleus PAare

ﬁxed. The cross section then depends on 8 independent

kinematic variables e.g. (E0,cos θl, Eπ,Ωπ,ΩN, MB).

One sees that one produces an (unobserved) residual sys-

tem with invariant mass MBand kinetic energy

TB=qM2

B+ (pB)2−MB,(5)

with the momentum of the system (or equivalently the

missing momentum pm) given by

−pm≡pB=q−kπ−kN.(6)

Energy conservation means that

ω+MA=MB+TB+Eπ+EN.(7)

Given a value for MBthe above equations may be solved

for EN, the explicit expression is given in Ref. [42].

In a direct knockout reaction we probe missing mo-

menta of the order of the Fermi momentum kFsuch that

TB.k2

2(A−1)MN

,(8)

when the mass of the residual system is of the order of

(A−1)MN. The kinetic energy becomes negligible for

large nuclei. We may hence simplify by neglecting the

small recoil energy of the residual system, TBin Eq. (7).

Model-dependence comes in when considering the val-

ues that MBcan take. We consider the interaction with

a nucleon within the RMF shell model, for which the

initial-state nucleus is described as a Slater determinant

of single-particle orbitals. The single particle states are

characterized by isospin projection, a principal quantum

number n, relativistic angular momentum κand the pro-

jection of total angular momentum mj. Due to spher-

ical symmetry the mjstates for ﬁxed n, κ are energy-

degenerate. The single-particle energy En,κ is the en-

ergy needed to excite a nucleon from an (n, κ) state to

the continuum. As we neglect TBwe thus have

TN=ω−Tπ−En,κ.(9)

This approach implies that the residual system is left

in an internally excited state with invariant mass MB=

En,κ +MA−MN. Neglecting the nuclear recoil, summing

over the possible (n, κ) states and integrating over the

outgoing nucleon energy the cross section becomes

d8σ(E)

dE0dΩdEπdΩπdΩN

=G2

Fcos2θc

2 (2π)8

EMNkπLµν X

kN,κHµν

n,κ.

(10)

The total angular momentum of a single particle state

is j=|κ| − 1/2 and the orbital angular momentum is

given by

l=κfor κ > 0

−(κ+ 1) for κ < 0(11)

For the ground state in carbon we assume the lowest

energy proton and neutron orbitals are fully occupied.

These are the s1/2shell (n= 1, κ =−1) and p3/2shell

(n= 1, κ =−2).

This shell model treatment is known to be a ﬁrst ap-

proximation to the missing-energy distribution. Exper-

imental data obtained in coincidence experiments, e.g.

(e, e0p), show that the discrete states obtain a width, cen-

tered around the expected mean-ﬁeld values [43]. This

can be implemented empirically by smearing the shell-

model states with a Gaussian or Lorentzian [44–46]. Ad-

ditionally, correlations beyond the mean ﬁeld, both long-

and short-range, lead to a partial occupation of the shell

model states. This may be taken into account by includ-

ing spectroscopic factors [44,47]. The missing strength

then appears at larger missing energies and momenta [48]

and can be taken into account in factorized approaches,

notably in Ref. [49] for electron-induced pion production.

B. Nucleon distortion in the RDWIA

The hadron tensor for the interaction with a shell with

angular momentum κis

Hµν

κ=Nκ

2j+ 1

mj,sN

[Jµ(mj, sN, Qµ, kµ

N, kµ

π)]†Jν(mj, sN, Qµ, kµ

N, kµ

π)

(12)

where sNand mjare the projections of the spin of the

ﬁnal-state nucleon and the angular momentum of the

bound state, we average over the 2j+ 1 possible states

for mj. The occupation of the state is Nκ, which within

the shell model picture is also 2j+ 1.

To make approximations to the hadron current clear, it

is instructive to ﬁrst consider the most general expression

for the single-nucleon current in momentum space

Jν=1

(2π)3/2Zdp0

NZdp0

πψsN(p0

N,kN)φ∗(p0

π,kπ)

Oν(qµ, p0

N, p0

π, p0

m)ψmj

κ(p0

m=p0

N+p0

π−q).(13)

Here ψsN(p0

N,kN) and φ∗(p0

π,kπ) are the outgoing nu-

cleon and pion wavefunctions. These have ﬁxed asymp-

totic momenta kNand kπrespectively, and are func-

tions of the primed momenta p0

Nand p0

π. The bound

state wavefunction is ψκ, and the projections of spin and

angular momentum of the bound state are denoted by

superscripts sNand mjrespectively.

The outgoing nucleon and pion are energy eigen-

states, their asymptotic momenta ksatisfy the relation

E2=k2+M2. In the nuclear interior the particles are

not momentum-eigenstates, a momentum operator act-

ing on the wavefunctions yields the primed momenta. In

Eq. (13) the transition operator is hence a function of

the primed momenta, these are related by momentum

conservation q+p0

m=p0

π+p0

The full expression of Eq. (13) is computationally ex-

pensive, one has to compute nκ(2j+ 1) 6-dimensional

integrals for every point in the 8 dimensional phase

space. Moreover singularities can arise in the pole terms,

e.g. in pion exchange contributions, as in general p026=

k2=E2−M2. The singularities can be avoided by

using as energy of outgoing nucleon and pion in the

operator the energy derived from the primed momenta

E→E02=p02+M[31]. In this work we make an approx-

imation to the full expression by replacing the primed

momenta in the operator (but only in the operator) by

their asymptotic values

Oµ(q, p0

m, p0

N, p0

π)→ Oµ(q, pm, kN, kπ),(14)

with pm≡kπ+kN−q. We refer to this as the asymp-

totic approximation, sometimes called the local approxi-

mation [31], as it removes derivatives with respect to the

coordinates in r-space expressions. We are aware of a

limited number of calculations that use the full expres-

sion of Eq. (13), these where performed for fully exclusive

conditions for knockout from a speciﬁc shell in photon-

induced reactions [31,32]. These works seem to imply

that the full calculation leads to a slightly more smeared

out cross section, in particular for angular distributions,

compared to the asymptotic approximation. We plan to

utilize the full calculation, and investigate ambiguities in

the transition operator in future works.

With Eq. (14) one can reduce the expression of Eq. (13)

to a single 3-dimensional integral. If one writes the

momentum-space wavefunctions as the Fourier transform

of their coordinate space counterparts one can immedi-

ately perform the integrals over the primed momenta,

and momentum conservation leads to

Jν=Zdreiq·rφ∗(r,kπ)ψsN(r,kN)Oνψmj

κ(r).(15)

We will in this work always treat the pion as a plane

wave, the ﬁnal expression for the current in the RDWIA

used in this work is then given by

Jν=Zdrei(q−kπ)·rψsN(r,kN)Oνψmj

κ(r).(16)

It is clear that Eq. (15) allows to include a distorted

pion wavefunction without signiﬁcant increase of compu-

tational cost compared to Eq. (16). Instead, the problem

is to ﬁnd a suitable potential to treat the pion wave-

function. Empirical and microscopic optical potentials

derived from ﬁts to pion-nucleus elastic scattering are

available, but in these treatments any inelastic rescatter-

ing of the pion leads to a loss of ﬂux. Such potentials are

suitable to describe the process under exclusive condi-

tions, in which the missing energy of the residual system

is restricted to a narrow region. In neutrino experiments

such conditions are not met, instead certain rescattering

mechanisms (e.g. absorption) will lead to a reduction of

the signal, others (e.g. secondary nucleon knockout) do

not, and charge exchange reactions migrate pions from

one production channel to another. As such, an opti-

cal potential informed by elastic pion-nucleus scattering

would underestimate the total rates in the context of

neutrino scattering experiments. Contrary to this, the

results in which the pion is described by a plane wave in

most cases should be expected to overestimate rates in

neutrino experiments.

The nucleon states are scattering solutions of the

Dirac equation with the real Energy-Dependent RMF

(EDRMF) potential introduced in Ref. [19]. The

EDRMF potential is constructed by scaling the RMF

scalar and vector potentials as a function of the nucleon

energy, thereby implementing a softening of the poten-

tial with increasing energy. At low energies the potential

is identical to the RMF potential used to compute the

bound state wavefunctions, thereby the orthogonality of

initial and ﬁnal states is ensured when the momentum

content of bound and scattering state could potentially

overlap. This ensures speciﬁcally that the Pauli principle

is satisﬁed [39]. At high energies, cross sections computed

with the EDRMF are similar to those obtained with the

real part of optical potentials constrained by nucleon-

nucleus scattering as shown in Ref. [41]. We consider

the EDRMF potential suitable to describe interactions

in which the outgoing nucleon remains undetected (or

is not used in the deﬁnition of the experimental signal),

as is the case in neutrino induced pion production cross

sections that we consider.

To gauge the eﬀect of nucleon distortion we compare

the RDWIA calculations with the relativistic plane-wave

impulse approximation (RPWIA) where the ﬁnal state

nucleon is described by a plane wave. In this case the

asymptotic evaluation of the operator, Eq. (14), is of

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Assessingthetheory-datatensioninneutrino-inducedchargedpionproduction:theeectofnal-statenucleondistortionA.Nikolakopoulos,1,R.Gonzalez-Jimenez,2N.Jachowicz,3andJ.M.Udas21TheoreticalPhysicsDepartment,Fermilab,BataviaIL60510,USA2GrupodeFsicaNuclear,DepartamentodeEstructuradelaMateria,FsicaT...

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Assessing the theory-data tension in neutrino-induced charged pion production the eect of nal-state nucleon distortion A. Nikolakopoulos1R. Gonz alez-Jim enez2N. Jachowicz3and J. M. Ud as2

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