# Mehta’s eigenvectors for the finite Hartely transform.

Fethi Bouzeffour

###### Abstract.

This paper presents a novel approach for evaluating analytical eigenfunctions
of the finite Hartley transform. The approach is based on the use of
$N=1/2$-supersymmetric quantum mechanics as a fundamental tool, which builds
on the key observation that the Hartley transform commutes with the
supercharge operator. Using the intertwining operator between the Hartley
transform and the finite Hartley transform, our approach provides an
overcomplete basis of eigenvectors expressed in terms of supersymmetric
Hermite polynomials.
2000MSC: 33D45, 42A38

##  1\. Introduction

The finite Fourier transform $\Phi^{(N)}$ maps a function $f$ on
$\mathbb{Z}/N\mathbb{Z}$ to its Fourier coefficients, which are given by the
formula [15]

(1.1) $(\Phi^{(N)}f)(r)=\frac{1}{\sqrt{N}}\sum_{s=0}^{N-1}\exp\left(\frac{2\pi
i}{N}rs\right)f(s),\qquad r\in\mathbb{Z}/N\mathbb{Z}.$

Equivalently, if we identify the $N$-periodic function $f$ on $\mathbb{Z}$
with the vector

$\big{(}f(0),\,f(1),\,\dots,f(N-1)\big{)}$

in $\mathbb{C}^{N}$ , then $\Phi^{(N)}$ is a unitary operator on
$\mathbb{C}^{N}$ with matrix elements $\Phi_{rs}^{(N)}$ given by

$\Phi_{rs}^{(N)}=\frac{1}{\sqrt{N}}\exp(\frac{2\pi i}{N}rs),\quad
r,\,s=0,\,1,\,\dots,\,N-1.$

In his seminal paper [13], Mehta investigated the eigenvalue problem of the
finite Fourier transform $\Phi^{(N)}$ and found a set of eigenvectors
$F_{n}^{(N)}$ analytically. These eigenvectors satisfy the following
eigenvalue problem:

(1.2) $\sum_{s=0}^{N-1}\Phi_{rs}^{(N)}F_{n}^{(N)}(s)={\rm
i}^{n}F_{n}^{(N)}(r)\quad r=0,1,\ldots,N-1,$

where $F_{n}^{(N)}$ are given by the expression

(1.3)
$F_{n}^{(N)}(r):=\sum_{k=-\infty}^{\infty}e^{-\frac{\pi}{N}\left(kN+r\right)^{2}}H_{n}\Big{(}\sqrt{\tfrac{2\pi}{N}}(kN+r)\Big{)},\quad
r=0,1,\ldots,N-1,$

Here, $H_{n}\left(x\right)$ denotes the Hermite polynomial of degree $n$ in
$x$.
It was conjectured by Mehta [13] that the $F_{n}^{(N)}$ ($n=0,1,\ldots,N-1$ if
$N$ is odd and $n=0,1,\ldots,N-2,N$ if $N$ is even) are linearly independent,
but this problem has not been any further considered until now. As shown by
Ruzzi [19], these systems are in general not orthogonal. While the literature
has extensively discussed eigenvectors of the finite Fourier transform, with
various approaches and techniques proposed (see, for example [13, 14, 2, 3, 4]
and references therein), the analytical derivation of eigenvectors for the
finite Fourier transform remains an unsolved problem.
Recently, Zhedanov studied some algebraic properties of two finite-dimensional
intertwining operators associated with the DFT matrix. These operators are
represented by matrices $X$ and $Y$ of the same size, $N\times N$, such that
the intertwining relations are given by

(1.4) $Y\,\Phi=\Phi\,X,\qquad X\,\Phi=-\,\Phi\,Y.$

######  Proposition 1.1 (Zhedanov).

The operators $X$ and $Y$ provide a representation of the Askey-Wilson algebra
with commutation relations

$\displaystyle
X^{2}Y+YX^{2}-\left(q+q^{-1}\right)XYX=-\left(q-q^{-1}\right)^{2}Y,$
$\displaystyle
Y^{2}X+XY^{2}-\left(q+q^{-1}\right)YXY=-\left(q-q^{-1}\right)^{2}X.$

In this paper, we consider the finite Hartley transform $\mathcal{H}^{(N)}$, a
closed discrete transform that maps a function $f$ on $\mathbb{Z}/N\mathbb{Z}$
to its Hartley coefficients. The Hartley coefficients are given by the formula

(1.5)
$(\mathcal{H}^{(N)}f)(r)=\frac{1}{\sqrt{N}}\sum_{s=0}^{N-1}\big{(}\cos\left(\frac{2\pi
rs}{N}\right)-\sin\left(\frac{2\pi rs}{N}\right)\big{)}f(s),\qquad
r\in\mathbb{Z}/N\mathbb{Z}.$

The finite Hartley transform shares many important properties with the finite
Fourier transform, including linearity, invertibility, and Parseval’s
identity. These transforms are widely used in various areas of mathematics,
physics, and engineering, such as signal processing, data analysis, and number
theory. Compared to the finite Fourier transform, the finite Hartley transform
has some unique advantages, such as being real-valued and symmetric, which
makes it particularly useful for certain applications, such as image
processing. However, the finite Fourier transform is more commonly used and
has a richer theory and wider range of applications.
More specifically, we are concerned with the analytic solutions of following
eigenvalue problem:

(1.6)
$\sum_{s=0}^{N-1}\mathcal{H}_{rs}^{(N)}F_{n}^{(N)}(s)=\lambda_{n}F_{n}^{(N)}(r),\quad
r=0,1,\ldots,N-1,$

where the matrix elements $\mathcal{H}_{rs}^{(N)}$ are given by

$\mathcal{H}_{rs}^{(N)}=\frac{1}{\sqrt{N}}\big{(}\cos\left(\frac{2\pi
rs}{N}\right)-\sin\left(\frac{2\pi rs}{N}\right)\big{)}.$

The finite Hartley transform can be considered as a discrete analogue of the
well-known continuous Hartley, which is defined by

$(\mathscr{H}\psi)(\lambda)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\psi(x)\big{(}\cos(\lambda
x)-\sin(\lambda x)\big{)}\,dx.$

In this work, we present a method for constructing families of polynomials
with transformation properties similar to the integral Hartley transform. Our
approach is based on a crucial observation that the Hartley transform commutes
with the supercharge operator of $N=1/2$-supersymmetric quantum mechanics,
which was previously studied in [18]. Readers interested in further
information on supersymmetric quantum mechanics can refer to [16, 17, 18, 8].
Using this property, we construct eigenfunctions of the finite Hartley
transform expressed in terms of supersymmetric Hermite polynomials [18], using
the same method as earlier work by Mehta [13], and further developed by
Dahlquist [9], Matveev [14], and Atakishiyeva et al. [1, 15].
The paper is structured as follows. In Section 2, we provide an overview of
the Hartley transform and difference-differential operator, and recall general
properties of supersymmetric quantum mechanics with reflection. We establish a
connection between the Hartley transform and SUSY quantum mechanics, and use
it to obtain overcomplete bases for Hartley transform eigenvectors.
In Section 3, we leverage the idea presented by Matveev [14, Theorem 4.1] to
construct functions that exhibit favorable properties under the finite Hartley
transform. Specifically, we use these functions and their Hartley images to
obtain eigenfunctions of the finite Hartley transform.
Finally, in Section 4, we conclude the paper with a brief discussion of some
further research directions of interest.

##  2\. Hartley transform and Dunkl-SUSY Hermite polynomials

The Hartley transform is a linear operator defined on a suitable function
$\psi(x)$ as follows:

$(\mathscr{H}\psi)(\lambda)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\psi(x)\text{cas}(\lambda
x)\,dx,$

where the kernel of the integral, known as the cas function, is given by

(2.1)
$\text{cas}(x)=\cos(x)-\sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^{\binom{n}{2}}}{n!}x^{n},$

with $\binom{n}{2}=\frac{n(n-1)}{2}$ being the binomial coefficient.
The cas function (2.1) can be regarded as a generalization of the exponential
function $\exp$. This can be seen by considering a simple computation that
shows the cas function is the unique $C^{\infty}$-solution to the
differential-reflection problem.

(2.2) $\left\\{\begin{array}[]{l l}(\partial_{x}R)u(x)=\lambda u(x),\\\
u(0)=0.\end{array}\right.$

Here $\partial_{x}$ is the first order derivative and $R$ is the reflection
operator acting on the functions $f(x)$ as:

$(Rf)(x):=f(-x).$

Mapping and inversion properties of the Hartley transforms in
$L^{2}(\mathbb{R},dx)$, in [20] In order to find explicit eigenfunctions of
the Hartley transform, we can use the technique of finding a self-adjoint
operator with a simple spectrum that commutes with the Hartley transform.
To do so, let us start with the first order difference-differential operator
$\partial_{x}R$. By performing integration by parts, we get

$\displaystyle\int_{\mathbb{R}}(\partial_{x}Rf)(x)g(x)\,dx$
$\displaystyle=-\int_{\mathbb{R}}f(x)(R\partial_{x}g)(x)\,dx$
$\displaystyle=\int_{\mathbb{R}}f(x)(\partial_{x}R)g(x)\,dx,$

where $f,\,g\in S(\mathbb{R})$( $S(\mathbb{R})$ is the Schwartz space ).
Consequently, the Hartley transform satisfies the following intertwining
relations:

(2.3)
$\mathscr{H}\partial_{x}R=x\mathscr{H},\quad\mathscr{H}x=\partial_{x}R\mathscr{H}.$

Now, we define the following Dunkl-type operator $Q,$ which acts on
$S(\mathbb{R})$ as:

(2.4) $Q=\frac{1}{\sqrt{2}}(\partial_{x}R+x).$

Combining the two previous intertwining relations, we can easily see that the
Hartley transform commutes with the operator $Q$, that is

$Q\mathscr{H}=\mathscr{H}Q.$

Before going further into the physical interpretation of the operators $Q$,
let us first briefly summarize the main ideas of supersymmetric quantum
mechanics. Supersymmetric quantum mechanics is a framework that extends the
usual quantum mechanics to include both bosonic and fermionic degrees of
freedom. In this framework, a Hamiltonian $H$ is said to be supersymmetric if
there exist supercharges $Q$ and $Q^{\dagger}$ that satisfy the following
superalgebra relations hold:

(2.5) $[Q,H]=0,\quad[Q^{\dagger},H]=0,\quad H=\\{Q,Q^{\dagger}\\},$

where $[\cdot,\cdot]$ and $\\{\cdot,\cdot\\}$ denote the commutator and
anticommutator, respectively. The supercharges $Q$ and $Q^{\dagger}$ are
Hermitian conjugates of each other and act on the same Hilbert space as $H$.
The physical interpretation of the operators $Q$ and $Q^{\dagger}$ is that
they correspond to transformations that mix bosonic and fermionic degrees of
freedom. These transformations are useful in studying a wide range of physical
systems, including high-energy physics, condensed matter physics, and
statistical mechanics.
Note that, if the supercharge $Q$ is self-adjoint, i.e., $Q=Q^{\dagger}$, then
it follows from the above relations that

$H=2Q.$

In this case the system is said to be $N=1/2$ supersymmetric, since the
operator $Q$ transforms bosonic states into fermionic states and vice versa.
One-dimensional $N=1/2$ supersymmetric quantum mechanics can be realized by
taking the supercharge $Q$ defined in (2.4). The corresponding supersymmetric
Hamiltonian $H$ takes the form

(2.6)
$H=Q^{2}=-\frac{1}{2}\frac{d^{2}}{dx^{2}}-\frac{1}{2}\,x^{2}-\frac{1}{2}R.$

The potential term $\frac{1}{2}x^{2}$ comes from the harmonic oscillator
potential and the superpotential term $-\frac{1}{2}R$ arises from the presence
of reflection symmetry. The spectrum of $H$ is easily obtained by observing
that

(2.7) $H=H_{0}-\frac{1}{2}R,$

where $H_{0}$ is the one-dimensional harmonic oscillator given by

$H_{0}=-\frac{1}{2}\frac{d^{2}}{dx^{2}}-\frac{1}{2}\,x^{2}.$

It is well-known that the harmonic oscillator $H_{0}$ (or its self-adjoint
extension) possesses a discrete spectrum, and its corresponding eigenfunctions
are the Hermite functions. The eigenfunctions, denoted by $\psi_{n}(x)$, have
eigenvalues $\lambda_{n}=n+\frac{1}{2}$ and are expressed as:

(2.8)
$\psi_{n}(x)=\frac{1}{\sqrt{2^{n}n!\sqrt{\pi}}}e^{-\frac{1}{2}x^{2}}H_{n}(x),\quad
n\in\mathbb{N}_{0},$

where $H_{n}(x)$ denotes the $n$th Hermite polynomial.
Since

$R\psi_{n}(x)=(-1)^{n}\psi_{n}(x),$

it follows from (2.7) that

(2.9) $\displaystyle H\psi_{n}(x)=(n+\frac{1-(-1)^{n}}{2})\psi_{n}(x).$

Then the ground state wave function $\psi_{0}(x)$ is given by

$\psi_{0}(x)=\pi^{-1/4}e^{-x^{2}/2}$

and satisfies

$Q\psi_{0}=0.$

Let us now carry out the gauge transformation of $Q$ with the ground state
$\psi_{0}$. Put

$\widetilde{Q}=\psi_{0}^{-1}Q\psi_{0}.$

It is straightforward to see that

$\widetilde{Q}=\frac{1}{\sqrt{2}}\frac{d}{dx}R+\frac{1}{\sqrt{2}}x(1-R).$

So, in this way, one reduces the problem of funding the eigenfunctions of the
Hartley transform to one of solving the following difference-differential
equation

(2.10) $-u^{\prime}(-x)+x\big{(}u(x)-u(-x)\big{)}=\sqrt{2}\lambda u(x).$

To solve this equation, it is crucial to note that any function $u$ can be
expressed as the sum of an even function $u_{e}$ and an odd function $u_{o}$
as follows:

$\displaystyle u_{e}(x)$ $\displaystyle=\frac{1}{2}(u(x)+u(-x)),$
$\displaystyle u_{o}(x)$ $\displaystyle=\frac{1}{2}(u(x)-u(-x)).$

Here, $u_{e}$ and $u_{o}$ are respectively the even and odd parts of $u$. We
can rewrite the difference-differential equation (2.10) as a system of two
linear differential equations of first order:

(2.11) $\left\\{\begin{array}[]{l l}u^{\prime}_{e}=\lambda\sqrt{2}u_{o}\\\
u^{\prime}_{o}-2xu_{o}=-\lambda\sqrt{2}u_{e}.\end{array}\right.$

We can eliminate the function $u_{o}(x)$ from this system (2.11) and obtain a
second-order differential equation for $u_{e}(x)$:

(2.12) $u^{\prime\prime}_{e}(x)-2xu^{\prime}_{e}(x)=-2\lambda^{2}u_{e}(x).$

We make the change of variable $t=x^{2}$ and equation (2.12) becomes:

$tv^{\prime\prime}+(\frac{1}{2}-t)v^{\prime}=-\frac{\lambda^{2}}{2}v.$

Then the general even solution of (2.12) takes the form

$\displaystyle u_{e}(x)=A\ _{1}F_{1}\left(\begin{matrix}\
-\frac{\lambda^{2}}{2}\\\ \frac{1}{2}\end{matrix};x^{2}\right),$

where $A$ is constant and

$\ {}_{1}F_{1}\left(\begin{matrix}\ a\\\
b\end{matrix};z\right)=\sum_{n=0}^{\infty}\frac{(a)_{n}}{(b)_{n}}\frac{z^{n}}{n!}$

is Kummer’s confluent hypergeometric function [21]. From (2.11) we obtain the
solution for the function $u_{o}(x)$

$\displaystyle u_{o}(x)=-A\sqrt{2}\lambda x\ _{1}F_{1}\left(\begin{matrix}\
1-\frac{\lambda^{2}}{2}\\\ \frac{3}{2}\end{matrix};x^{2}\right).$

Therefore the general solution of (2.10)

$\displaystyle u(x)=A\ _{1}F_{1}\left(\begin{matrix}\
-\frac{\lambda^{2}}{2}\\\
\frac{1}{2}\end{matrix};x^{2}\right)-A\sqrt{2}\lambda x\
_{1}F_{1}\left(\begin{matrix}\ 1-\frac{\lambda^{2}}{2}\\\
\frac{3}{2}\end{matrix};x^{2}\right).$

It is straightforward to observe that polynomial solutions are only possible
if $\lambda=\pm\sqrt{2n},$ where $n=0,,1,,2,,\dots.$ In such cases, the first
term in the above equation is a polynomial of degree $2n$ and the second term
is a polynomial of degree $2n-1$. We denote the eigenfunction of $Q$
corresponding to the eigenvalue $\lambda_{n}=\pm\sqrt{2n}$ as
$\widehat{\psi}(x)$.
By using the well known explicit expressions of the Hermite polynomials in
terms of the confluent hypergeometric series:

$\displaystyle H_{2n}(x)$ $\displaystyle=(-1)^{n}{\frac{(2n)!}{n!}}\,\
_{1}F_{1}\left(\begin{matrix}\ -n\\\ \frac{1}{2}\end{matrix};x^{2}\right),$
$\displaystyle H_{2n+1}(x)$ $\displaystyle=(-1)^{n}{\frac{(2n+1)!}{n!}}\,2x\,\
_{1}F_{1}\left(\begin{matrix}\ -n\\\ \frac{3}{2}\end{matrix};x^{2}\right),$

we obtain the explicit expressions for these eigenfunctions given below.

(2.13) $\displaystyle\widehat{\psi}(x)=\kappa_{\pm
n}^{-1}e^{-x^{2}/2}\mathcal{H}_{\pm n}(x),\quad n\in\mathbb{N}_{0},$

where

(2.14) $\mathcal{H}_{\pm n}(x)=H_{2n}(x)\pm 2\sqrt{n}H_{2n-1}(x),\quad
n\in\mathbb{N}_{0}.$

The normalized constants $\kappa_{\pm n}$ are also chosen so that

(2.15) $\int_{\mathbb{R}}\widehat{\psi}^{2}_{\pm n}(x)\,dx=1.$

A simple computation shows that

(2.16) $\kappa_{n}=\kappa_{-n}=\pi^{\frac{1}{4}}2^{n+1/2}(2n!)^{1/2},\quad
n=0,\,1,\,2,\,\dots.$

The polynomials $\mathcal{H}_{n}(x)$ with $n\in\mathbb{Z}$ are called
supersymmetric Hermite polynomials. They satisfy the orthogonality relations

(2.17) $\int_{\mathbb{R}}\mathcal{H}_{\pm n}(x)\mathcal{H}_{\pm
m}(x)e^{-x^{2}}\,dx=\sqrt{\pi}2^{2n+1}(2n)!\,\delta_{nm},\quad
n,\,m\,\in\mathbb{N}_{0}.$

The system $\\{\widehat{\psi}_{n}(x)\\}_{n\in\mathbb{Z}}$ forms an orthonormal
set in $L^{2}(\mathbb{R},dx)$ and is complete, similar to the argument used to
show that the classical Hermite functions form a complete orthogonal set in
$L^{2}(\mathbb{R},dx)$. Moreover, the operator $Q$ is essentially self-
adjoint, and we have

(2.18) $Q\widehat{\psi}_{\pm n}(x)=\pm\sqrt{2n}\widehat{\psi}_{\pm
n}(x),\,n=0,\,1,\,2,\,\dots$

######  Theorem 2.1.

The eigenfunctions of the Hartley transform are given by
$\\{e^{-x^{2}/2}\mathcal{H}_{\pm n}(x)\\}_{n\in\mathbb{Z}}$. Moreover, we have

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\text{cas}(xy)e^{-y^{2}/2}\mathcal{H}_{n}(y)\,dy=(-1)^{n}\,e^{-x^{2}/2}\mathcal{H}_{n}(x),$

for all $n\in\mathbb{Z}$, where $\mathcal{H}_{n}(x)$ is the supersymmetric
Hermite polynomial of order $n$ given by

$\mathcal{H}_{\pm n}(x)=H_{2n}(x)\pm 2\sqrt{n}H_{2n-1}(x),\quad
n\in\mathbb{N}_{0}.$

##  3\. Metha’s eigenfunctions for the finite Hartley transform

Mehta’s eigenfunctions for the finite Hartley transform arise similarly as the
general relationship between the integral Fourier transform and the finite
Fourier transform and can be derived as an immediate consequence of a
generalized Poisson summation formula for the finite Hartley transform.
The Hartley series of a periodic real-valued function $f$ with period $a>0$ is
given by [11, §2]

(3.1) $f(x)=\sum_{n\in\mathbb{Z}}a_{n}cas(\frac{2\pi n}{a}x),$

where the coefficients $a_{n}$ are given explicitly by

(3.2) $a_{n}=\frac{1}{a}\int_{-a/2}^{a/2}f(x)\,cas(\frac{2\pi n}{a}x)\,dx.$

In order to proceed further, we will require the following generalized Poisson
summation formula for the Hartley transform.

######  Theorem 3.1.

Let $f\in S(\mathbb{R})$, we have

(3.3)
$ab\sum_{r\in\mathbb{Z}}f(b(ar+x))=\sqrt{2\pi}\sum_{m\in\mathbb{Z}}(\mathscr{H}f)\big{(}\frac{2m\pi}{ab}\big{)}\,cas(\frac{2m\pi
x}{a}).$

###### Proof.

The left-hand side of the equation is a periodic function with a period $a$,
while the right-hand side is its expression as the sum of its Hartley series.
∎

In particular for

$\displaystyle a=n,\quad x=j,\quad j=0,\,\dots,n-1,$ $\displaystyle
m=nr+k,\quad b=\sqrt{2\pi/N},\quad r\in\mathbb{Z},\quad k=0,\dots,\,n-1,$

the formula (3.3) takes the form

(3.4)
$\sum_{r\in\mathbb{Z}}f(b(ar+x))=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}\Big{\\{}\sum_{r\in\mathbb{Z}}(\mathscr{H}f)\big{(}\sqrt{\frac{2\pi}{N}}(Nr+k)\big{)}\Big{\\}}\,cas(\frac{2m\pi
x}{N}).$

We define a linear map $\mathscr{M}_{N}$ from the Schwartz space
$S(\mathbb{R})$ into the space of $N$-periodic functions on $\mathbb{Z}$ as
follows: for $f\in S(\mathbb{R})$ and $r\in\mathbb{Z}$,

(3.5)
$(\mathscr{M}_{N}f)(r)=\sum_{k=-\infty}^{\infty}\,f\Big{(}\sqrt{\tfrac{2\pi}{N}}\,(kN+r)\Big{)}.$

The finite Hartley transform pair for a $N$-periodic function $f$, is given by

(3.6) $\left\\{\begin{array}[]{l
l}(\mathcal{H}^{(N)})f(r):=N^{-1/2}\sum_{s=0}^{N-1}f(s)\,cas(\frac{2irs\pi}{N}),\\\
\\\
f(s):=N^{-1/2}\sum_{r=0}^{N-1}(\mathcal{H}^{(N)}f)(r)cas(\frac{2irs\pi}{N}).\end{array}\right.$

######  Theorem 3.2.

If $f,g\in S(\mathbb{R})$ are related by the integral Hartely transform

(3.7)
$g(y)\,=\,\frac{1}{\sqrt{2\pi}}\,\int_{\mathbb{R}}\,cas(xy)\,f(x)\,dx\,,$

and if $F:=\mathscr{M}_{N}f$, $G:=\mathscr{M}_{N}g$ then $F$ and $G$ are
related by the finite Hartely transform:

$G(r)=\frac{1}{\sqrt{N}}\sum_{s=0}^{N-1}cas(\frac{2irs\pi}{N})\,F(s).$

Let us now apply Theorem 3.2 to the case where $f$ and $g$ are given by :

$f(x):=e^{-x^{2}/2}\,\mathcal{H}_{n}(x),\qquad
g(y):=e^{-y^{2}/2}\,\mathcal{H}_{n}(y).$

Put

(3.8)
$g_{n}^{(N)}(r):=(\mathscr{M}_{N}f)(r)=\sum_{k=-\infty}^{\infty}\,e^{-\frac{\pi}{N}(kN+r)^{2}}\mathcal{H}_{n}\Big{(}\sqrt{\tfrac{2\pi}{N}}\,(kN+r)\Big{)}.$

######  Proposition 3.3.

The finite Hartley transform of the functions $g_{n}^{(N)}(r)$, is given by

(3.9) $(\mathcal{H}^{(N)}g_{n}^{(N)})(r)=(-1)^{n}\,g_{n}^{(N)}(r).$

##  4\. Concluding comments and outlook

Our work has demonstrated the connection between the supersymmetric Hermite
polynomials and the Hartley transform. Building upon earlier research by Mehta
[13], Dahlquist [9], Matveev [14], and Atakishiyeva et al. [1, 15], we have
employed a similar method to construct eigenfunctions of the finite Hartley
transform. Notably, in [15], the authors used a related technique to construct
$q$-extended eigenvectors of the finite Fourier transform, which were then
utilized to obtain eigenvectors of the finite Hartley transform.

An intriguing avenue for further exploration is the investigation of families
of $q$-polynomials from the Askey hierarchy of basic hypergeometric
polynomials [21] and their potential for possessing simple transformation
properties with respect to the finite Hartley transform. If such properties
exist, they could lead to novel $q$-variants of Mehta’s eigenfunctions (3.8)
of the finite Hartley transform.

Additionally, an area worth exploring is the study of symmetry properties
exhibited by finite-dimensional intertwining operators with the discrete
Hartley transform matrix, as discussed in [13, 14, 2, 3, 4, 5]. Another
interesting direction would involve investigating a discrete analogue of
$N=1/2$ supersymmetric quantum mechanics.

We plan to continue our research in both of these directions, with a
particular focus on constructing an explicit finite-difference equation for
the suitable eigenvectors (3.8) of the finite Hartley transform operator
$\mathcal{H}^{(N)}$, derived from supersymmetric quantum mechanics.
Analytically, this involves constructing a finite-difference operator $L$ that
commutes with $\mathcal{H}^{(N)}$, thereby enabling the decomposition of
eigenspaces of $\mathcal{H}^{(N)}$ through the spectral decomposition of $L$.

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> Fethi Bouzeffour
> Departemnt of Mathematics, King Saud University, P. O Box $2455$ Riyadh
> $11451$, Saud Arabia.
> email<EMAIL_ADDRESS>


# CEPC Research Report: Higgs Physics Analysis

Manqi Ruan  Institute of High Energy Physics, Chinese Academy of Science,
Beijing, China  Yaquan Fang  Institute of High Energy Physics, Chinese Academy
of Science, Beijing, China  Gang Li  Institute of High Energy Physics, Chinese
Academy of Science, Beijing, China  Dan Yu  Institute of High Energy Physics,
Chinese Academy of Science, Beijing, China

###### Abstract

In November 2018, the CEPC released the Conceptual Design Report (CDR) with
the physics potential investigation and the analysis about different detector
concepts. Since then, intensive researches on the Higgs Physics at the CEPC
have been progressed. The run at top thresholds are also taken into
consideration and the physics potential around the center-of-mass energy of
360GeV is studied. The $ttH$ channel information boost the precision of Higgs
width measurement by a factor of 2 with respect to 240 GeV Higgs Runs.
Different Higgs decay modes are analyzed in the CEPC Higgs factory at the
center-of-mass energy of 240GeV, including Higgs to b/c/g and Higgs to
$\tau\tau$, which improves the Higgs measurement. Advanced analysis
technologies are developed and applied in the analysis, such as jet
reconstruction and lepton identification in jets. New interpretations are
under developing to improve the performance.

The Circular Electron Positron Collider (CEPC) was proposed after the Higgs
boson discovery at the Large Hadron Collider (LHC). Compared with the LHC,
electron-positron colliders have significant advantages for the Higgs boson
property measurements. The Higgs boson signal is largely free of QCD
backgrounds and the signal to background ratio is significantly higher.
Moreover, Higgs boson candidates can be identified through the recoil mass
method without tagging the Higgs boson decay products, allowing for the
measurements of the Higgs boson width and couplings in a model-independent
way. With about one million Higgs bosons produced at the CEPC, many of the
major Higgs boson couplings can be measured with precisions about one order of
magnitude better than those achievable at the High Luminosity-LHC.

The CEPC Conceptual Design Report (CDR) was released[1] in November 2018. In
the CEPC-CDR, different detector concepts have been proposed. These concepts
can fulfill the physics requirements, confirmed by the dedicated simulation
and R&D program. The joint theory, detector design, and simulation studies
have given a clear demonstration of the CEPC physics capabilities: the CEPC
has an immense potential towards the precision Higgs and EW measurements, and
can provide complementary information in flavor physics.

More Higgs studies have been progressed after the CEPC-CDR delivery, including
run at top thresholds, differential measurements, new analysis technologies,
and new interpretations.

##  1 Run at top thresholds

The $ttH$ channel information boost the precision of Higgs width measurement
by a factor of 2 with respect to 240 GeV Higgs Runs[2]. The Table 1 shows the
combination of CEPC Higgs $\sigma\times Br$ and coupling measurement
precision.

Table 1: CEPC Higgs measurement precision | 240GeV, 5.6$ab^{-1}$ | 360GeV, 2$ab^{-1}$
---|---|---
| ZH | ZH |  $\nu\nu$H
any | 0.50% | 1% | -
$H\to bb$ | 0.27% | 0.63% | 0.76%
$H\to cc$ | 3.3% | 6.2% | 11%
$H\to gg$ | 1.3% | 2.4% | 3.2%
$H\to WW$ | 1.0% | 2.0% | 3.1%
$H\to ZZ$ | 7.9% | 14% | 15%
$H\to\tau\tau$ | 0.8% | 1.5% | 3%
$H\to\gamma\gamma$ | 5.7% | 8% | 11%
$H\to\mu\mu$ | 12% | 29% | 40%
$Br_{upper}$($H\to inv.$) | 0.2% | - | -
$\sigma(ZH)\times BR(H\to Z\gamma)$ | 16% | 25% | -
width | 2.9% | 1.4%

The couplings of Higgs to top quark are the key driver of the hierarchy
problem. The CEPC sensitivity to Higgs and top Effective Field Theory (EFT) is
analyzed using the minimal Higgs Top anomalous coupling EFT set[3]. The $tt$
threshold scan in CEPC can measure the top mass at a precision 1 order of
magnitude better than hadron colliders, estimated by constructing 1-dimension
likelihood combining the statistical power of all scan points[4]. Besides, it
provides the simultaneous measurement of top width, top Yukawa coupling and
the $\alpha_{s}$. A few proposals of luminosities and energies are also
tested.

##  2 The di-Higgs related physics

The studying of different two Higgs doublet model (2HDM) illustrates that the
future precision measurements of the Standard Model (SM) observables at the
proposed Z-factories and Higgs factories may have impacts on new physics
beyond the Standard Model. The achievable sensitivity of the CEPC to probe the
virtual effects of the 2HDM, including Type-II[5], Type-I[6], Type-L (lepton
specific) and Type-F (flipped Yukawa couplings)[7], has been examined. The
regions outside the contour lines in Figure 1 are accessible with 5$\sigma$
sensitivity. The discovery potential is explored based on the hypothetical
deviations in the precision data for the 2HDMs up to one-loop level. A large
part of the parameter space of the other types of 2HDMs can be distinguished
from the benchmark points of the target model. The impacts of loop corrections
are found to be significant in certain parameter regions.

Figure 1: 5$\sigma$ discovery regions outside the contour lines in the
$\cos(\beta-\alpha)$-$\tan\beta$ plane for CEPC at tree level (dashed) and
one-loop (solid). Red, green, blue and orange colors indicate the results of
Type-I, II, L and F, respectively. We choose $m_{H}=m_{A}=m_{H^{\pm}}=800GeV$
and $\sqrt{\lambda\nu^{2}}=300GeV$ for one-loop curves.

##  3 The measurement of the Higgs boson decaying to fermions

With the combination of detector hardwares and multivariable analysis
algorithms, the measurement efficiency of charm quark and bottom quark jet is
improved. Therefore, more effective evidence of the coupling between Higgs
boson and the first two generations of quarks can be found, and the coupling
strength between Higgs and the third generation heavy quarks can be measured
more accurately. The measurement of branching fraction of $H\to bb/cc/gg$ are
studied in $\mu\mu H$ and $eeH$ process[8], . The statistical uncertainty of
the signal cross section is estimated to be about 1% in the $H\to bb$ final
state, and approximately 5%-10% in the $H\to cc/gg$ final states. The
systematic uncertainties on the branching fraction measurements are also
studied, which are around 0.6%, 6% and 8% for $bb,cc$ and $gg$ final states
respectively. The high precision of this measurement benefits from the
distinct signature of events with the Higgs boson and clean background in
electron-positron collider, as well as the model independent analysis. This
study demonstrates the potential of precise measurement of the hadronic final
states of the Higgs boson decay at the CEPC, and will provide key information
to understand the Yukawa couplings between the Higgs boson and quarks, which
are predicted to be the origin of quarks’ masses in the standard model.

Figure 2: 3D-fit result projected on recoil mass distribution in $\mu\mu H$
(a) channel and $eeH$ channel (b), on B-likeness distribution in $\mu\mu H$
(c) channel and $eeH$ channel (d), and on C-likeness distribution in $\mu\mu
H$ (e) channel and $eeH$ channel (f)

##  4 Precise Higgs measurement in CEPC

By combining the analysis of different channels, the advantage of CEPC in
Higgs property measurement accuracy can be fully exploited, and the physics
potential can be studied quantitatively. The key physics measurement is
improved with innovative reconstruction methods and improved signal-background
discrimination. The Higgs invisible decay and $\tau$ decay, are studied and
corresponding articles have been published. Moreover, important studies
relevant to the Higgs are processed, including jet lepton identification, jet
reconstruction, and combined analysis technology.

###  4.1 $H\to invisible$ channel

The measurement potential of Higgs decay to an invisible channel on CEPC is
investigated[9]. In SM, the Higgs boson can only decay invisibly via $H\to
ZZ^{*}\to\nu\nu\nu\nu$ or dark matter, so any evidence of invisible Higgs
decay that exceeds BR ($H\to inv$.) will immediately point to a phenomenon
that is beyond the standard model (BSM). In this analysis, the upper limit of
BR ($H\to inv$) is estimated for three channels, including two leptonic
channels and one hadronic channel, under the assumption predicted by SM. With
the SM ZH production rate, the upper limit of BR ($H\to inv$) could reach
0.24% at the 95% confidence level, as shown in Figure3. The impact of the BMR
on the ZH ($Z\to qq,H\to inv$.) signal strength accuracy has been studied as
well, and the obtained curve shows that the analysis result would be one of
the indicators for the CEPC detector optimization study.

Figure 3: The likelihood profile of combination, where the green projective
line label out the location of 68%, 95% confidence level, which corresponds to
$-\Delta\log(L)=0.5,2onthey-axis$.

###  4.2 $H\to\tau\tau$ channel

In the $H\to\tau\tau$ analysis, the signal strength measurement is
studied[10]. Using the full simulation analysis, the CEPC is expected to
measure the signal strength to a relative accuracy of 0.8%. Multiple analysis
technologies and dedicated $\tau$ finding algorithms are developed. A precise
reconstruction of the impact parameter is essential for the $\tau$ events
identification, and the PFA oriented detector design and reconstruction are
critical for this analysis. The physics requirement on the mass resolution of
the Higgs boson with hadronic decay final states is also discussed, as shown
in Figure 4. It shows that the CEPC baseline design and reconstruction fulfill
the accuracy requirement of the $H\to\tau\tau$ signal strength.

Figure 4: The dependence of the accuracy of the qqH channel on the boson mass
resolution. The thicker red line indicates that the CEPC baseline BMR is 3.8%,
the other two red lines show the accuracy when the BMR drops to 6% and 8%.
(Signal: $ZH\to qq\tau\tau$, Background: $ZZ\to qq\tau\tau$,)

###  4.3 Jet reconstruction

Jet reconstruction is critical for the precision measurement of Higgs boson
properties and the electroweak observables at the CEPC[11]. The jet energy and
angular responses of benchmark 2- and 4-jet processes are analyzed with fully
simulated samples with the CEPC baseline detector geometry. As shown in Figure
5, the relative resolution of 3.5% and 1% are observed on the jet energy and
angular measurement for jets in the detector barrel region
($\left|\cos\theta\right|<0.6$) with energy greater than 60 GeV. Meanwhile,
the jet energy/angular scale can be controlled within 0.5/0.01%. The
differential dependences of the jet response on the jet direction and energy
are extracted. The impact on the jet responses induced by different jet
clustering algorithms and matching criteria are also analyzed, which yields a
relative difference of up to 8%.

Figure 5: (a) is the JER as a function of the GenJet energy of light-flavored
(uds) jets from semi-leptonic decay are divided into five fiducial regions.
(b-c) is the polar and azimuth angular resolutions as a function of the GenJet
$\cos\theta$ in several benchmark 2-jet events.

###  4.4 Jet lepton identification

Identifying the leptons inside jets is critical for the measurements of Higgs
boson and the flavor physics program at the CEPC[12]. Using the CEPC baseline
software, the identification performance of leptons generated inside a jet is
analyzed. The jet leptons are identified with typical efficiency and mis-
identification rate of 98% and 1% for energy higher than 2 GeV, with the $Z\to
bb$ process at 91.2 GeV center-of-mass energy. At the benchmark channel of the
CEPC flavor program of $B_{c}\to\tau\nu$ with $\tau\to e\nu\nu$, the electrons
are identified with inclusive efficiency times purity of 97%, providing
sufficient signal selection for the physics potential study. Compared to the
single leptons, these jet leptons identification efficiency degrades about
1-3%, and the mis-identification rate increases by less than 1% at the same
working point, as shown in Figure 6. The dependence between the performance of
the shower reconstruction and the lepton identification is quantified.

Figure 6: The efficiency and purity of electrons and muons in jets, comparing
with single particle case.

###  4.5 Global analysis

A global analysis is proposed to improve the precision of a set of branching
fractions of same resonance significantly, by taking all the cross talks among
all processes into account and benefiting from statistical character of the
multinomial distribution or binnomial distribution in the simplest case. It
was first used in $\tau$ decay branching fractions by the ALEPH collaboration
[13] and a good demonstration can also be found in [14] by the BESIII
collaboration. Usually only one signal and its background processes are
studied in a branching fraction measurement, which is a typical classification
problem of signal and background processes. If more than one decay modes from
same resonance are measured simultaneously, taking the Higgs decay as an
example, therefore two types of ”backgrounds” must be studied, one is the
backgrounds from other Standard Model (SM) processes, the other is the
contamination among all the decay modes of the Higgs (cross talks).

The main ingredients of the global analysis are determination of the
efficiency matrices, matrix manipulation, and minimization of $\chi^{2}$. The
first one could be accomplished with ML technique elegantly [15, 16]. The last
two are easily to be realized with some mature and widely used software tools.

The improvement of the precision of the Higgs decay branching fractions
brought by the global analysis approach could be explained with a few points.
First, the global analysis makes use of the full covariance of all counts,
which means more information is used to determine the fitting parameters.
Second, it deploys the law of multinomial distribution in the analysis.
Therefore, the statistical uncertainties are significantly smaller than those
of individual analysis, because they are calculated with produced numbers of
signals according to the multinomial distribution and then are transformed
into the ones of observed signals. Third, the global analysis introduces an
unitary constraint of the total branching fractions. In conclusion, the global
analysis approach adapts more information from data, statistical principle,
and prior of physics.

The preliminary results based on fast simulation show that the precision could
be improved by a factor of 2 in average [17].

##  5 Conclusion

The Higgs physics analysis at CEPC are progressing intensively. Different
reconstruction methods of physics objects are developed, and different
combinations of kinematic variables are applied in the analysis. More
international collaborations are involved in the CEPC R&D, and fruitful
achievements are expected.

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