# Derivations and linear functions
along rational functions

Eszter Gselmann  Institute of Mathematics
University of Debrecen
P. O. Box: 12.
Debrecen
H–4010
Hungary<EMAIL_ADDRESS>
###### Abstract.

The main purpose of this paper is to give characterization theorems on
derivations as well as on linear functions. Among others the following problem
will be investigated: Let $n\in\mathbb{Z}$, $f,g\colon\mathbb{R}\to\mathbb{R}$
be additive functions, $\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathbf{GL}_{2}(\mathbb{Q})$ be arbitrarily fixed, and
let us assume that the mapping

$\phi(x)=g\left(\frac{ax^{n}+b}{cx^{n}+d}\right)-\frac{x^{n-1}f(x)}{(cx^{n}+d)^{2}}\quad\left(x\in\mathbb{R},cx^{n}+d\neq
0\right)$

satisfies some regularity on its domain (e.g. (locally) boundedness,
continuity, measurability). Is it true that in this case the above functions
can be represented as a sum of a derivation and a linear function? Analogous
statements ensuring linearity will also be presented.

###### Key words and phrases:

derivation, linear function, polynomial function

###### 1991 Mathematics Subject Classification:

Primary 39B82; Secondary 39B72

This research has been supported by the Hungarian Scientific Research Fund
(OTKA) Grant NK 814 02 and by the TÁMOP 4.2.1./B-09/1/KONV-2010-0007 project
implemented through the New Hungary Development Plan co-financed by the
European Social Fund and the European Regional Development Fund.

##  1\. Introduction and preliminaries

Throughout this paper $\mathbb{N}$ denotes the set of the positive integers,
and $\mathbb{Z},\mathbb{Q}$, and $\mathbb{R}$ have the usual meaning.

The aim of this work is to prove characterization theorems on derivations as
well as on linear functions. Therefore, firstly we have to recall some
definitions and auxiliary results.

A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is called an _additive_
function if,

$f(x+y)=f(x)+f(y)$

holds for all $x,y\in\mathbb{R}$.

We say that an additive function $f:\mathbb{R}\rightarrow\mathbb{R}$ is a
_derivation_ if

$f(xy)=xf(y)+yf(x)$

is fulfilled for all $x,y\in\mathbb{R}$.

The additive function $f\colon\mathbb{R}\to\mathbb{R}$ is termed to be a
_linear function_ if $f$ is of the form

$f(x)=f(1)x\qquad\left(x\in\mathbb{R}\right).$

It is easy to see from the above definition that every derivation
$f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies equation

$f(x^{k})=kx^{k-1}f(x)\quad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right)$

for arbitrarily fixed $k\in\mathbb{Z}\setminus\left\\{0\right\\}$.
Furthermore, the converse is also true, in the following sense: if
$k\in\mathbb{Z}\setminus\left\\{0,1\right\\}$ is fixed and an additive
function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies $\left(\ast\right)$,
then $f$ is a derivation, see e.g., Jurkat [7], Kurepa [10], and
Kannappan–Kurepa [8].

Concerning linear functions, Jurkat [7] and, independently, Kurepa [10] proved
that every additive function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying

$f\left(\frac{1}{x}\right)=\frac{1}{x^{2}}f(x)\quad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right)$

has to be linear.

In [11] A. Nishiyama and S. Horinouchi investigated additive functions
$f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying the additional equation

$f(x^{n})=cx^{k}f(x^{m})\quad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right),$

where $c\in\mathbb{R}$ and $n,m,k\in\mathbb{Z}$ are arbitrarily fixed.

Furthermore, the above problem was generalized by Pl. Kannappan and S. Kurepa
in [8] by proving the following statement.

######  Theorem 1.

Let $f,g:\mathbb{R}\rightarrow\mathbb{R}$ be additive functions and
$n,m\in\mathbb{Z}\setminus\left\\{0\right\\}$, $n\neq m$. Suppose that

$f(x^{n})=x^{n-m}g(x^{m})$

holds for all $x\in\mathbb{R}\setminus\left\\{0\right\\}$. Then the functions
$F,G:\mathbb{R}\rightarrow\mathbb{R}$ defined by

$F(x)=f(x)-f(1)x\quad\text{and}\quad
G(x)=g(x)-g(1)x\quad\left(x\in\mathbb{R}\right)$

are derivations and $nF(x)=mG(x)$ is fulfilled for all $x\in\mathbb{R}$.

Here we remark that the ’approximate’ variant of the above theorem was dealt
with in [1].

Furthermore, in a series of papers (see [2, 3, 4, 5, 6]) F. Halter–Koch and L.
Reich proved several characterization theorems concerning derivations as well
as field homomorphisms.

Let $\mathbb{K}$ be a field containing $\mathbb{Q}$,
$n\in\mathbb{Z}\setminus\left\\{0\right\\}$, $\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathbf{GL}_{2}(\mathbb{Q})$ and let
$f,g\colon\mathbb{K}\to\mathbb{K}$ be additive function so that

$f\left(\frac{ax^{n}+b}{cx^{n}+d}\right)=\frac{x^{n-1}g(x)}{(cx^{n}+d)^{2}},$

respectively,

$f\left(\frac{ax^{n}+b}{cx^{n}+d}\right)=\frac{ag(x)^{n}+b}{cg(x)^{n}+d}$

holds for all possible values of $x$. In [4] it is proved that equation
$\left(\star\right)$ (under a mild condition) implies for the function $g$
that the function $G\colon\mathbb{K}\to\mathbb{K}$ defined by

$G(x)=g(x)-g(1)x\qquad\left(x\in\mathbb{K}\right)$

is a derivation. Furthermore, in [6] the authors succeed to prove that
equation $\left(\blacktriangle\right)$ furnishes that the mapping
$g(1)^{-1}\cdot g\colon\mathbb{K}\to\mathbb{K}$ is a field automorphism.

In this work we will extend the above mentioned results, and (among others)
the following problem will be dealt with. Let $n\in\mathbb{Z}$,
$f,g\colon\mathbb{R}\to\mathbb{R}$ be additive functions,
$\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathbf{GL}_{2}(\mathbb{Q})$ be arbitrarily fixed, and
let us assume that the mapping

$\phi(x)=g\left(\frac{ax^{n}+b}{cx^{n}+d}\right)-\frac{x^{n-1}f(x)}{(cx^{n}+d)^{2}}\quad\left(x\in\mathbb{R},cx^{n}+d\neq
0\right)$

satisfies some regularity on its domain (e.g. (locally) boundedness,
continuity, measurability). Is it true that in this case the above functions
can be represented as a sum of a derivation and a linear function?

During the second section the concept of multi-additive functions will be
used. Therefore, at this part we will list some definitions and statements
that we will use subsequently.

Let $G,H$ be abelian groups, let $h\in G$ be arbitrary and consider a function
$f:G\rightarrow H$. The _difference operator_ $\Delta_{h}$ with the span $h$
of the function $f$ is defined by

$\Delta_{h}f(x)=f(x+h)-f(x)\qquad\left(x\in G\right).$

The iterates $\Delta^{n}_{h}$ of $\Delta_{h}$, $n=0,1,\ldots$ are defined by
the recurrence

$\Delta^{0}_{h}f=f,\qquad\Delta^{n+1}_{h}f=\Delta_{h}\left(\Delta^{n}_{h}f\right)\qquad\left(n=0,1,\ldots\right)$

Furthermore, the superposition of several difference operators will be denoted
shortly

$\Delta_{h_{1}\ldots h_{n}}f=\Delta_{h_{1}}\ldots\Delta_{h_{n}}f,$

where $n\in\mathbb{N}$ and $h_{1},\ldots,h_{n}\in G$.

Let $n\in\mathbb{N}$ and $G,H$ be abelian groups. A function
$F:G^{n}\rightarrow H$ is called _$n$ –additive_ if, for every
$i\in\\{\,1\,,\,2\,,\,\dots\,,\,n\,\\}$ and for every
$x_{1},\ldots,x_{n},y_{i}\in G\,$,

$F\left(x_{1},\ldots,x_{i-1},x_{i}+y_{i},x_{i+1},\ldots,x_{n}\right)\\\
=F\left(x_{1},\ldots,x_{i-1},x_{i},x_{i+1},\ldots,x_{n}\right)+F\left(x_{1},\ldots,x_{i-1},y_{i},x_{i+1},\ldots,x_{n}\right),$

i.e., $F$ is additive in each of its variables $x_{i}\in G$, $i=1,\ldots,n$.
For the sake of brevity we use the notation $G^{0}=G$ and we call constant
functions from $G$ to $H$ $0$–additive functions. Let $F:G^{n}\rightarrow H$
be an arbitrary function. By the _diagonalization (or trace)_ of $F$ we
understand the function $f:G\rightarrow H$ arising from $F$ by putting all the
variables (from $G$) equal:

$f(x)=F(x,\ldots,x)\qquad\left(x\in G\right).$

It can be proved by induction that for any symmetric, $n$–additive function
$F:G^{n}\rightarrow H$ the equality

$\Delta_{y_{1},\ldots,y_{k}}f(x)=\left\\{\begin{array}[]{lcl}n!F(y_{1},\ldots,y_{n})&\text{for}&k=n\\\
0&\text{for}&k>n\end{array}\right.$

holds, whenever $x,y_{1},\ldots,y_{n}\in G$, where $f:G\rightarrow H$ denotes
the trace of the function $F$. This means that a symmetric, $n$–additive
function is uniquely determined by its trace.

The function $f:G\rightarrow H$ is called a _polynomial function_ of degree at
most $n$, where $n$ is a nonnegative integer, if

$\Delta_{y_{1},\ldots,y_{n+1}}f(x)=0$

is satisfied for all $x,y_{1},\ldots,y_{n+1}\in G$.

######  Theorem 2 (Kuczma [9], Székelyhidi [13]).

The function $p:G\rightarrow H$ is a polynomial at degree at most $n$ if and
only if there exist symmetric, $k$–additive functions $F_{k}:G^{k}\rightarrow
H$, $k=0,1,\ldots,n$ such that

$p(x)=\sum_{k=0}^{n}f_{k}(x)\qquad\left(x\in G\right),$

where $f_{k}$ denotes the trace of the function $F_{k}$, $k=0,1,\ldots,n$.
Furthermore, this expression for the function $p$ is unique in the sense that
the functions $F_{k}$, which are not identically zero, are uniquely
determined.

During the proof of our main theorem, the basic idea is to apply two
statements from [12, 13] concerning polynomial functions. Despite the fact
that the following results will be used only in case the topological group is
$\mathbb{R}$ and the topological linear space is also $\mathbb{R}$, here we
present the general case. The reason for it is, that the problem, we
investigate, can be formulated in more general circumstances. Perhaps in the
general case the two cited theorems of L. Székelyhidi would also play an
important role.

######  Theorem 3 (Székelyhidi [12, 13]).

Let $G$ be an abelian group and let $X$ be a locally convex topological linear
space. If a polynomial $p:G\rightarrow X$ is bounded on $G$, then it is
constant.

######  Theorem 4 (Székelyhidi [12, 13]).

Let $G$ be a topological abelian group which is generated by any neighbourhood
of the zero, and let $X$ be a topological linear space, and $p:G\rightarrow X$
be a polynomial function. Then the following statements hold.

  1. (i)

If $p:G\rightarrow X$ is continuous at a point, then it is continuous on $G$.

  2. (ii)

Assume that $G$ is locally compact and $X$ is locally convex. If
$p:G\rightarrow X$ is bounded on a measurable set of positive measure, then it
is continuous.

  3. (iii)

Suppose that $G$ is locally compact and $X$ is locally convex and locally
bounded. If $p:G\rightarrow X$ is measurable on a measurable set of positive
measure, then it is continuous.

##  2\. Main results

### Preparatory statements

In order to avoid superfluous repetitions, henceforth we will say that the
function in question is _locally regular_ on its domain, if at least one of
the following statements are fulfilled.

  1. (i)

bounded on a measurable set of positive measure;

  2. (ii)

continuous at a point;

  3. (iii)

there exists a set of positive Lebesgue measure so that the restriction of the
function in question is measurable in the sense of Lebesgue.

Furthermore, a function will be called _globally regular_ , if instead of
(ii),
(ii)’ continuous on its domain
holds.

Firstly we prove a simple lemma.

######  Lemma 5.

Let $\alpha\in\mathbb{R}$ be arbitrarily fixed and let us assume that for the
function $\phi\colon]0,+\infty[\to\mathbb{R}$ the following statements are
valid.

  1. (a)

the function $\phi$ is $\mathbb{Q}$-homogeneous of order $\alpha$, that is,

$\phi(rx)=r^{\alpha}\phi(x)\qquad\left(x\in]0,+\infty[,r\in\mathbb{Q}\cap]0,+\infty[\right).$

  2. (b)

the function $\phi$ is continuous at a point.

Then $\phi$ is continuous everywhere.

###### Proof.

Let us assume that the function $\phi$ is continuous at the point
$x_{0}\in]0,+\infty[$ and let $\tilde{x}\in]0,+\infty[$ be arbitrary. Then,
there exists a sequence of positive rational numbers
$(r_{n})_{n\in\mathbb{N}}$ so that
$\lim_{n\to\infty}r_{n}=\dfrac{\tilde{x}}{x_{0}}$. In this case the sequence
$\left(\frac{1}{r_{n}}\right)_{n\in\mathbb{N}}$ is also a sequence of positive
rational numbers and it converges to $\dfrac{x_{0}}{\tilde{x}}$. Due to
property (a),

$\frac{1}{r_{n}^{\alpha}}\phi(\tilde{x})=\phi\left(\frac{\tilde{x}}{r_{n}}\right)\qquad\left(n\in\mathbb{N}\right).$

Taking the limit $n\to\infty$, the left hand side converges to
$\left(\frac{x_{0}}{\tilde{x}}\right)^{\alpha}\phi(\tilde{x})$. Furthermore,
the sequence $\left(\frac{\tilde{x}}{r_{n}}\right)_{n\in\mathbb{N}}$ converges
to $x_{0}$, therefore the continuity of the function $\phi$ implies that the
right hand side tends to $\phi(x_{0})$ as $n\to\infty$. This implies that

$\left(\frac{x_{0}}{\tilde{x}}\right)^{\alpha}\phi(\tilde{x})=\phi\left(\frac{x_{0}}{\tilde{x}}\tilde{x}\right)$

is fulfilled. Since $\tilde{x}\in]0,+\infty[$ was arbitrary, we get that

$\phi(\lambda
x)=\lambda^{\alpha}\phi(x)\qquad\left(\lambda,x\in]0,+\infty[\right),$

which obviously implies the (everywhere) continuity of the function $\phi$.

∎

Furthermore, it is important pointing out the following fact. Fix
$\alpha\in\mathbb{R}$ and let $\phi\colon]0,+\infty[\to\mathbb{R}$ be a
$\mathbb{Q}$-homogeneous function of order $\alpha$. Suppose that $\phi$
fulfils property (i) or (iii) on the set of positive Lebesgue measure $D$.
Then, the $\mathbb{Q}$-homogeneity of the function $\phi$ implies that (i),
respectively (iii) holds for the function $\phi$ on the set $r\cdot D$ for all
$r\in\mathbb{Q}$.

######  Theorem 6.

Let $n,m\in\mathbb{Z}\setminus\left\\{0\right\\}$, $n\neq m$ so that $n=-m$ or
$\mathrm{sign}(n)=\mathrm{sign}(m)$, let further
$f,g\colon\mathbb{R}\to\mathbb{R}$ be additive functions. Define the function
$\phi\colon\mathbb{R}\setminus\left\\{0\right\\}\to\mathbb{R}$ by the formula

$\phi(x)=f\left(x^{n}\right)-x^{n-m}g\left(x^{m}\right)\qquad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right),$

and assume that $\phi$ is locally regular. Then, the functions
$F,G\colon\mathbb{R}\to\mathbb{R}$ defined by

$F(x)=f(x)-f(1)x\quad\text{and}\quad
G(x)=g(x)-g(1)x\qquad\left(x\in\mathbb{R}\right)$

are derivations and

$nF(x)=mG(x)$

holds for arbitrary $x\in\mathbb{R}$.

###### Proof.

Concerning the values of $n$ and $m$ we have to distinguish three cases. At
first, let us assume that $n,m>0$. There is no loss of generality in assuming
$n>m$. Define the function $\Phi$ on $\mathbb{R}^{n}$ by

$\Phi(x_{1},\ldots,x_{n})=f(x_{1}\cdots
x_{n})-\frac{1}{\binom{n}{m}}\sum_{\mathrm{card}(I)=m}\left(\prod_{j\in\\{1,2,\dots,n\\}\setminus
I}x_{j}\right)g\left(\prod_{i\in I}x_{i}\right)\\\
\left(x_{1},\ldots,x_{n}\in\mathbb{R}\right),$

where the summation is considered for all subsets $I$ of cardinality $m$ of
the index set $\left\\{1,2,\ldots,n\right\\}$. Due to the additivity of the
functions $f$ and $g$, the function $\Phi$ is a symmetric and $n$–additive
function. Furthermore, its trace, that is,

$\Phi(x,\ldots,x)=\phi(x)=f\left(x^{n}\right)-x^{n-m}g\left(x^{m}\right)\qquad\left(x\in\mathbb{R}\right)$

is a polynomial function. On the other hand $\phi$ is a locally regular
function. In view of Theorem 4, this means that $\phi$ is a continuous
polynomial function. Therefore, there exists $c\in\mathbb{R}$ such that

$\Phi(x_{1},\ldots,x_{n})=cx_{1}\cdots
x_{n}\quad\left(x_{1},\ldots,x_{n}\in\mathbb{R}\right),$

that is,

$\phi(x)=cx^{n}\quad\left(x\in\mathbb{R}\right).$

With the substitution $x=1$, we get $\phi(1)=c$. On the other hand, the
definition of the function $\phi$ yields that $\phi(1)=f(1)-g(1)$. Thus,

$f(x^{n})-x^{n-m}g(x^{m})=\left[f(1)-g(1)\right]x^{n}\qquad\left(x\in\mathbb{R}\right).$

Define the functions $F,G\colon\mathbb{R}\to\mathbb{R}$ by

$F(x)=f(x)-f(1)x\quad\text{and}\quad
G(x)=g(x)-g(1)x\qquad\left(x\in\mathbb{R}\right).$

Then the above identity yields that

$F\left(x^{n}\right)=x^{n-m}G(x^{m})\qquad\left(x\in\mathbb{R}\right).$

The statement of the theorem follows now from Theorem 1.

Secondly, let us assume that $n,m<0$. In this case we get that the function

$\phi(x)=f\left(x^{n}\right)-x^{n-m}g\left(x^{m}\right)\qquad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right)$

is locally regular on its domain. Let
$u\in\mathbb{R}\setminus\left\\{0\right\\}$, with the substitution
$x=\frac{1}{u}$ this yields that

$\phi\left(\frac{1}{u}\right)=f\left(u^{-n}\right)-u^{-(n-m)}g\left(u^{-m}\right)\qquad\left(u\in\mathbb{R}\setminus\left\\{0\right\\}\right).$

Since $-n,-m>0$, the results of the previous case can be applied for the
function

$\psi(u)=\phi\left(\frac{1}{u}\right)\qquad\left(u\in\mathbb{R}\setminus\left\\{0\right\\}\right),$

which is, due to the local regularity of $\phi$, also locally regular.

Finally, let us assume that $n=-m$. Without the loss of generality $m>0$ can
be assumed. In this case

$\phi(x)=f\left(x^{-m}\right)-x^{-2m}g\left(x^{m}\right)\qquad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right)$

is locally regular, or equivalently, the mapping

(1)
$\psi(x)=\phi\left(\sqrt[m]{x}\right)=f\left(\frac{1}{x}\right)-\frac{1}{x^{2}}g(x)\qquad\left(x>0\right)$

has the local regularity property. Let $u>0$ be arbitrary and let us
substitute $u(u+1)$ in place of $x$. Then

$\psi(u(u+1))=\phi\left(\sqrt[m]{u(u+1)}\right)\\\
=f\left(\frac{1}{u(u+1)}\right)-\frac{1}{u^{2}(u+1)^{2}}g\left(u(u+1)\right)\\\
\left(u>0\right).$

Using the additivity of the function $f$,

$\psi(u(u+1))=\phi\left(\sqrt[m]{u(u+1)}\right)\\\
=f\left(\frac{1}{u}\right)-f\left(\frac{1}{u+1}\right)-\frac{1}{u^{2}(u+1)^{2}}g\left(u(u+1)\right)\\\
\left(u>0\right).$

On the other hand,

$\psi(u)=\phi\left(\sqrt[m]{u}\right)=f\left(\frac{1}{u}\right)-\frac{1}{u^{2}}g(u)\qquad\left(u>0\right)$

and

$\psi(u+1)=\phi\left(\sqrt[m]{u+1}\right)=f\left(\frac{1}{u+1}\right)-\frac{1}{(u+1)^{2}}g(u+1)\qquad\left(u>0\right).$

Therefore,

$\psi(u(u+1))-\psi(u)+\psi(u+1)\\\
=\phi\left(\sqrt[m]{u(u+1)}\right)-\phi(\sqrt[m]{u})+\phi(\sqrt[m]{u+1})\\\
=f\left(\frac{1}{u}\right)-f\left(\frac{1}{u+1}\right)-\frac{1}{u^{2}(u+1)^{2}}g\left(u(u+1)\right)\\\
-f\left(\frac{1}{u}\right)+\frac{1}{u^{2}}g(u)+f\left(\frac{1}{u+1}\right)-\frac{1}{(u+1)^{2}}g(u+1)\qquad\left(u>0\right)$

Making use of the additivity of the function g, after rearrangement, we obtain
that

$\chi(u)=2ug(u)-g\left(u^{2}\right)\qquad\left(u>0\right),$

where

$\chi(u)=u^{2}(u+1)^{2}\left[\psi\left(u(u+1)\right)-\psi(u)+\psi(u+1)\right]+u^{2}g(1)\qquad(u>0).$

By our assumptions, the function $\phi$ is locally regular on
$\mathbb{R}\setminus\left\\{0\right\\}$ and due to the additivity of $f$ and
$g$, it is $\mathbb{Q}$-homogeneous of order $n$. Thus, by Lemma 5, $\phi$ is
globally regular on $\mathbb{R}\setminus\left\\{0\right\\}$. This implies that
$\psi$ is globally regular on $]0,+\infty[$, which means that $\chi$ is
locally regular. Due to the results of the first case this yields that the
function $G\colon\mathbb{R}\to\mathbb{R}$ defined by

$G(x)=g(x)-g(1)x\qquad\left(x\in\mathbb{R}\right)$

is a derivation. In view of (1), this implies that

$\phi\left(\sqrt[m]{x}\right)=f\left(\frac{1}{x}\right)-\frac{1}{x^{2}}\left[G(x)+g(1)x\right]\qquad\left(x>0\right),$

that is ,

$\phi\left(\sqrt[m]{x}\right)=f\left(\frac{1}{x}\right)+G\left(\frac{1}{x}\right)+g(1)\frac{1}{x}\qquad\left(x>0\right),$

since $G$ is a derivation. Let $u>0$, with the substitution $x=\frac{1}{u}$ we
get that

$\psi(u)=\phi\left(\sqrt[m]{\frac{1}{u}}\right)=f(u)+G(u)+g(1)u\qquad\left(u>0\right).$

Let us observe that the right hand side of this identity is an additive
function, being the sum of additive functions. Moreover, the left hand side is
locally regular, due to the local regularity of $\phi$. Thus $\psi$ is a
regular additive function, which means that there exists $c\in\mathbb{R}$ so
that

$f(u)+G(u)+g(1)u=cu\qquad\left(u\in\mathbb{R}\right).$

With $u=1$, $c=f(1)+g(1)$ can be obtained, therefore,

$f(u)=\left[f(1)+g(1)\right]u-g(1)u+G(u)\qquad\left(u\in\mathbb{R}\right),$

i.e.,

$f(u)=-G(u)+f(1)u\qquad\left(u\in\mathbb{R}\right).$

This means that the function $F\colon\mathbb{R}\to\mathbb{R}$ defined by

$F(x)=f(x)-f(1)x\qquad\left(x\in\mathbb{R}\right)$

is a derivation and

$F(x)=-G(x)\qquad\left(x\in\mathbb{R}\right)$

holds. ∎

######  Lemma 7.

Let $\kappa\in\mathbb{R}$, $n,m\in\mathbb{Z}$, $n\neq m$ and assume that
$f\colon\mathbb{R}\to\mathbb{R}$ is an additive function. Define the function
$\phi\colon\mathbb{R}\setminus\left\\{0\right\\}\to\mathbb{R}$ by

$\phi(x)=f\left(x^{n}\right)-\kappa
x^{n-m}f\left(x^{m}\right)\qquad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right)$

and assume that $\phi$ is locally regular. Then, the function
$F\colon\mathbb{R}\to\mathbb{R}$ defined by

$F(x)=f(x)-f(1)x\quad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right)$

is a derivation so that for any $x\in\mathbb{R}$

$\left(n-\kappa m\right)F(x)=0.$

###### Proof.

In view of the previous theorem, it is enough to deal with the case
$\mathrm{sign}(n)\neq\mathrm{sign}(m)$ and $n\neq-m$. Due to the definition of
the function $\phi$

$\phi\left(x^{n}\right)=f\left(x^{n^{2}}\right)-\kappa
x^{n(n-m)}f\left(x^{nm}\right)\qquad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right)$

and

$\phi\left(x^{m}\right)=f\left(x^{nm}\right)-\kappa
x^{m(n-m)}f\left(x^{m^{2}}\right)\qquad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right),$

therefore

$\phi\left(x^{n}\right)+\kappa
x^{n(n-m)}\phi\left(x^{m}\right)=f\left(x^{n^{2}}\right)-\kappa^{2}x^{n^{2}-m^{2}}f\left(x^{m^{2}}\right)\qquad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right).$

By our assumptions, $\phi$ is a locally regular function on
$\mathbb{R}\setminus\left\\{0\right\\}$. However, the additivity of $f$
implies that

$\phi(rx)=r^{n}\phi(x)\qquad\left(x\in\mathbb{R}\setminus\left\\{0\right\\},r\in\mathbb{Q}\setminus\left\\{0\right\\}\right).$

Using Lemma 5, we get that $\phi$ is globally regular. Therefore, the function

$\psi(x)=\phi\left(x^{n}\right)+\kappa
x^{n(n-m)}\phi\left(x^{m}\right)\qquad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right)$

is locally regular. Since $n^{2},m^{2}>0$ and $n^{2}\neq m^{2}$, the results
of the previous theorem can be applied (with the choice
$\psi(x)=\phi\left(x^{n}\right)+\kappa x^{n(n-m)}\phi\left(x^{m}\right)$ and
$g(x)=\kappa^{2}f(x)$) to obtain that

$f(x)=F(x)+f(1)x\qquad\left(x\in\mathbb{R}\right),$

where $F\colon\mathbb{R}\to\mathbb{R}$ is a derivation and

$nF(x)=m\kappa F(x)$

is also fulfilled for all $x\in\mathbb{R}$.

∎

From this lemma, the following corollary can be concluded immediately.

######  Corollary 8.

Let $r\in\mathbb{Q}\setminus\left\\{0,1\right\\}$ be arbitrarily fixed and
$f\colon\mathbb{R}\to\mathbb{R}$ be an additive function and define the
function by

$\phi(x)=f\left(x^{r}\right)-rx^{r-1}f\left(x\right)\qquad\left(x\in\mathbb{R},\,x>0\right),$

and assume that $\phi$ is locally regular. Then, the function
$F\colon\mathbb{R}\to\mathbb{R}$ defined by

$F(x)=f(x)-f(1)x\quad\left(x\in\mathbb{R}\right)$

is a derivation.

### Derivations along rational functions

In view of the results of the previous subsection, we are able to prove the
following

######  Theorem 9.

Let $n\in\mathbb{Z}\setminus\left\\{0\right\\}$ and
$\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathbf{GL}_{2}(\mathbb{Q})$ be such that

  1. –

if $c=0$, then $n\neq 1$;

  2. –

if $d=0$, then $n\neq-1$.

Let further $f,g\colon\mathbb{R}\to\mathbb{R}$ be additive functions and
define the function $\phi$ by

$\phi(x)=f\left(\frac{ax^{n}+b}{cx^{n}+d}\right)-\frac{x^{n-1}g(x)}{\left(cx^{n}+d\right)^{2}}\qquad\left(x\in\mathbb{R},\,cx^{n}+d\neq
0\right).$

Let us assume $\phi$ to be globally regular. Then, the functions
$F,G\colon\mathbb{R}\to\mathbb{R}$ defined by

$F(x)=f(x)-f(1)x\quad\text{and}\quad
G(x)=g(x)-g(1)x\quad\left(x\in\mathbb{R}\right)$

are derivations.

###### Proof.

Firstly, let us suppose that $c=0$. This means that the function

$\phi(x)=f\left(\frac{a}{d}x^{n}+\frac{b}{d}\right)-\frac{1}{d^{2}}x^{n-1}g(x)\qquad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right)$

is globally regular. In this case, the statement immediately follows from
Theorem 6.

Similarly, if $d=0$, then

$\phi(x)=f\left(\frac{a}{c}+\frac{b}{c}x^{-n}\right)-x^{-n-1}g(x)\qquad\left(x\in\mathbb{R}\setminus\left\\{0\right\\}\right)$

is globally regular. Therefore, due to Theorem 6, we obtain that the functions

$F(x)=f(x)-f(1)x\quad\text{and}\quad
G(x)=g(x)-g(1)x\qquad\left(x\in\mathbb{R}\right)$

are derivations.

Thus, henceforth $cd\neq 0$ can be assumed. Furthermore, due to the
$\mathbb{Q}$-homogeneity of the functions $f$ and $g$, $c=1$ can be supposed.
That is,

$\phi(x)=f\left(\frac{ax^{n}+b}{x^{n}+d}\right)-\frac{x^{n-1}g(x)}{\left(x^{n}+d\right)^{2}}\qquad\left(x\in\mathbb{R},\,x^{n}+d\neq
0\right).$

Since the function $f$ is additive,

$f\left(\frac{ax^{n}+b}{x^{n}+d}\right)=f(a)-f\left(\frac{D}{x^{n}+d}\right),\qquad\left(x\in\mathbb{R},\,x^{n}+d\neq
0\right),$

therefore,

(2)
$\phi(x)=f(a)-f\left(\frac{D}{x^{n}+d}\right)-\frac{x^{n-1}g(x)}{(x^{n}+d)^{2}}\qquad\left(x\in\mathbb{R},\,x^{n}+d\neq
0\right),$

where $D=\det\left(\begin{array}[]{cc}a&b\\\ 1&d\end{array}\right)$. Let us
observe that

$\frac{D}{x^{n}+d}=\frac{D}{d}-\frac{D}{\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)^{n}+d}$

holds for all $x\in\mathbb{R},\,x\neq 0,x^{n}+d\neq 0.$ Using this identity,
we receive

(3)
$\phi(x)=f(a)-f\left(\frac{D}{d}\right)+f\left(\frac{D}{\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)^{n}+d}\right)-\frac{x^{n-1}g(x)}{\left(x^{n}+d\right)^{2}}\\\
\left(x\in\mathbb{R},\,x\neq 0,x^{n}+d\neq 0\right),$

where the additivity of the function $f$ was also used. Let us replace $x$ by
$\sqrt[n]{d^{2}}\dfrac{1}{x}$ in (2) to acquire

$\phi\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)=f(a)-f\left(\frac{D}{\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)^{n}+d}\right)-\frac{\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)^{n-1}g\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)}{\left(\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)^{n}+d\right)^{2}}\\\
\left(x\in\mathbb{R},\,x\neq 0,x^{n}+d\neq 0\right).$

Since

$\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)^{n}+d=d\frac{1}{x^{n}}\left(x^{n}+d\right),$

the above identity yields that

$\phi\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)=f(a)-f\left(\frac{D}{\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)^{n}+d}\right)-\frac{\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)^{n-1}g\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)}{\left(d\frac{1}{x^{n}}\right)^{2}\left(\left(x^{n}+d\right)\right)^{2}}\\\
\left(x\in\mathbb{R},\,x\neq 0,x^{n}+d\neq 0\right).$

After some rearrangement, we arrive at

(4)
$\phi\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)=f(a)-f\left(\frac{D}{\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)^{n}+d}\right)-\frac{x^{n-1}\frac{1}{\sqrt[n]{d^{2}}}x^{2}g\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)}{\left(x^{n}+d\right)^{2}}\\\
\left(x\in\mathbb{R},\,x\neq 0,x^{n}+d\neq 0\right).$

In case we add (3) and (4) together,

$\phi(x)+\phi\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)=f\left(2a-\frac{D}{d}\right)-\frac{x^{n-1}}{\left(x^{n}+d\right)^{2}}\left[g(x)+x^{2}\frac{1}{\sqrt[n]{d^{2}}}g\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)\right]\\\
\left(x\in\mathbb{R},\,x\neq 0,x^{n}+d\neq 0\right).$

Let us define the functions

$h(x)=\frac{1}{\sqrt[n]{d^{2}}}g\left(\sqrt[n]{d^{2}}x\right)\qquad\left(x\in\mathbb{R}\right)$

and

$\psi(x)=-\frac{1}{x^{2}}\frac{\left(x^{n}+d\right)^{2}}{x^{n-1}}\left[\phi(x)+\phi\left(\sqrt[n]{d^{2}}\frac{1}{x}\right)-f\left(2a-\frac{D}{d}\right)\right]\\\
\left(x\in\mathbb{R},\,x\neq 0,x^{n}+d\neq 0\right).$

In this case

$\psi(x)=h\left(\frac{1}{x}\right)+\frac{1}{x^{2}}g(x)\quad\left(x\in\mathbb{R},\,x\neq
0,x^{n}+d\neq 0\right)$

holds. By our assumptions $\phi$ is a globally regular mapping, therefore the
function $\psi$ has the local regularity property. Due to Theorem 6, this
gives that the functions $F,G\colon\mathbb{R}\to\mathbb{R}$ defined by

$F(x)=f(x)-f(1)x\quad\text{and}\quad
G(x)=g(x)-g(1)x\quad\left(x\in\mathbb{R}\right)$

are derivations. ∎

### A characterization of linearity

Finally, in the last part of this paper we will present a characterization of
linearity. Just as in the proof of Theorem 9, Theorem 6 will again play an
important role.

######  Theorem 10.

Let $n\in\mathbb{N},\,n\neq 1$ and $f\colon\mathbb{R}\to\mathbb{R}$ be an
additive function. Define $\phi$ on $\mathbb{R}$ by

$\phi(x)=f\left(x^{n}\right)-f(x)^{n}\qquad\left(x\in\mathbb{R}\right).$

Let us assume that $\phi$ is locally regular. Then the function $f$ is linear,
that is,

$f(x)=f(1)x$

holds for all $x\in\mathbb{R}$.

###### Proof.

Let us define the function $\Phi\colon\mathbb{R}^{n}\to\mathbb{R}$ by

$\Phi(x_{1},\ldots,x_{n})=f\left(x_{1}\cdots x_{n}\right)-f(x_{1})\cdots
f(x_{n})\qquad\left(x_{1},\ldots,x_{n}\in\mathbb{R}\right).$

Due to the additivity of $f$, the function $\Phi$ is a symmetric, $n$-additive
function. Furthermore,

$\Phi(x,\ldots,x)=\phi(x)=f\left(x^{n}\right)-f(x)^{n}\qquad\left(x\in\mathbb{R}\right).$

From the local regularity of the function $\phi$ we immediately deduce that
$\phi$ is a continuous polynomial function. Consequently,

(5) $\Phi(x_{1},\ldots,x_{n})=cx_{1}\cdots
x_{n}\qquad\left(x_{1},\ldots,x_{n}\in\mathbb{R}\right)$

holds with a certain $c\in\mathbb{R}$. Due to the definition of the function
$\phi$, we have $\phi(1)=f(1)-f(1)^{n}$. On the other hand

$\phi(1)=\Phi(1,\ldots,1)=c.$

Hence $c=f(1)-f(1)^{n}$. Let $u\in\mathbb{R}$, with the substitution

$x_{1}=u,\qquad x_{i}=1\,\text{for $i\geq 2$},$

equation (5) yields that

$f(u)-f(u)f(1)^{n-1}=\left(f(1)-f(1)^{n}\right)u\qquad\left(u\in\mathbb{R}\right).$

In case $f(1)^{n-1}\neq 1$, this furnishes

$f(u)=f(1)u\qquad\left(u\in\mathbb{R}\right).$

If $f(1)^{n-1}=1$, then

$c=f(1)-f(1)^{n}=f(1)\left[1-f(1)^{n-1}\right]=0.$

Therefore equation (5) with the substitutions

$x_{1}=u,\quad x_{2}=v,\quad\text{and}\quad x_{i}=1\quad\text{for $i\geq
3$}\qquad\left(u,v\in\mathbb{R}\right)$

yields that

$f(uv)=f(u)f(v)f(1)^{n-2}\qquad\left(u,v\in\mathbb{R}\right),$

that is, $f(1)^{n-2}\cdot f$ is a non identically zero real homomorphism. In
view of Kuczma [9, Theorem 14.4.1.] this implies that

$f(1)^{n-2}f(u)=u\qquad\left(u\in\mathbb{R}\right).$

Since $f(1)^{n-2}\cdot f(1)=f(1)^{n-1}=1$,

$\frac{f(u)}{f(1)}=u$

holds for all $u\in\mathbb{R}$, that is, $f$ is a linear function, indeed. ∎

### Acknowledgements

This paper is dedicated to the 60th birthday of Professor László Székelyhidi
(University of Debrecen, Hungary). Furthermore, the author wishes to express
her gratitude to the two anonymous referees for their work.

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# Probing the $Zb\bar{b}$ coupling at the $Z$-pole of future lepton colliders

Bin Yan<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese Academy
of Sciences, Beijing 100049, China  C.-P. Yuan<EMAIL_ADDRESS>Department of
Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
Shu-Run Yuan<EMAIL_ADDRESS>School of Physics, Peking University,
Beijing 100871, China

###### Abstract

The determination of the $Zb\bar{b}$ coupling in experiments has been a long-
standing challenge, as the limited precision of off $Z$-pole measurements at
the LEP has resulted in two degenerate solutions remained to be resolved. In
this paper, we propose a novel method to probe the $Zb\bar{b}$ coupling by
measuring the forward-backward asymmetry of the bottom quark,
$A^{b}_{\text{{FB}}}$, in the $b\bar{b}$ system of the $e^{+}e^{-}\to
b\bar{b}\gamma$ and/or $e^{+}e^{-}\to b\bar{b}g$ processes at the $Z$-pole of
future lepton colliders. The additional hard photon or light jet radiation can
mimic the energy scanning of the $e^{+}e^{-}\to b\bar{b}$ process, and the
$A_{\rm FB}^{b}$ distribution from the $\gamma$-$Z$ interference process is
linearly sensitive to the $Zb\bar{b}$ coupling. By combining the expected
measurements of $R^{0}_{b}$ and $A^{0,b}_{\text{{FB}}}$ at the $Z$-pole at the
CEPC, the $A^{b}_{\text{{FB}}}$ distributions can break the degeneracy
observed at the LEP, leading to a unique determination of the $Zb\bar{b}$
coupling through $Z$-pole running alone.

††preprint:  MSUHEP-23-022

##  I Introduction

The forward-backward asymmetry of bottom quark ($A^{b}_{\text{FB}}$) at the
$Z$-pole ($A^{0,b}_{\text{FB}}$) from the Large Electron-Positron collider
(LEP) exhibits a long-standing discrepancy with the Standard Model (SM)
prediction, with a significance about 2.1$\sigma$ Zyla et al. (2020). This
anomaly can potentially be explained by intriguing new physics (NP) models
that involve a significant modification of the right-handed $Zb\bar{b}$
coupling while maintaining a SM-like left-handed $Zb\bar{b}$ coupling
Choudhury et al. (2002); Agashe et al. (2006); Liu et al. (2018); Crivellin et
al. (2021). Such modifications could arise from an approximate custodial
symmetry inherent in the underlying theory Agashe et al. (2006). Additionally,
the fact that the left-handed bottom quark and top quark belong to the same
electroweak doublet implies that any deviation in the $Zb\bar{b}$ coupling is
inevitably connected to top quark interactions Agashe et al. (2006); Peccei et
al. (1991); Haber and Logan (2000); Cao et al. (2017); Cao and Yan (2015); Cao
et al. (2020). Thus, precise measurements of the $Zb\bar{b}$ coupling are
crucial not only for testing NP models involving the bottom quark but also for
probing the properties of the top quark.

However, the $Zb\bar{b}$ coupling can not be determined uniquely due to the
quadratic dependence on this coupling of observables such as $A_{\rm
FB}^{0,b}$, $R^{0}_{b}$ (the ratio of the $Z\to b\bar{b}$ partial decay width
to the inclusive hadronic decay width at the LEP), and $A_{b}$ (the left-right
forward-backward asymmetry at the Stanford Linear Collider (SLC)) at the
$Z$-pole Schael et al. (2006). To break this degeneracy, the measurement of
$A_{\rm FB}^{b}$ in the off-$Z$ pole region becomes crucial as it allows for
discrimination among the four degenerate solutions by taking into account the
important $\gamma$-$Z$ interference effects, which are linearly dependent on
the $Zb\bar{b}$ coupling Choudhury et al. (2002). Unfortunately, due to
limitations in the statistics of off-$Z$ pole data from LEP, only the
degeneracy of the left-handed component coupling $Zb_{L}\bar{b}_{L}$ has been
broken, while the degeneracy of the right-handed component $Zb_{R}\bar{b}_{R}$
coupling remains unresolved.

Recent studies have proposed various approaches to address this degeneracy and
further investigate the anomalous $Zb\bar{b}$ couplings at the Large Hadron
Collider (LHC) and future colliders, with the aim of confirming or ruling out
the discrepancy between the $A^{0,b}_{\text{FB}}$ measurement and the SM
prediction at the $Z$-pole. For example, the axial-vector component of the
$Zb\bar{b}$ coupling can be determined through precision measurements of the
$gg\to Zh$ production cross section and exclusive $Z$-boson rare decays
$Z\to\Upsilon(ns)+\gamma$ at the LHC and high-luminosity LHC (HL-LHC) Yan and
Yuan (2021); Dong et al. (2022). On the other hand, the vector component of
the coupling can be constrained effectively by the electron single-spin
asymmetry in neutral current deeply inelastic scattering (DIS) process at HERA
and the upcoming Electron-Ion Collider (EIC) Yan et al. (2021). Additionally,
the application of additional jet charge information in DIS process at the EIC
can further improve the measurement of the $Zb\bar{b}$ coupling through the
jet charge weighted single-spin asymmetry Li et al. (2021). Furthermore, it
has been suggested in Ref. Bishara and Qian (2023) that the degeneracy of the
$Zb_{R}\bar{b}_{R}$ coupling could also be broken through the measurement of
the charge asymmetry of the final state $b$ and $\bar{b}$ jets in the
associated production of $Z$ boson and two $b$-jets at the HL-LHC.

At the same time, future lepton colliders such as the Circular Electron-
Position Collider (CEPC) Dong et al. (2018), International Linear Collider
(ILC) ILC (2013), and the Future Circular Collider (FCC-ee) Abada et al.
(2019) offer excellent opportunities for further investigation of the
$Zb\bar{b}$ coupling. With significant improvements in the statistical
uncertainties of $Z$-pole measurements, the discrepancy of the $A_{\rm
FB}^{0,b}$ at the LEP could be directly resolved at these future lepton
colliders. Additionally, the degeneracy of the $Zb_{R}\bar{b}_{R}$ coupling
could also be broken by measuring $A_{\rm FB}^{b}$ around $\sqrt{s}=240\sim
250~{}{\rm GeV}$ at these colliders, which is the primary collider energy of
the next generation lepton collider for studying the Higgs properties Gori et
al. (2016).

Figure 1:  Illustrative Feynman diagrams for the $e^{+}e^{-}\to
b\bar{b}\gamma$ and $e^{+}e^{-}\to b\bar{b}g$ productions at the future lepton
colliders.

In this paper, we suggest studying the $e^{+}e^{-}\to b\bar{b}\gamma$ and/or
$e^{+}e^{-}\to b\bar{b}g$ processes, as shown in Fig. 1, to break the
degeneracy of the $Zb_{R}\bar{b}_{R}$ coupling at the future lepton collider
at the $Z$-pole. By allowing for the emission of one hard photon or light jet,
the invariant mass of the $b\bar{b}$ system, denoted as $m_{b\bar{b}}$, can be
shifted away from the $Z$-pole, mimicking the energy scanning of the
$e^{+}e^{-}\to b\bar{b}$ process. Consequently, one can evaluate the forward-
backward asymmetry of the bottom quark in the center-of-mass (c.m.) frame of
the $b\bar{b}$ system, and the contribution from the $\gamma$-$Z$ interference
is linearly sensitive to the $Zb\bar{b}$ coupling. Due to the large amount of
data that will be collected around the $Z$-pole at the future lepton collider,
we demonstrate that by combining the $A^{b}_{\text{FB}}$ distributions from
the $b\bar{b}\gamma/g$ processes with the $R^{0}_{b}$ and
$A^{0,b}_{\text{FB}}$ measurements at the $Z$-pole, it is possible to uniquely
determine the $Zb\bar{b}$ coupling. Therefore, our results provide important
complementary information to other methods in the literature for probing the
$Zb\bar{b}$ interaction.

##  II Bottom Quark Forward-Backward Asymmetry

In this section, we discuss the bottom quark forward-backward asymmetry as a
function of $m_{b\bar{b}}$ at the parton-level through $e^{+}e^{-}\to
b\bar{b}\gamma$ and $e^{+}e^{-}\to b\bar{b}g$ processes at the $Z$-pole. In
the $e^{+}e^{-}\to b\bar{b}\gamma$ process, the photon can arise from both
initial state radiation (ISR) and final state radiation (FSR), while in the
$e^{+}e^{-}\to b\bar{b}g$ channel, gluon radiation is only available from FSR;
see Fig. 1. To illustrate the impact of the $Zb\bar{b}$ coupling on $A_{\rm
FB}^{b}$ as a function of $m_{b\bar{b}}$ at the $Z$-pole, we will use the
$e^{+}e^{-}\to b\bar{b}\gamma$ process as an example. The $Zb\bar{b}$
effective Lagrangian can be parametrized as

$\mathcal{L}_{eff}=\frac{g_{W}}{c_{W}}Z_{\mu}\left(\kappa_{L}g_{L}\bar{b}_{L}\gamma^{\mu}b_{L}+\kappa_{R}g_{R}\bar{b}_{R}\gamma^{\mu}b_{R}\right)\,,$
(1)

where $g_{W}$ is the $SU(2)_{L}$ gauge coupling, and $g_{L}=-1/2+s^{2}_{W}/3$
and $g_{R}=s^{2}_{W}/3$ are the left and right-handed components of
$Zb\bar{b}$ coupling in the SM, respectively. Here,
$s_{W}^{2}\equiv\sin\theta_{W}^{2}$, with $\theta_{W}$ being the weak mixing
angle. $\kappa_{L,R}$ are effective parameters which encode possible NP
effects in $Zb\bar{b}$ interaction and $\kappa_{L,R}=1$ for the SM.

Figure 2:  The $A^{b}_{\text{FB}}$ distribution of the $e^{+}e^{-}\to
b\bar{b}\gamma$ production at $Z$-pole in the SM. The “$Z$ propagator ISR”,
“$Z$ propagator FSR” and “$Z$ propagator ISR-FSR interference” correspond to
Fig. 1, 1 and their interference, while “$Z$ propagator” (Fig. 1\+ 1),
“$\gamma$ propagator” (Fig. 1\+ 1) and “$\gamma$-$Z$ propagator” are the
contributions from the mediator $Z$ boson, photon and their interference.

Figure 2 presents the $A_{\rm FB}^{b}$ distributions as a function of
$m_{b\bar{b}}$ for different parts of the $e^{+}e^{-}\to b\bar{b}\gamma$
process in the SM. We impose a requirement of $p_{T}^{\gamma}>10~{}{\rm GeV}$
for the transverse momentum of the photon to enhance the contribution of the
$\gamma$-$Z$ interference. From the orange line of Fig. 2, it is evident that
the dominant contribution to $A_{\rm FB}^{b}$ comes from the on-shell $Z$
production with a hard photon radiation from the final states (see Fig. 1,
labelled as “$Z$ propagator FSR” in Fig. 2). The value of $A_{\rm FB}^{b}$
from this channel increases with $m_{b\bar{b}}$ until $m_{b\bar{b}}\sim
30~{}\rm{GeV}$, after which it flattens out. The behavior at small
$m_{b\bar{b}}$ is due to the non-negligible effect of bottom quark mass, which
flips the helicity of the bottom quark. As a result, the production rate is
proportional to $m_{b}\sin\theta$, where $\theta$ is the polar angle of the
bottom quark in the $b\bar{b}$ system, so that $A_{\rm FB}^{b}$ becomes
smaller as $m_{b\bar{b}}$ decreases. The interference effects between ISR and
FSR with the mediator of the $Z$ boson (brown line, labelled as “$Z$
propagator ISR-FSR interference” in Fig. 2; the interference between Fig. 1
and Fig. 1) also contribute significantly to $A_{\rm FB}^{b}$, showing a
stronger dependence on $m_{b\bar{b}}$. This is because the $Z$-boson
propagator in the ISR process is $1/(m_{b\bar{b}}-m_{Z}^{2})$, cf. Fig. 1. It
is important to note that the $A_{\rm FB}^{b}$ from this interference is not a
result of the parity violation of the $Zb\bar{b}$ interaction. The size of
$A_{\rm FB}^{b}$ depends on the combination of the couplings
$g_{V}^{2}+g_{A}^{2}$, where $g_{V}=g_{L}+g_{R}$ and $g_{A}=g_{L}-g_{R}$, and
can be understood from the charge conjugation transformation for the final
states Brown et al. (1973). However, neither of the contributions discussed
above can break the degeneracy of the $Zb\bar{b}$ interaction owing to their
quadratic dependence on couplings. To achieve that, one must consider the
interference effects from the $\gamma$-$Z$ process (black line, labelled as
“$\gamma$-$Z$ interference” in Fig. 2; the interference between Fig. 1+Fig. 1
and Fig. 1+Fig. 1), which is linearly dependent on the $Zb\bar{b}$ coupling.

Figure 3: The forward and backward cross sections of $\gamma$-$Z$ interference
channel for the $e^{+}e^{-}\to b\bar{b}\gamma$ production at the $Z$-pole in
the SM.

To clarify the source of the $A_{\rm FB}^{b}$ in the $\gamma$-$Z$ interference
process, we present the forward and backward cross sections from the ISR + FSR
and the interference between the ISR and FSR in Fig. 3. This asymmetry is
related to the difference between the forward and backward cross sections, as
a result, it is dominated by the ISR + FSR contributions themselves (blue and
red lines), while the contributions from the interference between the ISR and
FSR are very small (orange and brown lines). This is because the asymmetry is
proportional to $g_{V}g_{V}^{e}$, where $g_{V}^{e}$ is the vector component of
the SM $Ze\bar{e}$ coupling, $g_{V}^{e}=-1/2+2s_{W}^{2}\simeq-0.038$, which is
nearly zero, and can be understood from the charge conjugation transformation
for the final states Brown et al. (1973). As a result, the interference term
becomes negligible for the $A_{\rm FB}^{b}$ due to the small value of
$g_{V}^{e}$. The situation for the $e^{+}e^{-}\to b\bar{b}g$ process is
similar to the $b\bar{b}\gamma$ channel and would be simpler, as only the on-
shell $Z$ production channel and the $\gamma$-$Z$ interference channel
contribute to the asymmetry.

Figure 4: The $A^{b}_{\text{FB}}$ distributions, with $\kappa_{L,R}=1$ (blue)
and $\kappa_{L}=-\kappa_{R}=1$ (red), of the $e^{+}e^{-}\to b\bar{b}\gamma$
(a) and $e^{+}e^{-}\to b\bar{b}g$ (b) productions at the $Z$-pole. The error
bar is the expected statistical uncertainty of measurement for each bin at the
CEPC.

Figure 4 presents the $A_{\rm FB}^{b}$ distributions from $e^{+}e^{-}\to
b\bar{b}\gamma$ (a) and $e^{+}e^{-}\to b\bar{b}g$ (b) processes at the
$Z$-pole. The blue lines represent the SM case ($\kappa_{L,R}=1$), while the
red lines correspond to the wrong-sign solution for the $Zb_{R}\bar{b}_{R}$
coupling ($\kappa_{L}=-\kappa_{R}=1$). Similar to the $b\bar{b}\gamma$
channel, we require a transverse momentum of the light jet
$p_{T}^{j}>20~{}{\rm GeV}$ for the $b\bar{b}g$ process. In the SM, the
$\gamma$-$Z$ interference gives a negative contribution to $A_{\rm FB}^{b}$
(see Fig. 2), as a result, the wrong-sign solution will enhance this
asymmetry. To assess the possibility of distinguishing between these two
parameter scenarios, we also show the expected uncertainty for each bin at the
CEPC running at the $Z$-pole by rescaling the uncertainties as follows:

$\delta(A^{b}_{\text{FB}})i=\sqrt{\frac{\sigma(e^{+}e^{-}\to
b\bar{b})}{\sigma(e^{+}e^{-}\to b\bar{b}\gamma/g)i}}\delta
A^{0,b}_{\text{FB}},$ (2)

where $\delta A^{0,b}_{\text{FB}}=5.4\times 10^{-5}$ is the projected
statistical uncertainty of $A_{\rm FB}^{0,b}$ at the CEPC Gori et al. (2016).
It is important to note that, due to the large statistical uncertainties for
each bin; see Fig. 4, we will neglect the systematic errors in the following
analysis.

Figure 5: The expected limits on the anomalous $Zb\bar{b}$ couplings
$\kappa_{L}$ and $\kappa_{R}$ from measuring the $A_{\rm FB}^{b}$
distributions of the $e^{+}e^{-}\to b\bar{b}\gamma$ (a) and $e^{+}e^{-}\to
b\bar{b}g$ (b) productions at the CEPC, running at the Z-pole, at the 95% C.L.
(cyan contours). The blue and red contours come from the expected $R^{0}_{b}$
and $A^{0,b}_{\text{FB}}$ measurements at the CEPC, respectively. Both the
statistical and systematic errors have been included for the $R_{b}^{0}$ and
$A_{\rm FB}^{0,b}$ measurements. We also assume the experimental values agree
with the SM predictions.

##  III Sensitivity at the CEPC

In this section, we utilize the $A_{\rm FB}^{b}$ distributions from
$e^{+}e^{-}\to b\bar{b}\gamma/g$ at the $Z$-pole to probe the $Zb\bar{b}$
coupling. By performing the pseudo experiments, we conduct a combined
$\chi^{2}$ analysis as,

$\chi^{2}=\sum_{i}\left[\frac{(A^{b}_{\text{FB}})^{\text{th}}_{i}-(A^{b}_{\text{FB}})^{\text{SM}}_{i}}{\delta(A^{b}_{\text{FB}})_{i}}\right]^{2}\,,$
(3)

where $(A_{\rm FB}^{b})_{i}^{\rm SM}$ is the forward-backward asymmetry of the
SM (i.e., $\kappa_{L,R}=1$) for the $i$-th bin. The statistical uncertainty
$\delta(A_{\rm FB}^{b})_{i}$ is estimated from the $A_{\rm FB}^{0,b}$
measurement at the $Z$-pole, see Eq. (2). For simplicity, we have assumed that
the experimental values of the $(A_{\rm FB}^{b})_{i}$ agree with the SM
predictions.

In Fig. 5, we show the expected limits (cyan contours) on the $Zb\bar{b}$
coupling from the $e^{+}e^{-}\to b\bar{b}\gamma$ (a) and $e^{+}e^{-}\to
b\bar{b}g$ (b) at the 95% confidence level (C.L.) at the CEPC running at the
$Z$-pole. To uniquely determine the $Zb\bar{b}$ coupling, we plot the expected
limits from the $A_{\rm FB}^{0,b}$ (red contours) and $R_{b}^{0}$ (blue
contours) at the CEPC in the same figure. Here, we focus on the parameter
space with $\kappa_{L}>0$ which has been verified by the off-$Z$ pole $A_{\rm
FB}^{b}$ measurements at the LEP. It clearly shows that the $Zb\bar{b}$
coupling could be determined uniquely with the measurements at the $Z$-pole
alone at the future lepton colliders.

##  IV Conclusions

In this paper, we propose a novel approach to probe the $Zb\bar{b}$ coupling
by studying the forward-backward asymmetry distributions as a function of
$m_{b\bar{b}}$ in the center-of-mass frame of the $b\bar{b}$ system from
$e^{+}e^{-}\to b\bar{b}\gamma$ and/or $e^{+}e^{-}\to b\bar{b}g$ processes at
the CEPC running at the $Z$-pole. We demonstrate that the $\gamma$-$Z$
interference effects play a crucial role in shaping the $A_{\rm FB}^{b}$
distributions and show these effects are linearly sensitive to the $Zb\bar{b}$
coupling parameters, $\kappa_{L,R}$, and provide complementary information to
the traditional measurements of $R^{0}_{b}$ and $A^{0,b}_{\text{FB}}$ at the
$Z$-pole, which primarily probe the quadratic terms of $\kappa_{L}$ and
$\kappa_{R}$. The forward-backward asymmetry distribution measurements from
these processes offers a straightforward and effective means of resolving the
sign degeneracy of $\kappa_{R}$ observed in precision electroweak data from
LEP and SLC experiments. This approach does not require any special
optimization, making it a promising avenue for studying the $Zb\bar{b}$
coupling at the future lepton colliders.

Acknowledgments. Bin Yan is supported by the IHEP under Contract No. E25153U1.
C.-P. Yuan is supported by the U.S. National Science Foundation under Grant
No. PHY-2013791 and the support from the Wu-Ki Tung endowed chair in particle
physics. Shu-Run Yuan is supported in part by the National Science Foundation
of China under Grants No.11725520, No.11675002 and No.12235001.

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