| Keywords [eng] |
BRST, symmetry, neutrinos, Grimus–Neufeld model, Majorana, Nielsen identities, gauge symmetry, gauge parameter, gauge dependence, self-energy, Standard model, Higgs, BRST transformations, BRST-exact Lagrangian |
| Abstract [eng] |
The goal of this work was to implement the Nielsen identities in the Grimus–Neufeld model and automate using computer algebra systems. The concepts required to understand the BRST formalism and the Grimus–Neufeld model are discussed to some degree. In sections 1 and 2, we discuss the Faddeev–Popov method, which leads us to the BRST symmetry, as well as the BRST formalism. Generalized Ward–Takahashi identities for the non-Abelian theory are introduced; we show how, from Lee identities, Nielsen identities are derived (these identities are derived in Quantum Chromodynamics (QCD) and quantum electrodynamics (QED) as examples). At the end of the literature review, that is in section 3, we discuss neutrinos and how their masses are generated, as well as the seesaw mechanism. We introduce the Higgs mechanism, which is the procedure of the mass generation of the gauge bosons via spontaneous symmetry breaking. We also introduce the Grimus–Neufeld model, which seeks to explain small neutrino masses by extending the Standard model with one heavy Majorana neutrino and an additional Higgs doublet. In section 4, we derive the BRST transformation of the Higgs and neutrino sectors in mass eigenstate basis and explictly construct the BRST-exact Lagrangian, which we need for calculating the Nielsen identities for the neutrino self-energy. Lastly, in section 5, we explicitly derive and calculate the Nielsen identities for the neutrino self-energy, and express the result in the Passarino-Veltman function basis. We also calculate the gauge dependence of the neutrino self-energy independently and compare the result with the expression obtained from the Nielsen identity. The results matched between both independent methods, which confirmed that the Nielsen identity for the neutrino self-energy implementation in the Grimus–Neufeld was successful. We also analyze the automatization of the Nielsen identities in the Grimus–Neufeld model to some degree. We discuss the plan of how to automate the 3-point vertex functions, using FeynRules packet, which are needed in order to calculate Nielsen identities for the neutrino self-energy: defining the classification of the BRST sources of the gauge parameter and the Higgs sector, the problem of the fermionic BRST sources classification is discussed; we show example of how to define BRST transformation and BRST-exact Lagrangian of the neutrino sector, which are needed in order to generate the Feynman rules for the mentioned 3-pointr vertex functions. The automatization of the Nielsen identities in the Grimus–Neufeld model was not achieved. |
