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-~ + O(~)
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(12-231)
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The symmetry SU(2) x U(l)yis broken but the symmetry under U(l)Q is preserved. This achieves the desired result, since one vector field coupled to the electric charge remains massless. The lagrangian reads
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2= -iAl'vAI'V - iBl'vBI'V
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+ [Re(i - g'$)R e + Le(i - ;' $ + g ;i A)Le
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- GiLeRe + tReLe) + e~f1 ]
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+ (Ol' - i;' BI' -
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TiAl'i
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TiAl'i ) - V( t )
(12-232)
where Al'v and Bl'v stand for the field strength tensors and Ti (i = 1,2, 3) are the Pauli matrices. The S U(2) symmetry prevents us from writing mass terms of the electron and muon, but it does not forbid the introduction of the coupling to the scalar field with coupling constants Ge and Gil" In order to understand the physical content of this model, let us use a unitary gauge. We use the parametrization
4>(x) ~ e','<'""{ :fiX))
(12-233)
QUANTUM FIELD THEORY
and perform the SU(2) gauge transformation
(X),~ ~ j(X)) ,
'(x) ( v
Equation (12-232) is expressed as
~A 2
(x) --+ ~ A' (x) 2 1"
e-i~,T'/2v (~o I' + ~ A .) ei~,T'/2v g 2
(12-234)
r: = e- i ,r'/2v L
B, R invariant
-!Al'vAI'V - !Bl'vW
[eR(i~ -
g'$)eR + Le(i - ;' fJ + g
~i A)Le
+ eReL - Ge vJ2p (-
-) 1;1 ;11' + eLeR + e~f1 + -ZUI'PU P
+ i(v + p [(g' BI' - gA!)2 + g2(A~All' + A~A21')J - v [(V +/)2J
(12-235)
The scalar (Higgs) field p has a mass 2f12. The electron and the muon have acquired masses equal to me = Gev/J2 and ml' = Gl'v/J2. The charged vector field + 1 (Al - A2) (12-236) WI'- = J2 I' + I I' is also massive, with
vg Mw=2
(12-237)
Finally, the quadratic form in A 3 and B is diagonalized by
ZI' = (g2
+ g,2)-1/2( -gA~ + g'BI')
AI' = (g2
so that
+ g,2) -1/2(gBI' + g' A~)
v - (g2 + g'2)1/2
(12-238)
(12-239)
MA=O
The leptonic interaction terms of ff' may then be rewritten in terms of the physical fields W , Z, and A:
(12-240)
NONABELIAN GAUGE FIELDS 623
In Eq. (12-240) we have introduced the Weinberg angle Ow, defined in such a way that tan Ow=g
(12-241)
e = g' cos Ow
= g sin Ow
The last term in (12-240) is the usual electromagnetic coupling to the field AI'" The first one has the form (12-176). Its coupling is related to Fermi's constant through
8Mfi,
2v 2
(12-242)
From the knowledge of G, we deduce lower bounds on the masses of W and Z:
Mw= - = -.~-GeV 2 sm Ow
M z = ,,-----=-----=-2 cos Ow
38 GeV
(12-243)
76 . 20 GeV ~ 76 GeV sm w
Mw<Mz
The coupling constants Ge and GI' are determined from the electron and muon masses:
(12-244)
GI'=mI'Ge~4xlO-4 me
This model involves a new type of coupling of Z I' to a neutral, parity violating weak current constructed from eL, eR, and Ve (and j1L,R, vl')' This is a feature of most renormalizable models of weak interactions. They introduce either a neutral current and a vector field coupled to it, or new leptons assumed to be heavy to comply with experimental facts, or both. In the former case, the explicit form of the current depends on the model, viz on the choice of representations for the various fields, etc. The Weinberg-Salam model incorporates in a natural way the electron-muon universality. Only Ge and GI' are sensitive to the nature of the lepton. On the other hand, the model does not provide any natural explanation for the electric charge quantization.
This is not the case for other models based on simple groups such as the Georgi-Glashow model (Sec. 12-5-3). As a model for weak and electromagnetic interactions, it is now ruled out by experiment since it does not incorporate neutral currents. Among the three components of the gauge field, two become massive (the analogs of W/ in the Salam-Weinberg model) while the last one remains massless (the photon). The benefit of dealing with a simple group is that the electric charge is quantized.
QUANTUM FIELD THEORY
12-6-2 Electron-Neutrino Cross Sections
To expose some striking consequences of the existence of neutral leptonic currents, we will now compute the elastic e - v cross sections to lowest order in the Weinberg-Salam model. The relevant diagrams are depicted in Fig. 12-13. In the limit where the incident neutrino energy is small as compared to the masses of the Wand Z bosons, we may content ourselves with the effective lagrangian
steff,int =
-}2 Gmvey~(1 -
ys)e][ey.(1 - Ys)v e]
+ (v.Y v. + vey v e)(2 sin 2 iJweRy.eR -
cos 2iJ weLy.ed}
(12-245)
where the first and the second terms represent the Wand Z contributions respectively. After a Fierz transformation on the first term
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