In this work, we describe the possible phenomenologies of right-handed neutrinos in two simple extensions of the standard model. In this regard, we briefly summarize gauge theory and the standard model as SU(3)C ×SU(2)L×U(1)Y. Given this setting, we first discuss the Type I pendulum mechanism with singlet right-handed gauge neutrinos of the Standard Model.
In this paper, we attempt to extend the Standard Model (SM) with right-handed neutrinos (RHNs). This can be achieved by adding a right-handed neutrino, which can be SM gauge singlet via Type-I tilting mechanism [3]. In this case, the charged leptons are mixed with the neutrinos with charged and neutral right-handed neutrinos [4].
SM issues as discussed above lead us to go beyond the Standard Model scenario in Section 7, where we introduce the right neutrino as a Majorana fermion. Following on from in Section 9, we extend the Standard Model from a RHN that is a SU(2)L triplet (type III postcard) and the corresponding breakdown phenomenology is also discussed.
Abelian Gauge Theory
- Global gauge invariance in complex spinor field
- Local gauge transformation of complex spinor field
- Quantum Electrodynamics (QED)
- Vector Current
- Axial Vector Current
To make Lagrangian local gauge invariant, we introduce a modified derivative (covariant derivative), Dµ, which covariantly transforms under the gauge transformation just like ψ. The construction of the covariant derivative is: Dµ≡∂µ−iQAµ. Now, if we replace ∂µ in the Lagrangian with Dµ, it has the form:. If we treat the new field as a physical photon field, we must add a term to the Lagrangian corresponding to its kinetic energy, and this term must be invariant with respect to the gauge transformation.
We have already noted that the first term is invariant under local gauge transformation because we defined the co-variant derivative accordingly. We have already seen that the Dirac Lagrangian is invariant under global gauge transformation as (ψ −→ eiQαψ). So by making the gauge parameter temporarily space-time dependent, we can arrive at the axial vector flow.
So from the invariance of the Lagrangian we can identify ∂μα as the gauge value Aμ as shown for the vector currents and obtain the correct chiral current. Similar to the right-chiral current, we can obtain the left-chiral current as jAμL =Qψ†LσμI2ψL. 22) So the two types of flows are different.
Non-Abelian Gauge Theory
In this subsection, we derive the interaction vertices of the leptonic fields under SU(2)L×U(1)Y gauge group. From SU(2)L×U(1)Y we also derive the rotation angle that gives rise to photon and Z-boson. The SU(2)L is a non-abelian gauge group that has three generators represented by Pauli spin matrices.
On the other hand, the weak overload subgroup U(1)Y has identity as a generator and the field of the real component is given by the vector boson Bµ. The weak hypercharge is different for left and right fermions, but is the same for doublets. To conserve charge, the covariant derivative of a field must carry the same electric charge as the field itself.
Here the well-defined mass eigenstate is not Bµ and Wµ3 but their orthogonal linear combination (one is massless photon field Aµ and the other is massive Z boson vector field Zµ). So the coefficient of neutrino current is zero and the coefficient of electric current is electric charge e. So the mixing angle is given by the ratio of the two independent group coupling constants, tanθW = gg1.
Electro-weak symmetry breaking and Higgs Mechanism
The field φ here is written in Unitary gauge, where we remove the Goldstone bosons, that is, the charged part φ+ gives the pair of charged Glodstones W± and the complex part of the neutral component becomes the neutral Glodstone. The mass of fermions in the standard model is generated by the Yukawa term. The guiding principle for any term in Lagrangian is that it must be real, gauge invariant and Lorentz invariant. If we assume that in addition to the usual left-handed neutrinos νl there are also right-handed neutrinos νR (which is not compatible with SM), then one can write the Dirac.
The Majorana mass expression can be constructed from νLalone, in which case we have a left-handed Majorana mass. 2mLνLcνL+h.c., (80) or only from νR, in which case we have a right-handed Majoran mass,. In this case, the mass terms are of Dirac type and derive from a Yukawa coupling of the form, .
In the Standard Model, if we want to include a neutrino mass term, then we must extend the Standard Model by adding a right-handed neutrino νR and a Yukawa coupling −Yνν¯LφνR+h.c. If in the Standard Model the Dirac mass term does not vanish, the source of the neutrino mass would only be the Majorana mass term. If the neutrino has Dirac mass as well as Majorana mass, then the Lagrangian for the total mass term will be,.
So, we have a neutrino mass on a scale mR of new physics and a very light neutrino, whose mass is mm2D. After electro-weak symmetry breaking of Higgs field, the neutrinos will gain mass, The Lagrangian of the total system is given by:. The covariant derivative Dµ can be replaced by ∂µ, because the added heavy right-handed neutrino is SM singlet and does not interact with the gauge field.
Here we consider: MN is the mass of right-handed neutrino, mh is the mass of Higgs, mn is the mass of SM neutrino ≈0. Similarly, there is another decay mode, in which the right-handed neutrino decays into one Z boson and one light neutrino (ν). In both graphs we can see that the decay width of the decay to hν is smaller than that of the other, although the mass of the Higgs boson is greater than the mass of the vector bosons, Z and W.
Therefore, we can conclude that if the mass of the right neutrino is of the order of TeV (such as 5 TeV or 20 TeV), there will not be any significant difference in the partial decay width or decay lifetime between them. On the other hand, as we have seen in the case of the partial decay width, the branching ratio of the decay in eW is larger than the decay in Zν (almost double from Figure 10 (a)) even if mz > mw.
Displaced Decays of right-handed neutrinos
In the case of figure 11, we have calculated the total decay width and plotted the Yukawa coupling (YN) and neutrino mass (MN).
Production of right-handed neutrinos
The type III seesaw model has, in addition to the SM fields, triples of SU(2) fermions with zero hypercharge Σ. In essence, we can obtain a type III oscillatory mechanism by replacing the type I fermionic singlet with a triplet. In terms of 2 × 2 triplet notation, the Lagrangian becomes:. 108) Where MΣ is the triplet mass matrix and YΣ is the Yukawa coupling matrix. If we consider the Higgs doublet as H = H+. 109) So, without the mass term Yukawa becomes,.
Note that there is no measurement interaction term in type-I seesaw, which is the difference between them. Like the type-I seesaw model, there are also possible decay states for the type-III model where the right-handed neutrino is a fermionic triplet.
Production
In this work, we started with the abelian gauge theory and continued to study non-abelian gauge theory. In the process, we study the weak sector of Standard Model asSU(2)L×U(U)Y gauge theory. Standard Model is a very successful theory with the Higgs mechanism giving mass to the W±, Z bosons, quarks and the charged leptons.
However, the Standard Model fails to produce a mass for neutrinos, which have recently been observed to be massive. A simple expansion of the SM with a singlet SM can be achieved by adding a right-handed neutrino. It not only creates mass for the neutrinos, but explains everything why one of the neutrinos will be very light and the other will remain heavy.
Although their decay phenomenology is very interesting, producing such inert particles is quite challenging because there is no direct gauge coupling. The production mechanism always depends on the mixing angle with the left-handed neutrinos and thus such cross sections are very low. This results in the high luminosity required to investigate such right-handed neutrinos in colliders, viz.
Next, we studied the expansion of SM with a SU(2) triplet and Y = 0 fermion, leading to a charged fermion pair and a neutral right-handed neutrino. Similar to type-I, type-III also decays to the gauge boson and other leptons via the mixing angles. The production diameter in this case is larger than type I right-handed neutrinos due to SU(2) gauge coupling.
In[1]:= (* partial decay width; branching ratio vs neutrino mass *). PlotLabel→"Branching Ratio vs Neutrino Mass Graph",*) LabelStyle→Black, PlotLegends→. 6] Introduction to the Standard Model of Particle Physics Pascal Paganini, Laboratoire Leprince Ringuet, Ecole Polytechnique, Palaiseau France. 8] QUARKS AND LEPTONS: An Introductory Course in Modern Particle Physics by Francis Halzen and Alan D.
