# A note on cohomology of Clifford algebras

Bikram Banerjee

###### Abstract

In this article we construct a cochain complex of a complex Clifford algebra
with coefficients in itself in a combinatorial fashion and we call the
corresponding cohomology by Clifford cohomology. We show that Clifford
cohomology controls the deformation of a complex Clifford algebra.

††2020 Mathematics Subject Classification : 16E40; 16S80††Keywords: Clifford
algebra, Deformation theory, Hochschild cohomology

##  1 Introduction

Clifford algebras were created by William K. Clifford who introduced a new
multiplication rule into Grassmann’s exterior algebra
$\bigwedge\mathbb{R}^{n}.$ A Clifford algebra is a unital associative algebra
and generalizes the real numbers, complex numbers and Hamilton’s quaternions
and plays important roles in geometry and theoretical physics. On the other
hand algebraic deformation theory of associative algebras was developed by M.
Gerstenhaber [2] [3]. Clifford algebras can be seen as deformations of
exterior algebras. Here by a deformation we will always mean a formal one. In
algebraic deformation theory of associative algebras the Hochschild cohomogy
group $HH^{\ast}(A,A)$ of an associative algebra $A$ with coefficients in $A$
plays the key role. In fact the second Hochschild cohomology group
$HH^{2}(A,A)$ controls the deformations of $A$ in the sense that the
triviality of it implies that any deformation of $A$ is equivalent to null
deformation and in such case $A$ is called rigid. In general Hochschild
cohomology groups are difficult to compute. Now it is well known that complex
Clifford algebras are rigid i.e. any deformation is equivalent to null
deformation. The Hochschild cohomology groups of complex Clifford algebras are
easy to compute due to the fact that a complex Clifford algebra is either
isomorphic to a complex central simple algebra or isomorphic to a direct sum
of two isomorphic complex central simple algebras.
In this article we construct a cochain complex of a complex Clifford algebra
$A$ with coefficients in $A$ in a combinatorial fashion and we call the
corresponding cohomology by Clifford cohomogy which will be denoted by
$H^{\ast}_{Cl(B)}(A)$. Here the symbol $`B$’ will be explained later. It turns
out that $H^{\ast}_{Cl(B)}(A)$ becomes isomorphic to $HH^{\ast}(A,A)$ and
consequently the deformations of $A$ is controlled by $H^{2}_{Cl(B)}(A).$
In section $2$ and $3$ we briefly recall some facts about Clifford algebras
and formal deformations of associative algebras respectively. In section $4$
we introduce Clifford cohomology.

##  2 Clifford algebra

We start by recalling something about Clifford algebras [5] [6]. There are
lots of literatures on Clifford algebras to be mentioned. Let $V$ be an $n$
dimensional complex vector space equipped with a non degenerated quadratic
form $q$ and $T(V)=\bigoplus_{k=0}^{\infty}\otimes^{k}V$ be the tensor algebra
over $V.$

######  Definition 2.1.

A Clifford algebra $C(n)$ over $V$ is defined to be the quotient algebra
$T(V)/I_{q},$ where $I_{q}$ is the two-sided ideal generated by elements of
the form $v\otimes v-q(v)$ for all vectors $v$ in $V.$ The product in $C(n)$
is called the Clifford product.

Now the corresponding bilinear form of $q$ is
$\beta_{q}(u,v)=\frac{1}{2}(q(u+v)-q(u)-q(v)).$ We note that

$q(u+v)=(u+v)^{2}=u^{2}+uv+vu+v^{2}=q(u)+uv+vu+q(v)$

and so $uv+vu=2\beta_{q}(u,v).$ Thus if $u$ and $v$ are orthogonal vectors
then $uv=-vu.$

Now if we choose an orthogonal basis $B=\\{v_{1},\cdots,v_{n}\\}$ of $V$ then
the set $\Im$ consisting of $2^{n}$ many elements given by
$\Im=\\{1,E_{m_{1}m_{2}\cdots m_{k}}\\}$, $1\leq k\leq n$ and $1\leq
m_{1}<m_{2}<\cdots<m_{k}\leq n$ form a basis of the vector space $C(n)$ where
$E_{m_{1}m_{2}\cdots m_{k}}=v_{m_{1}}v_{m_{2}}\cdots v_{m_{k}}$ (here right
side is the Clifford product of $v_{m_{1}},\cdots,v_{m_{k}}$) and thus $\dim
C(n)=2^{n}.$

Periodicity of Clifford algebras: Let $M(k)$ denotes the matrix algebra of
$k\times k$ complex matrices. It is known that there is an isomorphism between
$C(n+2)$ and $C(n)\otimes_{\mathbb{C}}M(2).$ Now from this isomorphism and
along with the fact that $C(0)\cong\mathbb{C}$ and
$C(1)\cong\mathbb{C}\oplus\mathbb{C}$ it follows that:

$\begin{array}[]{c l}C(2n)\cong M(2^{n})\\\ C(2n+1)\cong M(2^{n})\oplus
M(2^{n}).\\\ \end{array}$

##  3 Formal deformation of associative algebras and Hochschild cohomology

Let us start with a review of deformation theory of associative algebras [1]
[2] [3]. Throughout, $A$ will denote a finite dimensional associative algebra
over a field $k$ of characteristic zero, $k[[t]]$ be the formal power series
ring over $k$ and $A[[t]]$ is the formal power series over $A$ which is a
$k[[t]]$ algebra.

######  Definition 3.1.

A formal deformation of $A$ is a $k[[t]]$ bilinear map $\mu:A[[t]]\times
A[[t]]\rightarrow A[[t]]$ defined by
$\mu(a,b)=\mu_{0}(a,b)+\mu_{1}(a,b)t^{2}+\mu_{2}(a,b)t^{2}+\cdots$ for all
$a,b\in A,$ $\mu_{i}:A\otimes_{k}A\rightarrow A,$ $i=0,1,2,\cdots,$
$\mu_{0}(a,b)=ab$ (the multiplication in $A$) and satisfying

$\sum_{i+j=k.i,j\geq 0}(\mu_{i}(\mu_{j}(a,b)),c)-\mu_{i}(a,\mu_{j}(b,c))=0$

for all $a,$ $b,$ $c\in A$ and for each $k\geq 1.$

If one chooses $\mu_{i}=0$ for all $i\geq 1$ then the deformation of $A$ is
called null deformation.

Let $\mu=\mu_{0}+\mu_{1}t+\mu_{2}t^{2}+\cdots$ and
$\mu^{\prime}=\mu^{\prime}_{0}+\mu^{\prime}_{1}t+\mu^{\prime}_{2}t^{2}+\cdots$
be two deformations of $A.$ Now we say $\mu$ and $\mu^{\prime}$ are are
equivalent if there exists a $k[[t]]$ linear map $u:A[[t]]\rightarrow A[[t]]$
defined by $u=id_{A}+\phi_{1}t+\phi_{2}t^{2}+\cdots,$ $\phi_{i}\in
Hom_{k}(A,A),$ $i\in\mathbb{N},$ such that

$u\circ\mu=\mu^{\prime}\circ(u\otimes u).$

If every formal deformation of $A$ is equivalent to null deformation then $A$
is called rigid.

The main tool in studing deformation theory of an associative algebra $A$ is
the Hochschild cochain complex [4] $C^{\ast}(A,A):$

$0\rightarrow
C^{0}(A,A)\stackrel{{\scriptstyle\delta_{Hoch}}}{{\rightarrow}}\cdots\stackrel{{\scriptstyle\delta_{Hoch}}}{{\rightarrow}}C^{n}(A,A)\stackrel{{\scriptstyle\delta_{Hoch}}}{{\rightarrow}}C^{n+1}(A,A)\stackrel{{\scriptstyle\delta_{Hoch}}}{{\rightarrow}}\cdots$

where $C^{n}(A,A)=Hom_{k}(A^{\otimes n},A)$ is the space of Hochschild
$n$-cochains, i.e., the $n$-linear maps $f$ on a $A$ with values in $A.$ The
differential $\delta_{Hoch}:C^{n}(A,A)\rightarrow C^{n+1}(A,A)$ is defined by:

$(\delta_{Hoch}f)(a_{0}\otimes a_{1}\otimes\cdots\otimes
a_{n})=a_{0}f(a_{1}\otimes\cdots\otimes a_{n})$
$+\sum_{i=1}^{n}(-1)^{i}f(a_{0}\otimes\cdots\otimes
a_{i-1}a_{i}\otimes\cdots\otimes a_{n})+(-1)^{n+1}f(a_{0}\otimes\cdots\otimes
a_{n-1})a_{n}.$

It turns out that $\delta_{Hoch}^{2}=0$ and Hochschild cohomology of $A$ with
coefficients in $A$ is defined by
$HH^{\ast}(A,A)=H^{\ast}(C^{\ast}(A,A);\delta_{Hoch}).$
If $A$ is an associative algebra such that $HH^{2}(A,A)=0$ then it is a fact
that $A$ is rigid, i.e., all formal deformations of $A$ are equivalent to the
null deformation.

##  4 Clifford cohomology

In this section we will construct a cochain complex of a Clifford algebra
$C(n)$ over an $n$ dimensional complex vector space $V$ equipped with a non
degenerated quadratic form $q$, which we call  Clifford cochain complex. For
this firstly we will define a bilinear product $\overline{\circ}_{B}$ on
$C(n)$ for every choice of an ordered orthogonal basis $B$ of $V$ and then use
it to define the Clifford cochain complex. It turns out that if $B$ consists
of orthonormal vectors then the Clifford cochain complex of $C(n)$ coincides
with the Hochschild cochain complex.
We start with a finite set $X=\\{1,2,\cdots,n\\}.$ Let $\mathbb{S}$ be the set
of all finite sequences in $X$ and we define a $`2$-shuffle’, $sh_{i},$ on
$\\{m_{1},\cdots,m_{i},m_{i+1},\cdots,m_{k}\\}$ $\in\mathbb{S}$ by

$\\{m_{1},\cdots,m_{i},m_{i+1},\cdots,m_{k}\\}\stackrel{{\scriptstyle
sh_{i}}}{{\longrightarrow}}\\{m_{1},\cdots,m_{i+1},m_{i},\cdots,m_{k}\\}$

for $1\leq i\leq k-1.$
Now let $P(X)\uparrow$ be the set of all ordered subsets
$\\{m_{1},m_{2},\cdots,m_{k}\\}$ of $X$ such that $m_{1}<m_{2}<\cdots<m_{k},$
$1\leq m_{i}\leq n$ for $i=1,2,\cdots,k.$ We consider the empty set,
$\\{\quad\\},$ as a member of $P(X)\uparrow.$ Let us define a binary operation
$\circ$ on $P(X)\uparrow$ as follows:

$\\{m_{1},\cdots,m_{k}\\}\circ\\{t_{1},\cdots,t_{s}\\}=\\{l_{1},\cdots,l_{p}\\},$

if $\\{m_{1},\cdots,m_{k}\\}\neq\\{t_{1},\cdots,t_{s}\\};$ where
$\\{l_{1},\cdots,l_{p}\\}$ is obtained by applying the minimum number of
$2$-shuffles $\tau$ on the sequence $\\{m_{1},\cdots,m_{k},t_{1},$
$\cdots,t_{s}\\}$ to get a monotonic increasing sequence and then deleting
$m_{i}$ and $t_{j}$ if $m_{i}=t_{j},$ $1\leq i\leq k,$ $1\leq j\leq s.$ We
note that
$p=card(\\{m_{1},\cdots,m_{k}\\}\bigtriangleup\\{t_{1},\cdots,t_{s}\\})$ (here
$\bigtriangleup$ means symmetric difference of two sets). Let us explain this
by an example:
We take $X=\\{1,2,3,4\\}$ and we compute $\\{1,3,4\\}\circ\\{2,3\\}.$ We see
$\\{1,3,4\\}\circ\\{2,3\\}=\\{1,2,4\\}.$ This is obtained by successively
applying $2$-shuffles $sh_{3},sh_{2}$ and $sh_{4}$ on the sequence
$\\{1,3,4,2,3\\}$ to get $\\{1,2,3,3,4\\}$ and then finally deleting $3$ and
$3$ to get $\\{1,2,4\\}.$ Here $\tau=3.$
Clearly $\\{m_{1},\cdots,m_{k}\\}\circ\\{m_{1},\cdots,m_{k}\\}=\\{\quad\\},$
$1\leq k\leq n,$ is obtained by applying $\frac{k(k-1)}{2}$ many $2$-shuffles
on the sequence $\\{m_{1},\cdots,m_{k},m_{1},\cdots,m_{k}\\}$ and then by
deleting everything.
Let $B=\\{v_{1},\cdots,v_{n}\\}$ be an ordered orthogonal basis of an $n$
dimensional complex vector space $V$ and the set $\Im=\\{1,E_{m_{1}m_{2}\cdots
m_{k}}\\}$, $1\leq k\leq n$, $1\leq m_{1}<m_{2}<\cdots<m_{k}\leq n$; where
$E_{m_{1}m_{2}\cdots m_{k}}=v_{m_{1}}v_{m_{2}}\cdots v_{m_{k}}$, form a basis
of $C(n).$ Here $v_{m_{1}}v_{m_{2}}\cdots v_{m_{k}}$ denotes the Clifford
product of $v_{m_{1}},v_{m_{2}},\cdots,v_{m_{k}}.$ Now we define a map
$\circ_{B}:\Im\times\Im\rightarrow C(n)$ by

$E_{m_{1}m_{2}\cdots m_{k}}\circ_{B}E_{t_{1}t_{2}\cdots
t_{s}}=(-1)^{\tau}E_{l_{1}l_{2}\cdots l_{p}}$

if $\\{m_{1},\cdots,m_{k}\\}\neq\\{t_{1},\cdots,t_{s}\\},$

$E_{m_{1}m_{2}\cdots m_{k}}\circ_{B}E_{m_{1}m_{2}\cdots
m_{k}}=(-1)^{\frac{k(k-1)}{2}}$

and

$1\circ_{B}E_{m_{1}m_{2}\cdots m_{k}}=E_{m_{1}m_{2}\cdots
m_{k}}=E_{m_{1}m_{2}\cdots m_{k}}\circ_{B}1.$

Finally we extend $\circ_{B}$ bilinearly on $C(n)\times C(n)$ and get a linear
map $\overline{\circ}_{B}:C(n)\otimes C(n)\rightarrow C(n).$

######  Lemma 4.1.

If $B^{\prime}=\\{e_{1},e_{2},\cdots,e_{n}\\}$ is an ordered orthonormal basis
of $V$ then $\overline{\circ}_{B^{\prime}}$ coincides with the Clifford
product in $C(n).$

###### Proof.

Let $\overline{E}_{m_{1}m_{2}\cdots m_{k}}=e_{m_{1}}e_{m_{2}}\cdots
e_{m_{k}}.$ Now the lemma follows from the fact that

$\overline{E}_{m_{1}m_{2}\cdots
m_{k}}\circ_{B^{\prime}}\overline{E}_{t_{1}t_{2}\cdots
t_{s}}=(-1)^{\tau}\overline{E}_{l_{1}l_{2}\cdots
l_{p}}=e_{m_{1}}e_{m_{2}}\cdots e_{m_{k}}e_{t_{1}}e_{t_{2}}\cdots e_{t_{s}}.$

if $\\{m_{1},\cdots,m_{k}\\}\neq\\{t_{1},\cdots,t_{s}\\},$

$\overline{E}_{m_{1}m_{2}\cdots
m_{k}}\circ_{B^{\prime}}\overline{E}_{m_{1}m_{2}\cdots
m_{k}}=(-1)^{\frac{k(k-1)}{2}}=e_{m_{1}}e_{m_{2}}\cdots
e_{m_{k}}e_{m_{1}}e_{m_{2}}\cdots e_{m_{k}}$

and

$1\circ_{B^{\prime}}\overline{E}_{m_{1}m_{2}\cdots
m_{k}}=\overline{E}_{m_{1}m_{2}\cdots m_{k}}=\overline{E}_{m_{1}m_{2}\cdots
m_{k}}\circ_{B^{\prime}}1.$

∎

Now let $C^{q}(C(n))=Hom_{\mathbb{C}}(C(n)^{\otimes q},C(n)),$ for all $q\geq
0$ and $\delta_{Cl(B)}:C^{q}(C(n))\rightarrow C^{q+1}(C(n))$ be the coboundary
map defined by:

$(\delta_{Cl(B)}f)(a_{0}\otimes a_{1}\otimes\cdots\otimes
a_{q})=a_{0}\overline{\circ}_{B}f(a_{1}\otimes\cdots\otimes a_{q})$
$+\sum_{i=1}^{q}(-1)^{i}f(a_{0}\otimes\cdots\otimes
a_{i-1}\overline{\circ}_{B}a_{i}\otimes\cdots\otimes
a_{q})+(-1)^{q+1}f(a_{0}\otimes\cdots\otimes
a_{q-1})\overline{\circ}_{B}a_{q},$

where $f\in C^{q}(C(n)).$ We note that if
$B^{\prime}=\\{e_{1},e_{2},\cdots,e_{n}\\}$ is an ordered orthonormal basis of
$V$ then $\delta_{Cl(B^{\prime})}=\delta_{Hoch}.$

######  Lemma 4.2.

For any choice of an ordered orthogonal basis $B$ of $V,$
$\delta_{Cl(B)}^{2}=0.$

###### Proof.

Let $B=\\{v_{1},\cdots,v_{n}\\}$ be an ordered orthogonal basis and
$B^{\prime}=\\{e_{1},e_{2},\cdots,e_{n}\\}$ is an ordered orthonormal basis of
$V$, $E_{m_{1}m_{2}\cdots m_{k}}=v_{m_{1}}v_{m_{2}}$ $\cdots v_{m_{k}}$ and
$\overline{E}_{m_{1}m_{2}\cdots m_{k}}=e_{m_{1}}e_{m_{2}}\cdots e_{m_{k}}.$ We
take the vector space isomorphism $\psi:C(n)\rightarrow C(n)$ defined by

$\psi(\overline{E}_{m_{1}m_{2}\cdots m_{k}})=E_{m_{1}m_{2}\cdots m_{k}}$

$\psi(1)=1$

and consider the isomorphism $\psi^{\ast}:C^{q}(C(n))\rightarrow C^{q}(C(n))$
defined by:

$f\longmapsto\psi^{\ast}f,$

where

$\psi^{\ast}f(a_{0}\otimes a_{1}\otimes\cdots\otimes
a_{q-1})=\psi^{-1}[f(\psi^{\otimes q}(a_{0}\otimes a_{1}\otimes\cdots\otimes
a_{q-1}))].$

It is noted earlier that $\delta_{Cl(B^{\prime})}=\delta_{Hoch}$ and by Lemma
4.1, $\overline{\circ}_{B^{\prime}}$ coincides with the Clifford product in
$C(n).$ Now it is an easy check that the following diagram

${C^{q}(C(n))}$${C^{q+1}(C(n))}$${C^{q}(C(n))}$${C^{q+1}(C(n))}$$\scriptstyle{\delta_{Cl(B)}}$$\scriptstyle{\psi^{\ast}}$$\scriptstyle{\psi^{\ast}}$$\scriptstyle{\delta_{Hoch}}$

commutes for all $q\geq 0$ and therefore
$\psi^{\ast}(\delta_{Cl(B)}^{2}f)=\delta_{Hoch}^{2}(\psi^{\ast}f)=0,$ $f\in
C^{q}(C(n)).$ Finally as $\psi^{\ast}$ is an isomorphism so
$\delta_{Cl(B)}^{2}f=0.$ This completes the proof.

∎

We define the Clifford cochain complex $(C^{\ast}(C(n));\delta_{Cl(B)})$ of
the Clifford algebra $C(n)$ with respect to an ordered orthogonal base $B$ of
$V$ by:

$0\rightarrow
C^{0}(C(n))\stackrel{{\scriptstyle\delta_{Cl(B)}}}{{\rightarrow}}\cdots\stackrel{{\scriptstyle\delta_{Cl(B)}}}{{\rightarrow}}C^{q}(C(n))\stackrel{{\scriptstyle\delta_{Cl(B)}}}{{\rightarrow}}C^{q+1}(C(n))\stackrel{{\scriptstyle\delta_{Cl(B)}}}{{\rightarrow}}\cdots$

and the Clifford cohomology of $C(n)$ with respect to $B$ by:

$H^{\ast}_{Cl(B)}(C(n))=H^{\ast}(C^{\ast}(C(n));\delta_{Cl(B)}).$

######  Theorem 4.3.

For any ordered orthogonal base $B$ of $V,$

$H^{\ast}_{Cl(B)}(C(n))\cong HH^{\ast}(C(n),C(n)).$

###### Proof.

The proof at once follows from the commutativity of the diagram in Lemma 4.2
along with the fact that $\psi^{\ast}$ is an isomorphism. ∎

######  Corollory 4.4.

Up to isomophism Clifford cohomology groups of $C(n)$ are independent of the
choice of an ordered orthogonal base of $V.$

###### Proof.

It readily follows from the above Theorem. ∎

######  Remark 4.5.

As $H^{2}_{Cl(B)}(C(n))\cong HH^{2}(C(n),C(n))$ therefore the deformation of
the Clifford algebra $C(n)$ is controlled by $H^{2}_{Cl(B)}(C(n)).$

## References

  * [1] Doubek M., Markl M., Zima P., Deformation Theory (Lecture Notes), arXiv: 0705.3719
  * [2] Gerstenhaber M., Schack D.S., Algebraic cohomology and deformation theory, in Deformatin Theory of Algebras and Structures and Applications, Editors M. Hazewinkel and M. Gerstenhaber, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Vol. 247, Kluwer Academic Publishers Group, Dordrecht, 1988.
  * [3] Gerstenhaber M., On the deformation of rings and algebras, Ann. of Math, (2) 79 (1964), 59-103.
  * [4] Loday J-L., Cyclic homology, Grundlehren der Mathematischen Wissenschaften, Vol. 301, Springer-Verlag, Berlin, 1992.
  * [5] Lounesto P., Clifford Algebras and Spinors, Cambridge University Press, Cambridge, 1997.
  * [6] Lundholm D., Svensson L., Clifford algebra, Geometric algebra and applications, arXiv: 0907.5356

Department of Mathematics,
Ranaghat College, Ranaghat,
W. B. 741201, India.
E-mail<EMAIL_ADDRESS>


# Pathfinder for a High Statistics Search for Missing Energy In Gamma Cascades

James B. Dent<EMAIL_ADDRESS>Department of Physics, Sam Houston State
University, Huntsville, TX 77341, USA  Bhaskar Dutta  Mitchell Institute for
Fundamental Physics and Astronomy, Department of Physics and Astronomy, Texas
A&M University, College Station, TX 77843, USA  Andrew Jastram  Mitchell
Institute for Fundamental Physics and Astronomy, Department of Physics and
Astronomy, Texas A&M University, College Station, TX 77843, USA  Doojin Kim
Mitchell Institute for Fundamental Physics and Astronomy, Department of
Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA
Andrew Kubik  SNOLAB, Creighton Mine #9, 1039 Regional Road 24, Sudbury, ON
P3Y 1N2, Canada  Rupak Mahapatra  Mitchell Institute for Fundamental Physics
and Astronomy, Department of Physics and Astronomy, Texas A&M University,
College Station, TX 77843, USA  Surjeet Rajendran  Department of Physics &
Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA
Harikrishnan Ramani  Stanford Institute for Theoretical Physics, Stanford
University, Stanford, CA 94305, USA  Adrian Thompson  Mitchell Institute for
Fundamental Physics and Astronomy, Department of Physics and Astronomy, Texas
A&M University, College Station, TX 77843, USA  Shubham Verma  Mitchell
Institute for Fundamental Physics and Astronomy, Department of Physics and
Astronomy, Texas A&M University, College Station, TX 77843, USA

###### Abstract

We investigate the feasibility of a high statistics experiment to search for
invisible decay modes in nuclear gamma cascades using 200 kg of Cs(Tl)
scintillators that are presently available at Texas A&M. The experiment aims
to search for missing energy by robustly establishing the absence of a photon
in a well identified gamma cascade. We report on the experimental
demonstration of the energy resolution necessary for this search. Prior
explorations of this detector concept focused on baryonically coupled physics
that could be emitted in $E_{2}$ transitions. We point out that this protocol
can also search for particles that are coupled to photons by searching for the
conversion of a photon produced in a gamma cascade into a hidden particle.
Examples of these processes include the oscillation of a photon into a hidden
photon and the conversion of a photon into an axion-like-particle either in
the presence of a magnetic field or via the Primakoff process. This proof-of-
concept apparatus appears to have the ability to search for hitherto
unconstrained baryonically coupled scalars and pseudoscalars produced in
$E_{0}$ and $M_{0}$ transitions. If successfully implemented, this experiment
serves as a pathfinder for a larger detector with greater containment that can
thoroughly probe the existence of new particles with mass below 4 MeV that lie
in the poorly constrained supernova “trapping window” that exists between 100
keV and 30 MeV.

##  I Introduction

There are considerable theoretical motivations to search for light, weakly
coupled particles. These particles could constitute the dark matter or act as
mediators between the standard model (SM) and the dark sector Green and
Rajendran (2017). They also arise in several frameworks of physics beyond the
standard model that address theoretical puzzles such as the strong CP problem
Peccei and Quinn (1977); Weinberg (1978); Wilczek (1978) , the hierarchy
problem Graham et al. (2015), the cosmological constant problem Graham et al.
(2018, 2019), and the quantum nature of gravity. A significant impediment in
searching for such particles in a controlled laboratory setting is statistics:
the more weakly coupled a particle, the harder it is to produce it. As a
result, probes of such particles have largely centered around their effects on
astrophysical bodies and cosmology.

This is clearly unsatisfactory. First, these probes are limited to the parts
of parameter space where astrophysics and cosmology are well understood.
Together, these place significant constraints on light particles that interact
with electrons. However, particles that couple to baryons and are heavier than
$\sim 100$ keV are not well constrained Green and Rajendran (2017). Such
particles are too heavy to be produced in objects such as horizontal branch
stars (HB stars). They can be produced in supernovae and can be constrained if
they interact more weakly with the standard model than neutrinos, leading to
anomalous cooling. But, if their interactions with baryons are stronger than
neutrinos, these particles do not efficiently cool the star and are thus
unconstrained. Cosmological limits on such particles are also weak since the
baryon abundance drops significantly as the universe cools below $\sim$ GeV
energies. In concert with the heating of the standard model that occurs during
the QCD phase transition, the relative abundance of baryonically interacting
particles is suppressed. Second, these astrophysical and cosmological bounds
are not robust against minor changes to the model DeRocco et al. (2020).
Additional interactions between the standard model and these particles can
cause significant density-dependent effects. Given the enormous difference in
densities in the early universe, the interior of stars, and the laboratory, it
is not difficult to avoid these bounds when the underlying model is changed in
minor ways.

Given the need for laboratory methods to probe these particles, how can we
achieve the required statistics? One way to accomplish this task using hot
radioactive nuclear sources was discussed in Benato et al. (2019). In this
scheme, one searches for missing energy in a well identified gamma cascade.
Consider, for example, the beta decay of 60Co to 60Ni. With a branching
fraction of $99.9\%$ this decay populates the $4+$ excited state of 60Ni which
decays to the ground state by first emitting a 1.17 MeV gamma to decay to the
$2+$ state followed by a 1.33 MeV gamma to decay to the ground state. The idea
of Benato et al. (2019) is to identify the initial 1.17 MeV gamma and with
great efficiency tag the second 1.33 MeV gamma. If the second gamma was seen
at a lower rate than the expected efficiency of the detector, it could imply
the existence of a decay into new particles. The key parameters of this
experiment are: the containment efficiency of the gammas (to avoid missing the
second gamma), the energy resolution of the setup (to separate out the gammas
of interest) and the detector response time (enabling background rejection by
focusing only on events within a short $\sim$ ns interval). Another central
parameter is the nuclear source itself. While 60Co is a readily available
source, for instance, the fact that the two gammas are close to each other in
energy limits the ultimate reach of an experiment using this isotope. As
discussed in Benato et al. (2019), these limitations can be overcome using
24Na. However, the short half-life introduces experimental and logistical
complications. In this work we explore an expanded list of nuclei including
46Sc. This nucleus has a 83 day half-life, sufficient energy separation
between the two decay photons and importantly, the first photon is more
energetic (1.12 MeV) than the second (889 keV). These features make this a
promising isotope for this experiment.

The purpose of this paper is to further develop the experimental
implementation of the theoretical concepts proposed in Benato et al. (2019) in
a concrete experimental apparatus at Texas A&M university. This apparatus uses
CsI(Tl) crystals that act as a reasonably fast scintillating detector for the
produced gammas. We discuss the properties of these crystals, the trigger
logic and the optimal geometries necessary for maximum containment. We also
report on an experimental demonstration of the energy resolution needed for
this experiment. These inputs are used to estimate the sensitivity of the
apparatus. All of these are discussed in Section II. Following this, in
Section III, we consider a broader class of transitions than discussed in
Benato et al. (2019). This includes $M_{1}$ and $M_{0}$ transitions that are
particularly useful in searching for pseudoscalars such as axions and axion-
like-particles and $E_{0}$ transitions where the standard model produces
$e^{+}/e^{-}$ pairs that can be tagged far more effectively than gammas. In
Section IV, we estimate the reach of this setup for a variety of models, well
beyond the ones considered in Benato et al. (2019). We also discuss a new
process - in Benato et al. (2019), the new physics that was being searched for
involved particles that directly coupled to nuclei and were thus directly
produced in the decay. Here, we discuss another possibility - suppose the
decay yields the usual gamma cascade. But, it is possible that the gamma
oscillates into a dark particle before interacting with the standard model
resulting in missing energy. This phenomenon is possible for axion-like
particles in the presence of a background magnetic field or through the
Primakoff process. We discuss this new process as well in Section IV.

##  II Experimental Setup

The proposed experimental setup will utilize 2-5 tons of 5 kg,
$2^{\prime\prime}\times 2^{\prime\prime}\times 12^{\prime\prime}$ in dimension
CsI(Tl) crystals. CsI(Tl) has been chosen due to its high light yield, high
density and less hygroscopic nature (which makes it suitable to be used under
normal atmospheric conditions). The initial prototype will be an approximately
200 kg experiment with 36 crystals organized as a cube as shown in Fig. 1(a)
to provide uniform containment in a 4$\pi$ geometry. In our setup, each
CsI(Tl) crystal is wrapped with a Teflon tape for improved internal reflection
of the scintillating photons and then a layer of thin aluminized mylar film is
wrapped which reflects external light. One end of these crystals will be
coupled to conventional EMI photomultiplier tubes (PMT) for light collection.

The preliminary characterization of the available CsI(Tl) crystals was
performed using various radioactive sources primarily 22Na, 60Co and 137Cs. We
found that the two gammas from 60Co (at 1.17 MeV and 1.33 MeV) could not be
resolved using the $12^{\prime\prime}$ long crystals due to the length of the
crystals. The crystal length results in a smaller light collection efficiency
which would increase the relative contribution of electronic noise and hence
results in poor energy resolution Blucher et al. (1986). We used a half size
crystal (about $6^{\prime\prime}$ in length) to have an acceptable resolution
at MeV scale to resolve the two gammas from 60Co, as shown in Fig. 1(b). For
future upgrades we plan to use SiPMs as photodetetcors instead of PMTs, as
they have smaller profiles than the large PMTs. These facts also simplify the
complexity of the geometry. The ability to search for missing gammas depends
critically on how hermetic the set up is, thus any larger scale up will
utilize SiPMs. Other benefits of SiPMs include reduced crosstalk, lower dark
count, low afterpulse, outstanding photon detection efficiency, low voltage
operation, high gain and good signal to noise ratio. SiPMs are also
insensitive to magnetic fields, enabling their use in setups where magnetic
fields may be necessary. The cost of SiPMs was prohibitive for this 200 kg
prototype that was entirely built using internal funds. The end of the CsI(Tl)
crystal that will be attached to the PMT will be well polished and examined
for any possible defects and contamination. After selecting the crystals to be
used for the prototype assembly, each crystal is connected to a PMT and it is
then characterized for energy response primarily using 511 keV gammas from a
22Na radioactive source.

(a)

(b)

Figure 1: (a) Conceptual drawing of a 200-kg prototype experiment with each
CsI(Tl) scintillating crystal coupled to SiPM array at one end. (b) 60Co peak
observed in a half size crystal with a PMT coupled to one end.

Besides calibration, 22Na is also used for the estimation of detector
efficiency for detecting gammas from the radioactive sources. 22Na decays by
emitting a positron mostly into a 1275 keV level of 22Ne. These positrons
annihilate with the surrounding electrons, producing two 511 keV photons. If
we tag one of these two gammas and look for the other gamma in the same event,
we can perform an in-situ measurement of the gamma detection efficiency of the
setup. The new physics that is being searched for in this experiment will in
general couple very differently to nucleons and electrons. In particular, the
electron couplings are much more constrained. Thus if we see missing gammas in
the nuclear decay channel without any loss in the positron annihilation
channel, it will be convincing evidence of a genuine signal. We also propose
to use this setup to probe axion-like-particles by searching for the
disappearance of photons via the Primakoff process or in the presence of an
applied magnetic field. The verification of detector efficiency for this
channel requires a different strategy. In the case where the conversion
happens due to an external magnetic field, the signal can be turned off by
reducing the magnetic field and this can be used to prove the existence of new
physics. The Primakoff case is more challenging - the verification of the
signal could potentially be performed by utilizing the fact that the
conversion efficiency is independent of the photon energy whereas a loss
channel in the detector is likely to be energy-dependent.

Quantity | CsI(Tl) | NaI(Tl) | BGO
---|---|---|---
Emission spectrum max (nm) | 550 | 415 | 480
Density (g/cm3) | 4.51 | 3.67 | 7.13
Hygroscopic | Slightly | Yes | No
Refractive index (at max. emission) | 1.79 | 1.85 | 2.15
Radiation length (cm) | 1.85 | 2.59 | 1.12
Interaction length (g/cm2) | 167 | 152 | 156
Light yield (photons/keV$\gamma$) | 54 | 38 | 10
Decay time ($\mu$s) | 0.679 (fast), 3.34 (slow) | 0.250 | 0.300
Table 1: Properties of CsI(Tl) along with other scintillating materials
Blucher et al. (1986)

We plan to put the source at the center of this 200 kg detector assembly and
perform the efficiency measurements after each detector has been calibrated
for gamma energy. Table 1 lists the properties of CsI(Tl) crystals which will
be used in this experiment. The slow component of the decay time of CsI(Tl)
crystals is around 3 $\mu$s. To avoid pileups, we can select the activity of
the source to have an average time between decays of $\sim$ 5 $\mu$s, allowing
a trigger rate around 200 kHz. Pile ups can still be handled offline during
data analysis as long as the two events don’t occur within a $\mu$s. With this
trigger rate, we expect to take data for a year and analyze around $10^{13}$
decays/year. This will be phase 1 of this experiment. For data acquisition,
the signals from the PMTs are fed into a spectroscopic amplifier (CAEN model
N568E) which also performs pulse shaping and has a web interface to control
the gain and shaping time. The signal is then fed into a leading edge
discriminator with multiplicity option for the trigger (CAEN model N841). VME-
based 50 MHz analogue-to-digital converter with 64-channels (CAEN VX2740) is
used for signal digitization and recording. This also offers multi-channel
analysis for nuclear spectroscopy and it is thus well suited for our purpose.
The data will be analyzed with an Optimum-filter-based analysis package and
the ROOT analysis package Brun et al. (2003). Additionally, the detector and
laboratory geometry will be modeled, and backgrounds will be calibrated and
validated using the GEANT4 simulation toolkit using the Shielding Physics List
Agostinelli et al. (2003). Work along this direction has already been
initiated for a smaller 50 kg CsI(Tl) prototype geometry.

##  III Multipole Transitions

The transitions considered in this work can be broadly classified into two
types. In the first type, the standard model process is a beta/electron
capture (EC) decay followed by a gamma cascade of two or more photons all the
way to the daughter nucleus’ ground state. We use all but one of the photons
as the trigger and an event corresponds to exactly one photon in the cascade
being absent. The transition instead proceeds through the small but non-zero
branching fraction to a dark particle. The $E_{2}$ transitions considered in
Benato et al. (2019) as well as the $M_{1}$ transition in ${}^{65}\textrm{Ni}$
which are well suited for axion searches are of this type. For dipole and
higher transitions, the sensitivity to the rare dark decay is limited by the
containment of a single photon, i.e. the smallest branching fractions that can
be probed is given by,

$Br_{\rm dark}^{\rm lim}\approx\exp\left(-{\frac{R_{\rm det}}{\lambda_{\rm
abs}}}\right).\quad\textrm{ (for dipole and higher transitions)}$ (1)

where $\lambda_{\rm abs}$ is the mean free path of the photon in the
scintillator and $R_{\rm det}$ is the radius of a spherical detector. After
this branching fraction is reached, subsequent statistics only improve the
reach after background subtraction, and limits therefore scale as the square-
root of statistics.

This limitation is relaxed for particles produced in $E_{0}$ or $M_{0}$
transitions. These are rare monopole transitions which cause the decay of a
$0^{+}$ or $0^{-}$ to a ground state $0^{+}$. Due to angular momentum
conservation, this can never proceed through a single photon. In the standard
model, the $E_{0}$ transition proceeds through internal pair production
($e^{+}e^{-}$), internal conversion (single $e^{-}$) or through two photons
with a very small probability. $M_{0}$ transitions have never been observed
but are expected to proceed through emission of two internal electrons or
through a photon pair. The interaction of electrons with the scintillator is
classical, thus the probability of missing the transition entirely is
negligible. The limit on this transition is instead determined by the
probability of entirely missing these rare photon pairs.

$Br_{\rm dark}^{\rm lim}\approx Br_{\rm\gamma\gamma}\exp\left(-{\frac{R_{\rm
det}}{\lambda_{\rm abs}}}\right).\quad\textrm{(for monopole transitions)}$ (2)

We next discuss each transition in turn.

###  III.1 $E_{2}$ transitions

The $E_{2}$ transitions were treated in depth in Benato et al. (2019). We
expand on the number of candidate nuclei studied in this work. Candidate
nuclei which exhibit $E_{2}$ transitions are tabulated in Table 2. Nuclei in
which the second photon in the cascade has lower energy than the first trigger
photon, i.e. $E_{2}<E_{1}$, are preferred because the probability of mis-
tagging the second photon as the trigger photon due to Compton scattering is
negligible. Branching fractions to various models of dark particles are
presented in Section IV and model-dependent limit projections for $E_{2}$
transitions are made in Section V.

Candidate | $\tau_{\frac{1}{2}}$ |  $E_{2}<E_{1}$? |  $E_{\rm probe}$[MeV] |  $E_{\rm trigger}$[MeV]
---|---|---|---|---
207Bi | 31 year | Yes | 0.57 | 1.06
60Co | 30 years | No | 1.33 | 1.17
46Sc | 83 day | Yes | 0.89 | 1.12
48V | 16 day | Yes | 0.98 | 1.31
48Sc | 43.6 hr | Yes | 0.98 | 1.04 or 1.31
24Na | 15 hr | Yes | 1.37 | 2.75
Table 2: Candidate $E_{2}$ transitions

###  III.2 $M_{1}$ transitions

$M_{1}$ transitions have been considered in Avignone et al. (1988) in the
specific context of axion searches. The decay chain of the
${}^{65}\textrm{Ni}$ isotope with a half-life of 2.5 hours exhibits such a
transition. The parity odd-nature of these transitions makes them especially
suited for searches of ALPs coupled to nuclei. Branching fractions to ALPs are
presented in Section IV and model-dependent limit projections for $M_{1}$
transitions are made in Section V.

Candidate | $\tau_{\frac{1}{2}}$ | multipole |  $E_{\rm probe}$[MeV]
---|---|---|---
65Ni | 2.5 hour | $M_{1}$ | 1.11
90Nb | 14.6 hour | $E_{0}$ | 1.80
170Lu | 2 day | $M_{0}$ | 2.82
Table 3: Candidates for other transitions

###  III.3 $E_{0}$ transitions

In the standard model, a single photon transition from $0^{+}\rightarrow
0^{+}$ is strictly forbidden. This instead proceeds through $e^{+}e^{-}$
emission (also called internal pair production or IPP), internal conversion
followed by x-rays (IC) or through two photons (2$\gamma$). The IPP and IC are
the main decay branches. The electron loses energy classically (unlike the
photon), which makes containment much more efficient. Moreover, branching
fractions to invisibles such as scalars and millicharged particles are
enhanced. 90Nb, whose decay has been studied in detail in Pettersson et al.
(1968); Warburton and Alburger (1982), has a 15 hour lifetime and $\beta^{+}$
decays predominantly to the 3.58 MeV, $8^{+}$ state of 90Zr. This then
cascades down, and with small branching fraction populates the 1.76 MeV
$0^{+}$ state. The subsequent $E_{0}$ transition to the ground state has been
studied in detail in the context of 90Y $\beta^{-}$ decays D’Arienzo (2013).
The standard model branching fraction to two photons which forms the dominant
background due to containment is estimated to be only $1.8\times 10^{-4}$ of
the dominant $e^{+}e^{-}$ decay Schirmer et al. (1984). The branching
fractions to various models in $E_{0}$ transitions is presented in Section IV
and projections in Section V.

###  III.4 $M_{0}$ transitions

Magnetic monopole transitions in decays of $0^{-}\rightarrow 0^{+}$, have been
studied in a theoretical context in Krutov and Knyazkov (1970).
Experimentally, they have been studied in light nuclei Krasznahorkay et al. ;
Alburger (1978). The heavy nucleus 170Lu is the candidate nucleus we consider
for $M_{0}$ transitions. In $6\%$ of its decay via electron capture, it
populates the $2.82$ MeV $0^{-}$ state of 170Yb. The possibility of a standard
model transition directly from this state to the ground state was studied in
Kuhnert et al. (1993) and no events were detected. Instead, the decay proceeds
through other intermediate states with higher multipolarity. Unlike the other
multi-polar transitions considered in this text, the study of $M_{0}$
transitions could also be novel due to the prospect of observing the $M_{0}$
in the standard model as well. These transitions will have favorable branching
fractions to ALPs, the relevant branching fractions are discussed in Section
IV and projections for these transitions are discussed in Section V.

##  IV Models and Branching Fractions

In this section we describe the models considered and compute the relevant
branching fractions. Here we will restrict ourselves to a handful of popular
models and their associated effective operators, keeping the list short, yet
illustrating the breadth of models accessible to an experiment looking for
missing energy in nuclear transitions.

###  IV.1 Dark scalars coupling to nucleons

We first consider a dark scalar with an effective Yukawa coupling to nucleons
(specifically the proton for easy matrix element computation).

$\mathcal{L}\supset g_{N}\phi\bar{N}{N}$ (3)

In the UV-complete model, this effective interaction may come about through a
Yukawa coupling to heavy or light quarks or through couplings to gluons Knapen
et al. (2017); Dev et al. (2020). The effective coupling to protons can be
looked for in $E$-type transitions considered in this work, i.e. $E_{2}$ and
$E_{0}$ transitions. The branching fraction to a scalar, $\textrm{Br}_{\rm
miss}\equiv\frac{\Gamma_{\phi}}{\Gamma_{\rm SM}}$ is given by Benato et al.
(2019); Izaguirre et al. (2015),

$\displaystyle\textrm{Br}_{\rm miss}(\phi,E_{2})$
$\displaystyle=\frac{g_{p}^{2}}{2e^{2}}\left(1-\frac{m_{\phi}^{2}}{\omega^{2}}\right)^{\frac{5}{2}}$
$\displaystyle\textrm{Br}_{\rm miss}(\phi,E_{0})$
$\displaystyle=\frac{8\pi\omega^{5}}{\alpha\kappa(\omega,m_{e})}\frac{g_{p}^{2}}{e^{2}}\left(1-\frac{m_{\phi}^{2}}{\omega^{2}}\right)^{\frac{5}{2}}$
(4)

Here

$\displaystyle\kappa(\omega,m)$
$\displaystyle=b\left(\frac{\omega-2m}{\omega+2m}\right)\left(\omega-2m_{e}\right)^{3}\left(\omega+2m_{e}\right)^{2}$
$\displaystyle b(S)$
$\displaystyle=\frac{3\pi}{8}\left(1-S/4-S^{2}/8+S^{3}/16-S^{4}/64+5S^{5}/512\right)$
(5)

The larger branching fraction in the $E_{0}$ case is due to favorable
selection rules.

###  IV.2 Dark photons

We next consider a dark photon, $A^{\prime}$, coupled to the standard model
photon via kinetic mixing,

$\mathcal{L}\supset\epsilon
F^{\mu\nu}F^{\prime}_{\mu\nu}+m^{2}_{A^{\prime}}A^{\prime 2}$ (6)

The branching ratio to the $A^{\prime}$ is straightforward to compute for
$E_{2}$ transitions. It is just $\epsilon^{2}$ with a kinematic factor that
captures the effect of a non-zero $m_{A^{\prime}}$

$\displaystyle\textrm{Br}_{\rm miss}(\gamma^{\prime},E_{2})$
$\displaystyle=\epsilon^{2}\left(1-\frac{m_{A^{\prime}}^{2}}{\omega^{2}}\right)^{\frac{5}{2}}$
(7)

Note that the above expression is only valid in the limit where
$m_{A^{\prime}}\gg\frac{1}{\lambda_{\text{abs}}}$, the inverse mean-free-path.
In this regime, the transverse mode that couples to the SM can maximally
oscillate into the sterile transverse mode. In the opposite limit, i.e. for
very small dark photon masses, the dark photon decouples.

For $E_{0}$ transitions, the transverse mode is forbidden by selection rules.
The longitudinal mode behaves just like the scalar in Eqn. (5). However, it
decouples for small $m_{A^{\prime}}$ and hence the branching fraction scales
as $\left(\frac{m_{A^{\prime}}}{\omega}\right)^{2}$. Thus, we have,

$\displaystyle\textrm{Br}_{\rm miss}(\gamma^{\prime},E_{0})$
$\displaystyle=\frac{8\pi\omega^{5}}{\alpha\kappa(\omega,m_{e})}\epsilon^{2}\frac{m^{2}_{A^{\prime}}}{\omega^{2}}\left(1-\frac{m_{A^{\prime}}^{2}}{\omega^{2}}\right)^{\frac{5}{2}}$
(8)

###  IV.3 Millicharged particles

We next consider particles carrying fractional charge under the SM photon also
known as millicharge particles. This charge could arise either due to direct
charge under the SM U(1), or through charge under a dark photon that
kinetically mixes with the SM photon. The branching fractions for both cases
are identical. For both $E_{2}$ and $E_{0}$ transitions, we use the
calculation of matrix elements in internal pair production in the SM Snover
and Hurd (2003) and adapt it to milli-charge particles of mass $m_{Q}$ and
charge $Q$ in units of the electric charge. We find

$\displaystyle\textrm{Br}_{\rm miss}(\textrm{milli},E_{2})$
$\displaystyle=Q^{2}\frac{25\alpha}{9}\frac{\kappa(\omega,m_{Q})}{\omega^{5}}$
$\displaystyle\textrm{Br}_{\rm miss}(\textrm{milli},E_{0})$
$\displaystyle=Q^{2}\frac{\kappa(\omega,m_{Q})}{\kappa(\omega,m_{e})}$ (9)

###  IV.4 ALP-nucleon coupling

Axion-like particles should generically couple to nucleons through the “wind-
coupling”. The Lagrangian for the nucleon coupling can be written as,

$\mathcal{L}\supset\frac{1}{f_{aN}}\partial_{\mu}a\bar{N}\gamma^{\mu}\gamma^{5}N$
(10)

with $f_{aN}$ the relevant decay constant. Due to parity considerations,
magnetic-type transitions are best suited to produce ALPs according to the
following branching fraction Avignone III et al. (1988),

$\displaystyle\textrm{Br}_{\rm miss}(a,M_{1})\sim
0.13\left(\frac{\textrm{GeV}}{f_{aN}}\right)^{2}\left(1-\frac{m_{a}^{2}}{\omega^{2}}\right)^{\frac{3}{2}}$
(11)

Finally, we also consider the $M_{0}$ transition in the decay of 170Lu in this
work. The 2.82 MeV, $0^{-}$ state of 170Yb is populated in $6\%$ of all the
$\epsilon$ decay of 170Lu. This $0^{-}$ state could decay via an ALP to the
$0^{+}$ ground state. In the SM, it instead decays via $E_{2}/M_{1}$
transitions to intermediate states. In order to make an order of magnitude
estimate of the branching fraction to ALPs, we approximate the $M_{0}$ width
to ALPs to the $E_{0}$ width to scalars.

$\displaystyle\textrm{Br}_{\rm
miss}(a,M_{0})=\frac{\Gamma_{M_{0}}(2.82~{}\textrm{MeV})}{\Gamma_{E_{2}}(1.4~{}\textrm{MeV})}\approx\frac{50}{9\alpha}\frac{\rm
GeV}{f_{aN}}\frac{\left((2.82~{}\textrm{MeV})^{2}-m_{a}^{2}\right)^{\frac{5}{2}}}{(1.4~{}\textrm{MeV})^{5}}$
(12)

###  IV.5 ALP coupling to photons

ALPs coupled to photons can be produced in transitions via an internal pair
production like process, with an off-shell photon producing an ALP and an on-
shell photon. However, we identify a more efficient production mechanism: the
decaying photon converting into an ALP in the detector. We start with the
Lagrangian,

$\mathcal{L}\supset\frac{1}{4f_{a\gamma}}aF_{\mu\nu}\tilde{F}^{\mu\nu}\,.$
(13)

This operator gives rise to the well-known Primakoff scattering process, in
which a photon may coherently scatter with the atomic and nuclear electric
fields of an atom and convert into an ALP (Fig. 2, left). The differential
production cross-section is given by

$\frac{d\sigma_{P}}{d\theta}=\frac{1}{4f_{a\gamma}^{2}}\alpha
Z^{2}F^{2}(t)\frac{|\vec{p}_{a}|^{4}\sin^{3}\theta}{t^{2}}\,,$ (14)

where $\theta$ is the angle of ALP with respect to the momentum direction of
the incoming photon. The total cross-section in the elastic limit is Tsai
(1986)

$\sigma_{P}=\dfrac{\alpha}{4\pi
f_{a\gamma}^{2}}\bigg{[}Z^{2}\ln(184Z^{-1/3})+Z\ln(1194Z^{-2/3})+Z^{2}\bigg{(}\ln\Big{(}\dfrac{403A^{-1/3}{\rm{MeV}}}{m_{e}}\Big{)}-2\bigg{)}\bigg{]}\,.$
(15)

Decay photons that stream into the detector volume may then undergo Primakoff
conversion which gives to a probability of disappearance. This probability is
given by product of an absorption probability and an effective branching
fraction of photons that Primakoff-convert versus undergoing ordinary
absorption;

$P(\gamma\to
a)=\dfrac{\sigma_{P}}{\sigma_{P}+\sigma_{SM}}(1-e^{-\sigma_{SM}n\ell})=\dfrac{\sigma_{P}}{\sigma_{P}+\sigma_{SM}}(1-e^{-\ell/\lambda}).$
(16)

Here, $\sigma_{P}$ is the total Primakoff scattering cross-section,
$\sigma_{SM}$ is the total $\gamma$ absorption cross-section as a function of
the photon energy $E_{\gamma}$, $n$ is the atomic number density and $\ell$ is
the $\gamma$ path length. Alternatively, we can also express the exponent in
terms of the mean-free-path $\lambda=1/(n\sigma_{SM})$.

$a$$\gamma$$e^{-},N$$e^{-},N$$a$$e^{-}$$e^{-}$$\gamma$

Figure 2: Primakoff (left) and Compton-like (right) processes for photon-axion
conversion. Figure 3: Sensitivity on the ALP-photon coupling, derived from
Primakoff-disappearance, as a function of detector mass (bottom axis) and the
equivalent detector radius (top axis) assuming spherical geometry. The effect
of full containment is manifest when the sensitivity flattens out at the limit
of a sufficiently large detector.

For ALPs lighter than a few keV, it is favorable to have an external magnetic
field to do this conversion. There are two ways to situate this magnetic field
- one could construct a geometry where the source is surrounded by a vacuum
region ($\sim 1\textrm{ m}^{3}$) with a large magnetic field ($B_{\rm ext}$)
that is in turn hermetically covered by scintillators, or the magnetic field
and the scintillator material can be co-located. The former, while more
expensive, has better science reach since the photon can oscillate into an ALP
over the entire volume of the vacuum region as opposed to the latter case
where the conversion region is restricted to the mean free path of the photon
in the scintillator. In the case of a vacuum surrounding the source, the
probability for photons to convert into ALPs over a distance $L$ is given by,

$P(\gamma\rightarrow a)=4\frac{B_{\rm
ext}^{2}\omega^{2}}{m_{a}^{4}f_{a\gamma}^{2}}\sin^{2}\left(\frac{m_{a}^{2}L}{4\omega}\right).$
(17)

We will assume a magnetic field $B_{\rm ext}=10~{}\textrm{Tesla}$ with an
extent of $L=1~{}\textrm{meter}$, for $m_{a}<1\textrm{eV}$,

$P(\gamma\rightarrow a)=\left(\frac{B_{\rm
ext}L}{2f_{a\gamma}}\right)^{2}\approx 25\bigg{(}\frac{\rm
GeV}{f_{a\gamma}}\bigg{)}^{2}\quad[{\rm Full~{}Containment}].$ (18)

When the scintillator material and the external magnetic field are co-located,
the probability to convert to an ALP in the small mass limit is given by,

$P(\gamma\rightarrow a)=4\frac{B_{\rm
ext}^{2}\omega^{2}}{m_{a}^{4}f_{a\gamma}^{2}}\int
dle^{-\frac{l}{\lambda}}\frac{m_{a}^{4}l}{8\omega^{2}}\approx\left(\frac{B_{\rm
ext}\lambda}{\sqrt{2}f_{a\gamma}}\right)^{2}\quad[{\rm Prototype}].$ (19)

Finally, we summarize the model operators discussed above and the relevant
photon-disappearance branching ratios ($BR_{miss}$) in Table 1. In the next
section, we will discuss the search potential and projected reach for each
case.

Model | $\mathcal{L}_{\text{int}}$ | Transition | $\textrm{BR}_{\rm miss}$
---|---|---|---
Scalar (nucleon coupling) | $g_{p}\phi\bar{N}{N}$ | $E_{2}$ | $\frac{g_{p}^{2}}{2e^{2}}\left(1-\frac{m_{\phi}^{2}}{\omega^{2}}\right)^{\frac{5}{2}}$
$E_{0}$ | $\frac{8\pi\omega^{5}}{\alpha\kappa(\omega,m_{e})}\frac{g_{p}^{2}}{e^{2}}\left(1-\frac{m_{\phi}^{2}}{\omega^{2}}\right)^{\frac{5}{2}}$
Dark Photon | $\epsilon F^{\mu\nu}F^{\prime}_{\mu\nu}$ | $E_{2}$ | $\epsilon^{2}\left(1-\frac{m_{\phi}^{2}}{\omega^{2}}\right)^{\frac{5}{2}}$
$E_{0}$ | $\frac{8\pi\omega^{5}}{\alpha\kappa(\omega,m_{e})}\epsilon^{2}\frac{m^{2}_{A^{\prime}}}{\omega^{2}}(1-\frac{m_{A^{\prime}}^{2}}{\omega^{2}})^{\frac{5}{2}}$
Milli-charged Particle | $-Q\bar{\chi}\gamma^{\mu}A_{\mu}\chi$ | $E_{2}$ | $Q^{2}\frac{25\alpha}{9}\frac{\kappa(\omega,m_{Q})}{\omega^{5}}$
$E_{0}$ | $Q^{2}\frac{\kappa(\omega,m_{Q})}{\kappa(\omega,m_{e})}$
ALP (nucleon coupling) | $f_{aN}^{-1}\partial_{\mu}a\bar{N}\gamma^{\mu}\gamma^{5}N$ | $M_{1}$ | $0.13\left(\frac{\textrm{GeV}}{f_{aN}}\right)^{2}\left(1-\frac{m_{a}^{2}}{\omega^{2}}\right)^{\frac{3}{2}}$
$M_{0}$ | $\frac{50}{9\alpha}\frac{\rm GeV}{f_{aN}}\frac{\left(\omega^{2}-m_{a}^{2}\right)^{\frac{5}{2}}}{\omega^{5}}$
ALP (photon coupling) | $\frac{1}{4f_{a\gamma}}aF_{\mu\nu}\tilde{F}^{\mu\nu}$ | $E_{2}/M_{1}$ | $\dfrac{\sigma_{P}}{\sigma_{P}+\sigma_{SM}}(1-e^{-\ell/\lambda})$
Table 4: Summary of the operators and branching ratios for each model
considered.

##  V Projected Reach

In this section we estimate the potential reach of the proof-of-concept
apparatus at Texas A&M for the models considered in the previous section. We
also estimate the sensitivity of a full scale experiment. All of these models
are subject to constraints from existing laboratory results and
astrophysical/cosmological considerations. The astrophysical limits, primarily
arising from the cooling of HB stars and SN1987a Raffelt (1996) are highly
model-dependent. For example, these limits are completely evaded in models
where the particles either have a mass or coupling that depends upon the
environment (see, for example, Refs. Jaeckel et al. (2007); Khoury and Weltman
(2004); Masso and Redondo (2005, 2006); Dupays et al. (2007); Mohapatra and
Nasri (2007); Brax et al. (2007); DeRocco et al. (2020)). Similar caveats also
apply to cosmology, i.e. the $N_{\rm eff}$ constraints from BBN and CMB Berlin
and Hook (2020); DeRocco et al. (2020). While this experiment can search for
particles that are not subject to astrophysical and cosmological constraints,
this model dependence encourages the production of robust laboratory limits on
particles that are naively constrained by astrophysics and cosmology.

In Fig. 4 we show the projected reach over the parameter space for the scalar
coupling to nucleons. The top row illustrates projections from using 46Sc
decays, a candidate for $E_{2}$ transitions, and the bottom row illustrates
projections for 90Nb, a candidate for $E_{0}$ transitions. Whereas the left
column shows projections for 36 crystals, the right column exhibits
projections if full containment of the signal photon is achieved. For large
couplings in the range $g_{N}\sim 10^{-3}-10^{-1}$ and for $m_{\phi}>0.4$ MeV,
the main constraint comes from binding energies in nuclear matter (NM) Liu et
al. (2016). We also show astrophysical constraints from supernovae Dev et al.
(2020) in addition to those from HB stars Raffelt (1996). In dotted lines we
also show the limits on invisible kaon decays from E949 Artamonov et al.
(2008) arising from model-dependent couplings to the light quarks.

We see that the projected reach for an exposure to $10^{9}$ total decays,
either in the case of 46Sc or 90Nb sources, can already probe beyond the
existing NM limits. Comparing the top and bottom rows, we see that there is
superior reach for the $E_{0}$ transition in 90Nb (bottom row) compared to the
$E_{2}$ transition. This is entirely due to the projections being containment
limited in the $E_{2}$ transition in contrast to containment issues arising
only for the severely suppressed 2-photon decay in $E_{0}$ transitions. This
is the reason why $g_{N}$ scales as $N_{d}^{-\frac{1}{4}}$ (where $N_{d}$ is
the total number of decays) in the prototype setup for the $E_{2}$ transition
and scales as $N_{d}^{-\frac{1}{2}}$ for the other setups where containment is
not an issue. We also see this by comparing the left plots to those on the
right i.e. the effect of going from a 36-crystal prototype to an apparatus
that offers full containment; this exponentially reduces the background of
escaped gammas, thereby pushing the sensitivity curve. With $10^{16}$ decays
and full containment, the entire supernova trapping window is accessible.

In Fig. 5 we show the limits on the kinetic mixing parameter, $\epsilon$, for
the dark photon model with mass $m_{A^{\prime}}$. The dominant laboratory
bounds come from the NA64 experiment Banerjee et al. (2019). We show the
dominant astrophysical limits Chang et al. (2017); Hardy and Lasenby (2017)
and have not plotted the sub-dominant BBN/CMB constraints (since we regard
both of these as model dependent). The sensitivity reach here, shown again for
46Sc and 90Nb sources, suggests a strong science case for a full scale
experiment with $10^{16}$ decays of 90Nb.

The projected reach for millicharge particles with charge $Q$ and mass $m_{Q}$
is shown in Fig. 6. Existing laboratory limits arise from the SLAC millicharge
experiment Prinz et al. (1998), while astrophysical and cosmological limits
are obtained from Chang et al. (2018); Vogel and Redondo (2014); Berlin and
Hook (2020). A 90Nb source with full containment performs excellently here as
well. In Fig. 7, we show projections for the ALP-nucleon coupling, with
$M_{1}$ transitions in the decay chain of 65Ni in the top row and $M_{0}$
transitions in the decay chain of 170Lu in the bottom row. Existing limits
arise from SN1987a Chang et al. (2018) and beam dumps Ramani and Woolley
(2019). New parameter space can be probed with $10^{9}$ decays even in the
current setup with 170Lu nuclei. With full containment, parameter space below
the supernova cooling constraints can be reached with 170Lu owing to the
favorable selection rules which enhance the ALP decay branching fraction.

We finally discuss sensitivity of ALP coupling to photons. The projected
sensitivity via Primakoff conversion in the detector material is shown in Fig.
8 for both 207Bi and 60Co sources. We show both sources to illustrate that
while 60Co has a wider range of $m_{a}$ sensitivity ($E_{\text{probe}}=1.33$
MeV), a 207Bi source has deeper reach due to the higher absorption of the
207Bi probe gamma ($E_{\text{probe}}=0.57$ MeV). However, existing constraints
from beam dump experiments Bauer et al. (2017) limit the reach over new
parameter space to a full-containment scenario ($\sim 400$ crystals) with a
larger activity exposure ($\gtrsim 10^{16}$ decays). With full-containment,
the reach can cover the “cosmological triangle” Carenza et al. (2020); Depta
et al. (2020), which is still allowed by the astrophysical and beam dump data.
In Fig. 9, we also show the effect of using a magnetic field to induce the
coherent conversion of photons over the entire spatial extent of the field
instead of using atomic conversion in material. Due to the coherence condition
in Eq. (18), however, this does limit the reach to lighter ALP masses.
Although existing constraints cover the sensitive area of parameter space, a
large part of this comes from stellar cooling (HB stars) which are model-
dependent as discussed above.

Lastly, new parameter space for direct ALP-electron couplings can be probed.
Like in the $g_{a\gamma}$ case, this sensitivity is driven by photon-ALP
conversion in material, but this time through the Compton-like conversion
$\gamma\,e^{-}\rightarrow a\,e^{-}$. The reach with full containment in such a
scenario can be $g_{ae}\sim 10^{-4}$ for $m_{a}<0.75$ MeV. Since this may not
be competitive with existing laboratory limits, we omit the projections for
this work. However, a future experimental setup with a more active source may
be able to expand the sensitivity in this parameter space.

Figure 4: Reach for scalar coupled to nuclei in $E_{2}$ (top) and $E_{0}$
(bottom) transitions for current (left panel) and future (right panel)
scintillator configurations.

Figure 5: Reach for dark photons in $E_{2}$ (top) and $E_{0}$ (bottom)
transitions for current (left panel) and future (right panel) scintillator
configurations

Figure 6: Reach for millicharge particles in $E_{2}$ ($E_{0}$) top (bottom)
transitions for current (left panel) and future (right panel) scintillator
configurations

Figure 7: Reach for ALP coupled to protons for $M_{1}$ (top) and $M_{0}$
(bottom) transitions for current (left panel) and future (right panel)
scintillator configurations.

Figure 8: Reach for ALPs coupled to photons for $E_{2}$ photons converting to
axions via Primakoff scattering in the detector for current (left panel) and
future (right panel) scintillator configurations.

Figure 9: Reach for ALPs coupled to photons for $E_{2}$ photons converting to
axions in the presence of an external magnetic field $B_{\rm ext}=1$T for
current (left panel) and future (right panel) scintillator configurations.

##  VI Conclusions

The investigations performed in this paper using the 36 Cs(Tl) scintillators
available at Texas A&M show that a proof of concept experimental apparatus
using these scintillators can yield science results in nuclei with $E_{0}$ and
$M_{0}$ transitions, probing presently unconstrained parameter space for
scalars and pseudoscalars coupled to nucleons. The key limitation of this
apparatus for other multipole transitions is containment - with 36 crystals,
unlike the $E_{0}$ and $M_{0}$ transitions that produce difficult to miss
electrons and positrons, the gammas produced in other transitions can be
missed if the detector is not sufficiently big. A successful demonstration of
this proof of concept setup would sharpen the case for the construction of a
larger detector that is able to probe this broader class of transitions. These
investigations also support the case for modular development of this
experiment with an inner module that is made of crystal scintillators and an
outer module that could be constructed from liquid or plastic scintillators.
This is because of the fact that the required energy resolution requires
shorter scintillating modules to avoid loss of photons from absorption. The
inner modules of the detector, where the vast majority of the produced gammas
will be absorbed, thus need to be well instrumented. The outer modules, which
constitute the bulk of the volume of the detector, exist to observe rare
gammas that exit the inner module without interactions. Due to the low
probability associated with this possibility, the requirements on energy
resolution in the outer modules is more relaxed, enlarging the range of
detector material. This modular approach is thus likely to be cost effective
without sacrificing the science potential of the experiment.

The strong science case for laboratory probes of light, weakly coupled
particles warrants investigation of other technologies that could be used to
search for these particles using this protocol. For example, investments in
dark matter detection has resulted in the development of sensors that can
detect eV scale nuclear recoils with low background and excellent rejection
between nuclear and electron recoils. It would be interesting to incorporate
this technology into this experiment - with eV scale detection, the nuclear
recoil produced from a decay can be directly detected without the need for a
gamma cascade, enlarging the range of nuclei that could be used for this kind
of detection concept. In fact, this technology could be used to potentially
probe missing energy in decays involving electron capture - the nuclear recoil
from the electron capture can be observed and the experiment can search for
the missing Auger electrons/photons, both of which are difficult to miss. This
kind of experiment would be uniquely sensitive to new physics coupled to
electrons, complementing the probes of particles that couple to nucleons and
photons presented here.

###### Acknowledgements.

The work of RM, AJ and SV are supported by DOE Grant Nos DE-SC0018981, DE-
SC0017859, and DE-SC0021051, as well as internal funding from the Mitchell
Institute. JBD acknowledges support from the U.S. National Science Foundation
under Grant No. NSF PHY-1820801. The work of BD, DK and AT are supported by
the DOE Grant No. DE-SC0010813. SR is supported by the NSF under grant
PHY-1818899, the SQMS Quantum Center and DoE support for MAGIS. RM
acknowledges the critical contribution of the CsI crystals by Jim Alexander
from the CLEO experiment at Cornell University. HR is supported in part by NSF
Grant PHY-1720397 and the Gordon and Betty Moore Foundation Grant GBMF7946.

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# Phase-Shift Design and Channel Modeling for Focused Beams in IRS-Assisted
FSO Systems

Jung-Hoon Noh and Byungju Lee

###### Abstract

Interest in free-space optics (FSO) is rapidly growing as a potential solution
for the backhaul of next-generation mobile or low-orbit satellite
communications. Various techniques have been suggested for employing an
intelligent reflecting surface (IRS) in FSO systems, such as anomalous
reflection, power amplification, and beam splitting. It is possible to deliver
more power to the receiver (Rx) by collimating or focusing the reflected beam
at the Rx lens. In this study, we propose a phase-shift design of an IRS for
beam focusing. In addition, we propose a new pointing error model and an
outage performance analysis applicable when the beam width is comparable to or
less than the aperture size of the Rx. The analytical results are validated by
Monte Carlo simulations. This study provides essential preliminary results for
future researches that assume a focused beam in FSO systems.

###### Index Terms:

Intelligent reflecting surfaces, free space optical system, beam focusing,
pointing error

##  I Introduction

Optical wireless systems, such as free-space optical (FSO) systems, are a
potential technology for high-data-rate transmission with license-free
operation over a wide range of bandwidth in the optical spectrum [1]. In
contrast to radio frequency (RF), FSO systems are immune to electromagnetic
interference and have been considered a cost-effective solution for
terrestrial backhaul/fronthaul wireless applications for 5G and beyond 5G
networks. Recently, the use of an intelligent reflecting surface (IRS) in FSO
systems has attracted significant research interest [2-12]. An IRS is an
electromagnetic device with electronically controllable characteristics.
Various techniques can be used to manipulate the amplitude, phase, and
polarization of the incoming signal using IRS. For example, a power-amplifying
IRS module was proposed in [2]. The authors in [3] and [4] introduced power
splitting to multiple receivers (Rx) using IRS.

Anomalous reflection is a fundamental technique which freely adjusts the
direction of the reflected beam by manipulating the phase of the incoming beam
over the surface of the IRS. Through anomalous reflection using IRS, we can
alleviate the line-of-sight (LOS) requirements, which are essential for
establishing FSO links. For the anomalous reflection of an IRS-assisted FSO
system, [5] proposed a phase-shift design and pointing error investigation.
Moreover, analytical expressions for the end-to-end channel model [6] and
various performance metrics [7, 8, 9, 10, 11] were introduced in which both
atmospheric turbulence and pointing errors were considered.

Furthermore, there is increasing interest in techniques that not only control
the direction of the beam, but also its size [5, 12]. Using the IRS, we can
adjust the convergence of the reflected beam at the Rx. However, research
using IRS to create a focused beam is still in its infancy; not just the IRS
phase-shift designs, but research into the pointing error for the focused beam
remains an open problem [12]. While the focused beam using the IRS system may
deliver greater power to the Rx than a general reflected Gaussian beam under
perfect beam tracking, its performance can be more sensitive to pointing
errors.

In this study, we first propose a phase-shift design of the IRS that can
freely adjust the beam width as well as the direction of the beam. Then, we
present a pointing error model for the case where the beam width is similar
to, or smaller than, the diameter of the Rx lens. It should be noted that the
conventional Gaussian form pointing error model can be applied only if the
beam width is more than twice the diameter of the Rx aperture [13].
Furthermore, using the newly proposed pointing error model, the outage
probability of the system was investigated when the effects of the pointing
error and atmospheric loss were simultaneously considered. The proposed phase-
shift design and pointing error model were verified using simulation results.

Finally, we discuss how the outage performance is affected by the beam width,
atmospheric loss, transmitted power, and pointing error. These factors are
intricately intertwined, making it difficult for system designers to determine
the system parameters. However, the presented mathematical modeling can
facilitate performance analysis in terms of these factors for the focused
beam. In addition, it offers insight into system design. The size of the beam
can be determined to achieve the best performance in a combination of several
channel influences.

##  II System Model

###  II-A Power Density of Gaussian Beam

First, we present the electric field and power density distributions of the
transmitted Gaussian beam on the IRS. We consider the two-dimensional (2D)
coordinate system of the $yz$-plane, where the IRS exists on the $y$-axis.
Without loss of generality, it is assumed that both the centers of the IRS and
transmitted beam are at the origin, as illustrated in Fig. 1. The lengths of
the IRS line and the Rx lens lines are denoted as $2a_{r}$ and $2a_{l}$,
respectively. $\theta_{i}$ and $\theta_{r}$ are the incident and reflection
angles, respectively, at the IRS.

Figure 1: Illustration of focused beam shaping using IRS

Next, we define the electric field and power density of the Gaussian beam in
the positive $\hat{z}$ direction from the transmitter using
$\hat{y}\hat{z}$-coordinates. The electric field is expressed as follows:

$\displaystyle
E(\hat{y},\hat{z};w_{0})=E_{0}\sqrt{\frac{w_{0}}{w(\hat{z},w_{0})}}$
$\displaystyle\qquad\times\exp\left(-\left({\frac{\hat{y}^{2}}{w^{2}(\hat{z},w_{0})}}+j\psi(\hat{y},\hat{z},w_{0})\right)\right)$
(1)

where phase

$\psi_{\mathrm{G}}(\hat{y},\hat{z},w_{0})=-k\hat{z}-k\frac{\hat{y}^{2}}{2R(\hat{z},w_{0})}+\zeta(\hat{z}),$
(2)

$\hat{y}$ is the axis perpendicular to $\hat{z}$, $E_{0}$ denotes the electric
field at the origin, $w_{0}$ denotes the beam waist radius, $k=2\pi\lambda$ is
the wave number, $\lambda$ is the optical wavelength,
$R(\hat{z},w_{0})=\hat{z}(1+(z_{0}/\hat{z})^{2})$ is the curvature radius of
the beam’s wavefront at $\hat{z}$, $z_{0}=\pi w_{0}^{2}/\lambda$ is the
Rayleigh range, $\zeta(\hat{z})=\tan^{-1}\left(\hat{z}/z_{0}\right)$ is the
Gouy phase, and
$w(\hat{z},w_{0})=w_{0}\sqrt{1+\left(\hat{z}/z_{0}\right)^{2}}$ is the beam
width, where the intensity values fall to $1/e^{2}$ of their axial values on
the $\hat{z}$ plane. In addition, the power density distribution of the
Gaussian beam $I(\cdot)=|E(\cdot)|^{2}$ is given by

$I(\hat{y},\hat{z};w_{0})=\frac{2}{\sqrt{\pi}w(\hat{z},w_{0})}\exp\left(-\frac{2\hat{y}^{2}}{w^{2}(\hat{z},w_{0})}\right).$
(3)

Then, we express the electric field across the IRS as
$E(y\cos\theta_{i},d_{t2r}+y\sin\theta_{i};w_{0})$, and approximate the power
density distribution across the IRS as follows:

$I_{irs}(y,d_{t_{2}r};\theta_{i},w_{0})\\\
\approx\frac{\sqrt{2}\cos\theta_{i}}{\sqrt{\pi}w(d_{t_{2}r},w_{0})}\exp\left(-\frac{2cos^{2}\theta_{i}y^{2}}{w(d_{t_{2}r},w_{0})}\right),$
(4)

where the approximation arises when $d_{t2r}\gg y\sin\theta_{i}$.

###  II-B Channel Model

Next, we define the FSO communication link. This study assumes that the
intensity modulation and direct detection are employed where the on-off keying
(OOK) is used as the modulation scheme. Laser beams propagate through a
turbulence channel with additive white Gaussian noise (AWGN) in the presence
of pointing errors. The received optical signal is then converted into an
electrical signal at the photodetector. The received signal, $y$, can be
expressed as $y=\eta hx+n$, where $x$ is the transmitted signal, $\eta$ is the
effective photo-electric conversion ratio, $h$ is the channel gain, and $n$ is
the signal-independent AWGN with a variance of $N_{0}$.

The channel gain, $h$, can be modeled as $h=h_{l}h_{a}h_{p}$, where $h_{l}$
represents the attenuation caused by the scattering and absorption of
particles in the air, $h_{a}$ is the atmospheric fading loss factor, and
$h_{p}$ is the pointing error loss factor. Note that $h_{a}$ and $h_{p}$ are
both independent random variables, whereas $h_{l}$ is a constant given by
$h_{l}=\exp\left(-\sigma z\right)$ where $\sigma$ is attenuation coefficient
and $z$ is link distance.

Becuase the coherence time of FSO channels is considerably longer than the
typical bit interval, owing to the high data rates, the FSO channel is
commonly modeled as a slow-fading channel, where the outage probability
becomes a relevant metric in performance evaluation. Given the rate
requirement $R_{0}$, the outage probability is expressed as
$P_{out}=\mbox{Prob}(h<h_{0})$ where
$h_{0}=\sqrt{\left(2^{R_{0}}-1\right)\sigma_{n}^{2}/(\sqrt{2}P_{t}R_{0})^{2}}$,
and $P_{t}$ is the average transmitted optical power.

##  III Phase-Shift Design of IRS

This section presents the IRS phase-shift design for creating a reflected beam
focused on the Rx. By manipulating the phase of the reflected beam over the
IRS surface, we can control the convergence of the beam at the Rx. First, we
consider the case when the reflected beam is completely focused at the center
of the Rx. A focused beam can be created by allowing the beam reflected from
the IRS to have a phase profile of the virtual beam propagating from the
center of the Rx with the opposite direction.

Assuming a mirror-assisted system, we consider the virtual beam that comes
from the Rx lens, whose location is specified by $\theta_{r}$ and
$d_{r_{2}l}$. It is assumed that the virtual beam has the same power density
distribution as the original beam from the transmitter (Tx) on the IRS, as
follows:

$I_{irs}(y,d_{t_{2}r};\theta_{i},w_{0})=I_{irs}(y,d_{r_{2}l};\theta_{r},\tilde{w}_{0}),$
(5)

where $\tilde{w}_{0}$ is the beam waist of the virtual beam. As the solution
of the equation $w(d_{r_{2}l},\tilde{w}_{0})=w_{eq,\theta_{r}}$, we obtain

$\tilde{w}_{0}=\sqrt{\frac{w_{eq,r}^{2}}{2}-\frac{1}{2\pi}\sqrt{\pi^{2}w_{eq,r}^{4}-4d_{r_{2}l}^{2}\lambda^{2}}},$
(6)

where $w_{eq,\theta_{r}}=w_{irs}\cos\theta_{r}$ and
$w_{irs}=w(d_{t_{2}r},w_{0})/\cos\theta_{i}$ is the width of the transmitted
beam on the IRS.

Next, we construct the phase-shift profile $\Delta\psi(y)$ which is added to
the phase of reflected beam across the IRS. It is built by adding the phase of
the virtual beam with reversed direction, that is
$\psi\left(y\cos\theta_{r},-(d_{r_{2}l}+y\sin\theta_{r}),\tilde{w}_{0}\right)$,
and subtracting the phase of the incoming beam, that is
$\psi(y\cos\theta_{i},d_{t_{2}r}+y\sin\theta_{i},{w}_{0})$. The resulting
phase-shift profile across the IRS becomes

$\Delta\psi(y|\theta_{i},\theta_{r},w_{0},\tilde{w}_{0})=\pi-\psi\big{(}y\cos\theta_{i},d_{t_{2}r}+y\sin\theta_{i},{w}_{0}\big{)}\\\
+\psi\big{(}y\cos\theta_{r},-(d_{r_{2}l}+y\sin\theta_{r}),\tilde{w}_{0}\big{)}.$
(7)

Eventually, the reflected beam is focused at a distance $d_{r_{2}l}$ with the
beam width of $\tilde{w}_{0}$.

Furthermore, we consider the case when the beam width at the Rx is larger than
$\tilde{w}_{0}$ and smaller than $w_{irs}$. In this case, we assume that
reflected beam is focused at a greater distance by $d_{f}$ than the Rx, where
the virtual beam source is located. Then, we must find the virtual beam whose
width is $w_{l}$ at the RX lens of distance $d_{f}$, and $w_{irs}$ at the IRS
of distance $d_{r_{2}l}+d_{f}$. To obtain the exact $\tilde{w}_{0}$ of such a
beam, the following equations must be solved:

$\frac{w(d_{r_{2}l}+d_{f},\tilde{w}_{0})}{cos\theta_{r}}=\frac{w(d_{t_{2}r},w_{0})}{cos\theta_{i}}\,\mbox{
and }\,w(d_{f},\tilde{w}_{0})=w_{l}.$ (8)

However, the analytical solution of (8) is very complicated, because it
requires the roots of a general cubic equation to obtain $\tilde{w}_{0}$.
Instead of an exact analytical solution, we present an approximated solution.
Assuming the Rx is at a great distance from the IRS, i.e., $d_{r2l}\gg z_{R}$,
$\tilde{w}_{0}$ can be approximated as follows [14]:

$\tilde{w}_{0}\approx\frac{\lambda d_{r2l}}{\pi(w_{eq,\theta_{r}}-w_{l})}$ (9)

Note that (9) can be verified by (8).

Further, we compare the power density distributions of reflected beam at the
Rx, which are obtained by two different method: the geometric optics and the
Huygen-Fresnel principle. The power density of the geometric optics was
obtained by the beam propagated from virtual beam source to the Rx, at a
distance $d_{f}$ from the source. The Huygens-Fresnel principle treats every
point on the IRS as a secondary point source emitting a spherical wave [14].
Therefore, the electric field on the Rx is the superposition of the waves
originating from all secondary sources. This is expressed as follows:

$\displaystyle
E_{\mathrm{Rx}}(\tilde{y})=\frac{\varsigma}{j\sqrt{\lambda}}\int_{-a_{r}}^{a_{r}}E_{\mathrm{irs}}(y)\frac{\exp(jk|\mathbf{r}_{0}-\mathbf{r}_{1}|)}{\sqrt{|\mathbf{r}_{0}-\mathbf{r}_{1}|}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\exp(j\Delta\psi(y))dy,$
(9)

where $\varsigma=\sqrt{\cos\theta_{i}/\cos\theta_{r}}$,
$\mathbf{r}_{0}=\big{[}\begin{smallmatrix}\cos\theta_{r}&\sin\theta_{r}\\\
-\sin\theta_{r}&\cos\theta_{r}\end{smallmatrix}\big{]}\big{[}\begin{smallmatrix}\tilde{y}\\\
d_{r_{2}l}\end{smallmatrix}\big{]}$,
$\mathbf{r}_{1}=\big{[}\begin{smallmatrix}y\\\ 0\end{smallmatrix}\big{]}$, and
$\Delta\psi(y)$ is the phase-shift profile across the IRS, given by (7).

Figure 2: Power density distribution on Rx lens with $\theta_{i}=\pi/3$,
$\theta_{r}=\pi/6$, $w_{0}=1$ mm, $a_{l}=10$ cm, and
$a_{r}\in\\{10,\infty\\}$.

The results of this comparison are presented in Fig. 2. The received power
density computed from the Huygens-Fresnel integration matches well with the
geometric optics computation. Thus, the proposed phase-shift design for
focusing beams is verified. Expanding this result to three dimensions can be
achieved using a rotated astigmatic Gaussian beam. The results of 2D case in
the $yz$ plane can be repeated for the $xz$ plane, and eventually extended to
the rotated astigmatic Gaussian beam for the three-dimensional (3D) case.
Further details of extension to 3D system are demonstrated in [5], and are not
covered in this paper owing to the limitation of the length.

So far, we have introduced a method to focus the beam at the Rx using the IRS.
With the proposed phase-shift design, we can concentrate the transmitted power
on the Rx using the IRS. However, the focused beam is more sensitive to
pointing errors. The following section examines the effect of the pointing
error when the beam is focused, that is, when the beam width is similar to or
smaller than the Rx size.

##  IV Pointing Error Modeling

This section examines the effect of pointing errors on the quality of the FSO
channels when the beam is focused on a size similar to or smaller than the Rx
aperture. Note that the laser beam reflected from the IRS becomes an
astigmatic Gaussian beam at the Rx lens in the 3D space. However, we
considered only a general Gaussian beam. This is because the main purpose of
the pointing error modeling is to analyze the influence of pointing errors
according to the size of the beam rather than its shape. Based on the results
of the general Gaussian beam, we can expand it to an astigmatic Gaussian beam
in future work.

###  IV-A Channel Coefficient Modeling

First, we introduce the pointing error model and then derive the probability
density function (PDF) for two cases according to the beam width. The first
case is when the beam width is similar to the radius of Rx lens. This can be
approximated using the Gaussian error function, i.e., $\mbox{erf}(\cdot)$ as
follows:

$\displaystyle h_{p}(u;w_{l},a_{l})$
$\displaystyle=\int_{-a}^{a}\int_{-\sqrt{a_{l}^{2}-x^{2}}}^{\sqrt{a_{l}^{2}-x^{2}}}\frac{2}{\pi
w_{l}^{2}}$
$\displaystyle\qquad\qquad\times\exp{\left(-2\frac{(x-u)^{2}+y^{2}}{w_{l}^{2}}\right)}dydx$
(10)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\approx}}\int_{-\infty}^{\frac{\sqrt{\pi}a_{l}}{2}}\frac{\sqrt{2}E}{\sqrt{\pi
w_{l}^{2}}}\exp{\left(\frac{-2(x-u)^{2}}{w_{l}^{2}}\right)}dx$
$\displaystyle=\frac{E}{2}\left(\mbox{erf}\left(\frac{\sqrt{2}}{w_{l}}\left(\frac{\sqrt{\pi}}{2}a_{l}-|u|\right)\right)+1\right),$
(11)

where the beam propagates in the positive $z$ direction, the Rx lens is
located on the $x\\-y$-plane, $E$ is derived from the error function where
$E={\rm{erf}}\left(\sqrt{\pi}a_{l}/(\sqrt{2}w_{l})\right)$, $w_{l}$ is the
beam width at the Rx lens, and $u$ is the displacement of beam from the center
of the Rx. In (11), (a) follows from that the power of the beam outside the Rx
lens is negligibly small when the beam width is similar to or smaller than the
radius of Rx aperture.

Assuming a Gaussian fluctuation of the center of beam on both the horizontal
and vertical axes of the Rx, $u$ follows a Rayleigh distribution. The PDF of
the pointing error loss, $h_{p}$, is then given by

$f_{h_{p}}(h_{p})=\sqrt{\frac{\pi}{2}}w_{l}\lambda(h_{p})\exp{\left(v(h_{p})^{2}-\frac{\lambda(h_{p})^{2}}{2\sigma^{2}}\right)}$
(12)

where $\lambda(h_{p})=\frac{\pi}{2}a_{l}-\frac{w_{l}}{\sqrt{2}}v(h_{p})$,
$v(h_{p})=\mbox{erf}^{-1}\left(\frac{2}{E}h_{p}-1\right)$,
$\mbox{erf}^{-1}(\cdot)$ is the inverse of the $\mbox{erf}(\cdot)$, $0\leq
h_{p}\leq A_{0}$, and $A_{0}~{}=\frac{E(E+1)}{2}$ is the fraction of the
collected power at $r=0$.

Meanwhile, integrating (12) to obtain a performance metric such as a outage
probability may be limited in cases where $w_{l}\ll a_{l}$. Note that for
$w_{l}\ll a_{l}$, $E\approx 1$ and $A_{0}\approx 1$. Then, $f_{h_{p}}(A_{0})$
becomes infinite, and (12) cannot be integrated from $0$ to $A_{0}$. Thus, we
introduce an another pointing error model applicable for the case of $w_{l}\ll
a_{l}$ as follows:

$h_{p}(u;a_{l})=\mathbbm{1}_{A}(u)$ (13)

where $A=[-a_{l},a_{l}]$, $\mathbbm{1}_{A}(u)$ is an indicator function with
$\mathbbm{1}_{A}(u)=1$ only if $u\in A$, and $\mathbbm{1}_{A}(u)=0$ otherwise.
Then, the PDF can be expressed as follows:

$f_{h_{p}}\left(h_{p}\right)=F_{u}(a_{l})\delta(h_{p}-1)+(1-F_{u}(a_{l}))\delta(h_{p}),$
(14)

where $\delta(\cdot)$ is the impulse function, and $F_{u}(\cdot)$ is the
cumulative probability distribution (CDF) of displacement which is a Rayleigh
random variable. Note that (14) can be integrated.

Figure 3: Comparison of $h_{p}(u;w_{l},a_{l})$ modeling for different values
of $w(z)/a_{l}$.

The exact expression, (10), and proposed approximation model for $h_{p}(r)$
are plotted in Fig. 3 for the three case of $w_{l}/a_{l}\in\\{0.1,0.5,1.0\\}$.
For the approximate model of $h_{p}$, (11) is used when
$w_{l}/a_{l}\in\\{0.5,1.0\\}$, and (13) is used when $w_{l}/a_{l}=0.1$. In
addition, we compared the proposed approximation models of $h_{p}$ with
conventional Gaussian form modeling [13]. Note that the exact expression in
(10) and the proposed approximation model correspond for all three cases.
However, we can observe that the conventional approximation deviates from the
exact expression as $w_{l}/a_{l}$ decreases.

###  IV-B Outage Probability

In this section, we obtain the outage probability when considering atmospheric
and pointing error losses together with the quality of the FSO channel. Given
$f_{h_{p}}(h_{p})$, the PDF of the composite channel coefficient $h$ is
expressed as

$\displaystyle f_{h}(h)$ $\displaystyle=$ $\displaystyle\int
f_{h_{p}}\left(h_{p}\right)f_{h|h_{p}}\left(h|h_{p}\right)dh_{p}$ (15)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$
$\displaystyle\int^{A_{0}}_{0}f_{h_{p}}\left(h_{p}\right)\frac{1}{h_{p}h_{l}}f_{h_{a}}\left(\frac{h}{h_{p}h_{l}}\right)dh_{p}$
(16)

where (a) is derived from $h=h_{l}h_{p}h_{a}$. We can then compute the outage
probability as

$P_{out}=\int^{h_{0}}_{0}f_{h}(h)dh.$ (17)

By substituting $f_{h}(h)$ from (16) into (17) and changing the order of
integration, the outage probability becomes

$\displaystyle P_{out}$ $\displaystyle=$
$\displaystyle\int^{A_{0}}_{0}\frac{1}{h_{p}h_{l}}f_{h_{p}}\left(h_{p}\right)\int^{h_{0}}_{0}f_{h_{a}}\left(\frac{h}{h_{p}h_{l}}\right)dh\,dh_{p}$
(18) $\displaystyle=$
$\displaystyle\int^{A_{0}}_{0}f_{h_{p}}\left(h_{p}\right)F_{h_{a}}\left(\frac{h_{0}}{h_{p}h_{l}}\right)dh_{p}$

where $F_{h_{a}}(\cdot)$ is the CDF of turbulence-induced channel $h_{a}$.
Note that we consider both log-normal fading for weak turbulence channel and
Gamma-Gamma fading for strong turbulence channel.

For the lognormal fading, we have

$F_{h_{a}}\left(h_{a}\right)=\frac{1}{2}+\frac{1}{2}\mbox{erf}\left(\frac{\log\left(h_{a}\right)}{\sqrt{8\sigma^{2}}}\right),$
(19)

where $\sigma^{2}\approx\sigma_{R}^{2}/4$ is the variance of the log fading
amplitude, that is, $0.5\log(h_{a})$. $\sigma_{R}^{2}$ is the Rytov variance,
which is given by $\sigma_{R}^{2}=1.23C_{n}^{2}k^{7/6}d_{e2e}^{11/6}$ where
$C_{n}^{2}$ is the index of refraction structure parameter. For the Gamma-
Gamma fading, we have

$F_{h_{a}}\left(h_{a}\right)=\pi\mbox{csc}\left(\pi\gamma\right)\left[\frac{\left(\alpha\beta
h_{a}\right)^{\beta}{}_{1}F_{2}(\beta;1-\gamma,1+\beta;\alpha\beta
h_{a})}{\Gamma(\alpha)\Gamma(1-\gamma)\Gamma(\beta+1)}\right.\\\
-\left.\frac{\left(\alpha\beta
h_{a}\right)^{\alpha}{}_{1}F_{2}(\alpha;1+\gamma,1+\alpha;\alpha\beta
h_{a})}{\Gamma(\beta)\Gamma(1+\gamma)\Gamma(\alpha+1)}\right],$ (20)

where

$\displaystyle\alpha=$
$\displaystyle\left[{\exp\left({\tfrac{0.49\sigma_{R}^{2}}{(1+1.11\sigma_{R}^{12/5})^{7/6}}}\right)-1}\right]^{-1}\text{and}$
(21) $\displaystyle\beta=$
$\displaystyle\left[{\exp\left({\tfrac{0.51\sigma_{R}^{2}}{(1+0.69\sigma_{R}^{12/5})^{5/6}}}\right)-1}\right]^{-1},$
(22)

$\gamma=\alpha-\beta$, and ${}_{p}F_{q}$ is the generalized hypergeometric
function for integers $p$ and $q$ [15].

Then, the outage probability can be obtained by substituting
$f_{h_{p}}(h_{p})$ from (12) and (14), and $F_{h_{a}}(h_{a})$ from (19) and
(20) into (18), according to the pointing error model and turbulence-induced
fading models. Note that for (12), the outage probability cannot be computed
in closed form. However, it involves only one finite integral that can be
easily computed numerically.

Meanwhile, for (14), the outage probability can be obtained in closed form,
and it is given by

$\displaystyle P_{out}$ $\displaystyle=$ $\displaystyle
F_{u}(a_{l})\int^{1}_{0}\delta(h_{p}-1)F_{h_{a}}\left(\frac{h_{0}}{h_{p}h_{l}}\right)dh_{p}$
(23)
$\displaystyle+\left(1-F_{u}(a_{l})\right)\int^{1}_{0}\delta(h_{p})F_{h_{a}}\left(\frac{h_{0}}{h_{p}h_{l}}\right)dh_{p}$
$\displaystyle=$ $\displaystyle
F_{u}(a_{l})\left(F_{h_{a}}\left(\frac{h_{0}}{h_{l}}\right)-1\right)+1.$

Moreover, from (23), we can also identify that the outage probability reaches
a lower limit when $P_{t}$ approaches infinity as follows:

$P_{out}\geq\lim_{P_{t}\to\infty}P_{out}\\\ =\lim_{h_{0}\to
0}F_{r}(a)\left(F_{h_{a}}\left(\frac{h_{0}}{h_{l}}\right)-1\right)+1=1-F_{r}(a).$
(24)

Note that even if the power increases, the outage probability does not
decrease accordingly and gradually converges to the lower limit. In addition,
we observe that this lower limit of the outage probability depends only on the
probabilistic characteristics of the pointing error, regardless of the
atmospheric effect.

##  V Numerical Results

Figure 4: Outage probability comparison between analysis and simulation

We performed Monte Carlo simulation to verify the proposed pointing error
model and the outage probability. The simulation parameters are as follows.
For all simulation cases, we assumed that $\lambda=1550$ nm, and the total
length of the propagation path between the Tx to Rx via IRS, $d_{t_{2}l}$, was
1 km.

The weather conditions on FSO link are characterized by two cases: clear
weather with weak turbulence, and light fog with strong turbulence, as in
[13]. For the clear whether, we assumed that the visibility was $10$ km; thus
$h_{l}=0.9$. In the light foggy whether, the visibility was assumed to be
$0.5$ km, and $h_{l}=0.08$. The strength of turbulence was characterized by
$C_{n}^{2}$ and $\sigma_{R}^{2}$. We assumed that
$C_{n}^{2}=$5\text{\times}{10}^{-14}$$ and $\sigma_{R}^{2}=1$ for the weak
turbulence, and $C_{n}^{2}=$0.5\text{\times}{10}^{-14}$$ and
$\sigma_{R}^{2}=0.1$ for the strong turbulence.

Fig. 4 compares the outage probability of the analysis and simulation
according to the standard deviation of the beam’s displacement, $\sigma_{u}$,
and transmitted power $P_{t}$. The simulation and analytical results match
closely for both the proposed pointing error models, (11) and (13). In
addition, we observe that the outage probability reaches the lower limit,
which increases with $\sigma_{u}$. Moreover, the lower limit of the outage
probability was equivalent for both atmospheric models. Thus, (24) was
verified.

Fig. 5 shows that how the outage probability varies according to the beam
width and pointing error. We can obtain a better outage performance (lower
outage probability) with a more focused beam with a small $w_{l}/a_{l}$ when
the pointing error is marginal, with $\sigma_{u}=0.1$. However, when the
pointing error is significant, the more focused beam gets the worse outage
performance (high outage probability); when $w_{l}/a_{l}=0.1$ and
$\sigma_{u}=0.4$, the outage probability reaches the lower bound early about
at a power of $-15$ dBm and is larger than other cases of
$w_{l}/a_{l}\in\\{1.0,2.0\\}$ for the most of transmitted power ranges.

Furthermore, we can observe that the performance gain of $w_{l}/a=0.1$
compared to $w_{l}/a=1$ is less than $1$ dB when $\sigma_{u}=0.1$. Based on
this observation, it can be deduced that a focused beam smaller than size of
the Rx has no substantial benefit. We can transmit sufficient power with beam
whose beam width at the Rx is similar to the radius of the Rx aperture. If the
beam width is further reduced, it becomes more sensitive to pointing error and
only increases the impact of the pointing error.

##  VI Conclusion

Figure 5: Outage probability according to $w_{l}/a_{l}$ and $\sigma_{u}$. Note
that (13), (11), and conventional Gaussian form are used for each case of
$w_{l}/a_{l}\in\\{0.1,1.0,2.0\\}$, respectively.

In this paper, we proposed a phase-shift design of an IRS for focusing beams.
In addition, we introduced a new pointing error model and an outage
probability expression, applicable when the beam width is less than the Rx
aperture size, for cases where the conventional model cannot be applied. The
simulation validated the proposed phase-shift design, and the outage
probability expression was verified. The mathematical modeling presented in
this work can facilitate performance analysis according to various factors,
such as beam width, power, and pointing error for the focused beam. Thus, this
study provides essential preliminary research results for future studies that
assume a focused beam in FSO systems.

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