# Classical control, quantum circuits and linear logic
in enriched category theory

Mathys Rennela  Radboud University
Nijmegen, The Netherlands<EMAIL_ADDRESS>and  Sam Staton  Oxford
University
Oxford, The United Kingdom<EMAIL_ADDRESS>
###### Abstract.

We describe categorical models of a circuit-based (quantum) functional
programming language. We show that enriched categories play a crucial role.
Following earlier work on QWire by Paykin et al., we consider both a simple
first-order linear language for circuits, and a more powerful host language,
such that the circuit language is embedded inside the host language. Our
categorical semantics for the host language is standard, and involves
cartesian closed categories and monads. We interpret the circuit language not
in an ordinary category, but in a category that is enriched in the host
category. We show that this structure is also related to linear/non-linear
models. As an extended example, we recall an earlier result that the category
of W*-algebras is dcpo-enriched, and we use this model to extend the circuit
language with some recursive types.

###### Key words and phrases:

Enriched categories, categorical semantics, linear type theory, quantum
circuits, relative monad, quantum domain theory

###### 1991 Mathematics Subject Classification:

F.3.2 Semantics of Programming Languages

## Introduction

ClassicalQuantumcontrolmeasurements

One of the subtle points about quantum computation is the interaction between
classical control flow and quantum operations. One can measure a qubit,
destroying the qubit but producing a classical bit; this classical bit can
then be used to decide whether to apply quantum rotations to other qubits.
This kind of classical control can be neatly described in quantum circuits,
for example when one uses the measurement outcome of a qubit $a$ to
conditionally perform a gate $X$ on a qubit $b$:

This can be understood semantically in terms of mixed states, density
matrices, and completely positive maps. However, high level languages have
more elaborate data structures than bits: they have higher order functions and
mixed variance recursive types, and associated with these are elaborate
control structures such as higher order recursive functions. These are
important, paradigmatic ways of structuring programs. How should these high
level features be integrated with quantum computation? One option is to build
a semantic domain that accommodates both quantum computation and higher order
features. This is an aim of some categorical semantics of the quantum lambda
calculus [pagani-selinger-valiron-popl14, malherbe-scott-selinger] and of
prior work of the authors [rennela-mfps30, rennela-staton-mfps31]. This is a
fascinating direction, and sheds light, for example, on the structure of the
quantum teleportation algorithm (e.g. [pagani-selinger-valiron-popl14, Example
6]). However, the general connection between physics and higher-order quantum
functions is yet unclear. Although some recent progress has been made
[kissinger-uijlen-lics17], it is still unclear whether higher-order quantum
functions of this kind are useful for quantum algorithms. Another approach is
to understand a high level quantum programming language as an ordinary higher-
order functional language with extra features for building and running quantum
circuits. In this setting, quantum circuits form a first-order embedded domain
specific language within a conventional higher order language. This fits the
current state-of-the-art in interfaces to quantum hardware, and is the basis
of the languages Quipper [quipper] and LiQUi$|\rangle$[liquid]. This is the
approach that we study in this paper.

### Embedded languages and enriched categories

Our work revolves around a new calculus that we call ‘EWire’ (§1). It is a
minor generalization of the QWire language [qwire, randqwire]. QWire idealizes
some aspects of the architecture of Quipper and LiQUi$|\rangle$. The idea is
that we deal with a host language separated from an embedded circuit language.

  * •

The circuit language is a first order typed language. The types, called ‘wire
types’, include a type for qubits. The wire type system is linear to
accommodate the fact that qubits cannot be duplicated.

  * •

The host language is a higher order language. The types of the host language
do not include the wire types, there is not a type of qubits, and it is not a
linear type system. However, there is a special host type
$\mathrm{Circ}(W_{1},W_{2})$ associated to any pair of wire types $W_{1}$ and
$W_{2}$, whose inhabitants are the circuits with inputs of type $W_{1}$ and
outputs of type $W_{2}$.

Let us describe the circuit language in a nutshell: the very simple circuit
(Introduction) corresponds to the instruction below (2) in the circuit
language. Given two qubits $a$ and $b$, it measures the qubit $a$, stores the
result in a bit $x$ which is later used in the application of the classical-
controlled-$X$ gate and discards the bit $x$, then outputs the resulting qubit
$y$.

$\displaystyle-;a,b:\mathrm{qubit}\ \vdash\ C\ \
\stackrel{{\scriptstyle\text{def}}}{{=}}\ \ $ $\displaystyle
x\leftarrow\textbf{gate }\textbf{meas }a;(x,y)\leftarrow\textbf{gate
}(\textbf{bit-control }X)\,(x,b);$ $\displaystyle()\leftarrow\textbf{gate
}\textbf{discard }x;\textbf{output }y\ \qquad:\ \mathrm{qubit}$ (2)

The interface between the host language and the circuit language is set up in
terms of boxing and unboxing. For example, the instruction

$t\ \ \stackrel{{\scriptstyle\text{def}}}{{=}}\ \ \textbf{box
}(a,b)\Rightarrow C(a,b)\qquad\text{(where $C$ is as in \eqref{eqn:example})}$
(3)

creates a closed term of type Circ(qubit$\otimes$qubit,qubit) in the host
language. We recover the instruction $C$ in the circuit language (2) from the
boxed expression $t$ in the host language (3) by using the instruction
$\textbf{unbox }t\,w$ for some fresh wire $w$ of type qubit.

Also, it is possible to write a program that composes two circuits $C_{1}$ and
$C_{2}$ with the right input/output types, for example:

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[0,-1] *+¡.6em¿X @- [0,-1] @- [0,-1] @- [0,-1]
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*!R!¡.5em,0em¿=¡0em¿a *=¡1.8em,1.4em¿@- [0,-1] @= [0,-1] @= [0,-1] @= [0,-1] *!¡0em,.025em¿-=-¡.2em¿∙-1@= [-1,0]@= [0,-1]$w_{2}$<EMAIL_ADDRESS>@R=.7em *!R!¡.5em,0em¿=¡0em¿b& @- [0,-1] @- [0,-1]@- [0,-1]@- [0,-1] *+¡.6em¿X @- [0,-1] @- [0,-1] @- [0,-1]
*!R!¡.5em,0em¿=¡0em¿a *=¡1.8em,1.4em¿@- [0,-1] @= [0,-1] @= [0,-1] @= [0,-1] *!¡0em,.025em¿-=-¡.2em¿∙-1@= [-1,0]@= [0,-1]$w_{3}$

This is a program

$\text{comp }\ \stackrel{{\scriptstyle\text{def}}}{{=}}\
\lambda(C_{1},C_{2}).\ \textbf{box
}w_{1}\Rightarrow\big{(}w_{2}\leftarrow\textbf{unbox
}C_{1}w_{1};w_{3}\leftarrow\textbf{unbox }C_{2}w_{2};\textbf{output
}w_{3}\big{)}$

in the host language, associated with the type

$\text{comp}\ :\
\mathrm{Circ}(W_{1},W_{2})\times\mathrm{Circ}(W_{2},W_{3})\to\mathrm{Circ}(W_{1},W_{3})$
(4)

where $W_{i}$ is the type of the wire $w_{i}$ for $i\in\\{1,2,3\\}$.

Now, recall the idea of an enriched category, which is informally a category
such that the morphisms from $A$ to $B$ form an object of another category. In
Section LABEL:sec:cat-qwire, once we conceptualize types as objects and terms
as morphisms, we show that the embedding of the circuit language in the host
language is an instance of enriched category theory: the circuits (morphisms)
between wire types (objects) form a type (object) of the host language. The
host composition term in (4) is precisely composition in the sense of enriched
categories.

This enriched categorical framework is then incorporated in the definition of
categorical models of EWire (Definition LABEL:def:model-QWire), which is
associated to the following proposition, building on a partial normalization
procedure proposed in [qwire] for QWire expressions, which is shown to be
sound with respect to a semantics in completely positive maps [qwire, App.
A–B].

######  Proposition LABEL:prop:soundness (paraphrased).

The program equations that are used to justify the steps in the normalization
procedure in [qwire] are sound in any EWire model.

The idea of denoting a language for circuits within a symmetric monoidal
category is an established idea (e.g. [ghica-circuits]). The novel point here
is enrichment.

For a simple version of the model, wire types are understood as finite-
dimensional C*-algebras, and circuits are completely positive unital maps –
the accepted model of quantum computation. Host types are interpreted as sets,
and the type of all circuits is interpreted simply as the set of all circuits.
The category of sets supports higher order functions, which shows that it is
consistent for the host language to have higher order functions.

######  Proposition LABEL:prop:B2:fdcpu-model (paraphrased).

Taking finite-dimensional C*-algebras and completely positive unital maps as a
categorical model of the circuit language yields a categorical model of QWire
in which the types qubit and bit are respectively interpreted by the
C*-algebras $M_{2}$ and $\mathbb{C}\oplus\mathbb{C}$.

As with any higher order language, the set-theoretic model is not sufficient
to fully understand the nature of higher order functions. We envisage that
other semantic models (e.g. based on game semantics) will also fit the same
framework of enriched categories, so that our categorical framework provides a
sound description of the basic program equivalences that should hold in all
models. These equivalences play the same role that the $\beta$ and $\eta$
equivalences play in the pure lambda calculus. In other words, we are
developing a basic type theory for quantum computation.

### Linear/non-linear models

Quantum resources cannot be duplicated or discarded, whereas classical
resources can. This suggests a connection to linear logic [girard-LL], as many
others have observed [hasuo-hoshino, rios-selinger-qpl17, ross-phd-thesis,
malherbe-scott-selinger, lindenhovius-mislove-zamdzhiev, pagani-selinger-
valiron-popl14, geometry-parallelism]. In fact, enriched category theory is
closely related to linear logic through the following observation (see also
[enriched-effect-calculus, mellies-enriched-adjunctions, ms-state]).

######  Lemma LABEL:lemma:enriched-lnl (paraphrased).

The following data are equivalent:

  * •

A symmetric monoidal closed category enriched in a cartesian closed category,
with copowers.

  * •

A‘linear/non-linear’ (LNL) model in the sense of Benton [benton]: a symmetric
monoidal adjunction between a cartesian closed category and a symmetric
monoidal closed category.

Thus, from the semantic perspective, enriched category theory is a way to
study fragments and variations of linear logic. We show that every LNL model,
with some minor extra structure, induces an EWire model (see Proposition
LABEL:prop:Lwire-to-Ewire). We call these structures LNL-EWire models.

So LNL-EWire extends EWire with some additional connectives. However, although
these additional connectives have an established type theoretic syntax, their
meaning from the perspective of quantum physics is less clear at present. For
example, LNL assumes a first class linear function space, but categories of
completely positive unital maps (or CPTP maps) are typically not monoidal
closed. Nonetheless, there is a semantic way to understand their relevance,
through the following theorem, which is proved using a variation on Day’s
construction [day-convolution], following other recent work [rios-selinger-
qpl17, malherbe-scott-selinger, hearn-bunched-typing].

######  Theorem LABEL:th:Ewire-to-Lwire (paraphrased).

Every EWire model (with a locally presentable base) embeds in an LNL-EWire
model.

### Recursive types and recursive terms

Within our semantic model based on enriched categories, we can freely
accommodate various additional features in the host language, while keeping
the circuit language the same. For example, we could add recursion to the host
language, to model the idea of repeatedly trying quantum experiments, or
recursive types, to model arbitrary data types. This can be shown to be
consistent by modifying the simple model so that host types are interpreted as
directed complete partial orders (dcpo’s).

Many quantum algorithms are actually parameterized in the number of qubits
that they operate on. For example, the Quantum Fourier Transform (QFT) has a
uniform definition for any number of qubits by recursion, where $H$ is the
Hadamard gate $\frac{1}{\sqrt{2}}\left(\begin{smallmatrix}1&1\\\
1&-1\end{smallmatrix}\right)$ and $R_{n}$ is the $Z$ rotation gate
$\left(\begin{smallmatrix}1&0\\\ 0&e^{2\pi i/2^{n}}\end{smallmatrix}\right)$.

We formalize this by extending the circuit language with a wire type
$\mathrm{QList}$ of qubit-lists for which the following equivalence of types
holds:  $\mathrm{QList}\ \cong\ (\mathrm{qubit}\otimes\mathrm{QList})\oplus 1$
so that we can define a function
$\mathrm{fourier}:\mathrm{Circ}(\mathrm{QList},\mathrm{QList})$ In practice,
it will be useful for a circuit layout engine to know the number of qubits in
the lists, suggesting a dependent type such as
$\mathrm{fourier}:(n:\mathrm{Nat})\to\mathrm{Circ}(\mathrm{QList}(n),\mathrm{QList}(n))$
but we leave these kinds of elaboration of the type system to future work. The
categorical essence of recursive data types is algebraic compactness. In
short, one says that a category $\mathbf{C}$ is algebraically compact (for a
specific class of endofunctors) when every endofunctor
$F:\mathbf{C}\to\mathbf{C}$ has a canonical fixpoint, which is the initial
F-algebra [barr-algebraically]. In earlier work [rennela-mfps30], the first
author has shown that the category of W*-algebras is algebraically compact and
enriched in dcpo’s, and so this is a natural candidate for a semantics of the
language (Section LABEL:sec:ewire-domains). In brief: circuit types are
interpreted as W*-algebras, and circuits are interpreted as completely
positive sub-unital maps; host types are interpreted as dcpo’s; in particular
the collection of circuits $\mathrm{Circ}(W,W^{\prime})$ is interpreted as the
dcpo of completely positive sub-unital maps, with the Löwner order. In this
way, we provide a basic model for a quantum type theory with recursive types.
Relation to previous work. This paper expands and develops the paper [mfps]
presented at The Thirty-third Conference on the Mathematical Foundations of
Programming Semantics (MFPS XXXIII). This version includes numerous
elaborations, most notably Section LABEL:sec:LNL on linear/non-linear models
is entirely new.

##  1\. Functional programming and quantum circuits

We introduce a new calculus called EWire as a basis for analysing the basic
ideas of embedding a circuit language inside a host functional programming
language. EWire (for ‘embedded wire language’) is based on QWire [qwire]
(‘quantum wire language’), and we make the connection precise in Section
LABEL:sub:qwire. We discuss extensions of EWire in Section LABEL:sec:ewire-
domains. We assume two classes of basic wire types. • Classical wire types,
ranged over by $\mathbf{a},\mathbf{b},\dots$. The wire types exist in both the
circuit language and the host language. For example, the type of classical
bits, or Booleans. • Circuit-only wire types, ranged over by
$\alpha,\beta,\dots$. These wire types only exist in the circuit language. For
example, the type of qubits. From these basic types we build all wire types:
$W,W^{\prime}::=I\mid W\otimes
W^{\prime}~{}|~{}\mathbf{a}~{}|~{}\mathbf{b}~{}|~{}\alpha~{}|~{}\beta\dots$ We
isolate the classical wire types, which are the types not using any circuit-
only basic types: $V,V^{\prime}\ ::=\ I~{}|~{}V\otimes
V^{\prime}~{}|~{}\mathbf{a}~{}|~{}\mathbf{b}\dots$ We also assume a collection
$\mathcal{G}$ of basic gates, each assigned an input and an output wire type.
We write $\mathcal{G}(W_{\text{in}},W_{\text{out}})$ for the collection of
gates of input type $W_{\text{in}}$ and output type $W_{\text{out}}$. In
addition to the embedded circuit language, we consider a host language. This
is like Moggi’s monadic metalanguage [moggi] but with special types for the
classical wire types $\mathbf{a}$, $\mathbf{b}$ and a type
$\mathrm{Circ}(W,W^{\prime})$ of circuits for any wire types $W$ and
$W^{\prime}$. So the host types are $A,B\ ::=\ A\times B~{}|~{}1~{}|~{}A\to
B~{}|~{}T(A)~{}|~{}\mathrm{Circ}(W,W^{\prime})~{}|~{}\mathbf{a}~{}|~{}\mathbf{b}$
The monad $T$ is primarily to allow probabilistic computations, although one
might also add other side effects to the host language. Notice that every
classical wire type $V$ can be understood as a first order host type
$\left|V\right|$, according to the simple translation, called lifting:
$\left|V\otimes
V^{\prime}\right|\stackrel{{\scriptstyle\text{def}}}{{=}}\left|V\right|\times\left|V^{\prime}\right|\qquad\left|I\right|\stackrel{{\scriptstyle\text{def}}}{{=}}1\qquad\left|\mathbf{a}\right|\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbf{a}$

###  1.1. Circuit typing and host typing

To describe the well-formed terms, we consider two kinds of context: circuit
contexts $\Omega=(w_{1}:W_{1}\cdots w_{n}:W_{n})\qquad\text{ (for
}n\in\mathbb{N}\text{)}$ and host language contexts $\Gamma=(x_{1}:A_{1}\cdots
x_{m}:A_{m})\qquad\text{ (for }m\in\mathbb{N}\text{)}$ We will define a well-
formed circuit judgement $\Gamma;\Omega\vdash C:W$ and a well-formed host
language judgement $\Gamma\vdash t:A\text{.}$ Following Qwire [qwire], the
circuit language is based on patterns. Wires are organised in patterns given
by the grammar $p::=w\mid()\mid(p,p)$ associated to the following set of
rules, defining a ternary relation $\Omega\implies p:W$: ${tensy\vbox{\hbox
spread0.0pt{\hskip 0.0pt plus 0.0001fil\hbox{\kern
16.52768pt\hbox{$\displaystyle\penalty 1-$}}\hskip 0.0pt plus
0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=36.38869pt\hbox{}}}\hbox{\kern
0.0pt\hbox{$\displaystyle\cdot\implies():1$}}}}\qquad{tensy\vbox{\hbox
spread0.0pt{\hskip 0.0pt plus 0.0001fil\hbox{\kern
31.45586pt\hbox{$\displaystyle\penalty 1-$}}\hskip 0.0pt plus
0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=66.24504pt\hbox{}}}\hbox{\kern
0.0pt\hbox{$\displaystyle w:W\implies w:W$}}}}$ ${tensy\vbox{\hbox
spread0.0pt{\hskip 0.0pt plus 0.0001fil\hbox{$\displaystyle\penalty
1\Omega_{1}\implies p_{1}:W_{1}\qquad\Omega_{2}\implies p_{2}:W_{2}$}\hskip
0.0pt plus 0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=114.6401pt\hbox{}}}\hbox{\kern
3.19447pt\hbox{$\displaystyle\Omega_{1},\Omega_{2}\implies(p_{1},p_{2}):W_{1}\otimes
W_{2}$}}}}\qquad{tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus
0.0001fil\hbox{$\displaystyle\penalty
1\Omega_{1},w_{1}:W_{1},w_{2}:W_{2},\Omega_{2}\implies p:W$}\hskip 0.0pt plus
0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=126.68741pt\hbox{}}}\hbox{\kern
0.0pt\hbox{$\displaystyle\Omega_{1},w_{2}:W_{1},w_{1}:W_{2},\Omega_{2}\implies
p:W$}}}}$ Linear type theory for circuits. The first five term formation rules
are fairly standard for a linear type theory. These are the constructions for
sequencing circuits, one after another, and ending by outputting the wires,
for splitting a tensor-product type into its constituents, and the exchange
rule. The sixth rule includes the basic gates in the circuit language.
${tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus
0.0001fil\hbox{$\displaystyle\penalty 1\Gamma;\Omega_{1}\vdash
C_{1}:W_{1}\qquad\Omega\implies p:W_{1}\qquad\Gamma;\Omega,\Omega_{2}\vdash
C_{2}:W_{2}$}\hskip 0.0pt plus 0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=217.85579pt\hbox{}}}\hbox{\kern
54.11937pt\hbox{$\displaystyle\Gamma;\Omega_{1},\Omega_{2}\vdash p\leftarrow
C_{1};C_{2}:W_{2}$}}}}\qquad{tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus
0.0001fil\hbox{\kern 19.65285pt\hbox{$\displaystyle\penalty 1\Omega\implies
p:W$}}\hskip 0.0pt plus 0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=83.78114pt\hbox{}}}\hbox{\kern
0.0pt\hbox{$\displaystyle\Gamma;\Omega\vdash\textbf{output }p:W$}}}}$
${tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus
0.0001fil\hbox{$\displaystyle\penalty 1\Omega\implies
p:1\quad\Gamma;\Omega^{\prime}\vdash C:W$}\hskip 0.0pt plus
0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=101.23894pt\hbox{}}}\hbox{\kern
0.69444pt\hbox{$\displaystyle\Gamma;\Omega,\Omega^{\prime}\vdash()\leftarrow
p;C:W$}}}}\qquad{tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus
0.0001fil\hbox{$\displaystyle\penalty 1\Omega\implies p:W_{1}\otimes
W_{2}\quad\Gamma;w_{1}:W_{1},w_{2}:W_{2},\Omega^{\prime}\vdash C:W$}\hskip
0.0pt plus 0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=204.00632pt\hbox{}}}\hbox{\kern
39.62767pt\hbox{$\displaystyle\Gamma;\Omega,\Omega^{\prime}\vdash(w_{1},w_{2})\leftarrow
p;C:W$}}}}$ ${tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus
0.0001fil\hbox{$\displaystyle\penalty
1\Gamma;\Omega_{1},w_{1}:W_{1},w_{2}:W_{2},\Omega_{2}\vdash C:W$}\hskip 0.0pt
plus 0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=138.82419pt\hbox{}}}\hbox{\kern
0.0pt\hbox{$\displaystyle\Gamma;\Omega_{1},w_{2}:W_{2},w_{1}:W_{1},\Omega_{2}\vdash
C:W$}}}}$ ${tensy\vbox{\hbox spread0.0pt{\hskip 0.0pt plus
0.0001fil\hbox{$\displaystyle\penalty 1\Omega_{1}\implies
p_{1}:W_{1}\qquad\Omega_{2}\implies
p_{2}:W_{2}\qquad\Gamma;\Omega_{2},\Omega\vdash C:W$}\hskip 0.0pt plus
0.0001fil}\hbox{\hbox{\kern 0.0pt\vrule
height=0.25002pt,depth=0.25002pt,width=205.71901pt\hbox{\kern
3.00003pt${g\in\mathcal{G}(W_{1},W_{2})}$}}}\hbox{\kern
38.0357pt\hbox{$\displaystyle\Gamma;\Omega_{1},\Omega\vdash
p_{2}\leftarrow\textbf{gate }g\,p_{1};C:W$}}}}$
For example, qubit-based coin flipping is given by the following circuit:
$\text{flip}\stackrel{{\scriptstyle\text{def}}}{{=}}a\leftarrow\textbf{gate}~{}\mathrm{init}_{0}~{}();a^{\prime}\leftarrow\textbf{gate}~{}\mathrm{H}~{}a;b\leftarrow\textbf{gate}~{}\mathrm{meas}~{}a^{\prime};\textbf{output}~{}b$
(6) In quantum circuit notation, this would be notated:


# Dark matter detectors as a novel probe for light new physics

Anirban Majumdar<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Science Education and Research - Bhopal,
Bhopal Bypass Road, Bhauri, Bhopal 462066, India  D. K. Papoulias
<EMAIL_ADDRESS>Division of Theoretical Physics, University of Ioannina, GR
45110 Ioannina, Greece  Rahul Srivastava<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Science Education and Research - Bhopal,
Bhopal Bypass Road, Bhauri, Bhopal 462066, India

###### Abstract

We explore the prospect of constraining light mediators at the next generation
direct detection dark matter detectors through coherent elastic neutrino-
nucleus scattering (CE$\nu$NS) and elastic neutrino-electron scattering
(E$\nu$ES) measurements. Taking into account various details like the
quenching factor corrections, atomic binding effects, realistic backgrounds,
detection efficiency, energy resolution etc., we consider two representative
scenarios regarding detector specifications. For both scenarios, we obtain the
model-independent projected sensitivities for all possible Lorentz-invariant
interactions, namely scalar ($S$), pseudoscalar ($P$), vector ($V$), axial
vector ($A$) and tensor ($T$). For the case of vector interactions, we also
focus on two concrete examples: the well-known $U(1)_{B-L}$ and
$U(1)_{L_{\mu}-L_{\tau}}$ gauge symmetries. For all interaction channels
$X=\\{S,P,V,A,T\\}$, our results imply that the upcoming dark matter detectors
have the potential to place competitive constraints, improved by about 1 order
of magnitude compared to existing ones from dedicated CE$\nu$NS experiments,
XENON1T, beam dump experiments and collider probes.

##  1 Introduction

Although the Standard Model (SM) provides a rather successful description of
electroweak and strong interactions in nature, there is a list of shortcomings
that points to the need of new physics beyond the Standard Model (BSM).
Usually it is assumed that the scale of such new physics is larger than the
electroweak symmetry breaking scale and the new BSM particles are heavy. For
example, the existence of new neutral $Z^{\prime}$ gauge bosons under an extra
gauge symmetry is a common feature of many theories beyond the SM [1].
Extensive phenomenological studies at colliders [2] have been performed,
taking the $Z^{\prime}$ bosons to be heavier than electroweak scale.

However, there exist various motivated BSM extensions where the new physics is
desirable to be at the low scale. Light new physics, for example, can account
for the explanation of existing anomalies such as the longstanding anomalous
magnetic moment of muons [3, 4, 5]. Moreover, concerning the dark sector,
several dark matter models require the presence of light mediators to account
for dark matter self-interaction [6, 7] and/or the recent XENON1T anomaly [8,
9, 10]. Finally, the Peccei-Quinn solutions of the strong $CP$ problem also
imply the presence of a light Nambu-Goldstone boson called axion [11, 12, 13],
while neutrino mass models involving dynamical lepton number breaking often
lead to a light pseudoscalar boson called Majoron (Majorana neutrinos) or
Diracon (Dirac neutrinos) [14, 15].

While the implications of light vector and scalar mediators to the expected
signal rates had already been explored at direct detection dark matter
experiments [16, 17], after the recent observation of coherent elastic
neutrino-nucleus scattering (CE$\nu$NS) by the COHERENT experiment [18, 19,
20] there has been an intense interest for novel mediator investigations [21,
22, 23, 24, 25, 26, 27, 28]. In addition to COHERENT, data-driven constraints
also exist from upper limits on CE$\nu$NS, placed by the recent CONNIE [29]
and CONUS [30] measurements. Phenomenological studies focusing on various
$U^{\prime}(1)$ realizations such as the $L_{\mu}-L_{\tau}$ and $B-L$ gauge
symmetries have been explored using solar neutrinos [31, 32, 33], as well as
using supernova neutrinos taking also into account corrections from medium
effects [34]. It is interesting to note that the latter can induce a reduction
of the cross section due to Pauli blocking and therefore lead to less severe
constraints. Very recently, the expected modification to the neutrino floor
has been illustrated in the presence of vector and scalar mediators [35],
while the complementarity of CE$\nu$NS and direct detection dark matter
experiments was emphasized through a data-driven analysis of the neutrino
floor.

In addition to scalar and vector mediator scenarios, it is possible to explore
further Lorentz-invariant structures, such as scalar ($S$), pseudoscalar
($P$), vector ($V$), axial vector ($A$), or tensor ($T$) interactions in a
model-independent way through neutrino generalized interactions (NGIs).
Assuming heavy mediators, NGIs have been explored in Ref. [36] using CE$\nu$NS
data from COHERENT and in Ref. [37] through elastic neutrino-electron
scattering (E$\nu$ES) by analyzing the Borexino data. Reference [38] performed
a global NGI analysis in the light data from E$\nu$ES experiments, neutrino
deep inelastic scattering, and single-photon detection from electron-positron
collisions, while Ref. [39] pointed out the possibility of probing the Dirac
or Majorana neutrino nature through NGIs. Tensorial exotic interactions have
been explored in Ref. [40] using the TEXONO data as well as in Ref. [41] where
their connection to neutrino transition magnetic moments was emphasized.
Finally, constraints in the tensor parameter space were extracted from the
analysis of COHERENT data [42] and more recently by the CONUS Collaboration
[30].

The next generation direct detection dark matter experiments with multiton
mass scale and sub-keV capabilities are expected to become sensitive to
astrophysical neutrinos. Indeed, the XENON1T Collaboration has already
reported first results in its effort to identify a potential CE$\nu$NS
population–induced by 8B neutrinos of the solar flux–in the background data
[43]. Motivated by the upcoming large scale next generation direct detection
dark matter experiments such as XENONnT [44], in this work we consider the
possibility of exploring neutrino backgrounds using nuclear and electron
recoils. In particular, we are interested to explore the prospects of
constraining general neutrino interactions induced by novel mediators via
CE$\nu$NS and E$\nu$ES measurements at a future direct detection dark matter
experiment. Unlike previous studies–especially for the cases of pseudoscalar,
axial vector and tensor interaction channels–we allow the exchanged mediator
to be sufficiently light. Therefore, for all possible interaction channels,
the dependence on the mediator mass has been considered explicitly in the
cross sections. Our present results indicate that future dark matter detectors
will offer competitive constraints in the mass-coupling parameter space, being
also complementary to collider probes and DUNE.

The remainder of the paper has been organized as follows. In Sec. 2 we provide
the necessary formalism regarding CE$\nu$NS processes. We start with the
discussion of CE$\nu$NS within the SM and then discuss the new physics
contribution to it. In Sec. 3 we discuss the case of E$\nu$ES, again starting
with the SM discussion and ending with the new physics contribution. Our main
results are presented in Sec. 4 where we first discuss the model-independent
constraints on various types of new physics scenarios. We then, as examples,
take a few specific models with light mediators and further analyze our
constraints, comparing and contrasting them with constraints obtained from
other experimental probes. Our concluding remarks are summarized in Sec. 5.

##  2 Coherent Elastic Neutrino-Nucleus Scattering

In this section, we provide the basic formalism for the description of the
various CE$\nu$NS interaction channels considered in the present work, within
and beyond the SM. A model-independent analysis is performed by considering
all possible Lorentz-invariant interaction channels in the low-energy regime.
The latter can be conveniently classified in terms of the Lorentz and parity
transformations of the mediator particle i.e. whether the mediator transforms
as $S,P,V,A$, or $T$. One further advantage of such parametrization is that
interactions of any particle having a mixed transformation (e.g. SM $Z^{0}$
boson) can also be taken into account simply as a combination of $S,P,V,A,T$
interactions.

###  2.1 CE$\nu$NS within the SM

Assuming SM interactions only, for low and intermediate neutrino energies
$(E_{\nu}\ll M_{Z^{0}})$ CE$\nu$NS is accurately described in the context of
an effective four-fermion Fermi interaction Lagrangian [45]

$\mathscr{L}_{SM}=-2\sqrt{2}G_{F}\sum_{\begin{subarray}{c}f=u,d\\\
\alpha=e,\mu,\tau\end{subarray}}g_{\alpha,\alpha}^{f,P}\left[\bar{\nu}_{\alpha}\gamma^{\rho}L\nu_{\alpha}\right]\left[\bar{f}\gamma_{\rho}Pf\right]\,.$
(1)

Here, $P\equiv\\{L,R\\}$ stand for the chiral projection operators,
$f\equiv\\{u,d\\}$ represents the first generation quark and
$g_{\alpha,\alpha}^{f,P}$ is the $P$-handed coupling of the quark $f$ to the
SM $Z^{0}$ boson. The latter are expressed in terms of the weak mixing angle
$(\sin^{2}\theta_{W}=0.2387)$, as

$\displaystyle g_{\alpha,\alpha}^{u,L}$
$\displaystyle=\frac{1}{2}-\frac{2}{3}\sin^{2}\theta_{W}\,,\qquad~{}~{}g_{\alpha,\alpha}^{u,R}=-\frac{2}{3}\sin^{2}\theta_{W}\,,$
(2) $\displaystyle g_{\alpha,\alpha}^{d,L}$
$\displaystyle=-\frac{1}{2}+\frac{1}{3}\sin^{2}\theta_{W}\,,\qquad
g_{\alpha,\alpha}^{d,R}=\frac{1}{3}\sin^{2}\theta_{W}\,.$

At tree level111Subdominant radiative corrections are discussed in Ref. [46].,
the SM differential CE$\nu$NS cross section with respect to the nuclear recoil
energy $E_{nr}$ is given as [47]

$\left[\frac{d\sigma}{dE_{nr}}\right]_{SM}^{\nu
N}=\frac{G_{F}^{2}m_{N}}{\pi}\left({Q_{V}^{SM}}\right)^{2}\left(1-\frac{m_{N}E_{nr}}{2E_{\nu}^{2}}\right)\,,$
(3)

where $G_{F}$ is the Fermi constant and $m_{N}$ is the nuclear mass, while the
SM vector weak charge $Q_{V}^{SM}$ takes the form [48]

$Q_{V}^{SM}=\left[g_{p}^{V}Z+g_{n}^{V}N\right]F_{W}(q^{2})\,.$ (4)

Here, $Z$ and $N$ denote the number of protons and neutrons in the nucleus,
while the corresponding vector couplings for protons $(g_{p}^{V})$ and
neutrons $(g_{n}^{V})$ read [49]

$\displaystyle g_{p}^{V}=$ $\displaystyle
2\left(g_{\alpha,\alpha}^{u,L}+g_{\alpha,\alpha}^{u,R}\right)+\left(g_{\alpha,\alpha}^{d,L}+g_{\alpha,\alpha}^{d,R}\right)=1/2-2\sin^{2}{\theta_{W}}\,,$
(5) $\displaystyle g_{n}^{V}=$
$\displaystyle\left(g_{\alpha,\alpha}^{u,L}+g_{\alpha,\alpha}^{u,R}\right)+2\left(g_{\alpha,\alpha}^{d,L}+g_{\alpha,\alpha}^{d,R}\right)=-1/2\,.$

It is noteworthy that Eq.(3) is valid for sufficiently low momentum transfer
in order to satisfy the coherency condition $q\leq 1/R_{A}$ [48], with $R_{A}$
being the nuclear radius and $q^{2}=2m_{N}E_{nr}$ denoting the magnitude of
three-momentum transfer. Moreover, to account for the finite nuclear spatial
distribution, nuclear physics corrections are incorporated in Eq.(4) through
the weak nuclear form factor $F_{W}(q^{2})$. In our present study we adopt the
Helm parametrization, given as [50]

$F_{W}(q^{2})=\frac{3j_{1}(qR_{0})}{qR_{0}}\,e^{\left[-\frac{1}{2}(qs)^{2}\right]}\,,$
(6)

where $j_{1}(x)$ is the first-order spherical Bessel function, while the
diffraction radius is given by $R_{0}^{2}=\frac{5}{3}R^{2}-5s^{2}$ with the
nuclear radius and surface thickness taken to be $R=1.23\,A^{1/3}$ fm and
$s=0.9$ fm, respectively with $A=N+Z$ being the atomic mass number.

###  2.2 CE$\nu$NS contribution from light novel mediators

NGIs constitute a useful model-independent probe that can accommodate several
attractive BSM scenarios. By restricting ourselves to low-energy neutral-
current interactions (below the electroweak symmetry breaking scale), in this
work we consider general new physics interactions arising from the Lagrangian
[51, 36]

$\mathscr{L}_{NGI}=\frac{G_{F}}{\sqrt{2}}\sum_{\begin{subarray}{c}X=S,P,V,A,T\\\
f=u,d\\\
\alpha=e,\mu,\tau\end{subarray}}C_{\alpha,\alpha}^{f,P}\left[\bar{\nu}_{\alpha}\Gamma^{X}L\nu_{\alpha}\right]\left[\bar{f}\Gamma_{X}Pf\right]\,.$
(7)

Therefore, in what follows all possible Lorentz-invariant structures are taken
into account, i.e.
$\Gamma_{X}=\\{\mathbb{1},i\gamma_{5},\gamma_{\mu},\gamma_{\mu}\gamma_{5},\sigma_{\mu\nu}\\}$
(with $\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}])$, corresponding
to $X=\\{S,P,V,A,T\\}$ interactions, respectively. The dimensionless
coefficients $C_{\alpha,\alpha}^{f,P}$ measure the relative strength of the
new physics interaction $X$ and are of the order of
$(\sqrt{2}/G_{F})(g_{X}^{2}/(q^{2}+m_{X}^{2}))$ with $m_{X}$ and $g_{X}$ being
the mass of the exchanged light mediator and the corresponding coupling,
respectively. Throughout this work, we define the coupling $g_{X}=\sqrt{g_{\nu
X}g_{fX}}$; $f=\\{u,d\\}$ for CE$\nu$NS and $f=e$ for E$\nu$ES.

We proceed by relying on previous analyses [52, 53] that, in the limit of
vanishing momentum transfer, argued that the nucleonic matrix element of the
quark current is proportional to that of the corresponding nucleon current
$\langle N_{f}|\bar{q}\Gamma^{X}q|N_{i}\rangle\equiv\mathcal{F}^{X}(q)\langle
N_{f}|\bar{N}\Gamma^{X}N|N_{i}\rangle$, where $\mathcal{F}^{X}(q)$ is the form
factor calculated within the framework of nonperturbative low-energy QCD. The
individual interactions and their contribution to the CE$\nu$NS differential
cross section are listed in Table 1. For the different interactions $X$, the
relevant effective charges $Q_{X}$ entering the respective cross sections are
given by [53, 16, 54, 40]

$\displaystyle\begin{aligned} Q_{s}=&g_{\nu
S}\left(Z\sum_{q}g_{qS}\frac{m_{p}}{m_{q}}f^{p}_{q}+N\sum_{q}g_{qS}\frac{m_{n}}{m_{q}}f^{n}_{q}\right)F_{W}(q^{2})\,,\\\
Q_{P}=&g_{\nu
P}\left(Z\sum_{q}g_{qP}\frac{m_{p}}{m_{q}}h^{p}_{q}+N\sum_{q}g_{qP}\frac{m_{n}}{m_{q}}h^{n}_{q}\right)F_{W}(q^{2})\,,\\\
Q_{V}=&g_{\nu
V}\left[\left(2g_{uV}+g_{dV}\right)Z+\left(g_{uV}+2g_{dV}\right)N\right]F_{W}(q^{2})\,,\\\
Q_{A}=&g_{\nu
A}\left(ZS_{p}\sum_{q}g_{qA}\Delta^{p}_{q}+NS_{n}\sum_{q}g_{qA}\Delta^{n}_{q}\right)F_{W}(q^{2})\,,\\\
Q_{T}=&g_{\nu
T}\left(Z\sum_{q}g_{qT}\delta^{p}_{q}+N\sum_{q}g_{qT}\delta^{n}_{q}\right)F_{W}(q^{2})\,.\end{aligned}$
(8)

In the latter expressions, the hadronic structure parameters for the scalar
case ($S$) are $f_{u}^{p}=0.0208$, $f_{u}^{n}=0.0189$, $f_{d}^{p}=0.0411$, and
$f_{d}^{n}=0.0451$ [55], for the pseudoscalar case ($P$) are
$h^{p}_{u}=h^{n}_{u}=1.65$ and $h^{p}_{d}=h^{n}_{d}=0.375$ [56], and for the
tensor case ($T$) are $\delta^{p}_{u}=\delta^{n}_{d}=0.54$ and
$\delta^{p}_{d}=\delta^{n}_{u}=-0.23$ [55]. Finally, for the case of axial
vector ($A$) interactions the hadronic parameters read
$\Delta^{p}_{u}=\Delta^{n}_{u}=0.842$ and
$\Delta^{p}_{d}=\Delta^{n}_{d}=-0.427$ [57], while the spin expectation values
$(S_{p}\text{ and }S_{n})$ are nuclear model and isotope dependent. In the
present study we consider the following spin expectation values for protons
$(\mathrm{{}^{129}Xe}:S_{p}=0.010,~{}\mathrm{{}^{131}Xe}:S_{p}=-0.009)$ and
neutrons
$(\mathrm{{}^{129}Xe}:S_{n}=0.329,~{}\mathrm{{}^{131}Xe}:S_{n}=-0.272)$,
extracted from shell model nuclear structure calculations in Ref. [58]222Even-
even Xenon isotopes with $0^{+}$ ground state do not contribute to axial
vector interactions because of angular momentum conservation..

Mediator $\mathscr{L}_{X}$ Cross section Scalar $\left[\left(g_{\nu
S}\bar{\nu}_{R}\nu_{L}+\text{
{H.c.}}\right)+\sum_{q=\\{u,d\\}}g_{qS}\bar{q}q\right]S+\frac{1}{2}m_{S}^{2}S^{2}$
$\frac{m_{N}^{2}E_{nr}Q_{S}^{2}}{4\pi
E_{\nu}^{2}\left(q^{2}+m_{S}^{2}\right)^{2}}$ Pseudoscalar $\left[\left(g_{\nu
P}\bar{\nu}_{R}\gamma_{5}\nu_{L}+\text{
{H.c.}}\right)-i\sum_{q=\\{u,d\\}}g_{qP}\bar{q}\gamma_{5}q\right]P+\frac{1}{2}m_{P}^{2}P^{2}$
$\frac{m_{N}E_{nr}^{2}Q_{P}^{2}}{8\pi
E_{\nu}^{2}\left(q^{2}+m_{P}^{2}\right)^{2}}$ Vector $\left[g_{\nu
V}\bar{\nu}_{L}\gamma_{\mu}\nu_{L}+\sum_{q=\\{u,d\\}}g_{qV}\bar{q}\gamma_{\mu}q\right]V^{\mu}+\frac{1}{2}m_{V}^{2}V^{\mu}V_{\mu}$
$\left(1+\frac{Q_{V}}{\sqrt{2}G_{F}Q_{V}^{SM}\left(q^{2}+m_{V}^{2}\right)}\right)^{2}\left[\frac{d\sigma}{dE_{nr}}\right]_{SM}^{\nu
N}$ Axial vector $\left[g_{\nu
A}\bar{\nu}_{L}\gamma_{\mu}\gamma_{5}\nu_{L}-\sum_{q=\\{u,d\\}}g_{qA}\bar{q}\gamma_{\mu}\gamma_{5}q\right]A^{\mu}+\frac{1}{2}m_{A}^{2}A^{\mu}A_{\mu}$
$\frac{m_{N}Q_{A}^{2}(2E_{\nu}^{2}+m_{N}E_{nr})}{4\pi
E_{\nu}^{2}\left(q^{2}+m_{A}^{2}\right)^{2}}$ Tensor $\left[g_{\nu
T}\bar{\nu}_{R}\sigma_{\rho\delta}\nu_{L}-\sum_{q=\\{u,d\\}}g_{qT}\bar{q}\sigma_{\rho\delta}q\right]T^{\rho\delta}+\frac{1}{2}m_{T}^{2}T^{\rho\delta}T_{\rho\delta}$
$\frac{m_{N}Q_{T}^{2}(4E_{\nu}^{2}-m_{N}E_{nr})}{2\pi
E_{\nu}^{2}\left(q^{2}+m_{T}^{2}\right)^{2}}$

Table 1: Novel interactions $X=\\{S,P,V,A,T\\}$ and corresponding differential
CE$\nu$NS cross sections considered in the present work. Because of
interference, the $V$ interaction is the only case that includes the SM
contribution, while the $S,P,A,T$ cases acquire contributions from new physics
only, as shown in the cross sections (see the text for more details).

A few comments are in order. First, from Table 1 it becomes evident that for
the case of $S,P,A,T$ interactions there is an absence of interference with
the SM CE$\nu$NS cross section, in contrast to the vector mediator case where
the new physics contribution yields interference terms. For the sake of
clarity we should stress that, in principle, the axial vector interaction adds
incoherently to the SM CE$\nu$NS cross section, however the SM axial
contribution is significantly suppressed with respect to the vector one and
therefore neglected (see Ref. [42]). Moreover, as discussed in Ref. [36], the
existence of axial quark terms in the interaction Lagrangian (7) will lead to
pseudoscalar-scalar neutrino-quark couplings, the study of which will be
relaxed as it goes well beyond the scope of our work. Finally, the $P$ and $A$
terms are spin-dependent and therefore suppressed with respect to $S,V,T$
terms333In Ref. [16] the $P$ contribution to CE$\nu$NS was reported to be
vanishing based on the fact that the corresponding nucleon matrix element is
vanishing [59, 60].. For the latter issue, we will rely on the assumption that
the new physics interaction is controlled by the strength of the neutrino
coupling.

Focusing on light vector mediators only, in this work we will consider three
different models, determined by the charges of leptons $Q^{\prime}_{\ell}$ and
quarks $Q^{\prime}_{q}$ under the extra gauge symmetry ${U}(1)^{\prime}$ [61].
First, we consider a generic model where the new vector mediator $V$ couples
universally to all SM fermions, such that
$Q^{\prime}_{\ell}=Q^{\prime}_{q}=1$, corresponding to an effective vector
charge in Eq.(8) with $g_{qV}=g_{\nu V}$. Second, we focus on the $U_{B-L}$
extension of the SM, in which the anomaly cancellation conditions
$Q^{\prime}_{\ell}=-1,~{}Q^{\prime}_{q}=1/3$ imply an effective vector charge
with $g_{qV}=-g_{\nu V}/3$. Finally, we consider the gauged
$U_{L_{\mu}-L_{\tau}}$ symmetry, where the new vector mediator boson couples
directly only to muons and tauons with the absence of tree-level couplings to
quarks. In this model, contributions to CE$\nu$NS are possible at the one-loop
level, leading to an “effective” kinetic mixing between the new mediator and
the SM photon [62], and the corresponding cross section reads [28, 63]

$\displaystyle\begin{split}\left[\frac{d\sigma}{dE_{nr}}\right]_{L_{\mu}-L_{\tau}}^{\nu
N}=&\left(1\pm\frac{\alpha_{em}g_{\nu
V}g_{qV}\log\left(\frac{m_{\tau}^{2}}{m_{\mu}^{2}}\right)ZF_{W}(q^{2})}{3\sqrt{2}\pi
G_{F}Q_{V}^{SM}\left(q^{2}+m_{V}^{2}\right)}\right)^{2}\left[\frac{d\sigma}{dE_{nr}}\right]_{SM}^{\nu
N}\,,\end{split}$ (9)

where $\alpha_{em}$ is the fine structure constant, and $m_{\mu}$ and
$m_{\tau}$ denote the muon and tau masses, respectively444For a study focusing
on models with an extra $U_{B-2L_{\alpha}-L_{\beta}}$ and $U_{B-3L_{\alpha}}$
gauge symmetry, see Ref. [64]., while the plus (minus) sign accounts for
$\nu_{\mu}$-nucleus ($\nu_{\tau}$-nucleus) scattering.

##  3 Elastic Neutrino-Electron Scattering

Proceeding in an analogous way as in the case of CE$\nu$NS, in this section we
discuss the formalism describing E$\nu$ES within and beyond the SM.

###  3.1 E$\nu$ES through SM interaction Channel

E$\nu$ES is a well-understood weak flavored process where ( anti)neutrinos of
flavor $\alpha=\\{e,\mu,\tau\\}$ interact with electrons via elastic
scattering at low and intermediate energies, receiving contributions from both
neutral and charged currents for $\alpha=e$, and neutral current only for
$\alpha=\\{\mu,\tau\\}$. Within the framework of the SM, the Lagrangian
density corresponding to the process
$\overset{\scriptscriptstyle(-)}{\nu_{e}}+e^{-}\rightarrow\overset{\scriptscriptstyle(-)}{\nu_{e}}+e^{-}$,
reads555For the case of $\nu_{\mu,\tau}$–$e^{-}$ E$\nu$ES the Lagrangian
involves only neutral-current terms. [65]

$\displaystyle\begin{split}\mathscr{L}_{SM}(\overset{\scriptscriptstyle(-)}{\nu_{e}}+e^{-}\rightarrow\overset{\scriptscriptstyle(-)}{\nu_{e}}+e^{-})=&-\frac{G_{F}}{\sqrt{2}}\\{\left[\bar{\nu}_{e}\gamma^{\rho}\left(1-\gamma_{5}\right)e\right]\left[\bar{e}\gamma_{\rho}\left(1-\gamma_{5}\right)\nu_{e}\right]\\\
&+\left[\bar{\nu}_{e}\gamma^{\rho}\left(1-\gamma_{5}\right)\nu_{e}\right]\left[\bar{e}\gamma_{\rho}\left(g_{V}-g_{A}\gamma_{5}\right)e\right]\\}\,,\end{split}$
(10)

while at the tree level the corresponding differential cross section with
respect to the electron recoil energy $E_{er}$, is given as [66]

$\displaystyle\begin{split}\left[\frac{d\sigma_{\nu_{\alpha}}}{dE_{er}}\right]_{SM}^{\nu
e}=&\frac{G_{F}^{2}m_{e}}{2\pi}[(g_{V}+g_{A})^{2}+(g_{V}-g_{A})^{2}\left(1-\frac{E_{er}}{E_{\nu}}\right)^{2}-(g_{V}^{2}-g_{A}^{2})\frac{m_{e}E_{er}}{E_{\nu}^{2}}]\,.\end{split}$
(11)

In Eq.(11), $g_{V}$ and $g_{A}$ are vector and axial vector couplings
respectively, and take the form

$g_{V}=-\frac{1}{2}+2\sin^{2}\theta_{W}+\delta_{\alpha e},\qquad
g_{A}=-\frac{1}{2}+\delta_{\alpha e}\,,$ (12)

where the $\delta_{\alpha e}$ term is the Kronecker delta, which becomes
nonzero only for $\nu_{e}$–$e^{-}$ interactions. For the case of antineutrino
scattering, the E$\nu$ES cross section is given by Eq.(11) with the
substitution $g_{A}\to-g_{A}$.

###  3.2 E$\nu$ES through light novel mediators

Similar to the CE$\nu$NS case, here also we consider all possible E$\nu$ES
contributions arising from the Lorentz-invariant forms $X=\\{S,P,V,A,T\\}$.
For light vector and axial vector novel mediators, the total differential
cross sections can be achieved by replacing $g_{V}$ and $g_{A}$ from the SM
cross section as [67, 68]

$g^{\prime}_{V/A}=g_{V/A}+\frac{g_{\nu V/A}\cdot
g_{eV/A}}{4\sqrt{2}G_{F}(2m_{e}E_{er}+m_{V/A}^{2})}\,.$ (13)

It is interesting to notice that in the $U_{B-L}$ gauge extension the cross
section is also given by Eq.(13) since
$Q^{\prime}_{\ell}=Q^{\prime}_{\nu}=-1$, while in the universal case it holds
that $Q^{\prime}_{\ell}=Q^{\prime}_{\nu}=1$, hence both cases leading to
$g_{\nu V/A}=g_{eV/A}$ in Eq.(13). On the other hand, under
$U_{L_{\mu}-L_{\tau}}$ symmetry, the new light vector mediator contributes at
one-loop level for $\sigma_{\nu_{\mu}-e}$ and $\sigma_{\nu_{\tau}-e}$ only,
while $\sigma_{\nu_{e}-e}$ is vanishing666 Nonzero couplings contribute to
$\sigma_{\nu_{e}-e}$ at two-loop level through $Z_{0}-V$ mixing, which are
neglected in this work.. The relevant couplings read [62]

$g^{\prime}_{V}=g_{V}\pm\frac{\alpha_{em}}{3\sqrt{2}\pi
G_{F}}\log{\left(\frac{m^{2}_{\tau}}{m_{\mu}^{2}}\right)}\frac{g_{\nu V}\cdot
g_{eV}}{(2m_{e}E_{er}+m_{V}^{2})}\,,$ (14)

where, unlike the CE$\nu$NS case, the plus (minus) sign corresponds to
$\sigma_{\nu_{\tau}-e}$ $(\sigma_{\nu_{\mu}-e})$ scattering. The remaining
$S$, $P$, and $T$ cross section contributions add incoherently to the SM cross
section and have been previously written as [10, 69]

$\left[\frac{d\sigma_{\nu_{\alpha}}}{dE_{er}}\right]_{S}^{\nu
e}=\left[\frac{g^{2}_{\nu S}\cdot
g^{2}_{eS}}{4\pi(2m_{e}E_{er}+m_{S}^{2})^{2}}\right]\frac{m_{e}^{2}E_{er}}{E_{\nu}^{2}}\,,$
(15) $\left[\frac{d\sigma_{\nu_{\alpha}}}{dE_{er}}\right]_{P}^{\nu
e}=\left[\frac{g^{2}_{\nu P}\cdot
g^{2}_{eP}}{8\pi(2m_{e}E_{er}+m_{P}^{2})^{2}}\right]\frac{m_{e}E_{er}^{2}}{E_{\nu}^{2}}\,,$
(16) $\left[\frac{d\sigma_{\nu_{\alpha}}}{dE_{er}}\right]_{T}^{\nu
e}=\frac{m_{e}\cdot g^{2}_{\nu T}\cdot
g^{2}_{eT}}{2\pi(2m_{e}E_{er}+m_{T}^{2})^{2}}\cdot\left[1+2\left(1-\frac{E_{er}}{E_{\nu}}\right)+\left(1-\frac{E_{er}}{E_{\nu}}\right)^{2}-\frac{m_{e}E_{er}}{E_{\nu}^{2}}\right]\,.$
(17)

##  4 Results

###  4.1 CE$\nu$NS and E$\nu$ES events at direct detection dark matter
detectors

In our analysis we consider astrophysical neutrinos coming from the Sun [70],
the atmosphere [71], and diffuse supernovae (DSN) [72]. In our calculations we
have neglected the neutrino contributions coming from Earth, known as
geoneutrinos, as their induced events are expected to be overshadowed by
several orders of magnitude with respect to solar neutrinos [73, 74, 75]. For
the normalization of the different neutrino fluxes we use the recommended
conventions reported in Ref. [76] which are listed in Table 2.

Component |  $E^{max}_{\nu}$ (MeV) | Flux ($\mathrm{cm^{-2}s^{-1}}$)
---|---|---
$pp$ | $0.42341$ | $5.98\times 10^{10}$
$pep$ | $1.44$ | $1.44\times 10^{8}$
${}^{7}\mathrm{Be_{\text{High}}}$ | $0.8613$ | $4.35\times 10^{9}$
${}^{7}\mathrm{Be_{\text{Low}}}$ | $0.3843$ | $4.84\times 10^{8}$
8B | $16.36$ | $5.25\times 10^{6}$
$hep$ | $18.784$ | $7.98\times 10^{3}$
13N | $1.199$ | $2.78\times 10^{8}$
15O | $1.732$ | $2.05\times 10^{8}$
17F | $1.74$ | $5.29\times 10^{6}$
Atm. | $981.75$ | $10.5$
DSN | $91.201$ | $86$
Table 2: Neutrino end point energy and flux normalization for the different
astrophysical neutrino sources. The flux normalizations are taken from Ref.
[76].

For the interaction channel $X$, the differential event rate of CE$\nu$NS at a
given detector follows from the convolution of the differential cross section
with the neutrino energy distribution $d\Phi/dE_{\nu}$, as777An ideal detector
with perfect efficiency and resolution power is assumed. [48]

$\left[\frac{dR}{dE_{nr}}\right]_{X}^{\nu
N}=t_{run}N_{target}\mathcal{A}(E_{nr})\sum_{i}\int_{E_{\nu}^{min}}^{E_{\nu}^{max}}dE_{\nu}\,\frac{d\Phi_{i}(E_{\nu})}{dE_{\nu}}\left[\frac{d\sigma}{dE_{nr}}(E_{\nu},E_{nr})\right]_{X}^{\nu
N}\,,$ (18)

where $\mathcal{A}(E_{nr})$ is the detection efficiency, $t_{run}$ denotes the
exposure time and $N_{target}$ represents the number of target nuclei. In the
latter expression, the index $i$ runs over all the neutrino sources with
energy distribution $d\Phi_{i}/dE_{\nu}$ and $E_{\nu}^{max}$ denotes the
maximum neutrino energy of the $i$th source (see Table 2). Finally, the
minimum neutrino energy $E_{\nu}^{min}$ required to generate a nuclear recoil
with energy $E_{nr}$ is trivially obtained from the kinematics of the process
and reads

$E_{\nu}^{min}=\frac{1}{2}\left[E_{nr}+\sqrt{E_{nr}^{2}+2m_{N}E_{nr}}\right]\approx\sqrt{\frac{m_{N}E_{nr}}{2}}\,.$
(19)

In our analysis we also consider corrections from detector-specific
quantities. In particular, for the ionization xenon detectors considered here,
a significant amount of nuclear recoil energy $E_{nr}$ is lost into heat and
other dissipative energies, so that the actual energy measured by the detector
is an _electron equivalent_ energy $E_{er}$ [77]. In our calculation this
effect is taken into account through the quenching factor
$\mathcal{Q}_{f}(E_{nr})$, calculated on the basis of theoretical predictions
within Lindhard theory [78]

$\frac{E_{er}}{E_{nr}}=\mathcal{Q}_{f}{(E_{nr})}=\frac{kg(\gamma)}{1+kg(\gamma)}\,,$
(20)

where $g(\gamma)=3\gamma^{0.15}+0.7\gamma^{0.6}+\gamma$, with
$\gamma=11.5\cdot Z^{-\frac{7}{3}}E_{nr}(\mathrm{keV_{nr}})$ and $k=0.133\cdot
Z^{\frac{2}{3}}A^{-\frac{1}{2}}$. Let us stress that for the case of
germanium, low-energy corrections to the quenching factor can been accounted
for through the adiabatic correction of Lindhard theory, introduced in Ref.
[79]888Because of the lack of data for the xenon isotope we are interested in
this work, we have verified that our results remain unaffected even when
considering $\pm 1\sigma$ deviations from the adiabatic parameter
corresponding to germanium (see Table I of Ref. [79]). Improved quenching
factors at sub-keV energies are comprehensively discussed in Ref. [80].. After
incorporating the quenching factor corrections, the number of events in the
$j$th bin is written as

$\left[R_{j}\right]_{X}^{\nu
N}=\int_{E_{er}^{j}}^{E_{er}^{j+1}}dE_{er}\,\left[\frac{dR}{dE_{nr}}\right]_{X}^{\nu
N}\frac{1}{\mathcal{Q}_{f}}\left(1-\frac{E_{er}}{{\mathcal{Q}_{f}}}\frac{d\mathcal{Q}_{f}}{dE_{er}}\right)\,,$
(21)

At this point we turn our attention to E$\nu$ES. For the case of solar
neutrinos the differential number of events takes into account the effect of
neutrino oscillations in propagation and is given according to the expression
[81]

$\left[\frac{dR}{dE_{er}}\right]_{X}^{\nu
e}=t_{run}N_{target}\mathcal{A}(E_{er})\sum_{i=\mathrm{solar}}\int_{E_{\nu}^{min}}^{E_{\nu}^{max}}dE_{\nu}\frac{d\Phi^{\nu_{e}}_{i}(E_{\nu})}{dE_{\nu}}\left(P_{ee}{\left[\frac{d\sigma_{\nu_{e}}}{dE_{er}}\right]_{X}^{\nu
e}}+\overline{P_{ef}}{\left[\frac{d\sigma_{\nu_{f}}}{dE_{er}}\right]_{X}^{\nu
e}}\right)\,,$ (22)

with $f=\mu,\tau$ and the minimum neutrino energy being

$E_{\nu}^{min}=\frac{1}{2}\left[E_{er}+\sqrt{E_{er}^{2}+2m_{e}E_{er}}\right]\,.$
(23)

Since the E$\nu$ES cross section is not flavor blind as the CE$\nu$NS one,
Eq.(22) incorporates neutrino oscillations by weighting the flavored cross
section with the corresponding oscillation probability. For our purposes it is
sufficient to consider the averaged oscillation probability $P_{ee}(E_{\nu})$
in the two-flavor approximation, which we take from [82]. Notice that in the
second term of Eq.(22) the oscillation factors read
$\overline{P_{e\mu}}\equiv(1-P_{ee})\cos^{2}\theta_{23}$ and
$\overline{P_{e\tau}}\equiv(1-P_{ee})\sin^{2}\theta_{23}$ for incoming
$\nu_{\mu}$ and $\nu_{\tau}$ neutrinos, respectively999The value of the
atmospheric mixing angle ($\theta_{23}$) is taken from the best fit of the
2020 Valencia global-fit [83].. Let us note that for the case of atmospheric
and DSN neutrinos, oscillation effects are neglected. This is a reasonable
approximation given the fact that the expected E$\nu$ES event rates from
atmospheric and DSN neutrinos are suppressed by several orders of magnitude
compared to the dominant solar ones (see the discussion below). Therefore in
our statistical analysis this approximation is not expected to have any
quantitative effect. The differential number of E$\nu$ES events relevant to
atmospheric and DSN neutrinos is then calculated as

$\left[\frac{dR}{dE_{er}}\right]_{X}^{\nu
e}=t_{run}N_{target}\mathcal{A}(E_{er})\sum_{i=\mathrm{atm,DSN}}\int_{E_{\nu}^{min}}^{E_{\nu}^{max}}dE_{\nu}\frac{d\Phi_{i}^{\nu_{\alpha}}(E_{\nu})}{dE_{\nu}}\left[\frac{d\sigma_{\nu_{\alpha}}}{dE_{er}}\right]_{X}^{\nu
e}\,,\quad\nu_{\alpha}=\\{\nu_{e},\bar{\nu}_{e},\nu_{\mu,\tau},\bar{\nu}_{\mu,\tau}\\}\,.$
(24)

State |  SP (eV) |  State |  SP (eV) |  State |  SP (eV) |  State |  SP (eV)
---|---|---|---|---|---|---|---
${}^{1}s_{\frac{1}{2}}$ | $34759.3$ | ${}^{3}p_{\frac{3}{2}}$ | $1024.8$ | ${}^{4}p_{\frac{3}{2}}$ | $708.1$ | ${}^{5}p_{\frac{1}{2}}$ | $13.4$
${}^{2}s_{\frac{1}{2}}$ | $5509.8$ | ${}^{3}p_{\frac{1}{2}}$ | $961.2$ | ${}^{4}p_{\frac{1}{2}}$ | $162.8$ | ${}^{5}p_{\frac{3}{2}}$ | $12.0$
${}^{2}p_{\frac{3}{2}}$ | $5161.5$ | ${}^{3}d_{\frac{5}{2}}$ | $708.1$ | ${}^{4}d_{\frac{5}{2}}$ | $73.8$ |  |
${}^{2}p_{\frac{1}{2}}$ | $4835.6$ | ${}^{3}d_{\frac{3}{2}}$ | $694.9$ | ${}^{4}d_{\frac{3}{2}}$ | $71.7$ |  |
${}^{3}s_{\frac{1}{2}}$ | $1170.5$ | ${}^{4}s_{\frac{1}{2}}$ | $229.4$ | ${}^{5}s_{\frac{1}{2}}$ | $27.5$ |  |
Table 3:  Single-particle (SP) energies for the xenon atom derived from
Hartree-Fock calculations in Ref. [84].

Up to this point, for all E$\nu$ES interaction channels within and beyond the
SM, our discussion applies to the case where neutrinos scatter off free
electrons. However, the target electrons are not free, but rather bounded
inside the atom. Therefore, in order to perform realistic simulations of the
expected number of events at a given detector, atomic binding effects should
be also considered. To account for binding effects in our analysis, the free
E$\nu$ES cross section $(d\sigma_{\nu_{\alpha}}/dE_{er})_{\text{free}}$
defined in Eqs.(11)-(17) is weighted by a series of step functions introduced
in Ref. [84] as follows

$\left[\frac{d\sigma_{\nu_{\alpha}}}{dE_{er}}\right]_{X}^{\nu
e}=\frac{1}{Z}\sum_{i=1}^{Z}\Theta(E_{er}-B_{i})\,\left(\left[\frac{d\sigma_{\nu_{\alpha}}}{dE_{er}}\right]_{X}^{\nu
e}\right)_{\text{free}}\,.$ (25)

Here, $\Theta(x)$ is the Heaviside step function, while the quantity
$\sum_{i=1}^{Z}\Theta(E_{er}-B_{i})$ quantifies the number of electrons that
can be ionized by the recoil energy $E_{er}$ and $B_{i}$ represents the
binding energy of the $i$th electron in the atom. The single-particle atomic
level binding energy of electrons in a 131Xe atom is given in Table 3 [84].

---

(a) Differential event spectra for CE$\nu$NS (left) and E$\nu$ES (right)
processes.

(b) Integrated event spectra for CE$\nu$NS (left) and E$\nu$ES (right)
processes.

Figure 1: Differential (a) and integrated (b) event spectra expected at a
xenon target for CE$\nu$NS and E$\nu$ES processes as a function of the recoil
energy (threshold recoil energy) measured in $\mathrm{keV_{ee}}$ units. The
contributions from the different new physics interactions $X$ are calculated
assuming the benchmark values $g^{2}_{X}=10^{-4}$ and $m_{X}=1$ GeV (for
details see the text).

Assuming typical benchmark values $g_{X}^{2}=10^{-4}$ and $m_{X}=1$ GeV,
differential and integrated CE$\nu$NS (E$\nu$ES) event rates above threshold
for a xenon target (assuming average isotopic abundance) are shown in the left
(right) panel of Fig. 1 as functions of the recoil energy for all possible
interactions $X$. It is interesting to notice that, in general, the CE$\nu$NS
number of events are expected to exceed the E$\nu$ES rate by several orders of
magnitude. However, for a realistic next generation experiment operating with
typical threshold energies i.e. $E_{er}>0.1~{}\mathrm{keV_{ee}}$, this is not
strictly true. First, for $E_{er}>0.1~{}\mathrm{keV_{ee}}$ it becomes evident
that CE$\nu$NS-induced events will be dominated by 8B neutrinos while the
E$\nu$ES ones will be dominated by $pp$ neutrinos. Then, from a closer
inspection of the graphs it can be deduced that CE$\nu$NS and E$\nu$ES are
expected to generate a comparable number of events (at least) within the SM.
This can be understood as follows. Although the CE$\nu$NS cross section scales
with $\sim N^{2}$ as a consequence of the coherency in neutrino-nucleus
scattering, the E$\nu$ES-induced number of events is enhanced by an overall
multiplicative factor $Z$ coming due to the number of electron targets, while
the remaining difference, which is of the order of an overall factor $\sim N$,
is compensated by the relative difference of 8B and $pp$ flux normalization.
Before closing this discussion, let us note the impact of nuclear (atomic)
effects in the CE$\nu$NS (E$\nu$ES) rates. In particular, as can be seen from
the CE$\nu$NS spectral rates, the various dips occurring for
$E_{er}>25~{}\mathrm{keV_{ee}}$ are due to the minima of the nuclear form
factor (see Fig. 3 of Ref. [85]), while the slight suppression of the E$\nu$ES
rates occurring for recoil energies below $\sim 35~{}\mathrm{keV_{ee}}$
reflects the atomic binding effects (see Table 3).

###  4.2 Sensitivity on $S,P,V,A,T$ interactions

At this point we are interested to explore the projected sensitivities on the
($m_{X},g_{X}$) parameter space at a typical next generation experiment
looking for direct detection of dark matter. For estimating the projected
sensitivities of the next generation experiments to the various new
interaction channels $X=\\{S,P,V,A,T\\}$, we perform a spectral fit in terms
of the $\chi^{2}$ function [86]

$\displaystyle\chi^{2}(\xi,m_{X},g_{X})=2\sum_{i=1}^{n_{\mathrm{bins}}}\Biggl{[}R_{\mathrm{pred}}^{i}(\xi,m_{X},g_{X})-R_{\mathrm{exp}}^{i}$
$\displaystyle+R_{\mathrm{exp}}^{i}\ln\left(\frac{R_{\mathrm{exp}}^{i}}{R_{\mathrm{pred}}^{i}(\xi,m_{X},g_{X})}\right)\Biggr{]}+\left(\frac{\xi}{\sigma_{\text{sys}}}\right)^{2}\,.$
(26)

where $\xi$ denotes a nuisance parameter to account for the various systematic
uncertainties. The expected number of events in the $i$th bin is defined as
$R_{\mathrm{exp}}^{i}=R_{\text{SM}}^{i}+R_{\text{bg}}^{i}$ with
$R_{\text{bg}}^{i}$ being the number of background events, while the predicted
number of events in the presence of the interaction $X$ is
$R_{\mathrm{pred}}^{i}=(1+\xi)\left[R_{\text{X+SM}}^{i}(m_{X},g_{X})+R_{\text{bg}}^{i}\right]$.
At this point we should stress that in our E$\nu$ES-based statistical
analysis, the atmospheric and DSN fluxes are safely neglected. Indeed, this is
a reasonable approximation given the expected SM event rates depicted in the
right panel of Fig. 1. In the present work we will consider two representative
case studies accounting for different detector-specific quantities, namely an
“optimistic” and a “realistic” scenario. The two scenarios differ in the
details of the detector properties and effects that are taken into account.
One of our aims is to compare and contrast the results obtained from an ideal
vs realistic detector. The details of the two scenarios are as described
below.

  * •

Optimistic scenario: A xenon detector with a $1~{}\mathrm{ton\cdot yr}$
exposure is assumed. This is, however, a conservative choice that exceeds
slightly the $0.65~{}\mathrm{ton\cdot yr}$ achieved by XENON1T [87] and by a
factor 20 less than the exposure goal of $20~{}\mathrm{ton\cdot yr}$ at
XENONnT [44]. We stick to the latter choice to avoid potential overestimation
of our projected sensitivities, since for this scenario the detector
efficiency and energy resolution are not taken into consideration. For both
CE$\nu$NS and E$\nu$ES analyses, we consider a flat background taken to be 10%
of the SM events as well as a systematic uncertainty
$\sigma_{\text{sys}}=10\%$. For our spectral fit we consider 100 log-spaced
bins in the range $(0.1,100)~{}\mathrm{keV_{ee}}$ for the case of CE$\nu$NS
and in the range $(0.1,10^{4})~{}\mathrm{keV_{ee}}$ for the case of E$\nu$ES.
Let us clarify that our optimistic scenario assumes a threshold as low as
$0.1~{}\mathrm{keV_{ee}}$ and a perfect efficiency, which is motivated by the
“$S2$-only” analysis done in Ref. [88]. For the case of E$\nu$ES we have also
checked that our projected sensitivities remain the same if a narrower recoil
window $(0.1,100)~{}\mathrm{keV_{ee}}$ is chosen.

  * •

Realistic scenario: This scenario will also serve as reference point to
highlight and contrast the results expected from an ideal and a realistic
detector. We consider the planned $20~{}\mathrm{ton\cdot yr}$ exposure
foreseen at XENONnT, and we incorporate realistic backgrounds, efficiency and
resolution effects. In particular, for our CE$\nu$NS analysis we consider the
neutron background101010For this particular case, due to lack of information
we do not assign any uncertainty on the neutron background i.e., our predicted
number of events in this case reads
$R_{\mathrm{pred}}^{i}=(1+\xi)\left[R_{\text{X+SM}}^{i}(m_{X},g_{X})\right]+R_{\text{bg}}^{i}$.
We have checked, however, that our result remains essentially unaltered if a
background uncertainty as high as 10% is assumed. from Ref. [44] and the
efficiency from Ref. [89]. For our E$\nu$ES analysis, the background model
$B_{0}$ (appropriately scaled to account for the $20~{}\mathrm{ton\cdot yr}$
exposure) and the detection efficiency, are both taken from Ref. [87].
Moreover, our theoretical E$\nu$ES event rates are eventually
smeared111111This requires another integration of Eq.(22) over the true recoil
energy $E_{er}$ taking the resolution function to be a normalized Gaussian,
see Ref. [90]. by taking into account the energy resolution power of XENON1T,
that is $\sigma(E)=a\cdot\sqrt{E}+b\cdot E$, with $E$ being the reconstructed
recoil energy, $a=0.310~{}\sqrt{\mathrm{keV_{ee}}}$, and
$b=0.0037~{}\mathrm{keV_{ee}}$ [87]. We consider 17 linearly spaced bins in
the range $(4,52)~{}\mathrm{keV_{nr}}$ for our CE$\nu$NS analysis and 30
linearly spaced bins in the range $(0,30)~{}\mathrm{keV_{ee}}$ for our
E$\nu$ES analysis [44]. Finally, a 10% systematic uncertainty is assigned for
CE$\nu$NS as in the optimistic scenario, while for the case of E$\nu$ES a 3%
systematic uncertainty is taken to account for the detection uncertainty of
XENON1T.

---

(a) Optimistic scenario

(b) Realistic scenario

Figure 2: Projected sensitivities for the various $X=\\{S,P,V,A,T\\}$
interactions for CE$\nu$NS (left) and E$\nu$ES (right). The results are
presented at 90% C.L. Graphs correspond to the (a) optimistic and (b)
realistic scenarios.

---

(a) Optimistic scenario: For the universal vector (left) and the scalar
(right) mediators.

(b) Realistic scenario: For the universal vector (left) and the scalar (right)
mediators.

Figure 3: Projected sensitivity at 90% C.L. in the parameter space
$(m_{V},g_{V})$ for the universal vector mediator model (left) and in
$(m_{S},g_{S})$ for a scalar mediator model (right). Graphs correspond to the
(a) optimistic and (b) realistic scenarios. A comparison is given with
existing constraints from dedicated CE$\nu$NS experiments and XENON1T (see the
text).

The resulting projected sensitivities in the ($m_{X},g_{X}$) parameter space
are illustrated at 90% C.L. in the left and right panels of Fig. 2,
respectively, while upper (lower) panels correspond to the optimistic
(realistic) scenario. As can be seen, in a future CE$\nu$NS (E$\nu$ES)
measurement among the different interaction channels the scalar (tensor)
interaction will be constrained with maximum sensitivity, while for both
CE$\nu$NS and E$\nu$ES the pseudoscalar interaction will be the least
constrained. A direct comparison of CE$\nu$NS and E$\nu$ES sensitivities leads
to the conclusion that future E$\nu$ES measurements will be a more powerful
probe for the investigation of novel light mediators.

At this point it is interesting to compare our projected sensitivities with
existing constraints in the literature. First, by focusing on the universal
light vector mediator scenario, in the left panel of Fig. 3 we compare our
present results at 90% C.L. with existing constraints from the combined
analysis of COHERENT CsI+Ar data [63], the recent CONNIE [29] and CONUS [30]
data, the Borexino analysis of Ref. [91] as well as the anomalous magnetic
moment of the muon reported in Ref. [4]. Also shown are the corresponding 90%
C.L. constraints coming from the XENON1T excess using E$\nu$ES calculated in
this work. Our realistic scenario coincides with the XENON1T setup, hence the
improvement on our E$\nu$ES-based constraints is driven by the factor $\sim
20$ larger exposure considered in the present analysis, while the shape
difference is due to the XENON1T excess data (SM expectation) considered as
$R_{\text{exp}}$ for the XENON1T (realistic scenario) analysis. The Borexino
limits of Ref. [91] extend to larger mediator masses compared to our present
results, since the analysis has been performed considering a larger recoil
energy window, covering also CNO and 8B neutrinos. By comparing the upper and
lower graphs, we conclude that constraints from dedicated CE$\nu$NS
experiments will be overridden by the future CE$\nu$NS measurements at direct
detection dark matter detectors, given that the threshold and detection
efficiencies will improve and the backgrounds will become even better
understood. Finally, it becomes evident that the E$\nu$ES channel dominates
over CE$\nu$NS in the low mass region for $m_{V}\leq 2$MeV.

Similar conclusions are drawn for the case of a scalar mediator, as shown in
the right panel of Fig. 3. Available results can be found from CE$\nu$NS
analyses in Refs. [29, 30, 63] and from the interpretation of XENON1T excess
performed in this work (see also Refs. [8, 9]). Regarding the cases of $S,V,A$
interactions for both CE$\nu$NS and E$\nu$ES channels as well as for the case
of $P$ using E$\nu$ES only, results corresponding to the optimistic case can
be found in Ref. [16]. Let us note, however, that in the latter work the
sensitivities were obtained by assuming different experimental configurations
as well as by neglecting quenching and atomic effects for CE$\nu$NS and
E$\nu$ES respectively. Regarding the cases of $P$ and $T$ interactions,
preliminary results have been presented in Ref. [54] by analyzing the COHERENT
data. Again, the future sensitivities discussed here are about one order of
magnitude more stringent than other studies. Finally, to the best of our
knowledge, studies corresponding to our realistic case have not been done for
any kind of model-dependent or -independent light mediator.

---

(a) Optimistic scenario: For the $B-L$ (left) and $L_{\mu}-L_{\tau}$ (right)
models.

(b) Realistic scenario: For the $B-L$ (left) and $L_{\mu}-L_{\tau}$ (right)
models.

Figure 4: Projected exclusion curves at 90% C.L. obtained in the present work
using CE$\nu$NS and E$\nu$ES for the $B-L$ (left) and $L_{\mu}-L_{\tau}$
(right) model. Graphs correspond to the (a) optimistic and (b) realistic
scenarios. A comparison with relevant experimental constraints is also given
(see the text).

We finally turn our attention to the vector mediator interactions predicted
within the $U_{B-L}$ and $U_{L_{\mu}-L_{\tau}}$ gauge extensions. Before
proceeding with the discussion of our statistical analysis, let us provide
some clarifications regarding the calculation of the number of CE$\nu$NS and
E$\nu$ES events within the $L_{\mu}-L_{\tau}$ model. Since the $\nu_{e}-e^{-}$
coupling is vanishing, we neglect the first term in Eq.(22) and therefore a
large portion of the solar neutrino flux will not contribute to the expected
E$\nu$ES rates. Likewise, in our CE$\nu$NS-based analysis we consider only the
oscillated $\nu_{\mu,\tau}$ fluxes of solar neutrinos, while from the
atmospheric neutrino flux we consider only the relevant $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ components. For atmospheric neutrinos with sub-GeV
neutrinos, matter-oscillation effects are negligible [92]. Following the
procedure of Ref. [81] one sees that the oscillated fluxes reaching the
detector are:
$\Phi^{\nu_{e}}_{\text{atm}}\approx\widetilde{\Phi}^{\nu_{e}}_{\text{atm}}$,
$\Phi^{\nu_{\mu}}_{\text{atm}}\approx
2/3\,\widetilde{\Phi}^{\nu_{\mu}}_{\text{atm}}$ and
$\Phi^{\nu_{\tau}}_{\text{atm}}\approx
1/3\,\widetilde{\Phi}^{\nu_{\mu}}_{\text{atm}}$, where
$\widetilde{\Phi}^{\nu_{\alpha}}_{\text{atm}}$ denotes the unoscillated flux
reported in Ref. [71]. We have verified that our results remain unaffected
when atmospheric neutrino oscillations are explicitly taken into account.
Finally, regarding the DSN neutrino flux we assume that roughly
$\Phi^{\nu_{\mu}+\bar{\nu}_{\mu}}_{\text{DSN}}\approx\Phi^{\nu_{\tau}+\bar{\nu}_{\tau}}_{\text{DSN}}$.

Left and right panels of Fig. 4 illustrate the projected sensitivity at 90%
C.L. for the studied $B-L$ and $L_{\mu}-L_{\tau}$ model, respectively. The
results are shown for both CE$\nu$NS and E$\nu$ES at a future dark matter
direct detection experiment with the same general conclusions as discussed
previously. Moreover, as expected, the exclusion curves corresponding to the
$B-L$ model are more stringent. In order to compare with other experimental
probes, existing limits placed by dielectron resonances at ATLAS [93] and
electron beam-dump fixed target experiments [94, 95], as well as Dark Photon
searches at BaBar [96, 97], KLOE [98], CMS [99], and LHCb [100]121212BaBar,
KLOE, CMS and LHCb limits, recasted to the $B-L$ case are obtained using the
Darkcast software package., are superimposed. Further constraints from the
E$\nu$ES experiments TEXONO, GEMMA, BOREXINO, LSND and CHARM II can be found
in Ref. [101]. Also shown are constraints derived from the analysis of the
COHERENT data [63] and the XENON1T excess [9]. For the $L_{\mu}-L_{\tau}$
case, limits are available from recent analyses of Borexino data [32, 102],
from $4\mu$ searches at BaBar [103], CMS [104], and ATLAS [105, 106] as well
as recasted limits from LHCb [100]. We furthermore show the corresponding
sensitivity from the analysis of XENON1T excess performed in this work, by
following the procedure of Ref. [8]. As can be seen, our projected
sensitivities for CE$\nu$NS dominate in mass range $0.1\leq m_{V}\leq 1$ GeV,
being complementary to Babar and fixed target experiments. In the same vein,
our projected sensitivities obtained using E$\nu$ES, are complementary to
fixed target experiments and particularly relevant for $m_{V}\leq 1$ MeV.

Before closing, we should stress that astrophysics and cosmology might place
severe bounds to scalar and vector interactions [23]. Those follow mostly from
cosmological limits on the sum of neutrino masses, Supernova/stellar-cooling
arguments and sterile neutrino trapping as detailed in Refs. [9, 26] (see also
Ref. [107]). Given the large astrophysical uncertainties, such constraints
should be considered as order of magnitude estimations while possible
mechanisms to evade them are explained in Ref. [26]. Finally, it should be
noted that upcoming experiments also have the potential to probe light
mediator particles. However, our results provide about an order of magnitude
more constrained bounds than those predicted for other future experiments such
as neutrino trident interactions at DUNE [62], model-dependent constraints for
DARWIN [33], and constraints extracted from missing energy searches at
NA64$\mu$ [33] (for expected limits from future beam dump experiments such as
SHiP and FASER2 see Ref. [107]).

##  5 Conclusions

The new era of direct dark matter experiments with multiton mass scale and
sub-keV operation threshold makes them favorable facilities with promising
prospects for detecting astrophysical neutrino backgrounds to dark matter.
Prompted by the latter, we estimated the projected sensitivities to general
Lorentz-invariant $X=\\{S,P,V,A,T\\}$ interactions through CE$\nu$NS- and
E$\nu$ES-induced signals. With respect to light vector mediators, our study
also considered the case of well-known $U_{B-L}$ and $U_{L_{\mu}-L_{\tau}}$
anomaly-free models. To maximize the reliability of event rate simulations,
important corrections from detector-specific quantities were taken into
account such as the quenching factor and atomic binding effects for the case
of CE$\nu$NS and E$\nu$ES, respectively. Our statistical analysis was
performed under the assumption of two benchmark scenarios, which allowed us to
compare the maximum potential of a future direct dark matter detection
experiment with the expected sensitivities when current detector
specifications are explicitly accounted for. In the optimistic scenario we
assumed a rather low detection threshold of $0.1~{}\mathrm{keV_{ee}}$, which
would correspond to an ionization-only analysis with a 100% detection
efficiency. In our goal to make our analysis as quantitative as possible, in
the realistic scenario the effects of finite detection efficiency, nonflat
backgrounds, and energy resolutions have been incorporated. We furthermore
considered an exposure between $1\text{ and }20~{}\mathrm{ton\cdot yr}$ which
is readily achievable by the current and future xenon detector technologies.
Our present results imply that future CE$\nu$NS or E$\nu$ES measurements at a
direct detection dark matter experiment will not only become sensitive to
neutrinos coming (mainly) from the Sun, but will also offer competitive
constraints to existing ones from dedicated CE$\nu$NS and E$\nu$ES
experiments, if backgrounds are under control and sub-keV thresholds become
possible. We have furthermore illustrated that the expected sensitivities will
cover a large part of the parameter space, previously unexplored from collider
probes and beam dump experiments, improving upon the existing bounds by about
1 order of magnitude.

###### Acknowledgements.

The authors are indebted to D. Aristizabal-Sierra for carefully reading the
manuscript and for insightful comments. The authors also acknowledge P.
Martínez-Miravé, M. Williams, P. Ilten, K. Ni, F. Gao and J. Pienaar for
useful correspondence. The research of D.K.P. is co-financed by Greece and the
European Union (European Social Fund- ESF) through the Operational Programme
“Human Resources Development, Education and Lifelong Learning” in the context
of the project “Reinforcement of Postdoctoral Researchers - 2nd Cycle”
(MIS-5033021), implemented by the State Scholarships Foundation (IKY). The
work of RS is supported by the SERB, Government of India grant
SRG/2020/002303.

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