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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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Element

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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 -

~5)Ub)'

(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.