Machine Learning Exploration of Flavor Structuresin Froggatt-Nielsen models [ arXiv : 2303.xxxxx ]Satsuki Nishimura ( Kyushu Univ. ), Coh Miyao ( Kyushu Univ. ), Hajime Otsuka ( Kyushu Univ. )1. Abstract
・The Froggatt-Nielsen model explains the mass hierarchyof matter particles & the flavor mixtures.
・The large parameter space makes brute-force difficult.

・We efficiently find sets of parameters that simultaneously
reproduce the quark-lepton masses & flavor mixing.
・We show the CP symmetry of the lepton sector
can be significantly violated.

We use Machine Learning. (notably Reinforcement Learning)

2. MethodProblem) To find the appropriate parameters 𝑞 & 𝜙 ,a vast number of combinations must be searched.

2.1. Froggatt-Nielsen Model−9 ≤ 𝑞 ≤ 9 → 1920 ~ 1025 patterns

Yukawa lagrangian has additional 𝑈 1 flavor symmetry.
𝑢𝑑𝑛𝑛
𝑢 𝜙 𝑖𝑗 ത 𝑖 𝑐 𝑗𝑑 𝜙 𝑖𝑗 𝑖𝑗
𝐿Yuk = 𝑦𝑖𝑗
𝑄 𝐻 𝑢 + 𝑦𝑖𝑗
𝑄 𝐻𝑑𝑀𝑀

We focus on the application of Machine Learning (ML).

𝜈𝑙𝑛𝑛
𝜈 𝜙 𝑖𝑗 ത𝑖 𝑐 𝑗𝑙 𝜙 𝑖𝑗 𝑖+𝑦𝑖𝑗
𝐿 𝐻 𝑁 + 𝑦𝑖𝑗
𝐿 𝐻𝑙 𝑗𝑀𝑀2.2. Reinforcement Learning𝑁𝑛
1 𝑁 𝜙 𝑖𝑗
ഥ 𝑐𝑖 𝑁 𝑗 + h. c.
+ 𝑦𝑖𝑗𝑀𝑁2𝑀Reinforcement Learning (RL) is a method of ML.

Yukawa couplings 𝑦 are 𝑂 1 parameters.

Agent

1. observes the environment
2. chooses actions
3. gets rewards

𝐿Yuk is 𝑈 1 invariant ↔ sums of 𝑈 1 charges are zero𝑢𝑖ex. 𝑞 𝜙 𝑛𝑖𝑗 − 𝑞 𝑄−𝑞 𝐻 +𝑞 𝑢

𝑗=0

The agent autonomously findthe principle of actionthat maximizes the total reward.

When complex scalar field 𝜙 develops an expectation value 𝜙 ,Froggatt-Nielsen (FN) charges will lead to a hierarchical𝑢𝑛𝑖𝑗𝑢𝑢
structure of physical Yukawa couplings (ex., 𝑌𝑖𝑗 = 𝑦𝑖𝑗 𝜙 ).

3. Setting of Learning

4. Result from Reinforcement Learning

Agent : increasing or decreasing
any FN charge of matters by ±1( repeat up to 32 steps )In the case of normal ordering of neutrinos,

The closer those are to
the experimental values,
the more points the agent get.
Repeat the above for 100,000 episodes…
The agent find the principle on its own
to derive the FN charges that conforms
to the experimental values.

1.0E+05［MeV］Reward : “the masses of particles& the mixing matrix”( calculated from FN charges )Mass Hierarchy(u,c,t,d,s,b,e,mu,tau)

1.0E+03

CKM elements

PMNS elements

110.1

0.1

0.01

0.01

0.001

0.001

1.0E+01FN

1.0E-01
FN

SM

SM

FN

SM

quark-lepton masses & flavor mixing[ From NuFIT 5.2 (2022) ]are actually reproduced.

Moreover, we found the CP violation in lepton sector.
The non-zero Jarlskog invariant ↔ CP is violated. (yellow)
𝐽PMNS = −0.0423 , δCP = −0.517𝜋We have found other parameter sets (green),which also violate the CP in the lepton sector.

5. Conclusions and Future Works
1. In our work, we have also searched inverted hierarchy ofneutrinos, reproducing correct mass hierarchies and mixing.

1. This work do not derive CP violation in quark sector.
We will try that …
1.1. extend 𝑂 1 Yukawa couplings to complex.
1.2. introduce two complex scalar fields 𝜙1 , 𝜙2 .

2. In both of the orders, we found non-zero Jarlskog invariant.
This means that CP violation of lepton sector is violated.

2. We fix the scale of right-handed neutrino 𝑀 = 1015 GeV.
If this value can be changed,
more precise sets of parameters may be found.

3. We propose the Machine Learning (Reinforcement Learning)to search flavor models is one of effective methods.

3. It is expected that Reinforcement Learning gives
some predictions about neutrino mass order.