# Connectedness of a space of branched coverings with a periodic cycle

Laurent Bartholdi  Fachrichtung Mathematik+Informatik, Universität des
Saarlandes<EMAIL_ADDRESS>
(Date: April 13th, 2022)

###### Abstract.

We prove the connectedness of the following locus: the space of degree-$d$
branched self-coverings of $S^{2}$ with two critical points of order $d$, one
of which is $n$-periodic.

##  1\. Introduction

Consider the space $\mathscr{M}_{d}$ of degree-$d$ branched self-coverings of
the sphere $S^{2}$. Much is known about its topology [segal:rational], for
example it is connected, and its fundamental group is $\mathbb{Z}/2d$.
Interesting subspaces arise by imposing dynamical conditions; the one we will
focus on in this article is

$\mathscr{P}_{d,n}=\\{f\colon S^{2}\righttoleftarrow:f\text{ has two critical
points of order $d$, one of which has period exactly }n\\}.$

Our main result is:

######  Theorem A.

The space $\mathscr{P}_{d,n}$ is path connected.

An equivalent statement is that any two maps in $\mathscr{P}_{d,n}$ are
isotopic through maps within $\mathscr{P}_{d,n}$. In fact, the homotopy type
of $\mathscr{P}_{d,n}$ can be determined quite explicitly (see Proposition D).

This purely topological statement has a complex analytic avatar: endow $S^{2}$
with its complex structure, now written ${\mathbb{P}^{1}}$. In this manner,
${\mathbf{Rat}}_{d}=\\{f\in{\mathbb{C}}(z):\deg(f)=d\\}$ embeds into
$\mathscr{M}_{d}$. The locus $\mathscr{P}_{d,n}\cap{\mathbf{Rat}}_{d}$ is an
important “slice” of parameter space, whose connectedness was asked by Milnor
[milnor:quadratic]. In fact, the group of Möbius transformations acts on
${\mathbf{Rat}}_{d}$ by conjugation, preserving the locus
$\mathscr{P}_{d,n}\cap{\mathbf{Rat}}_{d}$, so this question may be studied in
the quotient space. Milnor proves that $\\{f\in{\mathbf{Rat}}_{d}:f\text{ has
two critical points of order }d\\}/\operatorname{PSL}_{2}({\mathbb{C}})$ is
isomorphic to ${\mathbb{C}}^{2}$, for instance by identifying the map $(\alpha
z^{d}+\beta)/(\gamma z^{d}+\delta)$ with the pair
$(\beta\gamma/(\alpha\delta-\beta\gamma),(\alpha^{d+1}\beta^{d-1}+\gamma^{d-1}\delta^{d+1})/(\alpha\delta-\beta\gamma)^{2d})$.
Then the image of $\mathscr{P}_{d,n}\cap{\mathbf{Rat}}_{d}$ in
${\mathbb{C}}^{2}$ is an algebraic curve called $\mathbf{Per}_{n}^{d}(0)$,
whose connectedness is a tantalizing open problem. Theorem A should be seen as
a solution to this problem in the topological context; since by
[segal:rational] the inclusion of ${\mathbf{Rat}}_{d}$ in $\mathscr{M}_{d}$ is
a homotopy equivalence up to dimension $d$, one may similarly hope:

######  Conjecture B.

The inclusion
$\mathscr{P}_{d,n}\cap{\mathbf{Rat}}_{d}\hookrightarrow\mathscr{P}_{d,n}$ is a
homotopy equivalence up to dimension $d$.

###  1.1. Spaces of marked maps

The general setting is a structure $\Pi=\langle A,B,C,F,\deg\rangle$
consisting of sets $A,B,C$ with $A\subset B$ and $A\subset C$, and maps
$F\colon C\to B$ and $\deg\colon C\to\\{1,2,\dots\\}$. The corresponding space
of marked maps is

$\mathscr{P}_{\Pi}\coloneqq\\{(f,b,c):b\colon B\hookrightarrow S^{2},c\colon
C\hookrightarrow S^{2},f\colon(S^{2},c(C))\to(S^{2},b(B))\text{ branched
covering},\\\ f\circ c=b\circ F,\;\deg_{c(x)}f=\deg(x),\;b\restriction
A=c\restriction A,\;f\text{ ramifies only above }b(B)\\}.$

For example, setting $A=\\{a_{0},\dots,a_{n-1}\\}$ and $B=A\cup\\{v\\}$ and
$C=A\cup\\{c\\}$ with $\deg(a_{0})=\deg(c)=d$ and $F(a_{i})=a_{i+1\bmod n}$
and $F(c)=b$ specifies precisely the space $\mathscr{P}_{d,n}$ introduced
above. The method of this article should serve to solve a variety of
connectedness problems about loci $\mathscr{P}_{\Pi}$.

Thurston’s theory of iteration of rational maps involves considering their
topological counterparts. A branched covering $f$ is _critically finite_ if
the forward orbit $P_{f}$ of its critical points is finite. Setting
$A=B=C=P_{f}$ and $F=f\restriction P_{f}$ recovers the space of all maps with
same post-critical behaviour as $f$, and his fundamental result implies that
the connected component of $f$ is contractible and contains at most one
holomorphic representative (with a combinatorial criterion to determine
whether there is one), unless the map is double covered by a homothety on the
torus. At the other extreme, $A=\emptyset$ amounts to considering branched
coverings with no dynamical constraint, and subsumes the classical Hurwitz
theory of coverings of surfaces [hurwitz:ramifiedsurfaces].

By abuse of notation, we will think of $B,C$ both as abstract sets (used to
define $\Pi$) and as variable subsets of $S^{2}$; and abbreviate

$\mathscr{P}_{\Pi}=\\{f\colon(S^{2},C)\to(S^{2},B):f\restriction
A=F\restriction A,\deg_{f}=\deg\\}.$

Kahn, Firsova and Selinger [firsova-kahn-selinger:deformation] consider,
following Mary Rees [rees:tr], a space of maps modeled on Teichmüller space,
which we can express as follows in our setting. Recall that the Teichmüller
space of a marked sphere consists of all its complex structures; for $B\subset
S^{2}$ we may define

$\mathcal{T}_{B}=\\{\tau\colon S^{2}\to{\mathbb{P}^{1}}\text{
homeomorphism}\\}/{\sim},$

with $m\circ\tau\circ h\sim\tau$ for every Möbius transformation $m$ and every
homeomorphism $h$ of $(S^{2},B)$ that is isotopic to the identity rel $B$;
henceforth written $h\in{\operatorname{Homeo}}_{0}(S^{2},B)$. Then _Rees
space_ is

$\mathscr{R}_{\Pi}=(\mathcal{T}_{B}\times\mathscr{P}_{\Pi})/{\sim},$

with $(\tau\circ h,f)\sim(\tau,h\circ f)\sim(\tau,f\circ h^{\prime})$ whenever
$h\colon(S^{2},B)\righttoleftarrow,h^{\prime}\colon(S^{2},C)\righttoleftarrow$
satisfy $h\circ f\simeq f\circ h^{\prime}$ rel $C$ and $h\simeq h^{\prime}$
rel $A$. Every choice of $f\in\mathscr{P}_{\Pi}$ yields a map
$\mathcal{T}_{B}\to\mathscr{R}_{\Pi}$, which is a covering on its image, the
group of deck transformations being the centralizer
$\\{h\in\mathbf{Mod}_{B}:h\circ f=f\circ h\text{ rel }A\\}$ of $f$. Theorem A
can be rephrased as the statement

######  Corollary C.

For $\Pi$ marking two order-$d$ critical points one of which in $n$-periodic,
the space $\mathscr{R}_{\Pi}$ is a quotient of Teichmüller space
$\mathcal{T}_{B}$ by a subgroup of $\operatorname{Mod}(B)$, so in particular
is a $K(\pi,1)$ space for some $\pi\leq\operatorname{Mod}(B)$.

There is a fibration $\mathscr{P}_{\Pi}\to\mathscr{Q}$, the space of
configurations of $B\cup_{A}C$ in $S^{2}$. Let $\mathscr{W}$ denote a fibre,
thought of as a space of branched coverings with specified marked points. The
group ${\operatorname{Homeo}}_{0}(S^{2},C)$ is contractible and acts freely on
$\mathscr{W}$ by pre-composition, with discrete quotient. From the long exact
sequence of homotopy groups,

######  Proposition D.

The space $\mathscr{W}$ is homotopy equivalent to a discrete set. The space
$\mathscr{P}_{d,n}$ is path connected;
$\pi_{1}(\mathscr{P}_{d,n})\cong\ker(\pi_{1}(\mathscr{Q})\twoheadrightarrow\pi_{0}(\mathscr{W}))$;
and $\pi_{q}(\mathscr{P}_{d,n})\cong\pi_{q}(\mathscr{Q})$ for all $q>1$.
Furthermore, $\pi_{1}(\mathscr{Q})$ is naturally identified with the equalizer
of the two natural (forgetful) maps
$\operatorname{Mod}(B)\to\operatorname{Mod}(A),\operatorname{Mod}(C)\to\operatorname{Mod}(A)$,
and $\pi_{0}(\mathscr{W})$ is naturally identified with
$\operatorname{Mod}(C)\times\\{1,\dots,d^{n-2}$.∎

Every $(\tau,f)\in\mathscr{R}_{\Pi}$ yields two complex structures on
$(S^{2},A)$: one locally given by $\tau$ and one by $\tau\circ f$ and denoted
$\sigma_{f}(\tau)$. The _rational Rees space_
$\mathscr{R}_{\Pi}^{\mathbf{Rat}}$ is the locus at which these complex
structures agree; so for $(\tau,f)\in\mathscr{R}_{\Pi}^{\mathbf{Rat}}$ we have
a commutative diagram

${(S^{2},C)}$${{\mathbb{P}^{1}}}$${(S^{2},B)}$${{\mathbb{P}^{1}}}$$\scriptstyle{\tau}$$\scriptstyle{f}$${\ulcorner\sim}$$\scriptstyle{r_{\tau,f}}$$\scriptstyle{\tau}$

that commutes up to homotopy, for some rational map
$r_{\tau,f}\in{\mathbf{Rat}}_{d}$, unique up to conjugation by a Möbius
transformation.

Let $\mathbf{MRat}$ denote the quotient of ${\mathbf{Rat}}={\mathbb{C}}(z)$ by
$\operatorname{PSL}_{2}({\mathbb{C}})$ acting under conjugation; and consider
the locus

$\mathbf{MRat}_{\Pi}=({\mathbf{Rat}}\cap\mathscr{P}_{\Pi})/\operatorname{PSL}_{2}({\mathbb{C}}).$

Note that every $f\in\mathbf{MRat}_{\Pi}$ has attached subsets
$A,B,C\subset{\mathbb{P}^{1}}$, well-defined up to a Möbius transformation.
Every choice of $\tau\in\mathcal{T}_{B}$ gives a natural map
$\mathscr{P}_{\Pi}\to\mathscr{R}_{\Pi}$, by $f\mapsto(\tau,f)$. The natural
map $\mathscr{R}_{\Pi}^{\mathbf{Rat}}\to\mathbf{MRat}_{\Pi}$, given by
$(\tau,f)\mapsto r_{\tau,f}$, is an isomorphism. There is finally a map
$\mathbf{MRat}_{\Pi}\to\mathcal{M}_{B}$, the moduli space of $(S^{2},B)$,
given by $r_{\tau,f}\mapsto\tau\restriction B$. Consider, to complete the
picture, Epstein’s equalizer space: it is the space

${\mathcal{D}\\!\mathit{ef}}_{\Pi,f}=\\{\tau\in\mathcal{T}_{B}:\tau\text{ and
}\sigma_{f}(\tau)\text{ have the same image in }\mathcal{T}_{A}\\}.$

Epstein’s transversality theory [epstein:transversality] shows that
${\mathcal{D}\\!\mathit{ef}}_{\Pi,f}$ is a submanifold of dimension
$\\#B-\\#A-3$. In summary, we have

${{\mathcal{D}\\!\mathit{ef}}_{\Pi,f}}$${\mathcal{T}_{B}}$${\makebox[0.0pt][r]{$\mathbf{MRat}_{\Pi}={}$}\mathscr{R}_{\Pi}^{\mathbf{Rat}}}$${\mathscr{R}_{\Pi}}$${\mathcal{M}_{\Pi}}$${\mathcal{M}_{B}}$$\scriptstyle{\tau\mapsto(\tau,f)}$$\scriptstyle{(\tau,f)\mapsto\tau\restriction
B}$

and our result shows that the top vertical maps are onto. The bottom left
vertical map is a finite-to-one map [firsova-kahn-selinger:deformation]*Lemma
4.5 and in many cases, in particular that of a degree-$d$ map with a marked
$n$-cycle [firsova-kahn-selinger:deformation]*Lemma 4.7 is actually a
bijection.

Previous literature concentrated on connectivity and contractibility of
${\mathcal{D}\\!\mathit{ef}}_{\Pi}$. The arguments in [firsova-kahn-
selinger:deformation, hironaka-koch:disconnected], proving that
${\mathcal{D}\\!\mathit{ef}}_{\Pi}$ is disconnected or at least not
contractible, rely on showing that the fundamental group of the punctured
Riemann surface $\mathbf{MRat}_{\Pi}$ is not isomorphic to the equalizer
$E\leq\operatorname{Mod}(B)$.

###  1.2. Sketch of proof

We reduce Theorem A to a discrete problem, by means of isotopy.

  1. (1)

Consider the fibration $\mathscr{P}_{\Pi}\to\mathscr{Q}$ and let $W$ be the
collection of isotopy classes of a fibre. In our concrete situation, we
imagine that sets $A,B,C$ of respective sizes $n,n+1,n+1$ are frozen on
$S^{2}$, and we consider the collection $W$ of isotopy classes of maps
$(S^{2},C)\to(S^{2},B)$ that permute $A$ cyclically. Thus $W$ is the
collection of isotopy classes of maps $f\in\mathscr{P}_{d,n}$ with these fixed
$A,B,C$.

  2. (2)

There are two commuting actions on $W$, by the pure mapping class groups
$\operatorname{Mod}(B)$ and $\operatorname{Mod}(C)$, respectively by post-
composition and pre-composition. The action of $\operatorname{Mod}(C)$ is free
with $d^{n-2}$ orbits.

  3. (3)

There is a single degree-$d$ covering with two order-$d$ marked critical
points; this is the “Hurwitz problem” for the data $(d,d)$. In fact, by
[thom:polynome]*Théorème (1) this holds more generally for any degree-$d$
covering with a critical point of order $d$. It follows that there is a single
$(\operatorname{Mod}(B)\times\operatorname{Mod}(C))$-orbit on $W$.

  4. (4)

Denote respectively by $\varepsilon_{B},\varepsilon_{C}$ the natural
restriction maps
$\operatorname{Mod}(B)\to\operatorname{Mod}(A),\operatorname{Mod}(C)\to\operatorname{Mod}(A)$.
The subgroup
$E=\\{(g,h)\in\operatorname{Mod}(B)\times\operatorname{Mod}(C):\varepsilon_{B}(g)=\varepsilon_{C}(h)\\}$
acts on $W$, and movements in $E$ preserve the cycle $A$. Connectedness of
$\mathscr{P}_{d,n}$ is therefore equivalent to transitivity of $E$ on $W$.

  5. (5)

To prove transitivity of $E$, fix a polynomial map $f\in\mathscr{P}_{d,n}$,
thereby assuming $B=C$. We consider two subgroups of $E$: the diagonal
$\Delta=\\{(g,g):g\in\operatorname{Mod}(B)\\}$, which acts on $W$ by
conjugation, and $P=\ker(\varepsilon_{B})$, the “point pushes” of the critical
value, which act on $W$ by post-composition. It suffices to show that every
element of $W$ may be written $pgfg^{-1}$ with $g\in\operatorname{Mod}(B)$ and
$p\in P$.

  6. (6)

Compute the “lifting” operation: let $L\leq\operatorname{Mod}(B)$ denote the
index-$d^{n-2}$ subgroup of liftable classes; namely all
$h\in\operatorname{Mod}(B)$ such that there exists
$\phi(h)\in\operatorname{Mod}(C)$ with $h\circ f\cong f\circ\phi(h)$. Enough
images under homomorphism $\phi\colon L\to\operatorname{Mod}(C)$ may be
written explicitly to show that all generators of $\operatorname{Mod}(B)$ may
be obtained from $P$ via $\phi$ and conjugation.

###  1.3. Acknowledgments

This work owes much to Jan Kiwi, who explained to me the relevance of group-
theoretic considerations to connectedness questions in parameter spaces during
an invitation to Santiago. I am very grateful to MSRI for providing an
extremely stimulating environment, and to Tanya Firsova, Eko Hironaka, Jeremy
Kahn, Sarah Koch and Curtis McMullen for lively discussions and generous
explanations of their work.

##  2\. Dynamical bisets and mapping class bisets

Let us consider the polynomial $f(z)=z^{d}+c$ for which the supporting rays
have angles $\\{1,2\\}/(d^{n}-1)$, and encode $f$ group-theoretically. This
means we let $A=\\{a_{0}=a_{n}=0,a_{1}=c,a_{2}=c^{d}+c,\dots\\}$ be the
length-$n$ critical cycle of $f$; set $B=C=A\cup\\{\infty\\}$; fix a basepoint
$*$ in ${\mathbb{C}}\setminus A$ near $\infty$; let
$\pi=\pi_{1}({\mathbb{C}}\setminus A,*)$ be the fundamental group; and choose
“lollipop” generators $\gamma_{1},\dots,\gamma_{n},\gamma_{\infty}$ of $\pi$:
the generator $\gamma_{i}$ follows the external ray with angle
$d^{i}/(d^{n}-1)$ towards $a_{i}$, encircles it counterclockwise, and returns
back to $*$, while the generator $\gamma_{\infty}$ is a small (on the sphere)
counterclockwise loop around $\infty$. Note that one of the generators of
$\pi$ is redundant, and we have

$\pi=\langle\gamma_{1},\dots,\gamma_{n},\gamma_{\infty}\mid\gamma_{\infty}\gamma_{n}\cdots\gamma_{1}=1\rangle.$

The “iterated monodromy group” theory [nekrashevych:ssg] exncodes $f$ as a
_biset_ : a set $\mathfrak{B}(f)$ with two commuting $\pi$-actions:

$\mathfrak{B}(f)=\\{\alpha\colon[0,1]\to{\mathbb{C}}\setminus
A:\alpha(0)=f(\alpha(1))=*\\}/{\sim},$

with the left action by pre-concatenation of a loop, and the right action by
post-concatenation with the unique lift of a loop that starts where $\alpha$
ends. The left action on $B_{f}$ is free with $d$ orbits; so $B_{f}$ may be
written $\pi\times\\{x_{1},\dots,x_{d}\\}$ by choosing a system of orbit
representatives. A natural choice consists of short, counterclockwise paths
around $\infty$ from $*$ to all its preimages; for concreteness if $*\gg 0$ we
may choose $x_{j}(t)=\exp(\log(*)(d+1-dt)/d+2\pi ijt/d)$.

It is then straightforward to trace paths and their lifts, so as to express
the right action of $\pi$ on $B_{f}$; it is

$\displaystyle x_{1}\cdot\gamma_{1}$
$\displaystyle=\gamma_{\infty}\gamma_{n}\cdot x_{2},$ $\displaystyle
x_{i}\cdot\gamma_{1}$ $\displaystyle=x_{i+1}\text{ if }1<i<d,$ $\displaystyle
x_{d}\cdot\gamma_{1}$ $\displaystyle=\gamma_{\infty}^{-1}\cdot x_{1},$
$\displaystyle x_{1}\cdot\gamma_{j+1}$ $\displaystyle=\gamma_{j}\cdot x_{1},$
$\displaystyle x_{i}\cdot\gamma_{j+1}$ $\displaystyle=x_{i}\text{ if }1<i\leq
d\text{ and }1\leq j\leq n,$ $\displaystyle x_{1}\cdot\gamma_{\infty}$
$\displaystyle=\gamma_{\infty}\cdot x_{d},$ $\displaystyle
x_{i+1}\cdot\gamma_{\infty}$ $\displaystyle=x_{i}\text{ if }1\leq i<d.$

This is the natural degree-$d$ generalization of the recursion defining the
group $\mathfrak{K}(0^{n-1})$ from [bartholdi-n:mandelbrot1]; see also
[poirier:portraits].

Note that a different choice of orbit representatives
$\\{x_{1}^{\prime},\dots,x_{d}^{\prime}\\}$ would give different formulas for
the right action; the object $B_{f}$, considered up to isomorphism of
$\pi$-$\pi$-bisets, is a complete invariant of $f$ but its presentation relies
on choices.

Let us make this a bit more precise. A _presentation_ of a biset
$B_{f^{\prime}}$, for a map $f^{\prime}\in W$, is a $d\times(n+1)$ matrix with
in entry $(i,j)$ a pair $(g,k)\in\pi\times\\{1,\dots,d\\}$, describing the
relation $x_{i}\cdot\gamma_{j}=g\cdot x_{k}$; here we use the convention
$n+1=\infty$. A permutation of the $i$ and $k$ defines an isomorphic
presentation (this amounts to reordering the orbit representatives); and for
every choice of $g_{1},\dots,g_{d}\in\pi$ the replacement of every $(g,k)$ at
position $(i,j)$ with $(g_{i}gg_{k}^{-1},k)$ also defines an isomorphic
presentation (this amounts to replacing the orbit representatives
$(x_{1},\dots,x_{d})$ with $(g_{1}x_{1},\dots,g_{d}x_{d})$).

Furthermore, the left and right actions on $W$ may be expressed in terms of
these presentations: pre-composition by $\psi$ amounts to replacing each
$(g,k)$ by $(\psi(g),k)$; while post-composition by $\phi$ amounts to using
the table to rewrite $x_{i}\cdot\phi(\gamma_{j})$ in the form $g\cdot x_{k}$
and recording the result in a new table.

Now, $\operatorname{Mod}(B)$ is generated by a collection of full Dehn twists
between elements of $B$, and acts on $\pi$ by outer automorphisms. We choose
as generators the $\tau_{i,j}$ for $1\leq i<j\leq\infty$ which, geometrically,
push $a_{i}$ and $a_{j}$ closer while avoiding all other paths $\gamma_{k}$,
and twist them fully around each other. The action of $\tau_{i,j}$ on $\pi$
may be written concretely as follows: set
$\alpha=\gamma_{j-1}\gamma_{j-2}\cdots\gamma_{i+1}$; then

$\tau_{i,j}(\gamma_{i})=\gamma_{i}^{\alpha^{-1}\gamma_{j}\alpha\gamma_{i}},\qquad\tau_{i,j}(\gamma_{j})=\gamma_{j}^{\alpha\gamma_{i}\alpha^{-1}},\qquad\tau_{i,j}(\gamma_{k})=\gamma_{k}.$

The subgroup of point pushes is $P=\langle\tau_{i,\infty}:1\leq i\leq
n\rangle$, and its liftable elements include $\tau_{i,\infty}^{d}$ as well as
$\tau_{1,\infty}$ and its conjugates.

Once all these choices are made, it is straightforward to check the following
identities:

######  Lemma 2.1.

In $W$ we have the identities

(1) $\displaystyle\tau_{1,\infty}\cdot f$
$\displaystyle=f\cdot\tau_{n,\infty},$ (2)
$\displaystyle\tau_{1,\infty}^{\tau_{i+1,\infty}}\cdot f$
$\displaystyle=f\cdot\tau_{i,\infty}\tau_{i,n}\tau_{n,\infty},$ (3)
$\displaystyle\tau_{i+1,\infty}^{d}\cdot f$
$\displaystyle=f\cdot\tau_{i,\infty}^{\tau_{i,n}},$ (4)
$\displaystyle\tau_{i+1,j+1}\cdot f$ $\displaystyle=f\cdot\tau_{i,j},$

for all $1\leq i,j\leq n-1$.

###### Proof.

We only consider (2), the other ones being checked in a similar but easier
manner. Set $\phi=\tau_{1,\infty}^{\tau_{i+1},\infty}$; then, writing
$\delta=\gamma_{i}\gamma_{i-1}\cdots\gamma_{1}$ and
$\beta=\gamma_{i+1}^{\delta}\gamma_{\infty}$, we have

$\displaystyle\phi(\gamma_{1})$
$\displaystyle=\gamma_{1}^{\gamma_{\infty}^{\beta}},$
$\displaystyle\phi(\gamma_{i+1})$
$\displaystyle=\gamma_{i+1}^{\delta\gamma_{\infty}^{-\gamma_{1}^{\beta^{-1}}}\gamma_{\infty}\delta^{-1}},$
$\displaystyle\phi(\gamma_{\infty})$
$\displaystyle=\gamma_{\infty}^{\beta\gamma_{1}\gamma_{\infty}^{-\gamma_{1}^{\beta^{-1}}\gamma_{\infty}}\beta},$

all other generators being fixed. The biset $B(\phi\circ f)$ of the post-
composition of $f$ with $\phi$ is then presented as follows in a basis
$(y_{1},\dots,y_{d})$, with $\epsilon=\gamma_{i}\cdots\gamma_{1}$ and
$\zeta=\gamma_{n-1}\cdots\gamma_{i+1}$ so
$\gamma_{\infty}\gamma_{n}\zeta\gamma_{i}\epsilon=1$:

$\mathfrak{B}(\phi\circ
f):\begin{cases}y_{1}\cdot\gamma_{1}&=\gamma_{i}^{-\zeta^{-1}}\cdot y_{2},\\\
y_{d-1}\cdot\gamma_{1}&=\gamma_{i}^{-\zeta^{-1}\gamma_{\infty}}\cdot y_{d},\\\
y_{d}\cdot\gamma_{1}&=\gamma_{\infty}^{-1}\gamma_{i}^{\zeta^{-1}}\epsilon^{-1}\zeta^{-1}\cdot
y_{1},\\\ y_{1}\cdot\gamma_{i+1}&=\gamma_{i}^{\zeta^{-1}\epsilon^{-1}}\cdot
y_{1},\\\
y_{1}\cdot\gamma_{\infty}&=\zeta\epsilon\gamma_{i}^{-\zeta^{-1}}\gamma_{\infty}\cdot
y_{d},\\\ y_{2}\cdot\gamma_{\infty}&=\epsilon^{-1}\zeta^{-1}\cdot y_{1},\\\
y_{d}\cdot\gamma_{\infty}&=\gamma_{i}^{\zeta^{-1}\gamma_{\infty}}\cdot
y_{2},\end{cases}$

all other entries being as in the presentation of $\mathfrak{B}(f)$. On the
other hand, set $\psi=\tau_{i,\infty}\tau_{i,n}\tau_{n,\infty}$; we have

$\displaystyle\psi(\gamma_{i})$
$\displaystyle=\gamma_{i}^{\zeta^{-1}\epsilon^{-1}},$
$\displaystyle\psi(\gamma_{n})$
$\displaystyle=\gamma_{n}^{\epsilon^{-1}\zeta^{-1}},$
$\displaystyle\psi(\gamma_{\infty})$
$\displaystyle=\gamma_{\infty}^{\epsilon^{-1}\zeta^{-1}},$

all other generators being fixed. The biset $\mathfrak{B}(f\circ\psi)$ is
presented as follows in a basis $(z_{1},\dots,z_{d})$:

$\mathfrak{B}(f\circ\psi):\begin{cases}x_{1}\cdot\gamma_{1}&=(\gamma_{\infty}\gamma_{n})^{\epsilon^{-1}\zeta^{-1}}\cdot
x_{2},\\\
x_{d}\cdot\gamma_{1}&=\gamma_{\infty}^{-\epsilon^{-1}\zeta^{-1}}\cdot
x_{1},\\\ x_{1}\cdot\gamma_{i+1}&=\gamma_{i}^{\zeta^{-1}\epsilon^{-1}}\cdot
x_{1},\\\
x_{1}\cdot\gamma_{\infty}&=\gamma_{\infty}^{\epsilon^{-1}\zeta^{-1}}\cdot
x_{d},\\\ \end{cases}$

all other entries being as in the presentation of $\mathfrak{B}(f)$. Now to
prove that $\mathfrak{B}(\phi\circ f)$ and $\mathfrak{B}(f\circ\psi)$ are
isomorphic, it suffices to map the basis of the former into the latter, as
follows:

$\displaystyle y_{1}$ $\displaystyle\mapsto z_{1},$ $\displaystyle y_{k}$
$\displaystyle\mapsto\epsilon^{-1}\zeta^{-1}\cdot z_{k}\text{ for }1<k<d,$
$\displaystyle y_{d}$
$\displaystyle\mapsto\gamma_{i}^{\zeta^{-1}\gamma_{\infty}}\epsilon^{-1}\zeta^{-1}\cdot
z_{d},$

and to check that the right actions of $\pi$ on $\mathfrak{B}(\phi\circ f)$
and $\mathfrak{B}(f\circ\psi)$ are intertwined by this map. ∎

######  Corollary 2.2.

We have $W=Pf^{\operatorname{Mod}(B)}$, for $P$ the group of point pushes and
$\operatorname{Mod}(B)$ acting by conjugation.

###### Proof.

The solution to the Hurwitz problem implies
$W=\operatorname{Mod}(B)f^{\operatorname{Mod}(B)}$. Now consider the set
$M\subseteq\operatorname{Mod}(B)$ of all $m\in\operatorname{Mod}(B)$ with
$mf^{\operatorname{Mod}(B)}\subseteq Pf^{\operatorname{Mod}(B)}$. Clearly $M$
is closed under conjugation, since $P$ is normal in $\operatorname{Mod}(B)$.
Also, $M$ is a subgroup: if $m,n\in M$ and
$mf^{\operatorname{Mod}(B)},nf^{\operatorname{Mod}(B)}\subseteq
Pf^{\operatorname{Mod}(B)}$, then

$mnf^{\operatorname{Mod}(B)}\subseteq
m(Pf^{\operatorname{Mod}(B)})=mf^{\operatorname{Mod}(B)}P\subseteq
Pf^{\operatorname{Mod}(B)}P=P^{2}f^{\operatorname{Mod}(B)}=Pf^{\operatorname{Mod}(B)}.$

Consider $m\in M$, and assume that we have an identity $m\cdot f=f\cdot h$ in
$W$; then $h\cdot f=(f\cdot h)^{h^{-1}}=(m\cdot f)^{h^{-1}}=m^{h^{-1}}\cdot
f^{h^{-1}}\in Pf^{\operatorname{Mod}(B)}$; so $h\in M$.

Obviously we have $P\subseteq M$, namely $\tau_{i,\infty}\in M$ for all $i$.
Then, by Lemma 2.1(2), we have $\tau_{i,n}\in M$. Then Lemma 2.1(4) gives
$\tau_{i-1,n-1}\in M$, etc., so finally all $\tau_{i,j}\in M$. Therefore
$M=\operatorname{Mod}(B)$. ∎

##  3\. Proof of Theorem A

We expand on the sketch presented in §1.2. Consider first the map
$\tau\colon\mathscr{P}_{\Pi}\ni(f,b,c)\mapsto(b,c)$ sending a branched
covering to its marked sets $b,c$ on $S^{2}$. This map is evidently a
fibration, with fibre $\mathscr{W}$ consisting of all branched coverings $f$
with fixed $b,c$. We treat, from now on, $A,B,C$ as fixed subsets of $S^{2}$
with $A\subset B\cap C$, so $\mathscr{W}$ is the set of branched coverings
$(S^{2},C)\to(S^{2},B)$ that map $C$ to $B$ with combinatorics and degree
prescribed by $\Pi$.

We let $W$ denote the quotient of $\mathscr{W}$ under isotopy. For any two
branched coverings $f,f^{\prime}\in\mathscr{W}$ that are isotopic, there is a
unique homeomorphism $h\in{\operatorname{Homeo}}_{0}(S^{2},C)$ with
$f^{\prime}=f\circ h$; in other words, $W$ is the quotient of $\mathscr{W}$ by
the free action of ${\operatorname{Homeo}}_{0}(S^{2},C)$. Now the group
${\operatorname{Homeo}}_{0}(S^{2},C)$ is contractible, assuming $\\#C\geq 3$.
It follows from the long exact sequence of a fibration and Whitehead’s theorem
that $\mathscr{W}$ has the homotopy type of a discrete set, and (except in
dimension $\leq 1$) the homotopy type of $\mathscr{P}_{\Pi}$ is that of the
base $\mathscr{Q}$ of the fibration,

$\mathscr{Q}\coloneqq\\{(b\colon B\hookrightarrow S^{2},c\colon
C\hookrightarrow S^{2}):b\restriction A=c\restriction A\\}.$

We have thus proven Proposition D, modulo the main connectedness claim.

There are two commuting actions on $W$, by the pure mapping class groups
$\operatorname{Mod}(B)$ and $\operatorname{Mod}(C)$, respectively by post-
composition and pre-composition. The action of $\operatorname{Mod}(C)$ is free
with finitely many orbits. More precisely, for every $f\in W$ there is a
finite-index subgroup $L_{f}\leq\operatorname{Mod}(B)$ consisting of
“liftable” classes: all $\ell\in\operatorname{Mod}(B)$ for which there exists
$m\in\operatorname{Mod}(B)$ with $\ell\circ f\cong f\circ m$. We denote this
(necessarily unique) $m\in\operatorname{Mod}(C)$ by $\sigma_{f}(\ell)$,
defining thus a homomorphism $\sigma_{f}\colon L_{f}\to\operatorname{Mod}(C)$.

The long exact sequence of homotopy groups gives

${\pi_{1}(\mathscr{Q})}$${\pi_{0}(\mathscr{W})}$${\pi_{0}(\mathscr{P}_{\Pi})}$${\pi_{0}(\mathscr{Q}).}$

There is a natural map
$\varepsilon_{B}\colon\operatorname{Mod}(B)\twoheadrightarrow\operatorname{Mod}(A)$
induced by the inclusion $A\subset B$; and similarly
$\varepsilon_{C}\colon\operatorname{Mod}(C)\twoheadrightarrow\operatorname{Mod}(A)$.
The equalizer of these two maps is the subgroup

$E=\\{(g,h)\in\operatorname{Mod}(B)\times\operatorname{Mod}(C):\varepsilon_{B}(g)=\varepsilon_{C}(h)\\}$

and we naturally have $E\cong\pi_{1}(\mathscr{Q})$. We thus have

######  Lemma 3.1.

The space $\mathscr{P}_{\Pi}$ is path connected if and only if the action of
$E$ on $W$ is transitive.∎

###  3.1. Specifics for degree-$d$ bicritical maps with an $n$-cycle

All the above considerations applied to a general portrait $\Pi$. We now
consider the specific choice of a portrait $\Pi$ specifying an $n$-cycle
marked $A$ that contains a single order-$d$ critical point, and another
order-$d$ critical point marked in $C$, with its image marked in $B$.

######  Lemma 3.2.

The action of $\operatorname{Mod}(B)\times\operatorname{Mod}(C)$ on $W$ is
transitive.

###### Proof.

In considering the action of
$\operatorname{Mod}(B)\times\operatorname{Mod}(C)$, we are in effect ignoring
the condition that $A$ is an invariant $n$-cycle; that is, we are considering
the space of coverings of $(S^{2},B)$ with two order-$d$ critical values.
Hurwitz’s theory [hurwitz:ramifiedsurfaces] classifies its connected
components as follows: there is then a bijection between, on the one hand,
isomorphism classes of coverings with two order-$d$ critical values
$b_{1},b_{2}$; and on the other hand orbits of pairs $(\rho_{1},\rho_{2})$ of
$d$-cycles in $\operatorname{Sym}(d)$ with $\rho_{1}\rho_{2}=1$, under the
diagonal action of $\operatorname{Sym}(d)$ by conjugation. There is evidently
a single such orbit, since all $d$-cycles are conjugate. ∎

(As mentioned in the sketch, this fact holds more generally for any portrait
with a critical value of maximal order, by [thom:polynome]*Théorème (1).
Imposing that this critical value be fixed defines a polynomial slice.)

Let us now fix a polynomial map $f\in\mathscr{P}_{d,n}$, assuming $B=C$. We
consider two subgroups of $E$: the diagonal
$\Delta=\\{(g,g):g\in\operatorname{Mod}(B)\\}$, which acts on $W$ by
conjugation, and $P=\ker(\varepsilon_{B})$, the “point pushes” of the critical
value, which act on $W$ by post-composition. We have $P=\pi_{1}(S^{2}\setminus
A,b)$, and Birman’s exact sequence [birman:ses] gives

${1}$${P}$${\operatorname{Mod}(B)}$${\operatorname{Mod}(A)}$${1.}$$\scriptstyle{\varepsilon_{B}}$

It suffices to show that every element of $W$ may be written $pgfg^{-1}$ with
$g\in\operatorname{Mod}(B)$ and $p\in P$. Now this is precisely Corollary 2.2.

  * math


[27,$\dagger$]S. Patra, \dagger\dagger$\dagger$Corresponding author.
11institutetext: Department of Physics, University of the Basque Country
UPV/EHU, 48080 Bilbao, Spain 22institutetext: University of Bonn, 53115 Bonn,
Germany 33institutetext: Brookhaven National Laboratory, Upton, New York
11973, USA 44institutetext: Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090, Russian Federation 55institutetext: Faculty of Mathematics
and Physics, Charles University, 121 16 Prague, The Czech Republic
66institutetext: Chonnam National University, Gwangju 61186, South Korea
77institutetext: Chung-Ang University, Seoul 06974, South Korea
88institutetext: University of Cincinnati, Cincinnati, OH 45221, USA
99institutetext: Deutsches Elektronen-Synchrotron, 22607 Hamburg, Germany
1010institutetext: Duke University, Durham, NC 27708, USA 1111institutetext:
Institute of Theoretical and Applied Research (ITAR), Duy Tan University,
Hanoi 100000, Vietnam 1212institutetext: University of Florida, Gainesville,
FL 32611, USA 1313institutetext: Department of Physics, Fu Jen Catholic
University, Taipei 24205, Taiwan 1414institutetext: Key Laboratory of Nuclear
Physics and Ion-beam Application (MOE) and Institute of Modern Physics, Fudan
University, Shanghai 200443, PR China 1515institutetext: Justus-Liebig-
Universität Gießen, 35392 Gießen, Germany 1616institutetext: Gifu University,
Gifu 501-1193, Japan 1717institutetext: II. Physikalisches Institut, Georg-
August-Universität Göttingen, 37073 Göttingen, Germany 1818institutetext:
SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193,
Japan 1919institutetext: Gyeongsang National University, Jinju 52828, South
Korea 2020institutetext: Department of Physics and Institute of Natural
Sciences, Hanyang University, Seoul 04763, South Korea 2121institutetext:
University of Hawaii, Honolulu, HI 96822, USA 2222institutetext: High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
2323institutetext: J-PARC Branch, KEK Theory Center, High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801, Japan 2424institutetext:
National Research University Higher School of Economics, Moscow 101000,
Russian Federation 2525institutetext: Forschungszentrum Jülich, 52425 Jülich,
Germany 2626institutetext: IKERBASQUE, Basque Foundation for Science, 48013
Bilbao, Spain 2727institutetext: Indian Institute of Science Education and
Research Mohali, SAS Nagar, 140306, India 2828institutetext: Indian Institute
of Technology Bhubaneswar, Satya Nagar 751007, India 2929institutetext: Indian
Institute of Technology Guwahati, Assam 781039, India 3030institutetext:
Indian Institute of Technology Hyderabad, Telangana 502285, India
3131institutetext: Indian Institute of Technology Madras, Chennai 600036,
India 3232institutetext: Indiana University, Bloomington, IN 47408, USA
3333institutetext: Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, PR China 3434institutetext: Institute for High
Energy Physics, Protvino 142281, Russian Federation 3535institutetext:
Institute of High Energy Physics, Vienna 1050, Austria 3636institutetext: INFN
- Sezione di Napoli, I-80126 Napoli, Italy 3737institutetext: INFN - Sezione
di Roma Tre, I-00146 Roma, Italy 3838institutetext: INFN - Sezione di Torino,
I-10125 Torino, Italy 3939institutetext: Iowa State University, Ames, Iowa
50011, USA 4040institutetext: Advanced Science Research Center, Japan Atomic
Energy Agency, Naka 319-1195, Japan 4141institutetext: J. Stefan Institute,
1000 Ljubljana, Slovenia 4242institutetext: Institut für Experimentelle
Teilchenphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe, Germany
4343institutetext: Kavli Institute for the Physics and Mathematics of the
Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan
4444institutetext: Department of Physics, Faculty of Science, King Abdulaziz
University, Jeddah 21589, Saudi Arabia 4545institutetext: Kitasato University,
Sagamihara 252-0373, Japan 4646institutetext: Korea Institute of Science and
Technology Information, Daejeon 34141, South Korea 4747institutetext: Korea
University, Seoul 02841, South Korea 4848institutetext: Kyungpook National
University, Daegu 41566, South Korea 4949institutetext: Université Paris-
Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France 5050institutetext: P.N.
Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991,
Russian Federation 5151institutetext: Liaoning Normal University, Dalian
116029, China 5252institutetext: Faculty of Mathematics and Physics,
University of Ljubljana, 1000 Ljubljana, Slovenia 5353institutetext: Ludwig
Maximilians University, 80539 Munich, Germany 5454institutetext: Luther
College, Decorah, IA 52101, USA 5555institutetext: Malaviya National Institute
of Technology Jaipur, Jaipur 302017, India 5656institutetext: Faculty of
Chemistry and Chemical Engineering, University of Maribor, 2000 Maribor,
Slovenia 5757institutetext: Max-Planck-Institut für Physik, 80805 München,
Germany 5858institutetext: School of Physics, University of Melbourne,
Victoria 3010, Australia 5959institutetext: University of Mississippi,
University, MS 38677, USA 6060institutetext: University of Miyazaki, Miyazaki
889-2192, Japan 6161institutetext: Moscow Physical Engineering Institute,
Moscow 115409, Russian Federation 6262institutetext: Graduate School of
Science, Nagoya University, Nagoya 464-8602, Japan 6363institutetext:
Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan
6464institutetext: Università di Napoli Federico II, I-80126 Napoli, Italy
6565institutetext: Nara Women’s University, Nara 630-8506, Japan
6666institutetext: National Central University, Chung-li 32054, Taiwan
6767institutetext: National United University, Miao Li 36003, Taiwan
6868institutetext: Department of Physics, National Taiwan University, Taipei
10617, Taiwan 6969institutetext: H. Niewodniczanski Institute of Nuclear
Physics, Krakow 31-342, Poland 7070institutetext: Nippon Dental University,
Niigata 951-8580, Japan 7171institutetext: Niigata University, Niigata
950-2181, Japan 7272institutetext: Novosibirsk State University, Novosibirsk
630090, Russian Federation 7373institutetext: Osaka City University, Osaka
558-8585, Japan 7474institutetext: Pacific Northwest National Laboratory,
Richland, WA 99352, USA 7575institutetext: Panjab University, Chandigarh
160014, India 7676institutetext: University of Pittsburgh, Pittsburgh, PA
15260, USA 7777institutetext: Punjab Agricultural University, Ludhiana 141004,
India 7878institutetext: Research Center for Nuclear Physics, Osaka
University, Osaka 567-0047, Japan 7979institutetext: Meson Science Laboratory,
Cluster for Pioneering Research, RIKEN, Saitama 351-0198, Japan
8080institutetext: Department of Modern Physics and State Key Laboratory of
Particle Detection and Electronics, University of Science and Technology of
China, Hefei 230026, PR China 8181institutetext: Showa Pharmaceutical
University, Tokyo 194-8543, Japan 8282institutetext: Soochow University,
Suzhou 215006, China 8383institutetext: Soongsil University, Seoul 06978,
South Korea 8484institutetext: Sungkyunkwan University, Suwon 16419, South
Korea 8585institutetext: School of Physics, University of Sydney, New South
Wales 2006, Australia 8686institutetext: Department of Physics, Faculty of
Science, University of Tabuk, Tabuk 71451, Saudi Arabia 8787institutetext:
Tata Institute of Fundamental Research, Mumbai 400005, India
8888institutetext: Department of Physics, Technische Universität München,
85748 Garching, Germany 8989institutetext: School of Physics and Astronomy,
Tel Aviv University, Tel Aviv 69978, Israel 9090institutetext: Toho
University, Funabashi 274-8510, Japan 9191institutetext: Department of
Physics, Tohoku University, Sendai 980-8578, Japan 9292institutetext:
Earthquake Research Institute, University of Tokyo, Tokyo 113-0032, Japan
9393institutetext: Department of Physics, University of Tokyo, Tokyo 113-0033,
Japan 9494institutetext: Tokyo Institute of Technology, Tokyo 152-8550, Japan
9595institutetext: Tokyo Metropolitan University, Tokyo 192-0397, Japan
9696institutetext: Virginia Polytechnic Institute and State University,
Blacksburg, VA 24061, USA 9797institutetext: Wayne State University, Detroit,
MI 48202, USA 9898institutetext: Yamagata University, Yamagata 990-8560, Japan
9999institutetext: Yonsei University, Seoul 03722, South Korea

# Search for charged lepton flavor violating decays of $\Upsilon(1S)$

27 V. Bhardwaj 49 K. Trabelsi 22,18 I. Adachi 93 H. Aihara, 86,44 S. Al Said 3
D. M. Asner 8 H. Atmacan 24 T. Aushev, 86 R. Ayad 9 V. Babu, 28 S. Bahinipati
31 P. Behera, 34 K. Belous 59 J. Bennett 21 M. Bessner, 29 B. Bhuyan, 5 T.
Bilka 4,72 A. Bobrov, 24,50 D. Bodrov 29 J. Borah, 69 A. Bozek 56,41 M. Bračko
37 P. Branchini 21 T. E. Browder, 37 A. Budano 36,64 M. Campajola 5 D.
Červenkov, 13 M.-C. Chang 68 P. Chang 66 A. Chen, 20 B. G. Cheon, 50 K.
Chilikin 20 H. E. Cho 46 K. Cho 99 S.-J. Cho 7 S.-K. Choi 84 Y. Choi 39 S.
Choudhury 97 D. Cinabro 9 S. Cunliffe, 55 S. Das, 36,64 G. De Nardo, 37 G. De
Pietro 30 R. Dhamija 36,64 F. Di Capua 2 J. Dingfelder 5 Z. Doležal 11 T. V.
Dong 4,72 D. Epifanov, 9 T. Ferber, 17 A. Frey, 74 B. G. Fulsom 75 R. Garg 96
V. Gaur 4,72 N. Gabyshev 30 A. Giri, 42 P. Goldenzweig 37 E. Graziani, 76 T.
Gu, 22,18 T. Hara, 71 K. Hayasaka, 65 H. Hayashii 21 M. T. Hedges, 9 M.
Hernandez Villanueva 68 W.-S. Hou, 85 C.-L. Hsu 63,62 T. Iijima, 62 K. Inami
35 G. Inguglia 22,18 A. Ishikawa 22,18 R. Itoh, 73 M. Iwasaki 22 Y. Iwasaki 32
W. W. Jacobs, 19 E.-J. Jang, 14 S. Jia, 93 Y. Jin 6 K. K. Joo, 42 J. Kahn 87
A. B. Kaliyar, 43 K. H. Kang 9 G. Karyan, 45 T. Kawasaki, 57 C. Kiesling, 20
C. H. Kim, 83 D. Y. Kim 99 K.-H. Kim, 47 K. T. Kim 99 Y.-K. Kim, 8 K.
Kinoshita, 5 P. Kodyš, 45 T. Konno, 4,72 A. Korobov 56,41 S. Korpar 4,72 E.
Kovalenko 52,41 P. Križan 59 R. Kroeger 4,72 P. Krokovny, 53 T. Kuhr 55 M.
Kumar, 77 R. Kumar 97 K. Kumara 4,72,50 A. Kuzmin, 99 Y.-J. Kwon, 55 K.
Lalwani, 96 T. Lam 15 J. S. Lange 48 S. C. Lee, 51 C. H. Li, 48 J. Li 8 L. K.
Li 14 Y. Li, 14 Y. B. Li 57 L. Li Gioi, 31 J. Libby 53 K. Lieret 97,22 D.
Liventsev, 9 A. Martini, 92,78 M. Masuda 60 T. Matsuda 4,72,50 D. Matvienko
36,64 M. Merola, 42 F. Metzner 65 K. Miyabayashi 50,24 R. Mizuk, 87 G. B.
Mohanty 38 R. Mussa, 22,18 M. Nakao, 21 A. Natochii, 30 L. Nayak 89 M. Nayak 3
N. K. Nisar, 22,18 S. Nishida 71 K. Ogawa, 90 S. Ogawa 70,71 H. Ono, 93 Y.
Onuki 50 P. Oskin 50,61 P. Pakhlov, 24,50 G. Pakhlova 76 T. Pang 36 S. Pardi
48 H. Park 22 S.-H. Park 37 A. Passeri 88,57 S. Paul 54 T. K. Pedlar 41 R.
Pestotnik 96 L. E. Piilonen 52,41 T. Podobnik 24 V. Popov 25 E. Prencipe, 2 M.
T. Prim 9 M. Röhrken, 9 A. Rostomyan, 31 N. Rout 64 G. Russo, 39 D. Sahoo 30
S. Sandilya, 8 A. Sangal 52,41 L. Santelj 91 T. Sanuki 76 V. Savinov, 1,26 G.
Schnell, 35 C. Schwanda, 71 Y. Seino, 98 K. Senyo 58 M. E. Sevior, 34 M.
Shapkin 55 C. Sharma 14 C. P. Shen, 68 J.-G. Shiu, 4,72 B. Shwartz 57 F.
Simon, 75,$\star$ J. B. Singh, \star\star$\star$Also at University of
Petroleum and Energy Studies, Dehradun 248007, India 34 A. Sokolov, 50 E.
Solovieva, 41 M. Starič, 96 Z. S. Stottler 74 J. F. Strube 16,78 M. Sumihama,
95 T. Sumiyoshi, 81,23,79 M. Takizawa, 38 U. Tamponi 40 K. Tanida, 9 F.
Tenchini, 94 M. Uchida, 50,24 T. Uglov, 20 Y. Unno 22,18 S. Uno, 58 P. Urquijo
4,72 Y. Usov, 2 R. Van Tonder, 21 G. Varner 4,72 A. Vinokurova, 10 A. Vossen,
22 E. Waheed 67 C. H. Wang, 12 D. Wang 76 E. Wang 68 M.-Z. Wang 99 S.
Watanuki, 47 E. Won, 82 X. Xu 85 B. D. Yabsley 80 W. Yan, 47 S. B. Yang 9 H.
Ye 12 J. Yelton 47 J. H. Yin 33 C. Z. Yuan, 71 Y. Yusa 39 Y. Zhai 80 Z. P.
Zhang, 4,72 V. Zhilich 50 V. Zhukova 4,72 V. Zhulanov
<EMAIL_ADDRESS>
###### Abstract

We present a search for the charged lepton-flavor-violating decays
$\Upsilon(1S)\rightarrow\ell^{\pm}\ell^{\prime\mp}$ and radiative charged
lepton-flavour-violating decays
$\Upsilon(1S)\rightarrow\gamma\ell^{\pm}\ell^{\prime\mp}$
[$\ell,\ell^{\prime}=e,\mu,\tau$] using the 158 million $\Upsilon(2S)$ sample
collected by the Belle detector at the KEKB collider. This search uses
$\Upsilon(1S)$ mesons produced in
$\Upsilon(2S)\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)$ transitions. We do not
find any significant signal, so we provide upper limits on the branching
fractions at the 90% confidence level.

Belle Preprint 2022-02

KEK Preprint 2021-59

##  1 Introduction

Observations of neutrino oscillations neutrino  imply that the accidental
lepton family symmetry in the standard model (SM) Lagrangian is broken. The
minimal extension of the SM that can explain neutrino oscillations requires
the presence of a right-handed neutrino. In such a framework, the conservation
of individual lepton flavor is violated, and charged lepton-flavor-violating
(CLFV) transitions can occur, mediated by $W^{\pm}$ bosons and massive
neutrinos. However, the existence of such CLFV transitions would imply a
minimal value of $\mathcal{B}\left(\mu^{\pm}\to e^{\pm}\gamma\right)\sim
10^{-54}$ intro1 ; intro2 . Several new physics models inspired by grand
unified theories, such as supersymmetry and those predicting leptoquarks,
typically enhance decay rates of CLFV transitions gut1 ; gut2 .

The effective Lagrangian of new physics (NP) models can be expressed as the
sum of a dipole term, four-fermionic interactions, and a gluonic interaction
part. The Wilson coefficients of the NP operators can be determined via fits
to measurements of phenomena those involve CLFV interactions petrov . Several
classes of operator, such as vector, axial-vector, and tensor operators
involved in four-fermionic interactions, allow CLFV transitions. Precise
measurement of two-body vector meson CLFV decays allows one to probe the
vector and tensor operators effectively.

A few results have previously been published related to two-body CLFV
$\Upsilon(nS)\rightarrow\ell\ell^{\prime}$ [$\ell,\ell^{\prime}=e,\mu,\tau$
and $n=1,2,3$] decays cleo ; babar . Only the
$\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ decay has been studied, and no
$\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$ and $\Upsilon(1S)\rightarrow
e^{\pm}\tau^{\mp}$ results are available. CLFV $\Upsilon(1S)$ decays can be
studied with direct production or di-pion tagging of $\Upsilon(2S)$ decays.
Belle accumulated 6 ${\rm fb}^{-1}$ of data at $\Upsilon(1S)$ resonance,
corresponding to 119 million $\Upsilon(1S)$ events. However, it is difficult
to judiciously trigger the two-charged-particle final state of these events:
the sample is subject to extensive backgrounds, predominantly from QED
processes. Belle also accumulated 25 ${\rm fb}^{-1}$ of data at $\Upsilon(2S)$
resonance, corresponding to 28 million $\Upsilon(1S)$ produced in
$\Upsilon(2S)\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)$ decays. The four-particle
final state of $\Upsilon(2S)\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)$ allows for
a more efficient trigger and for the suppression of the QED background.
Therefore, we search for the
$\Upsilon(1S)\rightarrow\ell^{\pm}\ell^{\prime\mp}$ decays using the
$\Upsilon(2S)$ data sample.

Radiative lepton-flavor-violating (RLFV) transitions allow one to probe the
operators which are not easily accessible in the two-body decays petrov .
Using three-body vector meson RLFV decays, one can put constraints on the
corresponding Wilson coefficients of axial-vector, scalar, and pseudoscalar
operators. Thus, RLFV studies of $\Upsilon(nS)$ [$n=1,2,3$] provide
complementary access to the NP parameters. Currently, there are no existing
results available for the
$\Upsilon(nS)\rightarrow\gamma\ell^{\pm}\ell^{\prime\mp}$ decays. We perform
the first search for RLFV in
$\Upsilon(1S)\rightarrow\gamma\ell^{\pm}\ell^{\prime\mp}$ decays using the
$\Upsilon(2S)$ data sample.

##  2 Belle experiment

The world’s largest $\Upsilon(2S)$ sample, corresponding to 158 million
$\Upsilon(2S)$ events, was collected with the Belle detector at the KEKB
asymmetric-energy $e^{+}e^{-}$ collider kekb  operating at a center-of-mass of
energy ($\sqrt{s}$) of 10.02 GeV. We study the $e^{+}e^{-}\rightarrow
q\bar{q}~{}(q=u,d,s,c)$ background using the 80 ${\rm fb}^{-1}$ data sample
collected at 10.52 GeV.

The Belle detector is a large-solid-angle spectrometer, which includes a
silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an
array of aerogel threshold Cherenkov counters (ACC), time-of-flight
scintillation counters (TOF), and an electromagnetic calorimeter (ECL)
comprised of 8736 CsI(Tl) crystals located inside a superconducting solenoid
coil that provides a 1.5T magnetic field. An iron flux return located outside
the coil is instrumented to detect $K^{0}_{\rm L}$ mesons and identify muons
(KLM). The detector is described in detail elsewhere detector .

##  3 Event selection

We use the EVTGEN package evtgen , with QED final-state radiation simulated by
PHOTOS photos , for the generation of Monte Carlo (MC) simulation events. We
generate the signal events for two-body CLFV modes using both the vector to
two leptons decay model, VLL and the phase space decay model, PHSP. The
reconstruction efficiencies for the MC signal events generated with the VLL
model are smaller (by approximately 8%) than for the PHSP model. We will use
the MC signal events generated with the VLL decay model for two-body CLFV
decays to quote the most conservative upper limits. We use the PHSP model to
generate the signal events for RLFV modes. We are using TAUOLA tauola  or
PYTHIA pythia  for generating signal events for the subsequent decays of
$\tau$ leptons. A GEANT3-based geant  MC simulation is used to model the
response of the detector. Thus, dedicated MC samples are generated for
different signal modes to determine signal efficiencies and define selection
criteria. Background studies and the optimization of those criteria are
performed using an MC simulated sample of $\Upsilon(2S)$ events with a size
corresponding to the integrated luminosity. Dominant backgrounds arise from
$\Upsilon(2S)\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)$ decays with
$\Upsilon(1S)\rightarrow\ell^{\pm}\ell^{\mp}~{}[\ell=e,\mu,\tau]$. The MC
samples for these decays, corresponding to about 20 times of data sample
sizes, are used to study backgrounds. For two-body CLFV searches,
$\Upsilon(1S)$ candidates are reconstructed in the $e^{\pm}\mu^{\mp}$,
$\mu^{\pm}\tau^{\mp}$, and $e^{\pm}\tau^{\mp}$ final states. We reconstruct
$\tau$ in $\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$,
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ for the
$\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ decay and
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$,
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ for the
$\Upsilon(1S)\rightarrow e^{\pm}\tau^{\mp}$ decay, comprising 28% of $\tau$
decays. To avoid potential background from
$\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ ($\Upsilon(1S)\rightarrow e^{+}e^{-}$)
decays, we do not consider
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ ($\tau^{-}\rightarrow
e^{-}\bar{\nu}_{e}\nu_{\tau}$) decays for the
$\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ ($\Upsilon(1S)\rightarrow
e^{\pm}\tau^{\mp}$) mode. Similarly, for RLFV decays we reconstruct
$\Upsilon(1S)$ in $\gamma e^{\pm}\mu^{\mp}$, $\gamma\mu^{\pm}\tau^{\mp}$, and
$\gamma e^{\pm}\tau^{\mp}$ final states, where the $\tau$ is identified in
$\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$
($\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$) decay for the
$\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$
($\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$) study. Also, we
reconstruct $\Upsilon(1S)\rightarrow e^{+}e^{-}$ and
$\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ decays, which are used to validate and
calibrate the analysis. To validate the recoil $\Upsilon(1S)$ sample along
with muon and electron identifications, we measure the branching fractions of
$\Upsilon(1S)\rightarrow e^{+}e^{-}$, and
$\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ decays. As taus are mostly
reconstructed in the leptonic decays, validating lepton identification with
the high momentum leptons from $\Upsilon(1S)$ is also important for
$\Upsilon(1S)\rightarrow\ell^{\pm}\tau^{\mp}$ and
$\Upsilon(1S)\rightarrow\gamma\ell^{\pm}\tau^{\mp}$ decays. This analysis
follows a blind analysis procedure.

Charged tracks are required to originate from the vicinity of the interaction
point (IP). The distance of the closest approach to the IP is required to be
within 3.5 cm along the beam direction and within 1.5 cm in the transverse
plane. The combined information from the CDC, TOF, and ACC is used to identify
charged pions based on the pion likelihood ratio, ${\cal L}_{\pi}={\cal
P}_{\pi}/(\mathcal{P}_{\pi}+\mathcal{P}_{K})$, where $\mathcal{P}_{\pi}$ and
$\mathcal{P}_{K}$ are likelihood values for the pion and kaon hypotheses,
respectively pid . Pions are required to have ${\cal L}_{\pi}>0.6$, which has
an identification efficiency of 94%. Muon candidates are identified using a
likelihood ratio, ${\cal L}_{\mu}$, which is based on the difference between
the range of the track calculated from the particle momentum and that measured
in the KLM. This ratio includes the value of $\chi^{2}$ formed from the KLM
hit locations with respect to the extrapolated track. The muon identification
efficiency for the applied selection, ${\cal L}_{\mu}$ > 0.95, is 89%, with a
pion misidentification probability of 1.4% muid . Identification of electrons
uses an analogous likelihood ratio, ${\cal L}_{e}$, based on specific
ionization from the CDC, the ratio of the energy deposited in the ECL to the
momentum measured by the CDC and SVD combined, the shower shape in the ECL,
hit information from the ACC, and matching between the position of the charged
track and the ECL cluster. The electron identification efficiency for the
applied selection, ${\cal L}_{e}$ > 0.6, is 95%, with a pion misidentification
probability 0.3% eid . To recover the bremsstrahlung energy loss for electrons
and positrons, we include the energy from the photon(s) within 50 mrad of each
of the $e^{\pm}$ tracks, which improves the efficiency for true signal events
by 2.7%.

For $\Upsilon(1S)\rightarrow\ell^{\pm}\tau^{\mp}$ decay, most of the
background comes from $\Upsilon(1S)\rightarrow\tau^{+}\tau^{-}$ and
$\Upsilon(1S)\rightarrow\ell^{\pm}\ell^{\mp}$ decays. To suppress events
coming from $\Upsilon(2S)$ decays, other than
$\Upsilon(2S)\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)$, we define the recoil mass
of two pions as:

$M^{{\rm recoil}}_{\rm\pi\pi}=\sqrt{(E_{\rm
total}-E_{\rm\pi\pi})^{2}-|{\overrightarrow{p}}_{\pi\pi}|^{2}}$ (1)

Where $E_{\rm total}$, $E_{\rm\pi\pi}$, and $|{\overrightarrow{p}}_{\pi\pi}|$
are the total energy of the colliding $e^{+}e^{-}$ beams, the energy of the
two pions from $\Upsilon(2S)$ and the magnitude of the 3-momentum of the pion
pair, respectively in the center-of-mass (CM) frame. The distribution of
$M^{{\rm recoil}}_{\rm\pi\pi}$ is shown in Fig. 1. The $M^{{\rm
recoil}}_{\rm\pi\pi}$ distribution peaks at the $\Upsilon(1S)$ mass for signal
events, while it is flat for the combinatorial background. We consider the
events within $9.450<M^{{\rm recoil}}_{\rm\pi\pi}<9.466$ ${\rm GeV}/c^{2}$,
corresponding to a $\pm 3\sigma$ region around the nominal $\Upsilon(1S)$ mass
pdg .

Figure 1: $M^{{\rm recoil}}_{\rm\pi\pi}$ distribution for $\Upsilon(2S)$ data.
Events within the two vertical lines are selected.

To suppress the background from $e^{+}e^{-}\gamma$ and $\mu^{+}\mu^{-}\gamma$,
we remove the events with the cosine of the angle between the two pions in the
$\Upsilon(2S)$ rest frame ($\cos\theta_{\pi\pi}$) greater than $0.5$. We
define the visible tau momentum ($p^{\tau}_{\rm vis}$) as the sum of the
momentum carried by the daughter charged track(s) of $\tau$ in the lab frame.
We select the $\tau$ candidates with $p^{\tau}_{\rm vis}$ > 0.3 ${\rm GeV}/c$.
Furthermore, $\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ is
reconstructed with the invariant mass of the three-pion lower than 1.8 ${\rm
GeV}/c^{2}$ and energy in the lab frame greater than 2.6 GeV. These $\tau$
selections are wide enough to account for the missing momentum from neutrinos.
Also, we fit the three-pion vertex for the
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ decay and events with
fitted $\chi^{2}$ < 15 are selected to reduce combinatorial backgrounds. We
count the number of tracks identified as muons or electrons with energy in the
lab frame greater than 1 GeV as prompt muons ($N_{\rm\mu}$) and prompt
electrons ($N_{\rm e}$), respectively. For
$\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ decays, in order to reject the
background coming from the $\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ decay, we
select the events with $N_{\rm\mu}=1$ and $N_{\rm e}\leq 1$ ($N_{\rm e}=0$)
for the $\tau^{-}\rightarrow e^{-}\bar{\nu}_{e}\nu_{\tau}$
($\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$) reconstruction mode.
Similarly, for $\Upsilon(1S)\rightarrow e^{\pm}\tau^{\mp}$ decays, we select
the events with $N_{\rm e}=1$ and $N_{\rm\mu}\leq 1$ ($N_{\rm\mu}=0$) for
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$
($\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$) reconstruction mode. We
suppress a large number of $\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ and
$\Upsilon(1S)\rightarrow e^{+}e^{-}$ backgrounds by the selections of prompt
leptons.

For the $\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$ study, the distribution of
lepton pair invariant mass ($M_{\rm e\mu}$) for $\pi^{+}\pi^{-}$ recoil sample
in $\Upsilon(2S)$ data is shown in Fig. 2. We select the events with $M_{\rm
e\mu}$ within 9.09 to 9.65 ${\rm GeV}/c^{2}$ by selecting a $\pm 3\sigma$
region around the mean position and the $\Upsilon(1S)$ momentum in the lab
frame ($|{\overrightarrow{p}}_{e\mu}|$) less than 4.4 ${\rm GeV}/c$ to reduce
the $e^{+}e^{-}\rightarrow q\bar{q}$ events.

Figure 2: $M_{\rm e\mu}$ distribution of $\pi^{+}\pi^{-}$ recoil sample in
$\Upsilon(2S)$ data. Events within two perpendicular lines are selected for
$\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$ decay.

After applying all the selections, we find 3% of events with multiple
$\Upsilon(2S)$ candidates for $\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$
decay, and 4% for both of $\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ and
$\Upsilon(1S)\rightarrow e^{\pm}\tau^{\mp}$ decays. We performed a vertex fit
with the reconstructed charged tracks of $\Upsilon(2S)$, and the fitted
$\chi^{2}$ value has been used to select the best candidate among the multiple
$\Upsilon(2S)$ candidates. Best candidate selection efficiencies for
$\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$,
$\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$, and $\Upsilon(1S)\rightarrow
e^{\pm}\tau^{\mp}$ decays are 97%, 89%, and 91%, respectively.

For RLFV decays, there is an extra photon in the final state. Therefore, we
include all the selections which are used for the pion and lepton in the
corresponding two-body CLFV decay previously discussed. For
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\mu^{\mp}$ decay, $M_{\rm e\mu}$ and
$|{\overrightarrow{p}}_{e\mu}|$ of $\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$
decay will be replaced by $M_{\rm\gamma e\mu}$ and
$|{\overrightarrow{p}}_{\gamma e\mu}|$, respectively. In addition to the above
selections to the corresponding non-radiative mode, we select photons with
energy in the lab frame greater than 200 MeV to remove soft photons and beam
backgrounds. The photons used in the bremsstrahlung recovery are not
considered in reconstructing the radiative candidates. Inside the signal
search window, we find 3%, 8%, and 7% multiple $\Upsilon(2S)$ candidates for
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\mu^{\mp}$,
$\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$, and
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$ decays, respectively.
Multiplicity due to misreconstructed charged particles is handled using a
procedure similar to two-body CLFV decays. Multiplicity occurring from the
multiple photon candidates is removed by selecting the event randomly. Best
candidate selection efficiencies for $\Upsilon(1S)\rightarrow\gamma
e^{\pm}\mu^{\mp}$, $\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$, and
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$ are 93%, 84%, and 87%,
respectively.

##  4 Study of $\Upsilon(1S)\rightarrow e^{+}e^{-},\mu^{+}\mu^{-}$ decays as
calibration modes

To study the calibration modes, we select events with lepton pair invariant
mass ($M_{\rm\ell\ell}$) within 9.09 to 9.65 ${\rm GeV}/c^{2}$ and momentum of
the reconstructed $\Upsilon(1S)$ in the lab frame
($|{\overrightarrow{p}}_{\ell\ell}|$) less than 4.4 ${\rm GeV}/c$. To extract
the signal for $\Upsilon(1S)\rightarrow\ell^{\pm}\ell^{\mp}$ decays, we
perform an unbinned maximum likelihood (UML) fits to $\Delta
M=M_{\rm\pi\pi\ell\ell}-M_{\rm\ell\ell}$, where $\ell=e,\mu$. The signal
probability density function (PDF) used is a sum of two Gaussians sharing a
common mean. Backgrounds from all the sources are flat in the $\Delta M$
window and small compared to the signal yields. We fit the background with a
first-order Chebyshev polynomial. To account for any resolution difference
between data and MC, the mean and the width parameter of the primary Gaussian
($\sigma_{1}$) are floated in the fit, and the width of the secondary Gaussian
is set to $\sigma_{2}=k\times\sigma_{1}$, with the factor $k$ fixed from MC.

Figure 3: $\Delta M$ fit for $ee$ events (left) and $\mu\mu$ events (right).
The dotted red curves represent the signal PDFs and the dashed cyan lines
represent the background PDFs. The solid blue curves represent the overall fit
to data.

Expected signal efficiencies for $ee$ and $\mu\mu$ are estimated to be 28.3%
and 35.6%, respectively. Fig. 3 shows the fits to $\Upsilon(2S)$ data. The
signal yields obtained for the $ee$ and $\mu\mu$ final states are $191353\pm
467$ and $246255\pm 504$ events, respectively. The data-MC differences of
widths for the $e^{\pm}e^{\mp}$ and $\mu^{\pm}\mu^{\mp}$ final states are
estimated to be 12% and 16%, respectively. One can calculate the branching
fractions using the following relation:

$\mathcal{B}[\Upsilon(1S)\rightarrow\ell^{\pm}\ell^{\mp}]=\frac{N_{\rm
sig}}{N_{\Upsilon(2S)}\times\mathcal{B}[\Upsilon(2S)\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)]\times\epsilon}$
(2)

where, $N_{\Upsilon(2S)}$, $N_{\rm sig}$ and $\epsilon$ are the number of
$\Upsilon(2S)$ produced in $e^{+}e^{-}$ collision, signal yield in data and
the effective signal efficiency (after implementing all the systematic
corrections) respectively. Using Equation (2), the calculated branching
fractions including only statistical uncertainties for
$\Upsilon(1S)\rightarrow e^{+}e^{-}$ and
$\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ are $(2.40\pm 0.01)\times 10^{-2}$ and
$(2.46\pm 0.01)\times 10^{-2}$, respectively. These are consistent with the
world average values pdg . These results are discussed further in Section 8
after including the systematic uncertainty.

##  5 Signal extraction for two-body CLFV decays

###  5.1 $\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$ decay

We extract the signal yield from a UML fit to the $\Delta M$ variable. $\Delta
M$ should peak at the nominal mass difference between $\Upsilon(2S)$ and
$\Upsilon(1S)$, approximately 560 ${\rm MeV}/c^{2}$ pdg . A sum of two
Gaussians sharing a common mean has been used as the signal PDF. To estimate
the peaking background, the shape of the peaking background is considered to
have the same shape as the signal PDF. The $q\bar{q}$ backgrounds are flat,
and they are modeled with a first-order Chebyshev polynomial. The width of the
signal PDF in the data is fixed at the MC width, corrected by the average of
the data-MC difference for the $\mu\mu$ and $ee$ samples.

Figure 4: $\Delta M$ fit to $\Upsilon(2S)$ data for $\Upsilon(1S)\rightarrow
e^{\pm}\mu^{\mp}$ decay. The fitted signal PDF is represented by the filled
red region, the dashed cyan line represents the flat background and the dotted
magenta curve is the peaking background from lepton misidentification. The
solid blue curve represents the overall fit to data. The long-dashed red curve
represents the signal PDF corresponding to 5 hypothetical signal events.

A few $\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ ($\Upsilon(1S)\rightarrow
e^{+}e^{-}$) events mimic our signal when a $\mu^{\pm}$ ($e^{\pm}$) is
misidentified as an $e^{\pm}$ ($\mu^{\pm}$). The amount of
$\Upsilon(1S)\rightarrow\tau^{+}\tau^{-}$ background is estimated to be
negligible. The number of $\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ background
events is estimated to be $3.5\pm 0.4$ using a large MC sample. Such
backgrounds are difficult to remove completely. To estimate the background
from muon to electron misidentification in the data, we derive a correction
factor for electron misidentification efficiency of muons using an
$e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ sample collected at $\sqrt{s}=10.52$ GeV
with a tag-and-probe method. The data to MC correction factor for electron
misidentification efficiency is estimated to be $2.5\pm 0.5$, which leads to
an estimation of this peaking background yield of $8.8\pm 2.0$. The background
from electron to muon misidentification is expected to be consistent with zero
($0.1\pm 0.1$ events). Fitted distribution of $\Upsilon(2S)$ data is shown in
Fig. 4. To consider the peaking background, we include a fixed PDF of 8.8
events in the data fit (dotted magenta line), and uncertainty (2.0) will be
added in systematic uncertainty. The estimated signal efficiency for the
$\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$ mode is 32.5%. We finally obtain a
yield of $-1.3\pm 3.7$ signal events for the $\Upsilon(1S)\rightarrow
e^{\pm}\mu^{\mp}$ decay.

###  5.2 $\Upsilon(1S)\rightarrow\ell^{\pm}\tau^{\mp}$ decay

For $\mu\tau$ and $e\tau$ decays of $\Upsilon(1S)$, we extract the signal from
an UML fit to the recoil mass of $\pi\pi\ell$ ($M^{{\rm
recoil}}_{\rm\pi\pi\ell}$), where $\ell=\mu,e$. $M^{{\rm
recoil}}_{\rm\pi\pi\ell}$ can be defined by replacing $\pi\pi$ with
$\pi\pi\ell$ in Eq. (1). As $M^{{\rm recoil}}_{\rm\pi\pi\ell}$ is calculated
from all the particles from the $\Upsilon(2S)$ except the $\tau$, $M^{{\rm
recoil}}_{\rm\pi\pi\ell}$ should peak at the nominal $\tau$ mass (around 1.78
${\rm GeV}/c^{2}$) pdg . Signal events of $\mu\tau$ and $e\tau$ decays are
modeled with a sum of one Gaussian and one bifurcated Gaussian, sharing a
common mean.

Figure 5: $M^{{\rm recoil}}_{\rm\pi\pi\ell}$ fit to $\Upsilon(2S)$ data for
the $\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ decay (top) and the
$\Upsilon(1S)\rightarrow e^{\pm}\tau^{\mp}$ decay (bottom). The fitted signal
PDFs are represented by the filled red regions. The dotted magenta line
represents the contribution from $\mu\mu$ (or $ee$) background and the dashed
cyan line represents the $\tau\tau$ background. The solid blue curves
represent the overall fit to data. The long-dashed red curves represent the
signal PDFs corresponding to 20 hypothetical signal events. To make the small
signal yield in data visible, we add zoomed inset of the signal region.

We obtain a difference of 7% (27%) for the resolution between data and MC for
$M^{{\rm recoil}}_{\rm\pi\pi\mu}$ ($M^{{\rm recoil}}_{\rm\pi\pi e}$) using the
data-MC difference for the $\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$
($\Upsilon(1S)\rightarrow e^{+}e^{-}$) mode. The width of the
$\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ ($\Upsilon(1S)\rightarrow
e^{\pm}\tau^{\mp}$) signal PDF in the data is fixed from the MC width
corrected by the data-MC difference for the $M^{{\rm recoil}}_{\rm\pi\pi\mu}$
($M^{{\rm recoil}}_{\rm\pi\pi e}$) parameter.

For $\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ decays, the main backgrounds
come from $\Upsilon(1S)\rightarrow\tau^{+}\tau^{-}$ and
$\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ decays. For the $\tau\tau$ background,
a charged lepton or a pion from one of the tau decays is used as the signal
muon. In the $M^{{\rm recoil}}_{\rm\pi\pi\mu}$ distribution, missing neutrino
energy from the misidentified $\tau$ shifts such events away from the actual
$\tau$ mass. Thus, the $\tau\tau$ background increases exponentially, starting
near the nominal $\tau$ mass value. We model the $\tau\tau$ background using
the following exponential threshold PDF starting near $M_{\rm th}$ (in ${\rm
GeV}/c^{2}$),

$\mathcal{F}(M^{{\rm recoil}}_{\rm\pi\pi\ell};A,B,M_{\rm th})=\exp{[A(M^{{\rm
recoil}}_{\rm\pi\pi\ell}-M_{\rm th})+B(M^{{\rm recoil}}_{\rm\pi\pi\ell}-M_{\rm
th})^{2}]}$ (3)

where $A$ and $B$ are the two slope parameters of the $\tau\tau$ background
PDF. We try other fitting models and find the current model describes the
background the best. To get the proper shape of the $\tau\tau$ background PDF,
we use a large $\Upsilon(1S)\rightarrow\tau^{+}\tau^{-}$ MC sample. In the
data fit, $M_{\rm th}$ and $B$ are fixed from the MC background, and $A$ is
allowed to float. Fig. 5 shows the fitted distributions of data for
$\Upsilon(1S)\rightarrow\ell^{\pm}\tau^{\mp}$ decays. The background from
$\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ is obtained from a large MC sample.
The expected number of $\mu\mu$ backgrounds is less than the number of
$\tau\tau$ backgrounds, but it widely populates around the signal region: it
peaks at the lower mass value ($<1~{}{\rm GeV}/c^{2}$) and has a broad tail.
The PDF is presented by a sum of one bifurcated Gaussian and one threshold
function starting from 0 ${\rm GeV}/c^{2}$ using a large
$\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ sample. To fit the data, we float the
yield of the $\mu\mu$ background fixing the shape of the Gaussian of $\mu\mu$
background from MC corrected by the data-MC difference for the $M^{{\rm
recoil}}_{\rm\pi\pi\mu}$ parameter.

For $\Upsilon(1S)\rightarrow e^{\pm}\tau^{\mp}$ decays, potential backgrounds
arise from $\Upsilon(1S)\rightarrow\tau^{+}\tau^{-}$ and
$\Upsilon(1S)\rightarrow e^{+}e^{-}$ decays. These backgrounds are handled
using a procedure similar to that used for the $\tau\tau$ and $\mu\mu$
backgrounds to $\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ decays.

The expected peaking backgrounds in the $\Upsilon(2S)$ data for
$\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ and $\Upsilon(1S)\rightarrow
e^{\pm}\tau^{\mp}$ decays are estimated from MC to be $0.7\pm 4.1$ and $4.6\pm
5.4$, respectively. As no significant peaking background is found in the
$\Upsilon(2S)$ decay MC sample, we do not include a peaking background
component in the fit. Considering both the $\tau$ reconstruction modes, the
effective signal efficiency for $\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$
($\Upsilon(1S)\rightarrow e^{\pm}\tau^{\mp}$) decay is 8.8% (7.1%). In
$\Upsilon(2S)$ data, we find the yield of
$\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ and $\Upsilon(1S)\rightarrow
e^{\pm}\tau^{\mp}$ signals to be $-1.5\pm 4.3$ and $-3.5\pm 2.7$,
respectively. Hence, there is no evidence for
$\Upsilon(1S)\rightarrow\ell^{\pm}\tau^{\mp}$ transitions.

##  6 Signal extraction for RLFV decays

###  6.1 $\Upsilon(1S)\rightarrow\gamma e^{\pm}\mu^{\mp}$ decay

Our RLFV signal extraction procedure is very similar to that used for the
corresponding non-radiative transition. We perform an UML fit to the mass
difference $\Delta M=M_{\rm\pi\pi\gamma e\mu}-M_{\rm\gamma e\mu}$. The signal
PDF used is a sum of two Gaussians sharing a common mean. To estimate the
peaking background from leptonic decays of the $\Upsilon(1S)$, we use the same
shape as the signal PDF as the shape of the background PDF and a large
$\Upsilon(1S)\rightarrow\ell^{\pm}\ell^{\mp}$ MC sample is used to have a more
precise estimation. Other backgrounds are flat on the $\Delta M$ window and
modeled with a first-order Chebyshev polynomial. Fig. 6 shows the $\Delta M$
fit for $\Upsilon(2S)$ data. To fit the data, we fix the width of the signal
PDF from the MC signal width corrected by the average of data-MC difference
for $\Upsilon(1S)\rightarrow e^{+}e^{-}$ and
$\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ modes. The yield of peaking background
is estimated to be $0.1\pm 0.1$.

Figure 6: $\Delta M$ fit to $\Upsilon(2S)$ data for the
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\mu^{\mp}$ decay. The fitted signal PDF
is represented by the filled red region and the dashed cyan line represents
the background. The solid blue curve represents the overall fit to data. The
long-dashed red curve represents the signal PDF corresponding to 5
hypothetical signal events.

The signal efficiency for $\gamma e\mu$ decay is 24.6%. From the
$\Upsilon(2S)$ data fit, the signal yield for the
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\mu^{\mp}$ decay is estimated to be
$0.8\pm 1.5$.

###  6.2 $\Upsilon(1S)\rightarrow\gamma\ell^{\pm}\tau^{\mp}$ decay

To extract the signal for $\gamma\mu\tau$ and $\gamma e\tau$ decays, we define
the recoil mass of $\pi\pi\gamma\ell$ ($M^{{\rm
recoil}}_{\rm\pi\pi\ell\gamma}$) using Eq. (1). We perform a UML fit to
$M^{{\rm recoil}}_{\rm\pi\pi\ell\gamma}$ to extract the signal yield and
estimate efficiency. For signal events, $M^{{\rm
recoil}}_{\rm\pi\pi\ell\gamma}$ should peak at the nominal $\tau$ mass. A
Gaussian and a bifurcated Gaussian sum sharing a common mean is used to model
the signal events for $\Upsilon(1S)\rightarrow\gamma\ell^{\pm}\tau^{\mp}$
decays.

Figure 7: $M^{{\rm recoil}}_{\rm\pi\pi\ell\gamma}$ fit to $\Upsilon(2S)$ data
for the $\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$ decay (top) and the
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$ decay (bottom). The fitted
signal PDFs are represented by the filled red regions and the dashed cyan
lines represent the total background. The solid blue curves represent the
overall fit to data. The long-dashed red curves represent the signal PDFs
corresponding to 20 hypothetical signal events.

For both of the $\Upsilon(1S)\rightarrow\gamma\ell^{\pm}\tau^{\mp}$ decays,
the dominant background comes from $\tau\tau$ decays and hadronic decays of
the $\Upsilon(1S)$. The $\tau\tau$ background is treated using an approach
similar to that for the $\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ decay,
with the background shape as described by the Eq. (3). Also, we find some
background from the radiative hadronic decays of $\Upsilon(1S)$. We fit
hadronic and all other backgrounds with an exponential PDF. The expected yield
of peaking background events for
$\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$ decay is estimated to be
consistent with zero ($1.8\pm 5.1$). Similarly, the expected yield for
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$ decay is estimated to be
$-7.9\pm 7.4$. To fit the data, we fix all the parameters of the background
PDF (including the shape of the exponential PDF and fraction of $\tau\tau$
background) from MC except the $A$ parameter of the $\tau\tau$ PDF. The
fractions of $\tau\tau$ PDF for
$\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$ and
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$ decays are 58% and 86%,
respectively. The width of the signal PDF for
$\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$
($\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$) decay in the data is fixed
from MC signal width corrected by the data-MC difference for the $M^{{\rm
recoil}}_{\rm\pi\pi\mu}$ ($M^{{\rm recoil}}_{\rm\pi\pi e}$) parameter. In Fig.
7, we show the $M^{{\rm recoil}}_{\rm\pi\pi\ell\gamma}$ fits to $\Upsilon(2S)$
data. The effective signal efficiency for $\gamma\mu\tau$ ($\gamma e\tau$)
decay is 5.8% (5.0%). The fitted signal yield for
$\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$ decays in the
$\Upsilon(2S)$ data is estimated to be $2.1\pm 5.9$. Similarly, the signal
yield for $\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$ decay is obtained
to be $-9.5\pm 6.3$. Hence, there is no evidence for
$\Upsilon(1S)\rightarrow\gamma\ell^{\pm}\tau^{\mp}$ transitions.

##  7 Systematic uncertainty and correction

We calculate the systematic uncertainty from various sources such as the
number of $\Upsilon(2S)$, track reconstruction, photon reconstruction,
identification of pions from $\Upsilon(2S)$, lepton identification,
uncertainty in signal efficiency, secondary branching fraction, and the
fitting model.

The uncertainty on the number of $\Upsilon(2S)$ events was determined from a
study of hadronic decays to be 2.3% ny2s . Reconstruction efficiency of
charged particle tracks are studied using a partially reconstructed
${D^{*+}}\rightarrow D^{0}[K_{S}^{0}(\pi^{+}\pi^{-})\pi^{+}\pi^{-}]\pi^{+}$
decay sample with $p_{\rm T}>200$ ${\rm MeV}/c$. Systematic uncertainty per
track is estimated to be 0.35%. Due to correlation, uncertainties in charged
track finding are added linearly. The efficiency of photon reconstruction is
estimated with radiative Bhabha events, and the associated uncertainty is 2.0%
t2lg .

Uncertainty from pion identification in
$\Upsilon(2S)\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)$ reconstruction may affect
our results. In order to estimate it, we use the results of a dedicated study
based on the $D^{*+}\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$ decay. A
correction for the difference in efficiency (between data and signal MC) is
obtained from the same source. This correction is used to correct the
efficiency, and its uncertainty is included as the systematic uncertainty due
to pion identification. For all the decays, the efficiency correction factor
and systematic uncertainty from pion pair reconstruction are estimated to be
1.00 and 1.9%, respectively. For the electron identification with ${\cal
L}_{e}>0.6$ and the muon identification with ${\cal L}_{\mu}>0.95$, systematic
uncertainty are calculated from the comparison between data and MC for
$2\gamma\rightarrow ee/\mu\mu$ decays. We calculate an efficiency correction
factor and systematic uncertainty for all of the electrons and muons using the
same approach. For $\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ decay,
the systematic uncertainty due to pion identification is estimated using the
$D^{*+}$-based method described above. The efficiency correction factors
associated with the leptons pair reconstructions for $e\mu$, $\mu\tau$,
$e\tau$, $\gamma e\mu$, $\gamma\mu\tau$, and $\gamma e\tau$ decays are 0.99,
0.98, 0.97, 0.95, 0.94, and 0.97, respectively and corresponding systematic
uncertainties are 1.9%, 2.1%, 2.3%, 2.6%, 2.8%, and 2.5%, respectively.

Due to the limited number of generated MC signal events, there is an
uncertainty in the fitted number of signal events as well as in the signal
efficiency ($\epsilon$), and the corresponding uncertainty is included in the
systematic uncertainty.

Source | Systematic uncertainty(%)
---|---
$\mathcal{S}_{e\mu}$ | $\mathcal{S}_{\mu\tau}$ | $\mathcal{S}_{e\tau}$ | $\mathcal{S}_{\gamma e\mu}$ | $\mathcal{S}_{\gamma\mu\tau}$ | $\mathcal{S}_{\gamma e\tau}$ | $\mathcal{S}_{ee}$ | $\mathcal{S}_{\mu\mu}$
Number of $\Upsilon(2S)$ | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3 | 2.3
Track reconstruction | 1.4 | 1.5 | 1.5 | 1.4 | 1.4 | 1.4 | 1.4 | 1.4
Photon reconstruction | - | - | - | 2.0 | 2.0 | 2.0 | - | -
Reconstruction of $\pi^{+}\pi^{-}$ from $\Upsilon(2S)$ | 1.9 | 1.9 | 1.9 | 1.9 | 1.9 | 1.9 | 1.9 | 1.9
1st lepton identification | 1.6 | 1.1 | 1.5 | 1.7 | 2.0 | 1.7 | 1.7 | 1.2
2nd lepton identification | 1.1 | 1.8 | 1.7 | 2.0 | 1.9 | 1.8 | 1.7 | 1.2
MC statistics | 0.2 | 0.3 | 0.6 | 0.3 | 0.4 | 0.4 | 0.4 | 0.3
Secondary branching fractions | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5
Fitting model | 0.1 | 1.5 | 0.9 | 0.1 | 0.8 | 1.5 | 0.1 | 0.0
Total | 4.1 | 4.5 | 4.4 | 4.9 | 5.1 | 5.1 | 5.0 | 4.4
Table 1: Summary of the systematic uncertainties for the measurement of
branching fractions of the $\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$
($\mathcal{S}_{e\mu}$), $\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$
($\mathcal{S}_{\mu\tau}$), $\Upsilon(1S)\rightarrow e^{\pm}\tau^{\mp}$
($\mathcal{S}_{e\tau}$), $\Upsilon(1S)\rightarrow\gamma e^{\pm}\mu^{\mp}$
($\mathcal{S}_{\gamma e\mu}$),
$\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$
($\mathcal{S}_{\gamma\mu\tau}$), $\Upsilon(1S)\rightarrow\gamma
e^{\pm}\tau^{\mp}$ ($\mathcal{S}_{\gamma e\tau}$), $\Upsilon(1S)\rightarrow
e^{+}e^{-}$ ($\mathcal{S}_{ee}$), and $\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$
($\mathcal{S}_{\mu\mu}$) decays.

To obtain the final results, previously measured branching fractions of
$\Upsilon(2S)\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)$,
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$, $\tau^{-}\rightarrow
e^{-}\bar{\nu}_{e}\nu_{\tau}$, and
$\tau^{-}\rightarrow\pi^{-}\pi^{+}\pi^{-}\nu_{\tau}$ are used pdg . The
uncertainties in the world average secondary branching ratios are included as
systematic uncertainties.

We fix some parameters (such as the mean, width, and fractions of the two
Gaussians) of the signal PDF while fitting the data. The associated systematic
uncertainties are estimated by varying each of the fixed parameters by $\pm
1\sigma$ from their central values and repeating the fit. For
$\Upsilon(1S)\rightarrow e^{+}e^{-}$ and
$\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ decays, these PDF systematic
uncertainties are estimated to be 0.08% and 0.04%, respectively. In the
absence of significant signal events for the CLFV modes, we take the average
value of the control modes (0.06%) as the systematic uncertainty from the
signal PDF for each of the CLFV modes. Similarly, the systematic uncertainty
from the background PDF for $\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$,
$\Upsilon(1S)\rightarrow e^{\pm}\tau^{\mp}$,
$\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$, and
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$ are estimated to be 1.4%,
0.8%, 0.7%, and 1.4%, respectively. No parameters were fixed to estimate the
background for $\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$,
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\mu^{\mp}$, $\Upsilon(1S)\rightarrow
e^{+}e^{-}$, and $\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ modes. Therefore, for
the above decays, the systematic uncertainty from the background PDF is
estimated as zero. Due to correlation, systematic uncertainties from signal
PDF and background PDF are added linearly. Table 1 summarizes the systematic
uncertainties from various sources for all the modes. Systematic uncertainties
from the different sources are added in quadrature in order to get the total
systematic uncertainty for a particular signal mode. The systematic
uncertainty due to the uncertainty in the peaking background for
$\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$ ($8.8\pm 2.0$) is directly included
in the estimated upper limit of the branching fraction. For other modes, the
effect of possible peaking background lowers the upper limit, and we do not
consider it to report conservative upper limits.

##  8 Results

Using equation (2), the branching fractions are calculated as
$\mathcal{B}[\Upsilon(1S)\rightarrow e^{+}e^{-}]=(2.40\pm 0.01({\rm stat})\pm
0.12({\rm syst}))\times 10^{-2}$ and
$\mathcal{B}[\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}]=(2.46\pm 0.01({\rm
stat})\pm 0.11({\rm syst}))\times 10^{-2}$ which agree within $\pm 1\sigma$
with world average values pdg . All of the results for the branching fractions
of CLFV modes are dominated by statistical uncertainty. In the absence of
significant signal, we estimate the upper limits (UL) of the branching
fractions with a frequentist approach ul . One can calculate the UL of
branching fractions using the following relation:

$\mathcal{B}[\Upsilon(1S)\rightarrow\ell^{\pm}\ell^{\prime\mp}]<\frac{N_{\rm
sig}^{\rm
UL}}{N_{\Upsilon(2S)}\times\mathcal{B}[\Upsilon(2S)\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)]\times\epsilon}$
(4)

where $N_{\rm sig}^{\rm UL}$ is the UL on the signal yield after including
systematic uncertainty. We perform 5000 pseudo-experiments by generating the
fixed background from the final PDF and varying the yield of the input signal
within 1 to 20. We use the corresponding PDF that has been used to fit
$\Upsilon(2S)$ data for generating the data sets for pseudo-experiments. The
fraction of pseudo-experiments with a fitted yield greater than the estimated
signal yield in data has been taken as the confidence level (CL). Systematic
uncertainties of the CLFV modes are included by smearing the yield of the
pseudo-experiments within the fluctuations. For $\Upsilon(1S)\rightarrow
e^{\pm}\mu^{\mp}$ decay, the fitted signal yields of pseudo-experiments have
been smeared within the corresponding uncertainty of peaking background to
include the associated systematic uncertainty.

For $e\mu$, $\mu\tau$, and $e\tau$ decays, the central values of signal yields
are obtained as $-1.3$, $-1.4$, and $-3.5$, respectively. The fraction of
pseudo-experiments with any positive yield has been used to estimate the 90%
CL upper limits. Considering the number of $\Upsilon(2S)$ as 157.8 million and
$\mathcal{B}[\Upsilon(2S)\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)]$ as $(17.85\pm
0.26)$%, we calculate the ULs of branching fractions ($\mathcal{B}^{\rm UL}$)
by substituting $N_{\rm sig}^{\rm UL}$ in Eq. (4). The estimated ULs for
$\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$,
$\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$, and $\Upsilon(1S)\rightarrow
e^{\pm}\tau^{\mp}$ at 90% CL are $3.9\times 10^{-7}$, $2.7\times 10^{-6}$, and
$2.7\times 10^{-6}$ respectively. We summarize these results in Table 2.

Decay |  $\epsilon$ (%) | $N_{\rm sig}^{\rm fit}$ | $N_{\rm sig}^{\rm UL}$ | $\mathcal{B}^{\rm UL}$ | PDG result
---|---|---|---|---|---
$\Upsilon(1S)\rightarrow e^{\pm}\mu^{\mp}$ |  32.5 | $-1.3\pm 3.7$ | 3.6 | $3.9\times 10^{-7}$ | $-$
$\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$ |  8.8 | $-1.5\pm 4.3$ | 6.8 | $2.7\times 10^{-6}$ | $6.0\times 10^{-6}$
$\Upsilon(1S)\rightarrow e^{\pm}\tau^{\mp}$ |  7.1 | $-3.5\pm 2.7$ | 5.3 | $2.7\times 10^{-6}$ | $-$
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\mu^{\mp}$ |  24.6 | $+0.8\pm 1.5$ | 2.9 | $4.2\times 10^{-7}$ | $-$
$\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$ |  5.8 | $+2.1\pm 5.9$ | 10.0 | $6.1\times 10^{-6}$ | $-$
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$ |  5.0 | $-9.5\pm 6.3$ | 9.1 | $6.5\times 10^{-6}$ | $-$
Table 2: Results of searches for CLFV in $\Upsilon(1S)$ decays. Here, $N_{\rm
sig}^{\rm fit}$ is the fitted signal yield. $N_{\rm sig}^{\rm UL}$ and
$\mathcal{B}^{\rm UL}$ are, respectively, the upper limits of signal yield and
branching fraction at 90% CL.

For $\gamma e\mu$ and $\gamma\mu\tau$ decays, the central values of signal
yields are estimated to be 0.8 and 2.1, respectively. The estimated signal
yield for $\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$ decay is $-9.5$.
The fraction of pseudo-experiments with a signal yield greater than 0.8 (2.1)
has been taken as the CL for $\gamma e\mu$ ($\gamma\mu\tau$) decay. For
$\gamma e\tau$ decay, we treat the fraction of pseudo-experiments with a
positive signal yield as the CL. Table 2 summarizes the $N_{\rm sig}^{\rm UL}$
for RLFV modes at the 90% CL. Using $N_{\rm sig}^{\rm UL}$ in Eq. (4), the ULs
of branching fractions for $\Upsilon(1S)\rightarrow\gamma e^{\pm}\mu^{\mp}$,
$\Upsilon(1S)\rightarrow\gamma\mu^{\pm}\tau^{\mp}$, and
$\Upsilon(1S)\rightarrow\gamma e^{\pm}\tau^{\mp}$ are estimated to be
$4.2\times 10^{-7}$, $6.1\times 10^{-6}$, and $6.5\times 10^{-6}$,
respectively.

##  9 Summary

In this paper, we report the searches for charged lepton-flavor-violation in
$\Upsilon(1S)\rightarrow\ell^{\pm}\ell^{\prime\mp}$ decays and radiative
lepton-flavor-violation in
$\Upsilon(1S)\rightarrow\gamma\ell^{\pm}\ell^{\prime\mp}$ decays conducted at
the Belle experiment, where $\ell,\ell^{\prime}=e,\mu,\tau$. The searches are
based on the 28 million $\pi^{+}\pi^{-}\Upsilon(1S)$ decays produced from 25
${\rm fb}^{-1}$ of $e^{+}e^{-}$ collisions collected at the $\Upsilon(2S)$
resonance. We study the sources of possible background using a large
$\Upsilon(2S)$ MC sample. To validate the signal extraction procedure we
measure the branching fractions for $\Upsilon(1S)\rightarrow e^{+}e^{-}$ and
$\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}$ modes and find
$\mathcal{B}[\Upsilon(1S)\rightarrow e^{+}e^{-}]=(2.40\pm 0.01({\rm stat})\pm
0.12({\rm syst}))\times 10^{-2}$ and
$\mathcal{B}[\Upsilon(1S)\rightarrow\mu^{+}\mu^{-}]=(2.46\pm 0.01({\rm
stat})\pm 0.11({\rm syst}))\times 10^{-2}$, respectively. In the absence of
signal, we set upper limits on the branching fractions of the CLFV decays at
the 90% CL. The result for the $\Upsilon(1S)\rightarrow\mu^{\pm}\tau^{\mp}$
decay is 2.3 times more stringent than the previous result from the CLEO
collaboration cleo , while the remaining modes are searched for the first
time.

###### Acknowledgements.

We thank the KEKB group for the excellent operation of the accelerator; the
KEK cryogenics group for the efficient operation of the solenoid; and the KEK
computer group, and the Pacific Northwest National Laboratory (PNNL)
Environmental Molecular Sciences Laboratory (EMSL) computing group for strong
computing support; and the National Institute of Informatics, and Science
Information NETwork 5 (SINET5) for valuable network support. We acknowledge
support from the Ministry of Education, Culture, Sports, Science, and
Technology (MEXT) of Japan, the Japan Society for the Promotion of Science
(JSPS), and the Tau-Lepton Physics Research Center of Nagoya University; the
Australian Research Council including grants DP180102629, DP170102389,
DP170102204, DE220100462, DP150103061, FT130100303; Austrian Federal Ministry
of Education, Science and Research (FWF) and FWF Austrian Science Fund No. P
31361-N36; the National Natural Science Foundation of China under Contracts
No. 11675166, No. 11705209; No. 11975076; No. 12135005; No. 12175041; No.
12161141008; Key Research Program of Frontier Sciences, Chinese Academy of
Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; the Shanghai Science and
Technology Committee (STCSM) under Grant No. 19ZR1403000; the Ministry of
Education, Youth and Sports of the Czech Republic under Contract No. LTT17020;
the Czech Science Foundation Grant No. 22-18469S; Horizon 2020 ERC Advanced
Grant No. 884719 and ERC Starting Grant No. 947006 “InterLeptons” (European
Union); the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft, the
Excellence Cluster Universe, and the VolkswagenStiftung; the Department of
Atomic Energy (Project Identification No. RTI 4002) and the Department of
Science and Technology of India; the Istituto Nazionale di Fisica Nucleare of
Italy; National Research Foundation (NRF) of Korea Grant Nos.
2016R1D1A1B01010135, 2016R1D1A1B02012900, 2018R1A2B3003643,
2018R1A6A1A06024970, 2019K1A3A7A09033840, 2019R1I1A3A01058933,
2021R1A6A1A03043957, 2021R1F1A1060423, 2021R1F1A1064008; Radiation Science
Research Institute, Foreign Large-size Research Facility Application
Supporting project, the Global Science Experimental Data Hub Center of the
Korea Institute of Science and Technology Information and KREONET/GLORIAD; the
Polish Ministry of Science and Higher Education and the National Science
Center; the Ministry of Science and Higher Education of the Russian
Federation, Agreement 14.W03.31.0026, and the HSE University Basic Research
Program, Moscow; University of Tabuk research grants S-1440-0321, S-0256-1438,
and S-0280-1439 (Saudi Arabia); the Slovenian Research Agency Grant Nos.
J1-9124 and P1-0135; Ikerbasque, Basque Foundation for Science, Spain; the
Swiss National Science Foundation; the Ministry of Education and the Ministry
of Science and Technology of Taiwan; and the United States Department of
Energy and the National Science Foundation.

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