The Higgs mechanism

We now consider the earlier model (3.1.1) for a charged scalar field but impose invariance under local U 1) gauge transformations (Higgs, 1964 a, b, 1966). 

According to our previous discussion (Section 2.3.2) we must replace the derivative d, by the covariant derivative = d + ieA and add the kinetic term so that C of (3.1.1) becomes 

We can once again look for a minimum in the potential, and we find one if A 0. If 0 there is again a ring of degenerate ground states, whereas the symmetric ground state = 0 obtains if 0. 

The term involving A A is a great surprise since in a quantmn picture it looks as if the gauge vector field A has acquired a mass. Gauge invariance is, of course, still there since (3.2.6) must be equivalent to (3.2.1). However, the gauge transformations look a little more complicated in terms 

Since the theory does not change with any choice of the transformation function 9 x) in (3.2.2) let us choose 0 x) at each space-time point to equal the phase of (/ (x). Then in this gauge 

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Superconductivity provides an illustration of the Higgs mechanism. It is the property of materials that show no electrical resistance, usually at low temperatures. Such materials are capable to carry persistent currents. These currents effectively screen out magnetic flux, which is therefore zero in a superconductor (the Meisner effect). Another way of describing the Meisner effect is to say that the photons are effectively massive, as in the Higgs phenomenon. These conclusions can be shown to follow from the Lagrangian (46). In this instance it is sufficient to consider a static situation, i.e. d4 = 0, etc, leading to the Lagrangian... 

Recall that the Higgs mechanism depends on local gauge symmetry, which is only defined within general relativity theory. 

The physical vacuum is assumed to be defined by the Higgs mechanism, and the SU(2) x SU(2) covariant derivative is... 

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. 

From the foregoing, it becomes clear that fields and potentials are freely intermingled in the symmetry-broken Lagrangians of the Higgs mechanism. To close this section, we address the question of whether potentials are physical (Faraday and Maxwell) or mathematical (Heaviside) using the non-Abelian Stokes theorem for any gauge symmetry ... 

It is seen that the acquisition of mass by the photon is the result of an equation of superconductivity, and this is, of course, the basis of spontaneous symmetry breaking and the Higgs mechanism (Section XIV). Beltrami equations account for all these phenomena, and are foundational in nature. Note that the London equation (919) is not gauge-invariant on the U(l) level because aphysical gauge-invariant current is proportional to the vector potential, which, in the received view, is gauge-noninvariant. This is another flaw of U(l) electrodynamics in the... 

By Ohm s law, the resistance of the conducting medium vanishes, and the medium becomes a superconductor. The Higgs mechanism and spontaneous symmetry breaking were derived using the properties of superconductors. 

It can be seen that the photon mass is carried by, 4Vl 1 and Av(2 but not by 4Vl 1 . This result is also obtained by a different route using the Higgs mechanism in Ref. 42, and is also consistent with the fact that the mass associated with 4Vl 3 corresponds with the superheavy boson inferred by Crowell [42], reviewed in... 

In contemporary thought, the Higgs mechanism has acted in such a way as to produce a field component Bi with mass, specifically, a scalar field with mass that is gauge-invariant. Therefore, spontaneous symmetry breaking of the vacuum introduces fields with effective mass. 

The above is a pure gauge field theory. The Higgs mechanism on the U(l) level provides further sources of vacuum energy as discussed already. On the 0(3) level, the Higgs mechanism can also be applied, resulting in yet more sources of energy. 

It has been demonstrated already that local gauge transformation on this Lagrangian leads to Eq. (153), which contains new charge current density terms due to the Higgs mechanism. For our present purposes, however, it is clearer to use the locally invariant Lagrangian obtained from Eq. (325), specifically... 

The effect of the Higgs mechanism can be seen most clearly by minimizing the Lagrangian (251) with respect to A ... 

The starting Lagrangian on the U(l) level for a free particle, such as an electron, is the standard Lagrangian for the Higgs mechanism ... 

The Higgs mechanism has produced an additional rest energy ... 

The left-hand side is the nonrelativistic kinetic energy of one particle. It can be seen that the Higgs mechanism changes the classical nonrelativistic expression... 

The Schrodinger equation in the presence of the Higgs mechanism is therefore... 

The starting point of our derivation is the globally invariant 0(3) Lagrangian of the Higgs mechanism... 

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... 

The problem of the particle masses is resolved with the Higgs mechanism [77,78]. The principle idea of the Higgs mechanism is that the vacuum state does not reflect the full symmetry of the underlying Lagrangian. This concept is quite similar to spontaneous symmetry breaking for chiral molecules. 

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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