FERMILAB-PUB-22-745-T Form factor and model dependence in neutrino-nucleus cross section predictions Daniel Simons1Noah Steinberg2Alessandro Lovato3 4Yannick Meurice1Noemi Rocco2and Michael Wagman2

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FERMILAB-PUB-22-745-T

Form factor and model dependence in neutrino-nucleus cross section predictions

Daniel Simons,1Noah Steinberg,2Alessandro Lovato,3, 4 Yannick Meurice,1Noemi Rocco,2and Michael Wagman2

1Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA

2Theoretical Physics Department, Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60410, USA

3Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

4INFN-TIFPA Trento Institute of Fundamental Physics and Applications, Via Sommarive, 14, 38123 Trento, Italy

To achieve its design goals, the next generation of neutrino-oscillation accelerator experiments re-

quires percent-level predictions of neutrino-nucleus cross sections supplemented by robust estimates

of the theoretical uncertainties involved. The latter arise from both approximations in solving the

nuclear many-body problem and in the determination of the single- and few-nucleon quantities taken

as input by many-body methods. To quantify both types of uncertainty, we compute ﬂux-averaged

double-diﬀerential cross sections using the Green’s function Monte Carlo and spectral function meth-

ods as well as diﬀerent parameterizations of the nucleon axial form factors based on either deuterium

bubble-chamber data or lattice quantum chromodynamics calculations. The cross-section results are

compared with available experimental data from the MiniBooNE and T2K collaborations. We also

discuss the uncertainties associated with N→∆ transition form factors that enter the two-body

current operator. We quantify the relations between neutrino-nucleus cross section and nucleon

form factor uncertainties. These relations enable us to determine the form factor precision targets

required to achieve a given cross-section precision.

I. INTRODUCTION

The study of neutrino processes is driven by deep ques-

tions whose answers may profoundly change our under-

standing of physics. In particular, given that neutri-

nos have mass and mix, the accelerator-neutrino pro-

gram aims at precisely measuring the parameters that

characterize their oscillations, investigating the possible

existence of a fourth neutrino ﬂavor, and testing addi-

tional Beyond the Standard Model scenarios. The suc-

cess of these experiments rests on our ability to compute

neutrino-nucleus cross sections with quantiﬁed theoreti-

cal uncertainties [1]. The latter presently yield a sizable

contribution to the total error budget of oscillation pa-

rameters [2,3].

On the other hand, accelerator neutrino experiments

allow us access to aspects of nuclear dynamics that would

otherwise be diﬃcult to probe at electron-scattering fa-

cilities. The chief example is the axial form factor of

the nucleon. Its experimental determination dates back

to bubble chamber experiments carried out in the 70s

and 80s [4–8] and to electroweak single pion production

measured at ANL and BNL in the 80s [9–12]. A simple

dipole parameterization with axial mass MA∼1 GeV re-

produces single-nucleon data. In contrast, a larger value

MA∼1.2 GeV was required in order to make relativis-

tic Fermi Gas predictions for the neutrino-12C cross sec-

tions compatible with MiniBooNE data [13]. Note that

the spectral-function formalism, which includes the vast

majority of nuclear correlations, requires an even larger

value of MAto reproduce experimental data [14].

This apparent inconsistency appeared to be solved

by models that include two-body current operators, as

they can reproduce MiniBooNE and T2K data with

MA∼1 GeV [15–21]. However, these models are

somewhat simpliﬁed, as they are based on a mean-ﬁeld

description of nuclear dynamics. As in the one-body cur-

rent case, it may well be that when nuclear correlations

are accounted for there is room for larger values of MA.

This is conﬁrmed by recent Green’s function Monte Carlo

(GFMC) results, which provide a full account of nuclear

correlations and two-body current eﬀects [22]. However,

the non-relativistic nature of the GFMC hampers its ap-

plicability to neutrino accelerator experiments where the

neutrino ﬂux energy is of the order of a few GeV. The

spectral function (SF) method, relying on the factoriza-

tion of the ﬁnal hadronic state, allows for the inclusion of

relativistic eﬀects and exclusive channels while retaining

most of the important eﬀects coming from multi-nucleon

dynamics. The factorization scheme has been extended

to include one- and two-body current operators in a con-

sistent fashion as well as pion production amplitudes and

validated against electron scattering data [23,24]. The

spectral function of light and medium mass nuclei has

been recently computed exploiting quantum Monte Carlo

(QMC) techniques and shares with the GFMC the same

description of nuclear dynamics [25,26]. Comparing

the results obtained for lepton-nucleus scattering using

these two diﬀerent approaches enables a precise quan-

tiﬁcation of the uncertainties inherent to factorization

schemes that need to be accounted for when assessing

the total error of the theoretical calculations in neutrino

oscillation analysis.

There has been signiﬁcant recent progress in lattice

quantum chromodynamics (LQCD) calculations of nu-

cleon axial and vector form factors [27–35], which have

now been performed using approximately physical val-

ues of the quark masses as well as multiple lattice spac-

ings and volumes to enable continuum, inﬁnite-volume

extrapolations [30,34,35]. Vector form factor results

show encouraging consistency between LQCD and ex-

perimental determinations [29,34,36]. Excited-state ef-

fects involving Nπ states have been identiﬁed as a sig-

niﬁcant source of systematic uncertainty in LQCD cal-

arXiv:2210.02455v2 [hep-ph] 4 Nov 2022

culations of axial form factors [28,30,34,37–39]. More

sophisticated analysis strategies employed in recent cal-

culations [28,30,34], as well as variational methods used

to study ππ [40–44], Nπ [45–49], and N N [50–52] scatter-

ing and Nπ transition form factors [39] that enable Nπ

and other excited-state eﬀects to be explicitly subtracted

from future nucleon elastic form factor calculations, pro-

vide paths towards better quantifying and reducing these

challenging systematic uncertainties. Although signiﬁ-

cant future progress is expected, LQCD calculations of

nucleon axial form factors have progressed to a point

where it is important to understand the phenomenolog-

ical impact of current results and quantitatively estab-

lish what form factor precision is required to achieve the

cross-section precision needs of current and future neu-

trino oscillation experiments [53–55].

Current LQCD results predict a signiﬁcantly larger ax-

ial form factor at Q2∼1 GeV2than determinations from

deuterium bubble chamber data.1LQCD form factor

results were incorporated into the GENIE [58] neutrino

event generator and shown to lead to signiﬁcant diﬀer-

ences in neutrino-nucleus cross sections in Ref. [54]. The

interplay of these form factor changes with other aspects

of GENIE remains to be studied, and in particular the

“empirical MEC model” of two-body currents in GENIE

and tuned versions of this model used in neutrino oscil-

lation experiments [59–61] are obtained by ﬁtting cross-

section predictions to data while assuming a particular

axial form factor model for quasi-elastic contributions.

It is important to understand axial form factor eﬀects on

the event generators used to analyze current experiments,

but using such a data-driven approach to two-body cur-

rent contributions makes it diﬃcult to disentangle the ef-

fects and uncertainties associated with nucleon form fac-

tors from those associated with two-body currents and

other nuclear eﬀects.

The theoretically well-deﬁned separation between one-

and two-body currents in the GFMC and SF results of

this work, in conjunction with the study of nuclear model

dependence enabled by comparing two realistic many-

body methods, enables robust quantiﬁcation of the ef-

fects and uncertainties of nucleon axial form factors and

other single-nucleon inputs to neutrino-nucleus cross sec-

tions. Parameterizing the nucleon axial form factor using

the model-independent zexpansion [62–65] allows un-

certainty quantiﬁcation to be performed without intro-

ducing any dependence on assumptions about the shape

of the axial form factor. Relations between neutrino-

nucleus cross section uncertainties and zexpansion pa-

rameter uncertainties are determined below and used to

construct quantitative precision targets for future LQCD

calculations of nucleon axial form factors. The eﬀects

of N→∆ transition form factors uncertainties are also

studied.

1A similar trend has recently been obtained within continuum

Schwinger function methods [56,57].

This paper is organized as follows. Section II intro-

duces the formalism necessary to compute the inclusive

neutrino-nucleus cross section. Section III is dedicated to

the determination of the axial form factor and its uncer-

tainties estimation. Results are presented in Section IV,

while Section Vprovides concluding remarks and dis-

cusses the outlook for future work.

II. METHODS

In the one-boson exchange approximation, the

neutrino-nucleus double diﬀerential cross sections can be

written in the form

dσ

dTµdcos θµν/¯ν

=G2

2π

2Eν

[LCC RCC + 2LCLRCL

+LLLRLL +LTRT±2LT0RT0],

(1)

where the ±sign corresponds to ν(¯ν) scattering. We take

G=GFcos θc, with GF= 1.1664×10−5GeV−2[66] and

cos θc= 0.9740 [67] for charged current processes studied

here. The Lifactors only depend upon the kinematics of

the initial and ﬁnal state leptons, while the electroweak

response functions Riare deﬁned as linear combinations

of diﬀerent components of the hadronic response tensor

Rµν

Rµν =X

fh0|Jµ†|fihf|Jν|0iδ(E0+ω−Ef).(2)

where |0iis the nuclear ground state, |fiare all possible

ﬁnal states of the A-nucleon system and Jµis the nuclear

current operator. The response tensor contains all the in-

formation on the structure of the nuclear target, deﬁned

in terms of a sum over all transitions from the ground

state to any ﬁnal state, including states with additional

hadrons. In this work we consider the one- and two-body

contributions to the nuclear current operator

Jµ=X

jµ

i+X

j>i

jµ

ij .(3)

and we will exclusively focus on ﬁnal states involving only

nucleons, ignoring single-nucleon excitation processes.

Exact expressions for Liand Rican be found in Ref. [68].

Computing the electroweak response functions in the

energy regime relevant to oscillation experiments is a

highly-nontrivial task. Existing approaches diﬀer based

on approximation schemes and kinematic regimes of ap-

plicability. In this work, we will consider two diﬀerent

methods that model the initial target state in a simi-

lar fashion, but diﬀer in the treatment of the interaction

vertex and ﬁnal-state interactions: the Green’s function

Monte Carlo, and the spectral function approaches.

A. Green’s Function Monte Carlo

The response function in Eq. (2) is expressed in the

energy domain. It can be related to a matrix element of

electroweak currents in the time domain,

Rαβ(q, ω) = Zdtei(ω+E0)th0|J†

αe−iHt Jβ|0i,(4)

where a completeness relation has been inserted to carry

out the sum over the possible ﬁnal states of the A-nucleon

system. The non-relativistic nuclear Hamiltonian Hused

in these calculations is comprised of the Argonne v18 [69]

(AV18) nucleon-nucleon potentials plus the Illinois-7 [70]

(IL7) three-nucleon force. The Green’s function Monte

Carlo method uses an imaginary-time projection to distill

information on the energy dependence of the response

functions. In particular, the imaginary-time response is

obtained by replacing the real time entering Eq. (4) with

the imaginary time t→ −iτ. The Laplace transform

of the energy-dependent response, called the Euclidean

response function, is deﬁned by

Eαβ(q, τ ) = Z∞

ωth

dω e−ωτ Rαβ (q, ω),(5)

where ωth is the inelastic threshold. Bayesian tech-

niques, most notably maximum entropy [71,72], can be

used to retrieve the energy dependence of the response

functions from their Laplace transform. Following this

strategy, GFMC calculations have been successfully per-

formed to obtain the inclusive electroweak response func-

tion of light nuclei while fully retaining the complexity of

many-body correlations and associated electroweak cur-

rents [22,73,74]. Recently, algorithms based on artiﬁ-

cial neural networks have been developed to invert the

Laplace transform [75] and show better accuracy that

maximum entropy, especially in the low-energy transfer

region.

The ﬁve response functions entering the CC cross sec-

tion have been calculated with GFMC methods for mo-

mentum transfers in the range (100–700) MeV in steps

of 100 MeV. In order to compute the ﬂux-averaged cross

sections presented in Sec. IV that involve larger momen-

tum transfers, we capitalize on the scaling features of the

GFMC response functions discussed in Refs. [22,76,77]

to interpolate and safely extrapolate them at momentum

transfers |q|>700 MeV. The charge-changing weak cur-

rent is the sum of vector and axial components

Jµ=Jµ

V−Jµ

J±=Jx±iJy.(6)

These operators comprise both one- and two-body con-

tributions, whose expressions are reported in Ref. [68].

Here, we focus on the axial and pseudo-scalar one-body

contributions given by

A=−FA

2mN

τ±σ· {k, eiq·r}

jA=−FAτ±hσeiq·r−1

4m2

Nσ{k2, eiq·r}−{(σ·k)k, eiq·r}

−1

2σ·q{k, eiq·r} − 1

2q{(σ·k), eiq·r}+iq×keiq·ri

jµ

P=FP

2m2

τ±qµσ·qeiq·r,(7)

where mNis the nucleon mass and {·,·} denotes the anti-

commutator. The parametrization of the axial form fac-

tor FAhas historically involved models with and without

over-constraining theoretical assumptions and will be a

focus of this work in Sec. III A. For the pseudo-scalar

form factor, PCAC and pion-pole dominance relations

valid in leading order chiral perturbation theory [78,79]

can be used to relate FPto FAas

FP=2FAm2

π+Q2,(8)

where Q2=−q2=−(q0)2+q2and mπis the pion mass.

This relation is consistent with the pseudoscalar form

factor constraints extracted from precise measurements

of the muon-capture rate on hydrogen and 3He [80]. It

is also consistent with current LQCD results [30,34].

Precise future LQCD calculations will further test Eq. (8)

and provide independent determinations of FPwithout

theoretical assumptions on its relation to FA.

The isovector two-body currents can be separated into

model-dependent and -independent terms. Those asso-

ciated with pion exchange are denoted as model inde-

pendent; they are constrained by the continuity equa-

tion and do not contain any free parameters, since they

are determined directly from the nucleon-nucleon inter-

action. The expressions adopted in this work and re-

ported in Refs. [68,81] are consistent with the semi-

phenomenological AV18 interaction. The diagrams asso-

ciated with the intermediate excitation of a ∆ isobar are

purely transverse and generally referred to as model de-

pendent. Their expression can be found in Refs. [68,81],

it is important to mention that in order to be used in

the GFMC calculations, the static ∆ approximation has

to be adopted, i.e. the kinetic-energy contributions in

the denominator of the ∆ propagator are neglected. The

impact of this approximations will be further discussed

when comparing the GFMC and SF results. Among the

axial two-body current operators, the leading terms of

pionic range are those associated with excitation of ∆-

isobar resonances, despite being less relevant the two-

body operators associated with axial πNN contact in-

teractions are also accounted for, we refer the reader to

Refs [68,81] for a more detailed discussion of these con-

tributions.

B. Extended Factorization Scheme

At large values of momentum transfer (|q|&400

MeV), a factorization scheme can be used in which the

neutrino-nucleus scattering is approximated as an inco-

herent sum of scatterings with individual nucleons, and

the struck nucleon system is decoupled from the rest of

the ﬁnal state spectator system. In the quasielastic re-

gion, the dominant reaction mechanism is single nucleon

knockout. In this case, the ﬁnal state is factorized ac-

cording to

|fi=|p0i⊗|ΨA−1

f,pA−1i,(9)

where |p0iis the ﬁnal state nucleon produced at the

vertex, assumed to be in a plane wave state, and

|ΨA−1

f,pA−1idescribes the residual system, carrying mo-

mentum pA−1.

Inserting this factorization ansatz as well as a single-

nucleon completeness relation gives the matrix element

of the 1 body current operator as

hf|jµ|0i → X

[hΨA−1

f|⊗hk|]|0ihp|X

jµ

i|ki,(10)

where p=q+k. This ﬁrst piece of the matrix element

explicitly does not depend on the momentum transfer

and so can be computed using techniques in nuclear many

body theory. The second piece can be straightforwardly

computed once the currents jµ

iare speciﬁed as the single

nucleon states are just free Dirac spinors. Substituting

the last equation into Eq. (2), and exploiting momentum

conservation at the single nucleon vertex, allows us to

rewrite the one body contribution to the response tensor

Rµν

1b(q, ω) = Zd3k

(2π)3dEPh(k, E)m2

e(k)e(k+q)

×X

ihk|jµ†

i|k+qihk+q|jν

i|ki

×δ(˜ω+e(k)−e(p)),

(11)

where e(k) = pm2

N+k2. The factors mN/e(k) and

mN/e(k+q) are included to account for the covariant

normalization of the four spinors in the matrix elements

of the relativistic current. The energy transfer has been

replaced by ˜ω=ω−mN+E−e(k) to account for some of

the initial energy transfer going into the residual nuclear

system. Finally, the calculation of the one-nucleon spec-

tral function Ph(k, E) provides the probability of remov-

ing a nucleon with momentum kand leaving the residual

nucleus with an excitation energy E. Its derivation using

QMC techniques is discussed in Sec. II C.

The relativistic current used in Eq. (11) is given as a

sum of vector and axial terms which can be written as

jµ

V=F1γµ+iσµν qν

2mN

jµ

A=γµγ5FA+qµγ5FP

(12)

In the above, F1and F2are usual isovector and isoscalar

form factors which themselves are functions of the proton

and neutron electric and magnetic form factors. These

are highly constrained by electron scattering and we

adopt the Kelly parameterization in this work [82]. The

expression of the pseudo scalar form factor in terms of

the axial one is the same of Eq. (8). The details on the

diﬀerent parameterizations adopted for FAare given in

Secs. III.

To treat amplitudes involving two-nucleon currents,

the factorization ansatz of Eq. (9) can be generalized as

|ψA

fi→|pp0ia⊗ |ψA−2

fi.(13)

where |p p0ia=|p p0i−|p0piis the anti-symmetrized state

of two-plane waves with momentum pand p0. Following

the work presented in Refs. [23,83,84], the pure two-

body current component of the response tensor can be

written as

Rµν

2b (q, ω) = V

2ZdE d3k

(2π)3

d3k0

(2π)3

d3p

(2π)3

e(k)e(k0)e(p)e(p0)

×PNM

h(k,k0, E)X

ij hk k0|jµ

ij †|p p0iahp p0|jν

ij |k k0i

×δ(ω−E+ 2mN−e(p)−e(p0)) .(14)

In the above equation, the normalization volume for the

nuclear wave functions V=ρ/A with ρ= 3π2k3

F/2 de-

pends on the Fermi momentum of the nucleus, which

for 12C is taken to be kF= 225 MeV. The factor 1/2

accounts for the double counting that occurs in this no-

tation (the product of the two direct terms is equal to

the one of the two exchange terms). Diﬀerently from

Refs. [23], the two-nucleon spectral function adopted in

this work is not factorized into the product of two one-

nucleon spectral functions, see Sec. II C for a detailed

discussion.

The two-body CC operator is given by the sum of

four distinct interaction mechanisms, namely the pion

in ﬂight, seagull, pion-pole, and ∆ excitations [85,86]

jµ

CC = (jµ

pif )CC + (jµ

sea)CC + (jµ

pole)CC + (jµ

∆)CC.(15)

Detailed expressions for the ﬁrst four terms of Eq. (15)

can be found in Refs. [23,86]. Below, we only report the

two-body current terms involving a ∆-resonance in the

intermediate state, as we ﬁnd them to be the dominant

contribution. Because of the purely transverse nature

of this current, the form of its vector component is not

subject to current-conservation constraints and its ex-

pression is largely model dependent, as discussed in the

previous section. The current operator can be written as

[85,86]:

(jµ

∆)CC =3

fπNN f∗

π("−2

3τ(2)

±+(τ(1) ×τ(2))±

3

×FπNN (k0

π)FπN∆(k0

π)(jµ

a)(1)

−2

3τ(2)

±+(τ(1) ×τ(2))±

3±

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FERMILAB-PUB-22-745-TFormfactorandmodeldependenceinneutrino-nucleuscrosssectionpredictionsDanielSimons,1NoahSteinberg,2AlessandroLovato,3,4YannickMeurice,1NoemiRocco,2andMichaelWagman21DepartmentofPhysicsandAstronomy,UniversityofIowa,IowaCity,IA52242,USA2TheoreticalPhysicsDepartment,FermiNationalAcc...

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FERMILAB-PUB-22-745-T Form factor and model dependence in neutrino-nucleus cross section predictions Daniel Simons1Noah Steinberg2Alessandro Lovato3 4Yannick Meurice1Noemi Rocco2and Michael Wagman2

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