Quantum chromodynamic effects and CP violation in the Kobayashi-Maskawa model

Nuclear Physics B163 (1980) 289-311 © North-Holland Publishing Company

Q U A N T U M C H R O M O D Y N A M I C E F F E C T S AND CP V I O L A T I O N

IN THE KOBAYASHI-MASKAWA M O D E L

B. GUBERINA * and R.D. PECCEI Max-Planck-Institut ¢'ftr Physik und Astrophysik, Munich, Fed. Rep. Germany

Received 31 July 1979 (Revised 26 October 1979)

We study, in tile context of the Kobayashi-Maskawa model, the effects that quantum chromodynamics has on the predictions for CP violation in kaon decays. We derive an effective hamiltonian for AS = 1 transitions in which all-order corrections in the strong coupling constant a s are incorporated, including those arising from penguin diagrams. We examine, in particular, the predictions of this hamiltonian for the ratio of the CP- violating parameters l er/e I. Our results for this ratio are well within the present experimental bound, but are considerably smaller than the recent calculation of Gilman and Wise. The discrepancy between our results and those of Gilman and Wise arises because of the different way we have estimated the amplitude for K 0 decay. A discussion of the reliability of both our estimate and that of Gilman and Wise is included.

1. In t roduc t ion

The Kobayashi-Maskawa [1] SU(2) X U(1) model , wi th three generat ions o f

quarks arranged in lef t -handed doublets and right-handed singlets, is perhaps the

simplest gauge-theory model which incorporates the possibili ty o f CP violation. The

pioneering investigations o f Pakvasa and Sugawara [2], Maiani [3] and especially

Ellis, Gaillard and Nanopoulos [4] indicated that for strange-particle decays the

predict ions of this mode l reduced to the superweak [5] form of CP violation. This

conclusion has been challenged recent ly in an interest ing let ter by Gilman and Wise

[6]. These lat ter authors argued that quan tum ch romodynamic (QCD) gluonic

effects can produce impor tan t addit ional contr ibut ions , which do not fit in the

superweak pattern. Most interest ingly, if the est imates o f Gilman and Wise are cor-

rect, it might be possible, wi th a modera te increase in the accuracy o f the relevant

exper iments , to test the combined predict ions o f the Kobayashi-Maskawa mode l and QCD.

* Alexander von Humboldt Fellow, on leave of absence from the Rudjer Bogkovid Institute, Zagreb, Croatia, Yugoslavia.

289

290 B. Guberina, R.D. Peccei / QCD effects and CP violation

W s = ~ d

I gluon _

q ~ q

Fig. 1. Penguin diagrams that contribute for &S = 1 processes.

The observation of Gilman and Wise is based on the work of Shifman, Vainshtein 1 and Zakharov [7] on the 2u r = ~ rule in non-leptonic weak decays. Shifman et al.,

1 argued that the contribution of the pure 2d = ~ penguin diagrams, shown in fig. 1, although of order as, accounts for the bulk of the observed amplitudes in non-leptonic interactions. In the Kobayashi-Maskawa model the penguin diagrams are not purely real and thus can in principle give rise to a direct CP-violating contr ibution to the process K ° ~ 2n. If this penguin contribution is large it can cause additional CP-violating effects above those arising from the kaon mass matrix, in contrast to tile usual assumptions of the superweak theory. Gilman and Wise estimated the effects that the penguin contribution induces on the ratio of the CP-violating parameters [8] [e'/e I, which vanishes in the superweak theory. They found that although

1 l e'/e] is small, it is perilously close to the present experimental limit [8]: [e'/e[ N so and for some choices of parameters it even exceeds this limit.

The original calculation of Gilman and Wise was only to lowest order in as. Fur- thermore, they took as a phenomenological input the f ract ion, f , o f the total amplitude in K ° -~ 2rr which Shifman et al., a t t r ibuted to the penguin contribution. In view of how close to the present bound on [e'/e[ the result of Gilman and Wise came, it seemed worthwhile to us to re-examine this question in more detail by considering the effects of QCD to all orders in a s *. There are two good physical rea- sons that spurred us to undertake this improved calculation. (i) Higher-order QCD corrections in general enhance the contribution of the penguin diagrams in a different way for the real and imaginary parts. Roughly speaking, the imaginary part receives most of its enhancement from the region in k 2 space going from rnt 2 to rnc 2, while the real part receives its enhancement from the region going from m 2 t o / f l , with ~ a typical hadronic mass. (ii) The estimate of Shifman et al., of the importance of the penguin contr ibution is essentially concerned with how much the real part gets enhanced and is rather parameter dependent since it involves the region from rnc 2 to / f t . In contrast, the imaginary part enhancement arising from the region going from mt 2 to rnc 2 is more reliable. Thus, a direct calculation of the imaginary

,t After this paper was submitted we became aware of a similar all-order calculation of the AS = 1 effective weak hamiltonian by Gilman and Wise [13]. We have incorporated in the text, where appropriate, a comparison of our work with their recent calculation. See also ref. [14].

B. Guberina, R.D. Peccei / QCD effects and CP violation 291

part of the effective penguin hamiltonian, to all orders in C~s, avoids two problems at once. One need not assume that the real and imaginary parts get the same enhancement, and one need not rely on an estimate of the importance of the real penguin contributions to the total amplitude.

The plan o f this paper is as follows. In sect. 2 we derive the QCD corrected effective hamiltonian for the Kobayashi-Maskawa model for AS = 1 weak processes. Our derivation of this hamiltonian is along the lines developed by Shifman et al., but is somewhat more complicated since we must consider operator mixing in two regions now, instead of one. Having the effective hamiltonian in hand, we proceed in sect. 3 to examine C P violation in the K ° -+ 27r decay. We obtain an expression for the ratio of the CP-violating parameters l e ' /el which, in contrast to the work of Gilman and Wise, is essentially independent of the magnitude of the contribution of the real penguins to the decay K ° -+ 27r. Our results for this ratio, for a broad range of parameters, are well within the experimental bound. In this section we also discuss the reason why our results are so much smaller than those obtained by Gilman and Wise [6,13]. We point out that this discrepancy is principally caused by differing assumptions on how to evaluate the relevant K ° amplitudes once the effective hamiltonian is in hand. Our conclusions, as well as some final observations on the reliability of our results are contained in sect. 4. Some technical details are left to the appendix.

2. Effective hamiltonian for AS = 1 processes

We want to derive an effective hamiltonian for AS = 1 processes in the context of the Kobayashi-Maskawa mbdel. To lowest order in the weak interactions, for processes where the momentum transfer can be neglected with respect to M~v, the AS = 1 weak hamiltonian takes the usual current-current form

1 --~ H = - ' v /~ ]GFJ+uJ , (1)

where the currents J~ are given by

J~ = ~7~(1 - 7s) U ~ , (2a)

J~- : ~Tu( 1 - 75) Utff • (2b)

Here we have suppressed color indices and for each color the field ~ is a six-compo- nent field given by

= ( a , ~ , ~ , 3 , ~ , b - ) . (3)

The 6 × 6 matrix U has the block form


where the 3 × 3 matr ix C is the usual Kobayashi-Maskawa matrix

C1 --SLC3 --SIS3 1

C = s i c 2 CLC2C3 - s2s3 el6 c i c 2 s 3 + s 2 c 3 ei8 , (5)

LSlS2 ClS2C 3 + c2s3 ei5 ClS2S 3 - c2c3 ei6_]

with ci = cos Oi, s i = sin 0 i. For AS = 1 processes we need to retain only terms involving an s-quark in (1). Fur thermore , since we are only interested in processes where AC = AB = A T = 0, we may further drop terms that violate these selection rules in (1). Notice further that we have not included in (1) any con t r ibu t ion from neutral currents, since in the Kobayashi-Maskawa model these currents are flavor diagonal and hence cannot cont r ibute to AS = 1 processes. Effectively then, the AS = 1 weak hami l ton ian can be taken to be

= 1 H (°) - -x /~GF [a ldTu(1 - 3'5) ut77~(1 - 7s) s

+ a2d') 'u(1 - "Ts) ccTU(1 - "Ys) s + a3dTu(1 - 3'5) t t~ 'u( 1 - 7s) s] ,

where

(6)

al = - e l s i e 3 , (7a)

a2 = S l c2 [Cl c2c3 -- $2s3 e i8 ] , (7b)

a 3 = s i s 2 [ClS2C 3 + c2s3 eifi] . (7c)

Note that since the Glashow-Ill iopoulos-Maiani mechanism [9] is incorporated in the Kobayashi-Maskawa model one has

a 1 + a 2 + a 3 = 0 . (8)

Using eq. (6) as our starting point , we may now consider what correct ion QCD induces on this structure. To lowest order in a s one encounters the diagrams of fig. 2. These diagrams are all logarithmically divergent because o f our approxima- t ion of neglecting the weak propagator. If the weak propagator were to be restored,

~ gtuon(+Crossed graphs) ~gluon

(e) {bl

Fig. 2. Diagrams which give the lowest-order QCD corrections to the hamiltonian H ~0). The diagrams of fig. 2b are the penguin contributions.

B. Guberina, R.D. Peccei / QCD effects and CP violation 29 3

one would find that the relevant cut-off is provided by M~v. Of particular interest are the diagrams of fig. 2b which are essentially the penguin diagrams of fig. 1 in the limit as M~v ~ '~. In the limit in which the gluon momentum transfer k 2 is much larger than all the quark masses entering in the quark loop, it is easy to check that the penguin diagrams vanish, since their contribution is proportional to al + a2 + a3. However, for problems involving strange-particle decays k 2 < < mc 2, mt 2 and the can- cellation is not complete. This is essentially the crucial observation of Shifman et al. [7], which was exploited by Gihnan and Wise [6].

An explicit evaluation of the penguin diagrams, dropping terms of order rn~/MZw, with mi a quark mass, leads to the following effective AS = 1 hamiltonian arising from the penguin diagrams:

1 HPenguin = d a F ~s [dTv(l - 75) )kaS ~ ¢ f T # X a q f ]

{ , / X , a 3 1 n ~ t +a21nm---~c + a l l n k2 j . (9)

The above reflects the fact that for the c- and t-quark loops of fig. 1, the effective scale of the diagrams is set by the masses of the quarks instead of the momentum transfer. Using eq. (8) we may equally well write the quantity in the curly bracket in eq. (9) as

mt 2 me 2 { } = (al + a2) l n ~ y + al in / 7 " (10)

m e

This rearrangement allows one to interpret the penguin hamiltonian in more physical terms. For momentum transfer scales ranging above mt 2 there is no penguin contribution. For effective momentum transfers between rnt 2 and rn2c the t-quark loop can be neglected, but the c- and u-quark loops are active. Finally, for momentum transfers below mc 2 only the u-'quark loop is important. Of course the true hamiltonian is the sum of all these contributions.

The contribution of the diagrams of fig. 2a can also be readily evaluated, remem- bering that the effective cut-off in the logarithms is to be taken as A 2 -~ M~v. For our future development it is convenient to write/4 (°) as the sum of two distinct terms:

H(°) = --V/~GF [0189 + 04os] , (1 1)

where the operators O189 and O40s for diS = 1,2xC = zSB = AT = 0 processes are given by

3

O189 = ~ [~aiffTu(1 - 7s) qictivu( 1 - 7s) s i = 1

1 - - - ~aidTu( 1 - 7s) Xaqi77iTu( 1 - 75) kaS] , (12a)


3

0405 = ~ [2aid')',(1 - "/5) qiffi '~"( 1 - "/5) s i=1

1 - -

+ 7~aiclT~z( 1 -- ")'5) ~'aqiqiT~( 1 -- 3"5) )~a s] • (12b)

Here ql = u, q2 = c, q3 = t. These operators correspond precisely to the ~ = 1, AC= zSJR = A T = 0 projection o f operators which transform as the 189- and 405- dimensional representations of SU(6). A simple calculation shows that the diagrams of fig. 2a again yield an effective hamiltonian which just involves these same operators

H2 a .~, .~ [ a s , M~v )] = --V 2~'r L8 ~ ln--~- (80189 -- 40405_ . (13)

Here/22 is a typical hadronic scale which should be identified with the momentum transfer involved in the relevant processes to be studied.

The total hamiltonian to El(as) is given by the sum of H (°), H penguin and//'2a. There are three distinct types o f logarithms that enter in this hamiltonian ln(M~v/m2t), ln(m2t/m2c) and ln(mc2//22). Associated with these logarithms there are corresponding operators. For the ln(M~v/m2t) terms only the operators O189 and 0405 enter, while for the rest the penguin operators of eq. (9) are also relevant. However, we should comment that associated with the logarithms of the type ln(m2t/m2c) it is sensible to neglect the contributions of the t-quarks in the operators, since effectively these quarks do not play an active role. Similarly, for the In(me2//22) terms it is permissible to drop both the t- and c-quark contributions. Hence the effective hamiltonian to O(as) splits into three separate contribution which we can associate with momentum transfer regions ranging from M~v to mt 2, from mt 2 to m 2 and from mc 2 to/22.

One can proceed to analyze higher-order QCD corrections to the weak hamiltonian in the same vein, by studying more complicated Feynman diagrams. However, this approach is cumbersome and can be effectively bypassed by making use of renormalization group methods [10]. Indeed we could have phrased our discussion from the beginning in this more elegant way. Our insistence in looking at Feynman graphs was motivated only by trying to bring to the fore, in a perhaps more direct way, the existence of three relevant regions of interest. If one writes the weak hamiltonian in the form

H (°) = - X / ~ G F ~ C i O i , (14) i

with the Ci being c-number coefficients and Oi being the relevant operators, then a renormalization group analysis of the effects of QCD informs us that the final effective hamfltonian will have an analogous form, with perhaps more operators entering, but where the coefficient functions now depend on M~v //22, /22 being a relevant scale for the processes studied. In the particular case in question, where there are different


scales entering, the correct procedure for arriving at the final hamiltonian consist of deriving at each scale an effective hamiltonian which then provides the starting point hamiltonian for the next stage. This is the procedure adopted for the 4-quark model by Shifman et al. [7]. A justification for this rather pragmatic approach can be found in the interesting paper of Witten [11].

Let us begin by considering QCD corrections to the AS = 1 weak hamiltonian

I4 (0) = --X/~-~GF [O189 + 0405] (15)

in the region ranging from M 2 to mt 2. In this region the penguin operators are inef- fective and the only relevant operators which enter are again just O189 and O408- To calculate the QCD corrections to the coefficients

C189 = 1 , (1 6a)

C4os = 1 , (16b)

it suffices to compute the renormalization matrix appropriate to these operators. Because O189 and O4os transform differently under SU(6), these operators will not mix, so that under renormalization

[ 01891-~[~189 0 ]F0189l (17) O40sJ Z40s LO40sJ "

The renormalization factors Z189 and Z40s can be readily extracted from an analysis of the diagrams of fig. 2a. Evaluating these diagrams with a cut-off A, and remem- bering to include appropriate factors associated with the renormalization of the indi- vidual external quark lines, one finds

Z189 = 8 [ ~ l n A] , (18a)

Z4o5 = - 4 a[~-~ In a ] . (18b)

Defining, as usual, the anomalous dimension by

3 In Z % 7n - - 7n, (19)

3 In A 4n

a straightforward application of the renormalization group [1 O] yields the QCD corrected coefficients, for the region ranging to mt:

C189(mt) = C189(Kt)~189/2b ,

Here

C4o5 (rot) = C405 (Kt) v405/2b .

(20a)

(20b)

as(m?) g t ~s(M~) , (21)


and the QCD running coupling constant as(q 2) is given by the expression

as(q 2) = as(/a2) 1 + as(U2)(b/47r) ln(q2/u2) '

with

(22)

b = l l 2 - 5 n f , ( 2 3 )

where nf is the number of flavors. For the region in question, since all 6 quarks are active, b = 7.

Using eqs. (16) and (18) one finds

C 189(rot) = K 4/7 , (24a)

C4os(mt) : K t 2]7 . (24b)

We note parenthetically that the anomalous dimensions of the operators O1 s9 and O4os are precisely the same as the ones found in ref. [12] for the case of SU(4). This is not surprising, since these operators contain, respectively, the operators that transform as the 20- and 84-dimensional representation of SU(4). The only differ- ence, at this stage between the 4-quark and 6-quark model resides in the different value ofb for the two cases.

The effective QCD corrected hamiltonian at rot, which is to be used as the starting point for computing the QCD corrections to me is then *

H(mt) = - X / ~ G F [K4t /7 0189 + Kt2]704o5] • (25)

However, as we mentioned earlier, since we are now propagating to regions below

mt we may drop effectively from the definitions of Oa89 and O40s of eqs. (12) all terms containing t-quarks. In the region from mt 2 to mc 2 it is no longer true that the

operators O1 s9 and O40s just reproduce themselves, since as our lowest-order calculation has indicated in this region the penguin operators appear with non-vanishing coefficients. Let us define the penguin operators, appropriate to the 4-quark case, as

Op = (a I + a2) [dTu(1 - 3's) XaS " {~7~Xa u + ~TUXac

+ d"/SXa d + gTuX a s}] . (26)

An analysis of the diagrams of fig. 2, where now the four-fermion operator is not

Hweak but Op, indicates that this operator in turn produces, in higher order in %,

* In principle we should also split the region going from m t to m c into regions going from m t to m b and then from m b to mc. The only effect of this further splitting is to slightly change the anomalous dimension factors, since b changes when less flavors are active. We have neglected this rather unimportant effect. This effect is included in ref. [ 13 ].

B. Guberina, R.D. Peccei / QCD effectsand CP violation 297

other structures besides itself. Let us define three additional operators

Ap = (a 1 + a2) [dTu(1 - 75) XaS • (aTuTska u + cTUTsXa c

+ dvU3`sXa d + ~-TUYsXaS}] , (27)

()p = (a I + a2)[d3,u(1 7s) S" {ayUu +VyUc+dTUd+gyUs}] , (28)

Ap = (al + a2)[JYu(1 - 3'5)s • "[a3"U3"s u + V3"U3"sC

+ d3"U3"sd + gTU3`sS}] . (29)

Then a straightforward calculation shows that O189, 0405, Op, Ap, Op and Ap form a closed set of operators which mix among themselves. The renormalization matrix for these operators is readily computed and is displayed in detail in the appendix. We have

Z = % In A F . (30) 4n

The matrix P can be diagonalized by a similarity transformation

S - 1 P S = A , (31)

where

m i j = ~ i j ~ k j . (32)

The effective hamiltonian at mc is then readily calculated from that at mt. One has

H(mc) = --N//~GF (K4t /7 ~.. S l i (gc)k i /2b(s -1) i jO] t l

+ K t 2/7 ~ S2i(Kc)Xi /2b(s-1)qOi) • (33) 1]

He re

~s(mcb Kc eq(mt2 ) , (34)

and the set of operators Oj, / = 1 ... 6, are given by

O] = (O189, 0405, Op, Ap, Op,/Ip} . (35)

Note that in (33) b should be given by the value appropriate for 4 flavors, b - 25 3 '

Eventually, to complete the calculation we must propagate the hamiltonian (33) down to/~2. In this last region only the u-, d- and s-quarks are active. If we drop the c-quark contributions, it is possible to rewrite the operators appearing in eq. (35) in terms of the standard set of operators used by Shifman et al. [7]. Defining

qL = ~(1 -- 75) q , (36a)


qR = 1( 1 + 7s) q , (36b)

we consider the following set of operators

01 = JL')'uSLU-L'Y#UL -- jL')'#ULI~LT#SL , (37)

02 = JL')'#SLU-L')'#UL + NL"//.t UL~-L')'#SL

+ 2NL"/#$LNLT#dL + 2NL')'# SLS-L')'#$L , (38)

03 = dL'),#SLffL'),#UL + alL')'# HL~/-L'),'uSL

+ 2dLTuSLdLTUdL- 3alL7 u SLS-LTUSL , (39)

04 = dLT#SLffLT~UL + dL~#ULRL"/#SL -- jLT#SLJLT#dL , (40)

OS = dLTU Xa SL ' [t~RTUXa UR + dRTUXa dR + SRTUXa SR] , (41)

06 = NL')'uSL • [H-R"/UH R + NRT#dR + S-RT#SR] . (42)

It is easy to rewrite the set (35) in terms of the above operators. One finds

O189 = - 2 0 1 , (43a)

_ 2 + 4 0 3 + 4 (43b) 0405 - ~O2 5 0 4 ,

Op = 2Os + 402 - 8 0 , , (43c)

Ap = 2Os - 402 + 8 0 1 , (43d)

Op = 206 + Ot + 02 , (43e)

./1p = 2 0 6 - O1 - 0 2 • (43f)

Using eqs. (33) and (43) we may rewrite the hamiltonian at mc in the form

6

H(mc) = --N/~GF ~ A i O i . (44) i=l

The coefficient functions A i are displayed explicitly in the appendix in eqs. (A .7 ) - (A.12).

Our final task is to use the hamiltonian (44) as a starting point to compute the QCD corrections from the scale set by m2c down to/a 2. The operators O1 ,02 , Os and 06 get mixed in this last zone. However, both 03 and 04 do not mix since they carry distinct quantum numbers. The renormalization matrix for the operators O1,


02, O5 and 0 6 is given in the appendix:

Z - - % l n A F ' , (45) 47T

while for both 03 and 04 one finds

% In A ( - 4 ) (46) Z3, 4 = 4~

The matrix P ' can be diagonalized by a similarity transformation

T -1 P ' T = A' (47)

and its eigenvalues are given in the appendix. Armed with these eigenvalues and the matrices T -1 and T, it is now straightforward to evolve H(mc) down to g2, by following the same procedure as outlined before. The final hamiltonian for AS = 1 processes, which includes corrections to all orders in cq, is given by the formula

H(U) = --Wf2-GF ~ BiOi , (48) i

where the coefficients Bi are given by sum of products of the coefficients Ai with new coefficients obtained from the last zone of interest. Specifically we have

Bi = ~ Ci/A/• (49) J

The new coefficients C//are also given explicit ly in the appendix in eqs. ( A . 2 2 ) - (A.39). We should comment that the anomalous exponents di to which

K - a(U2) (50) ~(~n2~)

in the coefficients Cij is raised, are given by

di = X'i/2b, (51)

where l) here is that appropriate for the case of 3 flavors, b = 9. The final hamiltonian we have obtained is rather cumbersome because we have

had to evolve fromMaw down to/22 through three distinct regions, and in the last two regions there was considerable operator mixing. As a check in our hamiltonian we note that if Kt = Ke = K = 1 we recover the original weak hamiltonian, as far as the u-, d- and s-quarks are concerned. I f K = Kc = 1 we recover H(mt) , and i f K = 1 we recover H(rnc). In sect. 3 we shall use this hamiltonian to study CP violation in the decay K ° --> 2n. An additional comment is in order. We have checked that the effective hamiltonian (48), which we have derived, is in agreement with the recent all-order computat ion of Gilman and Wise [13]. Their final expression is given in terms of a different set of operators but, on changing to our basis, similar results to ours obtain.

300 B. Guberina, R.D. Peceei / QCD effects and CP violation

3. CP violation in K-decays

We want to study now the implication of the hamiltonian (48) for CP violation in strange particle decays. Because the resulting hamiltonian is apparently so complicated, it is extremely useful first to try to focus on what are the dominant terms. Eventually, for a final analysis, one may also want to include the effects of the terms dropped in the first instance. The operators O1 through 04 all have a (V - A) × (V - A) structure, while Os and 06 have a (V - A) × (V + A) structure. While the first type o f structure is invariant under a Fierz transformation, the latter produces an (S + P) × (S - P) structure. When considering matrix elements of the operators Oi between pseudoscalar states, there will be a considerable enhancement of the (S + P) × (S - P) operators relative to the (V - A) × (V - A) operators. This is the physical reason why the penguin operator, which includes the operator Os, although multiplied by coefficients which are of O(as) can give important contributions to non-leptonic decays [7].

We are going to be principally interested in the imaginary part of H(~l). Even though the coefficients that multiply the operators Ol, 02, Os and 06 for the imaginary part are comparable, in view of the above considerations it is permissible to focus only on the terms connected with Os and 06. Furthermore, it is easy to show that between color singlet states

06 = ~6Os • (52)

Given the fact that the coefficients that multiply 06 are typically more than a factor of two smaller than those that multiply Os, it is also sensible to neglect, as a first approximation, all terms connected with 06. We shall focus then on the approximate hamiltonian

Hs(U) = -V~GFBsOs • (53)

For comparison with the work of Gilman and Wise [6,13], it is also useful to retain in Bs terms which are purely real, although these terms play no role in our evaluation of CP-violating effects.

There are a few more reasonable approximations which we can effect on B5. Recall that

B 5 = ~ l C s j A j . (54) i

The dominant terms in the above expression will be those in which at least one of the coefficients in Cs] or A/do not vanish when Kc, Kt, K go to unity. Further- more, given that all the Cabibbo-like angles Oi are small, we may also drop terms which involve non-leading products of the various small angles. In this way we arrive at the following approximate expression for Hs (~):

Hs(/a) = X/~-GF (C1S1 C3aR + iSl C2S2S3 (sin 8) al} Os , (55)


where the coefficients a R and ai of the real and imaginary parts, respectively, are

given by:

aR = _ K o. 57KO.48 [0.0390K o. 80 __ 0.0334KO.42 _ 0.002 6K-O. 12 _ 0.00 30K-O. 30]

- 1Kt° '29Kc° '24 [0.0681K ° 's° + 0.0071K O'42 + 0.0708K - °A 2 _ 0.1460K -°" 30] ,

(56a)

al = _ ( K t o. 57 [0.0370KO.85 _ 0.0308KO.42 _ 0.0024KeO. 13 _ 0.0038KcO.3S]

+ K t o . 29 [ 0 . 0 1 2 4 K °" 85 + 0.0028Kc o. 42 + 0.0140Kc °" 13 __ 0.0292Kc °" 35] 3-

X (0.8509K °" 8o + 0.0091K °" 42 + 0.1222K -°" 12 + 0.0178K -°" 303- . ( 5 6 b )

We note that the real part of the above hamiltonian is essentially that given by Shifman et al. [7], except that we have broken down the contribution coming from M~v to mc 2 into two distinct terms proportional to /~ t "s7 and K °'48, respectively. The

magnitude of the real part of Hs(/a) is rather sensitive to what values one takes for K, since the terms in (56a) proportional to K vanish as K ~ 1. In contrast the imaginary part is much less sensitive to the value of K assumed. In table 1, we display the

values taken by aR and ai for a broad range of parameters. In calculating these values we have used for K the formula

- as(/12) - 1 + 9 as(/~2 ) in mc2 (57) K %(mc2) 4n /12 '

and have varied bo th / l and mc, taking as(/l 2) = 1. Although one should perhaps worry about trusting the QCD enhancements down to values where a s = 1, this is precisely the approach followed by Shifman et al. [7]. Once K is determined, Kc and K t, follow once values o f m t and M w are given. In table 1 we have taken mt as a variable parameter and fixed Mw = 80 GeV.

We note that the values of aR are essentially independent on what we assume for m t but are crucially dependent on the values of/a and mc taken. In fact, we see that the values ofaR, for different choices of parameters can vary by an order of magnitude. In contrast ai is much less parameter dependent, ranging by at most a factor of two over the range of parameters chosen. One should note that the parameters which make a R largest (small/l, large mc) make a I the smallest. We shall comment again below on this point.

To give a quantitative evaluation of CP violation we need to know what is the matrix element of the operator O5 for the decay K ° ~ 27r. Shifman, Vainshtein and Zakharov [7] have evaluated this matrix element in a valence quark approximation. It proves convenient to quote their answer in terms of the experimental amplitude for this decay. They find, approximately

V~GFCl S 1 c3(nn(I = 0) lOs [K °) "" -3 .4A [K ° ~ nn(I = 0)] . (58)


Table 1 Value of the coefficients a R and a I for a range of parameters

t~ (GeV) m c (GeV) m t (Ge V) a R a I

1 1.2 15 -0.0138 -0.1140 1 2 15 -0.0441 -0.0831 1 1.2 50 -0.0138 -0.1517 1 2 50 -0.0439 -0.1203 0.5 1.2 15 -0.0550 -0.0992 0.5 2 15 -0.0789 -0.0764 0.5 1.2 50 -0.0549 -0.1341 0.5 2 50 -0.0787 -0.1118 0.14 1.2 15 -0.1146 -0.0867 0.14 2 15 -0.1334 -0.0687 0.14 1.2 50 -0.1146 -0.1206 0.14 2 50 -0.1334 -0.1028

Thus the real part of the hami l ton ian Hs(/a ) gives a cont r ibu t ion to the decay ampli-

tude

(nTr(I = 0)IgsReal(/a) I K °) -~ - -3 .4aRA [K ° ~ 7rn(I = 0)] . (59)

To obta in the result of Shifman et al., in which the penguin diagrams give approxi-

mately 85% of the total ampli tude, requires aR --~ --0.25. A glance at table 1 indicates that this is a rather unrealistic expectat ion. In fact, to be fair, this is also the conclusion of Shifman et al. The rather large value o f a R used by them is not really

a QCD predict ion, bu t comes f rom a fit of hyperon decays. In view of the crucial dependence o f a R on K, these authors preferred to take a R as a free parameter.

For tuna te ly to find how much Hs(l~ ) contr ibutes to the imaginary part of the decay ampli tude, we are much less sensitive to the parameters used. Using eq. (58) we have that

(TrTr ( I = 0)IHIsm(/a) I K °) = a I is 1 c2s2s 3 sin 6N/~GF(ZrTr(I = 0)10 5 I K °)

t~c2s2s3 sin d )1 " . . . . (3.4ai A [K ° -+ 2n(I = 0)] . (60) [_ ClC3

The quan t i ty in the square brackets above is precisely the parameter ~, defined by Gilman and Wise [6], which enters in the analysis of CP violation. We obtain, thus,

_ c2s2s3 sin 6 (3.4ai) ~ c2s2s3 sin 6(3.4ai) , (61) ClC3

dropping small-angle corrections. For the broad ranges of parameters displayed in table 1 the coefficient o f c2s2s3 sin 6 in eq. (61) ranges be tween - 0 . 2 3 4 and

B. Guberina, R.D. Peccei / QCD effects and CP violation

Table 2 Values for ~/s2c2c 3 sin 6 for a range of parameters evaluated according to our formula, eq. (61), and the formula of Gilman and Wise, eq. (62), with f = 0.75

303

u(GeV) mc(GeV) mt(GeV) Present Gihnan and Wise determination ( f= 0.75)

1 1.2 15 -0.388 -6.196 1 2 15 0.283 -1.413 1 1.2 50 -0 .516 -8.245 1 2 50 -0.409 -2.055 0.5 1.2 15 -0 .337 -1.353 0.5 2 15 -0 .260 -0.745 0.5 1.2 50 -0 .456 -1 .832 0.5 2 50 -0 .380 -1.065 0.14 1.2 15 -0.295 -0 .567 0.14 2 15 -0 .234 -0 .386 0.14 1.2 50 -0 .410 -0 .789 0.14 2 50 =0.350 -0 .577

- 0 . 5 1 6 , w i th the lowes t value co r r e spond ing to pa rame te r s for which a R is maxi-

mal, aR = - -0 .1333 .

It is i m p o r t a n t to compare our eva lua t ion o f ~ w i th tha t o f G i lman and Wise

[6 ,13] . These au tho r s pre fe r red to rely no t on the es t imate o f S h i f m a n et al. [7]

for the m a t r i x e l em en t o f the ope ra to r O5 in the K ° ~ 27r process b u t r a t h e r on the

to ta l c o n t r i b u t i o n t ha t the pengu in diagrams make for A ( K ° -~ nn(I = 0)) . Le t f b e

the f r ac t ion o f this to t a l a m p l i t u d e t h a t can be a t t r i b u t e d to penguins , t h e n Gi lman

and Wise e s t ima te ~ as

~GW=--C2S2s3sin6[f~R ] • (62)

F r o m ref. [7] one has t ha t f - ~ 7 5 - 8 5 % . A d o p t i n g f o r f t h e value f = 0 .75 and using

our results * for a I and aR f rom table 1, we com pare in t ab le 2 the d e t e r m i n a t i o n of

given b y eq. ( 61 ) and t h a t given b y eq. (62) . It is clear f rom the table t h a t the val-

ues o f ~ o b t a i n e d b y G i l m a n and Wise are typ ica l ly m u c h larger t h a n ours and are

r a the r s t rong ly d e p e n d e n t on the scale p a r a m e t e r / l . Such a d e p e n d e n c e is no t sur-

pr is ing since aR, as we have seen, is s t r o n g l y / / - d e p e n d e n t . Rough ly speaking, using a

• We have mentioned earlier that as u decreases so doesai(a ). In ref. [131 the opposite behav- iour obtains. However, this fact is an artifact of the ways one calculates K, K c and K t. Here we have kept %(~) = 1 and varied # in eq. (57) to get K and then K e and K t. What Gilman and Wise do is to calculate their results for varying as0Z ) = 1.25, 1, 0.75 and employ the formula ~s(Q 2) = 12n/(33 - 2nf) ln(Q2/A 2) with A 2 = 0.1 GeV 2. For the purposes of table 2 we have used our results for a I and a R rather than those of Gilman and Wise. However, we believe that the qualitative trend is rather similar.


lowest-order estimate, one has

ai ln(mt/mc) a R - ha(me/u) . (63)

Hence for values o f / j near me one gets large values of ~. We believe that our estimate of ~ is more reliable since it is much less dependent

on/ j . Physically, one expects that measurable quantities should not depend on the normalization scale/j. In our way of calculating ~, we have implicitly assumed that the valence-quark estimate for the matrix element of Os is independent o f / j and hence our only/J-dependence arises from that in al . In as much as a I is not terribly /J-dependent, our result satisfies the physical criteria discussed above. However, in fairness, we should point out that a similar assumption for the matrix element of O5, coupled with our evaluation ofaR(/J) would give a rather large variation with/J for the real part of the K ° -+ 27r amplitude. It is true, however, that there are other contributions to the real part of the K ° -+ 27r amplitude besides those arising from the penguins. But typically the coefficients multiplying the other operators are not that rapidly varying with/J and it appears difficult to argue that, for the real amplitude, the/J-dependence of the penguin contr ibution is cancelled by that of the remaining contributions. We think it is perhaps more sensible to argue that the rapid dependence on/J o faR is perhaps an artifact of trusting a leading log calculation in a region where a s is large. After all the principal contr ibution for aR comes precisely from the region going from me 2 to/J2 and here non-leading contributions may be important. For a~, on the other hand, one is dealing with the region from rnt 2 to mZc and a leading log calculation may be more sensible.

As we discussed above, the result of Gilman and Wise for ~ is rather/J-dependent. However, this/J-dependence should perhaps not be too severely criticized. Basically, Gilman and Wise have taken a different approach to calculate ~. They trust that the penguin contributions dominate in K ° -~ 27r, thus f - ~ 0.75. However, they do not commit themselves to what value of/1 such a result applies and hence present a broad range of values for ~(/J). In contrast, we do not assume that the penguin con-

1 tributions explain the & / = 5 rule but try to calculate directly at least what is their contributions to the phase ~.

Having obtained an expression for ~, the value of the measured CP-violating parameters e and e' [8] can now be estimated straightforwardly. Using the notat ion of Gilman and Wise one has

[ern + 2~1 l el = 2x/2 ' (64)

[e'[ = x/~l~'[~-~ • (65)

In the above em is given by the ratio of the imaginary to real part of the kaon mass matrix. This parameter has been evaluated in the Kobayashi-Maskawa model by

B. Guberina, R.D. Peccei / QCD effects and CP violation

Table 3 Values of le'/e I for a range of parameters

305

mc(GeV) mt(GeV) le'/e I (0 2 = 5 °) le'/e I (0 2 = 15 °)

1.2 15 0.0039 0.0030 1.5 15 0.0042 0.0033 2 15 0.0044 0.0037 1.2 30 0.0025 0.0022 1.5 30 0.0028 0.0022 2 30 0.0032 0.0024 1.2 50 0.0015 0.0020 1.5 50 0.0018 0.0019 2 50 0.0023 0.0020

Here we have taken ~z = 0.7 GeV. For ~z = 1 GeV all values would increase by roughly 10%, while for ~ = 0.14 GeV all values would decrease by roughly 20%.

Ellis et al. [4], wi th the result *

e m = 2S2CzS 3 (sin 6) P(02, T/) ,

where

(66)

s 2 I 1 + I . . . . 1-77_] P(02, r?) - , (67)

c~r? + s 4 - 2s~c~ ~ In

2 2 with r~ = me~rot. The ratio IA2/Aol in eq. (65) is the ratio o f the decay ampli tudes for K ° ~ 2n in which the fina! pions are in a state of isospin 2 or 0. Exper imenta l ly ,

]Az /Ao I 1 ~ f 6 . Using our results for ~ we may write

]s2c2s 3 sin 6 [P(02, r/) + le] = l 7 2 - 3'dmlI ' (68)

= 1 3.4a I

-f~ p(o2, ~ + 3.4al I . (69)

As Gilman and Wise correct ly po in ted out, there is not much that can be said about

[e[ since b o t h 0 2 and (5 are adjustable parameters. However , in contrast to the results

o f Gi lman and Wise, 3.4a I is not comparable to P(02, 7) but typically an order o f

magni tude smaller than it. Thus we find that [ e ' /e[ is well be low the present experi-

mental bound [8] te'/~t ~ {o. Our results for this ratio, for a variety o f parameters ,

are displayed in table 3.

* A complete calculation should also include QCD enhancements for em. For our purposes it probably suffices to rely on the lowest-order estimate since l eml >> if 1.


4. Concluding remarks

We have evaluated QCD corrections to the AS = 1 weak hamiltonian in the Kobayashi-Maskawa model with a view of seeing whether the effects of CP violation in this model can be distinguished from the superweak model. Our results for the ratio I e ' /el , although non-vanishing, are quite small and well below the present level of experimental uncertainty. It is unclear to us whether experimental limits can be pushed down to test the predictions of this model for I e ' /el , although it certainly would be of importance to obtain confirmation of a departure from the superweak theory.

Our results for L e'/e[ are at variance with those of Gilman and Wise but we believe we have understood the reason for this discrepancy and feel that our results are reliable. As can be seen from table 3, our predictions for l e '/e[ vary by about a factor of 2 depending on different choices of parameters. There are, however, two further causes of uncertainty in our estimates which ought to be mentioned. Our results depend on the value assumed for the matrix element (TrTrl Os IK°). In this paper we have relied on the estimate of this matrix element by Shifinan, Vainshtein and Zakharov, who used a valence-quark approximation. It may well be that using other approaches the value of this matrix element may change. However, we would be surprised if the change were more than a factor of 2 or so *. Finally, we have dropped from our calculations both subdominant terms in Hs(/a), as well as all terms in which c- and t-quarks enter. We have checked that the retention of the contribution of other operators besides Os in Hs(/a ) does not affect our results for ~. Of more importance, perhaps, is the inclusion of operators in which cg and t~-structures enter. The effect of these kind o f terms is difficult to estimate. However, since these contributions to the amplitude are Zweig suppressed, a reasonable assumption [2,4] is that they should contribute to ~ an amount of the order

1 ~ ~ IC2S2S 3 sin 61 . ( 70 )

If this is indeed the case, our results would not be markedly affected. In conclusion we feel that the Kobayashi-Maskawa model remains a viable model

for CP violation. The inclusion o f QCD effects gives non-vanishing contributions to [e'/e[ which, unfortunately, appear to be tantalizingly small.

We are both grateful to Hector Rubinstein for some useful conversations. One of us (B.G.) would like to thank the Max-Planck-Institute for its hospitali ty.

* Dr. J. Donoghue has kindly informed one of us (RDP) that a bag model evaluation of the matrix element of 0 5 in kaon decay differs by about a factor of two from that in the valence- quark approximation.


Appendix

Operator mixing and effective AS = 1 hamiltonian for the Kobayashi-Maskawa model

The operators O189, O40s, Op, Ap, when one considers QCD corrections.

01s ~8 0 - 5 0 0 1

04osl 0 - 4 - ~ 0 0 Op L 0 0 - ~ 5 0 j ~ a S l n A 32 Ap ] 4~ 0 0 " ~ 9 - 5- ()p 0 0 - 5 3 0

11 LAp 0 0 -~ 0 0

The eigenvalues of the above matrix, which we numerically:

Op and Ap defined in the text in general mix Under renormalization one finds

0-

0 32 -y

0

0

0

O18~

O40!

Op

Ap

Op Ap

(A.1)

shall denote by F, are readily found

)k i = (14.0856; 8; 7 .0020;-2 .1948;-4; -5 .7817}. (A.2)

The matrix F can be diagonalized by a similarity transformation

S- 1 P-S = A. (A.3)

The required matrices S and S -1 which diagonalize F are given by

~-0 .210421 0 . 6 3 7 8 2 - 0 . 0 1 9 5 7 1 - 0 . 0 3 8 1 4 ] [ - 0 . 0 7 0 8 0 0 - 0 : 0 5 7 8 6 0 . 1 1 0 5 2 0 -0.29500 / [ 3.841540 1.90966 -0 .598580 -1.57684 /

S= [ 5.530180 -1.02972 - 1 . 3 7 8 4 2 0 0 . 9 8 6 5 6 1 ' (A.4)

~.9960100-0"6227911.702311 00 -~.69372]

i 0 0 0 . 0 6 3 4 3 0.11271 0.08535 0.04803] 1 0 - 0 . 1 2 5 0 0 0.12500 0.16667-0.16667 /

S_ 1 = 0 0 0.20248 -0.15414 -0.23481 0.308451 0 0 -0.05072 -0.07492 0.36411 0.24652[ " (A.5) 0 1 --0.04167 0.04167 --0.11111 0.111111 0 0 --0.21518 0.11635 --0.21466 0.39699]

Using these matrices and the procedure indicated in the text one can derive the effective hamiltonian H(mc). Expanding this hamiltonian in terms of the set of operators Oi, i = 1 ... 6, defined by Shifman et al. [7] and given explicitly in the text, one has

6

H(mc) = --x/fGF ~ AiOi , (A.6) i=1


where the coefficient functions A i are given by the following formulas:

A, = (al + a2)(Kt °'sv [-0.0178K~c "8s + 0.5000Kc °'48 - 0.4764Kc °.42

_ 0.0006Kc o. 13 _ 0.0052Kc °" 35] + K~-O. 29 [_0.0060KcO.8S + 0.0432KCO.42

+ 0.0030Kc o. 13 _ 0.0402Kg °' 3s] } + al { -Kt °" 57K°'48} , (A.7)

A : = (al + a : ) { K °'sT [0.0030K ° ' S s - 0.0216K T M - 0 .0014Kc ° ' i s

+ 0.0200Kc o. 3s] + KtO.29 [0.0010Kc o. ss + 0.0020KO.42 + 0.0082KzO. 13

- 0.1666Kc °'24 + 0.1554Kc°'3s]} + a I {1K~-°'29Kc 0-24} , (A.8)

j- 2 ~ - - 0 . 2 9 u - - 0 . 2 4 q . A3 =a l Li~,,-t ,~c J , (A.9)

2 - -0 .29 - -0 .243 A4 =a l {~Kt Kc j , (A.10)

A s = (al + a2){/~tt' 57 [_0.0370KO.aS + 0.0308KO.42 + 0.0024Kc o. 13

+ 0.0038Kc O'3s] + K t 0"29 [--0.0124K T M - 0.0028K °'42

- 0.0140Kc °'13 + 0.0292Kc°'35]} , (A.11)

A 6 = (a 1 + a2) {K~t" s7 [-0.0280K~c" ss + 0.0470K °'42 - 0 .0120Kc °" 13

_ 0.0070Kc O.3s ] + K t 0.29 [_0.0094K o-Ss _ 0.0042K o-42

+ 0.0674Kc o. 13 _ 0.0538KcO.3S]}. (A.12)

To obtain the final hamiltonian we must consider how the operators O1 through 06 mix in the final region from m2c to/~2. Because 03 and O4 transform like a 27- dimensional representation of SU(3) they will not mix with the other operators.

1 They also do not mix with each other since one of them is a A / = $ operator and _ 3 the other is a A / - ~ operator. For the operators O1, 02, 05 and 06 one finds that

under renormalization ]o 5 0 O1

2o o2 02 in A , (A. 13)

"+ 32 Os } 12 05

6 0 3

a result found already by Shifman et al. [7]. Denoting this matrix by F ' , one finds

B. Guberina, R.D. Peccei / QCD effects and CP violation 3 0 9

the eigenvalues

X~ = {14.4443; 7 . 5 1 1 8 ; - 2 . 1 5 2 2 ; - 5 . 3 5 9 4 } .

F ' can be diagonalized by the similarity transformation

T - 1 F ' T = A' .

The relevant matrices T and T -1 are given by

0.22055 -9 .18839 0.01512 0.29840q -0.38529 -1 .94819 0.41546 -14 .66226 I

T= 4.81475 2.50395 -0 .71742 -1 .78647 I 1 j 1 1 1

0.06476 -0 .01131 0.17672 0.130507 T__ 1 = -0 .10680 - 0 . 0 0 2 2 6 0.00364 0.00517~

0.02874 0.07894 -0 .17040 o.844521 " L 0.01330 -0 .06537 -0 .00996 o .o1981j

Forcompleteness we remarkthat underrenormalization

03 ~ a s In A ( - 4 ) 0 3 , 47r

(A.14)

(A.15)

(A.16)

(A.17)

C15 = 0.3118K ° ' 8 ° - 0.2674K °'42 - 0.0206K -°-12 - 0.0238K -° .3° , (A.24)

C12 =-O.0250K°'s°+o.2081K°'42+O.Ol19K-°'12-O.1950K -°'3°, (A.23)

Direct computation, using the above formulas, yields the following expressions for the coefficients Cij, which are non-vanishing:

Cll = 0.0143K °'8° + 0.9813K °'42 + 0.0004K -° '12 + 0.0040K - ° ' 3° , (A.22)

0 4 - + % l n A ( - 4 ) 0 4 . (A.19) 47r

Using the procedure described in the text one can construct the final hamiltonian for AS = 1 processes, which included all order corrections in a s. The structure of the hamiltonian is given by the formula

6

H=-V"2GF ~ BiOi, (A.20) i=1

where the coefficient functions B i are given by

6

Bi = ~ CiiA i • (A.21) /=1

(A.18)


C16 = 0 . 0 6 4 8 K ° ' 8 ° - 0 . 1 0 6 8 K °'42 + 0 .0287K - ° ' 12 + 0 .0133K - ° ' a ° , (A.25)

C21 = -O.O025K°'8°+O.O208K°'42+O.OO12K-°'12-O.O195K-°'3°, (A.26)

C22 = 0 .0044K ° '8° + 0 .0044K °'42 + 0 .0328K - ° ' l z + 0 .9584K - ° ' a ° , (A.27)

Czs = -O.0545K°'8°-O.OO57K°'42-O.O566K-O.12+O.1168K - ° ' 3 ° , (A.28)

C26 = - 0 . 0 1 1 3 K ° ' 8 ° - O . O O 2 2 K ° ' 4 Z + o . o 7 8 9 K - ° ' 1 2 - O . O 6 5 4 K - ° ' 3 ° , (A.29)

C33 = K - ° ' 2 2 , (A.30)

C44 = K - ° ' 22 , (A.31)

Csl = O.0390K°'8°-O.O334K°'42-O.OO26K-°'12-O.OO30K-°'3°, (A.32)

Cs2 =-O.0681K°'8°-O.OO71K°'42-O.O708K-°'12+O.1460K - ° ' 3 ° , (A.33)

6"55 =0.8509K°'8°+O.OO91K°'42+O.1222K-°'12+O.O178K -° '3° , (A.34)

C56 = O.1767K°'8°+O.OO37K°'42-O.1704K-°'12-O.OlOOK -° '3° , (A.35)

C61 = 0 . 0 2 8 8 K ° ' 8 ° - 0 . 0 4 7 5 K °'42 + 0 .0128K - ° ' 12 + 0 .0059K - ° ' a ° , (A.36)

C62 = - 0 . 0 5 0 3 K ° ' 8 ° - 0 . 0 1 0 1 K °'42 + 0 . 3 5 0 9 K - ° ' 1 2 - 0 . 2 9 0 5 K - ° ' 3 ° , (A.37)

C65 = 0.6283K°'8° + O.O130K°'42-O.6059K-° 'I2-O.O354K -° '3° , (A.38)

C66=O.1305K°'8°+O.OO52K°'42+O.8445K-°'12+O.O198K -° '3° (A.39)

References

[1] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [2] S. Pakvasa and H. Sugawara, Phys. Rev. D14 (1976) 305. [3] L. Maiani, Phys. Lett. 62B (1976) 183. [4] J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B109 (1976) 213. [5] L. Wolfenstein, Phys. Rev. Lett. 13 (1964) 562. [6] F. Gilman and M. Wise, Phys. Lett. 83B (1979) 83. [7] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, JETP Lett. 22 (1975) 55; Nucl. Phys.

B120 (1977) 316; JETP (Soy. Phys.) 45 (1977) 670. [8] K. Kleinknecht, Proc. 17th Int. Conf. on High-energy physics, London 1974, ed. J.R.

Smith (Science Research Council, Rutherford Laboratory 1974), vol. 3, p. 23. [9] S.L Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285.


[10] G. Altarelli, R.K. Ellis, L. Maiani and R. Petronzio, Nucl. Phys. B88 (1975) 215. [11] E. Witten, Nucl. Phys. B120 (1977) 387. [12] B.W. Lee and M.K. Gaillard, Phys. Rev. Lett. 33 (1974) 108;

G. Altarelli and L. Maiani, Phys. Lett. 52B (1976) 351. [13] F. Gilman and M. Wise, Phys. Rev. D, to be published. [14] Y. Prokhorov, JETP Lett., to be published.

Documents

Quantum chromodynamic effects and CP violation in the Kobayashi-Maskawa model