#  HUPD-2212 Effective theory for universal seesaw model and FCNC

Takuya Morozumi 12 Apriadi Salim Adam3
Yuta Kawamura4 Albertus Hariwangsa Panuluh15
Yusuke Shimizu12 and Kei Yamamoto6 1 Physics Program, Graduate School of
Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima
739-8526, Japan  2 Core of Research for the Energetic Universe, Hiroshima
University, Higashi-Hiroshima 739-8526, Japan  3 Research Center for Quantum
Physics, National Research and Innovation Agency (BRIN) , South Tangerang
15314, Indonesia  4 Yokkaichi, Mie Prefecture, Japan  5 Department of Physics
Education, Sanata Dharma University, Paingan, Maguwohardjo, Sleman, Yogyakarta
55282, Indonesia  6 Department of Global Environment Studies, Hiroshima
Institute of Technology, Hiroshima 731- 5193, Japan
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>yu-
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the quark sector of the universal seesaw model with SU(2)L $\times$
SU(2)R $\times$ $U(1)$. The model incorporates the seesaw mechanism with the
vector-like quarks (VLQs). The purpose of this work is to study the model with
the effective theory. After integrating the heavy five VLQs, we derive the
effective theory with four up-type quark and three down type quark. In this
work, the FCNC of Z boson for top quark and top′ quark is derived.

##  1 Introduction

Though the standard model is a very successful theory, the origin of the
flavor and mass hierarchy of quarks can be explained only by tuning the Yukawa
coupling [1]. For instance up quark mass and top quark mass are respectively
given as,

$\displaystyle m_{u}=y_{u}\frac{v}{\sqrt{2}}<m_{t}=y_{t}\frac{v}{\sqrt{2}}$
$\displaystyle y_{u}\simeq 1.25\times 10^{-5}y_{t},\quad y_{t}\simeq 0.99,$
(1)

where we use the top quark mass from the direct measurement
$m_{t}=172.69$(GeV) and the up quark mass from the $\overline{MS}$ scheme
$m_{u}(2{\rm GeV})=2.16$(MeV) [2]. We also use $v=246.22$(GeV) to derive
Yukawa coupling of the top quark. The universal seesaw model explains the
smallness of the mass of the up quark with a tiny ratio of SU(2)R breaking
scale and a SU(2) singlet vector-like quark (VLQ) mass $M_{U}$. The standard
model Yukawa coupling $y_{u}$ is given by the seesaw like formula,

$\displaystyle
y_{u}=y_{uL}\left(\frac{v_{R}}{\sqrt{2}M_{U}}\right)y_{uR}\simeq\left(\frac{v_{R}}{\sqrt{2}M_{U}}\right)\simeq
10^{-5},$ $\displaystyle y_{uL}\simeq y_{uR}\simeq O(1),$ (2)

where $y_{uL}$ and $y_{uR}$ are mixing type Yukawa couplings between the
ordinary quark and VLQ. We study the quark sector of the universal seesaw
model with SU(2)L $\times$ SU(2)R $\times$ U(1) [3, 4, 5, 6]. Our aim is to
construct the effective theory obtained after integrating VLQs with their
masses larger than SU(2)R breaking scale. Then, we study the flavor structure
of the effective theory. The same model was investigated with the full theory
[7, 8, 9].

##  2 The Lagrangian of the model

We first present the quark and gauge sector. The two SU(2) doublet Higgs
fields are introduced. $\phi_{L}$ stands for SU(2)L doublet and $\phi_{R}$ for
SU(2)R. Their vacuum expectation values (vevs) $v_{L}$ and $v_{R}$ repectively
break SU(2)L and SU(2)R. From the Higgs potential study, one can show that the
vevs satisfy $v_{R}\gg v_{L}$. The ordinary quarks $\psi_{L}$($\psi_{R}$) are
also SU(2)L (SU(2)R) doublets. The six VLQs ($U_{1}\sim U_{3},D_{1}\sim
D_{3}$) are also introduced and the Lagrangian is,

$\displaystyle{\cal L}_{Doublets}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\overline{\psi_{Li}}\left(i\not{\partial}-g_{L}\not{W}_{L}-g_{1}\frac{1}{6}\not{B}_{1}\right)\psi_{Li}+(L\rightarrow
R),$ $\displaystyle{\cal L}_{VLQ}$ $\displaystyle=$
$\displaystyle\sum_{I=1}^{3}\overline{U_{I}}\left(i\not{\partial}-g_{1}\not{B}_{1}\frac{2}{3}-M_{U_{I}}\right)U_{I}+\sum_{I=1}^{3}\overline{D_{I}}\left(i\not{\partial}+g_{1}\not{B}_{1}\frac{1}{3}-M_{D_{I}}\right)D_{I},$
$\displaystyle{\cal L}_{VLQ-Doublets}$ $\displaystyle=$
$\displaystyle-y^{u}_{LiJ}\overline{\psi_{iL}}\tilde{\phi}_{L}U_{JR}-y^{u}_{RiJ}\overline{\psi_{iR}}\tilde{\phi}_{R}U_{JL}-h.c.$
(3)
$\displaystyle-y^{d}_{LiJ}\overline{\psi_{iL}}\phi_{L}D_{JR}-y^{d}_{RiJ}\overline{\psi_{iR}}\phi_{R}D_{JL}-h.c.,$

where $g_{L}$, $g_{R}$ and $g_{1}$ denote the gauge couplings for SU(2)L,
SU(2)R and $U(1)$, respectively.

##  3 The mixing of the Neutral Gauge Bosons

The symmetry breaking of SU(2)${}_{L}\times$ SU(2)${}_{R}\times$ U(1) into
U(1)em leads to the following relation between weak eigenstates
$(W^{3}_{L},B_{1}.W^{3}_{R})$ and mass eigenstates $(Z,A,Z^{\prime})$ of the
three neutral gauge bosons,

$\displaystyle\begin{pmatrix}W_{L\mu}^{3}\\\ B_{1\mu}\\\
W_{R\mu}^{3}\end{pmatrix}=O_{23}(\theta_{W_{R}})O_{12}(-\theta_{W})O_{13}(\theta_{13})\begin{pmatrix}Z_{\mu}\\\
A_{\mu}\\\ Z^{\prime}_{\mu}\end{pmatrix},$ (4)

where the rotation matrices are $O_{23}(\theta)=\begin{pmatrix}1&0&0\\\
0&\cos\theta&-\sin\theta\\\
0&\sin\theta&\cos\theta\end{pmatrix},O_{12}(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta&0\\\
\sin\theta&\cos\theta&0\\\ 0&0&1\end{pmatrix}$,
$O_{13}(\theta)=\begin{pmatrix}\cos\theta&0&-\sin\theta\\\ 0&1&0\\\
\sin\theta&0&\cos\theta\end{pmatrix}$ and the mixing angles are given as
$\tan\theta_{W_{R}}=\frac{g_{1}}{g_{R}},\tan\theta_{W}=\frac{g^{\prime}}{g_{L}},\tan
2\theta_{13}=\frac{\sin^{2}\theta_{W_{R}}\sin
2\theta_{W_{R}}}{\sin\theta_{W}}\frac{v_{L}^{2}}{v_{R}^{2}}$. We note that
$\theta_{13}$ is suppressed by the ratio of vevs
$\frac{v_{L}^{2}}{v_{R}^{2}}$. With the mixing angles, the standard model-like
U(1)Y hypercharge and the electromagnetic charge are given by
$g^{\prime}=g_{1}\cos\theta_{W_{R}},e=g^{\prime}\cos\theta_{W}$. The isospin
current of Z boson is,

$\displaystyle{\cal L}_{{\rm Z}I_{3}}$ $\displaystyle=$
$\displaystyle-\left\\{\frac{1}{2\cos\theta_{W}}\left(\overline{u^{i}_{L}}\gamma_{\mu}u^{i}_{L}-\overline{d^{i}_{L}}\gamma_{\mu}d^{i}_{L}\right)\left(g_{L}\cos\theta_{13}+e\tan\theta_{W_{R}}\sin\theta_{13}\right)\right.$
(5)
$\displaystyle\left.+\frac{g_{R}}{2\cos\theta_{W_{R}}}\left(\overline{u^{i}_{R}}\gamma_{\mu}u^{i}_{R}-\overline{d^{i}_{R}}\gamma_{\mu}d^{i}_{R}\right)\sin\theta_{13}\right\\}Z^{\mu}.$

##  4 Effective Lagrangian

We integrate the five VLQs except a up-type VLQ denoted by $U_{3}$ which mass
parameter $M_{U3}$ is smaller than $v_{R}$. The following effective Lagrangian
is obtained,

$\displaystyle{\cal L}_{\rm eff}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\overline{\psi_{Li}}\left(i\not{\partial}-g_{L}\not{W}_{L}-g_{1}\frac{1}{6}\not{B}_{1}\right)\psi_{Li}+(L\rightarrow
R)$ (6) $\displaystyle-$
$\displaystyle\sum_{ij}\frac{v_{L}}{\sqrt{2}}\overline{u_{Li}}u_{Rj}y^{u}_{ij}-h.c.-\sum_{ij}\frac{v_{L}}{\sqrt{2}}\overline{d_{Li}}d_{Rj}y^{d}_{ij}-h.c.$
$\displaystyle-$
$\displaystyle\sum_{i}\frac{y^{u}_{Li3}v_{L}}{\sqrt{2}}\overline{u_{Li}}U_{R3}-\sum_{i}\frac{y^{u}_{Ri3}v_{R}}{\sqrt{2}}\overline{u_{Ri}}U_{L3}-h.c.-\overline{U}_{3}M_{U_{3}}U_{3},$

where we have ignored the terms suppressed by a factor
$\frac{v_{L}^{2}}{M_{X}^{2}}\ll\frac{v_{R}^{2}}{M_{X}^{2}}\ll 1$ and standard
model like Yukawa couplings for light quarks $u,d,c,s,b$ are given by,

$\displaystyle y^{u}_{ij}$ $\displaystyle=$
$\displaystyle-\sum_{\alpha=1}^{2}y^{u}_{Li\alpha}y^{u\ast}_{Rj\alpha}\frac{v_{R}}{\sqrt{2}M_{U\alpha}},\quad
y^{d}_{ij}=-\sum_{\alpha=1}^{3}y^{d}_{Li\alpha}y^{d\ast}_{Rj\alpha}\frac{v_{R}}{\sqrt{2}M_{D\alpha}},$
(7)

where we have substituted the vevs to two Higgs doublets. ${\cal L}_{\rm eff}$
includes four up-type quarks and their $4\times 4$ mass matrix is given by,

$\displaystyle{\cal L}_{\rm eff\ up-type\
mass}=-\begin{pmatrix}\overline{u_{Li}}&\overline{U_{L3}}\end{pmatrix}M_{{\cal
U}}\begin{pmatrix}u_{jR}\\\ {U}_{R3}\end{pmatrix},M_{{\cal
U}}=\begin{pmatrix}-\sum_{\alpha=1}^{2}\frac{{\bf y}^{u}_{L\alpha}{\bf
y}^{u\dagger}_{R\alpha}v_{L}v_{R}}{{2}M_{U\alpha}}&\frac{{\bf
y}^{u}_{L3}v_{L}}{\sqrt{2}}\\\ \frac{{\bf
y}^{u\dagger}_{R3}v_{R}}{\sqrt{2}}&M_{U_{3}}\end{pmatrix},$ (8)

where $\frac{{\bf
y}^{uT}_{L(R)3}v_{L}}{\sqrt{2}}=\begin{pmatrix}\frac{{y}^{u}_{L(R)13}v_{L}}{\sqrt{2}}&\frac{{y}^{u}_{L(R)23}v_{L}}{\sqrt{2}}&\frac{{y}^{u}_{L(R)33}v_{L}}{\sqrt{2}}\end{pmatrix}$.
We apply the following bi-unitary transformation on ${M}_{\cal U}$ with two
$3\times 3$ unitary matrices $V^{u}$ and $W^{u}$,

$\displaystyle\begin{pmatrix}V^{u\dagger}&0_{3\times 1}\\\ 0_{1\times
3}&1\end{pmatrix}{M}_{\cal U}\begin{pmatrix}W^{u}&0_{3\times 1}\\\ 0_{1\times
3}&1\end{pmatrix}=\begin{pmatrix}-V^{u\dagger}(\sum_{\alpha=1}^{2}\frac{{\bf
y}^{u}_{L\alpha}{\bf
y}^{u\dagger}_{R\alpha}v_{L}v_{R}}{{2}M_{U\alpha}})W^{u}&V^{u\dagger}\frac{{\bf
y}^{u}_{L3}v_{L}}{\sqrt{2}}\\\ \frac{{\bf
y}^{u\dagger}_{R3}v_{R}}{\sqrt{2}}W^{u}&M_{U_{3}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&\frac{|{\bf
y}^{u}_{L3}|v_{L}}{\sqrt{2}}\\\ 0&0&\frac{|{\bf
y}^{u}_{R3}|v_{R}}{\sqrt{2}}&M_{U_{3}}\end{pmatrix}+\begin{pmatrix}-V^{u\dagger}(\sum_{\alpha=1}^{2}\frac{{\bf
y}^{u}_{L\alpha}{\bf
y}^{u\dagger}_{R\alpha}v_{L}v_{R}}{{2}M_{U\alpha}})W^{u}&0_{3\times 1}\\\
0_{1\times 3}&0\end{pmatrix}.$ (9)

Below we diagonalize the first term of Eq.(9) to obtain the spectrum of the
heavier up-type quark. The second term is ignored since the contribution is
suppressed by $\frac{1}{M_{U\alpha}}$ $(\alpha=1,2)$. With the bi-orthogonal
transformation, it is diagonalized as,

$\displaystyle\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&\cos\theta_{R}&-\sin\theta_{R}\\\
0&0&\sin\theta_{R}&\cos\theta_{R}\end{pmatrix}\begin{pmatrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&\cos\theta_{l}&\sin\theta_{l}\\\
0&0&-\sin\theta_{l}&\cos\theta_{l}\end{pmatrix}$ (10)
$\displaystyle\times\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&\frac{|{\bf
y}^{u}_{L3}|v_{L}}{\sqrt{2}}\\\ 0&0&\frac{|{\bf
y}^{u}_{R3}|v_{R}}{\sqrt{2}}&M_{U_{3}}\end{pmatrix}\begin{pmatrix}1&0&0&0\\\
0&1&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{pmatrix}\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&\cos\theta_{R}&\sin\theta_{R}\\\
0&0&-\sin\theta_{R}&\cos\theta_{R}\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&\cos\theta_{L}&-\sin\theta_{L}\\\
0&0&\sin\theta_{L}&\cos\theta_{L}\end{pmatrix}\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ 0&0&m_{L}&0\\\
0&0&M_{U_{3}}&m_{R}\end{pmatrix}\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&\cos\theta_{R}&\sin\theta_{R}\\\
0&0&-\sin\theta_{R}&\cos\theta_{R}\end{pmatrix}=\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ 0&0&m_{t}&0\\\ 0&0&0&m_{t^{\prime}}\end{pmatrix},$ $\displaystyle
m_{t(t^{\prime})}=\frac{\sqrt{M_{U_{3}}^{2}+(m_{R}+m_{L})^{2}}}{2}\mp\frac{\sqrt{M_{U_{3}}^{2}+(m_{R}-m_{L})^{2}}}{2},$

where $m_{L}=\frac{|{\bf y}^{u}_{L3}|v_{L}}{\sqrt{2}},m_{R}=\frac{|{\bf
y}^{u}_{R3}|v_{R}}{\sqrt{2}},\theta_{L}=\theta_{R}-\theta_{l},\tan\theta_{l}=\frac{M_{U_{3}}}{m_{R}+m_{L}}$
and $\tan
2\theta_{R}=\frac{2M_{U_{3}}m_{R}}{m_{R}^{2}-m_{L}^{2}-M_{U_{3}}^{2}}.$

##  5 Z FCNC for top quark and its partner

One can compute the Z FCNC by rewriting left and right up type qaurks in SU(2)
doublet in terms of top quark $t$ and its partner $t^{\prime}$,

$\displaystyle u_{3R}$ $\displaystyle=$
$\displaystyle-\sin\theta_{R}t_{R}+\cos\theta_{R}t_{R}^{\prime},$
$\displaystyle u_{3L}$ $\displaystyle=$
$\displaystyle\cos\theta_{L}t_{L}+\sin\theta_{L}t_{L}^{\prime}.$ (11)

With Eq.(11) and Eq.(5), Z FCNC for top quark and top prime quark is,

$\displaystyle{\cal L}_{\rm Z\
I_{3}}^{tt^{\prime}}=-\frac{g_{R}\sin\theta_{13}Z^{\mu}}{2\cos\theta_{W_{R}}}\left\\{\sin^{2}\theta_{R}\overline{t_{R}}\gamma_{\mu}t_{R}-\sin\theta_{R}\cos\theta_{R}\left(\overline{t_{R}}\gamma_{\mu}t_{R}^{\prime}+h.c.\right)+\cos^{2}\theta_{R}\overline{t^{\prime}_{R}}\gamma_{\mu}t^{\prime}_{R}\right\\}$
$\displaystyle-\frac{Z^{\mu}(g_{L}\cos\theta_{13}+e\tan\theta_{W_{R}}\sin\theta_{13})}{2\cos\theta_{W}}\left\\{\cos^{2}\theta_{L}\overline{t_{L}}\gamma_{\mu}t_{L}+\sin\theta_{L}\cos\theta_{L}\left(\overline{t_{L}}\gamma_{\mu}t_{L}^{\prime}+h.c.\right)+\sin^{2}\theta_{L}\overline{t^{\prime}_{L}}\gamma_{\mu}t^{\prime}_{L}\right\\}.$

When $M_{U_{3}}\ll m_{R}$, one can show $\theta_{L}\simeq
0\ll\theta_{R}\simeq\frac{M_{U_{3}}}{m_{R}}\ll 1$ .

## Acknowledgment

We would like to thank organizers of KAON 2022. The work of T. M. is supported
by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number
JP17K05418 and JP16H03993.

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