# Operators commuting with complex symmetric weighted composition operators on
$H^{2}$

Sudip Ranjan Bhuia  Indian Statistical Institute, Statistics and Mathematics
Unit, 8th Mile, Mysore Road, Bangalore, 560059, India<EMAIL_ADDRESS>
(Date: ; )

###### Abstract.

In this paper we study weighted composition operators $W_{g,\psi}$ commuting
with complex symmetric weighted composition operators $W_{f,\varphi}$ on the
Hardy space $H^{2}(\mathbb{D})$. In particular, we give the descriptions of
the symbols $g$ and $\psi$ such that the inducing weighted composition
operator $W_{g,\psi}$ commutes with the complex symmetric weighted composition
operator $W_{f,\varphi}$ with the conjugation $\mathcal{J}$. Furthermore, we
subsequently demonstrate that these weighted composition operators are normal
and complex symmetric in accordance with the properties of the fixed point of
the associated symbol $\varphi$.

###### Key words and phrases:

Weighted composition operator, complex symmetric operator, reproducing kernel
Hilbert space

###### 2020 Mathematics Subject Classification:

Primary 47B20 ; Secondary 47A05, 47B38, 47B33

##  1\. Introduction and preliminaries

Let $\mathcal{B}(H)$ be the algebra of all bounded linear operators on a
separable complex Hilbert space $H$. Given a fixed operator
$T\in\mathcal{B}(H)$, we say an operator $S$ commutes with $T$ if $TS=ST$. The
set of all operators which commute with $T$, denoted $\\{T\\}^{\prime}$ that
is, $\\{T\\}^{\prime}=\left\\{S\in\mathcal{B}(H)\mathrel{\mathop{\mathchar
58\relax}}ST-TS=0\right\\}$. It is well known that the set $\\{T\\}^{\prime}$
forms a weakly closed algebra which is called the commutant of $T$.

Let $\mathbb{D}$ denote the open unit disk in the complex plane $\mathbb{C}$.
The Hardy space $H^{2}(\mathbb{D})$ is the Hilbert space of the analytic
functions $f$ on $\mathbb{D}$ having power series representations with square-
summable complex coefficients. That is,

$H^{2}(\mathbb{D})=\left\\{f\mathrel{\mathop{\mathchar
58\relax}}f(z)=\displaystyle\sum_{n=0}^{\infty}a_{n}z^{n}\quad\text{and}\quad\displaystyle\sum_{n=0}^{\infty}|a_{n}|^{2}<\infty\right\\}.$

The space $H^{2}(\mathbb{D})$ is a Hilbert space with the inner product given
by

$\langle f,g\rangle=\displaystyle\sum_{n=0}^{\infty}a_{n}\bar{b}_{n},$

where $f(z)=\displaystyle\sum_{n=0}^{\infty}a_{n}z^{n}$ and
$g(z)=\displaystyle\sum_{n=0}^{\infty}b_{n}z^{n}$ are in $H^{2}(\mathbb{D})$.

Let $f$ be an analytic function on $\mathbb{D}$. Then $f$ is in
$H^{2}(\mathbb{D})$ if and only if

$\displaystyle\sup_{0<r<1}\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{2}d\theta<\infty.$

Moreover, the norm of such $f$ is given by

$\|f\|^{2}=\displaystyle\sup_{0<r<1}\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{2}d\theta<\infty.$

We define the Hilbert space $L^{2}(\mathbb{T})$ by the space of the square
integrable functions on $\mathbb{T}$, the unit circle in complex plane with
respect to Lebesque measure, endowed with the inner product given by

$\langle
f,g\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}f(e^{i\theta})\overline{g(e^{i\theta})}d\theta,$

for all $f,g\in L^{2}(\mathbb{T})$.

The space $L^{\infty}(\mathbb{T})$ is the Banach space consisting of all
essentially bounded measurable functions on $\mathbb{T}$.

The Hardy space $H^{2}(\mathbb{D})$ on the open unit disc $\mathbb{D}$ is
identified with the closed subspace $H^{2}(\mathbb{T})$ of $L^{2}(\mathbb{T})$
consisting of functions $f$ on the boundary $\mathbb{T}$ whose negative
Fourier coefficients vanish; the identification is given by the radial limit

$\tilde{f}(e^{i\theta})\mathrel{\mathop{\mathchar
58\relax}}=\lim_{r\rightarrow-1}f(re^{i\theta})\quad\text{for almost every
}\theta\in[0,2\pi]$

for $f\in H^{2}(\mathbb{D})$.

Moreover, the reverse process is given by the Poisson integral formula

$f(re^{i\theta})=\frac{1}{2\pi}\int_{0}^{2\pi}\tilde{f}(e^{it})\frac{1-r^{2}}{1+r^{2}-2r\cos(\theta-t)},\quad
re^{i\theta}\in\mathbb{D}.$

Here $f$ is the harmonic extension of $\tilde{f}$ into the open unit disc
$\mathbb{D}$ given by the Poisson integral formula. We usually use the same
symbol $f$ for $\tilde{f}$ and write $H^{2}(\mathbb{D})=H^{2}(\mathbb{T})$
under the identification. We denote by $H^{\infty}(\mathbb{D})$ the space of
all functions that are analytic and bounded on $\mathbb{D}$. The space
$H^{\infty}(\mathbb{D})$ is a subspace of $H^{2}(\mathbb{D})$.

Let $\varphi$ be a holomorphic self map of $\mathbb{D}$ and let
$w\in\overline{\mathbb{D}}$. We say that $w$ is a fixed point [7, page 50] of
$\varphi$ if

$\lim_{r\rightarrow 1^{-}}\varphi(rw)=w.$

By a well known result [7, page 51], if $w\in\mathbb{T}$ is a fixed point of
$\varphi$, then

$\varphi^{\prime}(w)=\lim_{r\rightarrow 1^{-}}\varphi^{\prime}(rw),$

exists as a positive real number or $+\infty$. Now let $\varphi$ be an
automorphism of $\mathbb{D}$. We say that $\varphi$ is:

  1. (1)

elliptic if it has exactly one fixed point situated in $\mathbb{D}$,

  2. (2)

hyperbolic if it has two distinct fixed points in $\mathbb{T}$, and

  3. (3)

parabolic if there is only one fixed point in $\mathbb{T}$.

The Hardy space $H^{2}(\mathbb{D})$ is a reproducing kernel Hilbert space with
the kernel function $K_{w}$, where $K_{w}(z)=\frac{1}{1-\bar{w}z}$ for each
$w\in\mathbb{D}$ with the property that $\langle f,K_{w}\rangle=f(w)$ for each
$f\in H^{2}(\mathbb{D})$. The linear span of the reproducing kernels
$\\{K_{w}\mathrel{\mathop{\mathchar 58\relax}}w\in\mathbb{D}\\}$ is dense in
$H^{2}(\mathbb{D})$.

For a holomorphic self map $\varphi$ on $\mathbb{D}$ and a holomorphic
function $f$ on $\mathbb{D}$, the weighted composition operator
$W_{f,\varphi}$ on $H^{2}(\mathbb{D})$ is defined by
$W_{f,\varphi}g=f\cdot(g\circ\varphi)$ for all $g\in H^{2}(\mathbb{D})$. The
composition operator $C_{\varphi}$ is defined by $C_{\varphi}=W_{1,\varphi}$.
It is worth to note that $W_{f,\varphi}=M_{f}C_{\varphi}$ whenever $f\in
H^{\infty}$, where $M_{f}$ denotes the multiplication operator. The action of
the adjoint of weighted composition operator $W_{f,\varphi}$ on the kernel
function is given by

$W^{*}_{f,\varphi}K_{w}=\overline{f(w)}K_{\varphi(w)}.$

For more about composition operators, we refer the book [7].

######  Definition 1.1.

A conjugation on a separable complex Hilbert space $H$ is an anti-linear
operator $C$ on $H$ which satisfies the following conditions

  1. (1)

$C$ is isometric: $\langle Cx,Cy\rangle=\langle y,x\rangle$, $x,y\in H$,

  2. (2)

$C$ is involutive: $C^{2}=I$.

where $I$ is the identity operator on $H$.

######  Definition 1.2.

An anti-linear operator $C$ on a separable complex Hilbert space $H$ is a
conjugation if and only if it is both unitary and self-adjoint.

We say that $T$ is $C$-symmetric if $T=CT^{*}C$, and complex symmetric if
there exists a conjugation $C$ with respect to which $T$ is $C$-symmetric.

In the Hardy space $H^{2}(\mathbb{D})$, the conjugation operator
$\mathcal{J}\mathrel{\mathop{\mathchar 58\relax}}H^{2}(\mathbb{D})\rightarrow
H^{2}(\mathbb{D})$ is defined by $\mathcal{J}f(z)=\overline{f(\bar{z})}$ for
all $z\in\mathbb{D}$ and $f\in H^{2}(\mathbb{D})$.

For more details on complex symmetric operators readers are referred to [9,
10, 11, 12]. The study of complex symmetric weighted composition operators on
the Hardy space was initiated by Garcia and Hammond in [8]. In [13], authors
gave a classification of complex symmetric weighted composition operators with
respect to a special conjugation. Generally, providing information about the
operators that commute with a specific operator offers insights into the
operator’s structure. The commutant of a particular operator is a relatively
rare subject of study. C. C. Cowen’s research has focused on examining the
commutants of composition and specific Toeplitz operators, as indicated in
references [3] and [4]. Additionally, B. Cload has contributed findings
concerning the commutants of composition operators in [2]. In [6], authors
have studied the self-adjoint weighted composition operators o the Hardy space
of unit disc. E. Ko studied the commutant of self-adjoint weighted composition
operators in [14]. It would be interesting to classify weighted composition
operators commuting with complex symmetric weighted composition operators
$W_{f,\varphi}$ with certain conjugation on $H^{2}$. In section $2$, we give
description of the weighted composition operators that commute with complex
symmetric weighted composition operator with the conjugation $\mathcal{J}$.

######  Theorem 1.3.

[5, Theorem 6] Let $\gamma\in\mathbb{N}$. Let $f$ be in $H^{\infty}$ and let
$\varphi$ be an analytic map of the unit disc into itself. If the weighted
composition operator $W_{f,\varphi}$ is Hermitian on $H_{\gamma}(\mathbb{D})$
, then $f(0)$ and $\varphi^{\prime}(0)$ are real and

(1.1) $\varphi(z)=a_{0}+\frac{a_{1}z}{1-\bar{a}_{0}z}\quad\text{and}\quad
f(z)=\frac{c}{(1-\bar{a}_{0}z)^{\gamma}}$

where $a_{1}=\varphi^{\prime}(0)$, and $c=f(0)$.

Conversely, let $a_{0}\in\mathbb{D}$, and let $c$ and $a_{1}$ be real numbers.
If $\varphi(z)=a_{0}+\frac{a_{1}z}{1-\bar{a}_{0}z}$ maps the unit disc into
itself and $f(z)=\frac{c}{(1-\bar{a}_{0}z)^{\gamma}}$, then the weighted
composition operator $W_{f,\varphi}$ is Hermitian.

######  Theorem 1.4.

[14, Theorem 3.5] Let $g\in H^{\infty}$ and let $\psi$ be an analytic map of
$\mathbb{D}$ into itself. Assume that $W_{f,\varphi}$ is self-adjoint on
$H^{2}(\mathbb{D})$ and $\varphi$, not an elliptic automorphism, has a fix
point $b\in\mathbb{D}$. Then $W_{g,\psi}\in\\{W_{f,\varphi}\\}^{\prime}$ if
and only if $W_{g,\psi}$ has the following symbols; for $b\neq 0$,

$\psi(z)=d_{0}+\frac{d_{2}z}{1-d_{1}z}\quad\text{and}\quad
g(z)=g(b)\frac{d_{3}}{1-d_{1}z},$

where
$\psi(0)=d_{0}=\frac{(\alpha-1)b}{|b|^{2}\alpha-1},\,d_{1}=\frac{(\alpha-1)\bar{b}}{|b|^{2}\alpha-1},\,\psi^{\prime}(0)=d_{2}=\alpha\frac{(|b|^{2}-1)^{2}}{(|b|^{2}\alpha-1)^{2}},\,d_{3}=\frac{|b|^{2}-1}{|b|^{2}\alpha-1}$
for some $\alpha\in\mathbb{C}$ and for $b=0$,

$\psi(z)=\alpha z\quad\text{and}\quad g(z)=g(0)$

for some $\alpha\in\mathbb{C}$.

######  Theorem 1.5.

[13, Theorem 3.3] Let $\varphi$ be an analytic self-map of $\mathbb{D}$ and
$f\in H^{\infty}(\mathbb{D})$ be not identically zero. If the weighted
composition operator $W_{f,\varphi}$ is complex symmetric defined on
$H^{2}(\mathbb{D})$ with the conjugation $\mathcal{J}$, then

$f(z)=\frac{b}{1-a_{0}z}\quad\varphi(z)=a_{0}+\frac{a_{1}z}{1-a_{0}z},$

where $a_{0}=\varphi(0),\,a_{1}=\varphi^{\prime}(0)$, and $b=f(0)$.

Conversely, let $a_{0}\in\mathbb{D}$. If
$\varphi(z)=a_{0}+\frac{a_{1}z}{1-a_{0}z}$ maps the unit disc into itself and
$f(z)=\frac{b}{1-a_{0}z}$, then the weighted composition operator
$W_{f,\varphi}$ is complex symmetric with the conjugation $\mathcal{J}$.

######  Lemma 1.6.

[16] A linear fractional map $\phi$ of the form $\phi(z)=\frac{az+b}{cz+d}$;
$ad-bc\neq 0$, maps $\mathbb{D}$ into itself if and only if

(1.2) $|b\bar{d}-a\bar{c}|+|ad-bc|\leq|d|^{2}-|c|^{2}.$

######  Lemma 1.7.

[13, Lemma 4.8] Let $\varphi(z)=a_{0}+\frac{a_{1}z}{1-a_{0}z}$. Then $\varphi$
maps the open unit disc into itself if and only if $|a_{0}|<1$ and
$2|a_{0}+\bar{a}_{0}(a_{1}-a_{0}^{2})|\leq 1-|a_{1}-a_{0}^{2}|^{2}$.

In particular, when $a_{1}=a_{0}^{2}=0$, $\varphi$ maps the open unit disc
into itself if and only if $|a_{0}|\leq\frac{1}{2}$, and when
$a_{1}-a_{0}^{2}=\pm 1$, $\varphi$ maps the open unit disc into itself if and
only if $a_{0}$ is either real or purely imaginary.

######  Proposition 1.8.

[13, Proposition 4.4] Let $\varphi$ be an analytic self-map of $\mathbb{D}$
and let $f\in H^{\infty}(\mathbb{D})$ be not identically zero on $\mathbb{D}$,
where $\varphi(0)\neq 0,\,\varphi^{\prime}(0)\neq 0$, and
$\varphi(\lambda)=\lambda$ for some $\lambda\in\mathbb{D}$. If $W_{f,\varphi}$
is complex symmetric with the conjugation $\mathcal{J}$, then

$g_{j}(z)\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{1-\lambda
z}\left(\frac{\lambda-z}{1-\lambda z}\right)^{j}$

is an eigenvector of $W_{f,\varphi}$ with respect to the eigenvalue
$f(\lambda)\varphi^{\prime}(\lambda)^{j}$ for each non-negative integer $j$.

##  2\. Description of commutant

Throughout this section, we consider $W_{f,\varphi}$ is complex symmetric with
the conjugation $\mathcal{J}$ and $\varphi$ is not an identity map. Because if
$\varphi$ is an identity map then $W_{g,\psi}\in\\{W_{f,\varphi}\\}^{\prime}$
will always holds. That is, in view of Theorem 1.5, we will always assume that
$a_{1}\neq 1$.

######  Lemma 2.1.

Let $g\in H^{\infty}$ and $\psi$ be an analytic map of $\mathbb{D}$ into
itself. Assume that $W_{g,\psi}\in\\{W_{f,\varphi}\\}^{\prime}$, where
$W_{f,\varphi}$ is complex symmetric with the conjugation $\mathcal{J}$. If
$\varphi$ has a fixed point in $\mathbb{D}$, then $\psi$ has the same fixed
point with $\varphi$.

###### Proof.

Since $W_{f,\varphi}$ is $\mathcal{J}$-symmetric with the conjugation
$\mathcal{J}f(z)=\overline{f(\bar{z})}$, it follows from Theorem 1.5 that

(2.1) $f(z)=\frac{b}{1-a_{0}z},\quad\varphi(z)=a_{0}+\frac{a_{1}z}{1-a_{0}z},$

where $a_{0}=\varphi(0),\,a_{1}=\varphi^{\prime}(0)$, and $b=f(0)$.

Let $\lambda$ be a fixed point of $\varphi$ in $\mathbb{D}$. It is very clear
that if $a_{0}=0$, then $\lambda=0$. When $a_{0}\neq 0$, then the fixed point
$\lambda$ is given by

(2.2)
$\lambda=\frac{2a_{0}}{1+a_{0}^{2}-a_{1}\mp\sqrt{(1+a_{0}^{2}-a_{1})^{2}-4a_{0}^{2}}}.$

Case 1: If $\lambda=0$, then $\varphi(z)=a_{1}z$. Since
$W_{g,\psi}\in\\{W_{f,\varphi}\\}^{\prime}$, we have

$\psi(a_{1}z)=\psi(\varphi(z))=\varphi(\psi(z))=a_{1}\psi(z).$

Now consider the power series
$\psi(z)=\displaystyle\sum_{n=0}^{\infty}\beta_{n}z^{n}$, then we have

$\displaystyle\sum_{n=0}^{\infty}\beta_{n}a_{1}^{n}z^{n}=\sum_{n=0}^{\infty}a_{1}\beta_{n}z^{n}.$

Hence $\beta_{0}=a_{1}\beta_{0}$. Since $a_{1}\neq 1$, we have $\beta_{0}=0$
and this implies that $\psi(0)=0$.

Case 2: If $\lambda\neq 0$, then consider the kernel function $K_{\lambda}$,
that is, $K_{\lambda}(z)=\frac{1}{1-\bar{\lambda}z}$ for all $z\in\mathbb{D}$.
We know that
$W_{f,\varphi}^{*}K_{\lambda}=\overline{f(\lambda)}K_{\varphi(\lambda)}$ and
since $W_{g,\psi}\in\\{W_{f,\varphi}\\}^{\prime}$, we have

$W^{*}_{f,\varphi}W^{*}_{g,\psi}K_{\lambda}=W^{*}_{g,\psi}W^{*}_{f,\varphi}K_{\lambda}=\overline{f(\lambda)}W^{*}_{g,\psi}K_{\varphi(\lambda)}=\overline{f(\lambda)}W^{*}_{g,\psi}K_{\lambda}.$

Case 2(i): If $W^{*}_{g,\psi}K_{\lambda}=0$, then $K_{\lambda}$ is the
eigenvector of $W^{*}_{g,\psi}$ corresponding to the eigenvalue $0$. Since
$\varphi(\psi(z))=\psi(\varphi(z))$, we get
$\varphi(\psi(\lambda))=\psi(\varphi(\lambda))=\psi(\lambda)$, and this
implies that $\psi(\lambda)$ is a fixed point of $\varphi$, and hence
$\psi(\lambda)=0$, or $\lambda$. Now if $\psi(\lambda)=0$, then
$a_{0}=\varphi(0)=\varphi(\psi(\lambda))=\psi(\varphi(\lambda))=\psi(\lambda)=0$,
a contradiction to the fact that $a_{0}\neq 0$. Therefore
$\psi(\lambda)=\lambda$.

Case 2(ii): If $W^{*}_{g,\psi}K_{\lambda}\neq 0$, then
$W^{*}_{g,\psi}K_{\lambda}$ is an eigenvector of $W^{*}_{f,\varphi}$ with an
eigenvalue $\overline{f(\lambda)}$. Since $W_{f,\varphi}$ is
$\mathcal{J}$-symmetric, we have $\mathcal{J}W^{*}_{g,\psi}K_{\lambda}$ is an
eigenvector corresponding to the eigenvalue $f(\lambda)$. Therefore, by
Proposition 1.8, we get $\mathcal{J}W^{*}_{g,\psi}K_{\lambda}=\beta
K_{\bar{\lambda}}$ for some nonzero complex number $\beta$, and this implies
$g(\lambda)K_{\overline{\psi(\lambda)}}=\beta K_{\bar{\lambda}}$. Therefore,
$\psi(\lambda)=\lambda$. Thus $\lambda$ is a fixed point of $\psi$. This
completes the proof.

∎

######  Lemma 2.2.

Let $d_{0}=\frac{\lambda(\alpha-1)}{\lambda^{2}\alpha-1}=d_{1}$,
$d_{2}=\alpha\frac{(\lambda^{2}-1)^{2}}{(\lambda^{2}\alpha-1)^{2}}$, where
$\alpha$ is any complex number and $\lambda$ as in Equation 2.2. Then the
following are true

  1. (1)

$d_{1}+a_{0}(d_{2}-d_{0}d_{1})=a_{0}+d_{1}(a_{1}-a_{0}^{2})$.

  2. (2)

$\bar{d}_{0}(d_{2}-d_{0}^{2}-1)=-\bar{\lambda}(\lambda^{2}+1)\frac{|1-\alpha|^{2}}{|\lambda^{2}\alpha-1|^{2}}$.

###### Proof.

Proof of (1): To prove the relation in (1), it is enough to prove
$a_{0}(d_{2}-d_{0}d_{1}-1)-d_{1}(a_{1}-a_{0}^{2}-1)=0$. Now

$\begin{split}a_{0}(d_{2}-d_{0}d_{1}-1)&=a_{0}(d_{2}-d_{0}^{2}-1)\\\
&=a_{0}\left(\frac{\lambda^{2}-\alpha}{\lambda^{2}\alpha-1}-1\right)\\\
&=a_{0}\frac{(1-\alpha)(\lambda^{2}+1)}{\lambda^{2}\alpha-1}\end{split}$

and

$\begin{split}d_{1}(a_{1}-a_{0}^{2}-1)=\frac{\lambda(\alpha-1)(a_{1}-a_{0}^{2}-1)}{\lambda^{2}\alpha-1}.\end{split}$

Therefore,

$\begin{split}&a_{0}(d_{2}-d_{0}d_{1}-1)-d_{1}(a_{1}-a_{0}^{2}-1)\\\
&=a_{0}\frac{(1-\alpha)(\lambda^{2}+1)}{\lambda^{2}\alpha-1}-\frac{\lambda(\alpha-1)(a_{1}-a_{0}^{2}-1)}{\lambda^{2}\alpha-1}\\\
&=\frac{1-\alpha}{\lambda^{2}\alpha-1}\left[a_{0}(\lambda^{2}+1)+\lambda(a_{1}-a_{0}^{2}-1)\right]\\\
&=\frac{1-\alpha}{\lambda^{2}\alpha-1}\left[a_{0}(\lambda^{2}+1)-\lambda(1+a_{0}^{2}-a_{1})\right]\\\
&=\frac{1-\alpha}{\lambda^{2}\alpha-1}\left[a_{0}\Bigg{\\{}\left(\frac{1+a_{0}^{2}-a_{1}\pm\sqrt{(1+a_{0}^{2}-a_{1})^{2}-4a_{0}^{2}}}{2a_{0}}\right)^{2}+1\Bigg{\\}}\right.\\\
&\hskip
56.9055pt\left.-\frac{1+a_{0}^{2}-a_{1}\pm\sqrt{(1+a_{0}^{2}-a_{1})^{2}-4a_{0}^{2}}}{2a_{0}}(1+a_{0}^{2}-a_{1})\right]\\\
&=\frac{1-\alpha}{\lambda^{2}\alpha-1}\left[\frac{(1+a_{0}^{2}-a_{1})^{2}\pm(1+a_{0}^{2}-a_{1})\sqrt{(1+a_{0}^{2}-a_{1})^{2}-4a_{0}^{2}}}{2a_{0}}\right.\\\
&\hskip
56.9055pt\left.-\frac{1+a_{0}^{2}-a_{1}\pm\sqrt{(1+a_{0}^{2}-a_{1})^{2}-4a_{0}^{2}}}{2a_{0}}(1+a_{0}^{2}-a_{1})\right]\\\
&=0.\end{split}$

This completes the proof of (1).

Proof of (2): We have

(2.3)
$\begin{split}d_{2}-d_{0}^{2}-1&=\alpha\frac{(\lambda^{2}-1)^{2}}{(\lambda^{2}\alpha-1)^{2}}-\frac{\lambda^{2}(\alpha-1)^{2}}{(\lambda^{2}\alpha-1)^{2}}-1\\\
&=\frac{\alpha\lambda^{4}-2\lambda^{2}\alpha+\alpha-\lambda^{2}\alpha^{2}+2\lambda^{2}\alpha-\lambda^{2}}{(\lambda^{2}\alpha-1)^{2}}-1\\\
&=\frac{\alpha\lambda^{4}+\alpha-\lambda^{2}\alpha^{2}-\lambda^{2}}{(\lambda^{2}\alpha-1)^{2}}-1\\\
&=\frac{\lambda^{2}-\alpha}{\lambda^{2}\alpha-1}-1\\\
&=\frac{\lambda^{2}-\alpha-\lambda^{2}\alpha+1}{\lambda^{2}\alpha-1}\\\
&=\frac{(1-\alpha)(\lambda^{2}+1)}{\lambda^{2}\alpha-1}.\end{split}$

Therefore,

$\bar{d}_{0}(d_{2}-d_{0}^{2}-1)=\frac{\bar{\lambda}(\bar{\alpha}-1)}{\overline{\lambda^{2}\alpha-1}}\times\frac{(1-\alpha)(\lambda^{2}+1)}{\lambda^{2}\alpha-1}=-\bar{\lambda}(\lambda^{2}+1)\frac{|1-\alpha|^{2}}{|\lambda^{2}\alpha-1|^{2}}$

∎

######  Remark 2.3.

$\bar{d}_{0}(d_{2}-d_{0}^{2}-1)=d_{0}(\overline{d_{2}-d_{0}^{2}}-1)$ if and
only if $\lambda$ is real.

######  Theorem 2.4.

Let $g\in H^{\infty}$ and let $\psi$ be an analytic map of $\mathbb{D}$ into
itself. Assume that $W_{f,\varphi}$ is complex symmetric with the conjugation
$\mathcal{J}$ on $H^{2}(\mathbb{D})$ and $\varphi$, not an elliptic
automorphim, has a fixed point $\lambda$ in $\mathbb{D}$ . Then
$W_{g,\psi}\in\\{W_{f,\varphi}\\}^{\prime}$ if and only if $W_{g,\psi}$ has
the following symbol functions; for $\lambda\neq 0$,

$\psi(z)=\frac{(\lambda^{2}-\alpha)z+\lambda(\alpha-1)}{\lambda(1-\alpha)z+(\lambda^{2}\alpha-1)}\quad\text{and}\quad
g(z)=g(\lambda)\frac{\lambda^{2}-1}{\lambda(1-\alpha)z+(\lambda^{2}\alpha-1)}$

and for $\lambda=0$,

$\psi(z)=\alpha z\quad\text{and}\quad g(z)=g(0)$

for some $\alpha\in\mathbb{C}$.

###### Proof.

Let $\lambda\neq 0$. Since $W_{g,\psi}\in\\{W_{f,\varphi}\\}^{\prime}$, we
have

$\begin{split}W_{f,\varphi}W_{g,\psi}K_{\bar{\lambda}}&=W_{g,\psi}W_{f,\varphi}K_{\bar{\lambda}}\\\
&=W_{g,\psi}\mathcal{J}W^{*}_{f,\varphi}\mathcal{J}K_{\bar{\lambda}}\\\
&=W_{g,\psi}\mathcal{J}W^{*}_{f,\varphi}K_{\lambda}\\\
&=f(\lambda)W_{g,\psi}K_{\bar{\lambda}}.\end{split}$

If $W_{g,\psi}K_{\bar{\lambda}}=0$, then $K_{\bar{\lambda}}$ is an eigenvector
of $W_{g,\psi}$ corresponding to an eigenvalue $0$.

If $W_{g,\psi}K_{\bar{\lambda}}\neq 0$, then If $W_{g,\psi}K_{\bar{\lambda}}$
is an eigenvector of $W_{f,\varphi}$ corresponding to the eigenvalue
$f(\lambda)$. Then by Proposition 1.8, we have
$W_{g,\psi}K_{\bar{\lambda}}=\gamma_{1}K_{\bar{\lambda}}$ for some nonzero
$\gamma_{1}\in\mathbb{C}$. Again by Proposition 1.8, the function
$\eta=\frac{1}{1-\lambda z}\left(\frac{\lambda-z}{1-\lambda z}\right)$ is an
eigenvector of $W_{f,\varphi}$ corresponding to the eigenvalue
$f(\lambda)\varphi^{\prime}(\lambda)$. Since
$W_{g,\psi}\in\\{W_{f,\varphi}\\}^{\prime}$, we have

$W_{f,\varphi}W_{g,\psi}\eta=W_{g,\psi}W_{f,\varphi}\eta=f(\lambda)\varphi^{\prime}(\lambda)W_{g,\psi}\eta.$

This shows that $W_{g,\psi}\eta$ is an eigenvector corresponding to the
eigenvalue $f(\lambda)\varphi^{\prime}(\lambda)$. Therefore,
$W_{g,\psi}\eta=\gamma_{2}\eta$. Thus we have

(2.4) $\begin{split}W_{g,\psi}\frac{1}{1-\lambda
z}\left(\frac{\lambda-z}{1-\lambda z}\right)&=\gamma_{2}\frac{1}{1-\lambda
z}\left(\frac{\lambda-z}{1-\lambda z}\right)\\\
g(z)\frac{1}{1-\lambda\psi(z)}\left(\frac{\lambda-\psi(z)}{1-\lambda\psi(z)}\right)&=\gamma_{2}\frac{1}{1-\lambda
z}\left(\frac{\lambda-z}{1-\lambda z}\right)\\\
W_{g,\psi}K_{\bar{\lambda}}(z)\left(\frac{\lambda-\psi(z)}{1-\lambda\psi(z)}\right)&=\gamma_{2}\frac{1}{1-\lambda
z}\left(\frac{\lambda-z}{1-\lambda z}\right)\\\
\gamma_{1}K_{\bar{\lambda}}(z)\left(\frac{\lambda-\psi(z)}{1-\lambda\psi(z)}\right)&=\gamma_{2}\frac{1}{1-\lambda
z}\left(\frac{\lambda-z}{1-\lambda z}\right)\\\
\gamma_{1}K_{\bar{\lambda}}(z)\left(\frac{\lambda-\psi(z)}{1-\lambda\psi(z)}\right)&=\gamma_{2}K_{\bar{\lambda}}(z)\left(\frac{\lambda-z}{1-\lambda
z}\right)\\\
\left(\frac{\lambda-\psi(z)}{1-\lambda\psi(z)}\right)&=\alpha\left(\frac{\lambda-z}{1-\lambda
z}\right).\end{split}$

Thus we conclude that
$\psi(z)=\frac{(\lambda^{2}-\alpha)z+\lambda(\alpha-1)}{\lambda(1-\alpha)z+(\lambda^{2}\alpha-1)}$.

Since $W_{g,\psi}K_{\bar{\lambda}}=\gamma_{1}K_{\bar{\lambda}}$ for some
nonzero $\gamma_{1}\in\mathbb{C}$, for all $z\in\mathbb{D}$, we have

$\frac{g(z)}{1-\lambda\psi(z)}=\frac{\gamma_{1}}{1-\lambda z}.$

If we put $z=\lambda$, we have

$\frac{g(\lambda)}{1-\lambda\psi(\lambda)}=\frac{\gamma_{1}}{1-\lambda^{2}}.$

Thus by Lemma 2.1, we have $\gamma_{1}=g(\lambda)$. Therefore,

$\begin{split}g(z)&=g(\lambda)\frac{1-\lambda\psi(z)}{1-\lambda z}\\\
&=\frac{g(\lambda)}{1-\lambda z}(1-\lambda\psi(z))\\\
&=\frac{g(\lambda)}{1-\lambda
z}\left(1-\lambda\frac{(\lambda^{2}-\alpha)z+\lambda(\alpha-1)}{\lambda(1-\alpha)z+(\lambda^{2}\alpha-1)}\right)\\\
&=g(\lambda)\frac{\lambda^{2}-1}{\lambda(1-\alpha)z+(\lambda^{2}\alpha-1)}\end{split}$

Now set $d_{0}=\frac{\lambda(\alpha-1)}{\lambda^{2}\alpha-1}=d_{1}$,
$d_{2}=\alpha\frac{(\lambda^{2}-1)^{2}}{(\lambda^{2}\alpha-1)^{2}}$, and
$d_{3}=\frac{\lambda^{2}-1}{\lambda^{2}\alpha-1}$, then

(2.5) $\psi(z)=d_{0}+\frac{d_{2}z}{1-d_{1}z}\quad\text{and}\quad
g(z)=g(\lambda)\frac{d_{3}}{1-d_{1}z}.$

On the other hand, consider the converse assertions to be true. Since
$W_{g,\psi}W_{f,\varphi}=$ $M_{g(f\circ\psi)}C_{\varphi\circ\psi}$ and
$W_{f,\varphi}W_{g,\psi}=M_{f(g\circ\phi)}C_{\psi\circ\phi}$, it suffices to
show that $g(f\circ\psi)=f(g\circ\varphi)$ and
$\varphi\circ\psi=\psi\circ\varphi$. It is clear when $\lambda=0$. Assume
$\lambda\neq 0$.

(2.6)
$\begin{split}(g(f\circ\psi))(z)&=\frac{g(\lambda)d_{3}}{1-d_{1}z}\times\frac{f(0)}{1-a_{0}\psi(z)}\\\
&=\frac{g(\lambda)d_{3}}{1-d_{1}z}\times\frac{f(0)}{1-a_{0}\frac{d_{0}+\left(d_{2}-d_{0}d_{1}\right)z}{1-d_{1}z}}\\\
&=\frac{g(\lambda)d_{3}f(0)}{\left(1-a_{0}d_{0}\right)-\left[d_{1}+a_{0}\left(d_{2}-d_{0}d_{1}\right)\right]z}.\end{split}$

(2.7)
$\begin{split}(f(g\circ\varphi))(z)&=\frac{f(0)}{1-a_{0}z}\times\frac{g(\lambda)d_{3}}{1-d_{1}\varphi(z)}\\\
&=\frac{g(\lambda)d_{3}f(0)}{\left(1-a_{0}d_{1}\right)-\left[a_{0}+d_{1}\left(a_{1}-a_{0}^{2}\right)\right]z}\end{split}$

Therefore, by using (1) of Lemma 2.2, we conclude that
$(g(f\circ\psi))(z)=(f(g\circ\varphi))(z)$.

(2.8)
$\begin{split}(\varphi\circ\psi)(z)&=\frac{a_{0}+(a_{1}-a_{0}^{2})\psi(z)}{1-a_{0}\psi(z)}\\\
&=\frac{a_{0}+\left(a_{1}-a_{0}^{2}\right)d_{0}+\left[-a_{0}d_{1}+\left(a_{1}-a_{0}^{2}\right)\left(d_{2}-d_{0}d_{1}\right)\right]z}{1-a_{0}d_{0}+\left[-d_{1}-a_{0}\left(d_{2}-d_{0}d_{1}\right)\right]z}.\end{split}$

and

(2.9)
$\begin{split}(\psi\circ\varphi)(z)&=\frac{d_{0}+\left(d_{2}-d_{0}d_{1}\right)\varphi(z)}{1-d_{1}\varphi(z)}\\\
&=\frac{d_{0}+\left(d_{2}-d_{0}d_{1}\right)\frac{a_{0}+\left(a_{1}-a_{0}^{2}\right)z}{1-a_{0}z}}{1-d_{1}\frac{a_{0}+\left(a_{1}-a_{0}^{2}\right)z}{1-a_{0}z}}\\\
&=\frac{d_{0}+\left(d_{2}-d_{0}d_{1}\right)a_{0}+\left[-a_{0}d_{0}+\left(d_{2}-d_{0}d_{1}\right)\left(a_{1}-a_{0}^{2}\right)\right]z}{1-a_{0}d_{1}+\left[-a_{0}-d_{1}\left(a_{1}-a_{0}^{2}\right)\right]z}.\end{split}$

Again using (1) of Lemma 2.2, we obtain
$(\varphi\circ\psi)(z)=(\psi\circ\varphi)(z)$. This completes the proof.

∎

Using [15, Remark 4.8] and Remark 2.3, we can easily prove the following:

######  Corollary 2.5.

Let $g\in H^{\infty}$ and $\psi$ be an analytic map of $\mathbb{D}$ into
itself. Assume that $W_{g,\psi}\in\\{W_{f,\varphi}\\}^{\prime}$, where
$W_{f,\varphi}$ is complex symmetric with the conjugation $\mathcal{J}$. If
$\varphi$ is not an elliptic automorphism and has a fixed point, say $\lambda$
in $\mathbb{D}$. Then the following holds true:

  1. (1)

If $\lambda$ is real, then $W_{g,\psi}$ is normal.

  2. (2)

If both $\lambda$ and $\alpha$ are real, then $W_{g,\psi}$ is self-adjoint.

  3. (3)

If $|d_{0}|<1$ and $2|d_{0}+\bar{d}_{0}\left(d_{2}-d^{2}_{0}\right)|\leq
1-|d_{2}-d_{0}^{2}|$, then $W_{g,\psi}$ is $\mathcal{J}$-symmetric.

######  Corollary 2.6.

Let $g\in H^{\infty}$ and $\psi$ be an analytic map of $\mathbb{D}$ into
itself. Assume that $W_{g,\psi}\in\\{W_{f,\varphi}\\}^{\prime}$ where
$W_{f,\varphi}$ is complex symmetric with the conjugation $\mathcal{J}$. If
$\varphi$ has a fixed point, say $\lambda$ in $\mathbb{D}$, then the space
$\mathcal{M}=\\{h\in H^{2}\mathrel{\mathop{\mathchar
58\relax}}h(\lambda)=0\\}$ is invariant subspace for $W_{g,\psi}$.

###### Proof.

By Lemma 2.1, $\psi$ has the same fixed point $\lambda$. Therefore, for any
$h\in\mathcal{M}$

$W_{g,\psi}h(\lambda)=g(\lambda)h(\psi(\lambda))=g(\lambda)h(\lambda)=0$

∎

######  Remark 2.7.

  1. (1)

$\mathcal{M}=B_{\lambda}H^{2}$, where
$B_{\lambda}(z)=\frac{z-\lambda}{1-\bar{\lambda}z}$ and $B_{\lambda}H^{2}$ is
invariant for $C_{\psi}$. Therefore, by employing [1, Theorem 2.3], we can
deduce that $\frac{B_{\lambda}\circ\psi}{B_{\lambda}}$ belongs to the class
denoted as $\mathcal{S}(\mathbb{D})$. In this context,
$\mathcal{S}(\mathbb{D})$ refers to the collection of functions that are both
holomorphic and have a modulus bounded by one on the domain $\mathbb{D}$,
formally defined as:

$\mathcal{S}(\mathbb{D})=\\{g\in
H^{\infty}(\mathbb{D})\mathrel{\mathop{\mathchar
58\relax}}\|g\|_{\infty}\mathrel{\mathop{\mathchar
58\relax}}=\sup_{z\in\mathbb{D}}|g(z)|\leq 1\\}.$

This particular set is recognized as the Schur class, and its elements are
known as Schur functions.

  2. (2)

If we assume that the weight function $\psi$ is identically $1$, and the
composition operator $C_{\varphi}$ is complex symmetric, then the symbol
$\varphi$ possess a fix point. Therefore, we can relax the assumption that
$\varphi$ has a fixed point from our results.

Acknowledgements

The research of the author is supported by the NBHM postdoctoral fellowship,
Department of Atomic Energy (DAE), Government of India (File No:
0204/16(21)/2022/R&D-II/11995) and the INSPIRE grant of Dr. Srijan Sarkar
(Ref: DST/INSPIRE/04/2019/000769), Department of Science & Technology (DST),
Government of India.

## References

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#

Yuri Kulchitsky1,2
on behalf of the ATLAS collaboration
1Institute of Physics, National Academy of Sciences, Minsk, Belarus
2JINR, Dubna, Russia

#
Measurement of the underlying-event properties with the ATLAS detector

Yuri Kulchitsky1,2
on behalf of the ATLAS collaboration
1Institute of Physics, National Academy of Sciences, Minsk, Belarus
2JINR, Dubna, Russia

###### Abstract

A correct modelling of the underlying event in proton-proton collisions is
important for the proper simulation of kinematic distributions of high-energy
collisions. The ATLAS collaboration extended previous studies at 7 TeV with a
leading track or jet or Z boson by a new study of Drell-Yan events in 1.1 fb-1
of data collected at a center-of-mass energy of 7 TeV. In this new study the
distributions of several topological event-shape variables based on charged
particles are measured, both integrated and differential in the transverse
momentum of the Drell-Yan lepton pair. These measurements are sensitive to the
underlying-event as well as the onset of hard emissions. The results have been
compared with the predictions of several state-of-the-art MC generators. The
collaboration has also performed a first study of the number and transverse
momentum sum of charged particles as a function of transverse momentum and
azimuthal angle in a special data set taken with low beam currents at a
center-of-mass energy of 13 TeV. The results are compared to predictions of
several MC generators.

##  1 Introduction

The Large Hadron Collider (LHC) was primarily built to explore the mechanism
of electroweak symmetry breaking and to search for new physics beyond the
Standard Model (SM) in proton-proton collisions characterised by parton-parton
scatterings with a high momentum transfer. These parton-parton scatterings are
unavoidably accompanied by interactions between the proton remnants which are
often called the “underlying event” (UE) and have to be modelled well in order
to be able to measure high-momentum-transfer processes to high accuracy. Since
the UE is dominated by low-scale strong-force interactions, in which the
strong coupling strength diverges and perturbative methods of quantum
chromodynamics (QCD) lose predictivity, it is extremely difficult to predict
UE-sensitive observables from an ab-initio calculation in QCD. As a result,
one has to rely on models implemented in general-purpose Monte Carlo (MC)
event generators. Generators such as Herwig 7 [1], Pythia 8 [2], Sherpa [3],
and EPOS [4] contain multiple partonic interactions (MPI) as well as QCD
radiation in the initial and final state to describe the UE. A correct
modelling of the underlying event in pp-collisions is important for the proper
simulation of kinematic distributions of high-energy collisions.

Event-shape observables measured with the ATLAS detector [5] at the LHC using
charged particles in inclusive Z-boson events are presented in Ref. [6]. The
measurements are based on an integrated luminosity of 1.1 fb-1 of pp-
collisions at a centre-of-mass energy of $\sqrt{s}=7$ TeV. Charged-particle
distributions, excluding the lepton-antilepton pair from the Z-boson decay,
are measured in different ranges of transverse momentum of the Z boson. These
distributions include multiplicity, scalar sum of transverse momenta, beam
thrust, transverse thrust, spherocity, and F-parameter, which are in
particular sensitive to properties of the underlying event at small values of
the Z-boson transverse momentum. A detector-level measurement of track
distributions sensitive to the properties of the underlying event is presented
in Ref. [7]. It is based on an integrated luminosity of 151 $\mu$b-1 of pp-
collisions at a centre-of-mass energy of $\sqrt{s}=13$ TeV. Distributions of
the track multiplicity and of the track transverse momentum are measured in
regions of azimuthal angle defined with respect to the leading track. The
measured distributions are compared with predictions from Monte Carlo event
generators. Earlier the underlying events for $Z$-boson and jets at 7 TeV were
studied with ATLAS detector (see Ref. [8]).

##  2 Event-shape observables for Drell-Yan events

Events were selected by requiring a Z-boson candidate decaying to an
$e^{+}e^{-}$ or $\mu^{+}\mu^{-}$ pair. The charged-particle event-shape
observables beam thrust, transverse thrust, spherocity, and $F$-parameter as
defined below were measured in inclusive Z production. The observables were
calculated for primary charged particles with $p_{\rm T}>0.5$ GeV and
$|\eta|<2.5$. At small $p_{\rm T}(l^{+}l^{-})$ values, events are expected to
have low jet activity from the hard process and hence high sensitivity to UE
characteristics. At high $p_{\rm T}(l^{+}l^{-})$ values, the event is expected
to contain at least one jet of high transverse momentum recoiling against the
$l^{+}l^{-}$ system, which is expected to be reasonably described by
perturbative calculations of the hard process.

Distributions $f_{O}=1/N_{ev}\cdot dN/dO$ were measured for all selected
events, $N_{ev}$, for the following observables $O$. 1) The charged-particle
multiplicity, $N_{ch}$. 2) The scalar sum of transverse momenta of selected
charged particles, $\sum_{i}p_{\rm T,i}=\sum p_{\rm T}$. 3) The beam thrust,
$B$, as proposed in Refs. [11, 12, 13]. This is similar to $\sum p_{\rm T}$
except that in the sum over all charged particles the transverse momentum of
each particle is weighted by a factor depending on its $\eta$:
$B=\sum_{i}p_{\rm T,i}\cdot e^{-|\eta_{i}|}$. As a result, contributions from
particles in the forward and backward direction are suppressed with respect to
particles emitted at central $\eta$. 4) The transverse thrust, $T$, as
proposed in Ref. [14]: $T=\max_{\vec{n}_{\rm T}}\frac{\sum_{i}|\vec{p}_{\rm
T,i}\cdot\vec{n}_{\rm T}|}{\sum_{i}p_{\rm T,i}}$, where the sum runs over all
charged particles, and the thrust axis, $\vec{n}_{\rm T}$, maximises the
expression. The solution for $\vec{n}_{\rm T}$ is found iteratively following
the algorithm described in Ref. [15]. 5) The spherocity, $S$, as proposed in
Ref. [14]: $S=\frac{\pi^{2}}{4}\min_{\vec{n}=(n_{x},n_{y},0)^{\rm
T}}\left(\frac{\sum_{i}\left|\vec{p}_{\rm
T,i}\times\vec{n}\right|}{\sum_{i}\vec{p}_{\rm T,i}}\right)^{2}$, where the
sum runs over all charged particles and the vector $\vec{n}$ minimises the
expression. 6) The $F$-parameter defined as the ratio of the smaller and
larger eigenvalues, $\lambda_{1}$ and $\lambda_{2}$
$F=\frac{\lambda_{1}}{\lambda_{2}}$ of the transverse momentum tensor
$M^{lin}$ in Ref. [6]. Pencil-like events, e.g. containing two partons emitted
in opposite directions in the transverse plane, are characterised by values of
$S$, $T$, and $F$ close to 0, 1, and 0, respectively. The corresponding values
of these observables for spherical events, e.g. containing several partons
emitted isotropically, are close to 1, $2/\pi$, and 1, respectively. While the
event-shape observables $S$, $T$, and $F$ show very high correlations among
themselves, they are weakly correlated with $N_{ch}$, $\vec{p}_{\rm T}$, and
$B$. Each observable was determined in the following ranges of the transverse
momentum of the Z boson, $p_{\rm T}(l^{+}l^{−})$, calculated from the four-
momenta of the $l^{+}l^{−}$ pair: 0–6, 6–12, 1–25, and $\geq 25$ GeV. Events
at small $p_{\rm T}(l^{+}l^{−})$ are expected to be particularly sensitive to
the UE activity, while events with large $p_{\rm T}(l^{+}l^{−})$ values are
expected to contain significant contributions from jet production coming from
the hard-scattering process.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1:  Distributions of the event-shape variables (a) charged-particle
multiplicity, (b) summed transverse momenta, (c) beam thrust, (d) transverse
thrust, (e) spherocity, and (f) $F$-parameter measured in $Z\rightarrow
e^{+}e^{-}$ events for the different ranges of $p_{\rm T}(e^{+}e^{-})$. Taken
from Ref. [6].

For comparison with corrected distributions, three recent versions of MC event
generators were used to provide predictions for the signal at particle level:
Sherpa 2.2.0 with up to two additional partons at NLO and with three
additional partons at LO and taking the NLO matrix element calculations for
virtual contributions from OpenLoops [16] with the NNPDF 3.0 NNLO PDF set
[17]; Pythia 8.212 with LO matrix element calculations using the NNPDF2.3 LO
PDF set [18]; Herwig 7.0 [1] taking the NLO matrix element calculations for
real emissions from MadGraph [19] and for virtual contributions from OpenLoops
using the MMHT2014 PDF set [20].

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 2:  (a–d) Multiplicity, (e–h) summed transverse momenta and (i–l) beam
thrust distributions of charged particles for $Z\rightarrow e^{+}e^{-}$ for
the $p_{\rm T}(e^{+}e^{-})$ ranges compared to the predictions from the MC
generators. Taken from Ref. [6].

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 3:  (a–d) Transverse thrust, (e–h) spherocity and (i–l) $F$-parameter
distributions of charged particles for $Z\rightarrow e^{+}e^{-}$ for the
$p_{\rm T}(e^{+}e^{-})$ ranges compared to the predictions from the MC
generators. Taken from Ref. [6].

The results from the electron and muon channels are in a good agreement. The
Figures show only results for the electron-positron channel. Figure 1 shows
the unfolded electron channel results for the six observables in the various
$p_{\rm T}(l^{+}l^{-})$ ranges. As $p_{\rm T}(l^{+}l^{-})$ rises, i.e. as
recoiling jets emerge, the number of produced charged particles $N_{ch}$
increases, as do $\sum p_{\rm T}$ and beam thrust. Correspondingly, transverse
thrust moves towards higher values and spherocity towards smaller values as a
result of the increasing jettiness of the events. Figures 2 and 3 show the
individual event-shape observables for the electron channel compared to
predictions obtained with the most recent versions of three different MC
generators. In general, Pythia 8 and Herwig 7 agree better with the data than
does Sherpa. The $p_{\rm T}(l^{+}l^{-})<6$ GeV bin is expected to be
characterised by low jet activity from the hard matrix element and hence
should be particularly sensitive to UE characteristics. In this case, Pythia 8
shows very good agreement with the data in the event-shape observables that
are not very sensitive to the number of charged particles ($T$, $S$, and
$F$-parameter). The observables that depend explicitly on the number of
charged particles ($N_{ch}$, $\sum p_{\rm T}$, $B$) are less well described,
with none of the generators succeeding fully. In this case, the best agreement
is observed for Herwig 7 while Pythia 8 still performs better than Sherpa. Low
$N_{ch}$ and $\sum p_{\rm T}$ values represent a challenging region for all
three generators. This region might be particularly sensitive to the way beam-
remnant interactions are modelled in the MC generators. Similar observations
can be made for $p_{\rm T}(l^{+}l^{-})$ ranges 6–12 and 12–25 GeV. At low
values of $B$, the observable in which tracks with larger $|\eta|$ values
contribute less to the sum of the track transverse momenta, better agreement
of the generator predictions with the data is observed than at low $\sum
p_{\rm T}$. At $p_{\rm T}(l^{+}l^{-})\geq 25$ GeV the event is expected to
contain at least one jet of high transverse momentum recoiling against the Z
boson, which is expected to be well described by the hard matrix element. In
this case, one still observes significant deviations of the MC generators from
the measurement, where, depending on the observable, either Herwig 7 or Pythia
8 shows in general the best agreement. However, all three generators show
better agreement with data compared to the $p_{\rm T}(l^{+}l^{-})<6$ GeV
range.

(a)

(b)

(c)

(d)

Figure 4:  Comparison of detector-level data and MC predictions for (a, b)
$\langle d^{2}N_{\rm ch}/d\eta d\phi\rangle$ and (c, d) $\langle d^{2}\sum
p_{\rm T}/d\eta d\phi\rangle$ as a function of $p^{\rm lead}_{\rm T}$ in the
transverse (a, c) and toward (b, d) regions. Taken from Ref. [7].

##  3  Leading track underlying event distributions

A detector-level measurement of track distributions sensitive to the
properties of the underlying event is given in Ref. [7]. It is based on 9
million events collected using the ATLAS detector in pp collisions at 13 TeV,
corresponding to an integrated luminosity of 151 $\mu$b-1. The underlying
event (UE) is defined as the activity accompanying any hard scattering in a
collision event. This includes partons not participating in the hard-
scattering process (beam remnants), and additional scatters in the same pp
collision, termed multiple parton interactions (MPI). Initial and final state
gluon radiation (ISR, FSR) also contribute to the UE activity. The direction
of the leading track is used to define regions in the azimuthal plane that
have different sensitivity to the UE. The direction of the leading track is
used to define regions in the azimuthal plane that have different sensitivity
to the UE, a concept first used in [23]. The azimuthal angular difference
between tracks and the leading track
$|\Delta\phi|=|\phi-\phi^{lead}_{track}|$, is used to define the following
three azimuthal UE regions: 1) $|\Delta\phi|<60^{\circ}$, the toward region,
2) $60^{\circ}<|\Delta\phi|<120^{\circ}$, the transverse region, and 3)
$|\Delta\phi|>120^{\circ}$, the away region. The transverse region is
sensitive to the underlying event, since it is by construction perpendicular
to the direction of the hard scatter and hence it is expected to have a lower
level of activity from the hard scattering process compared to the away
region. The observables measured in this analysis are number of tracks per
unit of $\eta$–$\phi$, $\langle d^{2}N_{\rm ch}/d\eta d\phi\rangle$; and
scalar sum of track $p_{\rm T}$ per unit of $\eta$–$\phi$, $\langle d^{2}\sum
p_{\rm T}/d\eta d\phi\rangle$. The average density of particles, $\langle
d^{2}N_{\rm ch}/d\eta d\phi\rangle$, and average transverse momentum sum
density, $\langle d^{2}\sum p_{\rm T}/d\eta d\phi\rangle$, are constructed by
dividing the mean values by the corresponding area. The leading track is
included in the toward region distributions, unless otherwise stated. The
results are presented at detector-level, without any corrections (except in
MC, the width of the vertex distribution along the $Z$ axis is reweighted to
match the data). The uncertainty resulting from tracking inefficiency is
estimated by randomly dropping tracks according to the tracking efficiency
uncertainty determined as a function of track $p_{\rm T}$ and $\eta$. In Fig.
4 the average densities of track multiplicity and scalar $\sum p_{\rm T}$ are
shown. In the transverse region, both show a gradual increase, rising to an
approximately constant “plateau” for $p^{\rm lead}_{\rm T}>6$ GeV. For higher
values of $p^{\rm lead}_{\rm T}$, the toward and away regions include
contributions from jet-like activity, yielding gradually rising $\langle\sum
p_{\rm T}\rangle$ density. Among the MC models, Pythia 8 A14, Monash [2] and
Herwig++ [1] predictions are closest to data in the plateau part of the
transverse region, but none of the models describe the initial rise well. The
EPOS [4] generator predicts significantly less activity at higher $p^{\rm
lead}_{\rm T}$, indicating the absence of semi-hard MB events.

##  4 Summary

A correct modelling of the underlying event in pp-collisions is important for
the proper simulation of kinematic distributions of high-energy collisions.
Event-shape observables sensitive to the underlying event were measured in 1.1
fb-1 integrated luminosity of pp-collisions collected with the ATLAS detector
at the LHC at a centre-of-mass energy of 7 TeV. The distributions of several
topological event-shape variables based on charged particles are measured for
events containing an oppositely charged electron or muon pair with an
invariant mass close to the Z-boson mass. The collaboration has also performed
a first study of the number and transverse momentum sum of charged particles
as a function of transverse momentum and azimuthal angle in a special data set
taken with low beam currents with integrated luminosity of 151 $\mu$b-1 at a
center-of-mass energy of 13 TeV. The results are compared to predictions of
several MC generators. These are required for tuning of the soft part of Monte
Carlo simulation.

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