# The Neutrinoless Double Beta Decay in the Colored Zee-Babu Model

Shao-Long Chen<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton
Physics (MoE) and Institute of Particle Physics, Central China Normal
University, Wuhan 430079, China  Center for High Energy Physics, Peking
University, Beijing 100871, China  Yu-Qi Xiao<EMAIL_ADDRESS>Key
Laboratory of Quark and Lepton Physics (MoE) and Institute of Particle
Physics, Central China Normal University, Wuhan 430079, China

###### Abstract

We study the neutrinoless double beta decay in the colored Zee-Babu model. We
consider three cases of the colored Zee-Babu model with a leptoquark and a
diquark introduced. The neutrino masses are generated at two-loop level, and
the constraints given by tree-level flavor violation processes and muon
anomalous magnetic moment $(g-2)_{\mu}$ have been considered. In our numerical
analysis, we find that the standard light neutrino exchange contribution can
be canceled by new physics contribution under certain assumption and
condition, leading to a hidden neutrinoless double beta decay. The condition
can be examined comprehensively by future complementary searches with
different isotopes.

##  I Introduction

It is widely assumed that the tiny masses of neutrinos could be generated
radiatively where neutrinos are Majorana particles. The Majorana neutrino mass
models at two-loop level have been discussed in many previous works, e.g.,
Ref. [1, 2, 3, 4, 5, 6, 7, 8], among which the Zee-Babu model [3, 4] has
attracted much attention. The addition of new particles in loops can bring us
rich phenomena. However, whether neutrinos are Majorana particles or Dirac
particles still remains unknown. The search for the neutrinoless double beta
($0\nu\beta\beta$) decay is the promising way to get us out of this dilemma.

The $0\nu\beta\beta$ decay can be realized if neutrinos are Majorana
particles. If one only consider the standard light neutrino exchange, the
inverse half-life has the form

$\displaystyle\left[T_{1/2}^{0\nu\beta\beta}\right]^{-1}=G_{\nu}|\mathcal{M}_{\nu}|^{2}\dfrac{|\langle
m_{ee}\rangle|^{2}}{m_{e}^{2}}\,,$ (1)

where $G_{\nu}$ and $\mathcal{M}_{\nu}$ are the phase space factor (PSF) and
nuclear matrix element (NME), $\langle
m_{ee}\rangle\equiv\sum_{i}U_{ei}^{2}m_{i}$ is the effective neutrino mass,
and $m_{i}~{}(i=1,2,3)$ are the masses of neutrinos. The most stringent limit
on the $0\nu\beta\beta$ decay half-life in ${}^{136}{\rm{Xe}}$ isotope is
$T_{1/2}^{0\nu\beta\beta}>1.07\times 10^{26}~{}{\rm{yrs}}$ given by KamLAND-
Zen experiment [9]. They obtained a constraint of $|\langle
m_{ee}\rangle|<61-165~{}{\rm{meV}}$. The GERDA experiment has published their
result $T_{1/2}^{0\nu\beta\beta}>1.8\times 10^{26}~{}{\rm{yrs}}$ with isotope
${}^{76}{\rm{Ge}}$ leading to a similar bound $|\langle
m_{ee}\rangle|<79-180~{}{\rm{meV}}$ [10]. The future $0\nu\beta\beta$ decay
experiments CUPID-1T [11] and LEGEND-1000 [12] using 100Mo and 76Ge isotopes
can push the half-life to $10^{27}-10^{28}~{}{\rm{yrs}}$, leading to a
sensitivity to $|\langle m_{ee}\rangle|$ of around $15~{}{\rm{meV}}$.

In the effective field theory approach, the $0\nu\beta\beta$ decay can be
described in terms of effective low-energy operators [13, 14, 15, 16, 17, 18].
The contributions to $0\nu\beta\beta$ decay can be divided into long-range
mechanisms [19, 20, 21, 22] and short-range mechanisms [23, 24]. The long-
range mechanisms involve light neutrinos exchanged between two point-like
vertices, which contain the standard light neutrino exchange. The short-range
mechanisms involve the dim-9 effective interaction and are mediated by heavy
particles. The decomposition of the short-range operators at tree-level has
been completely listed in [25], and at one-loop level has been discussed in
[26]. The short-range mechanisms at the LHC have been considered in [27]. An
analysis of the standard light neutrino exchange and short-range mechanisms
has been given in [28].

In this work, we study three cases of the colored Zee-Babu (cZB) model with a
leptoquark and a diquark running in the loops. We consider the realization of
tiny neutrino mass and contribution to $0\nu\beta\beta$ decay, with the
constraints given by tree-level flavor violation processes and $(g-2)_{\mu}$
considered. The B physics anomalies and some other phenomena in the cZB model
have been explored in [29, 30, 31, 32, 33, 34, 35, 36]. The long-range
contributions given by the leptoquarks have been considered extensively in the
previous discussion [37, 38, 18]. We focus on the short-range impact on
neutrinoless double beta decay in this model. The simultaneous introduction of
the leptoquarks and diquarks in the cZB model can lead to short-range
contribution, which can interfere with the standard light neutrino exchange
contribution resulting in cancellation.

We organize our paper as follows. In Sec. II, we show the three cases in the
cZB model and briefly review the constraints on them. In Sec. III, we discuss
the $0\nu\beta\beta$ decay in each case, including short-range mechanisms and
standard light neutrino exchange. Finally, we give our conclusion in Sec. IV.

##  II The Model and Constraints

The colored Zee-Babu (cZB) model requires a leptoquark and a diquark to
generate the neutrino mass. The diquark is set to be a color sextet under
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ symmetry where the hypercharge $Y$
is set to be equal to $Q-I_{3}$. The color triplet is not considered because
the coupling of fermions and diquark is antisymmetric, which means the vertex
$\overline{d^{c}}d\omega$ is zero and cannot contribute to the neutrinoless
double beta decay process. The cZB model with a leptoquark (LQ) and a color
sextet diquark (DQ) has three cases :

case 1: a singlet LQ $S_{1}\sim(\overline{3},1,1/3)$ and a singlet DQ
$\omega_{1}\sim(6,1,-2/3)$,

case 2: a triplet LQ $S_{3}\sim(\overline{3},3,1/3)$ and a singlet DQ
$\omega_{1}\sim(6,1,-2/3)$,

case 3: a doublet LQ $\tilde{S}_{2}\sim(3,2,1/6)$ and a triplet DQ
$\omega_{3}\sim(6,3,1/3)$.

The corresponding quantum numbers of the particles in these cases are
summarized in Table 1.

SM Particles | Quantum Number | New Particles | Quantum Number
---|---|---|---
$\Phi=(\phi^{+},\phi^{0})^{T}$ | $(1,2,1/2)$ | $S_{1}$ | $(\overline{3},1,1/3)$
$Q_{L}=(U_{L},D_{L})^{T}$ | $(3,2,1/6)$ | $\tilde{S}_{2}$ | $(3,2,1/6)$
$L_{L}=(\nu_{L},E_{L})^{T}$ | $(1,2,-1/2)$ | $S_{3}$ | $(\overline{3},3,1/3)$
$U_{R}$ | $(3,1,+2/3)$ | $\omega_{1}$ | $(6,1,-2/3)$
$D_{R}$ | $(3,1,-1/3)$ | $\omega_{3}$ | $(6,3,1/3)$
$E_{R}$ | $(1,1,-1)$ |  |
Table 1: The corresponding quantum numbers of the SM particles and new
particles under the gauge symmetry $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$.

In the fermion weak eigenbasis, the Yukawa interactions of these cases can be
written as

$\displaystyle-\mathcal{L}_{Y1}\supset\,$ $\displaystyle
y^{ij}_{1SR}\overline{(U_{R}^{i\alpha})^{c}}E_{R}^{j}S_{1}^{\overline{\alpha}}+y^{ij}_{1SL}\overline{(Q_{L}^{i\alpha})^{c}}i\sigma^{2}L_{L}^{j}S_{1}^{\overline{\alpha}}+z^{ij}_{1SR}\overline{(U_{R}^{i\alpha})^{c}}D_{R}^{j\beta}S_{1}^{*\gamma}\epsilon^{\alpha\beta\gamma}$
$\displaystyle+z^{ij}_{1SL}\overline{(Q_{L}^{i\alpha})^{c}}i\sigma^{2}Q_{L}^{j\beta}S_{1}^{*\gamma}\epsilon^{\alpha\beta\gamma}+z^{ij}_{1\omega}\overline{(D_{R}^{i\alpha})^{c}}D_{R}^{j\beta}\omega_{1}^{*\overline{\alpha}\overline{\beta}}+\text{h.c.}\,,$
(2) $\displaystyle-\mathcal{L}_{Y2}\supset\,$ $\displaystyle
y^{ij}_{3S}\overline{(Q_{L}^{i\alpha})^{c}}i\sigma^{2}(\sigma^{k}S_{3}^{k\overline{\alpha}})L_{L}^{j}+z^{ij}_{3S}\overline{(Q_{L}^{i\alpha})^{c}}i\sigma^{2}(\sigma^{k}S_{3}^{*k\beta})^{T}Q_{L}^{j\gamma}\epsilon^{\alpha\beta\gamma}$
$\displaystyle+z^{ij}_{1\omega}\overline{(D_{R}^{i\alpha})^{c}}D_{R}^{j\beta}\omega_{1}^{*\overline{\alpha}\overline{\beta}}+\text{h.c.}\,,$
(3) $\displaystyle-\mathcal{L}_{Y3}\supset\,$ $\displaystyle
y^{ij}_{2S}\overline{D_{R}^{i\alpha}}(\tilde{S}_{2}^{\alpha})^{T}i\sigma^{2}L_{L}^{j}+z^{ij}_{3\omega}\overline{(Q_{L}^{i\alpha})^{c}}i\sigma^{2}(\sigma^{k}\omega_{3}^{*k\overline{\alpha}\overline{\beta}})^{T}Q_{L}^{j\beta}+\text{h.c.}\,,$
(4)

where

$\displaystyle\sigma^{k}S_{3}^{k}=\begin{pmatrix}S_{3}^{+1/3}&\sqrt{2}S_{3}^{+4/3}\\\
\sqrt{2}S_{3}^{-2/3}&-S_{3}^{+1/3}\end{pmatrix}\,,\quad\sigma^{k}\omega_{3}^{k}=\begin{pmatrix}\omega_{3}^{+1/3}&\sqrt{2}\omega_{3}^{+4/3}\\\
\sqrt{2}\omega_{3}^{-2/3}&-\omega_{3}^{+1/3}\end{pmatrix}\,,$ (5)

$\psi^{c}$ is the charge conjugate of $\psi$, $i$ and $j$ label fermion
generations, and $(\alpha,\beta,\gamma)$ denote the color of $SU(3)_{c}$. The
$\sigma^{k}~{}(k=1,2,3)$ are Pauli matrices, $S_{3}^{k}$ and $\omega_{3}^{k}$
are the componets of $S_{3}$ and $\omega_{3}$ under $SU(2)_{L}$ symmetry, and
$\epsilon^{\alpha\beta\gamma}$ is the Levi-Civita symbol. The Yukawa coupling
matrices $z_{1SL}$, $z_{1\omega}$, and $z_{3\omega}$ are symmetric, $z_{3S}$
is antisymmetric, while the other coupling matrices are arbitrary [39]. To
study the phenomenologies, we rewrite the Lagrangian in the mass eigenbasis
as,

$\displaystyle-\mathcal{L}_{Y1}\supset\,$ $\displaystyle
y^{ij}_{1SR}\overline{(U_{R}^{i\alpha})^{c}}E_{R}^{j}S_{1}^{\overline{\alpha}}+(V^{*}y_{1SL})^{ij}\overline{(U_{L}^{i\alpha})^{c}}E_{L}^{j}S_{1}^{\overline{\alpha}}$
$\displaystyle-(y_{1SL}U)^{ij}\overline{(D_{L}^{i\alpha})^{c}}\nu_{L}^{j}S_{1}^{\overline{\alpha}}+z^{ij}_{1SR}\overline{(U_{R}^{i\alpha})^{c}}D_{R}^{j\beta}S_{1}^{*\gamma}\epsilon^{\alpha\beta\gamma}$
$\displaystyle+2(V^{*}z_{1SL})^{ij}(\overline{U_{L}^{i\alpha})^{c}}D_{L}^{j\beta}S_{1}^{*\gamma}\epsilon^{\alpha\beta\gamma}+z^{ij}_{1\omega}\overline{(D_{R}^{i\alpha})^{c}}D_{R}^{j\beta}\omega^{*\overline{\alpha}\overline{\beta}}+\text{h.c.}\,,$
(6) $\displaystyle-\mathcal{L}_{Y2}\supset\,$
$\displaystyle\sqrt{2}(V^{*}y_{3S}U)^{ij}\overline{(U_{L}^{i\alpha})^{c}}S_{3}^{-2/3,\overline{\alpha}}\nu_{L}^{j}-(y_{3S}U)^{ij}\overline{(D_{L}^{i\alpha})^{c}}S_{3}^{+1/3,\overline{\alpha}}\nu_{L}^{j}$
$\displaystyle-(V^{*}y_{3S})^{ij}\overline{(U_{L}^{i\alpha})^{c}}S_{3}^{+1/3,\overline{\alpha}}E_{L}^{j}-\sqrt{2}y_{3S}^{ij}\overline{(D_{L}^{i\alpha})^{c}}S_{3}^{+4/3,\overline{\alpha}}E_{L}^{j}$
$\displaystyle-2(z_{3S}V^{\dagger})^{ij}\overline{(D_{L}^{i\alpha})^{c}}S_{3}^{-1/3,\beta}U_{L}^{j\gamma}\epsilon^{\alpha\beta\gamma}+\sqrt{2}(V^{*}z_{3S}V^{\dagger})^{ij}\overline{(U_{L}^{i\alpha})^{c}}S_{3}^{-4/3,\beta}U_{L}^{j\gamma}\epsilon^{\alpha\beta\gamma}$
$\displaystyle-\sqrt{2}z_{3S}^{ij}\overline{(D_{L}^{i\alpha})^{c}}S_{3}^{+2/3,\beta}D_{L}^{j\gamma}\epsilon^{\alpha\beta\gamma}+z^{ij}_{1\omega}\overline{(D_{R}^{i\alpha})^{c}}D_{R}^{j\beta}\omega_{1}^{*\overline{\alpha}\overline{\beta}}+\text{h.c.}\,,$
(7) $\displaystyle-\mathcal{L}_{Y3}\supset\,$
$\displaystyle(y_{2S}U)^{ij}\overline{D_{R}^{i\alpha}}\tilde{S}_{2}^{-1/3,\overline{\alpha}}\nu_{L}^{j}-y^{ij}_{2S}\overline{D_{R}^{i\alpha}}\tilde{S}_{2}^{+2/3,\overline{\alpha}}E_{L}^{j}$
$\displaystyle-2(z_{3\omega}V^{\dagger})^{ij}\overline{(D_{L}^{i\alpha})^{c}}\omega_{3}^{-1/3,\overline{\alpha}\overline{\beta}}U_{L}^{j\beta}+\sqrt{2}(V^{*}z_{3\omega}V^{\dagger})^{ij}\overline{(U_{L}^{i\alpha})^{c}}\omega_{3}^{-4/3,\overline{\alpha}\overline{\beta}}U_{L}^{j\beta}$
$\displaystyle-\sqrt{2}z_{3\omega}^{ij}\overline{(D_{L}^{i\alpha})^{c}}\omega_{3}^{+2/3,\overline{\alpha}\overline{\beta}}D_{L}^{j\beta}+\text{h.c.}\,.$
(8)

Here we follow the basis transition
$U_{L}^{j}\rightarrow(V^{\dagger})_{jk}U_{L}^{k},~{}D_{L}^{j}\rightarrow
D_{L}^{j},~{}E_{L}^{j}\rightarrow E_{L}$ and $\nu_{L}^{j}\rightarrow
U_{jk}\nu_{L}^{k}$ in [40], where $V$ is the Cabibbo-Kobayashi-Maskawa (CKM)
matrix, and $U$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. The
scalar potential involving the leptoquark and the diquark fields contains
cubic terms

$\displaystyle V_{1}$
$\displaystyle\supset\mu_{1}S_{1}^{\overline{\alpha}}S_{1}^{\overline{\beta}}\omega_{1}^{\alpha\beta}+\text{h.c.}\,,$
(9) $\displaystyle V_{2}$
$\displaystyle\supset\mu_{2}(S_{3}^{\overline{\alpha}})^{T}S_{3}^{\overline{\beta}}\omega_{1}^{\alpha\beta}+{\text{h.c.}}$
$\displaystyle=2\mu_{2}S_{3}^{-2/3,\overline{\alpha}}S_{3}^{+4/3,\overline{\beta}}\omega_{1}^{-2/3,\alpha\beta}+\mu_{2}S_{3}^{+1/3,\overline{\alpha}}S_{3}^{+1/3,\overline{\beta}}\omega_{1}^{-2/3,\alpha\beta}+{\text{h.c.}}\,,$
(10) $\displaystyle V_{3}$
$\displaystyle\supset\mu_{3}(\tilde{S}_{2}^{\alpha})^{T}(\sigma^{k}\omega_{3}^{k,\alpha\beta})^{*}i\sigma_{2}\tilde{S}_{2}^{\beta}+{\text{h.c.}}$
$\displaystyle=\mu_{3}\tilde{S}_{2}^{+2/3,\alpha}\omega_{3}^{-1/3,\overline{\alpha}\overline{\beta}}\tilde{S}_{2}^{-1/3,\beta}+\sqrt{2}\mu_{3}\tilde{S}_{2}^{-1/3,\alpha}\omega_{3}^{+2/3,\overline{\alpha}\overline{\beta}}\tilde{S}_{2}^{-1/3,\beta}$
$\displaystyle-\sqrt{2}\mu_{3}\tilde{S}_{2}^{+2/3,\alpha}\omega_{3}^{-4/3,\overline{\alpha}\overline{\beta}}\tilde{S}_{2}^{+2/3,\beta}+\mu_{3}\tilde{S}_{2}^{-1/3,\alpha}\omega_{3}^{-1/3,\overline{\alpha}\overline{\beta}}\tilde{S}_{2}^{+2/3,\beta}+{\text{h.c.}}\,.$
(11)

For simplicity, we assume the quartic couplings of leptoquark and the SM Higgs
doublet $\Phi$ to be vanishing for case 2 and case 3. In addition, the quartic
couplings of $\omega_{3}$ and $\Phi$ are also assumed to be negligible. The
leptoquark/diquark multiplets are then degenerate in mass, we denote the
masses of $S_{1}$, $S_{2}$, $S_{3}$, $\omega_{1}$, and $\omega_{3}$ as
$M_{S_{1}}$, $M_{S_{2}}$, $M_{S_{3}}$, $M_{\omega_{1}}$, and $M_{\omega_{3}}$,
respectively. In our numeraical analysis, the masses of the leptoquarks and
diquarks are taken as $M_{S}\gtrsim 1.5$ TeV and $M_{\omega}\gtrsim 8$ TeV,
which accords with the bounds given by the ATLAS and CMS collaboration [41,
42, 43, 44, 45, 46, 47].

###  II.1 Neutrino masses

$\nu_{L}$$\nu_{L}$$\omega_{1}^{-2/3}$$D_{R}$$D_{L}$$D_{R}$$D_{L}$$S_{1,3}^{+1/3}$$S_{1,3}^{+1/3}$$\times$$\times$

$\nu_{L}$$\nu_{L}$$\omega_{3}^{+2/3}$$D_{L}$$D_{R}$$D_{L}$$D_{R}$$\tilde{S}_{2}^{-1/3}$$\tilde{S}_{2}^{-1/3}$$\times$$\times$

Figure 1: The two-loop diagrams which generate tiny neutrino masses in the
colored Zee-Babu model. The left diagram corresponds to cases 1 and 2, and the
right corresponds to case 3.

In the colored Zee-Babu model with a leptoquark and a diquark, the neutrino
masses can be generated at two-loop level with down-type quarks running in the
loop as shown in Fig. 1. The left Feynman diagram corresponds to case 1 and
case 2, and the right one to case 3. The neutrino mass matrix elements in
flavor basis take the form [29]

$\displaystyle M_{\nu_{a}}^{kn}=$ $\displaystyle
24\mu_{a}[y_{bS(L)}^{T}]^{kl}m_{D^{l}}z_{c\omega}^{lm}m_{D^{m}}y_{bS(L)}^{mn}\mathcal{I}_{lm}$
(12) $\displaystyle=$ $\displaystyle
24\mu[y_{S(L)}^{T}]^{kl}m_{D^{l}}z_{\omega}^{lm}m_{D^{m}}y_{S(L)}^{mn}\mathcal{I}_{lm}\,,$
(13)

where $m_{D^{l}}$ is the mass of the $l$-th generation down-type quark. The
superscripts $k,l,m,n=1,2,3$ and the subscripts $a,b,c=1,2,3$ of the couplings
are neglected to keep the expression concise. The expression can apply to all
three cases. Note that the coupling $y_{S(L)}$ equals $y_{1SL}$ in case 1,
$y_{3S}$ in case 2, and $y_{2S}$ in case 3. The $\mathcal{I}_{lm}$ in Eq. (13)
is loop integral

$\displaystyle\mathcal{I}_{lm}=\int\dfrac{d^{4}k}{(2\pi)^{4}}\int\dfrac{d^{4}q}{(2\pi)^{4}}\dfrac{1}{q^{2}-m_{D^{l}}^{2}}\dfrac{1}{q^{2}-M_{S}^{2}}\dfrac{1}{k^{2}-M_{S}^{2}}\dfrac{1}{k^{2}-m_{D^{m}}^{2}}\dfrac{1}{(k-q)^{2}-M_{\omega}^{2}}\,,$
(14)

where $M_{S}$ denotes leptoquark $S_{i}$ mass and $M_{\omega}$ is diquark
$\omega_{i}$ mass in different cases. The integral can be simplified as [48]

$\displaystyle\mathcal{I}_{lm}\simeq\dfrac{1}{(16\pi^{2})^{2}}\dfrac{1}{M_{S}^{2}}\tilde{I}\bigg{(}\dfrac{M_{\omega}^{2}}{M_{S}^{2}}\bigg{)}\,,\quad\tilde{I}(r)$
$\displaystyle=-\int_{0}^{1}dx\int_{0}^{1-x}dy\dfrac{1}{x+(r-1)y+y^{2}}\ln{\dfrac{y(1-y)}{x+ry}}\,,$
(15)

with the $\tilde{I}(r)$ can be calculated through the numerical integral way.
The neutrino mass matrix can be diagonalized by the PMNS matrix $U$

$\displaystyle U^{T}{M}_{\nu}U=\hat{M}_{\nu}=\begin{pmatrix}m_{1}&0&0\\\
0&m_{2}&0\\\ 0&0&m_{3}\end{pmatrix}\,,$ (16)

where $m_{1,2,3}$ are the masses of active neutrinos $\nu_{1,2,3}$.

###  II.2 Neutron-antineutron oscillation and proton decay

$u$$d$$\overline{u}$$\overline{d}$$S^{+1/3}$$S^{+1/3}$$\omega^{-2/3}$$d$$\overline{d}$$\overline{n}$$n$

$u$$d$$e^{+}(\mu^{+})$$\overline{u}$$S^{+1/3}$$u$$u$$\pi^{0}$$p$

Figure 2: Feynman diagrams of neutron-antineutron oscillation (left) and
proton decay $p\to\pi^{0}e^{+}(\mu^{+})$ (right) in case 1 and case 2. For
case 1, $S^{+1/3}=S_{1}^{1/3}$ and $\omega^{-2/3}=\omega_{1}^{-2/3}$. For case
2, $S^{+1/3}=S_{3}^{+1/3}$ and $\omega^{-2/3}=\omega_{3}^{-2/3}$.

With the leptoquark coupling $z_{S}^{11}\neq 0$ ($z_{S}$ corresponds to
$z_{1SR}$ and $V^{*}z_{1SL}$ in case 1 and $V^{*}z_{3S}$ in case 2), case 1
and case 2 can lead to neutron-antineutron oscillation, as shown in Fig. 2.
The transition rate of neutron-antineutron oscillation $\tau^{-1}$ is
proportional to $|(z_{S}^{11})^{2}z_{\omega}^{11}|$. Using the current limit
$\tau\geq 4.7\times 10^{8}~{}{\rm{s}}$ given by the Super-Kamiokande (Super-K)
experiment [49], one can get the bounds

$\displaystyle\left|(z_{S}^{11})^{2}z_{\omega}^{11}\right|\leq 1.4\sim
1.7\times
10^{-16}\times\left[\dfrac{M_{S}^{4}M^{2}_{\omega}}{\mu\cdot{\rm{TeV}}^{5}}\right]\,.$
(17)

With $M_{S}\sim 1.5~{}{\rm{TeV}}$ and $M_{\omega}\sim 8~{}{\rm{TeV}}$,
$|(z_{S}^{11})^{2}z_{\omega}^{11}|\lesssim 10^{-14}$. The construction of
operators and detailed calculation of neutron-antineutron oscillation can be
found in [50, 51, 52, 53, 54, 55, 56, 57, 58]. However, the proton will decay
when $z_{S}^{11}$ and $y_{S}^{11}$ are both set nonzero. For example, the non-
zero couplings can contribute to the process
$p\rightarrow\pi^{0}e^{+}(\mu^{+})$ as shown in Fig. 2. With the experimental
limit $\tau/B(p\to\pi^{0}e^{+})>2.4\times 10^{34}~{}{\rm{yrs}}$ and
$\tau/B(p\to\pi^{0}\mu^{+})>1.6\times 10^{34}~{}{\rm{yrs}}$ given by Super-K
[59], there is a strict bound on the couplings with the matrix element inputs
from lattice [60, 61]

$\displaystyle|z_{S}^{11}y_{S}^{11}|\lesssim
10^{-26}\times\left[\dfrac{M_{S}}{\rm{TeV}}\right]^{2}\,.$ (18)

where $y_{S}$ denotes $V^{*}y_{1SL}$ and $y_{1SR}$ in case 1 and $V^{*}y_{3S}$
in case 2. One can find that if $M_{S}\sim 1.5~{}{\rm{TeV}}$ and
$y_{S}^{11}\sim 0.01-1$, then $z_{S(L/R)}^{11}$ needs to be at
$\mathcal{O}(10^{-26}-10^{-24})$ scale to avoid an inappropriate proton decay,
leading to an unobservable neutron-antineutron oscillation. So in the
discussion of cases 1 and 2, we assume $z_{S}=0$ to escape proton decay.

###  II.3 The texture setup and constraints

We show our texture zeros setup of the couplings matrices and list the bounds
on the couplings, which are related to $0\nu\beta\beta$ decay in this
subsection. The bounds are derived from muon anomalous magnetic moment and
tree level flavor violation processes with four-fermion interactions
considered.

####  II.3.1 Texture setup

The standard parameterization of the PMNS matrix is

$\displaystyle U=\begin{pmatrix}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{pmatrix}\begin{pmatrix}c_{13}&0&s_{13}e^{-i\delta}\\\
0&1&0\\\
-s_{13}e^{i\delta}&0&c_{13}\end{pmatrix}\begin{pmatrix}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}1&0&0\\\
0&e^{i\eta_{1}}&0\\\ 0&0&e^{i\eta_{2}}\end{pmatrix}\,,$ (19)

where $c_{ij}(s_{ij})$ denotes $\cos{\theta_{ij}}(\sin{\theta_{ij}})$,
$\delta$ is the CP phase, and $\eta_{1,2}$ are the extra phases if neutrino
are Majorana particles. The best fit values of these neutrino oscillation
parameters have been derived in [62, 63, 64, 65]. As there are no information
about the Majorana phases ranges, they can varies from 0 to $2\pi$ freely. To
evade constraints from various lepton flavor violation (LFV) processes, we
adopt the Yukawa coupling matrices in case 1 as

$\displaystyle
y_{1SL}=V^{T}\begin{pmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\\#}&0&0\\\
0&0&\\#\\\
\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}{\\#}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\\#}&\\#\
\end{pmatrix}\,,\quad
y_{1SR}=\begin{pmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\\#}&0&0\\\
0&0&0\\\
0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\\#}&0\end{pmatrix}\,,\quad
z_{1\omega}=\begin{pmatrix}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\\#}&0&0\\\
0&0&\\#\\\ 0&\\#&\\#\end{pmatrix}\,.$ (20)

The matrix $y_{1SL}$ and $z_{1\omega}$ are set to be complex and $y_{1SR}$ is
real. The contribution to neutrinoless double beta decay can survive when
couplings $(V^{*}y_{1SL})^{11}$ $\equiv y_{1SL}^{\prime 11}$, $y_{1SR}^{11}$
and $z_{\omega}^{11}$ (in blue) are set nonzero. Moreover, we have set
$y_{1SL}^{\prime 32}$, $y_{1SR}^{32}\neq 0$ (in red) to obtain the muon
anomalous magnetic moment $(g-2)_{\mu}$. The entries of couplings
$y_{1SL}^{\prime 23},y_{1SL}^{\prime 33},z_{1\omega}^{33},z_{1\omega}^{23}$
provide enough independent parameters to generate appropriate neutrino mass
matrix. It is noted that the first component of the neutrino mass matrix is
negligible under these entries with the constraint from the tree-level flavor
violation processes $|y_{1SL}^{\prime 11}|<0.12$, as shown in the following
subsection, leading to an inconspicuous standard light neutrino exchange
$0\nu\beta\beta$ decay. Hence we introduce $y_{1SL}^{\prime 31}$ (in teal) to
open the standard $0\nu\beta\beta$ decay.

The texture zeros setup of coupling matrices in case 2 and case 3 are similar
to those in case 1

$\displaystyle y_{3S}=V^{T}\begin{pmatrix}\\#&0&0\\\ 0&0&\\#\\\
\\#&\\#&\\#\end{pmatrix}\,,z_{1\omega}=\begin{pmatrix}\\#&0&0\\\ 0&0&\\#\\\
0&\\#&\\#\end{pmatrix}\,,y_{2S}=\begin{pmatrix}\\#&0&0\\\ 0&0&\\#\\\
\\#&\\#&\\#\end{pmatrix}\,,z_{3\omega}=\begin{pmatrix}\\#&0&0\\\ 0&0&\\#\\\
0&\\#&\\#\end{pmatrix}\,,$ (21)

with $V^{*}y_{3S}\equiv y_{3S}^{\prime}$. Though the coupling $|y_{2S}^{11}|$
could be $\mathcal{O}(1)$ in case 3, we still choose the same form of the
matrix to make the analysis consistently.

####  II.3.2 Constraints

The Yukawa couplings are constrained by tree-level flavor violation processes.
It is natural to work with effective field theory where the effective
Lagrangian can be described with four-fermion interaction operators as the new
particles are at TeV scale. The effective Lagrangian involving $S_{1}$
leptoquark reads [31]

$\displaystyle\mathcal{L}_{{\rm{eff}},1}=$
$\displaystyle\dfrac{1}{2M^{2}_{S_{1}}}\bigg{\\{}(V^{*}y_{1SL})^{*ki}(V^{*}y_{1SL})^{nj}[\overline{E^{i}}\gamma_{\mu}P_{L}E^{j}][\overline{U^{k}}\gamma^{\mu}P_{L}U^{n}]$
$\displaystyle+y_{1SL}^{*ki}y_{1SL}^{nj}[\overline{\nu^{i}}\gamma_{\mu}P_{L}\nu^{j}][\overline{D^{k}}\gamma^{\mu}P_{L}D^{n}]+y_{1SR}^{*ki}y_{1SR}^{nj}[\overline{E^{i}}\gamma_{\mu}P_{R}E^{j}][\overline{U^{k}}\gamma^{\mu}P_{R}U^{n}]$
$\displaystyle-\left[y_{1SL}^{*ki}(V^{*}y_{1SL})^{nj}[\overline{\nu^{i}}\gamma_{\mu}P_{L}E^{j}][\overline{D}^{k}\gamma^{\mu}P_{L}U^{n}]+{\rm{h.c.}}\right]$
$\displaystyle+\left[y_{1SR}^{*ki}y_{1SL}^{nj}[\overline{\nu^{i}}P_{R}E^{j}][\overline{D^{k}}P_{R}U^{n}]+{\rm{h.c.}}\right]$
$\displaystyle+\left[(V^{*}y_{1SL})^{*ki}y_{1SR}^{nj}[\overline{E^{i}}P_{R}E^{j}][\overline{U^{k}}P_{R}U^{n}]+{\rm{h.c.}}\right]$
$\displaystyle-\dfrac{1}{4}\left[y_{1SR}^{*ki}y_{1SL}^{nj}[\overline{\nu^{i}}\sigma_{\mu\nu}P_{R}E^{j}][\overline{D^{k}}\sigma^{\mu\nu}P_{R}U^{n}]+{\rm{h.c.}}\right]$
$\displaystyle-\dfrac{1}{4}\left[(V^{*}y_{1SL})^{*ki}y_{1SR}^{nj}[\overline{E^{i}}\sigma_{\mu\nu}P_{R}E^{j}][\overline{U^{k}}\sigma^{\mu\nu}P_{R}U^{n}]+{\rm{h.c.}}\right]\bigg{\\}}\,.$
(22)

The effective Lagrangians induced by $S_{3}$ and $\tilde{S}_{2}$ can be
written as

$\displaystyle\mathcal{L}_{{\rm{eff}},2}=$
$\displaystyle\dfrac{1}{2M_{S_{3}}^{2}}\bigg{\\{}(V^{*}y_{3S})^{*ki}(V^{*}y_{3S})^{nj}[\overline{E^{i}}\gamma^{\mu}P_{L}E^{j}][\overline{U^{k}}\gamma_{\mu}P_{L}U^{n}]$
$\displaystyle+2(V^{*}y_{3S})^{*ki}(V^{*}y_{3S})^{nj}[\overline{\nu^{i}}\gamma^{\mu}P_{L}\nu^{j}][\overline{U^{k}}\gamma_{\mu}P_{L}U^{n}]$
$\displaystyle+2y_{3S}^{*ki}y_{3S}^{nj}[\overline{E^{i}}\gamma^{\mu}P_{L}E^{j}][\overline{D^{k}}\gamma_{\mu}P_{L}D^{n}]$
$\displaystyle+y_{3S}^{*ki}y_{3S}^{nj}[\overline{\nu^{i}}\gamma^{\mu}P_{L}\nu^{j}][\overline{D^{k}}\gamma_{\mu}P_{L}D^{n}]$
$\displaystyle+\left[y_{3S}^{*ki}(V^{*}y_{3S})^{nj}[\overline{\nu^{i}\gamma_{\mu}P_{L}E^{j}}][\overline{D^{k}}\gamma_{\mu}P_{L}U^{n}]+{\rm{h.c.}}\right]\bigg{\\}}\,,$
(23) $\displaystyle\mathcal{L}_{{\rm{eff}},3}=$
$\displaystyle-\dfrac{y_{2S}^{*ki}y_{2S}^{nj}}{2M_{S_{2}}^{2}}\bigg{\\{}[\overline{E^{i}}\gamma^{\mu}P_{L}E^{j}][\overline{D^{k}}\gamma_{\mu}P_{R}D^{n}]+[\overline{\nu^{i}}\gamma^{\mu}P_{L}\nu^{j}][\overline{D^{k}}\gamma_{\mu}P_{R}D^{n}]\bigg{\\}}\,.$
(24)

The constraints on the Wilson coefficient $\epsilon^{ijkn}$ have been derived
in [66, 67, 68, 69], where

$\displaystyle|\epsilon^{ijkn}|={\rm{Const.}}\times\dfrac{\left|y_{S(L)}^{(\prime)ki}y_{S(L)}^{(\prime)nj}\right|}{4\sqrt{2}G_{F}M_{S}^{2}}\,,$
(25)

with the constant equals $1$ in case 1 and case 3 while the constant takes as
$1$ or $2$ in the case 2. Here we have taken the bounds from [69] and have
listed some of them in Table 2 which are related to the couplings that can
contribute to neutrinoless double beta decay process. However, one needs to
pay attention that bounds on the couplings $|y_{S(L)}^{\prime
ki}y_{S(L)}^{\prime nj}|$ can be derived from the numerical combined
calculation of the bounds on $|y_{S(L)}^{ki}y_{S(L)}^{nj}|$ and
$|y_{S(L)}^{\prime ki}y_{S(L)}^{nj}|$ in case 1 and case 2. Moreover, the
neutral meson mixing can be contributed by the diquarks. The
$B^{0}_{d}-\bar{B}^{0}_{d}$ mixing needs to be concerned under our texture
setup. The 95% allowed range of the couplings [70] is

$\displaystyle|z_{1(3)\omega}^{11}z_{1(3)\omega}^{33}|<4.6(2.3)\times
10^{-5}\times\left(\dfrac{M_{\omega 1(3)}}{{\rm{TeV}}}\right)^{2}\,.$ (26)

Coefficients | Constraints | Coefficients | Constraints | Coefficients | Constraints
---|---|---|---|---|---
$|y_{1SL}^{\prime 11}y_{1SL}^{\prime 11}|$ | $6.88\times 10^{-3}$ | $|y_{1SL}^{\prime 11}y_{1SL}^{\prime 31}|$ | $1.61\times 10^{-2}$ | $|y_{1SL}^{\prime 11}y_{1SL}^{\prime 32}|$ | $2.57\times 10^{-1}$
$|y_{1SL}^{\prime 11}y_{1SL}^{\prime 23}|$ | $3.10\times 10^{-4}$ | $|y_{1SL}^{\prime 11}y_{1SL}^{\prime 33}|$ | $2.41\times 10^{-1}$ | $|y_{1SR}^{11}y_{1SR}^{11}|$ | $6.5\times 10^{-1}$
$|y_{1SR}^{11}y_{1SL}^{\prime 32}|$ | $5.87$ | $|y_{1SR}^{11}y_{1SL}^{\prime 23}|$ | $5.87$ | $|y_{1SR}^{11}y_{1SL}^{\prime 33}|$ | $10.2$
$|y_{3S}^{\prime 11}y_{3S}^{\prime 11}|$ | $6.78\times 10^{-3}$ | $|y_{3S}^{\prime 11}y_{3S}^{\prime 31}|$ | $4.02\times 10^{-3}$ | $|y_{3S}^{\prime 11}y_{3S}^{\prime 32}|$ | $4.66\times 10^{-4}$
$|y_{3S}^{\prime 11}y_{3S}^{\prime 23}|$ | $3.10\times 10^{-4}$ | $|y_{3S}^{\prime 11}y_{3S}^{\prime 33}|$ | $1.38\times 10^{-1}$ |  |
$|y_{2S}^{11}y_{2S}^{11}|$ | $1.78$ | $|y_{2S}^{11}y_{2S}^{31}|$ | $1.32\times 10^{-2}$ | $|y_{2S}^{11}y_{2S}^{32}|$ | $1.32\times 10^{-2}$
$|y_{2S}^{11}y_{2S}^{23}|$ | $3.23\times 10^{-1}$ | $|y_{2S}^{11}y_{2S}^{33}|$ | $2.71\times 10^{-1}$ |  |
Table 2: The constraints on the couplings of the leptoquarks, $y_{1SL}^{\prime
11},y_{1SR}^{11},y_{3SL}^{\prime 11},y_{2S}^{11}$, which can contribute to
$0\nu\beta\beta$ decay. The bounds are taken from [69] with the unit of
$(M_{S}/{\rm{TeV}})^{2}$.

The muon anomalous magnetic moments $a_{\mu}=(g-2)_{\mu}/2$ can also give
information on the couplings. The latest result of muon anomalous magnetic
moment has been presented by Muon $g-2$ collaboration [71] as

$\displaystyle\Delta a_{\mu}\equiv
a_{\mu}^{\text{exp}}-a_{\mu}^{\text{SM}}=(2.51\pm 0.59)\times 10^{-9}\,,$ (27)

which has a 4.2 $\sigma$ discrepancy. The expression of muon anomalous
magnetic moment in case 1 can be simplified as [72]

$\displaystyle\Delta
a_{\mu}(S_{1})\simeq\dfrac{3m^{2}_{\mu}}{8\pi^{2}M_{S_{1}}^{2}}\dfrac{m_{t}}{m_{\mu}}{\rm{Re}}[y_{1SR}^{32}y_{1SL}^{\prime*32}]\left[\dfrac{1}{3}f_{1}\left(\dfrac{m_{t}^{2}}{m_{S_{1}}^{2}}\right)+\dfrac{2}{3}f_{2}\left(\dfrac{m_{t}^{2}}{m_{S_{1}}^{2}}\right)\right]\,.$
(28)

While for case 2 and 3, the contributions are

$\displaystyle\Delta
a_{\mu}(S_{3})\simeq\dfrac{3m^{2}_{\mu}}{8\pi^{2}M_{S_{3}}^{2}}\sum\limits_{q}|y_{3S}^{\prime
q2}|^{2}\left[\dfrac{1}{3}f_{3}\left(\dfrac{m_{q}^{2}}{m_{S_{3}}^{2}}\right)+\dfrac{2}{3}f_{4}\left(\dfrac{m_{q}^{2}}{m_{S_{3}}^{2}}\right)\right]\,,$
$\displaystyle\Delta
a_{\mu}(\tilde{S}_{2})\simeq\dfrac{3m^{2}_{\mu}}{8\pi^{2}M_{S_{2}}^{2}}\sum\limits_{q}|y_{2S}^{q2}|^{2}\left[\dfrac{2}{3}f_{3}\left(\dfrac{m_{q}^{2}}{m_{S_{2}}^{2}}\right)+\dfrac{1}{3}f_{4}\left(\dfrac{m_{q}^{2}}{m_{S_{2}}^{2}}\right)\right]\,,$
(29)

with the functions $f_{i}$ are defined in [72]. To explain the discrepancy,
the couplings in case 1 have the relation
${\rm{Re}}[y_{1SR}^{32}y_{1SL}^{\prime*32}]\sim 8.48\times 10^{-2}$ with the
leptoquark mass $M_{S_{1}}=1.5$ TeV. However, in case 2 and case 3, the
contribution to muon anomalous magnetic moment $\Delta a_{\mu}$ are negligible
since there is no chiral-enhancement $m_{q}/m_{\mu}$ for these two cases.

After taking accounting of all the constraints mentioned, the parameter
regions taken in our numerical analysis are

$\displaystyle{\rm{case~{}1:~{}}}$ $\displaystyle|y_{1SL}^{\prime
11}|<0.12\,,~{}|y_{1SL}^{\prime 31,32}|<0.3\,,~{}|y_{1SL}^{\prime
33}|<0.4\,,~{}|y_{1SL}^{\prime 23}|<0.005\,,$
$\displaystyle|y_{1SR}^{11}|<1.2\,,~{}{\rm{Re}}[y_{1SR}^{32}y_{1SL}^{\prime*32}]\sim
0.0848\,,~{}|z_{1\omega}^{11}|<1.5\,,~{}|z_{1\omega}^{33}|<0.001\,,$ (30)
$\displaystyle{\rm{case~{}2:~{}}}$ $\displaystyle|y_{3S}^{\prime
11}|<0.12\,,~{}|y_{3S}^{\prime 31}|<0.07\,,~{}|y_{3S}^{\prime
32}|<0.008\,,~{}|y_{3S}^{\prime 33}|<0.4\,,$ $\displaystyle|y_{3S}^{\prime
23}|<0.005\,,~{}|z_{1\omega}^{11}|<1.5\,,~{}|z_{1\omega}^{33}|<0.002\,,$ (31)
$\displaystyle{\rm{case~{}3:~{}}}$
$\displaystyle|y_{2S}^{11}|<1.5\,,~{}|y_{2S}^{31}|<0.01\,,~{}|y_{2S}^{32}|<0.01\,,~{}|y_{2S}^{33}|<0.3\,,$
$\displaystyle~{}|y_{2S}^{23}|<0.3\,,~{}|z_{3\omega}^{11}|<0.01\,,~{}|z_{3\omega}^{33}|<0.15\,,$
(32)

and the $z_{i\omega}^{23}(i=1,3)$ in each case are set to be
$|z_{i\omega}^{23}|<1.5$. We take $\mu=M_{S}=1.5~{}{\rm{TeV}}$ and
$M_{\omega}=8$ TeV in our following discussion.

##  III The Neutrinoless Double Beta Decay

The $0\nu\beta\beta$ decay can be divided into short-range and long-range
mechanisms. To study how short-range contributions impact the $0\nu\beta\beta$
decay in the cZB model, we briefly review the general formula of the short-
range mechanisms via the effective field theory approach and give numerical
analysis in this section.

###  III.1 The short-range $0\nu\beta\beta$ decay

The short-range $0\nu\beta\beta$ decay operator can be written as
$\mathcal{O}^{0\nu\beta\beta}\propto\overline{u}\overline{u}dd\overline{e}\overline{e}$,
a dim-9 operator. The scalar mediated tree-level topologies and the
decomposition of this operator has been listed in [25]. We follow the general
parameterization of effective short-range Lagrangian in [28, 23]

$\displaystyle\mathcal{L}_{SR}=\dfrac{G_{F}^{2}V_{ud}^{2}}{2m_{p}}\sum\limits_{X,Y,Z}$
$\displaystyle\bigg{(}\epsilon_{1}^{\chi}J_{X}J_{Y}j_{Z}+\epsilon_{2}^{\chi}J^{\mu\nu}_{X}J_{Y,\mu\nu}j_{Z}+\epsilon_{3}^{\chi}J^{\mu}_{X}J_{Y,\mu}j_{Z}$
$\displaystyle+\epsilon_{4}^{\chi}J^{\mu}_{X}J_{Y,\mu\nu}j^{\nu}+\epsilon_{5}^{\chi}J^{\mu}_{X}J_{Y}j_{\mu}\bigg{)}+{\text{h.c.}}$
(33) $\displaystyle=\dfrac{G_{F}^{2}V_{ud}^{2}}{2m_{p}}\sum\limits_{\chi,i}$
$\displaystyle\epsilon_{i}^{\chi}\mathcal{O}_{i,\chi}^{0\nu\beta\beta}+{\text{h.c.}}\,,$
(34)

where $G_{F}$ is the Fermi constant, $m_{p}$ is the proton mass, $V_{ud}$ is
the $ud$ component of the CKM matrix, and the dimensionless effective
couplings are defined as
$\epsilon_{i}^{\chi}=\epsilon_{i}^{XYZ}~{}(X,Y,Z=R/L)$. The $J$ and $j$,
respectively, denote the quark and electron currents as

$\begin{gathered}J_{R/L}=\overline{u}(1\pm\gamma_{5})d\,,\quad
J_{R,L}^{\mu}=\overline{u}\gamma^{\mu}(1\pm\gamma_{5})d\,,\quad
J_{R/L}^{\mu\nu}=\overline{u}\sigma^{\mu\nu}(1\pm\gamma_{5})d\,,\\\
j_{R/L}=\overline{e}(1\mp\gamma_{5})e^{c},\quad
j^{\mu}=\overline{e}\gamma^{\mu}\gamma_{5}e^{c}\,.\end{gathered}$ (35)

One can express the effective operators in terms of the quark and electron
currents as [25, 24]

$\begin{gathered}\mathcal{O}_{1,XYZ}^{0\nu\beta\beta}\equiv
J_{X}J_{Y}j_{Z}\,,\quad\mathcal{O}_{2,XYZ}^{0\nu\beta\beta}\equiv
J^{\mu\nu}_{X}J_{Y,\mu\nu}j_{Z}\,,\quad\mathcal{O}_{3,XYZ}^{0\nu\beta\beta}\equiv
J^{\mu}_{X}J_{Y,\mu}j_{Z}\,,\\\ \mathcal{O}_{4,XY}^{0\nu\beta\beta}\equiv
J^{\mu}_{X}J_{Y,\mu\nu}j^{\nu}\,,\quad\mathcal{O}_{5,XY}^{0\nu\beta\beta}\equiv
J^{\mu}_{X}J_{Y}j_{\mu}\,.\end{gathered}$ (36)

The following expression gives $0\nu\beta\beta$ decay inverse half-life
involving the short-range mechanism and light-neutrino exchange [28]

$\displaystyle\left[T^{0\nu\beta\beta}_{1/2}\right]^{-1}$
$\displaystyle=G_{11+}^{(0)}\left|\sum\limits_{i=1}^{3}\epsilon_{i}^{XYL}\mathcal{M}^{XY}_{i}+\epsilon_{\nu}\mathcal{M}_{\nu}\right|^{2}+G_{11+}^{(0)}\left|\sum\limits_{i=1}^{3}\epsilon_{i}^{XYR}\mathcal{M}^{XY}_{i}\right|^{2}+G_{66}^{(0)}\left|\sum\limits_{i=4}^{5}\epsilon_{i}^{XY}\mathcal{M}^{XY}_{i}\right|^{2}$
$\displaystyle+G_{16}^{(0)}\times
2{\text{Re}}\left[\left(\sum\limits_{i=1}^{3}\epsilon_{i}^{XYL}\mathcal{M}^{XY}_{i}-\sum\limits_{i=1}^{3}\epsilon_{i}^{XYR}\mathcal{M}^{XY}_{i}+\epsilon_{\nu}\mathcal{M}_{\nu}\right)\left(\sum\limits_{i=4}^{5}\epsilon_{i}^{XY}\mathcal{M}^{XY}_{i}\right)^{*}\right]$
$\displaystyle+G_{11-}^{(0)}\times
2{\text{Re}}\left[\left(\sum\limits_{i=1}^{3}\epsilon_{i}^{XYL}\mathcal{M}^{XY}_{i}+\epsilon_{\nu}\mathcal{M}_{\nu}\right)\left(\sum\limits_{i=1}^{3}\epsilon_{i}^{XYR}\mathcal{M}^{XY}_{i}\right)^{*}\right]\,,$
(37)

where $\epsilon_{i}^{\chi}$ are the effective couplings shown in Eq. (34),
$\mathcal{M}^{XY}_{i}$ are the NMEs with the short-range mechanism and
$\mathcal{M}_{\nu}$ is the NMEs with the light neutrino exchange. The
dimensionless parameters $\epsilon_{\nu}$ can be written as
$\epsilon_{\nu}=\langle m_{ee}\rangle/m_{e}$, where $\langle
m_{ee}\rangle\equiv\sum_{i}U_{ei}^{2}m_{i}$ is the effective Majorana neutrino
mass and $m_{e}$ is the electron mass. The $G_{11+}^{(0)}$, $G_{11-}^{(0)}$,
$G_{16}^{(0)}$, and $G_{66}^{(0)}$ are the phase space factors (PSFs) defined
in [73]. The PSFs numerical values of different isotopes are taken from [28],
as shown in Table 3. The first column is the lower limits for the decay half-
life of different isotopes [10, 74, 75, 76, 77, 9]. The Table 4 shows the
values of light neutrino exchange NME $\mathcal{M}_{\nu}$ and short-range
mechanism NMEs $\mathcal{M}^{XY}_{i}$ within the microscopic interacting boson
model [28]. We just list the values of $\mathcal{M}_{i}^{XX}$ since both of
the quark currents in the cZB models are right-handed, i.e., $X=Y=R$.

Isotope |  $T^{0\nu}_{1/2}$ [$10^{25}$ yrs] | $G_{11+}^{(0)}$ | $G_{11-}^{(0)}$ | $G_{16}^{(0)}$ | $G_{66}^{(0)}$
---|---|---|---|---|---
76Ge | 18 [10] | 2.360 | -0.280 | 0.870 | 1.320
82Se | 0.24 [74] | 10.19 | -0.712 | 2.925 | 5.450
100Mo | 0.15 [75] | 15.91 | -1.053 | 4.456 | 8.482
128Te | 0.011 [76] | 0.585 | -0.156 | 0.313 | 0.371
130Te | 2.2 [77] | 14.20 | -1.142 | 4.367 | 7.672
136Xe | 10.7 [9] | 14.56 | -1.197 | 4.524 | 7.876
Table 3: The lower limits for the decay half-life time of different isotopes from different experiments and the numerical values of the phase space factors (PSFs) in units of $10^{-15}~{}{\text{yr}^{-1}}$ [28]. Isotope | $\mathcal{M}_{1}^{XX}$ | $\mathcal{M}_{2}^{XX}$ | $\mathcal{M}_{3}^{XX}$ | $\mathcal{M}_{4}^{XX}$ | $\mathcal{M}_{5}^{XX}$ | $\mathcal{M}_{\nu}$
---|---|---|---|---|---|---
76Ge | 5300 | -174 | -200 | -158 | 202 | -6.64
82Se | 4030 | -144 | -171 | -134 | 114 | -5.46
100Mo | 12400 | -189 | -124 | -134 | 1230 | -5.27
128Te | 4410 | -134 | -154 | -130 | 205 | -4.80
130Te | 4030 | -122 | -141 | -109 | 187 | -4.40
136Xe | 3210 | -96.1 | -111 | -86.0 | 147 | -3.60
Table 4: The numerical values of the NMEs for short-range operators and light-
neutrino exchange. The values are calculated within the microscopic
interacting boson model and axial coupling quenched $g_{A}=1.0$ [28].

The effective operators in the three cases can be written as

$\displaystyle{\rm{case~{}1:}}\quad$
$\displaystyle(\overline{u_{L}}\overline{e_{L}})(\overline{u_{L}}\overline{e_{L}})(d_{R}d_{R})\,\rightarrow\dfrac{1}{48}\mathcal{O}_{1}^{RRL}-\dfrac{1}{192}\mathcal{O}_{2}^{RRL}\,,$
$\displaystyle(\overline{u_{L}}\overline{e_{L}})(\overline{u_{R}}\overline{e_{R}})(d_{R}d_{R})\,\rightarrow\dfrac{1}{96i}\mathcal{O}_{4}^{RR}-\dfrac{1}{48}\mathcal{O}_{5}^{RR}\,,$
$\displaystyle(\overline{u_{R}}\overline{e_{R}})(\overline{u_{R}}\overline{e_{R}})(d_{R}d_{R})\,\rightarrow-\dfrac{1}{48}\mathcal{O}_{3}^{RRR}\,,$
(38) $\displaystyle{\rm{case~{}2:}}\quad$
$\displaystyle(\overline{u_{L}}\overline{e_{L}})(\overline{u_{L}}\overline{e_{L}})(d_{R}d_{R})\,\rightarrow\dfrac{1}{48}\mathcal{O}_{1}^{RRL}-\dfrac{1}{192}\mathcal{O}_{2}^{RRL}\,,$
(39) $\displaystyle{\rm{case~{}3:}}\quad$
$\displaystyle(\overline{u_{L}}\overline{u_{L}})(d_{R}\overline{e_{L}})(d_{R}\overline{e_{L}})\,\rightarrow\dfrac{1}{48}\mathcal{O}_{1}^{RRL}-\dfrac{1}{192}\mathcal{O}_{2}^{RRL}\,.$
(40)

The corresponding Feynman diagrams in different cases are shown in Fig. 3.

$d_{R}$$d_{R}$$\omega_{1}$$S_{1}$$S_{1}$$u_{L}$$u_{L}$$e_{L}$$e_{L}$

$d_{R}$$d_{R}$$\omega_{1}$$S_{1}$$S_{1}$$u_{L}$$u_{R}$$e_{L}$$e_{R}$

$d_{R}$$d_{R}$$\omega_{1}$$S_{1}$$S_{1}$$u_{R}$$u_{R}$$e_{R}$$e_{R}$

$d_{R}$$d_{R}$$\omega_{1}$$S_{3}$$S_{3}$$u_{L}$$u_{L}$$e_{L}$$e_{L}$

$d_{R}$$d_{R}$$\omega_{3}$$\tilde{S}_{2}$$\tilde{S}_{2}$$u_{L}$$u_{L}$$e_{L}$$e_{L}$

Figure 3: The Feynman diagrams of neutrinoless double beta decay in the
colored Zee-Babu model. The first row corresponds to case 1. The two diagrams
in the second row coreespond to case 2 (left) and case 3 (right).

The effective couplings $\epsilon_{i}^{\chi}$ in different cases are

$\displaystyle{\rm{case~{}1:}}\quad$
$\displaystyle\epsilon_{1}^{RRL}=+\dfrac{1}{48}\dfrac{2m_{p}}{G_{F}^{2}V_{ud}^{2}}\dfrac{4(y_{1SL}^{\prime*11})^{2}z_{1\omega}^{11}\mu_{1}}{M_{S_{1}}^{4}M^{2}_{\omega_{1}}}\,,\quad\epsilon_{2}^{RRL}=-\dfrac{1}{4}\epsilon_{1}^{RRL}\,,$
$\displaystyle\epsilon_{3}^{RRR}=-\dfrac{1}{48}\dfrac{2m_{p}}{G_{F}^{2}V_{ud}^{2}}\dfrac{4(y_{1SR}^{*11})^{2}z_{1\omega}^{11}\mu_{1}}{M_{S_{1}}^{4}M^{2}_{\omega_{1}}}\,,$
$\displaystyle\epsilon_{4}^{RR}=+\dfrac{1}{96i}\dfrac{2m_{p}}{G_{F}^{2}V_{ud}^{2}}\dfrac{4y_{1SR}^{*11}y_{1SL}^{\prime*11}z_{1\omega}^{11}\mu_{1}}{M_{S_{1}}^{4}M^{2}_{\omega_{1}}}\,,\quad\epsilon_{5}^{RR}=-2i\epsilon_{4}^{RR}\,,$
(41) $\displaystyle{\rm{case~{}2:}}\quad$
$\displaystyle\epsilon_{1}^{RRL}=+\dfrac{1}{48}\dfrac{2m_{p}}{G_{F}^{2}V_{ud}^{2}}\dfrac{4(y_{3S}^{\prime*11})^{2}z_{1\omega}^{11}\mu_{2}}{M_{S_{3}}^{4}M^{2}_{\omega_{1}}}\,,\quad\epsilon_{2}^{RRL}=-\dfrac{1}{4}\epsilon_{1}^{RRL}\,,$
(42) $\displaystyle{\rm{case~{}3:}}\quad$
$\displaystyle\epsilon_{1}^{RRL}=-\dfrac{1}{48}\dfrac{2m_{p}}{G_{F}^{2}V_{ud}^{2}}\dfrac{4(y_{2S}^{*11})^{2}(V^{*}z_{3\omega}V^{\dagger})^{11}\mu_{3}}{M_{S_{2}}^{4}M^{2}_{\omega_{3}}}\,,\quad\epsilon_{2}^{RRL}=-\dfrac{1}{4}\epsilon_{1}^{RRL}\,.$
(43)

The effective neutrino mass can be related to the first component of the
neutrino mass matrix [78] and takes form as

$\displaystyle|\langle m_{ee}\rangle|$
$\displaystyle=\big{|}\sum\limits_{i}U_{ei}^{2}m_{i}\big{|}=\big{|}M_{\nu}^{11}\big{|}\simeq\dfrac{3\mu
m_{b}}{16\pi^{4}M_{S}^{2}}\tilde{I}\left(\dfrac{M_{\omega}^{2}}{M_{S}^{2}}\right)\left\\{m_{s}|y_{S(L)}^{21}y_{S(L)}^{31}z_{\omega}^{23}|+m_{b}|[y_{S(L)}^{31}]^{2}z_{\omega}^{33}|\right\\}\,.$
(44)

The term related to $m_{s}$ cannot be neglected because we have assumed that
there is a hierarchy among the couplings $z_{\omega}^{ij}$.

###  III.2 Numerical results

Before we give our numerical results, it is necessary to notice that the QCD
corrections can modify the NMEs [79, 80, 81, 82, 83, 84]. If we consider the
leading order QCD corrections and the numerical values of the RGE
$\mu$-evolution matrix elements with the same chiral quark currents [80]

$\displaystyle\hat{U}^{XX}_{(12)}=\begin{pmatrix}2.39&0.02\\\
-3.83&0.35\end{pmatrix}\,,\quad\hat{U}^{XX}_{(3)}=0.70\,,\quad\hat{U}^{XX}_{(45)}=\begin{pmatrix}0.35&-0.96i\\\
-0.06i&2.39\end{pmatrix}\,,$ (45)

the NMEs need to be recomposited as

$\displaystyle\mathcal{M}_{1}^{XX}\rightarrow\beta_{1}^{XX}$
$\displaystyle=2.39\mathcal{M}_{1}^{XX}-3.83\mathcal{M}_{2}^{XX}\,,$ (46)
$\displaystyle\mathcal{M}_{2}^{XX}\rightarrow\beta_{2}^{XX}$
$\displaystyle=0.02\mathcal{M}_{1}^{XX}+0.35\mathcal{M}_{2}^{XX}\,,$ (47)
$\displaystyle\mathcal{M}_{3}^{XX}\rightarrow\beta_{3}^{XX}$
$\displaystyle=0.70\mathcal{M}_{3}^{XX}\,,$ (48)
$\displaystyle\mathcal{M}_{4}^{XX}\rightarrow\beta_{4}^{XX}$
$\displaystyle=0.35\mathcal{M}_{4}^{XX}-0.06i\mathcal{M}_{5}^{XX}\,,$ (49)
$\displaystyle\mathcal{M}_{5}^{XX}\rightarrow\beta_{5}^{XX}$
$\displaystyle=-0.96i\mathcal{M}_{4}^{XX}+2.39\mathcal{M}_{5}^{XX}\,.$ (50)

The inverse half-life
$[T_{1/2}^{0\nu\beta\beta}]^{-1}(G_{jk},\epsilon_{i},\mathcal{M}_{i})$ have to
be replaced with
$[T_{1/2}^{0\nu\beta\beta}]^{-1}(G_{jk},\epsilon_{i},\beta_{i})$. After
considering the experimental values and substituting the numerical values of
the PSFs and the NMEs shown in Table 3 and 4, one can get the limits on the
couplings.

Figure 4: The contour with the effective neutrino mass $|\langle
m_{ee}\rangle|$ in the unit of ${\rm{eV}}$ and the couplings
$|z_{\omega}^{11}(y_{S}^{11})^{2}|$. The lines show the experimental bound of
$0\nu\beta\beta$ decay half-life in different cases and different isotopes.
The upper and lower corner regions are excluded by KamLAND-Zen and GERDA
experiments. The red and blue regions correspond to the experiments CUPID-1T
and LEGEND-1000. The corresponding experiments are shown in the legends. The
gray region is not valid in our numerical analysis. Here we take
$\mu=M_{S}=1.5~{}{\rm{TeV}}$ and $M_{\omega}=8~{}{\rm{TeV}}$. We assume that
$|y_{1SR}^{11}|=|y_{1SL}^{\prime 11}|$ in case 1 and denote $|y_{1SR}^{11}|$,
$|y_{1SL}^{\prime 11}|$, $|y_{3S}^{\prime 11}|$, and $|y_{2S}^{11}|$ as
$|y_{S}^{11}|$.

Figure 5: The contour with the effective neutrino mass $|\langle
m_{ee}\rangle|$ in the unit of ${\rm{eV}}$ and the couplings
$|z_{\omega}^{11}(y_{S}^{11})^{2}|$ in case 1. Here we take
$\mu=M_{S}=1.5~{}{\rm{TeV}}$ and $M_{\omega}=8~{}{\rm{TeV}}$. We assume that
$|y_{1SR}^{11}|=10|y_{1SL}^{\prime 11}|$(left) and
$|y_{1SR}^{11}|=50|y_{1SL}^{\prime 11}|$(right), where $|y_{1SR}^{11}|$ is
denoted as $|y_{S}^{11}|$.

We show the contour with the effective neutrino mass $|\langle m_{ee}\rangle|$
in the unit of ${\rm{eV}}$ and the couplings
$|z_{\omega}^{11}(y_{S}^{11})^{2}|$ in Fig. 4 and Fig. 5. The purple and green
regions (upper and lower corners) are excluded by the $0\nu\beta\beta$ decay
experiments KamLAND-Zen [9] and GERDA [10]. The red and blue regions
correspond to the survival areas for the experiments CUPID-1T [11] and
LEGEND-1000 [12] if we assume that no signals are found and set the half-
lifetime to $T_{1/2}>5.0\times 10^{25}$ yrs (CUPID) and $5.0\times 10^{26}$
yrs (LEGEND), whereas the inner darker areas relate to the sensitivities. We
show all the three cases in Fig. 4, with the assumption $|y_{1SL}^{\prime
11}|=|y_{1SR}^{11}|$ for case 1. Under this assumption, the term that contains
$\epsilon_{1,2}^{RRL}$ and $\epsilon_{\nu}$ dominants in the inverse half-life
expression. Due to the cancellation between the standard light neutrino
exchange contribution and the new physics contribution, the $0\nu\beta\beta$
decay could be hidden when the relations are linear or almost linear,

$\displaystyle(y_{S}^{*11})^{2}z_{\omega}^{11}\simeq
300\times\dfrac{\mathcal{M}_{\nu}}{\beta_{1}-\beta_{2}/4}\dfrac{\langle
m_{ee}\rangle}{\rm{eV}}\times\left(\dfrac{1.5~{}{\rm{TeV}}}{\mu}\right)\left(\dfrac{M_{S}}{1.5~{}{\rm{TeV}}}\right)^{4}\left(\dfrac{M_{\omega}}{8~{}{\rm{TeV}}}\right)^{2}\,,$
(51)

where the specific value of $\mathcal{M}_{\nu}/(\beta_{1}-\beta_{2}/4)$ varies
from isotope to isotope. For the current experiments, the relation can be
realized successfully due to the similar ratio value of
$\mathcal{M}_{\nu}/(\beta_{1}-\beta_{2}/4)$ in ${}^{76}{\rm{Ge}}$ and
${}^{136}{\rm{Xe}}$ isotopes. Things will be different and intriguing when the
next-generation experiments with ${}^{100}{\rm{Mo}}$ isotope, e.g. AMoRE-II
[85] and CUPID-1T, push the half-life to be order of $10^{26}~{}{\rm{yrs}}$.
The slope of the band with ${}^{100}{\rm{Mo}}$ is different from
${}^{76}{\rm{Ge}}$ or ${}^{136}{\rm{Xe}}$, leading to the overlap of survival
areas being narrowed. With the high sensitivity of the future CUPID-1T and
LEGEND-1000 experiments, the survival band can be examined comprehensively. If
there is no signal of $0\nu\beta\beta$ decays, the survival region will be
reduced to the overlap area. On the other hand, if we see the signals in one
experiment, the corresponding contour lines will be suitable. The other
experiment can help us search for the appropriate region of the lines.

We show the contour of case 1 with assumption $|y_{1SL}^{\prime
11}|\ll|y_{1SR}^{11}|\equiv|y_{S}^{11}|$ in Fig. 5. This assumption is natural
as the allowed regions have ten times difference, and the influence on the
neutrino mass is negligible. The survival region is elliptical instead. The
left panel refers to $|y_{1SR}^{11}|=10|y_{1SL}^{\prime 11}|$. One can find
that the constraint on the effective Majorana neutrino mass can be larger than
the one with only standard neutrino exchange considered $|\langle
m_{ee}\rangle|\lesssim 0.2~{}{\rm{eV}}$. The experiment with
${}^{100}{\rm{Mo}}$ isotopes can help to reduce the survival area which is
similar to what we discussed before. The right panel refers to
$|y_{1SR}^{11}|=50|y_{1SL}^{\prime 11}|$ and the effect of the combined
analysis with different experiments is not apparent. The bound on the
couplings given by experiments KamLAND-Zen and GERDA are

$\displaystyle|z_{\omega}^{11}(y_{S}^{11})^{2}|<1.8\times\left(\dfrac{1.5~{}{\text{TeV}}}{\mu}\right)\left(\dfrac{M_{S}}{1.5~{}\text{TeV}}\right)^{4}\left(\dfrac{M_{\omega}}{8~{}\text{TeV}}\right)^{2}\,.$
(52)

The limits will be more stringent in next generation $0\nu\beta\beta$ decay
experiments, which have the potential to restrict
$|z_{\omega}^{11}(y_{S}^{11})^{2}|$ at $\mathcal{O}({10^{-1}})$ scale.

##  IV Summary

In this paper, we have discussed the neutrinoless double beta decay in the
colored Zee-Babu model. We study all three cases for the colored Zee-Babu
model with a leptoquark and a diquark. The tiny neutrino masses are generated
at two-loop level, and neutrinoless double beta decay gets additional
contribution from the leptoquarks. We set some texture zeros for the Yukawa
coupling matrices to evade constraints from various lepton flavor violation
processes. We obtain the allowed regions of parameters after considering the
constraints given by tree-level flavor violation processed and charged lepton
anomalous magnetic moment.

We have discussed the short-range and standard neutrino exchange mechanisms of
neutrinoless double beta decay for each case. The short-range contribution can
be realized at tree-level. The general formula of the short-range
contributions via the effective field theory approach is briefly reviewed. We
adopt the values of nuclear matrix elements calculated with the microscopic
interacting boson model and consider the leading order QCD running correction.
We give numerical analysis for the three cases with the current experimental
results and sensitivities of next-generation experiments. We find that the
neutrinoless double beta decay can be hidden with a linear relation in all the
cases under certain conditions. The relation can be examined by future
$0\nu\beta\beta$ decay experiments. The complementary analysis of the
different isotope experiments can help reduce the overlap area of the survival
region.

Acknowledgements. This work is supported in part by the National Science
Foundation of China (12175082, 11775093).

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Let $A\in\mathcal{H}_{n}$ and let $B$ be the minimal strongly synchronizing
automaton such that there is an automorphism $\phi_{A}$ of the underlying
digraph of $B$ with $A$ the minimal representative of $H(B,\phi_{A})$. Let $D$
be a strongly synchronizing automaton which is obtained by repeated
applications of Lemma 4.13 to the automaton $B$. Let $\psi_{A}$ be the
automorphism of the underlying digraph of $D$ with $A$ the minimal
representative of $H(D,\psi_{A})$. A relabelling $D^{\prime}$ of $D$ is called
_a relabelling through shadows_ if, for the induced automorphism
$\psi^{\prime}_{A}$ of the underlying digraph of $D^{\prime}$, for any state
$q\in Q_{B}$ the set $S_{q,D}$ of shadow states of $q$ are all
$\omega$-equivalent in $H(D^{\prime},\psi^{\prime}_{A})$ to the state $q$.

######  Lemma 4.17.

Let $A\in\mathcal{H}_{n}$ and let $B$ be the minimal strongly synchronizing
automaton such that there is an automorphism $\phi_{A}$ of the underlying
digraph of $B$ with $A$ the minimal representative of $H(B,\phi_{A})$. Let $D$
be a strongly synchronizing automaton which is obtained by repeated
applications of Lemma 4.13 to the automaton $B$. Let $\psi_{A}$ be the
automorphism of underlying digraph of $D$ with $A$ the minimal representative
of $H(D,\psi_{A})$. Let $D^{\prime}$ be a relabelling through shadows of $D$,
$\psi_{A^{\prime}}$ be the induced automorphism of the underlying digraph of
$D^{\prime}$, and $A^{\prime}$ be the minimal representative of
$H(D^{\prime},\psi_{A^{\prime}})$. Then $A^{\prime}$ is conjugate to $A$ and
there is a strongly synchronizing automaton $B^{\prime}$ and an automorphism
$\phi_{A^{\prime}}$ of the underlying digraph $G_{B^{\prime}}$ so that
$A^{\prime}$ is the minimal representative of
$H(B^{\prime},\phi_{A^{\prime}})$, with $G_{B^{\prime}}$ equal to the
underlying digraph of $B$.

Thus, the minimal strongly synchronizing automaton $C$ on which $A^{\prime}$
acts is carried by a digraph $G_{C}$ that is a graph quotient of $G_{B}$ and
we have $|G_{C}|\leq|G_{B}|$.

###### Proof.

As the relabelling process employed is a relabelling through shadows, it
preserves the equivalence of the local maps induced by the graph automorphism
across all shadow states shadowing any particular original state of $B$. In
particular the collapse of each state with all of its shadow states results in
an automaton $B^{\prime}$ which still admits an automorphism
$\psi_{A^{\prime}}$ of its underlying graph (which graph is isomorphic to
$G_{B}$) so that $A^{\prime}$ is the minimal representative of
$H(B^{\prime},\psi_{A^{\prime}})$. The result now follows. $\Box$

######  Lemma 4.18.

Let $A\in\mathcal{H}_{n}$ and let $B$ be the minimal strongly synchronizing
automaton such that there is an automorphism $\phi_{A}$ of the underlying
digraph of $B$ with $A$ the minimal representative of $H(B,\phi_{A})$. Suppose
that all circuits in $B$ are on orbits of length $n$ under the action of $A$.
Let $p\in Q_{B}$, $i\in\mathbb{N}$ be less than or equal to the orbit length
of $p$, and $\gamma\in X_{n}^{\ast}$ be such that $(p,\gamma,p\phi^{i})$ is a
path in $B$ from $p$ to $p\phi^{i}$, then $(p,\gamma,p\phi^{i})$ is on an
orbit of length $n$ under $\phi$.

###### Proof.

Let $k$ be the orbit length of $p$ and let $r$ be the order of $i$ in the
additive group $\mathbb{Z}_{k}$. For $1\leq a<r$ write $\gamma_{a}$ for the
label of the path $(p,\gamma,p\phi^{i})\phi^{ia}$ and set $\gamma_{0}=\gamma$.
Write $\Gamma=\gamma_{0}\gamma_{1}\ldots\gamma_{r-1}$, then the circuit
$(p,\Gamma,p)$ is on an orbit of length $n$ by assumption. From this it
follows that the path $(p,\gamma,p\phi^{i})$ is also on an orbit of length
$n$. $\Box$

######  Lemma 4.19.

Let $A\in\mathcal{H}_{n}$ and suppose there is a minimal strongly
synchronizing automaton $B$ such that there is an automorphism $\phi_{A}$ of
the underlying digraph of $B$ with $A$ the minimal representative of
$H(B,\phi_{A})$ and so that all circuits in $B$ are on orbits of length $n$
under $\phi_{A}$. Let $s,t,p\in Q_{B}$ be such that
$\mathop{\mathrm{Letters}}(s,p)=\mathop{\mathrm{Letters}}(t,p)\neq\emptyset$.
Then by repeated application of Lemma 4.13 one may obtain from $B$ a strongly
synchronizing automaton $D$ with automorphism $\psi_{A}$ so that $A$ is the
minimal representative of $H(D,\psi_{A})$ and where the pair $(D,\psi_{A})$
satisfies the following conditions:

  1. (a)

there are no shadow states of any element in the orbit of $p$ in $D$ (that is,
each application of Lemma 4.13 creates no shadow states for elements in the
orbit of $p$),

  2. (b)

$\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(s,p)=\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(t,p)=\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(s,p),$

  3. (c)

for $u\in\\{s,t\\}$, for any shadow state $u^{\prime}$ of $u$ we have,

$\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(u,p)=\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(s,p)=\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(s,p),$

  4. (d)

for any $x\in\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(s,p)$, the length of the
orbits of the edges $(s,x,p)$, $(t,x,p)$ under the action of $\psi_{A}$ on $D$
is $n$.

###### Proof.

We proceed in a similar way to Lemma 4.14. If $|B|=1$, then we are done.
Therefore we assume that $|B|>1$.

Inductively define subsets of $Q_{B}$ as follows.

Set $Q_{B}(0,p)=\\{p\\}$. Assume that $Q_{B}(i,p)$ is defined for some
$i\in\mathbb{N}$. Define $Q_{B}(i+1,p)$ as follows. A state $q\in Q_{B}$
belongs to $Q_{B}(i+1,p)$, if there are elements $x_{0},x_{1},\ldots,x_{i}\in
X_{n}$, such that $\pi_{B}(x_{i}x_{i-1}\ldots x_{j},q)\in Q_{B}(j,p)$ for
$0\leq j\leq i$, and the path $(q,x_{i}x_{i-1}\ldots x_{0},p)$ is on an orbit
of length strictly less than $n$ under $\phi_{A}$.

As in the proof of Lemma 4.14, there is a $k\in\mathbb{N}$ such that
$Q_{B}(k+1,p)=\emptyset$. Set $k\in\mathbb{N}$ to be minimal such that
$Q_{B}(k+1,p)=\emptyset$. If $Q_{B}(1,p)\cap\\{s,t\\}=\emptyset$, then we are
done. Thus we may assume that at least one of $s,t$ belongs to $Q_{B}(1,p)$
(and so $k\geq 1$).

Observe that by Lemma 4.18 for any $j\in\mathbb{N}$, $p\phi^{j}$ is not an
element of $Q_{B}(i,p)$ for any $1\leq i\leq k$.

We now repeatedly apply Lemma 4.13, as in the proof of Lemma 4.14, until we
have an automaton $D$ such that $Q_{D}(1,p)\cap\\{s,t\\}=\emptyset$. We note
that since $p$ is always the single element of $Q_{D}(0,p)$, then, as for any
$j\in\mathbb{N}$, $p\phi^{j}$ is not an element of $Q_{B}(i,p)$ for any $1\leq
i\leq k$, we do not create a shadow state of $p$ or elements in the orbit of
$p$ in an application of Lemma 4.13.

We prove the base case to illustrate how the proof goes. Let $q\in
Q_{B}(k,p)$. We may find $x_{0},x_{1},\ldots,x_{k}$ such that for any $0\leq
j\leq k$, $\pi_{B}(q,x_{k}x_{k-1}\ldots x_{j})\in Q_{B}(j,p)$. Let $b$ be the
length of the orbit of the path $(q,x_{k}x_{k-1}\ldots x_{0},p)$. Then for any
state $u\in Q_{B}$ for which there is an edge $(u,x,q)$, the lowest common
multiple of the length of the orbit of $(u,x,q)$ and $b$ is $n$. We may now
apply Lemma 4.13 to form a new transducer $B^{\prime}$ by adding shadow states
of $q$. We may define the sets $Q_{B^{\prime}}(i,p)$ as before, noting that
$Q_{B^{\prime}}(k+1,p)=\emptyset$. This follows as for any edge $(u,x,q)$ in
$B$, the orbit of the edge $(u^{\prime},x,q^{\prime})$ in $B^{\prime}$ (for
$u^{\prime}$ and $q^{\prime}$ either equal to $u$ and $q$ or shadow states of
$u$ and $q$ respectively) is equal to the orbit of the edge $(u,x,q)$.
Moreover, as in the proof of Lemma 4.14, the number of paths
$(q^{\prime},x_{0}x_{1}\ldots x_{k},p)$, where $q^{\prime}$ is either $q$ or
one of its shadow states, witnessing that $q^{\prime}\in Q_{B^{\prime}}(k)$ is
strictly fewer than the witness paths for $q$ in $B$. Thus inductively
applying Lemma 4.13, we find an automaton, which we again denote $B^{\prime}$,
in which $|Q_{B^{\prime}}(k,p)|<|Q_{B}(k,p)|$ since neither $q$ nor any of its
shadow states belong to $Q_{B^{\prime}}(k,p)$. Thus, replacing $B^{\prime}$
with $B$ we may repeat the process.

Eventually we reach an automaton $D^{\prime}$ such that
$Q_{D}(1,p)\cap\\{s,t\\}=\emptyset$. Moreover, since, by Lemma 4.13, shadow
states transition identically to their original counterparts on edges into
states which have no shadow states added (and this transition mirrors the
transition in $B$), the automaton $D^{\prime}$ satisfies the requirements of
the lemma. $\Box$

In what follows, set notation $\operatorname{ol}_{\tau}(\star)$ to represent
the orbit length of $\star$ under the action of $\tau$ a digraph automorphism
of some digraph $G$, where $\star$ is a vertex, edge, or path in $G$.

For an automaton $C$ over alphabet $Y$ with $a,b\in Q_{C}$, recall that
$\mathop{\mathrm{E}}{\,\\!\\!}_{C}(a,b)$ represents the set of edges of
$G_{C}$ from $a$ to $b$, while

$\mathop{\mathrm{Letters}}{\,\\!\\!}_{C}(a,b):=\\{y\in
Y\mid\exists(a,y,b)\in\mathop{\mathrm{E}}{\,\\!\\!}_{C}(a,b)\\}$

represents the set of letters from $Y$ which are the labels of these edges.

######  Lemma 4.20.

Let $A\in\mathcal{H}_{n}$ and let $B$ be the minimal strongly synchronizing
automaton such that there is an automorphism $\phi_{A}$ of the underlying
digraph of $B$ with $A$ the minimal representative of $H(B,\phi_{A})$. Suppose
that all circuits in $B$ are on orbits of length $n$ under $\phi_{A}$. Let
$s,t,p\in Q_{B}$ be such that $s$ and $t$ belong to distinct orbits but have
the same orbit length under $\phi_{A}$, and so that
$\mathop{\mathrm{Letters}}(s,p)=\mathop{\mathrm{Letters}}(t,p)$. Then there is
$A^{\prime}\in\mathcal{H}_{n}$ and an automorphism $\phi_{A^{\prime}}$ of the
underlying digraph of $B$ so that the following hold:

  1. (a)

the element $A$ is conjugate to $A^{\prime}$ in $\mathcal{H}_{n}$, where
$A^{\prime}$ is the minimal representative of $H(B,\phi_{A^{\prime}})$; and,

  2. (b)

there is an $N\in\mathbb{N}$ such that for all
$x\in\mathop{\mathrm{Letters}}(s,p)$ the edges $(s,x,p)$ and $(t,x,p)$ have
orbit length $N$ under $\phi_{A^{\prime}}$.

###### Proof.

The strategy is to apply Lemma 4.19 to add shadow states to $B$ to obtain an
automaton $D$ and an automorphism $\psi_{A}$ of $D$, such that $H(D,\psi_{A})$
has minimal representative $A$, $\psi_{A}$ preserves the orbit length of the
state $p$, and, the orbit length of the edges from $s$ and $t$ into $p$ is
$n$. We then apply a relabelling through shadows of $D$ by Lemma 4.17 to
obtain a conjugate element $A^{\prime}$ to $A$ represented by a transducer
$H(B,\phi_{A^{\prime}})$ for $\phi_{A^{\prime}}$ an automorphism of the
underlying digraph $G_{B}$. The key ingredient is that the relabelling of $D$
is chosen such that the orbits of the edges of $s$ and $t$ into $p$ now all
have the same length under $\phi_{A^{\prime}}$. In order to find a relabelling
achieving this goal, we will need to track numerous integer constants.

Set $r=\operatorname{ol}_{\phi_{A}}(p)$ and determine $m$ so that
$mr=\operatorname{lcm}(\operatorname{ol}_{\phi_{A}}(s),\operatorname{ol}_{\phi_{A}}(p))=\operatorname{lcm}(\operatorname{ol}_{\phi_{A}}(t),\operatorname{ol}_{\phi_{A}}(p))$.
For any edge $(s,x,p)\in\mathop{\mathrm{E}}{\,\\!\\!}_{B}(s,p)$ we have
$(s,x,p)\phi_{A}^{k}\in\mathop{\mathrm{E}}{\,\\!\\!}_{B}(s,p)$ if and only if
$k$ is an integer multiple of $mr$. For each
$x\in\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(s,p)=\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(t,p)$,
determine integers $u_{x}$, $v_{x}$ so that
$u_{x}mr=\operatorname{ol}_{\phi_{A}}((s,x,p))$ and
$v_{x}mr=\operatorname{ol}_{\phi_{A}}((t,x,p))$. In particular, for any
$x\in\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(s,p)$, the number of edges in
$\mathop{\mathrm{E}}{\,\\!\\!}_{B}(s,p)$ which belong to the orbit of
$(s,x,p)$ under $\phi_{A}$ is precisely $u_{x}$, while $v_{x}$ defines the
analogous number for $(t,x,p)$. It follows that there are permutations
$\theta_{s}:\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(s,p)\to\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(s,p)$
and
$\theta_{t}:\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(t,p)\to\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(t,p)$
induced from the permutations of edges from $s$ to $p$ (and from $t$ to $p$
respectively) achieved by applying $\phi_{A}^{mr}$. In particular, for
$x\in\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(s,p)(=\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(t,p))$,
we have the cycle of $\theta_{s}$ containing $x$ has length $u_{x}$ and the
cycle of $\theta_{t}$ containing $x$ has length $v_{x}$.

Adding shadows

:

Apply Lemma 4.19 to the quadruple $(s,t,p,B)$ to obtain a strongly
synchronizing automaton $D$ and a corresponding automorphism $\psi_{A}$ of the
digraph $G_{D}$ underlying $D$ so that $A$ is the minimal representative of
$H(D,\psi_{A})$, and the conclusions of Lemma 4.19 are satisfied. In
particular, for any $x\in\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(s,p)$ we have
$\operatorname{ol}_{\psi_{A}}((s,x,p))=\operatorname{ol}_{\psi_{A}}((t,x,p))=n$.

We now determine various constants arising from the construction so far.

By construction, $r$ remains the length of the orbit of $p$ in $D$; however,
the orbit lengths of $s$ and $t$ have possibly been padded out with shadow
states. Determine $e$, $f$ so that
$emr=\operatorname{lcm}(\operatorname{ol}_{\psi_{A}}(s),\operatorname{ol}_{\psi_{A}}(p))$
and
$fmr=\operatorname{lcm}(\operatorname{ol}_{\psi_{A}}(t),\operatorname{ol}_{\psi_{A}}(p))$.

As
$\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(s,p)=\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(t,p)=\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(t,p)=\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(s,p)$
we will often use the notation
$\mathop{\mathrm{Letters}}{\,\\!\\!}_{\,}(s|\\!|t,p)$ for this set, although,
we might use one of the other names if we specifically wish to emphasise that
we are considering the action of $\phi_{A}$ or of $\psi_{A}$ in that case. Set
as well $\zeta:=|\mathop{\mathrm{Letters}}{\,\\!\\!}_{\,}(s|\\!|t,p)|$.

Determining constants and orbit blocks from
$\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(s,p)$ and
$\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(t,p)$

:

As above, consider the permutation of
$\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(s,p)$ induced by applying
$\psi_{A}^{emr}$ to $G_{D}$. Note that this permutation is $\theta_{s}^{e}$.
Similarly $\psi_{A}^{fmr}$ induces $\theta_{t}^{f}$ on
$\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(t,p)$. Set $q_{e}$ to be the order of
$\theta_{s}^{e}$, so that $\theta_{s}^{q_{e}e}$ is the identity permutation.
Analogously define $q_{f}$ to be the order of $\theta_{t}^{f}$. Note in
passing that $q_{e}=n/(emr)$ is the number of times the orbit of an edge of
the form $(s,x,p)$ intersects $\mathop{\mathrm{E}}{\,\\!\\!}_{D}(s,p)$ under
the action of $\langle\psi_{A}\rangle$ (and that this number is independent
from the choice of such edge), and similarly, $q_{f}=n/(fmr)$ counts the
cardinality of the intersection of the orbit of an edge of the form $(t,x,p)$
with $\mathop{\mathrm{E}}{\,\\!\\!}_{D}(t,p)$ under the action of
$\langle\psi_{A}\rangle$ (again, independent of the choice of such an edge).
It follows that all cycles of $\theta_{s}^{e}$ have length $q_{e}$ and that
all the cycles of $\theta_{t}^{f}$ have length $q_{f}$.

We note that for all $x\in\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(s,p)$ we
have $q_{e}e=\operatorname{lcm}(u_{x},e)$ and
$q_{f}f=\operatorname{lcm}(v_{x},f)$. Therefore, for any
$x\in\mathop{\mathrm{Letters}}{\,\\!\\!}_{\,}(s|\\!|t,p)$, we have
$q_{e}|u_{x}$ and $q_{f}|v_{x}$. Further, as $q_{e}emr=q_{f}fmr=n$ we have
$q_{e}e=q_{f}f$.

Let
$u=\gcd\\{u_{x}:x\in\mathop{\mathrm{Letters}}{\,\\!\\!}_{\,}(s|\\!|t,p)\\}$
and
$v=\gcd\\{v_{x}:x\in\mathop{\mathrm{Letters}}{\,\\!\\!}_{\,}(s|\\!|t,p)\\}$.
As $q_{e}|u_{x}$ and $q_{f}|v_{x}$ for all
$x\in\mathop{\mathrm{Letters}}{\,\\!\\!}_{\,}(s|\\!|t,p)$, we see that
$q_{e}|u$ and $q_{f}|v$. Let $\overline{u}$ and $\overline{v}$ be such that
$\overline{u}q_{e}=u$ and $\overline{v}q_{f}=v$. It also follows that
$\operatorname{lcm}(u,e)=q_{e}e=q_{f}f=\operatorname{lcm}(v,f)$.

Let $\tau\in\\{s,t\\}$ and set $\sim_{\tau,p}$ to be the equivalence relation
on the set of edges from $\tau$ to $p$, where two edges are equivalent under
$\sim_{\tau,p}$ if they are in the same orbit under $\psi_{B}$. Let
$X(\tau,p)$ be a transversal for this equivalence relation. It follows that

$\sum_{(s,x,p)\in X(s,p)}u_{x}=\sum_{(t,x,p)\in
X(t,p)}v_{x}=|\mathop{\mathrm{Letters}}{\,\\!\\!}_{\,}(s|\\!|t,p)|=\zeta$

so in particular we see that both $u$ and $v$ divide $\zeta$.

Set $w:=\operatorname{lcm}(u,v)$. Since $u$ and $v$ divide $\zeta$ we have
that $w|\zeta$. Let $\alpha$ be such that $\alpha w=\zeta$. Further, as $u|w$
and $v|w$ there are $\mu,\nu\in\mathbb{N}$ such that $w=\mu
u=\mu\overline{u}q_{e}$ and $w=\nu v=\nu\overline{v}q_{f}$. Further,
$q_{e}e=\operatorname{lcm}(u,e)|\operatorname{lcm}(w,e)$ and
$q_{f}f=\operatorname{lcm}(v,f)|\operatorname{lcm}(w,f)$. Since both $u$ and
$v$ divide $q_{e}e=q_{f}f$, it follows that $w|q_{e}e$ and so
$\operatorname{lcm}(w,e)=q_{e}e=q_{f}f=\operatorname{lcm}(w,f)$. In
particular, $wmr|q_{e}emr=n$ and $wmr|q_{f}fmr=n$ and we have
$\operatorname{lcm}(wmr,emr)=n=\operatorname{lcm}(wmr,fmr)$.

Building the relabelling

:

Our relabelling will be a relabelling through shadows which will create a
pattern of labels along an edge orbit that repeats after every $wmr$ steps
under iteration of the new automorphism $\psi_{A^{\prime}}$ of $G_{D}$.

To be a relabelling through shadows we will need that for any $0\leq i<mr$ and
integer $k$ that the local map at $s\psi_{A^{\prime}}^{i+kmr}$ agrees with the
local map at $s\psi_{A^{\prime}}^{i}$ (and similarly for the orbit of $t$).

We will now define a new labelling $\lambda_{D}:X_{n}\times Q_{D}\to X_{n}$.
Recall that a relabelling function is always induced from a vertex fixing
automorphism of the underlying digraph of an automaton in the collapse
sequence of the original. In our particular construction of $\lambda_{D}$, the
reader will see that the automorphism employed in its creation is simply a
vertex fixing automorphism of the underlying digraph $G_{D}$ of $D$.

Suppose $(l,x^{\prime},p^{\prime})$ is an edge of $D$ which does not belong to
the orbit of an edge $(\tau,x,p)$ for any pair
$(\tau,x)\in\\{s,t\\}\times\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(\tau,p)$.
In this case we set $(x^{\prime},l)\lambda_{B}=x^{\prime}$.

For the moment we focus on edges in the orbit of some edge $(s,x,p)$. The
edges in the orbit of $(t,x,p)$ are dealt with analogously.

As $|\mathop{\mathrm{Letters}}{\,\\!\\!}_{\,}(s|\\!|t,p)|=\zeta=w\alpha$ we
may partition $\mathop{\mathrm{Letters}}{\,\\!\\!}_{\,}(s|\\!|t,p)$ into
$\alpha$ blocks of size $w$, which partition we organise through some
labelling of the set’s elements as follows:

$\mathop{\mathrm{Letters}}{\,\\!\\!}_{\,}(s|\\!|t,p)=\\{x^{c}_{i}\mid 0\leq
i<w,0\leq c<\alpha\\},$

where here, the $c^{th}$ part, denoted $x^{c}$, has $w$ elements arranged as
the ordered sequence $(x^{c}_{i})_{0\leq i<w}.$

For $0\leq i<mr$ and $q=s\psi_{A}^{i}$ we define $(x,q)\lambda_{D}=x$. Let
$0\leq i<w$, $0\leq c<\alpha$ and determine $u_{i}\in X_{n}$ so that
$(s,x^{c}_{i},p)\psi_{A}^{mr}=(s\psi_{A}^{mr},u_{i},p\psi_{A}^{mr})$. Now set
$(u_{i},s\psi_{A}^{mr})\lambda_{D}:=x^{c}_{(i+1)\mod w}$.

Now, the fact that we are relabelling through shadows determines the rest of
the relabelling function $\lambda_{D}$, as the local functions of
$H(G_{D},\psi_{A^{\prime}})$ have to agree for each shadow state with that
occurring at the state being shadowed, that is, we need the local function of
the transducer $H(G_{D},\psi_{A^{\prime}})$ at any state
$s\psi_{A^{\prime}}^{a+mr}$ to agree with the local function at
$s\psi_{A^{\prime}}^{a}$ (where $mr$ suffices as that is the orbit length of
the pair $(s,p)$ in $G_{B}$ under $\phi_{A}$, and our relabelling only impacts
the labels of edges in the orbit of edges from $s$ to $p$ in $G_{D}$).

The following inductive definition of $\lambda_{D}$ enforces this agreement.

In particular, suppose $mr<a<emr$ and for all $x\in X_{n}$ and $0\leq i<a$ we
have $(x,s\psi_{A}^{i})\lambda_{D}$ defined. Suppose further that
$(s\psi_{A}^{a-1},u,p\psi_{A}^{a-1})$ is an edge of $G_{D}$, $v\in X_{n}$ with
$(s\psi_{A}^{a-1},u,p\psi_{A}^{a-1})\psi_{A}=(s\psi_{A}^{a},v,p\psi_{A}^{a})$,
and that $u^{\prime},v^{\prime},x,y\in X_{n}$ so that:

$\displaystyle(s\psi_{A}^{a-1},u)\lambda_{D}$ $\displaystyle=u^{\prime}$
$\displaystyle(s\psi_{A}^{a-mr-1},x)\lambda_{D}$ $\displaystyle=u^{\prime}$
$\displaystyle(s\psi_{A}^{a-mr-1},x,p\psi_{A}^{a-mr-1})\psi_{A}$
$\displaystyle=(s\psi_{A}^{a-mr},y,p\psi_{A}^{a-mr})$
$\displaystyle(s\psi_{A}^{a-mr},y)\lambda_{D}$ $\displaystyle=v^{\prime}$

then set $(s\psi_{A}^{a},v)\lambda_{D}:=v^{\prime}$.

We define $\lambda_{D}$ analogously on all edges in the orbit of an edge from
$t$ to $p$.

We now determine $\psi_{A^{\prime}}$ acting on $G_{D}$ by the rules that
$\psi_{A^{\prime}}$ agrees with $\psi_{A}$ on the vertices of $G_{D}$, and if
$(u,x,v),(u\psi_{A},y,v\psi_{A})$ are edges of $G_{D}$ so that
$(u,x,v)\psi_{A}=(u\psi_{A},y,v\psi_{A})$ then
$(u,(x,u)\lambda_{D},v)\psi_{A^{\prime}}=(u\psi_{A^{\prime}},(y,u\psi_{A^{\prime}})\lambda_{D},v\psi_{A^{\prime}})$.

Note in passing that while the orbit of any edge of $G_{D}$ from $s$ to $p$
(or, from $t$ to $p$, respectively) has length $n$ under $\psi_{A^{\prime}}$,
the pattern of labels taken by such an edge repeats every $wmr$ steps, and so,
as the vertex pair $(s,p)$ (resp. $(t,p)$) is on an orbit of length $emr$
(respectively $fmr$) and $\operatorname{lcm}(wmr,emr)=n$ (resp.
$\operatorname{lcm}(wmr,fmr)=n$) we see that in the induced automorphism
$\phi_{A^{\prime}}$ of $G_{B}$ (so that $A^{\prime}$ is represented by both
$H(G_{B},\phi_{A^{\prime}})$ and $H(G_{D},\psi_{A^{\prime}}))$ the orbit of
any edge from $s$ or $t$ to $p$ is now on an orbit of length $wmr$.

$\Box$

##  5 Conjugate to an $n$-cycle

A circuit of length $k$ in an automaton $B$ is carried by $k$ directed edges
in $G_{B}$ in some order $(e_{0},e_{1},\ldots,e_{k-1})$ where the target of
$e_{i}$ is the source of $e_{i+1}$ (indices modulo $k$) for each index $i$. In
the next lemma, an automorphism $\phi$ of $G_{B}$ carries a circuit $C$ to
itself if and only if the image of each edge of $C$ is itself under the
automorphism. Specifically, a “rotation” of a circuit is not the circuit
itself.

######  Lemma 5.1.

Let $A\in\mathcal{H}_{n}$ be an element of finite order. Let $B$ be a strongly
synchronizing automaton for which there is an automorphism $\phi_{A}$ of the
underlying digraph $G_{B}$ of $B$ with $A$ the minimal representative of
$H(G_{B},\phi_{A})$. Then every point of $X_{n}^{-\mathbb{N}}$ is on an orbit
of length $n$ under the action of $A$ if and only if all circuits in $B$ are
on orbits of length $n$ under the action of $\phi_{A}$.

###### Proof.

This proof follows straight-forwardly from the observation that the orbits of
circuits in $B$ under the action of $\phi_{A}$ correspond to the action of $A$
on periodic points of $X_{n}^{-\mathbb{N}}$. Now as periodic points are dense
in $X_{n}^{-\mathbb{N}}$, the following chain of equivalences is true: all
points of $X_{n}^{-\mathbb{N}}$ are on orbits of length $n$ under the action
of $A$ if and only if all periodic points of $X_{n}^{-\mathbb{N}}$ are on
orbits of length $n$ under the action of $A$ if and only if all circuits of
$B$ are on orbits of length $n$ under the action of $A$. $\Box$

######  Lemma 5.2.

Let $A\in\mathcal{H}_{n}$ be an element of finite order and let $B$ be the
minimal strongly synchronizing automaton such that $A$ acts as an automorphism
$\phi_{A}$ of $G_{B}$ the underlying digraph of $B$. Suppose that all circuits
in $B$ are on orbits of length $n$, then $A$ is conjugate to an $n$-cycle.

###### Proof.

We proceed by induction. In each iteration, we successively replace $A$ with a
conjugate $C$ that acts as an automorphism of the underlying digraph of a
smaller strongly synchronizing automaton.

By Lemma 4.6 we may assume, replacing $A$ with a conjugate if necessary, that
the amalgamation and collapse sequence of $B$ cohere. Thus, a pair of states
$s,t$ which distribute similarly over $Q_{B}$ satisfy,
$\mathop{\mathrm{Letters}}(s,p)=\mathop{\mathrm{Letters}}(t,p)$ for all $p\in
Q_{B}$.

Suppose that $|B|>1$ (as otherwise we are done). Since $B$ is strongly
synchronising, we may find a pair of distinct states $s,t$ which distribute
similarly over $Q_{B}$.

We consider two cases.

First suppose that $s$ and $t$ belong to the same orbit. Fix a state $p$ for
which there is an edge from $s$ (and so from $t$) into $p$.

We apply Lemma 4.19 to the triple $(s,t,p)$ to obtain a transducer
$D^{\prime\prime}$ and automorphism $\psi^{\prime}_{A}$ such that there are no
shadow states for elements in the orbit of $p$, edges from $s$ and $t$ into
$p$ are on orbits of length $n$.

Now since there are no shadow states for elements in the orbit of $p$ in
$D^{\prime\prime}$, we may repeatedly apply Lemma 4.19 to $s,t$ and states in
the orbits of $p$ in turn, until we obtain an automaton $D^{\prime}$ and
automorphism $\psi_{A}$ which has no shadow states for elements in the orbit
of $p$, such that
$\mathop{\mathrm{Letters}}{\,\\!\\!}_{D^{\prime}}(s,p\psi_{A}^{i})=\mathop{\mathrm{Letters}}{\,\\!\\!}_{D^{\prime}}(t,p\psi_{A}^{i})=\mathop{\mathrm{Letters}}{\,\\!\\!}_{B}(s,p\phi_{A}^{i})$
for all $i\in\mathbb{N}$, and such that any edge from $s$ or $t$ into a state
in the orbit of $p$ has orbit length $n$.

Now we apply Lemma 4.7 to the automaton $D^{\prime}$. This results in an
automaton $D$ and conjugate automorphism $\psi_{A^{\prime}}$ with the
following properties. For $r$ minimal such that, for any $i,j\in\mathbb{N}$,
$\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(s\psi_{A^{\prime}}^{i},p\psi_{A^{\prime}}^{j})=\mathop{\mathrm{Letters}}{\,\\!\\!}_{D}(s\psi_{A^{\prime}}^{i+r},p\psi_{A^{\prime}}^{j+r})$,
then, the labels of the edges
$(s\psi_{A^{\prime}}^{i},x,p\psi_{A^{\prime}}^{j})\psi_{A^{\prime}}$ and
$(s\psi_{A^{\prime}}^{i+r},x,p\psi_{A^{\prime}}^{j})\psi_{A^{\prime}}$
coincide. Notice that, by minimality of $r$, Shadow states remain
$\omega$-equivalent to the state they shadow and $t$ is an element of the
orbit of $s$ under the action of $\psi_{A^{\prime}}^{r}$.

We now apply Lemma 4.17 to collapse down to the automaton $B$ with a conjugate
automorphism $\phi_{A^{\prime}}$. The conjugate automorphism
$\phi_{A^{\prime}}$ has the following properties. For edges not belonging to
the orbit of an edge from $s$ or $t$ into a state not in the orbit of $p$, the
action of $\phi_{A^{\prime}}$ coincides with the action of $A$. By
construction of $D$, the label of the edges
$(s\phi_{A^{\prime}}^{i},x,p\phi_{A^{\prime}}^{j})\phi_{A^{\prime}}$ and
$(t\phi_{A^{\prime}}^{i},x,t\phi_{A^{\prime}}^{j})\phi_{A^{\prime}}$ coincide
for any $i,j\in\mathbb{N}$.

We now repeat this process across all states of $B$ which have an edge from
$s$. Thus we end up up with a conjugate automorphism $\psi_{C}$ such that the
label of the edges $(s\psi_{C}^{i},x,q)\psi_{C}$ and
$(t\phi_{C}^{i},x,q)\psi_{C}$ coincide, for any state $q$ and any incoming
edge from $s$ and $t$ into $q$ labelled $x$.

Let $B^{\prime}$ be the automaton obtained from $B$ by identifying the pair of
states $(s\psi_{C}^{i},t\psi_{C}^{i})$ for all $i\in\mathbb{N}$. Since
$\psi_{C}$ induces the same action on labels for corresponding edges from
elements $s\psi_{C}^{i}$ and $t\psi_{C}^{i}$, then there is an induced action
$\phi_{C}$ of $\psi_{C}$ on the underlying digraph of $B^{\prime}$. That is
there is an element $C\in\mathcal{H}_{n}$ which is a conjugate of $A$ such
that $C$ is the minimal representative of $H(B^{\prime},\phi_{C})$ an
$H(B,\psi_{C})$.

Now consider the case that $s,t$ belong to distinct orbits. We may assume that
the orbit lengths of $s$ and $t$ coincide otherwise we may find a
$\tau^{\prime}$ distinct from $s$ and $t$ which has the same orbit length and
agrees with one of $s$ or $t$ on $Q_{B}$. Whereby we apply the previous case
to the pair $(s,\tau)$ or $(t,\tau)$.

Fix a state $p\in Q_{B}$ with an edge from $s$ (and so from $t$). We apply
Lemma 4.20 to obtain a conjugate automorphism $\phi_{A^{\prime\prime}}$ of the
underlying digraph of $B$ such that all edges from $s$ and $t$ into $p$ have
the same orbit length.

We repeat the process along all states of $B$ with an incoming edge from $s$.
This yields a conjugate automorphism $\phi_{A^{\prime}}$ of $B$, whereby, for
a given state $q\in Q_{B}$ all edges from $s$ and $t$ into $q$ have the same
orbit length under $\phi_{A^{\prime}}$.

We now repeatedly apply Lemma 4.9 to the triple $(s,t,B,\phi_{A^{\prime}})$ to
obtain a conjugate automorphism $\psi_{C}$ of $B$ which satisfies the
following. For any pair of edges $(s,x,q)$ and $(t,x,q)$, $(s,x,q)\psi_{C}$ an
$(t,x,q)\psi_{C}$ have the same labels. This means, we may once more identify
the pair of states $(s\psi_{C}^{i},t\psi_{C}^{i})$ to obtain an action
$\phi_{C}$ of $C$, the minimal representative of $H(B,\psi_{C})$, on a smaller
automaton $B^{\prime}$.

If $|C|>1$, then as $C$ is conjugate to $A$ we may now repeat the process with
$C$ instead of $A$.

Eventually we end up with the single state transducer.

$\Box$

We recall that by Theorem 3.5, for an element $A\in\mathcal{H}_{n}$ of finite
order, there is a strongly synchronizing automaton $B$ on which $A$ acts as an
automorphism $\phi_{A}$ of the underlying digraph of $B$ so that
$H(B,\phi_{A})$ has minimal representative $A$.

######  Theorem 5.3.

Let $A\in\mathcal{H}_{n}$ be an element of finite order. Then $A$ is conjugate
to an $n$-cycle if and only if every element of $X_{n}^{-\mathbb{N}}$ is on an
orbit of length $n$ under the action of $A$ if and only if for any strongly
synchronizing automaton $B$ on which $A$ acts as an automorphism $\phi_{A}$ of
the underlying digraph of $B$, every circuit of $B$ is on an orbit of length
$n$ under the action of $A$.

###### Proof.

The equivalences follow from lemmas 5.1 and 5.2. $\Box$

###  5.1 An Example

In this section we work through an example that illustrates the key ideas of
the proof.

Consider the automaton $A$ of Figure 7 which we encountered already in Example
3.6.

$a_{0}$$a_{1}$$a_{2}$$a_{3}$$1|4$
$0|5$$2|0$$3|1$$4|2$$5|3$$4|2$$5|3$$0|4$$1|5$$2|1$$3|0$$1|4$$0|5$$4|2$
$5|3$$2|1$$3|0$$1|5$$0|4$$2|0$ $3|1$$4|2$$5|3$ Figure 7: The element $A$

This is an element of $\mathcal{H}_{6}$ of order $6$, where every point in the
Cantor space $X_{6}^{-\mathbb{N}}$ is on an orbit of length $6$ under the
action of $A$. Following the construction in Subsection 3.3.1 (see Example
3.6), the minimal strongly synchronising automaton $B$ which admits an
automorphism $\phi_{A}$ of $G_{B}$ such that $H(B,\phi_{A})$ has minimal
representative $A$, is as depicted in Figure 8, where each drawn edge
represents two edges with labels as listed; the map $\phi_{A}$ on the vertices
on $G_{B}$ is the permutation which in cycle notation is

$(p_{0}\;p_{1}\;p_{2})(q_{0}\;q_{1}\;q_{2});$

the action of $\phi_{A}$ on the vertices and edges of $G_{B}$ is uniquely
determined from the fact that $A$ is the minimal representative of
$H(G_{B},\phi_{A})$. We refer to the vertices $q_{0},q_{1},q_{2}$ as the
vertices of the “inner triangle” and the vertices $p_{0},p_{1},p_{2}$ as the
vertices of the “outer triangle”.

$p_{0}$$q_{0}$$q_{2}$$q_{1}$$p_{1}$$p_{2}$$1$
$0$$2$$3$$4$$5$$4$$5$$0$$1$$2$$3$$1$$0$$4$
$5$$2$$3$$1$$0$$2$$3$$4$$5$$2$$3$$5$$4$$1$$0$$0$$1$$3$$2$$4$$5$ Figure 8:
Minimal automaton witnessing finite order of $A$.

We notice that the automaton $B$ has the property that its synchronizing and
amalgamation sequences cohere. In particular both reduce to the single vertex
with $6$ looped edges after $2$ steps.

The fact that every circuit of $G_{B}$ is on orbit of length $6$ can be seen
as follows. A circuit of $G_{B}$ which is not formed by repeating the circuit
(or a cyclic rotation of it) $p_{0}\to p_{1}\to p_{2}\to p_{0}$ a finite
number of times, must have an edge leaving a vertex in the inner triangle —
any such edge has orbit length $6$.

Therefore $A$ satisfies the hypothesis of Theorem 5.3. We work through Lemma
5.2 to find an element of $\mathcal{H}_{6}$ which conjugates $A$ to a
$6$-cycle.

In the first step we find two states of $G_{B}$ which can be collapsed i.e.
which distribute similarly over $Q_{B}$. We may take the pair $(p_{0},q_{0})$
(any other valid pair belongs to the orbit of this one). The orbits of $p_{0}$
and $q_{0}$ are distinct so we are in the second case of Lemma 5.2. Now all
edges leaving any vertex in the orbit of $q_{0}$ have orbit length $6$,
whereas the edges edges from $p_{0}$ to $q_{0}$ and $p_{0}$ to $q_{2}$ have
orbit length $3$ while the edges from $p_{0}$ to $p_{1}$ have orbit length
$6$. Thus we apply Lemma 4.20. We add shadow states using Lemma4.19. Focusing
on the vertex $q_{0}$ as our vertex $p$ (in the notation of Lemma4.19), we see
that

$\displaystyle Q(0,q_{0})$ $\displaystyle=\\{q_{0}\\}$ $\displaystyle
Q(1,q_{0})$ $\displaystyle=\\{p_{0},p_{1}\\}$ $\displaystyle Q(2,q_{0})$
$\displaystyle=\emptyset.$

The last follows since any incoming edge to a vertex on the outer triangle is
on an orbit of length $6$.

We may take either $p_{0}$ or $p_{1}$ as the heavy state (since they belong to
the same orbit). Our divisibility constant is $3$ (the orbit lengths of the
two orbits of edges from the outer triangle into the inner triangle, i.e. the
edge from $p_{0}$ into $q_{0}$ represents one such orbit, and the edge from
$p_{1}$ into $q_{0}$ represents the other); the number $n^{\prime}$ is
precisely $6$ – since every incoming edge into either $p_{0}$ or $p_{1}$ has
orbit length $6$. (We note as an aside that since the edge from $p_{1}$ to
$q_{0}$ is in the orbit of the edge from $p_{0}$ to $q_{2}$, we only need one
round of adding shadow states in order to fix the orbit lengths of both of
these edges, using more words, we need not consider $p=q_{2}$ as a separate
case).

Our new automaton $B^{\prime}$ will have shadow states
$p^{\prime}_{0},p^{\prime}_{1}$ and $p^{\prime}_{2}$ as is as depicted in
Figure 9. There is a lift $\psi_{A}$ of $\phi_{A}$ to $G_{B^{\prime}}$. The
action of $\psi_{A^{\prime}}$ is uniquely determined by the facts that the
orbit of $p_{0}$ under $\psi_{A^{\prime}}$ is
$(p_{0}\;p_{1}\;p_{2}\;p^{\prime}_{0}\;p^{\prime}_{1}\;p^{\prime}_{2})$ and
$H(B^{\prime},\psi_{A})$ has minimal representative $A$.

$p^{\prime}_{0}$$p_{0}$$q_{0}$$q_{2}$$q_{1}$$p_{1}$$p^{\prime}_{1}$$p_{2}$$p^{\prime}_{2}$$1$
$0$$2$$3$$4$$5$$1$$0$$2$$3$$4$$5$$4$$5$$0$$1$$2$$3$$1$$0$$4$
$5$$2$$3$$1$$0$$2$$3$$4$$5$$2$$3$$5$$4$$1$$0$$2$$3$$5$$4$$1$$0$$1$$0$$3$$2$$4$$5$$1$$0$$3$$2$$4$$5$
Figure 9: Adding shadows to form $B^{\prime}$.

We can now apply Lemma 4.20 to the orbit of the edge from $p_{2}$ to $q_{0}$
and from $p_{1}$ to $q_{0}$. Notice that since the orbit lengths of edges
leaving the inner triangle is $6$, the relabelling map of Lemma 4.20 will
simply wrap around the orbits of the relevant edges from $p_{2}$ and $p_{1}$
to increase their orbit lengths after re-identifying shadow states to $6$.
This can be achieved by relabelling such that the actions on letters of orbits
in the edge $(p_{0},\\{1,0\\},q_{0})$ mirrors the action on the corresponding
edge in the orbit of $(q_{0},\\{0,1\\},q_{0})$ (similarly for the pair
$(p_{1},\\{1,0\\},q_{0}\\}$ and $(q_{1},\\{1,0\\},q_{0})$. One such
relabelling is that induced by the vertex fixing automorphism of
$G_{B^{\prime}}$ that swaps the edges from $p_{1}$ to $q_{1}$, the edges from
$p_{2}$ to $q_{2}$, the edges from $p^{\prime}_{0}$ to $q_{0}$; the edges from
$p_{1}$ to $q_{0}$, from $p_{2}$ to $q_{1}$ and from $p^{\prime}_{0}$ to
$q_{2}$. This gives rise to the element $C$ in figure 10.

$c_{0}$$c_{1}$$c_{3}$$c_{5}$$c_{4}$$c_{2}$$1|0$$0|1$$2|3$$3|2$$4|4$$5|5$$4|4$$0|0$$1|1$$2|2$$3|3$$5|5$$0|0$$4|4$$5|5$$2|2$$3|3$$1|1$$1|1$$0|0$$2|2$$4|4$$5|5$$3|3$$2|2$$3|3$$5|4$$4|5$$1|0$$0|1$$0|0$$1|1$$3|2$$2|3$$4|5$$5|4$
Figure 10: Conjugator $C$.

The reader can verify that the conjugate of $A$ by $C$ is the automaton $D$ to
the left of Figure 11.

$a_{1}$$a_{3}$$a_{5}$$4|2$$5|3$$0|4$$1|5$$2|1$$3|0$$1|4$$0|5$$4|2$
$5|3$$2|1$$3|0$$1|5$$0|4$$2|0$$3|1$$4|2$$5|3$$a_{1}$$a_{3}$$a_{5}$$4$$5$$0$$1$$2$$3$$1$$0$$4$
$5$$2$$3$$1$$0$$2$$3$$4$$5$ Figure 11: Conjugate of $A$ by relabelling map
$C$.

The automaton $E$ to the right of Figure 11 admits an automorphism $\phi_{D}$
of its underlying digraph such that $H(E,\phi_{D})$ has minimal representative
$D$. The map $\phi_{D}$ is uniquely determined by the fact that
$H(E,\phi_{D})$ has minimal representative $D$. Notice that all edges of
$G_{D}$ are on orbits of length $6$ and the collapse and amalgamation
sequences of $G_{D}$ coincide. Following Lemma 5.2, we find a pair of vertices
which distribute similarly over $Q_{D}$, any pair of distinct vertices works —
we choose $(a_{1},a_{3})$. Now we are in the first case of Lemma 5.2 and the
relabelling protocol we apply is that given by Lemma 4.7. Essentially we want
to relabel such that the action of $a_{1}$ and $a_{3}$ on $X_{6}$ coincide
along their orbits. A relabelling that achieves this is obtained by swapping
the edged between $a_{1}$ and $a_{5}$ and between $a_{5}$ and $a_{3}$. This
relabelling gives rise to the conjugator $F$ in figure 12

$a_{1}$$a_{3}$$a_{5}$$4|5$$5|4$$0|0$$1|1$$2|3$$3|3$$1|1$$0|0$$4|4$
$5|5$$2|2$$3|3$$1|1$$0|0$$2|3$$3|2$$4|4$$5|5$ Figure 12: Conjugator $F$.

The reader can verify that conjugating $D$ by $F$ results in the single state
transducer corresponding to the $6$-cycle $(0\;4\;2\;1\;5\;3)$.

Therefore, the element $CF$ of $\mathcal{H}_{6}$ conjugates $A$ to the
$6$-cycle $(0\;4\;2\;1\;5\;3)$.

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