# lambeq: An Efficient High-Level Python
Library for Quantum NLP

Dimitri Kartsaklis  Ian Fan  Richie Yeung  Anna Pearson
Robin Lorenz  Alexis Toumi  Giovanni de Felice
Konstantinos Meichanetzidis  Stephen Clark  Bob Coecke
Cambridge Quantum Computing
17 Beaumont Street, Oxford, OX1 2NA, UK
{dimitri.kartsaklis;ian.fan;richie.yeung;anna.pearson;
robin.lorenz;alexis.toumi;giovanni.defelice;
<EMAIL_ADDRESS>

###### Abstract

We present lambeq, the first high-level Python library for Quantum Natural
Language Processing (QNLP). The open-source toolkit offers a detailed
hierarchy of modules and classes implementing all stages of a pipeline for
converting sentences to string diagrams, tensor networks, and quantum circuits
ready to be used on a quantum computer. lambeq supports syntactic parsing,
rewriting and simplification of string diagrams, ansatz creation and
manipulation, as well as a number of compositional models for preparing
quantum-friendly representations of sentences, employing various degrees of
syntax sensitivity. We present the generic architecture and describe the most
important modules in detail, demonstrating the usage with illustrative
examples. Further, we test the toolkit in practice by using it to perform a
number of experiments on simple NLP tasks, implementing both classical and
quantum pipelines.

##  1 Introduction

Quantum Natural Language Processing (QNLP) is a very young area of research,
aimed at the design and implementation of NLP models that exploit certain
quantum phenomena such as superposition, entanglement, and interference to
perform language-related tasks on quantum hardware. The advent of the first
quantum machines, known as noisy intermediate-scale quantum (NISQ) computers,
has already allowed researchers to make the first small steps towards
exploring practical QNLP, by training models and running simple NLP
experiments on quantum hardware (Meichanetzidis et al., 2020; Lorenz et al.,
2021). Despite the limited capabilities of the current quantum machines, this
early work is important in helping us understand better the process, the
technicalities, and the unique nature of this new computational paradigm. At
this stage, getting more hands-on experience is crucial in closing the gap
that exists between theory and practice, and eventually leading to a point
where practical real-world QNLP applications will become a reality.

With that goal in mind, we introduce lambeq111Stylised
‘$\lambda\textrm{ambeq}$’, pronounced “lambek”. The name is a tribute to
mathematician Joachim Lambek (1922-2014), whose seminal work lay at the
intersection of mathematics, logic, and
linguistics.222https://github.com/CQCL/lambeq, an open-source, modular,
extensible high-level Python library, which provides the necessary tools for
implementing a pipeline for experimental QNLP. At a high level, the library
allows the conversion of any sentence to a quantum circuit, based on a given
compositional model and certain parameterisation and choices of ansätze. This
first version of the library includes three compositional models, each using
syntactic information to a different degree: a bag-of-words model with no
syntactic information present, a word-sequence model which respects the order
of words, and a fully syntax-based model following the compositional
distributional framework from Coecke et al. (2010), often dubbed
DisCoCat333DIStributional COmpositional CATegorical.. It should be noted that
lambeq is extensible, and in practice it can accommodate any compositional
model that can encode sentences as string diagrams, lambeq’s native data
structure.444String diagrams in lambeq are close to tensor networks (a popular
structure in quantum physics), but richer from a representation point of view.
See Section 4 for more details. Further compositional models are currently
being developed and will be released in the near future as part of a new
version of the toolkit.

In general, lambeq loosely follows the pipeline employed in the first small-
scale NLP experiments on quantum hardware by Meichanetzidis et al. (2020) and
Lorenz et al. (2021). Figure 1 shows the processing of a sentence in more
detail, the main stages of which are the following:

Figure 1: Pipeline implemented by lambeq.

  1. 1.

Depending on the selected compositional model, a syntax tree for the sentence
might be obtained by calling a statistical CCG555Combinatory Categorial
Grammar (Steedman, 2000). parser. lambeq is equipped with a detailed API that
greatly simplifies this process, and is shipped with support for a state-of-
the-art parser (Yoshikawa et al., 2017).

  2. 2.

Internally, the parse tree is converted into a string diagram, following the
theory and practice described in Yeung and Kartsaklis (2021). The string
diagram can be seen as an abstract representation of the sentence reflecting
the relationships between the words as defined by the compositional model of
choice, independently of any implementation decisions that take place at a
lower level. For storing and manipulating string diagrams, lambeq uses as a
backend DisCoPy (de Felice et al., 2020), a specialised Python library
designed for this purpose.

  3. 3.

The string diagram can be simplified or otherwise transformed by the
application of rewriting rules; these can be used for example to remove
specific interactions between words that might be considered redundant for the
task at hand, or in order to make the computation more amenable to
implementation on a quantum processing unit.

  4. 4.

Finally, the resulting string diagram can be converted into a concrete quantum
circuit (or a tensor network in the case of a “classical” experiment666In this
paper we loosely use the term “quantum experiment/pipeline” to refer to an
experiment or pipeline that uses quantum circuits on quantum or classical
hardware. This is as opposed to a “classical experiment/pipeline”, which
mainly involves training tensor networks on classical hardware.), based on a
specific parameterisation scheme and concrete choices of ansätze. lambeq
features an extensible class hierarchy containing a selection of pre-defined
ansätze, appropriate for both classical and quantum experiments.

After Step 4, the output of the pipeline (quantum circuit or tensor network)
is ready to be used for training. In the case of a fully quantum pipeline, the
quantum circuit will be processed by a quantum compiler and subsequently
uploaded onto a quantum computer, while in the classical case the tensor
network will be passed to an ML or optimisation library, such as
PyTorch777https://pytorch.org or JAX888https://github.com/google/jax. This
first version of lambeq does not include any optimisation or training features
of its own, which is one of our future goals (for more details see Section 8).

We demonstrate the usage of the toolkit in practice by performing classical
and quantum experiments on the meaning classification dataset introduced by
Lorenz et al. (2021). Specifically, we implement:

  * •

a classical experiment, in which the model encodes sentences as tensor
networks and is trained using PyTorch;

  * •

a classical simulation of a quantum pipeline, where sentences are encoded as
quantum circuits with JAX used as part of the training backend;

  * •

a noiseless shot-based quantum simulation experiment executed on
qiskit’s999https://qiskit.org Aer simulator using
tket101010https://github.com/CQCL/pytket, CQC’s quantum compiler.

  * •

a noisy shot-based quantum simulation experiment, again on qiskit’s Aer
simulator using tket.

The rest of the paper is structured as follows: Section 2 puts this work in
context by providing a summary of related research; Section 3 explains
lambeq’s design goals; Section 4 discusses string diagrams, lambeq’s method
for representing sentences; Section 5 describes the two external components on
which lambeq is based, namely the statistical parser and DisCoPy; Section 6
outlines the software architecture and discusses the main modules and classes;
Section 7 describes the experiments, and discusses the results; and finally,
Section 8 summarises our future goals with regard to subsequent versions of
the toolkit.

##  2 Background

Bringing the promise of a substantial leap forward in computational power,
quantum computing has generated rapidly growing interest in recent years, with
scientists and engineers forming an active and dedicated research community.
Promising results and applications in quantum computing can be found in a wide
range of topics, such as cryptography (Pirandola et al., 2020), chemistry (Cao
et al., 2019), and biomedicine (Cao et al., 2018). One of the areas that has
attracted a lot of attention is quantum machine learning (Schuld and
Petruccione, 2018), with some work on quantum neural networks; see for example
(Gupta and Zia, 2001; Beer et al., 2020).

On language-related tasks, Bausch et al. (2021) employ Grover search to
achieve superpolynomial speedups for parsing, while Wiebe et al. (2019)
present a Fock-space representation of language, which, together with a
Harmony optimisation method, allows them to solve NLP problems on a quantum
computer. In Gallego and Orus (2017), parse trees are interpreted as
information coarse-graining tensor networks where it is also proposed that
they can be instantiated as quantum circuits. Ramesh and Vinay (2003) provide
quantum speedups for string-matching. There is also work on quantum-inspired
classical models incorporating features of quantum theory, such as its
inherent probabilistic nature (Basile and Tamburini, 2017) or the existence of
many-body entangled states (Chen et al., 2020).

From an experimental NLP point of view, Meichanetzidis et al. (2020) provided
the first proof of concept that practical QNLP is in principle possible in the
NISQ era, by training a classifier on a small dataset of 16 sentences. In
follow-up work, Lorenz et al. (2021) scaled up the training on datasets of
100-130 sentences, demonstrating convergence of the models and statistically
significant results over a random baseline, thus conducting the first complete
small-scale NLP experiments on quantum hardware.

The idea of representing language with string diagrams and monoidal categories
first appeared in Coecke et al. (2010), in the context of DisCoCat – a
compositional model of natural language whose mathematical foundations
provided a connection to quantum mechanics. In later work, this connection was
exploited so that aspects of the DisCoCat model were leveraged to obtain a
quadratic speedup (Zeng and Coecke, 2016). Further theoretical work
(Meichanetzidis et al., 2021) lays the foundations for implementations on NISQ
devices.

Finally, while lambeq is the first programming toolkit specifically aimed at
QNLP research, there are already a few products available for generic quantum
machine learning development. Examples include TensorFlow Quantum (Broughton
et al., 2020), Google’s library for developing hybrid quantum-classical ML
models, and Pennylane (Bergholm et al., 2018), a cross-platform Python library
for quantum machine learning and automatic differentiation developed by
Xanadu.

##  3 Design goals

In this section we briefly discuss the main design goals of the toolkit.
lambeq is written in Python, which due to its fast prototyping abilities,
extensibility, and the support it receives from a large and dedicated
community, is currently perhaps the most popular programming language in
academia and scientific computing. This choice allows lambeq to greatly
benefit from Python’s open-source ecosystem and the various freely available
scientific and numerical libraries such as numpy, scipy, PyTorch, JAX and many
more.

The toolkit is open-source, published under the Apache 2.0
licence111111https://www.apache.org/licenses/LICENSE-2.0. This allows code
transparency, better communication with users and easier reporting of issues.
More importantly, members of the expanding QNLP community will be able to make
their own contributions and write their own extensions. By open-sourcing
lambeq, we are aiming at active participation of QNLP researchers as part of a
focused effort for extending and improving the available functionality.

One of the main design choices for the toolkit was for it to be highly
modular. We keep the coupling between the various modules as low as possible,
so that each of them could be used independently of the others, providing
flexibility and extensibility. Extensibility, in particular, is another major
design goal, and is supported by detailed object-oriented design and a
flexible class hierarchy, which allows easy addition of basic components such
as compositional models, ansätze, and rewrite rules.

Finally, the toolkit was designed with interoperability in mind. For example,
adding a new parser boils down to encapsulating the appropriate calls into a
single wrapper class, for which a detailed API is provided. Further, exporting
trees and diagrams into JSON format is inherently supported by both lambeq and
DisCoPy, simplifying communication with other applications. With regard to
quantum hardware, DisCoPy features conversion to tket format, a compiler for
optimising and manipulating platform-agnostic quantum circuits. For
experiments on classical hardware, users have the ability to export the
diagrams in Google’s tensor network
format121212https://github.com/google/TensorNetwork, and use one of the
PyTorch or TensorFlow backends for manipulation and training.

##  4 String diagrams

###  4.1 Motivation and connection to tensor networks

“Programming” a quantum computer requires from developers the ability to
manipulate quantum gates (which can be seen as the “atomic” units of
computation in this paradigm) in order to create quantum circuits, which can
be further grouped into higher-order constructions. Working at such a low
level compares to writing assembly in a classical computer, and is extremely
hard for humans – especially on NLP tasks which contain many levels of
abstractions.

In order to simplify NLP design on quantum hardware, lambeq represents
sentences as string diagrams (Figure 2(a)). This choice stems from the fact
that a string diagram expresses computations in a monoidal category131313For
an introduction to monoidal categories, see Baez and Stay (2010); Coecke and
Paquette (2011)., an abstraction well-suited to model the way a quantum
computer works and processes data. In fact, monoidal categories have
previously been used to recast the entire framework of quantum mechanics at a
higher conceptual level, forming the field that is now known as categorical
quantum mechanics (Abramsky and Coecke, 2004). From a more practical point of
view, a string diagram can be seen as an enriched tensor network (Orús, 2014;
Pestun and Vlassopoulos, 2017) (Figure 2(b)), a mathematical structure with
many applications in quantum physics. Compared to tensor networks, string
diagrams have some additional convenient properties, for example they respect
the order of words, and allow easy rewriting/modification of their structure
(see for example Section 6.2).

(a)

(b)

Figure 2: String diagram and corresponding tensor network.

String diagrams and tensor networks constitute an ideal abstract
representation of the compositional relations between the words in a sentence,
in the sense that they remain close to quantum circuits, yet are independent
of any low-level decisions (such as choice of quantum gates and construction
of circuits representing words and sentences) that might vary depending on
design choices and the type of quantum hardware that the experiment is running
on. More information regarding the diagrammatic representation of quantum
processes can be found in Coecke and Kissinger (2017).

###  4.2 Pregroup grammars

lambeq’s string diagrams are equipped with types, which show the interactions
between the words in a sentence according to the pregroup grammar formalism
(Lambek, 2008). In a pregroup grammar, each type $p$ has a left ($p^{l}$) and
a right ($p^{r}$) adjoint, for which the following hold:

$p^{l}\cdot p\to 1\to p\cdot p^{l}\quad\quad\quad\quad p\cdot p^{r}\to 1\to
p^{r}\cdot p$ (1)

When annotated with pregroup types, the diagram in Figure 2(a) takes the
following form:

Note that each wire in the sentence is labelled with an atomic type or an
adjoint. In the above, $n$ corresponds to a noun or a noun phrase, and $s$ to
a sentence. The adjoints $n^{r}$ and $n^{l}$ indicate that a noun is expected
on the left or the right of the specific word, respectively. Thus, the
composite type $n\cdot n^{l}$ of the determiner “a” means that it is a word
that expects a noun on its right in order to return a noun phrase.

The transition from pregroups to vector space semantics is achieved by a
mapping that sends atomic types to vector spaces ($n$ to $N$ and $s$ to $S$)
and composite types to tensor product spaces (e.g. $n^{r}\cdot s\cdot
n^{l}\cdot n^{l}$ to $N\otimes S\otimes N\otimes N$) (Kartsaklis et al.,
2016). Therefore, each word can be seen as a specific state in the
corresponding space defined by its grammatical type, i.e. a tensor, the order
of which is determined by the number of wires emanating from the corresponding
box. The cups ($\cup$) denote tensor contractions. A concrete instantiation of
the diagram requires the assignment of dimensions (which in the quantum case
amounts to fixing the number of qubits) for each vector space corresponding to
an atomic type. More details about parameterisation can be found in Section
6.3.

##  5 Main components

At the highest possible level, lambeq can be loosely seen as a case of
middleware, sitting in between a statistical parser and DisCoPy, as shown in
Figure 3. In practice, though, its purpose greatly exceeds the strict
definition of middleware as a mere means of communication between
applications, since it includes extensive functionality of its own, designed
to support language-related tasks. Furthermore, DisCoPy supports the entire
range of lambeq’s components with low-level functionality and does not merely
provide an output format, as might be implied in the figure. The following
sections describe in detail the role of the two main components.

Figure 3: lambeq as middleware.

###  5.1 CCG parser

lambeq’s string diagrams are based on a pregroup grammar (Section 4.2) to keep
track of the types and the interactions between the words in a sentence. When
a detailed syntactic derivation is required (as in the case of DisCoCat), a
syntax tree needs to be provided by a statistical parser. However, since the
pregroup grammar formalism is not particularly well-known in the NLP
community, there is currently no wide-coverage pregroup parser that can
automatically provide the syntactic derivations. To address this problem,
Yeung and Kartsaklis (2021) recently provided a functorial passage from a
derivation in the closest alternative grammar formalism, namely Combinatory
Categorial Grammar (CCG) (Steedman, 2000), to a string diagram which
faithfully encodes the syntactic structure of the sentence in a pregroup-like
form. Due to the availability of many robust CCG parsing tools (for example,
see Clark and Curran (2007)) this allowed the conversion of large corpora with
sentences of arbitrary length and syntactic structure into pregroup and
DisCoCat form, solving a long-standing problem.141414As an example, the
DisCoCat version of the book “Alice in Wonderland” can be found at
https://qnlp.cambridgequantum.com/downloads.html; see also Figure 12.

lambeq does not use its own statistical CCG parser, but instead it applies the
approach of Yeung and Kartsaklis to implement a detailed interface that allows
connection to one of the many external CCG parsing tools that are currently
available. By default, lambeq is shipped with support for DepCCG (Yoshikawa et
al., 2017), a state-of-the-art efficient parser which comes with a convenient
Python interface.

###  5.2 DisCoPy

While the parser provides lambeq’s input,
DisCoPy151515https://github.com/oxford-quantum-group/discopy (de Felice et
al., 2020) is lambeq’s underlying engine, the component where all the low-
level processing takes place. At its core, DisCoPy is a Python library that
allows computation with monoidal categories. The main data structure is that
of a monoidal diagram, or string diagram, which is the format that lambeq uses
internally to encode a sentence. DisCoPy makes this easy, by offering many
language-related features, such as support for pregroup grammars and functors
for implementing compositional models such as DisCoCat. Furthermore, from a
quantum computing perspective, DisCoPy provides abstractions for creating all
standard quantum gates and building quantum circuits, which are used by lambeq
in the final stages of the pipeline in Figure 1.

##  6 Detailed architecture

Both the statistical parser and DisCoPy are integrated seamlessly into a
unified architecture (Figure 4), implementing all stages of the generic QNLP
pipeline, as was given in Figure 1. In the following sections we describe each
in detail, introducing all important lambeq modules along the way. More
details about lambeq’s API can be found in the toolkit’s
documentation,161616lambeq documentation is available at
cqcl.github.io/lambeq. DisCoPy documentation and usage examples can be found
at https://discopy.readthedocs.io/. along with extensive usage examples in the
form of Jupyter notebooks.

Figure 4: lambeq’s architecture and main modules.

###  6.1 Sentence input

The first part of the process in lambeq, given a sentence, is to convert it
into a string diagram. In order to obtain a DisCoCat-like output, we first use
the ccg2discocat module, which, in turn, calls the parser, obtains a CCG
derivation for the sentence, and converts it into a string diagram. Example A
uses the default DepCCG parser in order to produce the DisCoCat diagram of
Figure 2(a) for the sentence “John gave Mary a flower”.171717To see the code
examples, we refer the reader to the appendix. Other external parsers can be
made available to lambeq by extending the CCGParser class in order to create a
wrapper subclass that encapsulates the necessary calls and translates the
respective parser’s output into CCGTree format.

DisCoCat is not the only compositional model that lambeq supports. In fact,
any compositional scheme that manifests sentences as string diagrams/tensor
networks can be added to the toolkit via the module readers. For example, the
spiders_reader instance of the LinearReader class represents a sentence as a
“bag-of-words”, composing the words using a spider (a commutative operation),
while cups_reader composes words in sequence, from left to right, generating a
“tensor train” (Example A).

(a) spiders_reader

(b) cups_reader

Writing new compositional models entails the extension of the Reader class and
the implementation of the method sentence2diagram in the subclass using
DisCoPy calls.

Finally, lambeq features a CCGBankParser class, which allows conversion of the
entire CCGBank corpus (Hockenmaier and Steedman, 2007) into string diagrams.
CCGBank consists of 49,000 human-annotated CCG syntax trees, converted from
the original Penn Treebank into CCG form. Having a gold standard corpus of
string diagrams allows various supervised learning scenarios involving
automatic diagram generation. Example A shows how to convert Section 00 of
CCGBank into string diagrams.

###  6.2 Rewriting

Syntactic derivations in pregroup form can become extremely complicated, which
may lead to excessive use of hardware resources and prohibitively long
training times. The purpose of the rewrite module is to provide a means to the
user to address some of these problems, via rewriting rules that simplify the
string diagram. As an example, consider the sentence “John walks in the park”,
the string diagram of which is the following:

Note that the representation of the preposition is a tensor of order 5 in the
“classical” case, or a state of 5 quantum systems in the quantum case.
Applying the prepositional_phrase rewriting rule to the diagram (Example A)
takes advantage of the underlying compact-closed monoidal structure, by using
a “cap” ($\cap$) to bridge the discontinued subject noun wire within the
preposition tensor:

In the simplified diagram, the order of the preposition tensor is reduced by
2, which at least for a classical experiment is a substantial improvement.
This example clearly demonstrates the flexibility of string diagrams compared
to simple tensor networks, which was one of the main reasons for choosing them
as lambeq’s representation format. lambeq comes with a number of standard
rewrite rules covering auxiliary verbs, connectors, adverbs, determiners and
prepositional phrases (Table 1).

Rule | Description
---|---
auxiliary | Removes auxiliary verbs (such as “do”) by replacing them with caps.
connector | Removes sentence connectors (such as “that”) by replacing them with caps.
determiner | Removes determiners (such as “the”) by replacing them with caps.
postadverb, preadverb | Simplifies adverbs by passing through the noun wire transparently
| using a cap.
prepositional_phrase | Simplifies the preposition in a prepositional phrase by passing through
| the noun wire transparently using a cap.
Table 1: Rewriting rules.

###  6.3 Parameterisation

Up to this point of the pipeline, a sentence is still represented as a string
diagram, independent of any low-level decisions such as tensor dimensions or
specific quantum gate choices. This abstract form can be turned into a
concrete quantum circuit or tensor network by applying ansätze. An ansatz can
be seen as a map that determines choices such as the number of qubits that
every wire of the string diagram is associated with and the concrete
parameterised quantum states that correspond to each word. In lambeq, ansätze
can be added by extending one of the classes TensorAnsatz or CircuitAnsatz,
depending on the type of the experiment. For the quantum case, the library
comes equipped with the class IQPAnsatz, which turns the string diagram into a
standard IQP circuit181818Instantaneous Quantum Polynomial – a circuit which
interleaves layers of Hadamard gates with diagonal unitaries (Havlíček et al.,
2019).. For instance, the code in Example A produces the circuit in Figure 6
by assigning 1 qubit to the noun type and 1 qubit to the sentence type.

Figure 6: Quantum circuit for the sentence “John walks in the park” in qiskit
format. Noun and sentence types have been assigned 1 qubit each.

In the case of a classical experiment, parameterising with the TensorAnsatz
class with $d_{n}=4$ for the base dimension of the noun space, and $d_{s}=2$
as the dimension of the sentence space, produces the following tensor network:

Note that in classical experiments of this kind, the tensors associated with
certain words, such as conjunctions, can become extremely large. In some
cases, the order of these tensors can be 12 or even higher ($d^{12}$ elements,
where $d$ is the base dimension), which makes efficient execution of the
experiment impossible. In order to address this problem, lambeq includes
ansätze for converting tensors into various forms of matrix product states
(MPSs). Figure 7 shows the output of Example A, where the original tensors
have been split into smaller tensors linked with spiders (SpidersAnsatz) (a)
or standard contractions (MPSAnsatz) (b).

(a) SpiderAnsatz (maximum tensor order = 2).

(b) MPSAnsatz (maximum tensor order = 3).

Figure 7: String diagrams using matrix product states.

###  6.4 Optimisation/training

Although lambeq does not yet include a native machine learning module, it
seamlessly collaborates with standard ML and optimisation libraries such as
PyTorch and JAX. The documentation of the toolkit includes Jupyter notebooks
that demonstrate a variety of classical and quantum pipelines (see also
Section 7).

##  7 lambeq in practice

In this section we apply lambeq to a simple sentence classification task,
demonstrating its usage in four different scenarios: A classical pipeline,
where the sentences are encoded as tensor networks; a classical simulation of
a quantum pipeline, where sentences are encoded as quantum circuits but the
optimisation and training takes place on classical hardware; a shot-based
simulation of a quantum pipeline, without using a noise model, running on
qiskit’s Aer simulator; and a noisy shot-based simulation of a quantum
pipeline, again on Aer simulator. The code for the experiments can be found in
the examples folder of the toolkit’s documentation in the form of Jupyter
notebooks.

For our experiments we use the meaning classification dataset from Lorenz et
al. (2021), in which the goal is to classify simple sentences (such as
“skillful programmer creates software” and “chef prepares delicious meal”)
into two categories, food or IT. The dataset consists of 130 sentences created
using a simple context-free grammar. We proceed to provide details about the
parameterisation and results of the four experiments.

#### Classical pipeline

The first step in all the experiments is to convert the sentences into string
diagrams with the ccg2discocat module. For the classical pipeline, we chose to
apply SpiderAnsatz (Figure 7(a)), assigning a dimensionality of 2 to both the
noun and sentence spaces. We use the PyTorch backend of Google’s tensor
network to perform the training, with the Adam optimiser and standard binary
cross entropy as the loss. Figure 8 gives plots for the accuracy and loss on
the training and development sets. Perfect accuracy is also achieved on the
test set.

Figure 8: Classical pipeline results.

#### Classical simulation of quantum pipeline

In this scenario the string diagrams are converted into quantum circuits using
IQPAnsatz, with 1 qubit assigned to both the noun and sentence spaces. The
number of IQP layers were set to 1. For optimisation we use a gradient-
approximation technique, known as Simultaneous Perturbation Stochastic
Approximation (SPSA) (Spall, 1992). The reason for this choice is that in a
variational quantum circuit context like here, proper backpropagation requires
some form of “circuit differentiation” that would in turn have to be evaluated
on a quantum computer – something very costly from a practical perspective.
SPSA provides a less effective but acceptable choice for the purposes of these
experiments. The main prediction functions are “just-in-time” compiled with
JAX in order to improve speed. Results are shown in Figure 9. Note that this
configuration requires about 4 times more iterations than the previous one in
order to converge; however this is eventually achieved. Once again, we obtain
perfect accuracy on the test set.

Figure 9: Results for the “classical” simulation of a quantum pipeline.

#### Noiseless shot-based simulation of quantum pipeline

For this experiment we repeat the configuration of the classical simulation
pipeline, but this time we evaluate the model on qiskit’s Aer simulator that
is available through tket’s Python interface, pytket. The simulation is
performed without using a noise model. SPSA is again used as the optimisation
algorithm. Figure 10 shows that while there is a lot of fluctuation and
instability at the early stages of training, the model eventually converges
successfully. Accuracy on the test set is again perfect, although it is worth
noting that performance can vary considerably between different runs.

Figure 10: Results for the noiseless shot-based simulation of a quantum
pipeline.

#### Noisy shot-based simulation of quantum pipeline

Finally, we use again the Aer simulator to run a noisy simulation, which is
the closest we can get to an actual run on real quantum hardware. Excluding
the noise model, the setting is identical with the one of the noiseless run.
In Figure 11 we can see that even more iterations were required for the model
to converge, due to the noisy nature of the simulation. However, perfect
accuracy was achieved once more on the test set.

Figure 11: Results for the noisy shot-based simulation of a quantum pipeline.

###  7.1 Other uses of lambeq

Below are some links to work that has previously been carried out by CQC’s
QNLP team using the toolkit, before this first official release:

  * •

An early version of the toolkit has been used for the small-scale experiments
on quantum hardware by Lorenz et al. (2021), implementing the pipeline of
Figure 1.

  * •

A web tool191919https://qnlp.cambridgequantum.com/generate.html based on
lambeq’s ccg2discocat module was made available in the summer of 2021,
allowing the conversion of any sentence into a string diagram or quantum
circuit, and supporting various output formats.

  * •

Yeung and Kartsaklis (2021) used lambeq in order to create the first large-
scale DisCoCat resource including more than 3,000 string diagrams, parsing the
book “Alice in Wonderland” into DisCoCat
form202020https://qnlp.cambridgequantum.com/downloads.html (Figure 12).

Figure 12: The first few sentences of “Alice in Wonderland” in diagrammatic
form. The resource was created using lambeq’s ccg2discocat module.

##  8 Conclusions and future work

We introduced lambeq, the first high-level open-source Python toolkit for
quantum natural language processing. This first release includes abstractions
and tools for implementing all the necessary stages of a pipeline that
converts sentences into quantum circuits and tensor networks. We briefly
outline our plans for the future.

As mentioned before, providing more choices of compositional models and
ansätze, for both quantum and classical experimental pipelines, is one of our
top priorities and currently work in progress. Support for additional parsers
will also be included in one of the future versions of the package.
Specifically, lambeq will be extended with the transformer-based CCG parser
described in Clark (2021), which currently significantly improves the state-
of-the-art.

A more ambitious direction we plan to follow is the introduction of a
“quantum-friendly” optimisation/training module, providing easy access to
machine learning algorithms, objectives, and techniques suitable to quantum
hardware. This is a long-term goal that is more likely to be fulfilled in
stages. A priority is to integrate into lambeq the diagrammatic
differentiation features that were recently introduced in DisCoPy (Toumi et
al., 2021), taking one step closer to a fully automated quantum machine
learning pipeline.

Another longer-term goal is the addition of functionality that supports
discourse-related tasks at the paragraph or document level, according to the
DisCoCirc model (Coecke, 2021). The idea there is to encode an entire document
into a quantum circuit, where nouns are dynamically modified by the
interaction with functional words, such as verbs and adjectives, as the text
flows from one sentence to another.

Finally, one of the main directions of our team is the active support,
maintenance and further contributions to the development of DisCoPy, lambeq’s
low-level backend.

### Acknowledgements

We would like to thank our colleagues in Cambridge Quantum Computing for their
help and support. We are especially grateful to the Oxford QNLP team for all
those valuable discussions, insights and exchange of ideas during the past
year. The name of the toolkit is a tribute to the great Joachim Lambek, whom
Bob Coecke had the pleasure to know personally, and even live in his house for
some months as a postdoc in Montreal. In 2004 Bob was presenting categorical
quantum mechanics at the McGill category theory seminar; when Lambek saw
quantum teleportation with diagrams, he pointed to the screen and said: “This
is grammar!”. And right he was!

We acknowledge the use of IBM Quantum services for this work. The views
expressed are those of the authors, and do not reflect the official policy or
position of IBM or the IBM Quantum team.

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##  Appendix A Appendix

Example 1. Using ccg2discocat.

⬇

1from lambeq.ccg2discocat import DepCCGParser

2

3# Parse the sentence and convert it into a string diagram

4depccg_parser = DepCCGParser()

5diagram = depccg_parser.sentence2diagram(’John gave Mary a flower’)

6

7diagram.draw()

Example 2. SpidersReader and CupsReader.

⬇

1from lambeq.reader import cups_reader, spiders_reader

2

3sentence = ’John gave Mary a flower’

4

5# Create string diagrams based on spiders and cups reader

6spiders_diagram = spiders_reader.sentence2diagram(sentence)

7cups_diagram = cups_reader.sentence2diagram(sentence)

8

9spiders_diagram.draw()

10cups_diagram.draw()

Example 3. CCGBankParser

⬇

1from lambeq.ccg2discocat import CCGBankParser

2

3# Replace path below with the root folder of CCGBank in your system.

4# The sections must be located in <root>/data/AUTO

5parser = CCGBankParser(root=’/ccgbank’)

6diagrams_section_00 = parser.section2diagrams(section_id=0)

Example 4. Rewriting.

⬇

1from lambeq.ccg2discocat import DepCCGParser

2from lambeq.rewrite import Rewriter

3

4# Parse the sentence

5diagram = depccg_parser.sentence2diagram(’John walks in the park’)

6

7# Apply rewrite rule for prepositional phrases

8rewriter = Rewriter([’prepositional_phrase’])

9rewritten_diagram = rewriter(diagram)

10

11rewritten_diagram.draw()

Example 5. Generating a quantum circuit.

⬇

1from lambeq.ccg2discocat import DepCCGParser

2from lambeq.circuit import IQPAnsatz

3from lambeq.core.types import AtomicType

4

5# Define atomic types

6N = AtomicType.NOUN

7S = AtomicType.SENTENCE

8

9# Get a string diagram

10depccg_parser = DepCCGParser()

11diagram = depccg_parser.sentence2diagram(’John walks in the park’)

12

13# Convert string diagram to quantum circuit

14ansatz = IQPAnsatz({N: 1, S: 1}, n_layers=1)

15discopy_circuit = ansatz(diagram)

16discopy_circuit.draw()

17

18# Convert to tket form

19tket_circuit = discopy_circuit.to_tk()

Example 6. Tensor ansätze.

⬇

1from lambeq import ccg2discocat

2from lambeq.tensor import TensorAnsatz, MPSAnsatz, SpiderAnsatz

3from lambeq.core.types import AtomicType

4from discopy import Dim

5

6# Define atomic types

7N = AtomicType.NOUN

8S = AtomicType.SENTENCE

9

10# Parse the sentence

11depccg_parser = ccg2discocat.DepCCGParser()

12diagram = depccg_parser.sentence2diagram(’John walks in the park’)

13

14# Standard tensor diagram (no splitting of tensors)

15std_diagram = TensorAnsatz({N: Dim(4), S: Dim(2)})(diagram)

16

17# MPS tensor diagram

18mps_diagram = MPSAnsatz({N: Dim(4), S: Dim(2)}, bond_dim=4)(diagram)

19

20# Spiders diagram

21spider_diagram = SpiderAnsatz({N: Dim(4), S: Dim(2)})(diagram)

22

23std_diagram.draw()

24mps_diagram.draw()

25spider_diagram.draw()


# Toward a search for axion-like particles at the LNLS

L. Angel1,2<EMAIL_ADDRESS>P. Arias3<EMAIL_ADDRESS>C.
O. Dib4<EMAIL_ADDRESS>A. S. de Jesus1,2<EMAIL_ADDRESS>S.
Kuleshov5,6<EMAIL_ADDRESS>V. Kozhuharov7<EMAIL_ADDRESS>L. Lin8<EMAIL_ADDRESS>M. Lindner9<EMAIL_ADDRESS>F. S. Queiroz1,2,6
<EMAIL_ADDRESS>R. C. Silva1,2<EMAIL_ADDRESS>Y.
Villamizar10<EMAIL_ADDRESS>1 Departamento de Física Teórica e
Experimental, Universidade Federal do Rio Grande do Norte, 59078-970, Natal,
Rio Grande do Norte, Brasil  2 International Institute of Physics,
Universidade Federal do Rio Grande do Norte, Campus Universitário, Lagoa Nova,
Natal, Rio Grande do Norte, 59078-970, Brasil  3 Departamento de Física,
Universidad de Santiago de Chile, Casilla 307, Santiago, Chile  4 Departamento
de Física and CCTVal, Universidad Técnica Federico Santa María, Valparaíso
2340000, Valparaíso, Chile  5 Center for Theoretical and Experimental Particle
Physics, Facultad de Ciencias Exactas, Universidad Andres Bello, Fernandez
Concha 700, Santiago, Chile  6 Millennium Institute for Subatomic Physics at
High-Energy Frontier (SAPHIR), Fernandez Concha 700, Santiago, Chile  7
Faculty of Physics, Sofia University, 5 J. Bourchier Blvd., 1164, Sofia,
Bulgaria  8 Laboratório Nacional de Luz Síncrotron - LNLS, Caixa Postal 6192,
CEP 13084-971, Campinas, São Paulo, Brazil  9 Max Planck Institut fur
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  10 Centro de
Ciências Naturais e Humanas, Universidade Federal do ABC, 09210-580, Santo
André, São Paulo, Brasil

###### Abstract

Axion-Like Particles (ALPs) appear in several dark sector studies. They have
gained increasing attention from the theoretical and experimental community.
In this work, we propose the first search for ALPs to be conducted at the
Brazilian Synchrotron Light Laboratory (LNLS). In this work, we derive the
projected sensitivity of a proposed experiment for the production of ALPs via
the channel $e^{+}e^{-}\to a\gamma$. We show that such an experiment could
probe ALP masses between $1-55\,\mbox{MeV}$, and ALP-electron couplings down
to $g_{aee}=2-6\times 10^{-4}\,\mbox{GeV}^{-1}$ depending on the energy beam,
thickness of the target, and background assumptions. Therefore, this quest
would cover an unexplored region of parameter space for experiments of this
kind, constitute a promising probe for dark sectors, and potentially become
the first Latin-American dark sector detector.

##  I Introduction

The Standard Model (SM) of particle physics has successfully described
numerous observed phenomena in nature. However, there remain some missing
pieces to the puzzle. One common extension to the SM is the inclusion of new
neutral bosons, such as vector, scalar, and pseudoscalar particles. Among the
pseudoscalar extensions, the axion is one of the most well-known, having been
first introduced in the literature in 1977 [1, 2] as a solution to the strong
CP problem in QCD. By introducing a new global symmetry, the Peccei-Quinn
symmetry, which is spontaneously broken at some high energy scale and
explicitly broken due to instanton effects, the mechanism can resolve the
Strong CP problem and predict the existence of a new particle, the axion.
Alongside axions, new pseudoscalar particles, known as axion-like particles
(ALPs), have also emerged as possible extensions to the SM.

Analogously to axions, ALPs have an effective coupling to photons and possibly
fermions. However, for ALPs, their coupling constant and mass are independent
parameters. Their popularity arises from several aspects: if they are light
enough, they can be produced non-thermally in the early universe (e.g., via
the misalignment mechanism proposed by Dine and Fischler in 1982 [3]), without
the need for messenger particles, and potentially account for the entire dark
matter content observed today within a wide parameter space [4, 5]. Slim ALPs
in a higher mass range, with masses below or around MeV, have been extensively
sought after at the intensity frontier [6].

In the intermediate mass range, from MeV up to a few GeV, ALPs have received
increasing attention. Their potential to alleviate the tension observed in
measuring the anomalous electron and muon magnetic moments [7, 8, 9] and as
messengers between the SM and invisible (dark matter) sectors [10, 11] have
been recognized. Constraints on this mass range are mainly provided by
colliders, such as BaBar [12], CLEO [13], LEP [14], LHC, and CLIC [7], as well
as beam dump experiments.

So far, the most studied coupling for ALPs is the one with two photons, which
is highly constrained. However, the coupling to fermions still presents an
important gap, especially for coupling constants $g_{aee}\lesssim 10^{-2}$
GeV-1, for masses in the GeV range. Supernovae (SN) are potential sources of
ALPs, and they provide strong constraints on several couplings and mass ranges
[15, 16, 17]. ALPs in the MeV mass range that feature a coupling to photons
are mainly produced through the Primakoff effect, and they later decay,
producing a gamma ray flux that sets the upper bound
$g_{a\gamma\gamma}<10^{-11}$ GeV-1 for a mass $m_{a}\sim 10$ MeV [18, 19].

However, if ALPs couple to electrons, they can be produced in the core of a
supernova mainly through electron Bremsstrahlung, or if their mass is greater
than 10 MeV, through electron-positron annihilation. For sufficiently weak
couplings, the ALPs can escape the supernova and decay on their way to Earth.
This production mechanism can constrain the parameter space to
$g_{aee}>5\times 10^{-7}$ GeV-1 for $m_{a}\sim 120$ MeV [20]. Despite these
severe restrictions, it is important to check the aforementioned parameter
space with laboratory experiments that have controlled sources and
backgrounds, as the astrophysical environments are not fully modeled in some
cases, with significant improvements currently underway [21].

In this study, our objective is to develop a theoretical proposal for an ALP
search, and evaluate new physics reach of Search for Dark Sector (SeDS), a
detector for the dark matter sector [22] at the UVX Synchrotron. UVX is a
second-generation synchrotron light source that used accelerate electrons to
$1.37$ GeV [23, 24, 25], but it has recently been decommissioned. UVX has been
succeeded by Sirius, a fourth-generation storage ring [26, 27, 28]. Certain
subsystems of UVX may be reutilized to develop a new $1-3$ GeV positron
accelerator, which would give rise to SeDS, a small-scale fixed target
experiment aimed at searching for dark sectors, including ALPs. The design of
the proposed detector is illustrated in Fig. 1.

The production process will occur through electron-positron annihilation
processes, specifically via channels of the type $e^{+}e^{-}\to a\gamma$ as
illustrated in Fig. 2. The SeDS experiment will feature a positron beam
consisting of 10 bunches of $10^{9}$ positrons on target (POT) per second,
with energies ranging from $1$ to $3\,\mbox{GeV}$ directed at a diamond target
with a thickness of $d=100-500\,\mu\mbox{m}$. It is well-known that with such
an intense beam, a portion of the beam will pass through the target. To
address this issue, a magnetic dipole will be added to sweep the non-
interacting positron beam away from the calorimeters. This dipole magnet will
function as a filter, reducing the number of events in the electromagnetic
calorimeter (ECal). Furthermore, a spectrometer is expected to be installed
between the gap in the dipole magnet to detect possible positrons that emitted
would radiate bremsstrahlung photons, suppressing this background source. On
the other hand, since the photons are not affected by the magnetic dipole,
they will hit the calorimeter, be measured, and used to reconstruct the ALP
mass using the missing mass technique. This setup is expected to yield
accurate and precise measurements of the ALP mass. We will show that SeDS has
the potential to significantly contribute to the search for ALPs, and to
deepen our understanding of the possible dark sectors.

The missing mass technique is based on 4-momentum conservation, depending only
on the initial parameters of the setup and knowledge about the outgoing photon
4-momentum. With the missing mass technique, it is possible to predict the
mass of the ALP to be used in the calculations, given by

$\displaystyle
M^{2}_{\text{miss}}=(p_{e^{-}}+p_{\text{beam}}-p_{\gamma})^{2},$ (1)

where $p_{e^{-}}$, $p_{\text{beam}}$ and $p_{\gamma}$ are the 4-momentum of
the target’s electrons, positron beam and outgoing photon, respectively.

The rest of this article is organized as follows. In Section II we describe
the model for the ALP we consider. In Section III we analyze and discuss the
observability and bounds of the ALP. Section IV contains our conclusions.

Figure 1: Experimental scheme of the SeDS experiment. In this scheme, we show
the main parts of the experiment, such as the beam source in yellow, the
diamond target in orange, the dipole magnet in blue, the spectrometer in
green, and the electromagnetic calorimeter in vermilion.

##  II  The Model

As our model, we consider a straightforward case of ALPs that interact only
with photons and electrons111Actually, even if we remove the photon
interaction at tree level, a similar term would be induced at loop level [7].,
resulting in the following lagrangian,

$\displaystyle\mathcal{L}\supset\frac{1}{2}(\partial_{\mu}a)(\partial^{\mu}a)-\frac{1}{2}m_{a}a^{2}+\frac{1}{4}g_{a\gamma\gamma}aF_{\mu\nu}\tilde{F}^{\mu\nu}$
$\displaystyle+\frac{g_{aee}}{2}(\partial_{\mu}a)(\bar{e}\gamma^{\mu}\gamma^{5}e),$
(2)

where $F_{\mu\nu}$ is the electromagnetic strength tensor,
$\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ is
the dual tensor, $m_{a}$ is the ALP mass, $g_{a\gamma\gamma}$ is the effective
coupling of the ALPs with the photon and $g_{aee}$ being the effective
coupling of the axial interaction of the ALPs with the electron, both with
dimension of inverse energy.

Since the ALP only interacts with photons and electrons, it has two main decay
channels, $a\to\gamma\gamma$ and $a\to e^{+}e^{-}$. The decay widths of these
channels are given by,

$\displaystyle\Gamma_{a\to\gamma\gamma}$ $\displaystyle=$
$\displaystyle\frac{g_{a\gamma\gamma}^{2}m_{a}^{3}}{64\pi};$ (3)
$\displaystyle\Gamma_{a\to e^{+}e^{-}}$ $\displaystyle=$
$\displaystyle\frac{g_{aee}^{2}}{8\pi}m_{e}^{2}m_{a}\sqrt{1-\frac{4m_{e}^{2}}{m_{a}^{2}}}.$
(4)

(a) Electron mediated channel.

(b) Photon mediated channel.

Figure 2: Feynman diagrams of the channels that contribute to the ALP
production via electron-positron annihilation into a photon and the ALP
($e^{+}e^{-}\to a\gamma$).

We remind the reader that our proposal is based on the search for ALPs via the
process $e^{+}e^{-}\to a\gamma$. There are two different channels that can
produce ALPs with this signature, represented by the diagrams in Fig. 2. The
total cross-section of these channels are given by
$\sigma_{T}=\sigma_{a\gamma}+\sigma_{ae}+\sigma_{int}$, with,

$\displaystyle\sigma_{a\gamma}$ $\displaystyle=$
$\displaystyle\alpha_{em}g_{a\gamma\gamma}^{2}\frac{(s+2m_{e}^{2})(s-m_{a}^{2})^{3}}{24\beta
s^{4}},$ (5) $\displaystyle\sigma_{ae}$ $\displaystyle=$
$\displaystyle\alpha_{em}g_{aee}^{2}m_{e}^{2}\frac{-2m_{a}^{2}\beta
s+(s^{2}+m_{a}^{4}-4m_{a}^{2}m_{e}^{2})\log\frac{1+\beta}{1-\beta}}{2(s-m_{a}^{2})s^{2}\beta^{2}},$
(6) $\displaystyle\sigma_{int}$ $\displaystyle=$
$\displaystyle\alpha_{em}g_{a\gamma\gamma}g_{aee}m_{e}^{2}\frac{(s-m_{a}^{2})^{2}}{2\beta^{2}s^{3}}\log\frac{1+\beta}{1-\beta},$
(7)

where $\sqrt{s}=\sqrt{2m_{e}(m_{e}+E_{beam})}$ is the center of mass (CM)
energy, $m_{e}$ is the electron mass, $\beta=\sqrt{1-4m_{e}^{2}/s}$ and
$\alpha_{em}=e^{2}/4\pi$ is the electromagnetic fine structure constant. The
first equation ($\sigma_{a\gamma}$) is the contribution from the diagram in
Fig. 2(a), representing the cross-section of the photon-mediated channel as a
function of $m_{a}$, whilst the second equation ($\sigma_{ae}$) is for the
diagram in Fig. 2(b). The last equation is the interference term between the
two contributions.

It is important to note that ALPs can be produced through other channels, such
as Bremsstrahlung with the target ($e^{+}+N\to e^{+}+N+a$) and Primakoff
production originating from a secondary photon that collides with the target
($\gamma+N\to N+a$). However, these channels produce a distinct signal from
the previously discussed ones, and the missing mass technique described in Eq.
(1) cannot be employed in the same manner since there are no photons in the
final state. Hence, these channels do not represent a feasible signal for our
setup. An important ingredient of dark sector searches is the decay length.
One needs to know whether the ALP will decay inside or outside the detector.

(a) Photon energy for $E_{beam}=1\,\mbox{GeV}$.

(b) Photon energy for $E_{beam}=3\,\mbox{GeV}$.

Figure 3: Energy of the photon produced along with the ALP as a function of
the opening angle between the photon and the beam axis in the Laboratory frame
for different configurations of the ALP mass. As expected, the photon’s energy
gets smaller as larger masses are assumed.

###  II.1 Decay Length

The first method presented in this work is the so-called invisible search,
which is concentrated on the visible counterpart of the signal, i.e., the
photon in the final state. However, depending on the detector setup and
properties of the ALP, the decay products of the ALP can be measured, if it
decays inside the detector. The latter is known as visible search. A key
quantity in this case is the decay length, which is found to be,

$L_{a}=\gamma\beta c\tau_{a}\approx\frac{E_{a}}{m_{a}\Gamma_{a}},$ (8)

where $\gamma=E_{a}/m_{a}$ is the Lorentz factor, $\tau_{a}=\Gamma_{a}^{-1}$
is the lifetime of the ALP, which is inversely related to the decay width
presented in Eq. (4), $\beta=\sqrt{1-4m_{e}^{2}/s}$, and $E_{a}$ being the
energy of the ALP. In order to determine the decay length of the ALP to assess
whether it decays inside or outside the detector, we need to know the energy
of the ALP, $E_{a}$. The energy of the ALP is related to the outgoing photon
energy,

$E_{a}=\frac{s}{2m_{e}}-E_{\gamma L},$ (9)

where $E_{\gamma L}$ is the energy of the photon in the laboratory system,
which depends on the opening angle of the photon in relation to the beam axis,
and the mass of the ALP as follows,

$E_{\gamma
L}=\left(1-\frac{m_{a}^{2}}{s}\right)m_{e}\frac{1+\beta\cos^{2}\theta_{L}\sqrt{1+\tan^{2}\theta_{L}}}{1-\beta^{2}\cos^{2}\theta_{L}},$
(10)

where $\beta$ is the velocity of the CM in the laboratory frame and
$\theta_{L}$ is the angle between the photon and the beam axis.

In order to clearly demonstrate how the energy and angle of the photon
detected in the calorimeter are influenced by the ALP mass, we present the
results from Eq. (10) for two different beam energies in Fig. 3 from where it
can be inferred that the energy of the outgoing photon increases as the
opening angle decreases. Furthermore, in order to accurately determine the
mass of the ALP produced in the collision, it is crucial to use a calorimeter
with a good energy resolution, especially for photons detected at a high
opening angle.

The energy measurement of an electromagnetic calorimeter is determined by the
energy released in the detector material through ionization and excitation
processes, which is proportional to the energy of the incident particle. The
thickness of the absorber layers in radiation lengths affects the energy
resolution of an electromagnetic calorimeter, with a smaller thickness leading
to a larger number of detected particles and a better energy resolution [29].
To achieve precise determination of the photon energy and ALP mass, it is
important to reduce the thickness of the absorber layers. A Bismuth Germanate
(BGO) calorimeter has achieved an energy resolution of
$2\%/\sqrt{E{\rm(GeV)}}$ [30, 31], which is sufficient for our scientific
goal. For example, a $100$ MeV photon would yield a 6% energy resolution. A
Lutetium Yttrium Orthosilicate (LYSO) calorimeter one could improve this
energy resolution by a factor of two [32, 33, 34] though.

Having that in mind, we use the measured $E_{\gamma L}$, combined with
knowledge of the beam energy to derive $E_{a}$, and subsequently determine the
decay length of the ALP using Eq. (8) for $E_{beam}=1\,\mbox{GeV}$ (Fig. 4(a))
and $E_{beam}=3\,\mbox{GeV}$ (Fig. 4(b)). In Fig. 4, we present the decay
length of the ALP as a function of the photon angle in the laboratory frame
$\theta_{L}$, assuming a coupling of the ALP with the electron of
$g_{aee}=10^{-3}\,\mbox{GeV}^{-1}$, and various values of $m_{a}$ within the
region of interest. Notably, for this configuration, the decay length of the
ALP is greater than 20 meters, resulting in the ALP being invisible.
Therefore, our proposal is designed to detect invisible ALPs. Having
established the search for invisible ALPs we will address the production cross
section.

(a) Decay length for $E_{beam}=1\,\mbox{GeV}$.

(b) Decay length for $E_{beam}=3\,\mbox{GeV}$.

Figure 4: Decay length of the ALP as a function of the photon angle
$\theta_{L}$ in the laboratory frame, taking $g_{aee}=10^{-3}$ for
$E_{beam}=1\,\mbox{GeV}$ (Fig. 4(a)) and $E_{beam}=3\,\mbox{GeV}$ (Fig. 4(b)).
Notably, the ALP will decay outside the detector.

##  III  Discussion

Figure 5: Cross-section of the photon (solid lines) and electron (dashed
lines) mediated processes of ALP production as a function of the ALP mass
$m_{a}$. We considered $g_{a\gamma\gamma}=g_{aee}=1$ GeV-1 and beam energies
of 1 GeV and 3 GeV. The drop in the cross section occurs due to kinematics for
$m_{a}=32$ MeV and $m_{a}=55$ MeV, respectively.

Upon analyzing the total cross-section, it has been observed that the
interference term is suppressed by a factor of $m_{e}^{2}$. As a result, the
contribution of the interference term to the total cross-section is at most
3.5% compared to the case where no interference is present. Thus, we will
ignore it and focus on investigating the two ALPs production cross sections,
namely $\sigma_{a\gamma}$ and $\sigma_{ae}$, independently.

The contributions of the photon and electron mediated channels can be seen in
Fig. 5, where we plotted $\sigma_{a\gamma}$ and $\sigma_{ae}$ for the beam
energies of $E_{beam}=1\,\mbox{GeV}$ and $E_{beam}=3\,\mbox{GeV}$. The
contribution of the photon (electron) channel is presented in solid (dashed)
lines. We assumed $g_{a\gamma\gamma}=g_{aee}=1$ GeV-1. Although we acknowledge
that $g_{a\gamma\gamma}=1$ GeV-1 is excluded by observations, we aim to
emphasize the significance of $g_{a\gamma\gamma}$ in the production of ALPs
via $e^{+}e^{-}$ annihilations. Furthermore, when $m_{a}\simeq\sqrt{s}$, we
find a divergence at $\sigma_{ae}$, as shown in the denominator of Eq. (6).

When the couplings have comparable magnitudes, the contribution from the
photon-mediated process dominates over most of the parameter space. However,
as the mass of the ALP approaches the center of mass energy, the electron-
mediated process becomes the dominant channel. The center of mass energy is
determined by $\sqrt{s}=\sqrt{2E_{\text{beam}}m_{e}}$ and is equal to 31.9 and
55.3 MeV for beam energies of 1 and 3 GeV, respectively. Hence, when
$g_{a\gamma\gamma}\ll g_{aee}$ the ALP production is governed by $g_{aee}$.
Having in mind the restrictive bounds on $g_{a\gamma\gamma}$, we will assume
that $g_{aee}$ is sufficiently larger than $g_{a\gamma\gamma}$, hereafter.
That said, we will concentrate our work on $g_{aee}$ in what follows.

###  III.1 Current and Projected Bounds on the model

There are several limits on ALPS stemming from beam dumps experiments [35, 36,
37], cosmology [38], and astrophysical sources [20, 39]. Most of these
constraints arise from astrophysical sources. However, probing such
interactions in accelerators would be desirable as it would be subject to
smaller systematic uncertainties. In this study, we will focus on the
couplings between ALPs and electrons, specifically $g_{aee}$, which can be
directly probed using $e^{+}e^{-}$ collisions. To assess the relevance of our
proposal, we will put our findings into perspective with existing and
projected accelerator searches. We will outline the relevant bounds on this
coupling below.

DELPHI-LEP: The DELPHI experiment was a detector inside the Large Electron
Positron (LEP) collider at CERN [40]. A revision of the data from the DELPHI
experiment on monophoton processes of the type $e^{+}e^{-}\to X\gamma$
resulted in $g_{aee}<8\times 10\,\mbox{GeV}^{-1}$. The excluded region is
shown in Fig. 6.

BaBar: BaBar is a $e^{+}e^{-}$ collider [41]. The BaBar Collaboration analyzed
the production of dark photon in the process $e^{+}e^{-}\to A^{\prime}\gamma$
where $A^{\prime}$ represents the dark photon, decaying into
$e^{+}e^{-},\mu^{+}\mu^{-}$ [42]. Recasting this bound, we find
$g_{aee}<0.6\,\mbox{GeV}^{-1}$ for $1\,\mbox{MeV}<m_{a}<10\,\mbox{GeV}$. This
constraint is exhibited in Fig. 6.

Belle-II: The Belle-II experiment [43] has a similar setup to BaBar, being
able to search for visible and invisible ALPs. The projected bounds coming
from Belle-II are initially on $g_{a\gamma\gamma}$, but they can be translated
to $g_{aee}$. In this work, we considered the Belle-II projections for the
luminosity of 20 fb-1 and 50 ab-1. This high luminosity makes this projection
produce the most compelling constraints covering most regions of parameter
space, being the best probe for ALPs in the range of hundreds of MeV to 10
GeV. The projection for both luminosity configurations is given in Fig. 6.

NA64: The NA64 experiment [44] consists of a beam of electrons with an energy
of 100 GeV dumped at a fixed target. The experiment goal is to search for Dark
Photons via the process $e^{-}+Z\to e^{-}+Z+A^{\prime}$, where $A^{\prime}$ is
the dark photon that can decay visibly and invisibly, being able to analyze
data from the decays $A^{\prime}\to e^{+}e^{-},\gamma\gamma$. The results of
NA64 were used to search for ALPs [35]. As no excess of events was observed, a
bound was derived on the ALP-electron couplings, as shown in Fig. 6.

PADME: The PADME experiment features a positron beam with variable energy of
$200-550$ MeV directed at an active target. Measuring the energy of the final
states, we can constrain the presence of invisible particles such as ALPs
[45]. This search achieved an integrated luminosity of $4\times 10^{13}$ POT
with no positive signal. A new run is planned with $4\times 10^{16}$ POT [9].
The current and projected bounds are exhibited in Fig. 6.

###  III.2 SM Background Processes

Standard model processes can generate a signal that resembles $e^{+}e^{-}\to
a\gamma$. However, it is crucial to note that in the framework we are
considering, the ALP has a long decay length and therefore decays outside the
detector, resulting in signal events characterized by a single photon in the
electromagnetic calorimeter. By using the energy of the beam, we can calculate
the ALP mass using Eq. (1).

The primary background processes arise from
$\sigma(e^{+}e^{-}\to\gamma\gamma)=0.93$ mb, $\sigma(e^{+}Z\to e^{+}\gamma
Z)=2.2\times 10^{3}$ mb, $\sigma(e^{+}e^{-}\to e^{+}e^{-}\gamma)=77$ mb, and
$\sigma(e^{+}e^{-}\to\gamma\gamma\gamma)=0.02$ mb for a positron beam energy
of $1$ GeV. Among them, the largest cross-section originates from photon
bremsstrahlung. However, this background can be effectively reduced because
the photon from bremsstrahlung is almost collinear with the beam and the
charged particle will be guided to the spectrometer via the electric dipole.
Besides, a positron-electron veto system to further control this background
could also be installed to further control this background [46].

Similarly, the background arising from $e^{+}e^{-}\gamma$ production can be
managed. Although its cross section is not particularly large, the
$\gamma\gamma\gamma$ production presents a challenging background. Due to the
phase space, there is a lack of symmetry, which could result in a false signal
in the calorimeter, where one high-energy photon is detected, and two low-
energy photons would pass undetected. This scenario leads to no significant
peak in $M_{miss}^{2}$. Consequently, it is challenging to reduce this
background by applying cuts on the energy and angular distribution of the
calorimeter. Notice that this case is different from the two-photon
annihilation production, where there is symmetry in the direction of the
photons.

We want to emphasize to the reader that this work is a theoretical proposal
for an ALP search using a 3 GeV positron accelerator called SeDS, which is
planned to be constructed using parts of a partly decommissioned 1.37 GeV
accelerator at the LNLS. Our objective is to estimate the projected
sensitivity of such an accelerator without considering detector effects but to
demonstrate the physics potential of what could be the first Latin-American
dark sector detector. In the following section, we explain how we obtained the
projected limits.

(a) Sensitivity for $E_{beam}=1\,\mbox{GeV}$.

(b) Sensitivity for $E_{beam}=3\,\mbox{GeV}$.

Figure 6: Current and projected bounds for the $g_{aee}$ coupling in ALP
searches for the case where $g_{a\gamma\gamma}=0$. SeDS’ sensitivity for
$N_{back}\sim 0$ is presented in the green region, while the current and
projected limits considered here were taken from [9], where the solid (dashed)
gray (black) lines represent the current (projected) limits from DELPHI,
Babar, Belle-II, NA64, and PADME.

(a) Sensitivity for $E_{beam}=1\,\mbox{GeV}$.

(b) Sensitivity for $E_{beam}=3\,\mbox{GeV}$.

Figure 7: Current and projected bounds for the $g_{aee}$ coupling in ALP
searches for the case where $g_{a\gamma\gamma}=0$. SeDS’ sensitivity for
$N_{back}\sim N_{sig}$ is presented in the green region. The current and
projected limits considered here were taken from [9], where the solid (dashed)
gray (black) lines represent the current (projected) limits from DELPHI,
Babar, Belle-II, NA64, and PADME.

(a) Sensitivity for $E_{beam}=1\,\mbox{GeV}$.

(b) Sensitivity for $E_{beam}=3\,\mbox{GeV}$.

Figure 8: Same as Figures 6 and 7, but with $N_{back}\sim 10N_{signal}$.

###  III.3 Limits from SeDS

An important quantity to assess the sensitivity of such an accelerator is the
luminosity. The instantaneous luminosity of a positron beam impinging on a
target is given by,

$L_{\text{inst}}=\frac{N_{\text{P.O.T}}}{\text{s}}N_{A}\frac{Z\rho d}{A},$
(11)

where $N_{\text{P.O.T}}/\text{s}=10^{10}$ is the number of positrons on the
target per second considering the properties of the decommissioned
accelerator. Assuming a diamond target with a thickness of $d=100-500\,\mu$m,
we can calculate the instantaneous luminosity knowing that $N_{A}$ is the
Avogadro number, $Z=6$ is the atomic number of the diamond target, $\rho=3,51$
g/cm3 is the density of diamond and $A=12.01$ g is the carbon’s gram-molecular
weight.

We consider different setups. One uses a 1 GeV positron beam, and the other
uses a 3 GeV positron beam. With $E_{beam}=1$ GeV and $d=100\,\mu$m, we find
$\mathcal{L}=821\,\mbox{pb}^{-1}$ for 90 days of data taking, and
$\mathcal{L}=3331\,\mbox{pb}^{-1}$ for a one-year of data taking. For
$E_{beam}=3$ GeV and $d=500\,\mu$m, we find $\mathcal{L}=4106\,\mbox{pb}^{-1}$
for 90 days of data taking, and $\mathcal{L}=16650\,\mbox{pb}^{-1}$ for a one-
year of data taking.

Knowing the luminosity of the experiment, we can calculate the number of
signal events and delimit the region of parameter space in which an ALP could
be observed in the $\\{m_{a},g_{aee}\\}$ plane assuming $g_{a\gamma\gamma}=0$.

In Fig. 6, we present the regions at 95% C.L. assuming that the number of
signal events is dominant over the number of background events ($N_{back}\sim
0$). In Fig. 7, we consider the number of background events, which includes
the irreducible background $(e^{+}e^{-}\to\gamma\gamma\gamma)$, to be similar
to the number of signal events. Lastly, in Fig. 8, we assume the number of
background events to be 10 times larger than the number of signal events.
These figures demonstrate that SeDS is a promising probe for directly
measuring ALP couplings to electrons, even in a very conservative scenario. It
is important to note that these sensitivity projections do not take into
account detector effects, but rather serve to demonstrate the physics
potential of SeDS as the first Latin-American dark sector detector. In the
following section, we explain in detail how we obtained these projected
limits.

Fig. 6(a) accounts the projected sensitivity of SeDS for the case
$E_{beam}=1\,\mbox{GeV}$ and $d=100\,\mu\mbox{m}$. In this setup, SeDS probes
ALP-electron coupling down to $g_{aee}\sim 6\times 10^{-4}\,\mbox{GeV}^{-1}$
for masses up to $m_{a}\sim 32\,\mbox{MeV}$. Fig. 6(b) exhibits the
sensitivity for $E_{beam}=3\,\mbox{GeV}$ and $d=500\,\mu\mbox{m}$. In the
latter setup, SeDS could probe ALP-electron coupling down to $g_{aee}\sim
2.3\times 10^{-4}\,\mbox{GeV}^{-1}$ covering larger ALP masses up to
$m_{a}\approx 55\,\mbox{MeV}$ due to the larger center of mass energy. It is
noteworthy that the higher luminosity and center of mass energy of the latter
configuration can achieve better results than the first one. In both cases,
SeDS presents a promising prospect in the search for an electrophilic axion-
like particle and projects better results than other experiments of its kind.

However, it is important to note that the sensitivity presented in Fig. 6 is
the result of an optimistic analysis in which the experiment’s background can
be reduced as much as possible, ensuring that the signal is considerably
higher than the background. To present a more realistic scenario of SeDS’s
sensitivity for the electron-mediated channel, Fig. 7 shows the case where
$N_{back}\sim N_{signal}$.

The first scenario considered is the setup with $E_{beam}=1\,\mbox{GeV}$ and
$d=100\,\mu\mbox{m}$, presented in Fig. 7(a). In this case, SeDS can probe the
ALP-electron coupling for values of $g_{aee}>9\times 10^{-4}\,\mbox{GeV}^{-1}$
with $\mathcal{L}=3300\,\mbox{pb}^{-1}$. In the second experimental setup
where $E_{beam}=3\,\mbox{GeV}$ and $d=500\,\mu\mbox{m}$, SeDS can reach an
ALP-electron coupling of $g_{aee}>6.5\times 10^{-4}\,\mbox{GeV}^{-1}$. As
expected, SeDS’s sensitivity is reduced when there is a non-zero background.
Nevertheless, SeDS’s projected sensitivity is still an improvement compared to
the results of similar experiments.

In a similar vein, we consider a very conservative scenario, we repeat the
exercise above assuming the number of background events to be ten times larger
than the signal events, $N_{sig}=10\,N_{back}$, in Fig. 8(a) and Fig. 8(b).
Even so, SeDS covers an unexplored region of parameter space.

##  IV  Conclusion

In this work, we have proposed the first direct search for axion-like
particles at the LNLS using a detector named SeDS, which consists of a $1-3$
GeV accelerator impinging on a target planned to be constructed using
subsystems of UVX, a $1.37$ GeV accelerator which has been recently
decommissioned. We have concluded that SeDS could place competitive bounds on
the coupling of axion-like particle coupling to electrons. In particular, in a
conservative scenario, we have shown that SeDS could surpass current and
projected experiments in the mass range of 1-30 MeV using a $1$ GeV positron
beam. In an optimistic configuration with a $3$ GeV positron beam, SeDS could
potentially probe axion-like particles for masses up to 55 MeV and outperform
current and projected experiments of the kind, probing $g_{aee}$ down to
$3\times 10^{-4}\,\mbox{GeV}^{-1}$.

###### Acknowledgements.

We would like to thank Harry Westfahl, Daniel Tavares, and Narcizo Neto from
the LNLS for the hospitality and providing useful information about the
accelerator.

  * •

Funding: Simons Foundation (Award Number:1023171-RC), FAPESP Grant
2018/25225-9, 2021/01089-1, 2023/01197-4, ICTP-SAIFR FAPESP Grants
2021/14335-0, CNPq Grant 307130/2021-5, ANID-Programa Milenio-code
ICN2019_044, CAPES grant 88887.485509/2020-00, ANID-Programa Milenio-code
ICN2019_044, ANID PIA/APOYO AFB180002 (Chile), ANID/CONICYT FONDECYT Regular
1221463 and FONDECYT Regular 1210131. Y.V. expresses gratitude to São Paulo
Research Foundation (FAPESP) under Grant No. 2018/25225-9 and 2023/01197-4.

  * •

No conflict of interest to be reported.

  * •

Authors’ contributions: Álvaro S. de Jesus and Farinaldo S. Queiroz conceived
the study, contributed to the writing and mathematical computations of the
manuscript. Paola Arias, Claudio O. Dib, Sergey Kuleshov, Venelin Kozhuharov,
Manfred Lindner, Liu Lin, Yoxara Villamizar, Lucia Angel, and Ricardo C. Silva
contributed to the writing and phenomenology of the work.

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