# Decorated Linear Relations: Extending Gaussian Probability with
Uninformative Priors

Dario Stein  iHub, Radboud University Nijmegen, The Netherlands

###### Abstract

We introduce _extended Gaussian distributions_ as a precise and principled way
of combining Gaussian probability uninformative priors, which indicate
complete absence of information. To give an extended Gaussian distribution on
a finite-dimensional vector space $X$ is to give a subspace $D$, along which
no information is known, together with a Gaussian distribution on the quotient
$X/D$. We show that the class of extended Gaussians remains closed under
taking conditional distributions. We then introduce _decorated linear maps and
relations_ as a general framework to combine probability with nondeterminism
on vector spaces, which includes extended Gaussians as a special case. This
enables us to apply methods from categorical logic to probability, and make
connections to the semantics of probabilistic programs with exact
conditioning.

###### keywords:

probabilistic programming, probability theory, Bayesian inference, category
theory, Markov categories

††volume: NN††journal: Electronic Notes in Theoretical Informatics and
Computer Science††thanks: Email<EMAIL_ADDRESS>
##  1 Introduction

Gaussian (multivariate normal) distributions are a small but expressive family
of probability distributions with a variety of applications, from ridge
regression over Kálmán filters to Gaussian process regression. An important
feature is that Gaussians are self-conjugate, that is conditional
distributions of Gaussians are themselves Gaussian. For example, in a
regression problem, we may put Gaussian priors on the regression coefficients
$a_{i}\sim\mathcal{N}(0,\sigma)$111throughout, $\sigma$ denotes the _variance_
(and not standard deviation) for consistency with the covariance matrices
$\Sigma$ in the multivariate case. We make no further mentions of standard
deviations. and let $y=a_{i}x_{i}+\ldots+a_{b}x_{n}$. We can then make noisy
or exact observations for datapoints $(\vec{x}_{i},y_{i})_{i=1,\ldots,n}$ and
obtain Gaussian posteriors over both the regression coefficients and the
predicted value.

An important question is then how to model complete _absence of information_
in such a setting, say over coefficients or data. Conditioning on such absent
information should have no effect. However, no Gaussian distribution
$\mathcal{N}(\mu,\sigma)$ is ever completely uninformative, because it is
integrable and introduces a bias towards its mean. Intuitively, the
uninformative prior over the real line should be the limit of
$\mathcal{N}(\mu,\sigma)$ for $\sigma\to\infty$. However, this limit cannot be
taken in any ordinary sense because it assigns measure $0$ to every bounded
set (it converges weakly to the zero measure, which is not a probability
measure). The Lebesgue measure $\lambda$ is another candidate, however it is
not a probability measure because it fails to be normalized. Nonetheless, so
called _improper priors_ like the Lebesgue measure are frequently used in
density computations, because _after conditioning_ , they may result in a
well-defined and normalized posterior (e.g. [10]). We prefer to give a formal
account of this phenomenon which completely avoids non-normalization:

In Section 2, we present a theory of _extended Gaussian distributions_ , which
faithfully include ordinary Gaussian distributions while adding in
uninformative priors. Extended Gaussian variables can be added, scaled and
conditioned just like ordinary Gaussians. We generalize the natural duality
between precision and covariance for ordinary Gaussians to the extended case
(Section 2.2).

In Section 3, we generalize extended Gaussian distributions to a category
$\mathsf{GaussEx}$ of _extended Gaussian maps_ between vector spaces.
$\mathsf{GaussEx}$ combines features from Gaussian probability as well as
nondeterminism, which is modelled by linear relations. This is surprising
because probability and nondeterminism are difficult to combine in general
[24]. We show that $\mathsf{GaussEx}$ is an instance of a general
construction: For a suitable type of ‘decoration’ $S$, we define the
categories $\mathsf{Lin}_{S}$ of _decorated linear maps_ and
$\mathsf{LinRel}_{S}$ of _decorated linear relations_ , of which linear
relations, affine relations, ordinary Gaussians and extended Gaussians are
obtained for different choices of $S$. This joint formalism lets us elegantly
establish relationships between those categories, such as the support of a
Gaussian, which is an affine subspace (Section 3.3).

Because extended Gaussians are inherently not measure-theoretic, we will use
the language of categorical probability theory [6, 9] to talk about random
variables, distributions and conditioning in an abstract and unified way. In
technical terms, our main theorem is that $\mathsf{GaussEx}$ is a Markov
category which has all conditionals (Theorem 3.11).

Recent applications of categorical probability theory include [12, 13]. The
categorical framework has a canonical counterpart in the compositional
semantics of probabilistic programs [19, 21]. In [22], we treat _exact
conditioning_ of Gaussian random variables from a programming-languages
viewpoint. The study of uninformative priors is very natural in this context,
because it asks for a unit for the conditioning operation. We will discuss
this more in Section 4 but take care to keep the developments of Section 2
self-contained and avoid categorical language there.

##  2 Extended Gaussian Distributions

We begin with an informal introduction to extended Gaussians, before defining
them formally in Section 2.1. In a nutshell, an extended Gaussian on a vector
space $X$ is a Gaussian distribution on a quotient $X/D$. For an overview of
linear algebra terminology, see Section 6.1.

Representations: A _Gaussian distribution_ on $\mathbb{R}^{n}$ can be written
uniquely as $\mathcal{N}(\mu,\Sigma)$ where $\mu\in\mathbb{R}^{n}$ is its
_mean_ and $\Sigma\in\mathbb{R}^{n\times n}$ is a symmetric positive-
semidefinite matrix called its _covariance matrix_. The _support_ of
$\mathcal{N}(\mu,\Sigma)$ is the affine subspace $\mu+\mathsf{col}(\Sigma)$
where $\mathsf{col}(\Sigma)$ is the column space (image) of $\Sigma$. Gaussian
distributions transform as follows under linear maps: If
$A\in\mathbb{R}^{m\times n}$, then

$A_{*}\mathcal{N}(\mu,\Sigma)=\mathcal{N}(A\mu,A\Sigma A^{T})$

is its pushforward Gaussian distribution on $\mathbb{R}^{m}$. For an
introduction to Gaussian probability see e.g. [23].

We now wish to add certain forms of limits to Gaussian distributions, which
allow us to express ignorance along a certain subspace $D$. An _extended
Gaussian distribution on $\mathbb{R}^{n}$_ is an entity which can be written
as

$\mathcal{N}(\mu,\Sigma)+D$ (1)

where $D\subseteq\mathbb{R}^{n}$ is a vector subspace called _locus of
nondeterminism_. Extended Gaussian distributions transform as
$A_{*}(\mathcal{N}(\mu,\Sigma)+D)=\mathcal{N}(A\mu,A\Sigma A^{T})+A[D]$ where
$A[D]=\\{Ax:x\in D\\}$ is the image subspace. The intuition is that
nondeterministic (uninformative) noise is present in the distribution along
the subspace $D$. For example, the extended Gaussian distribution
$\mathcal{N}(0,0)+\mathbb{R}$ (which we’ll simply write as $\mathbb{R}$)
expresses a ‘uniform distribution’ over the real line; it is the distribution
of a point about which we have no information.

Translation invariance and non-uniqueness: An important consequence of the
uninformativeness is that the representation (1) can no longer be unique. For
example, the uniform distribution on $\mathbb{R}$ should be translation
invariant, because adding a constant to an unknown point results in an unknown
point. We can thus, for example, expect the following three representations of
extended Gaussians to denote the same distribution:

$\mathcal{N}(10,0)+\mathbb{R}=\mathbb{R}=\mathcal{N}(0,10)+\mathbb{R}$

Let us consider a more complex example involving two correlated random
variables:

######  Example 2.1.

Let $X_{1},X_{2}$ be independent normal variables, to which we add some
uniform noise along the diagonal $D=\\{(y,y):y\in\mathbb{R}\\}$; that is we
let $Y\sim\mathbb{R}$ and define

$Z_{1}=X_{1}+Y,\quad Z_{2}=X_{2}+Y$

The extended Gaussian vector $(Z_{1},Z_{2})$ can be decomposed into a sum (see
Figure 1)

$\displaystyle(Z_{1},Z_{2})$
$\displaystyle=\underbrace{(X_{1}+Y,X_{1}+Y)}_{\in D}+(0,X_{2}-X_{1})$

where the first summand lies in $D$ and the second one takes values in the
complement $0\times\mathbb{R}$. By translation invariance, the first
contribution gets absorbed by $D$. The remaining contribution
$(0,X_{2}-X_{1})$ has variance $2$ in the second coordinate. We can therefore
conclude that following are both valid representations of the joint
distribution of $(Z_{1},Z_{2})$:

$\mathcal{N}\left(\begin{pmatrix}0\\\ 0\end{pmatrix},\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\right)+D=\mathcal{N}\left(\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}0&0\\\ 0&2\end{pmatrix}\right)$ (2)

Figure 1: Decomposing an extended Gaussian with locus of nondeterminism $D$

Conditionals: Unlike $X_{1},X_{2}$, the variables $Z_{1},Z_{2}$ in Example 2.1
are no longer independent, because they are coupled via $Y$. In fact, one can
show that $Z_{2}$ given $Z_{1}$ has the conditional distribution

$Z_{2}|Z_{1}\sim\mathcal{N}(Z_{1},2)$

The expression $Z_{1}\mapsto\mathcal{N}(Z_{1},2)$ appearing in the conditional
is an example of an _extended Gaussian map_ , which we consider systematically
in Section 3. In Theorem 3.11, we prove that that conditional distributions of
extended Gaussians are again extended Gaussian. We now proceed to define
extended Gaussians formally.

###  2.1 Definition of Extended Gaussians

For a first definition of extended Gaussians, we start from the
representatives in (1) and quotient by an equivalence relation for when they
shall denote the same distribution:

######  Definition 2.2 (Preliminary Definition).

An extended Gaussian distribution on $\mathbb{R}^{n}$ is an equivalence class
of pairs $(D,\psi)$ where $D\subseteq\mathbb{R}^{n}$ is a vector subspace and
$\psi$ is a Gaussian distribution on $\mathbb{R}^{n}$. We identify two such
pairs $(D,\psi),(D^{\prime},\psi^{\prime})$ if and only if $D=D^{\prime}$ and
the pushforward distributions $(p_{K})_{*}\psi$ and $(p_{K})_{*}\psi^{\prime}$
agree for some (equivalently any) direct sum decomposition $D\oplus
K=\mathbb{R}^{n}$, where $p_{K}$ is the projection endomorphism onto $K$.

######  Example 2.3.

In Example 2.1, we considered the particular complement $K=0\times\mathbb{R}$
of $D$. The projection onto $K$ is given by the matrix

$A=\begin{pmatrix}0&0\\\ -1&1\end{pmatrix}$

From this we obtain the desired equality (2) by verifying that

$A_{*}\left[\mathcal{N}\left(\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\right)\right]=\mathcal{N}\left(\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}0&0\\\ 0&2\end{pmatrix}\right)$

The choice of complement is in no way canonical. A cleaner, high-level
definition goes as follows: _An extended Gaussian distribution on $X$ with
locus of nondeterminism $D$ is an ordinary Gaussian distribution on the
quotient space $X/D$_ (Definition 2.5). The preliminary definition 2.2 is then
obtained by identifying $\mathbb{R}^{n}/D$ with the complement $K$. However,
because the choice of complement is non-canonical, it is preferable to work
with the quotient directly. However, the abstract definition makes it
necessary to re-introduce Gaussian distributions on more general vector spaces
than just $\mathbb{R}^{n}$ in a coordinate-free way. This development is well-
known, and we refer to e.g. [23, Section 1] for an overview:

All vector spaces $X$ are henceforth assumed finite-dimensional. We write
$X^{*}$ for the dual space of $X$, consisting of all linear maps
$X\to\mathbb{R}$. By a _form_ on $X$, we mean a _symmetric bilinear map_
$\Omega:X\times X\to\mathbb{R}$. The kernel of $\Omega$ is the set

$\ker(\Omega)\stackrel{{\scriptstyle\text{def}}}{{=}}\\{x\in
X:\Omega(x,-)=0\\}=\\{y\in X:\Omega(-,y)=0\\}$

We call $\Omega$ _nondegenerate_ if $\ker(\Omega)=0$. Note that $\Omega$ can
be curried as $\omega:X\to X^{*}$ with $\omega(x)=\Omega(x,-)$. The notation
is consistent in that $\ker(\Omega)=\ker(\omega)$, and $\Omega$ is
nondegenerate if and only if $\omega:X\to X^{*}$ is an isomorphism. If
$\Omega$ has kernel $K$, then the quotient form
$\widetilde{\Omega}:(X/K)\times(X/K)\to\mathbb{R}$ is well-defined and
nondegenerate.

A form $\Omega:X\times X\to\mathbb{R}$ is called _positive semidefinite_ if
$\Omega(x,x)\geq 0$ for all $x\in X$, and _positive definite_ if
$\Omega(x,x)>0$ for all $x\neq 0$. A positive semidefinite form is positive
definite if and only if it is nondegenerate.

The correct coordinate-free version of the covariance matrix is that of a
covariance form $\Sigma:X^{*}\times X^{*}\to\mathbb{R}$ on the dual space.
Given a random variable $U$ on the space $X$, and linear functions
$f,g:X\to\mathbb{R}$, we compute the covariance

$\Sigma(f,g)\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{E}[f(U)g(U)]-\mathbb{E}[f(U)]\mathbb{E}[g(U)]$

This expression is symmetric, bilinear and positive semidefinite. A Gaussian
distribution is fully determined by its mean and covariance form, which
motivates the following definition:

######  Definition 2.4.

A Gaussian distribution on a vector space $X$ is a pair written
$\mathcal{N}(\mu,\Sigma)$ of a mean $\mu\in X$ and a positive semidefinite
form $\Sigma:X^{*}\times X^{*}\to\mathbb{R}$. If $f:X\to Y$ is a linear map,
the Gaussian distribution pushes forward to $\mathcal{N}(f(\mu),f_{*}\Sigma)$
where $(f_{*}\Sigma)(g,h)=\Sigma(gf,hf)$.

We can now give the following concise definition of extended Gaussians.

######  Definition 2.5.

An extended Gaussian distribution on a vector space $X$ is a pair $(D,\psi)$
where $D\subseteq X$ is a vector subspace and $\psi$ is a Gaussian
distribution on $X/D$.

Every linear map $f:X\to Y$ induces a linear map $f_{D}:X/D\to Y/f[D]$, and we
take the pushforward of $(D,\psi)$ to be $(f[D],(f_{D})_{*}\psi)$.

###  2.2 Precision and Duality

We will show that our definition of extended Gaussians fits into an elegant
duality between forms on a space and its dual. This lets us convert between
two equivalent representations, using a _covariance form_ or a _precision
form_ , which are convenient for different purposes:

Probability distributions and probability densities are dual to each other.
Distributions naturally push forward, and consequently the covariance form
must be defined on the _dual space_ $X^{*}$. On the other hand, _density
functions_ pull back. If the covariance matrix $\Sigma$ is nonsingular, the
Gaussian distribution $\mathcal{N}(\mu,\Sigma)$ has a Lebesgue density

$f(x)\propto\exp\left(-\frac{1}{2}(x-\mu)^{T}\Omega(x-\mu)\right)$ (3)

where $\Omega=\Sigma^{-1}$ is called the _precision matrix_. If $\Sigma$ is
singular, $\mathcal{N}(\mu,\Sigma)$ will only admit a density on its support
$S$, and the precision is only defined on that subspace. The coordinate-free
formulation of the precision-covariance correspondence is given by the
following duality statement (where for simplicity we assume centered
distributions, that is mean zero):

######  Theorem 2.6.

The following pieces of data are equivalent for every vector space $X$

  1. (1)

a form $\Sigma:X^{*}\times X^{*}\to\mathbb{R}$

  2. (2)

a subspace $S\subseteq X$ and a nondegenerate form $\Omega:S\times
S\to\mathbb{R}$

Recall that the evaluation pairing $X^{*}\times X\to\mathbb{R}$ induces a
duality between the subspaces of $X$ and $X^{*}$ called _annihilators_ , which
we here denote $(-)^{\bot}$. For subspaces $D\subseteq X$ and $F\subseteq
X^{*}$, the subspaces $D^{\bot}\subseteq X^{*},F^{\bot}\subseteq X$ are
defined as

$\displaystyle D^{\bot}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{f\in
X^{*}:f|_{D}=0\\},\qquad
F^{\bot}\stackrel{{\scriptstyle\text{def}}}{{=}}\\{x\in X:\forall f\in
F,f(x)=0\\}$ (4)

Taking annihilators is order-reversing and involutive; we list further
properties under Proposition 6.3.

In Theorem 2.6, the subspace $S$ is taken to be annihilator of the kernel
$K=\ker(\Sigma)$, that is $S=\\{x:\forall f\in K,f(x)=0\\}$. This recovers the
familiar support $S=\mathsf{col}(\Sigma)$ for covariance matrices. We may
think of the form $\Omega$ as taking the value infinity outside of $S$ (which
corresponds to vanishing a density under (3)).

Extended Gaussians admit a generalized and in fact more symmetric version of
this correspondence:

######  Theorem 2.7.

The following pieces of data are equivalent for every vector space $X$

  1. (1)

pairs $\langle S,\Omega\rangle$ of a subspace $S\subseteq X$ and
$\Omega:S\times S\to\mathbb{R}$

  2. (2)

pairs $\langle F,\Sigma\rangle$ of a subspace $F\subseteq X^{*}$ and
$\Sigma:F\times F\to\mathbb{R}$

######  Proof 2.8.

  1. (1)

Given $\langle S,\Omega\rangle$, let $D=\ker(\Omega)\subseteq S$ and define
$F=D^{\bot}$ and $K=S^{\bot}$.

Form the nondegenerate quotient form
$\widetilde{\Omega}:(S/D)\times(S/D)\to\mathbb{R}$. Its currying
$\widetilde{\omega}:(S/D)\to(S/D)^{*}$ is an isomorphism. Making use of the
canonical isomorphisms described in Proposition 6.3,

${{(S/D)^{*}}}$${{S/D}}$${{D^{\bot}/S^{\bot}}}$${{K^{\bot}/F^{\bot}}}$${{F/K}}$${{(F/K)^{*}}}$$\scriptstyle{\widetilde{\omega}^{-1}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\widetilde{\sigma}}$
(5)

we obtain an iso $\widetilde{\sigma}:(F/K)\to(F/K)^{*}$, which is the same
thing as a bilinear form $\Sigma:F\times F\to\mathbb{R}$ with kernel $K$.

  2. (2)

Conversely, given $\langle F,\Sigma\rangle$, let $K=\ker(\Sigma)$ and define
$D=F^{\bot}$ and $S=K^{\bot}$. Turn $\Sigma$ into an iso
$\widetilde{\sigma}:(F/K)\to(F/K)^{*}$, then reading the diagram (5) backwards
uniquely defines the iso $\widetilde{\omega}$ and hence the form $\Omega$ with
kernel $D$.

The constructions are clearly inverses to each other. It is furthermore easy
to see that the correspondence takes positive semidefinite forms to positive
semidefinite forms.

A (centered) extended Gaussian thus has a _covariance representation_ which is
a pair $\langle F,\Sigma\rangle$ with $F\subseteq X^{*}$ and $\Sigma$ a
positive semidefinite form on $F$, and a _precision representation_ $\langle
S,\Omega\rangle$ with $S\subseteq X$ and $\Omega$ positive semidefinite on
$S$. The covariance representation is convenient for computing pushforwards,
while the precision representation generalizes density functions and is useful
for conditioning. Note that the locus of nondeterminism of an extended
Gaussian is $D=\ker(\Omega)$, that is, unlike for ordinary Gaussians,
vanishing precision is now allowed! The proof of Theorem 2.7 is reminiscent of
the construction of the Moore-Penrose pseudoinverse, whose relevance to
Gaussian probability is well-known (e.g. [15]).

######  Definition 2.9.

We define the _uniform distribution_ $u_{X}$ on $X$ to be the unique extended
Gaussian distribution whose locus of nondeterminism is all of $X$. Its
precision representation is the zero form $X\times X\to\mathbb{R}$; its
covariance representation is the zero form on the trivial subspace $0\subseteq
X^{*}$.

##  3 A Category of Extended Gaussian maps

It is convenient to generalize Gaussian distributions on $X$ to _Gaussian
maps_ $X\to Y$, which are linear functions with Gaussian noise, informally
written $x\mapsto f(x)+\psi$, where $f:X\to Y$ is linear and $\psi$ is a
Gaussian distribution on $Y$ (independent of $x$). This allows us to discuss
distributions, linear maps and conditionals within a single formalism.

Gaussian maps can be composed in sequence and in parallel, where the noise is
pushed forward and accumulated appropriately. Formally, we defines a symmetric
monoidal category $\mathsf{Gauss}$ (due to [9, Ch. 6]) as follows: Objects are
vector spaces $X$, and morphisms are Gaussian maps $(f+\psi)$ between them. On
objects, the tensor is cartesian product, $X\otimes Y=X\times Y$, and the
categorical structure is given by

$\displaystyle(f+\psi)\circ(g+\varphi)=fg+\psi+f_{*}\varphi\qquad(f+\psi)\otimes(f^{\prime}+\psi^{\prime})=f\times
f^{\prime}+\psi\otimes\psi^{\prime}$

where $\psi\otimes\psi^{\prime}$ denotes the product distribution of
Gaussians. Furthermore, we can copy and discard information using the linear
maps $\mathsf{cpy}_{X}:X\to X\times X,x\mapsto(x,x)$ and
$\mathsf{del}_{X}:X\to 0$. This gives $\mathsf{Gauss}$ the structure of a
_Markov category_ [9], that is a categorical model of probability theory. We
can recover Gaussian distributions as Gaussian maps $0\to X$ out of the
trivial vector space.

In this section, we show that the construction of $\mathsf{Gauss}$ arises as
an instance of a general notion of _decorated linear map_ $x\mapsto f(x)+s$.
We then give a recipe to combine decorated linear maps with nondeterminism to
obtain _decorated linear relations_. This subsumes our earlier construction of
extended Gaussian distributions, as well as serving as a definition for
_extended Gaussian maps_ (Definition 3.7).

In Section 3.3, we use the generality of this construction to establish the
following diagram of identity-on-objects functors between Markov categories,
where the bottom row consists of decorated linear maps and the top row of
decorated linear relations:

${{\mathsf{LinRel}^{+}}}$${{\mathsf{AffRel}^{+}}}$${\mathsf{GaussEx}}$${\mathsf{Vec}}$${\mathsf{Aff}}$${\mathsf{Gauss}}$$\scriptstyle{\mathsf{supp}}$
(6)

In all categories of this diagram, the objects are finite-dimensional vector
spaces $X$. Recall that a _linear relation_ between vector spaces is a
relation $R\subseteq X\times Y$ that is also a vector subspace; an _affine
relation_ is a relation that is also an affine subspace (see Section 6.1). The
remaining Markov categories in the diagram are defined as

  * •

$\mathsf{Vec}$: vector spaces and linear functions

  * •

$\mathsf{Aff}$: vector spaces and affine-linear functions

  * •

$\mathsf{LinRel}^{+}$: vector spaces and left-total linear relations

  * •

$\mathsf{AffRel}^{+}$: vector spaces and left-total affine relations

The functor $\mathsf{supp}$ collapses an Gaussian distributions to their
supports, which are affine subspaces.

###  3.1 Decorated Linear Maps

Let $S:\mathsf{Vec}\to\mathsf{CMon}$ be a functor from the category of vector
spaces into the category of commutative monoids and monoid homomorphisms. We
think of elements $s\in S(Y)$ as “decorations” for linear maps into $Y$, and
thus call $S$ a decoration functor.

######  Definition 3.1.

We define a category $\mathsf{Lin}_{S}$ of $S$-decorated linear maps as
follows:

  1. (1)

Objects are vector spaces $X$

  2. (2)

Morphisms are pairs $(f,s)$ where $f:X\to Y$ is a linear map and $s\in S(Y)$

  3. (3)

Composition is defined as follows: if $g:X\to Y,f:Y\to Z$, define

$(f,t)\circ(g,s)=(fg,t+S(f)(s))$

Note that addition takes place in the commutative monoid $S(Z)$.

There is a faithful inclusion $J:\mathsf{Vec}\to\mathsf{Lin}_{S}$ sending $f$
to $(f,0)$. The functor $U:\mathsf{Lin}_{S}\to\mathsf{Vec}$ which forgets the
decoration is an opfibration; a decorated map $(f,s):X\to Y$ is opcartesian if
and only if $s$ is a unit in the monoid $S(Y)$. In fact, $\mathsf{Lin}_{S}$ is
precisely the (op-)Grothendieck construction for $S$ seen as a functor
$\mathsf{Vec}\to\mathsf{Cat}$.

We argue that $\mathsf{Lin}_{S}$ has the structure of a symmetric monoidal
category with the tensor $X\otimes Y=X\times Y$ on objects. For this, we first
observe that $S$ is automatically lax monoidal; for this we define natural
maps $\oplus:S(X)\times S(Y)\to S(X\times Y)$ given as follows: For $(s,t)\in
S(X)\times S(Y)$, let $s\oplus t=S(i_{X})(s)+S(i_{Y})(t)$ where $i_{X}:X\to
X\times Y,i_{Y}:Y\to X\times Y$ are the biproduct inclusions. We can now
define the tensor of decorated map as $(f,s)\otimes(g,t)=(f\times g,s\oplus
t)$. The monoidal category $\mathsf{Lin}_{S}$ is in general not cartesian; it
does however inherit copy and delete maps from $\mathsf{Vec}$. The category
$\mathsf{Lin}_{S}$ is a Markov category if and only if deleting is natural,
i.e. $S(0)\cong 0$.

######  Example 3.2.

We reconstruct the bottom row of (6) for the following decoration functors:

  1. (1)

For $S(X)=0$, $\mathsf{Lin}_{S}$ is equivalent to $\mathsf{Vec}$.

  2. (2)

For $S(X)=X$, $\mathsf{Lin}_{S}$ is equivalent to $\mathsf{Aff}$. A map $X\to
Y$ consists of a pair $(f,y)$ with $f:X\to Y$ linear and $y\in Y$.

  3. (3)

Define $\mathsf{Cov}(X)=\\{\sigma:X^{*}\times X^{*}\to\mathbb{R}\text{
positive semidefinite form}\\}$ to be the set of covariance forms. This is a
commutative monoid under pointwise addition, and is functorial via
$\mathsf{Cov}(f)(\sigma)(g,h)=\sigma(gf,hf)$. Let
$S(X)=X\times\mathsf{Cov}(X)$, then $\mathsf{Lin}_{S}$ is precisely
$\mathsf{Gauss}$.

###  3.2 Decorated Linear Relations

Given a decoration functor $S:\mathsf{Vec}\to\mathsf{CMon}$, we define
$S$-decorated linear relations by maintaining a locus of nondeterminism $D$
similar to Definition 2.5.

######  Definition 3.3.

We define a category $\mathsf{LinRel}_{S}$ as follows:

  1. (1)

objects are vector spaces $X$

  2. (2)

morphisms in $\mathsf{LinRel}_{S}(X,Y)$ are triples $(D,f,s)$ where
$D\subseteq Y$ is a vector subspace, $f:X\to Y/D$ is a linear map and $s\in
S(Y/D)$.

Intuitively, the subspace $D$ represents the direction of complete ignorance,
so we only decorate the quotient.

Composition of $(D,g,s):X\to Y$ and $(E,f,t):Y\to Z$ is slightly more
involved: We first define the composite subspace $F$ as $E+f[D]$, which is
well-defined. The composite $Y\xrightarrow{f}Z/E\to Z/F$ vanishes on $D$ and
so descends to $\tilde{f}:Y/D\to Z/F$. We define the composite as

$(E,f,t)\circ(D,g,s)=(F,\tilde{f}g,S(\tilde{f})(s)+S(Z/E\to Z/F)(t))$

To understand the name ‘decorated linear relation’, we consider the case $S=0$
where we obtain that $\mathsf{LinRel}_{S}$ is equivalent to
$\mathsf{LinRel}^{+}$. The key observation is the following:

######  Proposition 3.4.

To give a left-total linear relation $R\subseteq X\times Y$ is to give
subspace $D\subseteq Y$ and a linear function $f:X\to Y/D$ to the quotient.

The correspondence is fully spelled out in Proposition 6.1. This means we can
think of a left-total relation $R$ as a linear function with nondeterministic
noise $x\mapsto f(x)+D$ along some subspace $D$. This is similar to a
decorated linear map, however the choice of the linear map $f$ is no longer
unique, so further quotienting is required, as we discuss now:

As a quotient: We can demystify the composition of $\mathsf{LinRel}_{S}$ by
first constructing an auxiliary category and then quotienting it by a
congruence. We first consider the decoration functor
$\mathsf{Sub}:\mathsf{Vec}\to\mathsf{CMon}$ defined by
$\mathsf{Sub}(X)=\\{D\subseteq X\text{ vector subspace }\\}$. Each
$\mathsf{Sub}(X)$ is a commutative monoid under Minkowski addition $D+E$, and
the functorial action for $f:X\to Y$ is direct image $D\mapsto f[D]$, which is
a monoid homomorphism. Now we consider the category
$\mathsf{Lin}_{S\times\mathsf{Sub}}$ where morphisms $X\to Y$ are by
definition triples $(f,s,D)$ with $s\in S(Y)$ and $D\subseteq Y$, and
composition is $(f,t,E)\circ(g,s,D)=(fg,t+S(f)(s),E+f[D])$. This has all the
data for $\mathsf{LinRel}_{S}$ and explains why the composite subspace is
formed the way it is.

However some pieces of data need to be identified if they agree on the
quotient by $D$: We define an equivalence relation
$(f_{1},s_{1},D_{1})\approx(f_{2},s_{2},D_{2})$ if $D_{1}=D_{2}$ are the same
subspace $D$, and $\pi_{Y/D}f_{1}=\pi_{Y/D}f_{2}$ and
$S(\pi_{Y/D})(s_{1})=S(\pi_{Y/D})(s_{2})\in S(Y/V)$ where $\pi_{Y/D}:Y\to Y/D$
is the quotient map.

######  Proposition 3.5.

The relation $\approx$ is a congruence relation, and $\mathsf{LinRel}_{S}$ is
the quotient of $\mathsf{Lin}_{S\times\mathsf{Sub}}$ under $\approx$.
$\mathsf{LinRel}_{S}$ is symmetric monoidal and inherits the copy and delete
maps from $\mathsf{Vec}$. It is a Markov category if $S(0)\cong 0$.

######  Proof 3.6.

See Appendix (Section 6.3).

######  Definition 3.7.

We define the Markov category $\mathsf{GaussEx}$ to be
$\mathsf{LinRel}_{X\times\mathsf{Cov}}$.

In the style of Proposition 6.1, we can show that
$\mathsf{LinRel}_{X}\cong\mathsf{AffRel}^{+}$. This way the $\mathsf{LinRel}$
construction applied to Example 3.2 gives rise to the top row of (6).

###  3.3 Relationships between the Constructions

The constructions $\mathsf{Lin}$ and $\mathsf{LinRel}$ are themselves
functorial:

######  Proposition 3.8.

Let $S,T:\mathsf{Vec}\to\mathsf{CMon}$ be decoration functors and $\alpha:S\to
T$ a natural transformation. Then we have induced monoidal identity-on-objects
functors

$\displaystyle F_{\alpha}$
$\displaystyle:\mathsf{Lin}_{S}\to\mathsf{Lin}_{T},(f,s)\mapsto(f,\alpha(s))$
$\displaystyle G_{\alpha}$
$\displaystyle:\mathsf{LinRel}_{S}\to\mathsf{LinRel}_{T},(D,f,s)\mapsto(D,f,\alpha(s))$

which preserve copy and delete structure.

This proposition accounts for all functors in the diagram (6) except for
$\mathsf{supp}$. For every decoration functor $S$, we obtain an inclusion
functor $\mathsf{Lin}_{S}\to\mathsf{LinRel}_{S}$ by choosing locus of
nondeterminism $D=0$, that is forming the composite

${{\mathsf{Lin}_{S}}}$${{\mathsf{Lin}_{S\times\mathsf{Sub}}}}$${{\mathsf{LinRel}_{S}}}$$\scriptstyle{F_{\mathsf{id}\times
0}}$

From a decorated linear relation, we can extract its locus of nondeterminism
by forgetting the decoration via the following composite:

${{\mathsf{LinRel}_{S}}}$${{\mathsf{LinRel}_{0}\cong\mathsf{LinRel}^{+}}}$$\scriptstyle{G_{0}}$

Recall that the support of a Gaussian distribution $\mathcal{N}(\mu,\Sigma)$
on $\mathbb{R}^{n}$ is the affine space $\mu+\mathsf{col}(\Sigma)$. This
construction can be extended functorially to $\mathsf{Gauss}$ as follows:

######  Proposition 3.9.

We have a functor $\mathsf{supp}:\mathsf{Gauss}\to\mathsf{AffRel}^{+}$ which
takes the Gaussian noise to its support. Concretely on $\mathbb{R}^{n}$,
$\mathsf{supp}(x\mapsto Ax+\mathcal{N}(\mu,\Sigma))=(x\mapsto
Ax+\mu+\mathsf{col}(\Sigma))$.

######  Proof 3.10.

We construct $\mathsf{supp}$ as the composite

${{\mathsf{Lin}_{X\times\mathsf{Cov}}}}$${{\mathsf{Lin}_{X\times\mathsf{Sub}}}}$${{\mathsf{LinRel}_{X}\cong\mathsf{AffRel}^{+}}}$$\scriptstyle{F_{\mathsf{id}_{X}\times\alpha}}$

where $\alpha_{X}:\mathsf{Cov}(X)\to\mathsf{Sub}(X)$ is takes the covariance
form $\sigma$ to the annihilator of its kernel. Naturality is nontrivial and
can be shown using the Cholesky composition, which makes use of the positive
semidefiniteness of $\sigma$.

###  3.4 Conditioning

Our presentation of extended Gaussians is not measure-theoretic. Categorical
probability theory [9, 6, 21] is a general language of probability which
allows one to precisely state concepts such as determinism, independence and
support in an abstract way without relying on a measure-theoretic formalism.
An important notion is that of a _conditional distribution_ [9, Section 11],
which means we can break the sampling of a joint distribution $(X,Y)\sim\psi$
into two parts: We first sample $X\sim\psi_{X}$ from the marginal, and then
sample $Y$ dependently on $X$, $Y\sim\psi_{|X}(X)$. Conditional distributions
are extended to _conditionals_ of arbitrary morphisms $f:A\to X\otimes Y$ in
[9, Definition 11.5], which is a morphism $f_{|X}:X\otimes A\to Y$ satisfying
the equation $f(x,y|a)=f_{|X}(y|x,a)f_{X}(y|a)$ in the appropriate categorical
sense.

Conditionals in $\mathsf{Gauss}$ exist and are given by the usual conditional
distributions [9, 11.8]. $\mathsf{LinRel}^{+}$ also has conditionals, which
are essentially given by re-ordering a relation $R\subseteq A\times(X\times
Y)$ to a relation $R_{|X}\subseteq(X\times A)\times Y$. This $R_{|X}$ may not
be left-total yet, but any linear relation can be extended to a left-total one
outside of its domain.

We will now show that extended Gaussians have conditionals too: We proceed by
picking a convenient complement to the locus of nondeterminism $D$, which lets
reduce the proof to the existence of conditionals in $\mathsf{Gauss}$ and
$\mathsf{LinRel}^{+}$ separately. In fact, this strategy works for general
decorated linear relations:

######  Theorem 3.11.

If $\mathsf{Lin}_{S}$ has conditionals, so does $\mathsf{LinRel}_{S}$.

######  Proof 3.12.

Let $\varphi\in\mathsf{LinRel}_{S}(A,X\times Y)$ be represented modulo
$\approx$ by $(f,\mu,D)$ with $\mu\in S(X\times Y)$. By Lemma 6.5, there
exists a complement $K\subseteq X\times Y$ of $D$ such that $K_{X}$ is a
complement of $D_{X}$ in $X$. We then take $P_{K},P_{D}:X\times Y\to X\times
Y$ to be the projection endomorphisms, and replace the representative $\mu$ by
$S(P_{K})(\mu)$ and $f$ by $P_{K}f$ without affecting $\varphi$.

Now we consider $(f,\mu)\in\mathsf{Lin}_{S}(A,X\times Y)$ and find a
conditional $(g,\psi)\in\mathsf{Lin}_{S}(X\times A,Y)$. This means we can
obtain $(X_{1},Y_{1})\sim f(a)+\mu$ as

$\displaystyle X_{1}$ $\displaystyle\sim f_{X}(a)+\mu_{X}$ $\displaystyle
Y_{1}$ $\displaystyle\sim g(x,a)+\psi$

Similarly we can use conditionals in $\mathsf{LinRel}^{+}$ to find a linear
function $h:X\to Y$ and a subspace $H\subseteq Y$ such that $(X_{2},Y_{2})\sim
D$ can be obtained as

$\displaystyle X_{2}$ $\displaystyle\sim D_{X}$ $\displaystyle Y_{2}$
$\displaystyle\sim h(X_{2})+H$

Thus a joint sample $(X,Y)\sim\varphi(a)$ can be obtained as follows

$\displaystyle X_{1}$ $\displaystyle\sim f_{X}(a)+\mu_{X}$ $\displaystyle
X_{2}$ $\displaystyle\sim D_{X}$ $\displaystyle X$ $\displaystyle=X_{1}+X_{2}$
$\displaystyle Y_{1}$ $\displaystyle\sim g(X_{1},a)+\psi$ $\displaystyle
Y_{2}$ $\displaystyle\sim h(X_{2})+H$ $\displaystyle Y$
$\displaystyle=Y_{1}+Y_{2}$

Because we have chosen $K$ such that $K_{X}\oplus D_{X}=X$, we can extract the
values of $X_{1},X_{2}$ from $X$ via the projections $P_{K_{X}},P_{D_{X}}:X\to
X$ as $X_{1}=P_{K_{X}}(X)$ and $X_{2}=P_{D_{X}}(X)$. We can thus read off a
conditional for $\varphi$ by combining the two individual conditionals, namely

$\varphi|_{X}(x,a)=g(P_{K_{X}}(x),a)+h(P_{D_{X}}(x))+\psi+H$

######  Corollary 3.13.

$\mathsf{GaussEx}$ has all conditionals.

For example, the procedure of Theorem 3.11 recovers Example 2.1
systematically: In order to find a conditional distribution for the joint
distribution
$(Z_{1},Z_{2})\sim\mathcal{N}(0,I_{2})+\\{(x,x):x\in\mathbb{R}\\}$ on
$\mathbb{R}\times\mathbb{R}$, we choose the particular complement
$K=0\times\mathbb{R}$ of the diagonal and obtain desired decomposition

$\displaystyle Z_{1}$ $\displaystyle\sim\mathbb{R}$ $\displaystyle Z_{2}$
$\displaystyle\sim Z_{1}+\mathcal{N}(0,2)$

Conditioning on equality: In the context of Gaussian probability, we can
condition two random variables $U,V$ to be _exactly equal_ [22] by introducing
an auxiliary variable $Z=U-V$ for their difference, and computing the
conditional distribution $(U,V)|Z=0$. For example, if $U,V\sim N(0,1)$ are
independent standard normal variables, then the conditional distribution
$U|Z=0$ is $\mathcal{N}(0,0.5)$. Note that the variance of $U$ has shrunk
after conditioning, because $V$ is itself concentrated around $0$ and thus
induces a stronger concentration in $U$.

We can now formally show that our uniform extended Gaussians $u_{X}$
(Definition 2.9) are really uninformative in the sense that conditioning on
equality with a uniform variable does not change the prior:

######  Proposition 3.14.

For every extended Gaussian prior $\psi$ on $X$, if $U\sim\psi$ and $V\sim
u_{X}$, then $U|(U=V)$ still has distribution $\psi$.

######  Proof 3.15.

We introduce an auxiliary random variable $Z=U-V$ and show that the following
two joint distributions over $(U,V,Z)$ are equal:

$\begin{Bmatrix}U\sim\psi\\\ V\sim u_{X}\\\
Z=U-V\end{Bmatrix}=\begin{Bmatrix}Z\sim u_{X}\\\ U\sim\psi\\\
V=U-Z\end{Bmatrix}$

The right-hand side lets us read off the conditional on $Z$ immediately.
Conditioning on equality now means setting $Z=0$, after which we obtain
$U\sim\psi,V=U$.

##  4 Outlook and Discussion

We have defined a sound mathematical model for reasoning about Gaussian
distributions together with uninformative priors, and gained a new and
generalized perspective on linear relations. We gave a construction to combine
probability and nondeterminism on vector spaces, which is generally not
possible in a seamless way [11, 24].

The original motivation for this work comes from programming language theory
and the semantics of probabilistic programming [22]. While in statistics
literature, noisy observations are seen as the primary operation of interest,
our focus on singular covariances is natural from a logical or programming
perspective: Copying and conditioning are fundamental operations, however they
do lead to highly singular distributions. In [20], we defined a programming
language for Gaussian probability with a first-class exact conditioning
construct. In this setting, the existence of a uniform distribution is simply
asking for a unit for the conditioning operation. Further connections between
probability and logic arise both through the analogy with unification in logic
programming [18] and the relationship to linear relations (Section 3.3).
Linear relations have have been used in graphical reasoning and signal-flow
diagrams [5, 2, 3].

We believe that $\mathsf{GaussEx}$ solves the outstanding characterization of
$\mathsf{Cond}(\mathsf{Gauss})$ in [22]. The category
$\mathsf{Cond}(\mathsf{GaussEx})$ can express both logical (solving linear
equations) and probabilistic computation (conditioning Gaussians). We believe
that it is a hypergraph category [8], with conditioning as multiplication and
uniform distributions as units. [7] have suggested hypergraph categories as
the right setting for inference, and celebrated algorithms such as message-
passing inference can be formulated fully abstractly there [17]. We believe
that categorical probability theory is a compelling language to go beyond
measure-theoretic foundations and talking about statistical models and
conditioning in a high-level way. The systematic connection with probabilistic
programming languages is elaborated in [21].

We hope that extended Gaussians will also be useful outside of theoretical
computer science. Here we have introduced them primarily from an algebraic and
category-theoretic viewpoint and shown a duality theorem (Theorem 2.7) linking
them to certain types of quadratic forms. We would like to explore further
connections to statistics [16] and functional analysis: An important step is
to topologize the homsets of $\mathsf{GaussEx}$ and characterize which in
which ways Gaussian distributions can converge to extended Gaussians. Ideally
this would exhibit the construction of extended Gaussians as a form of
topological completion. The aspect of considering improper priors as limits of
normalized ones is treated in [1, 4].

It also seems interesting to consider extended Gaussians under the ‘principle
of transformation groups’ (e.g. [14]). In [22, VI], we remark that
$\mathsf{Gauss}$ is essentially presented as a PROP by the invariance of the
standard normal distribution under the orthogonal group $O(n)$. We expect the
uniform distributions to be presented by invariance under all of
$\mathsf{GL}(n)$.

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##  5 Acknowledgements

It has been useful to discuss this work with many people. Particular thanks go
to Tobias Fritz, Bart Jacobs, Dusko Pavlovic, Sam Staton and Alexander
Terenin.

##  6 Appendix

###  6.1 Glossary: Linear Algebra

All vector spaces are assumed finite dimensional. For vector subspaces
$U,V\subseteq X$, their _Minkowski sum_ is the subspace $U+V=\\{u+v:u\in
U,v\in V\\}$. If furthermore $U\cap V=0$, we call their sum a _direct sum_ and
write $U\oplus V$. A _complement_ of $U$ is a subspace $V$ such that $U\oplus
V=X$. An _affine subspace_ $W\subseteq X$ is a subset of the form $x+U$ for
some $x\in X$ and a (unique) vector subspace $U\subseteq X$. The space $W$ is
called a _coset_ of $U$ and the cosets of $U$ organize into the quotient
vector space $X/U=\\{x+U:x\in X\\}$.

An affine-linear map $f:X\to Y$ between vector spaces is a map of the form
$f(x)=g(x)+y$ for some linear function $g:X\to Y$ and $y\in Y$. Vector spaces
and affine-linear maps form a category $\mathsf{Aff}$.

A _linear relation_ $R\subseteq X\times Y$ is a relation which is also a
vector subspace of $X\times Y$. An _affine relation_ $R\subseteq X\times Y$ is
a relation which is also an affine subspace of $X\times Y$. We write
$R(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\\{y\in Y:(x,y)\in R\\}$. A
relation $R\subseteq X\times Y$ is _left-total_ if $R(x)\neq\emptyset$ for all
$x\in X$.

Linear relations, affine relations and left-total relations are closed under
the usual composition of relations. We denote by $\mathsf{LinRel}^{+}$ and
$\mathsf{AffRel}^{+}$ the categories whose objects are vector spaces, and
morphisms are left-total linear and affine relations respectively. Those
categories are Markov categories (left-totality must be assumed for discarding
to be natural).

The following characterization underlies Definition 3.3: Every left-total
linear relation can be written as a ‘linear map with nondeterministic noise’
$x\mapsto f(x)+D$.

######  Proposition 6.1.

Let $R\subseteq X\times Y$ be a left-total linear relation. Then

  1. (1)

$R(0)$ is a vector subspace of $Y$

  2. (2)

$R(x)$ is a coset of $R(0)$ for every $x\in X$

  3. (3)

the assignment $x\mapsto R(x)$ is a well-defined linear map $X\to Y/R(0)$

  4. (4)

every linear map $X\to Y/D$ is of that form for a unique left-total linear
relation

######  Proof 6.2.

For 1, consider $y,y^{\prime}\in R(0)$ (by assumption nonempty), then by
linearity of $R$

$(0,y)\in R,(0,y^{\prime})\in R\Rightarrow(0,\alpha y+\beta y^{\prime})\in R$

so $R(0)$ is a vector subspace. For 2, we can find some $w\in R(x)$ and wish
to show that $R(x)=w+R(0)$. Indeed if $y\in R(x)$ then $(x,y)-(x,w)=(0,y-w)\in
R$ so $y-w\in R(0)$, hence $y\in w+R(0)$. Conversely for all $z\in R(0)$ we
have $(x,w+z)=(x,w)+(0,z)\in R$ so $w+z\in R(x)$. This completes the proof
that $R(x)$ is a coset. For 3, the previous point shows that the map
$\rho:x\mapsto R(x)$ is a well-defined map $X\to Y/R(0)$. It remains to show
it is linear. That is, if $w\in R(x)$ and $z\in R(y)$ then $\alpha w+\beta
z\in R(\alpha x+\beta y)$. This follows immediately from the linearity of $R$.
For the last point 4, given a linear map $f:X\to Y/V$ we construct the
relation

$(x,y)\in R\Leftrightarrow y\in f(x)$

which is left-total because $f(x)\neq\emptyset$. To see that $R$ is linear,
let $(x,y)\in R,(x^{\prime},y^{\prime})\in R$ meaning $y-z\in V$ and
$y^{\prime}-z\in V$ for representatives $z,z^{\prime}$ of
$f(x),f(x^{\prime})$. Linearity of $f$ means that $\alpha z+\beta z^{\prime}$
is a representative of $f(\alpha x+\beta x^{\prime})$. Thus

$\alpha y+\beta y^{\prime}-(\alpha z+\beta
z^{\prime})=\alpha(y-z)+\beta(y^{\prime}-z^{\prime})\in V$

#### Annihilators

######  Proposition 6.3.

  1. (1)

Taking annihilators is order-reversing and involutive

  2. (2)

If $D\subseteq S\subseteq X$, then $S^{\bot}\subseteq D^{\bot}\subseteq X^{*}$
and we have a canonical isomorphism

$(S/D)^{*}\cong D^{\bot}/S^{\bot}$ (7)

and similarly for $K\subseteq F\subseteq X^{*}$, we have

$(F/K)^{*}\cong K^{\bot}/F^{\bot}$ (8)

  3. (3)

We have

$(V+W)^{\bot}=V^{\bot}\cap W^{\bot}$ $(F\cap W)^{\bot}=F^{\bot}+G^{\bot}$

If $D\subseteq X$ and $f:X\to Y$, then

$(f[D])^{\bot}=\\{g\in Y^{*}:gf\in D^{\bot}\\}$

If $U\subseteq X,V\subseteq Y$, we have a canonical isomorphism

$(U\times V)^{\bot}\cong U^{\bot}\times V^{\bot}$

######  Proof 6.4.

Standard. An explicit description of the canonical iso (7) is given as
follows.

  1. (1)

We define $\alpha:D^{\bot}/S^{\bot}\to(S/D)^{*}$ as follows. If $f\in
D^{\bot}$, then $f$ is a function $X\to\mathbb{R}$ such that $f|_{D}=0$. The
restriction $f|_{S}:S\to\mathbb{R}$ thus descends to the quotient
$S/D\to\mathbb{R}$, and we let $\widetilde{\alpha}(f)=f|_{S}$. To check this
is well-defined, notice that the kernel of $\widetilde{\alpha}$ consists of
those $f\in X^{*}$ such that $f|_{S}=0$, that is $S^{\bot}$.

  2. (2)

We define $\alpha^{-1}:(S/D)^{*}\to D^{\bot}/S^{\bot}$ as follows. An element
$f\in(S/D)^{*}$ is a function $f:S\to\mathbb{R}$ with $S|_{D}=0$. Find any
extension of $f$ to a linear function $\bar{f}:X\to\mathbb{R}$ (such an
extension exists because $S$ is a retract of $X$). Then still
$\bar{f}|_{D}=0$, so $\bar{f}\in D^{\bot}$. It remains to show that the choice
of extension does not matter in the quotient $D^{\bot}/S^{\bot}$. Indeed if
$\bar{f}_{2}$ is another extension, then $(\bar{f}-\bar{f}_{2})|_{S}=f-f=0$,
hence $(\bar{f}-\bar{f}_{2})\in S^{\bot}$.

###  6.2 Conditioning

The proof of the existence of conditionals in $\mathsf{LinRel}_{S}$ proceeds
by picking a good complement to the locus of nondeterminism $D\subseteq
X\times Y$ as follows:

######  Lemma 6.5.

Let $V\subseteq X\times Y$ be a vector subspace, and let $V_{X}\subseteq X$ be
its projection. Then there exists a complement $K\subseteq X\times Y$ of $V$
such that $K_{X}$ is a complement of $V_{X}$.

######  Proof 6.6.

We give an explicit construction, where in fact we can choose $K$ to be a
cartesian product of subspaces $U\times W$. Define

$V_{X}=\\{x:(x,y)\in V\\}\qquad H=\\{y:(0,y)\in V\\}$

We argue that if $U\oplus V_{X}=X$ and $W\oplus H=Y$, then $(U\times W)\oplus
V=X\times Y$. First we prove that $(U\times W)\cap V=0$: Indeed, if $(u,w)\in
V$ for $u\in U,w\in W$, then $u\in V_{X}$, but that implies $u=0$. So we know
$(0,w)\in V$, i.e. $w\in H$. Thus $w=0$.

It remains to show that we can write every $(x,y)$ as $(u+v_{1},w+v_{2})$ with
$u\in U,w\in W$ and $(v_{1},v_{2})\in V$.

  1. (1)

We can write $x=u+v_{1}$ with $u\in U$ and $v_{1}\in V_{X}$.

  2. (2)

We claim that there exists a $b\in W$ such that $(v_{1},b)\in V$. Because
$v_{1}\in V_{X}$, there exists some $b^{\prime}\in Y$ such that
$(v_{1},b^{\prime})\in V$. We now decompose $b^{\prime}=b+h$ for $b\in W,h\in
H$. By definition of $H$, we have $(0,h)\in V$, so
$(v_{1},b)=(v_{1},b^{\prime})-(0,h)\in V$.

  3. (3)

Write $y=w^{\prime}+h$ with $w^{\prime}\in W,h\in H$ and define
$w=w^{\prime}-b$ and $v_{2}=h+b$. Then we have $w\in W$ and
$(v_{1},v_{2})=(v_{1},b)+(0,h)\in V$, and as desired

$(u,w)+(v_{1},v_{2})=(x,w^{\prime}-b+h+b)=(x,w^{\prime}+h)=(x,y).$

###  6.3 Composition and Congruence

For the construction of $\mathsf{LinRel}_{S}$, it remains to check that the
relation $\approx$ is a monoidal congruence on
$\mathsf{Lin}_{S\times\mathsf{Sub}}$. Recall that $(f,s,U)\approx(g,t,U)$ if
and only if $\pi f=\pi g$ and $S(\pi)(s)=S(\pi)(t)$ where $\pi=\pi_{Y/U}:Y\to
Y/U$ is the quotient map.

Transitivity: Let $(f,r,V)\approx(g,s,V)\approx(h,t,V)$ meaning $\pi f=\pi
g=\pi h$ and $S(\pi)(r)=S(\pi)(s)=S(\pi)(t)$. Then clearly also
$(f,r,V)\approx(h,t,V)$.

Congruence: Let $f_{i}:Y\to Z$, $g_{i}:X\to Y$ and $U\subseteq Y,V\subseteq Z$
be given for $i=1,2$, and assume that $(f_{1},r_{1},V)\approx(f_{2},r_{2},V)$
and $(g_{1},s_{1},U)\approx(g_{2},s_{2},U)$. We need to show that
$(f_{1}g_{1},r_{1}+S(f_{1})(s_{1}),V+f_{1}[U])\approx(f_{2}g_{2},r_{2}+S(f_{2})(s_{2}),V+f_{2}[U])$.

Firstly, we need that $W=V+f_{1}[U]=V+f_{2}[U]$ is well-defined. Let
$\pi=\pi_{Y/V}$ then by assumption $\pi f_{1}=\pi f_{2}$, so
$f_{1}(y)-f_{2}(y)\in V$ for all $y\in Y$. Hence $f_{1}[U]+V=f_{2}[U]+V$.

Now, we need to show that $f_{1}g_{1}(x)-f_{2}g_{2}(x)\in W$. We know that
$f_{1}(x)-f_{2}(x)\in V$ and $g_{1}(y)-g_{2}(y)\in U$, hence

$\displaystyle
f_{2}(g_{2}(x))-f_{1}(g_{1}(x))=\underbrace{f_{1}(g_{2}(x)-g_{1}(x))}_{\in
f_{1}[U]}+\underbrace{(f_{2}(g_{2}(x))-f_{1}(g_{2}(x)))}_{\in V}\in W$

For the decorations, we know by assumption that
$S(\pi_{Y/U})(s_{1})=S(\pi_{Y/U})(s_{2})$ and
$S(\pi_{Z/V})(r_{1})=S(\pi_{Z/V})(r_{2})$. We need to show that
$S(\pi_{Z/W})(r_{1}+S(f_{1})(s_{1}))=S(\pi_{Z/W})(r_{2}+S(f_{2})(s_{2}))$.

The composites $Y\xrightarrow{f_{i}}Z\xrightarrow{\pi_{Z/W}}Z/W$ are equal for
$i=1,2$ and vanish on $U$, thus descending to a map $\tilde{f}:Y/U\to Z/W$,
and we obtain

$\displaystyle S(\pi_{Z/W})(S(f_{1})(s_{1}))$
$\displaystyle=S(\pi_{Z/W}f_{1})(s_{1})$
$\displaystyle=S(\tilde{f}\pi_{Y/U})(s_{1})$
$\displaystyle=S(\tilde{f})(S(\pi_{Y/U})(s_{1}))$
$\displaystyle=S(\tilde{f})(S(\pi_{Y/U})(s_{2}))$
$\displaystyle=S(\tilde{f}\pi_{Y/U})(s_{2})$
$\displaystyle=S(\pi_{Z/W}f_{2})(s_{2})$
$\displaystyle=S(\pi_{Z/W})(S(f_{2})(s_{2}))$

The desired proposition now follows easily from the additivity of
$S(\pi_{Z/W})(-)$.

Tensor: Let $f_{i}:X^{\prime}\to X$, $g_{i}:Y^{\prime}\to Y$ and $U\subseteq
X,V\subseteq Y$ be given for $i=1,2$ and assume
$(f_{1},s_{1},U)\approx(f_{2},s_{2},U)$ and
$(g_{1},t_{1},V)\approx(g_{2},t_{2},V)$. We need to show that $(f_{1}\times
g_{1},s_{1}\oplus t_{1},U\times V)\approx(f_{2}\times g_{2},s_{2}\oplus
t_{2},U\times V)$.

It is immediate that $(f_{1}(x),g_{1}(y))-(f_{2}(x),g_{2}(x))\in U\times V$
for all $x\in X^{\prime},y\in Y^{\prime}$. For the sum of decorations, we
chase them around the diagram

${X}$${{X\times Y}}$${{(X\times Y)/(U\times V)}}$${{X\times
Y}}$${Y}$${{X/U}}$${{(X/U)\times(Y/V)}}$${{Y/V}}$$\scriptstyle{\pi_{X/U}}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{\pi_{Y/V}}$$\scriptstyle{i_{X}}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{i_{Y}}$${\cong}$

to notice that $S(\pi)(S(i_{X})(s_{i}))$ depends only on
$S(\pi_{X/U})(s_{i})$.


aainstitutetext: Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, Chinabbinstitutetext: School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China

# Migdal effect of Phonon-mediated neutrino nucleus scattering in
semiconductor detectors

Yu-Feng<EMAIL_ADDRESS>a,b Shuo-yu<EMAIL_ADDRESS>(corresponding author)

###### Abstract

In this work, we introduce the theoretical framework of the phonon-mediated
Migdal effect for neutrino-nucleus scattering in semiconductors, considering
both the Standard Model and the presence of the neutrino magnetic moment. We
calculate the rate of electron excitation resulting from the Migdal effect and
observe a substantial coherent enhancement compared to ordinary neutrino-
electron scattering. Furthermore, we provide numerical sensitivities for
detecting the Migdal effect and constraints on the neutrino magnetic moment,
utilizing an experimental setup with reactor neutrinos.

##  1 Introduction

The search for dark matter has seen tremendous progress in the past decade
SuperCDMS:2018mne ; EDELWEISS:2020fxc ; SENSEI:2020dpa ; LZ:2022ufs ;
XENON:2018voc ; XENON:2020fgj  and the direct detection experiments play
crucial roles in searching for the dark matter elastic scattering on targets.
There have been notable advances in the direct detection techniques with
various target materials such as semiconductors Liang:2021zkg ; Liang:2018bdb
; Graham:2012su ; Essig:2015cda ; Hochberg:2016sqx , superfluid helium
Caputo:2019xum ; Caputo:2019cyg ; Knapen:2016cue  and superconductors
Hochberg:2015pha ; Hochberg:2016ajh . The direct detection experiments have
shown a high degree of sensitivity to the nuclear recoil signals lower than
100 keV with rather high target masses and excellent shielding from cosmic ray
muons. With these excellent features, the direct detection experiments, which
share many similarities with underground neutrino detectors, can serve as
useful tools for the precise test of the electroweak theory Papoulias:2017qdn
; Cadeddu:2018izq , measuring the neutrino electromagnetic properties
Cadeddu:2018dux ; Cadeddu:2019eta ; Kim:2021lun  and searching for the
evidence of nonstandard neutrino interactions beyond the Standard Model (SM)
Coloma:2017ncl ; Liao:2017uzy ; Denton:2018xmq ; AristizabalSierra:2018eqm ;
Giunti:2019xpr . Furthermore, direct detection experiments with extremely low
energy thresholds like SENSEI SENSEI:2020dpa  and EDELWEISS EDELWEISS:2017uga
may serve as excellent platforms to explore novel neutrino interactions at the
very low energy scale.

The Migdal effect was first proposed by A.B. Migdal in 1939 migdal:1939svj  as
the process where the emission of ionized electrons from suddenly struck
nucleus. Just recently in 2017, the authors of Ref. Ibe:2017yqa  have provided
a theoretical clarification that the Migdal effect can be triggered more
easily than expected. Subsequently, the theoretical description of the Migdal
effect has been significantly improved Essig:2019xkx ; Bell:2019egg ;
Dolan:2017xbu ; Berghaus:2022pbu . In addition, the Migdal effect has garnered
considerable attention in multiple physical fields including dark matter
detection, nuclear physics and particle physics in recent years
Gelmini:2020xir ; Knapen:2020aky ; Liang:2020ryg ; Liang:2019nnx ;
Liang:2022xbu ; Essig:2019xkx . Although a direct observation of the Migdal
effect in the nucleus collision has not been achieved, various attempts and
proposals to conduct the measurements are already under way, utilizing either
the neutron nucleus scattering Berghaus:2022pbu ; Bell:2021ihi ;
Araujo:2022wjh ; Cox:2022ekg  or the dark matter direct detection experiments
EDELWEISS:2022ktt ; LUX:2018akb ; CDEX:2019hzn ; EDELWEISS:2019vjv ;
XENON:2019zpr ; SENSEI:2020dpa ; Essig:2022dfa . Neutrinos can be a potential
tool to explore the parameter space of the Migdal effect, because the theory
of standard neutrino-nucleus scattering is well established Drukier:1984vhf ;
Patton:2012jr , and meanwhile the electron excitation from the Migdal effect
allows for distinguishing the corresponding events from coherent elastic
neutrino-nucleus scattering (CE$\nu$NS) without the Migdal effect
Tiffenberg:2017aac , which is a major component of the neutrino scattering
signal at such low energies. However, the corresponding Migdal excitation
still benefits from the coherent enhancement effect of the CE$\nu$NS process.

Semiconductor materials have significantly lower ionization thresholds of
$\mathcal{O}(1)\;{\rm eV}$ compared to the typical ionization threshold in
atoms of the order of $\mathcal{O}(10)\;{\rm eV}$. The electron excitation
signals induced by CE$\nu$NS can also be amplified with the Neganov-Trofimov-
Luke effect Neganov:1985 ; Luke:1976jap , which can help to probe various low-
energy phenomena related to the neutrino-nucleus interactions. Therefore,
semiconductors are considered as ideal targets for exploring the Migdal
effect. Extensive efforts have been made to improve the theoretical
understanding of the Migdal effect in semiconductors at low energy
Liang:2020ryg ; Liang:2019nnx ; Liang:2022xbu ; Essig:2019xkx . In particular,
the authors of Ref. Liang:2022xbu  have developed a multi-phonon model to
describe the Migdal effect induced by non-relativistic dark matter particles
in semiconductors at low energies, where the contribution of phonons cannot be
neglected. However, a proper theoretical description of the Migdal effect
induced by relativistic neutrinos in semiconductors, which is also an
important observation goal in the dark matter direct detection experiments, is
still lacking. In this work, we will extend the framework of phonon-mediated
Migdal effect in Ref. Liang:2022xbu  to neutrino-nucleus scattering in
semiconductor and calculate the corresponding event rate in silicon with
reactor neutrino flux. The contribution of the Migdal effect with reactor
neutrinos are investigated and we present the corresponding sensitivity with
silicon as target. We will also explore the contribution of the Migdal effect
in the presence of the neutrino magnetic moment Cisneros:1970nq ;
Barbieri:1988nh ; Ma:2001mr ; Beda:2012zz ; Super-Kamiokande:2004wqk ;
XENON:2020rca , which is known to exhibit significant enhancement at such a
low energy region.

This paper is organized as follows. Section II provides a complete description
of the theoretical framework of the neutrino-induced phonon-mediated Migdal
effect and the corresponding Migdal excitation rate in silicon with reactor
neutrinos as a source. In Section III, we analyze the associated sensitivity
of the Migdal excitation as well as the detection potential of the neutrino
magnetic moment with the Migdal effect based on the silicon target. Finally we
give the concluding remarks and discussions in Section IV.

##  2 Neutrino induced Migdal effect as a multi-phonon process

In this section we derive the formulation of the cross section contributed by
the Migdal effect induced by the CE$\nu$NS in semiconductor materials under
the multi-phonon framework. Since the energy region we are concerned is a few
tens of eV and the collective behavior of electrons and nuclei can not be
neglected, the concept of a freely-recoiling nucleus no longer holds. However,
the corresponding description in the context of phonons can be naturally
extended to such low energies. To describe the neutrino induced Migdal effect
at this energy region, the effects of the incident neutrino can be
approximately regarded as a perturbation on the lattice system of the target
material Liang:2021zkg . This approach has already been theoretically
illustrated in Ref. Li:2023jhep , which utilizes the framework of the non-
relativistic effective field theory (NR EFT) to describe the elastic neutrino-
electron scattering (E$\nu$ES). The Lagrangian for CE$\nu$NS induced by
reactor electron neutrinos can be written as

$\mathcal{L}_{\rm CE\nu NS}=-i\frac{\sqrt{2}}{2}G_{\rm
F}[g_{n}NF_{N}^{2}(\bm{q})+g_{p}ZF_{Z}^{2}(\bm{q})]{\nu}_{e}\gamma^{\mu}(1-\gamma^{5})\bar{\nu}_{e}\bar{\mathcal{N}}\gamma_{\mu}\mathcal{N}\,,$
(2.1)

where $\mathcal{N}$ is the nucleus and $\bar{\nu}_{e}$ is the electron
neutrino field from the nuclear fission of reactors. The constants to describe
the couplings between the neutrino and neutron and proton are $g_{n}=1/2$ and
$g_{p}=-1/2+2\sin^{2}\theta_{W}$ respectively. Since the energy region we are
concerned is lower than 1 keV, the coherence effects plays an important role
in enhancing the interaction cross section and the neutron number $N$ and
proton number $Z$ coherently contribute to the scattering process. The form
factor $F_{N,Z}^{2}(\bm{q})$ can be also approximately considered as 1 at this
energy region. With a similar way with that described in Ref Li:2023jhep , the
above Lagrangian can be written as an effective one in the energy region that
we are concerned

$\mathcal{L}_{\rm eff}=-i\frac{\sqrt{2}}{2}G_{\rm
F}[N-Z(1-4\sin^{2}\theta_{W})]\nu_{e,L}\gamma^{0}\bar{\nu}_{e,L}\phi_{+}^{\dagger}\phi_{+}\,,$
(2.2)

where $\phi_{+}$ is the non-relativistic effective operator of the target
nucleus. It can be proved that the above effective Lagrangian is identical for
all flavor neutrinos. From this expression, we can calculate the corresponding
effective potential that contributes to the perturbation on the nucleus system

$\mathcal{V}_{\rm CE\nu NS}=-i\frac{\sqrt{2}}{2}G_{\rm
F}[N-Z(1-4\sin^{2}\theta_{W})]\,.$ (2.3)

Following the Feynman rules for the multi-phonon process summarized in Ref
Liang:2021zkg , the amplitude of the neutrino induced Migdal effect as a
multi-phonon process in the framework of the zero-temperature QFT can be
written as

$\displaystyle i\mathcal{M}$ $\displaystyle=(-i)\mathcal{V}_{\rm CE\nu
NS}(\bm{q})Ne^{-W(\bm{q})}\prod_{s=1}^{n}\left(\frac{-i\bm{q}\cdot\bm{\epsilon}_{\bm{k}_{s},\alpha_{s}}}{\sqrt{2Nm_{N}\omega_{\bm{k}_{s},\alpha_{s}}}}\right)$
(2.4)
$\displaystyle\times\sum_{\bm{G}^{\prime}}\sum_{\bm{k},\alpha}\left(\frac{-i\bm{q}\cdot\bm{\epsilon}_{\bm{k},\alpha}}{\sqrt{Nm_{N}}}\right)\left\\{\left[\frac{i}{(\varepsilon_{i}-\varepsilon_{j})^{2}-\omega^{2}_{\bm{k},\alpha}}\right]\left[\frac{i(\bm{k}+\bm{G}^{\prime})\cdot\bm{\epsilon}_{\bm{k},\alpha}}{\sqrt{Nm_{N}}}\right]\right.$
$\displaystyle\left.\times\left[\frac{-iNZ_{ion}4\pi\alpha_{e}}{V|\bm{k}+\bm{G}^{\prime}|^{2}}\right]<i|e^{i(\bm{k}+\bm{G}^{\prime})\cdot\hat{\bm{x}}}|j>\right\\}\sum_{\bm{G}}\delta_{\sum_{s}\bm{k}_{s}+\bm{G}+\bm{k},\bm{G}}\,,$

where $m_{N}$ is the mass of the target nucleus and $e^{-W(\bm{q})}$ is the
Debye-Waller factor Debye:1913adp ; Waller:1923zfp  that describes the effect
of the quantum and thermal uncertainties of the lattice. The first line
accounts for the process where the perturbation of the incident neutrino
excites a multi-phonon excitation on the nucleus system with $\bm{q}$ is the
transferred momentum. The remaining part accounts for the process where the
phonon induced by the incident neutrino leads to the excitation of a band
electron and explains the electron excitation in the Migdal effect in the view
of a multi-phonon model. The momentum of electron is written in the form of a
momentum $\bm{k}$ within a Bloch Zone (BZ) and reciprocal lattice vectors
$\bm{G}$ and $\bm{G}^{\prime}$. Since the target material we have considered
is the semiconductor, the band gap $\varepsilon_{i}-\varepsilon_{j}$ is much
larger than the phonon eigenenergies $\omega_{\bm{k},\alpha}$ and the
contraction relation
$\sum_{\alpha}(\bm{q}\cdot\bm{\epsilon}_{\bm{k},\alpha})\cdot[(\bm{k}+\bm{G}^{\prime})\cdot\bm{\epsilon}_{\bm{k},\alpha}]=\bm{q}\cdot(\bm{k}+\bm{G}^{\prime})$
can be applied to simplify the amplitude

$\displaystyle i\mathcal{M}=$ $\displaystyle(-i\mathcal{V}_{\rm CE\nu
NS}(\bm{q})Ne^{-W(\bm{q})}\prod_{s=1}^{n}\left(\frac{-i\bm{q}\cdot\bm{\epsilon}_{\bm{k}_{s},\alpha_{s}}}{\sqrt{2Nm_{N}\omega_{\bm{k}_{s},\alpha_{s}}}}\right)$
(2.5) $\displaystyle\times$
$\displaystyle\sum_{\bm{G}^{\prime}}\sum_{\bm{k}}\left(\frac{\bm{q}\cdot(\bm{k}+\bm{G}^{\prime})}{m_{N}}\right)\left[\frac{1}{(\varepsilon_{i}-\varepsilon_{j})^{2}}\right]\left[\frac{Z_{ion}4\pi\alpha_{e}}{V|\bm{k}+\bm{G}^{\prime}|^{2}}\right]<i|e^{i(\bm{k}+\bm{G}^{\prime})\cdot\hat{\bm{x}}}|j>$
$\displaystyle\times$
$\displaystyle\sum_{\bm{G}}\delta_{\sum_{s}\bm{k}_{s}+\bm{G}+\bm{k},\bm{G}}\,.$

With the relations of the neutrino flux $\Phi(E_{\nu})$ and recoil energy
$E_{R}(\bm{q})=q^{2}/(2m_{N})$, the resulting event rate for the neutrino
induced Migdal effect can be written as

$\displaystyle R$
$\displaystyle=\frac{N}{V^{2}}Z^{2}\sum_{\bm{q}}2|{\mathcal{V}_{\rm CE\nu
NS}}|^{2}\int\frac{\mathop{}\\!\mathrm{d}\omega}{\omega^{4}}\frac{\mathop{}\\!\mathrm{d}\Phi(E_{\nu})}{\mathop{}\\!\mathrm{d}E_{\nu}}\frac{\mathop{}\\!\mathrm{d}E_{\nu}\mathop{}\\!\mathrm{d}\cos\theta}{2}$
(2.6)
$\displaystyle\times\sum_{\left\\{n_{i}\right\\}}\frac{e^{-\frac{E_{R}(\bm{q})}{3N\omega_{1}}}}{n_{1}!}\left(\frac{E_{R}(\bm{q})}{3N\omega_{1}}\right)^{n_{1}}\dots\dots\frac{e^{-\frac{E_{R}(\bm{q})}{3N\omega_{{3N}}}}}{n_{3N}!}\left(\frac{E_{R}(\bm{q})}{3N\omega_{3N}}\right)^{n_{3N}}\delta(\sum_{i=1}^{3N}n_{i}\omega_{1}+\omega_{pp^{\prime}})$
$\displaystyle\times\sum_{\bm{G}^{\prime},\bm{G}^{\prime\prime}}\sum_{\bm{k}^{\prime}\in
1\rm{BZ}}\frac{\bm{q}\cdot(\bm{k}+\bm{G}^{\prime})}{m_{N}}\frac{\bm{q}\cdot(\bm{k}+\bm{G}^{\prime\prime})}{m_{N}}\rm{Im}\left[\epsilon^{-1}_{\bm{G}^{\prime},\bm{G}^{\prime\prime}}(\bm{k},\omega)\right]\tilde{V}_{\bm{G}^{\prime},\bm{G}^{\prime\prime}}^{cou}(\bm{k})\,,$

where $n_{i}$ describes the occupation number of the energy $\omega_{i}$ and
$\rm{Im}[\epsilon^{-1}_{\bm{G}^{\prime},\bm{G}^{\prime\prime}}(\bm{k},\omega)]$
is the energy loss function of the material. The number of the valence
electrons $Z$ depends on the material and $N$ is the number of nuclei in the
target material. The effective Coulomb potential is defined as
$\tilde{V}_{\bm{G}^{\prime},\bm{G}^{\prime\prime}}^{cou}(\bm{k})={4\pi\alpha}/{(|\bm{k}+\bm{G}^{\prime}|\cdot|\bm{k}+\bm{G}^{\prime\prime}|)}$
and $\omega$ is the deposited energy in the lattice. The second line is the
scattering function $S(q,E)$, which can be rewritten in the following form

$\displaystyle
S(q,E)=e^{-E_{R}(\bm{q})\sum_{i}\frac{1}{N\omega_{i}}}\sum_{+\infty}^{n=0}\frac{E_{R}(\bm{q})^{n}}{n!}T_{n}(-E_{\nu}^{\prime}+E_{\nu}-\omega)\,,$
(2.7)

where

$\displaystyle T_{n}(\omega)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{+\infty}f(t)^{n}e^{-i\omega
t}\mathop{}\\!\mathrm{d}t\,,$ (2.8) $\displaystyle f(t)$
$\displaystyle=\sum_{i=1}^{3N}\frac{1}{3N}\frac{e^{i\omega_{i}t}}{\omega_{i}}\,.$

The scattering function $S(q,E)$ is a combined Poisson distribution and it can
be proved that it converges to a Gaussian form at large $q$ region with the
center at $E_{R}(q)$ Liang:2022xbu . To further simplify the calculation, we
define $E=-E_{\nu}^{\prime}+E_{\nu}-\omega$ and with
$E_{\nu}^{\prime}=\sqrt{E_{\nu}^{2}-2qE_{\nu}\cos\theta+q^{2}}$ the
integration of $T_{n}(-E_{\nu}^{\prime}+E_{\nu}-\omega)$ can be rewritten as

$\displaystyle\int\mathop{}\\!\mathrm{d}\cos\theta
T_{n}(-\sqrt{E_{\nu}^{2}-2qE_{\nu}\cos\theta+q^{2}}+E_{\nu}-\omega)$ (2.9)
$\displaystyle=$
$\displaystyle\int_{0}^{2E_{\nu}-q-\omega(q>E_{\nu}),\,q-\omega(q<E_{\nu})}\mathop{}\\!\mathrm{d}E\left|\frac{\mathop{}\\!\mathrm{d}\cos\theta}{\mathop{}\\!\mathrm{d}E}\right|T_{n}(E)$
$\displaystyle=$
$\displaystyle\int_{0}^{2E_{\nu}-q-\omega(q>E_{\nu}),\,q-\omega(q<E_{\nu})}\mathop{}\\!\mathrm{d}E\frac{-E-\omega+E_{\nu}}{E_{\nu}q}T_{n}(E)\,,$

It can be found that $E_{R}(q)=q^{2}/2M$ is much smaller than the upper limit
of the above integration under most conditions and the Gaussian function
degenerates into a delta function $\delta(E-q^{2}/2M)$ as a result. Therefore,
the event rate can be simplified as

$\displaystyle R$ $\displaystyle=\frac{2\alpha NZ^{2}}{3\pi
m_{N}^{2}\Omega}|{\mathcal{V}_{\rm CE\nu
NS}}|^{2}\int{q^{3}\mathop{}\\!\mathrm{d}q}\int\frac{\mathop{}\\!\mathrm{d}\omega}{\omega^{4}}\mathcal{F}(\omega)\frac{\mathop{}\\!\mathrm{d}\Phi(E_{\nu})}{\mathop{}\\!\mathrm{d}E_{\nu}}{\mathop{}\\!\mathrm{d}E_{\nu}}$
(2.10)
$\displaystyle\times\int_{0}^{2E_{\nu}-q-\omega(q>E_{\nu}),\,q-\omega(q<E_{\nu})}\frac{-E-\omega+E_{\nu}}{E_{\nu}q}\delta(E-q^{2}/2M)\mathop{}\\!\mathrm{d}E\,,$

where we have employed the the assumption that the target material is
isotropic and $\Omega$ is the volume of the unit cell. Following Ref.
Liang:2020ryg , it is convenient to define the non-dimensional averaged energy
loss function $\mathcal{F}(\omega)$, which takes the form of

$\mathcal{F}(\omega)=\sum_{\bm{G}^{\prime},\bm{G}^{\prime\prime}}\int_{1\rm{BZ}}\frac{\Omega\mathop{}\\!\mathrm{d}^{3}{k}}{(2\pi)^{3}}\frac{(\bm{k}+\bm{G}^{\prime})\cdot(\bm{k}+\bm{G}^{\prime\prime})}{|\bm{k}+\bm{G}^{\prime}|\cdot|\bm{k}+\bm{G}^{\prime\prime}|}\rm{Im}\left[\epsilon^{-1}_{\bm{G}^{\prime},\bm{G}^{\prime\prime}}(\bm{k},\omega)\right]\,.$
(2.11)

Since the energies of the present neutrino source range from a few hundreds of
keV to tens of MeV, it is reasonable to make the assumption that $E_{\nu}\gg
q^{2}/2M$ and $E_{\nu}\gg\omega$ and then the event rate can be written as

$\displaystyle R$ $\displaystyle=\frac{\alpha NZ^{2}}{3\pi
m_{N}^{2}\Omega}G_{F}^{2}[N-Z(1-4\sin^{2}\theta_{W})]^{2}\int\frac{\mathop{}\\!\mathrm{d}\Phi(E_{\nu})}{\mathop{}\\!\mathrm{d}E_{\nu}}\mathop{}\\!\mathrm{d}E_{\nu}\int
q^{3}\mathop{}\\!\mathrm{d}q\int\frac{\mathop{}\\!\mathrm{d}\omega}{\omega^{4}}\mathcal{F}(\omega)\,.$
(2.12)

It should be noted that the event rate contributed from the Migdal effect
suffers from electromagnetic suppression, which makes it significantly lower
than the CE$\nu$NS event rate without the electron excitation. However, the
excitation of the additional electron from the Migdal effect is expected to be
observed in silicon CCD detectors like SENSEI SENSEI:2020dpa ;
Tiffenberg:2017aac  and the CE$\nu$NS signals without the electron excitation
can be distinguished from those from the Migdal effect. Therefore, the primary
source of the backgrounds for the neutrino induced Migdal events is
anticipated to be the E$\nu$ES signal, given that the detectors are usually
well shielded from cosmic rays and the neutrino flux dominates the signal
sources of dark matter direct detection experiments. Compared to the E$\nu$ES
signal, the Migdal event rate benefits from the coherent effect and is
therefore expected to be significantly improved, making the corresponding
detection substantially easier. In addition, by deploying the semiconductor
material in an electric field charges will drift through the material, which
results in the fully conversion of the work done by the electric field into
phonons based on the Neganov-Trofimov-Luke effect Neganov:1985 ; Luke:1976jap
. The electron excitation signals induced by the Migdal effect can therefore
be dramatically amplified. With these unique features, the Migdal excitation
can serve as a valuable tool for the low energy phenomenology in the context
of neutrino-nucleus interaction. Specifically, in this work we will apply to
the study of the neutrino magnetic moment, which has attracted considerable
efforts over the past few decades. The effective potential contributed by the
neutrino magnetic moment in CE$\nu$NS can be written in a similar way with
that in E$\nu$ES, which can be derived as Li:2023jhep

$\displaystyle\mathcal{V}_{\rm
mag}=Z\mu_{\nu_{e}}\frac{E_{\nu}}{m_{e}}\frac{8\pi\alpha}{q^{2}}\,,$ (2.13)

where $\mu_{\nu_{e}}$ is the magnetic moment of the electron neutrino. Since
in the model we have employed the scattering process is regarded as a
perturbation on the lattice system, we can calculate the event rate from the
contribution of the neutrino magnetic moment by simply replacing the potential
of the standard neutrino-nucleus scatting $\mathcal{V}_{\rm CE\nu NS}$ with
$\mathcal{V}_{\rm mag}$ in the above derivation as

$\displaystyle R$ $\displaystyle=\mu_{\nu_{e}}^{2}\frac{128\pi
N\alpha^{3}Z^{4}}{3m_{N}^{2}m_{e}^{2}\Omega}\int
E_{\nu}^{2}\frac{\mathop{}\\!\mathrm{d}\Phi(E_{\nu})}{\mathop{}\\!\mathrm{d}E_{\nu}}\mathop{}\\!\mathrm{d}E_{\nu}\int\frac{\mathop{}\\!\mathrm{d}q}{q}\int\frac{\mathop{}\\!\mathrm{d}\omega}{\omega^{4}}\mathcal{F}(\omega)\,.$
(2.14)

Figure 1: The differential excitation rates in the silicon target induced by
reactor neutrinos with the neutrino magnetic moment
$\mu_{\nu}=10^{-10}\mu_{\rm{B}}$ (red) and in the SM (blue). The neutrino
source of Taishan-1 reactor and a distance of 30 meters are employed for the
numerical calculations JUNO:2020ijm . The total excitation rates with the
Migdal effect are shown with the solid lines and those from E$\nu$ES in dash
lines. The maximal mean energies for one and two ionization electron signals
are shown with vertical dash dotted lines.

For illustration, we display the differential excitation rates induced by
reactor neutrinos via the E$\nu$ES and multi-phonon Migdal processes in Fig.
1. We present the results for the silicon target under the SM (blue) and in
the presence of the neutrino magnetic moment (red), which highlight the great
potential of the multi-phonon Migdal process to explore new physics beyond the
SM. The contributions from the Migdal effect are shown in solid lines and
those from E$\nu$ES are shown in dash lines. For illustration, we choose the
Taishan-1 reactor as the neutrino source and employ the distance at 30 meters,
which is a similar set with that of Taishan Antineutrino Observatory (TAO)
JUNO:2020ijm  with the inverse-beta-decay interaction as the primary target.
It should be noticed that the Migdal process exhibits a rapid decrease in the
excitation rate with the increasing of the excited energy, limiting the
capability of detectors to probe the relevant physics unless they can achieve
an exceedingly low threshold. We also label the maximal mean energies for one
and two ionization electron signals respectively, by using the mean energy per
electron-hole pair $\epsilon=3.8$ eV, the band gap energy for silicon $E_{\rm
gap}=1.2$ eV from Ref. Berghaus:2022pbu , and the relation

$Q=1+\large\lfloor\frac{\omega-{E}_{\rm gap}}{\epsilon}\large\rfloor\,,$
(2.15)

where $Q$ is the number of ionization electrons. As depicted in Fig 1, the
coherent effect significantly augments the excitation rates of the multi-
phonon Migdal process both in the SM and in the presence of the neutrino
magnetic moment. It should also be noted that the shape of the differential
excitation spectra of multi-phonon Migdal process remains the same in the SM
and in the presence of the neutrino magnetic moment, since the only energy
dependent parts of Eq. (2.12) and Eq. (2.14) are from the last integrals of
the averaged energy loss function $\mathcal{F}(\omega)$. All the other
components are calculated independently and contributed as an constant.

##  3 Numerical Sensitivities and Constraints

Figure 2: The sensitivities of the Migdal effect under different levels of the
energy threshold and exposure. the 5$\sigma$ discovery potential and 90$\%$
C.L. limit for the Migdal effect in blue and red dash lines respectively. The
vertical dash dotted lines are shown for the thresholds to observe all the
events with more than two or three ionization electron signals. Figure 3: The
necessary exposure to achieve a sensitivity level of the neutrino magnetic
moment at 90$\%$ is illustrated for various energy thresholds in the presence
of the Migdal effect. The contours for the exposures of 1 kg$\cdot$yr, 10
kg$\cdot$yr and 100 kg$\cdot$yr in are shown with red, orange and blue dash
lines respectively. The vertical dash dotted lines are shown for the energy
thresholds with more than two or three ionization electron signals.

In this section, we provide an analysis of numerical sensitivities on the
Migdal effect and the corresponding constraints on the neutrino magnetic
moment in the presence of the Migdal effect. We conduct our calculation based
on an idealized experiment with silicon CCD where the electron recoil signals
can be distinguished from those from CE$\nu$ES by identifying the presence of
additional electron excitations SENSEI:2020dpa ; Tiffenberg:2017aac . In the
following analysis, we restrict our attention to the events with more than two
ionization signals (larger than 5 eV) and assume zero background within this
energy region. We have excluded the events with one single ionization
electron, as this energy region includes substantial amount of background
signals that have not been extensively researched. The same as in Fig 1, we
consider the Taishan-1 reactor with a thermal power of 4.6 $\rm GW_{\rm th}$,
and a distance of 30 meters for the detector location JUNO:2020ijm . To
explore the projected constraints of the Migdal effect and the neutrino
magnetic moment, we employ the standard least squares method with the
$\chi^{2}$ function as

$\chi^{2}=\sum_{i=1}^{n}\left(\frac{N_{i}^{\rm exp}-(1+\epsilon_{\rm
exp})N_{i}^{\rm pred}}{\sigma_{i}}\right)^{2}+\left(\frac{\epsilon_{\rm
exp}}{\sigma_{\rm exp}}\right)^{2}\,,$ (3.16)

where $\sigma_{i}^{2}=N_{i}^{\rm exp}$, $N_{i}^{\rm exp}$ is the pseudo event
number of the signal of the considered experiment in the $i$th energy bin, and
$N_{i}^{\rm pred}$ is the predicted event number. $\epsilon_{\rm exp}$ is the
simplified nuisance parameter which quantifies the total detection uncertainty
with $\sigma_{\rm exp}=5\%$ being the expected standard deviation. The current
semiconductor detectors are extremely sensitive to electron ionization signals
and it is a convenient approximation to employ an ideal efficiency of 100% for
illustration.

In Fig. 2, we illustrate the sensitivities of the Migdal effect under
different levels of the energy threshold and exposure. We present the
5$\sigma$ discovery potential and 90$\%$ confidence level (C.L.) limit for the
Migdal effect in blue and red dash lines respectively. The vertical dash
dotted lines are shown for the thresholds to observe all the events with more
than two or three ionization electron signals. One can observe that with a
energy threshold of 8.8 eV, which enables us to detect all events with more
than two ionization signals, we need an exposure of few kg$\cdot$yr to reach
the limit of 90$\%$ C.L., while an exposure of a few tens of kg$\cdot$yr is
needed to confirm the discovery at 5$\sigma$. Thus for the present detector
technology with less than one to a few kg$\cdot$yr of exposure SENSEI:2020dpa
; SuperCDMS:2018mne , at least a partially detection of events with two
ionization signals is required to confirm the discovery of the Migdal effect.
In the future, the silicon based CCD detectors could reach the exposure level
of several tens of kg$\cdot$yr or even one hundred SuperCDMS:2018mne , which
will give more opportunity to probe the current prediction of the Migdal
effect. A better sensitivity can also be achieved by lowering the threshold of
ionization signals, but in this case the requirement on the CCD readout and
background filtering would be more challenging EDELWEISS:2017uga ;
EDELWEISS:2020fxc ; EDELWEISS:2022ktt .

In Fig. 3 the necessary exposure to achieve a sensitivity level of the
neutrino magnetic moment at 90$\%$ is illustrated for various energy
thresholds in the presence of the Migdal effect. We illustrate the contours
for the exposures of 1 kg$\cdot$yr, 10 kg$\cdot$yr and 100 kg$\cdot$yr in red,
orange and blue dash lines respectively. The vertical dash dotted lines are
shown for the thresholds to observe all the events with more than two or three
ionization electron signals. Notably, with the contribution of the Migdal
effect, a silicon-based detector with an exposure of 1 kg$\cdot$yr and
threshold of more than three ionization signals (above 10 eV), which is
possible for the present detector technology, can readily reach the constraint
level that is comparable to those of LUZ-ZEPLIN AtzoriCorona:2022jeb ;
Giunti:2023yha  or XENON1T XENON:2020rca . Moreover, we also observe from Fig.
3 that reducing the threshold form 20 eV to 5 eV is equivalent to increasing
the exposure by approximately ten fold, which implies the necessity of
detecting the low energy events in order to fully utilize the potential of
detectors.

##  4 Conclusion

In this work we have presented for the first time a framework of the phonon-
mediated Migdal effect of the neutrino nucleus scattering in the
semiconductors. We have calculated the neutrino-induced Migdal excitation
rates in semiconductors with the multi-phonon model in the SM and in the
presence of the neutrino magnetic moment. We have shown that the contribution
from the Migdal effect is dramatically enhanced by the coherent effect of the
neutrino nucleus scattering. Meanwhile, the Migdal excitation can be
distinguished from the CE$\nu$ES process without the Migdal effect by
observing the presence of additional electron excitations. Consequently, these
excellent features make the Migdal excitation an effective tool to investigate
the new physics phenomena at low energy scales.

Utilizing the new theoretical calculation, we have performed a numerical
simulation on the sensitivities of the Migdal effect and neutrino magnetic
moment with reactor neutrinos and presented the detection potential under
different detection energy thresholds and exposures, which demonstrate the
excellent prospect for testing the Migdal effect in reactor neutrino
experiments and further provide a unique frontier to probe the new physics
beyond the SM.

###### Acknowledgements.

The authors are grateful to Dr. Zhengliang Liang for helpful discussions. This
work was supported in part by National Natural Science Foundation of China
under Grant Nos. 12075255, 12075254 and 11835013.

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