Final state radiation from high and ultrahigh energy neutrino interactions
Ryan Plestid
1
,*
and Bei Zhou
2,3
,
†
1
Walter Burke Institute for Theoretical Physics,
California Institute of Technology
,
Pasadena, California 91125, USA
2
Theoretical Physics Department,
Fermi National Accelerator Laboratory
, Batavia, Illinois 60510, USA
3
Kavli Institute for Cosmological Physics,
University of Chicago
, Chicago, Illinois 60637, USA
(Received 19 April 2024; accepted 3 January 2025; published 5 February 2025)
Charged leptons produced by high-energy and ultrahigh-energy neutrinos have a substantial
probability of emitting prompt internal bremsstrahlung
ν
l
þ
N
→
l
þ
X
þ
γ
. This can have important
consequences for neutrino detection. We discuss observable consequences at high- and ultrahigh-energy
neutrino telescopes and the Large Hadron Collider
’
s(LHC
’
s) Forward Physics Facility. Logarithmic
enhancements can be substantial (e.g.,
∼
20%
) when either the charged lepton
’
s energy or the rest of the
cascade is measured. We comment on final state radiation
’
s impacts on measuring the inelasticity
distribution,
ν
=
̄
ν
flux ratio, throughgoing muons, and double-bang signatures for high-energy neutrino
observation. Furthermore, for ultrahigh-energy neutrino observation, we find that final state radiation
increases the overall detectable energy by as much as 20%, affects flavor measurements, and decreases
the energy of both Earth-emergent tau leptons and regenerated tau neutrinos. Many of these have
significant impacts on measuring neutrin
o fluxes and spectra. Finally, for the LHC
’
s Forward Physics
Facility, we find that final state radiation will im
pact future extractions of strange quark parton
distribution functions. Final state radiation should b
e included in future analyses at neutrino telescopes
and the Forward Physics Facility.
DOI:
10.1103/PhysRevD.111.043007
I. INTRODUCTION
High-energy (HE, i.e.,
∼
100
GeV
–
100
PeV) and
ultrahigh-energy (UHE, i.e.,
≳
100
PeV) neutrinos are
important for both neutrino astrophysics and multimessenger
astronomy
[1
–
6]
. They provide a unique window into
extreme astrophysical environments. They also probe neu-
trino interactions at high center-of-mass energies, and offer
valuable tests of the Standard Model of particle physics
[1,7,8]
. For example, measurements of neutrino-nucleon
scattering at TeV energies and above probe quantum
chromodynamics (QCD) at small
x
[9
–
12]
and offer
unique opportunities to constrain strange-quark distribu-
tion functions inside the nucleon
[13
–
17]
. Measurements
of neutrino interactions at high energies can be obtained
using naturally occurring neutrino fluxes with neutrino
telescopes
[18
–
23]
. Similar measurements can also be
performed at the Large Hadron Collider
’
s (LHC
’
s)
Forward Physics Facility (FPF), which is exposed to
neutrinos with energies as high as a few TeV
[24,25]
.
Furthermore, measurements of high- and ultrahigh-energy
neutrinos (including energy spectra
[26
–
32]
, flavor
ratios
[33
–
37]
, and arrival directionality
[38
–
40]
and
timing
[41
–
43]
) offer new probes of neutrino properties
in a largely untested energy range and offer discovery
opportunities for physics beyond the Standard Model
[2]
.
For both astrophysics and particle physics applications,
a precise description of neutrino interaction and detection
plays a central role. Statistical samples have grown, and
will grow, with more and more HE and UHE neutrino
telescopes being built or proposed (e.g., Refs.
[44
–
52]
;see
Ref.
[1]
for a comprehensive overview) and the prospect
of new data from the LHC
’
sFPF
[25]
. Consequently,
precision goals are becoming increasingly stringent. This
has motivated the study of subleading QCD corrections to
neutrino-nucleus cross sections, with quoted uncertainties
as small as
∼
1%
[53
–
55]
. Other work has focused on
subdominant interaction channels, such as
W
-boson and
trident production
[17,56
–
61]
.
In this work, we consider subleading corrections to
charged-current deep-inelastic scattering (CCDIS) stem-
ming from the emission of hard on-shell photons, i.e.,
final state radiation (FSR). This has been previously
discussed in the context of the inelasticity distribution
*
Contact author: rplestid@caltech.edu
†
Contact author: beizhou@fnal.gov
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
’
s title, journal citation,
and DOI. Funded by SCOAP
3
.
PHYSICAL REVIEW D
111,
043007 (2025)
2470-0010
=
2025
=
111(4)
=
043007(15)
043007-1
Published by the American Physical Society
for accelerator
[62]
and astrophysical
[63]
neutrinos.
We revisit FSR
’
s impact on the inelasticity distribution,
and study many other phe
nomenological impacts for HE
and UHE neutrino observation for the first time. As we
discuss below, for neutrino energies above 100 GeV,
quantum electrodynamic (QED)
radiative corrections can
be sizeable when observables are sensitive to FSR
[63]
,
i.e., when the charged lepton (
l
) emits an energetic
photon (
γ
) on top of the CCDIS,
ν
l
þ
N
→
l
þ
X
þ
γ
.
Here, we label the nucleon by
N
and final state hadrons
by
X
, and the relevant cross section is inclusive with
respect to final state hadrons.
In Fig.
1
, we show a schematic of
ν
μ
scattering
(CCDIS
þ
FSR) and detection as an example. (FSR also
significantly impacts
ν
τ
scattering and detection, as dis-
cussed below.) High-energy muons travel macroscopic
distances in matter (e.g., tens of km), whereas photons
produce electromagnetic cascades over a distance scale set
by the radiation length,
X
0
∼
10
cm, similar to the had-
ronic cascades. In neutrino telescopes, muons are
“
stripped
bare
”
of any prompt radiation, because the photon is
absorbed in a local cascade, whereas the muon itself leaves
a long track. In contrast to jetlike observables, the
experimentally observed muon energy does not contain
the prompt photon.
For many detection channels of HE and UHE neutrinos,
the charged lepton and the primary (hadronic) cascade are
separable. In these scenarios, relevant observables are not
inclusive with respect to FSR, which is enhanced by
Sudakov double logarithms. This is especially true for
observables with large statistical samples that are binned as
a function of energy, e.g., inelasticity distributions
[64]
.
Therefore, radiative corrections must be carefully consid-
ered when discussing noninclusive observables for
sub-TeV to EeV neutrino detection. For a quick estimate,
consider the Sudakov factor
1
(see, e.g., Ref.
[65]
)
F
S
ð
s; E
min
Þ
∼
exp
−
α
2
π
log
s
m
2
l
log
E
2
l
E
2
min
;
ð
1
Þ
which gives the probability to
not
radiate any photons above
E
min
in a collision with center-of-mass energy
ffiffiffi
s
p
and final
state charged-lepton energy
E
l
. Taking
l
as the muon (
μ
),
E
min
≃
1
10
E
μ
,and
s
≃
2
E
ν
m
N
(
m
N
is the nucleon mass) with
E
ν
¼
10
TeV, we find
F
S
∼
0
.
9
. This implies that roughly
10% of all events will contain some prompt real and
energetic photon radiation. Since the photon is absorbed
into the cascade rather than being associated with the track,
this distorts the energy distributions of the charged lepton
and the rest of the cascade. This can be compared to the
probability of a high-energy muon undergoing hard brems-
strahlung within one radiation length, which is given
parametrically by
ð
m
e
=m
μ
Þ
2
∼
2
.
4
×
10
−
5
[66]
.
In what follows, we will discuss QED radiative cor-
rections in their simplest form (cf. the discussion in
Refs.
[62,63]
), focusing on FSR. We make use of splitting
functions, working at fixed order, and capturing the
doubly logarithmically enhanced corrections. These are
the most important radiative corrections, and are large
enough to impact current HE and UHE neutrino experi-
ments. A full
O
ð
α
Þ
account of radiative corrections
demands a one-loop electroweak calculation, in addition
to the treatment of real final state photons
[67
–
75]
.This
level of accuracy (
∼
1%
) is, however, extraneous for
neutrino telescopes given their current statistical samples,
and detector resolutions. Generalizations beyond our
fixed-order treatment, i.e., to include leading-log resum-
mation, are straightforward, but unnecessary for applica-
tions where a
∼
10%
accuracy is sufficient. Since our focus
is on QED rather than QCD, we will always assume that
the QCD-based neutrino CCDIS cross section is given.
We make use of toy parametrizations for illustration, and
realistic QCD calculations on the isoscalar target (i.e.,
averaging over proton and neutron)
[55]
when discussing
the numerical impact on observables.
The rest of the paper is organized as follows: in Sec.
II
,
we review relevant facts about neutrino interactions and
QED splitting functions. In Sec.
III
, we discuss the impact
of FSR on the energies of the neutrino interaction final
states. We also discuss the impacts, including inelasticity
distributions, distinguishing neutrinos and antineutrinos,
throughgoing muons, and double-bang signatures, in HE
neutrino telescopes. In Sec.
IV
, we discuss UHE neutrino
detection and observables that are impacted by FSR,
including flavor measurements, Earth-emergent tau lepton,
FIG. 1. Effect of final state radiation on HE neutrino scattering
(CCDIS
þ
FSR) and detection, taking
ν
μ
as an example. A
ν
μ
produces a high-energy muon, which emits a photon during
CCDIS with a nucleon (bottom left). The photon is absorbed in
the cascade, while the muon is observed as a track. In general, any
measurements that can separate the charged lepton and cascade
are subject to logarithmically enhanced QED radiative correc-
tions. Tau neutrino detection is also significantly impacted by
FSR, as discussed below.
1
There is a factor of two difference in the argument of the
exponential relative to the classic Sudakov factor since there is
one lepton, rather than two, in the final state.
RYAN PLESTID and BEI ZHOU
PHYS. REV. D
111,
043007 (2025)
043007-2
and
ν
τ
regeneration. In Sec.
V
, we discuss FSR
’
s impacts on
neutrino flux and spectrum measurements. In Sec.
VI
,we
comment on similar phenomenology at the FPF. Finally, in
Sec.
VII
, we summarize our findings.
II. CHARGED-CURRENT NEUTRINO
SCATTERING AND FINAL STATE RADIATION
In what follows, we will have in mind charged-current
(CC) neutrino (or antineutrino) scattering on nucleons.
We therefore consider the interaction
ν
l
þ
N
→
X
þð
l
−
þ
n
γ
Þ
;
ð
2
Þ
where
N
is a nucleon,
X
is any hadronic final state, and we
have allowed for
n
final state photons emitted from the
charged lepton. (Note that the photon emission from quarks
is not important because the detectors cannot distinguish
the hadronic shower and electromagnetic shower. Photon
emission from the intermediate
W
boson is also negligible
because it is suppressed by the heavy
W
mass.) In this
paper, we consider
n
¼
0
and
n
¼
1
. Convenient kinematic
variables are Bjorken
x
and the inelasticity
y
,
x
¼
Q
2
2
p
N
·
Q
¼
ð
p
X
−
p
N
Þ
2
2
p
N
·
ð
p
X
−
p
N
Þ
;
ð
3
Þ
y
¼
E
X
E
ν
¼
p
N
·
E
X
p
N
·
p
ν
:
ð
4
Þ
Notice that both variables can be defined independent
of the charged lepton or photon kinematics. For tree-level
scattering, suppressing terms of order
M=E
ν
and
m
l
=E
ν
,
the cross section can be written as
[76]
d
2
σ
ð
0
Þ
ν
;
̄
ν
d
x
d
y
¼
G
F
ME
ν
π
ð
1
þ
Q
2
=M
2
W
Þ
2
×
½
y
2
F
1
þð
1
−
y
Þ
F
2
xy
ð
1
−
y=
2
Þ
F
3
;
ð
5
Þ
where
F
i
¼
F
i
ð
x; Q
2
Þ
are structure functions and the
“
þ
”
is for
ν
and the
“
−
”
is for
̄
ν
. Predictions for the DIS cross
sections can be found in, e.g., Refs.
[10,53
–
55,77
–
79]
.In
what follows, we only need d
σ
ð
0
Þ
=
d
y
.
Let us next consider
n
¼
1
photon in the final state. At
leading-logarithmic accuracy, FSR factorizes leg-by-leg.
Any radiation from the hadronic parts of the diagram will
be captured in the cascade. Since the cascade is inclusive
with respect to hadronic
þ
electromagnetic energy deposi-
tion, the Kinoshita-Lee-Nauenberg (KLN) theorem
[80,81]
guarantees that hadronic FSR does not generate any
logarithmically enhanced QED radiative corrections. The
only
“
large
”
QED effects are, therefore, those involving
FSR off the charged-lepton leg, and these can be computed
at leading-log accuracy using splitting functions.
Whenever the probability of emitting radiation is sub-
stantially less than one, we may obtain accurate estimates
without a full resummation of the leading logarithms. For
this purpose, a fixed-order calculation is sufficient. The
relevant distribution function, or
l
→
l
γ
,isgivenby
[65,82]
P
l
→
l
γ
ð
z
Þ¼
α
2
π
log
s
m
2
l
ð
1
þ
z
2
Þ
½
1
−
z
þ
þ
3
2
δ
ð
1
−
z
Þ
;
ð
6
Þ
where (
1
−
z
) is the fraction of the charged lepton
’
s
momentum carried away by the photon. The
“
1
=
ð
1
−
z
Þ
”
behavior
2
of the splitting function is characteristic of
bremsstrahlung and encodes the soft photon singularity.
The logarithmic enhancement in Eq.
(6)
stems from
integrating over
R
d
p
2
T
=p
2
T
from
p
T
∼
m
l
to
p
T
∼
s
.A
detailed discussion of an
“
improved
”
approximation
scheme can be found in Ref.
[62]
[replacing
s
by
s
ð
1
−
y
þ
xy
Þ
, with
x
being the Feynman
x
and
y
the
inelasticity]. However, at leading-log accuracy, this
improvement does not affect the results. For simplicity,
we use Eq.
(6)
as written, but refer the interested reader to
Sec. Vof Ref.
[62]
for more details. Note that at leading-log
accuracy, log
ð
s
ð
1
−
y
þ
xy
Þ
m
2
Þ
≃
log
ð
s
m
2
Þ
for high- and ultrahigh-
energy neutrinos.
Let us write d
σ
¼
d
σ
ð
0
Þ
þ
d
σ
ð
1
Þ
þ
, where d
σ
ð
1
Þ
includes all
O
ð
α
Þ
corrections to the differential cross
section. The correction to the cross section contains a piece
due to internal bremsstrahlung, and a virtual correction.
These pieces conspire to ensure that inclusive observables
(e.g., at fixed hadronic energy transfer) contain no large
kinematic logarithms. If we consider a leptonic variable,
for example,
E
l
, then the
O
ð
α
Þ
logarithmically enhanced
corrections to the cross section are given by
d
σ
ð
1
Þ
d
E
l
¼
α
2
π
Z
d
y
Z
d
z
d
σ
ð
0
Þ
d
y
δ
ð
E
l
−
ð
1
−
y
Þ
zE
ν
Þ
× log
s
m
2
l
1
þ
z
2
½
1
−
z
þ
þ
3
2
δ
ð
1
−
z
Þ
:
ð
7
Þ
This quantity is infrared (IR)-safe (but
not collinear
safe).
When considering distributions binned as a function of
E
l
,
then the bin width in the charged-lepton energy serves as an
effective IR-cutoff scale in estimating the size of double-
logarithmic enhancements. In our numerical estimates, we
take
α
¼
α
ð
M
Z
Þ
with
α
−
1
ð
M
Z
Þ¼
129
[83]
.
Before proceeding, let us note that more sophisticated
techniques have been developed for the resummation of
final state photons. Standard parton shower tools can
automatically include final state radiation provided that
the appropriate settings are
“
turned on.
”
In what follows,
we will use Eq.
(7)
for all of our numerical illustrations,
and fixed-order treatments are likely sufficient for most
2
We use
1
=
ð
1
−
z
Þ
and
1
=
½
1
−
z
þ
interchangeably for this
specific colloquial remark, but implement the plus distribution in
our calculations.
FINAL STATE RADIATION FROM HIGH AND ULTRAHIGH
...
PHYS. REV. D
111,
043007 (2025)
043007-3
(if not all) applications at neutrino telescopes. If
∼
1%
accuracy is required, then a full QED calculation is
warranted (see, e.g.,
[74,75]
for a discussion in the context
of GeV neutrino scattering)
—
many
O
ð
α
Þ
effects (without
logarithmic enhancement) are not captured by parton
showers. Nevertheless, if it proves more convenient in
an experimental analysis, Eq.
(7)
can be replaced with an
equivalent parton shower description.
For illustration
’
s sake, let us make use of IceCube
’
s
simplified parametrization of d
σ
ð
0
Þ
=
d
y
[64]
,
1
σ
ð
0
Þ
d
σ
ð
0
Þ
d
y
¼
C
ð
ε
0
;
λ
0
Þð
1
þ
ε
ð
1
−
y
Þ
2
Þ
y
λ
0
−
1
;
ð
8
Þ
where
ε
0
is in practice determined by specifying the mean
inelasticity
h
y
i
0
and using the formula
ε
0
¼
−
ðð
λ
0
þ
2
Þð
λ
0
þ
3
ÞÞðð
λ
þ
1
Þh
y
i
0
−
λ
0
Þ
2
ðð
λ
0
þ
3
Þh
y
i
0
−
λ
0
Þ
:
ð
9
Þ
The constant
C
is fixed by normalization.
In Fig.
2
, we show the effect of FSR computed using
Eq.
(7)
for input parameters of
h
y
i
0
¼
0
.
35
and
λ
0
¼
1
.
00
,
and these input parameters correspond to realistic choices
for
E
ν
≃
100
TeV
[64]
. Having discussed the general
formalism we use (i.e., splitting functions at fixed order
in
α
) and identified corrections that are sizeable, we now
turn to applications at neutrino telescopes.
III. HIGH-ENERGY NEUTRINO OBSERVATION
In this section, we discuss how FSR impacts the
observation of HE neutrinos (
100
GeV
≲
E
ν
≲
100
PeV).
High-energy neutrino telescopes detect neutrinos via
two basic topologies:
“
tracks
”
and
“
cascades
”
(cf. Fig.
1
).
Tracks are formed by muons, which travel macroscopic
distances. Cascades include hadronic and electromagnetic
energy deposition in a shower that is localized over a few
hadronic interaction lengths. Events produced by HE
neutrinos can originate from either inside or outside (i.e.,
throughgoing muons) the detector. When FSR is emitted, it
transfers energy from the track part of the topology to the
cascade. More generally, for instance, in the case of the
“
double-bang
”
signature that is used to detect tau neutrinos,
FSR distorts the ratio of leptonic and hadronic energy
estimators.
Therefore, in what follows, we distinguish two parts of
energy: the
charged-lepton energy
,
E
l
, and the
shower
energy
,
E
shower
, which combines the energies of the FSR
photon and hadronic shower.
3
The sum,
E
shower
þ
E
l
,
can be used as an estimator for the neutrino energy
E
ν
.
Whenever an observable measures
E
shower
or
E
l
separately,
however, QED corrections can be enhanced by large FSR
kinematic logarithms and be sizeable, as shown below.
In Fig.
3
, we show the average relative shift in
the charged-lepton energy (
h
Δ
E
l
i
=E
l
) and the shower
energy (
h
Δ
E
shower
i
=E
shower
) due to FSR. By definition,
h
Δ
E
l
iþh
Δ
E
shower
i¼
0
. The calculation is based on real-
istic neutrino CCDIS d
σ
ð
0
Þ
=
d
y
distributions based on QCD
and the isoscalar target
[55,84]
. For leptons, the shifts can
be as large as
∼
5%
, whereas for the shower energy the
0.0
0.2
0.4
0.6
0.8
1.0
–0.5
0.0
0.5
1.0
1.5
2.0
FIG. 2. Illustration of the effect of FSR on the inelasticity
distribution using Eq.
(8)
for input parameters of
h
y
i
0
¼
0
.
35
,
λ
0
¼
1
.
00
, and
E
ν
¼
100
TeV. In the leading-log approximation,
the effect of FSR is to migrate strength from smaller
y
to larger
y
without influencing the normalization of the distribution.
FIG. 3. Average relative shift in energy due to FSR for the final
state charged-lepton energy (curves below zero) and shower
energy, i.e.,
“
the rest
”
E
ν
−
E
l
(curves above zero), from neutrino
CCDIS. For example, taking a HE muon neutrino in IceCube, the
lepton energy corresponds to the track, and the rest corresponds
to the shower. Curves are plotted as a function of the parent
neutrino energy (
E
ν
).
3
The hadronic shower also contains a large amount of
electromagnetic activity due to, e.g.,
π
0
→
γγ
.
RYAN PLESTID and BEI ZHOU
PHYS. REV. D
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043007-4
estimated energy can shift by as much as
∼
25%
. This is a
consequence of the tree-level distribution being asymmetric
between the leptonic and hadronic energy. Moreover, it is
important to note that the shifts in the shower energies will
be further enhanced by
≃
10
–
20%
in the realistic exper-
imental settings, because electromagnetic showers have
more light yields than hadronic showers
[85]
. Therefore,
Fig.
3
demonstrates that QED FSR can substantially distort
experimental observables at a level that is relevant for
existing neutrino telescopes. Notice that the relative energy
shifts increase with energy, which reflects the increased
probability of FSR for high- vs low-energy leptons.
Convenient scaling relations for charged-lepton energies
are given as follows:
h
Δ
E
μ
i
E
μ
≃
4
.
6%
þ
0
.
0075
×log
10
E
ν
10
10
GeV
;
ð
10
Þ
h
Δ
E
τ
i
E
τ
≃
3
.
7%
þ
0
.
0075
×log
10
E
ν
10
10
GeV
:
ð
11
Þ
The effects of FSR are enhanced for muons relative to tau
leptons due to the log
ð
s=m
2
l
Þ
collinear enhancement.
Finally, notice that there is a difference between
ν
and
ν
;
which we will comment on in Sec.
III A 1
.
A. Inelasticity measurements and neutrino telescopes
As we have discussed above, a simple exclusive observ-
able that is sensitive to FSR is the differential cross section
d
σ
=
d
E
μ
in muon-neutrino CC scattering. At leading order
in
α
(i.e., without FSR), this observable is trivially related to
the inelasticity distribution d
σ
ð
0
Þ
=
d
y
. As is immediately
evident from Eq.
(7)
, this simple correspondence is violated
once FSR is included. The experimental definition used as
an estimate of the inelasticity is
y
exp
≡
E
shower
E
track
þ
E
shower
¼
y
QCD
þ
E
γ
E
ν
;
ð
12
Þ
where
y
QCD
¼
E
X
=E
ν
, and
E
γ
is the energy deposited by
FSR from the lepton leg.
In Fig.
4
, we plot the absolute (
Δ
y
avg
, left panel) and
relative (
Δ
y
avg
=y
avg
, right panel) shifts in the average
experimental inelasticity
h
y
expt
i
, defined as
Δ
y
avg
≡
h
y
exp
i
−
h
y
QCD
i¼h
E
γ
i
=E
ν
.Thisisausefulquantitativemeasureof
FSR. The trends of the curves are related to that of Fig.
3
.
Since
h
E
γ
i
>
0
,
Δ
y
avg
is always positive. The increase in
Δ
y
avg
with
E
ν
occurs both because of increasing logarithmic
enhancements, log
ð
s=m
2
l
Þ
, and because
h
y
QCD
i
decreases as
E
ν
increases. The relative shift due to FSR,
Δ
y
avg
=y
avg
,can
be as large as 25%. Again, it is important to note that the
shifts in
y
avg
will be further enhanced by
≃
10%
–
20%
in
the realistic experimental settings, because electromagnetic
FIG. 4. Shifts in the experimentally measured average inelasticity,
y
avg
¼
1
−
h
E
μ
i
=E
ν
, from neutrino CCDIS due to the FSR photon
taking part of the energy from the charged lepton to the hadronic side, as a function of the parent neutrino energy (
E
ν
).
Left panel
: the
shift,
Δ
y
avg
≡
h
E
γ
i
=E
ν
.
Right panel
: the relative shift,
Δ
y
avg
=y
avg
. We do not include experimental efficiencies as a function of
y
in the
estimate of
y
avg
[64]
. We also do not account for the difference in light yield between hadronic and electromagnetic showers, which will
further enhance the shifts by
≃
10%
–
20%
[85]
.
FINAL STATE RADIATION FROM HIGH AND ULTRAHIGH
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PHYS. REV. D
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showers have more light yields than hadronic showers
[85]
.
The difference in the light yield also affects the inference of
the parent neutrino energy from the measured total energy,
although it is minor.
Both Figs.
3
and
4
demonstrate that the impact of FSR is
substantial and influences observables at a level that
is relevant for ongoing and near-term experiments. This
is important because the inelasticity distribution has many
useful applications. For instance, the differing dependence
of Eq.
(5)
on
y
for
ν
vs
̄
ν
allows one to statistically infer the
flux ratio
Φ
ν
=
Φ
̄
ν
, which has interesting applications both
for astrophysics and for inferring neutrino mixing param-
eters. In addition to application to astrophysics and neutrino
physics, the inelasticity distribution also provides interest-
ing constraints on QCD distributions, such as the charm
quark fraction of nucleons. In what follows (Secs.
III A 1
and
III A 2
), we discuss the impact of FSR on these
applications, in turn.
1. Measuring
ν
=
̄
ν
flux ratio
The ratio of neutrinos to antineutrinos is strongly
correlated with the atmospheric muon charge ratio. The
excess of
μ
þ
vs
μ
−
stems from the excess of protons in
cosmic rays and the steeply falling cosmic ray spectrum
[86,87]
. Since neutrinos and muons both come from weak
decays of hadrons (primarily
π
and
K
), measurements
of the
ν
=
̄
ν
flux ratio provide complementary information
on cosmic rays and an interesting cross-check on the
μ
þ
to
μ
−
ratio. Furthermore, measuring the astrophysical
ν
=
̄
ν
flux ratio is an important and powerful tool for testing the
source properties and neutrino properties.
The inelasticity distribution is a useful discriminator of
ν
vs
̄
ν
due to the sign difference multiplying the structure
function
F
3
in Eq.
(5)
. At the quark level, this is because
neutrinos and antineutrinos are sensitive to different quark
flavors with different densities, especially at large Bjorken
x
values, where valence quarks dominate, leading to different
cross sections (e.g., Fig. 1 of Ref.
[59]
) and inelasticity
distributions (e.g., Fig. 8 of Ref.
[64]
). A consequence of
these differing inelasticity distributions is that FSR induces
different shifts in
y
avg
.
We illustrate this point in Fig.
4
, where the shift in
y
avg
is
due to FSR. The difference between
ν
and
̄
ν
decreases as
neutrino energy increases as smaller Bjorken
x
is favored,
where sea quarks are more dominant, and almost vanishes
for
E
ν
≳
10
6
GeV. Since atmospheric neutrino fluxes are
larger than astrophysical ones, only the atmospheric
ν
=
̄
ν
flux ratio has been measured
[64]
, but the astrophysical
ν
=
̄
ν
flux ratio could potentially be measured in the future.
In Fig.
5
, we illustrate how IceCube collaboration
measured the
ν
=
̄
ν
flux ratio using the experimental inelas-
ticity defined above
[64]
and how FSR would affect the
measurement. For each starting muon event, its inelasticity
and total energy (
E
vis
, close to the parent neutrino energy)
are measured. The 2650 events collected over five years are
distributed across five different
E
vis
bins (Fig.
5
only shows
the lowest-energy bin), and within each
E
vis
bin, the events
are further distributed into ten
y
exp
bins. Then, the
ν
=
̄
ν
flux
ratio is measured by fitting the data with theoretical
predictions of
ν
and
̄
ν
components, including cross sections,
d
σ
=
d
y
QCD
, and fluxes. The histograms in Fig.
5
are
generated purely from theory, including FSR, with no
detector effects included, and the data are simulated based
on the same statistics of Ref.
[64]
. In reality, statistics in the
bins with the smallest and largest
y
exp
are suppressed due to
detector effects (see Fig. 7 of Ref.
[64]
). However, our Fig.
5
shows that the largest distortion of FSR on extracting the
ν
=
̄
ν
flux ratio also comes from the bins with the smallest
and largest
y
exp
(see also Fig.
2
). One could, in principle,
perform a Poissonian likelihood fit to the data with and
without FSR, and we find it gives unrealistic estimates.
Therefore, it is crucial to include the proprietary exper-
imental information, such as efficiencies, energy smearing,
and missing energy, to get a realistic estimate of FSR
’
s
impact on extracting the
ν
=
̄
ν
flux ratio.
Here, we attempt a simplified fit (but less sensitive to the
experimental details) of the
ν
=
̄
ν
flux ratio using the mean
inelasticity measurement plotted in Fig. 8 of Ref.
[64]
.For
the theoretical input, we take the CSMS
y
avg
ð
E
ν
Þ
curves
from the same figure and multiply them by the factors given
FIG. 5. Illustration of how FSR can impact determinations of
the
ν
=
̄
ν
flux ratio. We have generated pseudodata, based on the
top solid histogram, that mimic the IceCube Collaboration
’
s
measurements of the
ν
=
̄
ν
flux ratio using inelasticity distributions
of starting muon events
[64]
. The histograms are produced
assuming the same statistics as in Ref.
[64]
. The dashed histo-
grams are derived from QCD-based neutrino-isoscalar CCDIS
inelasticity distributions
[55,84]
, which are corrected by FSR
[Eq.
(7)
] to get the solid histograms.
RYAN PLESTID and BEI ZHOU
PHYS. REV. D
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043007 (2025)
043007-6
in Fig.
4
to get the FSR corrected curves. Next, we sum the
ν
and
̄
ν
curves weighted by the corresponding cross sections
[53]
and atmospheric neutrino fluxes,
4
with the
ν
=
̄
ν
flux
ratio as a free-floating parameter. We then extract the
ν
=
̄
ν
flux ratio by fitting the theory, for the cases without FSR and
with FSR, to the data. The extracted value of the
ν
=
̄
ν
flux
ratio is then divided by the input value of the
̄
ν
=
ν
ratio
(
“
prediction
”
), i.e.,
R
fit
¼ð
ν
=
̄
ν
Þ
fit
=
ð
ν
=
̄
ν
Þ
prediction
. We find
that FSR shifts the inferred
ν
=
̄
ν
flux ratio,
R
fit
, by about 5%
relative to the case of not including FSR. This is to be
compared with
0
.
77
þ
0
.
44
−
0
.
25
from the previous IceCube meas-
urement
[64]
. We note again that it is necessary to include
the experimental effects to get a more rigorous estimate
and direct comparison with the previous measurement.
The experimental uncertainty will decrease quickly thanks
to the increasing data and especially future telescopes
(e.g., IceCube-Gen2
[46]
) that probe higher energies, where
FSR
’
s effect is more prominent.
2. Applications to fundamental physics
The
ν
=
̄
ν
flux ratio is also interesting from the perspec-
tive of both neutrino oscillation and hadronic physics.
Inelasticity is a useful tool for constraining the mass
hierarchy, the atmospheric mixing angle
θ
23
, and the
CP
-violating phase
δ
CP
[90,91]
. It also gives valuable
information on the strange quark parton-distribution func-
tions (PDFs) of nucleons
[64]
.
Reference
[90]
concludes that a complete
ν
=
̄
ν
separation
would enhance sensitivity to
CP
-violation and the mass
hierarchy by a factor of 2
–
3; however, detector resolution
effects limit the improvement in practice to
∼
1
.
25
, i.e.,
25%. Reference
[91]
reaches similar conclusions, estimat-
ing the ultimate impact of inelasticity separation as a
∼
40%
improvement in the sensitivity to the mass hierarchy. This
issue has been recently revisited in Ref.
[92]
, where the
authors reach similar conclusions. As we discussed above,
the asymmetric nature of d
σ
=
d
y
means that the
ν
=
̄
ν
flux
ratio is unusually sensitive to radiative corrections, and so
FSR may affect extractions of neutrino oscillation param-
eters from atmospheric neutrino data.
Another application of inelasticity to fundamental phys-
ics is statistical charm tagging. Strange sea quarks give rise
to sizeable charm production rates, since these are propor-
tional to the Cabibbo
–
Kobayashi
–
Maskawa matrix ele-
ments
j
V
cs
j
2
≫
j
V
uc
j
2
. Cutting on the inelasticity can
supply
“
charm-rich
”
samples. For example, IceCube
’
s
analysis estimates that more than
1
=
3
of all events with
y
expt
>
0
.
8
contain charm mesons
[64]
. The effect of FSR is
to shuffle energy from the leptonic system into the hadronic
system, thereby increasing
y
for all CCDIS events.
Therefore, when attempting to statistically identify charm
production rates, it is important to account for increased
valence quark contributions that
“
leak
”
into the signal
window due to FSR. A related discussion of charm
production in the context of more granular detectors at
the LHC
’
s FPF is presented further below.
B. Throughgoing muons and parent neutrino energy
In the above discussion, we have largely focused on
events in which shower and track energies are separately
observable. This is a good description of so-called
“
starting
”
events, but most HE neutrino events originate outside the
active volume of the detector. In this case, any FSR would
be absorbed outside the detector, along with the hadronic
shower, and be missing from the energy budget. The effect
of QED FSR is, therefore, to systematically bias muon
tracks to lower energies.
Throughgoing muon tracks are associated with muons
produced externally to the detector, but pass through the
active volume, leaving a track of Cherenkov radiation. Such
events are important for high-energy neutrinos since the
fluxes at these energies are much smaller, and the effective
volume over which a throughgoing muon can be produced
is an order of magnitude larger than the active volume
of the detector. Including FSR will lower the energy of
throughgoing muons by as much as
∼
5%
, and should be
included when estimating the parent neutrino energy.
C. Impact on the double-bang signature from
ν
τ
In addition to the muon track events discussed above,
another measurement impacted by FSR is the
“
double-
bang
”
signature. This is induced by CCDIS of
ν
τ
=
̄
ν
τ
at
E
ν
≳
10
5
TeV, with the first bang/shower formed by the
hadronic cascade and the second (separated) bang/shower
formed by the tau lepton decay
[93,94]
.
When FSR happens on
ν
τ
=
̄
ν
τ
CCDIS, the photon takes
energy from the tau lepton (second bang) to the hadronic
cascade (first bang) (Fig.
3
). The distortion in the energy
balance between the two bangs will distort the inference of
the parent
ν
τ
=
̄
ν
τ
energy. Moreover, a reduction of the tau
lepton energy also decreases the separation between the
two bangs, both spatially and temporally, which would
make it harder to identify the double-bang signature.
Before discussing UHE detection in the next section,
we briefly comment on the detection of
ν
e
=
̄
ν
e
CC events.
Incident
ν
e
and
̄
ν
e
produce high-energy
e
∓
. Unlike muons,
electrons travel relatively short distances (on the order of
the radiation length
X
0
∼
30
cm) and are contained in the
same cascade as the hadronic shower. Since photons initiate
electromagnetic showers over the same length scale, both
the electron
and
the photon are contained within a single
cascade. These events are, therefore, inclusive with respect
to FSR and do not have any large kinematic logarithms, as
is guaranteed by the KLN theorem. For the same reasons,
FSR from
ν
e
=
̄
ν
e
CC does not affect the balance between
hadronic and lepton showers
[95]
.
4
The atmospheric neutrino fluxes are calculated using the
Hillas-Gaisser H3a cosmic ray model
[88]
and the
Sybill2.3c
hadronic interaction model
[89]
.
FINAL STATE RADIATION FROM HIGH AND ULTRAHIGH
...
PHYS. REV. D
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043007-7
IV. ULTRAHIGH-ENERGY NEUTRINO
OBSERVATIONS
In this section, we discuss the impact of FSR on UHE
neutrino observation. Most of the discussions are based on
our results in Fig.
3
and Eqs.
(10)
and
(11)
.
There are two major observational strategies for the
detection of UHE neutrinos
[1]
. The first approach involves
measurements of coherent Askaryan radiation (nanosecond
radio flashes) emitted by the electromagnetic showers
generated from UHE neutrino interactions in dense media,
such as ice
[1,96]
. Experiments pursuing this detection
strategy include ANITA
[97]
, ARIANNA
[98]
,PUEO
[48]
,
the IceCube-Gen2 radio array
[46]
, etc. The second strategy
is to detect Earth-emergent charged (mainly tau) leptons
from UHE neutrino interactions within the Earth or other
material outside the actively instrumented detector volume.
Experiments pursuing this approach are PUEO
[48]
,
POEMMA
[50]
, TAMBO
[51]
, GRAND
[45]
, etc. For a
comprehensive overview of UHE neutrino experiments,
see Ref.
[1]
.
A. Askaryan radio detectors
1. Overall detectable energy
For the in-ice Askaryan radio detectors, FSR increases
the overall detectable energy of the detector, which also
effectively lowers the detection energy threshold. For
ν
τ
=
̄
ν
τ
CCDIS, the energy loss of the tau lepton is not as strong as
the muon or electron due to its heavy mass, and it could
deposit only a small fraction or even none of its energy to
the detector
[99]
. Therefore, FSR could increase the overall
detectable energy of
ν
τ
=
̄
ν
τ
CCDIS by up to 20% (Fig.
3
), as
the photon deposits nearly all of its energy in the detector.
For
ν
μ
=
̄
ν
μ
CCDIS, the enhancement from FSR is less, as the
muon energy loss is stronger than
τ
, and a detector-level
simulation is needed to quantify the effect. For
ν
e
=
̄
ν
e
CCDIS, there is no enhancement from FSR, as the electrons
also deposit nearly all the energy in the detector.
2. Flavor measurement
Recent work has explored the possibility of using a
detailed substructure of Askaryan radiation to separate
charged lepton and hadronic showers from one another
[100,101]
. The new method proposed in Ref.
[101]
involves measuring the combined
ν
μ
&
̄
ν
μ
þ
ν
τ
&
̄
ν
τ
flux
by identifying events with at least one displaced shower
(generated by tau leptons or muons while propagating in
ice) relative to the primary shower from the neutrino CC
interaction vertex. A complementary strategy to identify
ν
e
&
̄
ν
e
-induced events was also explored, wherein one
takes advantage of the elongated shower produced by an
electron from the
ν
e
CC interaction due to the Landau-
Pomeranchuk-Migdal effect. The observable signal would
then involve several slightly displaced subshowers.
Collectively, these features elongate the particle profile
of the shower and its radio emission, compared to the more
compact hadronic showers generated in other types of
interactions (e.g., neutral current events).
Both the
ν
μ
&
̄
ν
μ
þ
ν
τ
&
̄
ν
τ
and
ν
e
&
̄
ν
e
detection chan-
nels are affected by FSR. For muon and tau neutrinos, FSR
reduces the detectability of displaced showers by reducing
the energy of the charged lepton (see Fig. 3 of Ref.
[101]
). As
a concrete example, the detection efficiency curves in Fig. 3
of Ref.
[101]
would shift to the right by
≃
3%
–
5%
, depending
on the neutrino energy and flavor. For the
ν
e
&
̄
ν
e
channel,
FSR does not impact the signal because both electrons and
photons initiate electromagnetic showers. However, FSR
increases the rate of the background events originating from
the CCinteractionsof
ν
μ
=
̄
ν
μ
=
ν
τ
=
̄
ν
τ
withthesubshower (from
muon/tau lepton) very close to the primary shower, thus
mimicking the signal. Without considering FSR, Ref.
[101]
concludes that the background rate is negligible. With the
inclusion of FSR, the photon would create a detectable
subshower by itself, leading to a higher background rate.
Moreover, the method proposed in Ref.
[101]
relies on
“...
subtle features in the waveform shape
...”
that are classified
with a deep neural network. Elongated showers induced by
FSR photons may distort these waveforms and introduce an
uncontrolled systematic uncertainty. The relevant infrared
cutoff is the energy at which a photon would induce a
Landau
–
Pomeranchuk
–
Migdal (LPM) elongated shower.
We take this threshold as
E
LPM
¼
3
×
10
16
eV based on
Refs.
[66,101]
. Using Eq.
(1)
,wefinda
∼
15%
probability of
FSR per flavor for a
3
×
10
17
eV neutrino, and the effect
grows proportional to log
ð
E
ν
=E
LPM
Þ
. Assuming comparable
fluxes of
ν
τ
∶
ν
μ
∶
ν
e
then gives a
∼
30%
background from
ν
μ
=
̄
ν
μ
=
ν
τ
=
̄
ν
τ
interactions. Finally, it would also be useful to
include photonuclear interaction, which is larger than pair
production above
10
20
eV, and direct pair production and
electronuclear interaction, which are larger than bremsstrah-
lung above
10
20
eV
[102]
.
B. Air shower detectors
For the detection strategy involving Earth-emergent
leptons, the impacts of FSR are straightforward to estimate.
As the FSR photon and the hadronic shower are both
absorbed in the Earth or mountain where the neutrino
interacts, the detected charged-lepton energy will decrease
by the amount given in Eqs.
(10)
and
(11)
. This process is
analogous to the throughgoing muon track in HE neutrino
telescopes. Not including the FSR then leads to an under-
estimation of the parent neutrino energy by
∼
5%
. Although
this is smaller than the energy resolution of upcoming UHE
neutrino detectors (e.g., Refs.
[45,48,50,51,103]
), this is a
guaranteed correction from the Standard Model, so it
should be included in energy estimators sooner or later.
Finally, FSR also affects the
ν
τ
regeneration process
[104
–
106]
, in a manner similar to the Earth-emergent
RYAN PLESTID and BEI ZHOU
PHYS. REV. D
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043007 (2025)
043007-8
leptons discussed above. A new feature for
ν
τ
regeneration
is the possibility of multiple neutrino interactions, each of
which may include FSR.
5
Taking FSR into account
decreases
the regenerated
ν
τ
flux by lowering the tau
energy, which simultaneously
increases
the flux since
lower-energy taus have a longer absorption length within
the Earth. It would be interesting to incorporate FSR into
simulations of
ν
τ
regeneration and understand the interplay
of these two effects.
V. IMPLICATION ON FLUX AND SPECTRUM
MEASUREMENTS
Any bias in the total detectable energy due to FSR
discussed above will be amplified when measuring the
neutrino flux due to the steeply falling spectrum, i.e.,
ð
1
−
δ
E
ν
Þ
Γ
≃
1
−
Γ
×
δ
E
ν
;
ð
13
Þ
where
−
Γ
is the spectrum index and is usually
≤−
2
. That
is to say, not including FSR will underestimate the neutrino
flux by
Γ
×
δ
E
ν
. For example, in the case of UHE
ν
τ
=
̄
ν
τ
discussed in Sec.
IVA 1
and assuming
Γ
¼
3
,the
underestimation could be as much as
20%
×
3
¼
60%
. For
another example, in the case of throughgoing muons
[107]
(Sec.
IV B
) or Earth-emergent tau leptons (Sec.
III B
), the
underestimation could be as much as
5%
×
3
≃
15%
.
FSR also impacts the spectral shape measurements
because its effects depend on the parent neutrino energy
(Fig.
3
). We estimate this impact as follows. To measure
a neutrino spectrum
F
ν
∝
E
−
Γ
ν
between
E
ν
;
1
and
E
ν
;
2
(
E
ν
;
2
>E
ν
;
2
), the spectral index
−
Γ
could be determined by
−
Γ
¼
log
10
F
ν
;
2
−
log
10
F
ν
;
1
log
10
E
ν
;
2
−
log
10
E
ν
;
1
ð
14
Þ
without FSR, where
F
ν
;
1
and
F
ν
;
2
are the neutrino fluxes at
E
ν
;
1
and
E
ν
;
2
, respectively. However, with FSR,
E
ν
;
1
and
E
ν
;
2
are shifted by
δ
E
ν
;
1
and
δ
E
ν
;
1
, respectively, and as a
result,
F
ν
;
1
and
F
ν
;
2
are shifted by
δ
F
ν
;
1
¼
Γ
×
δ
E
ν
;
1
and
δ
F
ν
;
1
¼
Γ
×
δ
E
ν
;
2
, respectively.
For spectral-shape measurements using only charged
leptons (e.g., throughgoing muons
[107]
or Earth-emergent
tau leptons), the measured spectral index becomes
−
Γ
0
¼
−
Γ
−
1
ln
10
Γ
δ
E
ν
;
2
−
Γ
δ
E
ν
;
1
log
10
E
ν
;
2
−
log
10
E
ν
;
1
≃−
Γ
−
0
.
0075
Γ
ln
10
;
ð
15
Þ
which means that
−
Γ
>
−
Γ
0
. Therefore, not including FSR
will result in a measured spectral index (
−
Γ
0
) that is softer
than the true spectrum (
−
Γ
). Since
Γ
∼
2
–
3
, the distortion is
∼
0
.
01
, which is much smaller than the current measure-
ment uncertainty (e.g., IceCube measured
Γ
¼
−
2
.
37
þ
0
.
09
−
0
.
09
using throughgoing muon tracks
[107]
). Note that the
above estimate does not account for the smearing from
the neutrino energy to the muon deposit energy, which
needs a detailed detector-level simulation.
VI. COLLIDER NEUTRINOS
The impact of FSR is detector and observable dependent,
depending on energy thresholds and analysis choices. The
detectors at the LHC
’
s FPF are very different from the HE
and UHE neutrino telescopes discussed above. Therefore,
although the FPF will be exposed to neutrinos in the TeV
range, similar to the kinematics considered above, their
treatment warrants a separate discussion.
The FPF currently has two operational neutrino detectors
(FASER
ν
[108,109]
and SND
[110,111]
), and one pro-
posed but currently unfunded module (FLArE
[112]
) that is
relevant to the current discussion. Successor experiments to
both FASER and SND have been proposed
[25]
. FASER
ν
expects on the order of
2
×
10
4
CC-
μ
events
[113,114]
,
which is sufficiently large to consider a differential meas-
urement of d
σ
=
d
y
in analogy with IceCube. A similar event
rate is expected at SND
[113]
, and FLArE, having a larger
fiducial volume, will obtain roughly ten times the statistics
of FASER
ν
at a fixed beam energy. Both FASER
ν
and
SND employ nuclear emulsion detectors
[108,110]
,
whereas the FLArE proposal involves a liquid argon time
projection chamber
[112]
.
Nuclear emulsion detectors have submicron vertex
resolution, however photons only become visible in the
detector after converting to
e
þ
e
−
pairs. This takes place, on
average, within roughly one radiation length. The majority,
if not all, of CCDIS events involve neutral pions and etas in
the final state that decay promptly to two photons. It is,
therefore, nearly impossible to distinguish an FSR photon
from photons produced via
π
0
;
η
→
γγ
. Therefore, despite
their impressive vertex resolution, FSR is still best thought
of as increasing the apparent energy of a hadronic cascade.
Nuclear emulsion detectors can, however, isolate the
primary
e
track from a vertex, since its ionization track is
connected directly to the interaction point. This enables
these detectors to construct exclusive observables, such
as d
σ
=
d
E
e
, in much the same way as neutrino telescopes
construct d
σ
=
d
E
μ
and d
σ
=
d
E
τ
. In FLArE, only muon tracks
will be distinguishable, and the analysis presented above
applies to IceCube.
At the neutrino energies of interest for the FPF, i.e.,
0.1
–
2 TeV, radiative corrections for muons are of similar
size to those discussed for neutrino telescopes above. If
electron tracks can be reliably separated from the cascade
of hadronic and electromagnetic energy deposition, then
radiative corrections to electron neutrino inelasticity
5
We note that FSR
’
s impact on tau decay,
τ
→
ν
τ
X
þ
γ
, is not
enhanced by a large logarithm, log
ð
E
ν
=m
τ
Þ
, since the Lorentz
invariant energy scale is set by
m
τ
.
FINAL STATE RADIATION FROM HIGH AND ULTRAHIGH
...
PHYS. REV. D
111,
043007 (2025)
043007-9