Impact of a large ƒB on rare kaon decays

Physics Letters B 278 ( 1992 ) 330-336 North-Holland

PHYSICS LETTERS B

Impact of a largefB on rare kaon decays C.Q. Geng and P. Turcotte Laboratoire de Physique Nucl#aire, Universit# de Montreal, C.P. 6128, Succ. A, Montreal, PQ, Canada H3C 3.17

Received 1 November 1991; revised manuscript received 13 January 1992

We examine the effects of a large B-meson decay constant fs which has been obtained by recent lattice QCD calculations on rare K decays. We find that the short-distance contributions to the branching ratios of CP conserving decays such as KL--*Iali and K +~n+v9 are much smaller than previously expected and estimated to be in the tighter ranges of (0.2-1.5)× 10-9 and (0.41.6) × I0 -1°, respectively, whereas that of CP violating ones KL--,n°e+e- and KLan°V9 are enhanced with fa=250+ 50 MeV, me= 1.2-1.8 GeV and mr= 90-200 GeV.

It is well known that the rare K decays such as KL--*l.tlTt, K+--*n+vg, KL--*•°e+e - and KL--,n°V9 involving flavor-changing neutral current interactions are forbidden in the lowest order but they can occur radiatively through the one-loop diagrams. In the standard model the predictions on these rare K decays depend on the tquark mass and the elements Vii ( i = d , s) o f the Cabibbo-Kobayashi-Maskawa ( C K M ) mixing matrix [1] which are constrained by the measurements o f the C K M parameters I Vcbl and I Vub//Vcbl, the C P violating parameter ~ in the K ° - K ° system and the Bd-Bd o -o mixing. Apart from the errors from the above experimental measurements, for a given mr, the main uncertainties o f the C K M elements arise from the various hadronic matrix elements such as the B-meson decay constant fB and bag parameters BK and BB o f K and B mesons, respectively. Recently, several lattice Q C D calculations [ 2 - 4 ] have shown that fB would he by a factor two or more higher than the conventional values quoted in the literature [ 5 ]. It should be noted that large values o f f s have been also obtained in other strong interaction models such as the Q C D sum rule [ 6 ] and the potential model [ 7 ]. As emphasized in ref. [ 8 ] the result o f a largefs has a strong impact on the elements of the C K M matrix and leads to much larger C P violating effects in B decays such as the C P violating asymmetry in B--.J/~-I-Ks decay than previously estimated [ 9 ]. In this letter we study its implications on the short-distance contributions to both C P violating and conserving rare kaon decays. We will show that the upper bounds of the branching ratios for C P conserving decays such as K ÷ ~ n + v 9 and the short-distance part o f KL~ ~tft decrease significantly and change weakly by the variation o f the t-quark mass whereas for the C P violating modes, K L ~ n ° e + e - and KL--,n°Vg, both upper and lower bounds increase for a higher value offB. Because of the smallness o f the branching ratios for the CP conserving decays, we will include the uncertainty from the mass of the charm quark in our discussions. We use the Wolfenstein parametrization [ 10 ] o f the C K M matrix. It contains three unknown parameters A, p and q constrained by the experimental data o n Vcb , [ Vub/Vcb [, ~. and the Ba-Ba o -o mixing parameter defined as )ca = A M ~ F , which give [ 11 ] [ Vcb [ =A22=0.044__+ 0.007,

(1)

I Vub/Vcb 1 = 2 ~

(2)

330

=0.11 + 0 . 0 4 ,

0370-2693/92/$ 05.00 © 1992 Elsevier SciencePublishers B.V. All rights reserved.

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MKf2KBK'2A22%l[--tlccB(xc)+rlctB(xc, xt)+thtA2,~*(1--p)B(xt)]

I¢l = ~

= ( 2 . 2 7 2 + 0 . 0 2 2 ) X 10 -3 , 2 GF

2

(3) 2 6

Xd= -~2 Ma~B~znrlaMw A 2 [ (1-p)E +rlZ]B(xt)=0.71+_O.11 ,

(4)

respectively, where

B(x,)=

xi (

3 - 9x~

1+ (x,_l)---- +

6xiz In xi'~

x, xj ( ( x2 - Sxj + 4 ! ln xj 3 ) B(x~, xj) = - ~ - \ (xj - 1 )2(xj - x ~ ) - 2( 1 - x ~ ) ( 1 -xj) + (x~-~xj) ,

(5)

2 2 with x, = m~/Mw, i = c, t. The value of I Vcb I in eq. ( 1 ) is obtained by combining the results from the inclusive and exclusive decays while in eq. (2) we have included the theoretical error, quoted in ref. [ 8 ]. The constraints from I~l in eq. (3) and Xd in eq. (4) depend very strongly on the inputs ofBK a n d f 2 B B , respectively, which have large theoretical uncertainties. For the bag parameters, estimates of BK vary in the range from 0.4 to 1.5 and BB is expected to be O ( 1 ). As ref. [ 8 ], we take

BK=0.8 +0.2,

Ba---1.

(6)

We note that the uncertainty from Ba can be absorbed infn. The factor fn has been estimated based on various Q C D models ~. But the results vary considerably. It was widely accepted in calculating the Bd-Bd° -o mixing parameter in the literature that 100
(7)

As comparison we also include lower values offB, fB = 1 3 0 + 4 0 M e V ,

(8)

in our discussions. The remaining parameters in eqs. (3) and (4) are the same as that in ref. [ 13 ] except the values of Mw = 80.6 + 0.4 GeV and zB = ( 1.18 + 0.11 ) × 10- t 2 s . We now describe our strategy of fitting the data. We use the ;(2 minimization program M I N U I T . For a given t-quark mass we fit the three parameters A, p and r/with the four constraints given by eqs. ( 1 ) - ( 4 ) . In our numerical evaluations, the errors from the various parameters are combined in quadrature. In fig. 1 we show the best fits 2 =~min 2 Imt within the range 89 < m t < 200 GeV. The C K M parameter A, p and t/corresponding to the best fits for different values of m t are displayed in fig. 2 with the solid curves. The dotted and dashed curves in fig. 2 are the upper and lower bounds of the parameters, respectively, for values o f z 2 equal to one standard deviation above Z2in. F r o m fig. 1 we see that the perfect fits, i.e., Z2in = 0 , occur at m t = 135 and 138 GeV for fB-- 130 + 40 MeV and 250 + 50 MeV respectively. We emphasize that the higher (lower) fB in eq. (7) (eq. (8) ) favors positive (negative) p and slightly larger r/while in all cases the parameter A is not far from its center value as shown in fig. 2. We note that the discontinuity of the upper bound o f p at mt = 162 GeV in fig. 2a results from ~ For reviews, see refs. [ 8,12 ]. 331

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0.8

f -2

0.6 0.4 0.2

Fig. 1. The best fits ,&. as function of m, for fa=250 k 50 MeV (solid curve) and& = 130 k 40 MeV (dashed curve).

~~~

125

150

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100

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200

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O 100

I

I

I

I

125

150

175 ml

I

,

200

(GeV)

0

100

Fig. 2. Values of parameters A, p and q allowed by fits as function of m, with m,= 1.5 GeV and (a) fs= 130 k 40 and (b) fB= 250 k 50 MeV. The solid curves correspond to the best fits, &.. The dashed and dotted curves are boundaries taking into account a change in x2 by one unit with respect to &..

the appearance of a minimum with a positive value ofp. Next we study the consequences of the tits for rare kaon decays. Due to the prospects of significantly improving the precision in the study of rare processes in the K system at BNL-AGS, CERN, Fermilab, KEK and the future kaon factory, it is very important to know the prediction as well as its uncertainty of each process in the standard model. We shall concentrate on the following four crucial rare modes: KL+ up, K+ -+rr+vij, KL+rroe-eand K,_-+rrOti. Recently the short-distance contributions to these decays and their connections in the CKM parameters have been extensively considered in refs. [ 13,141. However, in these studies the B-decay constantf, was assumed to be less than 200 MeV. As shown in fig. 2, the factor fB plays an important role in determining each of the CKM parameters in the fits, especially the sign of p. 332

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Therefore a largef, such as the value in eq. (7) will certainly change the previous conclusions on the rare kaon decays. Before presenting our results, we briefly review the formulas for the standard model contributions to the decay branching ratios. For Kc-+pP the branching ratio due to the short-distance contribution from one loop diagrams is [ 131 Br(KL+pp)s,=4.06x

lo-‘OA4( 1-P)~C:(X,)

,

(9)

where the function C,(x,) is given in ref. [ 13 1. The charm quark contribution is about four orders of magnitude smaller than the unitarity limit from the two-photon intermediate state and thus it has been neglected. For a given m,, the branching ratio depends only on A2 ( 1 -p). For three generations of neutrinos the branching ratio of the decay K+ +tr+vO is given by [ 13-l 71 Br(K+-+rc+vV)=7.01 x lo-’ xQ&,

1,

[I( -xe_x

2 2xc

c

B,QxQ

(?

+A4A8q21 C”(X,, xa) I2

1

In x, -qBx, In x,) + fqzxc In x,-+x,

+A214( 1 -p)C,(x,,

xa)

>

(10)

9

where

(11) and the QCD correction factors [ 16- 18 ] Viz=

12n {$K&/23&,6/25( 1 _~;db/‘S)+ +;2/=K;;/=( Ly,(m,) ln x,

1 _&,lW5)

+K;~25[fK$23(l-K;($3)+&K;~23(1-K&3/23)]), qB=

12a a,(m,lnx,.

[1-K:d”(2-K&/“)]

,

q&Q= (y (;2)zlnx s c

[Kp,“25-K;i25(2-K,y..‘23)] Q

,

(12)

where K,,= a,( tj) /a,( 0. In this decay the c- and t-quark contributions are of the same order of magnitude and the branching ratio depends on both A2( 1 -p) and A2q. For the direct CPviolating parts of K,_+n’e+e- and K,+n”W, we have [ 13 ] Br(KL+rroe+e-),i,=2.6X10-‘4A4~2(C$+C~)

(13)

and Br(K~+&V)~~r=4.61X10-“A4~2]C,(~~,0)(2,

(14)

where Cv and C, are defined in eqs. (3.20) and (3.2 1) of ref. [ 131 #’ and the charm quark contribution in eq. ( 14) has been neglected. ** In eqs. (3.2 1) of ref. [ 131, the factor - 17 in the function F, (xJ is replaced by

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We now use our fits and eqs. ( 9 ) - ( 1 4 ) to study the rare kaon decays. For each decay we first obtain the predictions for the best fits a n d then we look for the ranges of the decay branching ratios with all possible C K M parameters by allowing a change in Z ~ by one unit with respect to the best fits X2i, for a given m t. With m e = 1.5 GeV and AQCD= 150 MeV the results are presented in figs. 3 and 4 for fB = 130 + 40 MeV and 250 + 50 MeV, respectively. We remark that the results of the lower value offa here are consistent with that in ref. [ 13 ] where a slightly different set of the parameter inputs and a different method of the fit were used. As can be seen from figs. 3 a n d 4 for the branching ratios of C P conserving decays: K L ~ t l l a n d K ÷ - , n + v g , with the higherfa in eq. (7), the upper b o u n d s and the predictions with respect to ,~min 2 are much smaller than the values with the lowerfa in eq. (8) and, moreover, the allowed ranges become tighter a n d less sensitive to the change of the t-quark mass. Explicitly, we find the following branching ratios: Br(KL--'~tli)SD = (0.2-0.5) × 10 -9 to (0.6-1.3) X 10 -9 , B r ( K + - * n + v g ) = ( 0 . 5 - 0 . 8 ) × 10 -1° to (0.7-1.2) × l 0 - ' °

(15)

forfs = 250 _+50 MeV a n d mc = 1.5 GeV with mt = 90-200 GeV while the corresponding branching ratios for the lower values, fa = 130 _+40 MeV, are given by Br(KL--,~tft)sD = (0.2-1.2) X 10 -9 to ( 0 . 6 - 5 . 2 ) × l 0 -9 , B r ( K + - - , n + v g ) = (0.5-1.7) × l 0 -1° to ( 0 . 8 - 3 . 9 ) × 10 -1° .

(16)

We note that the lower b o u n d s in eq. (16) are lower than that predicted in ref. [ 13 ]. The m a i n reason for this is that the parameter Amin=0.9 was used in ref. [13] and here, as shown in fig. 2a, it can be as low as 0.73. However, there is hardly any useful b o u n d on mr, that can be extracted from the new measurements [ 19 ] on KL--*~t at K E K a n d B N L - A G S for both lower and higherfn ~3. ~3 A lower bound, mt>~115 GeV extracted from the study of the long-distancecontributions on KL--*btQin ref. [20] was based on the center value of an early measurement [Br(KL--,ktli) = (8.4_+ 1.1 ) × 10 -9 ] at KEK [21 ]. 6

4 3

1' .

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o

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150

175 200 m, (OeV)

1 0

.....i T " i 100

125

,

I

150

,

r

,

I

175 200 m t (GeV)

Fig. 3. The short-distancecontributions to the branching ratios of KL--,p.Ji;K+~n+vg; KL~n°e+e-; and KL--*7~°V9as function of mr with me= 1.5 GeV andfB= 130+40. The legend is the same as in fig. 2. 334

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6

,.~. :3.

1' -.c.

5

3 4

3

& 2

rn

% 1

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26 March 1992

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,~

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%

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Fig. 4. Same as fig. 3 but withfa=250 + 50 MeV. Conversely, for the larger value offB, the direct CP violating contributions to the branching ratios in KL--, n°e + e and K L ~ n°V9 decays are enhanced especially for the lower bounds. It is very interesting to see that, for the higher )ca, the decay KL-,x°e+e - is dominated by the direct CP violating part #4. But due to the potential background coming from the decay mode KL~Y'/e+e - and the smallness o f the branching ratio, it is a challenge for the experiments to isolate this direct CP violating contribution. The above significant effects on both CP conserving and violating parts o f rare kaon decays arise from the use o f the constraint on the C K M parameters in eq. (4) given by the B°-l] ° mixing where the parameterfB enters. Clearly, a precision measurement on rare kaon decays could provide another or even substitute this constraint. For example, the precision measurement on the K+--.n+v9 mode at a future phase o f E787 at B N L - A G S or kaon factory [23 ] would lead to such a goal since the standard model prediction in (10) is very clean. This, in turn, would help to justify the case o f h a v i n g f a in eq. (7) or eq. (8). Since the predicted branching ratios o f the CP conserving decays are much smaller than previously estimated and weakly depending on m t for having a largefB, it is necessary to check the uncertainty from the charm quark on the results. We have performed the fits with various me. We find that the branching ratios o f the decays K + ~ n + v 9 and KL--~l.tlTt (KL--,n°e+e - and KL---~n°Vg) increase (decrease) for increasing mc where the changes of the branching ratios in K L ~ ~tft and CP violating decays mainly arise from the changes o f p and r/parameters in the fitting, respectively. However, the rnc dependences of the branching ratios are all weak. Explicitly, for fB= 250 + 50 MeV, by allowing 1.2 ~
0 . 4 × 10-1°~
1.2× 1 0 - 1 2 ~ B r ( K L - , x ° e + e - )dir ~ 8 . 6 X 10 -12 ,

0 . 8 × 10 - I I ~
(17)

Finally, we remark that (a) with a even largerfa than the value in eq. (7) the ranges o f the first two branching ~4 The CP conserving and indirect CP violating contributions to the branching ratio in KL--,n°e+e- are estimated to be O ( 10-13) and 1.5 × 10-12 respectively (see ref. [22 ] and references therein). 335

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ratios in eq. ( 17 ) b e c o m e e v e n tighter w h e r e a s the last two o n e s are m o r e e n h a n c e d ; ( b ) i f we change the c e n t e r v a l u e o f BK in eq. ( 6 ) to 0.7 or 0.9, the CP c o n s e r v i n g decays are not m u c h a f f e c t e d but the d i r e c t CP v i o l a t i n g c o n t r i b u t i o n s to the d e c a y b r a n c h i n g ratios o f KL-+/t°e + e - a n d KL--~/t°v9 increase by 25% o r d e c r e a s e by 15%; a n d ( c ) o u r results are i n s e n s i t i v e to the c h o i c e o f the Q C D scale AQCD. We t h a n k G. B61anger, S. G o d f r e y , D. L o n d o n , Y. K u n o a n d P.J. O ' D o n n e l l for discussions. W e t h a n k R. M e n g for f o r w a r d i n g the p a p e r by H a r r i s a n d R o s n e r [24] after the c o m p l e t i o n o f the p r e s e n t letter. In t h e i r paper, the d e c a y K + - ~ x + v 9 has b e e n s t u d i e d w i t h f e = 340 MeV. T h i s w o r k was s u p p o r t e d in part by the N a t u r a l Science a n d E n g i n e e r i n g R e s e a r c h C o u n c i l o f C a n a d a a n d by F o n d s F C A R du Qu6bec.

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