Higgs field

Regarding A as a connection , we consider [A A A] as the self-dual part of the curvature of A. Similarly, 4> can be regarded as a Higgs field , and the equation /i = 0 represents the Seiberg-Witten equation. 

The scalar field that is the object of their greatest desires and most ardent research is called the Higgs field. It arises as a logical necessity in their grand unification of the forces of nature. However, the cosmic scalar field mentioned above is of a very different nature, for it is infinitely lighter than the Higgs field. 

The extension of U(l) x SU(2) electroweak theory to SU(2) x SU(2) elec-troweak theory succeeds in describing the empirically measured masses of the weakly interacting vector bosons, and predicts a novel massive boson that was been detected in 1999 [92]. The SU(2) x SU(2) theory is developed initially with one Higgs field for both parts of the twisted bundle [93], and is further developed later in this section. 

The theory so far is incomplete, however, because it has two SU(2) algebras that both act on the same Fermi spinor fields, and only one Higgs mechanism is used to compute the vacuum expectations for both fields. To improve the theory, consider that each SU(2) acts on separate spinor field doublets and that there are two Higgs fields that compute separate physical vacua for each SU(2) sector independently. The Higgs fields will give 2x2 vacuum diagonal expectations. If two entries in each of these matrices are equal, the resulting massive fermions in each of the two spinor doublets are identical. If the spin in one doublet... 

The gauge potentials and are 2x2 Hermitian traceless matrices, and the Higgs fields and % are also 2x2 matrices. These expectations are real-valued, and the nonzero contributions of the Higgs field on the physical vacuum are given by the diagonal matrix entries [95] ... 

To calculate the photon masses, define the Higgs field by a small expansion around the vacuum expectations... 

The contraction of the generators a11 and cr with the Higgs field matrix and right and left fields gives... 

The second Higgs field acts in such a way that if the vacuum expectation value is zero, ( ) = 0, then the symmetry breaking mechanism effectively collapses to the Higgs mechanism of the standard SU(2) x U(l) electroweak theory. The result is a vector electromagnetic gauge theory 0(3)/> and a broken chiral SU(2) weak interaction theory. The mass of the vector boson sector is in the A(3) boson plus the W and Z° particles. 

Furthermore, the general method presented in this chapter applies directly to solving the full Maxwell equations with currents. It can also be used to construct exact classical solutions of Yang-Mills equations with Higgs fields and their generalizations. Generically, the method developed in this chapter can be efficiently applied to any conformally invariant wave equation, on the solution set of which a covariant representation of the conformal algebra in Eq. (15) is realized. 

Consider an extended standard model to determine what form the electromagnetic and weak interactions assume on the physical vacuum defined by the Higgs mechanism. Such a theory would then be 5(7(2) x 5(7(2). We will at first consider such a theory with one Higgs field. The covariant derivative will then be... 

Here G"v and F v are elements of the field strength tensors for the two SU(2) principal bundles. So far the theory is entirely parallel to the basic standard model of electroweak intereactions. In further work the Dirac and Yukawa Lagrangians that couple the Higgs field to the leptons and quarks will be included. It will then be pointed out how this will modify the B field. The < )4 field may be written according to a small displacement in the vacuum energy ... 

The Lie algebraic indices are implied. The Higgs field is described by the harmonic oscillator equation where the field has the mass Mh — 1.0 TeV/c2. On the physical vacuum the gauge fields are... 

To take this theory further would be to embed it into an 517(4) gauge theory. The gauge potentials are described by 4 x 4 traceless Hermitian matrices and the Dirac spinor has 16 components. The neutrality of the photon is then given by the sum over charges, which vanishes by the tracelessness of the theory. The Higgs field is described by a 4 x 4 matrix of entries. 

If the coupling constant Y is comparable to the coupling constant g, then the Fermi expectation energies of the Fermions occur at the mean value for the Higgs field (4>0)- In this case the vacuum expectation of the vacuum is proportional to the identity matrix. This means that the masses acquired by the right chiral plus left chiral gauge bosons A + /iM are zero, while the left chiral minus right chiral... 

We also need to examine the photon masses. We define the Higgs field by a small expansion around the vacuum expectations q1 = E,1 + (qj) an(l q2 = t2 + (qo)-The contraction of the generators cr1 and a2 with the Higgs field matrix and right and left fields gives... 

However, if the mass of these bosons is given according to Higgs fields that determine mass spectra-spectra for gauge bosons, similar to what occurs with the Fermi masses given in Eq. (45), this estimate may be adjusted upward. 

Within the Higgs mechanism two complex scalar boson fields 4> x) and (t x) are introduced. These two fields form a SU(2)i, doublet, which is called the Higgs field (x). For this field a Lagrange density of the type... 

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