# Theory of QED radiative corrections to neutrino scattering at accelerator
energies

Oleksandr Tomalak  University of Kentucky, Department of Physics and
Astronomy, Lexington, KY 40506 USA  Fermilab, Theoretical Physics Department,
Batavia, IL 60510, USA  Theoretical Division, Los Alamos National Laboratory,
Los Alamos, NM 87545, USA  Qing Chen  University of Kentucky, Department of
Physics and Astronomy, Lexington, KY 40506 USA  Interdisciplinary Center for
Theoretical Study, University of Science and Technology of China, Hefei, Anhui
230026, China  Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026,
China  Richard J. Hill  University of Kentucky, Department of Physics and
Astronomy, Lexington, KY 40506 USA  Fermilab, Theoretical Physics Department,
Batavia, IL 60510, USA  Kevin S. McFarland  University of Rochester,
Department of Physics and Astronomy, Rochester, NY 14627, USA  Clarence Wret
University of Rochester, Department of Physics and Astronomy, Rochester, NY
14627, USA

###### Abstract

Control over quantum electrodynamics (QED) radiative corrections is critical
for precise determination of neutrino oscillation probabilities from observed
(anti)neutrino detection rates. It is particularly important to understand any
difference between such corrections for different flavors of (anti)neutrinos
in charged-current interactions. We provide theoretical foundations for
calculating these corrections. Using effective field theory, the corrections
are shown to factorize into soft, collinear, and hard functions. The soft and
collinear functions contain large logarithms in perturbation theory but are
computable from QED. The hard function parametrizes hadronic structure but is
free from large logarithms. Using a simple model for the hard function, we
investigate the numerical impact of QED corrections in charged-current
(anti)neutrino-nucleon elastic cross sections and cross-section ratios at GeV
energies. We consider the implications of mass singularity theorems that
govern the lepton-mass dependence of cross sections for sufficiently inclusive
observables and demonstrate the cancellation of leading hadronic and nuclear
corrections in phenomenologically relevant observables.

###### Contents

  1. 1 Introduction
  2. 2 Experimental signatures of events with photons
  3. 3 Static limit in (anti)neutrino-nucleon scattering
    1. 3.1 Lagrangian and leading-order cross section
    2. 3.2 Charged-current elastic process with one soft photon
    3. 3.3 Charged-current elastic process with one soft or collinear photon
    4. 3.4 Inclusive cross section in the static limit
  4. 4 Factorization
    1. 4.1 Factorization in the static limit
    2. 4.2 Factorization beyond the static limit
    3. 4.3 Soft function
    4. 4.4 Collinear function
    5. 4.5 Hard function
      1. 4.5.1 Invariant amplitudes
      2. 4.5.2 Hadronic model
      3. 4.5.3 Matching and renormalization group evolution
    6. 4.6 Factorization for inelastic processes
  5. 5 Inclusive cross sections
    1. 5.1 Hadronic model for hard real photon emission
    2. 5.2 Expansion in small lepton mass
  6. 6 Phenomenological applications
    1. 6.1 Cross section with charged lepton and soft photons
    2. 6.2 Electron-jet observable
    3. 6.3 Electronlike muon-jet events
    4. 6.4 Energetic photons isolated from charged lepton jets
    5. 6.5 Inclusive observables
    6. 6.6 Cross-section ratios
    7. 6.7 Leading nuclear effects
    8. 6.8 Comparison to (anti)neutrino-nucleus quasielastic data
  7. 7 Summary and outlook

##  1 Introduction

The program of studying neutrino oscillations to answer fundamental questions
about neutrinos requires precise interaction cross sections. In particular,
the ratio of cross sections for electron versus muon (anti)neutrinos is
critical to phenomenology and difficult to constrain by direct experimental
measurements. This ratio is impacted by electromagnetic radiative corrections
[1, 2], and it is thus imperative to compute these corrections and to
constrain their uncertainties. In this paper, we construct the theoretical
framework to determine radiative corrections to neutrino cross sections and
cross-section ratios.111For a concise summary of the theoretical formalism and
implications for $\nu_{e}/\nu_{\mu}$ ratios see Ref. [3]. We consider the
experimental conditions under which quantum electrodynamics (QED) radiative
corrections must be computed and discuss the practical implications for
neutrino oscillation analyses.

We formulate the problem of QED radiative corrections to (anti)neutrino
scattering processes in the framework of effective field theory. This allows a
clean separation between corrections that are rigorously computable within QED
and corrections that depend on hadronic physics. When the (anti)neutrino
energy $E_{\nu}$ is much larger than the charged lepton mass $m_{\ell}$ (i.e.,
$E_{\nu}\gg m_{\ell}$) the former, soft and collinear, corrections contain
flavor-dependent large logarithm enhancements, $\ln(E_{\nu}/m_{\ell})$; these
corrections depend on detailed experimental conditions but can be computed
perturbatively. The latter, hard, corrections are subject to hadronic
uncertainty but are independent of the charged lepton mass and cancel in
ratios of cross sections for different lepton flavors (i.e., electron and
muon) involving the same hadronic kinematics. Using effective field theory
methods, we resum large logarithms in perturbation theory and estimate the
uncertainty from neglected higher-order corrections. We investigate the
insensitivity of certain cross-section ratios to hadronic uncertainty at the
nucleon level. While the detailed incorporation of nucleon-level corrections
into nuclear cross sections is beyond the scope of this work, we also discuss
how nuclear effects impact these flavor ratios.

The paper is organized as follows. Section 2 provides an overview of the
experimental setups and analysis strategies for current and near-future
experiments with GeV energy (anti)neutrino beams, which are the primary focus
of our work. Section 3 studies the limit of a nonrelativistic nucleon,
$m_{\ell}\ll E_{\nu}\ll M$, where $M$ denotes the nucleon mass. This formal
limit is used to introduce a class of observables that incorporates real
photon radiation and to motivate the more general theoretical formalism that
follows. In Sec. 4, we discuss factorization for hard scattering exclusive
processes. We establish the relevant factorization theorem and evaluate the
separate contributions of soft, collinear, and hard photons. Section 5
considers the inclusion of noncollinear energetic photons in experimental
observables. Section 6 illustrates the phenomenological consequences of our
analysis; we study observables representing electron- and muon-induced
electromagnetic showers at neutrino detectors, present cross sections and
cross-section ratios, and compare them with existing experimental data.
Section 7 contains a summary and outlook.

##  2 Experimental signatures of events with photons

Our focus is on accelerator neutrino experiments with $E_{\nu}\sim
1{\rm\leavevmode\nobreak\ GeV}$. In the typical setup, a beam consisting
primarily of muon flavor neutrinos or antineutrinos is created, and one of the
goals in oscillation experiments is to determine the rate at which electron
flavor (anti)neutrinos are observed in the beam after propagating some
distance. In a simplified description, near detectors are used to constrain
(anti)neutrino flux and muon (anti)neutrino cross sections, while theory is
needed to determine the effect of differences between electron and muon
(anti)neutrino cross sections.

We consider two basic classes of observables: exclusive and inclusive.
Exclusive observables are subdivided into two general classes: first, the
charged-current elastic process accompanied by only soft photons of energy
smaller than some value $\Delta E$; second, the same as the first, but
including also energetic collinear photons, within a cone of angular size
$\Delta\theta$. In both cases, distributions are computed with respect to the
invariant momentum transfer $Q^{2}$ between the initial (anti)neutrino and the
final-state lepton jet. For small $\Delta E$ and small $\Delta\theta$, $Q^{2}$
may be identified with the hadronic momentum transfer for elastic
(anti)neutrino-nucleon scattering. For the inclusive case, we consider the
class of observables that also includes energetic noncollinear photons in the
cross section.

The parameter $\Delta E$ is a soft-photon cutoff parameter: photons with
energy smaller than $\Delta E$ are assumed to be unseen by the detector. The
appropriate choice of $\Delta E$ depends on the specific process and
observable under consideration and on properties of the detector. For example,
photons that interact repeatedly by Compton scattering in a diffuse way
throughout the detector, and photons that destructively interact by
$e^{+}e^{-}$ pair production and therefore appear spatially and directionally
local within the detector, will leave very different signatures. Photons in
the former class are effectively “invisible” as separate objects because they
do not leave a localized signature, although they might contribute to a
calorimetric measurement of nonlepton energy in some detectors. In polystyrene
scintillator or water, Compton scattering dominates the photon cross section
from energies between $20$ keV and $30$ or $25$ MeV, respectively, whereas in
liquid argon detectors, Compton scattering dominates photon interactions only
for energies between $150$ keV and $12$ MeV [4]. Therefore, $\Delta E$ will be
between ten and tens of MeV for experiments using these detectors.

The appropriate choice of the parameter $\Delta\theta$ depends on the flavor
of the (anti)neutrino and the detector technology. In nearly all detector
technologies deployed in oscillation experiments at accelerator energies,
electrons and positrons ($e^{\mp}$) from $\nu_{e}$ and $\bar{\nu}_{e}$
charged-current interactions are observed through electromagnetic showers,
i.e., a cascade of bremsstrahlung photons from electrons and positrons and
$e^{+}e^{-}$ pair production from these photons. Radiated photons that are
roughly collinear with the initial $e^{-}$ or $e^{+}$ will generally not be
distinguished from this electromagnetic shower and must be included as part of
the observable. For a given charged lepton direction, we consider this
observable to consist of all events with a specified total lepton jet energy,
after summing the charged lepton energy and the energy of photons within a
cone of angular size $\Delta\theta$. To estimate the appropriate
$\Delta\theta$, we consider the Molière radius of the electromagnetic shower
initiated by the primary $e^{\pm}$ and the length of the mean shower maximum
and find the angle which would place the photon within the Molière radius at
shower maximum [5]. This angle decreases logarithmically with the primary
$e^{\pm}$ energy. Consider for example a typical $500$ MeV $e^{\pm}$ energy;
for polystyrene scintillator, water, and liquid argon, the angle is
$9^{\circ}$, $10^{\circ}$, and $16^{\circ}$, respectively. For all three
materials, the angle is a factor $\sim 2$ lower at $3$ GeV $e^{\pm}$ energies.
Accordingly, a range of $\Delta\theta$ is interesting to consider: from
$5^{\circ}$ for the NO$\nu$A experiment, to $10^{\circ}$ for T2K, Hyper-K, and
DUNE, to $20^{\circ}$ for the short-baseline neutrino program at Fermilab [6].
For low-density tracking detectors in a magnetic field, such as in the
existing T2K and proposed DUNE near detectors, photons would generally not be
clustered together with the primary $e^{\pm}$.

In a tracking target detector, such as segmented scintillator, a gaseous
tracking detector in a magnetic field, or in a liquid argon time projection
chamber, the energy of final-state muons from $\nu_{\mu}$ or $\bar{\nu}_{\mu}$
interactions is typically determined using the range of the muon in the
detector or its curvature in the magnetic field. Accordingly, collinear
photons would usually not contribute to the reconstructed muon energy. On the
other hand, in a water Cherenkov detector, photons would contribute to the
observed energy associated with a muon if their angle from the muon direction
were consistent with multiple scattering over the range of the muon. We
determine the relevant cone size by computing the Gaussian core of multiple
scattering [5] for a muon. A similar criterion would hold for identification
of a photon in the detector; the photon typically would not be reconstructed
as distinct from the muon if it is emitted along the path of the muon. The
resulting angles are similar in the relevant materials (polystyrene
scintillator, water and liquid argon), with water 10% larger than polystyrene
and liquid argon another 10% larger. The angles depend weakly on the kinetic
energy of the muon, with angles of $\approx 2^{\circ}$ for the few hundred MeV
kinetic energies relevant for T2K in scintillator, dropping to $\approx
1.5^{\circ}$ at few GeV energies. Accordingly, a cone of size
$\Delta\theta=2^{\circ}$ around the muon is a reasonable separation between
photons in Cherenkov detectors222This analysis omits an important detail for
large water Cherenkov detectors such as Super-Kamiokande, where large
photosensors and scattering of Cherenkov light play significant roles in the
angular separation between adjacent features [7]. Such effects in a real
Cherenkov detector could result in merging of angular features on scales
significantly larger than the $2^{\circ}$ inherent resolution calculated
above, but a detailed detector simulation would be required to assess this
effect. that would be identified as collinear with muons and those which could
be seen as separate particles in the reconstruction.

##  3 Static limit in (anti)neutrino-nucleon scattering

Figure 1: Kinematics of charged-current elastic scattering for neutrino (left)
and antineutrino (right).

The tree-level neutrino scattering process is displayed in Fig. 1:

$\displaystyle\nu_{\ell}(p)+n(k)\to\ell^{-}(p^{\prime})+p(k^{\prime})\,.$ (1)

We consider also the related antineutrino process:

$\displaystyle\bar{\nu}_{\ell}(p)+p(k)\to\ell^{+}(p^{\prime})+n(k^{\prime})\,.$
(2)

It is instructive to first study charged-current elastic scattering in the
formal static limit, $m_{\ell}\ll E_{\nu}\ll M$. This is an important exactly
calculable limit for the more general case involving nontrivial hadronic
structure, analogous to the well-known McKinley-Feshbach correction for
electron-proton scattering [8, 9]. Beyond this formal utility, however, the
static limit captures the leading logarithmically enhanced contributions to
radiative corrections in perturbation theory and thus correctly describes the
leading logarithm approximation for flavor ratios of interest to neutrino
oscillation experiments. In the following subsections, we compute this
benchmark cross section.

###  3.1 Lagrangian and leading-order cross section

The static limit $M\to\infty$, as well as $E_{\nu}/M$ power corrections, may
be computed systematically in nonrelativistic (NR) effective field theory. The
appropriate effective theory consists of NRQED for relativistic charged
leptons and nonrelativistic nucleons [10, 11, 12, 13, 14], supplemented by the
four-fermion interaction between leptons and heavy nucleons:

$\displaystyle{\cal L}_{\rm eff}$ $\displaystyle=-\sqrt{2}\mathrm{G}_{\rm
F}V_{ud}\,\bar{\ell}\gamma^{\mu}\mathrm{P}_{\mathrm{L}}\nu_{\ell}\,\bar{h}_{v}^{(p)}\gamma_{\mu}\big{[}c_{V}+c_{A}\gamma_{5}\big{]}h_{v}^{(n)}+{\rm
h.c.}\,,$ (3)

with $\mathrm{P}_{\mathrm{L}}=\left(1-\gamma_{5}\right)/2$. Here $h_{v}^{(p)}$
and $h_{v}^{(n)}$ denote heavy particle fields for the proton and neutron,
respectively [10, 15, 16, 17]; subscript $v$ denotes the laboratory frame
reference vector $v^{\mu}=(1,0,0,0)$. We have expressed the operator
coefficients in terms of the scale-independent Fermi constant $\mathrm{G}_{\rm
F}$ and the Cabibbo-Kobayashi-Maskawa(CKM) matrix element $V_{ud}$. At tree
level, the effective operator coefficients are related to familiar nucleon
structure parameters: $c_{V}\to g_{V}\approx 1$ and $c_{A}\to
g_{A}\approx-1.27$. Beyond tree level, we define operators and coefficients in
the $\overline{\rm MS}$ scheme at renormalization scale $\mu$. The anomalous
dimension of both effective operators is readily calculated:

$\displaystyle\frac{\mathrm{d}\ln
c_{i}}{\mathrm{d}\ln\mu^{2}}=-\frac{3}{8}\frac{\alpha}{\pi}+{\cal
O}\left(\alpha^{2}\right).$ (4)

The tree-level differential cross section for the processes (1) and (2) is
given by [18, 19]

$\displaystyle\frac{\mathrm{d}\sigma_{\rm LO}}{\mathrm{d}Q^{2}}$
$\displaystyle=\frac{\mathrm{G}_{\rm
F}^{2}|V_{ud}|^{2}}{2\pi}\bigg{[}c_{V}^{2}+c_{A}^{2}-(c_{V}^{2}-c_{A}^{2})\frac{Q^{2}+m^{2}_{\ell}}{4E_{\nu}^{2}}\bigg{]}\,,$
(5)

where $Q^{2}=-(p^{\prime}-p)^{2}$ is the momentum transfer between initial and
final lepton states. The total cross section at leading order is

$\displaystyle\sigma_{\rm LO}$ $\displaystyle=\frac{\mathrm{G}_{\rm
F}^{2}|V_{ud}|^{2}}{\pi}\left(c_{V}^{2}+3c_{A}^{2}\right)E_{\nu}\sqrt{E_{\nu}^{2}-m_{\ell}^{2}}\,.$
(6)

Note that Eqs. (5) and (6) are valid for arbitrary lepton mass; below we will
focus on the limit $m_{\ell}/E_{\nu}\to 0$. The cross sections for neutrino
and antineutrino scattering are identical in the static limit.

###  3.2 Charged-current elastic process with one soft photon

Figure 2: Ratio of one-loop cross section to the tree-level result in the
static limit, for scattering of electron (anti)neutrinos (left plot) and muon
(anti)neutrinos (right plot). The observable including only the radiation of
one soft photon of energy below $\Delta E=20\leavevmode\nobreak\ \mathrm{MeV}$
is shown by the blue solid curve. The correction including also radiation of
one collinear hard photon is represented by the red dashed line, with cone
sizes $\Delta\theta=10^{\circ}$ and $2^{\circ}$ appropriate for electrons and
muons, respectively. Inclusive cross sections are shown by the green dotted
line. All curves are shown with the choice of the scale corresponding to the
physical nucleon mass $\mu=M$.

In the static limit, radiative corrections to the tree-level result in Eq. (5)
can be computed using the Feynman rules of the effective Lagrangian of Eq.
(3). Consider first the ${\cal O}(\alpha)$ correction to the observable that
does not distinguish between the elastic process and the process with
radiation of one photon with energy below $\Delta E$:

$\mathrm{d}\sigma^{\rm soft}=\left[1+\frac{\alpha}{\pi}\delta^{\rm
soft}\right]\mathrm{d}\sigma_{\mathrm{LO}}\,.$ (7)

The correction $\delta^{\rm soft}$ is the sum of virtual corrections
$\delta_{v}$ and real soft-photon corrections $\delta_{s}$. Note that the
superscript “soft” here denotes the observable including radiation of only
soft photons and does not imply restriction to the soft region of virtual
diagrams. In the limit of small charged lepton mass $m_{\ell}/E_{\nu}\ll 1$,
we have

$\displaystyle\delta_{v}$
$\displaystyle=2\left(1-\ln\frac{2E_{\nu}}{m_{\ell}}\right)\ln\frac{m_{\ell}}{\lambda_{\gamma}}-\ln^{2}\frac{2E_{\nu}}{m_{\ell}}+\ln\frac{2E_{\nu}}{m_{\ell}}+\frac{3}{4}\ln\frac{\mu^{2}}{m^{2}_{\ell}}+\frac{5}{6}\pi^{2}\,,$
(8) $\displaystyle\delta_{s}$
$\displaystyle=2\left(1-\ln\frac{2E_{\nu}}{m_{\ell}}\right)\ln\frac{\lambda_{\gamma}}{2\Delta
E}-\ln^{2}\frac{2E_{\nu}}{m_{\ell}}+\ln\frac{2E_{\nu}}{m_{\ell}}+1-\frac{1}{6}\pi^{2}\,,$
(9)

with the photon mass regulator $\lambda_{\gamma}$. The total correction is
thus

$\displaystyle\delta^{\rm
soft}=\delta_{s}+\delta_{v}=2\left(1-\ln\frac{2E_{\nu}}{m_{\ell}}\right)\ln\frac{E_{\nu}}{\Delta
E}+\frac{3}{4}\ln\frac{\mu^{2}}{m^{2}_{\ell}}+1+\frac{2}{3}\pi^{2}\,,$ (10)

where, as required, the result does not depend on the photon mass regulator
$\lambda_{\gamma}$ [20, 21, 22, 23]. This correction is shown as a function of
(anti)neutrino energy by the blue solid line in Fig. 2.333It is interesting to
note the cancellation of terms in Eq. (10) involving double logarithms in the
lepton mass, $\sim\ln^{2}(m_{\ell})$. This is a consequence of the
Kinoshita–Lee–Nauenberg (KLN) theorem [20, 21, 22, 23] and finiteness at
$m_{\ell}\to 0$ of the cross section including real collinear radiation. The
cross-section corrections are larger in the case of the electron flavor, which
is subject to larger kinematic logarithms.

###  3.3 Charged-current elastic process with one soft or collinear photon

Let us similarly consider the charged-current elastic observable that includes
energetic collinear radiation. For definiteness, the cone observable is
defined by fixing the charged lepton direction and allowing collinear
radiation within $\Delta\theta$ of this fixed direction. An explicit
computation yields the relative correction $\delta^{\mathrm{jet}}$, including
soft and virtual parts as above, to the leading-order cross section:

$\displaystyle\mathrm{d}\sigma^{\mathrm{jet}}$
$\displaystyle=\left(1+\frac{\alpha}{\pi}\delta^{\mathrm{jet}}\right)\mathrm{d}\sigma^{\mathrm{LO}},$
(11) $\displaystyle\delta^{\mathrm{jet}}$
$\displaystyle=2\left(1-\ln\frac{2E_{\nu}}{m_{\ell}}\right)\ln\frac{E_{\nu}}{\Delta
E}+\frac{3}{4}\ln\frac{\mu^{2}}{m^{2}_{\ell}}+\frac{13}{4}+\frac{2\pi^{2}}{3}+\left(\ln\left(1+\eta^{2}\right)-\frac{\eta^{2}}{1+\eta^{2}}\right)\ln\frac{E_{\nu}}{\Delta
E}-\left(\mathrm{tan}^{-1}\eta\right)^{2}\,$
$\displaystyle+\frac{1}{2}\mathrm{Li}_{2}\left(-\eta^{2}\right)+\frac{1}{4}\ln^{2}\left(1+\eta^{2}\right)+\frac{1}{4}\left(\frac{2}{1+\eta^{2}}-\frac{1}{\eta^{2}}-3\right)\ln\left(1+\eta^{2}\right)-\left(1+\frac{1}{1+\eta^{2}}\right)\frac{\mathrm{tan}^{-1}\eta}{\eta}\,,$
(12)

with the dimensionless parameter $\eta=\Delta\theta E_{\nu}/m_{\ell}$. At
$\Delta\theta\to 0$, the cross section reduces smoothly to the soft-photon-
only result, $\delta^{\rm jet}\to\delta^{\rm soft}$. In the limit of very
small charged lepton mass $m_{\ell}/E_{\nu}\ll\Delta\theta$, the cross section
becomes

$\delta^{\mathrm{jet}}\underset{m_{\ell}\ll\Delta\theta
E_{\nu}}{\longrightarrow}\left(1+\ln\frac{\left(\Delta\theta\right)^{2}}{4}\right)\ln\frac{E_{\nu}}{\Delta
E}-\frac{3}{4}\ln\frac{\left(\Delta\theta\right)^{2}}{4}+\frac{3}{4}\ln\frac{\mu^{2}}{4E^{2}_{\nu}}+\frac{\pi^{2}}{3}+\frac{13}{4}.$
(13)

Note that, in contrast to Eq. (10), the collinear singularity $\sim\ln
m_{\ell}$ in Eq. (13) is regulated by the jet size $\Delta\theta$. Flavor
dependence of radiative corrections enters via the parameter $\eta$ appearing
in Eq. (12) but cancels in the massless limit $m_{\ell}\to 0$, in agreement
with the KLN theorem [20, 21, 22, 23]. The correction to the jet observable in
Eq. (12) is shown as a function of (anti)neutrino energy by the red dashed
line in Fig. 2.

###  3.4 Inclusive cross section in the static limit

Finally, let us describe the contribution of noncollinear energetic photons.
Integrating over the total phase space for photons above $\Delta E$ in energy,
we obtain the real hard-photon contribution:

$\displaystyle\sigma_{h}=\frac{\alpha}{\pi}\delta_{h}\sigma_{\mathrm{LO}}\,,\qquad\delta_{h}=-2\left(1-\ln\frac{2E_{\nu}}{m_{\ell}}\right)\ln\frac{E_{\nu}}{\Delta
E}-\frac{3}{2}\ln\frac{2E_{\nu}}{m_{\ell}}+\frac{13}{4}-\frac{\pi^{2}}{3}.$
(14)

The resulting inclusive cross section $\sigma^{\rm tot}$ is factorizable:

$\sigma^{\rm tot}=\left(1+\frac{\alpha}{\pi}\delta^{\rm
tot}\right)\sigma_{\mathrm{LO}}.$ (15)

In the limit of small charged lepton mass $m_{\ell}/E_{\nu}\ll 1$, the
correction $\delta^{\rm tot}$ does not depend on lepton flavor and is given by

$\displaystyle\delta^{\rm tot}$
$\displaystyle=\delta_{v}+\delta_{s}+\delta_{h}=\frac{3}{4}\ln\frac{\mu^{2}}{4E^{2}_{\nu}}+\frac{17}{4}+\frac{\pi^{2}}{3}.$
(16)

This simple calculation provides another example (cf. [24]) of the
cancellation of Sudakov double logarithms [25] considering the inclusive cross
section with both soft and hard radiation. We present the ratio of the
inclusive cross section to the tree-level result in Fig. 2 by the green dotted
line and observe a cancellation of virtual and real contributions entering the
cross section with opposite signs. Lepton-mass corrections enter with
$m^{2}_{\ell}$ suppression in agreement with the KLN theorem [20, 21, 22,
23].444The correction to the total cross section of Eq. (6) with a finite
charged lepton mass can be written in terms of the charged lepton velocity
$\beta$ as
$\delta=\left(4+\frac{2}{\beta}\ln\frac{1+\beta}{1-\beta}\right)\ln\frac{1+\beta}{2\beta}+\frac{4}{\beta}\mathrm{Li}_{2}\frac{1-\beta}{1+\beta}+\frac{7-16\beta+3\beta^{2}}{8\beta}\ln\frac{1+\beta}{1-\beta}+\frac{3}{4}\ln\frac{\mu^{2}}{m^{2}_{\ell}}+\frac{17}{4}+\frac{\pi^{2}}{3\beta}.$
(17)

In the static limit, there is no dependence on hadronic structure (beyond the
numerical values of $c_{V}$ and $c_{A}$), and hence the entire calculation may
be performed perturbatively. For the general case, where $E_{\nu}$ is not
small compared to hadronic scales, it is imperative to disentangle the hard
scale, containing nonperturbative hadronic physics, from the perturbative soft
and collinear scales.

##  4 Factorization

The process under consideration is charged-current (anti)neutrino-nucleon
elastic scattering: $\nu_{\ell}n\to\ell^{-}p(\gamma)$,
$\bar{\nu}_{\ell}p\to\ell^{+}n(\gamma)$, where “$(\gamma)$” denotes possible
photon emission. Our default observable includes only soft or collinear
radiation of real photons, with the precise definition for soft and collinear
given in Sec. 2. Throughout the analysis, we consider fixed initial- and
final-state nucleon momenta. In a final step for phenomenological
applications, we may integrate over the final-state nucleon kinematics.

For both virtual and real radiative corrections to the tree-level process, we
may decompose the total momentum integration into regions. It is convenient to
work in the light-cone basis, decomposing an arbitrary four-vector $L^{\mu}$
as

$\displaystyle L^{\mu}=n\cdot L\frac{\bar{n}^{\mu}}{2}+\bar{n}\cdot
L\frac{n^{\mu}}{2}+L_{\perp}^{\mu}\equiv
L_{+}^{\mu}+L_{-}^{\mu}+L_{\perp}^{\mu}\leftrightarrow(n\cdot L,\,\bar{n}\cdot
L,\,L_{\perp})\,,$ (18)

where $n^{\mu}$ and $\bar{n}^{\mu}$ are conventional four-vectors satisfying
$n^{2}=\bar{n}^{2}=0$ and $n\cdot\bar{n}=2$. For a charged lepton moving in
the $z$ direction, we choose $n^{\mu}=(1,0,0,1)$ and
$\bar{n}^{\mu}=(1,0,0,-1)$, with the four-velocity of the initial-state
nucleon in its rest frame $v^{\mu}=(n^{\mu}+\bar{n}^{\mu})/2=(1,0,0,0)$. The
relevant momentum regions are

$\displaystyle{\rm soft:}\quad$ $\displaystyle
L^{\mu}\sim\Lambda(\lambda,\lambda,\lambda)\,,$ $\displaystyle{\rm
collinear:}\quad$ $\displaystyle
L^{\mu}\sim\Lambda(\lambda,1,\sqrt{\lambda})\,,$ $\displaystyle{\rm
hard:}\quad$ $\displaystyle L^{\mu}\sim\Lambda(1,1,1)\,,$ (19)

where the hard scale is $\Lambda\sim M\sim E_{\nu}\sim Q$. The dimensionless
expansion parameter $\lambda\sim\Delta E/\Lambda$ is determined by
experimental conditions that dictate the fraction of energy allowed in soft
radiation. The lepton mass satisfies $m_{\ell}\lesssim\sqrt{\lambda}\Lambda$,
and the jet angular resolution satisfies $\Delta\theta\lesssim\sqrt{\lambda}$.

In our default factorization analysis represented by Eq. (28) below, we employ
the formal power counting $m_{\ell}/\Lambda_{\rm hard}\ll 1$. For the muon, an
alternative counting $\Delta E\ll m_{\mu}\sim\Lambda_{\rm hard}$ would group
the muon mass with other “hard” scales, resulting in a simpler soft-hard
factorization formula without collinear function. Since “large logarithms” are
not extremely large for the muon, it may seem preferable to adopt this simpler
description. However, muon power corrections are well described by our power-
counting analysis and are numerically small compared to other uncertainties.
Moreover, important phenomenological features, such as the insensitivity of
flavor ratios to hadronic and nuclear uncertainties, can be readily seen by
adopting a unified description of electron and muon observables. A more
detailed effective theory analysis of soft-hard versus soft-collinear-hard
factorization is left to future work.

###  4.1 Factorization in the static limit

$\displaystyle=\parbox{113.81102pt}{\includegraphics[width=86.72267pt]{hard.pdf}}+\parbox{113.81102pt}{\includegraphics[width=86.72267pt]{soft.pdf}}+\parbox{113.81102pt}{\includegraphics[width=86.72267pt]{collinear.pdf}}$
Figure 3: Decomposition of full theory diagram into hard, soft and collinear
momentum regions.

In preparation for the more general case, let us examine the contributions
from different momentum regions to the cross section in the static limit. As
an example, consider the one-loop virtual correction depicted in Fig. 3 (in
Feynman gauge, with photon mass regulator $\lambda_{\gamma}$):

$\displaystyle{\cal M}$
$\displaystyle=ie^{2}\int\frac{d^{d}L}{(2\pi)^{d}}\frac{1}{L^{2}+2L\cdot
p}\frac{1}{-v\cdot
L}\frac{1}{L^{2}-\lambda_{\gamma}^{2}}\,v\\!\\!\\!/\big{(}L\\!\\!\\!/+p\\!\\!\\!/+m_{\ell}\big{)}\Gamma_{\ell}\otimes\Gamma_{h}\,,$
(20)

where the tree-level amplitude from Eq. (3) is conveniently expressed as the
product of leptonic $\Gamma_{\ell}$ and hadronic $\Gamma_{h}$ Dirac
structures: ${\cal M}^{\rm tree}=\Gamma_{\ell}\otimes\Gamma_{h}$. The hard
contribution to the scattering amplitude, ${\cal M}_{\mathrm{H}}$, is given in
dimensional regularization by setting $m_{\ell}=0$:

$\displaystyle{\cal M}_{\mathrm{H}}$
$\displaystyle=ie^{2}\int\frac{d^{d}L}{(2\pi)^{d}}\frac{1}{L^{2}+2L\cdot
p}\frac{1}{-v\cdot
L}\frac{1}{L^{2}}\,v\\!\\!\\!/\big{(}L\\!\\!\\!/+p\\!\\!\\!/\big{)}\Gamma_{\ell}\otimes\Gamma_{h}\bigg{|}_{p^{2}=0}\,.$
(21)

The soft contribution ${\cal M}_{\mathrm{S}}$ is

$\displaystyle{\cal M}_{\mathrm{S}}$
$\displaystyle=ie^{2}\int\frac{d^{d}L}{(2\pi)^{d}}\frac{1}{2L\cdot
p}\frac{1}{-v\cdot
L}\frac{1}{L^{2}-\lambda^{2}}\,v\\!\\!\\!/\big{(}p\\!\\!\\!/+m_{\ell}\big{)}\Gamma_{\ell}\otimes\Gamma_{h}\bigg{|}_{p^{2}=m_{\ell}^{2}}\,.$
(22)

Finally, the remaining collinear contribution ${\cal M}_{\mathrm{J}}$ is

$\displaystyle{\cal M}_{\mathrm{J}}$
$\displaystyle=ie^{2}\int\frac{d^{d}L}{(2\pi)^{d}}\frac{1}{L^{2}+2L\cdot
p}\frac{1}{-v\cdot
L_{-}}\frac{1}{L^{2}}\,v\\!\\!\\!/\big{(}L\\!\\!\\!/+p\\!\\!\\!/+m_{\ell}\big{)}\Gamma_{\ell}\otimes\Gamma_{h}\bigg{|}_{p^{2}=m_{\ell}^{2}}\,.$
(23)

It is straightforward to evaluate these integrals explicitly and to include
the soft and collinear contributions to wave-function renormalization. The
radiative corrections to the cross section of Eq. (7) are found to be

$\displaystyle\delta^{\rm soft}$
$\displaystyle=\delta_{\mathrm{S}}+\delta_{\mathrm{J}}+\delta_{\mathrm{H}}\,,$
(24)

where (after $\overline{\rm MS}$ renormalization 555The bare and renormalized
couplings are related as $\frac{e_{\rm
bare}^{2}}{4\pi}(4\pi)^{\epsilon}e^{-\gamma_{E}\epsilon}=\alpha_{\rm
bare}=\mu^{2\epsilon}\alpha(\mu)[1+{\cal O}(\alpha)]$. and including real
soft-photon radiation) the hard ($\delta_{\mathrm{H}}$), soft
$(\delta_{\mathrm{S}}$) and collinear ($\delta_{\mathrm{J}})$ contributions
are, respectively,

$\displaystyle\delta_{\mathrm{H}}$
$\displaystyle=-\frac{1}{4}\ln^{2}\frac{4E^{2}_{\nu}}{\mu^{2}}+\frac{1}{2}\ln\frac{4E^{2}_{\nu}}{\mu^{2}}-1+\frac{19\pi^{2}}{24}\,,$
$\displaystyle\delta_{\mathrm{S}}$
$\displaystyle=\left(1-\ln\frac{2E_{\nu}}{m_{\ell}}\right)\left(\ln\frac{\mu^{2}}{\left(\Delta
E\right)^{2}}+\ln\frac{2E_{\nu}}{m_{\ell}}\right)+1-\frac{\pi^{2}}{6}\,,$
$\displaystyle\delta_{\mathrm{J}}$
$\displaystyle=\frac{1}{4}\ln^{2}\frac{\mu^{2}}{m^{2}_{\ell}}+\frac{1}{4}\ln\frac{\mu^{2}}{m^{2}_{\ell}}+1+\frac{\pi^{2}}{24}\,.$
(25)

For the observable that includes energetic photon emission within angle
$\Delta\theta$ of the charged lepton direction, radiative corrections in Eq.
(12) are given by

$\displaystyle\delta^{\rm jet}$
$\displaystyle=\delta_{\mathrm{S}}+\delta_{\mathrm{J}}+\delta_{\mathrm{J}}^{\prime}+\delta_{\mathrm{H}}\,,$
(26)

where the additional contribution from the radiation of the real photon is
obtained from collinear photon Feynman diagrams and is given explicitly from
Eqs. (12) and (10) by

$\displaystyle\delta_{\mathrm{J}}^{\prime}=\delta^{\rm jet}-\delta^{\rm soft}$
$\displaystyle=\frac{9}{4}+\left(\ln\left(1+\eta^{2}\right)-\frac{\eta^{2}}{1+\eta^{2}}\right)\ln\frac{E_{\nu}}{\Delta
E}-\left(\mathrm{tan}^{-1}\eta\right)^{2}-\left(1+\frac{1}{1+\eta^{2}}\right)\frac{\mathrm{tan}^{-1}\eta}{\eta}\,$
$\displaystyle+\frac{1}{2}\mathrm{Li}_{2}\left(-\eta^{2}\right)+\frac{1}{4}\ln^{2}\left(1+\eta^{2}\right)+\frac{1}{4}\left(\frac{2}{1+\eta^{2}}-\frac{1}{\eta^{2}}-3\right)\ln\left(1+\eta^{2}\right)\,.$
(27)

We see that $\delta_{\mathrm{H}}$ contains no large logarithms when $\mu$ is
of order of the hard scale, $\mu\sim E_{\nu}={\cal O}(\Lambda)$. Similarly,
$\delta_{\mathrm{J}}$ contains no large logarithms when $\mu$ is of order of
the collinear scale, $\mu\sim m_{\ell}={\cal O}(\sqrt{\lambda}\Lambda)$. The
correction $\delta_{\mathrm{J}}^{\prime}$ contains a logarithm
$\ln(E_{\nu}/\Delta E)$, as required for cancellation of $\ln{m_{\ell}}$
singularities [22, 23] in the limit $m_{\ell}\to 0$. The expected eikonal
logarithms remain in $\delta_{\mathrm{S}}$ when $\mu$ is of order of the soft
scale, $\mu\sim\Delta E={\cal O}(\lambda\Lambda)$. We have thus achieved a
scale separation, isolating the contributions from each of the relevant hard,
collinear and soft momentum regions. Below, we state the general factorization
theorem beyond the static limit. The jet and soft functions are
straightforward generalizations of the static limit quantities computed above
and reduce to $\delta_{\mathrm{S}}$, $\delta_{\mathrm{J}}$, and
$\delta_{\mathrm{J}}^{\prime}$ in the appropriate limits. The hard function
becomes a nonperturbative quantity that we describe using phenomenological
form factors. In the formal static limit $E_{\nu}/M\to 0$ (and
$m_{\ell}/E_{\nu}\sim\Delta\theta\ll 1)$, the hard function would reduce to
$\delta_{\mathrm{H}}$ in Eq. (25), including the appropriate combination of
Wilson coefficients $c_{V}$ and $c_{A}$ from Eq. (3).

###  4.2 Factorization beyond the static limit

The above analysis of momentum regions, formalized in soft-collinear effective
theory (SCET), does not rely on the static limit and can be straightforwardly
generalized. The key difference for $E_{\nu}\sim M$ is that the hard function
becomes sensitive to hadronic structure and cannot be computed in perturbation
theory. Including also the general kinematic dependence in the soft and
collinear factors, we have the following factorization theorem for the cross
section:666Our model of Sec. 4.5.2 provides an explicit demonstration of the
factorization theorem at one-loop order. It reproduces correctly the soft and
collinear functions, leaving a hard function independent of IR scales.

$\frac{d\sigma}{dQ^{2}}\propto
H\left(\frac{E_{\nu}}{M},\frac{Q^{2}}{M^{2}},\frac{\mu}{M}\right)\bigg{[}J\left(\frac{\mu}{m_{\ell}}\right)R\left(\frac{\mu}{m_{\ell}},v_{\ell}\cdot
v_{p}\right)S\left(\frac{\mu}{\Delta E},v_{\ell}\cdot v_{p},v\cdot
v_{\ell},v\cdot v_{p}\right)\\\ +\int_{0}^{1-\frac{\Delta E}{E_{\ell}^{\rm
tree}}}dx\,j\left(\frac{\mu}{m_{\ell}},x,v\cdot
v_{\ell}\Delta\theta\right)R\left(\frac{\mu}{m_{\ell}},xv_{\ell}\cdot
v_{p}\right)S\left(\frac{\mu}{\Delta E},x\,v_{\ell}\cdot v_{p},x\,v\cdot
v_{\ell},v\cdot v_{p}\right)\bigg{]}\,,$ (28)

valid up to power corrections in $\lambda$. Here $v_{\ell}$ and $v_{p}$ denote
the charged lepton and proton four-velocities, respectively, entering the
tree-level process, $Q^{2}=-(k^{\prime}-k)^{2}=2M^{2}(v_{n}\cdot v_{p}-1)$ is
the momentum transfer between initial and final nucleon states, $E_{\ell}^{\rm
tree}=E_{\nu}+M-v\cdot k^{\prime}=m_{\ell}v\cdot v_{\ell}$ is the lepton
energy for the tree-level process, and $x=E_{\ell}/E_{\ell}^{\rm tree}$
denotes the fraction of the total jet energy carried by the charged lepton
(the total jet energy is defined as the energy carried by the charged lepton
plus collinear photons). The quantities $\Delta E$ and $\Delta\theta$ denote
soft energy and angular acceptance parameters depending on experimental
conditions, and $\mu$ defines the renormalization scale. For completeness, we
have included the remainder function $R$ in the factorization formula [17];
this function relates the running electromagnetic coupling in the QED theory
with and without the dynamical charged lepton $\ell$ and contributes
numerically small corrections starting at two-loop order. All energy
components are taken in the laboratory frame. To obtain various observable
predictions, the expression in Eq. (28), for appropriate choices of $\Delta E$
and $\Delta\theta$, can be integrated with respect to $Q^{2}$. The second term
in square brackets, involving $j$, is omitted for observables with soft-photon
radiation only.777For simplicity, we define the jet to contain a single
charged lepton plus any number of collinear or soft photons, ignoring
contributions of additional $e^{+}e^{-}$ pairs generated by “internal”
conversion of a virtual photon; such effects are of two-loop order. The region
of $x$ integration, $x=0..1-\Delta E/E_{\ell}^{\rm tree}$ results from
$x=m_{\ell}/E_{\ell}^{\rm tree}..1$, upon omitting the region near $x=1$ that
is contained in the soft-photon term and noticing that the region at $x=0$ is
power suppressed. The soft function multiplying $j$ corresponds to the charged
lepton of energy $E_{\ell}=xE_{\ell}^{\rm tree}$; the replacement $v\cdot
v_{\ell}\to xv\cdot v_{\ell}$ is by definition exact, and the replacement
$v_{\ell}\cdot v_{p}\to xv_{\ell}\cdot v_{p}$ is valid up to power
corrections. The integration region near $x=1$, corresponding to soft photons,
is described exactly up to power corrections in our explicit expressions for
$j$, which use only the approximation $m_{\ell}/E_{\ell}\ll 1$.

The factorization formula of Eq. (28) is valid up to power-suppressed
contributions in the power-counting parameter $\lambda$: $\lambda\sim\Delta
E/\Lambda_{\mathrm{hard}}\sim
m_{\ell}^{2}/\Lambda_{\mathrm{hard}}^{2}\sim\left(\Delta\theta\right)^{2}$,
with the hard scale $\Lambda_{\mathrm{hard}}\sim M\sim E_{\nu}\sim Q$ in
general. In this paper, we include lepton-mass corrections in tree-level
cross-section expressions. Consequently, all neglected power-suppressed
contributions are additionally suppressed by electromagnetic coupling constant
$\alpha$, although logarithmic enhancements are still possible. The power-
counting parameter turns out to be at or below the percent size at accelerator
neutrino experiments for all three determining quantities, as one can estimate
from discussions of experimental details in Sec. 2. For example, for photon
energy resolution $\Delta E=20\leavevmode\nobreak\ \mathrm{MeV}$ and cone size
$\Delta\theta=10^{\circ}$, we have $\Delta E/\Lambda_{\rm hard}\sim 0.02$,
$\left(\Delta\theta\right)^{2}\sim 0.03$,
$m_{e}^{2}/\Lambda_{\mathrm{hard}}^{2}\sim 3\times 10^{-7}$, and
$m_{\mu}^{2}/\Lambda_{\mathrm{hard}}^{2}\sim 0.01$, assuming
$\Lambda_{\mathrm{hard}}\sim 1\leavevmode\nobreak\ \mathrm{GeV}$.

In the following sections, we compute the separate soft ($S$), collinear
($J$), and hard ($H$) functions appearing in the factorization formula. We
then collect results and compute several illustrative observables.

###  4.3 Soft function

The soft function appearing in Eq. (28) accounts for physics below the energy
scale of particle masses. It is independent of hadronic structure and depends
only on particle velocities, electric charges, the energy cutoff variable
$\Delta E$ and the renormalization scale $\mu$. Including virtual soft
corrections and real soft-photon radiation with energy below $\Delta E$ yields
the process-independent soft function through one-loop order [17]:

$\displaystyle S\left(\frac{\mu}{\Delta E},\,v_{\ell}\cdot v_{p},\,v\cdot
v_{\ell},\,v\cdot
v_{p}\right)=1+\frac{\alpha}{\pi}\bigg{[}2\bigg{(}1-v_{\ell}\cdot
v_{p}f(v_{\ell}\cdot v_{p})\bigg{)}\ln\frac{\mu}{2\Delta
E}+G\left(v_{\ell}\cdot v_{p},v\cdot v_{\ell},v\cdot v_{p}\right)\bigg{]},$
(29)

where $v^{\mu}$ defines the laboratory frame in which $\Delta E$ is measured
[by default $v^{\mu}=(1,0,0,0)$], $v_{\ell}^{\mu}=p^{\prime\mu}/m_{\ell}$ and
$v_{p}^{\mu}$ are the charged lepton and proton velocity vectors,
respectively, and the functions $f$ and $G$ are given, respectively, by [26,
17]

$\displaystyle f(w)$ $\displaystyle=\frac{\ln w_{+}}{\sqrt{w^{2}-1}}\,,$
$\displaystyle G(w,x,y)$
$\displaystyle=\frac{x}{\sqrt{x^{2}-1}}\ln{x_{+}}+\frac{y}{\sqrt{y^{2}-1}}\ln{y_{+}}+\frac{w}{\sqrt{w^{2}-1}}\bigg{[}\ln^{2}x_{+}-\ln^{2}y_{+}$
$\displaystyle\quad+{\rm
Li}_{2}\left(1-\frac{x_{+}}{\sqrt{w^{2}-1}}(w_{+}x-y)\right)+{\rm
Li}_{2}\left(1-\frac{x_{-}}{\sqrt{w^{2}-1}}(w_{+}x-y)\right)$
$\displaystyle\quad-{\rm
Li}_{2}\left(1-\frac{y_{+}}{\sqrt{w^{2}-1}}(x-w_{-}y)\right)-{\rm
Li}_{2}\left(1-\frac{y_{-}}{\sqrt{w^{2}-1}}(x-w_{-}y)\right)\bigg{]}\,,$ (30)

with $a_{\pm}=a\pm\sqrt{a^{2}-1}$. The soft function of Eq. (29) applies to
both the neutrino process in Eq. (1), where $v_{p}$ denotes the final-state
proton velocity, and to the antineutrino process in Eq. (2), where $v_{p}$
denotes the initial-state proton velocity. The soft function is infrared (IR)
finite and is universal for the different operator structures contributing to
the hard function in Eq. (28). We remark that for $v\cdot v_{p}\to 1$, i.e.,
when the proton is at rest in the laboratory frame, the soft function reduces
to [24]

$\displaystyle S\left(\frac{\mu}{\Delta E},\,w,\,w,\,1\right)=1$
$\displaystyle+\frac{\alpha}{\pi}\bigg{[}1+2\left(1-\frac{1}{2\beta}\ln\frac{1+\beta}{1-\beta}\right)\ln\frac{\mu}{2\Delta
E}+\frac{1}{2\beta}\ln\frac{1+\beta}{1-\beta}\left(1+\ln\frac{1+\beta}{4w\beta^{2}}\right)\,$
$\displaystyle+\frac{1}{\beta}\left(\mathrm{Li}_{2}\frac{1-\beta}{1+\beta}-\frac{\pi^{2}}{6}\right)\bigg{]},$
(31)

where $\beta=\sqrt{1-1/w^{2}}$ is the final lepton velocity in the proton rest
frame. In particular, the limit (31) applies for the case of antineutrino-
proton scattering with $\Delta E$ measured in the proton rest frame.

Large logarithms appear in the limit of small charged lepton mass. For
example, in the proton rest frame,

$\displaystyle S\left(\frac{\mu}{\Delta
E},\,\frac{E_{\ell}}{m_{\ell}},\,\frac{E_{\ell}}{m_{\ell}},\,1\right)\underset{E_{\ell}\gg
m_{\ell}}{\longrightarrow}1+\frac{\alpha}{\pi}\left[-\ln^{2}\frac{2E_{\ell}}{m_{\ell}}+\ln\frac{2E_{\ell}}{m_{\ell}}+2\left(1-\ln\frac{2E_{\ell}}{m_{\ell}}\right)\ln\frac{\mu}{2\Delta
E}+1-\frac{\pi^{2}}{6}\right]\,,$ (32)

with the recoil charged lepton energy in the laboratory frame $E_{\ell}$. For
GeV (anti)neutrino energies, the double-logarithmic corrections for the
electron case, $\ell=e$, are large and higher-order contributions must be
included for percent-level accuracy. The result in Eq. (32) coincides with the
one-loop soft correction $\delta_{\mathrm{S}}$ in the static nucleon limit
with an ultrarelativistic lepton, Eq. (25).

The renormalization group evolution of the soft function is described through
two-loop order by [27, 28, 17]

$\frac{\mathrm{d}S\left(\mu\right)}{\mathrm{d}\ln\mu^{2}}=\gamma_{\mathrm{S}}\left(\mu\right)S\left(\mu\right),\qquad\gamma_{\mathrm{S}}\left(\mu\right)=\left(\gamma_{0}+\gamma_{1}\frac{\alpha\left(\mu\right)}{\pi}\right)\left(1-\frac{1}{2\beta}\ln\frac{1+\beta}{1-\beta}\right)\frac{\alpha\left(\mu\right)}{\pi}\,,$
(33)

with $\gamma_{0}=1$ and $\gamma_{1}=-5n_{\ell}/9$, including $n_{\ell}$
virtual dynamical charged leptons. Exponentiating the one-loop result, we
evaluate the complete resummed soft function through ${\cal O}(\alpha)$ in the
counting $\alpha\ln^{2}(2E_{\ell}/m_{\ell})={\cal O}(1)$ [i.e.,
$\ln(2E_{\ell}/m_{\ell})={\cal O}(\alpha^{-1/2})$]. Some expressions for the
next order in $\alpha$ can be found in Refs. [29, 30, 31, 32, 17].

###  4.4 Collinear function

The collinear contributions represented by $J$ and $j(x)$ in Eq. (28) describe
physics at energy scales of order of the charged lepton mass. Like the soft
function, these contributions are independent of hadronic structure. As
discussed in Sec. 2, energetic photons radiated within a cone of size
$\Delta\theta$ around the final lepton direction can be included in the
definition of the observable depending on the lepton flavor and detector
details.

Consider first the case when no energetic collinear radiation is emitted. We
have derived the collinear function both by momentum region decomposition of
the virtual correction, and by exploiting Feynman rules in the collinear
sector of SCET. The one-loop result is given by

$\displaystyle
J\left(\frac{\mu}{m_{\ell}}\right)=1+\frac{\alpha}{4\pi}\left(\ln^{2}\frac{\mu^{2}}{m_{\ell}^{2}}+\ln\frac{\mu^{2}}{m_{\ell}^{2}}+4+\frac{\pi^{2}}{6}\right)\,.$
(34)

Since the jet function is independent of hadronic structure, the result of Eq.
(34) is identical to the one-loop correction $\delta_{\mathrm{J}}$ in the
static nucleon limit with ultrarelativistic lepton, Eq. (25). Some two-loop
results for collinear functions in QED were derived in Refs. [33, 34, 35, 17].

Jet observables include energetic collinear photons. Through one-loop order,
the contribution to the cross section is given by ($\eta\equiv\Delta\theta
E_{\ell}^{\rm tree}/m_{\ell}$)

$\displaystyle
j\left(\frac{\mu}{m_{\ell}},x,\eta\right)=\frac{\alpha}{\pi}\left[\frac{1}{2}\frac{1+x^{2}}{1-x}\ln(1+x^{2}\eta^{2})-\frac{x}{1-x}\frac{x^{2}\eta^{2}}{1+{x^{2}\eta^{2}}}\right]\,.$
(35)

The expression in Eq. (35) is obtained using SCET Feynman rules. After
integrating over $x$, the contribution is identical to $\delta_{J}^{\prime}$
of Eq. (27) in the static limit (when $E_{\ell}^{\rm tree}=E_{\nu}$):

$\displaystyle\int_{0}^{1-\Delta E/E_{\ell}^{\rm
tree}}\mathrm{d}x\,j\left(\frac{\mu}{m_{\ell}},x,\eta\right)$
$\displaystyle=\frac{\alpha}{\pi}\left[\frac{1}{2}\mathrm{Li}_{2}\left(-\eta^{2}\right)+\frac{1}{4}\ln^{2}\left(1+\eta^{2}\right)+\frac{1}{4}\left(\frac{2}{1+\eta^{2}}-\frac{1}{\eta^{2}}-3\right)\ln\left(1+\eta^{2}\right)+\frac{9}{4}\right.\,$
$\displaystyle\left.+\left(\ln\left(1+\eta^{2}\right)-\frac{\eta^{2}}{1+\eta^{2}}\right)\ln\frac{E_{\ell}^{\rm
tree}}{\Delta
E}-\left(\mathrm{tan}^{-1}\eta\right)^{2}-\left(1+\frac{1}{1+\eta^{2}}\right)\frac{\mathrm{tan}^{-1}\eta}{\eta}\right]\,.$
(36)

We remark that, in the limit of small charged lepton mass,

$\displaystyle
j\left(\frac{\mu}{m_{\ell}},x,\eta\right)\underset{m_{\ell}\ll\Delta\theta
E_{\ell}^{\rm
tree}}{\longrightarrow}\frac{\alpha}{\pi}\left(\frac{1+x^{2}}{1-x}\ln\frac{xE_{\ell}^{\rm
tree}\Delta\theta}{m_{\ell}}-\frac{x}{1-x}\right)\,,$ (37)

where the coefficient of the logarithm is identified with the well-known
splitting function [36, 37, 38, 39, 40, 41, 42].

Figure 4: Differential cross section $\mathrm{d}\sigma^{\gamma}$ for one
radiated photon of energy above $\Delta E=20\leavevmode\nobreak\ \mathrm{MeV}$
within the angle $\Delta\theta$ to the lepton direction, divided by the tree-
level charged-current elastic cross section $\mathrm{d}\sigma_{\mathrm{LO}}$.
The ratio $\mathrm{d}\sigma^{\gamma}/\mathrm{d}\sigma_{\mathrm{LO}}$ is
computed using Eq. (36) for the scattering of electron (anti)neutrinos (left
plot) and muon (anti)neutrinos (right plot). The bottom, middle, and top
curves correspond to electromagnetic jet energy 600 MeV, 2 GeV, and 6 GeV,
respectively.

Figure 5: Derivative of $\mathrm{d}\sigma^{\gamma}$ in Fig. 5 with respect to
$\Delta\theta$, divided by the tree-level charged-current elastic cross
section $\mathrm{d}\sigma_{\mathrm{LO}}$.

Muon and electron jets for accelerator neutrino energies are quite different
objects. Photon radiation is highly collimated in the direction of the
outgoing lepton for charged-current processes with electron flavor, but only a
relatively small contribution is contained within a $1-2^{\circ}$ cone for the
muon flavor. We illustrate this dependence of radiated events on the jet angle
by plotting Eq. (36), and the derivative of Eq. (36) with respect to
$\Delta\theta$, in Figs. 5 and 5, respectively.

Two limiting cases are of interest. The limit of very small charged lepton
mass gives (recall $\eta=\Delta\theta E_{\ell}^{\rm tree}/m_{\ell}$)

$\int_{0}^{1-\Delta E/E_{\ell}^{\rm
tree}}\mathrm{d}x\,j\left(\frac{\mu}{m_{\ell}},x,\eta\right)\underset{\eta\gg
1}{\longrightarrow}\frac{\alpha}{\pi}\left[\left(2\ln\eta-1\right)\ln\frac{E_{\ell}^{\rm
tree}}{\Delta
E}-\frac{3}{2}\ln\eta-\frac{\pi^{2}}{3}+\frac{9}{4}+\frac{\pi}{2\eta}+{\cal
O}\left(\frac{1}{\eta^{2}}\right)\right].$ (38)

Adding the virtual soft and collinear corrections, the jet angular size
regulates the collinear singularity when
$m_{\ell}\ll\Delta\theta\left(E_{\ell}+E_{\gamma}\right)$, as observed for the
static limit case in Eq. (13). For example, $\eta\gtrsim 10$ is realized for
$E_{\ell}^{\rm tree}\gtrsim 30\leavevmode\nobreak\ \mathrm{MeV}$ for electrons
and $E_{\ell}^{\rm tree}\gtrsim 6\leavevmode\nobreak\ \mathrm{GeV}$ for muons
when the cone angle is $\Delta\theta=10^{\circ}$. In the opposite limit of
very narrow jet, the collinear radiation is suppressed by phase space and
vanishes as

$\int_{0}^{1-\Delta E/E_{\ell}^{\rm
tree}}\mathrm{d}x\,j\left(\frac{\mu}{m_{\ell}},x,\eta\right)\underset{\eta\ll
1}{\longrightarrow}\frac{\alpha}{\pi}\left[\frac{\eta^{2}}{24}+\left(\ln\frac{E_{\ell}^{\rm
tree}}{\Delta E}-\frac{23}{10}\right)\frac{\eta^{4}}{2}+{\cal
O}\left(\eta^{6}\right)\right].$ (39)

The two-loop renormalization group evolution of the collinear function is
given by [43, 44, 17]

$\displaystyle\frac{\mathrm{d}J\left(\mu\right)}{\mathrm{d}\ln\mu^{2}}$
$\displaystyle=\gamma_{\mathrm{J}}\left(\mu\right)J\left(\mu\right)\,,$
$\displaystyle\gamma_{\mathrm{J}}\left(\mu\right)$
$\displaystyle=\left(2\ln\frac{\mu^{2}}{m_{\ell}^{2}}+1\right)\left(\gamma_{0}+\gamma_{1}\frac{\alpha\left(\mu\right)}{\pi}\right)\frac{\alpha\left(\mu\right)}{4\pi}+\gamma_{2}\left(\frac{\alpha\left(\mu\right)}{4\pi}\right)^{2},$
(40)

where
$\gamma_{2}=\frac{3}{2}+\frac{50n_{\ell}}{27}-2\left(1+\frac{n_{\ell}}{3}\right)\pi^{2}+24\zeta_{3}$
with the number of dynamical charged leptons in the theory $n_{\ell}$.

In the factorization theorem of Eq. (28), the soft and jet functions appear as

$\displaystyle JR(1)S(1)+\int\mathrm{d}x\,j(x)R(x)S(x)\,,$ (41)

where the $x$ integration limits depend on the observable. In place of Eq.
(41), we use the simplified form
$JR(1)S(1)\exp\big{[}\int\mathrm{d}y\,j_{1}(y)\big{]}$ in the phenomenological
analysis, where $j_{1}(x)$ denotes the one-loop contribution. This
exponentiation of the one-photon collinear correction accounts for the
potentially enhanced leading phase-space double logarithms
$\left(\frac{2\alpha}{\pi}\ln\frac{\Delta E}{E_{\ell}}\ln\frac{\Delta\theta
E_{\ell}}{m_{\ell}}\right)^{n_{c}}$ arising from the radiation of two or more
($n_{c}$) collinear photons.

###  4.5 Hard function

The hard function $H$ appearing in Eq. (28) represents a matching coefficient
when the full theory including hadronic physics888For definiteness, we
consider the full theory as $n_{f}=3$ flavor quantum chromodynamics (QCD) in
the presence of four-fermion electroweak operators [45], i.e., the Standard
Model after integrating out electroweak vector bosons, $W^{\pm}$, $Z^{0}$,
Higgs field, $h$, and heavy quarks $t$, $b$, and $c$. is matched onto soft-
collinear effective theory (where hadronic physics is integrated out). The
matching is performed by equating the amplitudes displayed in Fig. 1, computed
in both full theory and effective theory, accounting for tree-level and one-
loop corrections. We begin in Sec. 4.5.1 by parametrizing the hard function by
invariant amplitudes. In Sec. 4.5.2, we then introduce a default model for the
nonperturbative matching condition. Finally, we consider the renormalization
group evolution for the hard matching coefficient within the effective theory
in Sec. 4.5.3.

####  4.5.1 Invariant amplitudes

At leading power, charged lepton masses may be ignored in the hard matching
condition. The matrix element of the charged-current elastic process with
massless charged lepton can then be expressed as999We use the shorthand
notation
$\bar{\ell}^{-}(\dots)\nu_{\ell}=\bar{u}^{(\ell)}(p^{\prime})(\dots)u^{(\nu)}(p)$
and
$\overline{\bar{\nu}}_{\ell}(\dots)\ell^{+}=\bar{v}^{(\nu)}(p^{\prime})(\dots)v^{(\ell)}(p)$
for the usual Dirac spinors with momentum assignment in Fig. 1.

$\displaystyle
T_{\nu_{\ell}n\to\ell^{-}p}=\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}\,\bar{\ell}^{-}\gamma^{\mu}\mathrm{P}_{\mathrm{L}}\nu_{\ell}\,\bar{p}\left(f_{1}\gamma_{\mu}+f_{2}\frac{i\sigma_{\mu\rho}q^{\rho}}{2M}+f_{A}\gamma_{\mu}\gamma_{5}-f^{3}_{A}\frac{{K}_{\mu}}{M}\gamma_{5}\right)n\,,$
$\displaystyle
T_{\bar{\nu}_{\ell}p\to\ell^{+}n}=\sqrt{2}\mathrm{G}_{\mathrm{F}}V^{*}_{ud}\,\overline{\bar{\nu}}_{\ell}\gamma^{\mu}\mathrm{P}_{\mathrm{L}}\ell^{+}\,\bar{n}\left(\bar{f}_{1}\gamma_{\mu}+\bar{f}_{2}\frac{i\sigma_{\mu\rho}q^{\rho}}{2M}+\bar{f}_{A}\gamma_{\mu}\gamma_{5}+\bar{f}^{3}_{A}\frac{{K}_{\mu}}{M}\gamma_{5}\right)p\,.$
(42)

Here $q=p-p^{\prime}=k^{\prime}-k$, $K=(k+k^{\prime})/2$,
$\mathrm{G}_{\mathrm{F}}$ is the Fermi constant, $V_{ud}$ is the CKM matrix
element and $M=(M_{p}+M_{n})/2$ is the average nucleon mass. The four
independent invariant amplitudes are functions of two kinematic variables:
$Q^{2}=-t=-q^{2}$ and
$\nu=\left(s-u\right)/4=ME_{\nu}-\left(Q^{2}+m_{\ell}^{2}\right)/4$, where
$s$, $t$, and $u$ are the usual Mandelstam invariants. For the antineutrino
case, the invariant amplitudes are given by
$\bar{f}_{i}\left(\nu+i0,Q^{2}\right)=f_{i}\left(-\nu-i0,Q^{2}\right)^{*}$,
where $f_{i}$ stands for one of $f_{1}$, $f_{2}$, $f_{A}$, and $f_{A}^{3}$.
The quantities $f_{i}$ are ultraviolet (UV) finite and IR divergent when
virtual QED corrections are included. These IR divergences cancel in the
matching between full theory and effective theory.

Using the representation of Eq. (42), the charged-current elastic cross
section (without radiation) in the laboratory frame is expressed in terms of
invariant amplitudes as [46]101010We neglect the relative difference in
nucleon masses, $(M_{n}-M_{p})/(M_{n}+M_{p})$, and electroweak power
corrections suppressed by the $W$-boson mass, $Q^{2}/M_{W}^{2}$; these effects
contribute at the permille level.

$\frac{d\sigma}{dQ^{2}}(E_{\nu},Q^{2})=\frac{\mathrm{G}_{\mathrm{F}}^{2}|V_{ud}|^{2}}{2\pi}\frac{M^{2}}{E_{\nu}^{2}}\left[\left(\tau+r^{2}\right)A(\nu,\leavevmode\nobreak\
Q^{2})-\frac{\nu}{M^{2}}B(\nu,\leavevmode\nobreak\
Q^{2})+\frac{\nu^{2}}{M^{4}}\frac{C(\nu,\leavevmode\nobreak\
Q^{2})}{1+\tau}\right]\,,$ (43)

where $\tau=Q^{2}/(4M^{2})$ and $r=m_{\ell}/(2M)$. The quantities $A$, $B$,
and $C$ are given, respectively, by111111The sign in front of $f_{A}^{3}$ in
$B$ differs from Ref. [46] for antineutrino scattering, as noted in Ref. [47].

$\displaystyle A$
$\displaystyle=\tau|g_{M}|^{2}-|g_{E}|^{2}+(1+\tau)|f_{A}|^{2}-\tau(1+\tau)|f_{A}^{3}|^{2}-r^{2}\left(|g_{M}|^{2}+|f_{A}+2F_{P}|^{2}-4\left(1+\tau\right)F_{P}^{2}\right)\,,$
$\displaystyle B$ $\displaystyle=\mathrm{Re}[\pm 4\tau
f^{*}_{A}g_{M}-2r^{2}\left(f_{A}-2\tau F_{P}\right)^{*}f_{A}^{3}]\,,$
$\displaystyle C$
$\displaystyle=\tau|g_{M}|^{2}+|g_{E}|^{2}+(1+\tau)|f_{A}|^{2}+\tau(1+\tau)|f_{A}^{3}|^{2}\,.$
(44)

For numerical results in this paper, we include all mass corrections within
the hadronic model of Sec. 4.5.2.121212In terms of Eq. (28), the matching
calculation includes power-suppressed terms in the hard function. This
procedure does not achieve a complete scale separation beyond the leading
power (both hard and collinear momentum regions contribute power-suppressed
terms), but the complete lepton-mass dependence in the hadronic model is
retained through ${\cal O}(\alpha)$. At tree level, these corrections are
accounted for by including the pseudoscalar form factor in the expressions for
$A$, $B$ and $C$ and retaining the lepton-mass dependence in kinematic
prefactors. We will use a standard ansatz (partially conserved axial current
and the assumption of pion pole dominance) for the pseudoscalar form factor:
$F_{P}(Q^{2})=2M^{2}F_{A}(Q^{2})/\left(m_{\pi}^{2}+Q^{2}\right)$. Electric and
magnetic amplitudes $g_{E}$ and $g_{M}$ are defined from $f_{1}$ and $f_{2}$
as, respectively,

$\displaystyle g_{E}=f_{1}-\tau f_{2},\qquad g_{M}=f_{1}+f_{2}.$ (45)

For antineutrino-proton scattering, the sign in the first term of $B$ is
negative.

When one-loop radiative corrections are included, the full-theory cross
section in Eq. (43) receives contributions originating from soft, collinear,
and hard regions of virtual photon momentum. The soft and collinear region
contributions are reproduced by the effective theory and cancel in the
matching; in the computation of observables, these contributions are replaced
by the complete soft and collinear functions computed within the effective
theory, incorporating real photon radiation and perturbative resummation. The
hard region contribution determines the hard function appearing in the
factorization formula of Eq. (28).

####  4.5.2 Hadronic model

The matching condition depends on the nonperturbative quantities $f_{1}$,
$f_{2}$, $f_{A}$, and $f_{A}^{3}$ appearing in Eq. (42). At the leading order
in $\alpha$, these invariant amplitudes are determined by quark current
operators taken between nucleon states. Assuming isospin symmetry, they can be
expressed in terms of isovector electromagnetic form factors
${F}_{V1,V2}={F}_{1,2}^{p}-{F}_{1,2}^{n}$, extracted from electron scattering
data with constraints from atomic spectroscopy [48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73,
74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92,
93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109,
110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122], and the
axial form factor ${F}_{\mathrm{A}}$ as [46]

$f_{1}(\nu,Q^{2})\to F_{V1}(Q^{2})\,,\qquad f_{2}(\nu,Q^{2})\to
F_{V2}(Q^{2})\,,\qquad f_{A}(\nu,Q^{2})\to F_{A}(Q^{2})\,,\qquad
f^{3}_{A}(\nu,Q^{2})\to 0.$ (46)

Radiative corrections modify each of the four invariant amplitudes and
introduce nonperturbative information beyond the form factors
$F_{1,2,A}(Q^{2})$. In principle, sufficiently precise (anti)neutrino-nucleon
scattering measurements could be used to extract this information [123, 124,
125, 126, 127, 128, 129, 47, 130, 131, 132, 133]. Alternatively, future
lattice QCD calculations could perform a first-principles evaluation starting
from the quark-level Lagrangian [134, 135, 136, 137, 138, 139, 140, 141].
Since neither of these options is currently available, we use a simple
hadronic model to represent the ${\cal O}(\alpha)$ corrections to the
invariant amplitudes in the matching condition for purposes of illustration.
The model is based on a form-factor insertion ansatz whereby point-particle
Feynman-t’Hooft diagrams are dressed with on-shell form factors at the
hadronic vertices. This procedure is illustrated in Fig. 6. This hadronic
model generalizes a commonly used estimate of two-photon exchange corrections
in elastic lepton-proton scattering [142, 143, 144, 145].

Figure 6: Hadronic model for hard matching at one-loop level. The photon is
exchanged between the charged lepton and nucleon lines.

The resulting vertex correction to the tree-level charged-current elastic
process on nucleons is given by [suppressing the prefactor
$\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}$ common with the tree-level expression
in Eq. (42)]

$\displaystyle T^{v}_{\nu_{\ell}n\to\ell^{-}p}$
$\displaystyle=e^{2}\int\frac{\mathrm{d}^{d}L}{\left(2\pi\right)^{d}}\bar{\ell}\gamma_{\mu}\frac{-p\\!\\!\\!/^{\prime}-L\\!\\!\\!/}{\left(L+p^{\prime}\right)^{2}-m_{\ell}^{2}}\gamma^{\sigma}\mathrm{P}_{\mathrm{L}}\nu_{\ell}\mathrm{\Pi}^{\mu\nu}\left(L\right)\bar{p}\left(\Gamma^{p}_{\nu}\frac{k\\!\\!\\!/^{\prime}-L\\!\\!\\!/+M}{\left(L-k^{\prime}\right)^{2}-M^{2}}\Gamma_{\sigma}+\Gamma_{\sigma}\frac{k\\!\\!\\!/+L\\!\\!\\!/+M}{\left(L+k\right)^{2}-M^{2}}\Gamma^{n}_{\nu}\right)n\,,$
$\displaystyle T^{v}_{\bar{\nu}_{\ell}p\to\ell^{+}n}$
$\displaystyle=e^{2}\int\frac{\mathrm{d}^{d}L}{\left(2\pi\right)^{d}}\bar{\bar{\nu}}_{\ell}\gamma^{\sigma}\mathrm{P}_{\mathrm{L}}\frac{p\\!\\!\\!/^{\prime}+L\\!\\!\\!/}{\left(L+p^{\prime}\right)^{2}-m_{\ell}^{2}}\gamma_{\mu}\bar{\ell}\mathrm{\Pi}^{\mu\nu}\left(L\right)\bar{n}\left(\Gamma^{n}_{\nu}\frac{k\\!\\!\\!/^{\prime}-L\\!\\!\\!/+M}{\left(L-k^{\prime}\right)^{2}-M^{2}}\Gamma_{\sigma}+\Gamma_{\sigma}\frac{k\\!\\!\\!/+L\\!\\!\\!/+M}{\left(L+k\right)^{2}-M^{2}}\Gamma^{p}_{\nu}\right)p\,.$
(47)

Here

$\displaystyle\mathrm{\Pi}^{\mu\nu}\left(L\right)$
$\displaystyle=\frac{i}{L^{2}-\lambda_{\gamma}^{2}}\left(-g^{\mu\nu}+\left(1-\xi_{\gamma}\right)\frac{L^{\mu}L^{\nu}}{L^{2}-a\xi_{\gamma}\lambda_{\gamma}^{2}}\right),$
(48)

is the momentum-space photon propagator with the photon mass regulator
$\lambda_{\gamma}$, the gauge-fixing parameter $\xi_{\gamma}$, and an
arbitrary constant $a$, and

$\Gamma^{N}_{\nu}=\gamma_{\nu}{F}_{1}^{N}\left(-L^{2}\right)+\frac{i\sigma_{\nu\rho}L^{\rho}}{2M}{F}_{2}^{N}\left(-L^{2}\right),$
(49)

is the electromagnetic vertex, where $N$ denotes proton ($p$) or neutron
($n$), while ${F}_{1}^{N}$ and ${F}_{2}^{N}$ are Dirac and Pauli
electromagnetic form factors, respectively. The charged-current weak vertex is

$\Gamma_{\sigma}=\gamma_{\sigma}{F}_{V1}\left(Q^{2}\right)+\frac{i\sigma_{\sigma\rho}q^{\rho}}{2M}{F}_{V2}\left(Q^{2}\right)+\gamma_{\sigma}\gamma_{5}{F}_{\mathrm{A}}\left(Q^{2}\right)+\frac{q_{\sigma}}{M}\gamma_{5}\mathrm{F}_{P}\left(Q^{2}\right)\,.$
(50)

We remark that a naive implementation of the form-factor insertion ansatz
would involve form factors evaluated at momentum transfer $(q\pm L)^{2}$
rather than $q^{2}$, as in Eq. (50). Such an ansatz would violate
electromagnetic gauge invariance, giving rise to spurious collinear
singularities.131313For example, current conservation requires a cancellation
between amplitudes for photon emission from the charged lepton line and from
the proton line. The former contribution would involve $F_{i}(-q^{2})$, but
the latter would involve $F_{i}(-(q\pm L)^{2})$, spoiling the cancellation. We
thus adopt Eq. (50) as our default model. For this model calculation of
virtual diagrams with Eq. (47), exploiting the on-shell vertex of Eq. (49), we
express the Dirac and Pauli form factors as [recall $M=(M_{p}+M_{n})/{2}$ is
the average nucleon mass and $\tau={Q^{2}}/({4M^{2}})$]

$\displaystyle{F}^{p,n}_{\mathrm{1}}\left(Q^{2}\right)$
$\displaystyle=\frac{G^{p,n}_{\mathrm{E}}+\tau
G^{p,n}_{\mathrm{M}}}{1+\tau},\quad{F}^{p,n}_{\mathrm{2}}\left(Q^{2}\right)=\frac{G^{p,n}_{\mathrm{M}}-G^{p,n}_{\mathrm{E}}}{1+\tau},$
(51)

and employ a simple dipole form for the proton electric, proton magnetic, and
neutron magnetic form factors, and an ansatz for the neutron electric form
factor constrained by its charge radius [120, 121] at low $Q^{2}$ and by $\sim
Q^{-4}$ perturbative QCD behavior at high $Q^{2}$ [146, 147, 148]:

$\displaystyle
G^{p}_{\mathrm{E}}\left(Q^{2}\right)=\frac{G^{p}_{\mathrm{M}}\left(Q^{2}\right)}{\mu_{p}}=\frac{G^{n}_{\mathrm{M}}\left(Q^{2}\right)}{\mu_{n}}=\frac{1}{\left(1+\frac{Q^{2}}{\Lambda^{2}}\right)^{2}}\,,\quad
G^{n}_{\mathrm{E}}\left(Q^{2}\right)$
$\displaystyle=\frac{-<r^{2}_{\mathrm{E}}>Q^{2}}{6\left(1+b\tau\right)}G^{p}_{\mathrm{E}}\left(Q^{2}\right)\,,$
(52)

with $<r^{2}_{\mathrm{E}}>=-0.1161\leavevmode\nobreak\ \mathrm{fm}^{2}$,
$\mu_{p}=2.7928$, $\mu_{n}=-1.9130$, $\Lambda^{2}=0.71\,{\rm GeV}^{2}$, and
$b=4.6$. We have performed the evaluation of the UV-finite amplitudes from
Eqs. (47) in $d=4$ and cross-checked imaginary parts of all terms in Eqs. (47)
by independent calculation exploiting unitarity.141414In contrast to elastic
charged lepton-proton scattering, the high-energy behavior of imaginary parts
in the charged-current elastic process does not allow us to write down
unsubtracted dispersion relations for any of the invariant amplitudes.
Performing crossing and charge conjugation, the amplitudes of (anti)neutrino-
nucleon elastic scattering $\nu_{\ell}n\to\ell^{-}p$
($\overline{\nu}_{\ell}p\to\ell^{+}n$) are not related to amplitudes of a
(anti)neutrino scattering reaction. Thus, we cannot use crossing relations for
amplitudes in the same channel to suppress the high-energy behavior and obtain
convergence of dispersive integrals. This section represents a model for the
real part of the nucleon contribution and a discussion of uncertainties owing
to parametric inputs and neglected inelastic contributions.

The amplitudes in the hadronic model can be represented as

$\displaystyle f_{1}(\nu,Q^{2})$
$\displaystyle=\sqrt{Z_{\ell}Z_{h}^{(p)}}\left(F_{V1}(Q^{2})+f^{v}_{1}(\nu,Q^{2})\right)\,,$
$\displaystyle f_{2}(\nu,Q^{2})$
$\displaystyle=\sqrt{Z_{\ell}Z_{h}^{(p)}}\left(F_{V2}(Q^{2})+f^{v}_{2}(\nu,Q^{2})\right)\,,$
$\displaystyle f_{A}(\nu,Q^{2})$
$\displaystyle=\sqrt{Z_{\ell}Z_{h}^{(p)}}\left(F_{A}(Q^{2})+f^{v}_{A}(\nu,Q^{2})\right)\,,$
$\displaystyle f^{3}_{A}(\nu,Q^{2})$
$\displaystyle=\sqrt{Z_{\ell}Z_{h}^{(p)}}\left(f^{3}_{A}\right)^{v}(\nu,Q^{2})\,,$
(53)

where the charged lepton field renormalization factor $Z_{\ell}$ is given by
the standard QED expression:

$Z_{\ell}=1-\frac{\alpha}{4\pi}\left(\ln\frac{\mu^{2}}{m_{\ell}^{2}}+2\ln\frac{\lambda_{\gamma}^{2}}{m_{\ell}^{2}}+4\right)+\frac{\alpha}{4\pi}\left(1-\xi_{\gamma}\right)\left(\ln\frac{\mu^{2}}{\lambda_{\gamma}^{2}}+1+\frac{a\xi_{\gamma}\ln(a\xi_{\gamma})}{1-a\xi_{\gamma}}\right)\,.$
(54)

For the external proton, we include the on-shell wave-function renormalization
constant of the heavy spin-1/2 fermion (cf. Ref. [17] for the analogous
discussion of Born amplitudes in electron-proton scattering)

$Z_{h}^{(p)}=1+\frac{\alpha}{2\pi}\ln\frac{\mu^{2}}{\lambda_{\gamma}^{2}}+\frac{\alpha}{4\pi}\left(1-\xi_{\gamma}\right)\left(\ln\frac{\mu^{2}}{\lambda_{\gamma}^{2}}+1+\frac{a\xi_{\gamma}\ln(a\xi_{\gamma})}{1-a\xi_{\gamma}}\right)\,.$
(55)

Amplitudes in Eqs. (53) do not depend on the arbitrary regularization
parameter $a$ and have correct IR behavior. The expressions in Eqs. (53)
provide a conventional, and model-independent, definition of charged-current
“Born” form factors $F_{1,2,A}(Q^{2})$, once the functions $f_{1,2,A}^{v}$ are
specified. For definiteness, we take $\mu=M$. In the absence of $CP$ violation
(which we neglect), we have defined the overall phase of the amplitudes in Eq.
(42) such that the Born form factors are real. We remark that the separation
of the gauge-independent hard matching contribution into “Born” and “non-Born”
terms is necessarily QED gauge dependent; for definiteness, we specify
Feynman-’t Hooft gauge, i.e., $\xi_{\gamma}=1$.151515Alternatively, one can
define gauge-independent “Born” form factors by moving local gauge-dependent
virtual contributions to field renormalization factors of the external proton
and neutron. Such a prescription differs from our default calculation by small
structure-dependent contributions to the nucleon self-energies, which are
independent of the lepton mass. Our hadronic model identifies $f_{1,2,A}^{v}$
with the contributions from Eqs. (47). For $F_{V1,V2}(Q^{2})$, we use the
isospin rotation of Born form factors161616We have neglected isospin-violating
effects in the relation of charged-current form factors to electromagnetic
form factors and in the omission of second-class current contributions to the
tree-level process [149, 150, 151, 152]. Isospin-breaking effects are expected
from constituent quark model estimates to be of permille level [153, 154, 155,
156]. According to a chiral perturbation theory-based calculation [157], they
can reach the percent level for the magnetic form factor; while this is
potentially significant compared to the precision of electron-proton
scattering data, the tree-level uncertainty of the charged-current process is
dominated by axial form factor. Radiative corrections are neglected in
existing extractions of $F_{A}(Q^{2})$. extracted from experimental $e-p$ and
$e-n$ scattering data, measurements of the neutron scattering length, and $\mu
H$ spectroscopy data [158], i.e.,
$F_{V1,2}(Q^{2})=F^{p}_{1,2}(Q^{2})-F^{n}_{1,2}(Q^{2})$, where $p$ stands for
the proton and $n$ stands for the neutron. For $F_{A}(Q^{2})$, we use the Born
form factor extracted from experimental $\nu_{\mu}-n$ scattering data [131].

Figure 7: Uncertainties from the hard function on the ratio of the
differential cross section, with only one soft photon radiated, to the tree-
level result in $\nu_{e}n\to e^{-}p$ (left plots) and $\bar{\nu}_{e}p\to
e^{+}n$ (right plots). For initial (anti)neutrino energies
$E_{\nu}=600\leavevmode\nobreak\ \mathrm{MeV}$ (upper plots) and
$E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$ (lower plots), the soft-photon
energy cutoff is $\Delta E=25\leavevmode\nobreak\ \mathrm{MeV}$ and $\Delta
E=10\leavevmode\nobreak\ \mathrm{MeV}$, respectively. The uncertainty
propagated from the tree-level form factors is shown by the red dashed line.
The difference obtained by varying nucleon electromagnetic form-factor
parameter $\Lambda^{2}$ as $\Lambda^{2}\to 1.1\Lambda^{2}$ in the loop diagram
is shown by the green dash-dotted line. As discussed in the text, the total
contribution of $G_{E}^{n}$ and $G_{M}^{n}$ at the neutron electromagnetic
vertex and the variation of the proton electromagnetic vertex obtained by
adding $G_{E}^{p}\to G_{E}^{p}+G_{E}^{n}$ and $G_{M}^{p}\to
G_{M}^{p}+G_{M}^{n}$ represent an estimate for neglected contributions in the
hadronic model. These variations are shown by the black dotted line and by the
black fine-dotted line, respectively. The sum of all variations in quadrature
is shown by the blue solid line.  Figure 8: The same as Fig. 7 but for the
muon flavor.

For the phenomenological analysis, we consider the following assignment of
uncertainty to the hard function matching condition. Consider first the Born
form factors $F_{1,2,A}(Q^{2})$. For tree-level amplitudes (but not for the
calculation of loop diagrams), these form factors and corresponding
uncertainties are taken as default fits of data with
$Q^{2}<1\leavevmode\nobreak\ \mathrm{GeV}^{2}$ from the analysis of scattering
and spectroscopy data [131, 158], taking the isospin-decomposed fit for the
vector form factors; all form factors are presented in $z$-expansion form
while uncertainties are described by the covariance matrix of $z$-expansion
parameters. In ratios to the leading-order cross section, this dominant source
of uncertainty is largely reduced. Second, for the amplitudes
$f^{v}_{1,2,A}(\nu,Q^{2})$, we consider separate variations in the hadronic
model to account for parametric inputs and for neglected contributions. Note
that the hadronic model involves form factors $F_{1,2,A}(Q^{2})$ appearing
inside virtual loop diagrams; here we use a simpler dipole ansatz for
electromagnetic form factors as specified in Eqs. (52). For the parametric
inputs, we vary the nucleon electromagnetic form factors appearing in loop
diagrams by shifting the pole parameter $\Lambda^{2}$ in Eqs. (52) as
$\Lambda^{2}\to(1\pm 0.1)\Lambda^{2}$; this range covers experimentally
allowed values for electromagnetic form factors [158]. Model dependence due to
the insertion of on-shell hadronic vertices and the neglect of inelastic
intermediate states is represented by a simple ansatz that adds the neutron
electric and magnetic form factors to each of the neutron and proton
electromagnetic vertices.171717This ansatz is motivated by the observation
that two-photon exchange corrections in elastic electron-proton scattering
from the proton Pauli form factor were shown to have a similar size to
inelastic contributions [159]. Based on this ansatz, relative contributions to
the uncertainty of the cross-section ratio to the tree-level result from the
hard function are displayed for the electron flavor in Fig. 7 and for the muon
flavor in Fig. 8. Each separate source of uncertainty is at permille level or
even below.

Our definition of Born form factors in Eqs. (53) corresponds to a conventional
$\overline{\rm MS}$ scheme at scale $\mu=M$ [17] and $\xi_{\gamma}=1$. As
discussed in Ref. [17], a variety of other schemes have been used to extract
Born form factors in the literature [142, 122] but differ at a level not
resolved by data. To account for this discrepancy, we add in quadrature a
perturbative uncertainty estimated by varying the scale of matching,
$\mathrm{min}\left(Q^{2},M^{2},E_{\nu}^{2}\right)/2\lesssim\mu^{2}\lesssim
2\mathrm{max}\left(Q^{2},M^{2},E_{\nu}^{2}\right)$, and assuming that the
experimental extraction $F_{i}(Q^{2})$ represents the Born amplitude at
$\mu=M$.

####  4.5.3 Matching and renormalization group evolution

The hard function $H$ in Eq. (28) is determined by matching the factorization
formula to the cross section of Eq. (43) evaluated in the hadronic model.
Contributions from soft and collinear momentum regions in the hadronic model
are the same, by construction, as the corresponding one-loop expressions in
the effective theory derived in Secs. 4.3 and 4.4. By choosing the hard
matching scale $\mu_{\mathrm{H}}\sim\Lambda_{\mathrm{hard}}\sim M\sim Q\sim
E_{\nu}$, we ensure that $H$ is free from large logarithms. In order to resum
large logarithms in the final observable cross section, we evolve the hard
function to the low scale $\mu_{\mathrm{L}}$, writing

$\mathrm{d}\sigma\sim
H\left(\mu_{\mathrm{H}}\right)\left[\frac{H\left(\mu_{\mathrm{L}}\right)}{H\left(\mu_{\mathrm{H}}\right)}\right]J\left(\mu_{\mathrm{L}}\right)S\left(\mu_{\mathrm{L}}\right).$
(56)

Performing the running at two-loop level with the anomalous dimension
$\gamma_{\mathrm{H}}=-\gamma_{\mathrm{J}}-\gamma_{\mathrm{S}}$, with
$\gamma_{\mathrm{S}}$ and $\gamma_{\mathrm{J}}$ from Eqs. (33) and (40),
respectively, we resum leading and subleading large logarithms and obtain
$H\left(\mu_{\mathrm{L}}\right)$ at renormalization scale
$\mu_{\mathrm{L}}\ll\mu_{\mathrm{H}}$. The low scale could be further resolved
into separate soft and collinear scales,
$\mu_{\mathrm{S}}\sim\lambda\Lambda_{\rm hard}$ and
$\mu_{\mathrm{J}}\sim\lambda^{\frac{1}{2}}\Lambda_{\rm hard}$, writing
$J(\mu_{\mathrm{L}})=J(\mu_{\mathrm{J}})[J(\mu_{\mathrm{L}})/J(\mu_{\mathrm{J}})]$
and
$S(\mu_{\mathrm{L}})=S(\mu_{\mathrm{S}})[S(\mu_{\mathrm{L}})/S(\mu_{\mathrm{S}})]$.
Since the impact of such further resummation is numerically small, we use a
common scale $\mu_{\mathrm{S}}=\mu_{\mathrm{J}}=\mu_{\mathrm{L}}$ for
simplicity. Residual perturbative uncertainty is estimated by varying
$\mathrm{min}\left(m_{\ell}^{2},\left(\Delta
E\right)^{2}\right)/2\lesssim\mu_{\mathrm{L}}^{2}\lesssim
2\,\mathrm{max}\left(m_{\ell}^{2},\left(\Delta E\right)^{2}\right)$ and
$\mathrm{min}\left(Q^{2},M^{2},E_{\nu}^{2}\right)/2\lesssim\mu_{\mathrm{H}}^{2}\lesssim
2\mathrm{max}\left(Q^{2},M^{2},E_{\nu}^{2}\right)$.

Figure 9: Ratio of the differential cross section, with only soft-photon
radiation, to the tree-level result in $\nu_{e}n\to e^{-}p$ (left plots) and
$\bar{\nu}_{e}p\to e^{+}n$ (right plots). For incident (anti)neutrino energies
$E_{\nu}=600\leavevmode\nobreak\ \mathrm{MeV}$ (upper plots) and
$E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$ (lower plots), the soft-photon
energy cutoff is $\Delta E=25\leavevmode\nobreak\ \mathrm{MeV}$ and $\Delta
E=10\leavevmode\nobreak\ \mathrm{MeV}$, respectively. The resummed result
including all corrections of ${\cal O}\left(\alpha\right)$ is shown by the
blue solid lines. It is compared to the resummation result including only the
leading logarithms, shown by the green dotted lines, and to the result
including subleading logarithms of order ${\cal O}\left(\sqrt{\alpha}\right)$,
shown by the red dashed lines. To illustrate the net effect of resummation, we
also present the fixed-order result using the same hadronic model by the black
dash-dotted lines. Hadronic uncertainty, described in Sec. 4.5.2, is displayed
for the fixed-order calculation. Only perturbative uncertainty, estimated by
scale variation as described in the text, is presented for the first three
lines.

To illustrate convergence of the perturbative expansion, we consider the
theory with one dynamical charged lepton $n_{\ell}=1$, i.e., electron loops
only.181818For numerical results in following sections, we specify
$n_{\ell}=0$ below the electron mass scale, $n_{\ell}=1$ between electron and
muon mass scales, and $n_{\ell}=2$ above. The electromagnetic coupling
constant at the scale $\mu$ is related to its on-shell value
$\alpha_{0}\approx 1/137.036$ as [17]

$\alpha=\alpha_{0}\left[1+\frac{1}{3}\ln\frac{\mu^{2}}{m_{e}^{2}}\left(\frac{\alpha_{0}}{\pi}\right)+\left(\frac{1}{9}\ln^{2}\frac{\mu^{2}}{m_{e}^{2}}+\frac{1}{4}\ln\frac{\mu^{2}}{m_{e}^{2}}+\frac{15}{16}\right)\left(\frac{\alpha_{0}}{\pi}\right)^{2}\right].$
(57)

The running of the hard function from the hard scale $\mu_{\mathrm{H}}$ to the
soft scale $\mu_{\mathrm{L}}$ can be solved analytically and can be expressed
in terms of the charged lepton energy in the proton rest frame,
$\varepsilon_{\ell}=m_{\ell}\left(v_{\ell}\cdot v_{p}\right)$, as

$\displaystyle\ln\frac{H\left(\mu_{\mathrm{L}}\right)}{H\left(\mu_{\mathrm{H}}\right)}$
$\displaystyle=\gamma_{0}\ln\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}\left(\frac{1}{4}\ln\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}+\frac{1}{2}\ln\frac{\mu_{\mathrm{L}}^{2}}{4\varepsilon_{\ell}^{2}}+\frac{5}{4}\right)\left(\frac{\alpha_{0}}{\pi}\right)+\frac{\gamma_{0}}{18}\ln\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}\left(\ln^{2}\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}+3\ln\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}+6\ln\frac{\mu_{\mathrm{L}}^{2}}{m_{e}^{2}}\right)\left(\frac{\alpha_{0}}{\pi}\right)^{2}$
$\displaystyle+\frac{\gamma_{0}}{6}\ln\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}\left(\ln\frac{\mu_{\mathrm{H}}^{2}}{4\varepsilon_{\ell}^{2}}\ln\frac{\mu_{\mathrm{L}}^{2}}{m_{e}^{2}}+\frac{1}{2}\ln\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}\ln\frac{m_{e}^{2}}{4\varepsilon^{2}_{\ell}}+\frac{1}{4}\ln\frac{\mu_{\mathrm{H}}^{2}}{m_{e}^{2}}+\frac{1}{4}\ln\frac{\mu_{\mathrm{L}}^{2}}{m_{e}^{2}}-\frac{3\gamma_{1}}{2\gamma_{0}}\ln\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}-\frac{3\gamma_{1}}{\gamma_{0}}\ln\frac{\mu_{\mathrm{L}}^{2}}{4\varepsilon_{\ell}^{2}}\right)\left(\frac{\alpha_{0}}{\pi}\right)^{2}$
$\displaystyle+\frac{\gamma_{0}}{18}\ln\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}\left(\frac{1}{4}\ln^{3}\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}+\frac{1}{2}\ln\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}\ln^{2}\frac{\mu_{\mathrm{L}}^{2}}{m_{e}^{2}}+\frac{1}{3}\ln^{2}\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}\ln\frac{\mu_{\mathrm{L}}^{2}}{4\varepsilon^{2}_{\ell}}+\frac{2}{3}\ln^{2}\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}\ln\frac{\mu_{\mathrm{L}}^{2}}{m_{e}^{2}}\right)\left(\frac{\alpha_{0}}{\pi}\right)^{3}$
$\displaystyle+\frac{\gamma_{0}}{18}\ln\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}\ln\frac{\mu_{\mathrm{L}}^{2}}{m_{e}^{2}}\left(\ln\frac{\mu_{\mathrm{L}}^{2}}{m_{e}^{2}}\ln\frac{\mu_{\mathrm{L}}^{2}}{4\varepsilon^{2}_{\ell}}+\ln\frac{\mu_{\mathrm{H}}^{2}}{\mu_{\mathrm{L}}^{2}}\ln\frac{\mu_{\mathrm{L}}^{2}}{4\varepsilon^{2}_{\ell}}\right)\left(\frac{\alpha_{0}}{\pi}\right)^{3}+{\cal
O}\left(\alpha_{0}^{\frac{3}{2}}\right)\,,$ (58)

where we keep terms contributing through order $\alpha^{1}$, when
$\alpha\ln^{2}(\mu_{\mathrm{H}}/\mu_{\mathrm{L}})={\cal O}(1)$. We illustrate
the convergence of the perturbation theory by evaluating the cross section for
the electron flavor charged-current elastic process, including only soft
radiation, in Fig. 9. The successive curves represent resummations correct
through ${\cal O}(1)$, ${\cal O}(\sqrt{\alpha})$, and ${\cal O}(\alpha)$,
respectively, with perturbative uncertainties of order ${\cal
O}(\sqrt{\alpha})$, ${\cal O}(\alpha)$, and ${\cal O}(\alpha\sqrt{\alpha})$;
resummation corrections beyond ${\cal O}(\sqrt{\alpha})$ are numerically
small. We also compare our result to the fixed-order calculation of the same
process.

###  4.6 Factorization for inelastic processes

The factorization formalism extends naturally to other exclusive processes at
the nucleon level, for example radiation of noncollinear hard photons
($\nu_{\ell}N\to\ell N\gamma$) or pion emission ($\nu_{\ell}N\to\ell N\pi$).
In each case, soft and jet functions are perturbatively calculable, depending
only on particle masses and velocities and kinematic variables. Hadronic
physics is encoded in a basis of hard functions. Semi-inclusive processes that
sum over hadronic channels ($\nu_{\ell}N\to\ell X$) can be similarly analyzed.
Such processes are left to future work.

##  5 Inclusive cross sections

In this section, we describe the inclusion of noncollinear hard photons to
scattering cross sections and discuss model-independent properties of
inclusive cross sections on nucleons and nuclei.

###  5.1 Hadronic model for hard real photon emission

Let us consider the process $\nu_{\ell}n\to\ell^{-}p(\gamma)$ [or the
analogous antineutrino process $\bar{\nu}_{\ell}p\to\ell^{+}n(\gamma)$]
including arbitrary photon kinematics. To describe the radiation of
noncollinear hard photons, we exploit the same hadronic model as in Sec.
4.5.2. Recall that our prescription is equivalent to employing free-particle
propagators for intermediate states, with the electroweak vertex evaluated
using external-leg kinematics, as in Eq. (50). This ansatz ensures
electromagnetic gauge invariance and avoids spurious collinear singularities
of a naive form-factor insertion model. In the phenomenological analysis, we
demonstrate how such spurious singularities would impact cross-section
predictions for near-collinear kinematics (Fig. 13). For generic,
noncollinear, kinematics, the difference between our default gauge invariant
and a local non-gauge-invariant model, where the electroweak vertex is
evaluated in kinematics of a local field theory, can be interpreted as a
simple measure of hadronic model dependence. This difference is illustrated in
Figs. 14 and 15.

###  5.2 Expansion in small lepton mass

We have observed previously that sufficiently inclusive observables have a
finite limit at vanishing lepton mass $m_{\ell}\to 0$. In the static limit,
this behavior is seen explicitly in the jet observable at fixed $\Delta\theta$
[cf. Eq. (13)] and in the inclusive cross section [cf. Eq. (16)]. In general,
the $m_{\ell}\to 0$ limit must be finite for any observable including real
photon radiation that is kinematically degenerate with the charged lepton in
the absence of radiation [20, 21, 22, 23].

The finiteness of the $m_{\ell}\to 0$ limit has important implications for
flavor ratios of charged-current (anti)neutrino cross sections. Consider,
e.g., the total inclusive cross section for $\nu_{\ell}n\to\ell^{-}p(\gamma)$
as a function of the lepton mass:

$\displaystyle\sigma(m_{\ell})=A+B_{0}\frac{m_{\ell}^{2}}{\Lambda^{2}}+B_{1}\frac{m_{\ell}^{2}}{\Lambda^{2}}\ln\frac{m_{\ell}^{2}}{\Lambda^{2}}+\dots\,,$
(59)

where $\Lambda$ denotes a conventional hard scale and the ellipsis denotes
terms of order $m_{\ell}^{4}$ and terms containing
$m_{\ell}^{2}\ln^{2}(m_{\ell})$ that arise from two-loop QED corrections; such
terms are negligibly small in the numerical analysis. Here $A$, $B_{0}$, and
$B_{1}$ are independent of $m_{\ell}$ but depend on the neutrino energy and
hadronic parameters. The ratio of muon to electron cross sections is then

$\displaystyle\frac{\sigma(m_{\mu})}{\sigma(m_{e})}$ $\displaystyle=1+{\cal
B}_{0}\frac{m_{\mu}^{2}}{\Lambda^{2}}+{\cal
B}_{1}\frac{m_{\mu}^{2}}{\Lambda^{2}}\ln\frac{m_{\mu}^{2}}{\Lambda^{2}}+{\cal
O}\left(\frac{m_{e}^{2}}{\Lambda^{2}},\alpha^{2}\frac{m_{\mu}^{2}}{\Lambda^{2}}\ln^{2}\frac{m_{\mu}}{\Lambda},\frac{m_{\mu}^{4}}{\Lambda^{4}}\right)\,,$
(60)

where we have written ${\cal B}_{0}=B_{0}/A$ and ${\cal B}_{1}=B_{1}/A$.

As an explicit example, the static-limit cross section through one-loop order
is

$\displaystyle\sigma(m_{\ell})=\sigma_{\rm
LO}\bigg{\\{}1+\frac{\alpha}{\pi}\left[\frac{3}{4}\ln\frac{\mu^{2}}{4E_{\nu}^{2}}+\frac{17}{4}+\frac{\pi^{2}}{3}+\frac{m_{\ell}^{2}}{E_{\nu}^{2}}\left(\frac{3}{4}\ln\frac{4E_{\nu}^{2}}{m_{\ell}^{2}}+\frac{19}{8}+\frac{\pi^{2}}{6}\right)+{\cal
O}(m_{\ell}^{4})\right]\bigg{\\}}\,.$ (61)

Choosing $\Lambda=2E_{\nu}$ and accounting for the $m_{\ell}$ dependence of
$\sigma_{\rm LO}$, we find

$\displaystyle{\cal
B}_{0}=-2+\frac{\alpha}{\pi}\left(\frac{19}{2}+\frac{2\pi^{2}}{3}\right)\,,\quad{\cal
B}_{1}=-\frac{3\alpha}{\pi}\,.$ (62)

The general case, beyond the static limit, is treated similarly but with
different numerical values of ${\cal B}_{0}$ and ${\cal B}_{1}$. For example,
at an illustrative neutrino energy $E_{\nu}=2\,{\rm GeV}$, and setting
$\Lambda=1\,{\rm GeV}$, integrating the total charged-current elastic cross
section of Eq. (43) yields

$\displaystyle{\cal B}_{0}(E_{\nu}=2\,{\rm GeV})=-0.28+{\cal
O}(\alpha,\epsilon_{\rm nuc})\,,\quad{\cal B}_{1}(E_{\nu}=2\,{\rm GeV})={\cal
O}(\alpha,\epsilon_{\rm nuc})\,,$ (63)

where $\epsilon_{\rm nuc}$ denotes a possible nuclear correction for
applications to bound nucleons. We will compute the ${\cal O}(\alpha)$
contributions within our hadronic model as part of the phenomenological
analysis in Sec. 6.5. We also consider ${\cal O}(\epsilon_{\rm nuc})$
corrections in a standard nuclear model in Sec. 6.7.

##  6 Phenomenological applications

Figure 10: Differential cross-section ratio to the tree-level result for
electron (anti)neutrino charged-current elastic scattering on the nucleon
accompanied by soft radiation, as a function of $Q^{2}$ for a few different
soft-photon energy cutoffs $\Delta E$. Left and right plots are for neutrino
and antineutrino scattering, respectively. Upper and lower plots are for
$E_{\nu}=600\leavevmode\nobreak\ \mathrm{MeV}$ and
$E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$, respectively. The curves show
the results for $\Delta E=10,\,20$, and $40\,{\rm MeV}$.  Figure 11: The same
as Fig. 10 but for the muon flavor.

In this section, we examine in detail several variations of the observables
introduced above. The observables are representative of typical event classes
for charged-current elastic (anti)neutrino scattering with final-state
electron or muon, with choices for real photons determined by typical detector
capabilities, as discussed in Sec. 2. In the following Sec. 6.1, we consider a
“minimal” observable containing just the charged lepton and soft radiation.
This type of observable is difficult to realize with typical neutrino
detectors that cannot cleanly separate electrons from energetic photons;
however, it provides a contrast with more inclusive cross sections, and also
is similar to well-studied observables in elastic electron-proton scattering.
Section 6.2 considers the default observable for electronlike jet events.
Section 6.3 discusses observables for muon flavor and studies scattering with
energetic photons, where muon (anti)neutrino events can be misidentified as
electron flavor. Section 6.4 considers the emission of noncollinear hard
photons. Section 6.5 includes such noncollinear hard-photon emission in the
evaluation of inclusive observables. Section 6.6 considers the ratio of
electron versus muon cross sections and discusses the extent to which hadronic
uncertainties cancel in the ratio. Leading nuclear effects are considered in
Sec. 6.7. Finally, we compare the size of radiative corrections with existing
experimental $\nu_{\mu}$ and $\overline{\nu}_{\mu}$ data in Sec. 6.8.

###  6.1 Cross section with charged lepton and soft photons

In all observables that involve only soft or collinear radiation, the cross
section is given by Eq. (28). The soft function $S$ is given by Eq. (29). The
hard function at the hard renormalization scale is computed as in Sec. 4.5,
and the running between hard and soft renormalization scales is determined as
in Sec. 4.5.3. The collinear function depends on the specification of jet
observable. For measurements that separate charged leptons from the
accompanying collinear radiation, only soft real radiation is relevant and the
collinear function is given by $J$ in Eq. (34). Precisely this observable has
been used in Sec. 4.5 to illustrate the impact of hadronic uncertainties in
Figs. 7 and 8, and to illustrate the convergence of resummed perturbation
theory in Fig. 9. The ratio of this cross section to the tree-level result in
the nonrelativistic limit was shown as a function of (anti)neutrino energy by
the solid blue curve in Fig. 2, where a default soft-photon energy cutoff
$\Delta E=20\,{\rm MeV}$ was used. The complete relativistic case, for a range
of $\Delta E$ values, is shown for neutrino-neutron and antineutrino-proton
scattering with electron flavor in Fig. 10 and with muon flavor in Fig. 11.
Results are displayed for typical accelerator (anti)neutrino energies
$E_{\nu}=600\leavevmode\nobreak\ \mathrm{MeV}$ and
$E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$. In these figures, $Q^{2}$ can be
identified as the momentum transfer between nucleons:
$Q^{2}=-(k-k^{\prime})^{2}$.

###  6.2 Electron-jet observable

Figure 12: Ratio of the differential cross section to the tree-level result
for $\nu_{e}$ ($\bar{\nu}_{e}$) including only soft or both soft and collinear
photon emission. Left and right plots are for neutrino and antineutrino
scattering, respectively. Upper and lower plots are for
$E_{\nu}=600\leavevmode\nobreak\ \mathrm{MeV}$ and
$E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$, with soft-photon energy cutoff
$\Delta E=25\leavevmode\nobreak\ \mathrm{MeV}$ and $\Delta
E=10\leavevmode\nobreak\ \mathrm{MeV}$, respectively. The lower blue dash-
dotted band corresponds to curves in Fig. 10. The upper pink dashed band
includes also real photons within a cone of size $\Delta\theta=10^{\circ}$.
The difference between the green dashed line and unity represents the
uncertainty of the tree-level cross section itself; this uncertainty largely
cancels in the displayed cross-section ratios.

In most practical neutrino detectors at energies of accelerator oscillation
experiments, final-state electrons with and without nearly collinear photons
cannot be distinguished. Neutrino detectors are typically homogeneous and are
the target material in which the (anti)neutrinos themselves interact. In
traveling through the detector, the final-state electrons lose energy by
bremsstrahlung, which also results in nearly collinear photons. These photons,
in turn, interact by producing $e^{+}e^{-}$ pairs, which then deposit energy
near the primary electron.

To account for the energy deposited by real collinear photons, we must modify
the observable from Sec. 6.1 that allowed only soft real photon emission. For
a fixed direction of charged lepton momentum, we consider all events where
either the photon is soft, i.e., has energy below $\Delta E$, or the photon is
energetic and collinear, i.e., has energy above $\Delta E$ and is radiated
within the angle $\Delta\theta$ to the electron direction. Such events are not
distinguished from charged-current elastic events without radiation. We remark
that this definition is similar to, but differs slightly from, a common scheme
in the QCD literature due to Sterman and Weinberg (SW) [160], where the angle
is specified relative to the direction of total energy flow. Such fine-grained
details are beyond the precision of the current generation of neutrino
experiments, but could be resolved using a more elaborate detector simulation.
Integrating over the phase space of energetic photons within the angle
$\Delta\theta$ to the electron direction, we obtain an additional contribution
to the collinear function in the factorization formula, given at one-loop
order by $j(x)$ in Eq. (35). As discussed in Sec. 4.4, in order to resum
leading large logarithms from multiple collinear photon emission, we
exponentiate the collinear photon correction [161, 162, 163].

Figure 12 shows the correction to unpolarized cross sections as a function of
hadronic momentum transfer, $Q^{2}=-(k-k^{\prime})^{2}$, over the kinematic
region of the elastic $2\to 2$ process. For reference, the plot includes the
curve from Sec. 6.1 with the observable describing only soft-photon radiation.
Also shown in the figure as a deviation from unity is the uncertainty of the
tree-level process, dominated by the poorly constrained axial-vector form
factor. This uncertainty dominates the error budget of the absolute cross
section but cancels between the numerator and denominator in various cross-
section ratios. The impact of this tree-level uncertainty and of radiative
corrections, on critical $\nu_{e}$ versus $\nu_{\mu}$ cross-section ratios
will be examined below.

###  6.3 Electronlike muon-jet events

Figure 13: Ratio of unpolarized radiative charged-current-elastic-like
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$ differential cross sections to the tree-
level result with collinear photon energy above the threshold for final muon
to be misidentified as electron, as described in the text. Reference scenarios
with $E_{\gamma}>E_{\mu}-m_{\mu}$ and $E_{\gamma}>200\,{\rm MeV}$ are shown
for comparison. Plots are for (anti)neutrino beam energies
$E_{\nu}=600\leavevmode\nobreak\ \mathrm{MeV}$ (upper plots) and
$E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$ (lower plots). The calculation
based on our default electroweak vertex is labeled as default model. The
calculation when the electroweak vertex is taken as in diagrams with local
interactions is labeled as local model.

Compared to the case of electrons, bremsstrahlung from muons is a rarer
process because of the larger muon mass. It is precisely this difference
between muons, which slowly lose energy by ionization and collision processes,
and electrons, which lose energy by bremsstrahlung, that is used as the
primary way to distinguish electron and muon (anti)neutrino interactions from
each other in neutrino oscillation experiments. Therefore, events with muons
and energetic collinear photons can be misidentified as electron events. For
this confusion to happen, the photon must carry a significant fraction of the
primary muon energy because low-energy collinear photons are consistent with
collisional processes where muons create $\gamma$ rays in the detector. To
accurately predict this confusion, a detailed detector simulation is required.
However, we may estimate the effect in a simplistic model in which a collinear
muon and photon will only be confused with an electron if the range of the
muon in the detector is less than the range of an electron with energy equal
to the sum of the photon energy and muon kinetic energy. Because the electron
shower range at high energies grows only logarithmically with electron energy,
but muon range grows nearly linearly, the fraction of the energy that must be
carried by the collinear photon will grow with muon energy. By comparing the
average length of electron showers [164, 5] to muon range [165], we have
developed an empirical parametrization of the collinear photon energies that
would not cause this confusion:
$E_{\gamma}\lesssim\tilde{E}\left\\{0.95-0.75/[(\tilde{E}/1\leavevmode\nobreak\
{\rm GeV})+0.85]\right\\}$, where $\tilde{E}=E_{\nu}-{Q^{2}}/{2M}-m_{\mu}$.
This parametrization is independent of the choice of detector materials
considered in this work – scintillator, water, and argon – and is valid for
$E_{c}\ll\tilde{E}\lesssim 5$ GeV, where $E_{c}$ is the electron critical
energy in the target medium, i.e., the energy above which electron energy loss
is dominated by bremsstrahlung [166]. $E_{c}$ is approximately $90$ MeV in
polystyrene scintillator, $75$ MeV in water, and $32$ MeV in liquid argon.

A quantity of interest is the cross section for muon neutrino and antineutrino
events with collinear photon energies large enough so that the final-state
muon could be mistaken for an electron and therefore lead to flavor
misidentification. This is a potential concern, especially since electron
(anti)neutrino events are much less common than muon (anti)neutrino events in
the relevant (anti)neutrino beams. The ratio of this cross section to the
tree-level charged-current elastic cross section is shown as the solid curve
in Fig. 13, using our default hadronic model (“default model” in the figure).
As the figure illustrates, the probability for such misidentification is
small, of order one part in $10^{4}$, and this is a subleading effect in the
current generation of experiments. This effect is potentially much larger for
experiments whose particle identification algorithms are sensitive to lower-
energy collinear photons than were presumed to cause misidentification in this
study. To illustrate the dependence on the assumed photon energy threshold for
misidentification as electron (anti)neutrinos, the figure also shows the cross
section for the cases where $E_{\gamma}>E_{\mu}-m_{\mu}$ and
$E_{\gamma}>200\leavevmode\nobreak\ \mathrm{MeV}$. As remarked in Sec. 4.5, a
naive form-factor insertion model for the hard matching condition violates
electromagnetic current conservation, leading, e.g., to spurious
$\sim\ln(m_{\ell})$ collinear singularities. We illustrate the importance of
ensuring this conservation condition by comparing our default hadronic model
to the naive form-factor insertion model (“local model” in the figure).

###  6.4 Energetic photons isolated from charged lepton jets

Figure 14: Ratio of unpolarized radiative charged-current-elastic-like
$\nu_{e}$ and $\bar{\nu}_{e}$ differential cross sections to the tree-level
result with hard photons outside the jet cone. Curves show
$E_{\gamma}>20\leavevmode\nobreak\ \mathrm{MeV}$, $40$, and $80$ MeV, for the
angle between the electron and photon $\theta>10^{\circ}$. Plots are for
(anti)neutrino beam energies $E_{\nu}=600\leavevmode\nobreak\ \mathrm{MeV}$
(upper plots) and $E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$ (lower plots).
The calculation based on our default electroweak vertex is labeled as default
model. The calculation when the electroweak vertex is taken as in diagrams
with local interactions is labeled as local model.  Figure 15: The same as
Fig. 14 but for the muon flavor charged-current-elastic-like cross section,
with $\theta>2^{\circ}$.

Hard photons outside of electron jets can be misidentified as coming from
$\pi^{0}\to\gamma\gamma$ and may be removed from many oscillation analyses in
an attempt to reduce such backgrounds. In the case of low-energy oscillation
experiments, attempts to select quasielastic muon neutrino or antineutrino
interactions may also remove events with a visible photon that is not
collinear with the muon. We estimate the relative contributions of such events
in the following.

To evaluate one-photon events with energy above $\Delta E$ outside the cone of
angle $\Delta\theta$, we use the hadronic model of Secs. 4.5.2 and 5.1,
labeled default model in our figures. Recall that this model differs from a
naive “form-factor insertion” ansatz by the choice of form-factor arguments:
$Q^{2}$ at the weak vertex is determined by the external nucleon leg
kinematics. In the phase space of photons outside the collinear region, the
naive model does not give rise to spurious $\sim\ln{m_{\ell}}$ singularities,
and we may use the difference between this local model and our default
hadronic model as a crude measure of model dependence. Note that the cross
section including these one-photon events is exclusive of the jet cross
section of Sec. 6.2 with only one photon, for the same $\Delta E$ and
$\Delta\theta$.

The results in Figs. 14 and 15 for photon energy thresholds $\Delta E$ of $20$
MeV, $40$, and $80$ MeV show that the cross section for hard photons outside
of electron jets is of order $1\%$ to several $\%$ of the tree-level cross
section and potentially significant for current and future neutrino
oscillation experiments.

###  6.5 Inclusive observables

Figure 16: Ratio of the inclusive differential cross section including
noncollinear emission of one hard photon to the tree-level result is presented
as a function of $Q^{2}$. Electron neutrino-neutron scattering is shown on
left plots and antineutrino-proton scattering on right plots. For comparison,
the result including only soft photon radiation with energy below $\Delta E$
is shown by the black dash-dotted lines and light blue band. The
$E_{\ell}+E_{\gamma}$ energy spectrum is shown by the red dashed lines; the
spectrum of events reconstructed from the energy within the
$\Delta\theta=10^{\circ}$ cone is shown by the dark-blue dashed double-dotted
lines and blue band; and the energy spectrum reconstructed from the electron
energy is shown by the green dotted lines. (Anti)neutrino beam energies are
taken as $E_{\nu}=600\leavevmode\nobreak\ \mathrm{MeV}$ (upper plots) and
$E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$ (lower plots), and the soft-
photon energy cutoff is $\Delta E=25\leavevmode\nobreak\ \mathrm{MeV}$ and
$\Delta E=10\leavevmode\nobreak\ \mathrm{MeV}$, respectively. $Q^{2}$ is
defined according to the spectrum in figures. The displayed $Q^{2}$ range for
$600\leavevmode\nobreak\ \mathrm{MeV}$ (anti)neutrino beam corresponds to the
maximum for tree-level elastic kinematics. For $2\leavevmode\nobreak\
\mathrm{GeV}$ (anti)neutrinos, the displayed $Q^{2}$ range is truncated to
focus on the phenomenologically relevant regime. Central value cross sections
for the entire $Q^{2}$ region are displayed in Fig. 18.  Figure 17: The same
as Fig. 16 but for the muon flavor. The cone observable is not presented since
the radiation inside the cone is negligible for the muon flavor.  Figure 18:
Tree-level differential cross section for the (anti)neutrino beam energies
$E_{\nu}=600\leavevmode\nobreak\ \mathrm{MeV}$ and
$E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$ is compared to the spectra of
radiated events, with photons above the energy $\Delta
E=25\leavevmode\nobreak\ \mathrm{MeV}$ and $\Delta E=10\leavevmode\nobreak\
\mathrm{MeV}$, respectively, when $Q^{2}$ is defined from the lepton energy
$E_{\ell}$ or from the sum of lepton and photon energies
$E_{\ell}+E_{\gamma}$. Results are shown for electron and muon flavors, for
neutrino-neutron scattering on the left plot and for antineutrino-proton
scattering on the right plot. The flavor difference of tree-level cross
sections is unresolved on this figure over the wide range of allowed
kinematics; see upper curves on all plots. We also present the spectrum of
events reconstructed from the energy within the cone of angle
$\Delta\theta=10^{\circ}$ for the electron flavor. $Q^{2}$ and its range are
defined according to the spectrum in figures. Integrals over all $Q^{2}$
values for $E_{\ell}$, energy in 10∘ cone, and $E_{\ell}+E_{\gamma}$ spectra
are the same.

Sections 6.1, 6.2, and 6.3 restricted attention to event classes that excluded
noncollinear hard photons. In Sec. 6.4, we considered the contribution of
noncollinear hard photons to the cross section for charged-current
(anti)neutrino-nucleon scattering, finding that such events can contribute
significantly in current and future neutrino experiments. These photons may
not be explicitly reconstructed but, if not vetoed, would contribute to the
observed event rate and would affect (anti)neutrino energy reconstruction
based solely on the visible energy in the leptonic shower.

For measurements that include (i.e., do not veto) such hard photons, the
relevant cross section involves integration over the full phase space of
radiated photons. For such observables, we perform a calculation within our
default hadronic model described in Secs. 4.5.2 and 5.1. In contrast to Secs.
6.1 and 6.2, we compute at fixed order in QED perturbation theory. We assign
the same tree-level form-factor uncertainty and the same uncertainty
assignment for virtual contributions (cf. Sec. 6.4 regarding uncertainty of
the hard real photon contribution). Figures 16 and 17 provide results for
distributions when the independent variable $Q^{2}$ is defined using different
measures of leptonic energy. For example, in the case of electrons, $Q^{2}$
may be defined using the electron energy itself, $Q^{2}=2M(E_{\nu}-E_{\ell})$,
or using the sum of electron and photon energies,
$Q^{2}=2M(E_{\nu}-E_{\ell}-E_{\gamma})$, or perhaps as in a realistic
experiment, using the energy reconstructed within a $\Delta\theta=10^{\circ}$
cone whose axis is around the electron direction. Note that for the $E_{\ell}$
spectrum, parameter $\Delta E$ does not enter: the observable sums over all
“unobserved” photons, not just those below a threshold $\Delta E$. Similarly
for the $E_{\ell}+E_{\gamma}$ spectrum, $\Delta E$ does not enter, since the
spectrum is equivalent to the spectrum in hadronic momentum transfer $Q^{2}$,
which sums over all possible photon energies.191919Parameter $\Delta E$ enters
as a regulator parameter during the calculation, separating an IR-divergent
soft region from the remainder, but is taken negligibly small for the final
results. For the “energy in cone” observable, we use $\Delta
E=20\leavevmode\nobreak\ \mathrm{MeV}$ to specify the selection, assuming that
photons below this value are not reconstructed.

Note that the allowed region of $Q^{2}$ is different for different spectra and
may extend beyond the tree-level kinematic limit. In particular, the
$E_{\ell}$ spectrum extends up to $Q^{2}=2M\left(E_{\nu}-m_{\ell}\right)$, in
contrast to the $E_{\ell}+E_{\gamma}$ spectrum, which extends only to the
tree-level kinematic limit. We illustrate these regions and provide
corresponding cross sections for neutrino-neutron and antineutrino-proton
scattering in Fig. 18.

###  6.6 Cross-section ratios

Figure 19: Electron over muon flavor ratios of unpolarized differential cross
section in neutrino-neutron (left plots) and antineutrino-proton (right
plots). Blue solid lines and blue band, tree level; black dash-dotted lines
and light blue band, including radiation of soft photons with energy below
$\Delta E$; red dashed lines and pink band, including radiation of soft
photons with energy below $\Delta E$ and including radiation of collinear
photons for the electron flavor as described in Sec. 6.2 but not for the muon
flavor. For (anti)neutrino beam energies $E_{\nu}=600\leavevmode\nobreak\
\mathrm{MeV}$ (upper plots) and $E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$
(lower plots), the soft-photon energy cutoff is $\Delta
E=25\leavevmode\nobreak\ \mathrm{MeV}$ and $\Delta E=10\leavevmode\nobreak\
\mathrm{MeV}$, respectively. Figure 20: Antineutrino-proton over neutrino-
neutron differential cross-section double ratios to the tree-level results
with electron (left plots) and muon (right plots) flavors including the
radiation of soft photons with energy below $\Delta E$, shown by the black
dash-dotted lines and light blue band, and accounting also for the radiation
of collinear photons in case of electron flavor as described in Sec. 6.2,
shown by the red dashed lines and pink band. For (anti)neutrino beam energies
$E_{\nu}=600\leavevmode\nobreak\ \mathrm{MeV}$ (upper plots) and
$E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$ (lower plots), the soft-photon
energy cutoff is $\Delta E=25\leavevmode\nobreak\ \mathrm{MeV}$ and $\Delta
E=10\leavevmode\nobreak\ \mathrm{MeV}$, respectively. Differences between the
double ratios with and without collinear photons arise from subleading effects
and are indistinguishable in the figure (see text). The relative uncertainty
of the tree-level antineutrino over neutrino cross-section ratio is
represented by the green dotted line as a deviation from unity. Figure 21:
(Inclusive) Electron over muon flavor ratios of unpolarized differential cross
sections in neutrino-neutron (left plots) and antineutrino-proton (right
plots) scattering at tree level, shown by the blue solid lines; the ratio of
$E_{\ell}$ spectra is shown by the green dotted lines; the ratio of spectrum
of events reconstructed from the energy within the $10^{\circ}$ cone for the
electron flavor to the $E_{\ell}$ spectrum for the muon flavor is shown by the
dark-blue dashed double-dotted lines and blue band; and the ratio of
$E_{\ell}+E_{\gamma}$ energy spectra is shown by the red dashed lines.
(Anti)neutrino beam energies are taken as $E_{\nu}=600\leavevmode\nobreak\
\mathrm{MeV}$ (upper plots) and $E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$
(lower plots); the soft-photon energy cutoff for jet events with electron
(anti)neutrinos is $\Delta E=25\leavevmode\nobreak\ \mathrm{MeV}$ and $\Delta
E=10\leavevmode\nobreak\ \mathrm{MeV}$, respectively. $Q^{2}$ is defined
according to the spectrum in figures.  Figure 22: (Inclusive) The same as Fig.
20 but for inclusive observables as specified in the caption to Fig. 21.

We have noted that, after the inclusion of radiative corrections, the
uncertainty of absolute cross sections for either electron or muon flavor, and
for either neutrino or antineutrino, is dominated by hadronic uncertainties of
the tree-level process. Cross-section ratios, for electron versus muon and for
neutrino versus antineutrino, are critical to oscillation analyses. Within our
formalism, we can systematically investigate the impact of radiative
corrections on these ratios, determining the absolute size of corrections, and
the extent to which hadronic uncertainties cancel.

Figure 19 displays the cross-section ratio of electron to muon flavor,
comparing the tree-level case to the default observable for exclusive electron
and muon charged-current events. For comparison, we show also the ratio of the
soft-photon-only cross sections. For observables taken at the same value of
hadronic $Q^{2}$, the hard function $H$ cancels identically in the ratio in
our leading-power analysis. The remaining soft and jet functions depend on the
charged lepton mass and the kinematic parameter $\Delta\theta$.

Figure 20 displays the double ratio of $\sigma/\sigma^{\rm LO}$ for neutrino
versus antineutrino events. The deviation from unity is small compared to the
magnitude of the total radiative correction to either neutrino or antineutrino
separately, reflecting the structure of the factorization formula, as follows.
First, for a given charged lepton mass, the jet function is identical when the
same event selection is used. Second, the soft function differs only in the
kinematics of the proton, and identical leading logarithmically enhanced terms
appear for neutrino and antineutrino cases at the same charged lepton
kinematics (the soft functions become identical in the static limit; cf. Sec.
3). Finally, deviations of the ratio from unity from the hard function $H$
also do not contain large logarithm enhancements. Note that unlike the case of
$e$ versus $\mu$ cross sections for either neutrino or antineutrino, hadronic
uncertainties in the hard function do not cancel. Since only the ratio of
antineutrino over neutrino cross sections but not the double ratio can be
accessed experimentally, we present the relative uncertainty of the tree-level
ratio as a deviation from unity in Fig. 20. This uncertainty from the hard
function presently limits our knowledge of antineutrino over neutrino cross-
section ratios and can be reduced only with improved knowledge of elementary
(anti)neutrino-nucleon scattering amplitudes.

Figures 21 and 22 show the analog of Figs. 19 and 20, but calculated for the
inclusive event sample, as discussed in Sec. 6.5. For each spectrum here,
$Q^{2}$ is reconstructed from the corresponding energy as described in Sec.
6.5. For this case, there is no exact cancellation between hard functions in
electron versus muon cross sections, since the events for a given value of
(reconstructed) $Q^{2}$ do not necessarily correspond to the same nucleon
kinematics. Moreover, the double ratio deviates significantly from unity for
the $E_{\ell}$ spectrum, for which there is no direct correspondence to the
events of the $E_{\ell}+E_{\gamma}$ spectrum.

###  6.7 Leading nuclear effects

In current neutrino detectors, most scattering events take place on nucleons
bound inside nuclei. In the following, we estimate the relative importance of
leading nuclear corrections. We represent these corrections by an “impulse”
model, where the nucleon-level process is described by assigning modified
kinematics to the initial-state nucleon, and by enforcing constraints on the
final-state nucleon kinematics. This model does not represent a complete
description of complicated nuclear dynamics.202020For work on nuclear
structure effects for interaction cross sections, and further references, see
Refs. [167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180,
181, 182, 183, 184, 185]. However for GeV (anti)neutrino energies and for the
nuclei of interest (e.g., carbon, oxygen, and argon), it is expected to
capture the leading effects and is sufficient for our purposes. In particular,
we demonstrate that the dominant corrections cancel in critical cross-section
ratios, in accordance with the general arguments in Sec. 5.2. We evaluate
separately the effects due to the binding/removal energy, Fermi motion, and
Pauli blocking. A detailed comparison to recent developments in nuclear
physics would require more elaborate calculation including final-state
interactions, long-range correlations, and meson exchange currents. Some of
these effects are taken into account in modern neutrino event generators [186,
187, 188, 189, 190, 191, 192, 193].

According to the prescription of Smith and Moniz [194], a bound nucleon is
described by changing the free-nucleon calculation by the replacement of the
initial nucleon mass $M\to M-E_{b}$.212121The precise replacement ansatz
depends on a conventional choice of hadronic amplitude basis and of
independent kinematic variables [130]. This choice does not affect the present
discussion. Since the limit $E_{b}\to 0$ is smooth, the leading term in the
expansion is of order $E_{b}/M$ and cancels in the flavor ratio when lepton-
mass corrections are ignored. For the relevant range of $E_{b}$, of order
${\rm few}\times 10\,{\rm MeV}$, binding effects on the absolute cross
sections are $\sim{\rm few}\times 1\,\%$ or below.

To estimate the corrections due to Fermi motion, we boost to the rest frame of
the initial-state struck nucleon. Instead of the lab frame (anti)neutrino
energy $E_{\nu}$, the nucleon in its rest frame sees a neutrino with energy

$\sqrt{E_{\nu}^{2}\sin^{2}\theta+\left(1+\frac{\vec{k}^{2}}{M^{2}}\right)\left(E_{\nu}\cos\theta-|\vec{k}|\right)^{2}},$
(64)

and the nuclear cross section is given by an average over the initial nucleon
angle $\theta$ and summation over all nucleon states up to the Fermi momentum,
$|\vec{k}|\leq k_{F}$ (for numerical evaluations, we take a value for the
${}^{40}\mathrm{Ar}$ nucleus $k_{F}\approx 240\leavevmode\nobreak\
\mathrm{MeV}$). Consequently, Fermi motion results in relative corrections of
order $k_{F}^{2}/M^{2}$ and $k_{F}^{2}/E_{\nu}^{2}$. Such corrections could
become large at very small (anti)neutrino energies close to the muon
production threshold, which we illustrate in Table 1. For flavor ratios,
relative changes are much smaller than for cross sections themselves. A
similar effect applies to all distributions presented in this paper for the
(anti)neutrino energies of interest.

|  $E_{\nu}$, GeV
---
$\frac{\delta\sigma}{\sigma}$, % |  $\left(\frac{\delta\left({\sigma_{e}}/{\sigma_{\mu}}\right)}{{\sigma_{e}}/{\sigma_{\mu}}}\right)_{\mathrm{LO}}$, % |  $\left(\frac{\delta\left({\sigma_{e}}/{\sigma_{\mu}}\right)}{{\sigma_{e}}/{\sigma_{\mu}}}\right)_{\mathrm{NLO}}$, %
$0.6$ | $\lesssim 2.5$ | $\lesssim 0.06$ | $\lesssim 0.05$
$2.0$ | $\lesssim 0.3$ |  $\lesssim 0.003$ | $\lesssim 0.01$
Table 1: Relative effects of the Fermi motion on the total unpolarized cross
sections and flavor ratios at leading (LO) and next-to-leading (NLO) orders
for (anti)neutrino beam energies $600$ MeV and $2$ GeV. Upper limits for both
neutrino and antineutrino (and for both muon and electron flavor in the second
column) are shown in this table.

Pauli blocking effects similarly scale as $k_{F}^{2}$, but because they are
concentrated at low $Q^{2}$ where the cross section is sizable, the impact on
total cross sections can be significant. In Table 2, we present the effect on
the cross section as well as the change of flavor ratios. We represent the
Pauli blocking effect by retaining only that region of phase space where the
final-state nucleon momentum has $|\vec{k}|>k_{F}$ [194]. This effect only
slightly depends on the (anti)neutrino flavor at energies much larger than the
muon mass. Going to lower (anti)neutrino energies, the effect increases [195,
196, 197]. Pauli blocking corrections to flavor ratios are below or comparable
to hadronic uncertainties propagated from total inclusive cross sections.

|  $E_{\nu}$, GeV
---
|  $\frac{\delta\sigma}{\sigma}$, % |  $\left(\frac{\delta\left({\sigma_{e}}/{\sigma_{\mu}}\right)}{{\sigma_{e}}/{\sigma_{\mu}}}\right)_{\mathrm{LO}}$, % |  $\Bigg{|}\left(\frac{\delta\left({\sigma_{e}}/{\sigma_{\mu}}\right)}{{\sigma_{e}}/{\sigma_{\mu}}}\right)_{\mathrm{NLO}}\Bigg{|}$, %
$0.6$ |  | $\nu$
---
$\bar{\nu}$
| $-13$
---
$-37$
| $0.15$
---
$0.21$
| $\lesssim 0.2$
---
$\lesssim 0.2$
$2.0$ |  | $\nu$
---
$\bar{\nu}$
| $-11$
---
$-18$
| $-0.002$
---
$0.009$
| $\lesssim 0.01$
---
$\lesssim 0.01$
Table 2: Relative effects of Pauli blocking on the total unpolarized cross
sections and flavor ratios for (anti)neutrino beam energies $600$ MeV and $2$
GeV.

In summary, while nuclear effects can have a sizable impact on absolute cross
sections, they do not significantly change flavor ratios of total inclusive
cross sections and of spectra with respect to the hadronic momentum transfer.
This statement remains true in the presence of radiative corrections,
following from the results in Sec. 6.5. Returning to the discussion of Sec.
5.2, we may summarize the impact of nuclear corrections on the flavor ratio as
a shift in ${\cal B}_{0}(E_{\nu}=2\,{\rm GeV})=-0.28\to-0.32$ [130], for the
combined effects of binding energy, Fermi motion and Pauli blocking.

###  6.8 Comparison to (anti)neutrino-nucleus quasielastic data

Our results predict significant radiative corrections to the charged-current
elastic scattering process. These corrections are especially important in
$\nu_{e}$ versus $\nu_{\mu}$ flavor ratios owing to the sparsity of $\nu_{e}$
cross-section measurements, the potentially larger contributions for $\nu_{e}$
versus $\nu_{\mu}$, and the direct impact of corrections on
$\nu_{\mu}\to\nu_{e}$ oscillation analyses [3]. However, radiative corrections
are also present in the more-abundant $\nu_{\mu}$ scattering data. The elastic
scattering process has been studied with muon flavor neutrinos and
antineutrinos scattering off free protons [198], weakly bound light nuclei
[123, 124, 126, 199, 127, 200, 201, 131], and tightly bound medium-sized
nuclei [202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215,
216, 217, 218, 219, 220]. Here, we examine how radiative corrections affect
the interpretation of these measurements, which were analyzed using tree-level
cross-section models.

We consider radiative corrections in several large datasets, all of which have
been collected using hydrocarbon targets. A complication in implementing the
radiative corrections is that, for measurements on nuclei, the same observable
hadronic final state can originate from different nucleon-level processes:
events originating from nucleon-level elastic scattering are observed together
with other processes such as multinucleon knockout, or pion production
followed by pion absorption in the target nucleus. The collection of these
events is referred to as “CCQE-like” or “CC0$\pi$” in the literature. In this
study of CC0$\pi$ $\nu_{\mu}$ and $\overline{\nu}_{\mu}$ interactions, we
define CC$0\pi$ to be an event with a single appropriately charged muon and no
other charged leptons, no mesons, and any number of nucleons in the final
state, after final-state interactions in the nucleus have occurred. To compare
the effects from radiative corrections with experimental data, we embed our
predictions into neutrino event generators that include a default model for
nuclear effects and predictions for inelastic feed-down into the signal.

Figure 23: Estimated cross-section ratio to the tree-level result for the
CC$0\pi$ process in T2K, NO$\nu$A, DUNE, and MINERvA (LE and ME flux
configurations) experiments after averaging over the corresponding neutrino
flux, as a function of $Q^{2}=-\left(p_{\nu}-p_{\mu}\right)^{2}$. See text for
details.

We implement radiative corrections as follows. For each neutrino energy, we
compute the unpolarized differential neutrino-nucleon cross section for
$\nu_{\mu}$ (and $\overline{\nu}_{\mu}$) scattering, with $Q^{2}$ defined
using lepton kinematics [i.e., $Q^{2}\equiv 2M(E_{\nu}-E_{\ell})$]. We form
the ratio of this result to the tree-level cross section. For
$E_{\nu}=600\,{\rm MeV}$ and $2\,{\rm GeV}$, this ratio is displayed in Fig.
17, labeled “$E_{\ell}$ spectrum”. In the sum over initial-state nucleons of
the nuclear model, radiative corrections are applied by multiplying the tree-
level result by this ratio, in the rest frame of each struck nucleon. A flux
average is then performed over neutrino energy for the relevant neutrino beam.
The logarithmically enhanced contributions to radiative corrections,
$\sim\ln(E_{\nu}^{2}/m_{\mu}^{2})$, depend only on lepton kinematics and are
independent of hadronic structure. We therefore apply the same
$Q^{2}$-dependent correction determined from “true CCQE” events to all events
that pass a CC0$\pi$ selection. For multinucleon events in the nuclear model,
the struck nucleon (neutron for $\nu$, proton for $\overline{\nu}$) is the one
considered participating in the interaction, and radiative corrections are
again applied in the rest frame of the struck nucleon. In this evaluation for
muon flavor, we neglect events in the phase-space region outside elastic
kinematics (cf. Fig. 18); explicit evaluation shows that such events change
muon (electron) flavor cross sections in the relevant energy range at permille
(percent) level or below.

Neutrino interactions are generated with the NEUT [221] and GENIE [222]
neutrino-event generators222222NEUT 5.5.0 with Valencia model 1p1h and 2p2h
and GENIEv3 G18_10a_02_11a. for the MINERvA, NO$\nu$A, and T2K muon
(anti)neutrino fluxes on a plastic scintillator (CH) target, and for DUNE
[223] on a ${}^{40}\text{Ar}$ target, using NUISANCE [224]. For MINERvA, both
the low-energy (LE) and medium-energy (ME) neutrino flux configurations are
included. NEUT and GENIE show very similar features, and only predictions
using NEUT are presented in the following figures. The resulting ratios to
tree-level cross sections for CC0$\pi$ events from the T2K, NO$\nu$A, MINERvA,
and DUNE experiments are shown in Fig. 23. A $4-7\leavevmode\nobreak\ \%$
suppression is observed at low $Q^{2}$ for the NO$\nu$A, MINERvA, and DUNE
experiments, all of which have significant neutrino flux in the
$E_{\nu}>1\leavevmode\nobreak\ \text{GeV}$ region.

Figure 24: The effect of radiative corrections to the tree-level prediction
for the $\cos\theta_{\mu}=0.94-0.98$ bin of the T2K ND280 $\nu_{\mu}$ (left)
and $\overline{\nu}_{\mu}$ (right) CCQE-like data [218] is shown for the cross
section per nucleon of a CH target.

Figure 25: The effect of radiative corrections to the tree-level prediction
for the $\cos\theta_{\mu}=0.60-0.70$ bin of the T2K ND280 $\nu_{\mu}$ (left)
and $\overline{\nu}_{\mu}$ (right) CCQE-like data [218] is shown for the cross
section per nucleon of the CH target.

Figure 26: The effect of radiative corrections to the tree-level prediction
for one slice of the MINERvA LE $\nu_{\mu}$ CCQE-like data [213] (left) and
$\overline{\nu}_{\mu}$ CCQE-like data [225] (right) is shown for the cross
section per nucleon of the CH target.

Figure 27: The effect of radiative corrections to the tree-level prediction
for the same slice of the MINERvA ME $\nu_{\mu}$ [216] (left) and
$\bar{\nu}_{\mu}$ [226, 227] (right) CCQE-like data is shown for the cross
section per nucleon of the CH target.

The kinematic dependence of radiative corrections is compared to experimental
data in Figs. 24-27. For the T2K experiment, the data are presented as a
function of the momentum of the muon, $p_{\mu}$, in a fixed range of the
cosine of the lepton scattering angle relative to the neutrino beam direction,
$\cos\theta_{\mu}$. We choose illustrative $\cos\theta_{\mu}$ bins close to
the peak in the $\cos\theta_{\mu}$ distribution from Ref. [218]. For the
MINERvA experiment, the data are presented as a function of the transverse
projection of the muon momentum along the neutrino beam direction, $p_{t}$, in
a fixed range of the longitudinal projection $p_{||}$ (we choose illustrative
$p_{||}$ bins close to the broad peak in the $p_{||}$ distributions from Refs.
[213, 225]). For the comparison to T2K $\nu_{\mu}$ and $\overline{\nu}_{\mu}$
data in Figs. 24 and 25, experimental error bars are large compared to the
small radiative corrections displayed in Fig. 23. For the LE MINERvA data,
opposite effects are observed in the low-$p_{t}$ and high-$p_{t}$ regions,
corresponding to the low-$Q^{2}$ and high-$Q^{2}$ regions of Fig. 23. For the
ME MINERvA data, the effects are enhanced further at low $p_{t}$ due to the
larger neutrino energy (cf. Fig. 23); as for the LE MINERvA data, the effect
for $\overline{\nu}_{\mu}$ is stronger than for $\nu_{\mu}$.

Previous studies have used the kinematic dependence of data-theory comparisons
to constrain uncertain nucleon and nuclear parameters, e.g., providing
measurements of axial nucleon form factors [123, 124, 125, 126, 127, 128, 129,
47, 130, 131, 132, 133, 198], multinucleon processes, or other nuclear effects
[228, 229]. The corrections to tree-level predictions in this work can be used
to determine the impact of radiative corrections on these measurements and
will be important for the interpretation of future precise measurements with
neutrinos.

##  7 Summary and outlook

In this paper, we have developed the framework for radiative corrections in
charged-current (anti)neutrino-nucleon elastic scattering at GeV
(anti)neutrino energies. Exploiting effective field theories, we have shown
that scattering cross sections factorize into soft, collinear, and hard
functions. The soft and collinear functions contain flavor-dependent large
logarithm enhancements and depend on detailed experimental conditions but can
be computed perturbatively. The hard function is subject to hadronic
uncertainty but is independent of the charged lepton mass and cancels in
ratios of cross sections for different lepton flavors involving the same
hadronic kinematics.

We have provided analytic expressions for the soft function and its small
lepton-mass limit. We have performed the first calculations of the collinear
function separately for virtual corrections and for the radiation of one
collinear photon, and have provided small-angle and small lepton-mass limits.
We have illustrated the factorization theorem in an exactly calculable model
corresponding to a nonrelativistic nucleon and ultrarelativistic lepton.

We have expressed the hard function for arbitrary kinematics in terms of a
model-independent amplitude decomposition for charged-current (anti)neutrino-
nucleon elastic scattering. We have determined the invariant amplitudes in a