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whereas the W boson contributes
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zg u(pe)yP(l - Ys)v(Pv)
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gP" - kPk"/M'iv k2 _ M'iv u(Pv)y,,(1- ys)u(Pll)
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QUANTUM FIELD THEORY
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Figure 12-9 Muon decay (a) in the Fermi theory and (b) as mediated by an intermediate vector boson.
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The two amplitudes coincide at energies such that k 2 == (Pe
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+ Pv/
MW, provided
(12-180)
Similarly the Born approximation to a scattering amplitude such as vii --+ vii would be reduced at high energy by a factor MW/s with respect to the Fermi amplitude. The violations of unitarity have seemingly been eliminated. Actually, the amplitude vii --+ W + W - still has a bad behavior. This points to the necessity of introducing more fields and more couplings. The dynamics of these massive vector fields have now to be prescribed. Particularly troublesome is the question of renormalizability in view of the large momentum behavior of the propagator:
MW)-l constant Mw k Mr,. However, we remember that in quantum electrodynamics the introduction of a photon mass has not spoiled renormalizability. If we adhere to the prejudice that more symmetry in a theory reduces the number of divergences, it is suggested to consider the WI' as a member of a set of gauge fields. A candidate for a symmetry group is SU(2) and the local in variance is of course explicitly broken by the mass terms of the W. This would lead to a universal coupling of the W. Since SU(2) gauge fields must form a triplet, a third neutral vector field has to be introduced. Such a model, suitably amended, will be shown to be renormalizable.
( gl'v -
kl'~V)(k2 -
For completeness, we mention a different historical reason for introducing massive vector fields. It was suggested in the early 1960s to base the theory of strong interactions on a gauge principle. The invariance group was SU(3) x U(I), corresponding to the eightfold symmetry and baryonic charge conservation. The vector bosons-massive gauge fields-were identified with the existing p, K*, OJ, and <p particles. An interesting property of such a model is that the forces were attractive between particles of antiparallel isospins and repulsive for parallel isospins, a generalization of the electromagnetic attraction between opposite charges. Such a property is in agreement with experimental facts at low energy. To see this, let us compute the elastic scattering amplitude of two scalar particles belonging to the real representations (1) and (2) of a simple Lie group. The lowest-order contribution (the exchange
NONABELIAN GAUGE FIELDS
XiX J 'Y
Figure 12-10 A scattering amplitude to lowest order.
of the vector field as shown in Fig. 12-10) comes from the following terms of the lagrangian:
(12-181)
where and reads
T(l)a antisymmetric a = 1, ... , r
(12-182)
The quantity
X. PYO
TM)aTJ~)a
must be projected on irreducible representations. To this end, we introduce the Clebsch-Gordan matrices for the product of the representations (1) and (2). If nl and n2 are the dimensions of the representations, the nl x n2 matrices M(t)A satisfy the orthogonality and completion relations tr
(M(I)AM(I')~ = (j1t'(jAB
L M~ftAM~!: =
(j ,(jppo
(12-183)
and transform according to the (t) representation
(12-184)
It is then easy to see that
X. Pyo =
L Td:)aTwa
a,A,t
M~~AT;;}aM~)p~Tft~t
(12-185)
in terms of the Casimir operators for the representations (1) and (2) and' (t) [see (12-119)]. The value of the Casimir operator increases with the dimension of the representation. For instance, in the case of SU(2), the representation of isospin I has C(I) = I(I + 1). The desired property follows from the positivity of(s - U)/(M2 - t) in the physical region. This calculation of a crossing matrix, relative here to internal degrees of freedom, is analogous to the Fierz reshuffling of Chap. 3. This model, considered by Sakurai, also predicts the widths of vector bosons. Neglect the 4> - w mixing (which improves the computation) and couple the (n, K, 1']) and (p, K*, 4 octets in a
QUANTUM FIELD THEORY
gauge-invariant way. The p-wave amplitude of the elastic nn, nK, and KK channels in the Born approximation reads
where q is the center of mass three-momentum, and a = ~, 1, and 2 for the three channels respectively. A unitary amplitude tl may be constructed in the form
: 71 1 - ip: 71
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