# The Varchenko Determinant of an Oriented Matroid

Hery Randriamaro This research was funded by my mother
Lot II B 32 bis Faravohitra, 101 Antananarivo, Madagascar
e-mail<EMAIL_ADDRESS>
The Varchenko Determinant of an Oriented Matroid

Hery Randriamaro1

1 This research was funded by the mother of the author
Lot II B 32 bis Faravohitra, 101 Antananarivo, Madagascar
<EMAIL_ADDRESS>
Abstract
Varchenko introduced in 1993 a distance function on the chambers of a
hyperplane arrangement that gave rise to a determinant whose entry in position
$(C,D)$ is the distance between the chambers $C$ and $D$, and computed that
determinant. In 2017, Aguiar and Mahajan provided a generalization of that
distance function, and computed the corresponding determinant. This article
extends their distance function to the topes of an oriented matroid, and
computes the determinant thus defined. Oriented matroids have the nice
property to be abstractions of some mathematical structures including
hyperplane and sphere arrangements, polytopes, directed graphs, and even
chirality in molecular chemistry. Independently and with another method,
Hochstättler and Welker also computed in 2019 the same determinant.

Keywords: Pseudohyperplane Arrangement, Distance, Determinant

MSC Number: 05B20, 05E10, 15A15, 52C35

##  1 Introduction

Varchenko introduced and computed a determinant defined from hyperplane
arrangements [16, § 1]. That determinant already appeared earlier in the
implicit form of that of a symmetric bilinear form on a Verma module over a
$\mathbb{C}$-algebra [15, § 1]. It also plays a key role to prove the
realizability of variant models of quon algebras in quantum mechanics [12, Th.
4.2], [13, Prop. 2.1]. Since then, there have been some attempts to provide
cleaner proofs of the result of Varchenko, and even more refined determinants
but still defined from hyperplane arrangements [1, § 8], [8, Th. 4.5], [14,
Th. 1.3]. Besides, the associated matrix has been investigated from several
angles. For hyperplane arrangements in semigeneral position, Gao and Zhang
computed its diagonal form [7, Th. 2], and for braid arrangements, Denham and
Hanlon studied its Smith normal form [4, Th. 3.3] while Hanlon and Stanley
computed its nullspace [9, Th. 3.3]. _This article studies the extension of
the Varchenko determinant to oriented matroids_. Folkman began to work on
oriented matroids by 1967, but unfortunately died before publishing his
theory. Lawrence completed that theory, and published the results in a joint
paper with Folkman in 1978 [6]. Independently, Bland and Las Vergnas developed
the same notion of oriented matroids [3]. Those latter are abstractions for
different mathematical objects such as directed graphs [2, § 1.1] and convex
polytopes [2, § 9].

Let $\mathbb{E}:=\\{+,-,0\\}$, and for $n\in\mathbb{N}^{*}$ denote by $u_{i}$
the $i^{\text{th}}$ component of a covector $u$ in $\mathbb{E}^{n}$. Equip
$\mathbb{E}^{n}$ with the binary operation $\circ$ as follows: if
$u,v\in\mathbb{E}^{n}$, $u\circ v$ is the vector $w$ of $\mathbb{E}^{n}$ such
that $\displaystyle w_{i}:=\begin{cases}u_{i}&\text{if}\ u_{i}\neq 0,\\\
v_{i}&\text{otherwise}\end{cases}$. Moreover, define the function
$\mathrm{S}:\mathbb{E}^{n}\times\mathbb{E}^{n}\rightarrow 2^{[n]}$ by
$\mathrm{S}(u,v):=\big{\\{}i\in[n]\ \big{|}\ u_{i}=-v_{i}\neq 0\big{\\}}$. We
use the covector axioms for the definition of an oriented matroid, that is, an
oriented matroid is a subset $\mathsf{M}$ of $\mathbb{E}^{n}$ satisfying the
conditions

  * •

$(0,\dots,0)\in\mathsf{M}$,

  * •

if $u\in\mathsf{M}$, then $-u\in\mathsf{M}$,

  * •

if $u,v\in\mathsf{M}$, then $u\circ v\in\mathsf{M}$,

  * •

for each pair $u,v\in\mathsf{M}$ and every $j\in\mathrm{S}(u,v)$, there exists
$w\in\mathsf{M}$ such that

$w_{j}=0\quad\text{and}\quad\forall i\in[n]\setminus\mathrm{S}(u,v):\
w_{i}=(u\circ v)_{i}.$

An integer $i\in[n]$ is a loop for an oriented matroid $\mathsf{M}$ if
$u_{i}=0$ for every $u\in\mathsf{M}$. _For simplicity, we assume that all
oriented matroids considered in this article are loop-free_. The covector set
$\mathsf{M}$ forms a poset with the partial order $\leq$ defined by $u\leq
v\,\Longleftrightarrow\,\forall i\in[n]:\,u_{i}\in\\{0,v_{i}\\}$. The rank of
an oriented matroid $\mathsf{M}$ is the length of a maximal chain in
$(\mathsf{M},\leq)$. A covector $u$ of $\mathsf{M}$ is called a tope if
$u_{i}\neq 0$ for every $i\in[n]$. Denote the set formed by the topes of
$\mathsf{M}$ by $T_{\mathsf{M}}$. The topal fiber of $\mathsf{M}$ relative to
a set $I\subseteq[n]$ and a covector $u\in\mathsf{M}$ such that $u_{i}\neq 0$
for every $i\in[n]\setminus I$ is the set of covectors
$\mathsf{M}_{I,u}:=\\{v\in\mathsf{M}\ |\ \forall i\in[n]\setminus
I:\,v_{i}=u_{i}\\}$. Let $R_{n}$ be the polynomial ring
$\mathbb{Z}\big{[}a_{i}^{+},a_{i}^{-}\ \big{|}\ i\in[n]\big{]}$ with variables
$a_{i}^{+},a_{i}^{-}$. Define the Aguiar-Mahajan distance
$\mathrm{v}:T_{\mathsf{M}}\cap\mathsf{M}_{I,u}\times
T_{\mathsf{M}}\cap\mathsf{M}_{I,u}\rightarrow R_{n}$ on the topes in
$\mathsf{M}_{I,u}$ by

$\mathrm{v}(v,v)=1\quad\text{and}\quad\mathrm{v}(v,w)=\prod_{i\in\mathrm{S}(v,w)\cap
I}a_{i}^{v_{i}}\ \text{if}\ v\neq w.$

Remark that the name distance is kept as the authors originally called it so
for central hyperplane arrangements [1, § 8.1]. The Varchenko determinant of a
topal fiber $\mathsf{M}_{I,u}$ is

$V_{\mathsf{M}_{I,u}}:=\big{|}\mathrm{v}(v,w)\big{|}_{v,w\in
T_{\mathsf{M}}\cap\mathsf{M}_{I,u}}.$

It naturally becomes the Varchenko determinant of the oriented matroid
$\mathsf{M}$ if $I=[n]$. The weight of a covector $u\in\mathsf{M}\setminus
T_{\mathsf{M}}$ is the monomial
$\displaystyle\mathrm{b}_{u}:=\prod_{\begin{subarray}{c}i\in[n]\\\
u_{i}=0\end{subarray}}a_{i}^{+}a_{i}^{-}$. For $i\in[n]$, define the
$i^{\text{th}}$ boundary $\partial_{i}v$ of a tope $v$ of $\mathsf{M}$ by the
set $\\{u\in\mathsf{M}\ |\ u\leq v,\,u_{i}=0\\}$. The multiplicity of a
covector $u\in\mathsf{M}\setminus T_{\mathsf{M}}$ such that $u_{i}=0$ is the
integer $\displaystyle\beta_{u}:=\frac{\\#\\{t\in T_{\mathsf{M}}\ |\
\max\partial_{i}t=u\\}}{2}$. We will see in Theorem 4.3 that $\beta_{u}$ is
independent of the chosen $i$. _We can now state the main result of this
article_.

######  Theorem 1.1.

Let $\mathsf{M}$ be an oriented matroid in $\mathbb{E}^{n}$, $I$ a subset of
$[n]$, and $u$ a covector of $\mathsf{M}$ such that $u_{i}\neq 0$ for every
$i\in[n]\setminus I$. Then, the Varchenko determinant of the topal fiber
$\mathsf{M}_{I,u}$ is

$V_{\mathsf{M}_{I,u}}=\prod_{v\in\mathsf{\mathsf{M}_{I,u}}\setminus
T_{\mathsf{M}}}(1-\mathrm{b}_{v})^{\beta_{v}}.$

For an oriented matroid and under the condition $a_{i}^{+}=a_{i}^{-}$, we
recover the determinant computed by Hochstättler and Welker with tools from
oriented matroid topology [10, Th. 1]. Besides, Olzhabayev and Zhang obtained
$V_{\mathsf{M}_{I,u}}$, for $(\mathsf{M}_{I,u},\leq)$ being poset isomorphic
to a configuration of pseudolines, from the diagonal form of the associated
matrix [11, Th. 1.2].

This article is organized as follows: We begin by recalling the topological
representation theorem in Section 2. That theorem allows to obtain the
Varchenko determinant of an oriented matroid from that of a pseudohyperplane
arrangement. To compute that latter, we are inspired by the proof strategy of
Aguiar and Mahajan for central hyperplane arrangements [1, Th. 8.11] by
extending a Witt identity to pseudohyperplane arrangements in Section 3, and
by using that extension to determine the Varchenko determinant of a
pseudohyperplane arrangement in Section 4. We finally prove Theorem 1.1 in
Section 5.

##  2 Topological Representation Theorem

We describe the topological representation theorem, and the Varchenko
determinant of a pseudohyperplane arrangement. For $d\in\mathbb{N}^{*}$, a
subsphere of a sphere $\mathbb{S}^{d}$ is a subset of $\mathbb{S}^{d}$ which
is homeomorphic to $\mathbb{S}^{d-1}$.

######  Lemma 2.1.

[2, Lem. 5.1.1] Letting $d\in\mathbb{N}^{*}$, for a subsphere $S$ of
$\mathbb{S}^{d}$ the following conditions are equivalent:

  1. 1.

$S$ is homeomorphic to the equator
$\big{\\{}(x_{1},\dots,x_{d+1})\in\mathbb{S}^{d}\ |\ x_{d+1}=0\big{\\}}$,

  2. 2.

$S$ is homeomorphic to some piecewise-linearly embedded subsphere,

  3. 3.

the closure of each connected component of $\mathbb{S}^{d}\setminus S$ is
homeomorphic to the $d$-ball.

A subsphere of $\mathbb{S}^{d}$ satisfying the conditions in Lemma 2.1 is
called a pseudosphere of $\mathbb{S}^{d}$. For a pseudosphere $S$ of
$\mathbb{S}^{d}$, denote by $S^{+}$ and $S^{-}$ both connected components of
$\mathbb{S}^{d}\setminus S$ called sides of $S$. A finite set
$\mathcal{S}=\\{S_{i}\\}_{i\in[n]}$ of pseudospheres in $\mathbb{S}^{d}$ is
called a pseudosphere arrangement if the following conditions hold:

  1. 1.

$\displaystyle S_{I}:=\bigcap_{i\in I}S_{i}$ is a sphere for all
$I\subseteq[n]$.

  2. 2.

For $I\subseteq[n]$ and $i\in[n]$, if $S_{I}\nsubseteq S_{i}$, then $S_{I}\cap
S_{i}$ is a pseudosphere in $S_{I}$ with sides $S_{I}\cap S_{i}^{+}$ and
$S_{I}\cap S_{i}^{-}$.

  3. 3.

The intersection of an arbitrary set of closed sides in $\mathbb{S}^{d}$ is
either a sphere or a ball.

For a pseudosphere arrangement $\mathcal{S}=\\{S_{i}\\}_{i\in[n]}$, define the
sign functions $\sigma_{i}:\mathbb{S}^{d}\rightarrow\mathbb{E}$ and
$\sigma_{\mathcal{S}}:\mathbb{S}^{d}\rightarrow\mathbb{E}^{n}$ respectively,
for $x\in\mathbb{S}^{d}$, by

$\sigma_{i}(x):=\begin{cases}+&\text{if}\ x\in S_{i}^{+}\\\ -&\text{if}\ x\in
S_{i}^{-}\\\ 0&\text{if}\ x\in
S_{i}\end{cases}\quad\text{and}\quad\sigma_{\mathcal{S}}(x):=\big{(}\sigma_{1}(x),\dots,\sigma_{n}(x)\big{)}.$

Consider a pseudosphere arrangement $\mathcal{S}=\\{S_{i}\\}_{i\in[n]}$ in
$\mathbb{S}^{d}$. Its rank is $\displaystyle d-\dim\bigcap_{i\in[n]}S_{i}$
assuming that $\dim\emptyset=-1$. It is essential if
$\displaystyle\bigcap_{i\in[n]}S_{i}=\emptyset$, and centrally symmetric if
$-S_{i}=S_{i}$ for every $i\in[n]$. Besides, denote by
$\mathsf{M}_{\mathcal{S}}$ the subset $\big{\\{}\sigma_{\mathcal{S}}(x)\ |\
x\in\mathbb{S}^{d}\big{\\}}\cup\big{\\{}(0,\dots,0)\big{\\}}$ of
$\mathbb{E}^{n}$.

######  Theorem 2.2.

[2, Topological Representation Theorem 5.2.1] Let $d$ be a nonnegative integer
and $\mathsf{M}$ a subset of $\mathbb{E}^{n}$. Then the following conditions
are equivalent:

  1. 1.

$\mathsf{M}$ is an oriented matroid of rank $d+1$.

  2. 2.

For some pseudosphere arrangement $\mathcal{S}=\\{S_{i}\\}_{i\in[n]}$ in
$\mathbb{S}^{d+1+k}$ such that $\displaystyle\dim\bigcap_{i\in[n]}S_{i}=k$, we
have $\mathsf{M}_{\mathcal{S}}=\mathsf{M}$.

  3. 3.

There exists an essential and centrally symmetric pseudosphere arrangement
$\mathcal{S}$ in $\mathbb{S}^{d}$ such that its induced cell complex is
shellable and $\mathsf{M}=\mathsf{M}_{\mathcal{S}}$.

Björner et al. defined the pseudohyperplane arrangements in real projective
spaces [2, § 5.2]. For us it is more convenient to define them on Euclidean
spaces like Deshpande did [5, Def 3.11]. For $d\in\mathbb{N}^{*}$, a
pseudohyperplane in $\mathbb{R}^{d+1}$ is set $H\subseteq\mathbb{R}^{d+1}$
such that $H$ is homeomorphic to $\mathbb{R}^{d}$, and $S=H\cap\mathbb{S}^{d}$
is a pseudosphere of a sphere $\mathbb{S}^{d}\subseteq\mathbb{R}^{d+1}$.
Denote by $H^{+}$ and $H^{-}$ both connected components of
$\mathbb{R}^{d+1}\setminus H$ such that $S^{+}\subseteq H^{+}$. A
pseudohyperplane arrangement is a finite set $\mathcal{H}$ of
pseudohyperplanes in $\mathbb{R}^{d+1}$ such that $\\{H\cap\mathbb{S}^{d}\ |\
H\in\mathcal{H}\\}$ is a centrally symmetric pseudosphere arrangement in
$\mathbb{S}^{d}$. For a pseudohyperplane arrangement
$\mathcal{H}=\\{H_{i}\\}_{i\in[n]}$, define the sign functions
$\epsilon_{i}:\mathbb{R}^{d+1}\rightarrow\mathbb{E}$ and
$\epsilon_{\mathcal{H}}:\mathbb{R}^{d+1}\rightarrow\mathbb{E}^{n}$
respectively by

$\epsilon_{i}(x):=\begin{cases}+&\text{if}\ x\in H_{i}^{+}\\\ -&\text{if}\
x\in H_{i}^{-}\\\ 0&\text{if}\ x\in
H_{i}\end{cases}\quad\text{and}\quad\epsilon_{\mathcal{H}}(x):=\big{(}\epsilon_{1}(x),\dots,\epsilon_{n}(x)\big{)}.$

Let $\mathsf{M}_{\mathcal{H}}$ be the subset
$\big{\\{}\epsilon_{\mathcal{H}}(x)\ |\ x\in\mathbb{R}^{d+1}\big{\\}}$ of
$\mathbb{E}^{n}$. _Denoting by $\mathcal{S}$ the pseudosphere arrangement
$\\{H\cap\mathbb{S}^{d}\ |\ H\in\mathcal{H}\\}$, we then have
$\mathsf{M}_{\mathcal{H}}=\mathsf{M}_{\mathcal{S}}$_. Hence, we deduce the
following corollary.

######  Corollary 2.3.

Let $d$ be a positive integer and $\mathsf{M}$ a subset of $\mathbb{E}^{n}$.
Then the following conditions are equivalent:

  1. 1.

$\mathsf{M}$ is an oriented matroid of rank $d$.

  2. 2.

For some pseudohyperplane arrangement $\mathcal{H}=\\{H_{i}\\}_{i\in[n]}$ in
$\mathbb{R}^{d+k}$ with $\displaystyle\dim\bigcap_{i\in[n]}H_{i}=k$, we have
$\mathsf{M}_{\mathcal{H}}=\mathsf{M}$.

  3. 3.

There exists a pseudohyperplane arrangement $\mathcal{H}$ in $\mathbb{R}^{d}$
such that $\mathsf{M}=\mathsf{M}_{\mathcal{H}}$.

###### Proof.

Use the topological representation theorem, the definition of a
pseudohyperplane arrangement, and the fact that, for any pseudosphere
arrangement $\mathcal{S}$ in $\mathbb{S}^{d-1}$, there exists a
pseudohyperplane arrangement $\mathcal{H}$ in $\mathbb{R}^{d}$ such that
$\mathcal{S}=\\{H\cap\mathbb{S}^{d-1}\ |\ H\in\mathcal{H}\\}$ and
$\mathsf{M}_{\mathcal{H}}=\mathsf{M}_{\mathcal{S}}$. ∎

A subset $F$ of $\mathbb{R}^{d}$ is a face of a pseudohyperplane arrangement
$\mathcal{H}$ if there exists a covector $u$ in $\mathsf{M}_{\mathcal{H}}$
such that $F:=\epsilon_{\mathcal{H}}^{-1}(u)$. Denote the set formed by the
faces of $\mathcal{H}$ by $F_{\mathcal{H}}$. It is a poset with partial order
$\preceq$ defined by $F\preceq
G\,\Longleftrightarrow\,\epsilon_{\mathcal{H}}(F)\leq\epsilon_{\mathcal{H}}(G)$.
_We observe that $(F_{\mathcal{H}},\preceq)$ is poset isomorphic to
$(\mathsf{M}_{\mathcal{H}},\leq)$_. A chamber of $\mathcal{H}$ is a face $C$
such that $\epsilon_{\mathcal{H}}(C)$ contains no $0$. Denote the set formed
by the chambers of $\mathcal{H}$ by $C_{\mathcal{H}}$. Consider a
pseudohyperplane arrangement $\mathcal{H}=\\{H_{i}\\}_{i\in[n]}$ in
$\mathbb{R}^{d}$, and let $R_{\mathcal{H}}$ be the polynomial ring
$\mathbb{Z}\big{[}a_{i}^{+},a_{i}^{-}\ \big{|}\ i\in[n]\big{]}$ with variables
$a_{i}^{+},a_{i}^{-}$. Define the Aguiar-Mahajan distance
$v:C_{\mathcal{H}}\times C_{\mathcal{H}}\rightarrow R_{\mathcal{H}}$ on
chambers by

$v(C,C)=1\quad\text{and}\quad
v(C,D)=\prod_{i\in\mathrm{S}\big{(}\epsilon_{\mathcal{H}}(C),\epsilon_{\mathcal{H}}(D)\big{)}}a_{i}^{\epsilon_{i}(C)}\,\
\text{if}\,\ C\neq D.$

The Varchenko determinant of the pseudohyperplane arrangement $\mathcal{H}$ is

$V_{\mathcal{H}}:=\big{|}v(D,C)\big{|}_{C,D\in C_{\mathcal{H}}}.$

The weight of a face $F\in F_{\mathcal{H}}\setminus C_{\mathcal{H}}$ is the
monomial $\displaystyle b_{F}:=\prod_{\begin{subarray}{c}i\in[n]\\\
\epsilon_{i}(F)=0\end{subarray}}a_{i}^{+}a_{i}^{-}$, and its multiplicity the
integer $\displaystyle\beta_{F}:=\frac{\\#\\{C\in C_{\mathcal{H}}\ |\
\overline{C}\cap H_{i}=F\\}}{2}$ with $\epsilon_{i}(F)=0$. One can easily see
that $b_{F}=\mathrm{b}_{\epsilon_{\mathcal{H}}(F)}$ and
$\beta_{F}=\beta_{\epsilon_{\mathcal{H}}(F)}$ if one considers the oriented
matroid $\mathsf{M}_{\mathcal{H}}$. _We can now state the intermediate result
which will allow to prove Theorem 1.1_.

######  Theorem 2.4.

Let $\mathcal{H}$ be a pseudohyperplane arrangement in $\mathbb{R}^{d}$. Then,

$\displaystyle V_{\mathcal{H}}=\prod_{F\in F_{\mathcal{H}}\setminus
C_{\mathcal{H}}}(1-b_{F})^{\beta_{F}}.$

_From the isomorphism between $(F_{\mathcal{H}},\preceq)$ and
$(\mathsf{M}_{\mathcal{H}},\leq)$, we naturally have
$V_{\mathcal{H}}=V_{\mathsf{M}_{\mathcal{H}}}$_.

######  Example.

Consider the oriented matroid $\mathsf{M}=\mathsf{M}_{\mathcal{S}}$ where
$\mathcal{S}$ is the non-Pappus arrangement of $10$ pseudocircles [5, Ex.
3.12]. There exist $i\in[10]$, that we can assume to be $10$, and a covector
$u\in\mathsf{M}$ such that the fiber $\mathsf{M}_{[9],u}$ is poset isomorphic
to the face poset of the non-Pappus configuration having $9$ pseudolines in
Figure 1. It is known that there exists no hyperplane arrangement whose face
poset is isomorphic to that of the non-Pappus configuration [17, Ex. 7.28]. We
use that latter to computer the Varchenko determinant of $\mathsf{M}_{[9],u}$.
For simplicity, we assume that the variables associated to that the
pseudolines are all equal to $a$. Regarding its faces, the non-Pappus
configuration has $33$ chambers, then $\mathsf{M}_{[9],u}$ contains $33$
topes. Moreover, it has $43$ lines with weight $a^{2}$ and multiplicity $1$,
$8$ points with weight $a^{6}$ and multiplicity $1$, and $7$ points with
weight $a^{4}$ and multiplicity $0$. Hence we obtain

$V_{\mathsf{M}_{[9],u}}=(1-a^{2})^{43}(1-a^{6})^{8}.$

Figure 1: Non-Pappus Configuration

##  3 Generalized Witt Identity

We extend the Witt identity [1, Proposition 7.30] to pseudohyperplane
arrangements, and use that extension to prove a key equality on the chambers.
The faces of a pseudohyperplane arrangement $\mathcal{H}$ form a monoid with
the multiplication defined, for $F,G\in F_{\mathcal{H}}$, by

$FG:=\epsilon_{\mathcal{H}}^{-1}\big{(}\epsilon_{\mathcal{H}}(F)\circ\epsilon_{\mathcal{H}}(G)\big{)}.$

A nested face of $\mathcal{H}$ is a pair $(F,G)$ of faces in $F_{\mathcal{H}}$
such that $F\prec G$. Denote by $F_{\mathcal{H}}^{(F,G)}$ the set of faces
$\\{E\in F_{\mathcal{H}}\ |\ F\preceq E\preceq G\\}$. Moreover, letting
$\displaystyle c_{\mathcal{H}}:=\dim\bigcap_{H\in\mathcal{H}}H$, the rank of a
face $F\in F_{\mathcal{H}}$ is $\mathrm{rk}\,F:=\dim F-c_{\mathcal{H}}$. The
opposite of a face $F\in F_{\mathcal{H}}$ is the face $\tilde{F}$ of
$\mathcal{H}$ such that
$\epsilon_{\mathcal{H}}(\tilde{F})=-\epsilon_{\mathcal{H}}(F)$. For a face $F$
of $\mathcal{H}=\\{H_{i}\\}_{i\in[n]}$, let $\mathcal{H}_{F}$ be the
pseudohyperplane arrangement $\big{\\{}H_{i}\in\mathcal{H}\ \big{|}\
\epsilon_{i}(F)=0\big{\\}}$. Let $\chi$ be the function Euler characteristic
of the shape of a topological space. Furthermore, assign a variable $x_{C}$ to
each chamber $C\in C_{\mathcal{H}}$.

######  Proposition 3.1.

Let $\mathcal{H}$ be a pseudohyperplane arrangement in $\mathbb{R}^{d}$, $D\in
C_{\mathcal{H}}$, and $(A,D)$ a nested face of $\mathcal{H}$. Then,

$\sum_{F\in
F_{\mathcal{H}}^{(A,D)}}(-1)^{\mathrm{rk}\,F}\sum_{\begin{subarray}{c}C\in
C_{\mathcal{H}}\\\
FC=D\end{subarray}}x_{C}\,=\,(-1)^{\mathrm{rk}\,D}\sum_{\begin{subarray}{c}C\in
C_{\mathcal{H}}\\\ AC=A\tilde{D}\end{subarray}}x_{C}.$

###### Proof.

We have $\displaystyle\sum_{F\in
F_{\mathcal{H}}^{(A,D)}}(-1)^{\mathrm{rk}\,F}\sum_{\begin{subarray}{c}C\in
C_{\mathcal{H}}\\\ FC=D\end{subarray}}x_{C}=\sum_{C\in
C_{\mathcal{H}}}\Big{(}\sum_{\begin{subarray}{c}F\in
F_{\mathcal{H}}^{(A,D)}\\\
FC=D\end{subarray}}(-1)^{\mathrm{rk}\,F}\Big{)}x_{C}$.

  * •

If $\epsilon_{\mathcal{H}_{A}}(C)=\epsilon_{\mathcal{H}_{A}}(A\tilde{D})$,
then $\displaystyle\sum_{\begin{subarray}{c}F\in F_{\mathcal{H}}^{(A,D)}\\\
FC=D\end{subarray}}(-1)^{\mathrm{rk}\,F}=(-1)^{\mathrm{rk}\,D}$.

Denote by $F_{\mathcal{H}}^{(A,\centerdot)}$ the set of faces $\\{F\in
F_{\mathcal{H}}\ |\ A\preceq F\\}=\big{\\{}F\in F_{\mathcal{H}}\ \big{|}\
\epsilon_{\mathcal{H}\setminus\mathcal{H}_{A}}(F)=\epsilon_{\mathcal{H}\setminus\mathcal{H}_{A}}(A)\big{\\}}$.
Let $f:F_{\mathcal{H}}^{(A,\centerdot)}\rightarrow F_{\mathcal{H}_{A}}$ be the
bijection such that, if $F\in F_{\mathcal{H}}^{(A,\centerdot)}$, then $f(F)$
is the face of $\mathcal{H}_{A}$ such that
$\epsilon_{\mathcal{H}_{A}}\big{(}f(F)\big{)}=\epsilon_{\mathcal{H}_{A}}(F)$.

  * •

If $\epsilon_{\mathcal{H}_{A}}(C)=\epsilon_{\mathcal{H}_{A}}(D)$, then

$\displaystyle\sum_{\begin{subarray}{c}F\in F_{\mathcal{H}}^{(A,D)}\\\
FC=D\end{subarray}}(-1)^{\mathrm{rk}\,F}$
$\displaystyle=(-1)^{-c_{\mathcal{H}}}\sum_{F\in
F_{\mathcal{H}}^{(A,D)}}(-1)^{\dim F}$
$\displaystyle=(-1)^{c_{\mathcal{H}}}\sum_{F\in
f\big{(}F_{\mathcal{H}}^{(A,D)}\big{)}}(-1)^{\dim F}$
$\displaystyle=(-1)^{c_{\mathcal{H}}}\chi\big{(}\overline{f(D)}\big{)}$
$\displaystyle=0.$

  * •

The case
$\epsilon_{\mathcal{H}_{A}}(C)\notin\big{\\{}\epsilon_{\mathcal{H}_{A}}(D),\epsilon_{\mathcal{H}_{A}}(A\tilde{D})\big{\\}}$
remains. Assume that $\epsilon_{\mathcal{H}_{A}}(D)=(+,\dots,+)$, and define
the pseudohyperplane arrangement
$\mathcal{H}_{A}(C):=\big{\\{}H_{i}\in\mathcal{H}_{A}\ \big{|}\
\epsilon_{i}(C)=-\big{\\}}$. Moreover, if $E\in C_{\mathcal{H}_{A}(C)}$, let
$F_{\mathcal{H}_{A}(C)}^{(d-1,E)}:=\\{F\in F_{\mathcal{H}_{A}(C)}\ |\ F\preceq
E,\,\dim F=d-1\\}$. If $\\#\mathcal{H}_{A}(C)>1$, then

$\forall F\in F_{\mathcal{H}_{A}(C)}^{(d-1,E)},\,\exists F^{\prime}\in
F_{\mathcal{H}_{A}(C)}^{(d-1,E)}\setminus\\{F\\}:\
\dim\mathrm{int}(\overline{F}\cap\overline{F^{\prime}})=d-2.$

We obtain,

$\displaystyle\sum_{\begin{subarray}{c}F\in F_{\mathcal{H}}^{(A,D)}\\\
FC=D\end{subarray}}(-1)^{\mathrm{rk}\,F}$
$\displaystyle=(-1)^{-c_{\mathcal{H}}}\sum_{\begin{subarray}{c}F\in
F_{\mathcal{H}}^{(A,D)}\\\
\epsilon_{\mathcal{H}_{A}(C)}(F)\,=\,(+,\dots,+)\end{subarray}}(-1)^{\dim F}$
$\displaystyle=(-1)^{c_{\mathcal{H}}}\sum_{\begin{subarray}{c}F\in
f\big{(}F_{\mathcal{H}}^{(A,D)}\big{)}\\\
\epsilon_{\mathcal{H}_{A}(C)}(F)\,=\,(+,\dots,+)\end{subarray}}(-1)^{\dim F}$
$\displaystyle=(-1)^{c_{\mathcal{H}}}\chi\Big{(}\overline{f(D)}\setminus\bigcup_{F\in
F_{\mathcal{H}_{A}(C)}^{(d-1,f(D))}}\overline{F}\Big{)}$
$\displaystyle=(-1)^{c_{\mathcal{H}}}\Big{(}\chi\big{(}\overline{f(D)}\big{)}-\chi\big{(}\bigcup_{F\in
F_{\mathcal{H}_{A}(C)}^{(d-1,f(D))}}\overline{F}\big{)}\Big{)}$
$\displaystyle=0.$

Hence $\displaystyle\sum_{F\in
F_{\mathcal{H}}^{(A,D)}}(-1)^{\mathrm{rk}\,F}\sum_{\begin{subarray}{c}C\in
C_{\mathcal{H}}\\\
FC=D\end{subarray}}x_{C}\,=\,(-1)^{\mathrm{rk}\,D}\sum_{\begin{subarray}{c}C\in
C_{\mathcal{H}}\\\
\epsilon_{\mathcal{H}_{A}}(C)\,=\,\epsilon_{\mathcal{H}_{A}}(A\tilde{D})\end{subarray}}x_{C}\,=\,(-1)^{\mathrm{rk}\,D}\sum_{\begin{subarray}{c}C\in
C_{\mathcal{H}}\\\ AC=A\tilde{D}\end{subarray}}x_{C}$. ∎

######  Lemma 3.2.

Let $\mathcal{H}$ be a pseudohyperplane arrangement in $\mathbb{R}^{d}$,
$C,D\in C_{\mathcal{H}}$, and $F\in F_{\mathcal{H}}$ such that $F\preceq C$.
Then,

$v(C,D)=v(C,FD)\,v(FD,D).$

###### Proof.

Assume that $\mathcal{H}=\\{H_{i}\\}_{i\in[n]}$, and for any chambers $F,G\in
C_{\mathcal{H}}$, denote the set of half-spaces containing $F$ but not $G$ by
$\mathscr{H}_{F,G}:=\big{\\{}H_{i}^{\epsilon_{i}(F)}\ \big{|}\
i\in[n]:\,\epsilon_{i}(F)\neq\epsilon_{i}(G)\big{\\}}$. We have

$\mathscr{H}_{C,FD}=\big{\\{}H_{i}^{\epsilon_{i}(C)}\ \big{|}\
H_{i}\in\mathcal{A}_{F},\,\epsilon_{i}(C)\neq\epsilon_{i}(FD)\big{\\}}=\big{\\{}H_{i}^{\epsilon_{i}(C)}\
\big{|}\
H_{i}\in\mathcal{A}_{F},\,\epsilon_{i}(C)\neq\epsilon_{i}(D)\big{\\}},$

and also

$\mathscr{H}_{FD,D}=\big{\\{}H_{i}^{\epsilon_{i}(FD)}\ \big{|}\
H_{i}\in\mathcal{A}\setminus\mathcal{A}_{F},\,\epsilon_{i}(FD)\neq\epsilon_{i}(D)\big{\\}}=\big{\\{}H_{i}^{\epsilon_{i}(C)}\
\big{|}\
H_{i}\in\mathcal{A}\setminus\mathcal{A}_{F},\,\epsilon_{i}(C)\neq\epsilon_{i}(D)\big{\\}}.$

Then, $\mathscr{H}_{C,D}=\mathscr{H}_{C,FD}\sqcup\mathscr{H}_{FD,D}$. ∎

The module of $R_{\mathcal{H}}$-linear combinations of chambers in
$C_{\mathcal{H}}$ is $\displaystyle M_{\mathcal{H}}:=\Big{\\{}\sum_{C\in
C_{\mathcal{H}}}x_{C}C\ \Big{|}\ x_{C}\in R_{\mathcal{H}}\Big{\\}}$.

Let $\\{C^{*}\\}_{C\in C_{\mathcal{H}}}$ be the dual basis of the basis
$C_{\mathcal{H}}$. Define the linear map
$\gamma_{\mathcal{H}}:M_{\mathcal{H}}\rightarrow M_{\mathcal{H}}^{*}$, for
every $D\in C_{\mathcal{H}}$, by

$\gamma_{\mathcal{H}}(D):=\sum_{C\in C_{\mathcal{H}}}v(D,C)\,C^{*}.$

Let $B_{\mathcal{H}}$ be the extension ring
$\displaystyle\bigg{\\{}\frac{p}{\displaystyle\prod_{F\in
F_{\mathcal{H}}\setminus C_{\mathcal{H}}}(1-b_{F})^{k_{F}}}\ \bigg{|}\ p\in
R_{\mathcal{H}},\,k_{F}\in\mathbb{N}\bigg{\\}}$ of $R_{\mathcal{H}}$.

######  Proposition 3.3.

Let $\mathcal{H}$ be a pseudohyperplane arrangement in $\mathbb{R}^{d}$, and
$D\in C_{\mathcal{H}}$. Then,

$D^{*}=\sum_{C\in
C_{\mathcal{H}}}x_{C}\,\gamma_{\mathcal{H}}(C)\quad\text{with}\quad x_{C}\in
B_{\mathcal{H}}.$

###### Proof.

The proof is similar to the proof of [1, Proposition 8.13]. For a nested face
$(A,D)$, let

$\displaystyle m(A,D):=\sum_{\begin{subarray}{c}C\in C_{\mathcal{H}}\\\
AC=D\end{subarray}}v(D,C)\,C^{*}\in M_{\mathcal{H}}^{*}.$

One proves by backward induction that $\displaystyle m(A,D)=\sum_{C\in
C_{\mathcal{H}}}x_{C}\,\gamma_{\mathcal{H}}(C)$ with $x_{C}\in
B_{\mathcal{H}}$. We obviously have $m(D,D)=\gamma_{\mathcal{H}}(D)$. Then,
Proposition 3.1 applied to $x_{C}=v(D,C)\,C^{*}$ in addition to Lemma 3.2
yield

$\sum_{F\in
F_{\mathcal{H}}^{(A,D)}}(-1)^{\mathrm{rk}\,F}m(F,D)=(-1)^{\mathrm{rk}\,D}\sum_{\begin{subarray}{c}C\in
C_{\mathcal{H}}\\\
AC=A\tilde{D}\end{subarray}}v(D,C)\,C^{*}=(-1)^{\mathrm{rk}\,D}\,v(D,A\tilde{D})\,m(A,A\tilde{D}).$

Hence, $\displaystyle
m(A,D)-(-1)^{\mathrm{rk}\,D-\mathrm{rk}\,A}\,v(D,A\tilde{D})\,m(A,A\tilde{D})=\sum_{F\in
F_{\mathcal{H}}^{(A,D)}\setminus\\{A\\}}(-1)^{\mathrm{rk}\,F-\mathrm{rk}\,A+1}m(F,D)$.
By induction hypothesis, for every $C\in C_{\mathcal{H}}$, there exists
$a_{C}\in B_{\mathcal{H}}$, such that

$\sum_{F\in
F_{\mathcal{H}}^{(A,D)}\setminus\\{A\\}}(-1)^{\mathrm{rk}\,F-\mathrm{rk}\,A+1}m(F,D)=\sum_{C\in
C_{\mathcal{H}}}a_{C}\,\gamma_{\mathcal{H}}(C).$

Since $A\preceq A\tilde{D}$ and $A(\widetilde{A\tilde{D}})=D$, by replacing
$D$ with $A\tilde{D}$, there exists also $e_{C}\in B_{\mathcal{H}}$ for every
$C\in C_{\mathcal{H}}$ such that $\displaystyle
m(A,A\tilde{D})-(-1)^{\mathrm{rk}\,A\tilde{D}-\mathrm{rk}\,A}\,v(A\tilde{D},D)\,m(A,D)=\sum_{C\in
C_{\mathcal{H}}}e_{C}\,\gamma_{\mathcal{H}}(C)$. Therefore,

$m(A,D)=\sum_{C\in
C_{\mathcal{H}}}\frac{a_{C}+(-1)^{\mathrm{rk}\,D-\mathrm{rk}\,A}\,v(D,A\tilde{D})\,e_{C}}{1-b_{A}}\gamma_{\mathcal{H}}(C).$

Hence, $\displaystyle
D^{*}=m\Big{(}\bigcap_{H\in\mathcal{H}}H,\,D\Big{)}=\sum_{C\in
C_{\mathcal{H}}}x_{C}\,\gamma_{\mathcal{H}}(C)$ with $x_{C}\in
B_{\mathcal{H}}$. ∎

##  4 Determinant Computing

We finally compute the Varchenko determinant of a pseudohyperplane arrangement
by first proving that it has the form $\displaystyle\prod_{F\in
F_{\mathcal{H}}\setminus C_{\mathcal{H}}}(1-b_{F})^{l_{F}}$, and by
determining $l_{F}$ for each face $F$. The Varchenko matrix of a
pseudohyperplane arrangement $\mathcal{H}$ is
$\mathbf{M}_{\mathcal{H}}:=\big{(}v(D,C)\big{)}_{C,D\in C_{\mathcal{H}}}$.

######  Proposition 4.1.

Let $\mathcal{H}$ be a pseudohyperplane arrangement in $\mathbb{R}^{d}$. For a
face $F\in F_{\mathcal{H}}\setminus C_{\mathcal{H}}$, there exists a
nonnegative integer $l_{F}$ such that

$\det\mathbf{M}_{\mathcal{H}}=\prod_{F\in F_{\mathcal{H}}\setminus
C_{\mathcal{H}}}(1-b_{F})^{l_{F}}.$

###### Proof.

It is clear that $\mathbf{M}_{\mathcal{H}}$ is the matrix representation of
$\gamma_{\mathcal{H}}$. The determinant of $\mathbf{M}_{\mathcal{H}}$ is a
polynomial in $R_{\mathcal{H}}$ with constant term $1$, so
$\mathbf{M}_{\mathcal{H}}$ is invertible. From Proposition 3.3, we know that,
for every chamber $D\in C_{\mathcal{H}}$, there exist $x_{C}\in
B_{\mathcal{H}}$ such that $\displaystyle D^{*}=\sum_{C\in
C_{\mathcal{H}}}x_{C}\gamma_{\mathcal{H}}(C)$. Hence,

$\gamma_{\mathcal{H}}^{-1}(D^{*})=\sum_{C\in
C_{\mathcal{H}}}x_{C}\,C\quad\text{with}\quad x_{C}\in B_{\mathcal{H}}.$

As the matrix representation of $\gamma_{\mathcal{H}}^{-1}$ is
$\mathbf{M}_{\mathcal{H}}^{-1}$, each entry of $\mathbf{M}_{\mathcal{H}}^{-1}$
is then an element of $B_{\mathcal{H}}$. To finish, note that
$\displaystyle\mathbf{M}_{\mathcal{H}}^{-1}=\frac{\mathrm{adj}(\mathbf{M}_{\mathcal{H}})}{\det\mathbf{M}_{\mathcal{H}}}$,
where each entry of $\mathrm{adj}(\mathbf{M}_{\mathcal{H}})$ is a polynomial
in $R_{\mathcal{H}}$. Then, the only possibility is
$\det\mathbf{M}_{\mathcal{H}}$ has the form $\displaystyle k\prod_{F\in
F_{\mathcal{H}}\setminus C_{\mathcal{H}}}(1-b_{F})^{l_{F}}$, with
$k\in\mathbb{Z}$. As the constant term of $\det\mathbf{M}_{\mathcal{H}}$ is
$1$, we deduce that $k=1$. ∎

Let $\mathcal{K}$ be a subset of a pseudohyperplane arrangement $\mathcal{H}$.
Define an apartment of $\mathcal{H}$ to be a chamber of $\mathcal{K}$. Denote
the set formed by the apartments of $\mathcal{H}$ by $K_{\mathcal{H}}$. The
restriction of $\mathcal{H}$ to $K\in K_{\mathcal{H}}$ is the pseudohyperplane
arrangement $\mathcal{H}^{K}=\\{H_{i}\in\mathcal{H}\ |\ H_{i}\cap
K\neq\emptyset\\}$. The sets formed by the faces and the chambers in $K$ are
$F_{\mathcal{H}}^{K}:=\\{F\in F_{\mathcal{H}}\ |\ F\subseteq K\\}$ and
$C_{\mathcal{H}}^{K}:=C_{\mathcal{H}}\cap F_{\mathcal{H}}^{K}$ respectively.
Moreover, the Varchenko matrix of $\mathcal{H}$ restricted to
$C_{\mathcal{H}}^{K}$ is

$\mathbf{M}_{\mathcal{H}}^{K}:=\big{(}v(D,C)\big{)}_{C,D\in
C_{\mathcal{H}}^{K}}.$

######  Corollary 4.2.

Let $\mathcal{H}$ a pseudohyperplane arrangement in $\mathbb{R}^{d}$, and
$K\in K_{\mathcal{H}}$. Then,

$\det\mathbf{M}_{\mathcal{H}}^{K}=\prod_{F\in F_{\mathcal{H}}^{K}\setminus
C_{\mathcal{H}}^{K}}(1-b_{F})^{l_{F}}.$

###### Proof.

Set $a_{i}^{+}=a_{i}^{-}=0$ for every
$H_{i}\in\mathcal{H}\setminus\mathcal{H}^{K}$. Then,

$v(C,D)\neq 0\quad\Longleftrightarrow\quad C=D\,\ \text{or}\,\ C,D\in
C_{\mathcal{H}}^{K}\,\ \text{or}\,\ C,D\in C_{\mathcal{H}}^{\tilde{K}}.$

Hence, for a suitable chamber indexing of the rows and the columns, we obtain

$\mathbf{M}_{\mathcal{H}}=\mathbf{M}_{\mathcal{H}}^{K}\oplus\mathbf{M}_{\mathcal{H}}^{\tilde{K}}\oplus\mathbf{I},$

where $\mathbf{I}$ is the identity matrix of order
$\\#C_{\mathcal{H}}-2\,\\#C_{\mathcal{H}}^{K}$. Using Proposition 4.1, we get

$\displaystyle\det\mathbf{M}_{\mathcal{H}}^{K}\oplus\mathbf{M}_{\mathcal{H}}^{\tilde{K}}$
$\displaystyle=\prod_{F\in F_{\mathcal{H}}^{K}\setminus
C_{\mathcal{H}}^{K}}(1-b_{F})^{l_{F}}\ \prod_{G\in
F_{\mathcal{H}}^{\tilde{K}}\setminus
C_{\mathcal{H}}^{\tilde{K}}}(1-b_{G})^{l_{G}}$ $\displaystyle=\prod_{F\in
F_{\mathcal{H}}^{K}\setminus
C_{\mathcal{H}}^{K}}(1-b_{F})^{l_{F}}(1-b_{\tilde{F}})^{l_{\tilde{F}}}$
$\displaystyle=\prod_{F\in F_{\mathcal{H}}^{K}\setminus
C_{\mathcal{H}}^{K}}(1-b_{F})^{l_{F}+l_{\tilde{F}}}.$

Since $\mathbf{M}_{\mathcal{H}}^{K}=\mathbf{M}_{\mathcal{H}}^{\tilde{K}}$ for
a suitable row and column indexing, then $l_{F}=l_{\tilde{F}}$. ∎

For $F\in F_{\mathcal{H}}$, and $H_{i}\in\mathcal{H}$ such that $F\subseteq
H_{i}$, let $\displaystyle\beta_{F}^{H_{i}}:=\frac{\\#\\{C\in C_{\mathcal{H}}\
|\ \overline{C}\cap H_{i}=F\\}}{2}$.

######  Theorem 4.3.

Let $\mathcal{H}$ be a pseudohyperplane arrangement in $\mathbb{R}^{d}$, and
$F\in F_{\mathcal{H}}$. Then, $\beta_{F}^{H_{i}}$ has the same value
$\beta_{F}$ for every $H_{i}\in\mathcal{H}_{F}$, and

$\det\mathbf{M}_{\mathcal{H}}=\prod_{F\in F_{\mathcal{H}}\setminus
C_{\mathcal{H}}}(1-b_{F})^{\beta_{F}}.$

###### Proof.

Take a face $E\in F_{\mathcal{H}}\setminus C_{\mathcal{H}}$. There exists an
apartment $K\in K_{\mathcal{H}}$ such that

$\displaystyle E=\bigcap_{H_{i}\in\mathcal{H}^{K}}H_{i}\cap K.$

We prove by backward induction on the dimension of $E$ that

$\forall H_{i},H_{j}\in\mathcal{H}_{E}:\
\beta_{E}^{H_{i}}=\beta_{E}^{H_{j}}=\beta_{E}\quad\text{and}\quad\det\mathbf{M}_{\mathcal{H}}^{K}=\prod_{F\in
F_{\mathcal{H}}^{K}\setminus C_{\mathcal{H}}^{K}}(1-b_{F})^{\beta_{F}}.$

Remark that $\displaystyle\beta_{F}^{H_{i}}=\frac{\\#\\{C\in
C_{\mathcal{H}}^{K}\ |\ \overline{C}\cap H_{i}=F\\}}{2}$. It is clear that, if
$\dim E=d-1$, then

$\beta_{E}=1\quad\text{and}\quad\det\mathbf{M}_{\mathcal{H}}^{K}=1-b_{E}.$

If $\dim E<d-1$, by induction hypothesis,

$\det\mathbf{M}_{\mathcal{H}}^{K}=(1-b_{E})^{l_{E}}\,\prod_{F\,\in\,(F_{\mathcal{H}}^{K}\setminus
C_{\mathcal{H}}^{K})\setminus\\{E\\}}(1-b_{F})^{\beta_{F}}.$

The leading monomial in $\det\mathbf{M}_{\mathcal{H}}^{K}$ is
$\displaystyle(-1)^{\frac{\\#C_{\mathcal{H}}^{K}}{2}}\prod_{C\in
C_{\mathcal{H}}^{K}}\mathrm{v}(C,E\tilde{C})=\big{(}-\prod_{H_{i}\in\mathcal{H}_{E}}a_{i}^{+}a_{i}^{-}\big{)}^{\frac{\\#C_{\mathcal{H}}^{K}}{2}}$.
Then, comparing the exponent of $a_{i}^{+}a_{i}^{-}$, we get $\displaystyle
l_{E}\,=\,\frac{\\#C_{\mathcal{H}}^{K}}{2}-\sum_{\begin{subarray}{c}F\,\in\,(F_{\mathcal{H}}^{K}\setminus
C_{\mathcal{H}}^{K})\setminus\\{E\\}\\\ F\subseteq
H_{i}\end{subarray}}\beta_{F}^{H_{i}}\,=\,\beta_{E}^{H_{i}}$. ∎

##  5 Proof of Theorem 1.1

From Corollary 2.3 and Theorem 2.4, we deduce that for any oriented matroid
$\mathsf{M}$ of rank $d$ included in $\mathbb{E}^{n}$, there exists a
pseudohyperplane arrangement $\mathcal{H}=\\{H_{i}\\}_{i\in[n]}$ in
$\mathbb{S}^{d}$ such that $V_{\mathsf{M}}=\det\mathbf{M}_{\mathcal{H}}$. _The
corresponding apartment and restriction to a fiber $\mathsf{M}_{I,u}$ are
respectively_

$K=\epsilon_{\mathcal{H}}^{-1}\Big{(}\big{\\{}v\in\mathsf{M}\ |\ \forall
i\in[n]\setminus
I:\,\epsilon_{i}(v)=\epsilon_{i}(u)\big{\\}}\Big{)}\quad\text{and}\quad\mathcal{H}^{K}=\\{H_{i}\\}_{i\in
I}$

_which means that $(\mathsf{M}_{I,u},\leq)$ is poset isomorphic to
$(F_{\mathcal{H}}^{K},\preceq)$_. Assuming that $a_{i}^{+}=a_{i}^{-}=0$ if
$i\in[n]\setminus I$, for a suitable row and column indexing we get
$\mathbf{M}_{\mathcal{H}}=\mathbf{M}_{\mathcal{H}}^{K}\oplus\mathbf{M}_{\mathcal{H}}^{K}\oplus\mathbf{I}$,
where $\mathbf{I}$ is the identity matrix of order
$\\#C_{\mathcal{H}}-2\,\\#C_{\mathcal{H}}^{K}$. Since
$V_{\mathsf{M}_{I,u}}=\det\mathbf{M}_{\mathcal{H}}^{K}$, we finally obtain

$V_{\mathsf{M}_{I,u}}=\prod_{F\in F_{\mathcal{H}}^{K}\setminus
C_{\mathcal{H}}}(1-b_{F})^{2\beta_{F}}=\prod_{v\in\mathsf{\mathsf{M}_{I,u}}\setminus
T_{\mathsf{M}}}(1-\mathrm{b}_{v})^{2\beta_{v}}=V_{\mathsf{M}_{I,u}}^{2}.$

$\blacksquare$

## References

  * [1] M. Aguiar, S. Mahajan, Topics in Hyperplane Arrangements, Math. Surveys Monogr. (226) (2017).
  * [2] A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G. Ziegler, Oriented Matroids, Encyclopedia Math. Appl. (46) $2^{\text{nd}}$ Ed. (1999).
  * [3] R. Bland, M. Las Vergnas, Orientability of Matroids, J. Combin. Theory Ser. B (24) (1978), 94–123.
  * [4] G. Denham, P. Hanlon, Some Algebraic Properties of the Schechtman-Varchenko Bilinear Forms, New Perspectives in Geometric Combinatorics, MSRI Publications 38, 1999.
  * [5] P. Deshpande, On Arrangements of Pseudohyperplanes, Proc. Indian Acad. Sci. Math. Sci. (126) 3 (2016), 399–420.
  * [6] J. Folkman, J. Lawrence, Oriented Matroids, J. Combin. Theory Ser. B (25) (1978), 199–236.
  * [7] Y. Gao, Y. Zhang, Diagonal Form of the Varchenko Matrices, J. Algebraic Combin. (48) (2018), 351–368.
  * [8] R. Gente, The Varchenko Matrix for Cones, PhD Thesis, Philipps-Universität Marburg, 2013.
  * [9] P. Hanlon, R. Stanley, A $q$-Deformation of a Trivial Symmetric Group Action, Trans. Amer. Math. Soc. (350) 11 (1998), 4445–4459.
  * [10] W. Hochstättler, V. Welker, The Varchenko Determinant for Oriented Matroids, Math. Z. (293) 3-4 (2019), 1415–1430.
  * [11] A. Olzhabayev, Y. Zhang, Diagonal Form of the Varchenko Matrices for Oriented Matroids, arXiv: 2001.06460.
  * [12] H. Randriamaro, A Deformed Quon Algebra, Commun. Math. (27) (2019), 103–112.
  * [13] H. Randriamaro, A Multiparametric Quon Algebra, Bull. Iranian. Math. Soc. (2020), DOI: https://doi.org/10.1007/s41980-020-00414-5.
  * [14] H. Randriamaro, The Varchenko Determinant for Apartments, Results Math. (75) 3 (2020) 86.
  * [15] V. Schechtman, A. Varchenko, Quantum Groups and Homology of Local Systems, Algebraic Geometry and Analytic Geometry, Springer, 1990.
  * [16] A. Varchenko, Bilinear Form of Real Configuration of Hyperplanes, Adv. Math. (97) 1 (1993), 110–144.
  * [17] G. Ziegler, Lectures on Polytopes, Grad. Texts in Math. (152) $7^{\text{th}}$ Pr. (2007).


# QED medium effects in (anti)neutrino-nucleus and electron-nucleus
scattering: elastic scattering on nucleons

Oleksandr Tomalak<EMAIL_ADDRESS>Theoretical Division, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  Ivan Vitev<EMAIL_ADDRESS>Theoretical
Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

(August 28, 2024)

Interpretation of current and future neutrino oscillation and electron
scattering experiments requires knowledge of lepton-nucleon and lepton-nucleus
interactions at the percent level. We study the exchange of photons between
charged particles and the nuclear medium for (anti)neutrino-, electron-, and
muon-induced reactions inside a large nucleus. While quantum electrodynamics
(QED)-medium contributions are formally suppressed by two powers of the
electromagnetic coupling constant $\alpha$ when compared to the leading-order
cross sections, low-energy modes and the nuclear size enhance the effect by
orders of magnitude. They require a proper infrared regularization, which we
implement as a screening of the electromagnetic interactions at atomic length
scales or above. We provide approximate analytic expressions for the
distortion of (anti)neutrino-nucleus and charged lepton-nucleus cross sections
and evaluate the QED-medium effects for realistic values of the screening
scale on the example of elastic scattering with nucleons inside the nucleus.
We find new permille- to percent-level effects, which were not considered in
either (anti)neutrino-nucleus or electron-nucleus scattering.

###### Contents

  1. 1 Introduction
  2. 2 QED medium effects in (anti)neutrino scattering
  3. 3 QED medium effects in electron scattering
  4. 4 Conclusions
  5. A Elastic lepton-nucleon scattering cross sections at tree level
  6. B QED medium corrections at different beam energies

##  1 Introduction

Electron, muon, and (anti)neutrino scattering on nucleons and nuclei provide
us with fundamental information about the complex internal structure and
dynamics of the target [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25]. This includes nucleon and nucleus
electromagnetic and electroweak form factors and radii, unpolarized and
polarized parton distribution functions, 3D tomography of the nucleon, and
parton interactions with the nuclear environment. Electromagnetic probes are
also at the heart of state-of-the-art electroweak measurements [26, 27, 28,
29], and utmost precision in our knowledge of (anti)neutrino-nucleus
scattering is paramount for the extraction of the neutrino oscillation
parameters [30, 31, 32, 33, 34, 35]. These research directions call for
improved description of the lepton-nucleus cross sections [36, 37, 38, 39, 40,
41, 42, 43, 44] and, ideally, the precise first-principles microscopic theory
of such interactions should be formulated and developed. The precision goal of
future experiments [45, 32, 35] cannot be met without accounting for the
exchange of virtual and real photons in (anti)neutrino-induced reactions [46,
47, 48]. Similarly, the exchange of photons between the electrically charged
leptons and nuclear medium in electron and (anti)neutrino scattering should be
taken into account and has not yet been studied in the literature. In this
paper, we take the first steps to address this outstanding question and show
that exchange of photons with the nuclear medium can induce permille- to
percent-level correction to the scattering cross sections through multiple
elastic scattering on nucleons in modern and near-future electron and neutrino
scattering experiments.

Understanding of particle scattering in matter is a problem of fundamental
importance to many fields of science. For continuous media, a formalism to
treat multiple interactions was developed by Molière more than seventy years
ago [49, 50]. In the context of quantum chromodynamics (QCD), derivation of
parton scattering in large nuclei via the exchange of gluons [51] was
motivated by efforts to understand the Cronin effect [52], the enhancement of
inclusive hadron production in proton-nucleus $p$A relative to proton-proton
$pp$ collisions at transverse momenta $\mathrm{p_{T}}\sim$ few GeV at Fermilab
fixed-target experiments. The approach was further refined to provide a boost-
invariant formulation of the scattering potential generated by the matter
constituents and applied to describe the broadening of back-to-back hadron
correlations [53] in deuterium-gold $d\mathrm{Au}$ reactions at the
Relativistic Heavy Ion Collider. Research on parton scattering in nuclear
matter is still ongoing [54, 55, 56]. Here, we focus instead on electron and
muon scattering off large nuclei.

The rest of our paper is organized as follows. In Section 2, we derive the
effects of collisional interactions in (anti)neutrino-nucleus scattering
within a soft-collinear effective field theory ($\mathrm{SCET}_{\mathrm{G}}$)
with Glauber photon exchanges between the charged leptons and the quantum
electrodynamics (QED) medium at leading power. We further perform calculations
without the effective field theory approximations and show phenomenological
results for the cross-section corrections for several nuclei of interest to
the neutrino and electron scattering communities. This formalism and results
are generalized to electron-nucleus scattering in Section 3, accounting for
the QED effects prior to the hard interaction. We provide a summary and
outlook in Section 4. Baseline lepton-nucleon scattering cross sections upon
which QED medium effects build are assembled in Appendix A. In Appendix B, we
present additional numerical examples for a wider range of beam energies
relevant to experiments from T2K/HyperK to MINERvA.

##  2 QED medium effects in (anti)neutrino scattering

In this Section, we follow the effective field theory (EFT) framework that was
used to derive QCD medium effects [57, 58] and formulate leading QED nuclear
medium contributions to the charged-current (anti)neutrino-nucleus scattering
cross sections.

The charged-current (anti)neutrino-nucleus scattering provides a clear
illustration of nuclear matter effects, which are induced by the exchange of
photons with a nuclear medium. Only the final-state charged lepton is subject
to such effects, while the four-fermion interactions (combined with QCD into
the Lagrangian ${\cal{L}_{\mathrm{F,QCD}}}$) can be always considered as the
hard scattering process. Following an elegant QCD derivation within the
$\mathrm{SCET}_{\mathrm{G}}$ effective field theory [57, 58, 59], we write
down the Lagrangian density ${\cal{L}}_{\mathrm{G}}$ for QED applications:

${\cal{L}}_{\mathrm{G}}={\cal{L}_{\mathrm{F,QCD}}}+\sum\limits_{p^{\prime},p^{\prime\prime}}e^{-i\left(p^{\prime}-p^{\prime\prime}\right)\cdot
x}\overline{\chi}_{n^{\prime},p^{\prime\prime}}i\left(n^{\prime}\cdot\partial+ieA^{\mathrm{G}}\right)\frac{\overline{n}\\!\\!\\!/\\!\,\,^{\prime}}{2}\chi_{n^{\prime},p^{\prime}}+{\cal{L}^{\prime}_{\mathrm{SCET}}}.$
(1)

Here, ${\cal{L}^{\prime}_{\mathrm{SCET}}}$ is the Lagrangian density for
power-suppressed operators and other sectors of QED that are not used in the
effective field theory calculation of this paper. The Glauber photon field is
$A_{\mathrm{G}}$: $\left(n^{\prime}\cdot A_{\mathrm{G}}\right)=2\pi
v\left(q\right)\delta\left(q_{0}\right)e^{iqz^{\prime}}/e$ (in the momentum
space) and is related to the static background potential $v$ arising from the
unit charge at the space-time point $z^{\prime}$. The electric charge is $e$,
and collinear spinors are $\chi$. Based on this Lagrangian, we write down the
leading-order scattering amplitude $\mathrm{T}^{\mathrm{LO}}_{\nu}$ of the
“hard” charged-current neutrino scattering process $J$ at the space-time point
$x_{0}$ as

$\mathrm{T}^{\mathrm{LO}}_{\nu}=\bar{\chi}_{n^{\prime},p^{\prime}}iJ\left(p^{\prime}\right)e^{ip^{\prime}\cdot
x_{0}},$ (2)

with the label momentum of the final-state charged lepton $p^{\prime}$ and the
corresponding collinear spinor $\chi_{n^{\prime},p^{\prime}}$. As usual,
$n^{\prime}_{\nu}$ and $\bar{n}^{\prime}_{\nu}$ are conventional four-vectors
that satisfy $\left(n^{\prime}\right)^{2}=\left(\bar{n}^{\prime}\right)^{2}=0$
and $n^{\prime}\cdot\bar{n}^{\prime}=2$. The direction of the charged-lepton
momentum is defined by the reference vector $n^{\prime}_{\nu}=(1,0,0,1)$. At
leading power, the unpolarized scattering cross section $\sigma_{\nu}$ is
determined by the squared matrix element as

$\sigma_{\nu}=c_{\nu}\left(\bar{n}^{\prime}\cdot
p^{\prime}\right)\mathrm{Tr}\left[\frac{n\\!\\!\\!/\\!\,\,^{\prime}}{2}J\left(p^{\prime}\right)J^{\dagger}\left(p^{\prime}\right)\right],$
(3)

where a constant $c_{\nu}$ depends on the kinematics of external
particles.111The explicit form for the tree-level elastic lepton-nucleon
scattering cross sections is presented in Appendix A.

Figure 1: The final-state charged lepton exchanges Glauber photons with the
nuclear medium. Diagrams with one and two photons contribute to leading
medium-induced corrections.

The electrically-charged medium serves as a source of the static
electromagnetic potential, which allows us to have the same expressions for
Glauber photons in an arbitrary $R_{\xi}$ gauge. Accounting for the exchange
of one and two Glauber photons (with the dominant perpendicular component of
the momentum: $q\sim\left(\lambda,\lambda,\sqrt{\lambda}\right)$ at small
expansion parameter $\lambda$) between the charged lepton and the nucleus
charge distribution $\rho$, as it is shown in Fig. 1 (taken from [58]), we
obtain $\mathrm{O}\left(\alpha^{2}\right)$ QED medium modification of the
(anti)neutrino scattering cross section $\delta\sigma_{\nu}$:

$\delta\sigma_{\nu}=\int\mathrm{d}z^{\prime}\rho\left(z^{\prime}\right)\int\frac{\mathrm{d}^{2}\vec{q}_{\perp}}{\left(2\pi\right)^{2}}|v\left(\vec{q}_{\perp}\right)|^{2}\left(\sigma_{\nu}\left(\vec{p}^{\prime}-\vec{q}_{\perp}\right)-\sigma_{\nu}\left(\vec{p}^{\prime}\right)\right).$
(4)

In Eq. (4), the integration goes along the charged-lepton direction
$z^{\prime}$, and we neglect the bending of the lepton inside the nucleus. The
$\vec{q}_{\perp}$ integration goes over the plane that is orthogonal to the
charged-lepton momentum. For typical Coulomb potential of the Glauber photon
in the momentum space
$v\left(\vec{q}_{\perp}\right)=e^{2}/\vec{q}^{2}_{\perp}$, the expression in
Eq. (4) diverges in the infrared. To remove this unphysical divergence, we
regularize the Coulomb potential by screening at atomic distances $R$ or above

$v\left(\vec{q}_{\perp}\right)=\frac{e^{2}}{\vec{q}_{\perp}^{2}+\zeta^{2}},\qquad\qquad\zeta
R\ll 1.$ (5)

In our calculations, we assume the same Woods-Saxon spherically-symmetric
distribution for neutrons and protons inside the nucleus, so that the charge
distribution is expressed as

$\rho\left(r\right)\sim\frac{1}{1+e^{\frac{r-R_{0}}{a}}},$ (6)

with the nucleus size parameter $R_{0}$ and $a=0.5\leavevmode\nobreak\
\mathrm{fm}$, and normalization to the charge of the nucleus $Z$:
$Z=\int\rho\left(r\right)\mathrm{d}^{3}r$ (or number of neutrons for the
neutron distribution). Subsequently, we average over all nucleons, as possible
scattering centers inside the nucleus. Only distributions of neutrons and
protons inside the nucleus determine the dependence of medium effects on the
nucleus under consideration. In this paper, we take nuclear radii from Ref.
[60] and collect them in Table 1.222Note that the root-mean-square (rms)
charge radius $R_{\mathrm{rms}}$ is related to the parameters $R_{0}$ and $a$
in Eq. (6) as
$R_{\mathrm{rms}}^{2}=12a^{2}\frac{\mathrm{Li}_{5}\left(-e^{\frac{R_{0}}{a}}\right)}{\mathrm{Li}_{3}\left(-e^{\frac{R_{0}}{a}}\right)}.$
(7)

| ${}^{2}_{1}\mathrm{H}$ | ${}^{12}_{6}\mathrm{C}$ | ${}^{16}_{8}\mathrm{O}$ | ${}^{40}_{18}\mathrm{Ar}$ | ${}^{56}_{26}\mathrm{Fe}$ | ${}^{208}_{82}\mathrm{Pb}$
---|---|---|---|---|---|---
$R_{\mathrm{rms}}\leavevmode\nobreak\ (\mathrm{fm})$ | $2.1421$ | $2.4702$ | $2.6991$ | $3.4274$ | $3.7377$ | $5.5012$
Table 1: Root-mean-square charge radii for various nuclei are presented [60].

Eq. (4) is obtained assuming the charged lepton to be collinear and massless
($m_{\ell}=0$). Besides this $\mathrm{SCET_{G}}$ calculation, we account for
all neglected power corrections by substituting into Eq. (2) exact expressions
for spinors of external particles and for the charged current $J$.

In this paper, we focus on the charged-current elastic (anti)neutrino-nucleon
scattering as the simplest charged-current process inside the nucleus. This
process depends on two kinematic variables: the (anti)neutrino energy
$E_{\nu}$ in the nucleon rest frame and the momentum transfer
$Q^{2}=-\left(p-p^{\prime}\right)^{2}$, which we determine from the lepton
kinematics solely, i.e., $p$ denotes the momentum of the initial lepton. At
leading power, the shifted cross section
$\sigma_{\nu}\left(\vec{p}^{\prime}-\vec{q}_{\perp}\right)$ in Eq. (4) can be
obtained by the replacement of the momentum transfer $Q^{2}$ in the tree-level
expression with

$Q^{2}\rightarrow
Q^{2}+q^{2}_{\perp}-2E_{\nu}q_{\perp}\sin\theta_{\ell}\cos\phi,$ (8)

where $E_{\nu}$ is the (anti)neutrino energy, $\theta_{\ell}$ is the lepton
scattering angle in the nucleon rest frame, and $\phi$ is the angle of
$\vec{q}_{\perp}$ in the orthogonal to the lepton momentum plane. The exact
QED treatment includes also $\vec{q}_{\perp}$, $\vec{q}_{||}$ (along the
lepton direction), lepton-mass, and kinematic corrections. For numerical
estimates, we take the default values for the nucleon form factors from Refs.
[61] and [62].

Figure 2: Relative single-nucleon (inside a nucleus) $\nu_{\mu}n\to\mu^{-}p$
cross-section correction induced by QED nuclear medium effects is shown as a
function of the momentum transfer $Q^{2}$ for the neutrino beam energy
$E_{\nu_{\mu}}=2\leavevmode\nobreak\ \mathrm{GeV}$. Results are presented for
two values of the regularization parameter $\zeta$ in Eq. (5):
$\zeta=10^{-10}\leavevmode\nobreak\ \mathrm{GeV}$ and
$\zeta=\frac{m_{e}\mathrm{Z}^{1/3}}{192}$. We compare the full QED result to
the $\mathrm{SCET}_{\mathrm{G}}$ “leading-power” approximation of Eqs. (4) and
(8). Figure 3: Same as Fig. 2 but for the antineutrino scattering
$\bar{\nu}_{\mu}p\to\mu^{+}n$.

As illustration, we present the relative distortion of charged-current elastic
cross sections per nucleon on ${}^{2}_{1}\mathrm{H}$, ${}^{12}_{6}\mathrm{C}$,
${}^{16}_{8}\mathrm{O}$, ${}^{40}_{18}\mathrm{Ar}$, ${}^{56}_{26}\mathrm{Fe}$,
and ${}^{208}_{82}\mathrm{Pb}$ nuclei for the neutrino scattering in Fig. 2
and for the antineutrino scattering in Fig. 3. We neglect the motion and
nuclear interactions of the initial and final nucleons and assume the
scattering to happen on individual nucleons. We choose the (anti)neutrino beam
energy $E_{\nu}=2\leavevmode\nobreak\ \mathrm{GeV}$, and vary the
regularization parameter between $\zeta=10^{-10}\leavevmode\nobreak\
\mathrm{GeV}$ and the atomic screening scale
$\zeta=\frac{m_{e}\mathrm{Z}^{1/3}}{192}$ [63]. The resulting correction does
not depend on the parameter $\zeta$ significantly, the variation of $\zeta$
from the atomic scale estimates the size of the uncertainty. At low momentum
transfer, when the cross section is large, medium effects are negative and can
reach a few permille. Going to a larger momentum transfer, the correction
crosses zero and increases up to the permille-level size. Contrary to neutrino
scattering, when the correction first decreases and then increases up to zero,
the correction in antineutrino scattering increases monotonically at low
momentum transfer due to a more rapid falloff of the tree-level cross section
with the momentum transfer. The same difference in tree-level cross sections
also explains why medium effects are larger for the antineutrino scattering
compared to the neutrino scattering . Remarkably, the
$\mathrm{SCET}_{\mathrm{G}}$ calculation is in very good agreement with the
complete QED result, at least above the lepton-mass scale at GeV energies.
Power corrections are relatively small but can be relevant for the precise
first-principles cross-section calculations at high momentum transfers and
(anti)neutrino energies close to the lepton production threshold. For lepton
energies much larger than its mass, the size of the correction at zero
momentum transfer can be easily estimated as
$\frac{2\alpha^{2}\mathrm{Z}^{4/3}}{\left(MR_{0}\right)^{2}}\left(1+\frac{E^{\prime}_{\ell}}{M}\right)^{2}\ln\frac{\zeta}{E_{\ell}^{\prime}}$
(with the energy of the final-state charged lepton $E_{\ell}^{\prime}$ and the
nucleon mass $M$), which demonstrates a logarithmic enhancement due to the
separation between atomic energy scale and energies in the experiment. This
simple expression estimates the order of magnitude of the correction to be at
permille level or below for incoming (anti)neutrino energy $E_{\nu}\lesssim
3\leavevmode\nobreak\ \mathrm{GeV}$ and nuclei of interest. QED nuclear medium
effects do not contain logarithms of the small lepton mass. Consequently, the
corresponding corrections are (anti)neutrino flavor universal at GeV energies
of the incoming beam. Phenomenological applications to lower (anti)neutrino
energies and to nuclear beta decay are left to future work.

##  3 QED medium effects in electron scattering

In this Section, we generalize the developed formalism to the scattering of
charged leptons. Both initial- and final-state leptons can interact with the
medium in this case. For the hard scattering $\mathrm{H}$ at the space-time
point $x_{0}$, we factor out spinors of initial and final states and obtain
the leading-order scattering amplitude $\mathrm{T}_{\ell}^{\mathrm{LO}}$:

$\mathrm{T}_{\ell}^{\mathrm{LO}}=\bar{\chi}_{n^{\prime},p^{\prime}}\mathrm{H}\left(p^{\prime},p\right)\chi_{n,p}e^{-i\left(p-p^{\prime}\right)\cdot
x_{0}},$ (9)

with the label momenta $p^{\prime},\leavevmode\nobreak\ p$ and light-cone
vectors $n^{\prime},\leavevmode\nobreak\ n$ for final and initial states,
respectively. For scattering of leptons, the unpolarized cross section
$\sigma_{\ell}$ at leading power is determined by the squared matrix element
as

$\sigma_{\ell}=c_{\ell}\left(\bar{n}^{\prime}\cdot
p^{\prime}\right)\left(\bar{n}\cdot
p\right)\mathrm{Tr}\left[\frac{n\\!\\!\\!/\\!\,\,^{\prime}}{2}\mathrm{H}\left(p^{\prime},p\right)\frac{n\\!\\!\\!/\\!\,\,}{2}\gamma_{0}\mathrm{H}^{\dagger}\left(p^{\prime},p\right)\gamma_{0}\right],$
(10)

where a constant $c_{\ell}$ depends on the kinematics of external particles.

Accounting for the exchange of Glauber photons with the nuclear medium, we
obtain $\mathrm{O}\left(\alpha^{2}\right)$ QED medium modification of the
lepton scattering cross section $\delta\sigma_{\ell}$:

$\delta\sigma_{\ell}=\int\frac{\mathrm{d}^{2}\vec{q}_{\perp}}{\left(2\pi\right)^{2}}|v\left(\vec{q}_{\perp}\right)|^{2}\left[\int\mathrm{d}z^{\prime}\rho\left(z^{\prime}\right)\left(\sigma_{\ell}\left(\vec{p}^{\prime}-\vec{q}_{\perp},\vec{p}\right)-\sigma_{\ell}\left(\vec{p}^{\prime},\vec{p}\right)\right)+\int\mathrm{d}z\rho\left(z\right)\left(\sigma_{\ell}\left(\vec{p}^{\prime},\vec{p}+\vec{q}_{\perp}\right)-\sigma_{\ell}\left(\vec{p}^{\prime},\vec{p}\right)\right)\right],$
(11)

where the integration goes along the initial- and final-state lepton
directions $z$ and $z^{\prime}$, respectively. We neglect the bending of
leptons inside the nucleus, and specify the hard scattering process as the
interaction with a larger momentum transfer. The interference of diagrams with
Glauber photons attached to the initial- and final-state charged leptons
vanishes exactly. Similar to (anti)neutrino scattering, we also include all
power corrections by performing an exact QED calculation.

For definiteness, we focus on the elastic lepton-nucleon scattering as the
hard process. As all elastic $2\to 2$ reactions, this process depends on two
kinematic variables: the incoming lepton energy $E_{\ell}$ in the nucleon rest
frame and the momentum transfer $Q^{2}=-\left(p-p^{\prime}\right)^{2}$, which
we determine from the lepton kinematics solely. At leading power, the shifted
cross sections
$\sigma_{\ell}\left(\vec{p}^{\prime}-\vec{q}_{\perp},\vec{p}\right)$ and
$\sigma_{\ell}\left(\vec{p}^{\prime},\vec{p}+\vec{q}_{\perp}\right)$ in Eq.
(11) can be obtained by the replacement of the momentum transfer $Q^{2}$ as

$\displaystyle Q^{2}$ $\displaystyle\to
Q^{2}+q^{2}_{\perp}-2p_{\ell}q_{\perp}\sin\theta_{\ell}\cos\phi,$ (12)
$\displaystyle Q^{2}$ $\displaystyle\to
Q^{2}+q^{2}_{\perp}-2p^{\prime}_{\ell}q_{\perp}\sin\theta_{\ell}\cos\phi,$
(13)

respectively. We consider the same inputs as for the (anti)neutrino-nucleon
scattering and average over all nucleons that serve as scattering centers
inside the nucleus. Assuming the scattering on individual nucleons and
neglecting motion and nuclear interactions of the target and recoil nucleons,
we present our results for the modification of elastic electron-nucleus cross
sections on the nucleus in Fig. 4.

Figure 4: Relative electron-nucleus cross-section correction induced by QED
nuclear medium effects is shown as a function of the momentum transfer $Q^{2}$
for the electron beam energy $E_{e}=2\leavevmode\nobreak\ \mathrm{GeV}$.
Results are presented for two values of the regularization parameter $\zeta$
in Eq. (5): $\zeta=10^{-10}\leavevmode\nobreak\ \mathrm{GeV}$ and
$\zeta=\frac{m_{e}\mathrm{Z}^{1/3}}{192}$. We compare the full QED result to
the $\mathrm{SCET}_{\mathrm{G}}$ “leading-power” approximation of Eqs. (11,
12), and (13).

Contrary to (anti)neutrino interaction, the scattering amplitude of charged
particles has $1/Q^{2}$ pole, which spoils a clear scale separation between
the hard and Glauber regions. The momentum transfer in the underlying process
can reach relatively small values, making the virtuality of the exchanged
Glauber photon of hard-scale size. For this kinematics, we switch between the
frameworks of Glauber photon exchanges from the initial- vs final-state
charged lepton. We study the same nuclear targets ${}^{2}_{1}\mathrm{H}$,
${}^{12}_{6}\mathrm{C}$, ${}^{16}_{8}\mathrm{O}$, ${}^{40}_{18}\mathrm{Ar}$,
${}^{54}_{26}\mathrm{Fe}$, and ${}^{208}_{82}\mathrm{Pb}$ as for the
(anti)neutrino scattering, and choose the electron beam energy
$E_{e}=2\leavevmode\nobreak\ \mathrm{GeV}$. Likewise, we vary the
regularization parameter $\zeta$ in Eq. (5) between
$10^{-10}\leavevmode\nobreak\ \mathrm{GeV}$ and the atomic scale
$\zeta=\frac{m_{e}Z^{1/3}}{192}$ [63]. The resulting correction does not
depend on the parameter $\zeta$ significantly, the variation of $\zeta$ from
the atomic scale indicates the size of the uncertainty. The effect grows with
the atomic number of the nucleus, as it is expected. As it is mentioned above,
the clear separation between hard and Glauber regions disappears going to low
momentum transfers and these regions overlap at sufficiently small $Q^{2}$.
Consequently, our correction is not well defined for $Q^{2}\lesssim
0.005-0.008\leavevmode\nobreak\ \mathrm{GeV}^{2}$ (formally obtained when the
difference of two momentum transfers in Eqs. (11) and (12) is of order
$10^{-3}Q^{2}$ and the slope of the correction significantly increases). In
this region, the leading-order cross section and the relative nuclear medium
effects diverge. Contrary to the (anti)neutrino-nucleus scattering, the
dependence of this correction on the momentum transfer is monotonically
decreasing, while the correction itself is positive definite. Power
corrections are relatively small but might be relevant for the extraction of
the nuclear spectral functions [64], medium effects from this paper have to be
included on top of the traditional radiative corrections and Coulomb effects,
and precise first-principles cross-section calculations, especially at
incoming lepton energies that are close to the threshold.

Account for the QED nuclear medium effects along the lines of the study
presented in this paper can shed new light on the charge radius discrepancy
between the elastic electron-proton scattering data of the recent high-
precision A1@MAMI result [12, 13] $R_{p}=0.879\pm 0.008\leavevmode\nobreak\
\mathrm{fm}$ and the PRad measurement [24], with the rms charge radius of the
proton $R_{p}=0.838^{+0.005+0.004}_{-0.004-0.003}$, the muonic [65, 66]
hydrogen Lamb shift extraction $R_{p}=0.84087\pm 0.00039\leavevmode\nobreak\
\mathrm{fm}$, and recent measurements with ordinary hydrogen [67, 68, 69].
However, our calculation with an approximate charge distribution works only
for an estimate of the corresponding effects and points to the reduction of
the scattering-data radius and toward agreement with other measurements.
Applying this correction to the generated pseudo-data with a floating
normalization, we obtain a reduction of the charge radius of order of the
proton-radius discrepancy at hundreds $\mathrm{MeV}$ and $\mathrm{GeV}$
electron beam energies. We leave a more elaborate analysis of the electron-
proton scattering data to future work.

##  4 Conclusions

In this work, we developed a method to evaluate the
$\mathrm{O}\left(\alpha^{2}\right)$ QED nuclear medium corrections to
(anti)neutrino-nucleus and charged lepton-nucleus scattering cross sections by
exploiting the $\mathrm{SCET}_{\mathrm{G}}$ effective field theory, and by the
full QED calculation in the light-cone basis. Physically, these are the
effects of multiple elastic scattering on nucleons in the nucleus along the
path of the charged-lepton propagation. At leading power, we found that the
radiation-free QED effects on (anti)neutrino-nucleus cross sections have the
same form as the QCD medium corrections to quark jets. We also included the
power-suppressed contributions to our numerical results. To generalize the
formalism to charged-lepton scattering, we identified the necessary
modifications for the exchange of photons from the incoming charged lepton and
pointed out that the interference terms between the exchange of photons from
the initial- and final-state charged leptons vanish, at least up to order
$\mathrm{O}\left(\alpha^{2}\right)$.

We presented numerical estimates on the example of the elastic scattering on
nucleons embedded in various nuclei of interest to the neutrino and electron
scattering communities. Our first-principles results provide thus far missing
permille- to percent-level corrections to lepton-nucleus scattering cross
sections. We found that QED medium effects in (anti)neutrino-nucleus
scattering depend on the kinematics with suppression in the forward (small
$Q^{2}$) direction and enhancement of the backward (large $Q^{2}$) scattering,
while these effects in electron-nucleus scattering lead to the enhancement,
which monotonically decreases from forward to backward scattering (increasing
the momentum transfer).

The results presented in this paper point to sizable QED medium effects for
high-Z nuclei. Our experience with strongly interacting systems indicates that
medium-induced radiative corrections can be even more significant than elastic
multiple scattering in nuclear matter alone [58, 70, 71, 72] and play a key
role in the modification of scattering cross sections. Calculations of the QED
medium effects in pion production, deep inelastic scattering [73], and
processes with bremsstrahlung are thus an important future priority that we
will pursue.

## Acknowledgments

O.T. thanks Richard Hill, Qing Chen, Kevin McFarland, and Clarence Wret for
collaboration on [47, 48], Richard Hill for suggestions regarding qualitative
description, Kevin McFarland for suggestions regarding electron scattering
section, and Weiyao Ke for useful discussions. The work is supported by the US
Department of Energy through the Los Alamos National Laboratory. Los Alamos
National Laboratory is operated by Triad National Security, LLC, for the
National Nuclear Security Administration of the U.S. Department of Energy
(Contract No. 89233218CNA000001). This research is funded by LANL’s Laboratory
Directed Research and Development (LDRD/PRD) program under project number
20210968PRD4. This research was supported in part by the National Science
Foundation under Grant No. NSF PHY-1748958. FeynCalc [74, 75], LoopTools [76],
Mathematica [77], and DataGraph were extremely useful in this work.

##  Appendix A Elastic lepton-nucleon scattering cross sections at tree level

We summarize the neutrino-neutron and antineutrino-proton charged-current
elastic scattering cross sections:

$\displaystyle\nu_{\ell}\left(p\right)n\left(k\right)$
$\displaystyle\to\ell^{-}\left(p^{\prime}\right)p\left(k^{\prime}\right),$
(14) $\displaystyle\bar{\nu}_{\ell}\left(p\right)p\left(k\right)$
$\displaystyle\to\ell^{+}\left(p^{\prime}\right)n\left(k^{\prime}\right),$
(15)

which get modified by the QED medium effects. At leading order in the QED
coupling constant, the unpolarized differential (anti)neutrino-nucleon
scattering cross section is conveniently expressed in terms of the structure-
dependent $A,B$, and $C$ parameters [78, 79, 80]

$\frac{d\sigma_{\nu}}{dQ^{2}}(Q^{2},E_{\nu})=\frac{c^{2}_{ud}}{16\pi}\frac{M^{2}}{E_{\nu}^{2}}\left[\left(\tau+r^{2}\right)A(Q^{2})-\nu
B(Q^{2})+\frac{\nu^{2}}{1+\tau}C(Q^{2})\right]\,,$ (16)

where $\tau=Q^{2}/\left(4M^{2}\right)$, $r=m_{\ell}/(2M)$, and
$\nu=E_{\nu}/M-\tau-r^{2}$. At leading order, the Wilson coefficients $c_{ud}$
of the four-fermion Lagrangian are given by
$2\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}$, where $\mathrm{G}_{\mathrm{F}}$ is
the Fermi coupling constant and $V_{ud}$ is the Cabibbo-Kobayashi-Maskawa
(CKM) matrix element. A more precise determination of the Wilson coefficients
is given in [81]. Assuming the isospin symmetry, the structure-dependent
factors $A,\leavevmode\nobreak\ B$, and $C$ are expressed in terms of Sachs
electric, $G^{V}_{E}$, and magnetic, $G^{V}_{M}$, isovector, axial, $F_{A}$,
and pseudoscalar, $F_{P}$, form factors as

$\displaystyle A$
$\displaystyle=\tau\left(G^{V}_{M}\right)^{2}-\left(G^{V}_{E}\right)^{2}+(1+\tau)F_{A}^{2}-r^{2}\left(\left(G^{V}_{M}\right)^{2}+F_{A}^{2}-4\tau
F_{P}^{2}+4F_{A}F_{P}\right)\,,$ (17) $\displaystyle B$
$\displaystyle=4\eta\tau F_{A}G^{V}_{M}\,,$ (18) $\displaystyle C$
$\displaystyle=\tau\left(G^{V}_{M}\right)^{2}+\left(G^{V}_{E}\right)^{2}+(1+\tau)F_{A}^{2}\,,$
(19)

where $\eta=+1$ corresponds to neutrino scattering $\nu_{\ell}n\to\ell^{-}p$
and $\eta=-1$ corresponds to antineutrino scattering
$\bar{\nu}_{\ell}p\to\ell^{+}n$. In the limit of isospin symmetry, when the
mass, $M$, for both nucleons is approximately the same, both the electric and
magnetic isovector form factors are given by the difference of proton and
neutron form factors, i.e., $G_{E,M}^{V}=G_{E,M}^{p}-G_{E,M}^{n}$. All the
form factors are functions of the momentum transfer
$Q^{2}=-\left(p-p^{\prime}\right)^{2}$. For electron and muon flavors, $Q^{2}$
increases from forward to backward directions with corresponding values
$Q^{2}_{\mathrm{-}}$ to $Q^{2}_{\mathrm{+}}$, respectively,

$Q^{2}_{\pm}=\frac{2ME^{2}_{\nu}}{M+2E_{\nu}}-4M^{2}\frac{M+E_{\nu}}{M+2E_{\nu}}r^{2}\pm\frac{4M^{2}E_{\nu}}{M+2E_{\nu}}\sqrt{\left(\frac{E_{\nu}}{2M}-r^{2}\right)^{2}-r^{2}}.$
(20)

Now, we consider the elastic scattering of an electrically charged lepton on
the nucleon target at rest $N$:

$\ell\left(p\right)N\left(k\right)\to\ell\left(p^{\prime}\right)N\left(k^{\prime}\right),$
(21)

which defines the laboratory frame as $k=(M,0)$, $p=(E_{\ell},\vec{p})$,
$k^{\prime}=(M+\frac{Q^{2}}{2M},\vec{p}-\vec{p}^{\prime})$,
$p^{\prime}=(E_{\ell}-\frac{Q^{2}}{2M},\vec{p}^{\prime})$, and the lepton
scattering angle is $\theta_{\mathrm{lab}}$. The unpolarized differential
cross section is conveniently expressed in terms of the nucleon Sachs electric
and magnetic form factors $G_{E}$ and $G_{M}$ as [82, 83]

$\frac{d\sigma_{\ell}}{dQ^{2}}(Q^{2},E_{\ell})=\frac{\pi\alpha^{2}}{2M^{2}\vec{p}^{2}}\frac{G_{M}^{2}+\frac{\varepsilon}{\tau}G_{E}^{2}}{1-\varepsilon_{\mathrm{T}}},$
(22)

with kinematic variables

$\varepsilon_{\mathrm{T}}=\frac{\nu_{\ell}^{2}-\tau(1+\tau)(1+2\varepsilon_{0})}{\nu_{\ell}^{2}+\tau(1+\tau)(1-2\varepsilon_{0})}\,,\qquad\quad\varepsilon=\varepsilon_{\mathrm{T}}+\varepsilon_{0}(1-\varepsilon_{\mathrm{T}})\,,\qquad\quad\varepsilon_{0}=\frac{2m_{\ell}^{2}}{Q^{2}}\,,\qquad\quad\nu_{\ell}=\frac{E_{\ell}}{M}-\tau.$
(23)

In the elastic lepton-nucleon scattering, $Q^{2}$ increases from $0$ for the
forward scattering to the maximally allowed value $Q^{2}_{\mathrm{max}}$:

$Q^{2}_{\mathrm{max}}=\frac{4M^{2}\left(E_{\ell}^{2}-m^{2}_{\ell}\right)}{M^{2}+2ME_{\ell}+m^{2}_{\ell}},$
(24)

for the backward scattering.

##  Appendix B QED medium corrections at different beam energies

In this Appendix, we provide calculations of QED nuclear medium corrections to
elastic scattering cross sections on nucleons inside the nuclei for different
beam energies relevant to the experiment. Specifically, the 600 MeV choice is
at the peak of T2K/HyperK [31, 33, 32] and close to the peak of short-baseline
experiments at Fermilab [84, 85, 86], and we show results in Figs. 5-7. We
further show the 4 GeV case in Figs. 8-10, which is close to the 3.5 GeV peak
of the MINERvA experiment [87, 88, 89, 90, 91, 92] and 2 times larger than our
default energy. At higher energy, $\mathrm{SCET}_{\mathrm{G}}$ agrees with
full QED calculation even better than at 2 GeV, while the lepton-mass effects
and kinematic modifications become important at lower energy. Note that
corrections in antineutrino scattering with $600\leavevmode\nobreak\
\mathrm{MeV}$ beam, cf. Fig. 6, can reach $\%$ level at backward angles when
the cross section itself is very small.

Figure 5: Same as Fig. 2 but for the incoming neutrino energy
$E_{\nu_{\mu}}=600\leavevmode\nobreak\ \mathrm{MeV}$.  Figure 6: Same as Fig.
3 but for the incoming antineutrino energy
$E_{\bar{\nu}_{\mu}}=600\leavevmode\nobreak\ \mathrm{MeV}$.  Figure 7: Same as
Fig. 4 but for the incoming electron energy $E_{e}=600\leavevmode\nobreak\
\mathrm{MeV}$.  Figure 8: Same as Fig. 2 but for the incoming neutrino energy
$E_{\nu_{\mu}}=4\leavevmode\nobreak\ \mathrm{GeV}$.  Figure 9: Same as Fig. 3
but for the incoming antineutrino energy
$E_{\bar{\nu}_{\mu}}=4\leavevmode\nobreak\ \mathrm{GeV}$.  Figure 10: Same as
Fig. 4 but for the incoming electron energy $E_{e}=4\leavevmode\nobreak\
\mathrm{GeV}$.

## References

  * Hofstadter [1956] R. Hofstadter, Rev. Mod. Phys. 28, 214 (1956).
  * Ernst  _et al._ [1960] F. J. Ernst, R. G. Sachs, and K. C. Wali, Phys. Rev. 119, 1105 (1960).
  * Hand  _et al._ [1963] L. N. Hand, D. G. Miller, and R. Wilson, Rev. Mod. Phys. 35, 335 (1963).
  * Dombey [1969] N. Dombey, Rev. Mod. Phys. 41, 236 (1969).
  * Akhiezer and Rekalo [1973] A. I. Akhiezer and M. P. Rekalo, Fiz. Elem. Chast. Atom. Yadra 4, 662 (1973).
  * Drechsel and Giannini [1989] D. Drechsel and M. M. Giannini, Rept. Prog. Phys. 52, 1083 (1989).
  * Boffi  _et al._ [1993] S. Boffi, C. Giusti, and F. D. Pacati, Phys. Rept. 226, 1 (1993).
  * Blomqvist  _et al._ [1998] K. I. Blomqvist _et al._ , Nucl. Instrum. Meth. A 403, 263 (1998).
  * Leemann  _et al._ [2001] C. W. Leemann, D. R. Douglas, and G. A. Krafft, Ann. Rev. Nucl. Part. Sci. 51, 413 (2001).
  * Arrington  _et al._ [2007] J. Arrington, C. D. Roberts, and J. M. Zanotti, J. Phys. G 34, S23 (2007), arXiv:nucl-th/0611050.
  * Perdrisat  _et al._ [2007] C. F. Perdrisat, V. Punjabi, and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 59, 694 (2007), arXiv:hep-ph/0612014.
  * Bernauer  _et al._ [2010] J. C. Bernauer _et al._ (A1), Phys. Rev. Lett. 105, 242001 (2010), arXiv:1007.5076 [nucl-ex].
  * Bernauer  _et al._ [2014] J. C. Bernauer _et al._ (A1), Phys. Rev. C 90, 015206 (2014), arXiv:1307.6227 [nucl-ex].
  * Bernauer [2010] J. C. Bernauer,  _Measurement of the elastic electron-proton cross section and separation of the electric and magnetic form factor in the Q 2 range from 0.004 to 1 (GeV/c)2_, Ph.D. thesis, Mainz U., Inst. Kernphys. (2010).
  * Zhan  _et al._ [2011] X. Zhan _et al._ , Phys. Lett. B 705, 59 (2011), arXiv:1102.0318 [nucl-ex].
  * Abrahamyan  _et al._ [2012] S. Abrahamyan _et al._ , Phys. Rev. Lett. 108, 112502 (2012), arXiv:1201.2568 [nucl-ex].
  * Accardi  _et al._ [2016] A. Accardi _et al._ , Eur. Phys. J. A 52, 268 (2016), arXiv:1212.1701 [nucl-ex].
  * Androic  _et al._ [2013] D. Androic _et al._ (Qweak), Phys. Rev. Lett. 111, 141803 (2013), arXiv:1307.5275 [nucl-ex].
  * Gilman  _et al._ [2013] R. Gilman _et al._ (MUSE), PSI report (2013), arXiv:1303.2160 [nucl-ex].
  * Punjabi  _et al._ [2015] V. Punjabi, C. F. Perdrisat, M. K. Jones, E. J. Brash, and C. E. Carlson, Eur. Phys. J. A 51, 79 (2015), arXiv:1503.01452 [nucl-ex].
  * Mosel [2016] U. Mosel, Ann. Rev. Nucl. Part. Sci. 66, 171 (2016), arXiv:1602.00696 [nucl-th].
  * Aschenauer  _et al._ [2019] E. C. Aschenauer, S. Fazio, J. H. Lee, H. Mantysaari, B. S. Page, B. Schenke, T. Ullrich, R. Venugopalan, and P. Zurita, Rept. Prog. Phys. 82, 024301 (2019), arXiv:1708.01527 [nucl-ex].
  * Adams  _et al._ [2018] B. Adams _et al._ , CERN report (2018), arXiv:1808.00848 [hep-ex].
  * Xiong  _et al._ [2019] W. Xiong _et al._ , Nature 575, 147 (2019).
  * Anderle  _et al._ [2021] D. P. Anderle _et al._ , Front. Phys. (Beijing) 16, 64701 (2021), arXiv:2102.09222 [nucl-ex].
  * Erler and Ramsey-Musolf [2005] J. Erler and M. J. Ramsey-Musolf, Phys. Rev. D 72, 073003 (2005), arXiv:hep-ph/0409169.
  * Anthony  _et al._ [2005] P. L. Anthony _et al._ (SLAC E158), Phys. Rev. Lett. 95, 081601 (2005), arXiv:hep-ex/0504049.
  * Kumar  _et al._ [2013] K. S. Kumar, S. Mantry, W. J. Marciano, and P. A. Souder, Ann. Rev. Nucl. Part. Sci. 63, 237 (2013), arXiv:1302.6263 [hep-ex].
  * Becker  _et al._ [2018] D. Becker _et al._ , Eur. Phys. J. A 54, 208 (2018), arXiv:1802.04759 [nucl-ex].
  * Adamson  _et al._ [2011] P. Adamson _et al._ (MINOS), Phys. Rev. Lett. 107, 181802 (2011), arXiv:1108.0015 [hep-ex].
  * Abe  _et al._ [2011] K. Abe _et al._ (T2K), Nucl. Instrum. Meth. A 659, 106 (2011), arXiv:1106.1238 [physics.ins-det].
  * Abe  _et al._ [2015] K. Abe _et al._ (Hyper-Kamiokande Proto-), PTEP 2015, 053C02 (2015), arXiv:1502.05199 [hep-ex].
  * Abe  _et al._ [2020] K. Abe _et al._ (T2K), Nature 580, 339 (2020), [Erratum: Nature 583, E16 (2020)], arXiv:1910.03887 [hep-ex].
  * Acero  _et al._ [2019] M. A. Acero _et al._ (NOvA), Phys. Rev. Lett. 123, 151803 (2019), arXiv:1906.04907 [hep-ex].
  * Abi  _et al._ [2020] B. Abi _et al._ (DUNE), Fermilab report (2020), arXiv:2002.03005 [hep-ex].
  * Jen  _et al._ [2014] C. M. Jen, A. Ankowski, O. Benhar, A. P. Furmanski, L. N. Kalousis, and C. Mariani, Phys. Rev. D 90, 093004 (2014), arXiv:1402.6651 [hep-ex].
  * Benhar  _et al._ [2017] O. Benhar, P. Huber, C. Mariani, and D. Meloni, Phys. Rept. 700, 1 (2017), arXiv:1501.06448 [nucl-th].
  * Ankowski  _et al._ [2016] A. M. Ankowski, O. Benhar, C. Mariani, and E. Vagnoni, Phys. Rev. D 93, 113004 (2016), arXiv:1603.01072 [hep-ph].
  * Ankowski and Mariani [2017] A. M. Ankowski and C. Mariani, J. Phys. G 44, 054001 (2017), arXiv:1609.00258 [hep-ph].
  * Katori and Martini [2018] T. Katori and M. Martini, J. Phys. G 45, 013001 (2018), arXiv:1611.07770 [hep-ph].
  * Alvarez-Ruso  _et al._ [2018] L. Alvarez-Ruso _et al._ (NuSTEC), Prog. Part. Nucl. Phys. 100, 1 (2018), arXiv:1706.03621 [hep-ph].
  * Nagu  _et al._ [2020] S. Nagu, J. Singh, J. Singh, and R. B. Singh, Nucl. Phys. B 951, 114888 (2020), arXiv:1905.13101 [hep-ph].
  * Rocco [2020] N. Rocco, Front. in Phys. 8, 116 (2020).
  * Khachatryan  _et al._ [2021] M. Khachatryan _et al._ (CLAS, e4v), Nature 599, 565 (2021).
  * Baussan  _et al._ [2014] E. Baussan _et al._ (ESSnuSB), Nucl. Phys. B 885, 127 (2014), arXiv:1309.7022 [hep-ex].
  * Day and McFarland [2012] M. Day and K. S. McFarland, Phys. Rev. D 86, 053003 (2012), arXiv:1206.6745 [hep-ph].
  * Tomalak  _et al._ [2022a] O. Tomalak, Q. Chen, R. J. Hill, and K. S. McFarland, Nature Commun. 13, 5286 (2022a).
  * Tomalak  _et al._ [2022b] O. Tomalak, Q. Chen, R. J. Hill, K. S. McFarland, and C. Wret, (2022b), arXiv:2204.11379 [hep-ph].
  * Moliere [1947] G. Moliere, Z. Naturforsch. A 2, 133 (1947).
  * Moliere [1948] G. Moliere, Z. Naturforsch. A 3, 78 (1948).
  * Gyulassy  _et al._ [2002] M. Gyulassy, P. Levai, and I. Vitev, Phys. Rev. D 66, 014005 (2002), arXiv:nucl-th/0201078.
  * Cronin  _et al._ [1975] J. W. Cronin, H. J. Frisch, M. J. Shochet, J. P. Boymond, R. Mermod, P. A. Piroue, and R. L. Sumner, Phys. Rev. D 11, 3105 (1975).
  * Qiu and Vitev [2003] J.-w. Qiu and I. Vitev, Phys. Lett. B 570, 161 (2003), arXiv:nucl-th/0306039.
  * Sadofyev  _et al._ [2021] A. V. Sadofyev, M. D. Sievert, and I. Vitev, Phys. Rev. D 104, 094044 (2021), arXiv:2104.09513 [hep-ph].
  * Ben  _et al._ [2022] F. G. Ben, A. V. Giannini, and M. V. T. Machado, J. Phys. G 49, 105005 (2022), arXiv:2201.13432 [hep-ph].
  * Barata  _et al._ [2022] J. a. Barata, A. V. Sadofyev, and C. A. Salgado, Phys. Rev. D 105, 114010 (2022), arXiv:2202.08847 [hep-ph].
  * Idilbi and Majumder [2009] A. Idilbi and A. Majumder, Phys. Rev. D 80, 054022 (2009), arXiv:0808.1087 [hep-ph].
  * Ovanesyan and Vitev [2011] G. Ovanesyan and I. Vitev, JHEP 06, 080 (2011), arXiv:1103.1074 [hep-ph].
  * Rothstein and Stewart [2016] I. Z. Rothstein and I. W. Stewart, JHEP 08, 025 (2016), arXiv:1601.04695 [hep-ph].
  * Marinova and Angeli [2022] K. Marinova and I. Angeli, “Nuclear charge radii,”  (2022).
  * Meyer  _et al._ [2016] A. S. Meyer, M. Betancourt, R. Gran, and R. J. Hill, Phys. Rev. D 93, 113015 (2016), arXiv:1603.03048 [hep-ph].
  * Borah  _et al._ [2020] K. Borah, R. J. Hill, G. Lee, and O. Tomalak, Phys. Rev. D 102, 074012 (2020), arXiv:2003.13640 [hep-ph].
  * Jackson [1998] J. D. Jackson,  _Classical Electrodynamics_ (Wiley, 1998).
  * Jiang  _et al._ [2022] L. Jiang _et al._ (Jefferson Lab Hall A), Phys. Rev. D 105, 112002 (2022), arXiv:2203.01748 [nucl-ex].
  * Pohl  _et al._ [2010] R. Pohl _et al._ , Nature 466, 213 (2010).
  * Antognini  _et al._ [2013] A. Antognini _et al._ , Science 339, 417 (2013).
  * Beyer  _et al._ [2017] A. Beyer _et al._ , Science 358, 79 (2017).
  * Bezginov  _et al._ [2019] N. Bezginov, T. Valdez, M. Horbatsch, A. Marsman, A. C. Vutha, and E. A. Hessels, Science 365, 1007 (2019).
  * Grinin  _et al._ [2020] A. Grinin, A. Matveev, D. C. Yost, L. Maisenbacher, V. Wirthl, R. Pohl, T. W. Hänsch, and T. Udem, Science 370, 1061 (2020).
  * Ovanesyan and Vitev [2012] G. Ovanesyan and I. Vitev, Phys. Lett. B 706, 371 (2012), arXiv:1109.5619 [hep-ph].
  * Fickinger  _et al._ [2013] M. Fickinger, G. Ovanesyan, and I. Vitev, JHEP 07, 059 (2013), arXiv:1304.3497 [hep-ph].
  * Kang  _et al._ [2017] Z.-B. Kang, F. Ringer, and I. Vitev, JHEP 03, 146 (2017), arXiv:1610.02043 [hep-ph].
  * Qiu and Vitev [2004] J.-W. Qiu and I. Vitev, Phys. Lett. B 587, 52 (2004), arXiv:hep-ph/0401062.
  * Mertig  _et al._ [1991] R. Mertig, M. Bohm, and A. Denner, Comput. Phys. Commun. 64, 345 (1991).
  * Shtabovenko  _et al._ [2016] V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 207, 432 (2016), arXiv:1601.01167 [hep-ph].
  * Hahn and Perez-Victoria [1999] T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. 118, 153 (1999), arXiv:hep-ph/9807565.
  * Inc. [2022] W. R. Inc., “Mathematica, Version 12.2.0.0,” (2022), champaign, IL, 2016\.
  * Llewellyn Smith [1972] C. H. Llewellyn Smith, Phys. Rept. 3, 261 (1972).
  * Formaggio and Zeller [2012] J. A. Formaggio and G. P. Zeller, Rev. Mod. Phys. 84, 1307 (2012), arXiv:1305.7513 [hep-ex].
  * Tomalak [2021] O. Tomalak, Phys. Rev. D 103, 013006 (2021), arXiv:2008.03527 [hep-ph].
  * Hill and Tomalak [2020] R. J. Hill and O. Tomalak, Phys. Lett. B 805, 135466 (2020), arXiv:1911.01493 [hep-ph].
  * Preedom and Tegen [1987] B. M. Preedom and R. Tegen, Phys. Rev. C 36, 2466 (1987).
  * Tomalak and Vanderhaeghen [2018] O. Tomalak and M. Vanderhaeghen, Eur. Phys. J. C 78, 514 (2018), arXiv:1803.05349 [hep-ph].
  * Antonello  _et al._ [2015] M. Antonello _et al._ (MicroBooNE, LAr1-ND, ICARUS-WA104), (2015), arXiv:1503.01520 [physics.ins-det].
  * McConkey [2017] N. McConkey (SBND), J. Phys. Conf. Ser. 888, 012148 (2017).
  * Machado  _et al._ [2019] P. A. Machado, O. Palamara, and D. W. Schmitz, Ann. Rev. Nucl. Part. Sci. 69, 363 (2019), arXiv:1903.04608 [hep-ex].
  * Fiorentini  _et al._ [2013] G. A. Fiorentini _et al._ (MINERvA), Phys. Rev. Lett. 111, 022502 (2013), arXiv:1305.2243 [hep-ex].
  * Rodrigues  _et al._ [2016] P. A. Rodrigues _et al._ (MINERvA), Phys. Rev. Lett. 116, 071802 (2016), [Addendum: Phys.Rev.Lett. 121, 209902 (2018)], arXiv:1511.05944 [hep-ex].
  * Cai  _et al._ [2020] T. Cai _et al._ (MINERvA), Phys. Rev. D 101, 092001 (2020), arXiv:1910.08658 [hep-ex].
  * Betancourt  _et al._ [2017] M. Betancourt _et al._ (MINERvA), Phys. Rev. Lett. 119, 082001 (2017), arXiv:1705.03791 [hep-ex].
  * Lu  _et al._ [2018] X. G. Lu _et al._ (MINERvA), Phys. Rev. Lett. 121, 022504 (2018), arXiv:1805.05486 [hep-ex].
  * Carneiro  _et al._ [2020] M. F. Carneiro _et al._ (MINERvA), Phys. Rev. Lett. 124, 121801 (2020), arXiv:1912.09890 [hep-ex].