The ΔI = 12 rule and violation of CP in the six-quark model

Volume 83B, number 1

PHYSICS LETTERS

23 April 1979

THE A I = 1/2 RULE A N D VIOLATION OF CP IN THE SIX-QUARK MODEL ¢r

Frederick J. GILMAN and Mark B. WISE 1 Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94305, USA Received 2 January 1979 Revised manuscript received 3 February 1979

The consequences for K° decay of the six-quark model with its natural poss~ility of incorporating a CP violating phase, 8, are investigated when a particular mechanism to give the AI = 1/2 rule is operative. The most important result is that the CP violation parameter e' is much larger than in previous analyses, and hence the theory has predictions for K° ~ rr~rdecay which are experimentally distinguishable from those of the superweak model.

In models of the weak interactions employing the gauge group [1 ] SU(2) × U(1) and with three or more lefthanded doublets, CP violation arises automatically: in addition to real Cabibbo-like mixing angles, there are one or more complex phases which cannot be swept into the fermion fields and which violate CP. Such is the case in a six-quark model, proposed for precisely this reason by Kobayashi and Maskawa [2], where there are three real angles, 0 i with i -- 1,2, 3 and one CP violating phase, 6. In this model the left-handed doublets are

where, with a standard choice o f the quark fields [2],

s'

SlC 2

ClC2C 3 - s2s3ei8

ClC2S 3 + s2c3ei6

b

\SlS 2

ClS2C 3 + c2s3ei6

ClS2S3 - c2c3ei8 /

,

(1)

and c i = cos Oi, s i = sin 0 i. Such a model has become popular with the increasing evidence for a fifth lepton [3], r, and more recently for a fifth quark, b, as a constituent for the upsilon family [4] of particles. A sixth quark, t, is expected, being necessary for the (generalized) GIM mechanism [5] as well as the cancellation of anomalies [61. The six-quark model with 6 4= 0 has been systematically studied by Ellis et al. [7] as the origin of CP violating effects in K 0 decay and elsewhere They found that the contribution to the K 0 - ~ 0 mass matrix corresponding to fig. 1, with the mixing expressed in eq. (1), leads to CP violation in K 0 decay in accord with experiment and in close approximation to predictions of the superweak model [8] for conrmon K decays. There is another feature of strange-particle decays, and in particular of K decays, which has detectable consequences for the pattern of CP violation, but whose effects so far have not been taken adequately into account. This is the A I = 1/2 rule. It had been hoped that strong interaction effects at short distances would sufficiently enhance the A I = 1/2 portion o f the non-leptonic weak interaction to explain the A I = 1/2 rule. However, detailed calculaWork supported in part by the Department of Energy under contract number EY-76-C-03-0515. I Supported in part by the National Research Council of Canada and Imperial Oil Ltd. 83

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W

u,c,t

W

,e,

u~c~t

<

d

s

i~

W

u,d

Fig. 1. Diagram contributing to the K°-K ° mass matrix.

>

~

~

u,d

Fig. 2. "Penguin diagram" contributing to weak decays.

tions using quantum chromodynamics (QCD) do not lead to a large enough enhancement [9]. It is claimed [10] that the answer lies in the amplitudes shown in fig. 2, sometimes called "Penguin diagrams". Since the gluon (g) carries no isospin, these amplitudes are pure A I = I/2. Although they appear at first glance to give a smaller contribution to the effective weak hamiltonian than the lowest-order current-current term, when their matrix elements in strange-particle decays are evaluated using light current-quark masses, very large contributions to decay amplitudes result. An extensive analysis [ 1 0 - 1 2 ] of both strange-meson and baryon decays supports the hypothesis that the magnitude of the amplitudes and the A1 = 1/2 rule are understandable on this basis. However, when the full six-quark model weak current with the mixing of eq. (1) is used to calculate the "Penguin diagrams" in fig. 2, they too will give a CP violating amplitude. In this paper we evaluate this contribution to K 0 decay and show it is of the same order as that arising from the standard analysis of fig. 1 for the CP violation parameter e. More importantly, the parameter e' is predicted to be much larger than previously in the six-quark model - sufficiently large so as to permit distinguishing the predictions from those of the superweak model. To be quantitative we first recall the contribution to the mass matrix given by fig. 1. Using eq. (1) with s 1 and s 3 treated as small quantities , i , direct calculation gives [7] ImM12/Re M12 = 2 s2c2s3 sin 6 P(02, r?) = e m ,

(2 a)

with s2(l + [7/(1 - ~)lln 7) - c~(~ + [r~/(1 - r/)lln r/) e ( 0 2 , 7) =

c4.+ s4-

t,/ l - , ) ] In,

,

(2b)

where r/= rn2c/rn2 t and M12 is the element of the K 0 - ~ 0 mass matrix defined by ,2 M12 = (K01HwlK °) + ~
(3)

On the other hand, direct calculation of fig. 2 gives a contribution to the effective weak hamiltonian density of the form ~(Penguin) = 2-1/2G(ots/127r) W

X [Sl c2 In (m2/# 2) + isls2c2s 3 sin ~ In (rn2//a 2) + Sl s2 In (rn2//a 2) - isls2c2s 3 sin 5 in (m2/~t2)] X (sTy(1 - 3'5) X~d) (uTVA~u + dq'v;Wd + ...) + h.c..

(4)

In eq. (4),/a is a typical hadronic mass, a s the strong interaction fine-structure constant, and u, d .... represent the corresponding quark fields in both ordinary space-time and in color space, where the matrices X~ operate. The first two terms in square brackets arise from fig. 2 with a c-quark loop, whereas the last two involve a t-quark. Terms ,1 The signs of the quark fields may be chosen so that 01,02, and 03 (but not necessarily 6 ) lie in the first quadrant and all sines and cosines of them are positive. ,2 We assume CPT invariance and adopt the conventions of Lee and Wu [ 13 ], except that we label the K° eigenstates by S and L. 84

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PHYSICS LETTERS

Volume 83B, number 1

involving m s and md, as well as higher powers of the sines of the small Cabibbo-like angles 01 and 03 have been neglected. Some effects due to higher-order QCD diagrams may be treated by renormalization group techniques. In as much as we will only be interested in the relative size of the real and imaginary terms in eq. (4), only differences in the magnitude of these multiplicative corrections for different terms need concern us. In the following we shall ignore differences between the leading logarithmic corrections to the terms in square brackets in eq. (4), just as they have also been ignored for the terms in the numerator and denominator of eq. (2). These are not expected to effect the character of our results. When 8 = 0 the effective "Penguin" hamiltonian is responsible for a fraction f of the real decay amplitude A(0~=0) for K 0 -+ 27r(I = 0) defined by (2rr (I = 0)IHw(8 = 0) IK 0) = A(6 =0)e i6 o ,

(5)

where 8 0 is the I = 0 strong interaction an phase shift. The claim [ 1 0 - 1 2] from detailed analysis of K 0 decays is that f is not far from unity. The total amplitude (apart from final-state an strong interactions) when 8 :/=0 is then A 0 = A (06=0) - ifA(08=O)s2c2 s 3 sin <5

in (m 2//~2) _ in (m2//a2)

~ A 0(8 =0)ei~

(6)

c 2 in (m2/~ 2) + s21n(m2t/la2 )

with the small quantity

=fs2c2s 3 sin8 - lnr~ in (m2/ 2)

-

s2 in

(7)

However, the standard convention in analyzing CP violation in K 0 decays is to take A 0 real [13]. This may be accomplished by redefining the phases of the K 0 and ~0 states * 3 : [K 0) _+ e-i~lK0), i~0) _+ eiq R0}, so that A (0~=0)ei~ -+A(0s=0) and Im (KOIMIK 0) -+ Im (e2i~(KOIMIK.O)} ~ Im (KOIM(K 0) + 2~ Re (K01MIK0). Therefore as a result

ImM12/ReM12 -+ ImM12/ReM12 + 2~ = e m + 2~.

(8)

The presence o f the 2~ term is not new - even the notation here is that of ref. [7]. What is new is that ~ as given in eq. (7) is not negligible compared to era, as given in eq. (2). The whole "Penguin" contribution previously was neglected because it was guessed that the In (m2/~t 2) terms in eq. (4) were instead o f the form m2/M 2 . Indeed, the "Penguin diagram" appears as fig. 2c in ref. [7], but its contribution to ~ is taken as small compared to em. How does all this effect the parameters of CP violation in K 0 -* 7rlr decay? In standard notation [13] e = i(Im P12 + i ImM12)/(½ (US - UL) + i ( m s - m L ) ) '

(9)

and since experimentally [14] 11 (F s - PL)I ~ Ims - roLl and IlmP12/ImM121 <~ 1/10, we neglect ImF12 (due to CP violating phase differences between the K 0 ~ rrTr(I = 0) decay and other modes) to obtain lel ~ IlmM121/21/21ms - mLI •

(10)

Recalling the result o f our phase redefinition, and using 2 ReM12 = m s - m L, it follows that ,4

lel = 2-3/21em+ 2~1 = 12-112s2c2s 3 sinfl'lP(O2,rT) +flnrl/(ln(m2/I.t 2) - s 2 lnr/)l .

(11)

The CP violation parameter [I 3] e' ~ 2 - 1 / 2 i e i(62-8°)A~1 I m A 2 ,

(12)

where A 0 and A 2 are the K 0 -+ rrrr decay amplitudes and 8 0 and 6 2 the strong interaction nTr phase shifts for final 43 We have also checked explicitly that if the phase convention on Ao is dropped, the predictions for physical observables in K° -~ 2~r decay remain exactly the same. 44 Our expression for e (in the absence of "Penguin diagrams") is a factor of two smaller than that in ref. [7 ], and consequently e'/e is a factor of two bigger than they would have. 85

Volume 83B, number 1

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isospin 0 and 2, respectively. Neither the CP violation arising from the mass matrix (eq. (2)) nor that from the "Penguin diagram" (eq. (4)) can contribute to the A I = 3/2 transition involved in the amplitude A 2. However, by virtue of the redefinition o f the K 0 and g 0 phases to make A 0 real, A 2 picks up a phase e - i ~ relative to A 0. Hence, le'l ~ 2-1/21~11A2/AoI.

(13)

Experimental information on K + and K 0 decays yields [13] IA2/AoI .~ 1/20. Numerically, we find that the two contributions to e from the mass matrix and "Penguin diagrams" are comparable. For example, with m c = 1.5 GeV, m t = 15 GeV,/~= 1 GeV, 02 = 15 °, and f = 0.75, the two terms in the second set of absolute value signs on the right-hand side o f eq. (11) are 5.2 (from em) and - 3 . 1 (from 2~). The difference in sign holds for all reasonable quark masses, mixing angles, and values o f f . Since sin 6 is adjustable to fit the experimental value of Lel, there is no critical test of the theory at this point. What is critical, and changed from previous estimates [7], is e'/e, which is independent of 6 and 03. From eqs. (11) and (13), 1

le'/el ~ ~6 12~/(em + 2~)1.

(14)

With the quark masses and other parameters used above, l e'/el ~ 1/13. Bigger values of 02, mc/g , or m t can lead to smaller values of Ie'/el: for example, with m t = 30 GeV and all other parameters left the same Ie'/el ~ 1/30, while changing/a to 0.2 GeV but leaving m t = 15 OeV gives l e'/eJ ~ 1/100. The phases of e and e' are calculable from eqs. (9) and (12), respectively. With sin 6 positive ,1 and the experimental values of m S - m E and F s - F L, the phase o f e is 43.8 -+0.2 °, in agreement with experiment [14]. Using the sign of ~ and the experimentally determined 30, 32, and the sign of ReA2/AO, the phase of e' is 37 -+6 ° [14]. Our predictions for e'/e are therefore almost real and positive. With e and e' nearly parallel in the complex plane, the experimental [14] value Ir~00/r~+ _ I = I(e - 2 e ' ) / (e + e' ) I = 1.008 +- 0.041 translates to ~e'/e = - 0.00 3 + 0.014. Given the uncertainties in 02, mt,/J, and the fraction,f, of the K -+ 27r amplitude given by the "Penguin diagram", we conclude that the model is not yet ruled out by experiment. However, a quantitative increase in the accuracy o f experimental determination of the CP violating K 0 ~ rrn decay parameter e', which now seems possible [15], would be capable o f distinguishing the six-quark model with a CP violating phase in the mixing matrix plus any sizable fraction o f the K 0 ~ 27r amplitude being due to "Penguin diagrams" from the superweak model [8] where all the CP violation in the K 0 system arises from the mass matrix and e' = 0. We are grateful to H. Harari and B. Winstein for useful and stimulating discussions. We thank L. Wolfenstein for constructive suggestions. [1] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Elementary particle physics: relativistic groups and analyticity, Nobel Symp. No. 8, ed. N. Svartholm Almquist and Wiksell, Stockholm, 1968) p. 367. [2] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [3] G.J. Feldman, rapporteur talk at the XIXth Intern. Conf. on High energy physics (Tokyo, Japan, 1978) unpublished; and SLAC-PUB-2224 contains a complete review, compilation, and references to original experiments on the tau lepton. [4] For a review of experiments and references to the original work see: L. Lederman, rapporteur talk at the XIXth Intern. Conf. on High energy physics (Tokyo, Japan, 1978), unpublished. [5] S.L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285.

86

[6] c. Bouchiat et al., Phys. Lett. 38B (1972) 519; D.J. Gross and R. Jackiw, Phys. Rev. D6 (1972) 477. [7] J. Ellis et al., Nucl. Phys. B109 (1976) 213. [8] L. Wolfenstein, Phys. Rev. Lett. 13 (1964) 562. [9] M.K. Gaillnrd and B.W. Lee, Phys. Rev. lett. 33 (1974) 108. [10] M.A. Shifman et al., JETP Lett. 22 (1975) 55; M.A. Shifman et al., Nucl. Phys. B120 (1977) 316. [11 ] M.A. Shifman et al., ITEP preprints ITEP-63, ITEP-64 (1976), unpublished. [12] M.K. Gaillard, Lecture at the SLAC Summer Institute on Particle physics (July 10-21, 1978) and Fermilab preprint FERMILAB-Conf-78/64-THY (1978), unpublished. [13] T.D. Lee and C.S. Wu, Ann. Rev. Nucl. Sci. 16 (1966) 511. [14] See, for example, K. Kleinknecht, in: Proc. xrvIIth Intern. Conf. on High energy physics (London, 1974), ed. J.R. Smith (Science Research Council, Rutherford Lab., 1974) p. 111-23. [15] B. Winstein, private communication.