The electric charge in multiplets of gauge theories

Abstract

The electrle charge ls a conserved quantlty as far as we know. It ls a property of matter which constltutes one of the most important milestones in Physics at the same level as conservation of energy. Its minimal value (without considering the quarks that cannot be isolated) has been experimentally determined as a quantized quantity since the nineteenth century and it is currently accepted as a constant (and exact value) of nature e = 1.602 176 634 x 10-19 C [1]. From a theoretical point of view, electric charge is recognized as the global conserved charge of the Quantum Electrodynamics (QED) symmetry group, U (1)Q. In this context, it is a Noether’s charge. However, at high energies QED is not the governing symmetry anymore, i.e. there is another gauge theory which explains the particle interactions above certain limit called electroweak energy scale: The Standard Model (SM). Thus, we have to understand how electric charges are predicted at this stage even though they are not defined at high energies; this is the main purpose of this work. SM ls a widely accepted theory thanks to its enormous success in predlctlng what would later be important discoveries such as the very existence of some particles, propertles of other particles already discovered and outcomes of some particle physics experiments. This gauge theory belongs to the symmetry group SU(3)C ⊗ SU(2)L ⊗ U(1)Y. For better understanding, it can be separated into (i) SU(3)C, which corresponds to Quantum Chromodynamics (QCD), theory developed in the early 70’s by Politzer [2], Wilczek and Gross [3] in order to describe strong interactions; and (ii) SU(2)L ⊗ U (1)Y that corresponds to Electroweak Standard Model (EWSM) developed in the middle 60’s by Weinberg, Salam and Glashow. The latter describes the electromagnetic and weak interactions in the same framework at high energies. The EWSM prototype was done by Glashow [4], while Weinberg and Salam [5,6] implemented the Higgs mechanism [7] to generate masses to all particles involved and succeeded in placing the model as a gauge field theory. It is in this EWSM gauge group that some of the most important predictions were made such as the existence of the vector bosons W and Z as well as the scalar Higgs boson, all of them confirmed experimentally some years later. All interactions are written to be a local gauge invariant in the Lagrangian density, which means that the unitary gauge transformations of all the fields are defined for that purpose. Also, within this electroweak Lagrangian there are all the interactions between the fields and their respective gauge bosons that conveying the forces. These gauge bosons (Wf, Wf, Wif and Bf for SU(2)L ⊗ U(1)Y) are massless until a process called Spontaneous Symmetry Breaking (SSB) takes place. In fact, all particles acquire mass after this process. Despite all the virtues of SM, it must be said that there are some reasons to believe that it is an incomplete theory and should be considered, at best, as an effective theory that describes particle physics only on energy scales that have been experimentally tested; that is, up to about 13 TeV c.m. in Run 2 of the LHC. Some of the items and questions that reveal the incompleteness of SM are the massive neutrinos, dark matter, matter-antimatter asymmetry, gravitational interaction, etc. and as a consequence, SM has to be extended to a more complete theory. In the search for these extensions it must be considered what types of new very massive particles could exist or what implications these could have on experiments at accessible energies. EWSM extensions are described by left-right symmetry groups like SU(2)L ⊗ SU(2)R ⊗ U(1)B-L or new chiral groups like SU(3)L ⊗ U(1)N. The symmetries inherent to these new theories occur at high energies and the massless particles represented as new multiplets Will be disjointed to generate massive physical fields when the symmetry breaks down to U(1)Q, at energies in which experiments are developed. That means that when energy decreases, a limit is evidenced when the symmetry is broken in a process called “Spontaneous Symmetry Breaking“ (SSB) as a result of Higgs mechanism. For example, above the electroweak scale in EWSM (100 ~ 1000 GeV), Lagrangian obeys the symmetry SU(2)L ⊗ U (1)Y where there are no masses at all and the hypercharge is the espective conserved charge121. After this breakdown the symmetry is that of the group U (1)Q where the electric charge is conserved. In the “breaking” process the particles acquire mass so that the number of degrees of freedom remains unchanged, according to the Goldstone theorem [8]. This work demonstrates that electric charge of all particles can be predicted from the very beginning, when the EWSM Lagrangian is defined, and we show how to do it from different symmetry groups. We are going to assign electric charges in gauge theories such as the minimal left-right theory that considers the conservation of parity from the beginning and belongs to the symmetry SU (2)L⊗SU(2)R⊗U(1)Y; the 331 theories which explain why there are three fermionic families and belongs to SU(3)L ⊗ U(1)N. Moreover, if we require a gauge theory that includes the previous two, we must take into account the model SU(3)R⊗U(1)X [9-12] and this is also analyzed. All these symmetries are built respecting the conservation of electric charge after the breakdown and this fact is used to elaborate charge operators for the different multiplets. In order to do that, we describe how the SSB occurs also in the gauge transformation operators getting the electric charges and how they are obtained for almost all multiplets involved in each symmetry group studied here. In order to achieve that, we have to mention that all of the mathematical deductions are formally discussed using classical unitary transformations for gauge theories as is customary in literature. One of the motivations for this work is that in publications where SM and extensions are discussed, it is not usual to read a description of the criteria used to choose the particle content inside the multiplets, presenting them a priori with their respective electric charges. Here, we suggest that the very process of obtaining electric charge is of the utmost importance in the multiplets construction at high energies (when breaking is not carried out yet) and on the other hand, charge operators contextualize the symmetry breaking process within the principle of electric charge conservation. To achieve this, the gauge transformation operators must be set up for each multiplet of the theory depending on the type of multiplet we are considering. So, the relation among the conserved charges of the symmetries involved in the breaking the electric charges are obtained is an application of the Gell-Mann Nishijima [13] formula applied to the electroweak interaction whose mathematical structure is analyzed in the context of a gauge theory. However, this formula is useful for column vector representations but it is not very clear for other types of representations. This work is focused on the latter. Then, the aim of this thesis is to achieve the electric charge operator for exotic multiplets in different gauge symmetries and demonstrate that it is not necessary to build them up from the subsequent interactions after the SSB. Therefore, the charge operators are the main issues in this work and its concept has to be examined. Since gauge theories are associated with local symmetries and they are generalized from global ones, they posses a conserved physical charge. These charges appear to be generators of their respective symmetry groups. In this way, electric charges are generators of the global U(1)Q symmetry group included in the QED. Besides, weak isospin and hypercharge constitute three generators of the SU(2)L ⊗ U (1)Y group of the electroweak theory in the Standard Model, or the nlne generators of SU(3)L ⊗ U(1)N. These conserved quantities are eigenvalues of Hermitian operators that commute with the Hamiltonian. For example, electric charges are eigenvalues of Q operator which commutes with Hamiltonian[3] [Q, H], and its expected values q are constants of the theory. Applied to a field particle QѰ = qѰ´, where q is the expected value of Q and ω is its eigenstate. On the other hand, in the quantization process of a field theory satisfying the unitarity conditions, it has demonstrated, see ref. [14], that charge operator must commute with linear and angular momentum operators. Also, the electric charge of a particle and its respective antiparticle have opposite signs, so that Q gets the total charge, Q = q (NѰ - Nώ), where N gives the particle and/or antiparticle count as seen in appendix (C). The charge operator varies depending on the multiplet representation to which it is applied. In this work Q is calculated for all multiples involved in the Standard Model (Chapter 1) as well as in three of its extensions (Chapters 2, 3 and 4).