# Investigating the fluxes and physics potential of LHC
neutrino experiments

Felix Kling<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  Toni Mäkelä<EMAIL_ADDRESS>National Centre for Nuclear Research, Pasteura 7, Warsaw, 02-093, Poland
Sebastian Trojanowski<EMAIL_ADDRESS>Astrocent, Nicolaus
Copernicus Astronomical Center Polish Academy of Sciences, ul. Rektorska 4,
00-614, Warsaw, Poland  National Centre for Nuclear Research, Pasteura 7,
Warsaw, 02-093, Poland

###### Abstract

The initiation of a novel neutrino physics program at the Large Hadron
Collider (LHC) and the purpose-built Forward Physics Facility (FPF) proposal
have motivated studies exploring the discovery potential of these searches.
This requires resolving degeneracies between new predictions and uncertainties
in modeling neutrino production in the forward kinematic region. The present
work investigates a broad selection of existing predictions for the parent
hadron spectra at FASER$\nu$ and the FPF to parameterize expected correlations
in the neutrino spectra produced in their decays and to determine the highest
achievable precision for their observation based on Fisher information. This
allows for setting constraints on various physics processes within and beyond
the Standard Model, including neutrino non-standard interactions. We also
illustrate how combining multiple neutrino observables could lead to
experimental confirmation of the enhanced-strangeness scenario proposed to
resolve the cosmic-ray muon puzzle already during the ongoing LHC Run 3.

††preprint: DESY-23-131

##  I Introduction

The subtle role of neutrinos in the Standard Model (SM) constantly motivates
measurements of their interactions across a broad energy spectrum, which also
remains essential for testing beyond the Standard Model (BSM) scenarios, cf.
Refs. Formaggio:2012cpf ; Proceedings:2019qno ; SajjadAthar:2021prg  for
reviews. The far-forward region of the Large Hadron Collider (LHC) is
particularly suitable for such studies DeRujula:1984pg ; Winter:1990ry ;
Vannucci:1993ud ; DeRujula:1992sn ; Park:2011gh ; Feng:2017uoz ; Beni:2020yfy
, as it offers a highly-collimated flux of the most energetic neutrinos ever
produced in a laboratory setup. A new neutrino physics program has recently
been initiated in this region with dedicated FASER FASER:2018ceo ;
FASER:2018bac ; FASER:2019dxq ; FASER:2020gpr ; FASER:2022hcn  and SND@LHC
SHiP:2020sos ; SNDLHC:2022ihg  experiments. Strikingly, this has already led
to the first observations of collider neutrinos FASER:2023zcr ; SNDLHC:2023pun
; CERN-FASER-CONF-2023-002 ; see also Refs. FASER:2021mtu ; Arakawa:2022rmp
for earlier analyses and discussion. The initial measurements pave the way for
further studies during the ongoing LHC Run 3, and in the future high-
luminosity LHC (HL-LHC) era in the proposed purpose-built Forward Physics
Facility (FPF) Anchordoqui:2021ghd ; Feng:2022inv .

While neutrinos in the SM interact via electroweak gauge bosons, their studies
can also indirectly teach us about QCD. This is due to their origin from
decays of various mesons produced in hadronic collisions. Due to the
uncertainties in modeling the parent hadron production at large
pseudorapidities, various theoretical predictions currently differ by as much
as an order of magnitude in the expected neutrino charged-current (CC) event
rates in the far-forward region of the LHC. Reducing these uncertainties is
among the primary goals of the new neutrino experimental program. This will
have far-reaching consequences for our understanding of strong interactions,
including parton distribution function (PDF) determination and non-
perturbative effects, and also broad implications for astroparticle physics
and BSM searches, cf. Refs. Anchordoqui:2021ghd ; Feng:2022inv ; Cruz-
Martinez:2023sdv ; LHCForwardPhysicsWorkingGroup:2016ote .

The dominant impact of modeling the parent hadron production is also expected
to generate notable correlations between neutrino spectra for different
flavors and at specific energy ranges. For instance, charm hadron decays
determine the forward tau neutrino flux and can contribute substantially to
the high-energy part of the electron and muon neutrino spectrum Kling:2021gos
. In this study, we propose to utilize these expected correlations to improve
the projected constraining power of the ongoing and future neutrino
measurements at the LHC.

To this end, we construct effective parameterization of the far-forward
neutrino spectra by interpolating between the leading predictions obtained
based on distinct modeling of the hadron production.111The code developed for
this study is available at https://github.com/makelat/forward-nu-flux-fit. The
charm-induced incident neutrino spectra used in the analysis were obtained
following the repository of the FPF Working Group 2. If these files are used
together with the code, relevant literature references should be cited, as
given in Table 1. We combine observations of interactions for different
neutrino flavors, energy, and pseudorapidity to determine the expected
precision of such analyses using the Hessian-based approach, similar to PDF
fits Schmidt:2018hvu . According to the Cram$\acute{\textrm{e}}$r-Rao bound,
this expected precision is given by the Fisher Information, which can be
easily computed cramer-rao ; cramer-rao2 . Despite existing uncertainties, a
multi-channel approach to studying $\nu$-induced events allows for identifying
new effects that cannot be easily mimicked by leading SM predictions of the
far-forward neutrino spectra or their combinations. This can be used to place
strong constraints or discover such phenomena. We illustrate this for an
enhanced strangeness production hypothesis with possible groundbreaking
implications for cosmic-ray physics Anchordoqui:2016oxy ; Baur:2019cpv ;
Anchordoqui:2022fpn  and for BSM-induced neutrino non-standard interactions
(NSI) that can also be probed this way at the LHC Kling:2020iar ;
Ismail:2020yqc ; Falkowski:2021bkq ; Batell:2021snh .

The paper is organized as follows. In Sec. II, we discuss our modeling and
provide projected bounds on the far-forward neutrino spectra. Sec. IV is
devoted to discussing applications of this methodology to constrain enhanced
strangeness production and BSM operators describing neutrino NSI. We conclude
in Sec. V. Further details about our statistical analysis are given in
Appendix A.

##  II Methodology

In our analysis, we first obtain a set of neutrino flux predictions to
determine the energy and pseudorapidity distribution of far-forward neutrinos
at the LHC. The latter distribution can be well described by the radial
distribution of events away from the beam collision axis. These predictions
are based on different Monte Carlo (MC) generators and other results in the
literature, as discussed below. We then define a parameterized flux model,
which is constructed from linear combinations of the individual predictions.
Using this input, we estimate an expected number of neutrino CC scattering
events in existing and proposed on-axis forward neutrino experiments at the
LHC. We discuss the necessary ingredients of this analysis in this section. We
then estimate how well the LHC neutrino experiments can constrain the flux
model on a statistical level and present the results in Sec. III.

###  II.1 Incident Neutrino Fluxes and Spectra

Neutrinos that can reach the far-forward detectors of our interest are
produced most abundantly near the ATLAS Interaction Point (IP). The meson
decays can be either prompt, e.g., for charm mesons, or displaced from the IP,
like for charged pions and kaons. In the latter case, the impact of the LHC
magnets and infrastructure must be considered in precise modeling. It
effectively suppresses neutrino production at distances larger than about
$100\,\textrm{m}$ away from the $pp$ collision point. Importantly, for LHC
neutrino energies, $E_{\nu}\sim\textrm{few hundred GeV}$, and the distance
between the IP and the detectors, $L\sim\textrm{few hundred meters}$, one
expects a negligible impact from neutrino oscillations unless it is enhanced
by BSM effects FASER:2019dxq . Hence, the measured neutrino spectra are
directly inherited from the parent hadrons.

Various hadrons contribute to the total neutrino flux measured in the far-
forward experiments, although the dominant contributions come from charged
pions, kaons, D-mesons, and charmed baryons, cf. Ref. Kling:2021gos  for
detailed discussion. The pion decays dominate the muon neutrino spectrum for
energies up to a few hundred GeV, while electron neutrinos with these energies
mostly come from kaon decays. Charm contributions might become important at
larger energies above TeV and they also determine the tau neutrino flux. Given
differences in modeling of the forward hadronic fluxes between charm and light
mesons, i.e., pions and kaons, we treat both contributions separately in our
analysis. Below, we briefly discuss the MC tools and predictions used in our
study, cf. Table 1 for a summary.

Light mesons ($\pi$, $K$) | Charm hadrons ($D$, $\Lambda_{c}$)
---|---
Name | Refs | Name | Refs
SIBYLL 2.3d | Ahn:2009wx ; Ahn:2011wt ; Riehn:2015oba ; Fedynitch:2018cbl  | SIBYLL 2.3d | Ahn:2009wx ; Ahn:2011wt ; Riehn:2015oba ; Fedynitch:2018cbl
EPOS-LHC | Pierog:2013ria  | BKRS | BKRSpaper
DPMJET 3.2019.1 | Roesler:2000he ; Fedynitch:2015kcn  | BDGJKR | Bai:2020ukz ; Bai:2021ira ; Bai:2022xad
QGSJET II-04 | Ostapchenko:2010vb  |  BKSS $k_{T}$ | Bhattacharya:2023zei
Pythia 8.2 (forward) | Fieg:2023kld  |  MS $k_{T}$ | Maciula:2022lzk
Table 1: A list of Monte Carlo tools and predictions with references used to
obtain far-forward neutrino spectra employed in our study. We treat pions,
kaons, and charm hadrons separately in the statistical analysis. See the text
for details.

Light mesons ($\pi$, $K$)

Light meson production in the forward kinematic region of the LHC cannot be
described within perturbative QCD (pQCD). Instead, it is typically modeled
using hadronic interaction models, many of which were originally designed for
cosmic-ray physics. In our analysis, we employ several most commonly used and
publicly available MC generators: EPOS-LHC Pierog:2013ria , DPMJET 3.2019.1
Roesler:2000he ; Fedynitch:2015kcn , QGSJET II-04 Ostapchenko:2010vb , and
SIBYLL 2.3d Ahn:2009wx ; Riehn:2015oba . We follow their implementation in the
CRMC package CRMC . We additionally use light meson spectra predictions
obtained with a new dedicated forward-physics Pythia 8.2 tune Fieg:2023kld .

Notably, these tools use different approaches to model forward hadron
production, and their variation incorporates a variety of underlying physics
effects, cf. Refs. LHCForwardPhysicsWorkingGroup:2016ote ; Albrecht:2021cxw
for reviews. The corresponding predictions form an envelope around the LHCf
data on neutral hadron spectra, although there remain sizable variations
between them, cf. Refs. LHCf:2012mtr ; LHCf:2015nel ; LHCf:2015rcj ;
Adriani:2023tyb  for comparison. The first forward muon FASER:2023zcr ;
SNDLHC:2023pun  and electron CERN-FASER-CONF-2023-002  neutrino data obtained
during the current LHC Run 3 show a broad overall agreement with theoretical
predictions that we use, albeit with large statistical uncertainties.

We treat pions and kaons independently in our analysis. To study the
robustness of our results, we have performed several numerical tests with a
limited set of only three MC generators out of the list of five above and
found similar bounds. However, we use the above complete MC generator list in
the following.

Charmed hadrons

Unlike light mesons, charm hadron production can also be described using pQCD.
In addition, many of the above generators do not treat forward charm
production, or it has not been validated and tuned to LHC data. For this
reason, we model the charmed hadron spectra differently in our study. We
consider predictions from SIBYLL 2.3d Ahn:2011wt ; Fedynitch:2018cbl  and,
additionally, use several recent results prepared specifically for the far-
forward neutrino searches at the LHC. We denote them in the following by
acronyms: BDGJKR Bai:2020ukz ; Bai:2021ira ; Bai:2022xad , BKRS BKRSpaper ,
BKSS $k_{T}$ Bhattacharya:2023zei , and MS $k_{T}$ Maciula:2022lzk .

Forward charm production in SIBYLL is modeled phenomenologically by replacing
the production of a strange pair $s\bar{s}$ by a charm $c\bar{c}$ pair with a
small probability determined by fitting to the data Ahn:2011wt . Instead, the
remaining predictions employ pQCD calculations of the charm production cross
section. The next-to-leading order (NLO) results are used to obtain the BKRS
and BDGJKR spectra within the collinear factorization approach. The former
calculation uses POWHEG Nason:2004rx ; Frixione:2007vw ; Frixione:2007nw ;
Alioli:2010xd  and the NNPDF3.1sx+LHCb set of parton distribution functions
(PDFs) with $\alpha_{s}=0.118$ at NLO+NLLx accuracy as input Ball:2017otu ;
Bertone:2018dse . The latter results, using the framework of Ref. Bai:2021ira
, are obtained with the PROSA FFNS PDF Zenaiev:2019ktw  with renormalization
and factorization scales proportional to transverse mass set by fitting to the
LHCb data. The BDGJKR predictions include additional Gaussian $k_{T}$ smearing
introduced to mimic the effect of the intrinsic transverse momentum of initial
state partons and soft gluon emissions. In contrast, the BKSS $k_{T}$ and MS
$k_{T}$ model these effects within the hybrid $k_{T}$ factorization approach
Dumitru:2005gt ; Deak:2009xt . The Kutak-Sapeta gluon unintegrated PDF (uPDF)
Kutak:2014wga  is used in this case.

An important effect on the forward charm hadron spectra is related to modeling
hadronization and fragmentation. The BDGJKR and MS $k_{T}$ results are based
on applying the Peterson fragmentation function (FF) Peterson:1982ak  by
assigning a fraction of the momentum of the parent charm quark to the final-
state hadron in the partonic center-of-mass frame and laboratory frame,
respectively. We note, however, that this calculation neglects the impact of
hadronization with beam remnants. Hence, in general, FFs are not expected to
be applicable in forward collisions at the LHC, cf. section 6.2.2 in Ref.
Feng:2022inv  for further discussion. In particular, using them implies that
charm hadrons are always less energetic than charm quarks, which reduces the
flux of high-energy neutrinos. In the MS $k_{T}$ case, additional
hadronization with beam remnants is also considered via a recombination
formalism which is sizeable for $D_{0}$ and $D^{\pm}$ mesons but negligible
for $D_{s}$. This effect dominates at high energies and for forward
rapidities. On the other hand, SIBYLL, BKRS, and BKSS $k_{T}$ predictions rely
on string fragmentation to include hadronization with beam remnants. The
latter two results employ the string fragmentation model implemented in Pythia
8.2 Sjostrand:2014zea .

###  II.2 Neutrino Flux Parameterization

The forward hadron spectra predictions mentioned above are used to obtain
neutrino spectra arising from the decays of the light mesons $\pi^{\pm}$,
$K^{\pm}$, $K^{0}_{L}$, $K^{0}_{S}$, and the charmed hadrons $D^{\pm}$,
$D^{0}$, $\overline{D}^{0}$, $D^{\pm}_{s}$, $\Lambda^{\pm}_{c}$. To treat
possible variations in the normalization and shape of the neutrino spectra, we
take the actual spectra used in our analysis as an interpolation (or
extrapolation) between these predictions. For simplicity, we neglect
subdominant production modes of neutrinos in hyperon and B-meson decays, as
well as secondary production modes in hadronic showers induced in the elements
of the LHC infrastructure away from the ATLAS IP.

To rescale the flux components and to obtain the corresponding binned spectra,
we define a model parametrizing the contributions of different predictions in
a weighted sum, resulting in a total sample. The parent hadrons are divided
into three classes: pions ($\pi$), kaons ($K$), and charmed hadrons ($c$),
each with a dedicated weight in the sum. Then with $p\in\\{\pi,K,c\\}$, we
employ $N_{p}$ predictions for the number of CC scattering events in the
detector in a given energy and radial bin, $G_{n\geq 0}^{(p)}$, by introducing
$N_{p}-1$ nuisance parameters $\lambda_{i\geq 1}^{(p)}$ to obtain the
interpolated prediction with the following expression

$m=\sum_{p\in\\{\pi,K,c\\}}\frac{1}{N_{p}}\left[G_{0}^{(p)}\left(1-\sum_{i=1}^{N_{p}-1}\lambda_{i}^{(p)}\right)+\sum_{i=1}^{N_{p}-1}G_{i}^{(p)}\left(1+N_{p}\lambda_{i}^{(p)}-\sum_{j=1}^{N_{p}-1}\lambda_{j}^{(p)}\right)\right].$
(1)

The model then reduces to the contribution of the $i\geq 1$th prediction
$G_{i}$ when $\lambda_{i}=1,\lambda_{j\neq i}=0$, while
$\lambda_{i}=-1~{}\forall~{}i$ returns the spectrum of $G_{0}$. Setting
$\lambda_{i}=0~{}\forall~{}i$ yields the average of all predictions, chosen as
the baseline for the discussion below. Note that such a setting is not
imperative for implementing the model calculation, and choosing the baseline
as a general set of parameter values is also possible. In particular, we will
discuss the result obtained for the SIBYLL baseline prediction in Sec. IV.1.

The effective description of the neutrino data obtained this way is
characterized by $12$ nuisance parameters, on top of additional free
parameters that we introduce when constraining specific new effects discussed
in Sec. IV. While future studies will keep refining the choice of the nuisance
parameters in analyses of this kind, the present work is the first
quantitative assessment of employing such parameterizations to study LHC
neutrinos. These are introduced to relate far-forward neutrino data to
fundamental hadronic physics, instead of treating neutrino spectra as fully
uncorrelated.

We then perform a likelihood-based analysis and estimate a minimal variance of
the model parameters via the Fisher information matrix, as dictated by the
Cram$\acute{\textrm{e}}$r-Rao bound cramer-rao ; cramer-rao2 ; see also Refs.
Cranmer:2014lly ; Brehmer:2016nyr ; Brehmer:2017lrt  for similar discussions
for other LHC data analyses. To this end, our procedure should reproduce the
projected most robust bounds to be obtained thanks to the data gathered in
considered experimental searches after profiling over nuisance parameters that
represent theoretical uncertainties. At the same time, we also comment on
expected deviations from this picture in the presence of finite efficiency
factors affecting the measurements. The results are, eventually, translated
into physically meaningful quantities for their interpretation. We provide
more details about a statistical analysis in Appendix A.

In the following, we will focus on the constraints on the combined neutrino
and antineutrino spectrum for each flavor, $\nu_{\ell}+\bar{\nu}_{\ell}$. We
note that the forward LHC detectors have capabilities to disentangle between
neutrinos and antineutrinos, especially for $\nu_{\mu}$. This allows for
measuring their spectra separately. We leave the discussion about the
potential consequences of such measurements for future studies while we
concentrate in this analysis on the dominant impact of meson decays that can
be well constrained by the combined spectra.

###  II.3 Neutrino Detection

The collimated flux of high-energy forward neutrinos produced at the LHC can
be detected in relatively small experiments that allow for detailed studies of
neutrino interactions. We will illustrate the prospects of these searches for
a selection of such ongoing and future proposed detectors.

FASER$\nu$

Focusing first on the current LHC Run 3, we will study the projected
capabilities of the FASER$\nu$ emulsion detector FASER:2019dxq ; FASER:2020gpr
. It consists of tungsten target material layers with a total mass of $1.1$
ton. These are interleaved with emulsion films with the transverse size of
$25~{}\textrm{cm}\times 30~{}\textrm{cm}$ that store information about the
tracks of charged particles produced in neutrino scatterings. High-energy
muons produced this way can travel through the entire detector, and their
momentum is measured in the FASER spectrometer placed downstream of the
emulsion detector. The excellent spatial resolution of emulsion films allows
for measuring $\nu_{\tau}$-induced tau lepton tracks with a few hundred GeV
energy and, therefore, study $\nu_{\tau}$ charged current (CC) interactions on
an event-by-event basis.

The expected vertex detection efficiency of FASER$\nu$ is of order $90\%$ for
the most energetic neutrinos produced at the LHC, while it decreases to about
$(30-40)\%$ for $E_{\nu}\sim 100~{}\textrm{GeV}$. We implement it following
Fig. 9 in Ref. FASER:2019dxq . We additionally employ a geometrical acceptance
factor of $80\%$ and lepton identification efficiencies of $86\%$ for muons
and $75\%$ for taus following that study. We assume that electrons can be
identified with nearly $100\%$ detection efficiency in emulsion due to their
expected showering. Considering all the effects above, we estimate that, e.g.,
one can identify a CC scattering of the 1 TeV muon neutrino with more than
$60\%$ efficiency in FASER$\nu$. In this analysis, we use $5$ energy bins per
decade in the likelihood analysis, which can reproduce expected $30\%$
neutrino energy resolution in this detector FASER:2019dxq . We assume
$\mathcal{L}=150~{}\textrm{fb}^{-1}$ of integrated luminosity in LHC Run 3.

FASER$\nu$2

The emulsion detector technology has also been proposed for the FASER$\nu$2
detector in the FPF. The assumed transverse size of $40~{}\textrm{cm}\times
40~{}\textrm{cm}$ and total tungsten mass of $20$ tons, as well as larger
integrated luminosity in the HL-LHC era, $\mathcal{L}=3~{}\textrm{ab}^{-1}$,
result in a significantly increased expected neutrino event statistics in this
detector, up to 1M muon neutrino CC scatterings Anchordoqui:2021ghd ;
Feng:2022inv . The larger detector size of FASER$\nu$2 permits better event
containment than in FASER$\nu$. This results in an expected improvement in
energy resolution. We, therefore, employ $10$ bins per decade of the incident
neutrino energy in this case. Similarly to FASER$\nu$, the neutrino detection
efficiency in FASER$\nu$2 will be flavor-dependent. Given the lack of detailed
efficiency studies for FASER$\nu$2, we present the results below assuming
$100\%$ efficiency. However, we also comment on the impact of employing
efficiency cuts similar to those discussed above for the currently operating
FASER$\nu$ detector.

FLArE

We also present the results for the proposed FLArE detector Batell:2021blf ;
Anchordoqui:2021ghd ; Feng:2022inv  employing liquid argon (LAr) time-
projection chamber (TPC) technology. FLArE will offer improved calorimetric
capabilities and dynamical information about events to disentangle neutrino-
induced signals from muon backgrounds. The outgoing muons from neutrino
interactions can be measured with a dedicated muon tagger and with the help of
the FASER2 spectrometer. Studying tau neutrinos might be more challenging in
this case due to the expected lower spatial resolution of LArTPCs than in
emulsion detectors. However, $\nu_{\tau}$-induced events can still be searched
for as fluctuations over the expected backgrounds from other neutrino flavors.
In the following, we assume $1~{}\textrm{m}\times 1~{}\textrm{m}$ transverse
area and 10-ton fiducial mass of the LAr target in FLArE, and the integrated
luminosity of $\mathcal{L}=3~{}\textrm{ab}^{-1}$. We take $100\%$ efficiency
for neutrino detection in FLArE while commenting on the case with a decreased
$50\%$ efficiency.

All the detectors discussed above are centered around the beam-collision axis.
Importantly, off-axis far-forward detectors have also been proposed, namely
the SND@LHC SHiP:2020sos ; SNDLHC:2022ihg  and AdvSND Anchordoqui:2021ghd ;
Feng:2022inv  experiments for the ongoing LHC Run 3 period and the HL-LHC era,
respectively. These extend pseudorapidity coverage of far-forward searches at
the LHC toward lower values of $\eta$. In the following, we focus on the on-
axis experiments and present representative results obtained for the ongoing
measurements in FASER$\nu$ and the proposed FASER$\nu$2 and FLArE searches. We
note, however, that additional data gathered off-axis may further improve the
projected constraints discussed below.

When modeling neutrino interactions in the detectors of our interest, we
convolute the neutrino flux with the interaction cross-sections predicted by
GENIE Andreopoulos:2009rq  as obtained in Ref. FASER:2019dxq . These results
are based on a Bodek-Yang model used to describe deep inelastic scattering
(DIS) events Yang:1998zb ; Bodek:2002vp . The alternative NNFS$\nu$ approach
has been recently discussed in Ref. Candido:2023utz , which generally agrees
with the Bodek-Yang model at TeV-scale energies, cf. also Refs. Xie:2023suk ;
Jeong:2023hwe  for other recent analyses. However, uncertainties in the
predicted scattering cross section up to a few percent for
$E_{\nu}\sim\textrm{TeV}$ have been reported that are driven by PDF
uncertainties Candido:2023utz . This is not expected to significantly affect
the interpretation of the results presented below for the ongoing FASER$\nu$
measurements. On the other hand, improved sensitivity of the FPF experiments
will allow us to reach the level of precision where PDF uncertainties are
anticipated to become important. In fact, by using additional kinematic
variables, the FPF is expected to constrain PDFs, especially for strange
quarks Anchordoqui:2021ghd ; Feng:2022inv . The proposed Electron-Ion Collider
(EIC) will further improve relevant bounds on up and down quark PDFs
Accardi:2012qut . The corresponding uncertainties should then be reduced
during the FPF data-taking period. In the following, we focus on the dominant
uncertainties affecting neutrino fluxes and spectra in the far-forward
kinematic region of the LHC related to the differences in parent hadron
spectra predictions. We leave the discussion of a joint fit considering both
production and interaction rate uncertainties for the future.

##  III Neutrino spectra and projected constraints

In the upper panels of Fig. 1, we illustrate single-differential neutrino
energy distributions for CC scattering events in the FLArE detector using
several combinations of the abovementioned MC predictions for parent meson
spectra. We present the results for all three neutrino flavors. We denote
different predictions by $\textrm{p}_{1}+\textrm{p}_{2}$ in the plots, where
$\textrm{p}_{i}$ stands for the prediction name, and $i=1$ and $2$ corresponds
to light and charm hadron spectra, respectively. In each case, the plots show
the combined neutrino and antineutrino spectra.

Figure 1: In the upper panels, the colorful histograms correspond to different
predictions of the combined energy distributions of neutrinos and
antineutrinos interacting via CC scatterings in FLArE, as indicated in the
plot. The left (central, right) panel corresponds to the electron (muon, tau)
neutrinos. An average of the predictions employed in the analysis gives the
baseline spectrum shown with a black solid line. The bottom panels illustrate
the expected Cramer-Rao uncertainty bands ($1\sigma$) on the baseline spectrum
as gray-shaded regions. The robustness of the obtained uncertainties against
varying event statistics is shown with purple and green histograms, where the
number of events is changed up and down by a factor of two.

As can be seen, various predictions agree remarkably well for the electron and
muon neutrinos with energies up to $E_{\nu}\sim 300~{}\textrm{GeV}$. In this
energy regime, an observed discrepancy between different MC results is about a
factor of $2$. This reflects a relatively better understanding of light meson
spectra production in the far-forward region of the LHC, and these mesons
dominate the $\nu_{e}$ and $\nu_{\mu}$ fluxes up to a few hundred GeV of
energy. Instead, the larger the neutrino energy becomes, the uncertainties
grow both for light mesons and especially for the possible charm hadron
contributions. The latter also determine the $\nu_{\tau}$ flux predictions
over the entire energy range. The charm-induced spectra currently show an
order-of-magnitude discrepancy between various predictions.

Focusing on the tau neutrino spectrum plot, we find that the lack of beam
remnant induced effects in hadronization, e.g. the beam drag effect in
modeling the $D_{s}$-meson production, suppresses the high-energy part of
charm-induced neutrino spectra. This is evident when comparing the BDGJKR and
MS $k_{T}$ predictions with the BKRS and BKSS $k_{T}$ results. We note that
even though the high-energy part of the BKSS $\textrm{k}_{\textrm{T}}$
spectrum is suppressed by considering gluon saturation, this prediction
remains the most optimistic in terms of the expected number of $\nu$-induced
events in the detector. The difference between this prediction and the least
optimistic MS $k_{T}$ result is the largest for the most energetic tau
neutrinos with $E_{\nu_{\tau}}\sim\textrm{few TeV}$. Furthermore, we have
verified that the uncertainties in the charm predictions also partially
propagate to the high-energy part of $\nu_{e}$ and $\nu_{\mu}$ spectra, adding
to uncertainties in determining light meson spectra.

We also show in the plots the baseline model prediction obtained as an average
of all the considered predictions, assuming equal weights. In the bottom
panels in Fig. 1, we assume that the baseline prediction correctly describes
the data to be gathered in the FPF. The gray-shaded regions illustrate the
projected statistical precision with which our flux model can be constrained
at $1\sigma$ level; see Appendix A for details of the statistical analysis.

The uncertainty bands found this way illustrate excellent precision in
constraining the neutrino spectra in the FPF experiments. This is especially
evident for muon neutrinos with energies $100~{}\textrm{GeV}\lesssim
E_{\nu_{\mu}}\lesssim 1~{}\textrm{TeV}$, as shown in the bottom central panel
in the figure. Due to the largest expected event statistics, the projected
bounds, in this case, are at the percent level. This translates into a narrow
gray uncertainty band over the baseline neutrino spectrum in the central upper
panel, which is barely visible in the plot. In particular, the FPF data will
allow for differentiating between the baseline hypothesis and specific MC
results presented in Fig. 1 with high precision.

Due to reduced event statistics, the uncertainty bands grow at the spectrum’s
low and high energy tails. The high-energy neutrinos with
$E_{\nu}\gtrsim\textrm{a few}\times\textrm{TeV}$ are more rarely produced at
the LHC. Instead, low-energy neutrinos with $E_{\nu}\lesssim
10~{}\textrm{GeV}$ are produced more isotropically and often miss far-forward
experiments. However, we find the projected uncertainty to be of order several
percent between these two regimes. This remains at the level of PDF
uncertainties affecting the neutrino DIS cross-section predictions, as
discussed above. This happens also for the electron neutrinos, for which the
expected number of events is only a factor of a few lower than for
$\nu_{\mu}$s. We show the electron neutrino uncertainty bands in the bottom
left panel.

The bottom right panel illustrates the results for the tau neutrinos. In this
case, the projected uncertainties are larger but, remarkably, also stay below
$5\%$ for $100~{}\textrm{GeV}\lesssim E_{\nu_{\tau}}\lesssim
3~{}\textrm{TeV}$. At first, this result might seem odd, given significantly
lower event statistics of $\nu_{\tau}$-induced events than for the other
neutrino flavor. However, we note that the analysis for the tau neutrinos
implicitly concerns the results obtained for both $\nu_{e}$ and $\nu_{\mu}$.
This is because the spectra of these neutrinos are also affected by the
forward charm production, especially in their high-energy tails. Possible
enhanced production of charm hadrons is then strongly constrained in this
energy regime by the electron and muon neutrino data, which then translates
into stronger bounds on $\nu_{\tau}$. Instead, in the low-energy part of the
spectrum, below $100~{}\textrm{GeV}$, both the tau neutrino flux is decreased,
and the correlation with the electron and muon neutrino spectra is lost. As a
result, the constraining power for $\nu_{\tau}$ in this energy regime is
significantly weaker.

We have also verified numerically that the expected uncertainty bands on the
$\nu_{\tau}$ energy spectrum depend only mildly on the choice of the baseline
spectrum. For instance, after switching to the baseline spectrum defined as
DPMJET($\pi$,$K$) + BKRS($c$) shown in red in Fig. 1, one finds reduced
uncertainties, by up to a factor of $2$, in some of the low-energy bins for
$E_{\nu_{\tau}}\lesssim 100~{}\textrm{GeV}$. The improvement in high-energy
bins is, however, much smaller, even though the new baseline spectrum predicts
a larger number of $\nu_{\tau}$-induced events up to
$E_{\nu_{\tau}}\sim\textrm{TeV}$. This additionally illustrates that the high-
energy tail of the tau neutrino spectrum is not only sensitive to the
$\nu_{\tau}$ spectrum, but the charm contribution to the spectra of other
neutrino flavors strongly constrains it too. The latter constraining power is
not significantly affected by changing the baseline spectrum. This is because
DPMJET predictions accidentally lie close to the average spectra for $\nu_{e}$
and $\nu_{\mu}$ over the entire energy range, as can be seen by comparing red
and black histograms in the left and central upper panels of Fig. 1.

In Fig. 1, we also illustrate the expected uncertainty bands for each neutrino
flavor that assume only $50\%$ of event statistics. We show this with purple
histograms in the bottom panels. As discussed above, this could correspond to
a more realistic treatment of the neutrino detection efficiency factors in
FLArE. Importantly, as can be seen, this has only a mild impact on the
expected constraining power of this experiment. Similarly, we present the
expected results for increased event statistics up to $200\%$ of events with
green histograms in the bottom panels. This could be due to increasing the
fiducial volume of the detector. Again, the predicted impact on the neutrino
spectrum uncertainty bands is relatively small. Hence, small variations in
efficiency factors or detector sizes in the FPF are not expected to affect the
neutrino physics program significantly.

Figure 2: The upper panel illustrates the combined neutrino and antineutrino
CC event scattering rates in FLArE. The results are shown for each neutrino
flavor in three radial bins, as indicated in the plot. The spectra are divided
by the corresponding bin area. The lower panel indicates the improvement in
uncertainty obtained by combining the information from three (red) or eight
(purple) radial bins.

However, adding spatial information about events can improve the neutrino
spectrum uncertainty bands. This allows for constraining double-differential
neutrino production cross section in the far-forward region of the LHC, which
takes into account additional information about the pseudorapidity
distribution on top of the previously discussed energy distribution. We
illustrate this in Fig. 2, in which the spatial distribution of neutrino
scattering events in FLArE is considered by virtue of radial bins. In the
upper panels, we show the neutrino interaction spectrum in three radial bins
defined as $R<0.1~{}\textrm{m}$, $0.1~{}\textrm{m}<R<0.25~{}\textrm{m}$, and
$R>0.25~{}\textrm{m}$, where $R$ is the radial distance away from the beam
collision axis. The detector is assumed to be centered around the beam
collision axis ($R=0$), and the last radial bin extends to the edges of the
detector transverse size defined by the square of size $1~{}\textrm{m}\times
1~{}\textrm{m}$. The spectra are normalized to the bin area to illustrate
better the concentration of neutrino-induced events around the beam collision
axis.222In the analysis below, we also use radial bins for the other
experiments that are defined as follows. For FASER$\nu$, with the smallest
transverse size, we use $R<0.06~{}\textrm{m}$,
$0.06~{}\textrm{m}<R<0.13\textrm{m}$, and $R>0.13~{}\textrm{m}$ up to the edge
of the detector. In the case of FASER$\nu$2, we define the bins differently:
$R<0.1~{}\textrm{m}$, $0.1~{}\textrm{m}<R<0.2~{}\textrm{m}$, and
$R>0.2~{}\textrm{m}$ up to the edge of the detector.

As shown with solid black lines in the upper panels, the central parts of the
detector ($R<0.1~{}\textrm{m}$) can constrain well the most uncertain high-
energy parts of the neutrino spectra. Instead, the outermost radial bin in
this energy regime is characterized by more than an order of magnitude lower
neutrino flux per unit area, as shown with yellow solid lines. This is,
however, compensated by a larger area of this radial bin when counting the
total number of events. Hence, each radial bin has similar constraining power
in our analysis in the high-energy tails of the distributions. Instead,
neutrinos with lower energies, below a few hundred GeV, are dominantly
constrained by the data gathered in the parts of the detector with a larger
total transverse area. This is understood as their parent mesons are often
less energetic and less forward-focused after production at the LHC.

Considering this spatial information further improves the FPF detectors’
constraining power. We illustrate this in the bottom panels of Fig. 2. In the
plots, gray-shaded regions correspond to the previously discussed results with
only one radial bin. In this case, only a single-differential distribution in
the energy of the neutrino production cross section is used to constrain
neutrino spectra. Instead, red and purple lines in the plots show the results
obtained for three or eight radial bins. As can be seen, adding spatial
information reduces the uncertainties to the sub-percent level for the muon
neutrinos with $100~{}\textrm{GeV}\lesssim E_{\nu_{\mu}}\lesssim\textrm{TeV}$.
A similar reduction is observed for the electron neutrinos. The improvement by
up to a factor of a few in the expected uncertainty band is also found in the
low- and high-energy tails of the respective neutrino spectra. Increasing the
number of radial bins further does not substantially improve the uncertainty
bands. This is due to reduced event statistics in each of the bins observed in
this case.

The baseline spectrum uncertainty for $\nu_{\tau}$s is, similarly, reduced
over the entire energy range by using spatial information. In particular, the
low-energy tail of the spectrum obtained for $E_{\nu_{\tau}}\sim\textrm{a few
tens of GeV}$ can now be better constrained. Charm-induced neutrinos are
characterized by a noticeably different pseudorapidity distribution than those
produced in decays of light mesons. The latter tend to be more collimated
around the beam collision axis, as dictated by their characteristic transverse
momentum, $p_{T}\sim m/p$, where $m$ is the hadron mass and $p$ is its total
momentum. Therefore, including information about the double-differential
distribution allows for better disentangling charm-induced excess of $\nu_{e}$
and $\nu_{\mu}$ scattering events over the dominant events associated with the
neutrino production in light meson decays. The improved charm constraining
power also reduces uncertainty bands on the $\nu_{\tau}$ spectrum.

Figure 3: Similar to Fig. 1, but a comparison of the baseline neutrino CC
scattering interaction rates obtained for FLArE, FASER$\nu$2, and FASER$\nu$
are shown, assuming luminosities of $150\mathrm{fb}^{-1}$ for FASER$\nu$ and
$3\mathrm{ab}^{-1}$ for the remainder. In the bottom panels, relevant
uncertainty bands are shown.

In Fig. 3, we show a comparison between the baseline neutrino spectra and
uncertainty bands obtained for FLArE and FASER$\nu$2 in the FPF and the
currently operating FASER$\nu$ detector. As can be seen, the FPF experiments
will offer more than two orders of magnitude larger neutrino event statistics
than FASER$\nu$. The highest number of events is expected for FASER$\nu$2,
which, according to the current design, has a larger target mass by a factor
of two than FLArE. Additional improvement comes from an increased tungsten
density with respect to LAr. This allows for concentrating the target mass
better around the beam collision axis, where high-energy neutrino flux is
collimated. Because of the larger transverse size of FLArE, a peak of the
expected neutrino spectrum in this detector is slightly shifted toward lower
energies when compared to emulsion detectors.

The increased event statistics in the FPF detectors translate into
significantly narrower uncertainty bands than for FASER$\nu$, as shown in the
bottom panels. These have been obtained assuming $3$ radial bins for each
detector. The relevant ranges of $R$ have been changed for each detector,
depending on its transverse size. Notably, the ongoing measurements in
FASER$\nu$ will be able to constrain the electron and muon neutrino spectra
with $\mathcal{O}(10\%)$ precision for the energy between a few hundred GeV
and several TeV. However, the uncertainties in determining the tau neutrino
flux will remain much larger. The FPF detectors are needed to reduce them to a
few percent level.

##  IV Physics Applications

As discussed above, detailed information about the neutrino flavor, energy
spectrum, and the spatial distribution of events in the detector will allow
one to differentiate between various predictions. It can also be used to
constrain other effects. Employing complete information about events allows
for better identification of the unique impact of such phenomena on the far-
forward neutrino data. We illustrate this below for two sample effects. One is
related to proposed enhanced strangeness production in hadronic collisions at
large energies and pseudorapidities. The other effect concerns potential NSI
contributions to neutrino event rates in the far-forward neutrino experiments
at the LHC.

###  IV.1 Enhanced Strangeness

Far-forward searches at the LHC are naturally connected to ultra-high energy
cosmic-ray (UHECR) physics. This is due to the sensitivity of both physics
programs to high-energy hadronic collisions and the importance of large
pseudorapidity regimes of such interactions. We have already shown how LHC
data can help differentiate between available MC generators that are also
routinely used in modeling cosmic ray (CR) air showers to tune them better in
the future. Here, we focus on the expected impact of these searches on
explaining anomalies in cosmic-ray data.

A striking example of such anomaly is the so-called muon puzzle first observed
in the Pierre Auger Observatory data PierreAuger:2014ucz ; PierreAuger:2016nfk
; PierreAuger:2021qsd . Other experimental collaborations subsequently
confirmed it, and the anomaly is currently considered to have a combined
statistical significance of $8\sigma$, cf. Ref. Albrecht:2021cxw  for review.
The anomaly is related to an apparent enhancement in muon rates at the level
of a few tens of percent in hadronic components of CR-induced showers. This
corresponds to high energies starting at $E\sim 10^{8}~{}\textrm{GeV}$, which
translates into the LHC energy scale in the CM frame for the $pp$ interaction.
The discrepancy between the observed and predicted muon rates grows higher
with increasing energy. It has been shown that the dominant explanation of the
anomaly is likely due to a reduced transfer of energy from a hadronic to an
electromagnetic component of the shower, e.g., by suppressing the neutral pion
production or decay rate in atmospheric air showers Allen:2013hfa .

Among the models proposed to accommodate such an effect, particularly
important is the enhanced strangeness hypothesis, in which suppressed pion to
kaon production ratio in the final state of high-energy $pp$ collisions is
assumed, cf. Refs. Anchordoqui:2016oxy ; Baur:2019cpv  for possible underlying
mechanisms. In a simple phenomenological approach, this can be achieved by
introducing a finite swapping probability that turns a fraction of pions into
kaons. A detailed study of this effect has been performed in Ref.
Anchordoqui:2022fpn . It has been shown that the relevant $\pi\to K$ swapping
fraction $f_{s}$ at the level of a few tens of percent can explain the
anomaly. To this end, and to be reconciled with other experimental data, the
swapping probability should primarily affect high-energy collisions in the
large pseudorapidity regime. Interestingly, hints of enhanced strangeness
production have also been found in the mid-rapidity region in the ALICE data
ALICE:2016fzo .

In the following, we will analyze a simple phenomenological model introduced
in Ref. Anchordoqui:2022fpn . In this case, in the presence of the non-zero
$f_{s}$ parameter, the number of neutrinos produced from pion decays in the
forward region of the LHC is reduced by a common energy-independent factor,
$N_{\pi\to\nu}\to(1-f_{s})\,N_{\pi\to\nu}$. Simultaneously, the number of
neutrinos produced in kaon decays is increased as
$N_{K\to\nu}\to(1+6.6\,f_{s})N_{K\to\nu}$. Here, a numerical factor of $6.6$
is related to a relative difference in the pion and kaon production rates at
large pseudorapidities at the LHC. It has been determined numerically to
reproduce best a complete treatment of the model, in which individual pions
are changed into kaons in simulations of the forward neutrino spectra. The
difference in the production rates of both mesons is due to their different
masses and quark compositions. Additional effects considered in these
simulations are due to finite kaon lifetimes and the change of $\pi^{0}$ into
$K^{0}_{S,L}$. In the latter case, the neutrino can only be produced after the
swapping, while the initial neutral pion would typically decay into two
photons. Assuming SIBYLL as a baseline MC generator, it has been shown that
introducing such a universal swapping fraction $f_{s}$ for collisions
characterized by projectile energies above PeV and pseudorapidities $|\eta|>4$
in the CM frame in CR air shower simulations allows for fitting the muon data.
This requires $f_{s}$ to lie between about $0.3$ and $0.8$, where larger
values are favored when the increasing primary energy is considered.

Such effects can be particularly prominent in the forward LHC neutrino data if
they change $\nu$ interaction rates in kinematic regions less affected by
variations in MC predictions. We illustrate this for the enhanced strangeness
effect in the upper left panel of Fig. 4 with two plots obtained for electron
and muon neutrinos. In the plots, we present green histograms representing the
expected neutrino CC event scattering rate in the FLArE detector obtained for
SIBYLL and $f_{s}=0.5$. This should be compared with black solid lines in the
plots representing the baseline scenario obtained for $f_{s}=0$. As can be
seen, the enhanced strangeness production, in this case, would manifestly
increase the electron neutrino event rates over the entire energy range,
especially for $E_{\nu_{e}}\lesssim 1~{}\textrm{TeV}$. This is due to the
dominant $\nu_{e}$ production mode in kaon decays. A similar enhancement is
predicted for muon neutrinos above $100~{}\textrm{GeV}$. Instead, for lower
energies, one expects a decrease in the $\nu_{\mu}$ event statistics, albeit
this is a less significant effect driven by a reduced number of forward-going
pions. Applying a non-zero swapping probability does not affect the tau
neutrino spectrum. A combined impact of these modifications of the neutrino
spectra measured in the far-forward region of the LHC provides a strong
signature of this effect, which cannot be easily reproduced by changing and
interpolating between various MC predictions in our analysis. To illustrate
this, we have added yellow-shaded prediction envelopes in the plots around the
baseline distributions that correspond to various MC results shown in Fig. 1.

Figure 4: Left: The top panels show the electron (left) and muon (right)
neutrino CC event scattering rates in FLArE obtained using SIBYLL as the
baseline MC generator and with three radial bins. The solid black histograms
correspond to $f_{s}=0$, while the dashed orange (blue, black) ones to
$f_{s}=0.01$, $0.003$, $0.001$. The latter remain barely distinguishable from
the $f_{s}=0$ baseline in the plots. These values of $f_{s}$ roughly
correspond to $1\sigma$ exclusion bounds obtained for FASER$\nu$ and FLARE
with 10% or 100% of total data. For FASER$\nu$, the efficiency factors arising
from geometry, energy dependence, and charged lepton identification have been
applied. The green histograms represent the $f_{s}=0.5$ case, for which the
cosmic-ray muon puzzle can be solved. The variations in the neutrino event
rate due to different MC predictions from Fig. 1 are shown with yellow-shaded
bands. The bottom panels zoom in on the uncertainty bands on the neutrino
spectrum, shown as gray-shaded bands similar to Fig. 1. Expected deviations
from the $f_{s}=0$ case are also shown as colorful lines that correspond to
the aforementioned exclusion bounds from FASER$\nu$ and FLArE. Right: The
$2\sigma$ constrained values (gray) for $f_{s}$ obtained using FLARE and
FASER$\nu$, also demonstrating the effect of choosing different predictions as
the baseline for the latter. These are compared to less constraining values
obtained for the discovery potential at FASER$\nu$ (turquoise), with and
without the information on tau neutrinos and high-energy contributions to the
$\nu_{e}$ spectrum. Notably, all of the predicted constraints cover the
$0.3<f_{s}<0.8$ region shown in dark green, i.e., the values of $f_{s}$
favored by the enhanced strangeness solution to the CR muon puzzle. The light
green band extending to lower values of $f_{s}\sim 0.005$ is added to indicate
that the effect might manifest in a more subtle way in $pp$ collisions at the
LHC.

We first note that essential bounds on the $f_{s}$ parameter will be obtained
thanks to the data gathered in FASER$\nu$ during the ongoing LHC Run 3. Using
the procedure outlined above, we have found that already within the next few
years, FASER$\nu$ will be able to constrain the enhanced strangeness
hypothesis up to the level of $f_{s}\simeq 0.013$ ($1\sigma$) assuming SIBYLL
as a baseline (measured) neutrino spectrum. These results only mildly depend
on the precise choice of the baseline spectrum. In particular, we have also
verified this for the spectra generated with the EPOS-LHC and QGSJET MC tools
and found similar expected bounds at the level of $f_{s}\simeq 0.013$ and
$0.012$, respectively. As discussed in Ref. Sciutto:2019pqs , these MC
generators predict either smaller or larger enhancement effects in the CR
shower data. Notably, regardless of the precise choice of the generator, the
constraining power of FASER$\nu$ significantly exceeds the preferred value of
$f_{s}\sim(0.3-0.8)$ obtained by fitting the UHECR data.

This motivates studying potential discovery prospects in FASER$\nu$. We have
tested them assuming that the neutrino data gathered in FASER$\nu$ will
correspond to SIBYLL predictions enhanced by an additional impact of the non-
zero $f_{s}$ parameter. We find in this case that the unique features of this
scenario differ from other SM predictions sufficiently strongly to allow for
excluding the $f_{s}=0$ hypothesis at the $5\sigma$ level for the swapping
fraction $f_{s}=0.06$ or so. We recall that this result has been obtained by
considering realistic FASER$\nu$ efficiency factors, as discussed in Sec.
II.3.

To obtain even more baseline-independent results, we similarly study the
discovery prospects for FASER$\nu$ focusing only on the muon neutrino data and
electron neutrinos with energies below $300~{}\textrm{GeV}$. This excludes
high-energy electron neutrinos and the tau neutrino data that are currently
subject to the largest theoretical uncertainties based on various MC
predictions, cf. Fig. 1 and yellow-shaded bands in the upper left panels of
Fig. 4. After limiting the dataset for the enhanced strangeness analysis this
way, we still find good discovery prospects in FASER$\nu$. The $f_{s}=0$
hypothesis will be then excluded at $5\sigma$ for $f_{s}\gtrsim 0.2$. This is
driven by the low-energy part of the $\nu_{e}$ spectrum, in which significant
deviations from all the MC predictions are expected for $f_{s}$ of order tens
of percent. The capabilities of FASER$\nu$ in probing this effect will be
further enhanced by combining the data gathered by this detector and the
SND@LHC experiment. We conclude that the ongoing far-forward neutrino physics
program at the LHC will be able to decisively test benchmark models predicting
a few tens of percent pion to kaon swapping fractions in forward collisions at
the relevant energy and probe this solution to the CR muon puzzle.

While LHC Run 3 searches will already place strong constraints on this
scenario, it is also possible that the swapping probability might not be a
constant factor. In particular, it can depend on the mass number of colliding
nuclei and become more substantial for increasing $A$, while it could be less
pronounced in $pp$ collisions Anchordoqui:2016oxy ; Anchordoqui:2022fpn . In
addition, the impact of energy and pseudorapidity dependence of $f_{s}$ on the
CR data has recently been studied in Ref. Sciutto:2023zuz . It has been shown
that introducing such dependence can, e.g., allow for solving the puzzle for
the linearly increasing $f_{s}$ parameter with growing energy. This would
predict smaller values of $f_{s}^{\textrm{(LHC)}}$ at LHC energies, while the
maximum value $f_{s}^{\textrm{(max)}}$ would still be large and substantially
modify the kaon production rate at higher energies. In the example discussed
therein, one can estimate $f_{s}^{\textrm{(LHC)}}\sim 0.005$ if
$f_{s}^{\textrm{(max)}}\sim 0.5$ is assumed. The muon puzzle can still be
solved in this case. It is then possible that only a more subtle impact of the
enhanced strangeness scenario could be seen in $pp$ collisions at the LHC.

Going beyond a few percent precision might be then crucial to probe this
scenario in the far-forward LHC searches. This will be possible with the
proposed FPF experiments. In the bottom left panels of Fig. 4, we show in gray
the expected uncertainty bands on the electron and muon neutrino spectra in
FLArE obtained similarly to Fig. 1. On top of this, we show the predicted
deviations for the pion-to-kaon swapping probability of $1.3\%$, $0.43\%$, and
$0.14\%$. These correspond to the FASER$\nu$ $1\sigma$ exclusion bound
discussed above and to FLArE constraints obtained with either $10\%$ of data
or the full dataset. As can be seen, within the first one to two years of data
taking, FLArE will surpass the ongoing LHC searches by a factor of a few in
probing the $f_{s}$ parameter. The improvement by about an order of magnitude
in $f_{s}$ is expected after the entire HL-LHC era such that sub-percent
values of this parameter will be tested.

We summarize the expected bounds on $f_{s}$ in the right panel of Fig. 4. In
the plot, we indicate with a dark green color the preferred range of values of
the $f_{s}$ parameter that could explain the CR muon puzzle. We put it between
$0.3$ and $0.8$ following the results present in Ref. Anchordoqui:2022fpn  for
concreteness. We also show in the plot an extended light green band towards
lower values of $f_{s}\sim 0.005$, which refers to the possible smaller
magnitude of this effect in $pp$ collisions at the LHC. On top of this, we
show in turquoise the $f_{s}$ ranges that can lead to discovery in the ongoing
FASER$\nu$ searches, based on either the full neutrino data or a limited
dataset to $\nu_{\mu}$ and low-energy electron neutrinos. We also present in
the figure a set of gray-shaded exclusion bands at $2\sigma$ obtained for
FASER$\nu$ with three different baseline MC generators and for FLArE with the
entire or limited data sets, as discussed above. The proposed FLArE experiment
will probe this scenario up to $\mathcal{O}(0.1\%)$ level in $f_{s}$, below
which barely any effect on the $pp$ final-state meson distribution is
expected.

###  IV.2 Neutrino Charged Current Non-Standard Interactions

One of the major developments of the far-forward neutrino physics program at
the LHC is the possibility of studying CC interactions of the tau neutrinos at
the TeV energy scale on an event-by-event basis. This is thanks to the
exceptional capabilities of the currently operating emulsion detectors that
could be further improved in the future in the FPF experiments. Below, we
discuss how these searches can help to constrain possible new physics
contributions to high-energy neutrino interactions, cf. Refs.
Bahraminasr:2020ssz ; Kling:2020iar ; Bakhti:2020szu ; Jodlowski:2020vhr ;
Ismail:2020yqc ; Falkowski:2021bkq ; Ismail:2021dyp ; Ansarifard:2021elw ;
Kelly:2021mcd ; Cheung:2021tmx ; Ansarifard:2021dju ; Cheung:2022oji ;
Aloni:2022ebm ; MammenAbraham:2023psg ; Cheung:2023gwm ; Asai:2023xxl  for
other studies regarding far-forward neutrinos and new physics.

In the SM, the CC neutrino scatterings off nuclei are driven by the $W$ boson
exchange. BSM contributions that could modify these interaction rates are
typically associated with new physics at the scale above the characteristic
momentum transfer in neutrino interactions at the LHC, especially if they go
beyond the SM-like V-A interactions that could be affected by pure neutrino-
philic species, cf. Ref. Kelly:2021mcd  for sample such analysis for forward
LHC searches. Therefore, a convenient way to describe such BSM-induced
interactions is via an effective field theory (EFT) approach. The typical
momentum transfer in CC DIS neutrino scatterings at the LHC,
$Q\sim\mathcal{O}(10~{}\textrm{GeV})$, and we require the new physics scale to
remain higher, $\Lambda\gg Q$, for the validity of the EFT.

The sensitivity reach of FASER$\nu$ to a number of such operators that could
arise, e.g., within the framework of the weak EFT Jenkins:2017jig ;
Falkowski:2019xoe ; Falkowski:2019kfn , has been studied in Ref.
Falkowski:2021bkq  and competitive exclusion bounds have been found for some
of them, primarily related to $\nu_{\tau}$-like CC scattering signatures.
Here, for illustration, we focus on two such right-handed operators that are
described by the following Lagrangian

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{2\,V_{ud}}{v^{2}}\times(\bar{u}\gamma^{\kappa}P_{R}d)\times$
$\displaystyle\left[\epsilon_{R}^{\mu\tau}\,(\bar{\ell}_{\mu}\gamma_{\kappa}P_{L}\nu_{\tau})+\epsilon_{R}^{\tau
e}\,(\bar{\ell}_{\tau}\gamma_{\kappa}P_{L}\nu_{e})\right],$

where we use $V_{ud}$ as the relevant entry of the Cabibbo-Kobayashi-Maskawa
(CKM) matrix, $v\simeq 246~{}\textrm{GeV}$ is the SM Higgs vacuum expectation
value, and $\epsilon_{R}^{\alpha\beta}$ are the respective Wilson coefficients
describing neutrino NSI.

The presence of neutrino NSI would affect both production and interaction
rates of neutrinos. We follow the discussion of Ref. Falkowski:2021bkq  and
apply the neutrino detection and production coefficients modified by new
physics contributions derived therein. In particular, it has been found that
these coefficients are not expected to vary significantly with the incident
neutrino energy in the range relevant to the far-forward LHC searches. Hence,
they are not strongly sensitive to precise modeling of the neutrino energy
spectrum. Still, new physics can lead to distinct features in the LHC data by
modifying the spectra for only selected neutrino flavors and parent mesons.

We extend the previous analysis by including the modeling of MC prediction
uncertainties, as discussed in Sec. II. The bounds presented below are
obtained after profiling over all the nuisance parameters describing the
neutrino spectra variations. These variations could a priori surpass the
impact of neutrino NSI and should be considered in estimating new physics
reach. As we present below, however, this effect does not significantly limit
the sensitivity of the FPF experiments, at least for EFT operators selected in
our analysis. In our analysis, we consider both the energy and spatial
distribution of events in the detectors. For the latter distribution, we
consider three radial bins for both FASER$\nu$ and FASER$\nu$2\. We focus on
the emulsion detectors with the best capabilities to study $\nu_{\tau}$
interactions.

We present the results of our analysis in Fig. 5. In the left panel, we show
gray-shaded uncertainty bands on the electron, muon, and tau-neutrino CC
scattering rates in FASER$\nu$2\. In this case, no impact of new physics has
been assumed. The baseline model is chosen to be an average of the
predictions, similar to the results discussed in Sec. III. On top of this, we
also present colorful lines representing predicted deviations from the
baseline scenario due to the presence of neutrino NSI. These have been
obtained by simultaneously changing both the Wilson coefficients mentioned
above and the nuisance parameters describing MC variations. We subsequently
profile over all the parameters besides either $\epsilon_{R}^{\mu\tau}$ or
$\epsilon_{R}^{\tau e}$.

Figure 5: Left: The uncertainties for the neutrino CC event scattering rates
at FASER$\nu$2, assuming 100% of the data collected and using three radial
bins, along with the NSI parameters $\epsilon_{\tau e}$ and
$\epsilon_{\mu\tau}$ set to the obtained constraints. Right: The projected
FASER$\nu$2 constraints are compared to those obtainable using only $10\%$ of
the expected data and those attainable with $100\%$ of the expected FASER$\nu$
data. Current bounds on the respective Wilson coefficients are shown with
gray-shaded bands.

The former Wilson coefficient $\epsilon_{R}^{\mu\tau}$ is related to the
operator which couples the (charged) muon and tau neutrino. It primarily
affects the neutrino production rate by inducing a non-zero branching fraction
for the process, $\pi\to\mu\nu_{\tau}$, which enhances the tau neutrino flux.
This operator could also induce CC scatterings of the tau neutrinos leading to
the final state muons, $\nu_{\tau}N\to\mu X$. Such a process would reduce the
number of events reconstructed as $\nu_{\tau}$-like CC scattering
interactions, given the lack of the final-state tau lepton $\tau$. However,
the net impact on the $\nu_{\tau}$ production rate, i.e., the increase of the
$\nu_{\tau}$ flux, is significantly more substantial. It is driven by a large
flux of parent pions that, otherwise, never produce tau neutrinos.

On the other hand, the Wilson coefficient $\epsilon_{R}^{\tau e}$ couples
$\nu_{e}$ and the tau lepton $\tau$. In this case, the impact on the
$\nu_{\tau}$-like detection rate is more significant, and it is determined by
the NSI-induced CC electron neutrino scatterings, $\nu_{e}N\to\tau X$, which
mimic interactions of the tau neutrinos. The presence of this operator does
not induce any additional significant production modes for the electron
neutrinos together with the tau lepton. The dominant such modes would be
associated with decays of charm hadrons and then related to operators
involving quarks from the second generation.

The projected bounds on both the coefficients considered individually that we
obtain at $1\sigma$ and for FASER$\nu$2 read: $|\epsilon_{R}^{\tau e}|<0.0158$
and $|\epsilon_{R}^{\mu\tau}|<0.0034$. The resulting deviations from the
baseline tau neutrino spectrum are at the $\mathcal{O}(1\%)$ level for the
$\tau e$ operator, as shown with the purple line in the top left panel of Fig.
5. They do not depend significantly on the incident neutrino energy. This is
because the corresponding impact of new physics on the tau neutrino detection
rate only mildly depends on $E_{\nu}$. Instead, in the $\mu\tau$ case, the
deviations from the baseline spectrum show clear energy dependence. Notably,
in the SM, pion decay contribution to the muon neutrino far-forward spectrum
at the LHC dominates at energies below a few hundred GeV. It is then this
energy regime in which one expects the most significant enhanced production of
$\nu_{\tau}$s from rare NSI-induced pion decays, which is the reason behind
the observed enhanced effect.

We note that the observation of new physics in interactions at
$E_{\nu_{\tau}}\sim$ few tens of GeV could be affected by a decreasing vertex
detection efficiency in emulsion for lowering energies FASER:2019dxq . In
order to estimate the impact of this effect on our NSI results, we have
additionally studied FASER$\nu$2 bounds after applying the relevant effect and
lepton detection efficiency. To this end, we have employed the same efficiency
functions as in FASER$\nu$, cf. Sec. II.3 for discussion. The projected bounds
found this way are about $20\%$ less strong for the $\tau e$ operator. The
weakening of the predicted constraints is more pronounced for the $\mu\tau$
operator. The excluded value grows by about $30\%$, as expected from a
stronger energy dependence of the NSI effect in this case. In general,
however, we find that both operators can be constrained well in FASER$\nu$2
even for decreasing detection efficiency at lower energies. The precise
constraining power will be further sensitive to PDF uncertainties, as
discussed in Sec. II.3.

In the central and bottom left panels of Fig. 5, we also show with colorful
lines the expected NSI-driven deviations from the baseline CC scattering rates
for the electron and muon neutrinos. As can be seen, these are significantly
smaller than for the tau neutrinos. The observed difference is due to much
larger expected scattering rates for $\nu_{e}$ and $\nu_{\mu}$ that are less
sensitive to small variations in the number of events than $\nu_{\tau}$s. We
note that the results of such analysis would be much different in the presence
of non-negligible neutrino oscillations in long-baseline neutrino experiments.
Instead, far-forward neutrino searches at the LHC combine capabilities of
short-baseline neutrino experiments with the potential to detect
$\nu_{\tau}$-induced CC scattering events directly.

The right panel of Fig. 5 corresponds to the results obtained after profiling
over all the nuisance parameters but without profiling over both the Wilson
coefficients. The projected bounds found this way are similar in constraining
power to the ones discussed above. At $90\%$ CL they read $|\epsilon^{\tau
e}_{R}|<0.026$ and $|\epsilon^{\mu\tau}_{R}|<0.0057$. Both considered EFT
operators affect the tau neutrino CC event scattering rate almost
independently. We also confirm this by finding that the relevant information
matrix is close to the diagonal. The expected constraining power of the far-
forward neutrino physics program at the LHC can be compared with other
searches. In the case of the $\tau e$ operator, the dominant such bounds of
$|\epsilon_{R}^{\tau e}|<0.12$ at $90\%$ CL have been derived in Ref.
Biggio:2009nt  based on past NOMAD constraints on $\nu_{e}$ oscillations into
$\nu_{\tau}$ NOMAD:2001xxt ; NOMAD:2003mqg . The $\mu\tau$ operator can be
currently best constrained by using the ratio of pion decay widths to the
electron and muon, $\Gamma(\pi\to e\nu_{e})/\Gamma(\pi\to\mu\nu_{\mu})$
Bhattacharya:2011qm ; ParticleDataGroup:2022pth . The bounds derived this way
are at the level of $|\epsilon_{R}^{\mu\tau}|<0.071$ at $90\%$ CL
Falkowski:2021bkq . As can be seen in the right panel of Fig. 5, the projected
FPF bounds can improve past limits by up to an order of magnitude and find new
leading limits already with the first $10\%$ of data. We additionally note
that in the presence of multiple Wilson coefficients describing non-vanishing
neutrino NSI, interesting cancellations can appear that might significantly
weaken these bounds in fine-tuned scenarios Bhattacharya:2011qm . In order to
better resolve such issues, measuring the final-state neutrino flavor remains
crucial, which further highlights the importance of neutrino NSI searches in
the FPF experiments.

We also comment on the importance of using double differential distributions
in these analyses. Given a relatively small transverse size of both FASER$\nu$
and FASER$\nu$2, we find only mild improvement in using three radial bins over
not considering the spatial distribution of events. However, going to larger
pseudorapidity regimes could visibly strengthen the bounds. We have
numerically studied this by extending the search to $1~{}\textrm{m}$ away from
the beam-collision axis, i.e., to the distance characteristic for FLArE. The
proposed AdvSND detector could extend this coverage even further. Based on our
analysis, we expect further $\mathcal{O}(10\%)$ improvement in the NSI bounds
on $\epsilon_{R}^{\mu\tau}$ and $\epsilon_{R}^{\tau e}$ from analyzing the
data in the full pseudorapidity range of the FPF experiments.

Finally, it is instructive to comment on an approximate scale of heavy new
physics species $\Lambda$, which could be involved in generating the low-
energy operators of our interest. This could be obtained by matching our
operators to the SMEFT operators above the electroweak (EW) scale
Cirigliano:2012ab ; Jenkins:2017jig ; Dekens:2019ept . In this case, off-
diagonal right-handed EFT operators receive only $\Lambda^{-4}$ corrections
Falkowski:2019xoe . The FASER$\nu$2 bounds found above could then be
translated into about $\Lambda=v/\epsilon^{1/4}\simeq 600~{}\textrm{GeV}$ and
$900~{}\textrm{GeV}$ at $90\%$CL for the $\tau e$ and $\mu\tau$ operators,
respectively.

##  V Conclusions

When estimating the discovery potential of a novel experimental program, it
always remains crucial to properly consider possible Standard Model effects
and related uncertainties that could mimic new phenomena. Breaking this
degeneracy is also essential for understanding the expected impact of the
recently started far-forward neutrino physics program at the LHC. In the
current work, we have made an important step in this direction.

We have proposed parameterizing the expected neutrino spectra by combining the
leading predictions based on various approaches to modeling forward parent
hadron spectra. The parameterized flux model obtained this way is
characterized by 12 nuisance parameters describing the variations in neutrino
spectrum normalization and shape. Importantly, these variations take into
account expected correlations between the neutrino spectra of different
flavors. We then estimated how well the current and proposed forward LHC
neutrino experiments can constrain this model. Our analysis considers
information about the neutrino charged-current interaction rates for different
flavors, energies, and pseudorapidities.

In particular, we have shown that the future Forward Physica Facility data
will allow for constraining the LHC neutrino fluxes up to even a sub-percent
level for $\nu_{e}$ and $\nu_{\mu}$, i.e., to precision at which additional
PDF uncertainties affecting neutrino interaction rates become important. These
will be reduced thanks to future EIC and FPF measurements. The FPF data will
then allow for differentiating between various MC predictions with high
precision. Instead, the expected uncertainty bands are of order few percent
for the tau neutrinos.

The forward LHC neutrino data will also allow for further improving the tunes
of the MC tools used to predict the parent hadron spectra. This will
profoundly affect our understanding of cosmic-ray physics, including the
possibility of solving the puzzling excess of the muon rate observed in CR-
induced air showers at ultrahigh energies. We have analyzed a recently
proposed solution to this problem based on the pion-to-kaon swapping among
products of high-energy $pp$ collisions at large pseudorapidities. Our study
shows that the currently operating FASER$\nu$ detector offers excellent
capabilities to probe this scenario within the next few years of LHC Run 3.
Future FPF searches could further improve relevant bounds on the swapping
fraction up to sub-percent precision.

New physics contributions to neutrino interactions can also be probed this
way. We have illustrated this for a $\nu_{\tau}$-like signature of CC
interactions for TeV-scale energies of incident neutrinos. These can be
measured on an event-by-event basis in the far-forward emulsion detectors at
the LHC. We have tested a scenario in which two Wilson coefficients describing
BSM right-handed couplings of quarks to charged leptons and neutrinos are
varied simultaneously. We show that the unique effect of new physics can be
identified by employing full forward LHC neutrino data to disentangle NSI from
variations in MC predictions attributed to an insufficient understanding of
the forward hadron production. We have shown that selected Wilson coefficients
can be then constrained in the future FASER$\nu$2 detector with up to about an
order of magnitude better precision than current bounds.

One can extend the current work to other physics analyses. This includes,
i.a., specific effects predicted to modify neutrino production rates, e.g.,
intrinsic charm Brodsky:1980pb ; Brodsky:2015fna  or gluon saturation at small
$x$ Gribov:1983ivg ; Mueller:1985wy  that will affect the charm-induced tau
neutrino spectrum in the far-forward kinematic region. New physics could also
non-trivially manifest itself in the LHC neutrino data if oscillations into
sterile neutrinos are present FASER:2019dxq , cf. also recent discussion about
the discovery prospects for neutrino-modulino oscillations Anchordoqui:2023qxv
. The onset of a new era of precision neutrino physics at the LHC offers
exciting opportunities to improve our understanding of hadronic interactions
and the physics of the most elusive among SM particles.

## Acknowledgements

We thank Weidong Bai, Atri Bhattacharya, Luca Buonocore, Yu Seon Jeong, Rafal
Maciuła, Mary Hall Reno, Luca Rottoli, Ina Sarcevic, Anna M. Stasto, and
Antoni Szczurek for helpful discussions and for sharing the files used to
obtain the charm-induced neutrino spectra. We would like to thank Luis
Anchordoqui, Anatoli Fedynitch, Max Fieg, Tanguy Pierog, Felix Riehn, Dennis
Soldin for useful discussions and comments on the manuscript. We are grateful
to the authors and maintainers of many open-source software packages,
including Rivet Buckley:2010ar ; Bierlich:2019rhm  and scikit-hep
Rodrigues:2020syo . FK acknowledges support by the Deutsche
Forschungsgemeinschaft under Germany’s Excellence Strategy - EXC 2121 Quantum
Universe - 390833306. TM and ST are supported by the National Science Centre,
Poland, research grant No. 2021/42/E/ST2/00031. ST is also supported by the
grant “AstroCeNT: Particle Astrophysics Science and Technology Centre” carried
out within the International Research Agendas programme of the Foundation for
Polish Science financed by the European Union under the European Regional
Development Fund. ST is additionally partly supported by the European Union’s
Horizon 2020 research and innovation program under grant agreement No 952480
(DarkWave).

##  Appendix A Application of the Cram$\acute{\textrm{e}}$r-Rao bound to
forward LHC neutrino measurements

As discussed in Sec. II.2, we interpolate between established predictions for
the forward neutrino spectra to obtain the expected number of neutrino
interaction events in each of the detectors considered in our study. Here, we
discuss further steps of our statistical analysis.

The observables in the binned histogram analysis are the numbers of events
$n_{i}$ observed in each $i$th bin. The likelihood function is obtained as a
product of the Poisson likelihoods for all bins

$L(\text{data}|\text{model})=\prod_{\text{bins}~{}i}\text{Pois}(n_{i}|N_{i})=\prod_{\text{bins}~{}i}\frac{N_{i}^{n_{i}}e^{-N_{i}}}{n_{i}!},$
(3)

where $N_{i}$ is the expected number of events per bin in the model. In the
following, we provide a function for the expected log-likelihood ratio $\log
r$, where the likelihood ratio with respect to the baseline model reads

$r(\lambda^{\pi},\lambda^{K},\lambda^{c})=\frac{L(\text{expected
data}|\lambda^{\pi},\lambda^{K},\lambda^{c})}{L(\text{expected
data}|\lambda^{\pi}=0,\lambda^{K}=0,\lambda^{c}=0)}$ (4)

with the expected data corresponding to
$\lambda^{\pi}=\lambda^{K}=\lambda^{c}=0$.

The expected likelihood ratio is approximated as

$-2\log r=-\frac{d^{2}\log
r}{d\lambda^{(i)}d\lambda^{(j)}}\Delta\lambda^{(i)}\Delta\lambda^{(j)}=I_{ij}\Delta\lambda^{(i)}\Delta\lambda^{(j)},$
(5)

where $(i),(j)$ run over all parent hadrons $\pi,K,c$ for all generators, and
$I_{ij}$ are the components of the Fisher Information matrix. By the
Cramér–Rao bound cramer-rao ; cramer-rao2 , the smallest uncertainty
achievable in the measurement is then obtained when the covariance matrix
$\mathrm{C}_{ij}=I_{ij}^{-1}$. To avoid introducing additional numerical
uncertainty in the computation of the Fisher information, the expected number
of events per bin in the model is generalized into a real positive parameter
in Eq. (3). The uncertainty bands for the neutrino spectra are obtained by
solving for the eigenvalues and -vectors of the information matrix. The model
is then varied from the baseline to the direction of each eigenvector
individually, and the uncertainty in each bin is obtained as the square root
of the quadratic sum of the differences of each variation to the baseline.
When using multiple radial bins, the uncertainty $\delta_{i}$ for each $i$-th
radial bin is first computed in the aforementioned way. These are then
combined as
$\delta_{\textrm{tot}}=\sqrt{\sum_{i}\delta_{i}^{2}}\left(\sum_{i}\delta_{i}\right)^{-1}$,
separately for all energy bins, yielding the total uncertainty shown in the
spectrum plots. In the present work, the uncertainties of all spectra are
reported at the $1\sigma$ level. Results corresponding to different
statistical significance are also provided in selected cases in Sec. IV.

We use a profiling procedure amounting to a parallel projection of a
generalized ellipsoid in the parameter space to estimate the constraints that
can be obtained for a parameter used in the model computation. To profile over
the $n$-th parameter in the information matrix $I$, the $n$-th column (or row)
of $I$, with the $n$-th entry removed, is taken as the vector $\mathbf{m}$
describing the mixing between the profiled parameter and the remainder. A
reduced information matrix $I^{\rm reduced}$ is attained by removing the
$n$-th column and row from $I$, and the profiled information matrix is given
by Brehmer:2016nyr

$I^{\rm profiled}=I^{\rm reduced}-\mathbf{m}\otimes\mathbf{m}/I_{nn}.$ (6)

The procedure is repeated to profile over multiple parameters, starting with
the information matrix resulting from the previous step. By profiling over all
but one parameter, the information matrix eventually reduces into a single
entry $a$, and the ultimate constraint for the remaining parameter is then
obtained as $a^{-1/2}$.

## References

  * (1) J. A. Formaggio and G. P. Zeller, “From eV to EeV: Neutrino Cross Sections Across Energy Scales,” Rev. Mod. Phys. 84 (2012) 1307–1341, arXiv:1305.7513 [hep-ex].
  * (2) Neutrino Non-Standard Interactions: A Status Report, vol. 2.  2019\.  arXiv:1907.00991 [hep-ph].
  * (3) M. Sajjad Athar et al., “Status and perspectives of neutrino physics,” Prog. Part. Nucl. Phys. 124 (2022) 103947, arXiv:2111.07586 [hep-ph].
  * (4) A. De Rujula and R. Ruckl, “Neutrino and muon physics in the collider mode of future accelerators,” in SSC Workshop: Superconducting Super Collider Fixed Target Physics, pp. 571–596.  5, 1984.
  * (5) K. Winter, “Detection of the tau-neutrino at the LHC,” in ECFA Large Hadron Collider (LHC) Workshop: Physics and Instrumentation, pp. 37–49.  1990\.
  * (6) F. Vannucci, “Neutrino physics at LHC / SSC,” in 4th International Symposium on Neutrino Telescopes.  3, 1993.
  * (7) A. De Rujula, E. Fernandez, and J. J. Gomez-Cadenas, “Neutrino fluxes at future hadron colliders,” Nucl. Phys. B 405 (1993) 80–108.
  * (8) H. Park, “The estimation of neutrino fluxes produced by proton-proton collisions at $\sqrt{s}=14$ TeV of the LHC,” JHEP 10 (2011) 092, arXiv:1110.1971 [hep-ex].
  * (9) J. L. Feng, I. Galon, F. Kling, and S. Trojanowski, “ForwArd Search ExpeRiment at the LHC,” Phys. Rev. D 97 (2018) no. 3, 035001, arXiv:1708.09389 [hep-ph].
  * (10) N. Beni et al., “Further studies on the physics potential of an experiment using LHC neutrinos,” J. Phys. G 47 (2020) no. 12, 125004, arXiv:2004.07828 [hep-ex].
  * (11) FASER Collaboration, A. Ariga et al., “Letter of Intent for FASER: ForwArd Search ExpeRiment at the LHC,” arXiv:1811.10243 [physics.ins-det].
  * (12) FASER Collaboration, A. Ariga et al., “Technical Proposal for FASER: ForwArd Search ExpeRiment at the LHC,” arXiv:1812.09139 [physics.ins-det].
  * (13) FASER Collaboration, H. Abreu et al., “Detecting and Studying High-Energy Collider Neutrinos with FASER at the LHC,” Eur. Phys. J. C 80 (2020) no. 1, 61, arXiv:1908.02310 [hep-ex].
  * (14) FASER Collaboration, H. Abreu et al., “Technical Proposal: FASERnu,” arXiv:2001.03073 [physics.ins-det].
  * (15) FASER Collaboration, H. Abreu et al., “The FASER Detector,” arXiv:2207.11427 [physics.ins-det].
  * (16) SHiP Collaboration, C. Ahdida et al., “SND@LHC,” arXiv:2002.08722 [physics.ins-det].
  * (17) SND@LHC Collaboration, G. Acampora et al., “SND@LHC: The Scattering and Neutrino Detector at the LHC,” arXiv:2210.02784 [hep-ex].
  * (18) FASER Collaboration, H. Abreu et al., “First Direct Observation of Collider Neutrinos with FASER at the LHC,” arXiv:2303.14185 [hep-ex].
  * (19) SND@LHC Collaboration, R. Albanese et al., “Observation of collider muon neutrinos with the SND@LHC experiment,” arXiv:2305.09383 [hep-ex].
  * (20) FASERcollaboration Collaboration, “Observation of high-energy electron neutrino interactions with FASER’s emulsion detector at the LHC,” tech. rep., CERN, Geneva, 2023.  https://cds.cern.ch/record/2868284.
  * (21) FASER Collaboration, H. Abreu et al., “First neutrino interaction candidates at the LHC,” Phys. Rev. D 104 (2021) no. 9, L091101, arXiv:2105.06197 [hep-ex].
  * (22) J. Arakawa, J. L. Feng, A. Ismail, F. Kling, and M. Waterbury, “Neutrino detection without neutrino detectors: Discovering collider neutrinos at FASER with electronic signals only,” Phys. Rev. D 106 (2022) no. 5, 052011, arXiv:2206.09932 [hep-ph].
  * (23) L. A. Anchordoqui et al., “The Forward Physics Facility: Sites, experiments, and physics potential,” Phys. Rept. 968 (2022) 1–50, arXiv:2109.10905 [hep-ph].
  * (24) J. L. Feng et al., “The Forward Physics Facility at the High-Luminosity LHC,” J. Phys. G 50 (2023) no. 3, 030501, arXiv:2203.05090 [hep-ex].
  * (25) J. M. Cruz-Martinez, M. Fieg, T. Giani, P. Krack, T. Mäkelä, T. Rabemananjara, and J. Rojo, “The LHC as a Neutrino-Ion Collider,” arXiv:2309.09581 [hep-ph].
  * (26) LHC Forward Physics Working Group Collaboration, K. Akiba et al., “LHC Forward Physics,” J. Phys. G 43 (2016) 110201, arXiv:1611.05079 [hep-ph].
  * (27) F. Kling and L. J. Nevay, “Forward neutrino fluxes at the LHC,” Phys. Rev. D 104 (2021) no. 11, 113008, arXiv:2105.08270 [hep-ph].
  * (28) C. Schmidt, J. Pumplin, C. P. Yuan, and P. Yuan, “Updating and optimizing error parton distribution function sets in the Hessian approach,” Phys. Rev. D 98 (2018) no. 9, 094005, arXiv:1806.07950 [hep-ph].
  * (29) R. C. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81 (1945) .
  * (30) H. Cramér, “Mathematical Methods of Statistics,” Princeton University Press (1946) .
  * (31) L. A. Anchordoqui, H. Goldberg, and T. J. Weiler, “Strange fireball as an explanation of the muon excess in Auger data,” Phys. Rev. D 95 (2017) no. 6, 063005, arXiv:1612.07328 [hep-ph].
  * (32) S. Baur, H. Dembinski, M. Perlin, T. Pierog, R. Ulrich, and K. Werner, “Core-corona effect in hadron collisions and muon production in air showers,” Phys. Rev. D 107 (2023) no. 9, 094031, arXiv:1902.09265 [hep-ph].
  * (33) L. A. Anchordoqui, C. G. Canal, F. Kling, S. J. Sciutto, and J. F. Soriano, “An explanation of the muon puzzle of ultrahigh-energy cosmic rays and the role of the Forward Physics Facility for model improvement,” JHEAp 34 (2022) 19–32, arXiv:2202.03095 [hep-ph].
  * (34) F. Kling, “Probing light gauge bosons in tau neutrino experiments,” Phys. Rev. D 102 (2020) no. 1, 015007, arXiv:2005.03594 [hep-ph].
  * (35) A. Ismail, R. Mammen Abraham, and F. Kling, “Neutral current neutrino interactions at FASER$\nu$,” Phys. Rev. D 103 (2021) no. 5, 056014, arXiv:2012.10500 [hep-ph].
  * (36) A. Falkowski, M. González-Alonso, J. Kopp, Y. Soreq, and Z. Tabrizi, “EFT at FASER$\nu$,” JHEP 10 (2021) 086, arXiv:2105.12136 [hep-ph].
  * (37) B. Batell, J. L. Feng, M. Fieg, A. Ismail, F. Kling, R. M. Abraham, and S. Trojanowski, “Hadrophilic dark sectors at the Forward Physics Facility,” Phys. Rev. D 105 (2022) no. 7, 075001, arXiv:2111.10343 [hep-ph].
  * (38) E.-J. Ahn, R. Engel, T. K. Gaisser, P. Lipari, and T. Stanev, “Cosmic ray interaction event generator SIBYLL 2.1,” Phys. Rev. D 80 (2009) 094003, arXiv:0906.4113 [hep-ph].
  * (39) E.-J. Ahn, R. Engel, T. K. Gaisser, P. Lipari, and T. Stanev, “Sibyll with charm,” in 16th International Symposium on Very High Energy Cosmic Ray Interactions.  2, 2011.  arXiv:1102.5705 [astro-ph.HE].
  * (40) F. Riehn, R. Engel, A. Fedynitch, T. K. Gaisser, and T. Stanev, “A new version of the event generator Sibyll,” PoS ICRC2015 (2016) 558, arXiv:1510.00568 [hep-ph].
  * (41) A. Fedynitch, F. Riehn, R. Engel, T. K. Gaisser, and T. Stanev, “Hadronic interaction model sibyll 2.3c and inclusive lepton fluxes,” Phys. Rev. D 100 (2019) no. 10, 103018, arXiv:1806.04140 [hep-ph].
  * (42) T. Pierog, I. Karpenko, J. M. Katzy, E. Yatsenko, and K. Werner, “EPOS LHC: Test of collective hadronization with data measured at the CERN Large Hadron Collider,” Phys. Rev. C 92 (2015) no. 3, 034906, arXiv:1306.0121 [hep-ph].
  * (43) L. Buonocore, F. Kling, L. Rottoli, and J. Sominka, “Predictions for neutrinos and new physics from forward heavy hadron production at the lhc,”. In preparation.
  * (44) S. Roesler, R. Engel, and J. Ranft, “The Monte Carlo event generator DPMJET-III,” in International Conference on Advanced Monte Carlo for Radiation Physics, Particle Transport Simulation and Applications (MC 2000), pp. 1033–1038.  12, 2000.  arXiv:hep-ph/0012252.
  * (45) A. Fedynitch, Cascade equations and hadronic interactions at very high energies.  PhD thesis, KIT, Karlsruhe, Dept. Phys., 11, 2015.
  * (46) W. Bai, M. Diwan, M. V. Garzelli, Y. S. Jeong, and M. H. Reno, “Far-forward neutrinos at the Large Hadron Collider,” JHEP 06 (2020) 032, arXiv:2002.03012 [hep-ph].
  * (47) W. Bai, M. Diwan, M. V. Garzelli, Y. S. Jeong, F. K. Kumar, and M. H. Reno, “Parton distribution function uncertainties in theoretical predictions for far-forward tau neutrinos at the Large Hadron Collider,” JHEP 06 (2022) 148, arXiv:2112.11605 [hep-ph].
  * (48) W. Bai, M. Diwan, M. V. Garzelli, Y. S. Jeong, K. Kumar, and M. H. Reno, “Forward production of prompt neutrinos from charm in the atmosphere and at high energy colliders,” arXiv:2212.07865 [hep-ph].
  * (49) S. Ostapchenko, “Monte Carlo treatment of hadronic interactions in enhanced Pomeron scheme: I. QGSJET-II model,” Phys. Rev. D 83 (2011) 014018, arXiv:1010.1869 [hep-ph].
  * (50) A. Bhattacharya, F. Kling, I. Sarcevic, and A. M. Stasto, “Forward Neutrinos from Charm at Large Hadron Collider,” arXiv:2306.01578 [hep-ph].
  * (51) M. Fieg, F. Kling, H. Schulz, and T. Sjöstrand, “Tuning Pythia for Forward Physics Experiments,” arXiv:2309.08604 [hep-ph].
  * (52) R. Maciula and A. Szczurek, “Far-forward production of charm mesons and neutrinos at forward physics facilities at the LHC and the intrinsic charm in the proton,” Phys. Rev. D 107 (2023) no. 3, 034002, arXiv:2210.08890 [hep-ph].
  * (53) C. Baus, T. Pierog, and R. Ulrich, “Cosmic Ray Monte Carlo (CRMC),”. https://web.ikp.kit.edu/rulrich/crmc.html.
  * (54) J. Albrecht et al., “The Muon Puzzle in cosmic-ray induced air showers and its connection to the Large Hadron Collider,” Astrophys. Space Sci. 367 (2022) no. 3, 27, arXiv:2105.06148 [astro-ph.HE].
  * (55) LHCf Collaboration, O. Adriani et al., “Measurement of forward neutral pion transverse momentum spectra for $\sqrt{s}$ = 7TeV proton-proton collisions at LHC,” Phys. Rev. D 86 (2012) 092001, arXiv:1205.4578 [hep-ex].
  * (56) LHCf Collaboration, O. Adriani et al., “Measurement of very forward neutron energy spectra for 7 TeV proton–proton collisions at the Large Hadron Collider,” Phys. Lett. B 750 (2015) 360–366, arXiv:1503.03505 [hep-ex].
  * (57) LHCf Collaboration, O. Adriani et al., “Measurements of longitudinal and transverse momentum distributions for neutral pions in the forward-rapidity region with the LHCf detector,” Phys. Rev. D 94 (2016) no. 3, 032007, arXiv:1507.08764 [hep-ex].
  * (58) O. Adriani et al., “Measurement of the forward $\eta$ meson production rate in p-p collisions at $\sqrt{s}$=13 TeV with the LHCf-Arm2 detector,” arXiv:2305.06633 [hep-ex].
  * (59) P. Nason, “A New method for combining NLO QCD with shower Monte Carlo algorithms,” JHEP 11 (2004) 040, arXiv:hep-ph/0409146.
  * (60) S. Frixione, P. Nason, and C. Oleari, “Matching NLO QCD computations with Parton Shower simulations: the POWHEG method,” JHEP 11 (2007) 070, arXiv:0709.2092 [hep-ph].
  * (61) S. Frixione, P. Nason, and G. Ridolfi, “A Positive-weight next-to-leading-order Monte Carlo for heavy flavour hadroproduction,” JHEP 09 (2007) 126, arXiv:0707.3088 [hep-ph].
  * (62) S. Alioli, P. Nason, C. Oleari, and E. Re, “A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX,” JHEP 06 (2010) 043, arXiv:1002.2581 [hep-ph].
  * (63) R. D. Ball, V. Bertone, M. Bonvini, S. Marzani, J. Rojo, and L. Rottoli, “Parton distributions with small-x resummation: evidence for BFKL dynamics in HERA data,” Eur. Phys. J. C 78 (2018) no. 4, 321, arXiv:1710.05935 [hep-ph].
  * (64) V. Bertone, R. Gauld, and J. Rojo, “Neutrino Telescopes as QCD Microscopes,” JHEP 01 (2019) 217, arXiv:1808.02034 [hep-ph].
  * (65) PROSA Collaboration, O. Zenaiev, M. V. Garzelli, K. Lipka, S. O. Moch, A. Cooper-Sarkar, F. Olness, A. Geiser, and G. Sigl, “Improved constraints on parton distributions using LHCb, ALICE and HERA heavy-flavour measurements and implications for the predictions for prompt atmospheric-neutrino fluxes,” JHEP 04 (2020) 118, arXiv:1911.13164 [hep-ph].
  * (66) A. Dumitru, A. Hayashigaki, and J. Jalilian-Marian, “The Color glass condensate and hadron production in the forward region,” Nucl. Phys. A 765 (2006) 464–482, arXiv:hep-ph/0506308.
  * (67) M. Deak, F. Hautmann, H. Jung, and K. Kutak, “Forward Jet Production at the Large Hadron Collider,” JHEP 09 (2009) 121, arXiv:0908.0538 [hep-ph].
  * (68) K. Kutak, “Hard scale dependent gluon density, saturation and forward-forward dijet production at the LHC,” Phys. Rev. D 91 (2015) no. 3, 034021, arXiv:1409.3822 [hep-ph].
  * (69) C. Peterson, D. Schlatter, I. Schmitt, and P. M. Zerwas, “Scaling Violations in Inclusive e+ e- Annihilation Spectra,” Phys. Rev. D 27 (1983) 105.
  * (70) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, and P. Z. Skands, “An introduction to PYTHIA 8.2,” Comput. Phys. Commun. 191 (2015) 159–177, arXiv:1410.3012 [hep-ph].
  * (71) K. Cranmer, “Practical Statistics for the LHC,” in 2011 European School of High-Energy Physics, pp. 267–308.  2014\.  arXiv:1503.07622 [physics.data-an].
  * (72) J. Brehmer, K. Cranmer, F. Kling, and T. Plehn, “Better Higgs boson measurements through information geometry,” Phys. Rev. D 95 (2017) no. 7, 073002, arXiv:1612.05261 [hep-ph].
  * (73) J. Brehmer, F. Kling, T. Plehn, and T. M. P. Tait, “Better Higgs-CP Tests Through Information Geometry,” Phys. Rev. D 97 (2018) no. 9, 095017, arXiv:1712.02350 [hep-ph].
  * (74) B. Batell, J. L. Feng, and S. Trojanowski, “Detecting Dark Matter with Far-Forward Emulsion and Liquid Argon Detectors at the LHC,” Phys. Rev. D 103 (2021) no. 7, 075023, arXiv:2101.10338 [hep-ph].
  * (75) C. Andreopoulos et al., “The GENIE Neutrino Monte Carlo Generator,” Nucl. Instrum. Meth. A 614 (2010) 87–104, arXiv:0905.2517 [hep-ph].
  * (76) U.-K. Yang and A. Bodek, “Parton distributions, $d/u$, and higher twist effects at high x,” Phys. Rev. Lett. 82 (1999) 2467–2470, arXiv:hep-ph/9809480.
  * (77) A. Bodek and U. K. Yang, “Modeling deep inelastic cross-sections in the few GeV region,” Nucl. Phys. B Proc. Suppl. 112 (2002) 70–76, arXiv:hep-ex/0203009.
  * (78) A. Candido, A. Garcia, G. Magni, T. Rabemananjara, J. Rojo, and R. Stegeman, “Neutrino Structure Functions from GeV to EeV Energies,” JHEP 05 (2023) 149, arXiv:2302.08527 [hep-ph].
  * (79) K. Xie, J. Gao, T. J. Hobbs, D. R. Stump, and C. P. Yuan, “High-energy neutrino deeply inelastic scattering cross sections from 100 GeV to 1000 EeV,” arXiv:2303.13607 [hep-ph].
  * (80) Y. S. Jeong and M. H. Reno, “Neutrino Cross Sections: Interface of shallow- and deep-inelastic scattering for collider neutrinos,” arXiv:2307.09241 [hep-ph].
  * (81) A. Accardi et al., “Electron Ion Collider: The Next QCD Frontier: Understanding the glue that binds us all,” Eur. Phys. J. A 52 (2016) no. 9, 268, arXiv:1212.1701 [nucl-ex].
  * (82) Pierre Auger Collaboration, A. Aab et al., “Muons in Air Showers at the Pierre Auger Observatory: Mean Number in Highly Inclined Events,” Phys. Rev. D 91 (2015) no. 3, 032003, arXiv:1408.1421 [astro-ph.HE]. [Erratum: Phys.Rev.D 91, 059901 (2015)].
  * (83) Pierre Auger Collaboration, A. Aab et al., “Testing Hadronic Interactions at Ultrahigh Energies with Air Showers Measured by the Pierre Auger Observatory,” Phys. Rev. Lett. 117 (2016) no. 19, 192001, arXiv:1610.08509 [hep-ex].
  * (84) Pierre Auger Collaboration, A. Aab et al., “Measurement of the Fluctuations in the Number of Muons in Extensive Air Showers with the Pierre Auger Observatory,” Phys. Rev. Lett. 126 (2021) no. 15, 152002, arXiv:2102.07797 [hep-ex].
  * (85) J. Allen and G. Farrar, “Testing models of new physics with UHE air shower observations,” in 33rd International Cosmic Ray Conference, p. 1182.  7, 2013.  arXiv:1307.7131 [astro-ph.HE].
  * (86) ALICE Collaboration, J. Adam et al., “Enhanced production of multi-strange hadrons in high-multiplicity proton-proton collisions,” Nature Phys. 13 (2017) 535–539, arXiv:1606.07424 [nucl-ex].
  * (87) S. J. Sciutto, “Air showers, hadronic models, and muon production,” EPJ Web Conf. 210 (2019) 02007, arXiv:1904.12056 [astro-ph.HE].
  * (88) S. J. Sciutto, L. A. Anchordoqui, C. Garcia Canal, F. Kling, and J. F. Soriano, “Gauging the cosmic ray muon puzzle with the Forward Physics Facility,” PoS ICRC2023 (2023) 388, arXiv:2307.08634 [hep-ph].
  * (89) M. Bahraminasr, P. Bakhti, and M. Rajaee, “Sensitivities to secret neutrino interaction at FASER$\nu$,” J. Phys. G 48 (2021) no. 9, 095001, arXiv:2003.09985 [hep-ph].
  * (90) P. Bakhti, Y. Farzan, and S. Pascoli, “Discovery potential of FASER$\nu$ with contained vertex and through-going events,” JHEP 04 (2021) 075, arXiv:2010.16312 [hep-ph].
  * (91) K. Jodłowski and S. Trojanowski, “Neutrino beam-dump experiment with FASER at the LHC,” JHEP 05 (2021) 191, arXiv:2011.04751 [hep-ph].
  * (92) A. Ismail, S. Jana, and R. M. Abraham, “Neutrino up-scattering via the dipole portal at forward LHC detectors,” Phys. Rev. D 105 (2022) no. 5, 055008, arXiv:2109.05032 [hep-ph].
  * (93) S. Ansarifard and Y. Farzan, “Neutral exotica at FASER$\nu$ and SND@LHC,” JHEP 02 (2022) 049, arXiv:2109.13962 [hep-ph].
  * (94) K. J. Kelly, F. Kling, D. Tuckler, and Y. Zhang, “Probing neutrino-portal dark matter at the Forward Physics Facility,” Phys. Rev. D 105 (2022) no. 7, 075026, arXiv:2111.05868 [hep-ph].
  * (95) K. Cheung, C. J. Ouseph, and T. Wang, “Non-standard neutrino and Z’ interactions at the FASER$\nu$ and the LHC,” JHEP 12 (2021) 209, arXiv:2111.08375 [hep-ph].
  * (96) S. Ansarifard and Y. Farzan, “Excess of tau events at SND@LHC, FASER$\nu$ and FASER$\nu$2,” Eur. Phys. J. C 82 (2022) no. 6, 568, arXiv:2112.08799 [hep-ph].
  * (97) K. Cheung and C. J. Ouseph, “Constraining the transitional magnetic dipole moment interaction between active and heavy neutrinos with the $Z^{\prime}$ at FASER$\nu$,” Eur. Phys. J. C 83 (2023) no. 7, 593, arXiv:2205.11077 [hep-ph].
  * (98) D. Aloni and A. Dery, “Revisiting leptonic non-unitarity in light of FASER$\nu$,” arXiv:2211.09638 [hep-ph].
  * (99) R. Mammen Abraham, S. Foroughi-Abari, F. Kling, and Y.-D. Tsai, “Neutrino Electromagnetic Properties and the Weak Mixing Angle at the LHC Forward Physics Facility,” arXiv:2301.10254 [hep-ph].
  * (100) K. Cheung, T. T. Q. Nguyen, and C. J. Ouseph, “Leptoquark Search at the Forward Physics Facility,” arXiv:2302.05461 [hep-ph].
  * (101) K. Asai, A. Das, J. Li, T. Nomura, and O. Seto, “Probing for chiral $Z^{\prime}$ gauge boson through scattering measurement experiments,” arXiv:2307.09737 [hep-ph].
  * (102) E. E. Jenkins, A. V. Manohar, and P. Stoffer, “Low-Energy Effective Field Theory below the Electroweak Scale: Operators and Matching,” JHEP 03 (2018) 016, arXiv:1709.04486 [hep-ph].
  * (103) A. Falkowski, M. González-Alonso, and Z. Tabrizi, “Reactor neutrino oscillations as constraints on Effective Field Theory,” JHEP 05 (2019) 173, arXiv:1901.04553 [hep-ph].
  * (104) A. Falkowski, M. González-Alonso, and Z. Tabrizi, “Consistent QFT description of non-standard neutrino interactions,” JHEP 11 (2020) 048, arXiv:1910.02971 [hep-ph].
  * (105) C. Biggio, M. Blennow, and E. Fernandez-Martinez, “General bounds on non-standard neutrino interactions,” JHEP 08 (2009) 090, arXiv:0907.0097 [hep-ph].
  * (106) NOMAD Collaboration, P. Astier et al., “Final NOMAD results on muon-neutrino —$>$ tau-neutrino and electron-neutrino —$>$ tau-neutrino oscillations including a new search for tau-neutrino appearance using hadronic tau decays,” Nucl. Phys. B 611 (2001) 3–39, arXiv:hep-ex/0106102.
  * (107) NOMAD Collaboration, P. Astier et al., “Search for nu(mu) —$>$ nu(e) oscillations in the NOMAD experiment,” Phys. Lett. B 570 (2003) 19–31, arXiv:hep-ex/0306037.
  * (108) T. Bhattacharya, V. Cirigliano, S. D. Cohen, A. Filipuzzi, M. Gonzalez-Alonso, M. L. Graesser, R. Gupta, and H.-W. Lin, “Probing Novel Scalar and Tensor Interactions from (Ultra)Cold Neutrons to the LHC,” Phys. Rev. D 85 (2012) 054512, arXiv:1110.6448 [hep-ph].
  * (109) Particle Data Group Collaboration, R. L. Workman et al., “Review of Particle Physics,” PTEP 2022 (2022) 083C01.
  * (110) V. Cirigliano, M. Gonzalez-Alonso, and M. L. Graesser, “Non-standard Charged Current Interactions: beta decays versus the LHC,” JHEP 02 (2013) 046, arXiv:1210.4553 [hep-ph].
  * (111) W. Dekens and P. Stoffer, “Low-energy effective field theory below the electroweak scale: matching at one loop,” JHEP 10 (2019) 197, arXiv:1908.05295 [hep-ph]. [Erratum: JHEP 11, 148 (2022)].
  * (112) S. J. Brodsky, P. Hoyer, C. Peterson, and N. Sakai, “The Intrinsic Charm of the Proton,” Phys. Lett. B 93 (1980) 451–455.
  * (113) S. J. Brodsky, A. Kusina, F. Lyonnet, I. Schienbein, H. Spiesberger, and R. Vogt, “A review of the intrinsic heavy quark content of the nucleon,” Adv. High Energy Phys. 2015 (2015) 231547, arXiv:1504.06287 [hep-ph].
  * (114) L. V. Gribov, E. M. Levin, and M. G. Ryskin, “Semihard Processes in QCD,” Phys. Rept. 100 (1983) 1–150.
  * (115) A. H. Mueller and J.-w. Qiu, “Gluon Recombination and Shadowing at Small Values of x,” Nucl. Phys. B 268 (1986) 427–452.
  * (116) L. A. Anchordoqui, I. Antoniadis, K. Benakli, J. Cunat, and D. Lust, “Searching for neutrino-modulino oscillations at the Forward Physics Facility,” arXiv:2308.11476 [hep-ph].
  * (117) A. Buckley, J. Butterworth, D. Grellscheid, H. Hoeth, L. Lonnblad, J. Monk, H. Schulz, and F. Siegert, “Rivet user manual,” Comput. Phys. Commun. 184 (2013) 2803–2819, arXiv:1003.0694 [hep-ph].
  * (118) C. Bierlich et al., “Robust Independent Validation of Experiment and Theory: Rivet version 3,” SciPost Phys. 8 (2020) 026, arXiv:1912.05451 [hep-ph].
  * (119) E. Rodrigues et al., “The Scikit HEP Project – overview and prospects,” EPJ Web Conf. 245 (2020) 06028, arXiv:2007.03577 [physics.comp-ph].