# On Abelian subvarieties of bounded degree
in a polarized Abelian variety

Lucio Guerra

###### Abstract

If $A$ is an Abelian variety, endowed with a polarization $L$, we study the
function $N_{A}(t)$ which counts the number of Abelian subvarieties $S$ in $A$
such that for the induced polarization $L|_{S}$ the Euler characteristic
$\chi(L|_{S})$ is bounded above by $t$. We give an estimate for the asymptotic
order of growth of this function.

##  1 Introduction

Let $A$ be an Abelian variety, endowed with a polarization $L$. (We view
polarizations as divisor classes, rather than line bundles.) Let $S$ be a
subvariety of $A$, of dimension $n$. The intersection number

$S\cdot L^{n}=(L|_{S})^{n}$

can be considered as a geometric degree of $S$, at least when $L$ is a very
ample divisor class. If $S$ is an Abelian subvariety of $A$, the geometric
form of the Riemann Roch theorem for Abelian varieties says that

$\frac{(L|_{S})^{n}}{n!}=\chi(L|_{S})$

is the Euler characteristic of the line bundle associated to the restricted
polarization. So, for an Abelian subvariety, the Euler characteristic
$\chi(L|_{S})$ can be considered as a ‘reduced degree’. Denote with the symbol
${\sf AS}_{A}$ the collection of all Abelian subvarieties of $A$. For every
positive integer or real number $t$, define

$N_{A}(t):=\\#\big{\\{}S\in{\sf AS}_{A}\mbox{ s.t. }\chi(L|_{S})\leq
t\big{\\}}\vskip 3.0pt plus 1.0pt minus 1.0pt.$

That this is a finite number was already known to Poincaré, at least for the
subcollection of subvarieties of dimension $n=1$. The general statement
follows from a result of Birkenhake and Lange [2], to the effect that the
collection of all Abelian subvarieties with bounded exponent in $A$ is finite.
In a couple of papers [3],[4] we studied the problem of giving an asymptotic
estimate for the finite number of elliptic curves with bounded degree in $A$.
The aim of the present work is to show that the method of the previous papers
can be adjusted to the general situation. The argument leads to make explicit
the relation between the rate of growth of the counting function $N_{A}(t)$
and the complete decomposition of $A$ up to isogeny into a product of simple
Abelian varieties.

Assume that the complete decomposition of $A$ is given by an isogeny

$A\approx B_{1}^{k_{1}}\times\cdots\times B_{q}^{k_{q}}$

where $B_{1},\ldots,B_{q}$ are simple Abelian varieties, each other non-
isogenous, and where $k_{1},\ldots,k_{q}$ are positive integers. Define the
integers: $k$, the maximum among the multiplicities $k_{j}$, and $h$, the
maximum rank of an endomorphism group $End(B_{j})$. In terms of these data, we
prove the following

######  Theorem 1.1.

There is an asymptotic estimate

$N_{A}(t)=O\left(t^{\,q(kh+2)(k-1)}\right).$

The main tools for the argument are: a thorough use of the complete
reducibility theorem in the relative version involving a given polarization
(cf. §2, especially Proposition 2.1), and a classical result in Number Theory
concerning the number of lattice points in a bounded region of Euclidean space
(cf. §3.3). The proof of Theorem 1.1 is given in the final §6.

##  2 Some remarks about reducibility

###  2.1 Reducibility with respect to a polarization

Let $A$ be an Abelian variety, with a polarization $L$. Let $S$ be an Abelian
subvariety of $A$. There is in $A$ an Abelian subvariety $S^{\prime}$ such
that

  * $-$

$S\cap S^{\prime}$ is a finite set and $S+S^{\prime}=A$, so that the sum map

$s:S\times S^{\prime}\rightarrow A$

is an isogeny, and

  * $-$

the pullback polarization from $A$ splits as

$s^{\ast}(L)=p^{\ast}(L|_{S})+(p^{\prime})^{\ast}(L|_{S^{\prime}})$

where $p,p^{\prime}$ denote the natural projections of the product $S\times
S^{\prime}$ to the factors $S,S^{\prime}$, respectively. The complementary
Abelian subvariety $S^{\prime}$ is uniquely determined.

###  2.2 On the degree of the sum isogeny

In the setting of the preceding section, the following holds:

$\deg(s)\;\chi(L)=\chi(L|_{S})\;\chi(L|_{S^{\prime}}).$

In fact, one has $\chi(s^{\ast}(L))=\deg(s)\,\chi(L)$ (the behavior of the
Euler characteristic under an etale finite morphism) and on the other hand
$\chi(s^{\ast}(L))=\chi(p^{\ast}(L|_{S})+(p^{\prime})^{\ast}(L|_{S^{\prime}}))=\chi(L|_{S})\,\chi(L|_{S^{\prime}})$
(by the Künneth formula for the cohomology of shaves). Furthermore, concerning
the degree $\deg(s)=\\#(S\cap S^{\prime})$, we point out the following:

######  Proposition 2.1.

For the degree of the sum isogeny in the reduciblity theorem, one has the
upper bound

$\deg(s)\leq\chi(L|_{S})^{2}.$

###### Proof.

When $L$ is a principal polarization on $A$, the relation between the types of
the induced polarizations $L|_{S}$ and $L|_{S^{\prime}}$ is completely
understood (see [1], 12.1.5, p. 366). If the type for the smaller dimension is
$(d_{1},\ldots,d_{m})$ then for the higher dimension the type is
$(1,\ldots,1,d_{1},\ldots,d_{m})$. It follows that
$\chi(L|_{S})=\chi(L|_{S^{\prime}})$, both equal to the product $d_{1}\cdots
d_{m}$, and therefore $\deg(s)=\chi(L|_{S})^{2}$ follows from the equation
above, with $\chi(L)=1$.

If $L$ is an arbitrary polarization, there is an isogeny $g:A\rightarrow B$
onto an Abelian variety $B$ having a principal polarization $M$ such that
$L=g^{\ast}(M)$ (see [1], 4.1.2, p. 71). If $S$ is an Abelian subvariety of
$A$, define $T:=g(A)$ and let $d_{S}$ be the degree of the restriction
$g_{S}:S\rightarrow T$. One has $\chi(L|_{S})=d_{S}\,\chi(M|_{T})$.

Let $T^{\prime}$ be the complement of $T$ in $B$ with respect to $M$. Then
$S^{\prime}$, the complement of $S$ in $A$ with respect to $L$, coincides with
$g^{-1}_{\,0}(T^{\prime})$, the connected component of $0$ in the pre-image
$g^{-1}(T^{\prime})$. In order to check this, it is enough to work with the
commutative diagram

$\begin{CD}S\times g^{-1}_{\,0}(T^{\prime})@>{s}>{}>A\\\ @V{\overline{g\times
g}}V{}V@V{}V{g}V\\\ T\times T^{\prime}@>{}>{t}>B\end{CD}$

where $s,t$ are the sum homomorphisms, and check that the pullback of $M$
splits on $S\times g^{-1}_{\,0}(T^{\prime})$. We have just seen that
$\\#(T\cap T^{\prime})=\chi(M|_{T})^{2}$ holds since $M$ is a principal
polarization. Then $\\#(S\cap S^{\prime})\leq\\#(S\cap g^{-1}(T^{\prime}))\leq
d_{S}\,\\#(T\cap T^{\prime})\leq{d_{S}}^{2}\chi(M|_{T})^{2}=\chi(L|_{S})^{2}$.
∎

Finally, here is an immediate consequence:

$\chi(L|_{S^{\prime}})\leq\chi(L)\,\chi(L|_{S}).$

###  2.3 Complete reducibility with respect to a polarization

As a consequence of the reducibility theorem quoted in §2.1 one has the
following theorem of complete reducibility into simple factors.

Let $A$ be an arbitrary Abelian variety, endowed with a polarization $L$.
There exist:

  * $-$

a sequence of simple Abelian varieties $B_{1},\ldots,B_{q}$, each other non-
isogenous,

  * $-$

a corresponding sequence of positive integers $k_{1},\ldots,k_{q}$,

  * $-$

and an isogeny

$s:B_{1}^{k_{1}}\times\cdots\times B_{q}^{k_{q}}\rightarrow A$

such that the pullback polarization $s^{\ast}(L)$ splits.

In general we say that a polarization on a product variety
$S_{1}\times\cdots\times S_{k}$ is a split polarization if it is of the form

$p_{1}^{\ast}(L_{1})+\cdots+p_{k}^{\ast}(L_{k})$

where $L_{i}$ is a polarization on $S_{i}$ and $p_{i}$ is the projection to
the $i$th factor.

In the theorem of complete reducibility, the collection of pairs
$(B_{j},k_{j})$ is determined from $A$ alone, up to isogeny, however the
isogeny $s$ that is described above also depends on the polarization $L$.

###  2.4 Behavior under isogenies

Let $A$ and $B$ be Abelian varieties, and let $g:B\rightarrow A$ be an
isogeny, of degree $d$. Let $L$ be a polarization on $A$ and consider on $B$
the pullback polarization $g^{\ast}(L)$. There is a one to one correspondence
${\sf AS}_{A}\overset{{}_{\sim}}{\longrightarrow}{\sf AS}_{B}.$ Given
$S\subset A$ the corresponding $S^{\ast}$ in $B$ is the connected component of
$0$ in the pre-image $\varphi^{-1}(S)$. The restricted isogeny
$S^{\ast}\rightarrow S$ has degree $d_{S}\leq d$ (in fact a divisor of $d$).
With respect to the polarizations one has
$\chi((g^{\ast}L)|_{S^{\ast}})=d_{S}\,\chi(L|_{S})$. So there is the chain of
inequalities $\chi(L|_{S})\leq\chi((g^{\ast}L)|_{S^{\ast}})\leq
d\,\chi(L|_{S})$. It follows that the functions counting Abelian subvarieties
in $A$ and in $B$ are related by the inequalities

$N_{A}(t)\leq N_{B}(dt){\rm\ \ \ and\ \ \ }N_{B}(t)\leq N_{A}(t)$

and therefore one has

$N_{A}(t)=O(t^{e})\mbox{ \; if and only if \; }N_{B}(t)=O(t^{e}).$

######  Remark 2.2.

Because of the invariance under isogeny of the asymptotic estimate for the
counting function, and because of the complete reducibility theorem with
respect to a polarization, in order to prove Theorem 1.1 it is enough to
confine oneself to the case of a product of simple Abelian varieties endowed
with a split polarization. This is what we do in the following.

##  3 Bounding homomorphisms

###  3.1 A quadratic form

Let $A,A^{\prime}$ be Abelian varities, of dimensions $n,n^{\prime}$, endowed
with polarizations $\Theta,\Theta^{\prime}$. Consider the function

$Hom(A,A^{\prime})\longrightarrow\mathbb{Z}$

which is defined by the assignment

$f\longmapsto\dfrac{\Theta^{n-1}}{(n-1)!}\cdot f^{\ast}\Theta^{\prime}.$

It is known that this is a quadratic form, always non-negative. positive
definite if $A$ is simple. The second and the third assertion are almost
apparent, by some intersection theory. The first assertion for the special
case $A=A^{\prime}$, i.e. for an endomorphism group $End(A)$, can be found for
instance in [1], §5.1, p. 117; and the general case can be reduced to this
special one by embedding the homomorphism group into a suitable endomorphism
group in such a way that the function defined above for homomorphisms can be
recovered, up to some scalar factor, as the restriction of the analogous
function defined for endomorphisms.

###  3.2 Homomorphisms in a hyperellipsoid

Assume now that $A$ is a simple Abelian variety. Define

$\Phi(t):=\\#\left\\{f\in Hom(A,A^{\prime})\mbox{ s.t.
}\dfrac{\Theta^{n-1}}{(n-1)!}\cdot f^{\ast}\Theta^{\prime}\leq
t^{2}\right\\}.$

Let $H$ be the rank of the homomorphism group $Hom(A,A^{\prime})$.

######  Proposition 3.1.

There is an asymptotic estimate

$\Phi(t)=O(t^{H}).$

###### Proof.

Choose an isomorphism $Hom(A,A^{\prime})\cong\mathbb{Z}^{H}$. The quadratic
form on $Hom(A,A^{\prime})$ corresponds to a quadratic form
$Q:\mathbb{Z}^{H}\rightarrow\mathbb{Z}$, and this is naturally extended to
$\widehat{Q}:\mathbb{R}^{H}\rightarrow\mathbb{R}$. As the original form is
positive definite, then $Q$ is positive definite and so is $\widehat{Q}$ too.
And so we have

$\Phi(t)=\\#\\{x\in\mathbb{Z}^{H}\mbox{ s.t. }Q(x)\leq t^{2}\\}.$

Now observe that $\\{x\in\mathbb{Z}^{H}\mbox{ s.t. }Q(x)\leq t^{2}\\}$ can be
written as

$t\cdot\\{y\in\mathbb{R}^{H}\mbox{ s.t. }\widehat{Q}(y)\leq
1\\}\cap\,\mathbb{Z}^{H}$

and that $\\{y\in\mathbb{R}^{H}\mbox{ s.t. }\widehat{Q}(y)\leq 1\\}$, a
hyperellipsoid, is compact and convex. So we are lead to a classical topic in
Number Theory, the problem of estimating the number of lattice points
belonging to a bounded region in the Euclidean space. Thus the conclusion
follows from the general result quoted below. ∎

###  3.3 Lattice points in a bounded region

Let $K$ be a subset of $\mathbb{R}^{n}$, compact convex and with Lipschitz
frontier. The function $\\#(tK\cap\mathbb{Z}^{n})$ counts the number of points
of the lattice $\mathbb{Z}^{n}$ which lie in a deformed region $tK$. There is
an asymptotic estimate in the form $\\#(tK\cap\mathbb{Z}^{n})={\rm
Vol}(K)\,t^{n}+O(t^{n-1})$ (see [5], Ch. 6, Lemma 2, p. 165), that we only use
in the coarse version

$\\#(tK\cap\mathbb{Z}^{n})=O(t^{n}).$

##  4 Subvarieties from homomorphisms

Let $A$ and $A^{\prime}$ be Abelian varieties, and consider the product
$A\times A^{\prime}$ with the natural projections $p,p^{\prime}$. Let $n$ be
the dimension of $A$. If $S$ is an Abelian subvariety in $A\times A^{\prime}$,
we denote by $p_{S},p^{\prime}_{S}$ the two projections restricted to $S$.

###  4.1 Parametrization

We describe the Abelian subvarieties $S$ such that the projection
$p_{S}:S\rightarrow A$ is surjective and finite, i.e. an isogeny. Define

${\sf AS}_{A\times A^{\prime}/\,p}:=\Big{\\{}S\in{\sf AS}_{A\times
A^{\prime}}\mbox{ s.t. }p{}_{S}:S\rightarrow A\mbox{ is an isogeny}\Big{\\}}.$

So assume that $p_{S}:S\rightarrow A$ is an isogeny. Take the inverse, or
dual, isogeny $\widehat{p_{S}}:A\rightarrow S$ and define the homomorphism
$f:A\rightarrow A^{\prime}$ as $f:={p^{\prime}_{S}}\circ\widehat{p_{S}}$. Let
$a:=\deg(p_{S})$. Then $\deg(\widehat{p_{S}})=a^{2n-1}$ (follows from
$p_{S}\circ\widehat{p_{S}}=a_{A}$). If $\widehat{p_{S}}$ is viewed as a map
$A\rightarrow A\times A^{\prime}$, it is given by $x\mapsto(ax,f(x))$. So we
can describe $S$ as the image of some homomorphism $A\rightarrow A\times
A^{\prime}$ of the form $(a,f)$.

For every integer $a>0$ and every homomorphism $f:A\rightarrow A^{\prime}$ we
denote by $C_{a,f}$ the image of the homomorphism $A\rightarrow A\times
A^{\prime}$ given by the pair $(a,f)$, and denote by the same symbol $(a,f)$
the induced homomorphism $A\rightarrow C_{a,f}$. The condition that

$\deg(a,f)=a^{2n-1}$

is necessary and sufficient for the projection $C_{a,f}\rightarrow A$ to be of
degree $a$, and in this case the inverse isogeny $A\rightarrow
C_{a,f}\hookrightarrow A\times A^{\prime}$ is naturally identified to $(a,f)$.
So we have a correspondence

$\Bigg{\\{}\mbox{\begin{tabular}[]{c}$(a,f)\in\mathbb{Z}_{>0}\times
Hom(A,A^{\prime})$\\\ s.t.
$\deg(a,f)=a^{2n-1}$\end{tabular}}\Bigg{\\}}\longrightarrow{\sf AS}_{A\times
A^{\prime}/\,p}\,.$

This is a bijective correspondence. The surjectivity is clear from the
discussion above. Assume that $(a,f)$ and $(a^{\prime},f^{\prime})$ determine
the same subvariety $C_{a,f}=C_{a^{\prime},f^{\prime}}$. (Taking the degree of
the first projection we already have $a=a^{\prime}$.) Taking the inverse of
the first projection we have $(a,f)=(a^{\prime},f^{\prime})$.

###  4.2 Degree

Assume that on $A\times A^{\prime}$ is given a split polarization
$L={p}^{\ast}\Theta+{p^{\prime}}{}^{\ast}\Theta^{\prime}$. For a subvariety
$S=C_{a,f}$ we now study the degree $S\cdot L^{n}=(L|_{S})^{n}$, for which we
give some estimate from below.

Pulling back to $A$, by means of $\widehat{p_{S}}:A\rightarrow S$, we obtain
$\widehat{p_{S}}^{\ast}(L|_{S})^{n}=a^{2n-1}(L|_{S})^{n}$. On the other hand,
since $\widehat{p_{S}}^{\ast}(L|_{S})=a^{\ast}\Theta+f^{\ast}\Theta^{\prime}$,
we have
$\widehat{p_{S}}^{\ast}(L|_{S})^{n}=(a^{2}\Theta+f^{\ast}\Theta^{\prime})^{n}$.
Here we replaced $a^{\ast}\Theta$ with $a^{2}\Theta$, in the computation of
the intersection number, because
$a^{\ast}(\Theta)\equiv\frac{a^{2}+a}{2}\,\Theta+\frac{a^{2}-a}{2}\,(-1)^{\ast}\Theta$
in $Pic(A)$ (see [1], Proposition 2.3.5, p. 33) and because
$\Theta\equiv(-1)^{\ast}\Theta$ in $H_{2n-2}(A,\mathbb{Z})$. It follows that

$a^{2n-1}(L|_{S})^{n}=(a^{2}\Theta+f^{\ast}\Theta^{\prime})^{n}=\sum_{i=0}^{n}\binom{n}{i}a^{2i}\,(\Theta^{i}\cdot
f^{\ast}{\Theta^{\prime}}^{n-i}).$

Furthermore, due to the ampleness of $\Theta$ and $\Theta^{\prime}$, one has

$\Theta^{i}\cdot f^{\ast}{\Theta^{\prime}}^{n-i}>0$

for $0\leq i\leq n$ (see [1], Lemma 4.3.2, p. 76). As a consequence, if we
drop from the right hand side all terms except the two with $i=n,n-1$, and
then divide by $n!$, we obtain the inequality

$\chi(L|_{S})=\frac{(L|_{S})^{n}}{n!}\geq
a\;\dfrac{\Theta^{n}}{n!}+\dfrac{1}{a}\;\dfrac{\Theta^{n-1}}{(n-1)!}\cdot
f^{\ast}\Theta^{\prime}.$

###  4.3 On subvarieties of bounded degree $-$ I

Assume now that $A$ is a simple Abelian variety. We study the set

${\sf AS}_{A\times A^{\prime}/\,p}:=\Big{\\{}S\in{\sf AS}_{A\times
A^{\prime}}\mbox{ s.t. }p{}_{S}:S\rightarrow A\mbox{ is an isogeny}\Big{\\}}$

and give an asymptotic estimate for the counting function

$N_{A\times A^{\prime}/\,p}(t):=\\#\Big{\\{}S\in{\sf AS}_{A\times
A^{\prime}/\,p}\mbox{ s.t. }\chi(L|_{S})\leq t\Big{\\}}.$

The estimate is going to be in terms of the quantity

$H:=\mbox{the rank of the group }Hom(A,A^{\prime}).$

######  Theorem 4.1.

In the present setting, with $A$ simple, one has

$N_{A\times A^{\prime}/\,p}(t)=O\left(t^{H+1}\right).$

###### Proof.

We apply the description in §4.1. The set ${\sf AS}_{A\times A^{\prime}/\,p}$
is bijective to the set of parameter data

$\Big{\\{}(a,f)\in\mathbb{Z}_{>0}\times Hom(A,A^{\prime})\mbox{ s.t.
}\deg(a,f)=a^{2n-1}\Big{\\}}.$

Thus $N_{A\times A^{\prime}/\,p}(t)$ is the number of elements of the set

$\Big{\\{}(a,f)\in\mathbb{Z}_{>0}\times Hom(A,A^{\prime})\ \mbox{s.t.}\
\deg(a,f)=a^{2n-1}\ \mbox{and}\ \dfrac{\deg C(a,f)}{n!}\leq t\Big{\\}}.$

Then we consider the description in §4.2. We have the inequality

$\frac{\deg C_{a,f}}{n!}\geq
a\;\dfrac{\Theta^{n}}{n!}+\dfrac{1}{a}\;\dfrac{\Theta^{n-1}}{(n-1)!}\cdot
f^{\ast}\Theta^{\prime}.$

Hence the codnition $\frac{\deg C_{a,f}}{n!}\leq t$ implies

$a\leq\dfrac{t}{M}\ \mbox{ and }\ \dfrac{\Theta^{n-1}}{(n-1)!}\cdot
f^{\ast}\Theta^{\prime}\leq a\,t,$

where $M:=\Theta^{n}/n!$. It follows that the set of parameter data above is a
subset of the product

$\Big{\\{}a\in\mathbb{Z}\mbox{ s.t. }1\leq
a\leq\frac{t}{M}\Big{\\}}\times\Big{\\{}f\in Hom(A,A^{\prime})\ \mbox{s.t.}\
\dfrac{\Theta^{n-1}}{(n-1)!}\cdot
f^{\ast}\Theta^{\prime}\leq\frac{t^{2}}{M}\Big{\\}}.$

Since from Proposition 3.1 we know that the cardinality of the set on the
right hand side of the product is a function of order $O(t^{H})$, it follows
that $N_{A\times A^{\prime}/\,p}(t)$ is bounded above by a function of order
$O(t)\,O(t^{H})$ as in the statement. ∎

##  5 Subvarieties of a product variety

Keeping the notation introduced in the preceding section, we extend the
description of subvarieties in the product variety $A\times A^{\prime}$.

###  5.1 Parametrization

We describe the Abelian subvarieties $S$ of $A\times A^{\prime}$ such that the
projection $p{}_{S}:S\rightarrow A$ is surjective.

Define $F:=Ker_{0}(p_{S})$ and view it as an Abelian subvariety of
$A^{\prime}$. Let $T$ in $S$ be an Abelian subvariety such that $F+T=S$ and
$F\cap T$ is finite. Then the sum map $s:F\times T\rightarrow S$ is an
isogeny, and the restricted projection $p_{T}:T\rightarrow A$ is an isogeny
too.

Assume that a polarization $L$ is given on $A\times A^{\prime}$. Because of
the reducibility theorem relative to a given polarization (cfr. §2.1), we can
choose $T$ in such a way that
$s^{\ast}(L|_{S})={q}^{\ast}(L|_{F})+{q^{\prime}}{}^{\ast}(L|_{T})$ where $q$
and $q^{\prime}$ denote the projections from $F\times T$ to $F$ and $T$,
respectively.

We say that two subvarieties $F$ and $T$ in $A\times A^{\prime}$ are
complementary with respect to $L$ when the sum map $s:F\times T\rightarrow
F+T$ is an isogeny and the pullback polarization $s^{\ast}(L|_{F+T})$ splits
as ${q}^{\ast}(L|_{F})+{q^{\prime}}^{\ast}(L|_{T})$.

There is a correspondence

$\left\\{\mbox{\begin{tabular}[]{c}$(F,T)\in{\sf AS}_{A^{\prime}}\times{\sf
AS}_{A\times A^{\prime}/\,p}$\\\ s.t. complementary w.r.t.
$L$\end{tabular}}\right\\}\longrightarrow\left\\{\mbox{\begin{tabular}[]{c}$S\in{\sf
AS}_{A\times A^{\prime}}$ s.t.\\\ $p_{S}:S\rightarrow A$
surjective\end{tabular}}\right\\}$ $\hskip 38.0pt(F,T)\hskip
10.0pt\longmapsto\hskip 10.0ptF+T$

and this is a bijective correspondence. That the correspondence is surjective
is clear from the construction. Given $S$, the choice of $F$ is unique,
because $Ker(p_{S})$ must be equal to $(T\cap F)+F$ and therefore
$Ker_{0}(p_{S})=F$. That the choice of $T$ is then unique is part of the
reducibility theorem.

###  5.2 Degree

Concerning the (reduced) degree of $S=F+T$ with respect to the polarization
$L=p^{\ast}\Theta+{p^{\prime}}{}^{\ast}\Theta^{\prime}$, i.e. the Euler
characteristic $\chi(L|_{S})$, we here recall what we know. Define

$a:=\deg(s:F\times T\rightarrow S)=\\#(F\cap T)=\deg(p{}_{T}:T\rightarrow A).$

From §2.2 we have the basic facts

$a\,\chi(L|_{S})=\chi(\Theta^{\prime}|_{F})\,\chi(L|_{T})\ \ \ \mbox{ and }\ \
\ \chi(L|_{T})\leq\chi(L|_{S})\,\chi(\Theta^{\prime}|_{F})\hskip 20.0pt$

and from §4.2 we take the inequality

$aM\leq\chi(L|_{T})$

where $M=\chi(\Theta)$, holding because $T$ is of the form $C_{a,f}$ with
$a=\deg(p{}_{T})$ as above.

###  5.3 On subvarieties of bounded degree $-$ II

Assume now that $A$ is a simple Abelian variety.

######  Theorem 5.1.

If $A$ is simple and if $N_{A^{\prime}}(t)=O(t^{E})$ then $N_{A\times
A^{\prime}}(t)=O\left(t^{E+2(H+1)}\right)$.

###### Proof.

Since $A$ is simple, the set ${\sf AS}_{A\times A^{\prime}}$ is the disjoint
union of ${\sf AS}_{A^{\prime}}$ and the complementary subset which, according
to §5.1, is bijective to the set of parameter data

$\Big{\\{}(F,T)\in{\sf AS}_{A^{\prime}}\times{\sf AS}_{A/\,p}\mbox{ \ s.t. $F$
and $T$ are complementary w.r.t. $L$}\Big{\\}}.$

The basic facts concerning the degree of $S=F+T$ expressed in terms of the
degree $a$ of the sum isogeny $F\times T\rightarrow S$ and the degrees of $F$
and $T$ are collected in §5.2. In the present setting we can show that

$\chi(L|_{S})\leq t\ \ \ \mbox{ implies }\ \ \ \chi(\Theta^{\prime}|_{F})\leq
t\ \ \mbox{ and }\ \ \chi(L|_{T})\leq t^{2}.$

We have $a\,\chi(L|_{S})=\chi(\Theta^{\prime}|_{F})\chi(L|_{T})$. So from
$\chi(L|_{T})\geq aM$ we obtain $\chi(\Theta^{\prime}|_{F})\leq t/M$ and from
$\chi(L|_{T})\leq\chi(L|_{S})\chi(\Theta^{\prime}|_{F})$ we obtain
$\chi(L|_{T})\leq t^{2}/M$. Note that in the weaker inequalities stated above
we dropped the constant $M$, just because it will not affect the exponent in
the estimate of the counting function $N_{A\times A^{\prime}}(t)$.

Summing everything up, we have

$N_{A\times A^{\prime}}(t)\leq
N_{A^{\prime}}(t)+N_{A^{\prime}}(t)\,{N}_{A\times A^{\prime}/\,p}(t^{2}).$

From Theorem 4.1 we know that

${N}_{A\times A^{\prime}/\,p}(t^{2})=O(t^{2(H+1)}).$

So, with the hypothesis that $N_{A^{\prime}}(t)=O(t^{E})$, the statement
follows. ∎

##  6 Subvarieties of a completely reduced product

Let $A=B^{k}$ be the $k$th self product of a simple Abelian variety $B$, of
dimension $n$, endowed with a split polarization
$L=p_{1}^{\ast}\Theta_{1}+\cdots+p_{k}^{\ast}\Theta_{k}$, where each
$\Theta_{i}$ is a polarization on $B$. The estimate for the counting function
$N_{B^{k}}(t)$ is in terms of $h$, the rank of the endomorphism group
$End(B)$.

######  Theorem 6.1.

There is an asymptotic estimate

$N_{B^{k}}(t)=O\left(t^{(kh+2)(k-1)}\right).$

###### Proof.

By induction on $k$. The initial case $k=1$ being obvious, assume $k\geq 2$.
Write $B^{k}=B\times B^{k-1}$ and let $p:B^{k}\rightarrow B$ and
$p^{\prime}:B^{k}\rightarrow B^{k-1}$ be the natural projections, with
$p:=p_{1}$ and $p^{\prime}:=(p_{2},\ldots,p_{k})$. The given polarization on
$B^{k}$ splits as $L=p^{\ast}\Theta+p^{\prime}{}^{\ast}\Theta^{\prime}$ where
$\Theta:=\Theta_{1}$ and where $\Theta^{\prime}$ is the split polarization on
$B^{k-1}$ obtained from $\Theta_{2},\ldots,\Theta_{k}$.

The group $Hom(B,B^{k-1})$ is of rank $H=(k-1)h$. By the induction hypothesis
we have $N_{B^{k-1}}(t)=O(t^{E})$ with $E=\left((k-1)h+2\right)(k-2)$. From
Theorem 5.1 we have $N_{B^{k}}(t)=(t^{E+2(H+1)})$. In the present setting
$E+2(H+1)=(kh+2)(k-1)$, and the statement is proven. ∎

Let $A=B_{1}^{k_{1}}\times\cdots\times B_{q}^{k_{q}}$ be a product of simple
Abelian varieties, each occurring with some multiplicity, and assume that
$B_{i}$ and $B_{j}$ are not isogenous for $i\neq j$.

Each simple Abelian subvariety of $A$ is contained in a unique $B_{j}^{k_{j}}$
and is then isogenous to $B_{j}$. An arbitrary Abelian subvariety of $A$ may
be written as $S=S_{1}\times\cdots\times S_{q}$ where

$S_{j}$ | = | the sum of all simple Abelian subvarieties of $S$ contained in $B_{j}^{k_{j}}$
---|---|---
|  | (and hence belonging to the isogeny class of $B_{j}$)
| = |  $S\cap B_{j}^{k_{j}}\,.$

Thus we have a correspondence

${\sf AS}_{B_{1}^{k_{1}}}\times\cdots\times{\sf
AS}_{B_{q}^{k_{q}}}\longrightarrow{\sf AS}_{A}$
$(S_{1},\ldots,S_{q})\longmapsto S_{1}\times\ldots\times S_{q}$

and clearly this is a bijective correspondence.

Let $L$ be a split polarization on $A$. It may be written as
$L=p_{1}^{\ast}(\Theta_{1})+\cdots+p_{q}^{\ast}(\Theta_{q})$ where each
$\Theta_{j}$ is a split polarization on $B_{j}^{k_{j}}$. If
$S=S_{1}\times\ldots\times S_{q}$ is an Abelian subvariety of $A$, as above,
then (the Künneth formula again)

$\chi(L|_{S})=\chi(\Theta_{1}|_{S_{1}})\cdots\chi(\Theta_{q}|_{S_{q}}).$

In the present setting, the estimate for the function $N_{A}(t)$, and its
proof, involve the following data:

$h_{j}$ | the rank of the endomorphism group $End(B_{j})$,
---|---
$h$ | the maximum among the ranks $h_{1},\ldots,h_{q}$,
$k$ | the maximum among the multiplicities $k_{1},\ldots,k_{q}$.

######  Theorem 6.2.

There is an asymptotic estimate

$N_{B_{1}^{k_{1}}\times\cdots\times
B_{q}^{k_{q}}}(t)=O\left(t^{\,q(kh+2)(k-1)}\right).$

###### Proof.

We have seen that the set ${\sf AS}_{A}$ is bijective to ${\sf
AS}_{B_{1}^{k_{1}}}\times\cdots\times{\sf AS}_{B_{q}^{k_{q}}}$ and from the
expression for $\chi(L|_{S})$ above it follows that $N_{A}(t)\leq
N_{B_{1}^{k_{1}}}(t)\cdots N_{B_{q}^{k_{q}}}(t)$. From Theorem 6.1 we have
$N_{B_{j}^{k_{j}}}(t)=O\left(t^{(k_{j}h_{j}+2)(k_{j}-1)}\right)$ and therefore
$N_{B_{1}^{k_{1}}}(t)\cdots
N_{B_{q}^{k_{q}}}(t)=O\left(t^{(k_{1}h_{1}+2)(k_{1}-1)+\cdots+(k_{q}h_{q}+2)(k_{q}-1)}\right)$,
and so the statement follows. ∎

According to Remark 2.2 the proof of Theorem 1.1 is now complete.

## References

  * [1] Ch. Birkenhake, H. Lange, Complex Abelian Varieties. Second edition. Grundlehren der Mathematischen Wissenschaften, 302. Springer-Verlag, Berlin, 2004.
  * [2] Ch. Birkenhake, H. Lange, The exponent of an abelian subvariety. Math. Ann. 290 (1991), no. 4, 801-814.
  * [3] L. Guerra, On elliptic curves of bounded degree in a polarized Abelian surface. Rend. Ist. Mat. Univ. Trieste 48 (2016), 495-508.
  * [4] L. Guerra, On elliptic curves of bounded degree in a polarized Abelian variety. Rend. Ist. Mat. Univ. Trieste 51 (2019), 105-123.
  * [5] D. Marcus, Number Fields, Springer-Verlag, 1995.

email<EMAIL_ADDRESS>


# Muon $g-2$ in $U(1)_{\mu-\tau}$ Symmetric Gauged Radiative Neutrino Mass
Model

Dong Woo Kang<EMAIL_ADDRESS>School of Physics, KIAS, Seoul 02455,
Korea  Jongkuk Kim<EMAIL_ADDRESS>School of Physics, KIAS, Seoul 02455,
Korea  Hiroshi Okada<EMAIL_ADDRESS>Asia Pacific Center for
Theoretical Physics (APCTP) - Headquarters San 31, Hyoja-dong, Nam-gu, Pohang
790-784, Korea  Department of Physics, Pohang University of Science and
Technology, Pohang 37673, Republic of Korea

###### Abstract

We explore muon anomalous magnetic moment (muon $g-2$) in a scotogenic
neutrino model with a gauged lepton number symmetry $U(1)_{\mu-\tau}$. In this
model, a dominant muon $g-2$ contribution comes not from an additional gauge
sector but from a Yukawa sector. In our numerical $\Delta\chi^{2}$ analysis,
we show that our model is in favor of normal hierarchy with some features. We
demonstrate one benchmark point, satisfying muon $g-2$ at the best fit value
$25.1\times 10^{-10}$.

††preprint: KIAS-P21026, APCTP Pre2021 - 016

##  I Introduction

Flavor dependent gauged $U(1)$ scenarios; especially $U(1)_{\mu-\tau}$
He:1990pn ; He:1991qd , are widely applied to explaining recent anomalies such
as muon anomalous magnetic moment (muon $g-2$ or $\Delta a_{\mu}$)
Davier:2010nc ; Davier:2017zfy ; Davier:2019can ; Borah:2021mri ; Qi:2021rhh ;
Singirala:2021gok ; Buras:2021btx ; Zhou:2021vnf ; Borah:2021jzu ;
Chen:2021vzk ; Zu:2021odn ; Huang:2021nkl ; Patra:2016shz ;
Altmannshofer:2016oaq  and rare $B$ meson decays $b\to s\ell\bar{\ell}$
Crivellin:2015mga ; Crivellin:2015lwa ; Cen:2021iwv ; Ko:2017yrd ;
Kumar:2020web  as beyond the Standard Model (SM), in addition to neutrino mass
models Baek:2015mna ; Nomura:2018cle ; Nomura:2018vfz ; Asai:2018ocx ;
Asai:2017ryy ; Lee:2017ekw . 111Flavor dependent gauge $U(1)$ scenarios can
modify the lepton universality, since a new gauge boson $Z^{\prime}$ does not
universally couple to lepton sector. See Chun:2018ibr .  In fact, the
longstanding muon $g-2$ anomaly was first discovered by BNL two decades ago
and is recently confirmed by FNAL. Combined two results are given by
Muong-2:2021ojo

$\Delta a_{\mu}^{\rm new}=(25.1\pm 5.9)\times 10^{-10}.$ (I.1)

The deviation from the SM expectation is 4.2$\sigma$ confidence level. One can
typically address this deviation with the contribution from additional gauge
boson with new neutral gauge boson mass $m_{Z^{\prime}}$ and coupling
$g^{\prime}$. Even though there are several constraints on this new gauge
sector such as neutrino trident bound; $m_{Z^{\prime}}/g^{\prime}\lesssim 550$
GeV, there exists allowed region Altmannshofer:2014pba .

In this paper, we propose a scotogenic neutrino model with $U(1)_{\mu-\tau}$
gauge symmetry that explains muon $g-2$ by the Yukawa sectors not by the new
gauge sector. In order to get the sizable muon $g-2$, we need a diagram
without chiral suppression and we introduce several exotic fermions and
scalars. These several exotic fermions can also play a role in radiative
neutrino mass generation at one-loop level Ma:2006km . Heavier neutral
fermions run inside the loop and the lightest one could be a promising dark
matter (DM) candidate. Finally, we show several features of our model by
demonstrating the numerical $\Delta\chi^{2}$ analysis.

This paper is organized as follows. In Sec. II, we present the model set up
and how to generate neutrino masses in normal and inverted hierarchy. In Sec.
III, we discuss constraints from lepton flavor violations (LFVs) and address
the muon anomalous magnetic moment. We carry out numerical $\Delta\chi^{2}$
analysis and present the allowed region satisfying the neutrino oscillation
data, LFVs and muon $g-2$. Conclusions and discussions are given in Sec. IV,
briefly mentioning a possibility of DM candidate and how to explain the
correct relic density, satisfying direct detection bounds.

##  II Model

| Leptons
---|---
Fermions |  $L_{L_{e}}$ |  $L_{L_{\mu}}$ |  $L_{L_{\tau}}$ |  $\ell_{R_{e}}$ |  $\ell_{R_{\mu}}$ |  $\ell_{R_{\tau}}$ |  $N_{R_{e}}$ |  $N_{R_{\mu}}$ |  $N_{R_{\tau}}$ |  $L^{\prime}$
$SU(2)_{L}$ | $\bm{2}$ | $\bm{2}$ | $\bm{2}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{1}$ | $\bm{2}$
$U(1)_{Y}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-\frac{1}{2}$ | $-1$ | $-1$ | $-1$ | $0$ | $0$ | $0$ | $-\frac{1}{2}$
$U(1)_{\mu-\tau}$ | $0$ | $1$ | $-1$ | $0$ | $1$ | $-1$ | $0$ | $1$ | $-1$ | $0$
$Z_{2}$ | $+$ | $+$ | $+$ | $+$ | $+$ | $+$ | $-$ | $-$ | $-$ | $-$
Table 1: Field contents of fermions and their charge assignments under $SU(2)_{L}\times U(1)_{Y}\times U(1)_{\mu-\tau}\times Z_{2}$, where $SU(3)_{C}$ is singlet. | VEV$\neq 0$ | VEV$=0$
---|---|---
Bosons |  $H$ |  $\varphi$ |  $\eta$ |  $\chi^{-}$
$SU(2)_{L}$ | $\bm{2}$ | $\bm{1}$ | $\bm{2}$ | $\bm{1}$
$U(1)_{Y}$ | $\frac{1}{2}$ | $0$ | $\frac{1}{2}$ | $-1$
$U(1)_{\mu-\tau}$ | $0$ | $1$ | $0$ | $1$
$Z_{2}$ | $+$ | $+$ | $-$ | $-$
Table 2: Field contents of bosons and their charge assignments under
$SU(2)_{L}\times U(1)_{Y}\times U(1)^{\prime}\times Z_{2}$, where $SU(3)_{C}$
is singlet.

In this section, we set up our model Lagrangian and focus on lepton sector and
Higgs sector which are crucial for generating neutrino mass and muon g-2
contribution. And we address detail discussion for the neutrino mass matrix.

###  II.1 Lepton Lagrangian

We introduce three right-handed neutral fermions
$[N_{R_{e}},N_{R_{\mu}},N_{R_{\tau}}]$ with $[0,1,-1]$ charges under
$U(1)_{\mu-\tau}$ symmetry and an isospin doublet vector-like lepton
$L^{\prime}\equiv[N^{\prime},E^{\prime}]^{T}$ with $0$ under $U(1)_{\mu-\tau}$
symmetry. In addition, we impose $Z_{2}$ odd for new fermions where even is
assigned to the SM fermions. The $Z_{2}$ symmetry plays a role in generating
the active neutrino mass matrix not at tree-level but one-loop level.
Furthermore, the lightest field with odd $Z_{2}$ can be a dark matter
candidate. Notice here that the $U(1)_{\mu-\tau}$ anomaly is independently
canceled among the SM fermions or $N_{R}$. In the bosonic sector, we introduce
extra bosons with two isospin singlets $\varphi,\,\chi^{-}$ and a doublet
$\eta$. These bosons respectively have $[1,1,0]$ charges under the
$U(1)_{\mu-\tau}$ symmetry. $\varphi$ is even under $Z_{2}$ and $\eta,\
\chi^{-}$ are odd under $Z_{2}$ and induce the active neutrino mass matrix
together with $N_{R}$. The field contents and assignments for new fermions and
the SM leptons are summarized in Tables 1. Each of $\varphi$ and the SM Higgs
denoted by $H$ has non-zero vacuum expectation values that are symbolized by
$\langle\varphi\rangle\equiv v_{\varphi}/\sqrt{2}$ and $\langle
H\rangle\equiv[0,v_{H}/\sqrt{2}]^{T}$, after spontaneous symmetry breaking of
$U(1)_{\mu-\tau}$ and electroweak symmetry. The bosonic field contents and
their assignments are summarized in Tables 2.

Under these symmetries, our renormalizable Lagrangian is given by

$\displaystyle-\mathcal{L}_{Y}$
$\displaystyle=M_{ee}\overline{N_{R_{e}}^{c}}N_{R_{e}}+M_{\mu\tau}(\overline{N_{R_{\mu}}^{c}}N_{R_{\tau}}+\overline{N_{R_{\tau}}^{c}}N_{R_{\mu}})+M_{L^{\prime}}\overline{L^{\prime}_{L}}L^{\prime}_{R}+\text{h.c.}$
$\displaystyle+y_{e}\overline{L_{L_{e}}}He_{R}+y_{\mu}\overline{L_{L_{\mu}}}H\mu_{R}+y_{\tau}\overline{L_{L_{\tau}}}H\tau_{R}+{\rm
h.c.}$
$\displaystyle+h_{e\mu}(\overline{N_{R_{e}}^{c}}N_{R_{\mu}}+\overline{N_{R_{\mu}}^{c}}N_{R_{e}})\varphi^{*}+h_{e\tau}(\overline{N_{R_{e}}^{c}}N_{R_{\tau}}+\overline{N_{R_{\tau}}^{c}}N_{R_{e}})\varphi+{\rm
h.c.}$
$\displaystyle+f_{e}\overline{L_{L_{e}}}(i\sigma_{2})\eta^{*}N_{R_{e}}+f_{\mu}\overline{L_{L_{\mu}}}(i\sigma_{2})\eta^{*}N_{R_{\mu}}+f_{\tau}\overline{L_{L_{\tau}}}(i\sigma_{2})\eta^{*}N_{R_{\tau}}$
$\displaystyle+g_{e}\overline{\ell_{R_{e}}}N_{R_{\mu}}^{c}\chi^{-}+g_{\mu}\overline{\ell_{R_{\mu}}}N_{R_{e}}^{c}\chi^{-}$
$\displaystyle+h\overline{L_{L_{\mu}}}L^{\prime
c}_{L}\chi^{-}+y^{\prime}\overline{L^{\prime}_{L}}(i\sigma_{2})H^{*}N_{R_{e}}+y^{\prime\prime}\overline{L^{\prime
c}_{R}}HN_{R_{e}}+{\rm h.c.},$ (II.1)

where $\sigma_{2}$ is the second Pauli matrix and the charged-lepton mass
matrix is diagonal due to the $\mu-\tau$ symmetry;
$\mathcal{M}_{\ell}=\frac{v_{H}}{\sqrt{2}}\text{diag}(|y_{e}|,|y_{\mu}|,|y_{\tau}|)\equiv(m_{e},m_{\mu},m_{\tau})$
after the phase redefinition. Therefore, the neutrino oscillation data is
induced via neutrino sector.

###  II.2 Higgs sector

Our Higgs potential is also given by

$\displaystyle{\cal V}$ $\displaystyle={\cal V}_{2}^{tri}+{\cal
V}_{4}^{tri}+\lambda_{0}(H^{\dagger}\eta)^{2}+\lambda^{\prime}_{0}\varphi^{*}(H^{T}i\sigma_{2}\eta)\chi^{-}+{\rm
h.c.},$ (II.2)

where we define $\eta\equiv[\eta^{+},(\eta_{R}+i\eta_{I})/\sqrt{2}]^{T}$,
$H\equiv[h^{+},(v_{H}+h_{0}+iz_{0})/\sqrt{2}]^{T}$,
$\varphi\equiv(v_{\varphi}+\varphi_{R}+iz_{\varphi})/\sqrt{2}$, and ${\cal
V}_{2}^{tri}$ and ${\cal V}_{4}^{tri}$ are respectively trivial quadratic and
quartic terms of the Higgs potential; ${\cal
V}_{2}^{tri}=\sum_{\phi=H,\varphi,\eta,\chi^{-}}\mu_{\phi}^{2}|\phi|^{2},\quad{\cal
V}_{4}^{tri}=\sum_{\phi^{\prime}\leq\phi}^{H,\varphi,\eta,\chi^{-}}\lambda_{\phi\phi^{\prime}}|\phi|^{2}|\phi^{\prime}|^{2}+\lambda^{\prime}_{H\eta}|H^{\dagger}\eta|^{2}.$
Notice here that $h^{+}$, $z_{0}$, and $z_{\varphi}$ are respectively absorbed
by the longitudinal degrees of freedom in gauge sectors. Consequently, we have
massive gauge bosons $W^{\pm},Z$ in the SM and $Z^{\prime}$ in the
$U(1)_{\mu-\tau}$ gauge symmetry. The $\lambda_{0}$ term plays an important
role in generating the non-vanishing neutrino mass matrix. In our model, the
neutrino mass matrix is proportional to the mass-squared difference between
$\eta_{R}$ and $\eta_{I}$; $m_{R}^{2}-m_{I}^{2}=\lambda_{0}v_{H}^{2}$, where
$m_{R,I}$ is the mass eigenstate of $\eta_{R,I}$ Ma:2006km . Even though there
is mixing between $\chi^{\pm}$ and $\eta^{\pm}$ from $\lambda^{\prime}_{0}$,
we suppose that the mixing is negligibly tiny. 222If we consider the large
mixing, we have a contribution to muon $g-2$ without chiral suppression, which
might lead to large muon $g-2$. However, it cannot be large enough. In fact,
we need large mass hierarchy between them, but it is forbidden by oblique
parameters Peskin:1991sw . Satisfying these conditions, we have found $\Delta
a_{\mu}\approx 10^{-13}$ at most in our numerical estimation.

###  II.3 Neutral fermion mass matrices

After the phase redefinition of the neutral fermions, the mass matrix in basis
of $[N_{R_{e}},N_{R_{\mu}},N_{R_{\tau}},N^{\prime c}_{L},N^{\prime}_{R}]$ is
found as follows:

$\displaystyle\mathcal{M}_{N}=\begin{pmatrix}M_{ee}&M_{e\mu}&M_{e\tau}&m^{\prime}&m^{\prime\prime}e^{i\zeta}\\\
M_{e\mu}&0&M_{\mu\tau}e^{i\xi}&0&0\\\ M_{e\tau}&M_{\mu\tau}e^{i\xi}&0&0&0\\\
m^{\prime}&0&0&0&M_{L^{\prime}}\\\
m^{\prime\prime}e^{i\zeta}&0&0&M_{L^{\prime}}&0\\\ \end{pmatrix},$ (II.3)

where $M_{e\mu}\equiv\frac{v_{\varphi}}{\sqrt{2}}|h_{e\mu}|$,
$M_{e\tau}\equiv\frac{v_{\varphi}}{\sqrt{2}}|h_{e\tau}|$ ,
$m^{\prime}\equiv\frac{v_{H}}{\sqrt{2}}|y^{\prime}|$,
$m^{\prime\prime}\equiv\frac{v_{H}}{\sqrt{2}}|y^{\prime\prime}|$ are real mass
parameters, while $\xi,\ \zeta$ are physical phases. The mass matrix
$\mathcal{M}_{N}$ is then diagonalized by introducing a unitary matrix $V$.
This matrix satisfies

$\displaystyle
V^{T}\mathcal{M}_{N}V\equiv\text{diag}(M_{1},M_{2},M_{3},M_{4},M_{5}).$ (II.4)

Here, the mass eigenstate $\psi_{R}$ is defined by
$N_{R_{i}}=\sum_{k=1}^{5}V_{ik}\psi_{R_{k}}$, and its mass eigenvalue is
defined by $M_{k}(k=1,2,3,4,5)$. Then, the valid Lagrangian is rewritten in
terms of mass eigenstates as follows 333Notice here that $h\bar{\nu}_{\mu
L}E^{\prime C}_{L}\chi^{-}$ does not contribute to the neutrino mass matrix,
since $E^{\prime C}_{L}$ cannot propagate in the loop. :

$\displaystyle-{\cal L}$
$\displaystyle=\bar{\nu}_{L_{a}}F_{ak}\psi_{R_{k}}(\eta_{R}-i\eta_{I})-\sqrt{2}\bar{\ell}_{L_{a}}F_{ak}\psi_{R_{k}}\eta^{-}+\bar{\ell}_{R_{a^{\prime}}}G_{a^{\prime}k}\psi_{R_{k}}^{C}\chi^{-}+H_{4k}\bar{\mu}_{L}\psi_{R_{k}}\chi^{-}+{\rm
h.c.},$ (II.5) $\displaystyle F_{ia}$
$\displaystyle=\frac{1}{\sqrt{2}}\sum_{j=1,2,3}\begin{pmatrix}f_{e}&0&0\\\
0&f_{\mu}&0\\\ 0&0&f_{\tau}\\\ \end{pmatrix}_{ij}V_{ja},\quad
G_{ia}=\sum_{j=1,2,3}\begin{pmatrix}0&g_{e}&0\\\ g_{\mu}&0&0\\\
\end{pmatrix}_{ij}V^{*}_{ja},\quad H_{4k}\equiv h\sum_{k=1}^{5}V_{4k},$ (II.6)

where $F$ is three by five matrix, and $G$ is two by five matrix, therefore
$a^{\prime}$ runs over $e,\mu$. The first term of Eq. (II.5) contributes to
the neutrino mass matrix, while the other terms induce LFVs, and muon $g-2$ as
can be seen below. The active neutrino mass matrix is given by Ma:2006km

$\displaystyle(m_{\nu})_{ij}$
$\displaystyle=\sum_{k=1}^{5}\frac{F_{ik}M_{k}F_{kj}^{T}}{2(4\pi)^{2}}\left[\frac{m^{2}_{R}}{m^{2}_{R}-M^{2}_{k}}\ln\frac{m^{2}_{R}}{M^{2}_{k}}-\frac{m^{2}_{I}}{m^{2}_{I}-M^{2}_{k}}\ln\frac{m^{2}_{I}}{M^{2}_{k}}\right]~{}$
$\displaystyle\simeq\frac{\lambda_{0}v^{2}_{H}}{(4\pi)^{2}}\sum_{k=1}^{5}\frac{F_{ik}M_{k}F_{kj}^{T}}{m_{0}^{2}-M^{2}_{k}}\left[1-\frac{M^{2}_{k}}{m^{2}_{0}-M^{2}_{k}}\ln\frac{m^{2}_{0}}{M^{2}_{k}}\right]~{},$
(II.7)

where we assume to be $\lambda_{0}v_{H}^{2}=m^{2}_{R}-m^{2}_{I}\ll
m_{0}^{2}\equiv(m_{R}^{2}+m_{I}^{2})/2$ in the second line, and $m_{\nu}$ is
diagonalzied by a unitary matrix $U_{\rm PMNS}$ Maki:1962mu ; $D_{\nu}\equiv
U_{\rm PMNS}^{T}m_{\nu}U_{\rm PMNS}$. Here, we define dimensionless neutrino
mass matrix as
$m_{\nu}\equiv(\lambda_{0}v_{H})\tilde{m}_{\nu}\equiv\kappa\tilde{m}_{\nu}$.
Since $\kappa$ does not depend on the flavor structure, we rewrite this
diagonalization in terms of dimensionless form
$(\tilde{D}_{\nu_{1}},\tilde{D}_{\nu_{2}},\tilde{D}_{\nu_{3}})\equiv U_{\rm
PMNS}^{T}\tilde{m}_{\nu}U_{\rm PMNS}$. Thus, we fix $\kappa$ by

$\displaystyle({\rm NH}):\ \kappa^{2}=\frac{|\Delta m_{\rm
atm}^{2}|}{\tilde{D}_{\nu_{3}}^{2}-\tilde{D}_{\nu_{1}}^{2}},\quad({\rm IH}):\
\kappa^{2}=\frac{|\Delta m_{\rm
atm}^{2}|}{\tilde{D}_{\nu_{2}}^{2}-\tilde{D}_{\nu_{3}}^{2}},$ (II.8)

where $\Delta m_{\rm atm}^{2}$ is the atmospheric neutrino mass-squared
difference. Here, NH and IH stand for the normal hierarchy and the inverted
hierarchy, respectively. Subsequently, the solar neutrino mass-squared
difference is depicted in terms of $\kappa$ as follows:

$\displaystyle\Delta m_{\rm
sol}^{2}={\kappa^{2}}({\tilde{D}_{\nu_{2}}^{2}-\tilde{D}_{\nu_{1}}^{2}}).$
(II.9)

This should be within the experimental value. The neutrinoless double beta
decay is also given by

$\displaystyle\langle
m_{ee}\rangle=\kappa\left|\tilde{D}_{\nu_{1}}\cos^{2}\theta_{12}\cos^{2}\theta_{13}+\tilde{D}_{\nu_{2}}\sin^{2}\theta_{12}\cos^{2}\theta_{13}e^{i\alpha_{2}}+\tilde{D}_{\nu_{3}}\sin^{2}\theta_{13}e^{i(\alpha_{3}-2\delta_{CP})}\right|,$
(II.10)

which may be able to observed by KamLAND-Zen in future KamLAND-Zen:2016pfg .

##  III  Results

###  III.1 Lepton flavor violations

Because of the flavor dependence in Eq.(II.5), we have lepton flavor violation
decay processes for example, $\mu\to e\gamma$ or $\bar{\mu}\to\bar{e}\gamma$.
These decay channels are originated from the terms of $GH^{\dagger}$ or
$HG^{\dagger}$. 444In our numerical analysis, we also have considered non-
dominant contributions of $GG^{\dagger}$ and $FF^{\dagger}$. The corresponding
branching ratio is given as follows Lindner:2016bgg ; CarcamoHernandez:2019ydc
; Baek:2016kud

$\displaystyle{\rm BR}(\bar{\mu}\to\bar{e}\gamma)$
$\displaystyle\simeq\frac{48\pi^{3}\alpha_{\rm
em}}{(4\pi)^{4}m_{\ell_{i}}^{2}G_{F}^{2}}|H_{4k}M_{k}G^{\dagger}_{k1}F(M_{k},m_{\chi})|^{2},$
(III.1) $\displaystyle{\rm BR}(\mu\to e\gamma)$
$\displaystyle\simeq\frac{48\pi^{3}\alpha_{\rm
em}}{(4\pi)^{4}m_{\ell_{i}}^{2}G_{F}^{2}}|G_{1k}M_{k}H^{\dagger}_{k4}F(M_{k},m_{\chi})|^{2},$
(III.2) $\displaystyle F(m_{1},m_{2})$
$\displaystyle=\int[dx]_{3}\frac{y}{(x^{2}-x)m^{2}_{\mu}+xm_{1}^{2}+(y+x)m_{2}^{2}},$
(III.3)

where the fine structure constant $\alpha_{\rm em}\simeq 1/135$ at muon mass
energy scale TheMEG:2016wtm , and the Fermi constant $G_{F}\simeq 1.17\times
10^{-5}$ GeV-2. The current experimental upper bound at 90% C.L. are
TheMEG:2016wtm ; Adam:2013mnn

$\displaystyle{\rm BR}(\mu\to e\gamma)={\rm
BR}(\bar{\mu}\to\bar{e}\gamma)<4.2\times 10^{-13}.$ (III.4)

###  III.2 Muon anomalous magnetic moment

In our model, we can achieve the required muon $g-2$ contribution through the
same interaction with LFVs. The muon $g-2$ contribution is given as

$\displaystyle\Delta a_{\mu}\approx-\frac{m_{\mu}}{(4\pi)^{2}}{\rm
Re}[(H_{4k}M_{k}G^{\dagger}_{k2}+G_{2k}M_{k}H^{\dagger}_{k4})F(M_{k},m_{\chi})]=-\frac{2m_{\mu}}{(4\pi)^{2}}{\rm
Re}[H_{4k}M_{k}G^{\dagger}_{k2}]F(M_{k},m_{\chi}).$ (III.5)

Recent experimental estimation Hanneke:2008tm  of muon $g-2$ indicates the
following value at $4.2\sigma$ Muong-2:2021ojo :

$\displaystyle\Delta a_{\mu}=a_{\mu}^{\rm{EXP}}-a_{\mu}^{\rm{SM}}=(25.1\pm
5.9)\times 10^{-10}.$ (III.6)

We fix the best fit (BF) value; $\Delta a_{\mu}=25.1\times 10^{-10}$, in our
numerical analysis below.

###  III.3 Numerical analysis

We show our numerical $\Delta\chi^{2}$ analysis to satisfy the neutrino
oscillation data, LFV ($\mu\to e\gamma$ only in our case) as well as muon
$g-2$. Here we fix $\Delta a_{\mu}$ as BF value $25.1\times 10^{-10}$, where
we adopt five known observables $\sin^{2}\theta_{12},\ \sin^{2}\theta_{23},\
\sin^{2}\theta_{13},\ \Delta m^{2}_{\rm atm},\ \Delta m^{2}_{\rm sol}$ in
Nufit 5.0 Esteban:2020cvm  at $3\sigma$ confidence level. Notice that we do
not include the Dirac CP phase in $\Delta\chi^{2}$ analysis because of big
ambiguity at $3\sigma$ interval. Furthermore, we employ Gaussian
approximations for charged-lepton masses. However since we have found the
allowed region within $4.59\lesssim\sqrt{\Delta\chi^{2}}$ in case of IH, it
does not satisfy the sizable muon $g-2$ whose maximum order is $10^{-12}$ in
our model. Thus, we focus on NH only. At first, we fix the range of our input
parameters as follows:

$\displaystyle\\{f_{e},f_{\mu},f_{\tau}\\}\in[10^{-5},10],\
\\{h,|g_{\mu}|\\}\in[1,10],\
\\{\xi,\zeta\\}\in[0,\pi],\quad\\{m^{\prime},m^{\prime\prime}\\}\in[1,617]\
{\rm GeV},$ (III.7)
$\displaystyle\\{M_{ee},M_{e\mu},M_{e\tau},M_{\mu\tau}\\}\in[10,10^{6}]\ {\rm
GeV},\quad\\{m_{\eta^{+}},M_{L^{\prime}},m_{\chi^{\pm}},m_{0}\\}\in[10^{2},10^{6}]\
{\rm GeV},$ (III.8)

where $0.9\times m_{0}\leq m_{\eta^{\pm}}\leq 1.2\times m_{0}$ to evade the
bound on oblique parameters, and $g_{e}=0$ to suppress the LFVs in our work.
555In the case of $g_{e}=0$, both BR$(\mu\to e\gamma)$ and muon $g-2$ are
proportional to Re[$hg_{\mu}^{*}$] that does not depend on the flavor
structure. Thus, we can take any values for $h$ and $g_{\mu}$ in our numerical
ranges as far as Re[$hg_{\mu}^{*}$] does not change. In our benchmark points
in Table 3, we set $h$ and $g_{\mu}$ as one of examples so as to be
Re[$hg_{\mu}^{*}$]$\sim$1.52. The upper bounds on $m^{\prime},\
m^{\prime\prime}$ come from $|y^{{}^{\prime}(^{\prime\prime})}|v_{H}/\sqrt{2}$
taking the limit of perturbation
$|y^{{}^{\prime}(^{\prime\prime})}|=\sqrt{4\pi}$ in addition to $\\{F,\ G,\
H\\}\in\sqrt{4\pi}$.

Figure 1: The scatter plots of sum of the neutrino masses $\sum m_{i}$ in
terms of $\sin^{2}\theta_{23}$ (left), neutrinoless double beta decay $\langle
m_{ee}\rangle$ in terms of $\sin^{2}\theta_{23}$ (center), and $\langle
m_{ee}\rangle$ in terms of $\sum m_{i}$ (right). Each of the color plots of
green, yellow, and red represents the region at $\sigma$ of $0$–$1$, $1$–$2$,
$2$–$3$, respectively. The each vertical line of black-solid and red-dotted
represents the best fit and 3$\sigma$ interval.

In Fig. 1, we show the scatter plots of sum of neutrino masses $\sum m_{i}$ in
terms of $\sin^{2}\theta_{23}$ (left), neutrinoless double beta decay $\langle
m_{ee}\rangle$ in terms of $\sin^{2}\theta_{23}$ (center), and $\langle
m_{ee}\rangle$ in terms of $\sum m_{i}$ (right). Here, each of the color plots
of green, yellow, and red represents the region at $\sigma$ of $0$–$1$,
$1$–$2$, $2$–$3$, respectively. After the $\Delta\chi^{2}$ analysis, as shown
in the figure, we find that $\sum m_{i}$ is allowed in the range of
$[72.9-80.4]$ [meV] at $1\sigma$ interval, $[71.8-81.6]$ [meV] at $2\sigma$
interval, $[71.7-82.1]$ [meV] at $3\sigma$ interval. $\langle m_{ee}\rangle$
is allowed in the range of $[11.1-14.6]$ [meV] at $1\sigma$ interval,
$[11.0-15.1]$ [meV] at $2\sigma$ interval, and $[10.8-15.1]$ [meV] at
$3\sigma$ interval.

| NH
---|---
$h$ | $-{1.005}$
$g_{\mu}$ | ${1.514+0.628i}$
$[f_{e},f_{\mu},f_{\tau}]$ | $[14.5,\ -7.78,\ -17.7]\times 10^{-5}$
$[\xi,\zeta]$ | $[328^{\circ},209^{\circ}]$
$[m_{0},m_{\chi^{\pm}},m_{\eta^{\pm}},]$ |  $[123,\ 165,\ 102]$ GeV
$\begin{pmatrix}M_{ee}&M_{e\mu}&M_{e\tau}&m^{\prime}&m^{\prime\prime}\\\ M_{e\mu}&0&M_{\mu\tau}&0&0\\\ M_{e\tau}&M_{\mu\tau}&0&0&0\\\ m^{\prime}&0&0&0&M_{L^{\prime}}\\\ m^{\prime\prime}&0&0&M_{L^{\prime}}&0\\\ \end{pmatrix}$ |  $\begin{pmatrix}40.2&2.15\times 10^{4}&8.12\times 10^{4}&587&-88.5\\\ 2.15\times 10^{4}&0&8.34\times 10^{3}&0&0\\\ 8.12\times 10^{4}&8.34\times 10^{3}&0&0&0\\\ 587&0&0&0&486\\\ -88.5&0&0&486&0\\\ \end{pmatrix}$GeV
${\rm BR}(\mu\to e\gamma)$ | $4.11\times 10^{-22}$
$\Delta m^{2}_{\rm atm}$ | $2.52\times 10^{-3}{\rm eV}^{2}$
$\Delta m^{2}_{\rm sol}$ | $7.44\times 10^{-5}{\rm eV}^{2}$
$\sin^{2}\theta_{12}$ | $0.307$
$\sin^{2}\theta_{23}$ | $0.573$
$\sin^{2}\theta_{13}$ | $0.0221$
$[\delta_{CP}^{\ell},\ \alpha_{21},\,\alpha_{31}]$ | $[0.191^{\circ},\,360^{\circ},\,0.405^{\circ}]$
$\sum m_{i}$ |  $77.5$ meV
$\langle m_{ee}\rangle$ |  $13.3$ meV
$\kappa$ |  $5.33$GeV
$\sqrt{\Delta\chi^{2}}$ | $0.391$
Table 3: Numerical benchmark points of our input parameters and observables in
NH, satisfying muon $g-2$ at the best fit value $25.1\times 10^{-10}$. Here,
the NH is taken such that $\sqrt{\Delta\chi^{2}}$ should be minimum.

Finally, we show a benchmark point for NH in Table 3, where it is taken such
that $\Delta\chi^{2}$ is minimum in our model. Notice here that all the phases
$\delta_{CP}$, $\alpha_{21},\ \alpha_{31}$ are very close to be zero through
our $\sqrt{\Delta\chi^{2}}$ analysis.

##  IV Conclusions and Discussions

We have proposed a gauged $U(1)_{\mu-\tau}$ neutrino mass model which can
address the muon $g-2$ through the Yukawa sector without relying on the new
gauge sector. The neutrino mass matrix is induced at one-loop level and
participated in five neutral fermions run inside the loop. Through our
numerical $\Delta\chi^{2}$ analysis, we have found that NH is in favor of our
model. The best fit value of $\Delta a_{\mu}$ is mainly obtained by
Re[$hg^{*}_{\mu}$] in case of $g_{e}=0$ for NH. IH does not satisfy the
sizable muon $g-2$, but we have allowed region at
$4.59\lesssim\sqrt{\Delta\chi^{2}}$ in neutrino oscillation data. Several
unique features are listed below:

  1. 1.

$\sum m_{i}$ is allowed in the range of $[72.9-80.4]$ [meV] at $1\sigma$
interval, $[71.8-81.6]$ [meV] at $2\sigma$ interval, $[71.7-82.1]$ [meV] at
$3\sigma$ interval.

  2. 2.

$\langle m_{ee}\rangle$ is allowed in the range of $[11.1-14.6]$ [meV] at
$1\sigma$ interval, $[11.0-15.1]$ [meV] at $2\sigma$ interval, and
$[10.8-15.1]$ [meV] at $3\sigma$ interval.

  3. 3.

All the phases $\delta_{CP}$, $\alpha_{21},\ \alpha_{31}$ are localized in
vicinity of zero through our ${\Delta\chi^{2}}$ analysis.

Before closing our discussion, we briefly mention a DM candidate. Basically,
we have two candidates; bosonic and fermionic one.
In case of bosonic DM candidate, the lightest particle of $\eta_{R}$ and
$\eta_{I}$ is the one. In case of fermionic DM candidate, $\psi_{R_{1}}$ is
the one. In the case of fermionic DM candidate, we do not need to consider the
bound on direct detection, because it does not interact with quark sector
directly. The dominant cross section arises from $G$ and $H$ terms that also
appear in muon $g-2$ and have s-wave dominant. But this contribution is at
most ${\cal O}(5\times 10^{-12})$ GeV-2 that is too small to resolve the
correct relic density. Thus, our promising DM candidate is bosonic. Here, let
us suppose $\eta_{R}$ to be DM, and we simply neglect any interactions coming
from Higgs potential in order to evade any bounds from direct detection
experiments. 666This assumption would be reasonable since these couplings are
highly suppressed by bound on direct detection searches. The coupling among DM
and Higgses are of the order $10^{-3}$ Kanemura:2010sh . Moreover, we assume
that the mass difference between $\eta_{R}$ and $\eta_{I}$ is more than 200
keV in order to evade the inelastic direct detection bounds via $Z$ boson
portal Barbieri:2006dq . Similar to the result of fermionic case, we cannot
rely on any contributions from Yukawa couplings because these constributions
are too tiny to explain the relic density. Thus, we have to make the use of
kinetic interactions from the SM. In this situation, nature of the DM is
seriously analyzed by e.g. Ref. Hambye:2009pw . The solutions are uniquely
found at the points of half of the Higgs mass; $\sim 63$ GeV and 534 GeV where
coannihilation processes such as $\eta_{R}\eta_{I}\to Z\to\bar{f}f$ are taken
in consideration.

Acknowledgments The work is supported in part by KIAS Individual Grants, Grant
No. PG074202 (JK) and No. PG076201 (DK) at Korea Institute for Advanced Study.
This research was supported by an appointment to the JRG Program at the APCTP
through the Science and Technology Promotion Fund and Lottery Fund of the
Korean Government. This was also supported by the Korean Local Governments -
Gyeongsangbuk-do Province and Pohang City (H.O.). H.O. is sincerely grateful
for all the KIAS members.

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