# Radiative (anti)neutrino energy spectra from muon, pion, and kaon decays

Oleksandr Tomalak  Theoretical Division, Los Alamos National Laboratory, Los
Alamos, NM 87545, USA

(August 27, 2024)

To describe low-energy (anti)neutrino fluxes in modern coherent elastic
neutrino-nucleus scattering experiments as well as high-energy fluxes in
precision-frontier projects such as the Enhanced NeUtrino BEams from kaon
Tagging (ENUBET) and the Neutrinos from STORed Muons (nuSTORM), we evaluate
(anti)neutrino energy spectra from radiative muon ($\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}(\gamma),\leavevmode\nobreak\ \mu^{+}\to
e^{+}{\nu}_{e}\bar{\nu}_{\mu}(\gamma)$), pion $\pi_{\ell 2}$
($\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}(\gamma),\leavevmode\nobreak\
\pi^{+}\to\mu^{+}{\nu}_{\mu}(\gamma)$), and kaon $K_{\ell 2}$
($K^{-}\to\mu^{-}\bar{\nu}_{\mu}(\gamma),\leavevmode\nobreak\
K^{+}\to\mu^{+}{\nu}_{\mu}(\gamma)$) decays. We compare detailed
$\mathrm{O}\left(\alpha\right)$ distributions to the well-known tree-level
results, investigate electron mass corrections and provide energy spectra in
analytic form. Radiative corrections introduce continuous and divergent near
the endpoint component on top of the monochromatic tree-level meson-decay
spectra which can change the flux-averaged cross section at $6\times 10^{-5}$
level for the scattering on ${}^{40}\mathrm{Ar}$ nucleus with (anti)neutrinos
from the pion decay at rest. Radiative effects modify the expected
(anti)neutrino fluxes from the muon decay around the peak region by $3-4$
permille which is a precision goal for the next-generation artificial neutrino
sources.

###### Contents

  1. 1 Introduction
  2. 2 Pion and kaon decays
    1. 2.1 Pion and kaon decays at tree level
    2. 2.2 Virtual corrections
    3. 2.3 Soft-photon bremsstrahlung
    4. 2.4 Contribution of hard photons
    5. 2.5 Decay spectra and integrated cross sections
  3. 3 Muon decay
    1. 3.1 Muon decay at tree level
    2. 3.2 Virtual corrections
    3. 3.3 Soft-photon bremsstrahlung
    4. 3.4 Contribution of hard photons
    5. 3.5 Decay spectra and integrated cross sections
  4. 4 Conclusions and Outlook

##  1 Introduction

Muon, pion, and kaon decays provide us basic information on the low-energy
effective field theories of the Standard Model and serve as main sources for
the corresponding low-energy constants, i.e., precise lifetime measurements
give us access to the Fermi coupling constant $\mathrm{G}_{\mathrm{F}}$, the
pion decay constant $f_{\pi}$, and the product of the kaon decay constant
$f_{K}$ with Cabibbo-Kobayashi-Maskawa (CKM) matrix element $V_{us}$. These
decays show an excellent agreement of modern experiments with precise Standard
Model predictions [1, 2, 3, 4, 5, 6, 7].

Muon decay is a pure leptonic process independent of hadron-structure
corrections. At leading order, muon decays to electron and two corresponding
(anti)neutrinos. Radiative decay is suppressed by the electromagnetic coupling
constant and is a sub-leading process for a sizable photon energy resolution.
The next subdominant channel contains also $e^{+}e^{-}$ pair besides electron
and neutrinos. Its branching is suppressed by 5 orders of magnitude compared
to the main decay mechanism (the branching ratio is $\left(3.4\pm
0.4\right)\times 10^{-5}$ [8, 9]). Other decay channels of the muon were not
observed experimentally. The dominant pion decay mode contains muon and muon
(anti)neutrino in the final state. The subdominant electronic channel is
helicity suppressed by 4 orders of magnitude [10]. The corresponding radiative
decays and decays accompanied with $e^{+}e^{-}$ pairs are suppressed by the
electromagnetic coupling constant. The other observed channel $\pi^{+}\to
e^{+}\nu_{e}\pi^{0}$ is suppressed by a small phase-space volume by a factor
$\left(1.036\pm 0.006\right)\times 10^{-8}$ [11, 9] at the level of decay
width.

Having a larger mass, kaons have a variety of decay modes. Pure hadronic, pure
leptonic, and semi-leptonic modes are allowed. The dominant leptonic decay
with the branching ratio $\left(63.56\pm 0.11\right)\leavevmode\nobreak\ \%$
has muon and the corresponding (anti)neutrino in the final state. Decays into
the neutral pion are not suppressed by the phase-space volume compared to
decays of $\pi^{\pm}$ and have branchings around 5 % for $K_{e3}$ mode
$K^{\pm}\to\pi^{0}e^{\pm}\nu_{e}$ and around 3.4 % for $K_{\mu 3}$ mode
$K^{\pm}\to\pi^{0}\mu^{\pm}\nu_{\mu}$. Other decay channels with electrons are
suppressed by five orders of magnitude.

Muon, pion, and kaon decay channels are at the heart of artificial
(anti)neutrino sources. At low energies, experiments dedicated to searches for
sterile neutrinos [12, 13, 14, 15, 16] and to measurements of the coherent
elastic neutrino-nucleus scattering cross sections [17, 18, 19, 20] are
performed with (anti)neutrinos from stopped muons and pions [21, 22, 23, 24,
25, 12, 26, 27, 13, 17, 28]. At higher energies, neutrino oscillation
experiments [29, 30, 31, 32, 33] are based on muons, pions, and kaons decaying
in flight [34, 35, 36, 37]. Achieving percent-level precision at low and high
energies requires an improved control over the (anti)neutrino production
mechanisms. In particular, radiative corrections to muon and meson decays can
distort (anti)neutrino energy spectra [38].

While radiative corrections in muon, pion, and kaon decays are relatively old
subject, only corrections to the total decay width and to distributions w.r.t.
charged lepton and photon variables are well described in the literature [39,
40, 41, 42, 43, 44, 45, 46, 47, 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, 50, 84, 85, 51, 86, 87, 88, 89, 90, 91, 92, 93, 94,
95, 96, 97, 98]. To quantify the (anti)neutrino fluxes, other set of
observables are of interest. In this work, we evaluate leading in the
electromagnetic coupling constant $\alpha$ corrections
($\mathrm{O}\left(\alpha\right)$) to the (anti)neutrino energy spectra and
compare our calculations to previous results for the muon decay [50, 93] in
the approximation of vanishing electron mass as well as with well-known
results for the total decay widths [42, 99, 83, 57, 66].

While uncertainties on the electron antineutrino fluxes could be reduced from
10 % to a few-% level relying on the well-known inverse beta decay reaction at
low energies [100, 101, 102, 103, 104, 105], the muon component has no
feasible “standard candles”. However, the precise theoretical knowledge of
muon and pion decays could allow us to control the muon over electron flavor
ratios of incoming (anti)neutrinos. Our work paves the way for such control at
the percent level and below.

An acknowledged need in improved neutrino-nucleus cross-section inputs for
modern and future neutrino oscillation experiments has motivated new cross-
section measurements with potentially known fluxes. The Enhanced NeUtrino
BEams from kaon Tagging (ENUBET) project aims to control the (anti)neutrino
fluxes by tagging charged leptons from decays of light mesons [106, 107],
kaons and pions in this case. Reconstruction of (anti)neutrino fluxes from
high-statistics charged lepton measurements could potentially reduce flux
uncertainties from 5-10 % level, when estimates are based on the
hadroproduction cross sections, to a percent level or even below. The other
method to precisely control (anti)neutrino fluxes is based on the decays of
large samples of stored muons when only two types of (anti)neutrinos are
present as decay products, and the absolute flux normalization could be
controlled by precise measurements of stored currents. This technique is the
basis for the Neutrinos from STORed Muons (nuSTORM) activity [108, 109, 110]
which will potentially allow percent-level cross-section measurements. The
target precision goals call for the improved theoretical description of muon
and light-meson decays.

In this work, we address the radiative corrections to (anti)neutrino energy
spectra from muon, pion, and kaon decays in the $\overline{\mathrm{MS}}$
renormalization scheme. The paper is organized as follows. In Section 2, we
describe the decay of charged pions and kaons. We provide details of tree-
level energy spectra as well as virtual and real contributions at leading
order in chiral expansion. We quantify uncertainties of resulting spectra,
present relative contributions to (anti)neutrino energy distributions from
radiative corrections, and estimate size of strong dynamics effects. In
Section 3, we calculate radiative corrections to (anti)neutrino energy spectra
from muon decay, study the relative size of $\mathrm{O}\left(\alpha\right)$
contributions compared to the leading-order result, and present electron-mass
corrections. We provide conclusions and outlook in Section 4.

##  2 Pion and kaon decays

Leptonic decays of charged light mesons is the well-established source of
monochromatic (anti)neutrinos when the (anti)neutrino energy depends only on
masses of the light meson and charged lepton. The radiation of photons adds
the smooth component to (anti)neutrino energy distributions. In this Section,
we calculate (anti)neutrino energy spectra from decays of light mesons in the
meson rest frame accounting for possible radiation of one photon and estimate
the size of structure-dependent contributions. We consider pion $\pi_{\ell
2}:\leavevmode\nobreak\ \pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)$
and kaon $K_{\ell 2}:\leavevmode\nobreak\
K^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)$ decays. Charge-conjugated
reactions $\pi^{+}\to\mu^{+}{\nu}_{\mu}(\gamma)$ and
$K^{+}\to\mu^{+}{\nu}_{\mu}(\gamma)$ are obtained by replacing antineutrinos
with neutrinos in all following discussions, while decays to the electron
flavor are given by replacing the muon mass with the electron mass. We derive
all results considering the example of the pion decay. For numerical
evaluations, we substitute the pion mass and pion low-energy constants with
the kaon mass and corresponding constants for the kaon.

###  2.1 Pion and kaon decays at tree level

At the level of quarks, the pion decay
$\pi^{-}\left(p_{\pi}\right)\to\mu^{-}\left(p_{\mu}\right)\bar{\nu}_{\mu}\left(k_{\bar{\nu}_{\mu}}\right)$
is described by the low-energy four-fermion Lagrangian

$\displaystyle{\cal L}_{\rm
eff}=-c^{ud}\bar{\mu}\gamma_{\nu}\mathrm{P}_{\mathrm{L}}\nu_{\mu}\,\bar{{u}}\gamma^{\nu}\mathrm{P}_{\mathrm{L}}d+\mathrm{h.c.},$
(1)

where the effective coupling constant is expressed in terms of the Fermi
coupling $\mathrm{G}_{\mathrm{F}}$ and the CKM matrix element $V_{ud}$ as
$c^{ud}=2\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}+\mathrm{O}\left(\frac{\alpha}{\pi}\right)$,
c.f. [94, 111] for a detailed determination at order
$\mathrm{O}\left(\alpha\right)$. For low-energy applications, mesons are
appropriate degrees of freedom with the corresponding leading-order Lagrangian

$\displaystyle{\cal L}_{\rm
eff}=-\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}f_{\pi}\bar{\mu}\gamma_{\nu}\mathrm{P}_{\mathrm{L}}\nu_{\mu}i\mathrm{D}^{\nu}\pi^{-}+\mathrm{h.c.},$
(2)

with the covariant derivative $\mathrm{D}^{\nu}=\partial^{\nu}+ieA^{\nu}$,
where $e^{2}=4\pi\alpha$, $A^{\nu}$ is the photon field, and the pion-decay
constant $f_{\pi}$ is defined from the matrix element of the quark current

$\displaystyle
c^{ud}<0|\bar{{u}}\gamma^{\nu}\mathrm{P}_{\mathrm{L}}d|\pi^{-}\left(p_{\pi}\right)>=-i\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}f_{\pi}p_{\pi}^{\nu}.$
(3)

Exploiting equations of motion, the leading-order Lagrangian is expressed as a
derivative-free expression

$\displaystyle{\cal L}_{\rm
eff}=-i\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}f_{\pi}m_{\mu}\bar{\mu}\mathrm{P}_{\mathrm{L}}\nu_{\mu}\pi^{-}+\mathrm{h.c.}.$
(4)

The squared matrix element at leading order
$|\mathrm{T}_{\mathrm{LO}}|^{2}=4\mathrm{G}_{\mathrm{F}}^{2}|V_{ud}|^{2}f^{2}_{\pi}m^{2}_{\mu}k_{\bar{\nu}_{\mu}}\cdot
p_{\mu}$ results in the following expression for the decay width
$\Gamma_{\mathrm{LO}}$ in the pion rest frame with the monochromatic
(anti)neutrino spectrum:

$\displaystyle\Gamma_{\mathrm{LO}}\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\right)=\frac{\mathrm{G}_{\mathrm{F}}^{2}|V_{ud}|^{2}f^{2}_{\pi}}{8\pi}m^{2}_{\mu}m_{\pi}\left(1-\frac{m^{2}_{\mu}}{m^{2}_{\pi}}\right)^{2}\int\delta\left(E_{\bar{\nu}_{\mu}}-\frac{m_{\pi}^{2}-m_{\mu}^{2}}{2m_{\pi}}\right)\mathrm{d}E_{\bar{\nu}_{\mu}},$
(5)

and well-known helicity suppression of the electron decay channel compared to
the dominant $\pi_{\mu 2}$ decay.

At leading order, the radiative pion decay
$\pi^{-}\left(p_{\pi}\right)\to\mu^{-}\left(p_{\mu}\right)\bar{\nu}_{\mu}\left(k_{\bar{\nu}_{\mu}}\right)\gamma\left(k_{\gamma}\right)$
is described by structure-independent gauge-invariant inner Bremsstrahlung
contribution $\mathrm{T}^{1\gamma}_{\mathrm{IB}}$ [70, 71, 72, 79]:

$\displaystyle\mathrm{T}^{1\gamma}_{\mathrm{IB}}=e\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}f_{\pi}m_{\mu}\bar{\mu}\left(\frac{\gamma^{\nu}k\\!\\!\\!/\\!\,\,_{\gamma}}{2p_{\mu}\cdot
k_{\gamma}}+\frac{p^{\nu}_{\mu}}{p_{\mu}\cdot
k_{\gamma}}-\frac{p_{\pi}^{\nu}}{p_{\pi}\cdot
k_{\gamma}}\right)\mathrm{P}_{\mathrm{L}}\nu_{\mu}\pi^{-}\varepsilon^{\star}_{\nu},$
(6)

where $k\\!\\!\\!/\\!\,\,\equiv k_{\mu}\gamma^{\mu}$ for any four-vector $k$.
At next order of the chiral expansion
$\mathrm{O}\left(\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\right)$, strong
dynamics contributes to the radiation of the real photon with a matrix element
$\mathrm{T}^{1\gamma}_{\mathrm{SD}}$ [83]:

$\displaystyle\mathrm{T}^{1\gamma}_{\mathrm{SD}}=ie\frac{\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}}{m_{\pi}}\left(F_{V}\varepsilon^{\nu\lambda\alpha\beta}\left(k_{\gamma}\right)_{\alpha}\left(p_{\pi}\right)_{\beta}+iF_{A}\left(k_{\gamma}\cdot
p_{\pi}g^{\lambda\nu}-k_{\gamma}^{\lambda}p_{\pi}^{\nu}\right)\right)\bar{\mu}\gamma_{\lambda}\mathrm{P}_{\mathrm{L}}\nu_{\mu}\varepsilon^{\star}_{\nu},$
(7)

where the nonperturbative vector $F_{V}$ and axial $F_{A}$ form factors are
defined from matrix elements of axial and vector quark currents, subtracting
the point-particle contribution described above, as

$\displaystyle<\gamma\left(k_{\gamma}\right)|\bar{{u}}\gamma^{\lambda}\gamma_{5}d|\pi^{-}\left(p_{\pi}\right)>$
$\displaystyle=\frac{2\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}}{c^{ud}}e\frac{F_{A}}{m_{\pi}}\left(k_{\gamma}\cdot
p_{\pi}g^{\lambda\nu}-k_{\gamma}^{\lambda}p_{\pi}^{\nu}\right)\varepsilon^{\star}_{\nu},$
(8)
$\displaystyle<\gamma\left(k_{\gamma}\right)|\bar{{u}}\gamma^{\lambda}d|\pi^{-}\left(p_{\pi}\right)>$
$\displaystyle=\frac{2\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}}{c^{ud}}ie\frac{F_{V}}{m_{\pi}}\varepsilon^{\nu\lambda\alpha\beta}\left(k_{\gamma}\right)_{\alpha}\left(p_{\pi}\right)_{\beta}\varepsilon^{\star}_{\nu}.$
(9)

Form factors are functions of the squared momentum transfer
$q^{2}=\left(p_{\pi}-k_{\gamma}\right)^{2}$. Up to
$\mathrm{O}\left(\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\right)$, these form
factors do not depend on the kinematics. Leading contributions in the chiral
expansion are expressed in terms of Gasser and Leutwyler low-energy constants
[112, 113] as [83]

$\displaystyle F_{V}$
$\displaystyle=\frac{m_{\pi}}{4\pi^{2}f_{\pi}}+\mathrm{O}\left(\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\right),$
(10) $\displaystyle F_{A}$
$\displaystyle=\frac{8m_{\pi}}{f_{\pi}}\left(L_{9}+L_{10}\right)+\mathrm{O}\left(\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\right),$
(11)

with the natural size of low-energy constants $L_{9},\leavevmode\nobreak\
L_{10}\sim\frac{1}{\left(4\pi\right)^{2}}$. The resulting radiation is
determined by the sum of these two contributions
$\mathrm{T}^{1\gamma}=\mathrm{T}^{1\gamma}_{\mathrm{IB}}+\mathrm{T}^{1\gamma}_{\mathrm{SD}}$.

In this work, we calculate the (anti)neutrino energy spectra from the inner
Bremsstrahlung $\mathrm{T}^{1\gamma}_{\mathrm{IB}}$ and consider the
structure-dependent contributions $\mathrm{T}^{1\gamma}_{\mathrm{SD}}$ from
$F_{V}$ and $F_{A}$ as an alternative error estimate vs
$\mathrm{O}\left(\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\right)$ ChPT power
counting. Note that the complete inclusion of radiative effects at
$\mathrm{O}\left(\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\right)$ will require
introducing an additional to Eq. (7) structure-dependent form factor for the
calculation of virtual corrections and will be
$\mathrm{O}\left(\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\right)$ uncertain
due to purely-known low-energy constants for electroweak pion loops at
$\mathrm{O}\left(\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\right)$ [96].

###  2.2 Virtual corrections

At leading order
$\mathrm{O}\left(\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\right)$ we are
interested in, virtual radiative corrections enter as a factor $f$ multiplying
the tree-level amplitude $\mathrm{T}_{\mathrm{LO}}$, i.e., virtual corrections
change the leading-order result as

$\displaystyle\mathrm{T}_{\mathrm{LO}}\to\left(1+\frac{\alpha}{\pi}f+\mathrm{O}\left(\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\frac{\alpha}{\pi}\right)\right)\mathrm{T}_{\mathrm{LO}}.$
(12)

Starting with the Lagrangian of Eq. (2), the factor $f$ is determined by field
renormalization constants for external charged particles
$Z_{\pi},\leavevmode\nobreak\ Z_{\mu}$ and diagrams with virtual photon
exchange between pion and muon lines as well as with exchange between the
contact interaction and charged particle lines $f^{v}$:

$\displaystyle
f=f^{v}+\frac{\pi}{\alpha}\left(\sqrt{Z_{\pi}Z_{\mu}}-1\right).$ (13)

Starting with the Lagrangian of Eq. (4), the same correction $f^{v}$ is
obtained from the diagram with virtual photon exchange between pion and muon
lines and QED counterterm for the muon mass [114].

The field renormalization factors are evaluated from the one-loop self
energies in the $\overline{\mathrm{MS}}$ renormalization scheme as [115, 116,
117, 118]

$\displaystyle Z_{\mu}$
$\displaystyle=1-\frac{\alpha}{4\pi}\frac{\xi_{\gamma}}{\varepsilon}-\frac{\alpha}{4\pi}\left(\ln\frac{\mu^{2}}{m_{\mu}^{2}}+2\ln\frac{\lambda^{2}}{m_{\mu}^{2}}+4\right)+\frac{\alpha}{4\pi}\left(1-\xi_{\gamma}\right)\left(\ln\frac{\mu^{2}}{\lambda^{2}}+1+\frac{a\xi_{\gamma}\ln
a\xi_{\gamma}}{1-a\xi_{\gamma}}\right),$ (14) $\displaystyle Z_{\pi}$
$\displaystyle=1-\frac{\alpha}{4\pi}\frac{1+\xi_{\gamma}}{\varepsilon}-\frac{\alpha}{4\pi}\left(\ln\frac{\mu^{4}}{m^{4}_{\pi}}+2\ln\frac{\lambda^{2}}{m^{2}_{\pi}}+4\right)+\frac{\alpha}{4\pi}\left(1-\xi_{\gamma}\right)\left(\ln\frac{\mu^{2}}{\lambda^{2}}+1+\frac{a\xi_{\gamma}\ln
a\xi_{\gamma}}{1-a\xi_{\gamma}}\right),$ (15)

with the renormalization scale in dimensional regularization $\mu$, the number
of dimensions $D=4-2\varepsilon$, the photon mass $\lambda$ regulates the
infrared divergence, $\xi_{\gamma}$ is the photon gauge-fixing parameter, and
$a$ is an arbitrary constant which enter the photon propagator in the momentum
space as

$\displaystyle\mathrm{\Pi}^{\mu\nu}\left(k_{\gamma}\right)=\frac{i}{k^{2}_{\gamma}-\lambda^{2}}\left(-g^{\mu\nu}+\left(1-\xi_{\gamma}\right)\frac{k_{\gamma}^{\mu}k_{\gamma}^{\nu}}{k^{2}_{\gamma}-a\xi_{\gamma}\lambda^{2}}\right).$
(16)

Adding the virtual contribution $f^{v}$, the dependence on parameters
$\xi_{\gamma}$ and $a$ cancels. The resulting correction is expressed in terms
of the pion and muon masses as

$\displaystyle
f=-\frac{7}{4}\ln\frac{\mu}{m_{\mu}}-\frac{3}{2}+\left(1+\frac{m_{\pi}^{2}+m_{\mu}^{2}}{m_{\pi}^{2}-m_{\mu}^{2}}\ln\frac{m_{\mu}}{m_{\pi}}\right)\ln\frac{m_{\pi}}{\lambda}-\frac{1}{2}\frac{m_{\pi}^{2}+m_{\mu}^{2}}{m_{\pi}^{2}-m_{\mu}^{2}}\ln\frac{m_{\mu}}{m_{\pi}}\left(1-\ln\frac{m_{\mu}}{m_{\pi}}\right).$
(17)

The virtual correction depends on the renormalization scale $\mu$. Therefore,
the low-energy constant in Eqs. (2) and (4) also depends on the scale. To
avoid new definitions or fixing the renormalization scale, we integrate over
the total phase-space of radiated photons and express all our results in terms
of the experimental decay width of the pion

$\displaystyle\Gamma^{\mathrm{exp}}_{\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)}=\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\right)+\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right).$
(18)

###  2.3 Soft-photon bremsstrahlung

Radiative decay with photons of arbitrary small energy cannot be
experimentally distinguished from the decay without the radiation.
Consequently, all events with photons below some energy cutoff
$k_{\gamma}\leq\varepsilon$ (in the pion rest frame) contribute to the
measured observables. This region of the phase space also cancels the
infrared-divergent contributions from virtual diagrams. The corresponding
radiative decay width factorizes in terms of the tree-level decay width of Eq.
(5) as

$\displaystyle\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\left(k_{\gamma}\leq\varepsilon\right)\right)=\frac{\alpha}{\pi}\delta_{s}\left(\varepsilon\right)\Gamma_{\mathrm{LO}}\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\right),$
(19)

with the universal for QED processes correction
$\delta_{s}\left(\varepsilon\right)$ [119, 120, 121, 122, 118]:

$\displaystyle\delta_{s}\left(\varepsilon\right)=\frac{1}{\beta}\left(\mathrm{Li}_{2}\frac{1-\beta}{1+\beta}-\frac{\pi^{2}}{6}\right)-\frac{2}{\beta}\left(\beta-\frac{1}{2}\ln\frac{1+\beta}{1-\beta}\right)\ln\frac{2\varepsilon}{\lambda}+\frac{1}{2\beta}\ln\frac{1+\beta}{1-\beta}\left(1+\ln\frac{\rho\left(1+\beta\right)}{4\beta^{2}}\right)+1\,,$
(20)

where $\beta$ is the velocity of the muon in the pion rest frame:
$\beta=\frac{m^{2}_{\pi}-m^{2}_{\mu}}{m^{2}_{\pi}+m^{2}_{\mu}}$ and
$\rho=\sqrt{1-\beta^{2}}=\frac{2m_{\pi}m_{\mu}}{m^{2}_{\pi}+m^{2}_{\mu}}$.

As a result, soft and virtual contributions multiply the monochromatic tree-
level spectrum of Eq. (5) with infrared-finite factor, i.e., independent of
the fictitious photon mass $\lambda$ [123, 124, 125, 126], as

$\displaystyle\frac{\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\right)+\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\left(k_{\gamma}\leq\varepsilon\right)\right)}{\Gamma_{\mathrm{LO}}\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\right)}=1+\frac{\alpha}{\pi}\left(\delta_{s}\left(\varepsilon\right)+2f\right).$
(21)

###  2.4 Contribution of hard photons

The squared matrix elements for the radiative pion decay amplitudes
$\mathrm{T}^{1\gamma}_{\mathrm{IB}},\leavevmode\nobreak\
\mathrm{T}^{1\gamma}_{\mathrm{SD}}$ introduced in Section 2.1 are expressed in
terms of Lorentz invariants as

$\displaystyle\frac{|\mathrm{T}^{1\gamma}_{\mathrm{IB}}|^{2}}{e^{2}|\mathrm{T}_{\mathrm{LO}}|^{2}}k_{\bar{\nu}_{\mu}}\cdot
p_{\mu}$ $\displaystyle=-\left(\frac{p_{\pi}}{p_{\pi}\cdot
k_{\gamma}}-\frac{p_{\mu}}{p_{\mu}\cdot
k_{\gamma}}\right)^{2}+\frac{k_{\bar{\nu}_{\mu}}\cdot p_{\mu}}{p_{\mu}\cdot
k_{\gamma}}-\frac{k_{\bar{\nu}_{\mu}}\cdot p_{\pi}}{p_{\pi}\cdot
k_{\gamma}}-m^{2}_{\mu}\frac{k_{\bar{\nu}_{\mu}}\cdot
k_{\gamma}}{\left(p_{\mu}\cdot
k_{\gamma}\right)^{2}}+\frac{k_{\bar{\nu}_{\mu}}\cdot k_{\gamma}}{p_{\mu}\cdot
k_{\gamma}}$ $\displaystyle+\frac{k_{\bar{\nu}_{\mu}}\cdot
k_{\gamma}p_{\pi}\cdot p_{\mu}}{p_{\mu}\cdot k_{\gamma}p_{\pi}\cdot
k_{\gamma}},$ (22)
$\displaystyle\frac{\left(\mathrm{T}^{1\gamma}_{\mathrm{IB}}\right)^{\star}\mathrm{T}^{1\gamma}_{\mathrm{SD}}+\left(\mathrm{T}^{1\gamma}_{\mathrm{SD}}\right)^{\star}\mathrm{T}^{1\gamma}_{\mathrm{IB}}}{e^{2}|\mathrm{T}_{\mathrm{LO}}|^{2}}$
$\displaystyle=\frac{2F_{A}}{f_{\pi}m_{\pi}}\left(\frac{m^{2}_{\pi}k_{\bar{\nu}_{\mu}}\cdot
k_{\gamma}}{p_{\pi}\cdot k_{\gamma}k_{\bar{\nu}_{\mu}}\cdot
p_{\mu}}-\frac{k_{\bar{\nu}_{\mu}}\cdot p_{\pi}}{k_{\bar{\nu}_{\mu}}\cdot
p_{\mu}}-\frac{k_{\bar{\nu}_{\mu}}\cdot k_{\gamma}p_{\pi}\cdot
p_{\mu}}{p_{\mu}\cdot k_{\gamma}k_{\bar{\nu}_{\mu}}\cdot
p_{\mu}}+\frac{p_{\pi}\cdot k_{\gamma}}{p_{\mu}\cdot k_{\gamma}}\right)$
$\displaystyle+\frac{2\left(F_{A}-F_{V}\right)}{f_{\pi}m_{\pi}}\frac{k_{\bar{\nu}_{\mu}}\cdot
k_{\gamma}p_{\pi}\cdot k_{\gamma}}{p_{\mu}\cdot
k_{\gamma}k_{\bar{\nu}_{\mu}}\cdot p_{\mu}},$ (23)
$\displaystyle\frac{|\mathrm{T}^{1\gamma}_{\mathrm{SD}}|^{2}}{e^{2}|\mathrm{T}_{\mathrm{LO}}|^{2}}k_{\bar{\nu}_{\mu}}\cdot
p_{\mu}$
$\displaystyle=\frac{2\left(F_{A}-F_{V}\right)^{2}}{f_{\pi}^{2}m_{\pi}^{2}m^{2}_{\mu}}k_{\bar{\nu}_{\mu}}\cdot
k_{\gamma}p_{\pi}\cdot k_{\gamma}p_{\mu}\cdot
p_{\pi}+\frac{2\left(F_{A}+F_{V}\right)^{2}}{f_{\pi}^{2}m_{\pi}^{2}m^{2}_{\mu}}p_{\mu}\cdot
k_{\gamma}p_{\pi}\cdot k_{\gamma}k_{\bar{\nu}_{\mu}}\cdot p_{\pi}$
$\displaystyle-\frac{2\left(F_{A}^{2}+F_{V}^{2}\right)}{f_{\pi}^{2}m^{2}_{\mu}}k_{\bar{\nu}_{\mu}}\cdot
k_{\gamma}p_{\mu}\cdot k_{\gamma}.$ (24)

Performing the full phase-space integration from
$|\mathrm{T}^{1\gamma}_{\mathrm{IB}}|^{2}$, we relate the exclusive decay
width of Eq. (21) to the experimental decay width of the pion
$\Gamma^{\mathrm{exp}}_{\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)}$
up to contributions of order
$\mathrm{O}\left(\frac{\alpha^{2}}{\pi^{2}},\leavevmode\nobreak\
\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\frac{\alpha}{\pi}\right)$:

$\displaystyle\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\right)+\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\left(k_{\gamma}\leq\varepsilon\right)\right)=\left(1-\frac{\alpha}{\pi}\delta\left(\varepsilon\right)\right)\Gamma^{\mathrm{exp}}_{\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)},$
(25)

with the scale-independent correction $\delta\left(\varepsilon\right)$ [71]:

$\displaystyle\delta\left(\varepsilon\right)$
$\displaystyle=\left(1+\frac{m_{\pi}^{2}+m_{\mu}^{2}}{m_{\pi}^{2}-m_{\mu}^{2}}\ln\frac{m_{\mu}}{m_{\pi}}\right)\ln\frac{4\varepsilon^{2}}{m^{2}_{\pi}}-2\ln\left(1-\frac{m^{2}_{\mu}}{m^{2}_{\pi}}\right)+\frac{11}{4}+\left(\frac{9}{8}-\frac{3}{8}\frac{\left(m_{\pi}^{2}+m_{\mu}^{2}\right)^{2}}{\left(m_{\pi}^{2}-m_{\mu}^{2}\right)^{2}}\right)\ln\frac{m_{\mu}}{m_{\pi}}$
$\displaystyle+\frac{m_{\pi}^{2}+m_{\mu}^{2}}{m_{\pi}^{2}-m_{\mu}^{2}}\left(\mathrm{Li}_{2}\frac{m^{2}_{\mu}}{m^{2}_{\pi}}-\frac{\pi^{2}}{6}+\frac{3}{8}\left(\ln\frac{m_{\mu}^{2}}{m^{2}_{\pi}}-1\right)\right).$
(26)

Note that the relative radiative correction to the total inclusive cross
section $\delta+\delta_{s}+2f$ has lepton-mass singularity

$\displaystyle\delta+\delta_{s}+2f\underset{m_{\mu}\ll
m_{\pi}}{\longrightarrow}3\frac{\alpha}{\pi}\ln\frac{m_{\mu}}{m_{\pi}},$ (27)

in agreement with [70, 71, 81, 127, 96]. This singularity is multiplied by the
lepton mass from the tree-level amplitude and is absent on the level of
physical observables according to Kinoshita, Lee, Naunberg theorem [123, 128,
125, 126].

All double-differential distributions can be derived by an appropriate change
of variables from the double-differential energy spectrum

$\displaystyle\frac{\mathrm{d}\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)}{\Gamma^{\mathrm{exp}}_{\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)}\mathrm{d}E_{\bar{\nu}_{\mu}}\mathrm{d}E_{\gamma}}=\frac{\alpha}{\pi}\frac{m_{\pi}}{2\omega^{2}_{\bar{\nu}_{\mu}}}\left(\frac{2\omega_{\bar{\nu}_{\mu}}}{\omega_{\bar{\nu}_{\mu}}-E_{\bar{\nu}_{\mu}}}\frac{1-\frac{E_{\bar{\nu}_{\mu}}}{m_{\pi}}}{E_{\gamma}}-\frac{\frac{m^{2}_{\mu}}{m^{2}_{\pi}}E_{\bar{\nu}_{\mu}}}{\left(\omega_{\bar{\nu}_{\mu}}-E_{\bar{\nu}_{\mu}}\right)^{2}}-\frac{1-\frac{E_{\gamma}}{m_{\pi}}}{\omega_{\bar{\nu}_{\mu}}-E_{\bar{\nu}_{\mu}}}-\frac{\omega_{\bar{\nu}_{\mu}}}{E_{\gamma}^{2}}-\frac{1}{m_{\pi}}\right),$
(28)

with the maximum allowed by kinematics value of the (anti)neutrino energy
$\omega_{\bar{\nu}_{\mu}}=\frac{m^{2}_{\pi}-m^{2}_{\mu}}{2m_{\pi}}$
corresponding to the process without radiation. The double-differential
distribution in Eq. (28) is in agreement with [83, 96].

The leading continuous (anti)neutrino spectrum from the pion decay is
expressed as

$\displaystyle\frac{\mathrm{d}\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)}{\Gamma^{\mathrm{exp}}_{\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)}\mathrm{d}E_{\bar{\nu}_{\mu}}}$
$\displaystyle=\frac{\alpha}{\pi}\frac{1}{\omega^{2}_{\bar{\nu}_{\mu}}}\left(\frac{1}{2}\frac{E^{2}_{\bar{\nu}_{\mu}}+\frac{m^{2}_{\mu}}{4}}{m_{\pi}-2E_{\bar{\nu}_{\mu}}}-2E_{\bar{\nu}_{\mu}}-\frac{m^{2}_{\mu}}{8m_{\pi}}-\frac{E_{\bar{\nu}_{\mu}}m^{2}_{\mu}}{4\left(m_{\pi}-2E_{\bar{\nu}_{\mu}}\right)^{2}}\right)$
$\displaystyle+\frac{\alpha}{\pi}\frac{1}{\omega^{2}_{\bar{\nu}_{\mu}}}\frac{2E^{2}_{\bar{\nu}_{\mu}}+\left(m_{\pi}-E_{\bar{\nu}_{\mu}}\right)\omega_{\bar{\nu}_{\mu}}\ln\left(1-\frac{2E_{\bar{\nu}_{\mu}}}{m_{\pi}}\right)}{\omega_{\bar{\nu}_{\mu}}-E_{\bar{\nu}_{\mu}}}+\mathrm{O}\left(\frac{\alpha^{2}}{\pi^{2}},\leavevmode\nobreak\
\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\frac{\alpha}{\pi}\right),$ (29)

with the range of the (anti)neutrino energy $0\leq
E_{\bar{\nu}_{\mu}}\leq\omega_{\bar{\nu}_{\mu}}-\varepsilon$. Uncertainties
due to the structure-dependent contributions are estimated from ChPT power
counting by a factor of order
$\mathrm{O}\left(\frac{m^{2}_{\pi}}{16\pi^{2}f^{2}_{\pi}}\right)$ as well as
from the leading
$\Gamma_{F}\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)$, linear in
form factors, and subleading
$\Gamma_{F^{2}}\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)$, quadratic
in form factors, contributions from the form factors $F_{V}$ and $F_{A}$:

$\displaystyle\frac{\mathrm{d}\Gamma_{F}\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)}{\Gamma^{\mathrm{exp}}_{\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)}\mathrm{d}E_{\bar{\nu}_{\mu}}}$
$\displaystyle=\frac{\alpha}{\pi}\frac{\omega_{\bar{\nu}_{\mu}}-E_{\bar{\nu}_{\mu}}}{\omega^{2}_{\bar{\nu}_{\mu}}}\frac{m_{\pi}}{f_{\pi}}\left(\frac{m_{\pi}-E_{\bar{\nu}_{\mu}}}{m_{\pi}-2E_{\bar{\nu}_{\mu}}}\frac{2E_{\bar{\nu}_{\mu}}}{m_{\pi}}+\ln\left(1-\frac{2E_{\bar{\nu}_{\mu}}}{m_{\pi}}\right)\right)F_{A}$
$\displaystyle+\frac{\alpha}{\pi}\frac{2}{3f_{\pi}}\frac{E_{\bar{\nu}_{\mu}}^{2}}{\omega^{2}_{\bar{\nu}_{\mu}}}\frac{\left(\omega_{\bar{\nu}_{\mu}}-E_{\bar{\nu}_{\mu}}\right)^{2}}{\left(m_{\pi}-2E_{\bar{\nu}_{\mu}}\right)^{2}}\left(1+\frac{2m_{\pi}}{m_{\pi}-2E_{\bar{\nu}_{\mu}}}\right)\left(F_{A}-F_{V}\right),$
(30)
$\displaystyle\frac{\mathrm{d}\Gamma_{F^{2}}\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)}{\Gamma^{\mathrm{exp}}_{\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)}\mathrm{d}E_{\bar{\nu}_{\mu}}}$
$\displaystyle=\frac{\alpha}{\pi}\frac{2m_{\pi}}{f_{\pi}^{2}m^{2}_{\mu}}\frac{E_{\bar{\nu}_{\mu}}^{2}}{\omega^{2}_{\bar{\nu}_{\mu}}}\frac{\left(\omega_{\bar{\nu}_{\mu}}-E_{\bar{\nu}_{\mu}}\right)^{3}}{\left(m_{\pi}-2E_{\bar{\nu}_{\mu}}\right)^{2}}\left(\left(m_{\pi}-E_{\bar{\nu}_{\mu}}\right)\left(F_{A}+F_{V}\right)^{2}-m_{\pi}\left(F^{2}_{A}+F^{2}_{V}\right)\right)$
$\displaystyle+\left(\left(E_{\bar{\nu}_{\mu}}-\frac{m^{2}_{\mu}}{2m_{\pi}}\right)\frac{E^{2}_{\bar{\nu}_{\mu}}}{3}+\left(m_{\pi}E_{\bar{\nu}_{\mu}}-\frac{m^{2}_{\mu}}{4}\right)\left(\frac{m_{\pi}}{2}-E_{\bar{\nu}_{\mu}}\right)+\frac{m^{2}_{\mu}m^{2}_{\pi}/8}{m_{\pi}-2E_{\bar{\nu}_{\mu}}}\right)$
$\displaystyle\times\frac{\alpha}{\pi}\frac{2m_{\pi}}{f_{\pi}^{2}m^{2}_{\mu}}\frac{E_{\bar{\nu}_{\mu}}}{\omega^{2}_{\bar{\nu}_{\mu}}}\frac{\left(\omega_{\bar{\nu}_{\mu}}-E_{\bar{\nu}_{\mu}}\right)^{3}}{\left(m_{\pi}-2E_{\bar{\nu}_{\mu}}\right)^{3}}\left(F_{A}-F_{V}\right)^{2}.$
(31)

###  2.5 Decay spectra and integrated cross sections

In the following Fig. 1, we compare the leading-order continuous component of
the (anti)neutrino energy spectrum
$\mathrm{d}\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)/\mathrm{d}E_{\bar{\nu}_{\mu}}$
to structure-dependent power-suppressed contributions from the form factors
$F_{V}$ and $F_{A}$:
$\mathrm{d}\Gamma_{F}\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)/\mathrm{d}E_{\bar{\nu}_{\mu}}$,
$\mathrm{d}\Gamma_{F^{2}}\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)/\mathrm{d}E_{\bar{\nu}_{\mu}}$
as well as to the power-counting error estimate. We take the ChPT low-energy
constants $L_{9}$ and $L_{10}$ from Refs. [129, 130, 131] and all other
numerical values for physical quantities from PDG [9].

Figure 1: Various contributions to the continuous (anti)neutrino energy
spectra from the decay of light mesons normalized by
$\Gamma^{\mathrm{exp}}_{\pi^{-},K^{-}\to\mu^{-}\bar{\nu}_{\mu}}/\omega_{\bar{\nu}_{\mu}}$.
Pion decay is presented on the left plot, kaon decay is shown on the right
plot. The leading-order (anti)neutrino energy spectrum
$\mathrm{d}\Gamma\left(\pi^{-},K^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)/\mathrm{d}E_{\bar{\nu}_{\mu}}$
is shown by blue solid line. It is compared to the strong dynamics
contributions
$\mathrm{d}|\Gamma_{F}\left(\pi^{-},K^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)|/\mathrm{d}E_{\bar{\nu}_{\mu}}$
(red dashed line) and
$\mathrm{d}\Gamma_{F^{2}}\left(\pi^{-},K^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)/\mathrm{d}E_{\bar{\nu}_{\mu}}$
(green dashed-dotted line) as well as to the power-counting error estimate
shown by black dotted line. This error estimate is performed multiplying the
contribution from non-eikonal terms of Eq. (22), i.e., beyond the first one,
with $\frac{m_{\pi,K}^{2}}{16\pi^{2}f^{2}_{\pi,K}}$.

The decay spectrum is peaked approaching the kinematics of the process without
radiation and behaves as
$1/\left(\omega_{\bar{\nu}_{\mu}}-E_{\bar{\nu}_{\mu}}\right)$ near the
endpoint. Integrating the (anti)neutrino energy spectrum up to
$E_{\bar{\nu}_{\mu}}=\frac{1}{2}\left(m_{\pi}-\frac{m^{2}_{\mu}}{m_{\pi}-2\varepsilon}\right)$
regulates the dependence on the artificial cutoff parameter $\varepsilon$,
c.f. Eq. (25), on the level of the decay width as well as in possible
convolutions.111The integration over the $\epsilon$-suppressed region
$\omega_{\bar{\nu}_{\mu}}-\varepsilon\leq
E_{\bar{\nu}_{\mu}}\leq\frac{1}{2}\left(m_{\pi}-\frac{m^{2}_{\mu}}{m_{\pi}-2\varepsilon}\right)$
contributes to observables as
$\frac{\mathrm{d}\Gamma^{\varepsilon}\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\gamma\right)}{\Gamma^{\mathrm{exp}}_{\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)}\mathrm{d}E_{\bar{\nu}_{\mu}}}=\frac{\alpha}{\pi}\left(1+\frac{m_{\pi}^{2}+m_{\mu}^{2}}{m_{\pi}^{2}-m_{\mu}^{2}}\ln\frac{m_{\mu}}{m_{\pi}}\right)\ln\frac{m^{2}_{\mu}}{m^{2}_{\pi}}\delta\left(E_{\bar{\nu}_{\mu}}-\left(\omega_{\bar{\nu}_{\mu}}-\left(1-\frac{\omega_{\bar{\nu}_{\mu}}}{m_{\pi}}\right)\varepsilon\right)\right).$
(32) Besides the end-point region, the (anti)neutrino energy spectrum is
suppressed by more than three orders of magnitude compared to the intensity of
the monochromatic line. ChPT power counting uncertainties are slightly larger
than the linear in form factors $F_{A}$ and $F_{V}$ contribution in the case
of pion decay and have a similar size in the case of kaon decay. For the kaon
decay, the quadratic contribution from strong dynamics to the (anti)neutrino
spectra exceeds the linear contribution indicating missing ingredients for the
consistent series expansion as well as on the pure convergence of
$\mathrm{SU}\left(3\right)$ perturbative series.

As an illustrative example, we compare the flux-averaged cross sections on
${}^{40}\mathrm{Ar}$ nucleus

$\displaystyle\sigma^{{}^{40}\mathrm{Ar}}_{\bar{\nu}_{\mu}}$
$\displaystyle=\frac{\int\sigma^{{}^{40}\mathrm{Ar}}_{{\bar{\nu}_{\mu}}}\left(E_{\bar{\nu}_{\mu}}\right)\frac{\mathrm{d}\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)\right)}{\mathrm{d}E_{\bar{\nu}_{\mu}}}\mathrm{d}E_{\bar{\nu}_{\mu}}}{\int\frac{\mathrm{d}\Gamma\left(\pi^{-}\to\mu^{-}\bar{\nu}_{\mu}\left(\gamma\right)\right)}{\mathrm{d}E_{\bar{\nu}_{\mu}}}\mathrm{d}E_{\bar{\nu}_{\mu}}},$
(33)

accounting for the radiation of one photon in the (anti)neutrino production
from the decay of the pion at rest
$\sigma_{\bar{\nu}_{\mu}}^{{}^{40}\mathrm{Ar}}$ to the integration over the
monochromatic spectrum
$\sigma^{{}^{40}\mathrm{Ar}}_{\mathrm{\bar{\nu}_{\mu},LO}}$ [132, 133, 134,
135, 38] with the same normalization in both cases:

$\displaystyle\sigma^{{}^{40}\mathrm{Ar}}_{\bar{\nu}_{\mu}}$
$\displaystyle=15.1867\times 10^{-40}\leavevmode\nobreak\ \mathrm{cm}^{2},$
(34) $\displaystyle\sigma^{{}^{40}\mathrm{Ar}}_{\bar{\nu}_{\mu},\mathrm{LO}}$
$\displaystyle=15.1875\times 10^{-40}\leavevmode\nobreak\ \mathrm{cm}^{2}.$
(35)

The resulting relative effect of radiative corrections is around $6\times
10^{-5}$. The continuous component of the neutrino energy spectrum from the
decay of light mesons can be safely neglected as soon as the flux is
normalized to the meson decay width accounting for all radiative events.

Effects of radiative corrections on accelerator (anti)neutrinos produced from
pions decaying in flight are also negligibly small. As a toy example, let us
consider a monochromatic beam of pions with energy $4.7\leavevmode\nobreak\
\mathrm{GeV}$. Such pions decay into (anti)neutrinos of $2\leavevmode\nobreak\
\mathrm{GeV}$ energy. In this setup, radiative (anti)neutrino energy spectrum
lowers the mean (anti)neutrino energy by negligible $3.2\times 10^{-5}$
relative correction, which is approximately independent of the off-axis angle
on the length scales of modern detectors in oscillation neutrino experiments.

##  3 Muon decay

Muons from the decay of pion are a source of (anti)neutrinos of both electron
and muon flavors. In this Section, we calculate (anti)neutrino energy spectra
from the radiative decay of muon $\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}(\gamma)$ and $\mu^{+}\to
e^{+}{\nu}_{e}\bar{\nu}_{\mu}(\gamma)$ in the muon rest frame accounting for
all electron-mass effects. Accounting for the isotropic invariance, neutrino
energy spectra in the muon rest frame do not depend on the polarization of the
decaying muon.

###  3.1 Muon decay at tree level

Muon decay $\mu^{-}\left(p_{\mu}\right)\to
e^{-}\left(k_{e}\right)\bar{\nu}_{e}\left(k_{\bar{\nu}_{e}}\right)\nu_{\mu}\left(k_{\nu_{\mu}}\right)$
at leading order is governed by the low-energy effective four-fermion
interaction with a scale-independent Fermi coupling constant
$\mathrm{G}_{\mathrm{F}}$ [136, 137, 138, 139, 111]

$\displaystyle{\cal L}_{\rm
eff}=-2\sqrt{2}\mathrm{G}_{\mathrm{F}}\bar{\nu}_{\mu}\gamma^{\mu}\mathrm{P}_{\mathrm{L}}\nu_{e}\,\bar{e}\gamma_{\mu}\mathrm{P}_{\mathrm{L}}\mu+\mathrm{h.c.}.$
(36)

Performing the phase-space integration, we obtain the double-differential
distribution w.r.t. the electron energy $E_{e}$ and (anti)neutrino energy
$E_{\nu_{\mu},{\bar{\nu}_{e}}}$ in terms of the squared matrix element at
leading order
$|\mathrm{T}_{\mathrm{LO}}|^{2}=64\mathrm{G}_{\mathrm{F}}^{2}p_{\mu}\cdot
k_{\bar{\nu}_{e}}k_{e}\cdot k_{\nu_{\mu}}$:

$\displaystyle\frac{\mathrm{d}\Gamma_{\mathrm{LO}}\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right)}{\mathrm{d}E_{\nu_{\mu},{\bar{\nu}_{e}}}\mathrm{d}E_{e}}=\frac{|\mathrm{T}_{\mathrm{LO}}|^{2}}{2^{6}\pi^{3}m_{\mu}}.$
(37)

Integration of this distribution over the kinematically allowed range of
electron energies

$\displaystyle\frac{q^{4}+m^{2}_{e}m^{2}_{\mu}}{2m_{\mu}q^{2}}\leq
E_{e}\leq\frac{m^{2}_{\mu}+m^{2}_{e}}{2m_{\mu}},$ (38)

results in the following (anti)neutrino energy spectra in the muon rest frame
for electron and muon flavors [140]:

$\displaystyle\frac{\mathrm{d}\Gamma_{\mathrm{LO}}\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right)}{\mathrm{d}E_{\bar{\nu}_{e}}}$
$\displaystyle=\frac{\mathrm{G}_{\mathrm{F}}^{2}}{2\pi^{3}}\frac{\left(q^{2}-m_{e}^{2}\right)^{2}}{q^{2}}E_{\bar{\nu}_{e}}^{2},$
(39) $\displaystyle\frac{\mathrm{d}\Gamma_{\mathrm{LO}}\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right)}{\mathrm{d}E_{\nu_{\mu}}}$
$\displaystyle=\frac{\mathrm{G}_{\mathrm{F}}^{2}}{\pi^{3}}\left(\frac{q^{2}-m_{e}^{2}}{q^{2}}\right)^{2}\left(\frac{q^{2}-m^{2}_{e}}{12}+\frac{q^{2}+2m^{2}_{e}}{6q^{2}}m_{\mu}\left(m_{\mu}-E_{\nu_{\mu}}\right)\right)E_{\nu_{\mu}}^{2},$
(40)

with the momentum transfer from the muon to the corresponding (anti)neutrino
$q^{2}=m^{2}_{\mu}-2m_{\mu}E_{\nu_{\mu},{\bar{\nu}_{e}}}$. The (anti)neutrino
energy range is given by $0\leq
E_{\nu_{\mu},{\bar{\nu}_{e}}}\leq\omega_{\nu}=\frac{m_{\mu}^{2}-m_{e}^{2}}{2m_{\mu}}$.

Integrating over the (anti)neutrino energy, we obtain the well-known muon
decay width at leading order

$\Gamma_{\mathrm{LO}}\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right)=\frac{\mathrm{G}_{\mathrm{F}}^{2}m^{5}_{\mu}}{192\pi^{3}}\left(1-8r^{2}-24r^{4}\ln
r+8r^{6}-r^{8}\right),$ (41)

with $r=\frac{m_{e}}{m_{\mu}}$.

At leading order, the radiative muon decay $\mu^{-}\left(p_{\mu}\right)\to
e^{-}\left(k_{e}\right)\bar{\nu}_{e}\left(k_{\bar{\nu}_{e}}\right){\nu}_{\mu}\left(k_{{\nu}_{\mu}}\right)\gamma\left(k_{\gamma}\right)$
is described by the Bremsstrahlung contribution $\mathrm{T}^{1\gamma}$ [40]:

$\displaystyle\mathrm{T}^{1\gamma}=-2\sqrt{2}\mathrm{G}_{\mathrm{F}}e\leavevmode\nobreak\
\bar{\nu}_{\mu}\gamma^{\mu}\mathrm{P}_{\mathrm{L}}\nu_{e}\,\left(\left(\frac{{k}^{\nu}_{e}}{k_{e}\cdot
k_{\gamma}}-\frac{{p}^{\nu}_{\mu}}{p_{\mu}\cdot
k_{\gamma}}\right)\bar{e}\gamma_{\mu}\mathrm{P}_{\mathrm{L}}\mu+\frac{1}{2}\bar{e}\left(\frac{\gamma^{\nu}k\\!\\!\\!/\\!\,\,_{\gamma}\gamma_{\mu}}{k_{e}\cdot
k_{\gamma}}+\frac{\gamma_{\mu}k\\!\\!\\!/\\!\,\,_{\gamma}\gamma^{\nu}}{p_{\mu}\cdot
k_{\gamma}}\right)\mathrm{P}_{\mathrm{L}}\mu\right)\varepsilon^{\star}_{\nu}.$
(42)

In the following Sections, we include virtual corrections to the muon decay
and perform the integration over the photon, electron, and (anti)neutrino
phase space.

###  3.2 Virtual corrections

It is convenient to express vertex corrections as a deviation of the charged-
lepton current $\delta\mathrm{J}^{\mathrm{L}}_{\nu}$ from the tree-level
expression
$\mathrm{J}^{\mathrm{L}}_{\nu}=\bar{e}\left(k_{e}\right)\gamma_{\nu}\mathrm{P}_{\mathrm{L}}\mu\left(p_{\mu}\right)$
as

$\displaystyle\delta\mathrm{J}^{\mathrm{L}}_{\nu}=e^{2}\int\frac{\mathrm{d}^{d}L}{(2\pi)^{d}}\frac{\bar{e}\left(k_{e}\right)\gamma^{\lambda}\left(k\\!\\!\\!/\\!\,\,_{e}-L\\!\\!\\!/\\!\,\,+m_{e}\right)\gamma_{\nu}\mathrm{P}_{\mathrm{L}}\left(p\\!\\!\\!/\\!\,\,_{\mu}-L\\!\\!\\!/\\!\,\,+m_{\mu}\right)\gamma^{\rho}\mu\left(p_{\mu}\right)}{\left((k_{e}-L)^{2}-m_{e}^{2}\right)\left((p_{\mu}-L)^{2}-m_{\mu}^{2}\right)}\mathrm{\Pi}_{\lambda\rho}\left(L\right),$
(43)

with the momentum-space photon propagator of Eq. (16). The corresponding field
renormalization factors for external charged leptons are given above in Eq.
(14).

Neglecting Lorentz structures whose contractions with the (anti)neutrino
current vanish at $m_{\nu}=0$, the resulting correction to the charged lepton
current is expressed as

$\displaystyle\left(\sqrt{Z_{e}Z_{\mu}}-1\right)\left(\mathrm{J}^{\mathrm{L}}\right)^{\nu}+\left(\delta\mathrm{J}^{\mathrm{L}}\right)^{\nu}=\frac{\alpha}{2\pi}\bar{e}\left(k_{e}\right)\left(g_{M}\gamma^{\nu}-f_{2}\frac{p_{\mu}^{\nu}+rk_{e}^{\nu}}{2m_{\mu}}-g^{5}_{M}\gamma^{\nu}\gamma_{5}-f^{5}_{2}\frac{p_{\mu}^{\nu}-rk_{e}^{\nu}}{2m_{\mu}}\gamma_{5}\right)\mu\left(p_{\mu}\right),$
(44)

in terms of the form factors $g_{M},\leavevmode\nobreak\
g^{5}_{M},\leavevmode\nobreak\ f_{2}$, and $f^{5}_{2}$:

$\displaystyle g^{(5)}_{M}\left(\eta,r,\beta\right)$
$\displaystyle=-1+\frac{1}{\beta}\left(\frac{1}{2}\left(2\beta-\ln\frac{1+\beta}{1-\beta}\right)\ln\frac{2m_{\mu}}{\lambda}+\frac{1}{2}\ln\frac{1+\beta}{1-\beta}\ln\frac{1+\beta}{\beta}-\ln\frac{r\sqrt{1-\beta}-\sqrt{1+\beta}}{r\sqrt{1+\beta}-\sqrt{1-\beta}}\frac{\ln
r}{2}\right.$
$\displaystyle+\left.\frac{3}{8}\ln\frac{1+\beta}{1-\beta}+\frac{\sqrt{1-\beta^{2}}}{8\eta}\ln\frac{1+\beta}{1-\beta}+\frac{1}{4}\ln\frac{1+\beta}{1-\beta}\ln\frac{2r-\left(1+r^{2}\right)\sqrt{1-\beta^{2}}}{r^{2}\left(1-\beta\right)}+\frac{\pi^{2}}{12}\right.$
$\displaystyle+\left.\frac{1}{2}\mathrm{Li}_{2}\frac{1-\beta}{1+\beta}-\frac{1}{2}\mathrm{Li}_{2}\left(\frac{\sqrt{1-\beta}}{\sqrt{1+\beta}}r\right)-\frac{1}{2}\mathrm{Li}_{2}\left(\frac{\sqrt{1-\beta}}{\sqrt{1+\beta}}\frac{1}{r}\right)-\frac{5}{16}\ln^{2}\frac{1+\beta}{1-\beta}-\frac{1}{4}\ln^{2}r\right)$
$\displaystyle+\frac{\sqrt{1-\beta^{2}}}{8\beta}\frac{\left(1+\eta
r\right)^{2}\left(1-\eta\sqrt{1-\beta^{2}}\right)}{2r-\left(1+r^{2}\right)\sqrt{1-\beta^{2}}}\ln\frac{1+\beta}{1-\beta}+\frac{12r-\left(7r^{2}+5\right)\sqrt{1-\beta^{2}}}{2r-\left(1+r^{2}\right)\sqrt{1-\beta^{2}}}\frac{\ln
r}{4}-\ln 2r,$ (45) $\displaystyle f^{(5)}_{2}\left(\eta,r,\beta\right)$
$\displaystyle=\frac{1}{2}\frac{\sqrt{1-\beta^{2}}}{\beta}\frac{1-\eta\sqrt{1-\beta^{2}}}{2r-\left(1+r^{2}\right)\sqrt{1-\beta^{2}}}\ln\frac{1+\beta}{1-\beta}+\frac{1-\eta
r}{1+\eta
r}\frac{\sqrt{1-\beta^{2}}}{2r-\left(1+r^{2}\right)\sqrt{1-\beta^{2}}}\ln r,$
(46)

with the velocity of the electron in the muon rest frame $\beta$, $\eta=1$ for
the form factors $g_{M},\leavevmode\nobreak\ f_{2}$ and $\eta=-1$ for the form
factors $g^{5}_{M},\leavevmode\nobreak\ f^{5}_{2}$.

As a cross-check of our calculation, we have verified the virtual one-loop QED
contribution to the muon decay width [70] in the limit of small electron mass
considering both neutrino and antineutrino energy spectra.

###  3.3 Soft-photon bremsstrahlung

As for the pion decay in Section 2.3, the radiative decay width with photons
below some energy cutoff $k_{\gamma}\leq\varepsilon$ (in the muon rest frame)
factorizes in terms of the tree-level decay widths of Eqs. (39) and (40) as

$\displaystyle\mathrm{d}\Gamma\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\gamma\left(k_{\gamma}\leq\varepsilon\right)\right)=\frac{\alpha}{\pi}\delta_{s}\left(\varepsilon\right)\mathrm{d}\Gamma_{\mathrm{LO}}\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right),$ (47)

with the universal soft-photon correction $\delta_{s}\left(\varepsilon\right)$
of Eq. (20) which depends on the electron velocity in the muon rest frame
$\beta$.

As a result, soft and virtual contributions multiply the tree-level spectra of
Eqs. (39) and (40) as

$\displaystyle\frac{\mathrm{d}\Gamma\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right)+\mathrm{d}\Gamma\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\gamma\left(k_{\gamma}\leq\varepsilon\right)\right)}{\mathrm{d}\Gamma_{\mathrm{LO}}\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right)}=1+\frac{\alpha}{\pi}\left[g_{M}+g^{5}_{M}+\delta_{s}\left(\varepsilon\right)\right.$
$\displaystyle\left.-\frac{rm^{2}_{\mu}}{4}\frac{\left(p_{\mu}-k_{e}\right)^{2}}{p_{\mu}\cdot
k_{\bar{\nu}_{e}}k_{e}\cdot
k_{\nu_{\mu}}}\left(g_{M}-g^{5}_{M}\right)+\left(\frac{r^{2}m^{2}_{\mu}}{4}\frac{\left(p_{\mu}-k_{e}\right)^{2}}{p_{\mu}\cdot
k_{\bar{\nu}_{e}}k_{e}\cdot k_{\nu_{\mu}}}-\frac{k_{e}\cdot
k_{\bar{\nu}_{e}}}{p_{\mu}\cdot
k_{\bar{\nu}_{e}}}\right)\left(\left(\frac{1+r}{2}\right)^{2}f_{2}+\left(\frac{1-r}{2}\right)^{2}f^{5}_{2}\right)\right].$
(48)

To obtain the (anti)neutrino energy spectrum, we integrate the corresponding
squared matrix element over the electron energy in the kinematically allowed
region of the process without radiation, c.f. Section 3.1 for technical steps.

###  3.4 Contribution of hard photons

The squared matrix elements for the radiative muon decay amplitude
$\mathrm{T}^{1\gamma}$ introduced in Section 3.1 is expressed in terms of
Lorentz invariants as

$\displaystyle\frac{|\mathrm{T}^{1\gamma}|^{2}}{e^{2}|\mathrm{T}_{\mathrm{LO}}|^{2}}$
$\displaystyle=-\left(\frac{p_{\mu}}{p_{\mu}\cdot
k_{\gamma}}-\frac{k_{e}}{k_{e}\cdot k_{\gamma}}\right)^{2}+\frac{p_{\mu}\cdot
k_{e}}{p_{\mu}\cdot k_{\gamma}k_{e}\cdot
k_{\gamma}}\left(\frac{k_{\nu_{\mu}}\cdot k_{\gamma}}{k_{e}\cdot
k_{\nu_{\mu}}}-\frac{k_{\bar{\nu}_{e}}\cdot k_{\gamma}}{p_{\mu}\cdot
k_{\bar{\nu}_{e}}}\right)+\frac{k_{\bar{\nu}_{e}}\cdot
k_{\gamma}}{p_{\mu}\cdot k_{\bar{\nu}_{e}}p_{\mu}\cdot
k_{\gamma}}-\frac{p_{\mu}\cdot k_{\nu_{\mu}}}{k_{e}\cdot
k_{\nu_{\mu}}p_{\mu}\cdot k_{\gamma}}$ $\displaystyle+\frac{1}{k_{e}\cdot
k_{\gamma}}-\frac{1}{p_{\mu}\cdot k_{\gamma}}+\frac{k_{\nu_{\mu}}\cdot
k_{\gamma}}{k_{e}\cdot k_{\nu_{\mu}}k_{e}\cdot k_{\gamma}}+\frac{k_{e}\cdot
k_{\bar{\nu}_{e}}}{p_{\mu}\cdot k_{\bar{\nu}_{e}}k_{e}\cdot
k_{\gamma}}+\frac{k_{\bar{\nu}_{e}}\cdot k_{\gamma}}{\left(p_{\mu}\cdot
k_{\gamma}\right)^{2}}\frac{m^{2}_{\mu}}{p_{\mu}\cdot
k_{\bar{\nu}_{e}}}-\frac{k_{\nu_{\mu}}\cdot k_{\gamma}}{\left(k_{e}\cdot
k_{\gamma}\right)^{2}}\frac{m^{2}_{e}}{k_{e}\cdot k_{\nu_{\mu}}}.$ (49)

To evaluate the radiation of photons above the small energy cutoff parameter
$k_{\gamma}\geq\varepsilon$, we follow the technique developed in [141] and
further exploited in [121, 122, 118]. This technique was introduced for $2\to
2$ scattering processes with radiation such as the elastic neutrino-electron
scattering with one charged particle in the initial state and one charged
particle in the final state. Performing the crossing of the initial neutral
particle, we generalize the calculation to the decay process with radiation.

For the muon decay, we introduce the four-vector
$l=p_{\mu}-k_{\nu_{\mu}}-k_{\bar{\nu}_{e}}=\left(l_{0},\vec{f}\right)$ and the
angle $\gamma$ between the photon momentum and the vector
$\vec{f},\leavevmode\nobreak\ f=|\vec{f}|$. The energy-momentum conservation
implies

$\displaystyle l^{2}-m^{2}_{e}=2k_{\gamma}\left(l_{0}-f\cos\gamma\right).$
(50)

Performing the integration over the electron momenta and angular variables of
(anti)neutrino of interest, we obtain the (anti)neutrino energy distribution

$\displaystyle\frac{\mathrm{d}\Gamma^{1\gamma}}{\mathrm{d}{E}_{\nu_{\mu}}\mathrm{d}{E}_{\bar{\nu}_{e}}}=\int\frac{|\mathrm{T}^{1\gamma}|^{2}}{2^{11}\pi^{6}m_{\mu}}\frac{l^{2}-m_{e}^{2}}{\left(l_{0}-f\cos\gamma\right)^{2}}\
f\mathrm{d}f\mathrm{d}\Omega_{\gamma}.$ (51)

With our choice of the integration order, the integrand in Eq. (51) does not
depend on the photon azimuthal angle resulting in one trivial integration.
Conveniently, we split the integration into two parts. There are no
restrictions on the photon phase space in the region $\mathrm{I}$:
$l^{2}-m^{2}_{e}\geq 2\varepsilon\left(l_{0}-f\cos\gamma\right)$. For the muon
neutrino energy spectrum, the range of kinematical variables in this region is
given by

$\displaystyle 0\leq
E_{\nu_{\mu}}\leq\frac{m_{\mu}}{2}-\frac{1}{2}\frac{m^{2}_{e}}{m_{\mu}-2\varepsilon},$
(52) $\displaystyle 0\leq
E_{\bar{\nu}_{e}}\leq\frac{m_{\mu}}{2}-\frac{1}{2}\frac{m^{2}_{e}}{m_{\mu}-2E_{\nu_{\mu}}-2\varepsilon},$
(53) $\displaystyle|E_{\nu_{\mu}}-E_{\bar{\nu}_{e}}|\leq
f\leq\mathrm{min}\left(E_{\nu_{\mu}}+E_{\bar{\nu}_{e}},\leavevmode\nobreak\
\sqrt{\left(l_{0}-\varepsilon\right)^{2}-m^{2}_{e}}-\varepsilon\right).$ (54)

To obtain the electron antineutrino energy spectrum, we have to substitute
$E_{\nu_{\mu}}\leftrightarrow E_{\bar{\nu}_{e}}$ in all kinematic ranges. In
the region $\mathrm{II}$: $l^{2}-m^{2}_{e}\leq
2\varepsilon\left(l_{0}-f\cos\gamma\right)$, the angle between the photon
momentum and the vector $\vec{f}$ is constrained as

$\displaystyle\cos\gamma\geq\frac{1}{f}\left(l_{0}-\frac{l^{2}-m_{e}^{2}}{2\varepsilon}\right).$
(55)

The phase space of the region $\mathrm{II}$ is close to the kinematics of the
process without Bremsstrahlung. Only the first term from Eq. (49) contributes
in this region. Moreover, only this factorizable term generates the dependence
on the photon energy cutoff $\varepsilon$ from the region $\mathrm{I}$, which
cancels adding the contribution of low-energy photons with
$k_{\gamma}\leq\varepsilon$. Considering the nonfactorizable radiation terms,
we safely set $\varepsilon=0$ in the beginning of the calculation.

We have reproduced the known contribution of photons with energy
$k_{\gamma}\geq\varepsilon$ to the muon decay width [142] in the limit of
small electron mass. As another cross-check of our calculation, we have
verified the Bremstrahlung $\mathrm{O}\left(\alpha\right)$ contribution to the
muon decay width [70] in the limit of small electron mass considering both
neutrino and antineutrino energy spectra.

###  3.5 Decay spectra and integrated cross sections

Combining all pieces together, we have reproduced
$\mathrm{O}\left(\alpha\right)$ contribution to the muon lifetime both for
small and finite electron masses [70, 42, 99, 57, 66, 98]

$\Gamma\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\left(\gamma\right)\right)=\Gamma_{\mathrm{LO}}\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right)+\frac{\alpha}{\pi}\left(\frac{25}{8}-\frac{\pi^{2}}{2}-\left(34+24\ln
r\right)r^{2}+16\pi^{2}r^{3}+...\right)\frac{\mathrm{G}_{\mathrm{F}}^{2}m^{5}_{\mu}}{192\pi^{3}}.$
(56)

We consider the effect of radiative corrections on neutrino energy spectra and
flux-averaged cross sections in the following. In the limit of vanishing
electron mass, we obtain radiative electron antineutrino and muon neutrino
energy spectra in agreement with QCD corrections to the lepton energy spectra
in decays of heavy quarks [80]

$\displaystyle\frac{\mathrm{d}\Gamma\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\left(\gamma\right)\right)}{\mathrm{d}\Gamma_{\mathrm{LO}}\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right)}$
$\displaystyle=-\frac{\alpha}{\pi}\left(\mathrm{Li}_{2}\frac{2E_{\bar{\nu}_{e}}}{m_{\mu}}+\frac{1}{2}\ln^{2}\left(1-\frac{2E_{\bar{\nu}_{e}}}{m_{\mu}}\right)+\frac{\pi^{2}}{3}-\frac{19}{24}+\frac{5}{24}\frac{m_{\mu}}{E_{\bar{\nu}_{e}}}\right.$
$\displaystyle\left.+\left(\frac{2}{3}+\frac{1}{3}\frac{m_{\mu}}{E_{\bar{\nu}_{e}}}+\frac{5}{48}\frac{m_{\mu}^{2}}{E^{2}_{\bar{\nu}_{e}}}\right)\ln\left(1-\frac{2E_{\bar{\nu}_{e}}}{m_{\mu}}\right)\right),$
(57) $\displaystyle\frac{\mathrm{d}\Gamma\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\left(\gamma\right)\right)}{\mathrm{d}\Gamma_{\mathrm{LO}}\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right)}$
$\displaystyle=-\frac{\alpha}{\pi}\left(\mathrm{Li}_{2}\frac{2E_{\nu_{\mu}}}{m_{\mu}}+\frac{1}{2}\ln^{2}\left(1-\frac{2E_{\nu_{\mu}}}{m_{\mu}}\right)+\frac{\pi^{2}}{3}+\frac{\frac{43}{6}\frac{E_{\nu_{\mu}}}{m_{\mu}}-\frac{51}{8}+\frac{41}{24}\frac{m_{\mu}}{E_{\nu_{\mu}}}}{3-\frac{4E_{\nu_{\mu}}}{m_{\mu}}}\right.$
$\displaystyle\left.-\frac{\frac{8}{3}\frac{E_{\nu_{\mu}}}{m_{\mu}}-\frac{7}{2}+\frac{3}{2}\frac{m_{\mu}}{E_{\nu_{\mu}}}-\frac{41}{48}\frac{m_{\mu}^{2}}{E^{2}_{\nu_{\mu}}}}{3-\frac{4E_{\nu_{\mu}}}{m_{\mu}}}\ln\left(1-\frac{2E_{\nu_{\mu}}}{m_{\mu}}\right)\right).$
(58)

In the following Fig. 2, we compare the leading-order (anti)neutrino energy
spectrum from the muon decay $\mathrm{d}\Gamma_{\mathrm{LO}}\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right)/\mathrm{d}E_{\nu_{\mu},{\bar{\nu}_{e}}}$ to
the contribution of radiative corrections at $\mathrm{O}\left(\alpha\right)$
level from Eqs. (57) and (58) in the limit of a massless electron as well as
to corrections in the electron mass at $\mathrm{O}\left(\alpha\right)$ level.

Figure 2: (Anti)neutrino energy spectrum from the decay of the negative muon
$\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}\left(\gamma\right)$ normalized by
$\Gamma_{\mathrm{LO}}\left(\mu^{-}\to
e^{-}\bar{\nu}_{e}\nu_{\mu}\right)/\omega_{\nu}$. Electron antineutrino energy
spectrum is presented on the left plot, muon neutrino energy spectrum is shown
on the right plot. The leading-order result
$\Gamma_{\mathrm{LO}}\left(\mu^{-}\to e^{-}\bar{\nu}_{e}\nu_{\mu}\right)$ is
shown by blue solid line. It is compared to the
$\mathrm{O}\left(\alpha\right)$ contribution of Eqs. (57) and (58) for
massless electron (red dashed line) and to the electron mass corrections at
$\mathrm{O}\left(\alpha\right)$ order (black dotted line).

As it is expected from the QED power counting, radiative corrections
contribute around $3-4$ permille in the dominant energy region of the
(anti)neutrino flux. (Anti)neutrino energy spectra are infrared-safe
observables which are free from $\ln\frac{m_{e}}{m_{\mu}}$ singularities
according to Kinoshita, Lee, Naunberg theorem [123, 128, 125, 126, 50, 143].
Electron mass corrections are numerically of the next order in $\alpha$
expansion or even below and can be safely neglected.

As an illustrative example, we compare the flux-averaged cross sections on
${}^{40}\mathrm{Ar}$ nucleus, with the averaging according to Eq. (33),
accounting for the radiation of one photon in the (anti)neutrino production
from the decay of the muon at rest $\sigma^{{}^{40}\mathrm{Ar}}$ to cross
sections averaged over the leading-order flux
$\sigma^{{}^{40}\mathrm{Ar}}_{\mathrm{LO}}$ [132, 133, 134, 135, 38]:

$\displaystyle\sigma_{\bar{\nu}_{e}}^{{}^{40}\mathrm{Ar}}$
$\displaystyle=17.484\times 10^{-40}\leavevmode\nobreak\
\mathrm{cm}^{2},\qquad$
$\displaystyle\sigma_{\nu_{\mu}}^{{}^{40}\mathrm{Ar}}=22.448\times
10^{-40}\leavevmode\nobreak\ \mathrm{cm}^{2},$ (59)
$\displaystyle\sigma_{\bar{\nu}_{e},\mathrm{LO}}^{{}^{40}\mathrm{Ar}}$
$\displaystyle=17.490\times 10^{-40}\leavevmode\nobreak\
\mathrm{cm}^{2},\qquad$
$\displaystyle\sigma^{{}^{40}\mathrm{Ar}}_{\nu_{\mu},\mathrm{LO}}=22.454\times
10^{-40}\leavevmode\nobreak\ \mathrm{cm}^{2}.$ (60)

The resulting relative effect of radiative corrections is around 3 permille
both for electron antineutrino and for muon neutrino. Considering the same
normalization for the leading-order and radiatively corrected spectrum, the
relative effect changes sign and reduces to 1 permille. Consequently,
achieving a control over the artificial neutrino flux at permille level
requires a careful account for radiative corrections from the muon decay.

##  4 Conclusions and Outlook

Motivated by forthcoming improvements in the precision of artificial neutrino
fluxes at low and high energies, we have performed a study of radiative
corrections to the main (anti)neutrino production channels, i.e., pion, kaon,
and muon decays. We have presented analytical expressions for (anti)neutrino
energy spectra and have compared leading-order fluxes to
$\mathrm{O}\left(\alpha\right)$ contributions. Since photons are not
controlled at the neutrino production sources, we include radiation of all
hard photons allowed by kinematics of the decay. Our results are important for
a sub-percent control over (anti)neutrino energy distributions. As an
illustrative example, we have demonstrated the effect of radiative corrections
on the scattering cross sections on ${}^{40}\mathrm{Ar}$ nucleus. All results
are obtained in the rest frame of the decaying particle but can be easily
boosted to the arbitrary rest frame.

Radiative corrections to decays of charged pseudoscalar mesons introduce a
continuous and divergent near the (anti)neutrino endpoint component on top of
the monochromatic tree-level spectra. However, the relative intensity of the
radiative tail is well below the strength of the monochromatic line. The flux-
averaged coherent elastic neutrino-nucleus scattering cross section changes by
negligibly small amount of $6\times 10^{-5}$ compared to the leading-order
prediction when the same normalization is used both for radiatively corrected
and tree-level predictions. Pion and kaon structure-dependent contributions
are suppressed at least by two orders of magnitude compared to the leading
point-like $\mathrm{O}\left(\alpha\right)$ QED effects and are irrelevant for
the (anti)neutrino production mechanism.

Radiative corrections to (anti)neutrino spectra from the muon decay are around
$3-4$ permille in the dominant region of the neutrino flux. Such effects
introduce the similar size of the distortion for the flux-averaged cross
sections on the nuclear target. The relative correction diverges near the
endpoint of the (anti)neutrino energy where the leading-order contribution
vanishes resulting into finite physical observables at the fixed order of the
perturbation theory.

(Anti)neutrino energy spectra are infrared-safe observables which are free
from collinear logarithms of $\ln\frac{m_{e}}{m_{\mu}}$ type. QED radiative
corrections to such observables are well estimated by QED power counting and
have a size of $\frac{\alpha}{\pi}\sim 2$ permille up to the factor of the
natural size. Our results for (anti)neutrino spectra from the muon decay
confirm this simple estimate. $\mathrm{O}\left(\alpha\right)$ electron-mass
effects are a few orders of magnitude below the leading contribution and can
be neglected for applications at the neutrino scattering facilities.

## Acknowledgments

O.T. thanks Ryan Plestid as well as Pedro Machado and Vishvas Pandey for
numerous discussions on [38] which has motivated the main part of this work,
Vincenzo Cirigliano and Emanuele Mereghetti for useful advices regarding
literature and presentation, Emanuele Mereghetti for reading the manuscript,
Richard Hill, Matthias Heller, and Marc Vanderhaeghen for technical
discussions while working on other projects, and Kevin McFarland for
suggestions regarding applications to accelerator neutrino sources. O. T.
acknowledges the Fermilab theory group for warm hospitality and support. O. T.
acknowledges the theory group of Institute for Nuclear Physics at Johannes
Gutenberg-Universität Mainz for warm hospitality and support. This work is
supported by the US Department of Energy through the Los Alamos National
Laboratory. Los Alamos National Laboratory is operated by Triad National
Security, LLC, for the National Nuclear Security Administration of U.S.
Department of Energy (Contract No. 89233218CNA000001). This research is funded
by LANL’s Laboratory Directed Research and Development (LDRD/PRD) program
under project number 20210968PRD4. The work of O.T. is partially supported by
the Visiting Scholars Award Program of the Universities Research Association.
FeynCalc [144, 145], LoopTools [146], Mathematica [147] and DataGraph were
extremely useful in this work.

## References

  * Ott and Pritchard [1971] R. J. Ott and T. W. Pritchard, Phys. Rev. D 3, 52 (1971).
  * Numao  _et al._ [1995] T. Numao, J. A. Macdonald, G. M. Marshall, A. Olin, and M. C. Fujiwara, Phys. Rev. D 52, 4855 (1995).
  * Koptev  _et al._ [1995] V. P. Koptev _et al._ , JETP Lett. 61, 877 (1995).
  * Ambrosino  _et al._ [2008] F. Ambrosino _et al._ (KLOE), JHEP 01, 073 (2008), arXiv:0712.1112 [hep-ex].
  * Barczyk  _et al._ [2008] A. Barczyk _et al._ (FAST), Phys. Lett. B 663, 172 (2008), arXiv:0707.3904 [hep-ex].
  * Webber  _et al._ [2011] D. M. Webber _et al._ (MuLan), Phys. Rev. Lett. 106, 041803 (2011), arXiv:1010.0991 [hep-ex].
  * Casella  _et al._ [2013] C. Casella _et al._ , Nucl. Instrum. Meth. A 700, 1 (2013).
  * Bertl  _et al._ [1985] W. H. Bertl _et al._ (SINDRUM), Nucl. Phys. B 260, 1 (1985).
  * Zyla  _et al._ [2020] P. A. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Aguilar-Arevalo  _et al._ [2015] A. Aguilar-Arevalo _et al._ (PiENu), Phys. Rev. Lett. 115, 071801 (2015), arXiv:1506.05845 [hep-ex].
  * Pocanic  _et al._ [2004] D. Pocanic _et al._ , Phys. Rev. Lett. 93, 181803 (2004), arXiv:hep-ex/0312030.
  * Aguilar-Arevalo  _et al._ [2001] A. Aguilar-Arevalo _et al._ (LSND), Phys. Rev. D 64, 112007 (2001), arXiv:hep-ex/0104049.
  * Berns  _et al._ [2013] H. Berns _et al._ (CAPTAIN), in  _Community Summer Study 2013: Snowmass on the Mississippi_ (2013) arXiv:1309.1740 [physics.ins-det].
  * Taylor  _et al._ [2021] C. E. Taylor _et al._ (CAPTAIN), Nucl. Instrum. Meth. A 1001, 165131 (2021), arXiv:2008.11422 [physics.ins-det].
  * Aguilar-Arevalo  _et al._ [2021] A. A. Aguilar-Arevalo _et al._ (CCM), (2021), arXiv:2105.14020 [hep-ex].
  * Athar  _et al._ [2021] M. S. Athar _et al._ , (2021), arXiv:2111.07586 [hep-ph].
  * Akimov  _et al._ [2017] D. Akimov _et al._ (COHERENT), Science 357, 1123 (2017), arXiv:1708.01294 [nucl-ex].
  * Akimov  _et al._ [2019] D. Akimov _et al._ (COHERENT), Phys. Rev. D 100, 115020 (2019), arXiv:1909.05913 [hep-ex].
  * Akimov  _et al._ [2021a] D. Akimov _et al._ (COHERENT), Phys. Rev. Lett. 126, 012002 (2021a), arXiv:2003.10630 [nucl-ex].
  * Barbeau  _et al._ [2021] P. S. Barbeau, Y. Efremenko, and K. Scholberg, (2021), arXiv:2111.07033 [hep-ex].
  * Athanassopoulos  _et al._ [1997] C. Athanassopoulos _et al._ (LSND), Nucl. Instrum. Meth. A 388, 149 (1997), arXiv:nucl-ex/9605002.
  * Maschuw [1998] R. Maschuw (KARMEN), Prog. Part. Nucl. Phys. 40, 183 (1998).
  * Conrad and Shaevitz [2010] J. M. Conrad and M. H. Shaevitz, Phys. Rev. Lett. 104, 141802 (2010), arXiv:0912.4079 [hep-ex].
  * Lazauskas and Volpe [2010] R. Lazauskas and C. Volpe, J. Phys. G 37, 125101 (2010), [Erratum: J.Phys.G 42, 059501 (2015)], arXiv:1004.0310 [hep-ph].
  * Alonso  _et al._ [2010] J. Alonso _et al._ , (2010), arXiv:1006.0260 [physics.ins-det].
  * Pagliaroli  _et al._ [2013] G. Pagliaroli, C. Lujan-Peschard, M. Mitra, and F. Vissani, Phys. Rev. Lett. 111, 022001 (2013), arXiv:1210.4225 [hep-ph].
  * Brice  _et al._ [2014] S. J. Brice _et al._ , Phys. Rev. D 89, 072004 (2014), arXiv:1311.5958 [physics.ins-det].
  * Baxter  _et al._ [2020] D. Baxter _et al._ , JHEP 02, 123 (2020), arXiv:1911.00762 [physics.ins-det].
  * Abe  _et al._ [2011] K. Abe _et al._ (T2K), Nucl. Instrum. Meth. A 659, 106 (2011), arXiv:1106.1238 [physics.ins-det].
  * Abe  _et al._ [2015] K. Abe _et al._ (Hyper-Kamiokande Proto-), PTEP 2015, 053C02 (2015), arXiv:1502.05199 [hep-ex].
  * Abe  _et al._ [2020] K. Abe _et al._ (T2K), Nature 580, 339 (2020), [Erratum: Nature 583, E16 (2020)], arXiv:1910.03887 [hep-ex].
  * Acero  _et al._ [2019] M. A. Acero _et al._ (NOvA), Phys. Rev. Lett. 123, 151803 (2019), arXiv:1906.04907 [hep-ex].
  * Abi  _et al._ [2020] B. Abi _et al._ (DUNE), (2020), arXiv:2002.03005 [hep-ex].
  * Kopp [2007] S. E. Kopp, Phys. Rept. 439, 101 (2007), arXiv:physics/0609129.
  * Abe  _et al._ [2013] K. Abe _et al._ (T2K), Phys. Rev. D 87, 012001 (2013), [Addendum: Phys.Rev.D 87, 019902 (2013)], arXiv:1211.0469 [hep-ex].
  * Adamson  _et al._ [2016] P. Adamson _et al._ , Nucl. Instrum. Meth. A 806, 279 (2016), arXiv:1507.06690 [physics.acc-ph].
  * Aliaga  _et al._ [2016] L. Aliaga _et al._ (MINERvA), Phys. Rev. D 94, 092005 (2016), [Addendum: Phys.Rev.D 95, 039903 (2017)], arXiv:1607.00704 [hep-ex].
  * Tomalak  _et al._ [2021] O. Tomalak, P. Machado, V. Pandey, and R. Plestid, JHEP 02, 097 (2021), arXiv:2011.05960 [hep-ph].
  * Behrends  _et al._ [1956] R. E. Behrends, R. J. Finkelstein, and A. Sirlin, Phys. Rev. 101, 866 (1956).
  * Kinoshita and Sirlin [1957] T. Kinoshita and A. Sirlin, Phys. Rev. 107, 593 (1957).
  * Berman [1958a] S. M. Berman, Phys. Rev. 112, 267 (1958a).
  * Kinoshita and Sirlin [1959a] T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959a).
  * Bailin [1964] D. Bailin, Phys. Rev. 135, B166 (1964).
  * Matsson [1969] L. Matsson, Nucl. Phys. B 12, 647 (1969).
  * Grotch [1968] H. Grotch, Phys. Rev. 168, 1872 (1968).
  * Roos and Sirlin [1971] M. Roos and A. Sirlin, Nucl. Phys. B 29, 296 (1971).
  * Girotti [1971] H. O. Girotti, Nucl. Phys. B 27, 327 (1971).
  * Ross [1973] D. A. Ross, Nucl. Phys. B 51, 116 (1973).
  * Mohammad and Donnachie [1976] I. Mohammad and A. Donnachie, (1976).
  * Greub  _et al._ [1994] C. Greub, D. Wyler, and W. Fetscher, Phys. Lett. B 324, 109 (1994), [Erratum: Phys.Lett.B 329, 526 (1994)], arXiv:hep-ph/9312301.
  * Czarnecki and Jezabek [1994] A. Czarnecki and M. Jezabek, Nucl. Phys. B 427, 3 (1994), arXiv:hep-ph/9402326.
  * Czarnecki  _et al._ [1995] A. Czarnecki, M. Jezabek, and J. H. Kuhn, Phys. Lett. B 346, 335 (1995), arXiv:hep-ph/9411282.
  * Konyshev and Polyubin [1995] V. Konyshev and I. Polyubin, JETP Lett. 62, 469 (1995), arXiv:hep-ph/9507232.
  * Bauberger [1997] S. Bauberger,  _Two-loop contributions to muon decay_ , Ph.D. thesis, Bielefeld U. (1997).
  * Ferroglia  _et al._ [1999] A. Ferroglia, G. Ossola, and A. Sirlin, Nucl. Phys. B 560, 23 (1999), arXiv:hep-ph/9905442.
  * van Ritbergen and Stuart [1999] T. van Ritbergen and R. G. Stuart, Phys. Rev. Lett. 82, 488 (1999), arXiv:hep-ph/9808283.
  * van Ritbergen and Stuart [2000] T. van Ritbergen and R. G. Stuart, Nucl. Phys. B 564, 343 (2000), arXiv:hep-ph/9904240.
  * Davydychev  _et al._ [2001] A. I. Davydychev, K. Schilcher, and H. Spiesberger, Eur. Phys. J. C 19, 99 (2001), arXiv:hep-ph/0011221.
  * Arbuzov [2002] A. B. Arbuzov, Phys. Lett. B 524, 99 (2002), [Erratum: Phys.Lett.B 535, 378–378 (2002)], arXiv:hep-ph/0110047.
  * Fischer  _et al._ [2003] M. Fischer, S. Groote, J. G. Korner, and M. C. Mauser, Phys. Rev. D 67, 113008 (2003), arXiv:hep-ph/0203048.
  * Davydychev  _et al._ [2002] A. I. Davydychev, K. Schilcher, and H. Spiesberger, in  _16th International Workshop on High Energy Physics and Quantum Field Theory (QFTHEP 2001)_ (2002) pp. 214–221, arXiv:hep-ph/0203125.
  * Arbuzov  _et al._ [2002] A. Arbuzov, A. Czarnecki, and A. Gaponenko, Phys. Rev. D 65, 113006 (2002), arXiv:hep-ph/0202102.
  * Arbuzov [2003] A. Arbuzov, JHEP 03, 063 (2003), arXiv:hep-ph/0206036.
  * Arbuzov and Scherbakova [2004] A. B. Arbuzov and E. S. Scherbakova, Phys. Lett. B 597, 285 (2004), arXiv:hep-ph/0404094.
  * Anastasiou  _et al._ [2007] C. Anastasiou, K. Melnikov, and F. Petriello, JHEP 09, 014 (2007), arXiv:hep-ph/0505069.
  * Pak and Czarnecki [2008] A. Pak and A. Czarnecki, Phys. Rev. Lett. 100, 241807 (2008), arXiv:0803.0960 [hep-ph].
  * Caola  _et al._ [2014] F. Caola, A. Czarnecki, Y. Liang, K. Melnikov, and R. Szafron, Phys. Rev. D 90, 053004 (2014), arXiv:1403.3386 [hep-ph].
  * Fael  _et al._ [2021] M. Fael, K. Schönwald, and M. Steinhauser, Phys. Rev. D 104, 016003 (2021), arXiv:2011.13654 [hep-ph].
  * Czakon  _et al._ [2021] M. Czakon, A. Czarnecki, and M. Dowling, Phys. Rev. D 103, L111301 (2021), arXiv:2104.05804 [hep-ph].
  * Berman [1958b] S. M. Berman, Phys. Rev. Lett. 1, 468 (1958b).
  * Kinoshita [1959] T. Kinoshita, Phys. Rev. Lett. 2, 477 (1959).
  * Ginsberg and Pratt [1963] E. S. Ginsberg and R. H. Pratt, Phys. Rev. 130, 2105 (1963).
  * Kamal [1964] A. N. Kamal, Nuovo Cim. 33, 1108 (1964).
  * Bailin and Bailin [1970] A. Bailin and D. Bailin, Nucl. Phys. B 17, 317 (1970).
  * Goldman and Wilson [1977] J. T. Goldman and W. J. Wilson, Phys. Rev. D 15, 709 (1977).
  * Marciano and Sirlin [1976] W. J. Marciano and A. Sirlin, Phys. Rev. Lett. 36, 1425 (1976).
  * Snellman [1968] H. Snellman, Nucl. Phys. B 5, 220 (1968).
  * Garcia and Queijeiro [1981] A. Garcia and A. Queijeiro, Phys. Rev. D 23, 1562 (1981).
  * Bryman  _et al._ [1982] D. A. Bryman, P. Depommier, and C. Leroy, Phys. Rept. 88, 151 (1982).
  * Jezabek and Kuhn [1989] M. Jezabek and J. H. Kuhn, Nucl. Phys. B 320, 20 (1989).
  * Holstein [1990] B. R. Holstein, Phys. Lett. B 244, 83 (1990).
  * Gabrielli [1993] E. Gabrielli, Phys. Lett. B 301, 409 (1993).
  * Bijnens  _et al._ [1993] J. Bijnens, G. Ecker, and J. Gasser, Nucl. Phys. B 396, 81 (1993), arXiv:hep-ph/9209261.
  * Marciano and Sirlin [1993] W. J. Marciano and A. Sirlin, Phys. Rev. Lett. 71, 3629 (1993).
  * Finkemeier [1994] M. Finkemeier, (1994), arXiv:hep-ph/9501286.
  * Finkemeier [1996] M. Finkemeier, Phys. Lett. B 387, 391 (1996), arXiv:hep-ph/9505434.
  * Neufeld and Rupertsberger [1996] H. Neufeld and H. Rupertsberger, Z. Phys. C 71, 131 (1996), arXiv:hep-ph/9506448.
  * Kuraev [1997] E. A. Kuraev, JETP Lett. 65, 127 (1997), arXiv:hep-ph/9611294.
  * Chen  _et al._ [1997] C. H. Chen, C. Q. Geng, and C. C. Lih, Phys. Rev. D 56, 6856 (1997), arXiv:hep-ph/9709447.
  * Chen  _et al._ [2008] C. H. Chen, C. Q. Geng, and C. C. Lih, Int. J. Mod. Phys. A 23, 3204 (2008), arXiv:0801.1074 [hep-ph].
  * Chen  _et al._ [2011] C.-H. Chen, C.-Q. Geng, and C.-C. Lih, Phys. Rev. D 83, 074001 (2011), arXiv:1006.2939 [hep-ph].
  * Cirigliano  _et al._ [2002] V. Cirigliano, M. Knecht, H. Neufeld, H. Rupertsberger, and P. Talavera, Eur. Phys. J. C 23, 121 (2002), arXiv:hep-ph/0110153.
  * Broncano and Mena [2003] A. Broncano and O. Mena, Eur. Phys. J. C 29, 197 (2003), arXiv:hep-ph/0203052.
  * Descotes-Genon and Moussallam [2005] S. Descotes-Genon and B. Moussallam, Eur. Phys. J. C 42, 403 (2005), arXiv:hep-ph/0505077.
  * Cirigliano and Rosell [2007a] V. Cirigliano and I. Rosell, Phys. Rev. Lett. 99, 231801 (2007a), arXiv:0707.3439 [hep-ph].
  * Cirigliano and Rosell [2007b] V. Cirigliano and I. Rosell, JHEP 10, 005 (2007b), arXiv:0707.4464 [hep-ph].
  * Cirigliano  _et al._ [2008] V. Cirigliano, M. Giannotti, and H. Neufeld, JHEP 11, 006 (2008), arXiv:0807.4507 [hep-ph].
  * Sirlin and Ferroglia [2013] A. Sirlin and A. Ferroglia, Rev. Mod. Phys. 85, 263 (2013), arXiv:1210.5296 [hep-ph].
  * Nir [1989] Y. Nir, Phys. Lett. B 221, 184 (1989).
  * Vogel and Beacom [1999] P. Vogel and J. F. Beacom, Phys. Rev. D 60, 053003 (1999), arXiv:hep-ph/9903554.
  * Kurylov  _et al._ [2003] A. Kurylov, M. J. Ramsey-Musolf, and P. Vogel, Phys. Rev. C 67, 035502 (2003), arXiv:hep-ph/0211306.
  * Fukugita and Kubota [2004] M. Fukugita and T. Kubota, Acta Phys. Polon. B 35, 1687 (2004), arXiv:hep-ph/0403149.
  * Raha  _et al._ [2012] U. Raha, F. Myhrer, and K. Kubodera, Phys. Rev. C 85, 045502 (2012), [Erratum: Phys.Rev.C 86, 039903 (2012)], arXiv:1112.2007 [hep-ph].
  * Ankowski [2019] A. M. Ankowski, J. Phys. Conf. Ser. 1216, 012015 (2019).
  * Akimov  _et al._ [2021b] D. Akimov _et al._ (COHERENT), JINST 16, P08048 (2021b), arXiv:2104.09605 [physics.ins-det].
  * Longhin  _et al._ [2015] A. Longhin, L. Ludovici, and F. Terranova, Eur. Phys. J. C 75, 155 (2015), arXiv:1412.5987 [hep-ex].
  * Meregaglia [2016] A. Meregaglia, JINST 11, C12040 (2016).
  * Kyberd  _et al._ [2012] P. Kyberd _et al._ (nuSTORM), (2012), arXiv:1206.0294 [hep-ex].
  * Adey  _et al._ [2013a] D. Adey _et al._ (nuSTORM), (2013a), arXiv:1308.6822 [physics.acc-ph].
  * Adey  _et al._ [2013b] D. Adey _et al._ , (2013b), arXiv:1305.1419 [physics.acc-ph].
  * Hill and Tomalak [2020] R. J. Hill and O. Tomalak, Phys. Lett. B 805, 135466 (2020), arXiv:1911.01493 [hep-ph].
  * Gasser and Leutwyler [1984] J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984).
  * Gasser and Leutwyler [1985] J. Gasser and H. Leutwyler, Nucl. Phys. B 250, 465 (1985).
  * Kinoshita [2016] T. Kinoshita, (2016), arXiv:1604.04891 [physics.hist-ph].
  * Vanderhaeghen  _et al._ [2000] M. Vanderhaeghen, J. M. Friedrich, D. Lhuillier, D. Marchand, L. Van Hoorebeke, and J. Van de Wiele, Phys. Rev. C 62, 025501 (2000), arXiv:hep-ph/0001100.
  * Heller  _et al._ [2018] M. Heller, O. Tomalak, and M. Vanderhaeghen, Phys. Rev. D 97, 076012 (2018), arXiv:1802.07174 [hep-ph].
  * Heller  _et al._ [2019] M. Heller, O. Tomalak, M. Vanderhaeghen, and S. Wu, Phys. Rev. D 100, 076013 (2019), arXiv:1906.02706 [hep-ph].
  * Tomalak and Hill [2020] O. Tomalak and R. J. Hill, Phys. Rev. D 101, 033006 (2020), arXiv:1907.03379 [hep-ph].
  * Lee and Sirlin [1964] T. D. Lee and A. Sirlin, Rev. Mod. Phys. 36, 666 (1964).
  * Aoki  _et al._ [1981] K.-i. Aoki, Z. Hioki, R. Kawabe, M. Konuma, and T. Muta, Prog. Theor. Phys. 65, 1001 (1981).
  * Sarantakos  _et al._ [1983] S. Sarantakos, A. Sirlin, and W. J. Marciano, Nucl. Phys. B 217, 84 (1983).
  * Passera [2001] M. Passera, Phys. Rev. D 64, 113002 (2001), arXiv:hep-ph/0011190.
  * Bloch and Nordsieck [1937] F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937).
  * Nakanishi [1958] N. Nakanishi, Prog. Theor. Phys. 19, 159 (1958).
  * Kinoshita [1962] T. Kinoshita, J. Math. Phys. 3, 650 (1962).
  * Lee and Nauenberg [1964] T. D. Lee and M. Nauenberg, Phys. Rev. 133, B1549 (1964).
  * Knecht  _et al._ [2000] M. Knecht, H. Neufeld, H. Rupertsberger, and P. Talavera, Eur. Phys. J. C 12, 469 (2000), arXiv:hep-ph/9909284.
  * Yennie  _et al._ [1961] D. R. Yennie, S. C. Frautschi, and H. Suura, Annals Phys. 13, 379 (1961).
  * Geng  _et al._ [2004] C. Q. Geng, I.-L. Ho, and T. H. Wu, Nucl. Phys. B 684, 281 (2004), arXiv:hep-ph/0306165.
  * Bijnens and Ecker [2014] J. Bijnens and G. Ecker, Ann. Rev. Nucl. Part. Sci. 64, 149 (2014), arXiv:1405.6488 [hep-ph].
  * Unterdorfer and Pichl [2008] R. Unterdorfer and H. Pichl, Eur. Phys. J. C 55, 273 (2008), arXiv:0801.2482 [hep-ph].
  * Payne  _et al._ [2019] C. G. Payne, S. Bacca, G. Hagen, W. Jiang, and T. Papenbrock, Phys. Rev. C 100, 061304 (2019), arXiv:1908.09739 [nucl-th].
  * Yang  _et al._ [2019] J. Yang, J. A. Hernandez, and J. Piekarewicz, Phys. Rev. C 100, 054301 (2019), arXiv:1908.10939 [nucl-th].
  * Van Dessel  _et al._ [2020] N. Van Dessel, V. Pandey, H. Ray, and N. Jachowicz, (2020), arXiv:2007.03658 [nucl-th].
  * Hoferichter  _et al._ [2020] M. Hoferichter, J. Menéndez, and A. Schwenk, Phys. Rev. D 102, 074018 (2020), arXiv:2007.08529 [hep-ph].
  * Fermi [1934] E. Fermi, Z. Phys. 88, 161 (1934).
  * Feynman and Gell-Mann [1958] R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958).
  * Arason  _et al._ [1992] H. Arason, D. J. Castano, B. Keszthelyi, S. Mikaelian, E. J. Piard, P. Ramond, and B. D. Wright, Phys. Rev. D 46, 3945 (1992).
  * Antonelli and Maiani [1981] F. Antonelli and L. Maiani, Nucl. Phys. B 186, 269 (1981).
  * Okun [1982] L. B. Okun,  _Leptons and Quarks: Special Edition Commemorating the Discovery of the Higgs Boson_ (North-Holland, Amsterdam, Netherlands, 1982).
  * Ram [1967] M. Ram, Phys. Rev. 155, 1539 (1967).
  * Kinoshita and Sirlin [1959b] T. Kinoshita and A. Sirlin, Phys. Rev. Lett. 2, 177 (1959b).
  * Sirlin [2011] A. Sirlin, Phys. Rev. D 84, 014021 (2011), arXiv:1105.2842 [hep-ph].
  * Mertig  _et al._ [1991] R. Mertig, M. Bohm, and A. Denner, Comput. Phys. Commun. 64, 345 (1991).
  * Shtabovenko  _et al._ [2016] V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 207, 432 (2016), arXiv:1601.01167 [hep-ph].
  * Hahn and Perez-Victoria [1999] T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. 118, 153 (1999), arXiv:hep-ph/9807565.
  * [147] W. R. Inc., “Mathematica, Version 12.2.0.0,” Champaign, IL, 2016.


#  $\Lambda$CDM model with a scalar perturbation vs. preferred direction of
the universe

Xin Li<EMAIL_ADDRESS>Hai-Nan Lin<EMAIL_ADDRESS>Sai Wang
<EMAIL_ADDRESS>Zhe Chang<EMAIL_ADDRESS>Institute of High Energy
Physics,
Theoretical Physics Center for Science Facilities,
Chinese Academy of Sciences, 100049 Beijing, China

###### Abstract

We present a scalar perturbation for the $\Lambda$CDM model, which breaks the
isotropic symmetry of the universe. Based on the Union2 data, the
least-$\chi^{2}$ fit of the scalar perturbed $\Lambda$CDM model shows that the
universe has a preferred direction $(l,b)=(287^{\circ}\pm
25^{\circ},11^{\circ}\pm 22^{\circ})$. The magnitude of scalar perturbation is
about $-2.3\times 10^{-5}$. The scalar perturbation for the $\Lambda$CDM model
implies a peculiar velocity, which is perpendicular to the radial direction.
We show that the maximum peculiar velocities at redshift $z=0.15$ and
$z=0.015$ equal to $73\pm 28\rm km\cdot s^{-1}$ and $1099\pm 427\rm km\cdot
s^{-1}$, respectively. They are compatible with the constraints on peculiar
velocity given by Planck Collaboration.

###### pacs:

98.80.-k,98.80.Jk

##  I Introduction

The standard cosmological model, i.e., the $\Lambda$CDM model Sahni ;
Padmanabhan  has been well established. It is consistent with several precise
astronomical observations that involve Wilkinson Microwave Anisotropy Probe
(WMAP) Komatsu , Planck satellite Planck1 , Supernovae Cosmology Project
Suzuki . One of the most important and basic assumptions of the $\Lambda$CDM
model states that the universe is homogeneous and isotropic on large scales.
However, such a principle faces challenges Perivolaropoulos . The Union2 SnIa
data hint that the universe has a preferred direction
$(l,b)=(309^{\circ},18^{\circ})$ in galactic coordinate system Antoniou .
Toward this direction, the universe has the maximum expansion velocity.
Astronomical observations Watkins  found that the dipole moment of the
peculiar velocity field on the direction $(l,b)=(287^{\circ}\pm
9^{\circ},8^{\circ}\pm 6^{\circ})$ in the scale of $50h^{-1}\rm Mpc$ has a
magnitude $407\pm 81\rm km\cdot s^{-1}$. This peculiar velocity is much larger
than the value $110\rm km\cdot s^{-1}$ given by WMAP5 WMAP5 . The recent
released data of Planck Collaboration show deviations from isotropy with a
level of significance ($\sim 3\sigma$) Planck2 . Planck Collaboration confirms
asymmetry of the power spectrums between two preferred opposite hemispheres.
These facts hint that the universe may have a preferred direction.

Many models have been proposed to resolve the asymmetric anomaly of the
astronomical observations. An incomplete and succinct list includes: an
imperfect fluid dark energy Koivisto1 , local void scenario Alexander ; Garcia
, noncommutative spacetime effect Akofor , anisotropic curvature in cosmology
Koivisto2 , and Finsler gravity scenario Chang .

In this paper, we present a scalar perturbation for the flat Friedmann-
Robertson-Walker (FRW) metric Weinberg . Based on the Union2 data, the
least-$\chi^{2}$ fit of the scalar perturbed $\Lambda$CDM model shows that the
universe has a preferred direction. In the scalar perturbed $\Lambda$CDM
model, the universe could be treated as a perfect fluid approximately. In
comoving frame, however, the fluid has a small velocity $v$. It could be
regarded as the peculiar velocity of the universe. The data of Planck
Collaboration gives severe constraints on the peculiar velocity Planck3 . For
the bulk flow of Local Group, it should be less than $254\rm km\cdot s^{-1}$.
For bulk flow of galaxy clusters at $z=0.15$, it should be less than $800\rm
km\cdot s^{-1}$.

The paper is organized as follows. In Sec. II, we present a scalar
perturbation for the FRW metric. Explicit relation between luminosity and
redshift is obtained. In Sec. III, we show a least-$\chi^{2}$ fit of the
scalar perturbed $\Lambda$CDM model to the Union2 SnIa data. The preferred
direction is found $(l,b)=(287^{\circ}\pm 25^{\circ},11^{\circ}\pm
22^{\circ})$. The magnitude of the scalar perturbation is at the scale of
$10^{-5}$. This perturbation implies a peculiar velocity with value $73\pm
28\rm km\cdot s^{-1}$ at $z=0.15$, and $1099\pm 427\rm km\cdot s^{-1}$ at
$z=0.015$. The conclusions and remarks are given in Sec. IV.

##  II Scalar perturbation for FRW metric

The FRW metric describes the homogeneous and isotropic universe. In order to
describe the deviation from isotropy, we try to add a scalar perturbation for
the FRW metric. The scalar perturbed FRW metric is of the form

$ds^{2}=(1-2\phi(\vec{x}))dt^{2}-a^{2}(t)(1+2\phi(\vec{x}))\delta_{ij}dx^{i}dx^{j}.$
(1)

It should be noticed that the scalar perturbation field $\phi(\vec{x})$ is
time-independent. And the scalar perturbation can be interpreted as a sort of
space-dependent spatial curvature. By setting the scale factor $a(t)=1$, one
can find that the spatial Ricci tensor of metric (1) is of the form

$R_{ij}=-\delta_{ij}\phi_{,k,k}.$ (2)

The nonvanishing components of Einstein tensor for the metric (1) are given as

$\displaystyle G^{0}_{0}$ $\displaystyle=$ $\displaystyle
3(1+2\phi)H^{2}-2a^{-2}\phi_{,i,i}~{},$ (3) $\displaystyle G^{i}_{j}$
$\displaystyle=$
$\displaystyle\delta_{ij}(1+2\phi)\left(H^{2}+2\frac{\ddot{a}}{a}\right)~{},$
(4) $\displaystyle G^{0}_{j}$ $\displaystyle=$ $\displaystyle-2H\phi_{,j}~{},$
(5)

where the commas denote the derivatives with respect to $x^{i}$, the dot
denotes the derivatives with respect to cosmic time $t$ and
$H\equiv\frac{\dot{a}}{a}$. The scalar perturbation breaks homogeneity and
isotropy of the universe. Since $\phi$ is a perturbation, the cosmic inventory
could be treated as a perfect fluid approximately. In comoving frame, however,
the fluid has a perturbed velocity $v$. The energy-momentum tensor is given by

$T^{\mu\nu}=(\rho+p)U^{\mu}U^{\nu}-pg^{\mu\nu},$ (6)

where $\rho$ and $p$ are the energy density and pressure density of the fluid,
respectively. Here, we set $U^{\mu}$ as $U^{0}=1,\frac{U^{i}}{U^{0}}\equiv
v^{i}$, to first order in $v$. In this paper, we just investigate low redshift
region of the universe, where the universe is dominated by matter and dark
energy. Thus, the nonvanishing components of energy-momentum tensor are given
as

$\displaystyle T^{0}_{0}$ $\displaystyle=$
$\displaystyle\rho_{m}+\rho_{de}~{},$ (7) $\displaystyle T^{0}_{i}$
$\displaystyle=$ $\displaystyle\rho_{m}v_{i}~{},$ (8) $\displaystyle
T^{i}_{j}$ $\displaystyle=$ $\displaystyle\delta^{i}_{j}\rho_{de}~{},$ (9)

where $\rho_{m}$ and $\rho_{de}$ denote the energy density of matter and dark
energy, respectively. Then, the Einstein field equation $G^{\mu}_{\nu}=8\pi
GT^{\mu}_{\nu}$ gives three independent equations

$\displaystyle(1+2\phi)H^{2}-\frac{2a^{-2}}{3}\phi_{,i,i}$ $\displaystyle=$
$\displaystyle\frac{8\pi G}{3}(\rho_{m}+\rho_{de})~{},$ (10)
$\displaystyle(1+2\phi)(H^{2}+2\frac{\ddot{a}}{a})$ $\displaystyle=$
$\displaystyle 8\pi G\rho_{de}~{},$ (11) $\displaystyle H\phi_{,j}$
$\displaystyle=$ $\displaystyle-4\pi G\rho_{m}v_{j}~{}.$ (12)

The energy-momentum conservation equation reads

$\frac{\partial T^{\mu}_{\nu}}{\partial
x^{\mu}}+\Gamma^{\mu}_{\alpha\mu}T^{\alpha}_{\nu}-\Gamma^{\alpha}_{\nu\mu}T^{\mu}_{\alpha}=0,$
(13)

where $\Gamma^{\mu}_{\alpha\mu}$ is the Christoffel symbol. Then, following
the theory of general relativity, we obtain the specific form of energy-
momentum conservation equation for matter and dark energy in the perturbed FRW
universe (1). It is as follows:

$\displaystyle\frac{\partial\rho_{m}}{\partial
t}+3H\rho_{m}+\frac{\partial\rho_{m}v^{i}}{\partial x^{i}}$ $\displaystyle=$
$\displaystyle 0,$ (14) $\displaystyle\frac{\partial\rho_{m}v_{i}}{\partial
t}+3H\rho_{m}v_{i}-\phi_{,i}\rho_{m}$ $\displaystyle=$ $\displaystyle 0,$ (15)
$\displaystyle\frac{\partial\rho_{de}}{\partial t}$ $\displaystyle=$
$\displaystyle 0,$ (16) $\displaystyle\frac{\partial\rho_{de}}{\partial
x^{i}}$ $\displaystyle=$ $\displaystyle 0.$ (17)

The equations (16) and (17) show that the energy density of dark energy
remaining constant in our model. By making use of the field equation (12), we
find from equation (14) that

$\frac{\partial(\rho_{m}a^{3})}{\partial t}=-aH\frac{\phi_{,i,i}}{4\pi G}.$
(19)

The solution of equation (19) reads

$\rho_{m}a^{3}=-\frac{\phi_{,i,i}}{4\pi G}(a-1)+\rho_{m0},$ (20)

where $\rho_{m0}$ denotes the energy density of matter at present. We have
already used the initial condition that the present energy density of matter
is constant to deduce continuity equation (20).

The light propagation satisfies $ds=0$, which gives

$\frac{dt}{a(t)}=(1+2\phi(\vec{x}))\delta_{ij}dx^{i}dx^{j}~{}.$ (21)

The right-hand side of the equation (21) is time-independent. During a very
short time, the location of a galaxy is unchanged. Then, we get

$\int^{t_{0}}_{t_{1}}\frac{dt}{a(t)}=\int^{t_{0}+\delta t_{0}}_{t_{1}+\delta
t_{1}}\frac{dt}{a(t)}~{}.$ (22)

Thus, the redshift $z$ of galaxy satisfies

$1+z=\frac{1}{a},$ (23)

where we have set the scale factor $a(t)$ to be 1 at present.

A particular form of $\phi(\vec{x})$ is necessary for deriving relation
between luminosity distance and redshift. The form of $\phi(\vec{x})$ is
determined by the perturbed energy-momentum tensor. However, the information
of the perturbed energy-momentum tensor is unknown. On the contrary, we choose
a specific form of $\phi$ to determine the perturbed energy-momentum tensor.
It is given as

$\phi=A\cos\theta~{},$ (24)

where $A$ is a dimensionless parameter and $\theta$ is the angle between
$\vec{r}$ and $z$-axis. By making use of (24), we reduce the equation (10) to

$\frac{da}{dt}=H_{0}a(1-A\cos\theta)\sqrt{\Omega_{m0}a^{-3}+1-\Omega_{m0}-\frac{4A\cos\theta}{3r^{2}H_{0}^{2}}a^{-3}},$
(25)

where $H_{0}$ is Hubble constant and $\Omega_{m0}\equiv 8\pi
G\rho_{m0}/(3H_{0}^{2})$ is the energy density parameter for matter at
present.

Combining the equations (21), (23), (24), (25) and (20), and using the
definition of luminosity distance Weinberg , we obtain the relation between
luminosity distance and redshift

$H_{0}d_{L}=(1+z)\int_{0}^{z}\frac{(1-A\cos\theta)dx}{\sqrt{\Omega_{m0}(1+x)^{3}+1-\Omega_{m0}-\frac{4A\cos\theta(1+x)^{5}}{3H_{0}^{2}d^{2}_{L0}}}}~{},$
(26)

where
$d_{L0}\equiv(1+z)\int_{0}^{z}\frac{dx}{H_{0}\sqrt{\Omega_{m0}(1+x)^{3}+1-\Omega_{m0}}}$.

##  III Numerical results

Our numerical studies are based on the Union2 SnIa data Amanullah . Our goal
is to find whether the universe has a preferred direction or not. We perform a
least-$\chi^{2}$ fit to the Union2 SnIa data

$\chi^{2}\equiv\sum^{557}_{i=1}\frac{(\mu_{th}-\mu_{obs})^{2}}{\sigma_{\mu}^{2}},$
(27)

where $\mu_{th}$ is theoretical distance modulus given by

$\mu_{th}=5\log_{10}\frac{d_{L}}{\rm Mpc}+25~{}.$ (28)

$\mu_{obs}$ and $\sigma_{\mu}$, given by the Union2 SnIa data, denote the
observed values of the distance modulus and the measurement errors,
respectively. The least-$\chi^{2}$ fit of the $\Lambda$CDM model gives
$\Omega_{m0}=0.27\pm 0.02$ and $H_{0}=70.00\pm 0.35$ $\rm km\cdot s^{-1}\cdot
Mpc^{-1}$. Before using our model to fit the Union2 SnIa data, we fix the
values of $\Omega_{m}$ and $H_{0}$ as their mean values. Such an approach is
valid for the scalar perturbed model, since it is just a perturbation for the
$\Lambda$CDM model. Then, the least-$\chi^{2}$ fit of the formula (26) gives
$A=(-2.34\pm 0.91)\times 10^{-5}$ and $(l,b)=(287^{\circ}\pm
25^{\circ},11^{\circ}\pm 22^{\circ})$. The preferred direction is plotted as
point G of Fig.1. The preferred directions given by other models are plotted
in Fig.1 for contrast. Kogut et al. Kogut1993  got $(l,b)=(276^{\circ}\pm
3^{\circ},30^{\circ}\pm 3^{\circ})$ is shown as point A, Antoniou et al.
Antoniou  got
$(l,b)=({309^{\circ}}^{+23^{\circ}}_{-3^{\circ}},{18^{\circ}}^{+11^{\circ}}_{-10^{\circ}})$
is shown as point B, Cai and Tuo Cai and Tuo2012  got
$(l,b)=({314^{\circ}}^{+20^{\circ}}_{-13^{\circ}},{28^{\circ}}^{+11^{\circ}}_{-33^{\circ}})$
is shown as point C, Kalus et al. Kalus:2013zu  got
$(l,b)=({325^{\circ}},{-19^{\circ}})$ is shown as point D, Cai et al.
Cai:2013lja  got $(l,b)=({306^{\circ}},{-13^{\circ}})$ is shown as point E,
Chang et al. Chang  got $(l,b)=(304^{\circ}\pm 43^{\circ},-27^{\circ}\pm
13^{\circ})$ is shown as point F. Within a level of significance ($1\sigma$),
it is shown in Fig.2 that our results are consistent with the one of Kogut et
al. Kogut1993 , Antoniou et al. Antoniou  and Cai et al. Cai and Tuo2012 .

Figure 1: The direction of preferred axis in galactic coordinate. The point G
denotes our result, namely, $(l,b)=(287^{\circ}\pm 25^{\circ},11^{\circ}\pm
22^{\circ})$, which is obtained by fixing the parameters $\Omega_{m}=0.27$ and
$H_{0}=70.00$ and doing the least-$\chi^{2}$ to the Union2 data for formula
(26). The results for preferred direction in other models are presented for
contrast. Point A denotes the result of Kogut et al. Kogut1993 , point B
denotes the result of Antoniou et al. Antoniou , point C denotes the result of
Cai and Tuo Cai and Tuo2012 , point D denotes the result of Kalus et al.
Kalus:2013zu , point E denotes the result of Cai et al. Cai:2013lja , point F
denotes the result of Chang et al. Chang . Figure 2: Contours figure for the
preferred direction. The contours enclose 68% and 95% confidence regions of
the scalar perturbed $\Lambda$CDM model.

The scalar perturbation not only breaks the isotropy symmetry of the universe
but also gives a peculiar velocity for the matter. By setting $\phi$ to be the
form of (24), we find from (12) that

$v\equiv\sqrt{|v_{i}v^{i}|}=a^{4}\frac{2H|A|\sin\theta}{3rH_{0}^{2}\Omega_{m0}}=\frac{2H|A|\sin\theta}{3H_{0}^{2}d_{L}\Omega_{m0}}(1+z)^{-3},$
(29)

where we have used the relation $d_{L}=(1+z)r$ to obtain the second equation.
Substituting the value of $H_{0}d_{L}$ and $H$ (given by the $\Lambda$CDM
model) into formula (29), and setting $\sin\theta=1$ (the velocity $v$ is
perpendicular to the preferred direction $(l,b)=(287^{\circ}\pm
25^{\circ},11^{\circ}\pm 22^{\circ})$), we could obtain the value of peculiar
velocity $v$ for a given redshift. At $z=0.15$, we get $v|_{z=0.15}\simeq
73\pm 28\rm km\cdot s^{-1}$. This peculiar velocity is compatible with the
result of Planck Collaboration Planck3 . it gives a upper limit $800\rm
km\cdot s^{-1}$ for peculiar velocity at $z=0.15$. It should be noticed that
the peculiar velocity $v$ grows with time. We should check that if the
peculiar velocity at the lowest redshift in the Union2 SnIa data is compatible
with the result of Planck Collaboration Planck3 . The lowest redshift in the
Union2 SnIa data is 0.015. At $z=0.015$, we get $v|_{z=0.015}\simeq 1099\pm
427\rm km\cdot s^{-1}$. Planck Collaboration Planck3  gives an upper limit on
the bulk flow for Local Group, which equals to $254\rm km\cdot s^{-1}$.
Though, our result for peculiar velocity at $z=0.015$ larger than $254\rm
km\cdot s^{-1}$, with a level of significance ($1\sigma$), it is still
compatible with the upper limit $800\rm km\cdot s^{-1}$ given by Planck
Collaboration. And our result for peculiar velocity at $z=0.015$ represents
the upper limit of peculiar velocity in the scalar perturbed $\Lambda$CDM
model.

##  IV Conclusions and remarks

We presented a scalar perturbation for the $\Lambda$CDM model that breaks the
isotropic symmetry of the universe. Setting the scalar perturbation of the
form $\phi=A\cos\theta$, we obtained a modified relation (26) between
luminosity distance and redshift. The least-$\chi^{2}$ fit to the Union2 SnIa
data showed that the universe has a preferred direction $(l,b)=(287^{\circ}\pm
25^{\circ},11^{\circ}\pm 22^{\circ})$, which is close to the results of Kogut
et al. and Antoniou et al. and Cai et al. Kogut1993 ; Antoniou ; Cai and
Tuo2012 . Also, the least-$\chi^{2}$ fit to the Union2 SnIa data showed that
the magnitude of scalar perturbation $A$ equals to $(-2.34\pm 0.91)\times
10^{-5}$. The scalar perturbation has the same magnitude with the level of CMB
anisotropy. The CMB anisotropy is a possible reason for the preferred
direction of the universe.

The peculiar velocity was obtained directly from the Einstein equation (12).
The numerical calculations showed that the peculiar $v|_{z=0.15}\simeq 73\pm
28\rm km\cdot s^{-1}$ and $v|_{z=0.015}\simeq 1099\pm 427\rm km\cdot s^{-1}$.
They are compatible with the results of Planck Collaboration Planck3 . It
should be noticed that the peculiar velocity we obtained is perpendicular to
the radial direction.

Bianchi cosmology Rosquist  has been studied for many years. It admits a set
of anisotropic metrics such as Kasner metric Misner . The three dimensional
space of Bianchi cosmology admits a set of Killing vectors $\xi_{i}^{(a)}$
which obey the following property

$\left(\frac{\partial\xi_{i}^{(c)}}{\partial
x^{k}}-\frac{\partial\xi_{k}^{(c)}}{\partial
x^{i}}\right)\xi_{(a)}^{i}\xi_{(b)}^{k}=C^{c}_{ab},$ (30)

where $C^{c}_{ab}$ is the structure constant of the symmetry group of the
space. The scalar perturbation field $\phi(\vec{x})$ completely destroys the
rotational symmetry of cosmic space. It means that no Killing vectors
corresponding to the symmetry group of three dimensional cosmic space. Thus,
there is no obvious relation between the Bianchi cosmology and our model.

###### Acknowledgements.

We would like to thank Y. G. Jiang for useful discussions. Project 11375203
and 11305181 supported by NSFC.

## References

  * (1) V. Sahni, Class. Quant. Grav. 19, 3435 (2002).
  * (2) T. Padmanabhan, Phys. Rept. 380, 235 (2003).
  * (3) E. Komatsu, et al. (WMAP Collaboration), Astrophys. J. Suppl. 192, 18 (2011).
  * (4) Planck Collaboration, arXiv:1303.5062.
  * (5) N. Suzuki, et al., Astrophys. J. 746, 85 (2012).
  * (6) L. Perivolaropoulos, arXiv:1104.0539.
  * (7) I. Antoniou and L. Perivolaropoulos, JCAP 1012, 012 (2012).
  * (8) R. Watkins, H. A. Feldman and M. J. Hudson, Mon. Not. Roy. Astron. Soc. 392, 743 (2009).
  * (9) G. Hinshaw, et al., Astrophys. J. Suppl. 180, 225 (2009).
  * (10) Planck Collaboration, arXiv:1303.5083.
  * (11) T. Koivisto and D. F. Mota, Phys. Rev. D 73, 083502 (2006).
  * (12) S. Alexander, T. Biswas, A. Notari and D. Vaid, JCAP 0909, 025 (2009).
  * (13) J. Garcia-Bellido and T. Haugboelle, JCAP 0804, 003 (2008).
  * (14) E. Akofor, et al., JHEP 0805, 092 (2008).
  * (15) T. S. Koivisto, D. F. Mota, M. Quartin and T. G. Zlosnik, Phys. Rev. D 83, 023509 (2011).
  * (16) Z. Chang, M.-H. Li, X. Li and S. Wang, Eur. Phys. J. C 73, 2459 (2013).
  * (17) S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York, 1972.
  * (18) Planck Collaboration, arXiv:1303.5090.
  * (19) S. Perlmutter, et al., Astrophys. J. 517, 565 (1999); A. G. Riess, et al., Astron. J. 116, 1009 (1998); Astron. J. 117, 707 (1999).
  * (20) R. Amanullah, et al., Astrophys. J. 716, 712 (2010).
  * (21) A. Kogut, et al., Astrophys. J. 419, 1 (1993).
  * (22) R.-G. Cai and Z.-L. Tuo, J. Cosmol. Astropart. Phys. 1202, 004 (2012).
  * (23) B. Kalus, et al., Astron. Astrophys. 553, A56 (2013).
  * (24) R. G. Cai, Y. Z. Ma, B. Tang and Z. L. Tuo, Phys. Rev. D 87, 123522 (2013).
  * (25) K. Rosquist and R. T. Jantzen, Phys. Rept. 166, 89 (1988).
  * (26) C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W. H. Freeman and Company, San Francisco, 1973.