Low-energy elastic (anti)neutrino&#x2212;nucleon scattering in covariant baryon chiral perturbation theory

Jin-Man Chen; Ze-Rui Liang; De-Liang Yao

doi:10.1007/s11467-024-1417-4

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Front. Phys. ›› 2024, Vol. 19 ›› Issue (6) : 64202. DOI: 10.1007/s11467-024-1417-4

RESEARCH ARTICLE

Low-energy elastic (anti)neutrino−nucleon scattering in covariant baryon chiral perturbation theory

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Abstract

The low-energy antineutrino- and neutrino−nucleon neutral current elastic scattering is studied within the framework of the relativistic SU(2) baryon chiral perturbation theory up to the order of $O (p^{3})$ . We have derived the model-independent hadronic amplitudes and extracted the form factors from them. It is found that differential cross sections $d σ / d Q^{2}$ for the processes of (anti)neutrino−proton scattering are in good agreement with the existing MiniBooNE data in the $Q^{2}$ region $[0.13, 0.20]$ GeV², where nuclear effects are expected to be negligible. For $Q^{2} \leq 0.13$ GeV², large deviation is observed, which is mainly owing to the sizeable Pauli blocking effect. Comparisons with the simulation data produced by the NuWro and GENIE Mento Carlo events generators are also discussed. The chiral results obtained in this work can be utilized as inputs in various nuclear models to achieve the goal of precise determination of the strangeness axial vector form factor, in particular when the low-energy MicroBooNE data are available in the near future.

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chiral perturbation theory / neutrino−nucleon scattering / form factors / chiral Lagrangians / one-loop amplitude / neutral weak current

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Jin-Man Chen, Ze-Rui Liang, De-Liang Yao. Low-energy elastic (anti)neutrino−nucleon scattering in covariant baryon chiral perturbation theory. Front. Phys., 2024, 19(6): 64202 https://doi.org/10.1007/s11467-024-1417-4

1 Introduction

Neutrino−nucleon scattering, as one of the fundamental processes of neutrino interactions with matter, has been extensively investigated for decades for its importance in our understanding of a wide variety of physical phenomena; see e.g. Refs. [24, 46] for reviews. A comprehensive knowledge of the neutrino−nucleon/nuclei interactions is important to achieve the precision goal of modern neutrino−oscillation experiments [1–3, 7, 9, 13, 14]. Furthermore, the neutral current scatterings play a dominant role in the thermal coupling between neutrinos and the stellar environment, thereby impacting the mean energies of various neutrino types in core-collapse supernovae [19, 35]. Amongst them, a fundamental process is the

ν N

neutral current elastic (NCE) scattering. As nucleons are usually embedded in nuclei, a good and deep understanding of the

ν N

NCE interaction is indispensable for a precise extraction of nuclear effects [40], which is one of the main sources of systematical uncertainty in accurate determination of neutrino properties. On the other hand, it plays an extremely important role in unveiling the electroweak properties of the nucleon. Unlike the charged current quasi elastic (CCQE)

ν N

interaction, which only involves isovector weak current, the NCE reaction is sensitive to both the isovector and isoscalar weak currents [8]. Therefore, it provides the possibility to explore the strange quarks, which exist as sea quarks in nucleons, through their isoscalar contribution to the NCE interaction.

Experimental efforts dedicated to the measurements of

(\bar{ν}) ν

−N NCE scatterings have been undertaken since the middle of 1980s [4–6, 44, 45]. One of the goals of these experiments is to determine the strangeness contribution to the nucleon spin

Δ s

[19], which is associated with the strangeness axial form factors

G_{A}^{s} (Q^{2})

Δ s = G_{A}^{s} (Q^{2} = 0)

, with

Q^{2}

being the momentum transfer squared between the initial and final nucleons. The E734 experiment at Brookhaven National Laboratory (BNL) measured the differential cross sections

d σ / d Q^{2}

for (anti)neutrino−proton NCE scattering in the momentum transfer region

0.45 ≲ Q^{2} ≲ 1.05

GeV², enabling them to analyze the strangeness contribution to axial vector form factor

G_{A}^{s} (Q^{2})

but with assumptions for its

Q^{2}

-behaviour [6]. In 2010 and 2015, the MiniBooNE experiment at Fermi National Accelerator Laboratory released data of the differential cross-sections for neutrino- and antineutrino-induced NCE scattering in a mineral oil (CH₂) based target system [4, 5], with a focus on the momentum transfer range below 2 GeV². Note that the experimental data close to zero momentum transfer (i.e.,

Q^{2} = 0

) are provided in the Ph.D. dissertation by Perevalov [44], a member of the MiniBooNE collaboration. It should be emphasized that the low

Q^{2}

data of differential cross sections have a complicated varying behavior due to nuclear effects, which deserves more experimental investigations. To that end, the MicroBooNE experiment in Argon is conducting the measurements of muon neutrino NCE scattering from protons and more data will be available for

0.1 ≲ Q^{2} ≲ 1

GeV² [45].

In the theoretical aspects, various phenomenological models are used to analyze the experimental data on

(\bar{ν}) ν

−N NCE scattering, mainly focusing on the extraction of the neutral current axial form factors. For instance, with the dipole parametrization [33], the strange quark form factors of the proton and the axial vector dipole mass are determined by refitting the BNL experimental data on neutrino−proton elastic scattering [27]. In Ref. [50], Sufian et al. studied the NC weak axial form factor

G_{A}^{Z} (Q^{2})

by using the data from MiniBooNE experiments on neutrino−nucleon scattering and the result of the strange quark axial charge from lattice QCD [38]. Therein, the model-dependent dipole parametrization is again employed for the

G_{A}^{Z} (Q^{2})

. In a very recent work [42], the MiniBooNE NCE data are included in their global fit and an improvement is established in constraining the

G_{A}^{s} (Q^{2})

. In addition to the dipole model, the

z

-expansion model is also chosen to parametrize the strangeness axial vector form factor. Here, we intend to carry out a model-independent calculation of the (

\bar{ν}

)

ν

−N NCE scattering within the framework of relativistic baryon chiral perturbation theory (ChPT).

ChPT [28, 29, 51] is the effective field theory of quantum chromodynamics (QCD) at low energies, and has been extensively used in hadron physics and nuclear physics; see e.g. Refs. [16, 17, 31, 47, 56] for reviews and applications. It was initially proposed for pure Goldstone bosons and later extended to describe interactions with nucleons [30]. As examples of applications to heavy hadrons, recent one-loop analyses of heavy charmed mesons and doubly charmed baryons in ChPT can be found e.g. in Refs. [37, 39]. Returning to the

(\bar{ν}) ν

−

N

scattering, systematical calculation in ChPT at one-loop order is rare. Low-energy theorems associated with neutrino induced one-pion production off the nucleon were derived using heavy baryon formalism in Ref. [15]. It is until 2018 that a first systematic study of charged-current weak pion production was carried out in relativistic baryon ChPT [54]. The study is later extended to the neutral-current weak pion production [55]. The relevant formalism can also be applied to investigate the NCE (

\bar{ν}

)

ν

−

N

scattering, which will provide useful information for the explanation of the existing MiniBooNE data and future MicroBooNE data in the low

Q^{2}

regime.

In this work, the hadronic amplitude involved in the NCE

(\bar{ν}) ν

−

N

scattering is calculated in covariant baryon ChPT up to

O (p^{3})

. The ultraviolet (UV) divergences are dealt with by using the modified minimal subtraction (

\bar{M S}

−1) scheme in dimensional regularization, while the notable power counting breaking (PCB) terms are removed by imposing the so-called extended-on-mass-shell (EOMS) scheme [25]. In accordance with Lorentz and isospin symmetries, the amplitude can be decomposed and expressed in terms of 6 structure functions: isovector (isoscalar) vector Dirac and Pauli form factors, isovector axial−vector and induced pseudoscalar form factors. Explicit expressions of the 6 form factors are obtained and are consistent with the results derived in previous literature [26, 48, 53].

In our numerical computation, the low energy constants (LECs) are either determined elsewhere or set to be of natural size. We then predict the differential cross sections for four physical NCE processes:

ν p \to ν p

\bar{ν} p \to \bar{ν} p

ν n \to ν n

and

\bar{ν} n \to \bar{ν} n

. It is found that our results of

d σ / d Q^{2}

ν p \to ν p

and

\bar{ν} p \to \bar{ν} p

are in good agreement with the MiniBooNE data in the

Q^{2}

region

\sim [0.13, 0.20]

GeV², where the ChPT calculation is expected to be reliable. Nevertheless, when

Q^{2} \leq 0.13

GeV², deviation between our predictions and the experimental data can be seen. We ascribe the deviation to the occurrence of nuclear effects. It is well known that Pauli blocking is the dominant nuclear effect for small momentum transfer. It is then tested that, indeed, our ChPT results can be fine tuned by incorporating Pauli blocking effect, whose value is estimated by nuclear models implemented in NuWro Monte Carlo events generator [32, 36]. For the NCE scattering on neutron, there are no experimental data so far. Therefore, we compare our ChPT results with the simulation data by NuWro, and large discrepancy is observed. Future experimental data for the neutron channel are appealed to explain this inconsistency, though the detection of neutron is challengeable. At last, total cross sections are also predicted and compared with the NuWro and GENIE [10] data. The ChPT results of the NCE scattering on the nucleon, we obtain here, can be used as inputs in various nuclear models to extract the strangeness contribution to the nucleon spin and the strangeness axial vector form factor, especially when the MicroBooNE data appear in the near future.

The manuscript is organized as follows. Section 2 presents the basics of

(\bar{ν}) ν

−

N

scattering, such as isospin and Lorentz decomposition. The calculation of the form factors in baryon ChPT is detailed in Section 3. In Section 4, numerical results of differential and total cross sections are obtained, and comparisons to the MiniBooNE, NuWro and GENIE data are discussed. Section 5 is our summary and outlook. Definition of the loop functions is given in Appendix A, while the explicit expressions of hadronic amplitudes and form factors are relegated to Appendices B & C, respectively.

2 Basics of elastic neutrino−nucleon scattering

The process of

(\bar{ν}) ν

−

N

NCE scattering can be generally described by

(1)

ℓ (q_{1}) + N (p_{1}) \to ℓ (q_{2}) + N (p_{2}),

where

ℓ = ν

\bar{ν}

N = p

n

. The four momenta of the incoming and outgoing particles are indicated in parentheses. The Lorentz-invariant Mandelstam variables are defined by

(2)

s = (p_{1} + q_{1})^{2}, t = (p_{1} - p_{2})^{2}, u = (p_{1} - q_{2})^{2},

fulfilling the constraint

s + t + u = 2 m_{N}^{2}

, with

m_{N}

being the physical mass of the nucleon. In the standard model (SM), this process is mediated by the vector

Z

boson with momentum

q = p_{2} - p_{1}

, which is illustrated in Fig.1. Note that the one-boson exchange approximation is assumed. The energy region we are interested in is

q^{2} ≪ M_{Z}^{2}

, where

M_{Z}

is the mass of

Z

-boson. Therefore, the scattering amplitude

M

can be written as

Fig.1 Kinematics of neutral current elastic neutrino-nucleon scattering under the one-boson exchange approximation.

Full size|PPT slide

(3)

M = - \frac{G_{F}}{\sqrt{2}} L_{μ} H^{μ},

where

G_{F} = 1.166 \times 10^{- 5} {G e V}^{- 2}

is the Fermi constant [52];

L_{μ}

and

H^{μ}

are leptonic and hadronic matrix elements, respectively. Expressions of the leptonic and hadronic parts read

(4)

\begin{aligned} L_{μ} & = \bar{ν} (q_{2}) γ_{μ} (1 - γ_{5}) ν (q_{1}), \\ H^{μ} & = ⟨ N (p_{2}) | J_{N C}^{μ} (0) | N (p_{1}) ⟩, \end{aligned}

where the neutral weak current is

(5)

J_{N C}^{μ} = (1 - 2 \sin^{2} θ_{W}) {\hat{V}}_{3}^{μ} - {\hat{A}}_{3}^{μ} - 2 \sin^{2} θ_{W} {\hat{V}}_{0}^{μ},

with

{\hat{V}}_{3}^{μ}

{\hat{A}}_{3}^{μ}

and

{\hat{V}}_{0}^{μ}

being isovector vector, isovector axial-vector and isoscalar vector currents, in order.

θ_{W}

stands for the Weinberg angle. The leptonic part can be derived straightforwardly in the SM, while the hadronic one is complicated due to the inner structure of the nucleon. For the hadronic amplitude, one can perform isospin decomposition in the following way:

(6)

H^{μ} = χ_{f}^{†} (\frac{τ_{a}}{2} H_{V}^{μ} + \frac{τ_{0}}{2} H_{S}^{μ}) χ_{i}, a = 3,

where

χ_{i}

and

χ_{f}

are isospin spinors of the initial and final nucleons, respectively. To be specific, one has

χ_{p} = (1, 0)^{T}

for the proton and

χ_{n} = (0, 1)^{T}

for the neutron.

H_{V}^{μ}

and

H_{S}^{μ}

stand for the isospin vector and scalar amplitudes, respectively. In view of Eq. (5), one readily has

(7)

\begin{aligned} H_{V}^{μ} & = (1 - 2 \sin^{2} θ_{W}) V_{V}^{μ} - A_{V}^{μ}, \\ H_{S}^{μ} & = - 2 \sin^{2} θ_{W} V_{S}^{μ}, \end{aligned}

with the matrix elements

V_{V, S}^{μ}

and

A_{V}^{μ}

expressed in terms of form factors [8, 18]:

(8)

\begin{aligned} V_{V}^{μ} = & \bar{u} (p_{2}) (γ^{μ} F_{1}^{V} + \frac{i}{2 m_{N}} σ^{μ ν} q_{ν} F_{2}^{V}) u (p_{1}), \\ A_{V}^{μ} = & \bar{u} (p_{2}) (γ^{μ} γ_{5} G_{A} + \frac{q^{μ}}{m} γ_{5} G_{P}) u (p_{1}), \\ V_{S}^{μ} = & \bar{u} (p_{2}) (γ^{μ} F_{1}^{S} + \frac{i}{2 m_{N}} σ^{μ ν} q_{ν} F_{2}^{S}) u (p_{1}) . \end{aligned}

In above,

\bar{u} (p_{2})

and

u (p_{1})

represent the spinors of the final and initial nucleons, respectively. There are six unknown form factors:

F_{1}^{V}

F_{2}^{V}

G_{A}

G_{P}

F_{1}^{S}

F_{2}^{S}

F_{1}^{V}

(

F_{1}^{S}

) and

F_{2}^{V}

(

F_{2}^{S}

) are called isovector (isoscalar) Dirac and Pauli form factors, respectively.

G_{A}

and

G_{P}

are axial and induced pseudoscalar form factors of the nucleon, respectively. Both

G_{A}

and

G_{P}

are isovector, due to the absence of isosinglet axial current. We will calculate the form factors in the framework of covariant baryon ChPT in Section 3.

In practice, it is often useful to reorganize the hadronic amplitude according to its Lorentz structure. That is,

(9)

H^{μ} \equiv \bar{u} (p_{2}) H^{μ} u (p_{1}),

(10)

H^{μ} = γ^{μ} F_{1} + \frac{i}{2 m_{N}} σ^{μ ν} q_{ν} F_{2} - γ^{μ} γ_{5} G_{A} - \frac{q^{μ}}{m_{N}} γ_{5} G_{P},

where the combined form factors

F_{1, 2}

and

G_{A, P}

are related to the ones in Eq. (8) via

(11)

F_{i} (t) = \cos 2 θ_{W} F_{i}^{V} (t) \frac{C_{3}}{2} - 2 \sin^{2} θ_{W} F_{i}^{S} (t) \frac{C_{0}}{2}, i = 1, 2,

(12)

G_{j} (t) = G_{j} (t) \frac{C_{3}}{2}, j = A, P .

For brevity, isospin factors have been defined, i.e.,

C_{3} = χ_{f}^{†} τ_{3} χ_{i}

and

C_{0} = χ_{f}^{†} τ_{0} χ_{i}

. In this way, the hadronic amplitude for a given physical process can be obtained by properly choosing the values of

C_{3}

and

C_{0}

. We show the isospin factors in Tab.1 for easy reference.

Tab.1 Isospin factors for physical processes.

Physical process	$C_{3}$	$C_{0}$

$ν + p \to ν + p$	1	1
$\bar{ν} + p \to \bar{ν} + p$	1	1
$ν + n \to ν + n$	$- 1$	1
$\bar{ν} + n \to \bar{ν} + n$	$- 1$	1

3 Hadronic form factors in baryon ChPT

The hadronic form factors encode dynamical information of strong interaction inside the nucleon. At low energies, the fundamental theory quantum chromodynamics (QCD) is not feasible anymore due to the non-perturbative nature of strong interaction. Instead, a popular and powerful tool for the study of the low-energy dynamics is ChPT, which is an effective field theory of QCD in this regime. In this section, we are going to calculate the form factors involved in the

(\bar{ν}) ν

−

N

scattering within the framework of relativistic baryon ChPT up to

O (p^{3})

by using EOMS scheme.

3.1 Chiral effective Lagrangian

The chiral effective Lagrangian pertinent to our calculation can be organized in the form as

(13)

L_{e f f} = L_{π N}^{(1)} + L_{π N}^{(2)} + L_{π N}^{(3)} + L_{π π}^{(2)} + L_{π π}^{(4)},

where the superscripts denote the chiral orders. The pieces

L_{π π}^{(i)}

(

i = 2, 4

) describe the purely pionic interactions, while the terms

L_{π N}^{(j)}

(

j = 1, 2, 3

) are constructed by further including the nucleon fields as explicit degrees of freedom.

The

π π

Lagrangians are given by [28]

(14)

L_{π π}^{(2)} = \frac{F^{2}}{4} ⟨ u_{μ} u^{μ} + χ_{+} ⟩,

(15)

L_{π π}^{(4)} = \frac{1}{8} ℓ_{4} ⟨ u_{μ} u^{μ} ⟩ ⟨ χ_{+} ⟩ + \frac{1}{16} (ℓ_{3} + ℓ_{4}) ⟨ χ_{+} ⟩^{2},

where

F

is the pion decay constant in the chiral limit and

ℓ_{i} (i = 3, 4)

are unknown LECs. The chiral operators in the

O (p^{4})

Lagrangian are required for the renormalization of the pion mass and decay constant. The so-called chiral vielbein reads

(16)

u_{μ} = i {u^{†} (\partial_{μ} - i r_{μ}) u - u (\partial_{μ} - i l_{μ}) u^{†}},

and the Goldstone pion fields are collected in the unitary

2 \times 2

matrix

u

(17)

u = \exp (i \frac{Φ}{\sqrt{2} F}), Φ = π^{a} τ^{a} = (\begin{matrix} π^{0} & \sqrt{2} π^{+} \\ \sqrt{2} π^{-} & - π^{0} \end{matrix}),

with

τ^{a}

the Pauli matrices. Furthermore,

l_{μ}

and

r_{μ}

are left- and right-handed external fields, respectively. The chiral blocks

χ_{\pm}

are defined by

(18)

χ_{\pm} = u^{†} χ u^{†} \pm u χ^{†} u, χ = d i a g (M^{2}, M^{2}),

where

M

is the pion mass at leading order.

The leading order

π N

effective Lagrangian reads [23]

(19)

L_{π N}^{(1)} = {\bar{Ψ}}_{N} {i D̸ - m + \frac{g}{2} u̸ γ_{5}} Ψ_{N},

with

m

and

g

being the nucleon mass and axial coupling constant in the chiral limit, respectively. The isodoublet

Ψ_{N}

comprises the proton and neutron fields, i.e.,

Ψ_{N} = (p, n)^{T}

. The covariant derivative acting on the nucleon fields is given by

(20)

D_{μ} = \partial_{μ} + Γ_{μ} - i {\hat{v}}_{μ}^{s},

(21)

Γ_{μ} = \frac{1}{2} {u^{†} (\partial_{μ} - i r_{μ}) u + u (\partial_{μ} - i l_{μ}) u^{†}} .

Here,

{\hat{v}}_{μ}^{s}

denotes the isosinglet vector field.

The full set of

π N

Lagrangians at

O (p^{2})

and

O (p^{3})

can be found e.g. in Ref. [23]. Here, we only display the terms relevant to our calculation:

(22)

\begin{aligned} L_{π N}^{(2)} = & \frac{1}{8 m} c_{6} {\bar{Ψ}}_{N} {F_{μ ν}^{+} σ^{μ ν}} Ψ_{N} \\ + \frac{1}{8 m} c_{7} {\bar{Ψ}}_{N} {⟨ F_{μ ν}^{+} ⟩ σ^{μ ν}} Ψ_{N} + \dots, \end{aligned}

(23)

\begin{aligned} L_{π N}^{(3)} = & {\bar{Ψ}}_{N} {\frac{i}{2 m} d_{6} [D^{μ}, {\tilde{F}}_{μ ν}^{+}] D^{ν} + h . c .} Ψ_{N} \\ + {\bar{Ψ}}_{N} {\frac{i}{2 m} d_{7} [D^{μ}, ⟨ F_{μ ν}^{+} ⟩] D^{ν} + h . c .} Ψ_{N} \\ + {\bar{Ψ}}_{N} {\frac{1}{2} d_{16} γ^{μ} γ_{5} ⟨ χ_{+} ⟩ u_{μ}} Ψ_{N} \\ + {\bar{Ψ}}_{N} {\frac{i}{2} d_{18} γ^{μ} γ_{5} [D_{μ}, χ_{-}]} Ψ_{N} \\ + {\bar{Ψ}}_{N} {\frac{1}{2} d_{22} γ^{μ} γ_{5} [D^{ν}, F_{μ ν}^{-}]} Ψ_{N} + \dots, \end{aligned}

where

c_{i} (i = 6, 7)

and

d_{j} (j = 6, 7, 16, 18, 22)

are unknown LECs, and

h . c .

refers to the hermitian conjugation. The field tensors, regarding the

l_{μ}

r_{μ}

and

{\hat{v}}_{μ}

external sources, are defined as follows:

(24)

F_{μ ν}^{+} = f_{μ ν}^{+} + 2 {\hat{v}}_{μ ν}^{(s)},

(25)

F_{μ ν}^{-} = f_{μ ν}^{-},

(26)

f_{μ ν}^{\pm} = u f_{μ ν}^{L} u^{†} \pm u^{†} f_{μ ν}^{R} u,

(27)

{\hat{v}}_{μ ν}^{(s)} = \partial_{μ} {\hat{v}}_{ν}^{(s)} - \partial_{ν} {\hat{v}}_{μ}^{(s)},

with the left- and right-handed external field strength tensors given by

(28)

f_{μ ν}^{L} = \partial_{μ} l_{ν} - \partial_{ν} l_{μ} - i [l_{μ}, l_{ν}],

(29)

f_{μ ν}^{R} = \partial_{μ} r_{ν} - \partial_{ν} r_{μ} - i [r_{μ}, r_{ν}] .

The traceless tensor

{\tilde{F}}_{μ ν}^{+}

can be obtained through

(30)

{\tilde{F}}_{μ ν}^{+} = F_{μ ν}^{+} - \frac{1}{2} ⟨ F_{μ ν}^{+} ⟩,

where

⟨ \dots ⟩

denotes the trace in the flavor space. The coupling of the

Z

boson is incorporated by setting

(31)

l_{μ} = \frac{g_{W}}{2 \cos θ_{W}} (- 2 \cos^{2} θ_{W}) Z_{μ} \frac{τ_{3}}{2},

(32)

r_{μ} = \frac{g_{W}}{2 \cos θ_{W}} (2 \sin^{2} θ_{W}) Z_{μ} \frac{τ_{3}}{2},

(33)

{\hat{v}}_{μ}^{(s)} = \frac{g_{W}}{2 \cos θ_{W}} (2 \sin^{2} θ_{W}) Z_{μ} \frac{τ_{0}}{2},

with

g_{W}

the weak coupling constant. The factor

g_{W} / (2 \cos θ_{W})

is extracted out from the hadronic part to define the Fermi constant

G_{F}

in Eq. (9), i.e.,

G_{F} =

\sqrt{2} g_{W}^{2} / (8 M_{Z}^{2} \cos^{2} θ_{W})

3.2 Form factors

With the Lagrangians specified in the previous subsection, we are now in the position to calculate the form factors up to and including

O (p^{3})

. Tree and leading one-loop Feynman diagrams are shown in Fig.2 and Fig.3, respectively. Diagrams with one-loop corrections on the external nucleon lines are not displayed explicitly, which will be taken into account by the wave function renormalization procedure.

Fig.2 Tree-level Feynman diagrams up to and including $O (p^{3})$ . The solid, dashed and wavy lines represent the nucleon, pions and the $Z$ boson, in order. The circled numbers indicate the chiral orders of the vertices.

Full size|PPT slide

Fig.3 One-loop Feynman diagrams up to and including $O (p^{3})$ . The solid, dashed and wavy lines represent the nucleon, pions and the $Z$ boson, in order. The circled numbers indicate the chiral orders of the vertices.

Full size|PPT slide

The obtained hadronic amplitudes are shown diagram by diagram in Appendix B, from which one can extract the form factors. The results of form factors are expressed as

(34)

\begin{aligned} F_{1}^{V} (t) = & 1 - 2 d_{6} t + F_{1}^{V, l o o p s} + F_{1}^{V, w f}, \\ F_{2}^{V} (t) = & c_{6} + 2 d_{6} t + F_{2}^{V, l o o p s} + F_{2}^{V, w f}, \\ F_{1}^{S} (t) = & 1 - 4 d_{7} t + F_{1}^{S, l o o p s} + F_{1}^{S, w f}, \\ F_{2}^{S} (t) = & (c_{6} + 2 c_{7}) + 4 d_{7} t + F_{2}^{S, l o o p s} + F_{2}^{S, w f}, \\ G_{A} (t) = & g + (4 d_{16} M^{2} + d_{22} t) + G_{A}^{l o o p s} + G_{A}^{w f}, \\ G_{P} (t) = & \frac{2 g m_{N}^{2}}{M^{2} - t} + \frac{4 m_{N}^{2} M^{2} (2 d_{16} - d_{18})}{M^{2} - t} \\ + \frac{4 g m_{N}^{2} M^{2} ℓ_{4}}{F^{2} (M^{2} - t)} - 2 m_{N}^{2} d_{22} \\ - \frac{4 g m_{N}^{2} M^{2} [M^{2} ℓ_{3} + (M^{2} - t) ℓ_{4}]}{F^{2} (M^{2} - t)^{2}} \\ + G_{P}^{l o o p s} + G_{P}^{w f}, \end{aligned}

where the expressions of loop contributions are relegated to Appendix C. Note that the induced pseudoscalar form factor does not contribute to the cross sections if the neutrino mass is zero, but we show it here for completeness. The last terms in the above equations account for the effect of wave function renormalization. Up to

O (p^{3})

, they are given by

(35)

\begin{aligned} T^{w f} = (T^{(a)} + T^{(b)}) (Z_{N} - 1) + \dots, \\ T \in {F_{1}^{V}, F_{2}^{V}, F_{1}^{S}, F_{2}^{S}, G_{A}, G_{P}}, \end{aligned}

where the ellipsis stands for the higher order terms beyond our accuracy.

Z_{N}

is the wave function renormalization constant of the nucleon at leading one-loop order

(36)

Z_{N} = 1 + δ_{Z_{N}}^{(2)} + O (p^{3}),

where the

O (p^{2})

term reads [21]

(37)

\begin{aligned} δ_{Z_{N}}^{(2)} = & \frac{3 g^{2}}{4 F_{π}^{2} (4 m_{N}^{2} - m_{π}^{2})} {4 m_{π}^{2} [A_{0} (m_{N}^{2}) \\ + (m_{π}^{2} - 3 m_{N}^{2}) B_{0} (m_{N}^{2}, m_{π}^{2}, m_{N}^{2}) - m_{N}^{2}] \\ + (12 m_{N}^{2} - 5 m_{π}^{2}) A_{0} (m_{π}^{2})} . \end{aligned}

For the definitions of the loop integrals

A_{0}

and

B_{0}

, the readers are referred to Appendix A. Note that the bare parameters

F

m

and

M

have been replaced by the corresponding physical ones

F_{π}

m_{N}

and

m_{π}

, since the resultant difference is of higher order beyond our accuracy.

The loop amplitudes are calculated by imposing the dimensional regularization, and the UV divergence in the loop integrals are subtracted with the so-called

\bar{M S} - 1

subtraction scheme. Actually, the UV divergence are cancelled out by the counter terms from the chiral effective Lagrangian. In practice, one splits the bare LECs into two parts: a finite piece and a divergent one. To be specific, the LECs showing up in Eq. (34) are separated in the following manner:

(38)

X = X^{r} + \frac{β_{X}}{16 π^{2}} R, X \in {g, c_{6, 7}, d_{6, 7, 16, 18, 22}, ℓ_{3, 4}},

where

R = 2 / (d - 4) + γ_{E} - 1 - \ln (4 π)

d

is the dimension of space-time,

γ_{E}

denotes Euler constant, and

β_{X}

is called beta functions. In order to remove the UV divergence from the loops, the

β_{X}

’s have to take the following values,

(39)

\begin{aligned} β_{g} = \frac{g (g^{2} - 2) m^{2}}{F^{2}}, β_{c_{6}} = β_{c_{7}} = 0, β_{d_{6}} = \frac{g^{2} - 1}{12 F^{2}}, \\ β_{d_{16}} = \frac{g (g^{2} - 1)}{4 F^{2}}, β_{d_{7}} = β_{d_{18}} = β_{d_{22}} = 0, \\ β_{ℓ_{3}} = - \frac{1}{4}, β_{ℓ_{4}} = 1 . \end{aligned}

It is pointed out in Ref. [30] that the PCB issue arises when the nucleons appear as internal lines in the loop diagrams. To restore power counting, here we employ the EOMS scheme, in which an extra finite renormalization is required. In our case, the PCB terms are those pieces, stemming from the loops, with chiral orders lower than

O (p^{3})

. As discussed in Ref. [25], those terms are polynomials of, e.g., pion masses, which are called infrared regular terms therein, and hence can be absorbed by shifting the LECs of the tree amplitudes. That is, in addition to Eq. (39), the UV-finite

O (p)

and

O (p^{2})

LECs are further split,

(40)

X^{r} = \tilde{X} + \frac{{\tilde{β}}_{X}}{16 π^{2}} R, X \in {g, c_{6, 7}},

with the beta functions

{\tilde{β}}_{X}

’s given by

(41)

\begin{aligned} {\tilde{β}}_{g} = - \frac{g (g^{2} - 2) {\bar{A}}_{0} (m^{2})}{F^{2}}, {\tilde{β}}_{c_{6}} = - \frac{5 g^{2} m^{2}}{F^{2}}, \\ {\tilde{β}}_{c_{7}} = \frac{4 g^{2} m^{2}}{F^{2}} . \end{aligned}

Here the function

{\bar{A}}_{0} (m^{2})

represents the UV-subtracted one-point loop integral. Namely,

{\bar{A}}_{0} (m^{2})

is obtained by removing the piece proportional to

R

in Eq. (A1).

4 Numerical results and discussion

4.1 Differential cross section

In the laboratory frame, the differential cross section with respect to the momentum transfer squared reads

(42)

\frac{d σ}{d Q^{2}} = \frac{{\bar{| M |}}^{2}}{64 π m_{N}^{2} E_{ν}^{2}},

where

Q^{2} \equiv - q^{2} = - t

E_{ν}

is the neutrino energy, and

m_{N}

is the physical nucleon mass. The spin-averaged amplitude squared can be written as

(43)

{\bar{| M |}}^{2} = \frac{G_{F}^{2}}{2} L_{μ ν} H^{μ ν},

with the leptonic tensor

(44)

L_{μ ν} = 8 (q_{1 μ} q_{2 ν} - g_{μ ν} q_{1} \cdot q_{2} + q_{1 ν} q_{2 μ} \pm i ϵ_{μ λ ν σ} q_{1}^{λ} q_{2}^{σ}),

where the “

+

” and “−” signs correspond to the neutrino and antineutrino cases, respectively. The hadronic tensor is given by

(45)

H_{μ ν} = \frac{1}{2} T r [({p̸}_{1} + m_{N}) {\tilde{H}}_{μ} ({p̸}_{2} + m_{N}) H_{ν}],

where

{\tilde{H}}_{μ} = γ_{0} H_{μ}^{†} γ_{0}

and

H_{μ}

is defined in Eq. (9). One can further contract the leptonic and hadronic tensors, which leads to the traditional form of the differential cross section [27, 40]:

(46)

\begin{aligned} \frac{d σ}{d Q^{2}} = & \frac{G_{F}^{2} m_{N}^{2}}{8 π E_{ν}^{2}} [A (Q^{2}) \pm \frac{(s - u)}{m_{N}^{2}} B (Q^{2}) \\ + \frac{(s - u)^{2}}{m_{N}^{4}} C (Q^{2})], \end{aligned}

where the scalar functions

A (Q^{2})

B (Q^{2})

, and

C (Q^{2})

are related to the form factors in Eq. (11) and Eq. (12) through

(47)

\begin{aligned} A (Q^{2}) \equiv & 4 η [G_{A}^{2} (Q^{2}) (1 + η) + 4 η F_{1} (Q^{2}) F_{2} (Q^{2}) \\ - (F_{1}^{2} (Q^{2}) - η F_{2}^{2} (Q^{2})) (1 - η)], \\ B (Q^{2}) \equiv & 4 η G_{A} (Q^{2}) (F_{1} (Q^{2}) + F_{2} (Q^{2})), \\ C (Q^{2}) \equiv & \frac{1}{4} [G_{A}^{2} (Q^{2}) + F_{1}^{2} (Q^{2}) + η F_{2}^{2} (Q^{2})], \end{aligned}

with

(s - u) = 4 m_{N} E_{ν} - Q^{2}

and

η = Q^{2} / 4 m_{N}^{2}

. It should be noted that the neutrino mass is zero and consequently the pseudoscalar form factor

G_{P}

is absent. With the above formulae, we are in the position to compute the cross sections for physical processes numerically.

In our numerical computation, we use

m_{N} = 938.8

MeV,

m_{π} = 138

MeV and

F_{π} = 92.21

MeV [52]. Values of the LECs relevant to NCE scattering are summarized in Tab.2. The values of

g

d_{16}

and

d_{22}

are taken from Ref. [53]. They appear in the axial form factor

G_{A}

and were pinned down by fitting to the corresponding lattice QCD data simulated at unphysical pion masses [53]. The LECs

c_{6}

and

c_{7}

have been determined by employing the empirical values of the anomalous magnetic moments of the nucleon (

κ_{p}

and

κ_{n}

) in Refs. [12, 43]. We set

d_{7} = - 0.49

, which is extracted from the neutron and proton electromagnetic radii in Ref. [26]. The

d_{6}

is set to be of natural size^†

¹⁾ In Ref. [26], $d_{6}$ is determined to be $d_{6} = - 0.70$ but without error. Therefore, we prefer to vary it in a range $[- 1.0, 1.0]$ , in accordance with the naturalness ansatz.

, following Ref. [54]. The renomalization scale

μ

in the loop integrals is fixed at the physical nucleon mass:

μ = m_{N}

. We do not assign values to the LECs only showing up in

G_{P}

, since they are irrelevant to the NCE scattering cross sections.

Tab.2 Values of the LECs involved in neutrino−nucleon NCE scattering. The axial coupling constant $g$ is dimensionless. The LECs $c_{i}$ and $d_{j}$ are in units of GeV⁻¹ and GeV⁻², respectively.

	LEC	Value	Source

$L_{π N}^{(1)}$	$g$	$1.13 \pm 0.01$	$G_{A}$ [53]
$L_{π N}^{(2)}$	$c_{6}$	$1.35 \pm 0.04$	$κ_{p}$ and $κ_{n}$ [12, 43]
	$c_{7}$	$- 2.68 \pm 0.08$	$κ_{p}$ and $κ_{n}$ [12, 43]
$L_{π N}^{(3)}$	$d_{6}$	$0.0 \pm 1.0$	−
	$d_{7}$	$- 0.49$	Electromagnetic radii [26]
	$d_{16}$	$- 0.83 \pm 0.03$	$G_{A}$ [53]
	$d_{22}$	$0.96 \pm 0.03$	$G_{A}$ [53]

The differential scattering cross sections for

ν

−

p

and

\bar{ν}

−

p

NCE scatterings are displayed in Fig.4 and Fig.5, respectively. Our ChPT calculation is expected to be reliable up to

Q^{2} = 0.2

GeV², indicated by the gray region in the figures. The first measurement of the above two processes was conducted by the BNL E734 experiment [6, 34]. Unfortunately, the BNL E734 data are for the

Q^{2}

region higher than

0.45

GeV², which are far beyond the validity scope of ChPT. The available experimental data for lower

Q^{2}

are from the MiniBooNE experiment [4, 5], which are represented by the dots with error bars in Fig.4 and Fig.5. The MiniBooNE data for cross sections

d σ / d Q^{2}

on CH₂ are measured at

E_{ν} = 0.8

GeV for

ν p

and at

E_{\bar{ν}} = 0.65

GeV for

\bar{ν} p

. Our ChPT results are computed with the same (anti)neutrino energy for easy comparison. The blue solid lines with bands stand for the resulting ChPT predictions, which are plotted up to

Q^{2} = 0.4

GeV², twice of the ChPT validity limit, to see the trend of the curves. The error bands are obtained by varying the values of the LECs within their 1−

σ

uncertainties. It can be seen that our results are in good agreement with the experimental data in the range

0.13 \leq Q^{2} \leq 0.2

GeV² within uncertainties. In particular, the data at

Q^{2} ≃ 0.13

GeV² are excellently described. Note that it is just a coincidence that the ChPT result of the

\bar{ν}

−

p

scattering agrees better with the MiniBooNE data than that of the

ν

−

p

scattering.

Fig.4 Differential cross section $d σ / d Q^{2}$ of $ν$ − $p$ NCE scattering, with the neutrino energy being $E_{ν} = 0.8$ GeV. The MiniBooNE data [4, 5] are marked by black dots with error bars. The NuWro data for free nucleon, bound nucleon with and without Pauli blocking effect are represented by orange diamonds, red crosses and green triangles in order. The blue solid line denotes the ChPT result up to $O (p^{3})$ , while the magenta dashed line stands for the sum of our ChPT prediction and Pauli blocking effect. The error bands are obtained by varying the LECs in their 1− $σ$ uncertainties. We use the abbreviation “PBE” for “Pauli blocking effect” in the figure.

Full size|PPT slide

Fig.5 Differential cross section $d σ / d Q^{2}$ of $\bar{ν}$ − $p$ NCE scattering, with the anti neutrino energy being $E_{\bar{ν}} = 0.65$ GeV. Other description is the same as Fig.4.

Full size|PPT slide

However, for the

Q^{2}

region lower than

\sim 0.13

GeV², our free-nucleon scattering prediction deviates from the MiniBooNE data, which are extracted from

ν

(

\bar{ν}

)−CH₂ scatterings where the nucleon are tightly bound. Such a dramatic deviation actually implies that the nuclear effects start to play an important role at low

Q^{2}

s. It is acknowledged that three main types of nucleon effects for bound nucleons, i.e., Fermi motion, Pauli blocking and final state interaction, may affect the cross section; see e.g. Refs. [7, 40] for more details. Amongst them, the Pauli blocking effect suppresses the cross section at low momentum transfer square, as shown e.g. in Fermi gas model [49]. The Pauli blocking effect has been already implemented in various neutrino events generators such as NUANCE [20] and NuWro [32, 36]. Here, we use NuWro to generate the Pauli blocking effect numerically^†, which is then added to the ChPT result. The sum of our ChPT result and the Pauli blocking effect is shown by the dashed lines with error bands in Fig.4 and Fig.5. It can be found that the theoretical prediction is now comparable with the MiniBooNE data for the

ν

−

p

scattering, provided that the abnormal data point at

Q^{2} = 0.05

GeV² is excluded. Nevertheless, the incorporation of the Pauli blocking effect in the

\bar{ν}

−

p

cross section is not sufficient to make the chiral result agree with the low-

Q^{2}

data at

\sim 0.067

GeV².

For comparison, the simulation results from NuWro event generator are also shown. The free-nucleon cross sections are marked by orange diamonds. For the cases that the target is CH₂ nucleus, cross sections with and without the Pauli blocking effect are represented by red crosses and green triangles. It can be observed that in the ChPT validity region, our results exhibit better agreement with the MiniBooNE data, compared to the NuWro simulation results.

Fig.6 displays the differential cross sections

[d σ / d Q^{2}]_{ν n \to ν n}

with

E_{ν} = 0.8

GeV and

[d σ / d Q^{2}]_{\bar{ν} n \to \bar{ν} n}

with

E_{\bar{ν}} = 0.65

GeV. Experimental data on the two processes are not available so far. For comparison, the ChPT and NuWro results are shown together. Large discrepancy between the two determinations arises for the neutron processes. Future cross section data from experiments for the neutron channels are required to explain this discrepancy, though it is a challenge due to the difficulties associated with neutron detection.

Fig.6 Differential cross sections $[d σ / d Q^{2}]_{ν n \to ν n}$ with $E_{ν} = 0.8$ GeV and $[d σ / d Q^{2}]_{\bar{ν} n \to \bar{ν} n}$ with $E_{\bar{ν}} = 0.65$ GeV. The ChPT and NuWro simulation results are represented by solid lines and blue diamonds, respectively. The red error bands are obtained by varying the LECs in their 1− $σ$ uncertainties.

Full size|PPT slide

4.2 Total cross section

The total cross section can be readily obtained by integrating the differential cross-section over all possible scattering directions. It describes the overall probability of interaction between the incident particle and the target particle during the scattering. Taking

ν

−

N

NCE scattering for example, the total cross section as a function of the center-of-mass (CM) energy

\sqrt{s}

can be written as

(48)

σ (\sqrt{s}) = \int \frac{d σ}{d Q^{2}} d Q^{2},

where the differential cross section

d σ / d Q^{2}

is given by Eq. (46). In CM frame, the relation between the scattering angle

θ

and the Mandelstam variables reads

(49)

\cos θ = 1 + \frac{2 s t}{(s - m_{N}^{2})^{2}} .

Since

t = - Q^{2}

and

s

is a Lorentz invariant, which guarantees

s = m_{N}^{2} + 2 E_{ν} m_{N}

, one can get that

(50)

Q^{2} = \frac{2 m_{N} E_{ν}^{2}}{2 E_{ν} + m_{N}} (1 - \cos θ), θ \in [0, π],

where

E_{ν}

is the neutrino energy in the laboratory frame. With the above equation, it is straightforward to perform the integration in Eq. (48). Moreover, the resultant total cross section is a function of

E_{ν}

, i.e.,

σ (E_{ν})

. The above derivation also holds true for anti-neutrino nucleon NCE scattering.

It is necessary to determine the maximal neutrino energy

E_{ν}^{m a x}

, up to which our ChPT model is trustworthy. For this purpose, the contour plot of

Q^{2}

with respect to

E_{ν}

and

\cos θ

is shown in Fig.7. The ChPT validity limit of

Q^{2}

is expected to be 0.2 GeV² as mentioned in the preceding subsection. Furthermore,

\cos θ

must be able to take any values in the whole region

[- 1, 1]

, such that the numerical accuracy of the total cross section, obtained by the integration in Eq. (48), is ensured. Therefore, an acceptable estimate of

E_{ν}^{m a x}

\sim 0.28

GeV in view of the

Q^{2} = 0.2

GeV² contour in Fig.7.

Fig.7 Contour plot of $Q^{2}$ with varying $E_{ν}$ and $\cos θ$ .

Full size|PPT slide

In Fig.8, our ChPT results of the total section

σ (E_{ν})

are shown order by order, together with the simulation data by NuWro [32, 36] and GENIE [10]. The vertical dash-dotted lines indicate the ChPT validity limit,

E_{ν}^{m a x} \sim 0.28

GeV, as discussed above. The NuWro data are generated for free nucleons. For the

ν p

and

\bar{ν} p

channels, a calculation with higher chiral orders makes the predictions closer to the NuWro data. The GENIE data are produced with

C^{12}

as the target nucleus, and have been merely divided by 6 since

C^{12}

is composed of 6 protons and 6 neutrons. Namely, the nuclear effects are not excluded for the GENIE data, leading to a sizeable deviation from our model and the NuWro data. For the

ν n

and

\bar{ν} n

channels, our ChPT predictions are consistent neither with the NuWro data nor with the GENIE data. Similar to the situation for differential cross sections, experimental data are needed to explain the deviation.

Fig.8 Total cross sections at different chiral orders. Our ChPT prediction is expected to be reliable up to $E_{ν (\bar{ν})}^{m a x} = 0.28$ GeV, indicated by the gray vertical line. For comparison, the simulation data produced by NuWro and GENIE events generators are also shown by red crosses and black diamonds, respectively.

Full size|PPT slide

5 Summary and outlook

The (anti)neutrino−nucleon NCE scattering has been systematically investigated for the first time within the framework of covariant baryon ChPT up to

O (p^{3})

in the low energy region. We have derived the model-independent hadronic amplitudes, from which explicit expressions of form factors are obtained. The UV divergences from the loops are removed by applying the

\bar{M S}

−1 subtraction scheme, while the PCB terms are handled by using the EOMS scheme. We fix the LECs either to the values determined in previous literature or to natural size in accordance with the naturalness ansatz. Finally, numerical results of the differential cross sections and total cross sections are predicted.

In the regime

Q^{2} \in [0.13, 0.20]

GeV², where nuclear effects are expected to be negligible, our ChPT predictions of

[d σ / d Q^{2}]_{ν p \to ν p}

and

[d σ / d Q^{2}]_{\bar{ν} p \to \bar{ν} p}

are well consistent with the MiniBooNE data within 1−

σ

uncertainties. For

Q^{2} \leq 0.13

GeV², the so-called Pauli blocking effect starts to make sizeable contribution. It is found that our ChPT results can be fine tuned by incorporating the Pauli blocking effect, which is estimated by nuclear models implemented in the NuWro event generator. Moreover, our fine-tuned results are better than the simulation data, produced by NuWro, in the sense that they are more closer to the experimental data in the low

Q^{2}

region. Unfortunately, there are no available experimental data for the neutron channels, due to the difficulties in neutron detection at experiments. Therefore, we prefer to compare our ChPT results of

[d σ / d Q^{2}]_{ν n \to ν n}

and

[d σ / d Q^{2}]_{\bar{ν} n \to \bar{ν} n}

with NuWro, but large discrepancy is observed unexpectedly. Future experimental data for these channels are needed to interpret the deviation. We also compute the total cross sections of the four physical NCE processes and confront them with the NuWro and GENIE data. Similar to the differential cross sections, our predictions of total cross sections agree with NuWro for the proton channels, but are incompatible for the neutron ones. Our ChPT calculation of the total cross section should be reliable for the (anti)neutrino energy lower than

0.28

GeV.

In the near future, low

Q^{2}

data on

ν

−

N

NCE cross section will be available from, for instance, the MicroBooNE experiment. The model-independent ChPT results obtained in this work would be helpful for a precise determination of the strangeness axial vector form factor and hence the strangeness contribution to the nucleon spin, since they can be used in various nuclear models as inputs. On the other hand, the obtained form factors can be readily applied to investigate the CCQE scattering and muon capture processes, where the induced pseudoscalar form factor is involved. Besides, the

Δ (1232)

resonance contribution may be incorporated to improve the

Q^{2}

-behaviour of the form factors through the complexity of the

Δ (1232)

propagator.

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Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

We would like to thank Prof. M. J. Vicente-Vacas for comments. This work was supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 12275076, 11905258, 12335002, and 12047503), the Science Fund for Distinguished Young Scholars of Hunan Province under Grant No. 2024JJ2007, and the Fundamental Research Funds for the Central Universities under Contract No. 531118010379. D. L. Y. appreciates the support of Peng Huan-Wu Visiting Professorship and the hospitality of Institute of Theoretical Physics at Chinese Academy of Sciences.

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Abstract
Graphical abstract
Keywords
Cite this article
1 Introduction
2 Basics of elastic neutrino−nucleon scattering
Fig.1 Kinematics of neutral current elastic neutrino-nucleon scattering under the one-boson exchange approximation.
Tab.1 Isospin factors for physical processes.
3 Hadronic form factors in baryon ChPT
3.1 Chiral effective Lagrangian
3.2 Form factors
Fig.2 Tree-level Feynman diagrams up to and including O(p3). The solid, dashed and wavy lines represent the nucleon, pions and the Z boson, in order. The circled numbers indicate the chiral orders of the vertices.
Fig.3 One-loop Feynman diagrams up to and including O(p3). The solid, dashed and wavy lines represent the nucleon, pions and the Z boson, in order. The circled numbers indicate the chiral orders of the vertices.
4 Numerical results and discussion
4.1 Differential cross section
Tab.2 Values of the LECs involved in neutrino−nucleon NCE scattering. The axial coupling constant g is dimensionless. The LECs ci and dj are in units of GeV−1 and GeV−2, respectively.
Fig.4 Differential cross section dσ /d Q2 of ν−p NCE scattering, with the neutrino energy being Eν=0.8 GeV. The MiniBooNE data [4, 5] are marked by black dots with error bars. The NuWro data for free nucleon, bound nucleon with and without Pauli blocking effect are represented by orange diamonds, red crosses and green triangles in order. The blue solid line denotes the ChPT result up to O(p3), while the magenta dashed line stands for the sum of our ChPT prediction and Pauli blocking effect. The error bands are obtained by varying the LECs in their 1−σ uncertainties. We use the abbreviation “PBE” for “Pauli blocking effect” in the figure.
Fig.5 Differential cross section dσ /d Q2 of ν ¯− p NCE scattering, with the anti neutrino energy being E ν ¯=0.65 GeV. Other description is the same as Fig.4.
Fig.6 Differential cross sections [dσ/dQ2]ν n→ν n with Eν=0.8 GeV and [ dσ /dQ 2]ν ¯ n→ν ¯n with Eν ¯=0.65 GeV. The ChPT and NuWro simulation results are represented by solid lines and blue diamonds, respectively. The red error bands are obtained by varying the LECs in their 1−σ uncertainties.
4.2 Total cross section
Fig.7 Contour plot of Q2 with varying Eν and cos⁡θ.
Fig.8 Total cross sections at different chiral orders. Our ChPT prediction is expected to be reliable up to Eν ( ν ¯)max= 0.28 GeV, indicated by the gray vertical line. For comparison, the simulation data produced by NuWro and GENIE events generators are also shown by red crosses and black diamonds, respectively.
5 Summary and outlook
References
Declarations
Acknowledgements
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Received	Accepted	Published
29 Mar 2024	28 Apr 2024	15 Dec 2024
Issue Date
19 Jun 2024

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Abstract

Graphical abstract

Keywords

Cite this article

1 Introduction

2 Basics of elastic neutrino−nucleon scattering

Fig.1 Kinematics of neutral current elastic neutrino-nucleon scattering under the one-boson exchange approximation.

Tab.1 Isospin factors for physical processes.

3 Hadronic form factors in baryon ChPT

3.1 Chiral effective Lagrangian

3.2 Form factors

Fig.2 Tree-level Feynman diagrams up to and including O(p3). The solid, dashed and wavy lines represent the nucleon, pions and the Z boson, in order. The circled numbers indicate the chiral orders of the vertices.

Fig.3 One-loop Feynman diagrams up to and including O(p3). The solid, dashed and wavy lines represent the nucleon, pions and the Z boson, in order. The circled numbers indicate the chiral orders of the vertices.

4 Numerical results and discussion

4.1 Differential cross section

Tab.2 Values of the LECs involved in neutrino−nucleon NCE scattering. The axial coupling constant g is dimensionless. The LECs ci and dj are in units of GeV−1 and GeV−2, respectively.

Fig.5 Differential cross section dσ/dQ2 of ν¯−p NCE scattering, with the anti neutrino energy being Eν¯=0.65 GeV. Other description is the same as Fig.4.

4.2 Total cross section

Fig.7 Contour plot of Q2 with varying Eν and cos⁡θ.

5 Summary and outlook

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References

Declarations

Acknowledgements

RIGHTS & PERMISSIONS

Fig.2 Tree-level Feynman diagrams up to and including $O (p^{3})$ . The solid, dashed and wavy lines represent the nucleon, pions and the $Z$ boson, in order. The circled numbers indicate the chiral orders of the vertices.

Fig.3 One-loop Feynman diagrams up to and including $O (p^{3})$ . The solid, dashed and wavy lines represent the nucleon, pions and the $Z$ boson, in order. The circled numbers indicate the chiral orders of the vertices.

Tab.2 Values of the LECs involved in neutrino−nucleon NCE scattering. The axial coupling constant $g$ is dimensionless. The LECs $c_{i}$ and $d_{j}$ are in units of GeV⁻¹ and GeV⁻², respectively.

Fig.5 Differential cross section $d σ / d Q^{2}$ of $\bar{ν}$ − $p$ NCE scattering, with the anti neutrino energy being $E_{\bar{ν}} = 0.65$ GeV. Other description is the same as Fig.4.

Fig.7 Contour plot of $Q^{2}$ with varying $E_{ν}$ and $\cos θ$ .