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6.2. The Higgs Field

It was at this dramatic moment that the idea of the scalar field was born anew, albeit in a different role.

If a vector particle has a mass m, this means that the energy density of the vector field A contains a term m2A2. The field equation for a free field describing massive vector particles is

Equation 6.2.1 (6.2.1)

Compare (6.2.1) with the corresponding equation for the scalar field (Section 5) and check the role of the m2A term. The energy density of the vector field now contains a term m2A2/2 in addition to the m2phi2/2 term that appears in the energy density of the scalar field. The quantum theory of an equation like this with an m2A term in the right-hand side leads nowhere, in contrast to the successful QED (electrodynamics with massless photons) case.

The new idea now consists of considering an initially massless A interacting with the charged scalar field phi.

When we say that phi has a "charge", we don't mean electric charge, but the charge that describes the interaction with the vector field A (for example, that associated with the W± or Z0 vector fields, or any other vector field other than the electromagnetic field).

The energy density of the scalar field contains the term

Equation 6.2.2 (6.2.2)

The interaction with A means that each space or time derivative is transformed into a combination involving A:

Equation 6.2.3 (6.2.3)

The sum of the squares of the derivatives yields the characteristic term (7)

Equation 6.2.4 (6.2.4)

This is, in effect, a mass term for the A field, if the average <phi2> is not zero.

Figure 1

Figure 1.

How can this be arranged? The idea proposed by Higgs was to introduce the potential

Equation 6.2.5 (6.2.5)

in the initial equation for the energy density of the scalar field in place of the simple

Equation 6.2.6 (6.2.6)

(see Figs. 1 and 2); in the simple case Fig. 2, the lowest state is, obviously, symmetric: phi = 0. But the first, more complicated case is chosen precisely in order to have the minima Vmin(phim) = 0 occur at phim = ± phi0. The more complicated case is still symmetric V(phi) = V(-phi), but the center of symmetry at phi = 0 is now a maximum. At least at low temperatures, the system must be in one of the two states phi = +phi0 or phi = -phi0 (see Fig. 1 above).

Figure 2

Figure 2.

In both cases (+, -), the equation for the vector field (return to the term e2 phi2A2) is in effect identical to the equation for a massive vector field:

Equation 6.2.7 (6.2.7)

The equation for the vector potential has all of the required experimentally verified properties: short range of action and large mass of the particles associated with A (the W and Z fields being examples of A). On the other hand, when it comes to the higher-order corrections and renormalization, it turns out quite magically that all of our difficulties are over! When we had "manually" inserted the mass of A as a constant, this led to nonrenormalizability, i.e., an unacceptable theory.

The same mass does not spoil the theory as long as it is due to interaction with the scalar field phi; the extra virtual processes with phi cancel out the virtual processes with A (with <phi2> neq 0); the theory is renormalizable and acceptable to highbrow theoreticians. Putting m in directly is called the "hard method" of introducing the mass; while the e sqrt[2<phi20>] method is called the "soft method". The soft method is logically impeccable, but it necessarily involves introducing a new field - a scalar field with the peculiar energy term V(phi) - in Fig. 1, the so-called Higgs scalar field.

Put in a nutshell, the story is really funny: first the scalar field was invented; it was then pushed out by the vector fields; but it finally turns out that the massive vector fields themselves need scalar Higgs fields in order to survive.

7 To be exact, this term is needed in the Lagrangian, not in the energy density. The difference is important for real calculations; however, in that case, one must read other papers not this one. As far as one's understanding of the general idea goes, this is splitting hairs. Back.

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