The Standard Model is the best description of nature so far. It has many successes in particle physics. But there are also some limitations. For example, we have already observed neutrino oscillation. The standard model can not give a proper description of this. Lepton flavour mixing is also a very big and interesting puzzle. We also observe parity violations in the weak sector. The standard model can not give any proper explanation of these observed phenomena. If we consider the particle content of the standard model, there is no good explanation for the nonexistence of right-handed neutrino. Mass and coupling hierarchies are also not explained. Looking at these problems, one of the most natural extensions of the Standard Model is Minimal Left-Right Symmetric Model. We will explain in the model, how these hierarchies are solved naturally and also a good candidate for explaining charged lepton flavour violation, Parity violation, and neutrino majorana mass which is see-saw compatible with the help of extended Higgs sector. Then we will explicitly work out the MDM contribution at one loop in the LR model. It can be used to give bounds on the energy scale of the theory with the help of the magnetic dipole moment of the CLFV process. In the LR model, we get contributions from the extended Higgs sector for MDM as well as CLFV. But it is not enough due to phenomenology or observations. Considering LR symmetric model as the most realistic and natural, as it is not excluded yet, we will try to find possible ways to save the model, especially focusing on the charged lepton flavour violation problem.

1. Charged Lepton Flavour Violation in
Left-Right Symmetric Model
Md. Samim-ul-Islam
. Exam Roll no 2605
. Reg no H-2477
Supervisor: Dr. Arshad Momen
. Co-supervisor: Dr. Talal Ahmed Chowdhury
Department of Theoretical Physics, University of Dhaka
1/24

2. Outline
Motivation
Standard Model
Charged Lepton Flavour Violation (CLFV)
Neutrino Physics
Left-Right Symmetric Model (LRSM)
Phenomenology and Possible extensions
Conclusion
Thank You for Your Kind Attention.
2/24

3. Motivation

4. Motivation
In 1998, the SuperKamiokande collaboration presents definitive
evidence for neutrino oscillations, implying at least two
massive neutrinos which also require new particle(s) not
predicted by the Standard Model. Mixing of neutrinos would
eventually transfer to charged lepton sector due to the shared
coupling between neutral and the charged leptons (ν − e − W ). To
accommodate massive neutrinos, we therefore require an extension
of the Standard Model.
Thus, from neutrinos oscillation, we can see that lepton
flavour violation (LFV) occurs in the neutral lepton sector.
So, why not LFV occurs in the charged lepton sector ?
3/24

5. Motivation
Present experimental limit
Present experimental limit of CLFV, for instance: as of 2016 the
Particle Data Group (MEG experiment), is
BR(µ → eγ) < 4.2 × 10−13
CLFV prediction from Standard Model
Within SM, CLFV is predicted to be non-zero but very much
small.
BR(µ → eγ) < 10−40
4/24

6. Standard Model

7. Standard Model
Standard Model of Particle Physics
Standard model is so far the best description of particle physics
that describes the fundamental particles and their interactions
(without gravity). The underlying mathematics is the quantum
field theory(+ gauge symmetry). To account for ‘Electroweak
symmetry’ ; we introduce a phase transition, Higgs transition
(SSB) and the gauge groups are defined with simple Lie groups.
Gauge structure:
SU(3)C × SU(2)L × U(1)Y → SU(3)C × U(1)EM
Fermion Representation: 3 ×
{QLi
(3, 2)1/6, URi
(3, 1)2/3, DRi
(3, 1)−1/3, LLi
(1, 2)1/2, ERi
(1, 1)−1}
Higgs Sector: SU(2)L doublet ϕ =
ϕ+
ϕ0
!
5/24

8. Discrepancies of SM with experiments
Although we get tremendous success of SM in many
experiments(e.g. AMM ∆ae = −0.88 × 10−12), but it is now well
understood that Standard model of particle physics is not
“complete” theory of nature.
Some Discrepancies SM vs Experiment
ˆ Neutrino (ν) mass and Lepton flavour violation (LFV)
ˆ CP violation(CPV) or Matter Anti-matter asymmetry
ˆ Anomalous magnetic moment (AMM) of Muon (µ)
ˆ Dark Matter
ˆ Hierarchy problems
Some new physics or Physics Beyond the SM will replace SM in
the High energy regime so that these effects come naturally.
6/24

9. Charged Lepton Flavour Violation
(CLFV)

10. Prospect of CLFV process in Standard Model
Flavour violation can only occur after the symmetry
breaking as mass splitting occurs (GIM mechanism)
between generations of fermion family resulting mass
eigenstates and flavour eigenstates related by unitary
transformation. This fact eventually implies that there is no
flavour changing neutral currents (FCNC) at the Lagrangian
level i.e. the flavour cannot change in any process at tree
level. Only flavour changing charged currents (FCCC) exists
at tree level:
Lcc =
g
√
2
[uLγµ
(V †
u Vd )dL + νLγµ
(V †
ν Ve)eL]W +
µ + H.c.
But neutrinos are massless thus Vν = 1 i.e.trivial, no mixing of
neutrino!! Thus, no possibilities of CLFV in SM.
7/24

11. Feynman amplitude of CLFV process
Feynman amplitude of the general photonic CLFV process
(li → lj γ) where off shell photon emission also contributes,
written in a form
Mphotonic(li → lj γ) =
uj (p − q)[γλ(A + Bγ5) + qλ(C + Dγ5) + iqνσλν(E + Fγ5)]ui (p)ϵλ
where A, B, C, D, E and F are some constants or form factors.
However, This general form can be simplified and we are
left with electric and magnetic dipole transition term (EDM
& MDM)only
Mphotonic,on−shell (li → lj γ) = Fuj (p − q)[iqν
σλν(1 + γ5)]ui (p)ϵλ
8/24

12. Effective Operators of CLFV process
Therefore, Mphotonic,on−shell (li → lj γ) correspond to a Non
renormalizable operator
Lphotonic,on−shell
5 =
µM
ij
2
ψi σµν
ψj Fµν +
µE
ij
2
ψi σµν
γ5ψj Fµν + H.c.
Since [ψ] = 3
2, [Fµν] = 2. Hence, the term is 5 dimensional while
space time is 4 dimensional. So, it can not occur (at tree level)
at Lagrangian of the theory.
From L5, the diagonal part of µM and µE generates AMM and
EDM of the fundamental particles respectively. The off diagonal
elements contribute to CLFV processes(like li → lj γ). So,
naturally any flavor non-diagonal coupling will activate CLFV
processes and at the same time yield contributions to AMM.
9/24

13. Neutrino Physics

14. Neutrinos in Standard Model
Massless neutrinos in Standard Model
Dirac and Majorana mass term in Particle physics is given by
LDirac = mDψ̄LψR + H.c.
LMajorana = MMψT
L C−1
ψL
ˆ As there is no right-handed neutrino (NR) in SM. Thus
neutrinos in SM can’t acquire Dirac mass.
ˆ As Majorana mass term explicitly breaks SU(2)L gauge
symmetry and lepton number symmetry U(1)L. That’s why
this is also forbidden.
Observation: Small neutrino mass and mixing. 10/24

15. Neutrinos in Beyond Standard Model
We can have both neutrino mass and mixing in following
ways:
ˆ Introduce SU(2) scalar triplet (majorana mass).
ˆ or Add gauge invariant Non renormalizable operator.
Both ways lead to same physics. The simplest Non
renormalizable operator is Dim-5 Weinberg operator:
L5 = cij (Lc
Li · f
H∗)(f
H† · LLj )
Λ
+ H.c.
This extension solves both problem and leads to a mechanism
called ’Seesaw’ mechanism.
Seesaw-I: Fermionic Singlet (NR) , Seesaw-II: Scalar Triplet (∆),
Seesaw-III: Fermionic Triplet (Σ).
11/24

16. Left-Right Symmetric Model
(LRSM)

17. Why LRSM is the most natural extension?
2HDM
The most simple way to extend Higgs sector is to add a scalar
doublet similar to SM higgs. But it is not Seesaw compatible!
The triplet SU(2) scalar(LR)
Seesaw compatibility implies SU(2) triplet scalar Higgs (∆).
To match the IR limit of LR model, that is, after SSB,
LR
SSB
−
−
→ SM, LR model needs a triplet SU(2) scalar!!
So, there are many natural simple extensions. But the most
natural extension accounting the neutrino physics is the
Left-Right Symmetric Model.
12/24

18. LRSM Gauge sector and Fermion content
Gauge symmetry:(only lepton sector relevant)
GLRSM = SU(2)L × SU(2)R × U(1)B−L
Fermion content:
LLi =
ν
e
!
Li
; LRi =
ν
e
!
Ri
; QLi =
u
d
!
Li
; QRi =
u
d
!
Ri
Under gauge group their representation is given by
{QLi
(2, 1,
1
3
), QRi
(1, 2,
1
3
), LLi
(2, 1, −1), LRi
(1, 2, −1)}
where QEM = T3L + T3R + 1
2 (B − L). Here, T3L,R are the
generators of SU(2)L,R group. 13/24

19. Higgs sector
Extended Higgs sector:
Bi-Doublet: Φ = (2, 2, 0) for generating Dirac mass from SSB.
Triplet (extended): ∆L : (3, 1, 2) and ∆R : (1, 3, 2) for
generating Majorana mass from SSB.
∆ =
1
√
2
δ+ δ++
δ0 − 1
√
2
δ+
!
Φ =
ϕ0
11 ϕ+
12
ϕ−
21 ϕ0
22
!
* Only the neutral parts acquire vev. Vev of Higgs fields:
⟨ϕ⟩0 =
1
√
2
k 0
0 k′
!
and ⟨∆L,R⟩0 =
1
√
2
0 0
vL,R 0
!
Reality condition(LR): vL ≪ k, k′ ≪ vR
14/24

20. Prospect of CLFV in LRSM
It is well understood that to occur CLFV process, we need
ˆ an effective vertex that bypasses tree level or GIM
suppression
ˆ and non trivial mixing matrix of lepton sector,
UPMNS = V †
ν Ve such that neutrino matrix, Vν ̸= 1.
Interacting Lagrangian from LR model for photonic CLFV
(li → lj γ ) process is given by
LUV
int = ψ̄γµ[GLPL + GRPR]ψW a
µ Ta + ψ̄[GLPL + GRPR]ψH
At IR limit, this interacting Lagrangian becomes
LIR
int =
emli
2 ψli
V†
iαVαjσµν[GLPL + GRPR]ψlj
Fµν + H.c.
15/24

21. Effective vertex of CLFV in LRSM
Therefore, we can see from LRSM that not only there exists
a non trivial mixing matrix (V †
iαVαj ) which is nothing but
UPMNS originate from seesaw compatibility but also allows
Dim 5 Effective vertex running BSM particles through 1
loop :
δΓµ
eff ∼ 1
M2
scale
(G∗
L GLterms + G∗
RGRterms) + m0
M2
scale
(G∗
RGLterms +
G∗
L GRterms)
16/24

22. Relevant diagrams for MDM :
We only consider the diagram with chiral flipping inside as
it has mass insertion inside loop which will enhance MDM
as well as CLFV and can neglect the diagram with chiral
flipping outside the loop due to negligible contribution
Relevant W-boson and Higgs loop diagrams
17/24

23. MDM contributions in LRSM
These are verified by PV integral tool, modern technique
for loop calculation.
µW = −1
2
em0Re(GW ∗
R GW
L )
16π2M2
W
R 1
0 dx1
R 1−x1
0 dx3
[3(1−x1)−sx1
2]
1−x1+rx1+s[x1(x1−1)+x1x3]
µH(a) =
em0Re(GH∗
R GH
L )
16π2M2
H
R 1
0 dx1
R 1−x1
0 dx3
x1
1−x1+rx1+s[x1(x1−1)+x1x3]
µH(b) =
em0Re(GH∗
R GH
L )
16π2M2
H
R 1
0 dx1
R 1−x1
0 dx3
x1−1
x1+r(1−x1)+s[x1(x1−1)+x1x3]
MDM contributions are
ˆ suppressed by boson mass
ˆ enhanced by (internal) fermion mass.
ˆ proportional to mixing of left and right sector of bosons.
18/24

24. Phenomenology and Possible
extensions

25. Phenomenology
Higgs Phenomenology
β terms from Higgs potential:
β1[Tr(ϕ∆Rϕ†∆†
L) + Tr(ϕ†∆Lϕ∆†
R)] + β2[Tr(ϕ̃∆Rϕ†∆†
L) +
Tr(ϕ̃†∆Lϕ∆†
R)] + β3[Tr(ϕ∆Rϕ̃†∆†
L) + Tr(ϕ†∆Lϕ̃∆†
R)]
Potential at vev must be extremal (minimal model):
β2 = (−β1kk′ − β3k′2 + vLvR(2ρ1 − ρ3))/k2
Reality condition(+ with seesaw condition) implies βi = 0.
Thus we get, vLvR(2ρ1 − ρ3) = 0 which is known as the
vev-seesaw relation.
Consequence
Due to Higgs phenomenology, we get that the mixing between ∆L
and ∆R are very small resulting almost no contribution from the
diagram with chiral flipping inside. Remaining contributions are
from the diagram with chiral flipping outside which is suppressed.
19/24

26. Types of LRSM
There are 2 types of LRSM based on origin of CPV -
ˆ Explicit CPV : Here CPV contributions are from higher UV
theory like GUT , SUSY after SSB.
LUV
D>4
SSB
−
−
→ LIR
D=4
That is, no contribution from Yukawa coupling of IR theory.
ˆ Implicit CPV : Here, CPV is implicit in IR theory. Thus, in
both model, the Higgs potential is real. Based on Yukawa
coupling implicit model can be further divided into -
∗ Manifest CPV : CPV term originate from Yukawa
couplings but vev of Higgs are real.
∗ Pseudo-Manifest CPV : CPV term originate from vev of
the Higgs field but Yukawa couplings are real and symmetric.
20/24

27. Immediate possible extensions
Possible extensions:
1.Triplet Fermion : We can have Type-III Seesaw compatible
LR model where femionic triplet BSM particles
ΣL = (Σ+, Σ0, Σ−)L : (3, 1, 0) and
ΣR = (Σ+, Σ0, Σ−)R : (1, 3, 0) are introduced. In these models,
SM like scalar doublets (ϕ) get vev which automatically
generates majorana DoF. No triplet scalar in needed.
2.Triplet Scalar+Triplet Fermion : Triplet Scalar is exactly the
same ∆L,R = (∆++, ∆+, ∆0)L,R which are Type-II Seesaw
compatible. We can study the combination of these two LR
models where with scalar triplets generating majorana from
fermionic doublets and also scalar bi-doublets getting vev
generating majorana from fermionic triplets. This will have an
enhanced scalar and fermionic sector which is exactly we need for
getting enhanced MDM and also for DM physics.
21/24

28. Immediate Possible extensions
Possible extensions:
3. 2 Scalar bi-doublet : Like 2 HDM of SM, we can extend to
2 bi-Higgs doublet models. Then couple both bi-doublets (Φ)to
theory and calculate contributions from both fields to match the
observation. In these extensions, we get more scalar particles
which can be studied and given bounds to match observed CLFV.
4.Manifest+Pseudo-Manifest+Exact LR Model : The
manifest LR provides more minimization conditions than the
number of vev’s leading to a fine tuning problem. On the
contrary, the pseudo-manifest LR leads to a light triplet Higgs
which is excluded by observation. With these in mind, for
minimal LR symmetric model to be realistic and natural,
combination of all type of LR model must be taken into account
and then use DM, AMM constraints. Then embed in Higher UV
model like GUT.
22/24

29. Conclusion

30. Conclusion
Although this is the general natural extension due to the
structure of MDM and EDM , Higgs phenomenology,
Higgs mass scale, reality relation etc. ; the minimal LR
model seems not giving any significant contribution to
enhance MDM and Branching ratio (BR). Thus can’t have
significant CLFV processes. But it is not also excluded by
experiment. So, we can extend or modify the model and try
to give bounds to MDM and BR beyond current limit (need
non-minimal LR). We can also have strong constraints from
Dark matter, AMM, CPV etc. on the BSM DOF.
23/24

31. References
[1] Tai-Pei Cheng and Ling-Fong Li:“Gauge theory of elementary
particle physics”, Oxford University Press, 1984.
[2] Rabindra N. Mohapatra and Palash B. Pal:“Massive neutrinos
in physics and astrophysics”, World Scientific Publishing Co.
Pte. Ltd, 1998. ISBN: 981-02-3373-6
[3] G. Senjanovic and R. N. Mohapatra, “Exact Left-Right
Symmetry and Spontaneous Violation of Parity,” Phys. Rev. D
12, 1502 (1975) doi:10.1103/PhysRevD.12.1502
[4] E. Corrigan, “LEFT-RIGHT-SYMMETRIC MODEL
BUILDING,” LU-TP-15-30.
[5] G. Ecker, W. Grimus and H. Neufeld, “The Neutron Electric
Dipole Moment in Left-right Symmetric Gauge Models”, Nucl.
Phys. B 229, 421-444 (1983).
doi:10.1016/0550-3213(83)90341-3
24/24

32. Thank You for Your Kind Attention.