Effects of renormalization group kernels
on the lightest neutrino mass in theType-I Seesaw Model
Nicholas J. Benoit ( 広島大学 Visitor, Graduated Sept 2022)
Collaboration with:
T. Morozumi ( 広大 , CORE-U); Y. Shimizu ( 広大 , CORE-U)
K. Takagi; A. Yuu
Based on:PTEP (2022), 11, 113B02
DOI: https://doi.org/10.1093/ptep/ptac138
arXiv: 2210.00165 [hep-ph]
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Summary of our work
We study Type-I Seesaw models in view of the lightest neutrino being massless
Point 1:
Both the 3-3 and the 3-2 Type-I Seesawmodels can predict a massless neutrino

Point 2:The massless neutrino can become
massive under Renormalization Groupeffects in the 3-2 model

3-2 model

3-3 model

Our result: With the simplest renormalization group equations and scale matching,
the lightest neutrino always remains massless in the 3-3 model.
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Introduction to neutrinos
Facts about neutrinos●Standard Model fermions●
Only weakly interacts (electric and color charge neutral)●
SM only includes left-handed neutrinos●
In the SM, the flavor for neutrinos are defined from the electroweak doublet●A typical example including neutrinos is beta-decay

Measure the energy ofthe emitted electron

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Introduction to neutrinos
Beyond the Standard Model●Neutrinos are measured to oscillate between flavors
Alpha can be a electron, muon, or tauon flavor.
Measureable over macroscopic distances(kilometers)●Flavor oscillations as described by quantum mechanics

Probability of appearance
for flavor beta from alpha

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Introduction to neutrinos
Beyond the Standard Model●Neutrinos are measured to oscillate between flavorsAlpha can be a electron, muon, or tauon flavor.
Measureable over macroscopic distances(kilometers)●

Flavor oscillations as described by quantum mechanics

The phase of the flavor oscillations depend on neutrinos having a mass

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Introduction to neutrinos
Beyond the Standard Model●
Neutrinos are measured to oscillate between flavors
Alpha can be a electron, muon, or tauon flavor.
Measureable over macroscopic distances(kilometers)●

Flavor oscillations as described by quantum mechanics– Derived from experimental measurements[Normal Hierarchy; NuFITv5.1, 2021]

Problem: How do we fit neutrino masses into the Standard Model?
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Introduction to neutrinos
Issue is the neutrino mass type is ambiguous●Recall neutrinos are neutral → allows for Majorana or Dirac mass
●So, we have a choice for the neutrino mass term
Dirac mass
Comes from the Higgs mechanism

Majorana mass
From an unknown mechanism

Similar to other Standard Modelparticles

New particles depend on themechanism

Introduces new right-handedneutrinos

Superscript c is for charge
conjugation(details in backup)

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Introduction to neutrinos
Issue is the neutrino mass type is ambiguous●Another point, the neutrino masses are tiny
– Upper limit on lightest mass by KATRIN (possible it is massless)– Mass-squared differences are small

It can be interesting to explain why neutrinos have a tiny mass[Charaters by Yuki Akimoto, higgstan.com]

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The Type-I seesaw model

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Introduction to the Type-I Seesaw Model
New physics with Majorana neutrinos●Type-I Seesaw extends the Standard Model particle content,[Minkowski, P. 1977],[Yanagida, T. 1979],[Gell-Mann, M. Etc 1979],[Glashow, S. L. 1980],[Mohapatra, R. N. 1980],[Fukugita, M. & Yanagida, T. 1986]

Addition of new right-handed fermions, which transform as SM gauge singlets

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Introduction to the Type-I Seesaw Model
New physics with Majorana neutrinos●Type-I Seesaw extends the Standard Model particle content,[Minkowski, P. 1977],[Yanagida, T. 1979],[Gell-Mann, M. Etc 1979],[Glashow, S. L. 1980],[Mohapatra, R. N. 1980],[Fukugita, M. & Yanagida, T. 1986]

Addition of new right-handed fermions, which transform as SM gauge singlets
These are Majorana fermions that are from a new mass scale

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Introduction to the Type-I Seesaw Model
New physics with Majorana neutrinos●Type-I Seesaw extends the Standard Model particle content,[Minkowski, P. 1977],[Yanagida, T. 1979],[Gell-Mann, M. Etc 1979],[Glashow, S. L. 1980],[Mohapatra, R. N. 1980],[Fukugita, M. & Yanagida, T. 1986]

Addition of new right-handed fermions, which transform as SM gauge singlets
These are Majorana fermions that are from a new mass scale
They also form a Dirac mass term with the SM neutinos via the Higgs

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Introduction to the Type-I Seesaw Model
New physics with Majorana neutrinos●
Type-I Seesaw extends the Standard Model particle content,●[Minkowski, P. 1977],[Yanagida, T. 1979],[Gell-Mann, M. Etc 1979],[Glashow, S. L. 1980],[Mohapatra, R. N. 1980],[Fukugita, M. & Yanagida, T. 1986]

Neutrinos masses, after electroweak breaking, are from an effective matrix

Majorana mass matrix from thenew right-handed fermions(details in backup)

The Yukawa matrix fromthe Dirac mass term

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Introduction to the Type-I Seesaw Model
New physics with Majorana neutrinos●
Type-I Seesaw extends the Standard Model particle content,●Neutrinos masses, after electroweak breaking, are from an effective matrix

Majorana mass matrix from thenew right-handed fermions●[Minkowski, P. 1977],[Yanagida, T. 1979],[Gell-Mann, M. Etc 1979],[Glashow, S. L. 1980],[Mohapatra, R. N. 1980],[Fukugita, M. & Yanagida, T. 1986]

The Yukawa matrix fromthe Dirac mass term

The mass of the right-handed fermions act to suppress the Yukawa couplings– Neutrinos have a tiny mass if the right-handed fermions are heavy
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Introduction to the Type-I Seesaw Model
Heavy right-handed fermions●To generate tiny neutrino masses the new fermions have mass near GUT scales,
We do notconsider whatoccurs above thecut-off scale

Thus a new mass scale is introduced into the theory
that is larger than the SM content

Majorana mass matrix from thenew right-handed fermions(details in backup)

The Yukawa matrix fromthe Dirac mass term

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Renormalization in the Type-I Seesaw Model
Why the renormalization group is needed●
Naively, the effective neutrino mass matrix was written as a constant across allenergies●●

Pratically, we need to consider the renormalization of UV divergances appearing
in different diagrams
The resulting renormalized quanties then depend on the energy scale being
considered

Problem: Neutrino masses are energy scale dependent and should be treated carefully(details in backup)

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Renormalization in the Type-I Seesaw Model
Why the renormalization group is needed●The solution is to use renormalization group equations●
At one-loop the Yukawa matrix and right-handed mass matrix scale as,[Antusch, S. Etc 2001],[Antusch, S. Etc 2003],[Antusch, S. Etc 2005]●

Given an initial condition we can solve for the neutrino masses at specificenergies(details on t will be discussed later)

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Renormalization in the Type-I Seesaw Model
Why the renormalization group is needed●A final point, recall the new right-handed fermions are heavy,

If we want to calculate the neutrino masses atenergies here, we should consider all the righthanded fermions to be decoupled●

Decoupling results in an effective theory with a new Dim-5 operator in theLagrangian,[Weinberg, S. 1979]

Energy scale dependance is now in the coefficient kappa
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Renormalization in the Type-I Seesaw Model
Why the renormalization group is needed●This means to do a proper calculation we need to consider the series of effective
theories, linked together by renormalization group equations

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An example,renormalization group running forthe 3-2 Type-I Seesaw model

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Motivation for our work
How many heavy, right-handed fermions?
●
In the Type-I Seesaw we choose the number of heavy, right-handed fermions●The minimal choice allowed by experiments is the 3-2 Type-I Seesaw model

3-2 model:

number of active neutrinos andthe number of right-handed fermions

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3-2 Model example
How many heavy, right-handed fermions?
●
In the Type-I Seesaw we choose the number of heavy, right-handed fermions●
The minimal choice allowed by experiments is the 3-2 Type-I Seesaw model

3-2 model:

number of active neutrinos andthe number of right-handed fermions

Full Theory

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3-2 Model example
How many heavy, right-handed fermions?
●
In the Type-I Seesaw we choose the number of heavy, right-handed fermions●The minimal choice allowed by experiments is the 3-2 Type-I Seesaw model

3-2 model:Full Theory

number of active neutrinos andthe number of right-handed fermionsEffective Theory a=2

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Effective Theory a=1

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3-2 Model example
The massless neutrino●The 3-2 model has a popular understanding that the lightest neutrino is massless●In the full theory region the neutrino mass are given as,

Only two Yukawa couplings because only
two heavy, right-handed fermions
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3-2 Model example
The massless neutrino●The 3-2 model has a popular understanding that the lightest neutrino is massless●To prove this we start at energy equal to the heaviest mass
– Then the heaviest right-handed fermion is integrated out and we match theeffective theory (2) with the full theory at tree level

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3-2 Model example
The massless neutrino●The 3-2 model has a popular understanding that the lightest neutrino is massless●Then we must solve RG equations for the effective theory region (2)Kernel solution to kappa
RG equation
Kernel solution toYukawa RG equation
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3-2 Model example
The massless neutrino●The 3-2 model has a popular understanding that the lightest neutrino is massless●Lastly, we match the effective theories in region (1) with region (2)Kernel solution toYukawa RG equation

Kernel solution to kappa RG equation
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3-2 Model example
The massless neutrino●
The 3-2 model has a popular understanding that the lightest neutrino is massless●In the effective theory a=1 region, the result depends on the RG equations
For one-loop
RG equations(details in backup)

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3-2 Model example
The massless neutrino●The 3-2 model has a popular understanding that the lightest neutrino is massless

However, this is not always true
1. Two-loop renormalization group running generates mass but very tiny...
[S. Davidson, G. Isidori and A. Strumia (2007)],[Z.-z. Xing and D. Zhang (2020)]

2. One-loop level matching generates mass
[S. Zhou (2021)]

The neutrino does not always remain massless(details in backup)

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What about different Type-I Seesaw models?

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Motivation for our work
Recall we can choose the number of right-handed fermions●
3-2 and 3-3 Type-I Seesaw models are the simplest to explain neutrino masses
number of active neutrinos andthe number of right-handed fermions

3-2 model

3-3 model

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Motivation for our work
The 3-3 model●
The usual 3-3 model predicts all active neutrinos are massive

Three Yukawa vectors produce three massive active neutrinos
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Motivation for our work
The 3-3 model●
The usual 3-3 model predicts all active neutrinos are massive

We assume the third Yukawa is asuperposition of the first two

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Motivation for our work
The 3-3 model●The usual 3-3 model predicts all active neutrinos are massive

We assume the third Yukawa is asuperposition of the first two

The effective mass matrix becomes similar to the 3-2 model
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Motivation for our work
The 3-3 model●
The 3-3 models often predict all active neutrinos are massive

How does this condition behave at the low energy scales? Is it similar to the 3-2 model?

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3-3 Type-I Seesaw Model
Running down to the low energies●
We solve the same 2 steps as the 3-2 model, but for three effective theories
1. We solve one-loop renormalization group equations down to the next heaviest
mass

M3 must be integrated out creating effective theory (3)2. Then we use tree-level matching across the heavy mass

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3-3 Type-I Seesaw Model
Running down to the low energies●
We stop at the mass of M1 and calculate the active neutrino masses●The result is an equation for the effective mass matrix
Expand kappa from the tree-level matching solution

A kernel function from the solution of thekappa renormalization group equation
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3-3 Type-I Seesaw Model
Running down to the low energies●
To study if the lightest active neutrino is still massless we substitute the runningsolutions,Three different kernel matrices that transform, two kinds of Yukawa vectors

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3-3 Type-I Seesaw Model
Running down to the low energies●
We can rewrite the effective mass interms of the transformed matrices,●The result is three transformed vectors, so we can conclude

The lightest active neutrino mass depends on the details of how
the kernel matrices transform the Yukawa vectors

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3-3 Type-I Seesaw Model
Simplest Kernels●
In the one-loop renormalization group equations, the simplest kernels only
depend on the Yukawa matrices of the neutrinos
From the Yukawa renormalizationgroup equation
Difference between the kernels
From the kappa renormalizationgroup equation●

The similarity of the kernels allows us to write the following relation,

where,
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3-3 Type-I Seesaw Model
Simplest Kernels●Using the relation for the kappa kernel modifies the effective mass matrix,●The result is two Yukawa vectors and no more transformations

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3-3 Type-I Seesaw Model
Simplest Kernels●This means, at low energies the lightest active neutrino remains massless– Proof only valid for the simplest kernels

Only generates 2 active neutrino masses, because no third Yukawa to generate mass

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Conclusion
3-2 model

We studied Type-I Seesaw models in view of the lightest neutrino being massless

1. One-loop renormalization group running:
2. Two-loop renormalization group running:
3. One-loop level matching:[S. Davidson, et al. (2007)],[Z.-z. Xing and D. Zhang (2020)]

3-3 model[S. Zhou (2021)]

1. One-loop renormalization group runningand simplest kernels:
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Backups

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Introduction to neutrinos
Mass type, the Dirac Equation and Majorana fermions●Solutions to the fermion wavefuntion from the Dirac equation depend on therepresentation of the gamma matrices,●●●Real solutions to the fermion wavefuntion can be found using the Majorana
represenation of the gamma matrices,

Then, which is called the Majorana condition
For a general repesenation the Majorana condition becomes,Fields satisfying the Majorana condition are Majorana fields and can have Majorana mass
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Introduction to the Type-I Seesaw Model
Mixing matrix and the effective neutrino mass●
Type-I Seesaw after electroweak symmetry breaking●To diagonalize the the masses we can introduce a unitary transformation,●Then we can find the effective neutrino mass matrix is,

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Introduction to the Type-I Seesaw Model
Direct searchs for light right-handed fermions●Aside: This is not always true, you can construct a theory with light right-handed
fermions assuming tiny lepton number violation– These light (low-scale) right-handed fermions and the lepton numberviolation processes are looked for at colider experiments●
Light right-handed fermions are possible under the following condition,●●

Omega an arbitrary matrix, so the right-handed fermions can be light and allow
for tiny neutrino effective masses– The matrix V is also tiny, and it mediates lepton number violation (mixing
between neutrinos and the right-handed fermions)This type of situation has not been found at colliders[ATLAS, 2019; CERN-EP-2019-071]
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Renormalization in the Type-I Seesaw Model
1-Loop Renormalization in the Type-I Seesaw●
As an example if we consider the bare fields to be,●

Then the renormalized fields are,●

At one-loop we use the self-interaction divergances to determine the Z values

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Renormalization in the Type-I Seesaw Model
1-Loop Renormalization in the Type-I Seesaw●
The simplest renormalization group equations are then,●With the solutions,

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3-2 Model example
The massless neutrino●
In the effective theory a=1 region, the result depends on the RG equations●
To show this we calculate the rank of the effective mass matrix,
n is the number of massive neutrinos●

We use the characteristic polynomial of a Hermite matrix to calculate the rank,

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3-2 Model example
The massless neutrino●
In the effective theory a=1 region, the result depends on the RG equations●
So the rank can be determined by calculating the invariants of the characteristic
polynomial and finding if any are zero●

Most important for us is invariant I0 , (the others are nonzero)

These two terms exactly cancel the third term

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3-2 Model example
The massless neutrino becomes massive●
In the two-loop renormalization diagrams like this exist that change the rank of
the effective mass matrix of the neutrinos,[S. Davidson, G. Isidori and A. Strumia (2007)],[Z.-z. Xing and D. Zhang (2020)]

New two-loop term that changes the rank ofkappa from 2 to 3
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3-3 Type-I Seesaw Model
Running down to the low energies●To study if the lightest active neutrino is still massless we substitute the runningsolutions,

We substitute the RG solution
for kappa in region (2)Also we substitute the RG solution forthe Yukawa matrix in region (2)岡山理科大学セミナー、22/11/28

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