1. Introduction
2. Standar Model

3. $U(1)_d Model$

   3.1 Lagrangian associated with the model $U(1)_d$

   3.2 Z' boson propagator

   3.3 Restrictions on the mass and kinetic mixing term of the Z boson

4. Anomalous magnetic moment of the muon

5. Partial Results

   5.1 Contribution of the Z′ boson to the anomalous magnetic moment of the muon at 1 loop

6. References

In this work, an analysis of the anomalous magnetic moment of the muon is presented in the context of an extended theoretical model that incorporates an additional $U(1)_d$ symmetry group to the Standard Model of particle physics.

The magnetic moment is a fundamental purely quantum parameter that measures how sensitive the spin of a particle is to an external electromagnetic field. Now, the relationship between the magnetic moment and the spin is given by a proportionality factor called the gyromagnetic factor ($g$). Treating “classically” (i.e., considering the process to tree order) the Dirac equation, one arrives at the value of $g$ must be equal to 2. However, by introducing the formalism of quantum field theory, all possible processes respecting the initial and final states must be taken into account. Thus, new contributions to this value will appear (hence the name “anomalous”). Moreover, these new contributions will be sensitive to particles existing in nature (including also unknown particles).

In recent years, subtle discrepancies of the experimental values of this observable (Muon g-2 Collaboration) with the theoretical predictions (White Paper) have been observed, which could be an indication of physics beyond the Standard Model. In addition to this, collaborations employing the use of computational methods such as Lattice QCD (BMW Collaboration) or experimental measurements in scattering processes (CMD3 Collaboration) have arrived at results that conflict with the same prediction of the Standard Model (White Paper), creating a theoretical ambiguity.

The $U(1)_d$ Model is a simple extension of the Standard Model by adding a new gauge field associated with a new gauge symmetry. This symmetry group has only a single generator, so it will be similar to the $U(1)_Y$ symmetry group of the Standard Model. Furthermore, a scenario is assumed for a massive dark photon in which the known quarks and leptons have no $U(1)_d$ charge. That is, there is no direct interaction between the fermions of the Standard Model and this dark photon.

In that order of ideas, this work focused on the implications of considering an extended $U(1)_d$ model in the context of the anomalous magnetic moment of the muon. That is, the detailed calculation of the contribution to 1 loop associated to the new field coming from the addition of the new symmetry group $U(1)_d$ was made taking into account all the theoretical predictions of the Standard Model (White Paper, CMD3 and BMW) and these results were compared with the experimental prediction of the Muon g-2 collaboration. Where, from this comparison, an analysis of how constrained the parameters of the new model are (or how feasible it is) for proper agreement between the theoretical predictions and the experimental measurement can be obtained.

The results of this research are expected to provide a deeper understanding of possible extensions of the Standard Model and their implications in particle physics, thus contributing to the advancement of knowledge in this field.

Keywords: anomalous magnetic moment, muon, Standard Model,$U(1)_d$ Model, symmetry group, Muon g-2.