# Quotients with respect to strongly $L$-subgyrogroups

Ying-Ying Jin  (Y.-Y. Jin) Department of General Required Courses, Guangzhou
Panyu Polytechnic, Guangzhou 511483, P.R. China<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Li-Hong Xie  (L.-H. Xie) School of Mathematics and
Computational Science, Wuyi University, Jiangmen 529020, P.R. China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

A topological gyrogroup is a gyrogroup endowed with a compatible topology such
that the multiplication is jointly continuous and the inverse is continuous.
In this paper, we study the quotient gyrogroups in topological gyrogroups with
respect to strongly $L$-subgyrogroups, and prove that let $(G,\tau,\oplus)$ be
a topological gyrogroup and $H$ a closed strongly $L$-subgyrogroup of $G$,
then the natural homomorphism $\pi$ from a topological gyrogroup $G$ to its
quotient topology on $G/H$ is an open and continuous mapping, and $G/H$ is a
homogeneous $T_{1}$-space. We also establish that for a locally compact
strongly $L$-subgyrogroup $H$ of a topological gyrogroup $G$, the natural
quotient mapping $\pi$ of $G$ onto the quotient space $G/H$ is a locally
perfect mapping. This leads us to some interesting results on how properties
of $G$ depend on the properties of $G/H$. Some classical results in
topological groups are generalized.

###### Key words and phrases:

Topological gyrogroup; quotient topology; locally perfect mappings; local
pseudocompactness; local paracompactness

###### 2010 Mathematics Subject Classification:

54H11; 22A30; 22A22; 20N05; 54H99

* The corresponding author.

This work is supported by the Natural Science Foundation of Guangdong Province
under Grant (Nos. 2020A1515110458, 2021A1515010381), the Innovation Project of
Department of Education of Guangdong Province (No. 2022KTSCX145), Scientific
research project of Guangzhou Panyu Polytechnic (No. 2022KJ02), and Natural
Science Project of Jiangmen City (No. 2021030102570004880).

##  1\. Introduction

In topological groups, the following general problem was considered by some
scholars. Let $G$ be a topological group, $H$ a closed subgroup of $G$, and
$G/H$ the quotient space. Suppose that both $H$ and $G/H$ belong to some nice
class of topological spaces. When can we conclude that $G$ is in the same
class? Graev proved that if $H$ and $G/H$ are metrizable, then $G$ is also
metrizable [12]. Serre established that if $H$ and $G/H$ are locally compact,
then the topological group $G$ is also locally compact [13]. In 2005,
Arhangel’ski established that the following result:

######  Theorem 1.1.

[2, Theorem 1.2] Suppose that $G$ is a topological group, $H$ a locally
compact subgroup of $G$, and $\pi:G\rightarrow G/H$ the natural quotient
mapping. Then there exists an open neighbourhood $U$ of the neutral element
$e$ such that $\pi(\overline{U})$ is closed in $G/H$ and the restriction of
$\pi$ to $\overline{U}$ is a perfect mapping of $\overline{U}$ onto the
subspace $\pi(\overline{U}).$

Also, this result leads to interesting conclusions on how certain topological
properties of $G$ depend on the properties of $G/H$, when $H$ is locally
compact [2].

In [16], Ungar studies a parametrization of the Lorentz transformation group.
This leads to the formation of gyrogroup theory, a rich subject in mathematics
(among others, the interested reader can consult [1, 11]). Loosely speaking, a
gyrogroup (see Definition 2.1) is a group-like structure in which the
associative law fails to satisfy. Recently, as a generalization of topological
groups Atiponrat [4] defined the concept of topological gyrogroups, which is a
gyrogroup with a topology such that its binary operation is jointly continuous
and the operation of taking the inverse is continuous. Some basic properties
of topological gyrogroups are studied by Atiponrat [4]. Specially, Atiponrat
[4] discovered that for a topological gyrogroup, $T_{0}$ and $T_{3}$ are
equivalent. It is worth noting that Cai, Lin and He in [9] proved that every
Hausdorff first countable topological gyrogroup is metrizable.

As a generalization of topological groups, it is natural to consider the
quotients of topological gyrogroups. In [7, 8], Bao and Lin considered when
the admissible $L$-subgyrogroup $H$ of a strongly topological gyrogroup $G$
with some topological properties, such as locally compact, submetrizable and
so on, the natural quotient mapping has some nice topological properties. The
purpose of this paper is to investigate the quotients of topological
gyrogroups. Some results in topological groups are improved.

The paper is organized as follows: In Section 2, we mainly introduce the
related concepts and conclusions which are required in this article. In
Section 3, we introduce the concept of strongly $L$-subgyrogroups and give
some topological properties of the quotient spaces with respect to strongly
$L$-subgyrogroups of topological gyrogroups are studied. We mainly show that:
Let $(G,\tau,\oplus)$ be a topological gyrogroup and $H$ a closed strongly
$L$-subgyrogroup of $G$; then the natural homomorphism $\pi$ from a
topological gyrogroup $G$ to its quotient topology on $G/H$ is an open and
continuous mapping, and $G/H$ is a homogeneous $T_{1}$-space(see Theorem 3.5).
In Section 4, the quotients with respect to locally compact strongly
$L$-subgyrogroups of topological groups are studied. It is shown that if $G$
is a topological gyrogroup, $H$ a locally compact strongly $L$-subgyrogroup of
$G$, and $\pi:G\rightarrow G/H$ is the natural quotient mapping of $G$ onto
the quotient space $G/H$, then there exists an open neighbourhood $U$ of the
neutral element $0$ such that $\pi(\overline{U})$ is closed in $G/H$ and the
restriction of $\pi$ to $\overline{U}$ is a perfect mapping of $\overline{U}$
onto the subspace $\pi(\overline{U})$(see Theorem 4.2).

All spaces in this paper satisfy the $T_{2}$-separation axiom. All mappings
are continuous and onto. The reader may consult [3, 10] for unstated notations
and terminology.

##  2\. Definitions and preliminaries

Let $G$ be a nonempty set, and let $\oplus:G\times G\rightarrow G$ be a binary
operation on $G$. Then the pair $(G,\oplus)$ is called a groupoid. A function
$f$ from a groupoid $(G_{1},\oplus_{1})$ to a groupoid $(G_{2},\oplus_{2})$ is
said to be a groupoid homomorphism if
$f(x_{1}\oplus_{1}x_{2})=f(x_{1})\oplus_{2}f(x_{2})$ for any elements
$x_{1},x_{2}\in G_{1}$. In addition, a bijective groupoid homomorphism from a
groupoid $(G,\oplus)$ to itself will be called a groupoid automorphism. We
will write $Aut(G,\oplus)$ for the set of all automorphisms of a groupoid
$(G,\oplus)$.

######  Definition 2.1.

[16, Definition 2.7] Let $(G,\oplus)$ be a nonempty groupoid. We say that
$(G,\oplus)$ or just $G$ (when it is clear from the context) is a gyrogroup if
the followings hold:

  1. ($G1$)

There is an identity element $e\in G$ such that

$e\oplus x=x=x\oplus e\text{~{}~{}~{}~{}~{}for all~{}~{}}x\in G.$

  2. ($G2$)

For each $x\in G$, there exists an inverse element $\ominus x\in G$ such that

$\ominus x\oplus x=e=x\oplus(\ominus x).$

  3. ($G3$)

For any $x,y\in G$, there exists an gyroautomorphism $\text{gyr}[x,y]\in
Aut(G,\oplus)$ such that

$x\oplus(y\oplus z)=(x\oplus y)\oplus\text{gyr}[x,y](z)$

for all $z\in G$;

  4. ($G4$)

For any $x,y\in G$, $\text{gyr}[x\oplus y,y]=\text{gyr}[x,y]$.

######  Definition 2.2.

[15] Let $(G,\oplus)$ be a gyrogroup. A nonempty subset $H$ of $G$ is called a
subgyrogroup, denoted by $H\leq G$, if the following statements hold:

  1. (1)

The restriction $\oplus|_{H\times H}$ is a binary operation on $H$, i.e.
$(H,\oplus|_{H\times H})$ is a groupoid;

  2. (2)

For any $x,y\in H$, the restriction of $\text{gyr}[x,y]$ to $H$,
$\text{gyr}[x,y]|_{H}:H\rightarrow\text{gyr}[x,y](H)$, is a bijective
homomorphism; and

  3. (3)

$(H,\oplus|_{H\times H})$ is a gyrogroup.

Furthermore, a subgyrogroup $H$ of $G$ is said to be an $L$-subgyrogroup [15],
denoted by $H\leq_{L}G$, if $\text{gyr}[a,h](H)=H$ for all $a\in G$ and $h\in
H$.

In this paper, $\text{gyr}[a,b]V$ denotes $\\{\text{gyr}[a,b]v:v\in V\\}$.

The following Proposition 2.3 below summarizes some algebraic properties of
gyrogroups.

######  Proposition 2.3.

[16, 17] Let $(G,\oplus)$ be a gyrogroup and $a,b,c\in G$. Then

  1. (1)

$\ominus(\ominus a)=a$ Involution of inversion

  2. (2)

$\ominus a\oplus(a\oplus b)=b$ Left cancellation law

  3. (3)

gyr$[a,b](c)=\ominus(a\oplus b)\oplus(a\oplus(b\oplus c))$ Gyrator identity

  4. (4)

$\ominus(a\oplus b)=\text{gyr}[a,b](\ominus b\ominus a)$ cf.
$(ab)^{-1}=b^{-1}a^{-1}$

  5. (5)

$(\ominus a\oplus b)\oplus\text{gyr}[\ominus a,b](\ominus b\oplus c)=\ominus
a\oplus c$ cf. $(a^{-1}b)(b^{-1}c)=a^{-1}c$

  6. (6)

$\text{gyr}[a,b]=\text{gyr}[\ominus b,\ominus a]$ Even property

  7. (7)

$\text{gyr}[a,b]=\text{gyr}^{-1}[b,a],\text{the inverse of gyr}[b,a]$
Inversive symmetry

######  Definition 2.4.

[4, Definition 1] A triple $(G,\tau,\oplus)$ is called a topological gyrogroup
if and only if

  1. (1)

$(G,\tau)$ is a topological space;

  2. (2)

$(G,\oplus)$ is a gyrogroup;

  3. (3)

The binary operation $\oplus:G\times G\rightarrow G$ is continuous where
$G\times G$ is endowed with the product topology and the operation of taking
the inverse $\ominus(\cdot):G\rightarrow G$, i.e. $x\rightarrow\ominus x$, is
continuous.

######  Definition 2.5.

[17, Definition 2.9] Let $(G,\oplus)$ be a gyrogroup with gyrogroup operation
(or, addition) $\oplus$. The gyrogroup cooperation (or, coaddition) $\boxplus$
is a second binary operation in $G$ given by the equation

$(\divideontimes)~{}~{}~{}~{}a\boxplus b=a\oplus\text{gyr}[a,\ominus b]b$

for all $a,b\in G$. The groupoid $(G,\boxplus)$ is called a cogyrogroup, and
is said to be the cogyrogroup associated with the gyrogroup $(G,\oplus)$.

Replacing $b$ by $\ominus b$ in $(\divideontimes)$, along with
$(\divideontimes)$ we have the identity

$a\boxminus b=a\ominus\text{gyr}[a,b]b$

for all $a,b\in G$, where we use the obvious notation, $a\boxminus
b=a\boxplus(\ominus b)$.

Let $(G,\oplus)$ be a gyrogroup, $x\in G$ and $A,B\subseteq G$. We write
$A\oplus B=\\{a\oplus b:a\in A,b\in B\\}$, $x\oplus A=\\{x\oplus a:a\in A\\}$
and $A\oplus x=\\{a\oplus x:a\in A\\}$.

######  Definition 2.6.

[16] Let $(G,\oplus)$ be a gyrogroup, and let $x\in G$. We define the  left
gyrotranslation  by $x$ to be the function $L_{x}:G\rightarrow G$ such that
$L_{x}(y)=x\oplus y$ for any $y\in G$. In addition, the  right gyrotranslation
by $x$ is defined to be the function $R_{x}(y)=y\oplus x$ for any $y\in G$.

######  Proposition 2.7.

[4, Proposition 3, Lemma 4 and Corollary 5] Let $(G,\oplus)$ be a topological
gyrogroup, $x\in G$ and $A,B\subseteq G$.

  1. (1)

The inverse mapping $inv:G\rightarrow G$, where $inv(x)=\ominus x$ for every
$x\in G$, is a homeomorphism;

  2. (2)

The left translation $L_{x}:G\rightarrow G$, where $L_{x}(y)=x\oplus y$ for
every $y\in G$, is a homeomorphism;

  3. (3)

The right translation $R_{x}:G\rightarrow G$, where $R_{x}(y)=y\oplus x$ for
every $y\in G$, is a homeomorphism;

  4. (4)

Let $x,y\in G$. Then $\text{gyr}[x,y]:G\rightarrow G$ is a homeomorphism;

  5. (5)

If $A$ is open in $G$, then $x\oplus A$, $A\oplus x$, $A\oplus B$, $B\oplus A$
and $\ominus A$ are all open in $G$.

Let $X$ be a topological space and let $A\subseteq X$. We denote the closure
$A$ in $X$ by $\overline{A}$.

######  Corollary 2.8.

Let $G$ be a topological gyrogroup, and $A$ be a subset of $G$. Then
$\overline{\ominus A}=\ominus\overline{A}$.

###### Proof.

It is known that for every homeomorphism $f:G\rightarrow G$,
$\overline{f(A)}=f(\overline{A})$ is true. So we can get $\overline{\ominus
A}=\ominus\overline{A}$ by Proposition 2.7 (1). ∎

######  Corollary 2.9.

Let $G$ be a topological gyrogroup, $A$ be a subset of $G$ and $g,h$ be points
of $G$. Then $\text{gyr}[g,h]\overline{A}=\overline{\text{gyr}[g,h]A}$.

###### Proof.

It is known that for every homeomorphism $f:G\rightarrow G$,
$f(\overline{A})=\overline{f(A)}$ is true. So we can get
$\text{gyr}[g,h]\overline{A}=\overline{\text{gyr}[g,h]A}$ by Proposition 2.7
(4). ∎

######  Proposition 2.10.

If $H$ is a subgyrogroup ($L$-subgyrogroup) of topological gyrogroup $G$, then
$\overline{H}$ is a subgyrogroup ($L$-subgyrogroup) of $G$.

###### Proof.

We show first that, for all $A,B\subseteq G$, we have
$\overline{A}\oplus\overline{B}\subseteq\overline{A\oplus B}$. By the
definition of a topological gyrogroup, the mapping $op_{2}$ is continuous
$G\times G\rightarrow G$. The set $\overline{A}\oplus\overline{B}$ can be
represented in the form $op_{2}(\overline{A}\times\overline{B})$. In the
product space $G\times G$ we have that
$\overline{A}\times\overline{B}=\overline{A\times B}$, and it follows, by
continuity of the mapping $op_{2}$, that $op_{2}(\overline{A\times
B})\subseteq\overline{op_{2}(A\times B)}=\overline{A\oplus B}$. Since
$op_{2}(\overline{A\times B})=\overline{A}\oplus\overline{B}$, we have shown
that $\overline{A}\oplus\overline{B}\subseteq\overline{A\oplus B}$. Let $H$ be
a subgyrogroup of $G$. Then $H\oplus H=H$ and $H^{-1}=H$. By the foregoing, we
have $\overline{H}\oplus\overline{H}\subseteq\overline{H\oplus H}$, and thus
$\overline{H}\oplus\overline{H}\subseteq\overline{H}$; this means that the set
$H$ is closed under the operation of $G$. On the other hand, Corollary 2.8
shows that $\overline{\ominus H}=\ominus\overline{H}$; from this it follows
that $\ominus\overline{H}=\overline{H}$, and thus the set $H$ is also closed
under taking inverses. We have shown that $H$ is a subgyrogroup of $G$.

We assume that the subgyrogroup $H$ is an $L$-subgyrogroup. Then, for all
$a\in G$ and $h\in H$ we have that $\text{gyr}[a,h](H)=H$, and hence, by
Corollary 2.9, that
$\text{gyr}[a,h]\overline{H}=\overline{\text{gyr}[a,h]H}=\overline{H}$. As a
consequence, $\overline{H}$ is an $L$-subgyrogroup. ∎

######  Proposition 2.11.

Every open subgyrogroup $H$ of a topological gyrogroup $G$ is closed in $G$.

###### Proof.

From Proposition 2.10 it follows that $\overline{H}$ is a subgyrogroup. Since
$H$ is an open set containing the identity $0$, one can easily show that
$\overline{H}\subseteq H\oplus H=H$, which implies that $H$ is closed in $G$.
In fact, take any $x\notin H$. Then $x\notin H\oplus H$. This implies that
$(\ominus H)\oplus x\cap H=\emptyset$. Since $H$ is an open set containing the
identity $0$ and $G$ is a topological gyrogroup, $(\ominus H)\oplus x$ is an
open set containing $x$. This implies that $x\notin\overline{H}$. ∎

######  Proposition 2.12.

If $H$ is a locally compact subgyrogroup of a topological gyrogroup $G$, then
$H$ is closed in $G$.

###### Proof.

Let $K$ be the closure of $H$ in $G$. Then $K$ is a subgyrogroup of $G$ by
Proposition 2.10. Since $H$ is a dense locally compact subspace of $K$, it
follows from [10, Theorem 3.3.9] that $H$ is open in $K$. However, an open
subgyrogroup of a topological gyrogroup is closed by Proposition 2.11.
Therefore, $H=K$, that is, $H$ is closed in $G$. ∎

##  3\. Quotients with respect to strongly $L$-subgyrogroups

To study the quotients of gyrogroups, Suksumran [14, Definition 9] defined a
normal subgyrogroup in a similar fashion in groups. Recall that a subgyrogroup
$N$ of a gyrogroup $G$ is normal in $G$, written $N\unlhd G$, if it is the
kernel of a gyrogroup homomorphism of $G$. Also, Suksumran obtained the
following result in the same paper:

######  Proposition 3.1.

[14, Proposition 38] Let $G$ be a gyrogroup. If $H$ is a subgyrogroup of $G$
such that

  1. (1)

$\text{gyr}[h,a]=\text{id}_{G}$ for all $h\in H,a\in G$;

  2. (2)

$\text{gyr}[b,a]H\subseteq H$ for all $b,a\in G$;

  3. (3)

$a\oplus H=H\oplus a$ for all $a\in G$,

then $H\unlhd G$, that is $H$ is a normal subgyrogroup of $G$.

Recall that a subgyrogroup $H$ of a gyrogroup $G$ is a $L$-subgyrogroup if
$\text{gyr}[a,h]H=H$ for all $h\in H,a\in G$. According to [14, Proposition
24] it follows that the condition “(2) $\text{gyr}[b,a]H\subseteq H$ for all
$b,a\in G$” in Proposition 3.1 implies the condition“$\text{gyr}[b,a]H=H$ for
all $b,a\in G$”. This leads us to define strongly $L$-subgyrogroups as follow:

######  Definition 3.2.

A subgyrogroup $H$ of a gyrogroup $G$ is said to be a strongly
$L$-subgyrogroup111In [6, Definition 3.9] it is called a strongly
subgyrogroup., denoted by $H\leqslant_{SL}G$, if $\text{gyr}[a,b](H)\subseteq
H$ for all $a,b\in G$.

######  Remark 3.3.

It is easy to see that each normal subgyrogroup is a strongly $L$-subgyrogroup
and each strongly $L$-subgyrogroup is an $L$-subgyrogroup. There exist
strongly $L$-subgyrogroups which are not normal subgyrogroups. Since group $G$
can be considered the special gyrogroup where $\text{gyr~{}}[a,b]=\text{id}$
for all $a,b\in G$. Each subgroup $H\subseteq G$ satisfies
$\text{gyr~{}}[a,b](H)=H$ for all $a,b\in G$, and obviously every subgroup
doesn’t have to be a normal subgroup. However, we don’t known whether there is
an $L$-subgyrogroup which is not a strongly $L$-subgyrogroup.

######  Proposition 3.4.

Let $G$ be a gyrogroup. If $H\leqslant_{SL}G$, then $a\oplus(b\oplus
H)=(a\oplus b)\oplus H$ for all $a,b\in G$.

###### Proof.

Let $a,b\in G$. By Definition 3.2, a direct computation gives $a\oplus(b\oplus
H)=(a\oplus b)\oplus\text{gyr~{}}[a,b](H)=(a\oplus b)\oplus H$. ∎

Let $(G,\tau,\oplus)$ be a topological gyrogroup and $H$ a strongly
$L$-subgyrogroup of $G$. It follows from [15, Theorem 20] that $G/H=\\{a\oplus
H:a\in G\\}$ is a partition of $G$. We denote by $\pi$ the mapping $a\mapsto
a\oplus H$ from $G$ onto $G/H$. Clearly, for each $a\in G$, we have we have
$\pi^{-1}(\pi(a))=a\oplus H$, for each $a\in G$. Denote by $\tau(G)$ the
topology of $G$. In the set $G/H$, we define a family $\tau(G/H)$ of subsets
as follows:

$\tau(G/H)=\\{O\subseteq G/H:\pi^{-1}(O)\in\tau(G)\\}.$

The following result improves [6, Theorem 3.13].

######  Theorem 3.5.

Let $(G,\tau,\oplus)$ be a topological gyrogroup and $H$ a closed strongly
$L$-subgyrogroup of $G$. Then the natural homomorphism $\pi$ from a
topological gyrogroup $G$ to its quotient topology on $G/H$ is an open and
continuous mapping, and $G/H$ is a homogeneous $T_{1}$-space.

###### Proof.

Since a strongly $L$-subgyrogroup of a gyrogroup $G$ is an $L$-subgyrogroup,
$\pi:G\rightarrow G/H$ is open and continuous by [5, Theorem 3.7].

Let us now prove the homogeneity of $G/H$. For any $a\in G$, define a mapping
$h_{a}$ of $G/H$ to itself by the rule $h_{a}(x\oplus H)=a\oplus(x\oplus H)$.
Since $a\oplus(x\oplus H)=(a\oplus x)\oplus H\in G/H$ by Proposition 3.4, this
definition is correct. Since $G$ is a gyrogroup, the mapping $h_{a}$ is
evidently a bijection of $G/H$ onto $G/H$. In fact, $h_{a}$ is a
homeomorphism. This can be seen from the following argument.

Take any $x\oplus H\in G/H$ and any open neighbourhood $U$ of $0$. Then
$\pi((x\oplus U)\oplus H)$ is a basic neighbourhood of $x\oplus H$ in $G/H$.
Similarly, the set $\pi(a\oplus((x\oplus U)\oplus H))$ is a basic
neighbourhood of $a\oplus(x\oplus H)$ in $G/H$. Since, obviously,
$h_{a}(\pi((x\oplus U)\oplus H))=\pi(a\oplus((x\oplus U)\oplus H))$, it easily
follows that $h_{a}$ is a homeomorphism. Now, for any given $x\oplus H$ and
$y\oplus H$ in $G/H$, we can take $a=y\boxminus x$. Then $h_{a}(x\oplus
H)=a\oplus(x\oplus H)=(a\oplus x)\oplus H=((y\boxminus x)\oplus x)\oplus
H=y\oplus H$, by Proposition 3.4. Hence, the quotient space $G/H$ is
homogeneous. It is a $T_{1}$-space, since all cosets $x\oplus H$ are closed in
$G$ and the mapping $\pi$ is quotient. ∎

In [4, Theorem 3], it was proved that every $T_{1}$ topological gyrogroup is
regular. In fact, this result holds for the quotient space $G/H$ of a
topological gyrogroup $G$ with respect to a strongly $L$-subgyrogroup $H$ of
$G$.

######  Lemma 3.6.

Suppose that $(G,\tau,\oplus)$ is a topological gyrogroup, $H$ is a closed
strongly $L$-subgyrogroup of $G$, it is the natural quotient mapping of $G$
onto the quotient space $G/H$, and let $U$ and $V$ be open neighbourhoods of
the neutral element 0 in $G$ such that $\ominus V\oplus V\subseteq U$. Then
$\overline{\pi(V)}\subseteq\pi(U)$.

###### Proof.

Take any $x\in G$ such that $\pi(x)\in\overline{\pi(V)}$. Since $V\oplus x$ is
an open neighbourhood of $x$ and the mapping $\pi$ is open, $\pi(V\oplus x)$
is an open neighbourhood of $\pi(x)$. Therefore, $\pi(V\oplus
x)\cap\pi(V)\neq\emptyset$. It follows that, for some $a\in V$ and $b\in V$,
we have $\pi(a\oplus x)=\pi(b)$, that is, $a\oplus x=b\oplus h$, for some
$h\in H$. Hence, $x=\ominus a\oplus(b\oplus h)=(\ominus a\oplus
b)\oplus\text{gyr}[\ominus a,b]h=(\ominus a\oplus b)\oplus h_{1}$, for some
$h_{1}\in H$. Since $\ominus a\oplus b\in\ominus V\oplus V\subseteq U$, we get
$x\in U\oplus H$. Therefore, $\pi(x)\in\pi(U\oplus H)=\pi(U).$ ∎

######  Theorem 3.7.

For any topological gyrogroup $(G,\tau,\oplus)$ and any closed strongly
$L$-subgyrogroup $H$ of $G$, the quotient space $G/H$ is regular.

###### Proof.

Let $\pi$ be the natural quotient mapping of $G$ onto the quotient space
$G/H$, and let $W$ be an arbitrary open neighbourhood of $\pi(0)$ in $G/H$,
where $0$ is the neutral element of $G.$ By the continuity of $\pi$, we can
find an open neighbourhood $U$ of 0 in $G$ such that $\pi(U)\subseteq W$.
Since $G$ is a topological gyrogroup, we can choose an open neighbourhood $V$
of 0 such that $\ominus V\oplus V\subseteq U$. Then, by Lemma 3.6,
$\overline{\pi(V)}\subseteq\pi(U)\subseteq W$. Since $\pi(V)$ is an open
neighbourhood of $\pi(0)$, the regularity of $G/H$ at the point $\pi(0)$ is
verified. Now it follows from the homogeneity of $G/H$ that the space $G/H$ is
regular. ∎

Here is an obvious statement which is sometimes quite useful.

######  Proposition 3.8.

Suppose that $(G,\tau,\oplus)$ is a topological gyrogroup, $H$ is a closed
strongly $L$-subgyrogroup of $G$, $\pi$ is the natural quotient mapping of $G$
onto the quotient space $G/H$, $a\in G$, $\lambda_{a}$ is the left translation
of $G$ by $a$ (that is, $\lambda_{a}(x)=a\oplus x$, for each $x\in G$), and
$h_{a}$ is the left translation of $G/H$ by $a$ (that is, $h_{a}(x\oplus
H)=a\oplus(x\oplus H)$, for each $x\oplus H\in G/H$). Then $h_{a}$ is
homeomorphisms of $G/H$, and $\pi\circ\lambda_{a}=h_{a}\circ\pi$.

######  Theorem 3.9.

Suppose that $(G,\tau,\oplus)$ is a topological gyrogroup, $H$ is a closed
strongly $L$-subgyrogroup of $G$, $X$ is a subgyrospace of $G$, $\pi$ is the
natural homomorphism of $G$ onto the quotient space $G/H$, and $Y=\pi(X)$.
Suppose also that the space $H$ and the subspace $Y$ of $G/H$ are first-
countable. Then $X$ is also first-countable.

###### Proof.

By Proposition 3.8, we can assume that the neutral element 0 of $G$ is in $X$
and, for the same reason, it suffices to verify that $X$ is first-countable at
0. Let us fix a sequence of symmetric open neighbourhoods $W_{n}$ of 0 in $G$
such that $W_{n+1}\oplus W_{n+1}\subseteq W_{n}$, for each $n\in\omega$, and
$\\{W_{n}\cap H:n\in\omega\\}$ is a base for the space $H$ at 0. We also fix a
sequence of open neighbourhoods $U_{n}$ of 0 in $G$ such that
$\\{\pi(U_{n})\cap Y:n\in\omega\\}$ is a base for $Y$ at $\pi(0)$. Now put
$B_{i,j}=W_{i}\cap U_{j}\cap X$, for $i,j\in\omega$. To finish the proof, it
suffices to establish the following:

Claim. The family $\eta=\\{B_{i,j}:i,j\in\omega\\}$ is a base for $X$ at 0.

Clearly, each $B_{i,j}$ is open in $X$ and contains 0. Now take any open
neighbourhood $O$ of 0 in $G$. Let us show that some element of $\eta$ is
contained in $O$. There exists an open neighbourhood $V$ of 0 in $G$ such that
$V\oplus V\subseteq O$. Choose $m\in\omega$ such that $W_{m}\cap H\subseteq
V$. Further, there exists $k\in\omega$ to such that

$\pi(U_{k})\cap Y\subseteq\pi(V\cap W_{m+1}).$

Let us verify that $B_{m+1,k}\subseteq O.$ Take any $z\in
B_{m+1,k}=W_{m+1}\cap U_{k}\cap X$. Then $z\in U_{k}\cap X\subseteq(V\cap
W_{m+1})\oplus H$, since $\pi(z)\in\pi(U_{k})\cap Y\subseteq\pi(V\cap
W_{m+1})$. However, $z$ does not belong to $W_{m+1}\oplus(G\setminus W_{m})$,
since $W_{m+1}\oplus W_{m+1}\subseteq W_{m}$ and $z\in W_{m+1}=\ominus
W_{m+1}$. Hence, $z\in(V\cap W_{m+1})\oplus(H\cap W_{m})$. Since $W_{m}\cap
H\subseteq V,$ we conclude that $z\in V\oplus V\subseteq O$. Thus,
$B_{m+1,k}\subseteq O$, and $\eta$ is a base for $X$ at 0\. Since $\eta$ is
countable, it follows that $X$ is first-countable at 0. ∎

######  Corollary 3.10.

Suppose that $(G,\tau,\oplus)$ is a topological gyrogroup and $H$ is a closed
strongly $L$-subgyrogroup of $G$. If the spaces $H$ and $G/H$ are first-
countable, then the space $G$ is also first-countable.

##  4\. Quotients with respect to locally compact subgyrogroups

In this section, we will establish that for a locally compact strongly
$L$-subgyrogroup $H$ in topological gyrogroup $G$, the natural quotient map
$\pi$ of $G$ onto the quotient space $G/H$ has some good properties locally.
This will lead us to some interesting results about how the properties of $G$
depend on the properties of $G/H$ when $H$ is locally compact.

The following result improves [7, Theorem 3.3].

######  Theorem 4.1.

Suppose that $(G,\tau,\oplus)$ is a topological gyrogroup, $H$ is a closed
strongly $L$-subgyrogroup of $G$, $P$ is a closed symmetric subset of $G$ such
that $P$ contains an open neighbourhood of the neutral element $0$ in $G$, and
that $\overline{P\oplus(P\oplus P)}\cap H$ is compact. Let $\pi:G\rightarrow
G/H$ be the natural quotient mapping of $G$ onto the quotient space $G/H$.
Then the restriction $f$ of $\pi$ to $P$ is a perfect mapping of $P$ onto the
subspace $\pi(P)$ of $G/H$.

###### Proof.

Clearly, $f$ is continuous. First, we claim that $f^{-1}(f(a))$ is compact for
each $a\in P$. Indeed, from the definition of $f$, we have
$f^{-1}(f(a))=(a\oplus H)\cap P$. Since the subspace $(a\oplus H)\cap P$ and
$H\cap(\ominus a)\oplus P)$ are homeomorphic, thus both of them are closed in
$G$, since $H$ is closed. Since $\ominus a\in\ominus P=P$, we have
$H\cap((\ominus a)\oplus P)\subseteq H\cap(P\oplus
P)\subseteq\overline{P\oplus(P\oplus P)}\cap H$. Hence, $H\cap((\ominus
a)\oplus P)$ is compact and so is the set $f^{-1}(f(a))$.

Next we prove that the mapping $f$ is closed. Let us fix any closed subset $M$
of $P$, and let $a$ be a point of $P$ such that $f(a)\notin f(M)$. Then
$f^{-1}(f(a))\cap M=\emptyset$, i.e., $((a\oplus H)\cap P)\cap M=\emptyset$.
In particular, $(a\oplus H)\cap\overline{P\oplus P}\cap M=\emptyset$, since
$M\subseteq P$ and $P$ contains $0$. Clearly, the subspace $(a\oplus
H)\cap\overline{P\oplus P}$ and $H\cap\overline{\ominus a\oplus(P\oplus
P)}\subseteq H\cap\overline{P\oplus(P\oplus P)}$ are homeomorphic. Thus from
the compactness of $H\cap\overline{P\oplus(P\oplus P)}$ it follows that
$(a\oplus H)\cap\overline{P\oplus P}$ is compact. Since $(a\oplus
H)\cap\overline{P\oplus P}\cap M=\emptyset$ and $M$ is closed, there exists an
open neighbourhood $W$ of $0$ in $G$ such that $W\oplus((a\oplus
H)\cap\overline{P\oplus P})\cap M=\emptyset$.

Note that the set $\pi(W\oplus a)$ is an open neighborhood of $f(a)$ in $G/H$,
since the quotient mapping $\pi$ is open and $W\oplus a$ is an open
neighbourhood of $a$. According to assumption we can take $W\subseteq P$ such
that $\ominus W=W$. We claim that $\pi(W\oplus a)\cap f(M)=\emptyset$, which
implies that $f(a)\notin\overline{f(M)}$. That means that $f(M)$ is closed in
$f(P)$.

If $\pi(W\oplus a)\cap f(M)\neq\emptyset$, then we have that
$\pi^{-1}(\pi(W\oplus a))\cap\pi^{-1}(f(M))\neq\emptyset$, i.e., $(W\oplus
a)\oplus H\cap M\oplus H\neq\emptyset$. Thus there are $y\in W$ and $m\in M$
such that $(y\oplus a)\oplus H=m\oplus H$. Since $H$ is a strongly
$L$-subgyrogroup, we have that $y\oplus(a\oplus H)=m\oplus H$ by Proposition
3.4. Then $a\oplus h=\ominus y\oplus m\in\overline{P\oplus P}$, for some $h\in
H$, since $\ominus y\in\ominus W=W\subseteq P$ and $m\in M\subseteq P$.
Besides, $a\oplus h\in a\oplus H$. Hence, $a\oplus h\in(a\oplus
H)\cap\overline{P\oplus P}$ and $m=y\oplus(a\oplus h)\in W\oplus((a\oplus
H)\cap\overline{P\oplus P})$. Thus, $m\in M\cap W\oplus((a\oplus
H)\cap\overline{P\oplus P})$, which contradicts $W\oplus((a\oplus
H)\cap\overline{P\oplus P})\cap M=\emptyset$. ∎

The following result improves [7, Theorem 3.4].

######  Theorem 4.2.

Suppose that $(G,\tau,\oplus)$ is a topological gyrogroup, $H$ is a locally
compact strongly $L$-subgyrogroup of $G$, and $\pi:G\rightarrow G/H$ is the
natural quotient mapping of $G$ onto the quotient space $G/H$. Then there
exists an open neighborhood $U$ of the identity element $0$ such that
$\pi(\overline{U})$ is closed in $G/H$ and the restriction of $\pi$ to
$\overline{U}$ is a perfect mapping from $\overline{U}$ onto the subspace
$\pi(\overline{U})$.

###### Proof.

First, $H$ is closed in $G$ by Proposition 2.12. Since $H$ is locally compact,
there exists an open neighbourhood $V$ of 0 in $G$ such that $\overline{V\cap
H}$ is compact. By the regularity of $G$, we can choose an open neighborhood
$W$ of 0 such that $\overline{W}\subseteq V$. Hence $\overline{W}\cap H$ is
compact. Let $U_{0}$ be an arbitrary symmetric open neighborhood of 0 such
that $U_{0}\oplus(U_{0}\oplus U_{0})\subseteq W$. By the joint continuity, we
have
$\overline{U_{0}}\oplus(\overline{U_{0}}\oplus\overline{U_{0}})\subseteq\overline{U_{0}\oplus(U_{0}\oplus
U_{0})}\subseteq\overline{W}$. Then the set $P=\overline{U_{0}}$ satisfies all
restrictions on $P$ in Theorem 4.1. It follows from Theorem 4.1 that the
restriction of $\pi$ to $P$ is a perfect mapping from $P$ onto the subspace
$\pi(P)$.

Since $\pi$ is an open mapping, the set $\pi(U_{0})$ is open in $G/H$. Since
$G/H$ is regular by Theorem 3.7, there exists an open neighborhood $V_{0}$ of
$\pi(0)$ in $G/H$ such that $\overline{V_{0}}\subseteq\pi(U_{0})$. Hence
$U=\pi^{-1}(V_{0})\cap U_{0}$ is an open neighborhood of 0 contained in $P$
such that the restriction $f$ of $\pi$ to $\overline{U}$ is a perfect mapping
from $\overline{U}$ onto the subspace $\pi(\overline{U})$. Furthermore,
$\pi(\overline{U})$ is closed in $\pi(P)$, and
$\pi(\overline{U})\subseteq\overline{V_{0}}\subseteq\pi(U_{0})\subseteq\pi(P)$.
Then $\pi(\overline{U})$ is closed in $\overline{V_{0}}$, so that
$\pi(\overline{U})$ is closed in $G/H$. ∎

Recall that a regular closed set in a space is the closure of an open subset
of this space.

######  Corollary 4.3.

Assume that $\mathscr{P}$ is a topological property preserved by preimages of
spaces under perfect mappings (in the class of completely regular spaces) and
also inherited by regular closed sets. Assume further that $(G,\tau,\oplus)$
is a topological gyrogroup, $H$ is a locally compact strongly $L$-subgyrogroup
of $G$, and the quotient space $G/H$ has the property $\mathscr{P}$. Then
there exists an open neighborhood $\overline{U}$ of the identity element 0
such that $\overline{U}$ has the property $\mathscr{P}$.

###### Proof.

This is an immediate corollary from Theorem 4.2. ∎

Given a space $X$ and a property $\mathscr{P}$, if each point $x$ of $X$ has
an open neighborhood $U(x)$ such that $\overline{U(x)}$ has $\mathscr{P}$,
then we say that $X$ has the property $\mathscr{P}$ $locally$ [3]. It is well
known that local compactness, countable compactness, pseudocompactness,
paracompactness, the Lindel$\ddot{o}$f property, $\sigma$-compactness,
$\check{C}$ech-completeness, the Hewitt-Nachbin completeness, and the property
of being a $k$-space are all inherited by regular closed sets and preserved by
perfect preimages, see [10, Sections 3.7, 3.10, 3.11]. This observation proves
the following statement:

######  Corollary 4.4.

Suppose that $G$ is a topological gyrogroup, and that $H$ is a locally compact
strongly $L$-subgyrogroup of $G$ such that the quotient space $G/H$ has some
of the following properties:

  1. (1)

$G/H$ is locally compact;

  2. (2)

$G/H$ is locally countably compact;

  3. (3)

$G/H$ is locally pseudocompact;

  4. (4)

$G/H$ is locally $\sigma$-compact;

  5. (5)

$G/H$ is locally paracompact;

  6. (6)

$G/H$ is locally Lindel$\ddot{o}$f;

  7. (7)

$G/H$ is locally $\check{C}$ech-complete; and

  8. (8)

$G/H$ is locally realcompact,

then $G$ also has the same property.

######  Corollary 4.5.

Let $(G,\tau,\oplus)$ be a topological gyrogroup, and let $H$ be a locally
compact strongly $L$-subgyrogroup. If the quotient space $G/H$ is a $k$-space,
then $G$ is also a $k$-space.

###### Proof.

This statement follows from Corollary 4.3, since the property of being a
$k$-space is invariant under taking perfect preimages and a locally $k$-space
is a $k$-space [10, Sections 3.3]. ∎

Recall that a space $X$ is said to be zero-dimensional if it has a base
consisting of sets which are both open and closed in $X$.

The following result improves [7, Theorem 3.5].

######  Theorem 4.6.

Suppose that $G$ is a zero-dimensional topological gyrogroup and that $H$ is a
locally compact strongly $L$-subgyrogroup of $G$. Then the quotient space
$G/H$ is also zero-dimensional.

###### Proof.

Let $\pi:G\rightarrow G/H$ be the natural quotient mapping of $G$ onto the
quotient space $G/H$. According to Theorem 4.2, we can fix an open
neighbourhood $U$ of the neutral element 0 of $G$ such that
$\pi(\overline{U})$ is closed in $G$ and the restriction of $\pi$ to
$\overline{U}$ is a perfect mapping of $U$ onto the subspace
$\pi(\overline{U})$. Take any open neighbourhood $W$ of $\pi(0)$ in $G/H$.
Since the space $G$ is zero-dimensional, we can fix an open and closed
neighbourhood $V$ of 0 such that $V\subseteq U\cap\pi^{-1}(W)$. Then $\pi(V)$
is an open subset of $G/H$, since the mapping $\pi$ is open. On the other
hand, $\pi(V)$ is closed in $G/H$, since the restriction of $\pi$ to
$\overline{U}$ is a closed mapping and $\pi(\overline{U})$ is closed in $G/H$.
Clearly, $\pi(V)\subseteq W$. Hence, $G/H$ is zero-dimensional. ∎

We recall that the tightness of a space $X$ is the minimal cardinal
$\tau\geq\omega$ with the property that for every point $x\in X$ and every set
$P\subseteq X$ with $x\in\overline{P}$, there exists a subset $Q$ of $P$ such
that $|Q|\leq\tau$ and $x\in\overline{Q}$.

######  Lemma 4.7.

[3, Proposition 4.7.16] Suppose that $f:X\rightarrow Y$ is a closed continuous
mapping of a regular space $X$ onto a space $Y$ of countable tightness.
Suppose further that the tightness of every fiber $f^{-1}(y)$, for $y\in Y$ ,
is countable. Then the tightness of $X$ is also countable.

The following result improves [7, Theorem 3.10].

######  Theorem 4.8.

Suppose that $G$ is a topological gyrogroup, and that $H$ is a locally compact
metrizable strongly $L$-subgyrogroup of $G$ such that tightness of the
quotient space $G/H$ is countable. Then the tightness of $G$ is also
countable.

###### Proof.

It follows from Theorem 4.2 that there exists an open neighbourhood $U$ of the
neutral element 0 in $G$ such that $\overline{U}$ is a preimage of a space of
countable tightness under a perfect mapping with metrizable fibers. Then, by
Lemma 4.7, the tightness of $\overline{U}$ is also countable. Since $U$ is a
non-empty open subset of the homogeneous space $G$, the tightness of $G$ is
countable. ∎

##  5\. Acknowledgements

We wish to thank the reviewers for many valuable advices, and all their
efforts in order to improve the paper.

## References

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# Modeling Local Search Metaheuristics Using Markov Decision Processes

Rubén Ruiz-Torrubiano<EMAIL_ADDRESS>IMC Krems University of Applied
Sciences, Piaristengasse 1, Krems, 3500, Austria

###### Abstract

Local search metaheuristics like tabu search or simulated annealing are
popular heuristic optimization algorithms for finding near-optimal solutions
for combinatorial optimization problems. However, it is still challenging for
researchers and practitioners to analyze their behaviour and systematically
choose one over a vast set of possible metaheuristics for the particular
problem at hand. In this paper, we introduce a theoretical framework based on
Markov Decision Processes (MDP) for analyzing local search metaheuristics.
This framework not only helps in providing convergence results for individual
algorithms, but also provides an explicit characterization of the exploration-
exploitation tradeoff and a theory-grounded guidance for practitioners for
choosing an appropriate metaheuristic for the problem at hand. We present this
framework in detail and show how to apply it in the case of hill climbing and
the simulated annealing algorithm.

###### keywords:

metaheuristics, combinatorial optimization, Markov Decision Process,
stochastic control problems

##  1 Introduction

Metaheuristics are high level algorithmic frameworks that provide guidance or
strategies for developing heuristic optimization algorithms [26]. Frequently,
these strategies use concepts from other disciplines as their underlying
inspiration, like the evolution process in biology, the process of cooling a
metallic alloy, the cooperative behaviour of flocks of birds, and many more
[25]. Despite being heuristic in nature, so that in general there is no
general theoretical guarantee that the algorithm will find the global optimum,
metaheurstics have demonstrated a tremendous success in practice, being
effectively used in applications like logistics (e.g. vehicle routing problem
[20]), network design [11], scheduling [23, 18, 10, 2], finance (e.g.
portfolio optimization [7]) and many others.

In general, it can be distiguished between local search (or single-solution)
metaheuristics like simulated annealing (SA) [17], tabu search (TS) [12],
variable neighborhood search (VSN) [22] and large neighborhood search (LNS)
[27], on the one side, and population-based metaheuristics on the other, like
genetic algorithms (GA) [14] particle swarm optimization (PSO) [16] or
estimation of distribution algorithms (EDA) [19]. Local search metaheuristics
typically start with a candidate solution $x_{0}$ (not necesarilly feasible)
and apply stochastic operators to select the next candidate solution from a
suitable chosen neighborhood or pool of individuals. By iteratively repeating
this process, local search metaheuristics are able to find near-optimal
solutions to the problem at hand efficiently, even if the search space is very
large. By contrast, population-based metaheuristics start with a group of
candidate solutions (often referred to as population)
$P=\\{x_{1},x_{2},\dots,x_{n}\\}$ and apply different types of operators to
generate new candidate solutions, updating the population in the process.

It is well acknowledged that there is a lack of strong theoretical frameworks
for understanding and analyzing the behaviour of metaheuristic algorithms [21,
4]. While theoretical results and convergence analysis do exist for individual
algorithms like genetic algorithms and simulated annealing [28, 29], existing
frameworks are either focused on one type of analysis (like convergence
analysis [21]), or they provide only weak results [4]. Moreover, they provide
no theory-grounded guidance for practitioners on which algorithm might be most
appropriate a particular task.

In this paper, we try to fill this gap by proposing a novel framework based on
Markov Decision Processes (MDP) as the underlying model for metaheuristic
algorithms. Specifically, in this study we focus on local search
metaheuristics, leaving population-based metaheuristics for future work. We
show that particular instances of local search metaheuristics can be modeled
as policies of a suitably defined MDP and, based on that, we derive different
coefficients to model their convergence and exploration-exploitation
behaviour. By introducing different parameters into these bounds and assigning
them a practical meaning, we show which properties have to be taken into
account for a given metaheuristic (policy) to perform well on a particular
class of problems.

This paper is structured as follows. In Section 2, we review previous relevant
work. Section 3 presents the MDP framework for local search metaheuristics and
applies it to the hill climbing algorithm as a first example. In Section 4 we
apply this framework to simulated annealing, showing how the relevant measures
of convergence and exploration-exploitation can be explicitly calculated.
Section 5 concludes the paper and outlines future work.

##  2 Previous Work

Mathematical analysis of optimization metaheuristics has been done previously
in the literature in a variety of ways. The work on the schema theorem by
Holland [13] can be considered as one of the first attempts at mathematical
formalization of metaheuristic algorithms. This result applied only to simple
GAs with one-point crossover and binary encoding. One of the most popular
variants of mathematical analysis was sparheaded by the Markov chain analysis
of genetic algorithms (GA) which was mostly done in the 1990s. For instance,
in early work the authors of [8] proved that GAs with an elitist selection
scheme converge to the optimum with probability 1. Suzuki [28, 29] proved
theoretical results based on Markov chain analysis for GAs focusing on
mutation probabilities and developed an early comparison with simulated
annealing (SA). In [5], the authors applied a non-stationary Markov model for
the convergence analysis of genetic algorithms. One of the main drawbacks of
using Markov chain theory for metaheuristic analysis is that Markov chains are
not expressive enough for handling important aspects of the search process in
metaheuristics. Critically, the reward structure and the different operators
used in the algorithm can only be expressed implicitly via the transition
matrix.

More recent approaches in the litereature use other models, like the
evolutionary Turing machine [9] and the nested partition algorithm [4].
However, most of these results focus on convergence analysis and do not handle
the exploration-exploitation dilemma explicitly. This question was addressed
in [1] by experimental means, but no theoretical analysis was provided.
Similarly, a case study based on elitist evolutionary algorihtms (concretely,
random univariate search and evolutionary programming) was presented in [3]
using a specially-tailored probabilistic framework. An extensive survey on the
exploration-exploitation tradeoff in metaheuristics is given in [30]. In [6],
a mathematical characterization of the exploration-exploitation tradeoff is
given in the form of the optimal contraction theorem. However, their
definition of exploration leaves out the possibility that information
collected during the search can be used for exploring other areas of the
search space.

In contrast to previous approaches, our framework applies Markov decision
processes (MDP) to the problem of modeling the convergence behaviour and the
exploration-exploitation tradeoff of local search metaheuristics. We consider
MDP to be a more suitable framework than the previously presented approaches
since the reward structure of the general search problem can be represented
explicitly. Moreover, transformations and sequences of candidate solutions can
be modeled explicitly by means of actions, which in MDP are naturally linked
to rewards. We therefore see a particular metaheuristic algorithm as a
stochastic agent that makes decisions about which candidate solution to visit
next based on the information obtained so far and the reward obtained. We
explore the implications of this modeling choice and we show in the following
sections that the convergence and the exploration-exploitation analysis of
metaheuristic algorithms can be calculated explicitly.

##  3 Markov Decision Processes for Local Search Metaheuristics

In this section we develop the mathematical machinery needed to define a MDP
model for local search metaheuristics. Our main insight is that we can model
any local search metaheuristic as a stochastic agent that executes some policy
on a suitably defined MDP. Critically, we note that even if the goal of a
specific metaheuristic is to find the optimum of an objective function, the
algorithm itself proceeds by maximizing the total reward over a (possibly)
infinite time horizon. Clearly, the rewards are directly connected with the
best values found so far of the objective function, but the stochastic agent
needs to proceed by gathering information about promising directions in the
search space (exploration-exploitation dilemma). Therefore, the exploration of
the search space is guided mainly by the maximization of the total reward.

###  3.1 Definitions

We start by introducing some preliminaries in the form of notation and
definitions. Then we specify the model in detail and describe how to define
metaheuristic policies using this framework. In the following, we will mainly
follow the conventions and definitions used in [15].

######  Definition 1 (Markov Decision Process).

A Markov Decision Process (MDP) is a tuple
$\langle\mathcal{S},\mathcal{A},\mathcal{P},\mathcal{R},\alpha\rangle$, where:

  * •

$\mathcal{S}$ is a set of states.

  * •

$\mathcal{A}(i)$ is a set of actions available in state $i\in\mathcal{S}$.

  * •

$\mathcal{P}(a)=\\{p_{ij}(a)\\}_{ij}$ is the state transition matrix, given
action $a$, where $p_{ij}(a)=\mathbb{P}\\{S_{t+1}=j|S_{t}=i,A_{t}=a\\}$.

  * •

$\mathcal{R}(a)$ is a reward function,
$\mathcal{R}_{s}(a)=\mathbb{E}[r_{t+1}|S_{i}=s,A_{t}=a]$.

  * •

$\alpha\in[0,1]$ is a discount factor.

In words, a MDP is a stochastic decision model where an agent makes decisions
on each time step about which action $a$ to take from a predefined action set
$\mathcal{A}(i)$, which might be different for each state
$i\in\mathcal{A}(i)$. Most importantly, the transition probabilities to a new
state $S_{t+1}$ depend only on the last state $S_{t}$, and not on the previous
history (i.e. the previously visited states). This is known as the _Markov
property_.

######  Definition 2.

A _policy_ $R$ is a decision rule $R=(\pi^{1},\pi^{2},\dots,\pi^{t},\dots)$,
that can be expressed as a probability distribution of the actions, given
states:

$\pi_{ia}^{t}=\mathbb{P}\\{A_{t}=a|S_{t}=s\\}$

If the policy $R$ is such that there exist an action $a$ such that
$\pi_{ia}^{t}=1$ for every state $i\in\mathcal{S}$, we say the the policy is
_deterministic_ , otherwise, we call the policy _stochastic_. If the policy
does not depend on time, we call this policy _stationary_ and simply write
$\pi_{ia}$.

The natural problem that now arises is how find policies that maximize the
rewards obtained in some precise sense. This is captured by means of an
utility function, which might be e.g. maximize the total expected or average
reward. A MDP can be classified according to its time horizon in finite or
infinite time MDP. Likewise, we can also distinguish between discrete and
continuous time MDPs. For local search metaheuristics, we focus on the MDP
formulation where:

  1. (a)

The time is discrete and the time horizon is infinite.

  2. (b)

The utility function is the total expected reward.

Regarding condition (a), we note that clearly the execution time of a
metaheuristic is always finite, but this finite time cannot be fixed
beforehand (except for the class of metaheuristics that are run for a fixed
number of iterations). This is modeled by having terminal states with no
outgoing transitions (i.e. the probability of transitioning to any other state
is 0, and $\mathcal{A}(i)=\emptyset$ for all terminal states
$i\in\mathcal{S}$). We argue for condition (b) that, in general, the main
strategy used by all metaheuristics is to search for the optimum _indirectly_
by exploring the search space $\mathcal{X}$ and exploiting promising regions
where an optimum $x^{*}\in\mathcal{X}$ is supposed to be. However, this is
done by _sampling_ from the search space in some specific sense and evaluating
the objective function $f$ on that sample, thus collecting information about
the target function value. If we interpret this sampling and evaluating
procedure as collecting _rewards_ , the agent (i.e. the metaheuristic) can be
validly interpreted as trying to maximize the total reward obtained.

###  3.2 Optimal policies

In general, we therefore model local search metaheuristic algorithms by means
of policies $R$. We define the reward of such a policy as the total
(discounted) expected reward:

$v_{i}^{\alpha}(R)=\sum_{t=1}^{\infty}\mathbb{E}_{i,R}\\{\alpha^{t-1}r_{t}\\}$
(1)

where $r_{t}$ denotes the collected actual reward at time step $t$, and the
expectation is over policy $R$, given initial state $i\in\mathcal{S}$, and
$\alpha$ is a discount factor that regulates the value of future rewards. In
the following, we will asume $\alpha=1$ so we can write:

$v_{i}(R)=\sum_{t=1}^{\infty}\sum_{j,a}\mathbb{P}\\{S_{t+1}=j|S_{t}=i,A_{t}=a\\}r_{j}(a)$
(2)

Let $\\{P(\pi^{t})\\}_{ij}$ denote the transition matrix of policy $R$, and
$\\{r(\pi^{r})\\}_{i}$ its reward vector. We have

$\displaystyle\\{P(\pi^{t})\\}_{ij}$
$\displaystyle=\sum_{a\in\mathcal{A}}p_{ij}^{t}(a)\pi_{ia}^{t}\text{ for all
}i,j\in\mathcal{S}$ (3) $\displaystyle\\{r(\pi^{r})\\}_{i}$
$\displaystyle=\sum_{a\in\mathcal{A}}r_{i}^{t}(a)\pi_{ia}^{t}\text{ for all
}i\in\mathcal{S}$ (4)

The reward of policy $R$ can be now expressed based on the transition
probabilities as follows:

$v^{\alpha}(R)=\sum_{t=1}^{\infty}\alpha^{t-1}P(\pi^{1})P(\pi^{2})\dots
P(\pi^{t-1})r(\pi^{t})$ (5)

######  Definition 3.

We call a policy $R_{*}$ _optimal_ if it is the best achievable policy
considering the reward function, i.e.

$v^{\alpha}(R_{*})=\operatorname{sup}_{R}v^{\alpha}(R)$

###  3.3 Local search metaheuristics as policies

We now formally represent an arbitrary local search metaheuristic as a policy
in a MDP as follows. In this study, we focus on binary strings of length $n$
as candidate solutions, therefore $\mathcal{X}\subseteq\\{0,1\\}^{n}$. From
now on, we denote by $\mathbf{x}\in\\{0,1\\}^{n}$ a binary vector which is a
candidate solution for the problem at hand, which might be feasible or not
(i.e. it can be that, during the search process, there is an $\mathbf{x}_{t}$
such that $\mathbf{x}_{t}\notin\mathcal{X}$). In the following, we will assume
a maximization problem, i.e. find $\mathbf{x}^{*}$ such that
$\mathbf{x}^{*}=\operatorname{argmax}_{\mathbf{x}\in\mathcal{X}}f(\mathbf{x})$.

######  Definition 4.

We define a _neighborhood_ $\mathcal{N}_{H}(\mathbf{x})$ as the set of
candidate solutions $\mathbf{x}^{\prime}\in\mathcal{X}$ which are neighbors of
$\mathbf{x}$ according to a criterium $H$, i.e.
$\mathcal{N}_{H}(\mathbf{x})=\\{x^{\prime}\in\mathcal{X}\,|\,H(\mathbf{x}^{\prime};\mathbf{x})=1\\}$.

Let $\mathbf{x}\in\mathcal{X}$ and
$\mathbf{x}^{\prime}\in\mathcal{N}_{H}(\mathbf{x})$. We denote by
$\mathbf{x}\mapsto\mathbf{x}^{\prime}$ the operator that transforms
$\mathbf{x}$ into $\mathbf{x}^{\prime}$. For example, the neighborhood of all
binary strings located at Hamming distance one from $\mathbf{x}$, would be
defined by
$\mathcal{N}_{H_{D}}(\mathbf{x})=\\{x^{\prime}\in\mathcal{X}\,|\,H_{D}(\mathbf{x}^{\prime};\mathbf{x})=1\\}$.
In this case, $\mathbf{x}\mapsto\mathbf{x}^{\prime}$ denotes the operator that
flips the one bit necessary to turn $\mathbf{x}$ into $\mathbf{x}^{\prime}$.

######  Definition 5 (Local search MDP).

Let our local search MDP be defined by:

  * •

$\mathcal{S}=\\{0,1,\dots,2^{n-1}\\}$. Any state $i\in\mathcal{S}$ can
trivially be converted into a candidate solution as the binary representation
of the integer $i$.

  * •

$\mathcal{A}(i)=\\{i\mapsto j\,|\,j\in\bigcup_{k}\mathcal{N}_{k}\\}$, the set
of all possible transformations in all neighborhoods applicable to the current
state $i\in\mathcal{S}$.

  * •

The transition probabilities, for all $i,j\in\mathcal{S}$,
$a\in\mathcal{A}(i)$:

$p_{ij}(a)=\begin{cases}1/|\mathcal{A}(i)|\text{, if }a=i\mapsto j\\\ 0\text{,
otherwise}\end{cases}$ (6)

  * •

The reward for state $j$ given action $a=i\mapsto j$ is defined by
$r_{i}(a)=f(j)-f(i)$, where $f$ is the objective function111Note the slight
abuse of notation here since $f$ is defined on $\mathcal{X}$ rather than
$\mathcal{S}$, but the conversion is trivial..

An instantiation of this MDP therefore starts at an arbitrary state
$i\in\mathcal{S}$ and visits a number of states according to the transition
probabilities before halting at time $T$. The algorithm terminates as soon as
it reaches a _terminal_ or _absorving_ state, i.e. a state $i\in\mathcal{S}$
where the transition probability of reaching any other state is 0
($p_{ij}(a)=0$ for all $i\neq j$, and $p_{ii}(a)=1$ otherwise). Note that the
rewards are chosen in such a way that, in general, they incentivize making
progress towards the optimum $\mathbf{x}^{*}$.

In this setting, we identify a policy $R$ with a particular local search
metaheuristic.

######  Example 1 (Hill climbing).

We define a policy $R_{HC}$ for the standard hill climbing algorithm as
follows:

  * •

$\mathcal{A}(i)=\\{i\mapsto j\,|\,j\in\mathcal{N}_{H_{D}}\\}$ as defined
before (we flip one bit), and

  * •

Let
$\mathcal{M}(i)=\\{j\in\mathcal{S}\,|\,j=\operatorname{argmax}f(j^{\prime}),a=i\mapsto
j,a\in\mathcal{A}(i)\\}$, then

$\pi_{ia}=\begin{cases}1/|\mathcal{M}(i)|\text{, if }j\in\mathcal{M}(i)\\\
0\text{, otherwise}\end{cases}$ (7)

Note that $R_{HC}$ is a deterministic policy except when $|\mathcal{M}(i)|>1$,
in which case ties are broken randomly.

In general, the MDP is defined in such a way that the balance between
exploration and exploitation becomes explicit in the definition of the policy.
Informally, the agent can either maximize the reward in the short run (thus
going into an exploitation phase), or temporarily accept negative rewards and
therefore going into a exploration phase. In the following definitions, we
define exploration and exploitation explicitly using these notions.

######  Definition 6 (Exploration and exploitation).

Let $a\in\mathcal{A}(i)$ be an action such that $a=i\mapsto j$. We say that
$a$ is an _exploration_ action if and only if $r_{j}(a)\leq 0$. Otherwise, we
call $a$ and _exploitation_ action.

######  Definition 7 (Exploration-exploitation function).

Let $\sigma:\mathcal{A}(i)\rightarrow\\{0,1\\}$ be a function such that
$\sigma(a)=1$ if and only if $a$ is an exploration action, and $\sigma(a)=0$
otherwise. We call $\sigma$ the _exploration-exploitation_ function.

We now state our main result which shows that any local search metaheuristic
can be represented by a policy where the balance between exploration and
exploitation becomes explicit.

######  Theorem 1 (Local search exploration-exploitation theorem).

Let $M$ be a local search MDP. For any local search metaheuristic $A$, there
exist a policy $R_{A}$ such that

$\pi_{ia}^{t}=\alpha_{i}^{A}(t)\mathbb{P}\\{\sigma(a)=1\,|\,S_{t}=i\\}+\beta_{i}^{A}(t)\mathbb{P}\\{\sigma(a)=0\,|\,S_{t}=i\\}$
(8)

###### Proof.

Let $\mathcal{M}_{t}^{+}(i)=\\{j\in\mathcal{S}\,|\,f(j)>f(i),a=i\mapsto
j,a\in\mathcal{A}(i),S_{t}=i\\}$ and
$\mathcal{M}_{t}^{-}(i)=\\{j\in\mathcal{S}\,|\,f(j)\leq f(i),a=i\mapsto
j,a\in\mathcal{A}(i),S_{t}=i\\}$ be the sets of improving and non-improving
states given action $a$, respectively. We have:

$\displaystyle\pi_{ia}^{t}$
$\displaystyle=\mathbb{P}\\{A_{t}=a\,|\,S_{t}=i\\}$
$\displaystyle=\mathbb{P}\\{A_{t}=a,j\in\mathcal{M}_{t}^{+}(i)\,|\,S_{t}=i\\}+\mathbb{P}\\{A_{t}=a,j\in\mathcal{M}_{t}^{-}(i)\,|\,S_{t}=i\\}$
$\displaystyle=\frac{|\mathcal{M}_{t}(i)^{+}|}{|\mathcal{M}_{t}(i)^{+}|+|\mathcal{M}_{t}(i)^{-}|}\mathbb{P}\\{\sigma(a)=0\,|\,S_{t}=i\\}+$
$\displaystyle+\frac{|\mathcal{M}_{t}(i)^{-}|}{|\mathcal{M}_{t}(i)^{+}|+|\mathcal{M}_{t}(i)^{-}|}\mathbb{P}\\{\sigma(a)=1\,|\,S_{t}=i\\}$
(9)

Therefore,

$\alpha_{i}^{A}(t)=\frac{|\mathcal{M}_{t}(i)^{-}|}{|\mathcal{M}_{t}(i)^{+}|+|\mathcal{M}_{t}(i)^{-}|}\text{,
and
}\beta_{i}^{A}(t)=\frac{|\mathcal{M}_{t}(i)^{+}|}{|\mathcal{M}_{t}(i)^{+}|+|\mathcal{M}_{t}(i)^{-}|}$

∎

Theorem 1 provides hints to several interesting measures. The first one is a
measure that can be used for convergence analysis and can be defined as
follows:

$\gamma^{A}_{i}(t):=\cfrac{\beta_{i}^{A}(t)}{\alpha_{i}^{A}(t)}=\cfrac{|\mathcal{M}_{t}(i)^{+}|}{|\mathcal{M}_{t}(i)^{-}|}$
(10)

We call $\gamma^{A}_{i}(t)$ the _convergence coefficient_ of local search
metaheuristic $A$. We can now study the convergence behaviour of local search
metaheuristics according to the limit of $\gamma^{A}_{i}(t)$ as
$t\rightarrow\infty$. If $\lim_{t\rightarrow\infty}\gamma^{A}_{t}(t)=0$, this
means that algorithm $A$ has converged, since the improving set becomes zero
faster than the non-improving set.

The second interesting measure representes an explicit way of evaluating the
balance between exploration and exploitation that a specific local search
metaheuristic provides. We define

$\delta^{A}_{ia}(t):=\cfrac{\mathbb{P}\\{\sigma(a)=1\,|\,S_{t}=i\\}}{\mathbb{P}\\{\sigma(a)=0\,|\,S_{t}=i\\}}$
(11)

Averaging over all actions and times, we get

$\delta^{A}_{i}=\sum_{t=0}^{\infty}\sum_{j\in\mathcal{A}(i)}p_{ij}(a)\delta^{A}_{ja}(t)=\frac{1}{|\mathcal{A}(i)|}\sum_{j\in\mathcal{A}(i)}\sum_{t=0}^{\infty}\delta_{ja}^{A}(t)$
(12)

We call $\delta^{A}_{i}$ the _exploration-exploitation coefficient_ of local
search metaheuristic $A$ conditioned on state $i\in\mathcal{S}$. In vector
notation, we write $\delta^{A}$ and call it the _exploration-exploitation
coefficient vector_. In general, there can be components of $\delta^{A}$ that
converge, and others that diverge. Therefore, one possibility would be to
assert the general exploration-exploitation behaviour of $A$ based on
$\delta_{*}^{A}=\max_{i}\delta_{i}^{A}$ and investigate the individual
components of $\delta^{A}$ in case a more detailed analysis is needed.

We can now define for each metaheuristic a measure of its exploration-
exploitation balance as follows:

######  Definition 8.

We say that $A$ is a _balanced_ local search metaheuristic if
$\delta^{A}_{i}=C$ for a constant $C>0,C\in\mathbb{R}$. Otherwise, if
$\delta^{A}_{i}$ diverges to $\infty$, we say that $A$ is _exploration-
oriented_. Finally, if $\delta^{A}_{i}=0$ we say that $A$ is _exploitation-
oriented_.

Using our local search MDP, we have transformed the problem of convergence
analysis and determining the exploration-exploitation balance of any
metaheuristic into the quantitative problem of calculating $\delta^{A}_{i}$
and the limit of $\gamma^{A}_{t}(t)$ as $t\rightarrow\infty$ . Let’s apply
this to the hill climbing heuristic already mentioned before.

######  Example 2.

In the case of hill climbing, we have for every $i\in\mathcal{S}$:

$\displaystyle\alpha_{i}^{A}(t)$ $\displaystyle=0$
$\displaystyle\beta_{i}^{A}(t)$ $\displaystyle=1/|\mathcal{M}(i)|$
$\displaystyle\gamma_{i}^{A}(t)$ $\displaystyle=0$ (13)

Since $\lim_{t\rightarrow\infty}\gamma^{A}_{i}(t)=0,\forall i\in\mathcal{S}$,
we have that hill climbing always converges to a (possibly local) optimum.
Regarding exploration-exploitation behaviour, we have
$\delta_{ia}(t)=0,\forall a\in\mathcal{A}(i),\forall i\in\mathcal{S}$. We
therefore obtain that $\delta_{*}^{A}=0$ and hill climbing is exploitation-
oriented.

In the next section, we apply the machinery defined to the simulated annealing
(SA) metaheuristic.

##  4 Simulated Annealing

Simulated annealing (SA) [17] is a local search metaheuristic inspired by the
procedure of cooling a molten solid according to a predefined cooling
schedule. The main idea is to start with a high temperature $T$, and decrease
this temperature in subsequent iterations with the goal of reaching the
minimum value of an energy function $E$. Transitions from the current
candidate solution that improve the objective value are accepted with
probability 1. Otherwise, the solution is accepted with a probability that
depends on the magnitude of difference between the current best objective
value and the value of the new candidate solution. In the initial, high-
temperature phase, the probability of accepting worse solutions is high
(emphasizes exploration), whereas in later stages lower temperatures also
decrease this probability (emphasizes exploitation). More formally, this
probability can be written as:

$\mathbb{P}\\{\mathbf{x}_{t+1}=\mathbf{x}^{\prime}\,|\,\mathbf{x}_{t}=\mathbf{x}\\}=\begin{cases}1\text{,
if }E(\mathbf{x^{\prime}})<E(\mathbf{x})\\\
\exp\left(\cfrac{-\left(E(\mathbf{x})-E(\mathbf{x^{\prime}})\right)}{T}\right)\text{,
otherwise}\end{cases}$ (14)

In our case, since we are solving by convention maximization problems, $E=-f$.
Usually, a geometric cooling scheme is used: $T_{new}=\alpha T_{current}$,
where $\alpha\in[0,1)$. Let $T_{0}$ denote the initial temperature (which is a
free parameter of the algorithm). Then the temperature at time $t$ is given by
$T=\alpha^{t}T_{0}$.

We now define a policy $R_{SA}$ to model the behaviour of SA as a stochastic
decision agent as follows:

  * •

Let $\mathcal{M}_{t}^{+}(i)$ and $\mathcal{M}_{t}^{-}(i)$ be defined as in
Theorem 1.

  * •

Let’s consider action $a=i\mapsto j$ and state $i\in\mathcal{S}$. Using
Theorem 1 we obtain

$\displaystyle\pi^{t}_{ia}$
$\displaystyle=\mathbb{P}\\{A_{t}=a\,|\,S_{t}=i\\}$
$\displaystyle=\frac{|\mathcal{M}_{t}(i)^{+}|}{|\mathcal{M}_{t}(i)^{+}|+|\mathcal{M}_{t}(i)^{-}|}+\frac{|\mathcal{M}_{t}(i)^{-}|}{|\mathcal{M}_{t}(i)^{+}|+|\mathcal{M}_{t}(i)^{-}|}\exp\left(-\frac{f(j)-f(i)}{\alpha^{t}T_{0}}\right)$
(15)

Now, the convergence coefficient can be calculated as follows:

$\lim_{t\rightarrow\infty}\gamma_{i}^{A}(t)=\lim_{t\rightarrow\infty}\cfrac{|\mathcal{M}_{t}(i)^{+}|}{|\mathcal{M}_{t}(i)^{-}|}\xrightarrow[t\rightarrow\infty]{}0$
(16)

Since when time approaches infinity, assuming a geometric cooling scheme, the
algorithm will only accept improving actions and there will be many more non-
improving than improving actions.

Let’s now calculate the exploration-exploitation coefficient:

$\delta_{i}^{A}=\frac{1}{|\mathcal{A}(i)|}\sum_{j\in\mathcal{A}(i)}\sum_{t=0}^{\infty}\delta_{ja}^{A}(t)=\frac{1}{|\mathcal{A}(i)|}\sum_{j\in\mathcal{A}(i)}\sum_{t=0}^{\infty}\exp\left(-\frac{f(j)-f(i)}{\alpha^{t}T_{0}}\right)$
(17)

which for all $i\in\mathcal{S}$ and $\alpha\in[0,1)$ converges to a positive
constant $C>0$, since:

$\cfrac{\exp\left(-\frac{f(j)-f(i)}{\alpha^{t+1}T_{0}}\right)}{\exp\left(-\frac{f(j)-f(i)}{\alpha^{t}T_{0}}\right)}=\exp\left(-\frac{(1-\alpha)(f(j)-f(i))}{\alpha^{t+1}T_{0}}\right)\xrightarrow[t\rightarrow\infty]{}0$
(18)

Therefore, simulated annealing is a balanced local search metaheuristic.

##  5 Conclusions

In this paper, we have presented a new framework for modeling the convergence
and exploration-exploitation behaviour of local search metaheuristics based on
Markov decision processes. This framework provides a novel and intuitive way
of characterizing essential properties of local search metaheuristis in a
principled way. We proved that any local search metaheuristic can be analyzed
using the tools provided and we showed how to apply the framework to two
examples: hill climbing and simulated annealing. In the case of hill climbing,
we showed that the framework correctly classifies it as exploitation-oriented,
whereas simulated annealing was classified as balanced, which is consistent
with the general consensus in the literature. Our study goes beyond the state-
of-the-art in metaheuristic analysis, since the tools we develop provide a
more expressive methodology than methods based on Markov chains and other
probabilistic approaches. As a consequence, the behaviour of local search
metaheuristics can be modeled in a more realistic way and the results obtained
are consistent with practice.

As future work, we plan to extend our results to other local search
metaheuristics like variable neighborhood search (VNS) [22] and large
neighborhood search (LNS) [24]. Additionally, the framework will be extended
to be able to handle population-based metaheuristics (PBMH) like genetic
algorithms (GA) [14], particle swarm optimization (PSO) [16] and estimation of
distribution algorithms (EDA) [19].

## Declarations

No funds, grants, or other support was received. The authors have no financial
or proprietary interests in any material discussed in this article.

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11institutetext: Departamento de Matemáticas, Física y Ciencias Tecnológicas,
Universidad Cardenal Herrera-CEU, CEU Universities, 46115 Alfara del
Patriarca, València, Spain

# Computing one-loop radiative corrections in $\tau\to\pi(K)\nu[\gamma]$ and
testing new physics

Ignasi Rosell 11<EMAIL_ADDRESS>
###### Abstract

Without a doubt the ratios $R_{\tau/P}\equiv\Gamma\left(\tau\to
P\nu_{\tau}[\gamma]\right)/\Gamma\left(P\to\mu\nu_{\mu}[\gamma]\right)$
($P=\pi,K$) give a convenient scenario to test the lepton universality, the
CKM unitarity and to search for non-standard interactions in $\tau$ decays.
Moreover, the previous theoretical estimation of these observables is more
than twenty-five years old and some assumptions of that estimation are
unrealistic. Therefore, an update of $R_{\tau/P}$ was timely. The one-loop
radiative corrections have been computed by considering an expansion of Chiral
Perturbation Theory including the lightest spin-one resonances and respecting
the short-distance behavior dictated by QCD. We have reported $\delta
R_{\tau/\pi}=(0.18\pm 0.57)\%$ and $\delta R_{\tau/K}=(0.97\pm 0.58)\%$, where
the uncertainties have been induced fundamentally by the estimation of the
counterterms. We have tested the lepton universality, the CKM unitarity and
have searched for new physics in $\tau$ decays. As a by-product, we have also
determined the theoretical radiative corrections to the $\tau\to
P\nu_{\tau}[\gamma]$ decay rates, $\delta_{\tau\pi}=-(0.24\pm 0.56)\%$ and
$\delta_{\tau K}=-(0.15\pm 0.57)\%$.

##  1 Introduction

Although within the Standard Model the three lepton families are expected to
couple to the electroweak gauge bosons with the same intensity $g_{\ell}$
($\ell=e,\,\mu,\,\tau)$, the so-called lepton universality (LU), and this has
been tested experimentally and most of the results have been compatible with
LU, a few anomalies observed in semileptonic $B$ meson decays Albrecht:2021tul
seem to indicate a small deviation from LU. Therefore, an interesting task is
the study of all the different observables testing LU, including low-energy
precision probes using pion, kaons and tau leptons, which currently provide
the most precise tests of LU Bryman:2021teu . Following this idea, we have
tested LU between the second and third families through the ratio ($P=\pi,K$)
Marciano ; DF ; nosotros

$R_{\tau/P}\equiv\frac{\Gamma\left(\tau\to
P\nu_{\tau}[\gamma]\right)}{\Gamma\left(P\to\mu\nu_{\mu}[\gamma]\right)}=\left|\frac{g_{\tau}}{g_{\mu}}\right|^{2}_{P}R_{\tau/P}^{0}(1+\delta
R_{\tau/P}),$ (1)

where $g_{\mu}=g_{\tau}$ according to LU, $\delta R_{\tau/P}$ denotes the
radiative corrections, and $R_{\tau/P}^{(0)}$ is the tree-level contribution
given by

$R_{\tau/P}^{(0)}=\frac{1}{2}\frac{M_{\tau}^{3}}{m_{\mu}^{2}m_{P}}\frac{(1-m_{P}^{2}/M_{\tau}^{2})^{2}}{(1-m_{\mu}^{2}/m_{P}^{2})^{2}},$
(2)

which is free from quark mixing angles and hadronic couplings.

In Ref. DF , more than twenty-five years ago, the first complete estimation
for $\delta R_{\tau/P}$ was given: $\delta R_{\tau/\pi}=(0.16\pm 0.14)\%$ and
$\delta R_{\tau/K}=(0.90\pm 0.22)\%$. However, there were important reasons to
analysis this observable again: the hadronic form factors of Ref. DF  are
different for virtual- and real-photon contributions, do not follow the
correct QCD high-energy behavior, violate analyticity, unitarity, and the
chiral limit at leading non-trivial orders; moreover, a cutoff is implemented
in order to regulate loop integrals, splitting unphysically short- and long-
distance contributions. Besides, errors reported in Ref. DF  are not
realistic, since they are of the order of an expected
$\mathcal{O}(e^{2}p^{2})$ Chiral Perturbation Theory (ChPT) calculation, whose
only model-dependence is usually the estimation of the counterterms, but it
does not allow the inclusion of the $\tau$ lepton.

Depending on the process at hand, different values of
$\left|g_{\tau}/g_{\mu}\right|$ can be found in the literature:

  1. 1.

$\Gamma(\tau\to P\nu_{\tau}[\gamma])/\Gamma(P\to\mu\nu_{\mu}[\gamma])$
($P=\pi,K$). Considering the values of $\delta R_{\tau/P}$ given in Ref. DF ,
the 2018 HFLAV analysis Amhis:2019ckw  reported
$\left|g_{\tau}/g_{\mu}\right|_{\pi}=0.9958\pm 0.0026$ and
$\left|g_{\tau}/g_{\mu}\right|_{K}=0.9879\pm 0.0063$, at $1.6\sigma$ and
$1.9\sigma$ of LU.

  2. 2.

$\Gamma(\tau\to e\bar{\nu}_{e}\nu_{\tau}[\gamma])/\Gamma(\mu\to
e\bar{\nu}_{e}\nu_{\mu}[\gamma])$. This purely leptonic extraction gives
$\left|g_{\tau}/g_{\mu}\right|=1.0010\pm 0.0014$ Amhis:2019ckw , at
$0.7\sigma$ of LU.

  3. 3.

$\Gamma(W\to\tau\nu_{\tau})/\Gamma(W\to\mu\nu_{\mu})$. The weighted average of
the $W$-boson decay determinations yields
$\left|g_{\tau}/g_{\mu}\right|=0.995\pm 0.006$ Aad:2020ayz ; CMS:2021qxj , at
$0.8\sigma$ of LU.

Thus, a new estimation of $\delta R_{\tau/P}$ was interesting to solve these
discrepancies, being the improvement of the theoretical Ansätze an important
ingredient.

In Ref. nosotros  we have presented a new next-to-leading calculation of
$\delta R_{\tau/P}$ considering a large-$N_{C}$ effective approach including
the lightest resonances RChT  and overcoming all the aforementioned
theoretical difficulties. Note that whereas $P$ decays can be analyzed
unambiguously by using Chiral Perturbation Theory (the Standard Model), being
the estimation of the local counterterms the only model dependence, $\tau$
decays must be studied considering a model-dependent effective approach
encoding the hadronization of the QCD currents: this is the reason why we have
considered in Ref. nosotros  the large-$N_{C}$ approach quoted previously RChT
.

The new calculation of $\delta R_{\tau/P}$ has been used not only to analyze
the LU in $\left|g_{\tau}/g_{\mu}\right|$, but also to report the theoretical
radiative corrections in individual $\tau\to P\nu_{\tau}[\gamma]$ decay rates,
to test the CKM unitarity via $\left|V_{us}/V_{ud}\right|$ in $\Gamma(\tau\to
K\nu_{\tau}[\gamma])/\Gamma(\tau\to\pi\nu_{\tau}[\gamma])$ or through
$\left|V_{us}\right|$ in $\Gamma(\tau\to K\nu_{\tau}[\gamma])$ and to search
for non-standard interactions on these decays. It is interesting to stress
that our results have been used in the recent 2021 HFLAV analysis
HFLAV:2022pwe  instead of Ref. DF .

##  2 The calculation

Despite their different approaches, $P\to\mu\nu_{\mu}[\gamma]$ and $\tau\to
P\nu_{\tau}[\gamma]$ decay rates can be organized similarly Marciano ;
nosotros ; CR :

$\displaystyle\Gamma_{P_{\mu 2[\gamma]}}=\,$
$\displaystyle\Gamma^{(0)}_{P_{\mu 2}}\\!\,S_{\rm
EW}\Bigg{\\{}1+\frac{\alpha}{\pi}\,F(m_{\mu}^{2}/m_{P}^{2})\Bigg{\\}}\Bigg{\\{}1-\frac{\alpha}{\pi}\Bigg{[}\frac{3}{2}\log\frac{m_{\rho}}{m_{P}}$
$\displaystyle+c_{1}^{(P)}+\frac{m_{\mu}^{2}}{m_{\rho}^{2}}\bigg{(}c_{2}^{(P)}\,\log\frac{m_{\rho}^{2}}{m_{\mu}^{2}}+c_{3}^{(P)}+c_{4}^{(P)}(m_{\mu}/m_{P})\bigg{)}-\frac{m_{P}^{2}}{m_{\rho}^{2}}\,\tilde{c}_{2}^{(P)}\,\log\frac{m_{\rho}^{2}}{m_{\mu}^{2}}\Bigg{]}\Bigg{\\}}\,,$
(3) $\displaystyle\Gamma_{\tau_{P2[\gamma]}}=\,$
$\displaystyle\Gamma^{(0)}_{\tau_{P2}}\,S_{\rm
EW}\Bigg{\\{}1+\frac{\alpha}{\pi}\,G(m_{P}^{2}/M_{\tau}^{2})\Bigg{\\}}\Bigg{\\{}1-\frac{3\alpha}{2\pi}\log\frac{m_{\rho}}{M_{\tau}}+\delta_{\tau
P}\big{|}_{\mathrm{rSD}}+\delta_{\tau P}\big{|}_{\mathrm{vSD}}\Bigg{\\}}\,,$
(4)

where $S_{\rm EW}=1.0232\simeq
1+\frac{2\,\alpha}{\pi}\log\frac{m_{Z}}{m_{\rho}}$ corresponds to the
(universal) leading short-distance electroweak correction Marciano  (canceling
in the ratio $R_{\tau/P}$), the first bracketed terms are the universal long-
distance or point-like correction DF ; nosotros ; Kinoshita:1959ha , the
second bracketed terms include the structure-dependent (SD) contributions and
$\Gamma^{(0)}_{P_{\mu 2}}$ and $\Gamma^{(0)}_{\tau_{P2}}$ are the decay rate
at leading order ($F_{\pi}\sim 92$ MeV),

$\Gamma^{(0)}_{P_{\mu
2}}=\frac{G_{F}^{2}|V_{uD}|^{2}F_{P}^{2}}{4\pi}\,m_{P}\,m_{\mu}^{2}\,\left(1-\frac{m_{\mu}^{2}}{m_{P}^{2}}\right)^{2}\,,\qquad\Gamma^{(0)}_{\tau_{P2}}=\frac{G_{F}^{2}|V_{uD}|^{2}F_{P}^{2}}{8\pi}M_{\tau}^{3}\left(1-\frac{m_{P}^{2}}{M_{\tau}^{2}}\right)^{2},$
(5)

being $D=d,s$ for $P=\pi,K$, respectively. Whereas for structure-dependent
contributions in $P_{\mu 2}$ the notation of Ref. Marciano  has been used and
the numerical values for $c_{n}^{(P)}$ in Ref. CR  have been considered, we
have separated the structure-dependent contributions in $\tau_{P2}$ into real-
photon (rSD) and virtual-photon (vSD) corrections: we have taken rSD
corrections from Ref. Guo:2010dv  and have performed a new calculation for vSD
ones.

Contribution | $\delta R_{\tau/\pi}$ | $\delta R_{\tau/K}$ | Ref.
---|---|---|---
SI | $+1.05\%$ | $+1.67\%$ | DF
rSD | $+0.15\%$ | $+(0.18\pm 0.05)\%$ | CR ; Guo:2010dv
vSD | $-(1.02\pm 0.57)\%$ | $-(0.88\pm 0.58)\%$ | nosotros ; CR
Total | $+(0.18\pm 0.57)\%$ | $+(0.97\pm 0.58)\%$ | nosotros
Table 1: Numerical values of the different contributions to $\delta
R_{\tau/P}$: Structure Independent (SI), real-photon Structure Dependent (rSD)
and virtual-photon Structure Dependent (vSD). Uncertainties are not given if
they are negligible for the level of accuracy of this analysis, that is, lower
than $0.01\%$.

The technicalities of the calculation are given in Ref. nosotros  and the
different photonic contributions are showed in Table 1 . The final result
reads nosotros

$\delta R_{\tau/\pi}\,=\,(0.18\pm 0.57)\%\,,\qquad\qquad\delta
R_{\tau/K}\,=\,(0.97\pm 0.58)\%\,.$ (6)

The most relevant source of uncertainty comes from the estimation of the
counterterms appearing in vSD contributions of $\tau\to P\nu_{\tau}$ ($\pm
0.57\%$ and $\pm 0.58\%$ for the pion and kaon case, respectively, see Table
1). This uncertainty has been estimated by considering the running of the
counterterms between $0.5\,$and $1.0\,$GeV ($\pm 0.52\%$, in a similar way to
Ref. Marciano ) and considering the effect of assuming a less general
resonance effective Lagrangian.111Let us stress the conservative estimation
followed here. Due to the different scales, counterterms in effective
approaches with resonances are expected to be lower than in ChPT, without
resonances. Neverthesless, with our estimation the counterterms affecting the
vSD corrections in $P_{\mu 2}$ and $\tau_{P2}$ are assumed to be of similar
size.

Although the final result reported in (6) is compatible with Ref. DF , $\delta
R_{\tau/\pi}=(0.16\pm 0.14)\%$ and $\delta R_{\tau/K}=(0.90\pm 0.22)\%$, the
comparison needs to be done carefully. Firstly, in our view uncertainties were
underestimated in Ref. DF , for they have approximately the expected size in a
purely Chiral Perturbation Theory computation, a much more model-independent
scenario. And secondly, we would like to stress again the inconsistencies in
Ref. DF : the hadronic form factors are different for real- and virtual-photon
corrections, do not satisfy the asymptotic behavior dictated by QCD, violate
analyticity, unitarity and the chiral limit at leading non-trivial orders, and
a cutoff is used, separating for no reason short- and long-distance
contributions. Moreover, as it is explained in Appendix B of Ref. nosotros ,
despite the fact that central values of (6) are very similar to the ones
reported in Ref. DF , central values of the different SD corrections are very
different within both analysis, as it can be seen in Table 2:

  1. 1.

Virtual corrections by Ref. DF  are cutoff-dependent, since, as it has been
spotlighted previously, long- and short-distance photonic contributions are
separated unphysically: figures with an asterisk are cutoff-dependent.

  2. 2.

The reported error in the radiative correction of Ref. DF  arises from
uncertainties in hadronic parameters of SD contributions and from variations
in $\mu_{\rm cut}$.

  3. 3.

For the SI contribution in Ref. DF  we have added to the result obtained in
the point-like approximation ($1.05\%$), the term coming from cutting off the
loops at $\mu_{\rm cut}$ ($-0.21\%$).

  4. 4.

Different contributions of $\delta R_{\tau/K}$ are not provided in Ref. DF ,
preventing a comparison.

Consequently, the agreement between the central values seems to be only a
coincidence.

Contribution | Ref. DF  [$\mu_{\rm cut}=1.5\leavevmode\nobreak\ $GeV] | Ref. nosotros
---|---|---
SI | $+0.84\%^{*}$ | $+1.05\%$
rSD | $+0.05\%$ | $+0.15\%$
vSD | $-0.49\%^{*}$ | $-(1.02\pm 0.57)\%$
short-distance | $-0.25\%^{*}$ | 0
Total | $+(0.16\pm 0.14)\%^{*}$ | $+(0.18\pm 0.57)\%$
Table 2: Comparison of photonic contributions to $\delta R_{\tau/\pi}$
nosotros . The asterisk indicates that the figure at hand depends on the
cutoff, which has been established at $\mu_{\rm cut}=1.5$ GeV DF .

##  3 Applications

As it has been explained in the introduction, there are different interesting
applications of our result nosotros :

  1. 1.

Radiative corrections in $\tau\to P\nu_{\tau}[\gamma]$ decay rates. We have
used our results to estimate the radiative corrections in the individual
$\tau_{P2[\gamma]}$ decays,
$\Gamma_{\tau_{P2[\gamma]}}=\Gamma^{(0)}_{\tau_{P2}}S_{\rm
EW}\left(1+\delta_{\tau P}\right)$, where $\delta_{\tau P}$ includes all SI
and SD radiative corrections of (4) and we have reported nosotros :

$\delta_{\tau\pi}\,=\,-(0.24\pm 0.56)\%\,,\qquad\qquad\delta_{\tau
K}\,=\,-(0.15\pm 0.57)\%\,.$ (7)

  2. 2.

Lepton universality test. Considering (1) and (6), the LU can be tested
nosotros ,

$\displaystyle\left|\frac{g_{\tau}}{g_{\mu}}\right|_{\pi}=0.9964\pm
0.0028_{\mathrm{th}}\pm 0.0025_{\mathrm{exp}}=0.9964\pm 0.0038\,,$
$\displaystyle\left|\frac{g_{\tau}}{g_{\mu}}\right|_{K}=0.9857\pm
0.0028_{\mathrm{th}}\pm 0.0072_{\mathrm{exp}}=0.9857\pm 0.0078\,,$ (8)

at $0.9\sigma$ and $1.8\sigma$ of LU ($g_{\tau}=g_{\mu}$), respectively. Note
that while experimental and theoretical uncertainties are of similar size in
the pion case, experimental ones dominate in the kaon case. We can compare
these results with the 2018 HFLAV analysis Amhis:2019ckw ,
$\left|g_{\tau}/g_{\mu}\right|_{\pi}=0.9958\pm 0.0026$ and
$\left|g_{\tau}/g_{\mu}\right|_{K}=0.9879\pm 0.0063$ (at $1.6\sigma$ and
$1.9\sigma$ of LU), where $\delta R_{\tau/P}$ was taken from Ref. DF . In the
recent HFLAV analysis of Ref. HFLAV:2022pwe , where our results of (6) have
been considered, the agreement is larger:
$\left|g_{\tau}/g_{\mu}\right|_{\pi}=0.9959\pm 0.0038$ and
$\left|g_{\tau}/g_{\mu}\right|_{K}=0.9855\pm 0.0075$ (at $1.1\sigma$ and
$1.9\sigma$ of LU).

  3. 3.

CKM unitarity test via $\left|V_{us}/V_{ud}\right|$. One can use the ratio

$\frac{\Gamma(\tau\to
K\nu_{\tau}[\gamma])}{\Gamma(\tau\to\pi\nu_{\tau}[\gamma])}=\frac{|V_{us}|^{2}F_{K}^{2}}{|V_{ud}|^{2}F_{\pi}^{2}}\frac{(1\\!-\\!m_{K}^{2}/M_{\tau}^{2})^{2}}{(1\\!-\\!m_{\pi}^{2}/M_{\tau}^{2})^{2}}\left(1\\!+\\!\delta\right)\\!,$
(9)

in order to extract $\left|V_{us}/V_{ud}\right|$. Considering that the
radiative correction in (9) can be extracted from (7), $\delta=\delta_{\tau
K}-\delta_{\tau\pi}=(0.10\pm 0.80)\%$, we have found nosotros :

$\displaystyle\bigg{|}\frac{V_{us}}{V_{ud}}\bigg{|}=0.2288\pm
0.0010_{\mathrm{th}}\pm 0.0017_{\mathrm{exp}}=0.2288\pm 0.0020\,,$ (10)

at $2.1\sigma$ of unitarity, being dominant the experimental uncertainties.
Despite the fact that this analysis was absent in the 2018 HFLAV analysis
Amhis:2019ckw , it has been included in the 2021 version HFLAV:2022pwe , where
our result of (6) has been considered and it has been reported
$|V_{us}/V_{ud}|=0.2289\pm 0.0019$, very similar to (10). Our result is also
consistent with Ref. Seng:2021nar , $|V_{us}/V_{ud}|=0.2291\pm 0.0009$,
obtained in the context of kaon semileptonic decays, where the larger
statistics allows lower uncertainties.

  4. 4.

CKM unitarity test via $\left|V_{us}\right|$. Alternatively, $|V_{us}|$ can be
obtained directly from the $\tau\to K\nu_{\tau}[\gamma]$ decay rate,
$\Gamma_{\tau_{K2[\gamma]}}=\Gamma^{(0)}_{\tau_{K2}}S_{\rm
EW}\left(1+\delta_{\tau K}\right)$. We have found nosotros :

$\displaystyle|V_{us}|=0.2220\pm 0.0008_{\mathrm{th}}\pm
0.0016_{\mathrm{exp}}=0.2220\pm 0.0018\,,$ (11)

at $2.6\sigma$ of unitarity and again being dominant the experimental
uncertainties. This result can be compared with the 2018 HFLAV analysis
Amhis:2019ckw , $|V_{us}|=0.2234\pm 0.0015$ and using Ref. DF , or the recent
2021 HFLAV analysis HFLAV:2022pwe , $|V_{us}|=0.2219\pm 0.0017$ and
considering our results. Again (11) is compatible with Ref. Seng:2021nar ,
$|V_{us}|=0.2231\pm 0.0006$, obtained using kaon semileptonic decays and,
consequently, with a better precision.

  5. 5.

New physics in $\tau\to P\nu_{\tau}[\gamma]$ decay rates. We have also used
our results to constrain non-SM interactions in $\tau\to P\nu_{\tau}[\gamma]$
decays:

$\Gamma(\tau\to
P\nu_{\tau}[\gamma])=\Gamma^{(0)}_{\tau_{P2}}\left|\frac{\widetilde{V}_{uD}}{V_{uD}}\right|^{2}S_{\rm
EW}\left(1+\delta_{\tau P}+2\Delta^{\tau P}\right)\,,$ (12)

being $D=d,s$ for $P=\pi,K$, respectively. $\Delta^{\tau P}$ contains the
leading-order new-physics corrections not present in
$\widetilde{V}_{uD}=(1+\epsilon^{e}_{L}+\epsilon^{e}_{R})V_{uD}$, directly
incorporated by considering $V_{uD}$ from nuclear $\beta$ decays EFTtaudecays1
; EFTtaudecays2 :

$\Delta^{\tau
P}=\epsilon^{\tau}_{L}-\epsilon^{e}_{L}-\epsilon^{\tau}_{R}-\epsilon^{e}_{R}-\frac{m_{P}^{2}}{M_{\tau}(m_{u}+m_{D})}\epsilon^{\tau}_{P}\,.$
(13)

Considering our estimations of $\delta_{\tau P}$ and $|V_{us}/V_{ud}|$ given
in (7) and (10), respectively, we have estimated nosotros :

$\Delta^{\tau\pi}\,=\,-(0.15\pm 0.72)\%\,,\qquad\qquad\Delta^{\tau
K}\,=\,-(0.36\pm 1.18)\%\,,$ (14)

to be compared to $\Delta^{\tau\pi}=-(0.15\pm 0.67)\%$ in Ref. EFTtaudecays1 ,
$\Delta^{\tau\pi}=-(0.12\pm 0.68)\%$ and $\Delta^{\tau K}=-(0.41\pm 0.93)\%$
in Ref. EFTtaudecays2 , and $\Delta^{\tau\pi}=-(0.09\pm 0.73)\%$ and
$\Delta^{\tau K}=-(0.2\pm 1.0)\%$ in Ref. Cirigliano:2021yto . All these
values are reported at a scale of $\mu=2$ GeV in the
$\overline{\mathrm{MS}}$-scheme. Note that all these determinations are
consistent with each other and compatible with the SM, $\Delta^{\tau P}=0$.

For all these determinations branching ratios and masses have been obtained
from the PDG PDG , $|V_{ud}|=0.97373\pm 0.00031$ from Ref.Hardy:2020qwl ,
$S_{\rm EW}=1.0232$ from Ref. Marciano  and meson decay constants from the
FLAG analysis Aoki:2019cca : $\sqrt{2}F_{\pi}=(130.2\pm
0.8)\leavevmode\nobreak\ $MeV, $\sqrt{2}F_{K}=(155.7\pm
0.3)\leavevmode\nobreak\ $MeV and $F_{K}/F_{\pi}=1.1932\pm 0.0019$, .

I wish to thank the organizers of the workshop for the pleasant conference. I
am also very grateful to M. A. Arroyo-Ureña, G. Hernández-Tomé, G. López-
Castro and P. Roig, since the work presented here has been done with them and
also for their helpful comments to prepare these proceedings. This work has
been supported in part by the Universidad Cardenal Herrera-CEU [INDI21/15]; by
the Generalitat Valenciana [PROMETEU/2021/071] and by the Spanish Government
[MCIN/AEI/10.13039/501100011033, Grant No. PID2020-114473GB-I00].

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