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(12-217) (12-218)
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The other arguments of rand r s , such as AI" 1], if, g, ... , have been omitted. We stress that r is the generating functional of Green functions of fluctuations around
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the un symmetric vacuum. This identity is important because we already know how to renormalize the symmetric theory. From Sec. 12-4-3,
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zV 2A, ZV 21], ... ; go, A4>o' J12 + bJ12)
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(12-219)
Consequently, the symmetric counterterms will also make the broken theory finite:
r R(q" c, v, A, ... , J12) = rreg (Z1'2 q"
Co, VO,
Zj/2 A, ... , J12
+ bJ12)
(12-220)
provided v and care renormalized so as to maintain the finiteness of c q, and the validity of (12-218): (12-221) The second point concerns the variation of J12. Power counting tells us that in the unbroken theory such a variation only requires a modification of the counterterm bJ12. The identity (12-217) shows that this modification is also sufficient to make the broken theory finite. In this case, we might think that a variation of J12 also requires a modification of the mass term Ml of the vector field, which would invalidate our proof. However, Eq. (12-217) says that this modification of M2 A comes only from the variation of v as a function of J12. In conclusion we may reach any point of the (m 2 , v) plane (Fig. 12-12) and renormalize the corresponding theory with the symmetric counterterms. Modifying only the J12 counterterm, we renormalize the spontaneously broken theory [c = 0, m 2 = 0, v given (point b on Fig. 12-12)]. See also Fig. 11-14 for plots of the (J12, m2) or (J12, v) planes. The corresponding normalization conditions of the various proper functions have not been explicited. They can be deduced from the identity (12-217) and from the normalization conditions of the symmetric functions. Instead of this intermediate renormalization, we may prefer more physical conditions, such as those defining the coupling constant as the value of the three-point function at some on-shell point, etc. Needless to say, these new normalization conditions must be in agreement with the identities derived from (12-217) and from those satisfied by rs.
Figure 12-12 Curves of constant
(m 2 , v) plane.
f1.2
and c in the
NONABELIAN GAUGE FIELDS
The method followed here is economical, since it appeals to the simpler symmetric theory in order to renormalize the spontaneously broken one. It is, however, possible to avoid any reference to the massless unbroken case. The problem then amounts to show that the Slavnov-Taylor identities satisfied by the spontaneously broken theory may be preserved by renormalization. The previous analysis has been carried out in the renormalizable gauge where the comparison between broken and unbroken theories is obvious. This gauge is not physically satisfactory since it exhibits unphysical features such as the massless modes of <Pr (m 2 ---> 0 when c ---> 0). However, the study of renormalization in the unitary gauge such as the one in Eq. (12-209) would be much more difficult, as the theory looks nonrenormalizable and contact with the symmetric case has been lost.
12-5-5 Gauge Independence and Unitarity of the S Matrix
We want to show that all unphysical states-fictitious Goldstone bosons, additional polarization states of the vector field, and Faddeev-Popov fields-do not actually contribute to S-matrix elements. All these unphysical fields have propagators with a pole at k 2 = O. It is tantamount to showing that this singularity does not contribute to intermediate states. A simple proof uses the gauge independence of the S matrix to introduce the 't Booft gauge (12-222) We assume that the group is simple and hence has a single coupling constant, that the scalar field cP belongs to a real representation, and that cp' is obtained after a translation cp = cp' + v, <cp') = O. This gauge has several merits and its invention by 't Booft was a major step in the theoretical developments. It breaks explicitly the global invariance. Consequently, the would-be Goldstone bosons acquire a mass matrix
- i(m 2 "')I1./3CP' I1.CP' /3 = - ;A (cp~ T aI1./3V/3)( cp' Y TaybVb)
(12-223)
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