# Augmenting conformers with structured state space models for online speech
recognition

###### Abstract

Online speech recognition, where the model only accesses context to the left,
is an important and challenging use case for ASR systems. In this work, we
investigate augmenting neural encoders for online ASR by incorporating
structured state-space sequence models (S4), which are a family of models that
provide a parameter-efficient way of accessing arbitrarily long left context.
We perform systematic ablation studies to compare variants of S4 models and
propose two novel approaches that combine them with convolutions. We find that
the most effective design is to stack a small S4 using real-valued recurrent
weights with a local convolution, allowing them to work complementarily. Our
best model achieves WERs of 4.01%/8.53% on test sets from Librispeech,
outperforming Conformers with extensively tuned convolution.

Index Terms—  Online ASR, causal model, state-space model, conformer

##  1 Introduction

The search of neural architectures for encoding audio features in automatic
speech recognition (ASR) has been an important and ongoing research direction.
The commonly used building blocks are DNNs [1], recurrent networks such as
LSTMs [2], convolutional neural networks (CNNs) [3], and multi-head self
attention (MHSA) [4]. These components have different inductive biases for
learning and characteristics for optimization. The recent trend is to leverage
capabilities of different components and carefully combine them. For example,
[5] and [6] proposed combining the global context as modeled by MHSA, and
local context as modeled by depthwise convolution (simply “convolution”
afterwards), to build state-of-the-art encoders for ASR, i.e., the conformers.

Recently, structured state-space sequence models (S4) [7] emerged as a
promising paradigm for sequence modeling. These models can be interpreted as a
combination of RNNs, CNNs, and classical state-space models (SSMs) [8]. For
simplicity, in this work we use “SSMs” as synonymous with this family of S4
models [7, 9, 10, 11, 12] as opposed to classical state-space models such as
the Kalman filter and Hidden Markov Models (HMMs). As sequence models for deep
learning, SSMs have been shown to have advantages such as being able to
capture long-term dependencies and enjoy time complexity sub-quadratic in
sequence length. Naturally, this raises the question of whether they can be
used to augment ASR encoders. There have been several studies in this
direction. [13] proposed an attention-free multi-head state-space (MH-SSM)
architecture that achieves competitive performance on offline ASR without
MHSA. Motivated by the CNN view of SSMs, [14] introduced the DSSformer for
offline ASR, where they replaced convolution components in the conformer
encoder with SSMs. Besides these encoder-focused studies, [15] used SSMs in
the attention-based decoder, where the per-layer MHSA is replaced with SSMs
while the cross-attention remains unchanged. All such work focused on offline
ASR where encoders have full context.

Our contributions In this paper, we investigate augmenting ASR encoders with
SSMs in both the offline setting, similar to [13] and [14], and the online
setting, which was not covered by previous studies. We systematically compared
the approach from [13], where the SSM is used as a drop-in replacement (DIR)
for convolution in conformers, with two novel approaches: one where the SSM is
stacked with a local convolution component (COM); another where the SSM is
used to reparameterized the finite-size kernel within the convolution
component (REP). In all three approaches, we present careful ablation studies
on various SSM design choices. We found that in online ASR, the DIR approach
produces word-error rates (WERs) on par with but no better that that from
conformers. On the other hand, both novel approaches that we propose produce
small but consistent WER reduction. In both online and offline settings, we
found that COM is the most effective approach. In addition, success of the REP
approach suggests that SSMs can be effective even when they are forced to
model local context, pointing to the need of further theoretical understanding
of their capabilities.

Fig. 1: Architecture of the augmented convolution module. GL=gated linear ;
LN=layer normalization; conv=convolution; BN=batch normalization. Inside the
dashed box are three approaches of using SSMs that we studied. DIR=drop-in
replacement, COM=combination, REP=reparameterization. Following conventions
from [5], the entire block shown here is referred to as the convolution
module, while we study ways to modify the convolution component within it.

##  2 State space models

In this work, we exclusively considered the S4D variants of SSMs [10], which
have been shown to be simpler to implement and cheaper to compute while being
as effective as older variants. We refer the readers to [10] for further
details and summarize the structure here. Each SSM has a dimension $N$, which
can be viewed as the hidden-state size. It contains 4 sets of trainable
parameters (using $H$ to denote input dimension): $N\times N$ weights
$\bm{A}$, $N\times H$ weights $\bm{B}$, $H\times N$ weights $\bm{C}$, and
residual weights $\bm{D}\in\mathbb{R}^{H\times H}$. Following [10], $\bm{B}$
are assumed to be uniformly $1$ and untrainable. For mapping a discrete-time
input $\\{\bm{x}_{k}\in\mathbb{R}^{H}\\}_{k=1,...,T}$ to a discrete-time
output $\\{\bm{y}_{k}\mathbb{R}^{H}\\}_{k=1,...,T}$ with a timestep $\Delta$,
this model is discretized via the zero-order hold method [9] as

$\displaystyle\bm{x}_{k}$
$\displaystyle=\bar{\bm{A}}\bm{x}_{k-1}+\bar{\bm{B}}\bm{u}_{k}\quad\quad\bm{y}_{k}=\bm{C}\bm{x}_{k}+\bm{D}\bm{u}_{k}$
$\displaystyle\bar{\bm{A}}=e^{\bm{A}\Delta}\quad\quad\bar{\bm{B}}=(\bar{\bm{A}}-\mathbb{I})\bm{A}^{-1}\bm{B}.$
(1)

These expressions reveal that computationally, SSMs can be seen as specially
parameterized linear recurrent neural networks (RNN) with recurrent weights
$\bar{\bm{A}}\in\mathbb{C}^{N\times N}$, input weights
$\bar{\bm{B}}\in\mathbb{C}^{N\times H}$, readout weights
$\bar{\bm{C}}\in\mathbb{C}^{H\times N}$, and residual weights
$\bm{D}\in\mathbb{R}^{H\times H}$. This allows their efficient computation via
scan algorithms [16]. Via the well-known duality between linear RNNs and time
convolution, their output $\\{\bm{y}_{k}\\}$ (before adding the residual part)
is equivalently the result of (causally) convolving the input sequence with a
size $T\times H$ kernel, given by

$\bar{\bm{K}}=[\bm{C}\bar{\bm{B}},\bm{C}\bar{\bm{A}}\bar{\bm{B}},...,\bm{C}\bar{\bm{A}}^{T-1}\bar{\bm{B}}].$
(2)

Note that $\bar{\bm{K}}$ can be computed for input sequences of arbitrary
length using a fixed number of parameters.

Recent interest in applying SSMs to sequence modeling arises from a series of
work that proposed effective ways to parameterize and initialize such models
[7, 9, 10]. In particular, [10] suggested restricting the $\bm{A}$ to the
space of diagonal matrices. In addition, to ensure that powers of
$\bar{\bm{A}}$ does not diverge, it is sufficient to ensure that $\bm{A}$ has
eigenvalues with negative real parts. We considered two parameterization-
initialization schemes from [10]. In the S4D-Lin scheme, $\bm{A}$ is a complex
diagonal matrix of size $N\times N$ (it thus contains $2N$ trainable
parameters, composed of real and imaginary components of the diagonal
entries). The $n=0,...,N-1$th diagonal entry is initialized at
$A_{n}^{lin}=-\frac{1}{2}+i\pi n.$. Both real and imaginary parts are
trainable in this scheme. Real parts are ensured negative by reparameterizing
them with a non-positive function (e.g., $Re(A_{n})=-\exp(x_{n})$, where
$x_{n}$ is trained). In the S4D-Real scheme, $\bm{A}$ is a real diagonal
matrix (with $N$ trainable parameters) with the $n$th diagonal entry
initialized at $A_{n}^{real}=-n-1$ and again ensured negative via
reparameterization.

We utilize SSMs as parts in a sequence-to-sequence encoder. During training
and inference, an input of size $B\times T\times H$ is passed to the SSM,
where $B$ is the batch size. We treat the feature dimension as $H$ one-
dimensional time series, and each passes through a SSM. Each of the $H$ SSMs
is parameterized by the 3-tuple
$(\text{diag}(\bm{A}^{(h)}),\bm{B}^{(h)},\bm{C}^{(h)})_{h=1,...,H}$, and we
assume $\bm{A}^{(h)}$ to be tied across $h=1,...,H$ (while $B^{(h)}$,
$C^{(h)}$ are not). For concreteness, for encoder dimension $H=512$ and $N=4$,
our SSMs have around 4000 parameters.

##  3 Augmenting ASR encoders with SSMs

The conformer [5] is the current state-of-the-art encoder architecture for
ASR. The baseline conformer encoder that we studied uses hyperparameter
settings of the Conformer (L) encoder from [5]. Each conformer layer consists
of a MHSA module that is efficient at capturing long-term dependency, and a
convolution module that is good at capturing local dependency. Previous
research on augmenting offline ASR encoders with SSMs [13] indicates that MHSA
modules are indispensable for state-of-the-art performance. Therefore, we did
not modify the MHSA module and restrict our analysis to augmenting the
convolution module with SSMs. Concretely, we considered three approaches (Fig.
1)

  * •

Drop-in Replacement (DIR). Here, the convolution component inside the module
is replaced by an SSM. As the SSM has unlimited left context, this endows the
modified module with unlimited left context. We note that this is the approach
taken in [9] for offline ASR, albeit with older SSM variants.

  * •

Combination (COM). Recent work utilizing SSMs for sequence modeling has
suggested a potential advantage of combining the SSM with local (small kernel
size) convolution [17, 18]. We therefore propose stacking them inside the
module. This approach similarly endows the module with unlimited left context.

  * •

Reparameterization (REP). As discussed above, the SSM can be viewed as a way
to parameterize a convolution kernel, $\tilde{\bm{K}}$, that can be match the
length of arbitrary inputs. This perspective suggests that, similar to how
conformers perform better when their convolution kernel size is tuned to some
finite value, performance of the SSM can be similarly improved by truncating
$\bm{\tilde{K}}$ along the time dimension. This essentially means that we are
using conformers but the (finite size) convolution kernel is reparameterized
by the SSM. Unlike the previous two approaches, this does not increase the
left context size of the module compared to conformers.

All approaches only modify the convolution component inside the convolution
module. All other settings are the same as those used in the baseline
conformer. The same modification is applied to all layers in the network. The
resultant encoder models are collectively referred to as SSMformers.

##  4 Experiments

To assess and compare quality of the various SSM-augmented conformers, we
perform both offline and online ASR on the Librispeech dataset [19]. We use
the RNN-Transducer model [20] equipped with a single-layer LSTM decoder for
all experiments, label-sync and frame-sync beam search with beam size $8$ are
used for decoding in the offline and online settings respectively. We compute
80-channel filterbank features from input audio and apply two layers of
2D-convolution subsampling [21]. The resulting audio features, of 40ms frame
rate, are fed to the encoder. Models are trained on all 970 hours of labeled
training set. SpecAugment [22] and variational noise [23] are used to control
overfitting. For the baseline conformer, we use the recommended configuration
with around 119M parameters from [5], with 17 layers, 8 attention heads,
convolution kernel size 32, encoder dimension 512, and relative positional
embedding [24]). In the offline scenario, the attention module has unlimited
left and right context. In the online scenario, the attention module only
accesses unlimited left context; right context was also removed from the
wav2vec convolution and all convolution modules (so the effective filter size
becomes 16). Hyperparameter tuning and early stopping used the average WER on
dev clean and dev other.

###  4.1 Offline ASR with SSMformers

Table 1: WER (%) of offline ASR using encoders augmented with DIR/COM approaches. All SSMformers used SSMs with the S4D-Real scheme. Approach | $N$ | dev | dev | test | test | test
---|---|---|---|---|---|---
clean | other | clean | other | avg.
Conformer | NA | 1.84 | 4.49 | 2.12 | 4.63 | 3.38
DIR | 2 | 1.87 | 4.43 | 2.08 | 4.61 | 3.35
4 | 1.89 | 4.54 | 2.05 | 4.63 | 3.34
COM | 2 | 1.83 | 4.33 | 2.00 | 4.45 | 3.23
4 | 1.84 | 4.33 | 2.02 | 4.39 | 3.21
32 | 1.78 | 4.36 | 1.96 | 4.33 | 3.15

Although we focus our analysis on the online scenario, we first test our
method in the offline setting. Results with DIR and COM with different SSM
dimensions are reported in Table 1. We observe that an SSMformer using the COM
approach achieves the state-of-the-art at offline ASR on Librispeech
(1.96%/4.33% on test clean/other, 119M params, Table 1), on par with a larger
model from a recently proposed SSM-augmented encoder model (1.91%/4.36%, 140M
params) [13]. Our conformer implementation is also on par with published
results [5] (ours: 2.12%/4.63%; reported: 2.1%/4.3%).

Table 2: WER (%) of online ASR using the drop-in replacement (DIR) approach. Scheme | $N$ | dev | dev | test | test | test
---|---|---|---|---|---|---
clean | other | clean | other | avg.
S4D-Real | 2 | 3.76 | 9.21 | 4.18 | 8.77 | 6.48
4 | 3.97 | 9.44 | 4.38 | 9.08 | 6.73
8 | 3.99 | 9.58 | 4.35 | 9.14 | 6.75
S4D-Lin | 2 | 3.90 | 9.53 | 4.40 | 9.23 | 6.82
4 | 3.96 | 9.47 | 4.25 | 9.06 | 6.66
8 | 3.96 | 9.37 | 4.29 | 8.97 | 6.63

###  4.2 Online ASR

We now focus on the online setting, which is a difficult yet important use
case not examined by previous research on encoders with SSMs. We provide
results from exhaustive ablation studies on SSM settings for each of the three
approaches. All models have $119\pm 1M$ parameters. Since our baseline
conformer has parameters tuned for offline but not online ASR, we performed a
sweep of its convolution kernel size (from 2 to 64) and used the best setting
(4) for comparison with SSMformers. All results are summarized in Table 6.

####  4.2.1 Drop-in replacement (DIR)

Results from ablation studies using the DIR approach are shown in Table 2. On
test sets, the most performant SSMformer achieves $4.18\%/8.77\%$, on par with
but no better than our tuned conformer ($4.15\%/8.70\%$). The most effective
SSM to use has the smallest $N$ that we tested. This is somewhat surprising,
as previous work on SSMs for ASR [9] used much larger SSM dimension (64),
which was in part motivated by theoretical results showing that SSMs with
diagonal weights are equivalent to those with non-diagonal weights at infinite
dimension (further discussion in [9]). Our results indicate that for online
ASR, a small-$N$ SSM is the most effective, despite having fewer parameters.

####  4.2.2 Combination (COM)

We next test stacking the SSM and convolution to see whether they can act
complimentarily. We first sweep through different convolution sizes (Table 3).
Adding the convolution resulted in a substantial WER reduction (relative 4-7%)
compared to having the SSM alone (DIR), improving test set WER from
4.38%/9.08% to 4.16%/8.67%. However, our experiments suggest that for the
combination to be effective, kernel size of the convolution needs to be
substantially reduced. Using a large-ish kernel size (16), the resultant model
in fact performs worse (SSMformer built with COM gives 4.85%/9.91%; conformer
using this size gives 4.46%/9.46%).

We next test different SSM settings in the COM approach. In these experiments,
we fixed the convolution kernel size to 2 and varied SSM parameterization and
dimension (Table 4). We found that, as in the DIR approach, the most
performant SSM here is S4D-Real with a small SSM dimension.

Table 3: WERs(%) of online ASR from encoders using different pre-SSM convolution kernel sizes. Both DIR and COM used the S4D-Real parameterization scheme with $N=4$ (using other $N$ worsened performance). Note that in COM models, the conv. size only describes the convolution kernel size; the SSMs they use have unlimited left context. Appr. | conv. | $N$ | dev | dev | test | test | test
---|---|---|---|---|---|---|---
size | clean | other | clean | other | avg.
DIR | NA | 2 | 3.76 | 9.21 | 4.18 | 8.77 | 6.48
NA | 4 | 3.97 | 9.44 | 4.38 | 9.08 | 6.73
COM | 2 | 4 | 3.67 | 9.10 | 4.16 | 8.67 | 6.42
4 | 4 | 3.75 | 9.46 | 4.27 | 8.93 | 6.60
8 | 4 | 3.75 | 9.35 | 4.17 | 8.98 | 6.58
16 | 4 | 4.40 | 10.19 | 4.85 | 9.91 | 7.38
Table 4: WERs(%) of online ASR from SSMformers augmented with the COM approach using different recurrent weights ($\bm{A}$). All models use a pre-SSM convolution with kernel size 2. Scheme | $N$ | dev | dev | test | test | test
---|---|---|---|---|---|---
clean | other | clean | other | avg.
S4D-Real | 2 | 3.69 | 9.14 | 4.01 | 8.53 | 6.27
S4D-Real | 4 | 3.67 | 9.10 | 4.16 | 8.67 | 6.42
S4D-Real | 8 | 3.85 | 9.29 | 4.29 | 8.97 | 6.63
S4D-Real | 16 | 3.70 | 9.19 | 4.27 | 8.74 | 6.51
S4D-Real | 32 | 3.77 | 9.00 | 4.24 | 8.63 | 6.44
S4D-Real | 64 | 3.70 | 8.99 | 4.18 | 8.77 | 6.48
S4D-Lin | 4 | 3.71 | 9.21 | 4.25 | 8.86 | 6.56
S4D-Lin | 32 | 3.71 | 9.27 | 4.19 | 8.93 | 6.56

####  4.2.3 Reparameterization (REP)

We now consider the final approach, where the SSM is used to reparameterize a
finite-size convolution kernel. In all experiments, we use SSMs with S4D-Real
and dimension 4. To parameterize a convolution kernel of size $L$ using an
SSM, we compute the kernel as

$\tilde{\bm{K}}(L)=[\bm{C}\bar{\bm{B}},\bm{C}\bar{\bm{A}}\bar{\bm{B}},...,\bm{C}\bar{\bm{A}}^{L-1}\bar{\bm{B}}].$
(3)

We also note that, while in principle one could stack one convolution
component using an SSM-parameterized kernel $\tilde{\bm{K}}(L)$ and another
one using a conventional kernel, we found that such stacking resulted in
performance on par with or worse than that of COM (results not presented
here). As shown in Table 5, the best REP model achieves $4.09\%/8.66\%$ on
test sets, slightly worse than the best COM model but also better than the
tuned conformer ($4.15\%/8.70\%$), despite having the same kernel size and
left context. Compared to DIR or COM, the advantage of this approach is that
it does not modify the forward pass of the encoder at test time. The model
simply needs to compute and cache the finite-size kernel $\tilde{\bm{K}}(L)$
and perform convolution with it as a vanilla conformer.

Table 5: WERs(%) of online ASR from encoders augmented using the reparameterization (REP) approach. Appr. | left | dev | dev | test | test | test
---|---|---|---|---|---|---
cont. | clean | other | clean | other | avg.
REP | $\infty$ | 3.97 | 9.44 | 4.38 | 9.08 | 6.73
32 | 3.86 | 9.35 | 4.21 | 8.97 | 6.59
16 | 3.79 | 9.33 | 4.26 | 8.95 | 6.61
8 | 3.75 | 9.02 | 4.09 | 8.66 | 6.38
| 4 | 3.80 | 9.21 | 4.18 | 8.83 | 6.51
| 2 | 3.79 | 9.04 | 4.17 | 8.71 | 6.44

###  4.3 Summary

In Table 6, we summarize WERs from the most performant SSMformer using each of
the approaches. The conformer shown here uses the best convolution kernel size
we found (size 4). Compared to this carefully tuned conformer, COM and REP
still manage small improvement on test sets, achieving WERs of $4.01\%/8.53\%$
and $4.09\%/8.66\%$, respectively. Our key empirical takeaways are

  * •

Compared to conformers with tuned convolution, SSMformers achieve a small but
consistent WER reduction in online ASR.

  * •

In all settings, S4D-Real is more effective than S4D-Lin.

  * •

In online ASR, SSMs with very small $N$ (2-4) have the best performance.
Larger $N$ is more effective for offline ASR.

  * •

How SSMs are incorporated into the encoder matters. We found COM $>$ REP $>$
DIR.

Table 6: Summary of online ASR on Librispeech. All results are from our implementations. The conformer uses the best setting for online ASR that we found (kernel size 4). Model | Appr. | dev | dev | test | test | test
---|---|---|---|---|---|---
clean | other | clean | other | avg.
Conformer | NA | 3.74 | 9.01 | 4.15 | 8.70 | 6.43
| DIR | 3.76 | 9.21 | 4.18 | 8.77 | 6.48
SSMformer | COM | 3.69 | 9.14 | 4.01 | 8.53 | 6.27
| REP | 3.75 | 9.02 | 4.09 | 8.66 | 6.38

##  5 Discussion

We systematically investigated ways to augment encoder models for online ASR
with SSMs. We proposed two new approaches, COM and REP, both of which resulted
in models performing at the state-of-the-art. The success of REP is somewhat
surprising, as traditionally SSMs are thought to excel at modeling long-range
dependencies [8, 7, 10], yet the REP approach prevents them from doing so.
This suggests that even for short-range dependencies, SSMs can have advantages
over simple convolution, pointing to the need of further theoretical research.

## References

  * [1] Geoffrey Hinton, Li Deng, Dong Yu, George E. Dahl, Abdel-rahman Mohamed, Navdeep Jaitly, Andrew Senior, Vincent Vanhoucke, Patrick Nguyen, Tara N. Sainath, and Brian Kingsbury,  “Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups,”  IEEE Signal Processing Magazine, 2012.

  * [2] Sepp Hochreiter and Jürgen Schmidhuber,  “Long Short-Term Memory,”  Neural Computation, 1997.
  * [3] Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner,  “Gradient-based learning applied to document recognition,”  Proceedings of the IEEE, 1998.
  * [4] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, L ukasz Kaiser, and Illia Polosukhin,  “Attention is all you need,”  in NeurIPS, 2017.
  * [5] Anmol Gulati, James Qin, Chung-Cheng Chiu, Niki Parmar, Yu Zhang, Jiahui Yu, Wei Han, Shibo Wang, Zhengdong Zhang, Yonghui Wu, et al.,  “Conformer: Convolution-augmented transformer for speech recognition,”  arXiv preprint arXiv:2005.08100, 2020.
  * [6] Yifan Peng, Siddharth Dalmia, Ian Lane, and Shinji Watanabe,  “Branchformer: Parallel mlp-attention architectures to capture local and global context for speech recognition and understanding,”  in ICML, 2022.
  * [7] Albert Gu, Karan Goel, and Christopher Ré,  “Efficiently modeling long sequences with structured state spaces,”  arXiv preprint arXiv:2111.00396, 2021.
  * [8] Albert Gu, Isys Johnson, Karan Goel, Khaled Saab, Tri Dao, Atri Rudra, and Christopher Ré,  “Combining recurrent, convolutional, and continuous-time models with linear state space layers,”  in NeurIPS, 2021.
  * [9] Ankit Gupta, Albert Gu, and Jonathan Berant,  “Diagonal state spaces are as effective as structured state spaces,”  NeurIPS, 2022.
  * [10] Albert Gu, Karan Goel, Ankit Gupta, and Christopher Ré,  “On the parameterization and initialization of diagonal state space models,”  NeurIPS, 2022.
  * [11] Xuezhe Ma, Chunting Zhou, Xiang Kong, Junxian He, Liangke Gui, Graham Neubig, Jonathan May, and Luke Zettlemoyer,  “Mega: Moving average equipped gated attention,”  in The Eleventh International Conference on Learning Representations, 2023.
  * [12] Jimmy TH Smith, Andrew Warrington, and Scott Linderman,  “Simplified state space layers for sequence modeling,”  in The Eleventh International Conference on Learning Representations, 2023.
  * [13] Yassir Fathullah, Chunyang Wu, Yuan Shangguan, Junteng Jia, Wenhan Xiong, Jay Mahadeokar, Chunxi Liu, Yangyang Shi, Ozlem Kalinli, Mike Seltzer, and Mark J. F. Gales,  “Multi-head state space model for speech recognition,”  in Interspeech, 2023.
  * [14] George Saon, Ankit Gupta, and Xiaodong Cui,  “Diagonal state space augmented transformers for speech recognition,”  in ICASSP, 2023.
  * [15] Koichi Miyazaki, Masato Murata, and Tomoki Koriyama,  “Structured state space decoder for speech recognition and synthesis,”  in ICASSP, 2023.
  * [16] Eric Martin and Chris Cundy,  “Parallelizing linear recurrent neural nets over sequence length,”  arXiv preprint arXiv:1709.04057, 2017.
  * [17] Michael Poli, Stefano Massaroli, Eric Nguyen, Daniel Y Fu, Tri Dao, Stephen Baccus, Yoshua Bengio, Stefano Ermon, and Christopher Ré,  “Hyena hierarchy: Towards larger convolutional language models,”  arXiv preprint arXiv:2302.10866, 2023.
  * [18] Tri Dao, Daniel Y Fu, Khaled K Saab, Armin W Thomas, Atri Rudra, and Christopher Ré,  “Hungry hungry hippos: Towards language modeling with state space models,”  arXiv preprint arXiv:2212.14052, 2022.
  * [19] Vassil Panayotov, Guoguo Chen, Daniel Povey, and Sanjeev Khudanpur,  “Librispeech: An ASR corpus based on public domain audio books,”  in ICASSP, 2015.
  * [20] Alex Graves,  “Sequence transduction with recurrent neural networks,”  arXiv preprint arXiv:1211.3711, 2012.
  * [21] Alexei Baevski, Yuhao Zhou, Abdelrahman Mohamed, and Michael Auli,  “wav2vec 2.0: A framework for self-supervised learning of speech representations,”  NeurIPS, 2020.
  * [22] Daniel S Park, William Chan, Yu Zhang, Chung-Cheng Chiu, Barret Zoph, Ekin D Cubuk, and Quoc V Le,  “Specaugment: A simple data augmentation method for automatic speech recognition,”  arXiv preprint arXiv:1904.08779, 2019.
  * [23] Alex Graves,  “Practical variational inference for neural networks,”  Advances in neural information processing systems, vol. 24, 2011\.
  * [24] Zihang Dai, Zhilin Yang, Yiming Yang, Jaime Carbonell, Quoc V Le, and Ruslan Salakhutdinov,  “Transformer-xl: Attentive language models beyond a fixed-length context,”  arXiv preprint arXiv:1901.02860, 2019.


#  Neutrino Rate Predictions for FASER
FASER Collaboration

Roshan Mammen Abraham  0000-0003-4678-3808 Department of Physics and
Astronomy, University of California, Irvine, CA 92697-4575, USA  John Anders
0000-0002-1846-0262 CERN, CH-1211 Geneva 23, Switzerland  Claire Antel
0000-0001-9683-0890 Département de Physique Nucléaire et Corpusculaire,
University of Geneva, CH-1211 Geneva 4, Switzerland  Akitaka Ariga
0000-0002-6832-2466 Albert Einstein Center for Fundamental Physics, Laboratory
for High Energy Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern,
Switzerland  Department of Physics, Chiba University, 1-33 Yayoi-cho Inage-ku,
263-8522 Chiba, Japan  Tomoko Ariga  0000-0001-9880-3562 Kyushu University,
Nishi-ku, 819-0395 Fukuoka, Japan  Jeremy Atkinson  0009-0003-3287-2196 Albert
Einstein Center for Fundamental Physics, Laboratory for High Energy Physics,
University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland  Florian U.
Bernlochner  0000-0001-8153-2719 Universität Bonn, Regina-Pacis-Weg 3, D-53113
Bonn, Germany  Tobias Boeckh  0009-0000-7721-2114 Universität Bonn, Regina-
Pacis-Weg 3, D-53113 Bonn, Germany  Jamie Boyd  0000-0001-7360-0726 CERN,
CH-1211 Geneva 23, Switzerland  Lydia Brenner  0000-0001-5350-7081 Nikhef
National Institute for Subatomic Physics, Science Park 105, 1098 XG Amsterdam,
Netherlands  Angela Burger  0000-0003-0685-4122 CERN, CH-1211 Geneva 23,
Switzerland  Franck Cadoux  Département de Physique Nucléaire et
Corpusculaire, University of Geneva, CH-1211 Geneva 4, Switzerland  Roberto
Cardella  0000-0002-3117-7277 Département de Physique Nucléaire et
Corpusculaire, University of Geneva, CH-1211 Geneva 4, Switzerland  David W.
Casper  0000-0002-7618-1683 Department of Physics and Astronomy, University of
California, Irvine, CA 92697-4575, USA  Charlotte Cavanagh
0009-0001-1146-5247 University of Liverpool, Liverpool L69 3BX, United Kingdom
Xin Chen  0000-0003-4027-3305 Department of Physics, Tsinghua University,
Beijing, China  Andrea Coccaro  0000-0003-2368-4559 INFN Sezione di Genova,
Via Dodecaneso, 33–16146, Genova, Italy  Stephane Débieux  Département de
Physique Nucléaire et Corpusculaire, University of Geneva, CH-1211 Geneva 4,
Switzerland  Monica D’Onofrio  0000-0003-2408-5099 University of Liverpool,
Liverpool L69 3BX, United Kingdom  Ansh Desai  0000-0002-5447-8304 University
of Oregon, Eugene, OR 97403, USA  Sergey Dmitrievsky  0000-0003-4247-8697
Affiliated with an international laboratory covered by a cooperation agreement
with CERN.  Sinead Eley  0009-0001-1320-2889 University of Liverpool,
Liverpool L69 3BX, United Kingdom  Yannick Favre  Département de Physique
Nucléaire et Corpusculaire, University of Geneva, CH-1211 Geneva 4,
Switzerland  Deion Fellers  0000-0002-0731-9562 University of Oregon, Eugene,
OR 97403, USA  Jonathan L. Feng  0000-0002-7713-2138 Department of Physics and
Astronomy, University of California, Irvine, CA 92697-4575, USA  Carlo Alberto
Fenoglio  0009-0007-7567-8763 Département de Physique Nucléaire et
Corpusculaire, University of Geneva, CH-1211 Geneva 4, Switzerland  Didier
Ferrere  0000-0002-5687-9240 Département de Physique Nucléaire et
Corpusculaire, University of Geneva, CH-1211 Geneva 4, Switzerland  Max Fieg
0000-0002-7027-6921 Department of Physics and Astronomy, University of
California, Irvine, CA 92697-4575, USA  Wissal Filali  0009-0008-6961-2335
Department of Physics, Tsinghua University, Beijing, China  Stephen Gibson
0000-0002-1236-9249 Royal Holloway, University of London, Egham, TW20 0EX,
United Kingdom  Sergio Gonzalez-Sevilla  0000-0003-4458-9403 Département de
Physique Nucléaire et Corpusculaire, University of Geneva, CH-1211 Geneva 4,
Switzerland  Yuri Gornushkin  0000-0003-3524-4032 Affiliated with an
international laboratory covered by a cooperation agreement with CERN.  Carl
Gwilliam  0000-0002-9401-5304 University of Liverpool, Liverpool L69 3BX,
United Kingdom  Daiki Hayakawa  0000-0003-4253-4484 Department of Physics,
Chiba University, 1-33 Yayoi-cho Inage-ku, 263-8522 Chiba, Japan  Shih-Chieh
Hsu  0000-0001-6214-8500 Department of Physics, University of Washington, PO
Box 351560, Seattle, WA 98195-1460, USA  Zhen Hu  0000-0001-8209-4343
Department of Physics, Tsinghua University, Beijing, China  Giuseppe Iacobucci
0000-0001-9965-5442 Département de Physique Nucléaire et Corpusculaire,
University of Geneva, CH-1211 Geneva 4, Switzerland  Tomohiro Inada
0000-0002-6923-9314 CERN, CH-1211 Geneva 23, Switzerland  Luca Iodice
0000-0002-3516-7121 Département de Physique Nucléaire et Corpusculaire,
University of Geneva, CH-1211 Geneva 4, Switzerland  Sune Jakobsen
0000-0002-6564-040X CERN, CH-1211 Geneva 23, Switzerland  Hans Joos
0000-0003-4313-4255 CERN, CH-1211 Geneva 23, Switzerland  II. Physikalisches
Institut, Universität Göttingen, Göttingen, Germany  Enrique Kajomovitz
0000-0002-8464-1790 Department of Physics and Astronomy, Technion—Israel
Institute of Technology, Haifa 32000, Israel  Hiroaki Kawahara
0009-0007-5657-9954 Kyushu University, Nishi-ku, 819-0395 Fukuoka, Japan  Alex
Keyken  0009-0001-4886-2924 Royal Holloway, University of London, Egham, TW20
0EX, United Kingdom  Felix Kling  0000-0002-3100-6144 Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Daniela Köck
0000-0002-9090-5502 University of Oregon, Eugene, OR 97403, USA  Pantelis
Kontaxakis  0000-0002-4860-5979 Département de Physique Nucléaire et
Corpusculaire, University of Geneva, CH-1211 Geneva 4, Switzerland  Umut Kose
0000-0001-5380-9354 ETH Zurich, 8092 Zurich, Switzerland  Rafaella Kotitsa
0000-0002-7886-2685 CERN, CH-1211 Geneva 23, Switzerland  Susanne Kuehn
0000-0001-5270-0920 CERN, CH-1211 Geneva 23, Switzerland  Thanushan Kugathasan
0000-0003-4631-5019 Département de Physique Nucléaire et Corpusculaire,
University of Geneva, CH-1211 Geneva 4, Switzerland  Helena Lefebvre
0000-0002-7394-2408 Royal Holloway, University of London, Egham, TW20 0EX,
United Kingdom  Lorne Levinson  0000-0003-4679-0485 Department of Particle
Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel
Ke Li  0000-0002-2545-0329 Department of Physics, University of Washington, PO
Box 351560, Seattle, WA 98195-1460, USA  Jinfeng Liu  0000-0001-6827-1729
Department of Physics, Tsinghua University, Beijing, China  Margaret S. Lutz
0000-0003-4515-0224 CERN, CH-1211 Geneva 23, Switzerland  Jack MacDonald
0000-0002-3150-3124 Institut für Physik, Universität Mainz, Mainz, Germany
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Corpusculaire, University of Geneva, CH-1211 Geneva 4, Switzerland  Fulvio
Martinelli  0000-0003-4221-5862 Département de Physique Nucléaire et
Corpusculaire, University of Geneva, CH-1211 Geneva 4, Switzerland  Lawson
McCoy  0009-0009-2741-3220 Department of Physics and Astronomy, University of
California, Irvine, CA 92697-4575, USA  Josh McFayden  0000-0001-9273-2564
Department of Physics & Astronomy, University of Sussex, Sussex House, Falmer,
Brighton, BN1 9RH, United Kingdom  Andrea Pizarro Medina  0000-0002-1024-5605
Département de Physique Nucléaire et Corpusculaire, University of Geneva,
CH-1211 Geneva 4, Switzerland  Matteo Milanesio  0000-0001-8778-9638
Département de Physique Nucléaire et Corpusculaire, University of Geneva,
CH-1211 Geneva 4, Switzerland  Théo Moretti  0000-0001-7065-1923 Département
de Physique Nucléaire et Corpusculaire, University of Geneva, CH-1211 Geneva
4, Switzerland  Magdalena Munker  0000-0003-2775-3291 Département de Physique
Nucléaire et Corpusculaire, University of Geneva, CH-1211 Geneva 4,
Switzerland  Mitsuhiro Nakamura  Nagoya University, Furo-cho, Chikusa-ku,
Nagoya 464-8602, Japan  Toshiyuki Nakano  Nagoya University, Furo-cho,
Chikusa-ku, Nagoya 464-8602, Japan  Friedemann Neuhaus  0000-0002-3819-2453
Institut für Physik, Universität Mainz, Mainz, Germany  Laurie Nevay
0000-0001-7225-9327 CERN, CH-1211 Geneva 23, Switzerland  Ken Ohashi
0009-0000-9494-8457 Albert Einstein Center for Fundamental Physics, Laboratory
for High Energy Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern,
Switzerland  Hidetoshi Otono  0000-0003-0760-5988 Kyushu University, Nishi-ku,
819-0395 Fukuoka, Japan  Hao Pang  0000-0002-1946-1769 Department of Physics,
Tsinghua University, Beijing, China  Lorenzo Paolozzi  0000-0002-9281-1972
Département de Physique Nucléaire et Corpusculaire, University of Geneva,
CH-1211 Geneva 4, Switzerland  CERN, CH-1211 Geneva 23, Switzerland  Brian
Petersen  0000-0002-7380-6123 CERN, CH-1211 Geneva 23, Switzerland  Markus
Prim  0000-0002-1407-7450 Universität Bonn, Regina-Pacis-Weg 3, D-53113 Bonn,
Germany  Michaela Queitsch-Maitland  0000-0003-4643-515X University of
Manchester, School of Physics and Astronomy, Schuster Building, Oxford Rd,
Manchester M13 9PL, United Kingdom  Hiroki Rokujo  0000-0002-3502-493X Nagoya
University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan  Elisa Ruiz-Choliz
0000-0002-2417-7121 Institut für Physik, Universität Mainz, Mainz, Germany
André Rubbia  0000-0002-5747-1001 ETH Zurich, 8092 Zurich, Switzerland  Jorge
Sabater-Iglesias  0000-0003-2328-1952 Département de Physique Nucléaire et
Corpusculaire, University of Geneva, CH-1211 Geneva 4, Switzerland  Osamu Sato
0000-0002-6307-7019 Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602,
Japan  Paola Scampoli  0000-0001-7500-2535 Albert Einstein Center for
Fundamental Physics, Laboratory for High Energy Physics, University of Bern,
Sidlerstrasse 5, CH-3012 Bern, Switzerland  Dipartimento di Fisica “Ettore
Pancini”, Università di Napoli Federico II, Complesso Universitario di Monte
S. Angelo, I-80126 Napoli, Italy  Kristof Schmieden  0000-0003-1978-4928
Institut für Physik, Universität Mainz, Mainz, Germany  Matthias Schott
0000-0002-4235-7265 Institut für Physik, Universität Mainz, Mainz, Germany
Anna Sfyrla  0000-0002-3003-9905 Département de Physique Nucléaire et
Corpusculaire, University of Geneva, CH-1211 Geneva 4, Switzerland  Mansoora
Shamim  0009-0002-3986-399X CERN, CH-1211 Geneva 23, Switzerland  Savannah
Shively  0000-0002-4691-3767 Department of Physics and Astronomy, University
of California, Irvine, CA 92697-4575, USA  Yosuke Takubo  0000-0002-3143-8510
Institute of Particle and Nuclear Studies, KEK, Oho 1-1, Tsukuba, Ibaraki
305-0801, Japan  Noshin Tarannum  0000-0002-3246-2686 Département de Physique
Nucléaire et Corpusculaire, University of Geneva, CH-1211 Geneva 4,
Switzerland  Ondrej Theiner  0000-0002-6558-7311 Département de Physique
Nucléaire et Corpusculaire, University of Geneva, CH-1211 Geneva 4,
Switzerland  Eric Torrence  0000-0003-2911-8910 University of Oregon, Eugene,
OR 97403, USA  Svetlana Vasina  0000-0003-2775-5721 Affiliated with an
international laboratory covered by a cooperation agreement with CERN.
Benedikt Vormwald  0000-0003-2607-7287 CERN, CH-1211 Geneva 23, Switzerland
Di Wang  0000-0002-0050-612X Department of Physics, Tsinghua University,
Beijing, China  Yuxiao Wang  0009-0004-1228-9849 Department of Physics,
Tsinghua University, Beijing, China  Eli Welch  0000-0001-6336-2912 Department
of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA
Samuel Zahorec  0009-0000-9729-0611 CERN, CH-1211 Geneva 23, Switzerland
Charles University, Faculty of Mathematics and Physics, Prague; Czech Republic
Stefano Zambito  0000-0002-4499-2545 Département de Physique Nucléaire et
Corpusculaire, University of Geneva, CH-1211 Geneva 4, Switzerland  Shunliang
Zhang  0009-0001-1971-8878 Department of Physics, Tsinghua University,
Beijing, China

###### Abstract

The Forward Search Experiment (FASER) at CERN’s Large Hadron Collider (LHC)
has recently directly detected the first collider neutrinos. Neutrinos play an
important role in all FASER analyses, either as signal or background, and it
is therefore essential to understand the neutrino event rates. In this study,
we update previous simulations and present prescriptions for theoretical
predictions of neutrino fluxes and cross sections, together with their
associated uncertainties. With these results, we discuss the potential for
possible measurements that could be carried out in the coming years with the
FASER neutrino data to be collected in LHC Run 3 and Run 4.

††preprint: CERN-EP-2024-036

© 2024 CERN for the benefit of the FASER Collaboration. Reproduction of this
article or parts of it is allowed as specified in the CC-BY-4.0 license.

##  I Introduction

The Forward Search Experiment (FASER) Feng:2017uoz ; FASER:2022hcn ;
FASER:2018bac ; FASER:2020gpr  at CERN’s Large Hadron Collider (LHC)
complements the large LHC detectors through its ability to directly detect
light, weakly-interacting particles FASER:2021cpr ; FASER:2021ljd . These
particles include the neutrinos of the Standard Model (SM), as well as
proposed new particles. FASER is located along the beam collision axis, 480 m
from the ATLAS interaction point (IP), and began taking beam collision data at
the beginning of LHC Run 3 in 2022.

With the 2022 data set, corresponding to an integrated luminosity of
$37~{}\text{fb}^{-1}$, FASER detected 153 muon neutrinos, the first collider
neutrinos to be directly detected FASER:2023zcr . FASER also observed the
first electron neutrino interactions at a collider CERN-FASER-CONF-2023-002
and set new limits on long-lived particles FASER:2023tle . The neutrinos were
the most energetic neutrinos ever directly detected from an artificial source.
They have been supplemented by an additional eight muon neutrinos detected by
the SND@LHC experiment SNDLHC:2023pun . These discoveries have opened up the
new field of collider neutrino physics.

In the coming years, FASER is expected to collect a total integrated
luminosity of $250~{}\text{fb}^{-1}$ in Run 3, and has recently been approved
to continue operating through Run 4, which is expected to increase the total
(Run 3 + Run 4) integrated luminosity to $930~{}\text{fb}^{-1}$ Run4LOI . The
large expected neutrino event rates, together with their energy and spatial
distributions, will have many implications, including the potential to
constrain neutrino scattering cross sections of all three flavors at unprobed
energies, measure forward hadron fluxes Kling:2023tgr , improve constraints on
parton distribution functions (PDFs) using deep-inelastic scattering (DIS)
data Cruz-Martinez:2023sdv , resolve longstanding puzzles in astroparticle
physics Anchordoqui:2022fpn , and test predictions for new physics
Anchordoqui:2021ghd ; Feng:2022inv .

Because neutrinos play an important role in all FASER analyses, either as the
signal or as a background to new particle searches, a detailed understanding
of neutrino event rates, as well as estimates of the associated uncertainties,
is required. Simulations of the neutrino event rate require a number of tools
and calculations that are not typically associated with colliders, since
neutrino interactions have never before played a role at colliders. In this
study, the neutrino fluxes and interactions are simulated in FASER and
described. These results will be the basis for upcoming FASER analyses.

The forward neutrino beam at the LHC mainly originates from the weak decay of
the lightest mesons and baryons of a given flavor (pions, kaons, hyperons,
$D$-mesons, and charm baryons). A variety of tools and calculations are
available to simulate the production of these particles. The resulting
neutrino flux at FASER can then be estimated using the fast neutrino flux
simulation introduced in Ref. Kling:2021gos . That original work considered an
LHC configuration (collision energy, magnet strength, and crossing angle)
resembling conditions at the end of Run 2. In this work, this simulation is
updated to the LHC configurations realized in Run 3 and expected in Run 4.

In addition, Ref. Kling:2021gos  only contains a very rough estimate of the
neutrino flux uncertainty. This uncertainty mainly originates from the
modeling of hadron production in the primary collision, and its description
was based on the event generators available at the time. Since then,
additional tools and calculations of the neutrino flux have been presented.
Here, these new developments are used to update neutrino flux predictions at
FASER and establish the corresponding uncertainties.

Beyond flux uncertainty, an additional source of uncertainty of the expected
number of neutrino events is associated with the modeling of neutrino
interactions. Although neutrino interaction cross sections in the multi-
hundred GeV region have traditionally been modeled using the Bodek-Yang model
Bodek:2002vp ; Bodek:2004pc ; Bodek:2010km , several new cross section models
based on next-to-leading-order (NLO) structure functions have become available
in recent years. These predictions will be compared, and the corresponding
cross-section uncertainties will be defined.

This paper is structured as follows. A brief review of the original neutrino
simulation Kling:2021gos  and assumptions about the LHC configurations are
presented in Sec. II. In Sec. III models for forward hadron production and
their implications for neutrino fluxes are compared, and in Sec. IV neutrino
interactions and the accompanying uncertainties are discussed. With the
provided neutrino flux and cross-section results, predictions for neutrino
event rates at FASER are presented in Sec. V. This includes energy and spatial
distributions for all three neutrino flavors. Additionally, potential
measurements that could be conducted with FASER in LHC Run 3 and Run 4 are
briefly discussed. The conclusions derived from this analysis are summarized
in Sec. VI. In the Appendix several forward charm production models are
compared to each other and to data.

##  II Simulation of Forward Neutrinos at the LHC

The beam of forward, high-energy neutrinos observable at FASER mainly
originates from the weak decays of hadrons that are produced at the ATLAS
interaction point.111Neutrinos can also be produced in downstream hadronic
showers resulting from collisions of primary hadrons with the LHC
infrastructure. However, given the typically lower energy and large spread of
hadrons in later stages of the shower, as well as the fact that these hadrons
are more likely to interact in the LHC infrastructure than to decay to
neutrinos, the resulting neutrino flux is subdominant. Indeed, as found in
Ref. Kling:2021gos , the contribution of such processes to the neutrino flux
is below the percent level at $\sim\text{TeV}$ energies for all flavors. Based
on this finding, this flux component is not considered in our neutrino flux
estimate here. This includes light hadrons (pions, kaons, and hyperons), which
are long-lived and decay inside the LHC vacuum beam pipe, and also charm
hadrons, which decay essentially promptly.

To obtain the neutrino flux, one needs to model the trajectory of the long-
lived hadrons through the LHC beam pipe and magnetic fields and also model the
decay of these hadrons into neutrinos. This is done using the fast neutrino
flux simulation introduced in Ref. Kling:2021gos . This fast neutrino flux
simulation (i) reads the forward hadron fluxes from HepMC files produced by
the Monte-Carlo (MC) event generator; (ii) propagates the long-lived hadrons
through the LHC beam pipe and magnetic fields; (iii) obtains the neutrinos
from decays of hadrons at multiple locations along their trajectory; and (iv)
stores the resulting neutrinos going through a sample plane at the FASER
location as a MC event sample. All parts of the outlined simulation are
implemented as a RIVET module Buckley:2010ar ; Bierlich:2019rhm . The results
of the fast simulation have been validated against the full simulation using
BDSIM,222BDSIM Nevay:2018zhp  is a code based on Geant4 Agostinelli:2002hh ,
ROOT, and CLHEP to create radiation transport models of accelerators that can
track all particles. It creates a Geant4 model with translation to a
curvilinear coordinate system that follows the accelerator, as well as more
accurate and faster tracking algorithms specific to the magnetic fields of an
accelerator. Custom component geometry can be combined with a library of
detailed LHC and generic magnet geometries to create complete accelerator
models tracking all particles using the full physics of Geant4, including in
the yokes of magnets. BDSIM tracking has be thoroughly validated and it is
used extensively in the accelerator community and at CERN. and the predictions
were found to be in good agreement Kling:2021gos . In particular, the
differences between the full and fast simulations are significantly smaller
than the differences between MC event generators.

The simulation of Ref. Kling:2021gos  assumed the LHC Run 2 configuration with
center-of-mass energy $\sqrt{s}=13~{}\text{TeV}$. For this work, the RIVET
module has been updated for center-of-mass energies of 13.6 TeV and 14 TeV for
Run 3 and Run 4, respectively. In particular, the strengths of the magnetic
fields were adjusted to the higher beam energies. In addition, the planned
major changes to the LHC infrastructure for Run 4 ZurbanoFernandez:2020cco
(including the geometry of the beam pipe, magnet configuration, and position
of the target neutral absorber) were incorporated, as illustrated in Fig. 1.
The magnet strengths as well as the aperture information are taken from the
BDSIM model, which is automatically prepared from the MAD-X MAD-X  optics
strengths in combination with a separate aperture model and detailed models of
many components.

The simulation of forward neutrinos also depends critically on the beam
crossing angle, which modifies the nominal line-of-sight (LOS), that is, the
LOS in the absence of a crossing angle, to the true, or actual, LOS, that is,
the LOS with the crossing angle included. The previous simulation used the
2018 Run 2 beam crossing half-angle of $\theta_{1/2}=150~{}\mu$rad vertically
upwards. For Run 3, the beam crossing half-angle at the ATLAS IP was 160
$\mu$rad downwards in 2022 and 2023, and it is expected to change to 160
$\mu$rad upwards for 2024 and 160 $\mu$rad horizontally for 2025.333The
crossing angle is also changed by a few 10 $\mu$rad during each physics fill,
but this has a negligible effect on the results for FASER and is not taken
into account in this work. For simplicity, the estimates labeled Run 3 in this
paper assume the 2022 and 2023 crossing half-angle $\theta_{1/2}=160~{}\mu$rad
vertically downwards. The changes expected for 2024 and 2025 do not
significantly modify the estimates of the number of neutrino interactions in
FASER. More details can be found in Ref. Run4LOI . This Run 3 crossing angle
shifts the true LOS downward by 7.7 cm from the nominal LOS. For Run 4, we
assume $\theta_{1/2}=250~{}\mu$rad TomasGarcia:2803611  in the horizontal
plane (away from the LHC ring), which shifts the true LOS 12 cm horizontally
from the nominal LOS.

Figure 1: Beam Pipe Geometry and Magnets. The boundaries of the LHC’s beam
pipe (black lines) and magnetic fields (gray shaded areas) assumed in the
geometric model for LHC Run 3 (left) and Run 4 (right). The $x$, $y$, and $z$
coordinates form a right-handed FASER coordinate system (see text) with the
ATLAS IP at the origin, and FASER at approximately $(0,0,480~{}\text{m})$. The
red lines show trajectories calculated by the updated simulation from Ref.
Kling:2021gos  of the outward-going 6.8 TeV proton beam at Run 3 (left) and
the 7 TeV proton beam at Run 4 (right). The markers show the tracking points
obtained using BDSIM. The TAN and TAS (TAXN and TAXS) are the LHC (high
luminosity LHC) target neutral and passive absorbers.

The FASER$\nu$ detector consists of 730 1.1-mm thick tungsten plates
interleaved with emulsion films, with a total target mass of 1.1 tonnes, and
transverse dimensions 25 cm wide and 30 cm high. In our calculations,
FASER$\nu$ is simulated by assuming a simplified detector that is 25 cm wide,
30 cm high, and 80 cm deep in the beam direction, and filled with 1.1 tonnes
of tungsten.

To specify the location of the detector, FASER uses a right-handed coordinate
system, with the positive $x$-axis pointing in the horizontal direction toward
the center of the LHC, the positive $y$-axis pointing vertically upward, and
the positive $z$-axis pointing from the ATLAS IP toward FASER. The nominal LOS
is at the origin $(x,y)=(0,0)$. The simplified detector’s location in the
transverse plane is matched to FASER$\nu$’s location during 2022/2023
FASER:2023zcr , which is centered at
$(x,y)=(1.0~{}\text{cm},-3.3~{}\text{cm})$. Given this location and the
2022/2023 beam crossing angle, FASER$\nu$ covers pseudorapidities $\eta>8.3$.
Note that the center of the FASER spectrometer, defined by the axis of
symmetry of the magnets, is at $(0,-1.2~{}\text{cm})$. For Run 4, the FASER
and FASER$\nu$ detectors are both assumed to be shifted 5.0 cm horizontally
away from the LHC relative to their 2022/2023 locations, which brings the
centers of these detectors closer to the true LOS Run4LOI . For Run 4, then,
the center of FASER$\nu$ is at $(6.0~{}\text{cm},-3.3~{}\text{cm})$, and the
center of FASER is at $(x,y)=(5.0~{}\text{cm},-1.2~{}\text{cm})$. With the
beam crossing angle discussed above, FASER$\nu$ covers pseudorapidities
$\eta>8.2$ in Run 4.

##  III Neutrino Fluxes and Uncertainties

As already discussed, the two primary components of the forward LHC neutrino
beam observable at FASER are neutrinos produced downstream in light hadron
decays and neutrinos produced promptly in charm hadron decays. Aside from
differences in the production location and the associated simulation
requirements, they also differ qualitatively in their theoretical modeling.
We, therefore, discuss them in turn.

In inelastic collisions at the LHC, forward light hadrons are commonly
produced. Most of these collisions are of a soft, low-scale nature, with a
characteristic energy scale $Q$ roughly equivalent to $\Lambda_{\text{QCD}}$,
and these collisions generally don’t result in the production of heavy or
large-transverse-momentum particles. The kinematics of these events falls
outside the scope of perturbative QCD’s applicability. As a result, these
events are often simulated using phenomenological hadronic interaction models.
These models vary significantly in several aspects, such as their underlying
theoretical framework and the methods they use to represent hadronization,
parton distributions, diffraction, and correlations. (For an overview, see
Table 2 of Ref. Albrecht:2021cxw .)

Several such tools have been developed for cosmic ray physics. The most up-to-
date event generators include EPOS-LHC Pierog:2013ria , SIBYLL 2.3d
Riehn:2019jet , and QGSJET 2.04 Ostapchenko:2010vb ; these version numbers are
implicit when omitted below. In addition, a new tune of Pythia 8.3 has
recently been presented that is specifically designed to describe forward
particle production at the LHC Fieg:2023kld ; this tune will be referred to as
PYTHIAforward.

All of the mentioned event generators have been tuned to or validated with a
variety of low- and high-energy accelerator data. These include measurements
of forward neutral hadron production at LHCf LHCf:2008lfy , a zero-degree
calorimeter with two detectors that are located about 140 m upstream and
downstream of ATLAS, covering pseudorapidities $|\eta|\gtrsim 8.8$. In Fig. 2,
the predictions of the event generators are compared to the forward photon
LHCf:2017fnw , $\eta$-meson Piparo:2023yam , and neutron LHCf:2018gbv  energy
spectra measured by LHCf at $\sqrt{s}=13~{}\text{TeV}$. Here the photons
originate primarily from neutral pion decay. No production model gives a
perfect fit to the data for all particles, energies, and pseudorapidities, but
the four event generators shown all provide fairly good descriptions of the
data, and together they form an envelope around most of the data.

Figure 2: Forward Particle Energy Spectra at LHCf and Model Predictions. The
energy spectra of forward photons LHCf:2017fnw  (upper left and upper center),
$\eta$-mesons Piparo:2023yam  (upper right), and neutrons LHCf:2018gbv  (lower
left, lower center, and lower right) measured by LHCf in different
pseudorapidity bins, compared to the predictions of the event generators EPOS-
LHC Pierog:2013ria , SIBYLL Riehn:2019jet , QGSJET Ostapchenko:2010vb , and
PYTHIAforward Fieg:2023kld . The shaded bands correspond to the spread of the
event generator predictions.

Most of the event generators only provide a central prediction, with no
measure of uncertainty. To define an associated uncertainty, we follow an
approach that is often adopted in astroparticle physics, that is, the spread
of event generator predictions is taken as an estimator of the production
uncertainty. In particular, we consider the spread in the four aforementioned
event generators, which have the best agreement with LHCf data: EPOS-LHC,
SIBYLL, QGSJET, and PYTHIAforward. This approach has the advantage that it
captures differences associated with the underlying physics modeling. It
should be noted that an alternative definition of uncertainties, using tuning
variations in Pythia 8.3, has been proposed in Ref. Fieg:2023kld . That study
found that the uncertainties obtained in this way are similar to those
obtained using the spread of event generators.

In Fig. 3, the combined energy spectra of charged-current(CC)-interacting
electron and muon neutrinos, summed with their corresponding anti-neutrinos,
that are produced in light hadron decays and interact in FASER$\nu$ in LHC Run
3 are shown in red. Following the above discussion, EPOS-LHC is used as the
central prediction and the envelope formed by EPOS-LHC, SIBYLL, QGSJET, and
PYTHIAforward is used to define an uncertainty band. These results depend on
the assumed interaction cross section. As will be discussed in detail in Sec.
IV, we use the neutrino interaction generator GENIE 3.4 GENIE:2021wox  to
determine the total neutrino interaction cross section.

Figure 3: Neutrino CC Interactions in FASER$\nu$. The energy spectrum of
electron neutrinos (left) and muon neutrinos (right) expected to have CC
interactions in FASER$\nu$ in LHC Run 3 with an integrated luminosity of
$250~{}\text{fb}^{-1}$. The component of neutrinos originating from light
(charm) hadron decays is shown in red (blue). The solid contours are the
central values, and the shaded regions show the corresponding uncertainties
(see text).

The situation is different for forward charm hadrons. Their production is, so
far, only included in some of the available hadronic interaction models. Ref.
Kling:2021gos  used SIBYLL 2.3d Riehn:2019jet , Pythia 8.2 Sjostrand:2014zea ,
and DPMJET 3.2019.1 DPMJET  (an update of Ref. Roesler:2000he ) and found that
their predictions may differ by large factors of ${\cal O}(10)$. In contrast
to light mesons, forward charm production can, in principle, be described
using perturbative QCD methods. Although several such predictions based on
analytic perturbative calculations have been presented Bai:2020ukz ;
Maciula:2022lzk ; Bhattacharya:2023zei , these often use approximate
descriptions of either the hard scattering or the hadronization that may
affect their reliability.

More recently, a new calculation was presented that simulates forward charm
production in a way that addresses the shortcomings of previous estimates. It
uses state-of-the-art QCD predictions for heavy hadron production that include
radiative corrections Buonocore:2023kna , as well as the sophisticated
modeling of hadronization implemented in MC generators. This simulation uses
POWHEG Nason:2004rx ; Frixione:2007vw ; Alioli:2010xd  with the
NNPDF3.1sx+LHCb PDF set Ball:2017otu ; Bertone:2018dse  to model charm
production at next-to-leading order in $\alpha_{s}$ with small-$x$ resummation
at next-to-leading logarithmic accuracy, and then matches it with Pythia 8.3
Bierlich:2022pfr  for parton showering and hadronization. This analysis also
includes an uncertainty estimate based on scale uncertainties. Additional
sources of uncertainties, for example, those arising from the modeling of
hadronization, were investigated, but were found to be within the scale
uncertainty band. We therefore use the scale variations to define the flux
uncertainties.

In Fig. 3, the predicted energy spectra of electron and muon neutrinos that
are produced in charm hadron decays and interact in FASER$\nu$ in LHC Run 3
are shown in blue. Following the above discussion, POWHEG + Pythia 8.3 with
default scales is used as the central prediction, and the resummation and
factorization scales are varied by a factor of two to define the uncertainty
band.

##  IV Neutrino Interaction Cross Sections

The number of neutrino events depends not only on the neutrino flux, but also
the neutrino interaction cross section. Given the typically large neutrino
energy $E_{\nu}>100~{}\text{GeV}$, most neutrino interactions at FASER can be
described as DIS. It is worth noting, however, that, especially at lower
energies $E_{\nu}<100~{}\text{GeV}$, there can be a substantial non-DIS
contribution. Following the notation of Ref. Candido:2023utz , the CC
neutrino-nucleon interaction cross section can be written as

$\displaystyle\frac{d\sigma_{\nu
N}}{dx\,dy}=\frac{G_{F}^{2}m_{N}E_{\nu}}{\pi(1+m_{W}^{2}/Q^{2})^{2}}\,\big{[}xy^{2}F_{1}+(1-y)F_{2}+xy(1-y/2)F_{3}]\
,$ (1)

where $x$ is the fraction of the nucleon’s momentum carried by the quark in
the initial state, $y$ is the fraction of the neutrino momentum transferred to
the hadronic system, $m_{N}$ and $m_{W}$ are the masses of the nucleon and $W$
boson, respectively, $Q^{2}=2E_{\nu}m_{N}xy$ is the transferred four-momentum,
and $F_{i}(x,Q^{2})$ are the structure functions of the proton. Different
models for the structure functions have been proposed in the literature, some
of which allow one to extend this formalism into the non-DIS regime.

Figure 4: Top: Muon neutrino (left) and anti-neutrino (right) CC interaction
cross sections as functions of the incoming neutrino energy, as obtained using
GENIE GENIE:2021wox , NNSFv Candido:2023utz , and CKMT+PCAC-NT Jeong:2023hwe ,
as indicated. For comparison, we show the neutral-current (NC) cross section
prediction of GENIE in dotted green. The electron neutrino cross section
prediction is approximately the same, differing by less than 0.1% for
$E_{\nu}>100~{}\text{GeV}$, and the tau neutrino cross section is $\lesssim
20\%$ smaller, due to the tau mass. The red-shaded bands show the NNSFv
uncertainty band. Bottom: Muon neutrino (left) and anti-neutrino (right) CC
cross sections normalized to the GENIE $W>2~{}\text{GeV}$ results. The gray
bands correspond to flat 6% uncertainties in the energy range
$100~{}\text{GeV}<E_{\nu}<3~{}\text{TeV}$.

The Bodek-Yang model Bodek:2002vp ; Bodek:2004pc ; Bodek:2010km  is widely
recognized as a phenomenological framework for describing inelastic neutrino-
nucleon scattering cross sections in the multi-GeV energy range. It is
implemented in GENIE GENIE:2021wox , a commonly-used neutrino interaction MC
generator, which has been extensively benchmarked in neutrino experiments in
the 0.1–100 GeV energy range and simulates both DIS and non-DIS contributions
to the cross section. The Bodek-Yang model builds upon structure functions
using effective leading-order GRV98 PDFs Gluck:1998xa . To account for mass,
higher-order QCD, and nuclear effects, various phenomenological corrections
are employed. In addition, modifications have been incorporated to extend its
applicability to the non-perturbative, low-$Q^{2}$ regime.

For FASER predictions, however, the Bodek-Yang model has certain limitations:
(i) it relies on obsolete PDFs that neglect recent constraints on proton and
nuclear structure obtained in the last 25 years; (ii) it only includes PDFs
for up, down, and strange quarks; (iii) it omits available higher-order QCD
calculations; (iv) it was primarily designed for the multi-GeV domain rather
than TeV neutrino energies; and (v) it lacks a systematic assessment of
uncertainties linked to its predictions.

A variety of other cross section calculations based on NLO structure functions
and modern PDF sets have been presented, primarily for applications in
astroparticle physics, including the CSMS Cooper-Sarkar:2011jtt  and BGR18
Bertone:2018dse ; Garcia:2020jwr  models, as well as cross sections based on
the CT18 Xie:2023suk  PDFs. More recently, additional models have been
presented that further extend the NLO structure function models to the low
$Q^{2}$ regime and hence extend their applicability to lower neutrino
energies. Examples of such models are NNSFv Candido:2023utz  and CKMT+PCAC-NT
Jeong:2023hwe . For NNSFv, the structure functions are determined by a data-
driven parameterization at low and moderate values of $Q^{2}$ matched to
perturbative QCD calculations at high $Q^{2}$. Notably, NNSFv also includes an
uncertainty estimate obtained in a data-driven way. CKMT+PCAC-NT uses a
structure function parameterization augmented by a correction to account for
the partial conservation of the axial-vector current, normalized to structure
functions evaluated at NLO order in QCD, and includes target mass and heavy
quark corrections. We note, however, that the NLO structure function models do
not describe neutrino scattering at low-$W$, where $W$ is the invariant mass
of the final-state hadronic system, and are not yet available in event
generators. We can, however, use them to validate the Bodek-Yang model
implemented in GENIE.

In Fig. 4, the neutrino CC interaction cross sections obtained by the
different approaches are compared, including the inclusive interaction cross
sections obtained using GENIE, and the predictions of GENIE, NNSFv, and
CKMT+PCAC-NT after the DIS selection cuts $Q>0.03~{}\text{GeV}$ and
$W>2~{}\text{GeV}$. The red-shaded band shows the NNSFv uncertainty, as
obtained in Ref. Candido:2023utz . There is general agreement between the
predictions of the Bodek-Yang model, as implemented in GENIE, with NNSFv and
CKMT+PCAC-NT, and the cross section uncertainties for neutrinos with energies
above $100~{}\text{GeV}$ are roughly at the 6% level. For our calculations of
the interacting neutrino rate, the default GENIE cross section is used. For
purposes of comparison, the NC cross sections predicted by GENIE are also
shown in Fig. 4.

##  V Neutrino Rates and Distributions

We now turn to our predictions for the forward neutrino spectra at the LHC
during Run 3 and Run 4. The hadron spectra are generated using the event
generators discussed in Sec. III, propagated down the beam pipe, and decayed
to produce a flux of neutrinos. The total neutrino cross section provided by
the GENIE implementation of the Bodek-Yang model, as discussed in Sec. IV, is
used to produce the energy spectra of CC neutrino interactions in FASER$\nu$.

Generators | FASER$\nu$ at Run 3 | FASER$\nu$ at Run 4
---|---|---
light hadrons | charm hadrons | $\nu_{e}+\bar{\nu}_{e}$ | $\nu_{\mu}+\bar{\nu}_{\mu}$ | $\nu_{\tau}+\bar{\nu}_{\tau}$ | $\nu_{e}+\bar{\nu}_{e}$ | $\nu_{\mu}+\bar{\nu}_{\mu}$ | $\nu_{\tau}+\bar{\nu}_{\tau}$
EPOS-LHC | – | 1149 | 7996 | – | 3382 | 23054 | –
SIBYLL 2.3d | – | 1126 | 7261 | – | 3404 | 21532 | –
QGSJET 2.04 | – | 1181 | 8126 | – | 3379 | 22501 | –
PYTHIAforward | – | 1008 | 7418 | – | 2925 | 20508 | –
– |  POWHEG Max | 1405 | 1373 | 76 | 4264 | 4068 | 255
– | POWHEG | 527 | 511 | 28 | 1537 | 1499 | 91
– |  POWHEG Min | 294 | 284 | 16 | 853 | 826 | 51
Combination | $1675_{-372}^{+911}$ | $8507_{-962}^{+992}$ | $28_{-12}^{+48}$ | $4919_{-1141}^{+2748}$ | $24553_{-3219}^{+2568}$ | $91_{-41}^{+163}$
Table 1: The expected number of CC neutrino interaction events occurring in
FASER$\nu$ during LHC Run 3 with $250~{}\text{fb}^{-1}$ and Run 4 with
$680~{}\text{fb}^{-1}$. The detector geometry and locations for Run 3 and Run
4 are as described in Sec. II, and results are shown for the various event
generators described in Sec. III. In the bottom row, for the combination, we
show the sum of the averages of the light hadron and charm hadron
contributions as the central prediction, and their spread as the uncertainty.

In Table 1 the total number of neutrinos interacting in FASER$\nu$ is shown
for each flavor in LHC Run 3 and Run 4. For neutrinos produced in light hadron
decay, results for EPOS-LHC, SIBYLL, QGSJET, and PYTHIAforward are displayed.
The results from these event generators agree within roughly 10%. For
neutrinos produced in charm hadron decay, results are shown for POWHEG+Pythia
8.3 and the scale variations discussed in Ref. Buonocore:2023kna , providing a
maximum, central, and minimum prediction for the charm hadron flux. The spread
in event rates is much larger for charm hadrons than for light hadrons, as
seen in the lower section of Table 1. The charm hadrons are the source of
approximately 30% of the $\nu_{e}$ event rate, 5% of the $\nu_{\mu}$ event
rate, and 100% of the $\nu_{\tau}$ event rate. Moreover, the fraction of
$\nu_{e}$ coming from charm hadron decay is large at higher neutrino energies
and is approximately 50% at $E_{\nu}=1~{}\text{TeV}$ and 90% at
$E_{\nu}=3~{}\text{TeV}$. In the bottom row, the central prediction is derived
by summing the EPOS-LHC contribution for light hadrons and POWHEG+Pythia 8.3
for charm hadrons, while their variation is used to estimate the uncertainty.
Overall, the $\nu_{e}$, $\nu_{\mu}$, and $\nu_{\tau}$ event rates are found to
be approximately 1700, 8500, and 30 in LHC Run 3 and 4900, 25000, and 90 in
LHC Run 4, respectively, with the uncertainty in each being dominated by the
uncertainty in charm hadron production.

In the upper part of Fig. 5, the interacting neutrino spectra is shown,
including uncertainties, for all flavors at FASER$\nu$ for an integrated
luminosity of $250~{}\text{fb}^{-1}$ for the Run 3 configuration. The central
prediction is obtained using EPOS-LHC for neutrinos from light hadrons and
POWHEG+Pythia 8.3 for charm hadrons. For light hadron production, the
uncertainty is defined as the spread of event generators EPOS-LHC, SIBYLL,
QGSJET, and PYTHIAforward. For charm hadrons, the error bands are obtained
using scale uncertainties from POWHEG+Pythia 8.3. The central panel shows the
same spectra, but normalized to the central prediction. The lower panel shows
the fraction of neutrinos produced in charm hadron decays.

Figure 5: The energy spectra of neutrinos interacting in FASER$\nu$ at LHC Run
3 with an integrated luminosity of $250~{}\text{fb}^{-1}$ for electron
neutrinos (solid red), muon neutrinos (dashed blue), and tau neutrinos (dotted
green). For each neutrino species, the central prediction is determined by the
decay of light (charm) hadrons as predicted by EPOS-LHC (POWHEG+Pythia 8.3)
with the interaction cross section provided by GENIE. The shaded bands are the
uncertainties due to the flux and do not include cross section uncertainties.
The upper panel shows the energy spectra. The central panel shows the same
data normalized to the central prediction. The lower panel shows the fraction
of neutrinos produced in charm hadron decays. Figure 6: The distribution in
the transverse plane in nominal coordinates of neutrinos interacting in LHC
Run 3 with an integrated luminosity of $250~{}\text{fb}^{-1}$ in FASER$\nu$
for electron (left), muon (middle), and tau (right) neutrinos, with light and
charm hadron production modeled with EPOS-LHC and POWHEG+Pythia 8.3,
respectively. The dashed black circle indicates the area covered by the FASER
spectrometer.

The spatial distribution in the transverse plane of neutrinos interacting in
FASER$\nu$ at LHC Run 3 with a total integrated luminosity of
$250~{}\text{fb}^{-1}$ is shown in Fig. 6. The three panels show results for
the three neutrino species, $\nu_{e}$, $\nu_{\mu}$, and $\nu_{\tau}$, from
left to right. EPOS-LHC and POWHEG+Pythia 8.3 for light and charm hadron
production are used, respectively. Here, the integration has been performed
over all energies. At relatively low energies, muon neutrinos, produced
primarily in pion decays, are more collimated near the LOS than the other
flavors; electron neutrinos, dominantly produced in kaon decays, are less
collimated; and tau neutrinos, produced in charm hadron decays, are the least
collimated. At $\sim\text{TeV}$ energies, muon neutrinos are produced
dominantly in kaon decays, electron neutrinos have significant contributions
from both kaon and charm hadron decay, and tau neutrinos are again produced
only in charm hadron decays, leading to the same relative ordering of
collimation.

In Fig. 7, the distribution of interacting neutrinos is shown in the
$(x,\text{Energy})$ plane, where $x$ is the horizontal spatial coordinate;
this plot would look similar for the vertical component. Here a detector is
assumed with the same material as FASER$\nu$, centered on the true LOS, but
with dimensions $1~{}\text{m}\times 1~{}\text{m}\times 1~{}\text{m}$, where
the larger transverse extension is chosen to show the distribution over a
larger range. As in Fig. 6, muon neutrinos are more collimated than electron
neutrinos, which are more collimated than tau neutrinos, but we also see that
the highest-energy neutrinos of each species are focused along the LOS.

Figure 7: The rate (in fb per bin) of neutrinos interacting in a
$1~{}\text{m}\times 1~{}\text{m}\times 1~{}\text{m}$ FASER$\nu$-like detector
in the $(x,\text{Energy})$ plane in LHC Run 3. The detector is centered on the
true LOS, so the LOS is at $x=0$. Results are shown for electron (left), muon
(middle), and tau (right) neutrinos with light and charm hadron production
modeled with EPOS-LHC and POWHEG+Pythia 8.3, respectively.

In Fig. 8 the binned event rate is shown for $\nu_{e}$, $\nu_{\mu}$, and
$\nu_{\tau}$ for Run 3 (upper panels) and Run 3 + Run 4 (lower panels). For
each energy bin, the colors show the composition in terms of the parent
hadron. Also shown for each bin is the statistical uncertainty, defined as
$\sqrt{N_{\rm bin}}$. For $\nu_{e}$ the event rate is dominated by kaon decays
at low energy, with charm decays becoming comparable at higher energies.
Hyperon decays also provide a non-negligible contribution to the $\nu_{e}$
event rate. For $\nu_{\mu}$ the event rate is dominated by pion decays at low
energies and kaon decays at high energies, with charm decays providing a sub-
leading component. Charm hadrons are the only hadrons that produce
$\nu_{\tau}$ at FASER$\nu$. In the absence of sizeable systematic
uncertainties, we find that Run 3 will provide sufficient number of events to
distinguish the contributions of the various parent hadrons to the neutrino
flux detected at FASER.

Figure 8: Binned energy spectra for electron (left), muon (center), and tau
(left) neutrinos interacting in FASER$\nu$ at LHC Run 3 with a total
integrated luminosity of 250 $\text{fb}^{-1}$ (upper panels) and at LHC Run 3
+ Run 4 with a total integrated luminosity 930 $\text{fb}^{-1}$ (lower
panels). For each bin, the neutrinos are separated by their parent hadrons:
pions (red), kaons (green), charm hadrons (blue), and hyperons (yellow).
Hadron production and decay of light (charm) hadrons is modeled by EPOS-LHC
(POWHEG+Pythia 8.3). Also shown are statistical errors per bin, defined as
$\sqrt{N_{\rm bin}}$, which demonstrate that Run 3 measurements will have
sufficient statistics to be sensitive to components of neutrinos from the
different parent hadrons.

In Fig. 9 the energy distribution of muon neutrinos with CC interactions in
FASER$\nu$ is shown. In Ref. FASER:2023zcr  the electronic detector components
of FASER were used to track the outgoing muon in CC $\nu_{\mu}$ interactions.
By measuring the curvature of the muon in the magnetic field of FASER, the
charge of the muon was identified for muons with energy below a TeV, allowing
one to distinguish $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ interactions. In Fig. 9,
the energy distributions are therefore separated into $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ for energies below 1 TeV. Above ${\cal O}(1~{}{\rm TeV})$
the charge measurement will become unreliable, so $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ are grouped together. Here, only neutrinos that interact
within the portion of FASER$\nu$ that is within the aperture of the FASER
spectrometer are included, so that the forward-going muon can be measured.
Also included is a $20\%$ efficiency for this signal, which is typical for
this measurement at FASER FASER:2023zcr . By measuring the muon’s charge,
measurements by the electronic detector components of FASER will be able to
distinguish $\nu_{\mu}$ from $\bar{\nu}_{\mu}$, providing a further probe of
forward hadron production.

Figure 9: The energy spectra of muon neutrinos and anti-neutrinos detected
through their CC interactions, using only the electronic detector components
of FASER in LHC Run 3 with a total integrated luminosity of
$250~{}\text{fb}^{-1}$ (left) and in LHC Run 3 + Run 4 with a total integrated
luminosity of $930~{}\text{fb}^{-1}$ (right). For each bin, the neutrinos are
separated by their parent hadrons, as indicated. FASER’s magnets enable charge
identification of the outgoing muons, which allows $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ separation for energies below a TeV.

##  VI Conclusions

The recent discovery of collider neutrinos at FASER has opened up the new
field of TeV laboratory neutrinos. Their large fluxes and relatively large
interaction cross sections imply large event rates, even with relatively small
detectors. To fully realize the potential of these neutrino events for both SM
and BSM physics Anchordoqui:2021ghd ; Feng:2022inv , it is necessary to have
accurate predictions of the forward neutrino spectrum.

The forward neutrino flux at the LHC is dominantly produced by the decays of
pions, kaons, hyperons, and charm hadrons. As the forward region is relatively
unprobed, there are sizeable uncertainties in the fluxes of these parent
hadrons. For neutrinos from light hadron decays, the central value is taken to
be the predictions of EPOS-LHC Pierog:2013ria , with the associated
uncertainty given by the spread in predictions from the event generators EPOS-
LHC Pierog:2013ria , Sibyll Riehn:2019jet , QGSJET Ostapchenko:2010vb , and
PYTHIAforward Fieg:2023kld . For neutrinos from charm hadron decays, the
central value is taken to be the results of POWHEG+Pythia 8.3 Alioli:2010xd ;
Bierlich:2022pfr , with the uncertainty given by the variation resulting from
varying the factorization and renormalization scales Buonocore:2023kna .

The corresponding neutrino fluxes are obtained using the dedicated fast
neutrino flux simulation of Ref. Kling:2021gos , which has been updated for
the Run 3 and Run 4 beam configurations. To produce the spectra of CC neutrino
interactions in FASER$\nu$, the Bodek-Yang model Bodek:2002vp ; Bodek:2004pc ;
Bodek:2010km  implemented in GENIE GENIE:2021wox  is used, which agrees with
more recent cross section calculations for TeV neutrinos, NNSFv
Candido:2023utz  and CKMT+PCAC-NT Jeong:2023hwe , to within $\lesssim 6\%$
over the range of energies of interest.

The expected neutrino event rates are presented in Table 1. These include
results for LHC Run 3 with an expected integrated luminosity of
$250~{}\text{fb}^{-1}$. In addition, given the recent approval of FASER for
LHC Run 4 Run4LOI , neutrino event distributions are simulated for LHC Run 4
with an expected total integrated luminosity of $680~{}\text{fb}^{-1}$. The
central values of the expected neutrino event rates for $\nu_{e}$,
$\nu_{\mu}$, and $\nu_{\tau}$ are 1700, 8500, and 30 in LHC Run 3 and 4900,
25000, and 90 in LHC Run 4. Such event rates imply percent-level statistical
uncertainties for electron and muon neutrino studies. For tau neutrinos, the
number of events that will be observed at FASER in the coming years will
greatly enhance the number that have been observed to date.

Last, in Figs. 8 and 9 results are presented for both Run 3 and Run 3 + Run 4
for the energy spectra of neutrinos interacting in FASER$\nu$, decomposed into
components based on the parent hadron species. It can be seen that statistical
uncertainties will be small enough that FASER$\nu$ will be sensitive not only
to the leading contributions, but also to sub-leading contributions. Provided
experimental systematic uncertainties are not dominant, these results imply
promising prospects for studying very high energy neutrinos, forward hadron
production, and their many related topics with FASER in the coming years at
LHC Run 3 and Run 4.

###### Acknowledgements.

We thank Luca Buonocore, Anatoli Fedynitch, Alfonso Garcia, Yu Seon Jeong,
Hallsie Reno, Luca Rottoli, and Dennis Soldin for useful discussions. We thank
the technical and administrative staff members at all FASER institutions for
their contributions to the success of the FASER project. This work was
supported in part by Heising-Simons Foundation Grant Nos. 2018-1135,
2019-1179, and 2020-1840, Simons Foundation Grant No. 623683, U.S. National
Science Foundation Grant Nos. PHY-2111427, PHY-2110929, and PHY-2110648, JSPS
KAKENHI Grants Nos. JP19H01909, JP20K23373, JP20H01919, JP20K04004, and
JP21H00082, BMBF Grant No. 05H20PDRC1, DFG EXC 2121 Quantum Universe Grant No.
390833306, ERC Consolidator Grant No. 101002690, Royal Society Grant No.
URF\R1\201519, UK Science and Technology Funding Councils Grant No. ST/
T505870/1, the National Natural Science Foundation of China, Tsinghua
University Initiative Scientific Research Program, and the Swiss National
Science Foundation. *

##  Appendix A Comparison of Forward Charm Production Models

As evident in Fig. 8, forward charm hadron production plays an important role
in determining neutrino event rates at FASER, contributing significantly to
$\nu_{e}$ and $\nu_{\mu}$ rates, and providing essentially all of the
$\nu_{\tau}$ rate. In this work, the central value for forward charm
production is taken to be the results of POWHEG+Pythia 8.3 Alioli:2010xd ;
Bierlich:2022pfr , with the uncertainty given by varying the factorization and
renormalization scales Buonocore:2023kna .

Here we compare the results of this prescription to the results from other
generators. In Fig. 10, we show results from POWHEG+Pythia 8.3, along with
results from SIBYLL 2.3d Riehn:2019jet , Pythia 8.3 Sjostrand:2014zea , and
DPMJET 3.2019.1 DPMJET . As noted by the author Anatoli , DPMJET was never
validated for charm production and is not intended to be used for forward
charm production. However, DPMJET is used by FLUKA Battistoni:2015epi , a
widely-used framework for propagating particles through the LHC infrastructure
and estimating event rates in forward detectors, and so it is instructive to
include it here for comparison.

In Fig. 10 (left), results are shown for charm hadron production for
pseudorapidities $\eta$ in the range 2 to 4.5, along with data from LHCb.
POWHEG+Pythia 8.3, SIBYLL, and Pythia give comparable predictions, and the
variation in these generators is well characterized by varying the
factorization and renormalization scales in POWHEG+Pythia 8.3, as prescribed
in this work. These results are also consistent with LHCb data. In contrast,
DPMJET deviates from the other three generators and predicts charm hadron
rates that are inconsistent with the data. This inconsistency may be
attributed to a number of aspects of the DPMJET model: (i) the assumption of
massless charm quarks in calculating the underlying matrix element for $gg\to
c\bar{c}$; (ii) the use of CT14LO PDFs that may overestimate the charm quark
content; and (iii) a $k$ factor ($\sim 2$) that, although not unreasonably
large in this context, significantly enhances the rate.

The results of these generators for the spectrum of electron neutrinos
detected at FASER$\nu$ is shown in Fig. 10 (right). Once again, POWHEG+Pythia
8.3, SIBYLL, and Pythia give comparable predictions, and the variation between
them is fairly well characterized by the uncertainty prescription for
POWHEG+Pythia 8.3. However, DPMJET predicts much larger event rates, differing
from the other prescriptions by a factor of 10 or even larger at the highest
energies. Until the discrepancy with LHCb data is understood and resolved,
projections for neutrino event rates originating from DPMJET charm hadron
predictions cannot be considered on a solid footing and may be overestimating
the flux by as much as an order of magnitude.

Figure 10: Comparison of Forward Charm Production Models. Left: Comparison of
estimates of charm hadron fluxes from POWHEG+Pythia 8.3, SIBYLL, Pythia, and
DPMJET for pseudorapidities $2<\eta<4.5$, along with data from LHCb for $pp$
collisions at center-of-mass energy $\sqrt{s}=13~{}\text{TeV}$ LHCb:2015swx .
The spectra for the $3<\eta<3.5$ and $4<\eta<4.5$ pseudorapidity regions are
rescaled by factors of 10 and 100, respectively, for display purposes. Right:
For the same models, the predicted energy spectra of electron neutrinos from
charm hadron decay that interact in FASER$\nu$ during Run 3 at
$\sqrt{s}=13.6~{}\text{TeV}$. The gray-shaded region in each panel is the
uncertainty envelope for the POWHEG+Pythia 8.3 prediction, as determined by
varying the factorization and renormalization scales.

## References

  * (1) 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].
  * (2) FASER Collaboration, “The FASER Detector,” arXiv:2207.11427 [physics.ins-det].
  * (3) FASER Collaboration, “Technical Proposal for FASER: ForwArd Search ExpeRiment at the LHC,” arXiv:1812.09139 [physics.ins-det].
  * (4) FASER Collaboration, “Technical Proposal: FASERnu,” arXiv:2001.03073 [physics.ins-det].
  * (5) FASER Collaboration, “The trigger and data acquisition system of the FASER experiment,” JINST 16 (2021) no. 12, P12028, arXiv:2110.15186 [physics.ins-det].
  * (6) FASER Collaboration, “The tracking detector of the FASER experiment,” Nucl. Instrum. Meth. A 1034 (2022) 166825, arXiv:2112.01116 [physics.ins-det].
  * (7) FASER Collaboration, “First Direct Observation of Collider Neutrinos with FASER at the LHC,” Phys. Rev. Lett. 131 (2023) no. 3, 031801, arXiv:2303.14185 [hep-ex].
  * (8) FASER 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.
  * (9) FASER Collaboration, “Search for Dark Photons with the FASER detector at the LHC,” Phys. Lett. B 848 (2024) 138378, arXiv:2308.05587 [hep-ex].
  * (10) SND@LHC Collaboration, “Observation of Collider Muon Neutrinos with the SND@LHC Experiment,” Phys. Rev. Lett. 131 (2023) no. 3, 031802, arXiv:2305.09383 [hep-ex].
  * (11) FASER Collaboration, “Request to run FASER in Run 4,” Tech. Rep. CERN-LHCC-2023-009, CERN, Geneva, November, 2023.  https://cds.cern.ch/record/2882503.
  * (12) F. Kling, T. Mäkelä, and S. Trojanowski, “Investigating the fluxes and physics potential of LHC neutrino experiments,” Phys. Rev. D 108 (2023) no. 9, 095020, arXiv:2309.10417 [hep-ph].
  * (13) 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].
  * (14) 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].
  * (15) 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].
  * (16) 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].
  * (17) 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].
  * (18) 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.
  * (19) A. Bodek, I. Park, and U.-k. Yang, “Improved low Q**2 model for neutrino and electron nucleon cross sections in few GeV region,” Nucl. Phys. B Proc. Suppl. 139 (2005) 113–118, arXiv:hep-ph/0411202.
  * (20) A. Bodek and U.-k. Yang, “Axial and Vector Structure Functions for Electron- and Neutrino- Nucleon Scattering Cross Sections at all $Q^{2}$ using Effective Leading order Parton Distribution Functions,” arXiv:1011.6592 [hep-ph].
  * (21) 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].
  * (22) C. Bierlich et al., “Robust Independent Validation of Experiment and Theory: Rivet version 3,” SciPost Phys. 8 (2020) 026, arXiv:1912.05451 [hep-ph].
  * (23) L. J. Nevay et al., “BDSIM: An accelerator tracking code with particle–matter interactions,” Comput. Phys. Commun. 252 (2020) 107200, arXiv:1808.10745 [physics.comp-ph].
  * (24) S. Agostinelli et al., “Geant4 – a simulation toolkit,” Nucl. Instrum. Meth. A 506 (2003) 250.
  * (25) I. Zurbano Fernandez et al., “High-Luminosity Large Hadron Collider (HL-LHC): Technical design report,” vol. 10/2020.  12, 2020.
  * (26) “MAD — Methodical Accelerator Design,”. https://madx.web.cern.ch/madx.
  * (27) R. Tomas Garcia et al., “HL-LHC Run 4 proton operational scenario,” tech. rep., CERN, Geneva, 2022.  https://cds.cern.ch/record/2803611.
  * (28) 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].
  * (29) 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].
  * (30) F. Riehn, R. Engel, A. Fedynitch, T. K. Gaisser, and T. Stanev, “Hadronic interaction model Sibyll 2.3d and extensive air showers,” Phys. Rev. D 102 (2020) no. 6, 063002, arXiv:1912.03300 [hep-ph].
  * (31) 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].
  * (32) M. Fieg, F. Kling, H. Schulz, and T. Sjöstrand, “Tuning pythia for forward physics experiments,” Phys. Rev. D 109 (2024) no. 1, 016010, arXiv:2309.08604 [hep-ph].
  * (33) LHCf Collaboration, “The LHCf detector at the CERN Large Hadron Collider,” JINST 3 (2008) S08006.
  * (34) LHCf Collaboration, “Measurement of forward photon production cross-section in proton-proton collisions at $\sqrt{s}$ = 13 TeV with the LHCf detector,” Phys. Lett. B 780 (2018) 233–239, arXiv:1703.07678 [hep-ex].
  * (35) LHCf Collaboration, “Measurement of the forward $\eta$ meson production rate in p-p collisions at $\sqrt{\textrm{s}}$ = 13 TeV with the LHCf-Arm2 detector,” JHEP 10 (2023) 169, arXiv:2305.06633 [hep-ex].
  * (36) LHCf Collaboration, “Measurement of inclusive forward neutron production cross section in proton-proton collisions at $\sqrt{s}=13$ TeV with the LHCf Arm2 detector,” JHEP 11 (2018) 073, arXiv:1808.09877 [hep-ex].
  * (37) GENIE Collaboration, “Hadronization model tuning in genie v3,” Phys. Rev. D 105 (2022) no. 1, 012009, arXiv:2106.05884 [hep-ph].
  * (38) 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].
  * (39) A. Fedynitch, “DPMJET,”. https://github.com/DPMJET.
  * (40) 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.
  * (41) 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].
  * (42) 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].
  * (43) A. Bhattacharya, F. Kling, I. Sarcevic, and A. M. Stasto, “Forward neutrinos from charm at the Large Hadron Collider,” Phys. Rev. D 109 (2024) no. 1, 014040, arXiv:2306.01578 [hep-ph].
  * (44) L. Buonocore, F. Kling, L. Rottoli, and J. Sominka, “Predictions for Neutrinos and New Physics from Forward Heavy Hadron Production at the LHC,” arXiv:2309.12793 [hep-ph].
  * (45) P. Nason, “A New method for combining NLO QCD with shower Monte Carlo algorithms,” JHEP 11 (2004) 040, arXiv:hep-ph/0409146.
  * (46) 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].
  * (47) 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].
  * (48) 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].
  * (49) V. Bertone, R. Gauld, and J. Rojo, “Neutrino Telescopes as QCD Microscopes,” JHEP 01 (2019) 217, arXiv:1808.02034 [hep-ph].
  * (50) C. Bierlich et al., “A comprehensive guide to the physics and usage of PYTHIA 8.3,” arXiv:2203.11601 [hep-ph].
  * (51) 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].
  * (52) Y. S. Jeong and M. H. Reno, “Neutrino cross sections: Interface of shallow- and deep-inelastic scattering for collider neutrinos,” Phys. Rev. D 108 (2023) no. 11, 113010, arXiv:2307.09241 [hep-ph].
  * (53) M. Glück, E. Reya, and A. Vogt, “Dynamical parton distributions revisited,” Eur. Phys. J. C 5 (1998) 461–470, arXiv:hep-ph/9806404.
  * (54) A. Cooper-Sarkar, P. Mertsch, and S. Sarkar, “The high energy neutrino cross-section in the Standard Model and its uncertainty,” JHEP 08 (2011) 042, arXiv:1106.3723 [hep-ph].
  * (55) A. Garcia, R. Gauld, A. Heijboer, and J. Rojo, “Complete predictions for high-energy neutrino propagation in matter,” JCAP 09 (2020) 025, arXiv:2004.04756 [hep-ph].
  * (56) 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].
  * (57) A. Fedynitch. Private communication.
  * (58) G. Battistoni et al., “Overview of the FLUKA code,” Annals Nucl. Energy 82 (2015) 10–18.
  * (59) LHCb Collaboration, “Measurements of prompt charm production cross-sections in $pp$ collisions at $\sqrt{s}=13$ TeV,” JHEP 03 (2016) 159, arXiv:1510.01707 [hep-ex]. [Erratum: JHEP 09, 013 (2016), Erratum: JHEP 05, 074 (2017)].