2 Gauge Theories and the Standard Model
27
Fig. 2.2 A Schrödinger
potential V (x) analogous to
the Higgs potential
+
x
V x
( )
d
h
with different energies (the difference being proportional to δ). Suppose now that
you have a sum of n equal terms in the potential, V =
i V (x i ). Then the
transition amplitude would be proportional to δ n and would vanish for infinite n: the
probability that all degrees of freedom together jump over the barrier vanishes. In
this example there is a discrete number of minimum points. The case of a continuum
of minima is obtained, always in the Schrödinger context, if we take
V = 1/2 μ
2 r
2
+ 1/4 λ(r
2 )
2 ,
(2.55)
with r = (x, y, z). Also in this case the ground state is unique: it is given by
a state with total orbital angular momentum zero, an s-wave state, whose wave
function only depends on |r|, independent of all angles. This is a superposition of
all directions with the same weight, analogous to what happened in the discrete
case. But again, if we replace a single vector r, with a vector field M(x), that is a
different vector at each point in space, the amplitude to go from a minimum state in
one direction to another in a different direction goes to zero in the limit of infinite
volume. In simple words, the vectors at all points in space have a vanishing small
amplitude to make a common rotation, all together at the same time. In the infinite
volume limit all vacua along each direction have the same energy and spontaneous
symmetry breaking can occur.
The massless Goldstone bosons correspond to a long range force. Unless the
massless particles are confined, as for the gluons in QCD, these long range forces
would be easily detectable. Thus, in the construction of the EW theory we cannot
accept massless physical scalar bosons. Fortunately, when spontaneous symmetry
breaking takes place in a gauge theory, the massless Goldstone modes exist, but they
are unphysical and disappear from the spectrum. Each of them becomes, in fact, the
third helicity state of a gauge boson that takes mass. This is the Higgs mechanism
(it should be called Englert-Brout-Higgs mechanism [18], because an equal merit
should be credited to the simultaneous paper by Englert and Brout). Consider, for
example, the simplest Higgs model described by the lagrangian
L = −
1
4
F
2
μν + |(∂ μ + ieA μ Q)φ|
2
+ μ
2 φ
∗ φ −
λ
2
(φ
∗ φ)
2 .
(2.56)
27
Fig. 2.2 A Schrödinger
potential V (x) analogous to
the Higgs potential
+
x
V x
( )
d
h
with different energies (the difference being proportional to δ). Suppose now that
you have a sum of n equal terms in the potential, V =
i V (x i ). Then the
transition amplitude would be proportional to δ n and would vanish for infinite n: the
probability that all degrees of freedom together jump over the barrier vanishes. In
this example there is a discrete number of minimum points. The case of a continuum
of minima is obtained, always in the Schrödinger context, if we take
V = 1/2 μ
2 r
2
+ 1/4 λ(r
2 )
2 ,
(2.55)
with r = (x, y, z). Also in this case the ground state is unique: it is given by
a state with total orbital angular momentum zero, an s-wave state, whose wave
function only depends on |r|, independent of all angles. This is a superposition of
all directions with the same weight, analogous to what happened in the discrete
case. But again, if we replace a single vector r, with a vector field M(x), that is a
different vector at each point in space, the amplitude to go from a minimum state in
one direction to another in a different direction goes to zero in the limit of infinite
volume. In simple words, the vectors at all points in space have a vanishing small
amplitude to make a common rotation, all together at the same time. In the infinite
volume limit all vacua along each direction have the same energy and spontaneous
symmetry breaking can occur.
The massless Goldstone bosons correspond to a long range force. Unless the
massless particles are confined, as for the gluons in QCD, these long range forces
would be easily detectable. Thus, in the construction of the EW theory we cannot
accept massless physical scalar bosons. Fortunately, when spontaneous symmetry
breaking takes place in a gauge theory, the massless Goldstone modes exist, but they
are unphysical and disappear from the spectrum. Each of them becomes, in fact, the
third helicity state of a gauge boson that takes mass. This is the Higgs mechanism
(it should be called Englert-Brout-Higgs mechanism [18], because an equal merit
should be credited to the simultaneous paper by Englert and Brout). Consider, for
example, the simplest Higgs model described by the lagrangian
L = −
1
4
F
2
μν + |(∂ μ + ieA μ Q)φ|
2
+ μ
2 φ
∗ φ −
λ
2
(φ
∗ φ)
2 .
(2.56)