Contribution of BSM Particles on Electric Dipole Moment of Charged Leptons in the Minimal Left-Right Symmetric Model

In this paper, the contribution of Beyond the Standard Model (BSM) particles on the Electric Dipole Moment (EDM) of charged lepton is calculated in a minimal Left-Right Symmetric Model. First-order quantum correction i.e., one loop correction, from relevant diagrams in the unitary gauge using Feynman parameterization is used because there is no tree-level contribution in Electric Dipole Moment due to non-renormalizability of the dipole term in the Lagrangian. This study mainly focuses on the BSM Higgs sector contribution to the EDM of charged leptons due to phenomenological interest. CP violation is directly connected to the EDM of fundamental particles. Matter-antimatter asymmetry is a mystery of the universe that demands CP symmetry to be violated. The standard model cannot explain the observed CP violation. Standard Model (SM) is so far the best description of nature. Despite having tremendous success, e.g., Electro-weak Symmetry Breaking and Higgs discovery, the Standard model is not complete. Moreover, experimentally, we observed small neutrino mass and neutrino mixing. The Seesaw Mechanism can explain neutrino mass smallness and mixing introducing a large Majorana mass. Left-right symmetric model is the most natural extension of SM which can accommodate the Seesaw mechanism. So EDM contributions from BSM particles in the Left-Right symmetric Model are a good probe of new physics beyond the SM. These contributions then can be used to do phenomenology and test the Model.


Introduction
There is a well-known problem in cosmology called Baryon asymmetry, also known as matter antimatter asymmetry, which is the imbalance between baryonic and anti-baryonic matter in the observable universe [1].CP violation, in the lepton sector, generates a matter-antimatter asymmetry through a process called leptogenesis [2].This could become the preferred explanation in the Standard Model for the matter-antimatter asymmetry of the universe if CP violation is experimentally confirmed in the lepton sector.If CP violation in the lepton sector is experimentally determined to be too small to account for matterantimatter asymmetry, some new physics beyond the SM would be required to explain additional sources of CP violation.New BSM particles will give new sources of CP violation as CP is not a symmetry of the ground.The standard model cannot fully explain the observed CP violation needed to explain the observed Baryon asymmetry [1].In the LR model, Baryon number conservation is broken at a high energy.After a phase transition by a new Higgs field, the ground is now (1)  symmetric [3].
In the LR model, we extend the Higgs sector such that right-handed neutrinos exist, form weak doublet, and get decoupled after this new Higgs transition.This will naturally give P violation and also increase the CP violation due to the extended Higgs sector.Extension of the Higgs sector is constrained due to the bound coming from EDM experiments and Dark matter problems.The most common way to extend the Higgs sector is to add a scalar doublet similar to SM Higgs.This is called the two-Higgs doublet model (2HDM) [4].Now, one can add either an active (coupled to other particles) or an inert (decoupled from other particles) Higgs doublet.Adding an inert Higgs doublet provides a viable Dark matter candidate [5].But this extension will not enhance CP violation.So, looking at both issues, it is natural to extend with an active Higgs doublet.From neutrino physics, we have observed neutrino oscillation and smallness of neutrino mass [6].This neutrino oscillation and smallness of mass are well studied and to get to this observation, a mechanism called the See-saw mechanism is needed [7].Thus, the Higgs sector needs to be extended in such a way that it can give rise to Majorana mass for neutrinos and enable the Seesaw mechanism.CP violation is proportional to the EDM.At the tree level, EDM contributions are zero as the EDM operator is non-renorm.That's why, the 1st quantum contribution (1-loop) of relevant diagrams to EDM is studied.Present experimental limit of EDM, as of 2020 the Particle Data Group (ACME II experiment) [?], is |  | < 0.11 × 10 −28  ⋅  Within SM, EDM is predicted to be non-zero but very small.

|𝑑 𝑒 | ≈ 10 −38 𝑒 ⋅ 𝑐𝑚
In this article, we are choosing the Left-Right symmetric model as the BSM model, where neutrino physics problems are solved naturally, and calculate the expression for EDM of charged leptons in this model at 1 st order (1-loop) QFT correction.That's why, we organized the article as: in the 2 nd section we addressed the issues of neutrino physics, and in the 3 rd section we discussed the model incorporating neutrino physics.Then, in the 4 th section, discussing the QFT structure of the EDM operator, we calculated the expressions for EDM of charged leptons in the 5 th section.Finally, we concluded in the 6 th section.

Neutrinos in SM
In the SM, neutrinos don't acquire masses and mixings.There are no right-handed neutrinos in the Standard model.  =       + ℎ. .
They could have Majorana mass.However, due to gauge symmetry, it is forbidden.Because Majorana mass term in the Lagrangian is which is clearly breaking the (2)  gauge symmetry and also Lepton nnumber(1)  symmetry.Thus, for gauge symmetry and the absence of right-handed neutrinos, neutrinos in the SM cannot acquire mass.

BSM neutrinos
To solve for the Dirac mass term, there is no other option than adding a right-handed neutrino.Majorana's mass term violates Lepton's number conservation.But (1)  is an accidental symmetry that occurs due to the fermionic structure and gauge interactions.There is no fundamental reason why it should be preserved.So, we can introduce aa(2) scalar triplet to the theory, after spontaneous symmetry breaking, it will produce the Majorana mass term.This term breaks lepton number symmetry [9][10][11][12].
If neutrinos are Majorana particles, a Majorana mass can arise as the low-energy realization of a higherenergy theory.The dimension 5 non-renormalizable operator is

Spontaneous symmetry breaking
The vacuum expectation values of the scalar fields are zero.To obtain the symmetry breaking, we insert a non-zero vacuum expectation value for Higgs scalars in the (3) and (4).
We assume the order of magnitude relation: 2 for Seesaw compatible theory.Just considering the (2)  , the righthanded gauge group breaking along with the (-) abelian group, generates (1)  , the standard model hypercharge symmetry.

Majorana mass matrix for neutral leptons
When the scalar field  gets vacuum expectation value, from   () in ( 4), neutrinos acquire Majorana mass terms.
The mass term for neutrino is defined by where   is the mass matrix of the neutrino.Thus, in our case, neutrinos having both Dirac and Majorana masses, the Dirac mass term mixes  − ℎ  but Majorana mass mixes   and (  )  , enlarging the freedom of the neutrino.Thus, putting all the 6-L handed freedom into the vector, the mass matrix for the neutrino (both Dirac and Majorana) is given by (for more details [17]) where   is the Dirac mass term of the neutrino.

Physical Higgs
Expanding the potential around the vev, we can construct the bilinear terms.These will give us mass matrices.For more details, [18] and [3] are recommended.These physical Higgs bosons are: four real scalars  0,1,2,3 0 and two real pseudo-scalars  1,2 0 from the mixing of neutral fields, four complex scalars, the singly charged Higgs  1,2 + and two doubly charged Higgs  1,2 ++ .

Form factors
The amplitude for a particle ( → ′) scattering from a heavy target ( → ′) is given by where  is the coupling constant,  = ′ −  =  − ′ is the momentum exchange and e,   is the quantum corrected vertex function.The   function appears in the S-matrix element for scattering of an electron from an external electromagnetic field.With the vertex correction, the amplitude is given by −(′)  (, ′)()   (′ − ) Wherparity-violating    (′ − ) is the Fourier transform of    ().In the lowe depends   =   .In general, depends   depend on , ′,   .For a parity violating theory, it can also depend on  5 .This object   transforms like a vector under Lorentz transformation.

Non-renormalizable EDM operator
The mass dimensions of  and   are, So, the dimension of EDM or MDM term in the Lagrangian is 5 and the spacetime dimension being 4, it is non-renormalizable.
The dipole operator beingnon-renormalizabler, cannot be added to the Lagrangian of the theory and so at the tree level EDM contribution is zero.With constructing effective vertex   and using Gordon identity, we get vertices   occurring level a fifth occurring can     and      5 ofifth occurringop level.There are three kinds of suppression: three-level suppression, loop-level suppression and mixing matrix suppression.These are the reasons why the EDM of a fundamental particle is very small.The allowed diagrams by Feynman rules are loops either with W-bosons or Higgs scalars.These diagrams are shown in Fig. 1.

Feynman rules
For fermion coupling to W-bosons and Higgs , we consider the interaction Lagrangian 2 )]  (7) 2 )] where  ,  and  ,  are matrices in flavour space of all fermions.These   and  fields are mass eigenstate and so they are hermitian.From these interaction terms we can calculate the couplings of bosons with fermions.All these Feynman rules are given below.

Feynman parameterization
The expression for the most general form of 1-loop contribution has denominators, the products of the denominators of the propagators of the particle in the loop.Feynman parameterization is a trick to combine these denominators.For two denominators, we can write the integral To generalize to  numbers of denominators, we can take derivatives w concerning a, b and use the induction.By plugging in the explicit denominator forms, we can arrange the denominator term  1  1 +  2  2 +  3  3 as ga eneral quadratic expression from where by shifting momentum variable we can remove the linear dependency in the ddenominatorand or.These parameters , , ⋯ that make the loop integral simpler are called Feynman parameters.For more details, references [19] and [20]

Calculation of the EDM Loop Contributions
We have already discussed the QFT structure related to EDM (at one loop).We will follow the manual given in [19] and [20] for loop calculations.The definition of dipole moment is considered from reference [15].We will use the unitary gauge to calculate the one loop W-loop diagram.where,  = ′ −  is the momentum provided by the photon.Conservation of momentum implies 2 ⋅  +  2 = 0 For EDM   = −( 2 = 0), we will set  2 = 0.This will simplify our calculation a lot.The trace of generators is due to non-abelian nature.From normalization [    ] = 1 2   .With the Feynman parameters, we can combine these three denominators.

Diagram 1: W-loop contributions
where, Now, define  ≡  +  2  −  3 .The advantage of this trick is that the numerator terms which are linear in , can be dropped due to the integral limit from −∞ to +∞.Now, define  = Now for the numerator terms, collecting only the relevterms for electric dipole moment (EDM vertex from 6), we get Here, we observe that, only the term linear in mass is contributing.This is the mass insertion we were expecting for the chiral flipping.Simplifying the numerator using Clifford algebra and   =   −  2   +  3   and then applying momentum conservation, we get   0 term We can ignore the   term because ( 2 −  1 ) is odd in  1 and  2 , whereas the denominator is even  1  2 .Under Wick rotation ( 0 →   0 ), the integral changes like, ignoring the subscript E, For our case, we need to choose  = 4,  = 0,  = 3 in the general loop integral formula (9) and then Wick r rotates Now, we can do the integration.Applying Gordon's identity and integrating out  1 and  2 , our loop contribution Now, we have to do the   2 terms.Similarly using Clifford algebra, momentum shift  =  −  2  +  3  and     →  2   4   , we get fromth   0 and   2 terms together to get the final expression.The 1   4 terms vanish.From the definition of EDM form factor,   =   ( 2 )     5 , and EDM definition   = −  ( 2 = 0), we get the electric dipole moment of a charged lepton  from W-boson contribution in Fig. 9.We first need to combine three denominators using Feynman parameters.In the numerator, considering only the relevterms for EDM, using Clifford algebra and shifting the momentum, we simplify to get where we have used the fact that nthe umerator terms with an odd arewer of  iarez internally case, also due to  1 and  2 being odd in the numerator (and even in the denominator),   term contributes zero.Now, similarly as we did in the W-loop case, after Wick rotation (10), integrating out the  integration using (9), we can get the final expression for loop contribution.Thus, integrating Feynman parameters  1 ,  2 and from the definition of EDM (  =   ( 2 )     5 ), we find the Higgs loop contribution where photon is coupled to internal charged boson in Fig. 10 as We first need to combine three denominators using Feynman parameters.
Now, with the same definition  ≡  +  2  −  3 , we can write the denominator as  =  2 −  +  where, Now, for the numerator terms, we can simplify using Clifford algebra and momentum shift to write   = − 0 (   *    )[(  + ′  )( 3 − 1) −   ( 2 −  1 )] 5   Similarly, due to symmetric integral, we can neglect odd  and   terms.With a similar procedure, from the definition of EDM form factor, we can find the EDM contribution from Higgs loops where the photon is coupled to internal fermion in Fig. 11 as

Sum over all possible internal states
In all the calculations (11), ( 12) and ( 13), we did not consider the mixing of the internal states.Diagrams with all possible internal fermions are needed to be summed.For example, consider the W-loop contribution to EDM, Fig. 9.The internal fermion is neutrino, we need to sum over all possible neutrino states in the loop.Consider,   as charged lepton Yukawa coupling.By rotating with a unitary matrix   , we go to na ew basis such that   is diagonal,  ̂ =       † = (  ,   ,   ) Similarly for neutrino, we have to rotate by unitary matrix   such that it diagonalizes nin the theeutrino mass matrix.So, either we can go to charged lepton's mass basis (then neutrinos are in superposition) or go to neutrinos mass basis (where charged leptons are in superposition).From (5thethe uneutrinohas na on zero mass matrix.Therefore, the mixing matrix   of the lepton sector is given by In a minimal LR symmetric model, the neutrino has mass and so it has a non-trivial mixing matrix.In the charged lepton mass basis, neutrinos in the loop are in a superposition of all mass eigen basis.Thus, the minimal LR model enables mixing in the vertex of this theory as   is a non-trivial matrix.

Conclusion
In this extended structure, from the relevant diagrams, the electric dipole moment of charged leptons contributed from BSM particles is studied.From ( 11), ( 12) and ( 13), we find that the dipole contributions dominate on mass insertions of internal fermions and bosons.Apart from that, these contributions also depend on the imaginary part of the product of the couplings (  *   * ).These dipole contributions can be used to give bounds on the new physics (BSM) and also test the model.For singly charged Higgs bosons, ATLAS reports a lower limit on mass of approximately 100 GeV.Possible Higgs signals from LHC give a limit on doubly charged Higgs boson mass of approximately 400-500 GeV.As a large scale of   is needed for neutrino physics, no signal of the LR symmetric model has so far detected but it is not even excluded.A possible way to extend the scalar contribution is to extend the Higgs sector with another bi-doublet.Then coupling both bi-doublets to the theory, contributions from both fields can be calculated to match the observations from experiments.In this extension, new scalar particles can be given bound to match observed CPV.Then new flavor structure can be introduced turning off the minimal flavor violation (MFV) consideration.These extensions can also be related to Dark matter problems and can be used to give bounds from Dark matter physics.

Acknowledgement
Special thanks to Dr. Talal Ahmed Chowdhury, Assistant Professor, Department of Physics, University of Dhaka, for his assistance.

Figure 1 . 6 4. 3
Figure 1.W-bosons (a and b) and Higgs bosons (c and d) are all possible general loop diagrams contributing to EDM form factor of charged leptons

Figure 2 .
Figure 2. W-boson and Higgs bosons loop contributing to EM charge form factors (photon coupled to external fermion)

Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .
Figure 4. Fermion-fermion-photon vertex are recommended.The general formula for doing such kind of loop integral is given by

1
the delta function, we can write the denominator as  =  2 −  +  where,