CP violation in kaon decays

1
Introduction.
Since 1949, when K mesons were discovered1 , kaon physics has represented one of the richest sources of information in the study of fundamental interactions. One of the first ideas, originated by the study of K meson production and decays, was the Gell–Mann2 and Pais3 hypothesis of the ‘strangeness’ as a new quantum number. Almost at the same time, the famous ‘θ − τ puzzle’4 was determinant in suggesting to Lee and Yang5 the revolutionary hypothesis of parity violation in weak interactions. Lately, in the sixties, K mesons played an important role in clarifying global symmetries of strong interactions, well before than QCD was proposed6 − 8 . In the mean time they had a relevant role also in the formulation of the Cabibbo theory9 , which unified weak interactions of strange and non–strange particles. Finally, around 1970, the suppression of flavor changing neutral currents in kaon decays was one of the main reason which pushed Glashow, Iliopoulos and Maiani10 to postulate the existence of the ‘charm’. Hypothesis which was lately confirmed opening the way to the unification of quark and lepton electro–weak interactions. In 1964 a completely unexpected revolution was determined by the Christenson, Cronin, Fitch and Turlay observation of KL → 2π decay11 , i.e. by the discovery of a very weak interaction non invariant under CP . Even if more than thirty years have passed by this famous experiment, the phenomenon of CP violation is still not completely clear and
2)
INFN, Laboratori Nazionali di Frascati P.O. Box 13, I–00044 Frascati, Italy
Abstract We review the Standard Model predictions of CP violation in kaon decays. We present an elementary introduction to Chiral Perturbation Theory, four–quark effective hamiltonians and the relation among them. Particular attention is devoted ¯f decays. to K → 3π , K → 2πγ and K → π f
5 K → 3π decays. 5.1 Amplitude decomposition. . . . . . . . . . . . 5.2 Strong re–scattering. . . . . . . . . . . . . . . 5.3 CP –violating observables. . . . . . . . . . . . 5.4 Estimates of CP violation. . . . . . . . . . . . 5.4.1 Charge asymmetries. s ǫ′+−0 and ǫX +−0 . . . . . 5.5 Interference measurements for η3π parameters. 6 K → ππγ decays. 6.1 Amplitude decomposition. . . . . . . . 6.1.1 CP –violating observables. . . . 6.1.2 K → ππγ amplitudes in CHPT. 6.2 Estimates of CP violation. . . . . . . . 1 . . . . . . . . . . . . . . . .
4 Chiral Perturbation Theory. 4.1 Non–linear realization of G. . . . . . . . . 4.2 Lowest–order lagrangians. . . . . . . . . . 4.2.1 The strong lagrangian. . . . . . . . 4.2.2 The non–leptonic weak lagrangian. 4.3 Generating functional at order p4 . . . . . . 4.3.1 O (p4 ) Strong counterterms. . . . . 4.3.2 The WZW functional. . . . . . . . 4.3.3 O (p4 ) Weak counterterms. . . . . . 4.4 Models for counterterms. . . . . . . . . . . 4.4.1 The factorization hypothesis of LW .
7 Decays with two photons in the final state. 65 7.1 K → γγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.2 KL → π 0 γγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.3 Charge asymmetry in K ± → π ± γγ . . . . . . . . . . . . . . . . . . . . . . . 70 8 Decays with two leptons in the final state. ¯. . . . . . . . . . . . . . . . . . . . . . 8.1 K → πf f ¯. . . . 8.1.1 Direct CP violation in KL → π 0 f f 0 ¯ 8.1.2 Indirect CP violation in KL → π f f . . . 8.1.3 CP –invariant contribution of KL → π 0 γγ 8.1.4 K ± → π ± l+ l− . . . . . . . . . . . . . . . 8.1.5 K + → π + ν ν ¯. . . . . . . . . . . . . . . . 8.2 K → l+ l− . . . . . . . . . . . . . . . . . . . . . 8.3 K → πlν. . . . . . . . . . . . . . . . . . . . . . 8.4 K → ππl+ l− . . . . . . . . . . . . . . . . . . . . 9 Conclusions. A Loop functions. References . . . . . . . . . . . . to KL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . → π 0 e+ e− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 73 74 75 77 78 79 79 82 82 83 85 86
To appear in International Journal of Modern Physics A
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Work supported in part by HCM, EEC–Contract No. CHRX–CT920026 (EURODAΦNE)
Contents
1 Introduction. 2 Phenomenology of CP violation in kaon decays. ¯ 0 system. . . . . . . 2.1 Time evolution of the K 0 − K 2.2 K → 2π decays. . . . . . . . . . . . . . . . . . . . 2.3 Semileptonic decays. . . . . . . . . . . . . . . . . 2.4 Charged–kaon decays. . . . . . . . . . . . . . . . 3 CP 3.1 3.2 3.3 violation in the Standard Model. The CKM matrix. . . . . . . . . . . . Four–quark hamiltonians. . . . . . . K → 2π parameters ǫ and ǫ′ . . . . . . |∆S |=2 3.3.1 Hef f and the estimate of ǫ. 3.3.2 The estimate of ǫ′ . . . . . . . 3.4 B decays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4 4 8 10 12 13 15 17 22 22 23 26 30 31 32 34 35 38 40 41 42 44 45 47 47 49 50 51 52 55 55 57 57 59 61 63