Fermionic masses

It has been assumed so far that the leptons are massless. They can acquire masses in an analogous fashion to the masses of the bosons via coupling to the same Higgs field. Within the standard model the coupling proposed by Weinberg [14] is employed, which is of the Yukawa t3q)e 

This interaction is SU(2)l x U(1)y symmetric. Upon spontaneous symmetry breaking, the electrons acquire mass. If we use the vacuum value of the Higgs field, the Lagrangian density reduces to 

Qe is a free parameter of the model, which is chosen to reproduce the electron mass according to 

Since the electron mass is also due to spontaneous symmetry breaking, its mass is expected to vanish at sufficiently high energies, just as discussed previously for the bosons. 

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It is concluded within the toy model above that the B(3) field, or more likely a pseudofield, is consistent with an extended SU(2) x 51/(2) model of electroweak interactions. A more complete formalism of the 51/(2) x SU 2) theory with fermion masses will yield more general results. A direct measurement of B(3 should have a major impact on the future of unified field theory and superstring theories. The first such measurement was reported in Ref. 14, (see also Refs. 6 and 7). 

If there is no independent external gauge field, this defines an incremental fermion mass operator Smwc2 = gy T such that... 

All these mixing angles are free parameters of the Standard Model and so, together with the fermion masses, have to be determined using the experimental data. 

Prom (5.1.2), we can rewrite 9wwH = 2(v Gr) / M. This shows that the Higgs coupling to the W is proportional to the square of the boson mass as compared with the linear dependence on the fermion mass... 

The most common description of relativistic quantum mechanics for Fermion systems, such as molecules, is the Dirac equation. The Dirac equation is a one-electron equation. In formulating this equation, the terms that arise are intrinsic electron spin, mass defect, spin couplings, and the Darwin term. The Darwin term can be viewed as the effect of an electron making a high-frequency oscillation around its mean position. 

Consider N fermions in a box with fixed energy levels E, (including rest-mass which itself includes internal excitation energy). To each E, there correspond co, distinct (degenerate) states, made up of gi internal states and 4n Vp2dp/h3 kinetic degrees of freedom, where p is momentum and V is the volume. 

So, high gets a contribution from fermion modes with Dirac eigenvalues from the interval Mi to the Pauli-Villars mass M, and Detiow considers eigenvalues less than Mi. The product of these determinants is independent on the scale2 Mi. But, we may calculate both of them only approximately. In (D. Diakonov et.al., 1986) it was demonstrated a weak dependence of the product on M in the wide range of Mi which serves as a check of the approximations. 

Consider again non-relativistic fermions. Their BCS spectrum (for homogeneous systems) is isotropic when the polarizing field drives apart the Fermi surfaces of spin-up and down fermions the phase space overlap is lost, the pair correlations are suppressed, and eventually disappear at the Chandrasekhar-Clogston limit. The LOFF phase allows for a finite center-of-mass momentum of Cooper pairs Q and the quasiparticle spectrum is of the form... 

Obviously, the CD is related to the existence of unfilled negative energy (including rest mass energy) states, but arises only because fermions are... 

The weak interaction contribution to hyperfine splitting is due to Z-boson exchange between the electron and muon in Fig. 6.7. Due to the large mass of the Z-boson this exchange is effectively described by the local four-fermion interaction Hamiltonian... 

The two parts of the twisted bundle are copies of SU(2) with a doublet fermion structure. One of the fermions has a very large mass, m = Yn (y)1 ), which is assumed to be unstable and not observed at low energies. So one sector of the twisted bundle is left with the same Abelian structure, but with a singlet fermion, meaning that the SU(2) gauge theory becomes defined by the algebra over the basis elements... 

The positive muon was discovered in cloud chamber photographs made by C D. Anderson and S.H. Neddermyer on Pike s Peak in 1935, and the negative muon almost simultaneously in cloud chamber photographs made by J.C. Street and E.C. Stevenson. These particles have long been called mu-mesons, but since they are fermions (spin h while all other mesons arc bosons, the name muon is preferred, as is their classification with the leptons because of their small rest mass, which is about 206 mr, where me is the mass of the electron. Another reason is their inability to interact with other particles through the nuclear forces. 

Particle groups, like fermions, can also be divided into the leptons (such as the electron) and the hadrons (such as the neutron and proton). The hadrons can interact via the nuclear or strong interaction while the leptons do not. (Both particle types can, however, interact via other forces, such as the electromagnetic force.) Figure 1.4 contains artistic conceptions of the standard model, a theory that describes these fundamental particles and their interactions. Examples of bosons, leptons, hadrons, their charges, and masses are given in Table 1.6. 

The middle term is a Proca Lagrangian for a massive photon. Here the mass of this photon is assumed to be larger than the masses of the W and W° bosons. The current / 31( is determined by the charged fermions with masses given by the Yukawa interactions with the Higgs held. These are yet to be explored. Now consider the term in the Euler-Lagrange equation... 

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