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In Chap. 9 renormalization was performed at a value of the field equal to M to insure that (11-52) Returning to scalar electrodynamics, we can repeat a similar calculation using the symmetry factors shown in Fig. 11-6 for the contribution of cPt to V. The result
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~_2. 1--J-12_ ...
(b) (c)
Figure 11-6 Relative weights of the one-Ioo!,> diagrams for the Green function with four </>1 fields.
QUANTUM FIELD THEORY
reads
A 2 22 V = 4! ( 1 + 2) +
(5 A2 + 64n2 3e 1152n2
2 2 2[ ( i + ~) ( 1 + 2) In M2 - (5
(11-53)
The calculation was carried out in the Landau gauge. The last diagram, Fig. 11-6d, gives a vanishing contribution at zero momentum. A factor three in diagram (c) originates in the trace of (g"v - k"k v/k 2 ). The remaining weights arise from the vertices.
It is consistent to assume that Aand e4 are comparable. Indeed, the A coupling
is generated by the extra divergence occurring at order e4 . Therefore we may neglect A2 as compared to e4 . In this way the potential V of Eq. (11-53) has a minimum for a nonvanishing value ~ == ;;,1 + ~,2 given by
A [ - - 11 -
(e2)2J + 3 (e 2)2 In -M2 = 0 4n 4n ~
(11-54)
If we choose the scale of such that M2
;;, that is,
d V/ A-we find the relation
- d 4 4>2=4>~
(11-55)
(11-56) giving indeed a value for A of order e4 . The parameters of the theory are now e and v'
A similar reasoning would not apply to the pure </>4 theory of Eq. (11-50). It is true that the potential V has a minimum for a value </>v such that (11-57) that is,
.J.ln~=
</>;
32 2 - - n +O(.J.) 3
(11-58)
However, the quantity .J. In (</>;/M2) is now sizable and we expect higher quantum corrections to be nonnegligible. Stated differently, it is not legitimate in this case to set </>; = M2, since this would imply a large value of .J. for which the perturbative series cannot be trusted.
For scalar electrodynamics spontaneous symmetry breaking does not imply the appearance of any massless boson. On the contrary, both the vector and the scalar particle acquire masses given by
SYMMETRIES
= e 2 <fJ;;
(11-59)
with the striking consequence that
m.p = O(e 2 ) m~
(11-60)
The reason for such a behavior will be studied in more detail in the following chapter.
We conclude this section with a discussion of the role of the dimensionality of space-time. A theorem due to Mermin and Wagner states that a continuous symmetry can only be spontaneously broken in a dimension larger than two. For a discrete symmetry this lower critical dimensionality is one. This is, in fact, well known since in quantum mechanics with finitely many degrees of freedom (corresponding to one-dimensional field theory) tunneling between degenerate classical minima allows for a unique symmetric ground state. Alternatively, we may consider the discrete analog of a field theory, the simplest example being the Ising model in statistical mechanics. Path integrals are replaced by sums of terms of the form e- E / kT where E is the energy of a configuration. For the Ising model E = -J L(ill (Ji(Jj where the sum runs over neighboring sites on a lattice and the discrete "spin" (Ji takes values of 1. Such a model admits a discrete symmetry corresponding to reversing all the spins (Ji -+ -(Ji. This symmetry is spontaneously broken in the low-temperature phase below a critical point in dimension two or higher, but no transition occurs in dimension one (Peierls, 1938). A similar model caIIed the classical Heisenberg model replaces the variables (Ji by unit vectors Si on a sphere. In this case a continuous O(n) group operates if Si has n components, and no spontaneous magnetization occurs below dimension three. The Mermin-Wagner theorem has been restated by Coleman in the framework of field theory. We may establish this property by showing that the spontaneous breakdown of a continuous symmetry would lead to a Goldstone boson. But in a two-dimensional space-time it is not possible to construct a massless scalar field operator. Indeed, the corresponding two-point Wightman function
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