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12.13 CP Invariance
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TABLE 12.1 A Summary of the Determination of the Elements Uqq of the Kobayashi-Maskawa Matrix
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f3-Decay, generalize (12.107) K -> 7Tep and semileptonic hyperon decays, generalize discussion following (12.101) b -> ue-Pe , look for B meson decays with no K's in final state: gives I VII" 12 < 0.0210,,, 12
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-> p.-c and D+-> j( e+pc , see (12.108); also constrained by the unitarity of V, which implies I V'd1 2 + I V,,1 2 + 10,,,1 2 = I, together with the information on Io,dl and 10,-,,1. The (long) lifetime of the B meson, Tn - 1O-- 12 secs, and I VII"I. From unitarity bounds
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CP Invariance
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To investigate CP invariance, we first compare the amplitude for a weak process, say, the quark scattering process ab -+ cd, with that for the antiparticle reaction ilb -+ cd. We take ab -+ cd to be the charged current interaction of Fig. 12.20a. The amplitude
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- (u cyP.{l- y5)Ucaua)(ubYp.{1- y 5)Ubd Ud )t - UcPdt(u cyP.{l- y5)u a )(udYp.{1 -
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(12.121)
since Ubtd = udt. ~ describes either ab -+ cd or cd -+ ilb (remembering the antiparticle description of 3). On the other hand, the amplitude ~' for the antiparticle process ilb -+ cd (or cd -+ ab) is
~' - (J}';,)tJP.bd
- U:~Udb(uayP.{l - y 5)uJ(ubYp.{1 - y5)U d );
that is,
~/=~t.
(12.122)
Weak Interactions
Fig. 12.20 The processes described by (a) the weak amplitude ~(ab -> cd) and (b) its hermitian conjugate.
This should not be surprising. It is demanded by the hermicity of the Hamiltonian. By glancing back at (4.6) and (4.17), we see that ~ is essentially the interaction Hamiltonian V for the process. The total interaction Hamiltonian must contain ~ + ~t, where ~ describes the i -+ f transition and ~t describes the f -+ i transition in the notation of 4. In Section 12.1, we have seen that weak interactions violate both P invariance and C invariance, but have indicated that invariance under the combined CP operation may hold. How do we verify that the theory is CP invariant We calculate from ~(ab -+ cd) of (12.121) the amplitude ~cP' describing the CP-transformed process, and see whether or not the Hamiltonian remains hermitian. If it does, that is, if
~cP= ~t,
then the theory is CP invariant. If it does not, then CP is violated. ~cP is obtained by substituting the CP-transformed Dirac spinors in (12.121):
p( ui)c'
= a, ... ,d
(12.123)
where U c are the charge-conjugate spinors of Section 5.4,
(12.124)
Clearly, to form ~cP' we need Uc and, also, to know how yl'(l - y5) transforms under C. In the standard representation of the y-matrices, we have [see (5.39)]
C-1yI'C
C- 1 y l'y 5C
_(yl')T,
+ (yl'y5)T.
(12.125)
EXERCISE 12.24 Verify (12.125) using (5.39).
12.14 CP Violation: The Neutral Kaon System
With the replacements (12.123), the first charged current of (12.121) becomes
(JtJc
Uca (i7Jc yl'(l - y5)( uJc
Ucau~C-lyl'(1- y5)CU~
= Ucaun yl'(1
+ y5)] T u~ ( - )Ucaua yl'(l + y5)u c.
(12.126)
The above procedure is exactly analogous to that used to obtain the charge-conjugate electromagnetic current, (5.40). The parity operation P = yO, see (5.62), and so
p-lyl'(1
+ y5)p
yl't(l _ y5),
see (5.9)-(5.11). Thus,
(Jt;,)cp
= (-
)Ucaua yl't(l - y5)u c'
and hence
~cp - UcPdt [u a yl'(1 - y5) ucl [ UbYI' (1 - y5) u d ] .
(12.127)
We can now compare ~cp with ~t of (12.122). Provided the elements of the matrix U are real, we find
~cp=~t,
and the theory is CP invariant. At the four-quark (u, d, c, s) level, this is the case, as the 2 X 2 matrix U, (12.106), is indeed real. However, with the advent of the b (and t) quarks, the matrix U becomes the 3 X 3 Kobayashi-Maskawa (KM) matrix. It now contains a complex phase factor e ilJ Then, in general, we have
~cp*~t,
and the theory necessarily violates CP invariance. In fact, a tiny CP violation had been established many years before the introduction of the KM matrix. The violation was discovered by observing the decays of neutral kaons. These particles offer a unique "window" through which to look for small CP violating effects. We discuss this next.