HUPD-2203

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# Renormalization group effects for a rank degenerate Yukawa matrix and the
fate of the massless neutrino

Nicholas J. Benoit<EMAIL_ADDRESS>Graduate School of Science,
Hiroshima University, Higashi-Hiroshima 739-8526, Japan  Takuya Morozumi
Physics Program, Graduate School of Advanced Science and Engineering,
Hiroshima University, Higashi-Hiroshima 739-8526, Japan
<EMAIL_ADDRESS><EMAIL_ADDRESS>Core of Research for
the Energetic Universe, Hiroshima University,
Higashi-Hiroshima 739-8526, Japan  Yusuke Shimizu  Kenta Takagi  Higashi-
Hiroshima, Hiroshima Prefecture, Japan  Akihiro Yuu  Yokkaichi, Mie
Prefecture, Japan

###### Abstract

The Type-I seesaw model is a common extension to the Standard Model that
describes neutrino masses. The Type-I seesaw introduces heavy right-handed
neutrinos with Majorana mass that transform as Standard Model electroweak
gauge singlets. We initially study a case with two right-handed neutrinos
called the 3-2 model. At an energy scale above the right-handed neutrinos, the
effective neutrino mass matrix is rank degenerate implying the lightest
neutrino is massless. After considering renormalization effects below the two
right-handed neutrinos, the effective neutrino mass matrix remains rank
degenerate. Next, we study a model with three right-handed neutrinos called
the 3-3 model. Above the energy scale of the three right-handed neutrinos, we
construct the effective neutrino mass matrix to be rank degenerate. After
solving for the renormalization effects to energies below the three right-
handed neutrinos, we find the rank of the effective neutrino mass matrix
depends on the kernel solutions of the renormalization group equations. We
prove for the simplest kernel solutions the effective neutrino mass matrix
remains rank degenerate.

##  1 Introduction

The knowledge of the absolute masses of neutrinos is still lacking despite the
precise determination of mass squared differences from neutrino oscillation
experimentsEsteban:2020cvm ; NuFit:2021 ; Aker:2021gma ; Planck:2018vyg . If
the neutrino mass type is Majorana, the lightest absolute mass is a source of
the uncertainty in the determination of the rate of neutrinoless double beta
decayFurry:1939qr ; Schechter:1981bd ; Nieves:1984sn ; Takasugi:1984xr ;
Dolinski:2019nrj . Actuality, there are models that allow for the lightest
neutrino to be masslessMa:1998zg . For those models, the neutrinoless double
beta decay rate becomes more predictive since the non-zero masses are directly
obtained from the mass squared differences and only a single Majorana phase is
allowed. In cosmology, the absolute neutrino masses are important to early
universe phenomena such as big-bang nucleosynthesis, and large scale structure
formationDolgov:2002wy . And we have previously discussed the role of the
absolute masses for neutrinos in the time evolution of lepton
numberAdam:2021qiq ; Adam:2021yvr . Thus, understanding the absolute masses of
neutrinos is important for neutrinoless double beta decay, cosmology, and the
time evolution of lepton number. The main goal of this paper is to study a
massive neutrino models’ stability, with the lightest neutrino being massless,
including 1-loop quantum corrections.

We consider the massive neutrino model called the Type-I seesaw
Mohapatra:1979ia ; Minkowski:1977sc ; Gell-Mann:1979vob ; Glashow:1979nm ;
Yanagida:1979as  first with two right-handed neutrinos, the 3-2 seesaw model;
then with three right-handed neutrinos, the 3-3 seesaw model111A model with
only one right-handed fermion is excluded by the measured mass squared
differences of oscillation experiments.. For the 3-2 seesaw model the neutrino
Dirac mass matrix is a 3 by 2 matrix and its rank is two. The 3-2 seesaw model
has been previously studied sometimes under the name the minimal Type-I seesaw
model Ma:1998zg ; King:1999mb ; King:2002nf ; Frampton:2002qc ; Endoh:2002wm ;
Shimizu:2017fgu ; Xing:2020ald ; Zhou:2021bqs . For the model with three
right-handed neutrinos, we assume a texture of the neutrino Dirac mass terms
to reduce the rank of the matrix to two. Both of those models lead to a
vanishing mass for the lightest neutrino, i.e., $m_{1}=0$ for normal
hierarchy. In the usual Type-I seesaw the right-handed neutrinos have a
significantly heavier mass than the Standard Model particle content to
reproduce small neutrino masses. This means the heavy right-handed neutrinos
are integrated out at low-energy scales222Different nomenclature is used in
literature to name the heavy right-handed neutrinos, for example right-handed
neutrinos or heavy neutrino singlets or simply heavy neutrinos.

In this paper, we integrate out the right-handed neutrinos within the tree
level matching approximation and take into account the one loop
renormalization group (RG) effectsAntusch:2002rr ; Antusch:2005gp . Then we
determine the matrix rank of the effective neutrino mass matrix. The
renormalization group running of neutrino masses has been studied previously
in literature with the rank of the 3-2 seesaw model being considered for the
one loop matching approximation Zhou:2021bqs  and two loop renormalization
group effectsDavidson:2006tg . However, we believe the stability of the 3-3
seesaw model with a matrix of rank two has not been examined against
renormalization group effects.

We organize this paper as follows. In section 2, we introduce the seesaw model
as a review. In section 3, we demonstrate the method we use to determine the
rank of a matrix. Then in section 4, we study the renormalization group
effects in the 3-2 seesaw model and calculate the rank of the effective
Majorana mass matrix. Next in section 5.1, we apply the techniques of section
4 for the 3-3 seesaw model with generic solutions to the renormalization group
equations. Then using specific renormalization group solutions, we consider
the rank of the effective mass matrix in section 5.2. Lastly, section 6 is
devoted to summary and discussion. Appendix A holds the details of the
renormalization group equations and solutions used in the main text.

##  2 Type-I seesaw model

We consider a general Type-I seesaw model with right-handed neutrinos $N_{n}$,
the standard model lepton doublets $l_{L}$, and the standard model Higgs
doublet $\widetilde{\phi}=i\tau_{2}\phi^{\ast}$ as an extension to the
Standard Model Mohapatra:1979ia ; Minkowski:1977sc ; Gell-Mann:1979vob ;
Glashow:1979nm ; Yanagida:1979as . The Lagrangian of the extended section
reads,

$\mathcal{L}_{N}=\frac{1}{2}\overline{N_{n}}i/\mspace{-9.0mu}\partial
N_{n}-\frac{1}{2}\overline{N_{n}}M_{R\,n}N_{n}-(y_{\nu
jn}\overline{l_{jL}}\widetilde{\phi}N_{n}+\text{h.c.}),$ (1)

where $N_{n}^{C}=N_{n}$ and $C$ denotes charge conjugation, and $n$ are
integer values that correspond to the number of the heavy right-handed
neutrinos. Under the Lagrangian of Eq.(1) the masses of the active neutrinos
are found by diagonalizing the effective Majorana mass matrix,

$m_{eff}=-\frac{v^{2}}{2}y_{\nu}\frac{1}{M_{R}}y^{T}_{\nu},$ (2)

where $v=246$ GeV is the Higgs vacuum expectation value. If we neglect quantum
effects captured by running the renormalization group equations, $m_{eff}$
does not change with the energy scale. However, if we want to reproduce the
masses of the active neutrinos to be in the eV region, as constrained by
experiments, the masses of $M_{R}$ are required to be near the GUT scale. This
means for practical experiments the right-handed neutrinos are integrated out,
down to the energy scale being considered Antusch:2002rr ; King:2000hk . After
integration, the Wilson coefficient of the so-called Weinberg operator,
$\kappa(\mu)$, is generated Weinberg:1979sa .

$\mathcal{O}_{5}=\frac{1}{4}\kappa_{gh}^{\ast}(\mu)\left(\overline{l^{C}_{L}}^{\,g}\cdot\phi\right)\left(l^{h}_{L}\cdot\phi\right)+\text{h.c.}$
(3)

As the operator is dimension five, it is suppressed by the masses of the
right-handed neutrinos $M_{R}$. We display the diagrams representing how the
Wilson coefficient of the Weinberg operator appears after integration in
figure 1.

Figure 1: Diagrams for the Type-I seesaw model at different energy scales. The
left diagram is for the full theory and the right diagram is for the effective
theory. By integrating out the right-handed neutrinos shown in the left
diagram, the coefficient of the Weinberg operator in the right diagram is
generated.

The quantum effects captured by the renormalization group equations mean the
effective mass matrix of Eq.(2) becomes energy scale dependent $m_{eff}(\mu)$.
In the Type-I seesaw model, the energy scale dependence is realized with the
equation,

$\overset{(a)}{m}_{eff}(\mu)=-\frac{v^{2}}{4}\overset{(a)}{\kappa}(\mu)-\frac{v^{2}}{2}\overset{(a)}{y_{\nu}}(\mu)\frac{1}{\overset{(a)}{M_{R}}(\mu)}\overset{(a)}{y_{\nu}}(\mu)^{T}$
(4)

where $\kappa(\mu)$, $y_{\nu}(\mu)$, and $M_{R}(\mu)$ depend on the energy
scale $\mu$ being calculated up to a cutoff scale $\Lambda$333We do not
consider what occurs for scales above $\Lambda$ to generate mass for the
right-handed neutrinos.. We have adopted the superscript notation to denote
the different renormalization energy regions from the work of Antusch
Antusch:2005gp , for which the lowest region is denoted as $a=1$.

##  3 Determination of the rank of the mass matrix

In this section, we present the method we use to determine the rank of the
effective Majorana mass matrix of Eq.(4). Precisely speaking, we are
interested in the rank of the Hermite matrix that determines the mass squared
eigenvalues for light neutrinos,

$\overset{(1)}{\mathcal{M}}=\overset{(1)}{m}_{eff}(\mu_{1})[\overset{(1)}{m}_{eff}(\mu_{1})]^{\dagger},$
(5)

where $\overset{(1)}{m}_{eff}$ denotes the effective Majorana mass matrix of
Eq.(4) for the light neutrinos obtained after integrating out all the right-
handed neutrinos. The $\mu_{1}$ is the mass scale of the lightest right-handed
Majorana neutrino. We can solve for the rank of the Hermite matrix using the
characteristic polynomial,

$\det{(\overset{(1)}{\mathcal{M}}-x)}=0.$ (6)

The eigenvalue $x$ satisfies the following equation,

$-x^{3}+x^{2}I_{2}-xI_{1}+I_{0}=0,$ (7)

where $I_{0},I_{1},\text{ and }I_{2}$ are calculated using the Faddeev-
LeVerrier algorithm,

$\displaystyle I_{2}$ $\displaystyle=\text{Tr}[\overset{(1)}{\mathcal{M}}],$
(8) $\displaystyle I_{1}$
$\displaystyle=\frac{1}{2}\left((\text{Tr}[\overset{(1)}{\mathcal{M}}])^{2}-\text{Tr}[(\overset{(1)}{\mathcal{M}})^{2}]\right),$
(9)
$\displaystyle\begin{split}I_{0}&=\frac{1}{6}\left((\operatorname{Tr}[\overset{(1)}{\mathcal{M}}])^{3}-3\operatorname{Tr}[\overset{(1)}{\mathcal{M}}]\operatorname{Tr}[(\overset{(1)}{\mathcal{M}})^{2}]+2\operatorname{Tr}[(\overset{(1)}{\mathcal{M}})^{3}]\right)=\det\overset{(1)}{\mathcal{M}}.\end{split}$
(10)

Since the characteristic polynomial is invariant under the unitary
transformation $\mathcal{M}^{\prime}=U^{\dagger}\mathcal{M}U$, the
coefficients; $I_{0}$, $I_{1}$, and $I_{2}$, are also invariants.

In order to prove the rank explicitly, we calculate the matrix invariants
$I_{0}$, $I_{1}$, and $I_{2}$ of the Hermite matrix in Eq.(5) using
Eqs.(8-10). Then rank of Eq.(5) is determined by subtracting the degree of the
polynomial with the superscript of the 0 root in Eq.(7)444We are assuming the
Hermite matrix is diagonalizable.. If $I_{0}=0$ and the other invariants are
non-zero the rank of Eq.(5) is $3-1=2$, and one of the masses squared is zero
with the others being massive. This is clear when the invariants are written
in terms of the eigenvalues of the Hermite matrix

$\displaystyle I_{2}$ $\displaystyle=m_{1}^{2}+m_{2}^{2}+m_{3}^{2},$ (11)
$\displaystyle I_{1}$
$\displaystyle=m_{1}^{2}m_{2}^{2}+m_{2}^{2}m_{3}^{2}+m_{1}^{2}m_{3}^{2},$ (12)
$\displaystyle I_{0}$ $\displaystyle=(m_{1}m_{2}m_{3})^{2}.$ (13)

Then, for example, to have $I_{0}=0$ any of the eigenvalues can equal zero,
but the other two eigenvalues must remain non-zero to have $I_{1}$ and $I_{2}$
as non-zero.

##  4 3-2 Type-I seesaw model renormalization group running

In this section, we solve the renormalization group equations for two
different energy regions of the 3-2 Type-I seesaw model. The 3-2 Type-I seesaw
model is the minimalist version of the Type-I seesaw that is not directly
excluded by neutrino experiments and is sometimes named the minimal Type-I
seesaw model (MSM)Ma:1998zg ; King:1999mb ; King:2002nf ; Frampton:2002qc ;
Endoh:2002wm ; Shimizu:2017fgu ; Xing:2020ald ; Zhou:2021bqs . This version
introduces two new right-handed neutrinos to the Standard Model that we label
as $N_{1}$ and $N_{2}$ following Eq.(1). We assume the masses of the right-
handed neutrinos are hierarchical, satisfying $M_{1}<M_{2}<\Lambda$ according
to figure 2.

Figure 2:  Illustration of the different energy regions being considered in
the 3-2 Type-I seesaw model. The highest energy region occurs above $M_{2}$
where no $\kappa$ term exists. The renormalization energy scale decreases from
right to the left. The matrices $\kappa(\mu)$, $y_{\nu}(\mu)$, and
$M_{R}(\mu)$ are scale dependent based on the $\beta$-functions
$\beta_{\kappa}$, $\beta_{y_{\nu}}$, and $\beta_{M_{R}}$ defined by the
corresponding renormalization group equations. To denote the different
renormalization energy regions we adopt the superscript notation from
Antusch:2005gp .

Consequently, the renormalization scales are defined as;

$t_{1}\equiv\log\frac{M_{2}}{M_{1}},\quad t\equiv\log\frac{M_{2}}{\mu}.$ (14)

At the initial renormalization scale of $t=0\;(\mu=M_{2})$, we go to the
diagonal basis for the right-handed mass matrix,

$M_{R}(0)=\operatorname{diag}(M_{1}(0),\,M_{2}(0)).$ (15)

In this basis, the neutrino Yukawa matrix has a $3\times 2$ structure,

$y_{\nu}(0)=\begin{pmatrix}y_{\nu 1}(0)&y_{\nu 2}(0)\end{pmatrix}.$ (16)

An important feature of the 3-2 Type-I seesaw occurs at or below the initial
renormalization scale of $t=0$. There only $M_{R}(0)$ and $y_{\nu}(0)$
contribute to the effective neutrino mass calculation using Eq.(4). The
resulting $3\times 3$ effective mass matrix $m_{eff}(0)$ is rank degenerate,
i.e. $\operatorname{Rank}{m_{eff}(0)}=2$; which, based on the discussions of
Eqs.(11-13), means the lightest neutrino mass eigenvalue is zero.

Slightly above $t=0$, the renormalization energies are smaller than the mass
of the right-handed neutrino $M_{2}$. Thus, we must integrate out the right-
handed neutrino $N_{2}$, which generates the following Wilson coefficient,

$\overset{(2)}{\kappa}(0)=2y_{\nu 2}(0)\frac{1}{M_{2}(0)}y_{\nu 2}^{T}(0)$
(17)

We use the usual method of tree-level matching to demand the effective masses
of the neutrinos $m_{eff}(\mu)$ are continuous over the energy regions.
Specifically, we demand that the $m_{eff}(\mu)$’s are equal before and after
$N_{2}$ is integrated out;

$\overset{(2)}{m_{eff}}(0)=m_{eff}(0),$ (18)

where from Eq.(4) and Eq.(17) we have,

$\displaystyle m_{eff}(0)=$
$\displaystyle-\frac{v^{2}}{2}y_{\nu}(0)\frac{1}{M_{R}(0)}y^{T}_{\nu}(0),$
(19) $\displaystyle\overset{(2)}{m_{eff}}(0)=$
$\displaystyle-\frac{v^{2}}{4}\overset{(2)}{\kappa}(0)-\frac{v^{2}}{2}\overset{(2)}{y_{\nu}}(0)\left(\overset{(2)}{M_{R}}(0)\right)^{-1}\left(\overset{(2)}{y_{\nu}}(0)\right)^{T},$
(20)

where $\overset{(2)}{M_{R}}(0)={M_{1}}(0)$. The Wilson coefficient
$\overset{(2)}{\kappa}(t)$, the neutrino Yukawa matrix
$\overset{(2)}{y_{\nu}}(t)$, and the remaining right-handed mass
$\overset{(2)}{M_{R}}(t)$ are energy scale dependent and are only valid for
the scales $0<t<t_{1}$, see figure 2. We calculate the corrections from the
energy scale dependence with the one-loop renormalization group equations of
Appendix A.

As an example, we write the one-loop renormalization group equation for
$\overset{(2)}{\kappa}(t)$;

$-16\pi^{2}\frac{d\overset{(2)}{\kappa}(t)}{dt}=\left(\overset{(2)}{y_{\nu}}(t)\overset{(2)}{y_{\nu}}(t)^{\dagger}\right)\overset{(2)}{\kappa}(t)+\overset{(2)}{\kappa}(t)\left(\overset{(2)}{y_{\nu}}(t)\overset{(2)}{y_{\nu}}(t)^{\dagger}\right)^{T}+\overset{(2)}{\alpha_{\kappa}}(t)\overset{(2)}{\kappa}(t),$
(21)

which has a solution at boundary of the lower energy scale $t_{1}$ given by,

$\overset{(2)}{\kappa}(t_{1})=e^{-\frac{1}{16\pi^{2}}\int_{0}^{t_{1}}ds\,\overset{(2)}{\alpha_{\kappa}}(s)}\overset{(2)}{W}(t_{1})\overset{(2)}{\kappa}(0)\overset{(2)}{W}(t_{1})^{T},$
(22)

where we define the solution kernel $\overset{(2)}{W}(t_{1})$ following
Eq.(97),

$\overset{(2)}{W}(t_{1})\equiv
T\exp\left[-\frac{1}{16\pi^{2}}\int_{0}^{t_{1}}(\overset{(2)}{y_{\nu}}(s)\overset{(2)}{y_{\nu}}(s)^{\dagger})ds\right].$
(23)

The $T$ denotes the exponential is ordered by decreasing energy scale. The
energy dependence for the terms $\overset{(2)}{y_{\nu}}(t)$ and
$\overset{(2)}{M_{R}}(t)$ are solved similarly using Eqs.(95-96). Now we have
solutions for $\overset{(2)}{\kappa}(t_{1})$, $\overset{(2)}{y_{\nu}}(t_{1})$,
and $\overset{(2)}{M_{R}}(t_{1})$ down to the energy scale $t=t_{1}$, which is
equivalent to saying $\mu=M_{1}$. At this scale we must integrate out the
remaining right-handed neutrino $N_{1}$ and take a tree-level matching,

$\displaystyle\overset{(1)}{m_{eff}}(t_{1})$
$\displaystyle=\overset{(2)}{m_{eff}}(t_{1})$ (24)
$\displaystyle\overset{(1)}{m_{eff}}(t_{1})$
$\displaystyle=-\frac{v^{2}}{4}\overset{(2)}{\kappa}(t_{1})-\frac{v^{2}}{2}\overset{(2)}{y_{\nu}}(t_{1})\left(\overset{(2)}{M_{R}}(t_{1})\right)^{-1}\left(\overset{(2)}{y_{\nu}}(t_{1})\right)^{T},$
(25)

that is similar to Eq.(18).

In contrast to the scale of $t=0$, for the scales $t>t_{1}$ we have integrated
out all right-handed neutrinos. Thus, the neutrino effective mass matrix only
depends on the Wilson coefficient,

$\overset{(1)}{m_{eff}}(t_{1})=-\frac{v^{2}}{4}\overset{(1)}{\kappa}(t_{1}).$
(26)

We could solve the renormalization group equation for
$\overset{(1)}{\kappa}(t_{1})$ down to energy scales of practical experiments
and apply the solution to different observables. However, above $t=t_{1}$, or
below $M_{1}$, the one-loop renormalization group equation of
$\overset{(1)}{\kappa}(t)$ from Eq.(89) can not change its rank555See section
3.3 of Antusch:2005gp  for a more detailed analysis on why this is the case.
Therefore, we focus on the rank of $\overset{(1)}{\kappa}(t_{1})$ to study if
the matrix rank of $\overset{(1)}{m_{eff}}(t_{1})$ is equal to the matrix rank
of the high energy $m_{eff}(0)$.

To determine the rank of the low energy mass matrix
$\overset{(1)}{m_{eff}}(t_{1})$ we investigate the parts of the Wilson
coefficient $\overset{(1)}{\kappa}(t_{1})$. The parts of
$\overset{(1)}{\kappa}(t_{1})$ are determined from the tree-level matching of
Eq.(25) resulting in,

$\overset{(1)}{\kappa}(t_{1})=2\overset{(2)}{y_{\nu}}(t_{1})\left(\overset{(2)}{M_{R}}(t_{1})\right)^{-1}\overset{(2)}{y_{\nu}}(t_{1})^{T}+\overset{(2)}{\kappa}(t_{1}).$
(27)

Then substituting the solutions to the renormalization group equation of
Eq.(22) for $\overset{(2)}{\kappa}(t_{1})$;

$\overset{(1)}{\kappa}(t_{1})=2\overset{(2)}{y_{\nu}}(t_{1})\left(\overset{(2)}{M_{R}}(t_{1})\right)^{-1}\overset{(2)}{y_{\nu}}(t_{1})^{T}+e^{-\frac{1}{16\pi^{2}}\int_{0}^{t_{1}}\overset{(2)}{\alpha_{\kappa}}(s)ds}\overset{(2)}{W}(t_{1})\overset{(2)}{\kappa}(0)\overset{(2)}{W}(t_{1})^{T}.$
(28)

At this point we note two important considerations. First, the contribution
from the running of the right-handed mass $\overset{(2)}{M_{R}}(t_{1})$ is a
scaling factor that will not affect the rank of the low energy mass matrix.
Second, the running of the Yukawa couplings are what determines the rank of
$\overset{(1)}{m_{eff}}(t)$ for reasons to be explained. We explicitly write
the Yukawa couplings in terms of the solutions to the renormalization group
equations up to the energy scale $\mu=M_{2}$ or $t=0$.

$\overset{(2)}{y_{\nu}}(t_{1})=e^{-\frac{1}{16\pi^{2}}\int^{t_{1}}_{0}\overset{(2)}{\alpha_{y_{\nu}}}(s)ds}\overset{(2)}{U}(t_{1})\overset{(2)}{y_{\nu}}(0)$
(29)

where the kernel function $\overset{(2)}{U}(t_{1})$ is the solution to the
Yukawa renormalization group equation of Eq.(90). Note, a subtle difference
between the kernel of Eq.(97) for $\overset{(2)}{\kappa}(t_{1})$ and the
kernel of Eq.(98) for $\overset{(2)}{y_{\nu}}(t_{1})$ is an additional
$\frac{3}{2}$ factor in the exponential. Then, we write the matrix
$\overset{(1)}{m_{eff}}(t_{1})=\overset{(1)}{\kappa}(t_{1})$ with the two
kernel functions from Eq.(29) and Eq.(17),

$\begin{split}\overset{(1)}{\kappa}(t_{1})&=2e^{-\frac{1}{8\pi^{2}}\int^{t_{1}}_{0}\overset{(2)}{\alpha_{y_{\nu}}}(s)ds}\left(\overset{(2)}{M_{R}}(t_{1})\right)^{-1}\overset{(2)}{U}(t_{1})y_{\nu
1}(0)y_{\nu 1}^{T}(0)\overset{(2)}{U}(t_{1})^{T}\\\
&\quad+2e^{-\frac{1}{16\pi^{2}}\int_{0}^{t_{1}}\overset{(2)}{\alpha_{\kappa}}(s)ds}\frac{1}{M_{2}(0)}\overset{(2)}{W}(t_{1})y_{\nu
2}(0)y_{\nu 2}^{T}(0)\overset{(2)}{W}(t_{1})^{T},\end{split}$ (30)

where we have also used Eq.(16) to write the Yukawa vectors explicitly.

Next, we focus on the relationship between the kernel functions and the Yukawa
vectors. In Eq.(31) the two kernel functions appear as
$\overset{(2)}{W}(t_{1})$ and $\overset{(2)}{U}(t_{1})$, and are square
matrices shaped by the number of rows in the Yukawa vectors $y_{\nu 1}(0)$ and
$y_{\nu 2}(0)$. Upon multiplication, the kernel functions act, separately, as
linear transformation matrices on the Yukawa vectors. To better illustrate the
rotation we rewrite the effective mass matrix of Eq.(30) to be,

$\overset{(1)}{m_{eff}}(t_{1})=X\,\overset{(2)}{U}(t_{1})y_{\nu 1}(0)y_{\nu
1}^{T}(0)\overset{(2)}{U}(t_{1})^{T}+Z\,\overset{(2)}{W}(t_{1})y_{\nu
2}(0)y_{\nu 2}^{T}(0)\overset{(2)}{W}(t_{1})^{T},$ (31)

where $X$ and $Z$ are scaling functions that do not contribute to the matrix
rank. They are given as,

$\displaystyle
X\equiv-\frac{v^{2}}{2}e^{-\frac{1}{8\pi^{2}}\int^{t_{1}}_{0}\overset{(2)}{\alpha_{y_{\nu}}}(s)ds}\left(\overset{(2)}{M_{R}}(t_{1})\right)^{-1}$
(32) $\displaystyle
Z\equiv-\frac{v^{2}}{2}e^{-\frac{1}{16\pi^{2}}\int_{0}^{t_{1}}\overset{(2)}{\alpha_{\kappa}}(s)ds}\frac{1}{M_{2}(0)}.$
(33)

Thus, we rewrite Eq.(31) after the linear transformations have occurred,

$\overset{(1)}{m_{eff}}(t_{1})=X\,y^{U}_{\nu 1}(t_{1})y^{U}_{\nu
1}(t_{1})^{T}+Z\,y^{W}_{\nu 2}(t_{1})y^{W}_{\nu 2}(t_{1})^{T},$ (34)

where the transformed vectors of,

$\displaystyle y^{U}_{\nu 1}(t_{1})=\overset{(2)}{U}(t_{1})y_{\nu 1}(0)$ (35)
$\displaystyle y^{W}_{\nu 2}(t_{1})=\overset{(2)}{W}(t_{1})y_{\nu 2}(0)$ (36)

are non-zero and assumed to be linearly independent of each other. With the
effective mass matrix given in Eq.(34), the mass squared eigenvalues of the
Hermite matrix
$\overset{(1)}{\mathcal{M}}=\overset{(1)}{m_{eff}}(t_{1})[\overset{(1)}{m_{eff}}(t_{1})]^{\dagger}$
has a zero eigenvalue. To prove this, we evaluate three invariants in
Eqs.(8-10) with the effective Majorana mass matrix for 3-2 model in Eq.(34).
The first two invariants $I_{1}\neq 0$ and $I_{2}\neq 0$ hold the expressions
below,

$\displaystyle\begin{split}I_{1}=&|X|^{2}|Z|^{2}\left\\{y^{W\ast}_{12}\left[(y^{W}_{12}y^{U}_{21}-y^{U}_{11}y^{W}_{22})y^{U\ast}_{21}+(y^{W}_{12}y^{U}_{31}-y^{U}_{11}y^{W}_{32})y^{U\ast}_{31}\right]\right.\\\
&\left.+y^{U\ast}_{11}\left[(y^{U}_{11}y^{W}_{22}-y^{W}_{12}y^{U}_{21})y^{W\ast}_{22}+(y^{U}_{11}y^{W}_{32}-y^{W}_{12}y^{U}_{31})y^{W\ast}_{32}\right]+|y^{W}_{22}y^{U}_{31}-y^{U}_{21}y^{W}_{32}|^{2}\right\\}^{2},\end{split}$
(37)
$\displaystyle\begin{split}I_{2}=&|X(y^{U}_{11})^{2}+Z(y^{W}_{12})^{2}|^{2}+2|Xy^{U}_{11}y^{U}_{21}+Zy^{W}_{12}y^{W}_{22}|^{2}+|X(y^{U}_{21})^{2}+Z(y^{W}_{22})^{2}|^{2}\\\
&+2|Xy^{U}_{11}y^{U}_{31}+Zy^{W}_{12}y^{W}_{32}|^{2}+|X(y^{U}_{31})^{2}+Z(y^{W}_{32})^{2}|^{2}+2|Xy^{U}_{21}y^{U}_{31}+Zy^{W}_{22}y^{W}_{32}|^{2},\end{split}$
(38)

where the element notation for Eqs.(35-36) is used, e.g.,
$y^{W}_{i2}=[y^{W}_{\nu 2}(t_{1})]_{i}$. Importantly, the invariant $I_{0}$
can be rewritten as the complex product
$I_{0}=\mathcal{I}\times\mathcal{I}^{\ast}$. Then $\mathcal{I}$ is written as
the matrix elements from $[\overset{(1)}{m_{eff}}(t_{1})]_{ij}=m_{ij}$,

$\mathcal{I}\equiv
m_{33}(m_{11}m_{22}-m_{21}m_{12})+m_{13}(m_{32}m_{21}-m_{31}m_{22})+m_{32}(m_{31}m_{12}-m_{32}m_{11}).$
(39)

After substitution of the matrix elements from Eq.(34) the terms of
$\mathcal{I}$ are,

$\displaystyle
m_{33}(m_{11}m_{22}-m_{21}m_{12})=\frac{v^{3}}{8}XZ\left(y^{W}_{12}y^{U}_{21}-y^{U}_{11}y^{W}_{22}\right)^{2}\left(X(y^{U}_{31})^{2}+Z(y^{W}_{32})^{2}\right),$
(40) $\displaystyle
m_{13}(m_{32}m_{21}-m_{31}m_{22})=-\frac{v^{3}}{8}XZ\left(y^{W}_{12}y^{U}_{21}-y^{U}_{11}y^{W}_{22}\right)\left(y^{U}_{21}y^{W}_{32}-y^{W}_{22}y^{U}_{31}\right)\left(Xy^{U}_{11}y^{U}_{31}+Zy^{W}_{12}y^{W}_{32}\right),$
(41) $\displaystyle
m_{32}(m_{31}m_{12}-m_{32}m_{11})=-\frac{v^{3}}{8}XZ\left(y^{W}_{12}y^{U}_{21}-y^{U}_{11}y^{W}_{22}\right)\left(y^{W}_{12}y^{U}_{31}-y^{U}_{11}y^{W}_{32}\right)\left(Xy^{U}_{21}y^{U}_{31}+Zy^{W}_{22}y^{W}_{32}\right).$
(42)

The addition of the last two Eqs.(41-42) exactly cancel the first Eq.(40),
which leads to $\mathcal{I}=0$ that proves the
$\operatorname{Rank}[\overset{(1)}{m_{eff}}(t_{1})]=2$. In other words, the
lightest mass eigenvalue is zero $\overset{(1)}{m_{lightest}}(t_{1})=0$ as
typical for 3-2 Seesaw models. This result does not depend on the details of
how the kernel matrices affect the Yukawa vectors. Then from consideration of
the invariants we conclude that,

$\operatorname{Rank}\left[\overset{(1)}{\kappa}(t_{1})\right]=\operatorname{Rank}\left[\overset{(1)}{m_{eff}}(t_{1})\right]=2.$
(43)

This result for the 3-2 model is the one-loop renormalization group equations
and tree-level matching. However, the result has been shown to not hold for
one-loop matching in supersymmetry Zhou:2021bqs  and two-loop renormalization
group equation for the Weinberg operator coefficient $\kappa$ Davidson:2006tg
. Inclusion of all two-loop renormalization group equations Ibarra:2018dib ;
Ibarra:2020eia  and complete one-loop level matching in the Standard Model
could be interesting. Here we use the one-loop equations and tree-level
matching to prepare, for section 5.1, on how a rank degenerate matrix remains
that way even after renormalization group effects are considered.

##  5 3-3 Type-I Seesaw Model Renormalization Group Running

###  5.1 Effective mass matrix with generic renormalization group kernel
solutions

In this section we consider the next simplest version of the Type-I seesaw
model. The Lagrangian of Eq.(1) remains the same, but now three right-handed
neutrinos are included $N_{1}$, $N_{2}$, and $N_{3}$. We assume the masses of
the right-handed neutrinos are hierarchical with $M_{3}>M_{2}>M_{1}$ as
illustrated in Fig 3. This model is described by a full theory at the energy
scale of $\mu>M_{3}$ up to some cutoff $\Lambda$.

Figure 3:  Illustration of the different energy regions being considered for
the 3-3 model. Similar to Fig 2, the highest energy region of the full theory
occurs between $M_{3}$ and $\Lambda$ where no $\kappa$ term exists. The
renormalization energy scale decreases from right to the left. The matrices
$\kappa(\mu)$, $y_{\nu}(\mu)$, and $M_{R}(\mu)$ are scale dependent based on
the $\beta$-functions $\beta_{\kappa}$, $\beta_{y_{\nu}}$, and $\beta_{M_{R}}$
defined by the corresponding renormalization group equations.

We are interested in the case when the effective mass matrix is rank
degenerate between the energy scale of $M_{3}<\mu\leq\Lambda$. Meaning we
create the condition of,

$\displaystyle m_{eff}$
$\displaystyle=-\frac{v^{2}}{2}y_{\nu}\frac{1}{M_{R}}y^{T}_{\nu},$ (44)
$\displaystyle\operatorname{Rank}\left[m_{eff}\right]$ $\displaystyle=2,$ (45)

by assuming the Yukawa matrix is also rank degenerate,

$\displaystyle y_{\nu ij}$ $\displaystyle=\begin{pmatrix}y_{\nu i1}&y_{\nu
i2}&y_{\nu i3}\end{pmatrix},$ (46) $\displaystyle y_{\nu 3}$
$\displaystyle=ay_{\nu 1}+by_{\nu 2},$ (47)
$\displaystyle\operatorname{Rank}\left[y_{\nu}\right]$ $\displaystyle=2.$ (48)

In Eq.(47), $a$ and $b$ are complex-numbers of arbitrary values. The effective
mass matrix of Eq.(44) can be rewritten in a similar form of the 3-2 model by
using our assumption of Eq.(47);

$m_{eff}=-\frac{v^{2}}{2}\begin{pmatrix}y_{\nu 1}&y_{\nu
2}\end{pmatrix}\begin{pmatrix}\frac{1}{M_{1}}+\frac{a^{2}}{M_{3}}&\frac{ab}{M_{3}}\\\
\frac{ab}{M_{3}}&\frac{1}{M_{2}}+\frac{b^{2}}{M_{3}}\end{pmatrix}\begin{pmatrix}y^{T}_{\nu
1}\\\ y^{T}_{\nu 2}\end{pmatrix},$ (49)

where we take the diagonal basis of the right-handed neutrino mass matrix
$M_{R}$. Our condition of $\operatorname{Rank}\left[m_{eff}\right]=2$ holds
for the entire energy region of $M_{3}<\mu\leq\Lambda$. We prove this by
considering the solutions of the renormalization group equations of
Eqs.(98-99),

$\displaystyle Y_{\nu i}(\mu)=U^{\prime}(\mu)y_{\nu
1}\left\\{[K^{-1}(\mu)]_{i1}+a[K^{-1}(\mu)]_{i3}\right\\}+U^{\prime}(\mu)y_{\nu
2}\left\\{[K^{-1}(\mu)]_{i2}+b[K^{-1}(\mu)]_{i3}\right\\},$ (50)
$\displaystyle Y_{\nu 3}(\mu)=A(\mu)Y_{\nu 1}(\mu)+B(\mu)Y_{\nu 2}(\mu),$ (51)

where, $i=1,2$. $U^{\prime}(\mu)$ and $K(\mu)$ denote $3\times 3$ evolution
matrices for Yukawa vectors and Majorana mass matrix respectively, for the
region $M_{3}<\mu\leq\Lambda$. $A(\mu)$ and $B(\mu)$ are constructed from
elements of $K^{-1}(\mu)$, $a$, and $b$. This results in Eq.(49) becoming,

$m_{eff}(\mu)=-\frac{v^{2}}{2}\begin{pmatrix}Y_{\nu 1}(\mu)&Y_{\nu
2}(\mu)\end{pmatrix}\begin{pmatrix}\frac{1}{M_{1}}+\frac{A^{2}(\mu)}{M_{3}}&\frac{A(\mu)B(\mu)}{M_{3}}\\\
\frac{A(\mu)B(\mu)}{M_{3}}&\frac{1}{M_{2}}+\frac{B^{2}(\mu)}{M_{3}}\end{pmatrix}\begin{pmatrix}Y^{T}_{\nu
1}(\mu)\\\ Y^{T}_{\nu 2}(\mu)\end{pmatrix},$ (52)

which maintains the condition of $\operatorname{Rank}\left[m_{eff}\right]=2$.

The difference between Eq.(49) of the 3-3 model and the 3-2 model is how the
effective mass matrix is rank degenerate. In the 3-2 model, the rank
degeneracy occurs because the third right-handed neutrino is absent. Whereas,
in the 3-3 model we impose an additional condition on the Yukawa coupling
$y_{\nu 3}$ between the heavy and light neutrino Eq.(47); such that $y_{\nu
3}$ is described by a linear combination of the other two Yukawa couplings,
leading to a rank degenerate effective mass matrix. As we saw in the 3-2 model
in section 4, the running of the renormalization group equations can not
create a rank three effective mass matrix. Therefore, the lightest neutrino
mass is zero even if we take into account the renormalization group effects.
Thus, considering Eq.(49) we are interested if the 3-3 model has the rank
degenerate effective mass matrix at low energy similar to the 3-2 model.

To begin, we study the renormalization group effects with the same processes
as the 3-2 model. We start at the high energy scale of $\mu=M_{3}$ or $t=0$
where,

$t\equiv\log\frac{M_{3}}{\mu},\quad t_{2}\equiv\log\frac{M_{3}}{M_{2}},\quad
t_{1}\equiv\log\frac{M_{3}}{M_{1}}.$ (53)

Without loss of generality, one can start with the diagonal basis for the
right-handed Majorana neutrinos, based on the discussion after Eq.(49). To go
to lower energies we must integrate out the heaviest neutrino $N_{3}$, which
generates the effective coefficient $\overset{(3)}{\kappa}(t)$. Matching of
the effective mass matrix across $M_{3}$ provides initial conditions of the
renormalization group equations for $\overset{(3)}{\kappa}(t)$,
$\overset{(3)}{M_{R}}(t)$, and $\overset{(3)}{y_{\nu}}(t)$. Resulting in
equations similar to the 3-2 model,

$\displaystyle\overset{(3)}{m_{eff}}(0)$ $\displaystyle=m_{eff}(0),$ (54a)
$\displaystyle-\frac{v^{2}}{4}\overset{(3)}{\kappa}(0)-\frac{v^{2}}{2}\overset{(3)}{y_{\nu}}(0)\left(\overset{(3)}{M_{R}}(0)\right)^{-1}\left(\overset{(3)}{y_{\nu}}(0)\right)^{T}$
$\displaystyle=-\frac{v^{2}}{2}y_{\nu}(0)\frac{1}{M_{R}(0)}y^{T}_{\nu}(0).$
(54b)

Using Eq.(54), one can determine $\overset{(3)}{\kappa}(0)$ as,

$\overset{(3)}{\kappa}(0)=2y_{\nu 3}(0)\frac{1}{M_{3}(0)}y^{T}_{\nu 3}(0),$
(55)

where $y_{\nu 3}(0)=ay_{\nu 1}(0)+by_{\nu 2}(0)$ from Eq.(47).

Next, we integrate down to the mass of $M_{2}$ by solving the renormalization
group equations in the same way as Eqs.(94-96). In contrast to the 3-2 model,
an additional energy scale remains between the masses of $M_{2}$ and $M_{1}$
before we reach the lowest scale of $\mu=M_{1}$. To account for this
additional region we first must integrate out the right-handed neutrino
$N_{2}$. The mass of $N_{2}$ is not the same as $M_{2}(0)$ due to
renormalization group corrections from the running of
$\overset{(3)}{M_{R}}(0)$ generating off-diagonal components in
$\overset{(3)}{M_{R}}(t_{2})$. So we diagonalize the matrix
$\overset{(3)}{M_{R}}(t_{2})$ with a Takagi decomposition Takagi:1925 ;
Autonne:1915 ; Hahn:2006hr ,

$\overset{(3)}{\mathbf{M}_{R}}(t_{2})=V^{T}\overset{(3)}{M_{R}}(t_{2})V,$ (56)

where $V$ is a $2\times 2$ unitary matrix. The diagonal matrix
$\overset{(3)}{\mathbf{M}_{R}}(t_{2})$ is
$\overset{(3)}{\mathbf{M}_{R\,ij}}(t_{2})=\delta_{ij}\overset{(3)}{\mathbf{M}_{i}}(t_{2})$.
In the diagonal basis $\overset{(3)}{\mathbf{M}_{R}}(t_{2})$, the Yukawa
matrix $\overset{(3)}{\mathbf{y}_{\nu}}(t_{2})$ is given by the $3\times 2$
matrix,

$\overset{(3)}{\mathbf{y}_{\nu}}(t_{2})=\overset{(3)}{{y}_{\nu}}(t_{2})V=\begin{pmatrix}\overset{(3)}{\mathbf{y}_{\nu
1}}(t_{2})&\overset{(3)}{\mathbf{y}_{\nu 2}}(t_{2})\end{pmatrix}.$ (57)

Then we integrate out the right-handed neutrino $N_{2}$ in the diagonal basis
and the effective mass matrix is matched across the energy of $M_{2}$ in a
manner similar to Eq.(54),

$\displaystyle\overset{(2)}{m_{eff}}(t_{2})$
$\displaystyle=\overset{(3)}{m_{eff}}(t_{2}),$ (58)
$\displaystyle-\frac{v^{2}}{4}\overset{(2)}{\kappa}(t_{2})-\frac{v^{2}}{2}\overset{(2)}{y_{\nu}}(t_{2})\left(\overset{(2)}{M_{R}}(t_{2})\right)^{-1}\overset{(2)}{y_{\nu}}(t_{2})^{T}$
$\displaystyle=-\frac{v^{2}}{4}\overset{(3)}{\kappa}(t_{2})-\frac{v^{2}}{2}\overset{(3)}{\mathbf{y}_{\nu}}(t_{2})\left(\overset{(3)}{\mathbf{M}_{R}}(t_{2})\right)^{-1}\overset{(3)}{\mathbf{y}_{\nu}}(t_{2})^{T},$
(59)

where $\overset{(2)}{M_{R}}(t_{2})=\overset{(3)}{\mathbf{M}_{1}}(t_{2})$ and
$\overset{(2)}{y_{\nu}}(t_{2})=\overset{(3)}{\mathbf{y}_{\nu 1}}(t_{2})$. In
the left-handed side of Eq.(58), a new Wilson coefficient
$\overset{(2)}{\kappa}(t_{2})$ is generated and is determined as,

$\overset{(2)}{\kappa}(t_{2})=2~{}\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})\left(\overset{(3)}{\mathbf{M}_{2}}(t_{2})\right)^{-1}\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})^{T}+\overset{(3)}{\kappa}(t_{2}).$ (60)

Lastly, we solve the renormalization group equations for
$\overset{(2)}{\kappa}(t)$, $\overset{(2)}{M_{R}}(t)$, and
$\overset{(2)}{y_{\nu}}(t)$ between the energies of $N_{2}$ and $N_{1}$. This
results in an effective mass matrix of,

$\overset{(1)}{m_{eff}}(t_{1})=-\frac{v^{2}}{4}\overset{(2)}{\kappa}(t_{1})-\frac{v^{2}}{2}\overset{(2)}{y_{\nu}}(t_{1})\left(\overset{(2)}{M_{R}}(t_{1})\right)^{-1}\overset{(2)}{y_{\nu}}(t_{1})^{T}.$
(61)

After which, $N_{1}$ would need to be integrated out to continue going down to
the low energy experimental region $\mu<M_{1}$. We are not interested in what
occurs for $\mu<M_{1}$ as it will not influence the rank of the effective mass
matrix $\overset{(1)}{m_{eff}}(t)$.

To investigate the rank of the effective mass matrix, we rewrite Eq.(61) by
substituting the kernel functions for the region $[t_{1},t_{2})$ explicitly.
From Eq.(95) one obtains,

$\overset{(2)}{y_{\nu}}(t_{1})=e^{-\frac{1}{16\pi^{2}}\int_{t_{2}}^{t_{1}}\overset{(2)}{\alpha_{y_{\nu}}}(s)ds}\overset{(2)}{U}(t_{1},t_{2})\overset{(2)}{y_{\nu}}(t_{2}).$
(62)

Then we explicitly write the kernel function which relates
$\overset{(2)}{\kappa}(t_{1})$ to $\overset{(2)}{\kappa}(t_{2})$,

$\displaystyle\overset{(2)}{\kappa}(t_{1})={}$ $\displaystyle
e^{-\frac{1}{16\pi^{2}}\int_{t_{2}}^{t_{1}}\overset{(2)}{\alpha_{\kappa}}(s)ds}\overset{(2)}{W}(t_{1}-t_{2})\overset{(2)}{\kappa}(t_{2})\overset{(2)}{W}(t_{1}-t_{2})^{T},$
(63)
$\displaystyle\begin{split}={}&\frac{4q}{v^{2}}\overset{(2)}{W}(t_{1}-t_{2})\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})^{T}\overset{(2)}{W}(t_{1}-t_{2})^{T}\\\
&+\frac{4x}{v^{2}}\overset{(2)}{W}(t_{1}-t_{2})\overset{(3)}{W}(t_{2})\overset{(3)}{y_{\nu}}(0)\begin{pmatrix}a^{2}&ab\\\
ab&b^{2}\end{pmatrix}\overset{(3)}{y_{\nu}}(0)^{T}\overset{(3)}{W}(t_{2})^{T}\overset{(2)}{W}(t_{1}-t_{2})^{T}.\end{split}$
(64)

We have gathered the scaling terms into the factors,

$\displaystyle
q\equiv\frac{v^{2}}{2\overset{(3)}{\mathbf{M}_{2}}(t_{2})}e^{-\frac{1}{16\pi^{2}}\int_{t_{2}}^{t_{1}}\overset{(2)}{\alpha_{\kappa}}(s)ds},$
(65) $\displaystyle
x\equiv\frac{v^{2}}{2M_{3}(0)}e^{-\frac{1}{16\pi^{2}}\int_{t_{2}}^{t_{1}}\overset{(2)}{\alpha_{\kappa}}(s)ds}e^{-\frac{1}{16\pi^{2}}\int_{0}^{t_{2}}\overset{(1)}{\alpha_{\kappa}}(s)ds},$
(66) $\displaystyle
z\equiv\frac{v^{2}}{2\overset{(2)}{M_{R}}(t_{1})}e^{-\frac{1}{8\pi^{2}}\int_{t_{2}}^{t_{1}}\overset{(2)}{\alpha_{y_{\nu}}}(s)ds},$
(67)

because they will not modify the rank of the effective mass matrix. After the
substitution of Eq.(62) and Eq.(64) into Eq.(61),
$\overset{(1)}{m_{eff}}(t_{1})$ is rewritten as,

$\begin{split}\overset{(1)}{m_{eff}}(t_{1})={}&-q\overset{(2)}{W}(t_{1}-t_{2})\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})^{T}\overset{(2)}{W}(t_{1}-t_{2})^{T}\\\
&-x\overset{(2)}{W}(t_{1}-t_{2})\overset{(3)}{W}(t_{2})\overset{(3)}{y_{\nu}}(0)\begin{pmatrix}a^{2}&ab\\\
ab&b^{2}\end{pmatrix}\overset{(3)}{y_{\nu}}(0)^{T}\overset{(3)}{W}(t_{2})^{T}\overset{(2)}{W}(t_{1}-t_{2})^{T}\\\
&-z\overset{(2)}{U}(t_{1}-t_{2})\overset{(2)}{y_{\nu}}(t_{2})\overset{(2)}{y_{\nu}}(t_{2})^{T}\overset{(2)}{U}(t_{1}-t_{2})^{T}.\end{split}$
(68)

With the effective mass matrix given in Eq.(68), the mass squared eigenvalues
of the Hermite matrix
$\overset{(1)}{\mathcal{M}}=\overset{(1)}{m_{eff}}(t_{1})[\overset{(1)}{m_{eff}}(t_{1})]^{\dagger}$
does not clearly have a zero eigenvalue. To prove this we evaluate the three
invariants in Eqs.(8-10). The first two invariants $I_{1}\neq 0$ and
$I_{2}\neq 0$ are non-zero in a manner similar to the Eqs.(37-38) of the 3-2
model. Recall, the invariant $I_{0}$ can be rewritten as the complex product
$I_{0}=\mathcal{I}\times\mathcal{I}^{\ast}$. Then $\mathcal{I}$ is written as
the matrix elements from $[\overset{(1)}{m_{eff}}(t_{1})]_{ij}=m_{ij}$ as in
Eq.(39),

$\displaystyle\begin{split}m_{33}(m_{11}m_{22}-m_{21}m_{12})=&{}\left[-q(y^{W}_{32})^{2}-x(ay^{WW}_{31}+by^{WW}_{32})^{2}-z(y^{U}_{11})^{2}\right]\\\
&\times\left\\{-\left[qy^{W}_{12}y^{W}_{22}+x(ay^{WW}_{11}+by^{WW}_{12})(ay^{WW}_{21}+by^{WW}_{22})+zy^{U}_{11}y^{U}_{21}\right]^{2}\right.\\\
&\qquad+[q(y^{W}_{12})^{2}+x(ay^{WW}_{11}+by^{WW}_{12})^{2}+(y^{U}_{11})^{2}]\\\
&\Bigl{.}\qquad\times[q(y^{W}_{22})^{2}+x(ay^{WW}_{21}+by^{WW}_{22})^{2}+z(y^{U}_{21})^{2}]\Bigr{\\}},\end{split}$
(69)
$\displaystyle\begin{split}m_{13}(m_{32}m_{21}-m_{31}m_{22})={}&\left[-qy^{W}_{12}y^{W}_{32}-x(ay^{WW}_{11}+by^{WW}_{12})(ay^{WW}_{31}+by^{WW}_{32})-zy^{U}_{11}y^{U}_{31}\right]\\\
&\,\left\\{-\left[(q(y^{W}_{22})^{2}+x(ay^{WW}_{21}+by^{WW}_{12})^{2}+z(y^{U}_{21})^{2})\right.\right.\\\
&\left.\qquad\times(qy^{W}_{12}y^{W}_{32}+x(ay^{WW}_{11}+by^{WW}_{12})(ay^{WW}_{31}+by^{WW}_{32})+zy^{U}_{11}y^{U}_{31})\right]\\\
&\quad+(qy^{W}_{12}y^{W}_{22}+x(ay^{WW}_{11}+by^{WW}_{12})(ay^{WW}_{21}+by^{WW}_{22})+zy^{U}_{11}y^{U}_{21})\\\
&\left.\quad\times(qy^{W}_{22}y^{W}_{32}+x(ay^{WW}_{21}+by^{WW}_{22})(ay^{WW}_{31}+by^{WW}_{32})+zy^{U}_{21}y^{U}_{31})\right\\},\end{split}$
(70)
$\displaystyle\begin{split}m_{32}(m_{31}m_{12}-m_{32}m_{11})={}&\left[-qy^{W}_{22}y^{W}_{32}-x(ay^{WW}_{21}+by^{WW}_{22})(ay^{WW}_{31}+by^{WW}_{32})-zy^{U}_{21}y^{U}_{31}\right]\\\
&\,\left\\{-\left[(q(y^{W}_{12})^{2}+x(ay^{WW}_{11}+by^{WW}_{12})^{2}+z(y^{U}_{11})^{2})\right.\right.\\\
&\left.\qquad\times(qy^{W}_{22}y^{W}_{32}+x(ay^{WW}_{21}+by^{WW}_{22})(ay^{WW}_{31}+by^{WW}_{32})+zy^{U}_{21}y^{U}_{31})\right]\\\
&\quad+(qy^{W}_{12}y^{W}_{22}+x(ay^{WW}_{11}+by^{WW}_{12})(ay^{WW}_{21}+by^{WW}_{22})+zy^{U}_{11}y^{U}_{21})\\\
&\left.\quad\times(qy^{W}_{12}y^{W}_{32}+x(ay^{WW}_{11}+by^{WW}_{12})(ay^{WW}_{31}+by^{WW}_{32})+zy^{U}_{11}y^{U}_{31})\right\\}.\end{split}$
(71)

We define the following Yukawa vectors from Eq.(68),

$\displaystyle y^{W}_{\nu 2}(t_{1})$
$\displaystyle=\overset{(2)}{W}(t_{1}-t_{2})\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2}),$ (72) $\displaystyle y^{WW}_{\nu j}(t_{1})$
$\displaystyle=\overset{(2)}{W}(t_{1}-t_{2})\overset{(3)}{W}(t_{2})\overset{(3)}{y_{\nu
j}}(0),\qquad\text{where }j=1,2,$ (73) $\displaystyle y^{U}_{\nu 1}(t_{1})$
$\displaystyle=\overset{(2)}{U}(t_{1}-t_{2})\overset{(2)}{y_{\nu}}(t_{2}).$
(74)

The matrix elements in the determinant of $m_{ij}$ in Eqs.(69-71), are
identified as;

$\displaystyle y^{WW}_{ij}=[y^{WW}_{\nu j}]_{i},$ $\displaystyle
y^{W}_{i2}=[y^{W}_{\nu 2}]_{i},$ $\displaystyle y^{U}_{i1}=[y^{U}_{\nu
1}]_{i},$ $\displaystyle\text{where }i=1,2,3.$ (75)

Unlike the results for the 3-2 seesaw model, the terms of Eq.(69) and Eq.(70)
do not cancel with Eq.(71) under addition, leading us to no definite
conclusions. This is because of the three different kernel matrix
transformations acting on the Yukawa vectors are not guarantied to act
similarly. Thus, for a definite conclusion on the rank of the effective mass
matrix we must investigate the details of the kernel matrices.

###  5.2 The rank of the effective mass matrix with specific renormalization
group kernel solutions

In this section, we examine the rank of the effective Majorana mass matrix
Eq.(68) at low energy. Since the rank depends on the kernel functions, one
must specify them. We examine the rank for the case that all the
renormalization group evolution is governed by the Yukawa couplings for
neutrinos666Any effects of the charged lepton Yukawa couplings will change our
proof and may alter the present result.. In appendix B we prove the relation
of Eq.(106), which we will use now;

$\begin{split}\overset{(3)}{W}(t_{2})\overset{(3)}{y_{\nu}}(0)&=T\exp\left[-\frac{1}{16\pi^{2}}\int_{0}^{t_{2}}(\overset{(3)}{y_{\nu}}(s)\overset{(3)}{y_{\nu}}(s)^{\dagger})ds\right]\overset{(3)}{y_{\nu}}(0)\\\
&=\overset{(3)}{y_{\nu}}(t_{2})T\exp\left[\frac{1}{32\pi^{2}}\int_{0}^{t_{2}}(\overset{(3)}{y_{\nu}}(s)^{\dagger}\overset{(3)}{y_{\nu}}(s))ds\right]e^{\frac{1}{16\pi^{2}}\int^{t_{2}}_{0}\overset{(3)}{\alpha_{y_{\nu}}}(s)ds}\\\
&=\overset{(3)}{y_{\nu}}(t_{2})\overset{(3)}{W_{R}}(t_{2})e^{\frac{1}{16\pi^{2}}\int^{t_{2}}_{0}\overset{(3)}{\alpha_{y_{\nu}}}(s)ds},\end{split}$
(76)

where we have defined,

$\overset{(3)}{W_{R}}(t_{2})=T\exp\left[\frac{1}{32\pi^{2}}\int_{0}^{t_{2}}(\overset{(3)}{y_{\nu}}(s)^{\dagger}\overset{(3)}{y_{\nu}}(s))ds\right].$
(77)

Using the reverse Kernel relation of Eq.(76), we rewrite the middle term of
Eq.(68) as follows;

$\begin{split}\overset{(1)}{m_{eff}}(t_{1})={}&-q\overset{(2)}{W}(t_{1}-t_{2})\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})^{T}\overset{(2)}{W}(t_{1}-t_{2})^{T}\\\
&-x^{\prime}\overset{(2)}{W}(t_{1}-t_{2})\overset{(3)}{y_{\nu}}(t_{2})\overset{(3)}{W_{R}}(t_{2})\begin{pmatrix}a^{2}&ab\\\
ab&b^{2}\end{pmatrix}(\overset{(3)}{W_{R}}(t_{2}))^{T}\overset{(3)}{y_{\nu}}(t_{2})^{T}\overset{(2)}{W}(t_{1}-t_{2})^{T}\\\
&-z\overset{(2)}{U}(t_{1}-t_{2})\overset{(2)}{y_{\nu}}(t_{2})\overset{(2)}{y_{\nu}}(t_{2})^{T}\overset{(2)}{U}(t_{1}-t_{2})^{T},\end{split}$
(78)

where
$x^{\prime}=xe^{\frac{1}{8\pi^{2}}\int^{t_{2}}_{0}\overset{(3)}{\alpha_{y_{\nu}}}(s)ds}$.
Next we continue focusing on the second line of Eq.(78) where
$\overset{(3)}{y_{\nu}}(t_{2})$ is a 3 by 2 matrix. We rewrite
$\overset{(3)}{y_{\nu}}(t_{2})$ following Eq.(57) that comes from
diagonalizing the right-handed mass matrix at $t_{2}$,

$\overset{(3)}{{y}_{\nu}}(t_{2})=\overset{(3)}{\mathbf{y}_{\nu}}(t_{2})V^{-1}=\begin{pmatrix}\overset{(3)}{\mathbf{y}_{\nu
1}}(t_{2})&\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})\end{pmatrix}V^{-1}=\begin{pmatrix}\overset{(2)}{y_{\nu}}(t_{2})&\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})\end{pmatrix}V^{-1}$ (79)

where we use the relation
$\overset{(2)}{{y}_{\nu}}(t_{2})=\overset{(3)}{\mathbf{y}_{\nu 1}}(t_{2})$.
Then one can rewrite Eq.(78) into the following form,

$\begin{split}\overset{(1)}{m_{eff}}(t_{1})={}&-q\overset{(2)}{W}(t_{1}-t_{2})\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})^{T}\overset{(2)}{W}(t_{1}-t_{2})^{T}\\\
&-x^{\prime}\overset{(2)}{W}(t_{1}-t_{2})\begin{pmatrix}\overset{(2)}{y_{\nu}}(t_{2})&\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})\end{pmatrix}A(t_{2})\begin{pmatrix}\overset{(2)}{y_{\nu}}(t_{2})^{T}\\\
\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})^{T}\end{pmatrix}\overset{(2)}{W}(t_{1}-t_{2})^{T}\\\
&-z\overset{(2)}{U}(t_{1}-t_{2})\overset{(2)}{y_{\nu}}(t_{2})\overset{(2)}{y_{\nu}}(t_{2})^{T}\overset{(2)}{U}(t_{1}-t_{2})^{T},\end{split}$
(80)

where $A$ is a two by two matrix defined as,

$A(t_{2})\equiv V^{-1}\overset{(3)}{W}_{R}(t_{2})\begin{pmatrix}a^{2}&ab\\\
ab&b^{2}\end{pmatrix}(V^{-1}\overset{(3)}{W}_{R}(t_{2}))^{T}.$ (81)

Similar to Eqs.(76, 77, 106), one can show the following relation,

$\overset{(2)}{W}(t_{1}-t_{2})\overset{(2)}{y_{\nu}}(t_{2})=\overset{(2)}{y_{\nu}}(t_{1})\overset{(2)}{W_{R}}(t_{1}-t_{2})e^{\frac{1}{16\pi^{2}}\int^{t_{1}}_{t_{2}}\overset{(2)}{\alpha_{y_{\nu}}}(s)ds},$
(82)

where $\overset{(2)}{W_{R}}(t_{1}-t_{2})$ is defined by,

$\overset{(2)}{W_{R}}(t_{1}-t_{2})=T\exp\left[\frac{1}{32\pi^{2}}\int_{t_{2}}^{t_{1}}(\overset{(2)}{y_{\nu}}(s)^{\dagger}\overset{(2)}{y_{\nu}}(s))ds\right].$
(83)

Further one defines,

$\overset{W}{\mathbf{y}_{\nu
2}}(t_{1})\equiv\overset{(2)}{W}(t_{1}-t_{2})\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2}).$ (84)

Combining Eq.(82) with Eq.(84), one obtains the following relation that can
rewrite the second line of Eq.(80).

$\overset{(2)}{W}(t_{1}-t_{2})\begin{pmatrix}\overset{(2)}{y_{\nu}}(t_{2})&\overset{(3)}{\mathbf{y}_{\nu
2}}(t_{2})\end{pmatrix}=\begin{pmatrix}\overset{(2)}{y_{\nu}}(t_{1})&\overset{W}{\mathbf{y}_{\nu
2}}(t_{1})\end{pmatrix}\begin{pmatrix}\overset{(2)}{W_{R}^{\prime}}(t_{1}-t_{2})&0\\\
0&1\end{pmatrix},$ (85)

where
$\overset{(2)}{W_{R}^{\prime}}(t_{1}-t_{2})=\overset{(2)}{W_{R}}(t_{1}-t_{2})e^{\frac{1}{16\pi^{2}}\int^{t_{1}}_{t_{2}}\overset{(2)}{\alpha_{y_{\nu}}}(s)ds}$.
Then $\overset{(1)}{m_{eff}}(t_{1})$ has the following form,

$\begin{split}\overset{(1)}{m_{eff}}(t_{1})={}&-q\,\overset{W}{\mathbf{y}_{\nu
2}}(t_{1})\overset{W}{\mathbf{y}_{\nu 2}}(t_{1})^{T}\\\
&-x^{\prime}\begin{pmatrix}\overset{(2)}{y_{\nu}}(t_{1})&\overset{W}{\mathbf{y}_{\nu
2}}(t_{1})\end{pmatrix}\begin{pmatrix}\overset{(2)}{W^{\prime}_{R}}(t_{1}-t_{2})&0\\\
0&1\end{pmatrix}A(t_{2})\begin{pmatrix}\overset{(2)}{W^{\prime}_{R}}(t_{1}-t_{2})&0\\\
0&1\end{pmatrix}\begin{pmatrix}\overset{(2)}{y_{\nu}}(t_{1})^{T}\\\
\overset{W}{\mathbf{y}_{\nu 2}}(t_{1})^{T}\end{pmatrix}\\\
&-\frac{v^{2}}{2\overset{(2)}{M_{R}}(t_{1})}\overset{(2)}{y_{\nu}}(t_{1})\overset{(2)}{y_{\nu}}(t_{1})^{T}\end{split}$
(86)

Lastly, we can see only two Yukawa vectors remain in Eq.(86),
$\overset{(2)}{y_{\nu}}(t_{1})$ and $\overset{W}{\mathbf{y}_{\nu 2}}(t_{1})$.
This proves that the third Yukawa vector is constructed as a linear
combination of the other two. Next, we rewrite $\overset{(1)}{m_{eff}}(t_{1})$
to highlight this result,

$\overset{(1)}{m_{eff}}(t_{1})=\begin{pmatrix}\overset{(2)}{y_{\nu}}(t_{1})&\overset{W}{\mathbf{y}_{\nu
2}}(t_{1})\end{pmatrix}\begin{pmatrix}\alpha(t_{1},t_{2})&\beta(t_{1},t_{2})\\\
\beta(t_{1},t_{2})&\gamma(t_{1},t_{2})\end{pmatrix}\begin{pmatrix}\overset{(2)}{y_{\nu}}(t_{1})^{T}\\\
\overset{W}{\mathbf{y}_{\nu 2}}(t_{1})^{T}\end{pmatrix}.$ (87)

As a result, for the simplest kernel functions the two Yukawa vectors lead us
to conclude that the effective mass matrix is rank degenerate,

$\operatorname{Rank}\left[\overset{(1)}{m_{eff}}(t_{1})\right]\leq 2.$ (88)

This means that the determinant of the matrix is zero, thus the lightest
active neutrino is massless. The implication is in the full theory energy
region the lightest active neutrino is massless, and even after evolution of
the renormalization group equations to below the scale of $M_{3}$ that same
neutrino is massless.

##  6 Conclusion

We studied the effect of the renormalization group equations on the rank of
the effective mass matrix for active Majorana neutrinos. The Type-I seesaw
models are examined with two and three right-handed neutrinos with
hierarchical masses. The right-handed neutrinos are integrated one by one at
their thresholds, and it generates the Weinberg operator. For the 3-2 model,
with two right-handed Majorana neutrinos, the rank of the effective mass
matrix is two at high energy scales; and the renormalization group equations
do not change the rank at the energy scale lower than the lightest right-
handed neutrino $N_{1}$. Thus, the 3-2 model predicts one massless active
neutrino at low energy. In the case of the 3-3 model, with three right-handed
neutrinos, the Yukawa matrix is assumed to be rank two above an energy scale
corresponding to the mass of the heaviest right-handed neutrino $N_{3}$. After
we solve the renormalization group equations to energies below the lightest
right-handed neutrino $N_{1}$, the rank of the effective mass matrix depends
on the kernel solutions of the renormalization group equations. For the
simplest kernel solutions, the rank of the effective mass matrix does not
change. Meaning the lightest neutrino would remain massless across all
renormalization energies, in spite of the different kernels.

We examine the reason for this result analytically. In general, at the energy
scale higher than that of the heaviest right-handed neutrino mass,
renormalization group equations do not change the rank of the effective
Majorana mass matrix. Similarly, in the energy region below the mass for the
lightest right-handed neutrino, the renormalization group equations for the
Wilson coefficient of the Weinberg operator do not the change its rank.
Therefore, the possible rank change may occur between the two energy regions
where some right-handed neutrinos are still active and part of the effective
Majorana mass matrix is described by the Wilson coefficient of the Weinberg
operator. For the 3-2 model, such energy region is unique ($M_{1}<E<M_{2}$).
At high energy, the effective Majorana mass matrix is written with two
linearly independent Yukawa vectors, which leads to the rank two effective
mass matrix. Effectively, the kernel solutions of the renormalization group
equations act as transformations on the two linearly independent Yukawa
vectors, and they change into another set of two linearly independent vectors.
Therefore, the effective Majorana mass matrix at low energy is also written
with the two linearly independent Yukawa vectors leading to the rank two
effective mass matrix.

In the framework of the 3-3 model with the rank degenerate Yukawa matrix, the
effective mass matrix is the rank two at high energy scale, similar to the 3-2
model. However, the form of the effective mass matrix is different since there
are contributions from three Yukawa vectors. Each of the three contributions
is initially written in terms of three Yukawa vectors, and they are two
linearly independent vectors and their superpositions. These three
contributions come from each right-handed neutrinos $N_{1}$-$N_{3}$. The
effect of renormalization group equations on the three vectors are different
since there are two distinct energy regions $M_{1}<E<M_{2}$ and
$M_{2}<E<M_{3}$. This leads to the rank of the effective mass matrix being
dependent on the details of the renormalization group kernel solutions. In the
simplest case, the kernel solutions only depend on the neutrino Yukawa
vectors. Thus, the transformations of the kernel solutions can not increase
the linearly vectors from two to three. We note this may not be true for
kernel solutions involving the charged lepton Yukawa vectors or for higher
order renormalization group equations. Both of those situations could be
considered in a future analysis.

## Acknowledgment

The work of T.M. is supported by Japan Society for the Promotion of Science
(JSPS) KAKENHI Grant Number JP17K05418. N.J.B, would like to express thanks to
the Japanese government Ministry of Education, Culture, Sports, Science, and
Technology (MEXT) for the financial support during the writing of this work.

##  Appendix A Renormalization Group Equations

Here we list the one-loop renormalization group (RG) equations for the type-I
seesaw models we are considering. We simplified the RG equations by ignoring
the Yukawa couplings for the quarks and charged the leptons. Inclusion of the
couplings does not change our results. We start with the equations for the
mass and coupling constant matrices unique to the type-I seesaw,
$\overset{(n)}{M_{R}}(\mu)$, $\overset{(n)}{\kappa}(\mu)$, and
$\overset{(n)}{y_{\nu}}(\mu)$;

$\displaystyle-16\pi^{2}\frac{d\overset{(n)}{\kappa}(t)}{dt}=\bigl{(}\overset{(n)}{y_{\nu}}(t)\overset{(n)}{y_{\nu}}(t)^{\dagger}\bigr{)}\kappa(t)+\overset{(n)}{\kappa}(t)\bigl{(}\overset{(n)}{y_{\nu}}(t)\overset{(n)}{y_{\nu}}(t)^{\dagger}\bigr{)}^{T}+\overset{(n)}{\alpha_{\kappa}}(t)\overset{(n)}{\kappa}(t),$
(89)
$\displaystyle-16\pi^{2}\frac{d\overset{(n)}{y_{\nu}}(t)}{dt}=\left(\frac{3}{2}(\overset{(n)}{y_{\nu}}(t)\overset{(n)}{y_{\nu}}(t)^{\dagger})+\overset{(n)}{\alpha_{y_{\nu}}}(t)\right)\overset{(n)}{y_{\nu}}(t),$
(90)
$\displaystyle-16\pi^{2}\frac{d\overset{(n)}{M_{R}}(t)}{dt}=\bigl{(}\overset{(n)}{y_{\nu}}(t)^{\dagger}\overset{(n)}{y_{\nu}}(t)\bigr{)}^{T}\overset{(n)}{M_{R}}(t)+\overset{(n)}{M_{R}}(t)\bigl{(}\overset{(n)}{y_{\nu}}(t)^{\dagger}\overset{(n)}{y_{\nu}}(t)\bigr{)},$
(91)

where the scalar functions $\overset{(n)}{\alpha_{y_{\nu}}}(t)$ and
$\overset{(n)}{\alpha_{\kappa}}(t)$ are given as;

$\displaystyle\overset{(n)}{\alpha_{y_{\nu}}}(t)\equiv\operatorname{Tr}\bigl{[}\overset{(n)}{y_{\nu}}(t)\overset{(n)}{y_{\nu}}(t)^{\dagger}\bigr{]}-\frac{9}{20}g_{1}^{2}-\frac{9}{4}g_{2}^{2},$
(92) $\displaystyle\overset{(n)}{\alpha_{\kappa}}(t)\equiv
2\operatorname{Tr}\bigl{[}\overset{(n)}{y_{\nu}}(t)\overset{(n)}{y_{\nu}}(t)^{\dagger}\bigr{]}-3g_{2}^{2}-\lambda.$
(93)

In a manner similar to sections 4 and 5.1, we use the superscript $n$ to
notate the different energy regions. The solutions to Eqs.(89, 90, 91) are
written in terms of a kernel function,

$\displaystyle\overset{(n)}{\kappa}(t_{f})=e^{-\frac{1}{16\pi^{2}}\int_{t_{i}}^{t_{f}}\overset{(n)}{\alpha_{\kappa}}(s)ds}\overset{(n)}{W}(t_{f}-t_{i})\overset{(n)}{\kappa}(t_{i})\overset{(n)}{W}(t_{f}-t_{i})^{T},$
(94)
$\displaystyle\overset{(n)}{y_{\nu}}(t_{f})=e^{-\frac{1}{16\pi^{2}}\int_{t_{i}}^{t_{f}}\overset{(n)}{\alpha_{y_{\nu}}}(s)ds}\overset{(n)}{U}(t_{f}-t_{i})\overset{(n)}{y_{\nu}}(t_{i}),$
(95)
$\displaystyle\overset{(n)}{M_{R}}(t_{f})=\overset{(n)}{K}(t_{f}-t_{i})\overset{(n)}{M_{R}}(t_{i})\overset{(n)}{K}(t_{f}-t_{i})^{T}.$
(96)

The kernel functions are defined as,

$\displaystyle\overset{(n)}{W}(t_{f}-t_{i})$ $\displaystyle\equiv
T\exp\left[-\frac{1}{16\pi^{2}}\int_{t_{i}}^{t_{f}}(\overset{(n)}{y_{\nu}}(s)\overset{(n)}{y_{\nu}}(s)^{\dagger})ds\right],$
(97) $\displaystyle\overset{(n)}{U}(t_{f}-t_{i})$ $\displaystyle\equiv
T\exp\left[-\frac{1}{16\pi^{2}}\int_{t_{i}}^{t_{f}}\frac{3}{2}(\overset{(n)}{y_{\nu}}(s)\overset{(n)}{y_{\nu}}(s)^{\dagger})ds\right],$
(98) $\displaystyle\overset{(n)}{K}(t_{f}-t_{i})$ $\displaystyle\equiv
T\exp\left[-\frac{1}{16\pi^{2}}\int_{t_{i}}^{t_{f}}(\overset{(n)}{y_{\nu}}(s)^{\dagger}\overset{(n)}{y_{\nu}}(s))^{T}ds\right],$
(99)

where, $T$ denotes the exponential ordered by decreasing energy scale.

##  Appendix B The proof of Eq.(76) and Eq.(82)

We prove the property of the kernel function for $\kappa$ on Yukawa matrix in
Eq.(76).

$\begin{split}\overset{(3)}{W}(t_{2})\overset{(3)}{y_{\nu}}(0)&=T\exp\left[-\frac{1}{16\pi^{2}}\int_{0}^{t_{2}}(\overset{(3)}{y_{\nu}}(s)\overset{(3)}{y_{\nu}}(s)^{\dagger})ds\right]\overset{(3)}{y_{\nu}}(0)\\\
&=\overset{(3)}{{y}_{\nu}}(t_{2})T\exp\left[\frac{1}{32\pi^{2}}\int_{0}^{t_{2}}(\overset{(3)}{y_{\nu}}(s)^{\dagger}\overset{(3)}{y_{\nu}}(s))ds\right]e^{\frac{1}{16\pi^{2}}\int^{t_{2}}_{0}\overset{(3)}{\alpha_{y_{\nu}}}(s)ds}.\end{split}$
(100)

To prove the relation, we discretize the evolution kernel for $\kappa$ from
Eq.(97) between the values $t_{i}=0$ and $t_{f}=N\epsilon$. For which, $N$ is
some arbitrary value larger than the discretization step $\epsilon$ and
greater than zero,

$T\exp\left[-\frac{1}{16\pi^{2}}\int_{0}^{N\epsilon}(\overset{(3)}{y_{\nu}}(s)\overset{(3)}{y_{\nu}}(s)^{\dagger})ds\right]\overset{(3)}{y_{\nu}}(0)\\\
=(1-\frac{1}{16\pi^{2}}y[N-1]y[N-1]^{\dagger}\epsilon)\dotsm(1-\frac{1}{16\pi^{2}}y[1]y[1]^{\dagger}\epsilon)(1-\frac{1}{16\pi^{2}}y[0]y[0]^{\dagger}\epsilon)y[0],$
(101)

where we denote $y[n]\equiv\overset{(3)}{y_{\nu}}(n\epsilon)$. The
discretization step $\epsilon$ is some small value, which approaches zero in
the continuum limit. Let us investigate the n th step of the Yukawa coupling
in Eq.(101) due to the kernel of kappa,

$\displaystyle(1-\frac{1}{16\pi^{2}}y[n]y[n]^{\dagger}\epsilon)y[n]$
$\displaystyle=(1-\frac{3}{32\pi^{2}}y[n]y[n]^{\dagger}\epsilon+\frac{1}{32\pi^{2}}y[n]y[n]^{\dagger}\epsilon)y[n]$
(102)
$\displaystyle=y[n+1](1+\frac{1}{32\pi^{2}}y[n]^{\dagger}y[n]\epsilon)(1+\frac{\alpha_{y_{\nu}}[n]}{16\pi^{2}}\epsilon),$
(103)

where $\alpha_{y_{\nu}}[n]\equiv\overset{(3)}{\alpha_{y_{\nu}}}(n\epsilon)$.
Eq.(103) is valid up to O($\epsilon$). To derive Eq.(103), we use the
descritized version of the running of the Yukawa coupling of Eq.(95 )

$(1+\frac{\alpha_{y_{\nu}}[n]}{16\pi^{2}}\epsilon)y[n+1]=(1-\frac{3}{32\pi^{2}}y[n]y[n]^{\dagger}\epsilon)y[n].$
(104)

This allows us to move the Yukawa coupling $y[n]$ from the right side to the
left side. Applying Eq.(103) on Eq.(101) multiple times leads to,

$(1-\frac{1}{16\pi^{2}}y[N-1]y[N-1]^{\dagger}\epsilon)\dotsm(1-\frac{1}{16\pi^{2}}y[1]y[1]^{\dagger}\epsilon)(1-\frac{1}{16\pi^{2}}y[0]y[0]^{\dagger}\epsilon)y[0]\\\
=y[N](1+\frac{1}{32\pi^{2}}y[N-1]^{\dagger}y[N-1]\epsilon)\dotsm(1+\frac{1}{32\pi^{2}}y[0]^{\dagger}y[0]\epsilon)\prod_{n=0}^{N-1}(1+\frac{\alpha_{y_{\nu}}[n]}{16\pi^{2}}\epsilon).$
(105)

Lastly, we take the continuum limit to arrive at,

$T\exp\left[-\frac{1}{16\pi^{2}}\int_{0}^{t_{2}}(\overset{(3)}{y_{\nu}}(s)\overset{(3)}{y_{\nu}}(s)^{\dagger})ds\right]\overset{(3)}{y_{\nu}}(0)\\\
=\overset{(3)}{y_{\nu}}(t_{2})T\exp\left[\frac{1}{32\pi^{2}}\int_{0}^{t_{2}}(\overset{(3)}{y_{\nu}}(s)^{\dagger}\overset{(3)}{y_{\nu}}(s))ds\right]e^{\frac{1}{16\pi^{2}}\int^{t_{2}}_{0}\overset{(3)}{\alpha_{y_{\nu}}}(s)ds}.$
(106)

One can also prove Eq.(82) similar to Eq.(76),

$T\exp\left[-\frac{1}{16\pi^{2}}\int_{t_{2}}^{t_{1}}(\overset{(2)}{y_{\nu}}(s)\overset{(2)}{y_{\nu}}(s)^{\dagger})ds\right]\overset{(2)}{y_{\nu}}(t_{2})\\\
=\overset{(2)}{y_{\nu}}(t_{1})T\exp\left[\frac{1}{32\pi^{2}}\int_{t_{2}}^{t_{1}}(\overset{(2)}{y_{\nu}}(s)^{\dagger}\overset{(2)}{y_{\nu}}(s))ds\right]e^{\frac{1}{16\pi^{2}}\int^{t_{1}}_{t_{2}}\overset{(2)}{\alpha_{y_{\nu}}}(s)ds}.$
(107)

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