# Polyanalytic Besov spaces and approximation by dilatations

Ali Abkar
Department of Pure Mathematics, Faculty of Science,
Imam Khomeini International University, Qazvin 34149, Iran

Email<EMAIL_ADDRESS>

Abstract. Using partial derivatives $\partial_{z}f$ and
$\partial_{\overline{z}}f$, we introduce Besov spaces of polyanalytic
functions on the unit disk and on the upper half-plane. We then prove that the
dilatations of each function in polyanalytic Besov spaces converge to the same
function in norm. This opens the way for the norm approximation of functions
in polyanalytic Besov spaces by polyanalytic polynomials.

Keywords: Mean approximation, dilatation, polyanalytic function, polyanalytic
Besov space

MSC(2020): 30E10, 30H20, 46E15

##  1 Introduction

Let $\Omega$ be a domain in the complex plane. A function
$f:\Omega\to\mathbb{C}$ is called $q-analytic$, for $q\in\mathbb{N}$, if $f\in
C^{q}(\Omega)$ and

$\big{(}\frac{\partial}{\partial\overline{z}}\big{)}^{q}f=\frac{1}{2^{q}}\Big{(}\frac{\partial}{\partial
x}+i\frac{\partial}{\partial y}\Big{)}^{q}f=0,\quad z=x+iy.$

A $q$-analytic function $f$ is called a polyanalytic function of degree $q$;
or briefly a polyanalytic function. It is well-known that any polyanalytic
function $f$ can be written as (see [4]):

$f(z)=h_{0}(z)+\overline{z}h_{1}(z)+\cdots+\overline{z}^{q-1}h_{q-1}(z),$ (1)

where $h_{k},\,0\leq k\leq q-1$, are called the analytic components of $f$.
Compared to analytic functions, polyanalytic functions might behave in a
different manner; for example, they may vanish on a curve without vanishing
identically in the whole domain (see [3]).

Let $L^{p}(\Omega,dA),\,0<p<\infty$, denote the space of measurable functions
on $\Omega$ equipped with the usual $L^{p}$ norm; here $dA$ is the area
measure on $\Omega$. The polynalytic Bergman space
$\mathcal{A}^{p}_{q}(\Omega)$ is defined as the space of $q$-analytic
functions $f$ on $\Omega$ which belong to $L^{p}(\Omega)$ as well. The norm of
$f\in\mathcal{A}^{p}_{q}(\Omega)$ is given by

$\|f\|_{\mathcal{A}^{p}_{q}(\Omega)}=\left(\int_{\Omega}|f(z)|^{p}dA(z)\right)^{1/p}<\infty.$

A weight is a positive integrable function $w:\Omega\to[0,\infty)$. We say
that $f$ belongs to the weighted polyanalytic Bergman space
$\mathcal{A}^{p}_{q,w}(\Omega)$ if

$\|f\|_{\mathcal{A}^{p}_{q,w}(\Omega)}=\left(\int_{\Omega}|f(z)|^{p}w(z)dA(z)\right)^{1/p}<\infty.$

Sometimes we use the short term (weighted) poly-Bergman space to refer to this
space.

In this paper, $\Omega$ is either the unit disk
$\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ or the upper half-plane
$\mathbb{C}^{+}$. Let us begin by the unit disk. We recall that the weighted
Dirichlet space $D^{p}_{w},\,0<p<\infty$, consists of analytic functions in
the unit disk for which

$\|f\|^{p}_{D^{p}_{w}}=|f(0)|^{p}+\int_{\mathbb{D}}|f^{\prime}(z)|^{p}w(z)dA(z)$

is finite. Similarly, the weighted analytic Besov space $B_{w}^{p}$ consists
of analytic functions $f$ on the unit disk for which the integral

$\int_{\mathbb{D}}(1-|z|^{2})^{p-2}|f^{\prime}(z)|^{p}w(z)dA(z)<\infty.$

In the following, we present a definition for the polyanalytic Besov space by
using partial differential operators $\partial_{z}f$ and
$\partial_{\overline{z}}f$ instead of $f^{\prime}(z)$. Let $f$ be a
$q$-analytic function on the unit disk, and recall the differential operators
($z=x+iy$):

$\partial_{z}f(z)=\frac{1}{2}\left(\frac{\partial f}{\partial
x}-i\frac{\partial f}{\partial
y}\right),\quad\partial_{\overline{z}}f(z)=\frac{1}{2}\left(\frac{\partial
f}{\partial x}+i\frac{\partial f}{\partial y}\right).$

For a weight function $w$, we say that a $q$-analytic function $f$ belongs to
the weighted polyanalytic Dirichlet space $\mathcal{D}^{p}_{q,w}$ if

$\int_{\mathbb{D}}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(z)dA(z)<\infty.$

The norm of a function in $\mathcal{D}^{p}_{q,w}$ is given by

$\|f\|_{\mathcal{D}^{p}_{q,w}}=\left(|f(0)|^{p}+\int_{\mathbb{D}}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(z)dA(z)\right)^{1/p}.$

When $p=2,\,\mathcal{D}^{2}_{q,w}$ is a Hilbert space of polyanalytic
functions whose inner product is

$\langle f,g\rangle_{\mathcal{D}^{2}_{q,w}}=\langle
f(0),\overline{g(0)}\rangle+\int_{\mathbb{D}}\Big{[}\partial_{z}f(z)\overline{\partial_{z}g(z)}+\partial_{\overline{z}}f(z)\overline{\partial_{\overline{z}}g(z)}\Big{]}w(z)dA(z).$

Similarly, we define the weighted polyanalytic Besov space
$\mathcal{B}_{q,w}^{p}$ as the space of $q$-analytic functions $f$ on
$\mathbb{D}$ for which

$\int_{\mathbb{D}}(1-|z|^{2})^{p-2}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(z)dA(z)<\infty.$

The norm of a function in $\mathcal{B}_{q,w}^{p}$ is given by

$\|f\|_{\mathcal{B}^{p}_{q,w}}=\left(|f(0)|^{p}+\int_{\mathbb{D}}(1-|z|^{2})^{p-2}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(z)dA(z)\right)^{1/p}.$

In case that $\Omega$ equals the upper half-plane
$\mathbb{C}^{+}=\\{z\in\mathbb{C}:\operatorname{Im}(z)>0\\}$, the definition
of polyanalytic Besov space on the upper half-plane is slightly different;
just replace $f(0)$ by $f(i)$ and $dA(z)$ by a suitable Gaussian weight
function (see §3). Note that when $q=1$ (which means that $f$ is analytic), we
have $\partial_{z}f(z)=f^{\prime}(z)$, and $\partial_{\overline{z}}f(z)=0$, so
that the polyanalytic Besov (resp. Dirichlet) space reduces to the analytic
Besov (resp. Dirichlet) space. Therefore, the spaces defined above are natural
generalizations of the classical Dirichlet and Besov spaces. We shall at times
refer to a polyanalytic Besov space as a poly-Besov space; likewise, a
polyanalytic Dirichlet space is referred to as poly-Dirichlet space.

It is a natural question in operator theory of function spaces to decide
whether the polynomials are dense in the space or not. To answer this
question, the most important strategy is to see if the dilatations

$f_{r}(z)=f(rz),\quad z\in\Omega,\,0<r<1,$

are convergent to $f$. If this holds true, and if each $f_{r}$ can be
approximated by the polynomials, then we are done. We should point out that
for weighted spaces with radial weights (weights that depend only to $|z|$)
the problem is rather well-known. In this paper we consider non-radial weights
$w:\Omega\to(0,\infty)$ that satisfy the following condition:
There is a constant $C>0$, a non-negative integer $k$, and an $r_{0}\in(0,1)$
such that

$r^{k}w\left(\frac{z}{r}\right)\leq Cw(z),\quad|z|<r,\,r_{0}\leq r<1.$ (2)

Under this condition, we prove that the dilatations $f_{r}$ converge to $f$ in
the norm of the weighted polyanalytic Besov space, as $r\to 1^{-}$. If,
moreover, we assume that the weight $w$ is chosen in such a way that the
polyanalytic polynomials (polynomials in $z$ and $\overline{z}$) are included
in the weighted Besov space, and if each $f_{r}$ can be approximated by the
polyanalytic polynomials, then we are able to approximate each $f$ in the
weighted Besov space by polynomials of the form

$p_{k,m}(z,\overline{z})=\sum_{i=0}^{k}\sum_{j=0}^{m}c_{i,j}z^{i}\overline{z}^{j},\quad
c_{i,j}\in\mathbb{C}.$

The theory of polyanalytic functions has applications in signal analysis, and
in Gabor frames. In particular, Hilbert spaces of polyanalytic functions were
extensively used to model physical and engineering problems. The Bergman
spaces of polyanalytic functions and the estimation of its reproducing kernels
were recently studied by Haimi and Hedenmalm ([6], [7]). A good account of
polyanalytic Fock spaces of entire functions can be found in the expository
article by L.D. Abreu and H.G. Feichtinger [3]. The subject has also attracted
interest within operator theory of function spaces (see [13], [14]).

##  2 Polyanalytic Besov spaces in the unit disk

We begin by approximation in polyanalytic Dirichlet spaces.

###### Theorem 2.1

Let $0<p<\infty$ and let the weight $w$ satisfies the condition (2). Then the
polyanalytic polynomials are dense in the weighted polyanalytic Dirichlet
space $\mathcal{D}_{q,w}^{p}$.

Proof. It is enough to show that $f_{r}\to f$ in the norm of
$\mathcal{D}_{q,w}^{p}$. To this end, it suffices to work with the following
semi-norm (the constant term does not play any role in approximation):

$\|f\|^{p}_{\mathcal{D}^{p}_{q,w}}=\int_{\mathbb{D}}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(z)dA(z).$

We also note that

$\|f_{r}-f\|^{p}_{\mathcal{D}^{p}_{q,w}}=\int_{\mathbb{D}}|\partial_{z}f_{r}(z)-\partial_{z}f(z)|^{p}w(z)dA(z)+\int_{\mathbb{D}}|\partial_{\overline{z}}f_{r}(z)-\partial_{\overline{z}}f(z)|^{p}w(z)dA(z).$
(3)

It is easy to see that

$\partial_{z}f_{r}(z)=r\partial_{z}f(rz),\quad\partial_{\overline{z}}f_{r}(z)=r\partial_{\overline{z}}f(rz).$

By a change of variable (replace $z$ by $z/r$) and using the assumption on the
weight function, we obtain

$\displaystyle\int_{\mathbb{D}}|\partial_{z}f_{r}(z)|^{p}w(z)dA(z)$
$\displaystyle=r^{p-k-2}\int_{r\mathbb{D}}|\partial_{z}f(z)|^{p}r^{k}w\left(\frac{z}{r}\right)dA(z)$
$\displaystyle\leq
Cr^{p-k-2}\int_{r\mathbb{D}}|\partial_{z}f(z)|^{p}w(z)dA(z).$

Therefore we can apply the dominated convergence theorem to get

$\limsup_{r\to
1^{-}}\int_{\mathbb{D}}|\partial_{z}f_{r}(z)|^{p}w(z)dA(z)\leq\int_{\mathbb{D}}|\partial_{z}f(z)|^{p}w(z)dA(z).$

This implies that

$\lim_{r\to
1^{-}}\int_{\mathbb{D}}|\partial_{z}f_{r}(z)-\partial_{z}f(z)|^{p}w(z)dA(z)=0,$

which means that the first term in $(\ref{dir-1})$ tends to zero. Similarly,
we may verify that

$\lim_{r\to
1^{-}}\int_{\mathbb{D}}|\partial_{\overline{z}}f_{r}(z)-\partial_{\overline{z}}f(z)|^{p}w(z)dA(z)=0,$

from which we obtain

$\lim_{r\to 1^{-}}\|f_{r}-f\|^{p}_{\mathcal{D}^{p}_{q,w}}=0.$

$\square$

We now prove a similar statement for polyanalytic Besov spaces.

###### Theorem 2.2

Let $2\leq p<\infty$ and let the weight $w$ satisfies the condition (2). Then
the polyanalytic polynomials are dense in the weighted polyanalytic Besov
space $\mathcal{B}^{p}_{q,w}$.

Proof. As in the preceding theorem, it suffices to verify that for each
$f\in\mathcal{B}^{p}_{q,w}$ we have

$\limsup_{r\to
1^{-}}\int_{\mathbb{D}}(1-|z|^{2})^{p-2}|\partial_{z}f_{r}(z)|^{p}w(z)dA(z)\leq\int_{\mathbb{D}}(1-|z|^{2})^{p-2}|\partial_{z}f(z)|^{p}w(z)dA(z),$
(4)

and

$\limsup_{r\to
1^{-}}\int_{\mathbb{D}}(1-|z|^{2})^{p-2}|\partial_{\overline{z}}f_{r}(z)|^{p}w(z)dA(z)\leq\int_{\mathbb{D}}(1-|z|^{2})^{p-2}|\partial_{\overline{z}}f(z)|^{p}w(z)dA(z).$
(5)

To this end, we work with

$\|f\|^{p}_{\mathcal{B}^{p}_{q,w}}=\int_{\mathbb{D}}(1-|z|^{2})^{p-2}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(z)dA(z).$

Therefore, by a change of variables we have

$\displaystyle\|\partial_{z}f_{r}\|^{p}_{\mathcal{B}^{p}_{q,w}}$
$\displaystyle=\int_{\mathbb{D}}(1-|z|^{2})^{p-2}|\partial_{z}f_{r}(z)|^{p}w(z)dA(z)$
$\displaystyle=r^{p-k-2}\int_{r\mathbb{D}}\left(\frac{r^{2}-|z|^{2}}{r^{2}}\right)^{p-2}|\partial_{z}f(z)|^{p}r^{k}w\left(\frac{z}{r}\right)dA(z).$

But the function

$r\mapsto\left(\frac{r^{2}-|z|^{2}}{r^{2}}\right)^{p-2}$

is increasing in $r$ when $p\geq 2$. Now, we apply the monotone convergence
theorem to the last integral, and then invoke the dominated convergence
theorem to obtain

$\limsup_{r\to
1^{-}}\int_{\mathbb{D}}(1-|z|^{2})^{p-2}|\partial_{z}f_{r}(z)|^{p}w(z)dA(z)\leq\int_{\mathbb{D}}(1-|z|^{2})^{p-2}|\partial_{z}f(z)|^{p}w(z)dA(z),$

which proves $(\ref{bes-1})$. The proof of $(\ref{bes-2})$ is similar.
$\square$

##  3 Polyanalytic Besov spaces in the upper half-plane

To get an idea of how to define polyanalytic Besov spaces in the upper half-
plane, we start by recalling the definition of analytic Bergman spaces on the
upper-half plane. An analytic function $f$ on
$\mathbb{C}^{+}=\\{z=x+iy\in\mathbb{C}:\operatorname{Im}(z)=y>0\\}$ belongs to
the Bergman space if

$\int_{\mathbb{C}^{+}}|f(z)|^{p}\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)<\infty,\quad\alpha\geq
0,\,\beta\geq 0.$

In the literature, one usually encounters this definition when $\beta=0$ (see
[5]). Therefore, it is natural to define the weighted poly-Bergman space of
the upper half-plane, denoted by $\mathcal{A}^{p}_{q,w}(\mathbb{C}^{+})$, as
the space of $q$-analytic functions on $\mathbb{C}^{+}$ for which

$\|f\|^{p}_{\mathcal{A}^{p}_{q,w}(\mathbb{C}^{+})}=\int_{\mathbb{C}^{+}}|f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)<\infty.$

This suggests that the weighted Dirichlet space and the weighted Besov space
of analytic functions should be normed, respectively, by

$\|f\|^{p}_{D^{p}_{\mathbb{C}^{+},w}}=|f(i)|^{p}+\int_{\mathbb{C}^{+}}|f^{\prime}(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z),$

and

$\|f\|_{B^{p}_{\mathbb{C}^{+},w}}^{p}=|f(i)|^{p}+\int_{\mathbb{C}^{+}}|f^{\prime}(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha+p-2}e^{-\beta|z|^{2}}dA(z).$

The advantage of this definition is that, as in the unit disk case, when $p=2$
the Besov space reduces to the Dirichlet space. In analogy with the above, we
now declare the weighted Besov space of polyanalytic functions as the space of
$q$-analytic functions on $\mathbb{C}^{+}$ such that

$\|f\|_{\mathcal{B}^{p}_{\mathbb{C}^{+},w}}^{p}=|f(i)|^{p}+\int_{\mathbb{C}^{+}}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(z)\operatorname{Im}(z)^{\alpha+p-2}e^{-\beta|z|^{2}}dA(z)$

is finite. By our definition, when $p=2$, we get to the Dirichlet space of
polyanalytic functions on the upper half-plane, i.e., polyanalytic functions
on the upper half-plane for which

$\|f\|_{\mathcal{D}^{p}_{\mathbb{C}^{+},w}}^{p}=|f(i)|^{p}+\int_{\mathbb{C}^{+}}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)<\infty.$

###### Theorem 3.1

Let $0<p<\infty$ and let $w$ satisfies the condition (2). Then for each
$f\in\mathcal{D}^{p}_{q,w}(\mathbb{C}^{+})$, the dilatations $f_{r}$ converge
to $f$ in norm.

Proof. As for approximation, it is enough to work with the following semi-
norm:

$\|f\|^{p}_{\mathcal{D}^{p}_{q,w}(\mathbb{C}^{+})}=\int_{\mathbb{C}^{+}}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z).$

We also note that

$\displaystyle\|f_{r}-f\|^{p}_{\mathcal{D}^{p}_{q,w}(\mathbb{C}^{+})}$
$\displaystyle=\int_{\mathbb{C}^{+}}|\partial_{z}f_{r}(z)-\partial_{z}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)$
$\displaystyle+\int_{\mathbb{C}^{+}}|\partial_{\overline{z}}f_{r}(z)-\partial_{\overline{z}}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z).$
(6)

Making a change of variable, and using the fact that

$\partial_{z}f_{r}(z)=r\partial_{z}f(rz),$

we obtain

$\displaystyle\int_{\mathbb{C}^{+}}|\partial_{z}f_{r}(z)|^{p}w(z)$
$\displaystyle\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)$
$\displaystyle=r^{p-(k+\alpha+2)}\int_{\mathbb{C}^{+}}|\partial_{z}f(z)|^{p}r^{k}w\left(\frac{z}{r}\right)\operatorname{Im}(z)^{\alpha}e^{\frac{-\beta|z|^{2}}{r^{2}}}dA(z)$
$\displaystyle\leq
Cr^{p-(k+\alpha+2)}\int_{\mathbb{C}^{+}}|\partial_{z}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{\frac{-\beta|z|^{2}}{r^{2}}}dA(z)$

Therefore the dominated convergence theorem applies to ensure

$\limsup_{r\to
1^{-}}\int_{\mathbb{C}^{+}}|\partial_{z}f_{r}(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)\leq\int_{\mathbb{C}^{+}}|\partial_{z}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z).$

This entails that the first term on the right-hand side of $(\ref{dir-up-1})$
tends to zero:

$\lim_{r\to
1^{-}}\int_{\mathbb{C}^{+}}|\partial_{z}f_{r}(z)-\partial_{z}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)=0.$

Similarly, one shows that the second term on the right-hand side of
$(\ref{dir-up-1})$ tends to zero, from which we obtain

$\lim_{r\to 1^{-}}\|f_{r}-f\|^{p}_{\mathcal{D}^{p}_{q,w}(\mathbb{C}^{+})}=0.$

$\square$

For the Bergman space, the problem can be settled more easily.

###### Corollary 3.2

Let $0<p<\infty$ and let $w$ satisfies the condition (2). Then for each
function $f\in\mathcal{A}^{p}_{q,w}(\mathbb{C}^{+})$, the dilatations $f_{r}$
converge to $f$ in norm.

Proof. Recalling the definition of norm in
$\mathcal{A}^{p}_{q,w}(\mathbb{C}^{+})$, we just note that

$\displaystyle\|f_{r}\|^{p}_{\mathcal{A}^{p}_{q,w}{(\mathbb{C}^{+})}}$
$\displaystyle=\int_{\mathbb{C}^{+}}|f(rz)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)$
$\displaystyle=\frac{1}{r^{\alpha+k+2}}\int_{\mathbb{C}^{+}}|f(z)|^{p}r^{k}w\left(\frac{z}{r}\right)\operatorname{Im}(z)^{\alpha}e^{\frac{-\beta|z|^{2}}{r^{2}}}dA(z)$
$\displaystyle=\frac{1}{r^{\alpha+k+2}}\int_{\mathbb{C}^{+}}|f(z)|^{p}r^{k}w\left(\frac{z}{r}\right)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}e^{\beta|z|^{2}(1-r^{-2})}dA(z).$

Recall that for $r_{0}<r<1$ we have

$e^{\beta|z|^{2}(1-r^{-2})}\leq 1,$

and hence

$\displaystyle\|f_{r}\|^{p}_{\mathcal{A}^{p}_{q,w}{(\mathbb{C}^{+})}}$
$\displaystyle=\frac{1}{r^{\alpha+k+2}}\int_{\mathbb{C}^{+}}|f(z)|^{p}r^{k}w\left(\frac{z}{r}\right)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}e^{\beta|z|^{2}(1-r^{-2})}dA(z)$
$\displaystyle\leq\frac{C}{r^{\alpha+k+2}}\int_{\mathbb{C}^{+}}|f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)<\infty.$

This means that the dominated convergence theorem can be applied; so that

$\lim_{r\to
1^{-}}\|f_{r}\|^{p}_{\mathcal{A}^{p}_{q,w}{(\mathbb{C}^{+})}}=\int_{\mathbb{C}^{+}}|f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)=\|f\|^{p}_{\mathcal{A}^{p}_{q,w}{(\mathbb{C}^{+})}},$

and finally $f_{r}\to f$ in $\mathcal{A}^{p}_{q,w}{(\mathbb{C}^{+})}$.
$\square$

We now state a similar approximation theorem for the polyanalytic Besov spaces
on $\mathbb{C}^{+}$.

###### Theorem 3.3

Let $2\leq p<\infty$ and let $w$ satisfies the condition (2). Then for each
$f\in\mathcal{B}^{p}_{q,w}(\mathbb{C}^{+})$, the dilatations $f_{r}$ are
convergent to $f$ in the norm.

Proof. Let $f\in\mathcal{B}^{p}_{q,w}(\mathbb{C}^{+})$, and let
$d\mu(z)=w(z)\operatorname{Im}(z)^{\alpha+p-2}e^{-\beta|z|^{2}}dA(z)$. We
recall that

$\|f\|^{p}_{\mathcal{B}^{p}_{q,w}(\mathbb{C}^{+})}=\int_{\mathbb{C}^{+}}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}d\mu(z).$

Therefore, by a change of variable we have

$\displaystyle\int_{\mathbb{C}^{+}}|\partial_{z}f_{r}(z)|^{p}d\mu(z)$
$\displaystyle=\frac{r^{p}}{r^{k+\alpha+p}}\int_{\mathbb{C}^{+}}|\partial_{z}f(z)|^{p}r^{k}w\left(\frac{z}{r}\right)\operatorname{Im}(z)^{\alpha+p-2}e^{\frac{-\beta|z|^{2}}{r^{2}}}dA(z)$
$\displaystyle\leq\frac{C}{r^{k+\alpha}}\int_{\mathbb{C}^{+}}|\partial_{z}f(z)|^{p}d\mu(z).$

Now, we apply the dominated convergence theorem to get

$\limsup_{r\to
1^{-}}\int_{\mathbb{C}^{+}}|\partial_{z}f_{r}(z)|^{p}d\mu(z)\leq\int_{\mathbb{C}^{+}}|\partial_{z}f(z)|^{p}d\mu(z).$

The last inequality shows that

$\lim_{r\to
1^{-}}\int_{\mathbb{C}^{+}}|\partial_{z}f_{r}(z)-\partial_{z}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha+p-2}e^{-\beta|z|^{2}}dA(z)=0.$

The proof that

$\lim_{r\to
1^{-}}\int_{\mathbb{C}^{+}}|\partial_{\overline{z}}f_{r}(z)-\partial_{\overline{z}}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha+p-2}e^{-\beta|z|^{2}}dA(z)=0$

is similar. These equalities show that

$\lim_{r\to 1^{-}}\|f_{r}-f\|^{p}_{\mathcal{B}^{p}_{q,w}(\mathbb{C}^{+})}=0.$

$\square$

###### Example 3.4

(a). Let $\beta$ be a positive number, and $n$ be a positive integer. Then
$w(z)=e^{-\beta|z|^{n}}$ satisfies the condition (2). Indeed, for each
$0<r<1$, we have

$w\left(\frac{z}{r}\right)=e^{\frac{-\beta|z|^{n}}{r^{n}}}\leq
e^{-\beta|z|^{n}}=w(z).$

(b). Consider the non-radial weight

$w(z)=e^{-\beta|\operatorname{Re}(z)|^{n}}=e^{-\beta|x|^{n}}.$

Again, we have $w(z/r)\leq w(z)$.
(c). In some instances the function $r\mapsto w(z/r)$ may not satisfy the
condition (LABEL:main-condition), however there might exist some positive
integer $k$ for which the function $r\mapsto r^{k}w(z/r)$ satisfies the
required condition. For instance if $w(z)=\exp(|z|)$, then

$\frac{d}{dr}w\left(\frac{z}{r}\right)=-\frac{|z|}{r^{2}}\exp\left(\frac{|z|}{r}\right)<0$

while

$\frac{d}{dr}\left[rw(\frac{z}{r})\right]=\left(1-\frac{|z|}{r}\right)\exp\left(\frac{|z|}{r}\right)>0,\quad|z|<r.$

Note also that $r^{2}e^{|z|/r}$ is increasing for $r>1/2$ since

$\frac{d}{dr}(r^{2}e^{|z|/r})=2re^{|z|/r}+r^{2}(\frac{-1}{r^{2}}e^{|z|/r})=(2r-1)e^{|z|/r}>0.$

##  4 Angular weights

In contrast to radial weights, let us assume that the weight function depends
only on the argument of $z$; that is, $w(re^{i\theta})=w(\theta)$. We may call
such weights angular weights. For example,

$w(z)=w(re^{i\theta})=(4\pi^{2}-\theta^{2})^{\alpha},\quad
0\leq\theta<2\pi,\,\,\alpha>0,$

is an angular weight in the unit disk. It seems that the study of angular
weights was overlooked in the literature. Here we provide some statements on
the approximation by polyanalytic polynomials in such weighted spaces. Let us
fix the following notations.

###### Theorem 4.1

Let $w:\mathbb{D}\to(0,\infty)$ be an angular weight satisfying

$\int_{0}^{2\pi}w(\theta)d\theta<\infty.$

Then the $q$-analytic polynomials are dense in $\mathcal{B}^{p}_{q,w},\,2\leq
p<\infty$.

Proof. We first note that for $z=re^{i\theta}$ we have
$w(z)=w(z/r)=w(\theta)$, so that by neglecting the constant term in the
definition of norm we have

$\displaystyle\|f_{r}\|^{p}_{\mathcal{B}^{p}_{q,w}}$
$\displaystyle=\int_{\mathbb{D}}(1-|z|^{2})^{p-2}\Big{[}|\partial_{z}f_{r}(z)|^{p}+|\partial_{\overline{z}}f_{r}(z)|^{p}\Big{]}w(z)dA(z)$
$\displaystyle=r^{p-2}\int_{r\mathbb{D}}\left(\frac{r^{2}-|z|^{2}}{r^{2}}\right)^{p-2}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(z)dA(z).$

This implies that

$\limsup_{r\to
1^{-}}\|f_{r}\|^{p}_{\mathcal{B}^{p}_{q,w}}=\|f\|^{p}_{\mathcal{B}^{p}_{q,w}},$

from which it follows that $f_{r}\to f$ in norm. But on the unit disk, each
$f_{r}$ can be approximated by $q$-analytic polynomials (this is a consequence
of $(\ref{rep1})$), from which the assertion follows. $\square$

We now consider weights that are multiples of a radial weight and an angular
weight.

###### Theorem 4.2

Let $w(se^{i\theta})=\omega(s)v(\theta)$ be a weight function on $\mathbb{D}$
where $\omega$ and $v$ satisfy

$\int_{0}^{1}s\omega(s)ds<\infty,\,\,\int_{0}^{2\pi}v(\theta)d\theta<\infty,$

and $r^{k}\omega(s/r)\leq C\omega(s)$ for some integer $k\geq 0$. Then the
$q$-analytic polynomials are dense in $\mathcal{B}^{p}_{q,w},\,2\leq
p<\infty$.

Proof. Again, we see that

$\displaystyle\|f_{r}\|^{p}_{\mathcal{B}^{p}_{q,w}}$
$\displaystyle=\int_{\mathbb{D}}(1-|z|^{2})^{p-2}\Big{[}|\partial_{z}f_{r}(z)|^{p}+|\partial_{\overline{z}}f_{r}(z)|^{p}\Big{]}w(s)v(\theta)dA(z)$
$\displaystyle=r^{p-2-k}\int_{r\mathbb{D}}\left(\frac{r^{2}-|z|^{2}}{r^{2}}\right)^{p-2}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}r^{k}w(s/r)v(\theta)dA(z)$
$\displaystyle\leq
Cr^{p-2-k}\int_{r\mathbb{D}}\left(\frac{r^{2}-|z|^{2}}{r^{2}}\right)^{p-2}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(s)v(\theta)dA(z).$

Therefore, the dominated convergence theorem applies;

$\limsup_{r\to
1^{-}}\|f_{r}\|^{p}_{\mathcal{B}^{p}_{q,w}}=\|f\|^{p}_{\mathcal{B}^{p}_{q,w}},$

from which the result follows. $\square$

The above two theorems have upper half-plane analogs as well.

###### Theorem 4.3

Let $0<p<\infty$ and let $w$ be an angular weight function on $\mathbb{C}^{+}$
satisfying

$\int_{0}^{\pi}w(\theta)d\theta<\infty.$

Then for each $f\in\mathcal{D}^{p}_{q,w}(\mathbb{C}^{+})$, the dilatations
$f_{r}$ converge to $f$ in norm.

Proof. We note that

$\displaystyle\|f_{r}-f\|^{p}_{\mathcal{D}^{p}_{q,w}(\mathbb{C}^{+})}$
$\displaystyle=\int_{\mathbb{C}^{+}}|\partial_{z}f_{r}(z)-\partial_{z}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)$
$\displaystyle+\int_{\mathbb{C}^{+}}|\partial_{\overline{z}}f_{r}(z)-\partial_{\overline{z}}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z).$
(7)

By replacing $z$ by $z/r$, and using the fact that
$\partial_{z}f_{r}(z)=r\partial_{z}f(rz),$ and $w(z)=w(z/r)$, we obtain

$\displaystyle\int_{\mathbb{C}^{+}}|\partial_{z}f_{r}(z)|^{p}w(z)$
$\displaystyle\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)$
$\displaystyle=r^{p-(\alpha+2)}\int_{\mathbb{C}^{+}}|\partial_{z}f(z)|^{p}w\left(\frac{z}{r}\right)\operatorname{Im}(z)^{\alpha}e^{\frac{-\beta|z|^{2}}{r^{2}}}dA(z)$
$\displaystyle\leq
r^{p-(\alpha+2)}\int_{\mathbb{C}^{+}}|\partial_{z}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z),$

Therefore the dominated convergence theorem applies to ensure

$\limsup_{r\to
1^{-}}\int_{\mathbb{C}^{+}}|\partial_{z}f_{r}(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)=\int_{\mathbb{C}^{+}}|\partial_{z}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z).$

This entails that the first term on the right-hand side of $(\ref{dir-up-
ang-1})$ tends to zero:

$\lim_{r\to
1^{-}}\int_{\mathbb{C}^{+}}|\partial_{z}f_{r}(z)-\partial_{z}f(z)|^{p}w(z)\operatorname{Im}(z)^{\alpha}e^{-\beta|z|^{2}}dA(z)=0.$

Similarly, one shows that the second term on the right-hand side of
$(\ref{dir-up-ang-1})$ tends to zero, from which we obtain

$\lim_{r\to 1^{-}}\|f_{r}-f\|^{p}_{\mathcal{D}^{p}_{q,w}(\mathbb{C}^{+})}=0.$

$\square$

###### Theorem 4.4

Let $w(se^{i\theta})=\omega(s)v(\theta)$ be an angular weight function on
$\mathbb{C}^{+}$ where $\omega$ and $v$ satisfy

$\int_{0}^{1}s\omega(s)ds<\infty,\,\,\int_{0}^{\pi}v(\theta)d\theta<\infty,$

and $r^{k}\omega(s/r)\leq C\omega(s)$ for some integer $k\geq 0$. Then for
each $f\in\mathcal{B}^{p}_{q,w}(\mathbb{C^{+}}),\,2\leq p<\infty$, the
dilatations $f_{r}$ converge to $f$ in norm.

Proof. Again, we see that

$\displaystyle\|f_{r}\|^{p}_{\mathcal{B}^{p}_{q,w}(\mathbb{C^{+}})}$
$\displaystyle=\int_{\mathbb{D}}(1-|z|^{2})^{p-2}\Big{[}|\partial_{z}f_{r}(z)|^{p}+|\partial_{\overline{z}}f_{r}(z)|^{p}\Big{]}w(s)v(\theta)dA(z)$
$\displaystyle=r^{p-2-k}\int_{r\mathbb{D}}\left(\frac{r^{2}-|z|^{2}}{r^{2}}\right)^{p-2}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}r^{k}w(s/r)v(\theta)dA(z)$
$\displaystyle\leq
Cr^{p-2-k}\int_{r\mathbb{D}}\left(\frac{r^{2}-|z|^{2}}{r^{2}}\right)^{p-2}\Big{[}|\partial_{z}f(z)|^{p}+|\partial_{\overline{z}}f(z)|^{p}\Big{]}w(s)v(\theta)dA(z).$

Therefore, the dominated convergence theorem applies;

$\limsup_{r\to
1^{-}}\|f_{r}\|^{p}_{\mathcal{B}^{p}_{q,w}(\mathbb{C^{+}})}\leq\|f\|^{p}_{\mathcal{B}^{p}_{q,w}(\mathbb{C^{+}})},$

from which the result follows. $\square$

##  5 Declarations

Ethical approval
Not applicable

Competing interests
The author declares no competing interests.

Authors contribution
Not applicable

Funding
Not applicable

Availability of data and materials
Data sharing is not applicable to this article as no data sets were generated
or analyzed during the current study.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-076 LHCb-PAPER-2019-016 November 29, 2019

Search for the lepton-flavour-violating decays
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ and
${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$

LHCb collaboration

Results are reported from a search for the rare decays
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ and
${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$, where the tau lepton is
reconstructed in the channel
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\nu}_{\tau}}$.
These processes are effectively forbidden in the standard model, but they can
potentially occur at detectable rates in models of new physics that can induce
lepton-flavor-violating decays. The search is based on a data sample
corresponding to 3$\mbox{\,fb}^{-1}$ of proton-proton collisions recorded by
the LHCb experiment in 2011 and 2012. The event yields observed in the signal
regions for both processes are consistent with the expected standard model
backgrounds. Because of the limited mass resolution arising from the
undetected tau neutrino, the ${B}^{0}_{s}$ and ${B}^{0}$ signal regions are
highly overlapping. Assuming no contribution from
${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$, the upper limit
${\mathcal{B}}\left(\mbox{${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$}\right)<4.2\times
10^{-5}$ is obtained at 95% confidence level. If no contribution from
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ is assumed, a limit
of
${\mathcal{B}}\left(\mbox{${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$}\right)<1.4\times
10^{-5}$ is obtained at 95% confidence level. These results represent the
first limit on
${\mathcal{B}}\left(\mbox{${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$}\right)$
and the most stringent limit on
${\mathcal{B}}\left(\mbox{${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$}\right)$.

Published in Phys. Rev. Lett. 123 (2019) 211801

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

Lepton-flavor-violating decays of mesons containing $b$ quarks, such as
${{B}^{0}}(b\overline{d})\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ and
${{B}^{0}_{s}}(b\overline{s})\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$, are
extremely suppressed in the Standard Model (SM), with expected branching
fractions of order 10-54 [1]. (The inclusion of charge-conjugate processes is
implied throughout this Letter.) These processes involve not only quantum
loops, but also neutrino oscillations. Signals at the level expected in the SM
lie far below current and foreseen experimental sensitivities. However, many
theoretical models proposed to explain possible experimental tensions observed
in other $B$-meson decays (discussed below) naturally allow for branching
fractions that are within current sensitivity. Among them, models containing a
heavy neutral gauge boson (${Z}^{{}^{\prime}}$) could lead to a
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ branching fraction of
up to $10^{-8}$ [2, 3] when only left-handed or right-handed couplings to
quarks are considered, or of the order of $10^{-6}$ [3] if both are allowed.
In models with either scalar or vector leptoquarks, the largest predictions
for the ${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ branching
fraction range from $10^{-9}$ to $10^{-5}$, depending on the assumed
leptoquark mass [4, 5, 6]. The three-site Pati-Salam gauge model favours
values for this branching fraction in the range 10-4–10-6 [7, 8].

The SM predicts that the electroweak couplings for the three lepton families
are universal, a result referred to as Lepton Flavor Universality (LFU).
Experimental tests of LFU performed using ${b}\rightarrow s\ell^{+}\ell^{-}$
and $b\rightarrow c\ell^{-}\overline{\nu}$ processes show tensions with
respect to the SM predictions for the observables $R_{{K}^{(*)}}$ [9, 10] and
${\cal R}(D^{(*)})$ [11]. For the latter, the observed discrepancy with
respect to the SM prediction is greater than 3 standard deviations. Because
theoretical models that can account for the possible LFU effects observed in
data often predict Lepton Flavor Violation (LFV) as well [12], searches for
LFV processes provide a powerful signature for probing these models.

An upper limit
${\mathcal{B}}\left(\mbox{${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$}\right)<2.2\times
10^{-5}$ at $90\%$ confidence level (CL) was obtained by the BaBar
collaboration [13]. There are currently no experimental results for the
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ mode.

This Letter reports results from the first search for the decay
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$, along with the most
stringent limit on the process
${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$. The analysis is
performed on data corresponding to an integrated luminosity of
$3\mbox{\,fb}^{-1}$ of proton-proton ($pp$) collisions, recorded with the LHCb
detector during the years 2011 and 2012 at centre-of-mass energies of $7$ and
$8\mathrm{\,Te\kern-1.00006ptV}$, respectively. The $\tau$ leptons are
reconstructed through the decay
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\nu}_{\tau}}$,
which mainly proceeds via the production of two intermediate resonances,
$a_{1}(1260)^{-}\rightarrow\pi^{+}\pi^{-}\pi^{-}$ and
$\rho(770)^{0}\rightarrow\pi^{+}\pi^{-}$ [14], which help in the signal
selection. In this mode, the $\tau$ decay vertex can be precisely
reconstructed, facilitating a good reconstruction of the $B$-meson invariant
mass despite the undetected neutrino. To avoid experimenter bias, the
$B$-meson invariant-mass signal region was not examined until the selection
and fit procedures were finalised. The signal yield is determined by
performing an unbinned maximum-likelihood fit to the reconstructed $B$-meson
invariant-mass distribution and is converted into a branching fraction using
the decay
${{B}^{0}}\\!\rightarrow{{D}^{-}}(\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}){{\pi}^{+}}$
as a normalisation channel.

The LHCb detector [15, 16] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The tracking system provides a measurement of the momentum, $p$,
of charged particles with a relative uncertainty varying from 0.5% at low
momentum to 1.0% at 200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum
distance of a track to a primary vertex (PV), the impact parameter (IP), is
measured with a resolution of $(15+29/p_{\mathrm{T}}){\,\upmu\mathrm{m}}$,
where $p_{\mathrm{T}}$ is the component of the momentum transverse to the
beam, in ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged
hadrons are distinguished using information from two ring-imaging Cherenkov
detectors. Photons, electrons and hadrons are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers.

The on-line event selection is performed by a trigger [17] consisting of a
hardware stage based on information from the calorimeter and muon systems,
followed by a software stage, which performs a full event reconstruction. At
the hardware trigger stage, signal candidates are required to have a muon with
high $p_{\mathrm{T}}$, while, for the normalisation sample, events are
required to have a hadron with high transverse energy in the calorimeters. The
software trigger requires a two-, three-, or four-track secondary vertex with
a significant displacement from any primary $pp$ interaction vertex. A
multivariate algorithm [18] is used to identify secondary vertices consistent
with the decay of a $b$ hadron. At least one charged particle must have a
transverse momentum
$p_{\mathrm{T}}>1.0\>(1.6){\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for muons
(hadrons), and must be inconsistent with originating from a PV.

Simulation is used to optimise the selection, determine the signal model for
the fit and obtain the selection efficiencies. In the simulation, $pp$
collisions are generated using Pythia [19, *Sjostrand:2007gs] with a specific
LHCb configuration [21]. The $\tau$ decay is simulated using the Tauola decay
library tuned with BaBar data [22], while the decays of all other unstable
particles are described by EvtGen [23]. Final-state radiation is accounted for
using Photos [24]. The interaction of the generated particles with the
detector, and its response, are implemented using the Geant4 toolkit [25,
*Agostinelli:2002hh], as described in Ref. [27].

Both signal and normalisation candidates are formed using tracks that are
inconsistent with originating from any PV. Candidate
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\nu}_{\tau}}$ and
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays are
reconstructed from three tracks forming a good-quality vertex and with
particle identification information corresponding to their assumed particle
hypotheses. Candidate
${{B}^{0}_{({s})}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ decays are formed
by combining a reconstructed $\tau$ lepton and an oppositely charged track
identified as a muon. A control sample of same-sign candidates, which are
formed by a $\tau$ lepton and a muon with identical charges, is also selected
to serve, during the selection process, as a proxy for the large component of
the background in which the muon and the tau candidate charges are
uncorrelated. For the normalisation mode,
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ candidates are made out of a
reconstructed $D$ meson and an oppositely charged track identified as a pion.
The decay vertex of the signal or normalisation $B$ candidate is determined
through a fit to all reconstructed particles in the decay chain [28], which is
required to be of good quality. The $B$-meson $p_{\mathrm{T}}$ is required to
be greater than $5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for both signal and
normalisation modes.

While the neutrino from the $\tau$ decay escapes detection, its momentum
vector can be constrained from the measured positions of the primary and
$\tau$ decay vertices, the momenta of the muon and the three pions, and the
trajectory of the muon. Then, by imposing the requirements that the mass of
the system formed from the three pions and the unobserved neutrino corresponds
to the mass of the tau lepton, and by requiring that the $B$ decay vertex lies
on the trajectories of the muon, of the tau lepton, and of the $B$ meson, the
invariant mass of the ${B}^{0}_{({s})}$ candidate can be determined
analytically up to a twofold ambiguity. Because of the quadratic nature of the
equation, the computed masses may be unphysical. This occurs in 32% of the
selected signal in the simulated event sample due to measurement resolutions
and in 48% of the same-sign candidates in data. These candidates are removed,
thereby improving the signal-to-background ratio. The solution whose
distribution shows the largest separation between signal and background is
used as the reconstructed $B$ invariant mass, $M_{B}$, in the analysis. The
distributions of $M_{B}$ for candidates satisfying the previously described
initial selection in the simulated signal samples and in the opposite-sign
control sample in the data are shown in Fig. 1.

Figure 1: Normalized distributions of the reconstructed invariant mass for
${B}^{0}_{s}$ and ${B}^{0}$ in simulated event samples and for same-sign
candidates in data, after applying the initial event selection (see the text).

To reduce the data to a manageable level and focus on the rejection of the
most difficult backgrounds, the low-mass region with
${M_{B}}<4{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is discarded. The signal
loss due to this requirement is negligible.

To further reduce the background, additional requirements, optimised with
same-sign candidates and simulated samples, are applied to the selected
${{B}^{0}_{({s})}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ decays. Taking
advantage of the resonant structure of the
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\nu}_{\tau}}$
decay, candidates with both combinations of oppositely charged pions with
invariant-masses below $550{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are
removed. Candidates with a three-pion invariant mass greater than
$1.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are discarded to veto the
background contribution due to
${{D}^{+}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ decays.

A set of isolation variables is used to reduce background from decays with
additional reconstructed particles. The first class of isolation variables
exploits the presence of activity in the calorimeter to identify the
contribution of neutral particles contained in a cone centred on the $B$ or
$\tau$ flight directions. The second class is based on the presence of
additional tracks consistent with originating from the $B$ or $\tau$ decay
vertices, or uses a multivariate classifier, trained on simulated data, to
discriminate against candidates whose decay products are compatible with
forming good-quality vertices with other tracks in the event. These variables
are combined using a Boosted Decision Tree (BDT) [29], trained on same-sign
candidates and simulated
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ decays. Candidates
with a BDT output compatible with that of background are discarded. A second
BDT is used to reduce to a negligible level the contribution of combinatorial
background, which extends over the whole mass range but dominates at higher
masses. It uses variables related to vertex quality and reconstructed particle
opening angles and is trained on samples of same-sign candidates with
${M_{B}}>6.2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and simulated
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ decays.

Some background processes, such as
${{B}^{0}_{({s})}}\\!\rightarrow{{D}^{-}_{({s})}}(\\!\rightarrow{\mu^{-}}{{\overline{\nu}}_{\mu}}){{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$,
have $M_{B}$ distributions peaking in the signal region. In these decays, the
three pions come from the $B$ decay vertex, and therefore the reconstructed
$B$ and $\tau$ decay vertices are very close. Discarding candidates with a
reconstructed $\tau$ decay-time significance lower than $1.8$ reduces this
type of background to a negligible level while keeping $\sim$75$\%$ of signal,
according to studies performed on simulation. All previously described
selection criteria also suppress a possible contribution from the
${{B}^{0}}\\!\rightarrow{a_{1}(1260)^{-}}{\mu^{+}}{{\nu}_{\mu}}$ mode, whose
selection efficiency is 60 times lower than that of the signal. Its rate is
currently unmeasured, but, given that the largest known $b\rightarrow u$
semileptonic decay branching fractions are of the order of $10^{-4}$, its
branching fraction is not expected to be much higher. Events from the decay
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}{{\nu}_{\tau}}$
passing the selection are also included as signal.

The selection procedure retains 17 746 candidates. According to studies based
on simulations, the remaining background is dominated by
${{B}^{0}_{({s})}}\\!\rightarrow{D}_{(s)}^{(*)}\mu\nu\text{X}$ decays.

The selection efficiencies for the signal and normalisation modes,
$\epsilon_{{{B}^{0}_{({s})}}\rightarrow{\tau}\mu}$ and $\epsilon_{B\rightarrow
D\pi}$, respectively, are estimated using simulation or, whenever possible,
data. The efficiency $\epsilon_{{{B}^{0}_{({s})}}\rightarrow{\tau}\mu}$
includes those for both
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\nu}_{\tau}}$ and
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}{{\nu}_{\tau}}$
decays, where the latter is weighted by the ratio of the two branching
fractions. The
${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}{{\nu}_{\tau}}$
channel contributes by $\sim 16\%$ to the extracted signal yield. The tracking
and particle identification efficiencies are determined using data [30, 31].
The trigger efficiency for the normalisation channel is estimated using a
trigger-unbiased subsample made of events which have been triggered
independently of the normalisation candidate. For the signal, muons from
${{{B}^{+}}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\\!\rightarrow{\mu^{+}}{\mu^{-}}){{K}^{+}}$ decays are used to
evaluate the muon trigger efficiency and corrections are applied to the
simulated signal samples. To account for differences between the control and
the signal samples, the efficiency is computed as a function of the muon
$p_{\mathrm{T}}$ and IP. Simulation as well as
${{{B}^{+}}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\\!\rightarrow{\mu^{+}}{\mu^{-}}){{K}^{+}}$ decays is used to
determine the software-trigger efficiency and its systematic uncertainty.

The signal yield for the normalisation mode is obtained from a fit to the
invariant-mass distribution of the
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ candidates. In the fit the
signal is modelled by the sum of two Crystal Ball (CB) [32] functions, with
tails on opposite sides, having common means and widths, but independent tail
parameters. The tail parameters are fixed to values determined from a fit to a
sample of
${{B}^{0}}\\!\rightarrow{{D}^{-}}(\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}){{\pi}^{+}}$
simulated decays, while all other parameters are left free. The small
background contribution is described by an exponential function. The measured
yield of the ${{B}^{0}}\\!\rightarrow{{D}^{-}}{{\pi}^{+}}$ mode is
$N^{\text{norm}}=22\;588\pm 176$ where the uncertainty is statistical only.

The ${{B}^{0}_{({s})}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ branching
fractions can be written as

${\mathcal{B}}\left(\mbox{${{B}^{0}_{({s})}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$}\right)=\alpha_{(s)}^{\text{norm}}\cdot
N^{\text{sig}}_{(s)},$ (1)

where $N^{\text{sig}}_{(s)}$ is the number of observed
${{B}^{0}_{({s})}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ decays and
$\alpha_{(s)}^{\text{norm}}$ a normalisation factor. The latter is defined by

$\alpha_{(s)}^{\text{norm}}=\frac{f_{{B}^{0}}}{f_{{B}^{0}_{(s)}}}\cdot\frac{{\mathcal{B}}\left(\mbox{${{B}^{0}}\\!\rightarrow{{D}^{-}}(\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}){{\pi}^{+}}$}\right)}{{\mathcal{B}}\left(\mbox{${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\nu}_{\tau}}$}\right)}\cdot\frac{\epsilon_{B\rightarrow
D\pi}}{\epsilon_{{{B}^{0}_{({s})}}\rightarrow{\tau}\mu}}\cdot\frac{1}{N^{\text{norm}}},$
(2)

using externally measured quantities: the ratio of $b$-quark hadronisation
fractions to ${B}^{0}_{s}$ and ${B}^{0}$ mesons,
$f_{{B}^{0}_{s}}/f_{{B}^{0}}=0.259\pm 0.015$ [33],
${\mathcal{B}}\left(\mbox{${{B}^{0}}\\!\rightarrow{{D}^{-}}(\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}){{\pi}^{+}}$}\right)=(2.26\pm
0.14)\times 10^{-4}$ [34] and
${\mathcal{B}}\left(\mbox{${\tau^{-}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\nu}_{\tau}}$}\right)=(9.02\pm
0.05)\%$ [34]. The measured values of $\alpha_{(s)}^{\text{norm}}$ for the
${B}^{0}_{s}$ and ${B}^{0}$ modes are, respectively,

$\begin{split}\alpha_{s}^{\text{norm}}&=(4.32\pm 0.19\pm 0.45\pm 0.36)\times
10^{-7}\text{ and}\\\ \alpha^{\text{norm}}&=(1.25\pm 0.06\pm 0.13\pm
0.08)\times 10^{-7},\end{split}$ (3)

where the three quoted uncertainties are the statistical uncertainty due to
the sizes of the signal and normalisation simulated and data samples, the
systematic uncertainty on the selection efficiencies (dominated by the trigger
efficiency contribution, $\sim$11$\%$) and the total uncertainty on the
externally measured quantities.

A final BDT is built to split the selected candidates into four samples with
different signal-to-background ratios. It combines 16 discriminating
variables, none of which are correlated with the $B$-meson invariant mass. The
most important ones are the invariant masses of the three-pion system and of
the two combinations of oppositely charged pions, the $B$-meson IP and flight
distance significances, and the output of the BDT based on isolation
variables. The output of the BDT is transformed to have a uniform distribution
between 0 and 1 for
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}(\\!\rightarrow{{\pi}^{\pm}}{{\pi}^{\mp}}{{\pi}^{\pm}}{{\nu}_{\tau}}){{\mu}^{\mp}}$
simulated decays. As a consequence, its distribution for the background peaks
at low BDT values. All samples are divided into four bins of equal width in
BDT output. Their distributions are shown in Fig. 2.

Figure 2: Final BDT output binned distributions for data and simulated signal
samples. The markers are displaced horizontally to improve visibility.

The signal yield is evaluated by performing a simultaneous unbinned maximum-
likelihood fit to the $M_{B}$ distributions in the range
$[4.6,5.8]{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ of the four samples
corresponding to different BDT bins. In each bin, the data are described by
the sum of a signal and a background component. The background shape is
modelled by the upper tail of a reversed CB function, whose peak position and
tail parameters are shared among BDT bins. For the determination of the
systematic uncertainties, different sets of constrained parameters or
alternative background models, such as the sum of two Gaussian functions, are
considered. The signal shapes are described by double-sided Hypatia functions
[35] whose parameters are initialized to the values obtained from a fit to the
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ and
${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ simulated samples and
allowed to vary within Gaussian constraints accounting for possible
discrepancies between data and simulation. The width of the Hypatia functions
are $\sim 330{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for both signal modes
in the most sensitive BDT bin. As the separation between
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ and
${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ signal shapes is limited,
two independent fits are performed while assuming the contribution of either
the ${B}^{0}_{s}$ or the ${B}^{0}$ signal only. The signal fractional yields
in each BDT bin are Gaussian constrained according to their expected values
and uncertainties. The fit result corresponding to the hypothesis of the
${B}^{0}_{s}$ signal only is shown in Fig. 3.

Figure 3: Distributions of the reconstructed $B$ invariant-mass in data in the
four final BDT bins with the projections of the fit for the ${B}^{0}_{s}$
signal-only hypothesis overlaid. The lower-part of each figure shows the
normalised residuals.

The fit procedure is validated by performing fits to a set of
pseudoexperiments where the mass distributions are randomly generated
according to the background model observed in the data. The pulls of all
fitted parameters are normally distributed except those of the signal yields
$N^{\text{sig}}$, which have the expected widths but exhibit a very small bias
of $-3\pm 1$ ($2\pm 2$) events for the ${B}^{0}_{s}$ (${B}^{0}$) mode. This
effect is accounted for by adding the bias to $N^{\text{sig}}$ in the
simultaneous fits to the four BDT regions for both ${B}^{0}$ and
${B}^{0}_{s}$. The obtained signal yields are

$\begin{split}N^{\text{sig}}_{\mbox{${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$}}&=-16\pm
38\text{ and}\\\
N^{\text{sig}}_{\mbox{${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$}}&=-65\pm
58,\end{split}$

where the uncertainties account for the statistical ones as well as those on
the signal and background shape parameters. They show no evidence of any
signal excess.

Using the calculated values of the normalisation factors
$\alpha^{\text{norm}}$ and $\alpha_{s}^{\text{norm}}$ from Eq. 1 together with
Eq. 3, the observed yields from the likelihood fits are translated into upper
limits on the branching fractions using the CLs method [36, 37]. The total
uncertainty on the normalisation factor is accounted for as an additional
Gaussian constraint in the simultaneous fit. Furthermore, a systematic
uncertainty on the signal yield of 34 (41) for the ${B}^{0}_{s}$ (${B}^{0}$)
mode, derived using different sets of constrained parameters or alternative
background models, is added to account for the uncertainties in the background
shape. The expected and observed CLs values as a function of the branching
fraction are shown in the Supplemental Material [38]. The corresponding limits
on the ${B}^{0}_{s}$ and ${B}^{0}$ branching fractions at $90\%$ and $95\%$ CL
are given in Table 1 assuming negligible contribution from the
${{B}^{0}}\\!\rightarrow{a_{1}(1260)^{-}}{\mu^{+}}{{\nu}_{\mu}}$ mode. A
possible residual contribution of this background would lower the expected
limits by
$\sim$16$\%\times({\mathcal{B}}(\mbox{${{B}^{0}}\\!\rightarrow{a_{1}(1260)^{-}}{\mu^{+}}{{\nu}_{\mu}}$})/10^{-4})$.

Table 1: Expected and observed $90\%$ and $95\%$ CL limits on the ${{B}^{0}_{({s})}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ branching fraction. Mode | Limit |  $90\%$ CL |  $95\%$ CL
---|---|---|---
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ | Observed | $3.4\times 10^{-5}$ | $4.2\times 10^{-5}$
| Expected | $3.9\times 10^{-5}$ | $4.7\times 10^{-5}$
${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ | Observed | $1.2\times 10^{-5}$ | $1.4\times 10^{-5}$
| Expected | $1.6\times 10^{-5}$ | $1.9\times 10^{-5}$

The impact of systematic uncertainties on the final limits is about 35$\%$,
dominated by the uncertainty on the background model.

These results represent the best upper limits to date. They constitute a
factor $\sim$2 improvement with respect to the BaBar result for the ${B}^{0}$
mode [13] and the first measurement for the ${B}^{0}_{s}$ mode. The allowed
range on the ${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$
branching fraction preferred by the three-site Pati-Salam model [7, 8] is
significantly reduced by the results presented in this Letter.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).

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## Supplemental material for LHCb-PAPER-2019-016

Figures 4 and 5 show the expected and observed CLs values as a function of the
${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ and
${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ branching fractions.

Figure 4: The expected and observed p-values derived with the CLs method as a
function of the ${{B}^{0}_{s}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$
branching fraction.  Figure 5: The expected and observed p-values derived with
the CLs method as a function of the
${{B}^{0}}\\!\rightarrow{{\tau}^{\pm}}{{\mu}^{\mp}}$ branching fraction.

LHCb collaboration

R. Aaij30, C. Abellán Beteta47, B. Adeva44, M. Adinolfi51, C.A. Aidala78, Z.
Ajaltouni8, S. Akar62, P. Albicocco21, J. Albrecht13, F. Alessio45, M.
Alexander56, A. Alfonso Albero43, G. Alkhazov36, P. Alvarez Cartelle58, A.A.
Alves Jr44, S. Amato2, Y. Amhis10, L. An20, L. Anderlini20, G. Andreassi46, M.
Andreotti19, J.E. Andrews63, F. Archilli21, J. Arnau Romeu9, A. Artamonov42,
M. Artuso65, K. Arzymatov40, E. Aslanides9, M. Atzeni47, B. Audurier25, S.
Bachmann15, J.J. Back53, S. Baker58, V. Balagura10,b, W. Baldini19,45, A.
Baranov40, R.J. Barlow59, S. Barsuk10, W. Barter58, M. Bartolini22,h, F.
Baryshnikov74, V. Batozskaya34, B. Batsukh65, A. Battig13, V. Battista46, A.
Bay46, F. Bedeschi27, I. Bediaga1, A. Beiter65, L.J. Bel30, V. Belavin40, S.
Belin25, N. Beliy4, V. Bellee46, K. Belous42, I. Belyaev37, G. Bencivenni21,
E. Ben-Haim11, S. Benson30, S. Beranek12, A. Berezhnoy38, R. Bernet47, D.
Berninghoff15, H.C. Bernstein65, E. Bertholet11, A. Bertolin26, C.
Betancourt47, F. Betti18,e, M.O. Bettler52, Ia. Bezshyiko47, S. Bhasin51, J.
Bhom32, M.S. Bieker13, S. Bifani50, P. Billoir11, A. Birnkraut13, A.
Bizzeti20,u, M. Bjørn60, M.P. Blago45, T. Blake53, F. Blanc46, S. Blusk65, D.
Bobulska56, V. Bocci29, O. Boente Garcia44, T. Boettcher61, A. Bondar41,w, N.
Bondar36, S. Borghi59,45, M. Borisyak40, M. Borsato15, M. Boubdir12, T.J.V.
Bowcock57, C. Bozzi19,45, S. Braun15, A. Brea Rodriguez44, M. Brodski45, J.
Brodzicka32, A. Brossa Gonzalo53, D. Brundu25,45, E. Buchanan51, A.
Buonaura47, C. Burr59, A. Bursche25, J.S. Butter30, J. Buytaert45, W.
Byczynski45, S. Cadeddu25, H. Cai69, R. Calabrese19,g, S. Cali21, R.
Calladine50, M. Calvi23,i, M. Calvo Gomez43,m, A. Camboni43,m, P. Campana21,
D.H. Campora Perez45, L. Capriotti18,e, A. Carbone18,e, G. Carboni28, R.
Cardinale22,h, A. Cardini25, P. Carniti23,i, K. Carvalho Akiba30, A. Casais
Vidal44, G. Casse57, M. Cattaneo45, G. Cavallero22, R. Cenci27,p, M.G.
Chapman51, M. Charles11,45, Ph. Charpentier45, G. Chatzikonstantinidis50, M.
Chefdeville7, V. Chekalina40, C. Chen3, S. Chen25, S.-G. Chitic45, V.
Chobanova44, M. Chrzaszcz45, A. Chubykin36, P. Ciambrone21, X. Cid Vidal44, G.
Ciezarek45, F. Cindolo18, P.E.L. Clarke55, M. Clemencic45, H.V. Cliff52, J.
Closier45, J.L. Cobbledick59, V. Coco45, J.A.B. Coelho10, J. Cogan9, E.
Cogneras8, L. Cojocariu35, P. Collins45, T. Colombo45, A. Comerma-Montells15,
A. Contu25, G. Coombs45, S. Coquereau43, G. Corti45, C.M. Costa Sobral53, B.
Couturier45, G.A. Cowan55, D.C. Craik61, A. Crocombe53, M. Cruz Torres1, R.
Currie55, C.L. Da Silva64, E. Dall’Occo30, J. Dalseno44,51, C. D’Ambrosio45,
A. Danilina37, P. d’Argent15, A. Davis59, O. De Aguiar Francisco45, K. De
Bruyn45, S. De Capua59, M. De Cian46, J.M. De Miranda1, L. De Paula2, M. De
Serio17,d, P. De Simone21, J.A. de Vries30, C.T. Dean56, W. Dean78, D.
Decamp7, L. Del Buono11, B. Delaney52, H.-P. Dembinski14, M. Demmer13, A.
Dendek33, D. Derkach75, O. Deschamps8, F. Desse10, F. Dettori25, B. Dey6, A.
Di Canto45, P. Di Nezza21, S. Didenko74, H. Dijkstra45, F. Dordei25, M.
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Ferreres Sole30, M. Ferro-Luzzi45, S. Filippov39, R.A. Fini17, M. Fiorini19,g,
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F. Fontanelli22,h, R. Forty45, V. Franco Lima57, M. Franco Sevilla63, M.
Frank45, C. Frei45, J. Fu24,q, W. Funk45, E. Gabriel55, A. Gallas Torreira44,
D. Galli18,e, S. Gallorini26, S. Gambetta55, Y. Gan3, M. Gandelman2, P.
Gandini24, Y. Gao3, L.M. Garcia Martin77, J. García Pardiñas47, B. Garcia
Plana44, J. Garra Tico52, L. Garrido43, D. Gascon43, C. Gaspar45, G. Gazzoni8,
D. Gerick15, E. Gersabeck59, M. Gersabeck59, T. Gershon53, D. Gerstel9, Ph.
Ghez7, V. Gibson52, O.G. Girard46, P. Gironella Gironell43, L. Giubega35, K.
Gizdov55, V.V. Gligorov11, C. Göbel67, D. Golubkov37, A. Golutvin58,74, A.
Gomes1,a, I.V. Gorelov38, C. Gotti23,i, E. Govorkova30, J.P. Grabowski15, R.
Graciani Diaz43, L.A. Granado Cardoso45, E. Graugés43, E. Graverini46, G.
Graziani20, A. Grecu35, R. Greim30, P. Griffith25, L. Grillo59, L. Gruber45,
B.R. Gruberg Cazon60, C. Gu3, E. Gushchin39, A. Guth12, Yu. Guz42,45, T.
Gys45, T. Hadavizadeh60, C. Hadjivasiliou8, G. Haefeli46, C. Haen45, S.C.
Haines52, P.M. Hamilton63, Q. Han6, X. Han15, T.H. Hancock60, S. Hansmann-
Menzemer15, N. Harnew60, T. Harrison57, C. Hasse45, M. Hatch45, J. He4, M.
Hecker58, K. Heijhoff30, K. Heinicke13, A. Heister13, K. Hennessy57, L.
Henry77, M. Heß71, J. Heuel12, A. Hicheur66, R. Hidalgo Charman59, D. Hill60,
M. Hilton59, P.H. Hopchev46, J. Hu15, W. Hu6, W. Huang4, Z.C. Huard62, W.
Hulsbergen30, T. Humair58, M. Hushchyn75, D. Hutchcroft57, D. Hynds30, P.
Ibis13, M. Idzik33, P. Ilten50, A. Inglessi36, A. Inyakin42, K. Ivshin36, R.
Jacobsson45, S. Jakobsen45, J. Jalocha60, E. Jans30, B.K. Jashal77, A.
Jawahery63, F. Jiang3, M. John60, D. Johnson45, C.R. Jones52, C. Joram45, B.
Jost45, N. Jurik60, S. Kandybei48, M. Karacson45, J.M. Kariuki51, S.
Karodia56, N. Kazeev75, M. Kecke15, F. Keizer52, M. Kelsey65, M. Kenzie52, T.
Ketel31, B. Khanji45, A. Kharisova76, C. Khurewathanakul46, K.E. Kim65, T.
Kirn12, V.S. Kirsebom46, S. Klaver21, K. Klimaszewski34, S. Koliiev49, M.
Kolpin15, A. Kondybayeva74, A. Konoplyannikov37, P. Kopciewicz33, R.
Kopecna15, P. Koppenburg30, I. Kostiuk30,49, O. Kot49, S. Kotriakhova36, M.
Kozeiha8, L. Kravchuk39, M. Kreps53, F. Kress58, S. Kretzschmar12, P.
Krokovny41,w, W. Krupa33, W. Krzemien34, W. Kucewicz32,l, M. Kucharczyk32, V.
Kudryavtsev41,w, G.J. Kunde64, A.K. Kuonen46, T. Kvaratskheliya37, D.
Lacarrere45, G. Lafferty59, A. Lai25, D. Lancierini47, G. Lanfranchi21, C.
Langenbruch12, T. Latham53, C. Lazzeroni50, R. Le Gac9, R. Lefèvre8, A.
Leflat38, F. Lemaitre45, O. Leroy9, T. Lesiak32, B. Leverington15, H. Li68,
P.-R. Li4,aa, X. Li64, Y. Li5, Z. Li65, X. Liang65, T. Likhomanenko73, R.
Lindner45, F. Lionetto47, V. Lisovskyi10, G. Liu68, X. Liu3, D. Loh53, A.
Loi25, J. Lomba Castro44, I. Longstaff56, J.H. Lopes2, G. Loustau47, G.H.
Lovell52, D. Lucchesi26,o, M. Lucio Martinez44, Y. Luo3, A. Lupato26, E.
Luppi19,g, O. Lupton53, A. Lusiani27,t, X. Lyu4, F. Machefert10, F. Maciuc35,
V. Macko46, P. Mackowiak13, S. Maddrell-Mander51, O. Maev36,45, K. Maguire59,
D. Maisuzenko36, M.W. Majewski33, S. Malde60, B. Malecki45, A. Malinin73, T.
Maltsev41,w, H. Malygina15, G. Manca25,f, G. Mancinelli9, D. Marangotto24,q,
J. Maratas8,v, J.F. Marchand7, U. Marconi18, C. Marin Benito10, M.
Marinangeli46, P. Marino46, J. Marks15, P.J. Marshall57, G. Martellotti29, L.
Martinazzoli45, M. Martinelli45,23,i, D. Martinez Santos44, F. Martinez
Vidal77, A. Massafferri1, M. Materok12, R. Matev45, A. Mathad47, Z. Mathe45,
V. Matiunin37, C. Matteuzzi23, K.R. Mattioli78, A. Mauri47, E. Maurice10,b, B.
Maurin46, M. McCann58,45, A. McNab59, R. McNulty16, J.V. Mead57, B. Meadows62,
C. Meaux9, N. Meinert71, D. Melnychuk34, M. Merk30, A. Merli24,q, E.
Michielin26, D.A. Milanes70, E. Millard53, M.-N. Minard7, O. Mineev37, L.
Minzoni19,g, D.S. Mitzel15, A. Mödden13, A. Mogini11, R.D. Moise58, T.
Mombächer13, I.A. Monroy70, S. Monteil8, M. Morandin26, G. Morello21, M.J.
Morello27,t, J. Moron33, A.B. Morris9, R. Mountain65, H. Mu3, F. Muheim55, M.
Mukherjee6, M. Mulder30, D. Müller45, J. Müller13, K. Müller47, V. Müller13,
C.H. Murphy60, D. Murray59, P. Naik51, T. Nakada46, R. Nandakumar54, A.
Nandi60, T. Nanut46, I. Nasteva2, M. Needham55, N. Neri24,q, S. Neubert15, N.
Neufeld45, R. Newcombe58, T.D. Nguyen46, C. Nguyen-Mau46,n, S. Nieswand12, R.
Niet13, N. Nikitin38, N.S. Nolte45, A. Oblakowska-Mucha33, V. Obraztsov42, S.
Ogilvy56, D.P. O’Hanlon18, R. Oldeman25,f, C.J.G. Onderwater72, J. D.
Osborn78, A. Ossowska32, J.M. Otalora Goicochea2, T. Ovsiannikova37, P.
Owen47, A. Oyanguren77, P.R. Pais46, T. Pajero27,t, A. Palano17, M. Palutan21,
G. Panshin76, A. Papanestis54, M. Pappagallo55, L.L. Pappalardo19,g, W.
Parker63, C. Parkes59,45, G. Passaleva20,45, A. Pastore17, M. Patel58, C.
Patrignani18,e, A. Pearce45, A. Pellegrino30, G. Penso29, M. Pepe Altarelli45,
S. Perazzini18, D. Pereima37, P. Perret8, L. Pescatore46, K. Petridis51, A.
Petrolini22,h, A. Petrov73, S. Petrucci55, M. Petruzzo24,q, B. Pietrzyk7, G.
Pietrzyk46, M. Pikies32, M. Pili60, D. Pinci29, J. Pinzino45, F. Pisani45, A.
Piucci15, V. Placinta35, S. Playfer55, J. Plews50, M. Plo Casasus44, F.
Polci11, M. Poli Lener21, M. Poliakova65, A. Poluektov9, N. Polukhina74,c, I.
Polyakov65, E. Polycarpo2, G.J. Pomery51, S. Ponce45, A. Popov42, D. Popov50,
S. Poslavskii42, E. Price51, C. Prouve44, V. Pugatch49, A. Puig Navarro47, H.
Pullen60, G. Punzi27,p, W. Qian4, J. Qin4, R. Quagliani11, B. Quintana8, N.V.
Raab16, B. Rachwal33, J.H. Rademacker51, M. Rama27, M. Ramos Pernas44, M.S.
Rangel2, F. Ratnikov40,75, G. Raven31, M. Ravonel Salzgeber45, M. Reboud7, F.
Redi46, S. Reichert13, F. Reiss11, C. Remon Alepuz77, Z. Ren3, V. Renaudin60,
S. Ricciardi54, S. Richards51, K. Rinnert57, P. Robbe10, A. Robert11, A.B.
Rodrigues46, E. Rodrigues62, J.A. Rodriguez Lopez70, M. Roehrken45, S.
Roiser45, A. Rollings60, V. Romanovskiy42, A. Romero Vidal44, J.D. Roth78, M.
Rotondo21, M.S. Rudolph65, T. Ruf45, J. Ruiz Vidal77, J.J. Saborido Silva44,
N. Sagidova36, B. Saitta25,f, V. Salustino Guimaraes67, C. Sanchez Gras30, C.
Sanchez Mayordomo77, B. Sanmartin Sedes44, R. Santacesaria29, C. Santamarina
Rios44, M. Santimaria21,45, E. Santovetti28,j, G. Sarpis59, A. Sarti21,k, C.
Satriano29,s, A. Satta28, M. Saur4, D. Savrina37,38, S. Schael12, M.
Schellenberg13, M. Schiller56, H. Schindler45, M. Schmelling14, T.
Schmelzer13, B. Schmidt45, O. Schneider46, A. Schopper45, H.F. Schreiner62, M.
Schubiger30, S. Schulte46, M.H. Schune10, R. Schwemmer45, B. Sciascia21, A.
Sciubba29,k, A. Semennikov37, E.S. Sepulveda11, A. Sergi50,45, N. Serra47, J.
Serrano9, L. Sestini26, A. Seuthe13, P. Seyfert45, M. Shapkin42, T. Shears57,
L. Shekhtman41,w, V. Shevchenko73, E. Shmanin74, B.G. Siddi19, R. Silva
Coutinho47, L. Silva de Oliveira2, G. Simi26,o, S. Simone17,d, I. Skiba19, N.
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4University of Chinese Academy of Sciences, Beijing, China
5Institute Of High Energy Physics (IHEP), Beijing, China
6Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
7Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
8Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
9Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
10LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
11LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
12I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
13Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
14Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
15Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
16School of Physics, University College Dublin, Dublin, Ireland
17INFN Sezione di Bari, Bari, Italy
18INFN Sezione di Bologna, Bologna, Italy
19INFN Sezione di Ferrara, Ferrara, Italy
20INFN Sezione di Firenze, Firenze, Italy
21INFN Laboratori Nazionali di Frascati, Frascati, Italy
22INFN Sezione di Genova, Genova, Italy
23INFN Sezione di Milano-Bicocca, Milano, Italy
24INFN Sezione di Milano, Milano, Italy
25INFN Sezione di Cagliari, Monserrato, Italy
26INFN Sezione di Padova, Padova, Italy
27INFN Sezione di Pisa, Pisa, Italy
28INFN Sezione di Roma Tor Vergata, Roma, Italy
29INFN Sezione di Roma La Sapienza, Roma, Italy
30Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
31Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
32Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
33AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
34National Center for Nuclear Research (NCBJ), Warsaw, Poland
35Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
36Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
37Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
38Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
39Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
40Yandex School of Data Analysis, Moscow, Russia
41Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
42Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
43ICCUB, Universitat de Barcelona, Barcelona, Spain
44Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
45European Organization for Nuclear Research (CERN), Geneva, Switzerland
46Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
47Physik-Institut, Universität Zürich, Zürich, Switzerland
48NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
49Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
50University of Birmingham, Birmingham, United Kingdom
51H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
52Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
53Department of Physics, University of Warwick, Coventry, United Kingdom
54STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
55School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
56School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
57Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
58Imperial College London, London, United Kingdom
59Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
60Department of Physics, University of Oxford, Oxford, United Kingdom
61Massachusetts Institute of Technology, Cambridge, MA, United States
62University of Cincinnati, Cincinnati, OH, United States
63University of Maryland, College Park, MD, United States
64Los Alamos National Laboratory (LANL), Los Alamos, United States
65Syracuse University, Syracuse, NY, United States
66Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
67Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
68South China Normal University, Guangzhou, China, associated to 3
69School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
70Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 11
71Institut für Physik, Universität Rostock, Rostock, Germany, associated to 15
72Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 30
73National Research Centre Kurchatov Institute, Moscow, Russia, associated to
37
74National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 37
75National Research University Higher School of Economics, Moscow, Russia,
associated to 40
76National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
37
77Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 43
78University of Michigan, Ann Arbor, United States, associated to 65

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
xINFN Sezione di Trieste, Trieste, Italy
ySchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
zPhysics and Micro Electronic College, Hunan University, Changsha City, China
aaLanzhou University, Lanzhou, China
†Deceased