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y5)y",u v and use (6.36). See Exercise 5.15.
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12.3 Use the Feynman rules of 6: u e describes an ingoing e- or an outgoing e+.
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Answers and Comments on the Exercises
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12.4 The I = 1 states of 14 0 and 14N* are Ipp) and 1f(lnp) + Ipn , respectively. Each proton can decay, so the 14 0 -+ 14N * amplitude contains the factor 2
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12.5 For a detailed discussion of how to obtain G from data on lifetimes of jJ-emitters, see, for example, Kallen (1964), Gasiorowicz (1967), or Commins (1973). 12.6 Using (12.15), we obtain M,t = Ii e 2 /8G == (37.3 GeV)2. In the standard model with sin2 Ow = i, we have M w = 2 X 37.3 GeV = 74.6 GeV. 12.7 Use the trace theorems of Section 6.4. The presence of y5 can be taken care of by using (6.23). An elegant derivation of (12.28) is given by Bjorken and Drell (1964), page 262. Equation (12.29) follows directly from (12.27) and (12.28). 12.8 f d 3 k dw O(w) c5(w 2 - Ik1 2) = f d 3 k/2w, with w 2 = Ik1 2, see Solution 6.7. 12.9 Use (12.29). 12.10 2k p' == (k 12.11
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because of two lepton versus three (ud) color decay modes. 12.13 The result is that tK == tw' 12.14 For a point interaction, s is the only dimensional variable. As a is proportional to G 2 , the product G 2 s is the only combination with the dimensions of a cross section. But a point interaction is an s-wave (l = 0) process for which a .s; constantjs, and so this simple approach must fail when s 10 5 GeV 2. 12.15 Use s == 2m e E v and (12.60).
Answers and Comments on the Exercises
12.17 Note the similarity to the nuclear f3-decay rate, Section 12.3. 12.18 To obtain the factor
+ Fr + F{n
is, use (9.32) and (12.75), and neglect strange quarks: H u + it) + Hd + J ) + H u + ii) + Hd + J ) 5
2( u
+ it + d + d )
12.21 Draw quark diagrams and identify the Cabibbo-favored and Cabibbosuppressed vertices. The amplitudes are in the ratio cos 2 0" : sin O"cos 0" : 2 sin 0". 12.22 Use the (Ilm)5 rule of Exercise 12.17. To determine r(Do), estimate the leptonic branching ratio as was done for the r-Iepton in Exercise 12.12.
CHAPTER 13 13.2
1 ~12 gi[..! '--!J. 4 3
2 + c 2 )T!J. v V A
2c V CA T!J. v] 2
where the first term in brackets is the average over the three helicity states of the X boson and the second brackets gives the sum over the fermion spin states performed as in (6.20), where T1 and T2 are the traces
Tr( y!J.lj.;yVIj.;') ,
We denote the X,fl.J2 four-momenta by q, k, k', respectively. From (6.93) and (6.23), we see that the CVCA term vanishes; the polarization sum is symmetric under J.L - P, and T2 is antisymmetric. In the X rest frame,
k = ~Mx(I;O,O,I),
- g!J.vT{"
and using (6.25), we find
Finally, from (4.37), we have
-~flI2) 647T12Mxfl~12dn 16~Mx ~;(c~+ d)4M;.
Note that if we were to consider the different polarization states, arately, then T2 does not vanish. We find the widths
r( ) ex:
+ d)(1 + cos 2 0) 2c V cA 2 cos 0 + d)2 sin2 0
where k = t M x(l; sin 0,0, cos 0). Check that these results embody angular momentum conservation for, say, W + -+ e + P.
Answers and Comments on the Exercises
13.3 Substitute C v = CA =~, M x = M z ' and gx = gicos O~j/ into (13.43). Calculate g from (12.15) with G = 1.166 X 10- 5 GeV 1 and M w = M z cos Ow. and hence show that f(Z -+ pji) = 159 MeV. 13.4 Use Table 13.2 to calculate d + d; for the qq modes, include a factor 3 for color. With three generations. f(Z) = 2.5 GeV (neglecting fermion masses). 13.5