# Balancing Asymmetric Dark Matter with Baryon Asymmetry by Sphaleron
Transitions***Presented at the 1st Electronic Conference on Universe, 22–28
February 2021; Available online: https://ecu2021.sciforum.net/.

Arnab Chaudhuria†††e-mail<EMAIL_ADDRESS>Maxim Yu.
Khlopovb‡‡‡e-mail<EMAIL_ADDRESS>
a Novosibirsk State University
Pirogova ul., 2, 630090 Novosibirsk, Russia
b Institute of Physics, Southern Federal University
Stachki 194 Rostov on Don 344090, Russia
and Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris,
France
and National Research Nuclear University "MEPHI", Moscow 115409, Russia

###### Abstract

The effect of the electroweak sphaleron transition in balance between baryon
excess and and the excess of stable quarks of 4th generation is studied in
this paper. Considering the non-violation of $SU(2)$ symmetry and the
conservation of electroweak and new charges and quantum numbers of the new
family, it makes possible sphaleron transitions between baryons, leptons and
4th family of leptons and quarks. In this paper, we have tried to established
a possible definite relationship between the value and sign of the 4th family
excess relative to baryon asymmetry. If $U$-type quarks are the lightest
quarks of the 4th family and sphaleron transitions provide excessive $\bar{U}$
antiquarks, asymmetric dark matter in the form of dark atom bound state of
($\bar{U}\bar{U}\bar{U}$) with primordial He nuclei is balanced with baryon
asymmetry.

##  1 Introduction

SPHALERON (or electroweak sphaleron) [1] and [2] is a static (time-
independent) solution to the electroweak field equations of the Standard Model
of the particle physics. It is involved in certain hypothetical processes that
violate baryon and lepton numbers. Such process cannot be represented by
perturbative methods (like Feynman Diagrams) and hence termed as non
perturbative.

It is possible for a sphaleron to convert baryons into antileptons and
antibaryons to leptons and thus violate the baryon number. If the density of
the sphaleron at some point high enough, they could wipe out any net excess of
baryons or anti-baryons. Sphalerons are also associated with saddle points. It
can be interpreted as the peak energy configuration. It is shown in [3, 4]
that because of this anomaly transition between vacua are associated with a
violation of Baryon (and lepton) numbers.

Even though a baryon excess can be created at the time of electroweak symmetry
breaking, it is preserved if the process happened through first order phase
transition. In second order phase transition, sphalerons can wipe out the
entire asymmetry created but in first order, only the asymmetry created in the
unbroken phase is wiped out.

The cosmological dark matter may contain dark atoms, in which new stable
charged particles are bound by ordinary Coulomb interaction, see [5, 6, 7, 8,
9]. Many models can exists in which heave $-2$ charged stable species are
predicted:

1\. AC-leptons, predicted in the extension of standard model, based on the
approach of almost-commutative geometry, see [10, Author2(year), 11, 12].

2\. Technileptons and anti-technibaryons in the framework of walking
technicolor models (WTC), see [13, 14, 15, 16, 17, 18].

3\. and, finally, stable ”heavy quark clusters” ($\bar{U}\bar{U}\bar{U}$)
formed by anti-U quark of 4th generation, see [19, 20, 21].

All these models also predict corresponding +2 charge particles. If these
positively charged particles remain free in the early Universe, they can
recombine with ordinary electrons in anomalous helium, which is strongly
constrained in the terrestrial matter. Therefore cosmological scenario should
provide a mechanism, which suppresses anomalous helium. There are two ways
this can be done:

1\. The abundance of anomalous helium in the Galaxy may be significant, but in
the terrestrial matter there exists a recombination mechanism suppressing this
abundance below experimental upper limits. The existence of a new strict U(1)
gauge symmetry, causing new Coulomb like long range interaction between
charged dark matter particles, is crucial for this mechanism. Therefore the
existence of dark radiation in the form of hidden photons is inevitable in
this approach.

2\. Free positively charged particles are already suppressed in the early
Universe and the abundance of anomalous helium in the Galaxy is negligible.

Dark matter production and the detailed study of sphaleron transition due to
walking technicolor model has already been conducted, see [17].

In this paper, we consider 4th generation of particle family with a stable
quark and neutrino family and study the sphaleron transition during
electroweak phase transition. We consider the simplest model which is charge
neutral and all the particles are stable. We also consider the simplest second
order phase tranistion. Other complicated process such as annihilation of
unstable particles and first order phase tranistion are beyond the scope of
this paper.

##  2 Calculations and Discussions

The main scope of our current activity is to deduce the relationship between
baryon excess and excess of stable fermions of 4th family. We can take for
definiteness that 4th neutrino and U-quark of 4th family are stable and
establish the relationship between their excess and excess of baryons and
leptons by sphaleron transitions. The aim is to find conditions at which
observed baryon excess corresponds to such an excess of $\bar{U}$ that
explains observed dark matter by $(\bar{U}\bar{U}\bar{U})He$ atoms.

Three elementary particle frames are established and we are interested in the
frame of a heavy quark and a heavy neutrino (neutrino with half the mass of Z
boson). A Dark matter model can be proposed which can arise from this model if
the baryon asymmetric universe with normal baryon also contains stable
antiquarks $\bar{U}$ of 4th generation.

Owing to the $\bar{{U}}$ excess only $-2$ charge or neutral hadrons are
present in the universe and ${}^{4}He$ after it is formed in Big Bang
nucleosynthesis completely screens $Q^{--}$ charged hadrons in composite
$[^{4}HeQ^{--}]$ “atoms”. These neutral primordial nuclear interacting objects
saturate the modern DM density and play an important role of the non-trivial
form of the strongly interacting dark matter.

In the given scenario with electric charge $q=2/3$, the direct experimental
lower limit for stable $U$-type quark is $m_{U}>220$ GeV. The stronger lower
limit can be estimated as $m_{U}g_{\star\epsilon}1$ TeV from search for
R-hadrons, which have similar features as single $U$ ($\bar{U}$) hadrons. Due
to the large chromo-Coulomb binding energy bound states for both particles and
antiparticles are formed (both doublets and triplets). Even though accelerator
physics excludes the formation of these bound states but they can be formed in
the early universe and strongly influence the evolution of the 4th generation
hadrons.

In the early universe an antibaryon asymmetry for the 4th generation of quarks
can be generated so that it might form a certain fraction of dark matter. It
might correspond to the modern dark matter density.

If $\epsilon(<1)$ is the fraction of dark matter then from observational data
we have $\Omega_{B}+\epsilon(1+\Omega_{DM})\approx 0.3$ which leads to
$\epsilon\approx(0.3-\Omega_{B})/\Omega_{DM}$. Introducing entropy density $s$
such that $r_{b}=n_{B}/s$ and $r_{\bar{U}}=n_{\bar{U}}/s$ it is obtained that
$r_{b}\sim 8*10^{-11}$ and $r_{\bar{U}}$ corresponds to the ${\bar{U}}$ excess
and is a function of $\epsilon$.

A new point to note: If new stable species belong to non-trivial
representation of electroweak $SU(2)$ group, sphaleron transition at high
temperature can provide a relationship between baryon asymmetry and excess of
‘-2 charged’ stable species.

If 4th family follows from string phenomenology, we have new charge associated
with 4th family fermions (which follows from rank 6 of E6 group, embedding
symmetry of the Standard model with rank 4) and this charge F should be used
instead of technibaryon number TB and technilepton number L’. In principle
there should be evidently only one conserved number F for the 4th family,
however, it may be easier to have analogy with WTC and assume two numbers FB
(for 4th quark) and L’ for (4th neutrino). Detailed calculations of WTC have
been done in [11, 17] and most of the terminology are kept same as the above
mentioned papers.

At some energy scale higher than the electroweak one, we assume the existence
of a mechanism leading to a 4th asymmetry in the Universe. Given that the 4th
and baryon number have a very similar nature such an asymmetry is very
plausible and can have a common origin.

The asymmetry generated by 4th generation above electroweak phase transition
can be destroyed by quantum anomalies. However, baryon, lepton and 4th number
are not conserved even though their differences are conserved. This fact
allows for a nonzero asymmetry to survive. The processes leading to such a
violation are termed “sphaleron” processes and at the present time are
negligible. However these processes were active during the time the Universe
had a temperature above or at the scale of the electroweak symmetry breaking.

At some points when as the Universe expands and its temperature falls the
quantum number violation ceases the effect. The relation among the particles
emerging from the process (SM + 4th generation):

$3(\mu_{u_{L}}+\mu_{d_{L}})+\mu+\mu_{U_{L}}+\mu_{D_{L}}+\mu_{L^{\prime}}=0$
(1)

$\mu$ is chemical potential summed over all standard model leptons and
$\mu_{L^{\prime}}$ is the new generation leptons (and neutrinos).
$\mu_{U_{L}}$ and $\mu_{D_{L}}$ are the chemical potential for the 4th
generation quarks. For detailed calculations and breakdown of each parameter,
see [17]. The number density follows, for fermions and bosons, respectively,
the general expression:

$n=g_{*}T^{3}\frac{\mu}{T}f(\frac{m}{T})$ (2)

and

$n=g_{*}T^{3}\frac{\mu}{T}g(\frac{m}{T})$ (3)

where $f$ and $g$ are hyperbolic functions and $g_{*}$ is the effective
degrees of freedom. They are given by:

$f(z)=\frac{1}{4\pi^{2}}\int_{0}^{\infty}x^{2}cosh^{-2}\left(\frac{1}{2}\sqrt{z^{2}+x^{2}}\right)dx$
(4)

and

$g(z)=\frac{1}{4\pi^{2}}\int_{0}^{\infty}x^{2}sinh^{-2}\left(\frac{1}{2}\sqrt{z^{2}+x^{2}}\right)dx$
(5)

The Baryon number density is defined as the following:

$B=\frac{n_{B}-n_{\bar{B}}}{gT^{2}/6}$ (6)

This normalization is used thorough out and since in the end the ratio is of
interest, this normalization cancels out.

Defining the parameter $\sigma$, respectively, for fermions and bosons:

$\sigma=6f\frac{m}{T^{*}}$ (7)

and

$\sigma=6g\frac{m}{T^{*}}$ (8)

where $T^{*}$ is the phase transition temperature and is given by the
following expression:

$T^{*}=\frac{2M_{W}(T^{*})}{\alpha_{W}ln(M_{pl}/T^{*})}B(\frac{\lambda}{\alpha_{W}})$
(9)

where $M_{W}$ is the mass of W-boson, $M_{pl}$ is the Planck mass and
$\lambda$ is the self-coupling. The function $B$ is derived from experiment
and takes the value from 1.5 to 2.7.

The new generation charge is calculated to be:

$FB=\frac{2}{3}(\sigma_{U_{L}}\mu_{U_{L}}+\sigma_{U_{L}}\mu_{D_{L}}+\sigma_{D_{L}}\mu_{D_{L}}).$
(10)

where $FB$ corresponds to the anti-U ($\bar{U}$) excess.

The standard model baryonic charge and leptonic charges are given by the
following expression:

$B=\left[(2+\sigma_{t})(\mu_{uL}+\mu_{uR})+3(\mu_{dL}+\mu_{dR})\right]$ (11)

and

$L=4\mu+6\mu_{W}$ (12)

where in Equation (11) the factor 3 of the down-type quarks is the number of
families. For the new lepton family, originating from 4th generation, we have:

$L^{\prime}=2(\sigma_{\nu^{\prime}}+\sigma_{U_{L}})\mu_{\nu^{\prime}L}+2\sigma_{U_{L}}\mu_{W}+(\sigma_{\nu^{\prime}}-\sigma_{U_{L}})\mu_{0}$
(13)

where $\nu^{\prime}$ is the new family of neutrinos originating from the
extension of standard model and $\mu_{0}$ is the chemical potential from the
Higgs sector in standard model.

In a second order phase transition we expect the violating processes to
persist below the phase transition and the equilibrium conditions are imposed
after the phase transition. If the phase diagram as function of temperature
and density of the 4th generation would be known a specific order of the
electroweak phase transition would be used. The ratio of the number densities
of 4th generation to the baryons is determined by:

$\frac{\Omega_{FB}}{\Omega_{B}}=\frac{3}{2}\frac{FB}{B}\frac{m_{FB}}{m_{p}}$
(14)

For second order phase transition, the electrical neutrality and zero chemical
potential of the Higgs. The ratio of the number density of number density of
the 4th generation to the baryon number density can be expressed as a function
of $L/B$ and $L^{\prime}/B$. In the limiting case, we get:

$-\frac{FB}{B}=\frac{\sigma_{U_{L}}}{3(18+\sigma_{\nu^{\prime}})}\left[(17+\sigma_{\nu^{\prime}})+\frac{(21+\sigma_{\nu^{\prime}})}{3}\frac{L}{B}+\frac{2}{3}\frac{9+5\sigma_{\nu^{\prime}}}{\sigma_{\nu^{\prime}}}\frac{L^{\prime}}{B}\right].$
(15)

Hence a possible relationship between the baryon excess and the excess of
$\bar{U}$ has been established in the framework of second order electroweak
phase transition.

##  3 Conclusions

It is clear from Equation (15) that a definite relationship between the value
and sign of 4th generation family excess relative to the baryon asymmetry due
to electroweak phase transition and possible sphaleron production has been
established. Dark matter candidates in the form of bounded dark atom can
emerge from this model due to the excess of $\bar{U}$ within primordial He
nuclei. We have considered only the lightest and most stable particles and
also took into account only the second order phase transition. The situation
can change more drastically if unstable particle and first order phase
transition were taken into account but those scenarios are beyond the scope of
this paper.

##  4 Author Contribution

Article by A.C. and M.K. The authors contributed equally to this work. All
authors have read and agreed to the published version of the manuscript.

##  5 Funding

The work of A.C. is funded by RSF Grant 19-42-02004. The research by M.K. was
financially supported by Southern Federal University, 2020 Project
VnGr/2020-03-IF.

##  6 Conflict of Interest

There has been no conflict of interest among the authors of this paper.

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# Learning Open Information Extraction of Implicit Relations from Reading
Comprehension Datasets

Jacob Beckerman
The Wharton School
University of Pennsylvania
<EMAIL_ADDRESS>
Theodore Christakis
Western Engineering
University of Western Ontario
<EMAIL_ADDRESS>

###### Abstract

The relationship between two entities in a sentence is often implied by word
order and common sense, rather than an explicit predicate. For example, it is
evident that “Fed chair Powell indicates rate hike” implies (Powell, is a, Fed
chair) and (Powell, works for, Fed). These tuples are just as significant as
the explicit-predicate tuple (Powell, indicates, rate hike), but have much
lower recall under traditional Open Information Extraction (OpenIE) systems.
Implicit tuples are our term for this type of extraction where the relation is
not present in the input sentence. There is very little OpenIE training data
available relative to other NLP tasks and none focused on implicit relations.
We develop an open source, parse-based tool for converting large reading
comprehension datasets to OpenIE datasets and release a dataset 35x larger
than previously available by sentence count. A baseline neural model trained
on this data outperforms previous methods on the implicit extraction task.

##  1 Introduction

Open Information Extraction (OpenIE) is the NLP task of generating (subject,
relation, object) tuples from unstructured text e.g. “Fed chair Powell
indicates rate hike” outputs (Powell, indicates, rate hike). The modifier
_open_ is used to contrast IE research in which the relation belongs to a
fixed set. OpenIE has been shown to be useful for several downstream
applications such as knowledge base construction Wities et al. (2017), textual
entailment Berant et al. (2011), and other natural language understanding
tasks Stanovsky et al. (2015). In our previous example an extraction was
missing: (Powell, works for, Fed). Implicit extractions are our term for this
type of tuple where the relation (“works for” in this example) is not
contained in the input sentence. In both colloquial and formal language, many
relations are evident without being explicitly stated. However, despite their
pervasiveness, there has not been prior work targeted at implicit predicates
in the general case. Implicit information extractors for some specific
implicit relations such as noun-mediated relations, numerical relations, and
others Pal and Mausam (2016); Saha et al. (2017); Saha and Mausam (2018) have
been researched. While specific extractors are important, there are a
multiplicity of implicit relation types and it would be intractable to
categorize and design extractors for each one.

Past general OpenIE systems have been plagued by low recall on implicit
relations Stanovsky et al. (2018). In OpenIE’s original application – web-
scale knowledge base construction – this low recall is tolerable because facts
are often restated in many ways Banko et al. (2007). However, in downstream
NLU applications an implied relationship may be significant and only stated
once Stanovsky et al. (2015).

The contribution of this work is twofold. In Section 4, we introduce our
parse-based conversion tool and convert two large reading comprehension
datasets into implicit OpenIE datasets. In Section 5 and 6, we train a simple
neural model on this data and compare to previous systems on precision-recall
curves using a new gold test set for implicit tuples.

##  2 Problem Statement

We suggest that OpenIE research focus on producing implicit relations where
the predicate is not contained in the input span. Formally, we define implicit
tuples as (subject, relation, object) tuples that:

  1. 1.

Have a subject and object word or phrase contained in the input sentence.

  2. 2.

Have a relation token(s) entailed by word order of the sentence but not
contained in it.

These “implicit” or “common sense” tuples reproduce the relation explicitly,
which may be important for downstream NLU applications using OpenIE as an
intermediate schema. For example, in Figure 1, the input sentence tells us
that the Norsemen swore fealty to Charles III under “their leader Rollo”. From
this our model outputs (The Norse leader, was, Rollo) despite the relation
never being contained in the input sentence. Our definition of implicit tuples
corresponds to the “frequently occurring recall errors” identified in previous
OpenIE systems Stanovsky et al. (2018): noun-mediated, sentence-level
inference, long sentence, nominalization, noisy informal, and PP-attachment.
We use the term implicit tuple to collectively refer to all of these
situations where the predicate is absent or very obfuscated.

##  3 Related Work

###  3.1 Traditional Methods

Due to space constraints, see Niklaus et al. Niklaus et al. (2018) for a
survey of of non-neural methods. Of these, several works have focused on
pattern-based implicit information extractors for noun-mediated relations,
numerical relations, and others Pal and Mausam (2016); Saha et al. (2017);
Saha and Mausam (2018). In this work we compare to OpenIE-4
111https://github.com/knowitall/openie, ClausIE Corro and Gemulla (2013),
ReVerb Fader et al. (2011), OLLIE Mausam et al. (2012), Stanford OpenIE Angeli
et al. (2015), and PropS Stanovsky et al. (2016).

###  3.2 Neural Network Methods

Stanovsky et al. Stanovsky et al. (2018) frame OpenIE as a BIO-tagging problem
and train an LSTM to tag an input sentence. Tuples can be derived from the
tagger, input, and BIO CFG parser. This method outperforms traditional
systems, though the tagging scheme inherently constrains the relations to be
part of the input sentence, prohibiting implicit relation extraction. Cui et
al. Cui et al. (2018) bootstrap (sentence, tuple) pairs from OpenIE-4 and
train a standard seq2seq with attention model using OpenNMT-py Klein et al.
(2017). The system is inhibited by its synthetic training data which is
bootstrapped from a rule-based system.

###  3.3 Dataset Conversion Methods

Due to the lack of large datasets for OpenIE, previous works have focused on
generating datasets from other tasks. These have included QA-SRL datasets
Stanovsky and Dagan (2016) and QAMR datasets Stanovsky et al. (2018). These
methods are limited by the size of the source training data which are an order
of magnitude smaller than existing reading comprehension datasets.

##  4 Dataset Conversion Method

Span-based Question-Answer datasets are a type of reading comprehension
dataset where each entry consists of a short passage, a question about the
passage, and an answer contained in the passage. The datasets used in this
work are the Stanford Question Answering Dataset (SQuADv1.1) Rajpurkar et al.
(2016) and NewsQA Trischler et al. (2017). These QA datasets were built to
require reasoning beyond simple pattern-recognition, which is exactly what we
desire for implicit OpenIE. Our goal is to convert the QA schema to OpenIE, as
was successfully done for NLI Demszky et al. (2018). The repository of
software and converted datasets is available at http://toAppear.

###  4.1 QA Pairs to OpenIE Tuples

We started by examining SQuAD and noticing that each answer, $A$, corresponds
to either the subject, relation, or object in an implicit extraction. The
corresponding question, $Q$, contains the other two parts, i.e. either the (1)
subject and relation, (2) subject and object, or (3) relation and object.
Which two pieces the question contains depends on the type of question. For
example, “who was… factoid” type questions contain the relation (“was”) and
object (the factoid), which means that the answer is the subject. In Figure 1,
“Who was Rollo” is recognized as a who was question and caught by the
whoParse() parser. Similarly, a question in the form of “When did person do
action” expresses a subject and a relation, with the answer containing the
object. For example, “When did Einstein emigrate to the US“ and answer 1933,
would convert to (Einstein, when did emigrate to the US, 1933). In cases like
these the relation might not be grammatically ideal, but nevertheless captures
the meaning of the input sentence.

In order to identify generic patterns, we build our parse-based tool on top of
a dependency parser Honnibal and Johnson (2015). It uses fifteen rules, with
the proper rule being identified and run based on the question type. The rule
then uses its pre-specified pattern to parse the input QA pair and output a
tuple. These fifteen rules are certainly not exhaustive, but cover around
eighty percent of the inputs. The tool ignores questions greater than 60
characters and complex questions it cannot parse, leaving a dataset smaller
than the original (see Table 1).

Each rule is on average forty lines of code that traverses a dependency parse
tree according to its pre-specified pattern, extracting the matching spans at
each step. A master function parse() determines which rule to apply based on
the question type which is categorized by nsubj presence, and the type of
question (who/what/etc.). Most questions contain an nsubj which makes the
parse task easier, as this will also be the subject of the tuple. We allow the
master parse() method try multiple rules. It first tries very specific rules
(e.g. a parser for how questions where no subject is identified), then falls
down to more generic rules. If no output is returned after all the methods are
tried we throw the QA pair out. Otherwise, we find the appropriate sentence in
the passage based on the index.

Figure 1: Tuple conversion and alignment process flow.

###  4.2 Sentence Alignment

Following QA to tuple conversion, the tuple must be aligned with a sentence in
the input passage. We segment the passage into sentences using periods as
delimiters. The sentence containing the answer is taken as the input sentence
for the tuple. Outputted sentences predominantly align with their tuple, but
some exhibit partial misalignment in the case of some multi-sentence reasoning
questions. 13.6% of questions require multi-sentence reasoning, so this is an
upper bound on the number of partially misaligned tuples/sentences Rajpurkar
et al. (2016). While there may be heuristics that can be used to check
alignment, we didn’t find a significant number of these misalignments and so
left them in the corpus. Figure 1 demonstrates the conversion process.

###  4.3 Tuple Examination

Examining a random subset of one hundred generated tuples in the combined
dataset we find 12 noun-mediated, 33 sentence-level inference, 11 long
sentence, 7 nominzalization, 0 noisy informal, 3 pp-attachment, 24 explicit,
and 10 partially misaligned. With 66% implicit relations, this dataset shows
promise in improving OpenIE’s recall on implicit relations.

##  5 Our model

Our implicit OpenIE extractor is implemented as a sequence to sequence model
with attention Bahdanau et al. (2014). We use a 2-Layer LSTM Encoder/Decoder
with 500 parameters, general attention, SGD optimizer with adaptive learning
rate, and 0.33 dropout Hochreiter and Schmidhuber (1997). The training
objective is to maximize the likelihood of the output tuple given the input
sentence. In the case of a sentence having multiple extractions, it appears in
the dataset once for each output tuple. At test time, beam search is used for
decoding to produce the top-10 outputs and an associated log likelihood value
for each tuple (used to generate the precision-recall curves in Section 7).

Source Data | Sentences | Train Tuples | Validation Tuples
---|---|---|---
NewsQA | 50880 | 56646 | -
SQuAD | 38773 | 51949 | -
Total | 89653 | 107595 | 1000

Table 1:  Dataset statistics.

##  6 Evaluation

We make use of the evaluation tool developed by Stanovsky and Dagan Stanovsky
and Dagan (2016) to test the precision and recall of our model against
previous methods. We make two changes to the tool as described below.

###  6.1 Creating a Gold Dataset

The test corpus contained no implicit data, so we re-annotate 300 tuples from
the CoNLL-2009 English training data to use as gold data. Both authors worked
on different sentence sets then pruned the other set to ensure only implicit
relations remained. We note that this is a different dataset than our training
data so should be a good test of generalizability; the training data consists
of Wikipedia and news articles, while the test data resembles corporate press
release headlines.

###  6.2 Matching function for implicit tuples

We implement a new matching function (i.e. the function that decides if a
generated tuple matches a gold tuple). The included matching functions used
BoW overlap or BLEU, which aren’t appropriate for implicit relations; our goal
is to assess whether the meaning of the predicted tuple matches the gold, not
the only tokens. For example, the if the gold relation is “is employed by” we
want to accept “works for”. Thus, we instead compute the cosine similarity of
the subject, relation, and object embeddings to our gold tuple. All three must
be above a threshold to evaluate as a match. The sequence embeddings are
computed by taking the average of the GloVe embeddings of each word (i.e. BoW
embedding) Pennington et al. (2014).

Figure 2: PR curve on our implicit tuples dataset. Figure 3: PR curve on the
explicit tuples dataset.

##  7 Results

The results on our implicit corpus are shown in Figure 2 (our method in blue).
For continuity with prior work, we also compare our model on the origional
corpus but using our new matching function in Figure 3.

Our model outperforms at every point in the implicit-tuples PR curve,
accomplishing our goal of increasing recall on implicit relations. Our system
performs poorly on explicit tuples, as we would expect considering our
training data. We tried creating a multi-task model, but found the model
either learned to produce implit or explicit tuples. Creating a multi-task
network would be ideal, though it is sufficient for production systems to use
both systems in tandem.

##  8 Conclusion

We created a large training corpus for implicit OpenIE extractors based on
SQuAD and NewsQA, trained a baseline on this dataset, and presented promising
results on implicit extraction. We see this as part of a larger body of work
in text-representation schemes which aim to represent meaning in a more
structured form than free text. Implicit information extraction goes further
than traditional OpenIE to elicit relations not contained in the original free
text. This allows maximally-shortened tuples where common sense relations are
made explicit. Our model should improve further as more QA datasets are
released and converted to OpenIE data using our conversion tool.

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# Searching for Muonphilic Dark Sectors with Proton Beams

Claudia Rella<EMAIL_ADDRESS>Département de Physique Théorique,
Université de Genève, CH-1211 Genève, Switzerland  Babette Döbrich  CERN,
CH-1211 Genève 23, Switzerland  Max-Planck-Institut für Physik (Werner-
Heisenberg-Institut), 80805 München, Germany  Tien-Tien Yu  Department of
Physics and Institute for Fundamental Science, University of Oregon, Eugene,
Oregon 97403, USA

###### Abstract

Proton beam-dump experiments are a high-intensity source of secondary muons
and provide an opportunity to probe muon-specific dark sectors. We adopt a
simplified-models framework for an exotic light scalar particle coupling
predominantly or exclusively to muons. Equipped with state-of-the-art muon
simulations, we compute the sensitivity reach in the parameter space
$(m_{S},\,g_{\mu})$ of the dark mediator, examining in detail the examples of
the experiment NA62 in beam-dump mode and the proposed experiment SHiP. We
find a significant yield of such exotics in the sub-GeV mass range. Our
projections are competitive with those of primary muon-beam experiments and
complementary to current constraints, spanning uncharted parameter space and
accessing new physics potentially responsible for the $(g-2)_{\mu}$ anomaly.

##  I Introduction

Although the Standard Model (SM) of particle physics has provided a
theoretically consistent description of all the known particles and their
interactions, with the exception of gravity, there are several experimental
observations that require the existence of new physics beyond the SM (BSM).
Such observations include neutrino oscillations, and therefore neutrino
masses, the existence of dark matter, as well as various experimental
anomalies, all of which hint at a potential dark sector consisting of
particles that do not interact with the known SM forces. A multitude of ways
to explore the dark sector Battaglieri:2017aum ; Beacham:2019nyx ;
Essig:2022yzw  has been proposed and investigated. In this work we will focus
on the use of beam-dump experiments as a promising probe of low-mass, weakly-
coupled BSM mediators.

Searching for BSM physics in beam-dump data is not a new idea, and this idea
was notably pursued in the past via neutrino experiments. However, the current
lack of new-physics discoveries at the highest energies, as explored at the
Large Hadron Collider at CERN, has given these old tactics a renewed interest.
Among others, two factors push this development. Firstly, a plethora of
possible BSM mediators at the MeV-GeV scale is motivated through the past and
recent work of theorists (see, e.g., Refs. Alekhin:2015byh ; Alexander:2016aln
; Battaglieri:2017aum  and references therein). Secondly, several experimental
anomalies suggest a connection to weakly-coupled, low-mass dark sectors.
Examples of such potential signatures have been found in flavor physics111An
overview of the current state of understanding of flavor anomalies has been
presented in a 2021 workshop at CERN flavor ., while measurements of decays of
8Be hint at a new $X17$ boson PhysRevLett.116.042501 . Another example is the
long-standing muon magnetic anomaly $(g-2)_{\mu}$, which describes the
currently observed discrepancy between the empirical measurement and the SM
prediction of the anomalous muon magnetic moment $a_{\mu}$. The most recent
combined measurements of $a_{\mu}$ from the Fermilab National Accelerator
Laboratory and the Brookhaven National Laboratory are in tension with
theoretical predictions at the level of $4.2\,\sigma$ PhysRevLett.126.141801 ,
where the experimental measurements give a larger value than the theory
prediction. Consequently, various studies have been performed to evaluate the
(often competing) sensitivities of experiments that can be run in beam-dump
mode Beacham:2019nyx  to the detection of exotic long-lived, low-mass muon-
specific particles. Arguably, the most direct way to investigate the
$(g-2)_{\mu}$ anomaly is via the experimental probe of new-physics mediators
which (exclusively) couple to muons Kinoshita:1990aj ; Muong-2:2006rrc ;
Lindner:2016bgg . To this end, a primary muon beam can be used, as it has been
proposed for, e.g., NA64-$\mu$ Gninenko:2014pea ; Araki:2014ona ; Chen:2017awl
; Chen:2018vkr ; PhysRevD.105.052006  at CERN, as well as $M^{3}$ Kahn:2018cqs
and FNAL-$\mu$ Chen:2017awl  at Fermilab, or for a future muon collider
Cesarotti:2022ttv .

In this paper we aim to highlight and investigate the possibility to achieve
competitive results in the detection of exotic particles radiated via
bremsstrahlung of the secondary muons produced at proton beam dumps. The
experimental setup is illustrated in Fig. 1. Protons are dumped into a thick
target and produce a muon shower. A secondary muon, before being possibly
stopped or deflected by magnets, can emit a light scalar, which in turn
travels a certain distance to the experiment’s decay volume. Therein the
scalar decays, either into a di-lepton final state as shown in the example in
Fig. 1 or into a pair of photons. The decay products are detected mainly
through their signature in a spectrometer and a calorimeter. Details are given
in Section V.

Figure 1: Schematic illustration of the framework/proposal for muonphilic
scalar detection studied in this paper. Scalars emerging from a muon shower in
a proton beam dump travel to a decay volume. The scalar’s decay products are
recorded through calorimetric and spectrometric information. Lengths are not
to scale.

Similar studies have been performed for the secondary muons produced at
electron beam-dump experiments Marsicano:2018vin . The idea of employing
secondary muons from proton beam dumps was mentioned in Ref. Batell:2016ove
and subsequently proposed and estimated for the specific case of the
SeaQuest/DarkQuest experiment in Ref. Berlin:2018pwi . With this work we
provide the missing sensitivity projections for two other major players that
can probe new physics in this way, that is, the NA62 experiment run in beam-
dump mode222A novel proposal for an experiment called “SHADOWS”, to be run in
conjunction with NA62 in beam-dump mode, has recently emerged Baldini:2021hfw
. Our studies could easily be extended to include this experiment. and the
proposed SHiP experiment. Crucially, for both scenarios, we aim to produce an
estimate which is realistically close to what can be expected in practice.

In Section II, we review two simplified models of muon-specific new-physics
light scalars, which can potentially explain the $(g-2)_{\mu}$ anomaly. In
Section III, we detail the production mechanism of the exotic scalars via muon
bremsstrahlung. In Section IV, we describe the modelling of the secondary muon
flux from the proton beam dump, which is used as input for the radiation of
the exotics. In Section V, we describe the specific experimental setups of
NA62 and SHiP. We include the experimental acceptance of the scalar’s daughter
particles, which results from a dedicated detector modelling. In Section VI,
we outline the procedure for the final sensitivity evaluation, and we show the
prospective reach in the exotic scalar parameter space of the two selected
proton beam-dump experiments NA62 and SHiP in the context of complementary
searches. Finally, we conclude and further discuss our results in Section VII.

##  II Theoretical Models of Muonphilic Dark Sectors

The current discrepancy between the Standard Model prediction and the
experimental observation of the anomalous magnetic moment of the muon
$a_{\mu}\equiv(g-2)_{\mu}/2$ is given by PhysRevLett.126.141801

${\Delta a_{\mu}\equiv a_{\mu}^{\rm obs}-a_{\mu}^{\rm SM}=(251\pm 59)\times
10^{-11}\,.}$ (1)

One possible resolution of such discrepancy is obtained by introducing new-
physics light particles that couple predominantly to muons. Note that
particles with parity-odd couplings to muons, i.e., pseudo-scalar and axial-
vector couplings, give a negative contribution to the predicted value of
$a_{\mu}$, which further exacerbates the discrepancy with the experimental
measurement. Thus, we focus our attention on scalar and vector particles,
which have parity-even couplings. A generic light scalar $S$ or vector $V$ can
couple to muons through the operators

$g_{\mu,\,S}S\bar{\mu}\mu~{}~{}{\rm(scalar)}\,,~{}~{}~{}~{}~{}g_{\mu,\,V}V_{\alpha}\bar{\mu}\gamma^{\alpha}\mu~{}~{}\rm{(vector)}\,,$
(2)

where $g_{\mu,\,S},\,g_{\mu,\,V}$ are the couplings between $S,\,V$ and muons,
respectively. These operators contribute at leading-order to the muon
anomalous magnetic moment as Leveille:1977rc

${\Delta
a_{\mu}^{S}=\frac{g_{\mu,\,S}^{2}}{8\pi^{2}}\int_{0}^{1}dz\frac{m_{\mu}^{2}(1-z)^{2}(1+z)}{m_{\mu}^{2}(1-z)^{2}+m_{S}^{2}z}\simeq
2.3\times 10^{-9}\left(\frac{g_{\mu,\,S}}{3.5\times
10^{-4}}\right)^{2}~{}~{}~{}(m_{S}\ll m_{\mu})\,,}$ (3)

and

${\Delta
a_{\mu}^{V}=\frac{g_{\mu,\,V}^{2}}{4\pi^{2}}\int_{0}^{1}dz\frac{m_{\mu}^{2}z(1-z)^{2}}{m_{\mu}^{2}(1-z)^{2}+m_{V}^{2}z}\simeq
2.1\times 10^{-9}\left(\frac{g_{\mu,\,V}}{5.0\times
10^{-4}}\right)^{2}~{}~{}~{}(m_{V}\ll m_{\mu})\,.}$ (4)

The addition of a new $U(1)$ gauge group is a natural extension of the SM and
results in a new vector boson. The most straightforward way to obtain this is
through the “kinetic-mixing” portal that mixes the $U(1)_{Y}$ of SM
hypercharge with a $U(1)_{D}$. Although the associated vector boson, known as
the dark photon, does provide a positive contribution to $a_{\mu}$, the
parameters needed to alleviate the $(g-2)_{\mu}$ anomaly are inconsistent with
direct searches for the dark photon (see, e.g., Ref. NA482:2015wmo ). Another
proposed vector model extends the SM with the anomaly-free
$U(1)_{L_{\mu}-L_{\tau}}$ with corresponding vector boson $Z^{\prime}$. This
new vector boson couples to $\mu,\,\tau$ leptons as well as their
corresponding neutrino flavors $\nu_{\mu},\,\nu_{\tau}$. Although the
$L_{\mu}-L_{\tau}$ vector boson He:1990pn ; Foot:1990mn ; He:1991qd  is an
attractive candidate, its primary decays are through neutrinos. Due to the
difficulty of reconstructing the so-called “open decays” involving neutrinos
in a beam-dump setup, we do not evaluate this scenario in what follows.
However, our analysis can be straightforwardly extended to include this
possibility in case these invisible decay modes can be studied experimentally.
For the remainder of this work, we will only consider scalar candidates.

As discussed above, a new light scalar particle coupling to muons will
increase the value of $a_{\mu}$ and thus help alleviate the tension in the
muon magnetic anomaly $(g-2)_{\mu}$. At low-energies, the relevant part of the
Lagrangian is given by

${{\cal
L}\supset\frac{1}{2}(\partial_{\alpha}S)^{2}-\frac{1}{2}m_{S}^{2}S^{2}-\sum_{\ell=e,\mu,\tau}g_{\ell}S\bar{\ell}\ell\,,}$
(5)

where $g_{\ell}$ is the coupling between the scalar $S$ and the leptons
$\ell=e,\,\mu,\,\tau$, and $m_{S}$ is the mass of $S$. Although the effective
Lagrangian in Eq. (5) does not respect the $SU(2)\times U(1)$ gauge symmetry
of the SM, it can be generalised to the effective dimension-5 operator ${\cal
O}_{5}=\frac{1}{\Lambda}(\bar{L}E)HS$, where $H$ is the SM Higgs doublet, and
$L,\,E$ are the lepton doublets and singlets, respectively. The operator
${\cal O}_{5}$ does respect the $SU(2)\times U(1)$ gauge symmetry, and it can
in turn be UV completed with, e.g., vector-like fermions Fox:2011qd  or
multiple Higgs states Chen:2015vqy ; Batell:2016ove ; PhysRevD.98.055026 . For
the purpose of this work we will remain agnostic about the UV-completion of
the simplified model in Eq. (5), and we will regard the couplings $g_{\ell}$,
$\ell=e,\,\mu,\,\tau$, of the exotic particle $S$ to leptons as free
parameters of the theory. From the model-building point of view, there are
substantial motivations to consider $g_{\ell}\propto m_{\ell}$, where
$m_{\ell}$ is the mass of the lepton. In particular, introducing new lepton-
specific scalar mediators below the electroweak scale can lead to large
flavor-changing neutral currents (FCNCs), which are strongly constrained by
null searches for $\mu-e$ conversion, $\mu\to 3e$, and $\mu\to e\gamma$
SINDRUMmu2e ; MEG:2016leq ; ParticleDataGroup:2020ssz . Requiring the mass
proportionality $g_{\ell}\propto m_{\ell}$, a condition known as minimal
flavor violation (MFV) Cirigliano:2005ck , avoids FCNCs. We will denote this
scenario as the “leptophilic model”. Alternatively, we can further impose
$g_{e}=g_{\tau}=0$, such that the scalar exclusively couples to muons. In this
case, suppression of the FCNCs can be ensured by, e.g., considering
radiatively generated FCNCs PhysRevD.98.055026 , which are shown to be
suppressed by small Yukawa couplings. We will denote this scenario as the
“muonphilic model”. This muon-only framework is particularly challenging to
probe but not unfeasible, as we will see, for collider and electron or proton
beam-dump searches.

In both the leptophilic and the muonphilic models, the width of the scalar
decay into leptons $S\rightarrow\ell^{+}\ell^{-}$ is given by

${\Gamma_{\ell^{+}\ell^{-}}=\frac{m_{S}}{8\pi}g_{\ell}^{2}\left(1-\frac{4m_{\ell}^{2}}{m_{S}^{2}}\right)^{3/2}\,,}$
(6)

while the decay into photons $S\rightarrow\gamma\gamma$ arises from the
effective scalar coupling to vector boson pairs at one-loop level, and its
decay width is given by

${\Gamma_{\gamma\gamma}=\frac{\alpha^{2}m_{S}^{3}}{64\pi^{3}}\left|\sum_{\ell=e,\mu,\tau}\frac{g_{\ell}}{m_{\ell}}x_{\ell}[1+(1-x_{\ell})f(x_{\ell})]\right|^{2},}$
(7)

where $\alpha=1/137$ is the fine-structure constant,
$x_{\ell}=4m_{\ell}^{2}/m_{S}^{2}$, and

${f(x)=\begin{cases}\arcsin^{2}({x^{-1/2}})\,,&~{}~{}x>1\\\
-\frac{1}{4}\left[\ln\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)-i\pi\right]^{2}\,,&~{}~{}x\leq
1\end{cases}\,.}$ (8)

Note that, in the muonphilic scenario, in which $g_{e}=g_{\tau}=0$, the decay
into photons results from the muon loop contribution only, which is

${\Gamma_{\gamma\gamma}^{\mu-{\rm
loop}}=\frac{\alpha^{2}m_{S}^{3}}{64\pi^{3}}\left|\frac{g_{\mu}}{m_{\mu}}x_{\mu}[1+(1-x_{\mu})f(x_{\mu})]\right|^{2}\,.}$
(9)

We denote by $\Gamma_{S}$ the total decay width of the scalar. For
$2m_{e}<m_{S}<2m_{\mu}$, the widths of the decay channels
$S\rightarrow\gamma\gamma$ and $S\rightarrow e^{+}e^{-}$ sum to give

${\Gamma_{S}=\begin{cases}\Gamma_{e^{+}e^{-}}+\Gamma_{\gamma\gamma}\simeq\Gamma_{e^{+}e^{-}}\,,&~{}~{}g_{\ell}\propto
m_{\ell}\\\ \Gamma_{\gamma\gamma}^{\mu-{\rm loop}}\,,&~{}~{}g_{\mu}\propto
m_{\mu},\,g_{e}=g_{\tau}=0\end{cases}\,,}$ (10)

while, for $2m_{\mu}<m_{S}<2m_{\tau}$, the scalar decay into muons
$S\rightarrow\mu^{+}\mu^{-}$ is dominant in both models, that is

${\Gamma_{S}=\begin{cases}\Gamma_{\mu^{+}\mu^{-}}+\Gamma_{e^{+}e^{-}}+\Gamma_{\gamma\gamma}\simeq\Gamma_{\mu^{+}\mu^{-}}\,,&~{}~{}g_{\ell}\propto
m_{\ell}\\\ \Gamma_{\mu^{+}\mu^{-}}+\Gamma_{\gamma\gamma}^{\mu-{\rm
loop}}\simeq\Gamma_{\mu^{+}\mu^{-}}\,,&~{}~{}g_{\mu}\propto
m_{\mu},\,g_{e}=g_{\tau}=0\end{cases}\,.}$ (11)

In what follows, we use the exact formulas for $\Gamma_{S}$. The corresponding
decay length of the scalar is given by

${L_{S}=\frac{E_{S}}{m_{S}}\frac{\beta_{S}}{\Gamma_{S}}\,,}$ (12)

where $E_{S}$ is the energy of the scalar and
$\beta_{S}=\sqrt{1-m_{S}^{2}/E_{S}^{2}}$ is its boost factor. As an example,
for a reference value of $E_{S}=30$ GeV, we have

$\displaystyle L_{S}$ $\displaystyle\simeq 60.7{~{}\rm
m}\times\left(\frac{1.0\times
10^{-4}}{g_{\mu}}\right)^{2}\times\left(\frac{100~{}{\rm
MeV}}{m_{S}}\right)^{2}~{}~{}~{}\text{leptophilic ($g_{\ell}\propto
m_{\ell}$)}\,,$ (13) $\displaystyle L_{S}$ $\displaystyle\simeq 61.2{~{}\rm
m}\times\left(\frac{9.0\times
10^{-4}}{g_{\mu}}\right)^{2}\times\left(\frac{100~{}{\rm
MeV}}{m_{S}}\right)^{2}~{}~{}~{}\text{muonphilic ($g_{\mu}\propto
m_{\mu},\,g_{e}=g_{\tau}=0$)}\,,$ (14)

where the parameters are chosen to give decay lengths comparable to the
minimum distance that the scalar must travel to reach the decay volume of SHiP
(NA62) at 53.2 (79.0) m.

##  III Scalar Production Cross-Section

The dominant production mechanism for the new light scalar state $S$ at beam
dumps is the $\mu+N\to\mu+N+S$ bremsstrahlung process shown in Fig. 2, where
the incident muon333Here, $\mu$ refers to both $\mu^{+}$ and $\mu^{-}$. $\mu$
exchanges a virtual photon $\gamma^{(*)}$ with the target nucleon $N$ and
radiates the exotic scalar $S$ PhysRevD.80.075018 .

Figure 2: Feynman diagrams corresponding to the radiative production of the
scalar $S$ from muon bremsstrahlung.

When the beam energy is much higher than the masses of both the beam and the
radiation, in our case the muon and the scalar, respectively, the signal
production cross-section can be estimated by the Weizsäcker-Williams (WW)
approximation Weizsacker:1934 ; Williams:1934 . In this scenario of highly-
boosted incoming muons, the intermediate virtual photons produced by the muons
scattering off the target nuclei are nearly on-shell and can be approximated
by real photons. Thus, the phase space integration of the full $2\rightarrow
3$ scattering process $\mu(p)+N(P_{i})\to\mu(p^{\prime})+N(P_{f})+S(k)$ can be
estimated via the simpler $2\rightarrow 2$ process
$\mu(p)+\gamma(q)\to\mu(p^{\prime})+S(k)$, evaluated at minimum virtuality
$t_{\rm min}\equiv q^{2}_{\rm min}$ and weighted by the effective photon flux
$\chi$. Applying the WW approximation scheme under the assumptions $E_{\mu}\gg
m_{\mu},m_{S}$, the differential production cross-section in the lab frame is
given by

${\frac{d\sigma}{dx}(\mu+N\to\mu+N+S)\simeq\frac{g_{\mu}^{2}\alpha^{2}}{12\pi}\chi\beta_{S}\beta_{\mu}\frac{x^{3}[m_{\mu}^{2}(3x^{2}-4x+4)+2m_{S}^{2}(1-x)]}{[m_{S}^{2}(1-x)+m_{\mu}^{2}x^{2}]^{2}}\,,}$
(15)

where $x=E_{S}/E_{\mu}$ is the fraction of the incoming muon energy $E_{\mu}$
taken by the emission of an exotic scalar of energy $E_{S}$. The boost factors
are $\beta_{S}=\sqrt{1-m_{S}^{2}/(xE_{\mu})^{2}}$ and
$\beta_{\mu}=\sqrt{1-m_{\mu}^{2}/E_{\mu}^{2}}$. The effective photon flux
$\chi$ is defined as

${\chi=\int_{t_{\rm min}}^{t_{\rm max}}dt\frac{t-t_{\rm
min}}{t^{2}}G_{2}(t)\,,}$ (16)

where the virtuality $t$ is the momentum transfer squared and $G_{2}(t)$ is
the combined atomic and nuclear electric form factor of the target atom, which
includes both elastic and inelastic contributions PhysRevD.80.075018 . More
precisely, we have $G_{2}(t)=G_{2}^{\rm el}(t)+G_{2}^{\rm in}(t)$, where the
elastic and inelastic components are given by

$\displaystyle G_{2}^{\rm el}(t)$
$\displaystyle=\left(\frac{a^{2}t}{1+a^{2}t}\right)^{2}Z^{2}\left(\frac{1}{1+t/d}\right)^{2}\,,$
(17) $\displaystyle G_{2}^{\rm in}(t)$ $\displaystyle=\left(\frac{a^{\prime
2}t}{1+a^{\prime
2}t}\right)^{2}Z\left(\frac{1+t\,(\mu_{p}^{2}-1)/(4m_{p}^{2})}{(1+t/(0.71~{}\rm{GeV}^{2}))^{4}}\right)^{2}\,~{}~{}~{}~{}\text{($t/m_{p}^{2}\ll
1$)}\,,$ (18)

where $a=111\,Z^{-1/3}/m_{e}$ in the Thomas-Fermi model, $d=0.164~{}{\rm
GeV}^{2}A^{-2/3}$, $a^{\prime}=773\,Z^{-2/3}/m_{e}$, and $\mu_{p}=2.79$.
$m_{e},\,m_{p}$ are the electron and proton masses, and $A,\,Z$ are the atomic
mass and number of the target material.

The normalised distribution of Eq. (15), which reproduces the expected energy
spectrum of the new scalar mediator, changes significantly with the scalar
mass. As shown in Fig. 3, the peak of the distribution shifts from low to high
values of $x$ as $m_{S}$ increases. Moreover, for each choice of $m_{S}$ and
for $E_{\mu}\gg m_{S}$, the distribution lies dominantly in the region of high
Lorentz factor $\gamma_{S}=(1-\beta_{S}^{2})^{-1/2}\gg 1$, which corresponds
to the right of the matching dashed vertical line in Fig. 3. Thus, for
sufficiently high values of the muon energy, the emission of the exotic scalar
particle $S$ happens primarily in the highly relativistic regime.

Figure 3: The solid curves are the distributions of the fraction $x$ of the
incoming muon energy taken by the emission of the exotic scalar $S$ for
various scalar masses. The dashed vertical lines correspond to the values of
$x$ where the Lorentz factor $\gamma_{S}=100$ for each $m_{S}$ and a reference
choice of $E_{\mu}=50\,\text{GeV}$. We show $m_{S}=0.01\;\text{GeV}$ in blue,
$m_{S}=0.10\;\text{GeV}$ in green, and $m_{S}=0.30\;\text{GeV}$ in red.

A refinement of the WW approximation follows from the observation that, in the
domain of highly relativistic particles, the radiation is dominantly collinear
with the beam. Then, the phase space integration in Eq. (16) can be further
facilitated, yielding the Improved Weizsäcker-Williams (IWW) approximation
Liu:2016mqv ; PhysRevD.8.3109 , which is implemented in the simplification of
the integration limits in the virtuality as

${\chi\simeq\int_{m_{S}^{4}/(4E_{\mu}^{2})}^{m_{S}^{2}+m_{\mu}^{2}}dt\frac{t-m_{S}^{4}/(4E_{\mu}^{2})}{t^{2}}G_{2}(t)\,.}$
(19)

We note that the WW approximated expression for the scalar production
differential cross-section in Eq. (15) and the improved effective photon flux
integration in Eq. (19) are only valid for muon energies well above the muon
and scalar masses, i.e. $E_{\mu}\gg m_{\mu},m_{S}$. To check the applicability
of the IWW approximation to our context, and to roughly estimate its accuracy,
we compute the normalised distribution in $x$ of the scalar production
differential cross-section $\frac{1}{\sigma}\frac{d\sigma}{dx}$ using
MadGraph5 Alwall:2014hca  (MG) for selected benchmark points in $m_{S}$ and
$E_{\mu}$ at fixed coupling $g_{\mu}$. We then compare the MG results with the
corresponding distributions computed using the IWW approximation in Eqs. (15),
(19). We find a discrepancy between the IWW and the MG computed cross-section
distributions, quantified as their relative difference with sign, which is
$\lesssim 20\%$ when $E_{\mu}\gtrsim{\rm max}(300\,m_{S},\,m_{\mu})$, and a
larger discrepancy for smaller values of the muon energy444Similar conclusions
are found in Ref. Liu:2016mqv , where a systematic comparison of the exact
tree-level calculation compared to the WW and IWW approximations is performed
for scalar production at electron beam dumps. An analogous study for vectors
is presented in Ref. PhysRevD.104.076012 .. We additionally verify that, for
such values of $E_{\mu}$, the assumption of highly relativistic scalars holds
at the corresponding values of $m_{S}$. Therefore, we adopt an energy cut on
the incoming muons spectrum at $E_{\mu}^{\rm cut}={\rm
max}(300\,m_{S},\,m_{\mu})$, which ensures that we are in the regime of
validity of the IWW approximation and that our estimates are conservative. We
emphasise here that the purpose of this work is to present the potential of
using the secondary muons produced in proton beam-dump experiments to probe
new light muon-specific scalars. We leave a more detailed analysis, which
should make use of an exact cross-section calculation, for future work.

##  IV Modelling the Muons from the Beam Dump

At the heart of the sensitivity projections described in Section VI lies a
thorough modelling of the original muon flux from the proton beam dump with
sufficiently high statistics, to which different approaches have been explored
SHiP:2019gcl ; Ghinescu:2021sjm . The dominant muon flux comes from decaying
mesons (mostly pions and kaons) produced in the proton collisions with the
dump material. The challenge to model this flux is manifold and arises from
the following requirements:

  1. 1.

A detailed understanding of the physical processes that eventually yield the
muons;

  2. 2.

An efficient Monte Carlo (MC) simulation of those physical processes which
does not suffer from statistical fluctuations.

The muon flux used in this study is produced via GEANT4 GEANT4:2002zbu  and
the relevant code is publicly available in Ref. stealex_2022_6352043 . The
simulated meson spectrum can be cross-checked against the secondary mesons
created in proton-nucleon collisions, as modelled in Ref. VanDijk:2641500
from measurements using thin targets. We are most interested in the shape of
the muon spectrum shortly after the creation of the muons, which closely
tracks the shape of the meson spectrum. Note that validating the actual muon
spectrum in the downstream part of the experiment is a much more challenging
task as the experiment’s magnets modify the shape of the muon spectrum away
from the mesons. Fortunately, we do not need to undertake such a validation of
the downstream spectrum as our study does not require it. After validation,
the simulated muons undergo a biasing mechanism, as described in Ref.
Ghinescu:2021sjm , in order to efficiently reproduce a muon sample up to the
highest muon energies. This biasing process boosts the muon statistics by more
than three orders of magnitude without increasing computing resources and
without altering the physics of a “vanilla” simulation555In previous studies
Rosenthal:ICAP2018-SUPAG05 , “vanilla” muon simulations were parametrised and
then re-sampled to gain statistics. (see Ref. Ghinescu:2021sjm  for details).
We note that the simulated experiment specifically mimics the features of the
NA62 dump, which has a different geometry and composition than the SHiP dump.
However, as shown in Fig. 2 of Ref. Ghinescu:2021sjm , the relevant muons
typically emerge at depths of about $50~{}\text{cm}$. Thus, the actual length
of the dump does not matter here as long as it is at least of a few meters.
Moreover, the cross-section for meson production with larger target nuclei can
be obtained with an appropriate scaling of the target material as $A^{2/3}$,
where $A$ is the atomic mass number666As described in Section V, the NA62
target material is Copper (atomic mass of about $59.19~{}\text{GeV}$), while
the SHiP target material is Molybdenum (atomic mass of about
$89.37~{}\text{GeV}$).. This implies that using the NA62-simulated input data
will result in a slight underestimation of the reach of SHiP.

The original and effective distributions of the muons’ total momentum are
shown in Fig. 4 for the experiments NA62 (left) and SHiP (right). The input
distribution, in yellow, is obtained from $400\,\text{GeV}$ protons at
production point via MC simulation, following the techniques presented in Ref.
Ghinescu:2021sjm , as described above. The distributions in green are obtained
from the input spectrum by imposing the geometrical acceptance of the incoming
muons and the radiated scalars, as clarified in Section V. Then, the
additional IWW condition $E_{\mu}\gtrsim{\rm max}(300\,m_{S},\,m_{\mu})$,
which is explained in Section III, is applied for benchmark values of the
scalar mass in order to produce the effective distributions in blue for
$m_{S}=0.05\,\text{GeV}$, red for $m_{S}=0.10\,\text{GeV}$, and violet for
$m_{S}=0.15\,\text{GeV}$.

(a) NA62 Experiment (b) SHiP Experiment

Figure 4: Distributions of the total momentum $P_{\mu}$ of the secondary muons
for the experiments (a) NA62 and (b) SHiP. We show the input spectra in
yellow, the spectra after geometrical acceptance in green, and the spectra
after geometrical acceptance and fixed-mass IWW energy cuts in blue
($m_{S}=0.05\,\text{GeV}$), red ($m_{S}=0.10\,\text{GeV}$), and violet
($m_{S}=0.15\,\text{GeV}$).

We observe that both the experimental geometry cut and the IWW constraint on
the muon energy primarily act on the low-momentum part of the spectrum, where
the original peak occurs, and thus progressively shift the mean of the
distribution to the right. The total percentage loss of events from the yellow
to the blue distributions is about $97\,\%$ for NA62 and about $95\,\%$ for
SHiP. Moreover, the relatively small overall geometrical acceptance of NA62
dominates over the IWW energy cut, while the SHiP spectrum is predominantly
shrunk by the IWW since its larger transversal extent allows to keep a bigger
portion of the low-energy muons. Despite the loss of events induced by the
constraints under consideration, we observe that, for both experiments, for a
minimum scalar mass of about $50\,\text{MeV}$, the average effective muon flux
after geometrical acceptance and energy cuts is of the order of
$10^{-5}\,\mu/\text{POT}$. Then, the large number of primary protons which can
be typically dumped777As described in Section V, we consider
$10^{19}\,\text{POT}$ for NA62 and $2\times 10^{20}\,\text{POT}$ for SHiP.
suggests how proton beam-dump experiments might indeed be competitive signal
production sources in the search for exotic muon-specific scalars compared to
“direct” muon-beam experiments such as NA64-$\mu$ Gninenko:2014pea , as well
as the proposed $M^{3}$ Kahn:2018cqs  and FNAL-$\mu$ Chen:2017awl  at
Fermilab.

The effective angular distributions of the secondary muons are shown in Fig. 5
for the experiments NA62 (left) and SHiP (right). A schematic representation
of the geometrical meaning of the angle $\theta$ of the muons’ trajectories
with respect to the proton beam axis is shown in Fig. 7 in Section VI. The
effective distributions in green, blue, red, and violet are obtained from the
original spectrum applying the same requirements detailed above for Fig. 4. We
do not display the input angular distribution, which peaks at about
$0.04\,\text{rad}$ and extends up to about $0.9\,\text{rad}$, for clarity of
the plots.

(a) NA62 Experiment (b) SHiP Experiment

Figure 5: Angular distributions of the secondary muons for the experiments (a)
NA62 and (b) SHiP. Here, $\theta$ is the angle with respect to the proton beam
axis. We show the spectra after geometrical acceptance in green, and the
spectra after geometrical acceptance and fixed-mass IWW energy cuts in blue
($m_{S}=0.05\,\text{GeV}$), red ($m_{S}=0.10\,\text{GeV}$), and violet
($m_{S}=0.15\,\text{GeV}$).

Note that the scalars’ maximum angular acceptance for NA62 is about
$18\,\text{mrad}$, while it is about $64\,\text{mrad}$ for SHiP, that is,
about $3.5$ times larger. Moreover, since we assume that the scalars
approximately travel along the direction of their parent muons, the given muon
angular spectra provide a qualitative forecast of the overall sensitivities of
the two experiments. As for the total momentum spectra in Fig. 4, the
effective angular distribution for the experiment SHiP is more strongly
affected by the IWW energy cuts than in the case of NA62. For both
experiments, the IWW effect is bigger for bigger angles. To understand the
relative difference of the angular spectra before and after the IWW energy
cut, we consider the mutual dependence between the momentum and angular
distributions analysed so far. To do so, we show the two-dimensional effective
muon distributions in the plane $(P_{\mu},\,\theta)$ in Fig. 6 for the
experiments NA62 (left) and SHiP (right) for a fixed $m_{S}=0.10\,\text{GeV}$.

(a) NA62 Experiment (b) SHiP Experiment

Figure 6: Effective distributions of the secondary muons for the experiments
(a) NA62 and (b) SHiP in the plane $(P_{\mu},\,\theta)$. For each experiment,
we show the spectrum after geometrical acceptance (left) and the spectrum
after geometrical acceptance and IWW energy constraint for
$m_{S}=0.10\,\text{GeV}$ (right). Increasing or decreasing the value of
$m_{S}$ will shift the vertical cut due to the IWW towards higher or lower
momenta, respectively.

For both experiments, the low-momentum vertical strip of the two-dimensional
distribution after geometrical acceptance is largest at high values of
$\theta$. Thus, the IWW energy cut will generally have a stronger effect for
bigger angles. Besides, the high-$\theta$ horizontal strip of the distribution
for SHiP is more densely concentrated at the low-momentum end than it is for
NA62, whose fixed-angle spectrum is more widely spread in momentum. Thus, the
SHiP effective spectrum undergoes a more substantial loss due to the IWW
constraint at each value of $\theta$ than the NA62 spectrum.

##  V Experimental Setups and Signatures

The simplified experimental setups of NA62 and SHiP used in our toy MCs are
described here, with particular attention given to the acceptance signatures
of the exotic scalar’s daughter particles, which result from a dedicated
detector modelling. A universal schematic diagram is shown in Fig. 1 in
Section I.

###  V.1 NA62 Experiment run as Proton Beam Dump

The primary goal of the NA62 experiment NA62:2017rwk  is to make a precise
measurement of the branching ratio
$\mathcal{B}$($K^{+}\rightarrow\pi^{+}\nu\bar{\nu}$). However, the experiment
is also sensitive to a variety of BSM scenarios, which appear either from the
kaon decays or when NA62 is run as a beam dump PBCNA62Massri . We model NA62
in a toy MC as follows. The beam-defining collimator for “regular” data-taking
is used to dump the beam. This collimator, which is called TAX and
$L_{\text{tg}}=3.2~{}\text{m}$ long, is the source of the secondary muons that
radiate the scalars. The dump material is Copper. We consider a total of
$10^{19}$ protons on target (POT), as proposed to be collected between the
CERN Long Shutdowns LS3 and LS4 PBCNA62 . After a distance of
$L_{\text{sp}}=75.8~{}\text{m}$ from the end of the TAX, the fiducial decay
region starts, and it is $L_{\text{dec}}=81.0~{}\text{m}$ long. The scalars
must decay within this region to produce potentially detectable daughter
particles. Four STRAW spectrometer chambers are placed at the end of the decay
volume and are followed by a Liquid Krypton Calorimeter (LKr). The detector,
composed of the spectrometer and the calorimeter, among others, has a total
length of $L_{\text{det}}=65.2~{}\text{m}$ and an effective transverse
acceptance area of about $2\times 2~{}\text{m}^{2}$. Successful tracks are
required to hit the detector components at a minimum distance of
$5~{}\text{cm}$ away from the walls of the vessel.

For the detection of the charged final states $\mu^{\pm},\,e^{\pm}$, we impose
the acceptance of both tracks in the first and last STRAW chambers and in the
LKr with a minimum individual track energy of $5~{}\text{GeV}$. For the
detection of the neutral final state $\gamma$, we require both photons from
the scalar decay to be resolved in the LKr at a minimum mutual distance of
$10~{}\text{cm}$. In addition, the photons need to be more than
$15~{}\text{cm}$ away from the LKr central hole, through which the beam-pipe
passes, with a minimum individual energy of $1~{}\text{GeV}$ and a minimum
combined energy of $3~{}\text{GeV}$. Following the current knowledge based on
about $10^{17}$ POT, available in the 2021 data PBCNA62Massri , we assume no
background limitations.

###  V.2 SHiP Proton Beam-Dump Experiment

The proposed SHiP experiment SHIP:2021tpn  specifically aims at searching for
a large number of hidden new-physics states, among which weakly-interacting
long-lived dark sector mediators. The beam dump is
$L_{\text{tg}}=11.2~{}\text{m}$ long and proposed to be made of Molybdenum. We
assume $2\times 10^{20}$ POT. After a distance of
$L_{\text{sp}}=42.0~{}\text{m}$ from the end of the target, the fiducial decay
region starts, and it is $L_{\text{dec}}=50.5~{}\text{m}$ long. Again, the
scalars must decay within this region to produce potentially detectable
daughter particles. Four STRAW spectrometer chambers are placed at the end of
the decay volume and are followed by a calorimeter. The detector, composed of
the spectrometer and the calorimeter, among else, has a total length of about
$L_{\text{det}}=10.3~{}\text{m}$. Again, successful tracks are required to hit
the detector components at a minimum distance of $5~{}\text{cm}$ away from the
walls of the vessel.

For the detection of the charged final states $\mu^{\pm},\,e^{\pm}$, we follow
the guidance provided in Ref. SHiP:2020vbd . We require the two tracks to hit
all spectrometer chambers and the calorimeter with a minimum energy of
$5~{}\text{GeV}$ each. The detection of the neutral final state $\gamma$ in
the SHiP calorimeter is modelled following Ref. Ahdida:2654870 . We require
both photons produced by a scalar decay to hit the calorimeter within an
effective elliptical acceptance area of about $5\times 10~{}\text{m}^{2}$.
Moreover, both photons should have an individual energy of at least
$1~{}\text{GeV}$, a combined energy exceeding $3~{}\text{GeV}$, and they
should be at least $10~{}\text{cm}$ apart. We highlight that the proposed SHiP
calorimeter has the potential to reconstruct the photon direction, which in
turn allows for the reconstruction of the scalar mass. We assume no background
limitations as it is inherently required by the SHiP design.

##  VI Sensitivity Projections

In contrast to the muon beam-dump setup Chen:2017awl , the secondary muons
generated by protons scattering on a thick target in a proton beam-dump
experiment possess a whole spectrum of initial positions and momenta. To take
these into account, the projected number of detected exotic signal events is
computed separately for each muon from the input spectrum described in Section
IV, and the individual contributions are then summed to give the overall
sensitivity projection in the parameter space $(m_{S},\,g_{\mu})$ of the
exotic scalar. We illustrate the basic geometrical quantities of interest in
the simplified sketch in Fig. 7.

Figure 7: Schematic illustration of the experimental geometry up to scalar
decay. $\theta$ is the angle between the muon trajectory and the proton beam
axis. $y^{(i)}$ is the muon penetration length along its trajectory. $z^{(i)}$
is the length travelled by the scalar from production to decay along its
trajectory. Lengths are not to scale.

Let $N_{\mu}$ be the number of muons in the spectrum. For
$i=1,\ldots,N_{\mu}$, we denote by $E_{\mu}^{(i)}$ the energy of the $i$-th
muon at a given penetration depth $y^{(i)}$ along its trajectory. See Fig. 7
for a schematic representation of $y^{(i)}$. As the muon penetrates the
target, its energy $E_{\mu}^{(i)}$, as a function of $y^{(i)}$, decreases from
the initial value $E_{\mu,\text{init}}^{(i)}$ at the muon production point
inside the target to the minimum value $E_{\mu,\text{min}}^{(i)}$ at the end
of the target. We disregard muons that are produced outside the target. At
each step in $E_{\mu}^{(i)}$, the differential cross-section for scalar
production via muon bremsstrahlung $\frac{d\sigma}{dx}$ is given by the IWW
approximated expression in Eq. (15) in Section III. Here,
$x=E_{S}/E_{\mu}^{(i)}$.

Despite the fact that the propagating muon will have the highest energy at its
creation point, the exotic scalar can, in principle, be produced at any point
along the muon penetration path inside the target. The scalar production point
differs from the parent muon production point by the penetration length
$y^{(i)}$, which can be expressed as a function of $E_{\mu}^{(i)}$. More
specifically, in the energy range that we are interested in, going from a few
GeV to a few 100 GeV, ionization dominates the other mechanisms through which
muons can lose their energy. In this regime, the stopping power $dE_{\mu}/dy$
is approximately constant with respect to the muon momentum Groom:2001kq , and
we denote by $\langle dE_{\mu}/dy\rangle$ the average muon energy loss per
unit of penetration length due to ionization. It follows that $y^{(i)}$ and
$E_{\mu}^{(i)}$ are simply related via

${y^{(i)}\simeq\frac{E_{\mu,\text{init}}^{(i)}-E_{\mu}^{(i)}}{\langle
dE_{\mu}/dy\rangle}\,.}$ (20)

We assume the scalar to be collinear to the parent muon, and we denote by
$z^{(i)}$ the distance it travels along its trajectory before decay. See Fig.
7 for a schematic representation of $z^{(i)}$. The acceptance range for the
scalar decay is set by the geometry of the fiducial decay volume of the
specific experiment being considered, and the limiting values
$z_{\text{min}}^{(i)}$ and $z_{\text{max}}^{(i)}$ are the extrapolated lengths
of the straight paths which go from the scalar production point in the target
to their closest and furthest intersections with the decay volume of the
experiment. The geometries of NA62 and SHiP are described in Section V. We
disregard scalars that are not projected to intersect the decay volume.

After radiating from the muon, at each step in $z^{(i)}$, the scalar decay
probability density function per unit length $P_{S}(z)$ is given by

${P_{S}(z)=\frac{\text{e}^{-z/L_{S}}}{L_{S}}\,,}$ (21)

where $L_{S}$ is the scalar decay length, as described in Section II alongside
the description of the contributing decay channels. Again, at each step in
$z^{(i)}$, we denote by $P_{d}(z)$ the experimental acceptance probability of
the scalar’s daughter particles conditional to their production at the given
point. Specifically, we require the scalar’s daughters to obey the conditions
described in Section V for the experiments NA62 and SHiP, which are encoded in
$P_{d}(z)$. We assume here full reconstruction efficiency after the imposed
cuts on the signal. Note that, in the case that all decay products are
accepted, i.e. $P_{d}(z)=1$, the scalar decay probability density $P_{S}(z)$
gives a total probability of scalar decay within the geometric acceptance
range from $z_{\text{min}}^{(i)}$ to $z_{\text{max}}^{(i)}$ of

${\int_{z_{\text{min}}^{(i)}}^{z_{\text{max}}^{(i)}}dz\,P_{S}(z)=\text{e}^{-z_{\text{min}}^{(i)}/L_{S}}-\text{e}^{-z_{\text{max}}^{(i)}/L_{S}}\,.}$
(22)

Finally, the total number of detected exotic signals for a given choice of
parameters $(m_{S},\,g_{\mu})$ is produced by the following composition of the
scalar production differential cross-section and the convolution of the scalar
decay probability density with the daughters’ experimental acceptance
efficiency

${N_{S}=\frac{n_{\rho}}{\langle
dE_{\mu}/dy\rangle}{\sum_{i=1}}^{N_{\mu}}}{\int_{E_{\mu,\text{min}}^{(i)}}^{E_{\mu,\text{init}}^{(i)}}dE_{\mu}\;{\int_{0}^{1}}dx\,\frac{d\sigma}{dx}\;{\int_{z_{\text{min}}^{(i)}}^{z_{\text{max}}^{(i)}}}dz\,P_{S}(z)\cdot
P_{d}(z)\,,}$ (23)

where $n_{\rho}$ is the number density of the target nuclei.

Along with the many geometric and experimental cuts outlined above, the
computational implementation of Eq. (23) is helped by the double cut on the
muon energy that is required in order to apply the IWW approximation for our
predictions, as described in Section III and furthermore analysed in Section
IV. The resulting conservative estimates of the number of signal events
$N_{S}$ for exotic scalar production and detection at the experiment NA62 run
in beam-dump mode and at the proposed SHiP beam-dump facility, for both the
leptophilic and muonphilic effective models of scalar couplings, that have
been produced with the present work, are translated into projected sensitivity
constraints in the parameter plane $(m_{S},\,g_{\mu})$. After cutting at the
di-muon mass $m_{S}=2m_{\mu}$, the corresponding exclusion contours at $95\%$
confidence level are presented in Fig. 8 alongside current prospects and
constraints from other experiments in the current literature.

(a) Leptophilic Model (b) Muonphilic Model

Figure 8: Projected and experimental constraints in the
$(m_{S},\,g_{\mu})$-plane for (a) the leptophilic model, where
$g_{\ell}\propto m_{\ell}$, and (b) the muonphilic model, where
$g_{\mu}\propto m_{\mu},\,g_{e}=g_{\tau}=0$. Our projections for NA62 and SHiP
are cut at the di-muon mass $m_{S}=2m_{\mu}$ and are shown as solid blue and
red lines, respectively. The gray shaded regions denote the strongest existing
bounds on the new scalar mediator by various beam-dump, collider, and
astrophysical probes. The gray non-continuous lines represent a selection of
current prospects, which are included for comparison. See the text for
details. We also show the $5\sigma$-excluded region (shaded gray) and
$2\sigma$-favored region (shaded green) for the anomalous $(g-2)_{\mu}$
measurement by Fermilab and Brookhaven PhysRevLett.126.141801 .

In Fig. 8, the strongest existing constraints are shown as gray shaded
regions. For the leptophilic model, they come from the beam-dump experiments
at Orsay DAVIER1989150  and E137 PhysRevD.38.3375 , as well as collider
searches by the BaBar collaboration BaBar:2020jma . For the muonphilic model,
they arise from the E137 search and from experimental limits set on the
anomalous cooling of SN1987A due to low-mass muonphilic scalars Croon:2020lrf
888More recent work on supernova bounds for muonphilic scalars can be found in
Ref. PhysRevLett.128.221103 .. We show the $2\sigma$-favored region for the
anomalous $(g-2)_{\mu}$ measurement by Fermilab and Brookhaven
PhysRevLett.126.141801  in solid shaded green and the corresponding
$5\sigma$-excluded region in solid shaded gray. For comparison, we include the
projections for the proposed BDX search Marsicano:2018vin  (gray dotted line),
which considers scalar bremsstrahlung from secondary muons coming from an
electron beam, and the projections for the scalar searches using a muon beam
on a dump at NA64-$\mu$ (gray dashed line) and Fermilab (gray dash-dotted
line) Chen:2017awl ; Chen:2018vkr ; Kahn:2018cqs . We remark that our
conservative projections for both NA62 (solid blue line) and SHiP (solid red
line) span uncharted parameter space for both models. In the muonphilic case,
they intersect a part of the $2\sigma$-favored region of parameter space which
has not been rejected by existing experimental bounds, thus accessing a new-
physics sector potentially responsible for the $(g-2)_{\mu}$ anomaly, as
discussed in Section I.

Note that the upper profile of the beam-dump prospects, corresponding to large
coupling $g_{\mu}$, is mostly determined by the experimental geometry, while
the lower contour, corresponding to very small coupling, has a different slope
for primary- and secondary-muons set-ups. The former typically have a rather
monochromatic muon spectrum, while the latter possess a wide range of angular
and energetic muon components. For a given mass $m_{S}$, the scalars radiated
from secondary muons are generally less boosted, giving rise to the difference
in the slope of the lower shape. Moreover, we stop our projections at the di-
muon threshold $m_{S}=2m_{\mu}$ for ease of comparison with previous
prospects. In principle, both NA62 and SHiP will have sensitivity to muon
final states, as described in Section V. Expanding our analysis to include
muon final states would allow us to extend our reach beyond the di-muon mass,
thus probing further unexplored parameter space. However, the IWW
approximation for the computation of the scalar production cross-section is
less robust in this region, and the evaluation of the sensitivity beyond the
di-muon threshold requires additional computational resources. We leave this
extension of our results for future work.

##  VII Conclusions

In this paper, we have explored the potential of using the secondary muons of
a proton beam-dump experiment as a practicable and competing alternative
signal production source in the search for muon-specific extensions of the SM.
BSM models of long-lived and low-mass scalar particles that are primarily or
exclusively coupled to muons are prompted on experimental and theoretical
grounds, among which is the standing discrepancy between the SM prediction and
the experimental observation of the anomalous magnetic moment of the muon. In
contrast to searches using a dedicated muon beam, we have proposed to exploit
the shower of secondary muons that are created by the reactions of protons on
target in a proton beam dump, as sketched in Fig. 1 in Section I.

After outlining the two simplified models of exotic scalar couplings to
leptons that we have denoted as leptophilic and muonphilic models in Section
II, and which can potentially alleviate, or even resolve, the $(g-2)_{\mu}$
anomaly, we have described the muon bremsstrahlung process which dominates the
scalar production mechanism. We have introduced and justified, within its
limits of applicability, the IWW approximation scheme for the computation of
the scalar production cross-section in Section III, which is used to obtain
the results presented in this work. For our purposes, and given the input
energy distribution of the secondary muons and the geometrical acceptance of
the experiments NA62 and SHiP under analysis, as closely inspected in Sections
IV and V, the IWW approximation provides a good estimate of the exact
solution. However, we expect that a more rigorous study incorporating the full
scalar production cross-section via complete MC simulations will improve upon
our conservative projections, especially above the di-muon mass threshold. We
leave such an improvement to future work. For the thorough modelling of the
original muon flux from the proton beam dump reviewed in Section IV, we have
relied on the state-of-the-art MC simulation provided in Ref.
stealex_2022_6352043 , which efficiently produces high statistics muon
samples. In Section VI, we have computed the sensitivity reach of the
currently running NA62 experiment in beam-dump mode as well as the proposed
SHiP beam dump. We have shown in Fig. 8 that proton beam dumps can be
competitive with primary muon-beam experiments in the probe for dark muon-
specific scalars. Indeed, our conservative projections for both NA62 and SHiP
cover unexplored parameter space for both the leptophilic and the muonphilic
models. Moreover, for scalars with exclusive coupling to muons, our prospects
intersect a part of the $2\sigma$-favored region of parameter space which has
not been rejected by existing experimental bounds, roughly centered around
$m_{S}\sim 160\,\text{MeV}$ and $g_{\mu}\sim 7\times 10^{-4}$. Our result
enlarges the class of BSM models that can be effectively probed by proton beam
dump experiments and, in particular, gives access to a new-physics sector
potentially responsible for the $(g-2)_{\mu}$ anomaly.

We remark that a major additional challenge for the proton beam-dump setting,
as compared to muon facilities, is a reliable modelling of the secondary muon
spectrum. Addressing this challenge entails the need for accurate
computational tools that are validated in experimental set-ups. Because NA62,
for example, has already collected a significant amount of beam-dump data in
2021 and is progressing swiftly, the work presented in this paper is
particularly timely DAlessandro:2021gmw .

Finally, we note that it could be interesting to interface the scalar
production process from secondary muons presented here with the ALPINIST
framework Jerhot:2022chi , where the model-dependent and the model-independent
components of the sensitivity evaluation for pseudo-scalar (ALP) production
and decay are separated. This allows a somewhat smooth integration of new
models with non-trivial coupling structures, such as the ones studied in this
paper, and detector geometries with optimised computational effort. The
currently missing ingredient in order to realise such an interface is the
secondary muon spectrum coming from proton beams with lower energies, although
this could in principle be obtained. We leave this venture to future work.

## Acknowledgments

We acknowledge the contributions of Samantha I. Davis and Marcel Rosenthal to
a previous version of the project. We are grateful to Stefan Ghinescu for his
effort and help with the production of the muon samples. We thank Joerg
Jaeckel for discussions that led to the project idea and Sam McDermott and
Brian Shuve for general discussions. We also thank Yiming Zhong for providing
the MadGraph files needed to simulate the cross-section and for his patience
in troubleshooting. This work has been supported by the Swiss National Centre
of Competence in Research SwissMAP (NCCR 51NF40-141869 The Mathematics of
Physics), the European Research Council under Grant No. ERC-2018-StG-802836
(AxScale project), and the National Science Foundation under Grant No. NSF
CAREER-1944826-PHY.

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