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Exclusive charmless non-leptonic decays of the Bd0 meson
Volume 218, number 4
PHYSICS LETTERS B
2 March 1989
EXCLUSIVE C H A R M L E S S N O N - L E P T O N I C DECAYS O F T H E B ° M E S O N Morimitsu T A N I M O T O
ScienceEducationLaboratory,EhimeUniversity,Matsuyama790,Japan Received 14 December 1988
The charmless non-leptonic decays of the B° meson such as B° o K n + and I~°Qare studied in order to test the standard model. The amplitudes from penguin diagrams are studied focusing on the top quark mass, the momentum transfer carried by the gluon, the form factors and the QCD parameter A~-g. The experimental upper bounds of the branching ratios given by ARGUS and CLEO are still high comparing with the predicted values of the standard model.
The rare decays o f the B meson are expected to provide a sensitive test of the standard model. The b-~s7 and b ~ s ~ + £ - processes have been studied intensively [ 1 ]. These processes are very interesting because they depend sensitively on the top quark mass, which may be greater than 50 GeV from some analyses [2] of the recent A R G U S and CLEO results on B°-l] ° mixing [3 ]. On the other hand, the charmless non-leptonic decays of the B meson are also important because the dominant process is expected to be the loop-induced b - , s g transition, the so-called penguin diagrams [ 4 ]. Here we concentrate on the exclusive charmless non-leptonic decay such as B ° - - , K - x + and I~°0 [ 5,6 ] because the A R G U S and CLEO groups have recently reported [7] upper bounds on the branching ratios of the order o f 10 -4, which are of the same order as the naive estimate given by Gavela et al. [ 5 ] and Chau et al. [6]. We examine in detail the amplitude o f these exclusive processes, which depend on the top quark mass, the m o m e n t u m transfer carried by the gluon, the form factors and the Q C D parameter A~-s, and then compare the predicted branching ratios with the experimental upper limits [ 7 ]. The penguin diagram for the b - , s transition induces the effective hamiltonian [6 ] as
HPc'='~2GFa-5~( V'sV*li)[gT~'(1-75)½2ab]((t';'12aq)=xf2Gv-~( ~ V,sV*I;) × [ - ~ (ggL, qg) ((lgy;,b~) + (ggy~qg) (q~y,,bg) + ~ (ggq~) (Q~bg) - 2 (ggq~) ((t~bg) ] +h.c.,
( 1)
where V,j is the KM matrix element [ 8 ], I, is the loop integral function, i runs over u, c and t, and greek indices denote the color label. In deriving eq. ( 1 ), the color algebra relation and a Fierz transformation have been used in order to evaluate the matrix elements of the hamiltonian. The loop integral function L is given as I
I,= M~MW-m~( f x( l -x) ln[m2-k2x(1-x) ] dx 0
I
- J x(1 - x ) In [ M 2 ( 1
\
-x) + m]x-k2x(1
-x)]
d x ) (1 +
m~/2M~v),
(2)
/ 0
where Mw and m, are the W boson mass and the ith type quark mass, respectively, and k 2 is the m o m e n t u m transfer carried by the gluon. Eq. (2) is derived by calculating the lowest order penguin diagrams in the 't H o o f t Feynman gauge. The factor m~/2M~v in the last parentheses is due to the charged Higgs ghost. Here we have 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
481
Volume 218, number 4
PHYSICS LETTERSB
2 March 1989
performed a lowest order calculation of the penguin diagram in order to make the m,-dependence clear. Later we shall discuss the renormalization corrections for the penguin coefficients. In the case of M~v >> m ] >> k 2, the integral function Ii reduces to the well-known result L ~ -~In (M 2 / m 2 ) [46]. But k 2 is not negligible here compared with m2 in the B meson system because the order of k 2 is m 2, and a critical difference exists between the time-like and space-like cases of the emitted gluon momentum. In the case of k2> 4m ~ (time-like), L has an imaginary component derived from the logarithmic integral. The function L is classified according to the value of k 2, using the functions S, F, and F2: I,-
Mw
,
M~v - m ?
(3)
(S_F, +F2)(I+rn2/2M2w),
where the function S is given as follows:
,
~
S = g In m T - ~ - 2 m ~ / 3 k Z - ~ r ( 1
(r-l~ 2 +2m2/k 2) ln_ r + 1] (k2<0),
S = ~ l n m : _ ~ _ 2 m ~ / 3 k 2 + ½ r ( 1 + ~Lr 2) a r c t a n ( 1 / r )
(0
2) l n ( r - l ] -\
S=-~lnm~;-~-Zm~;/3k2-~r(l+2mZ/k
\r+lJ
(4a)
i~nr(l+2m2/k2 )
(4b) (k2>~4m]),
(4c)
where r = x/I 1 - 4 m ~;/k21, and the functions F~ and Fz are given as (R+6+~)2
F , = ½ 1 n m T - 1 - ½ d - ¼ R ( l + d ) ln\R---~_-~--
-~(l+6-R)21n(m2/M2),
(4d)
F 2 = ] In m~ --~--~ (I + 3 ) ( 3 + 2 6 ) + 2 M ~ v / 3 k 2 9
( R + 6 + 1), + - ~ R [ ( l + 6 ) 2-M~v/k 2 ] l n \ ~ -~(l+6-R)
31n(m2/M2),
(4e)
for k2 < ( M w - m i ) 2 F~ = ½In m~ - 1 - ~ 6 - ½R( 1 + 6 ) [arctan ( ( 6 - 1 )/r) - arctan ( (6+ 1 )/r) ] - ~ [(1 +6)2-2M~v/k 2] ln(m~g/M2), 9
9
I
F2=~ In m T - ~ - g ( l + 6 ) ( 3 + 2 6 ) + 2 M 2 w / 3 k
(4f)
2-~R[ (1 + 6 ) 2 - M Z /k 2]
X [arctan ( ( 6 - 1 )/r) - arctan ( (6+ 1 ) / r ) ] - -~[ ( 1 + 6) 2 - 3M2/k 2] ( 1+6) ln(m2/m2),
(4g)
for k2>~ ( M w - m i ) 2. R = x / [ (1 + 6)2-4M2w/k21 and 6= (M2w -rn2, ) /k 2. Now we begin by calculating the partial decay width of the process B 0 __,K - x +. This decay occurs through the penguin process and the Born process as shown in figs. l a - l c . The process of fig. lc is expected to be very small because the KM matrix elements are doubly suppressed. So we neglect at first the process of fig. I c, but discuss the effect of this Born diagram later on. The amplitude of the process B°o-,K-n + is given as follows [6]:
482
Volume 218, number 4
PHYSICS LETTERS B
2 March 1989 5
,
0
k
l
S
5 U
J
J
b
'
I
J
(a)
/u
S U
b
~
E
I" •
U
3
(b)
(c)
Fig. 1. Contributions to B°~K-~ + through (a): time-like penguin, (b): space-like penguin and (c): Born processes. The dashed line and the wavy line denote W boson and gluon, respectively.
2m~.
A(K-~ + time-like)=
)
( m . +m~) ( m b - - m ~ )
)(
, Gv-z~ ~ VibV*Ii ~g A(K-n+ space-like)=-f-~
1+
2rn~
' "~
( mb + rr~-m~--md)) '
(5)
where a and g are given by the vacuum insertion approximation as follows: a= (K-I-gy.75u[0)
( n + lilY.biB°),
g = ( K - n + [ gy~,d [0) ( 0 l - a y . 7 5 b l B° ).
(6)
Following ref. [ 6 ], the calculated values o f a and g are
(mo
a=fK
[(ms+m~):--m~']FB+~(m~')'g=f~\mK+m~/ [(mK+m~)Z--m~s]Faf"ihi(m~)'
(7)
where the form factor F~+~ is assumed to be approximated by a single pole as F +. ~( q
2 ) = F + .(~0 ) ( 1 - q 2 / m ~ . )
-~,
(8)
but this q2 dependence is not crucial for our result, because q2=m2 is very small compared with mB*, 2 and F B+~(q 2 ) is expected to be almost F ~ ( 0 ). We expect F ~ ( 0 ) < 1, but its numerical value depends on the internal structure o f the B and ~ mesons. We find F B+~(0)= 0.333 using the quark model o f ref. [ 9 ] to determine the form factor at q 2 = 0 , which is successful in the D meson and B meson decays. The annihilation form factor F~,ih~ is given reliably by the Q C D calculation [ 10] as +
2,
F~_nnihi ( q 2 ) = i 1 6 z c a ~ f Z / q
(9 )
w h e r e f i s the decay constant o f the meson. Here we c o m m e n t on the physical parameters used in our analysis. The k2-dependence o f the as parameter is given in one-loop approximation as follows: a/-' =(11-~
Nf) l n ( k
2
2 /A~s)/4~,
(10)
with Nf= 4 and A ~ = 0.1-0.2 GeV [ 11 ]. The value o f k 2 will he given definitely in this two-body decay, but at present we are lacking enough knowledge concerning the hadron in order to estimate k 2 exactly, so we adopt the naive value given by G6rard et al. [ 12 ]: ~l m ~2 < ]k2[ < ~ m g"~.
(11)
We use fK = 0.17 GeV, derived from K - --*£ - + v~ [ 13 ], and take tentatively fB~ =fK because of the little experimental information on fB~. Since our result depends mainly on the value offK, the ambiguity on the value offB~ is not too serious in our analysis. The values of the K M matrix elements o f Vu~, V~ and V~b are 0.22, 1 and 0.044_+0.010 [ 14], respectively. On the other hand, the upper bound is only known for Vub [ 14], SO we parametrize as 483
Volume 218, number 4
PHYSICS LETTERS B
Vub=ptV~bl exp(i0)
2 March 1989
(p<0.14,0<0
(12)
Since the predicted branching ratio is proportional to I Vcb] 2, one should take into account a value of 45% for the error in our result. The values of the quark masses used in our analysis are as follows [ 11 ]: (rob, m~, ms, mu, rnd) = (5, 1.5, 0.175, 0.005, 0.009) GeV.
(13)
The fluctuations of these values do not give serious problems in our result. Using the decay amplitude in eq. (5), the branching ratio is given as Br(B ° ~ K - ~ + ) = F ( B ° - , K - K + )/r,o,,
(14)
with
1 1
F(B° ~ K - T t + ) = 8~ m---~-BI ( K -it+ [//pen I B° ) ]2pK~
(l 5)
where PK is the m o m e n t u m of the final K meson, and Ftot = (5.58 + 0.65 ) X 10-13 GeV, which is derived from the experimental value %, = ( 1.18 + 0.14 ) X 10- ~2 s [ 14 ], so the error of the predicted branching ratio originating from rB~ is at most + 10%. The calculation of the process B ° ~I~°~ is similar to the process B ° - * K - K +, but this process occurs only through the penguin diagram, so it is more interesting to investigate this process. In order to evaluate the matrix element of this process, we use the following relations as shown in ref. [ 5 ]: ( O [ ~ ? , s ] 0 ) = go e~*,,
(16)
where gol Qs[ = m ~ / f , with Qs= - ½ a n d f ~/4rc= 15, and e~ (K°L ~7,,b [ B ° )
=2(p,/rno)mBf~K(m2),
(17)
(9K°lgy,~sdl0)
=m,F( m 2) e*~,+ G( m2 ) ( eo.pK ) p,,/ m~:.
(18)
We assume the form factors to be of single pole type as
FB+K(q2)=FB+K(o)(I_ q 2/mB;) 2
-t
, F(qa)=G(qa)=iF(O)(1-q2/rn2~) -' ,
(19)
where rnB: and rn~, are the masses of the vector meson B} and the axial-vector meson K~ (1280), respectively. We take F B+K( 0 ) = 0.379 as given in ref. [ 9 ], and take tentatively F ( 0 ) = - ~ , which is not crucial for our result because F B+K(q2) >> F(q2). The decay amplitude of the process B ° __.~o~ is given as follows: ( I¢°* I Ho~ ] B° ) = A (I¢°0 time-like ) + A (I~°0 space-like), A(K% time-like)= ~
1
a~(
A (g2o, space.like) = ~ _1~ GF _ ~%-( 2 XfBa [rnB +
. "~ 16rn,
G v ~ - _ }-', V~bV~I,)
-3--'£-0 rnBp°FB+K(rn~)'
. 8 2rn~ V~,V,J,)-~(I-(rnb+m~)-(-ms
(mB/mKm,) (mB - - 2 x / ~
+ m 2) ~
]
P,F(m2).
+ma)) (20)
Since F ( m ~) << F BK( m 2), it is expected that the time-like amplitude dominates the process B ° ~ I~°0. We show in figs. 2 and 3 the predicted branching ratio versus m, for k = 3.5, 3.0, 2.5 GeV, where k is defined as k = I x / ~ l , and we take A ~ g = 0 . 1 0 GeV, Vcu=0.044 andf~:=0.17 GeV. These values are smaller than the rough estimates in refs. [5,6] as seen in figs. 2 and 3. Gavela et al. [5] and Chau et al. [6] obtained 0.7X 10 -4 and 1.0X 10 .4 for B ° - - , K - n +, respectively, and 0.5X 10 - 4 and 0.4X 10 -4 for B°--,I~° G respectively. These discrepancies come mainly from the value of FB+~(0) and FB+K(0), which are assumed to be 1 in refs. [5,6].
484
Volume 218, number 4
PHYSICS LETTERS B
B;10-4
2 March 1989
Or 9 B
./-~-/
/
')A
/
/
/
/ /
ELEO
/
/
/_ _i"__,~ _ / _ - / - -/-
ELEO
- "
'A k ~
3.0 2.5 3.5 6eV
3.0 3.5
ZSOe V
K-~
#°0
l(f 6
5o
1do
,
~
15o
eoo
mt
Fig. 2. Predicted branching ratio of Bd°---,K-n+ versus the top quark mass for k=2.5, 3.0 and 3.5 GeV. Arrows A and B denote the predicted values by Gavela el al. and Chau et al., respectively. For comparison, the prediction in the case ofFB+~(0)= 1 for k= 3 GeV is given by a dashed line. The experimental upper bound by CLEO is also shown.
1( 6
I
50
100
i 150
mt i 200 Ge V
Fig. 3. Predicted branching ratio of B° --,I~°0.The notations are the same as in fig. 2, where the arrow B is omitted because B is close to A. The prediction in the case ofF BK(0) = 1 for k= 3 GeV is given by a dashed line. The experimental upper bound of B+ ~K+0 by CLEO is shown.
However, Bauer et al. [ 9 ] gave a branching ratio o f 1.2 X 10 - 5 for B° --, I~°9, which is consistent with our result because the same value o f F ' + K ( 0 ) is used in our calculation. F o r comparison, we show the p r e d i c t e d values in the case ofF~+~(0) =FB+K(0) = 1 at k = 3 G e V by d a s h e d lines in figs. 2 and 3. It is r e m a r k a b l e that the m,-dependence o f the branching ratios is very mild if m~ > 50 GeV, in contrast with the k-dependence. This fact suggests that the m a i n c o n t r i b u t i o n to these decay a m p l i t u d e s is i n d u c e d from c h a r m quark exchange in the case o f a large top quark mass, The m a x i m u m branching ratios are given in the case o f k = 3 GeV, which is related to the value o f 2me. Since the top quark mass is expected to be in the region o f 3 0 - 2 0 0 G e V [11 ] a n d k to be 2.5-3.5 GeV, the branching ratios o f the processes B° ~ K - T t + and B°--,I~% are p r e d i c t e d to be ( 0 . 8 - 1 . 7 ) × 10 -5 a n d ( 0 . 5 - 1 . 2 ) X 10 -5, respectively. F o r B°--,K-Tt +, the experimental upper limits 2.8 × 10 -4 by A R G U S and 0.9 × 10 -4 by CLEO [ 7 ] are still high c o m p a r e d with the predicted values as seen in fig. 2. The u p p e r limit o f the decay B°--*I~°Q is also still high, 4 × l0 -4 given by A R G U S [7] and 0.8 × 10-4 which is the u p p e r limit o f the decay B + ~ K+@ given by CLEO [ 7 ] as seen in fig. 3. N o w we investigate the A m d e p e n d e n c e o f our result, because the penguin a m p l i t u d e is p r o p o r t i o n a l to as. I f we take A~g = 0.15 (0.20) GeV instead o f 0.10 GeV, the value o f a s increases about 14% (25%) in the region o f k = 3 GeV. Then the branching ratio increases about 30%-50%. On the other hand, the value o f I Vcb] has a relatively large e x p e r i m e n t a l error as already discussed, so the predicted branching ratios change depending on V~,b within + 45%. In our calculation we have neglected the Born process o f the process B ° - - , K - n + in fig. 1c. We can estimate this process if the K M m a t r i x element Vubis fixed. We take tentatively p = 0.14 and ~ = zr/2 following eq. ( 12 ), and then we find that the p r e d i c t e d branching ratio changes to the degree o f 21% (mt = 30 G e V ) - 2 3 % ( m, = 200 G e V ) at k = 3 GeV. I f p << 0.14 the branching ratio is hardly changed. Therefore we can safely neglect the Born process o f fig. lc in the decay B° ~ K - r c +. We have calculated the penguin d i a g r a m in a lowest o r d e r a p p r o x i m a t i o n . The leading logarithmic Q C D corrections o f the penguin coefficients have been done by Ponce [ 15 ]. The i m p r o v e d r e n o r m a l i z a t i o n group 485
Volume 218, number 4
PHYSICS LETTERS B
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calculation does n o t change in a significant way the b r a n c h i n g ratio, b u t m a y reduce o u r p r e d i c t e d b r a n c h i n g ratios by 10%-20%. We s u m m a r i z e o u r result as follows. We have calculated the processes B ° ~ K - n + a n d B ° ~ K ° 0 in the framework of the s t a n d a r d model. T h e b r a n c h i n g ratios o f these processes are p r e d i c t e d d e p e n d i n g o n mr, k 2, A~--~, F B+~( 0 ) , F B+K( 0 ) a n d Vcb. T h e p r e d i c t e d v a l u e s are s m a l l e r t h a n the p r e v i o u s estimates in refs. [ 5 , 6 ] . T h e mrd e p e n d e n c e is f o u n d to be m i l d in the region o f rn, > 50 GeV, a n d the k 2 - d e p e n d e n c e to be rather strong. It will be necessary to d e t e r m i n e the value o f k, which is fixed in o u r analysis to be in the region 2 . 5 - 3 . 5 GeV. It is f o u n d that the time-like p e n g u i n a m p l i t u d e d o m i n a t e s the processes B ° - - , K - n + a n d B ° ~I~°O. T h e space-like p e n g u i n c o n t r i b u t i o n s in the b r a n c h i n g ratios are a b o u t 25% a n d 2% in the processes B ° - - , K - n + a n d B °--, I~°~, respectively. T h e precise e s t i m a t e o f the f o r m factors a n d the r e n o r m a l i z a t i o n group calculation o f the p e n g u i n coefficients will i m p r o v e the reliability o f the theoretical predictions. A n e x p e r i m e n t a l test o f the s t a n d a r d m o d e l will be possible in the n e a r future, p r o b a b l y f r o m the d a t a f r o m A R G U S a n d CLEO. T h e a u t h o r w o u l d like to t h a n k Professor M. M a t s u d a for helpful discussions a n d careful r e a d i n g o f the manuscript.
References [ 1 ] P.J. O'Donnell, Phys. LeU. B 175 (1986) 369; N.G. Deshpande et al., Phys. Rev. Lett. 57 (1986) 1106; N.G. Deshpande et al., Phys. Rev. LeU. 59 ( 1987 ) 183; S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59 ( 1987 ) 180; J. Ellis and P.J. Franzini, CERN report CERN-TH. 4952/88 (1988). [2]J. Ellis, J.S. Hagelin and S. Rudaz, Phys. Len. B 192 (1987) 201; X-G. He and S. Pakvasa, Phys. Lett. B 194 (1987) 132; I.I. Bigi and A.I. Sanda, Phys. Lett. B 194 (1987) 307; V. Barger, T. Hart and D.V. Nanopoulos, Phys. Lett. B 194 (1987) 312. [3] ARGUS Collab., H. Albrecht et al., Phys. Lett. B 192 (1987) 245; CLEO Collab., R. Fulton et al., Report at XX1Vth Intern. Conf. on High energy physics (Munich, August 1988), report No. 0708B. [4] A.I. Vainshtein, V.I. Zakharov and M.A. Shifman, Soy. Phys. JETP 45 (1987) 670. [5] M.B. Gavela et al., Phys. Lett. B 254 (1985) 425. [ 6] L-L. Chau and H-Y. Cheng, Phys. Rev. Len. 53 (1984) 1037; 59 (1987) 958; Phys. Lett. B 165 ( 1985 ) 429. [7 ] ARGUS Collab., A. Golutvin, talk at XXIVtb Intern. Conf. on High energy physics (Munich, August 1988); CLEO Collab., A. Jawahery, talk at XX1Vth Intern. Conf. on High energy physics (Munich, August 1988). [8] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [9] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 29 (1985) 637; C 34 (1987) 103. [ 10] G.P. Lepage and S.J. Brodsky, Phys. Lett. B 87 (1979) 359. [ 11 ] W.J. Marciano, BNL report BNL-41489 (1988). [ 12] J.M. G6rard and W-S. Hou, Max-Planck report MPI-PAE/PTh 26/88 (1988). [ 13 ] Particle Data Group, G.P. Yost et al., Review of particle properties, Phys. Lett. B 204 ( 1988 ) 16. [ 14] K. Kleinknecht, talk at XXIVth Intern. Conf. on High energy physics (Munich, August 1988). [ 15 ] W.A. Ponce, Phys. Rev. D 23 ( 1981 ) 1134.
486
PHYSICS LETTERS B
2 March 1989
EXCLUSIVE C H A R M L E S S N O N - L E P T O N I C DECAYS O F T H E B ° M E S O N Morimitsu T A N I M O T O
ScienceEducationLaboratory,EhimeUniversity,Matsuyama790,Japan Received 14 December 1988
The charmless non-leptonic decays of the B° meson such as B° o K n + and I~°Qare studied in order to test the standard model. The amplitudes from penguin diagrams are studied focusing on the top quark mass, the momentum transfer carried by the gluon, the form factors and the QCD parameter A~-g. The experimental upper bounds of the branching ratios given by ARGUS and CLEO are still high comparing with the predicted values of the standard model.
The rare decays o f the B meson are expected to provide a sensitive test of the standard model. The b-~s7 and b ~ s ~ + £ - processes have been studied intensively [ 1 ]. These processes are very interesting because they depend sensitively on the top quark mass, which may be greater than 50 GeV from some analyses [2] of the recent A R G U S and CLEO results on B°-l] ° mixing [3 ]. On the other hand, the charmless non-leptonic decays of the B meson are also important because the dominant process is expected to be the loop-induced b - , s g transition, the so-called penguin diagrams [ 4 ]. Here we concentrate on the exclusive charmless non-leptonic decay such as B ° - - , K - x + and I~°0 [ 5,6 ] because the A R G U S and CLEO groups have recently reported [7] upper bounds on the branching ratios of the order o f 10 -4, which are of the same order as the naive estimate given by Gavela et al. [ 5 ] and Chau et al. [6]. We examine in detail the amplitude o f these exclusive processes, which depend on the top quark mass, the m o m e n t u m transfer carried by the gluon, the form factors and the Q C D parameter A~-s, and then compare the predicted branching ratios with the experimental upper limits [ 7 ]. The penguin diagram for the b - , s transition induces the effective hamiltonian [6 ] as
HPc'='~2GFa-5~( V'sV*li)[gT~'(1-75)½2ab]((t';'12aq)=xf2Gv-~( ~ V,sV*I;) × [ - ~ (ggL, qg) ((lgy;,b~) + (ggy~qg) (q~y,,bg) + ~ (ggq~) (Q~bg) - 2 (ggq~) ((t~bg) ] +h.c.,
( 1)
where V,j is the KM matrix element [ 8 ], I, is the loop integral function, i runs over u, c and t, and greek indices denote the color label. In deriving eq. ( 1 ), the color algebra relation and a Fierz transformation have been used in order to evaluate the matrix elements of the hamiltonian. The loop integral function L is given as I
I,= M~MW-m~( f x( l -x) ln[m2-k2x(1-x) ] dx 0
I
- J x(1 - x ) In [ M 2 ( 1
\
-x) + m]x-k2x(1
-x)]
d x ) (1 +
m~/2M~v),
(2)
/ 0
where Mw and m, are the W boson mass and the ith type quark mass, respectively, and k 2 is the m o m e n t u m transfer carried by the gluon. Eq. (2) is derived by calculating the lowest order penguin diagrams in the 't H o o f t Feynman gauge. The factor m~/2M~v in the last parentheses is due to the charged Higgs ghost. Here we have 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
481
Volume 218, number 4
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performed a lowest order calculation of the penguin diagram in order to make the m,-dependence clear. Later we shall discuss the renormalization corrections for the penguin coefficients. In the case of M~v >> m ] >> k 2, the integral function Ii reduces to the well-known result L ~ -~In (M 2 / m 2 ) [46]. But k 2 is not negligible here compared with m2 in the B meson system because the order of k 2 is m 2, and a critical difference exists between the time-like and space-like cases of the emitted gluon momentum. In the case of k2> 4m ~ (time-like), L has an imaginary component derived from the logarithmic integral. The function L is classified according to the value of k 2, using the functions S, F, and F2: I,-
Mw
,
M~v - m ?
(3)
(S_F, +F2)(I+rn2/2M2w),
where the function S is given as follows:
,
~
S = g In m T - ~ - 2 m ~ / 3 k Z - ~ r ( 1
(r-l~ 2 +2m2/k 2) ln_ r + 1] (k2<0),
S = ~ l n m : _ ~ _ 2 m ~ / 3 k 2 + ½ r ( 1 + ~Lr 2) a r c t a n ( 1 / r )
(0
2) l n ( r - l ] -\
S=-~lnm~;-~-Zm~;/3k2-~r(l+2mZ/k
\r+lJ
(4a)
i~nr(l+2m2/k2 )
(4b) (k2>~4m]),
(4c)
where r = x/I 1 - 4 m ~;/k21, and the functions F~ and Fz are given as (R+6+~)2
F , = ½ 1 n m T - 1 - ½ d - ¼ R ( l + d ) ln\R---~_-~--
-~(l+6-R)21n(m2/M2),
(4d)
F 2 = ] In m~ --~--~ (I + 3 ) ( 3 + 2 6 ) + 2 M ~ v / 3 k 2 9
( R + 6 + 1), + - ~ R [ ( l + 6 ) 2-M~v/k 2 ] l n \ ~ -~(l+6-R)
31n(m2/M2),
(4e)
for k2 < ( M w - m i ) 2 F~ = ½In m~ - 1 - ~ 6 - ½R( 1 + 6 ) [arctan ( ( 6 - 1 )/r) - arctan ( (6+ 1 )/r) ] - ~ [(1 +6)2-2M~v/k 2] ln(m~g/M2), 9
9
I
F2=~ In m T - ~ - g ( l + 6 ) ( 3 + 2 6 ) + 2 M 2 w / 3 k
(4f)
2-~R[ (1 + 6 ) 2 - M Z /k 2]
X [arctan ( ( 6 - 1 )/r) - arctan ( (6+ 1 ) / r ) ] - -~[ ( 1 + 6) 2 - 3M2/k 2] ( 1+6) ln(m2/m2),
(4g)
for k2>~ ( M w - m i ) 2. R = x / [ (1 + 6)2-4M2w/k21 and 6= (M2w -rn2, ) /k 2. Now we begin by calculating the partial decay width of the process B 0 __,K - x +. This decay occurs through the penguin process and the Born process as shown in figs. l a - l c . The process of fig. lc is expected to be very small because the KM matrix elements are doubly suppressed. So we neglect at first the process of fig. I c, but discuss the effect of this Born diagram later on. The amplitude of the process B°o-,K-n + is given as follows [6]:
482
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PHYSICS LETTERS B
2 March 1989 5
,
0
k
l
S
5 U
J
J
b
'
I
J
(a)
/u
S U
b
~
E
I" •
U
3
(b)
(c)
Fig. 1. Contributions to B°~K-~ + through (a): time-like penguin, (b): space-like penguin and (c): Born processes. The dashed line and the wavy line denote W boson and gluon, respectively.
2m~.
A(K-~ + time-like)=
)
( m . +m~) ( m b - - m ~ )
)(
, Gv-z~ ~ VibV*Ii ~g A(K-n+ space-like)=-f-~
1+
2rn~
' "~
( mb + rr~-m~--md)) '
(5)
where a and g are given by the vacuum insertion approximation as follows: a= (K-I-gy.75u[0)
( n + lilY.biB°),
g = ( K - n + [ gy~,d [0) ( 0 l - a y . 7 5 b l B° ).
(6)
Following ref. [ 6 ], the calculated values o f a and g are
(mo
a=fK
[(ms+m~):--m~']FB+~(m~')'g=f~\mK+m~/ [(mK+m~)Z--m~s]Faf"ihi(m~)'
(7)
where the form factor F~+~ is assumed to be approximated by a single pole as F +. ~( q
2 ) = F + .(~0 ) ( 1 - q 2 / m ~ . )
-~,
(8)
but this q2 dependence is not crucial for our result, because q2=m2 is very small compared with mB*, 2 and F B+~(q 2 ) is expected to be almost F ~ ( 0 ). We expect F ~ ( 0 ) < 1, but its numerical value depends on the internal structure o f the B and ~ mesons. We find F B+~(0)= 0.333 using the quark model o f ref. [ 9 ] to determine the form factor at q 2 = 0 , which is successful in the D meson and B meson decays. The annihilation form factor F~,ih~ is given reliably by the Q C D calculation [ 10] as +
2,
F~_nnihi ( q 2 ) = i 1 6 z c a ~ f Z / q
(9 )
w h e r e f i s the decay constant o f the meson. Here we c o m m e n t on the physical parameters used in our analysis. The k2-dependence o f the as parameter is given in one-loop approximation as follows: a/-' =(11-~
Nf) l n ( k
2
2 /A~s)/4~,
(10)
with Nf= 4 and A ~ = 0.1-0.2 GeV [ 11 ]. The value o f k 2 will he given definitely in this two-body decay, but at present we are lacking enough knowledge concerning the hadron in order to estimate k 2 exactly, so we adopt the naive value given by G6rard et al. [ 12 ]: ~l m ~2 < ]k2[ < ~ m g"~.
(11)
We use fK = 0.17 GeV, derived from K - --*£ - + v~ [ 13 ], and take tentatively fB~ =fK because of the little experimental information on fB~. Since our result depends mainly on the value offK, the ambiguity on the value offB~ is not too serious in our analysis. The values of the K M matrix elements o f Vu~, V~ and V~b are 0.22, 1 and 0.044_+0.010 [ 14], respectively. On the other hand, the upper bound is only known for Vub [ 14], SO we parametrize as 483
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Vub=ptV~bl exp(i0)
2 March 1989
(p<0.14,0<0
(12)
Since the predicted branching ratio is proportional to I Vcb] 2, one should take into account a value of 45% for the error in our result. The values of the quark masses used in our analysis are as follows [ 11 ]: (rob, m~, ms, mu, rnd) = (5, 1.5, 0.175, 0.005, 0.009) GeV.
(13)
The fluctuations of these values do not give serious problems in our result. Using the decay amplitude in eq. (5), the branching ratio is given as Br(B ° ~ K - ~ + ) = F ( B ° - , K - K + )/r,o,,
(14)
with
1 1
F(B° ~ K - T t + ) = 8~ m---~-BI ( K -it+ [//pen I B° ) ]2pK~
(l 5)
where PK is the m o m e n t u m of the final K meson, and Ftot = (5.58 + 0.65 ) X 10-13 GeV, which is derived from the experimental value %, = ( 1.18 + 0.14 ) X 10- ~2 s [ 14 ], so the error of the predicted branching ratio originating from rB~ is at most + 10%. The calculation of the process B ° ~I~°~ is similar to the process B ° - * K - K +, but this process occurs only through the penguin diagram, so it is more interesting to investigate this process. In order to evaluate the matrix element of this process, we use the following relations as shown in ref. [ 5 ]: ( O [ ~ ? , s ] 0 ) = go e~*,,
(16)
where gol Qs[ = m ~ / f , with Qs= - ½ a n d f ~/4rc= 15, and e~ (K°L ~7,,b [ B ° )
=2(p,/rno)mBf~K(m2),
(17)
(9K°lgy,~sdl0)
=m,F( m 2) e*~,+ G( m2 ) ( eo.pK ) p,,/ m~:.
(18)
We assume the form factors to be of single pole type as
FB+K(q2)=FB+K(o)(I_ q 2/mB;) 2
-t
, F(qa)=G(qa)=iF(O)(1-q2/rn2~) -' ,
(19)
where rnB: and rn~, are the masses of the vector meson B} and the axial-vector meson K~ (1280), respectively. We take F B+K( 0 ) = 0.379 as given in ref. [ 9 ], and take tentatively F ( 0 ) = - ~ , which is not crucial for our result because F B+K(q2) >> F(q2). The decay amplitude of the process B ° __.~o~ is given as follows: ( I¢°* I Ho~ ] B° ) = A (I¢°0 time-like ) + A (I~°0 space-like), A(K% time-like)= ~
1
a~(
A (g2o, space.like) = ~ _1~ GF _ ~%-( 2 XfBa [rnB +
. "~ 16rn,
G v ~ - _ }-', V~bV~I,)
-3--'£-0 rnBp°FB+K(rn~)'
. 8 2rn~ V~,V,J,)-~(I-(rnb+m~)-(-ms
(mB/mKm,) (mB - - 2 x / ~
+ m 2) ~
]
P,F(m2).
+ma)) (20)
Since F ( m ~) << F BK( m 2), it is expected that the time-like amplitude dominates the process B ° ~ I~°0. We show in figs. 2 and 3 the predicted branching ratio versus m, for k = 3.5, 3.0, 2.5 GeV, where k is defined as k = I x / ~ l , and we take A ~ g = 0 . 1 0 GeV, Vcu=0.044 andf~:=0.17 GeV. These values are smaller than the rough estimates in refs. [5,6] as seen in figs. 2 and 3. Gavela et al. [5] and Chau et al. [6] obtained 0.7X 10 -4 and 1.0X 10 .4 for B ° - - , K - n +, respectively, and 0.5X 10 - 4 and 0.4X 10 -4 for B°--,I~° G respectively. These discrepancies come mainly from the value of FB+~(0) and FB+K(0), which are assumed to be 1 in refs. [5,6].
484
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B;10-4
2 March 1989
Or 9 B
./-~-/
/
')A
/
/
/
/ /
ELEO
/
/
/_ _i"__,~ _ / _ - / - -/-
ELEO
- "
'A k ~
3.0 2.5 3.5 6eV
3.0 3.5
ZSOe V
K-~
#°0
l(f 6
5o
1do
,
~
15o
eoo
mt
Fig. 2. Predicted branching ratio of Bd°---,K-n+ versus the top quark mass for k=2.5, 3.0 and 3.5 GeV. Arrows A and B denote the predicted values by Gavela el al. and Chau et al., respectively. For comparison, the prediction in the case ofFB+~(0)= 1 for k= 3 GeV is given by a dashed line. The experimental upper bound by CLEO is also shown.
1( 6
I
50
100
i 150
mt i 200 Ge V
Fig. 3. Predicted branching ratio of B° --,I~°0.The notations are the same as in fig. 2, where the arrow B is omitted because B is close to A. The prediction in the case ofF BK(0) = 1 for k= 3 GeV is given by a dashed line. The experimental upper bound of B+ ~K+0 by CLEO is shown.
However, Bauer et al. [ 9 ] gave a branching ratio o f 1.2 X 10 - 5 for B° --, I~°9, which is consistent with our result because the same value o f F ' + K ( 0 ) is used in our calculation. F o r comparison, we show the p r e d i c t e d values in the case ofF~+~(0) =FB+K(0) = 1 at k = 3 G e V by d a s h e d lines in figs. 2 and 3. It is r e m a r k a b l e that the m,-dependence o f the branching ratios is very mild if m~ > 50 GeV, in contrast with the k-dependence. This fact suggests that the m a i n c o n t r i b u t i o n to these decay a m p l i t u d e s is i n d u c e d from c h a r m quark exchange in the case o f a large top quark mass, The m a x i m u m branching ratios are given in the case o f k = 3 GeV, which is related to the value o f 2me. Since the top quark mass is expected to be in the region o f 3 0 - 2 0 0 G e V [11 ] a n d k to be 2.5-3.5 GeV, the branching ratios o f the processes B° ~ K - T t + and B°--,I~% are p r e d i c t e d to be ( 0 . 8 - 1 . 7 ) × 10 -5 a n d ( 0 . 5 - 1 . 2 ) X 10 -5, respectively. F o r B°--,K-Tt +, the experimental upper limits 2.8 × 10 -4 by A R G U S and 0.9 × 10 -4 by CLEO [ 7 ] are still high c o m p a r e d with the predicted values as seen in fig. 2. The u p p e r limit o f the decay B°--*I~°Q is also still high, 4 × l0 -4 given by A R G U S [7] and 0.8 × 10-4 which is the u p p e r limit o f the decay B + ~ K+@ given by CLEO [ 7 ] as seen in fig. 3. N o w we investigate the A m d e p e n d e n c e o f our result, because the penguin a m p l i t u d e is p r o p o r t i o n a l to as. I f we take A~g = 0.15 (0.20) GeV instead o f 0.10 GeV, the value o f a s increases about 14% (25%) in the region o f k = 3 GeV. Then the branching ratio increases about 30%-50%. On the other hand, the value o f I Vcb] has a relatively large e x p e r i m e n t a l error as already discussed, so the predicted branching ratios change depending on V~,b within + 45%. In our calculation we have neglected the Born process o f the process B ° - - , K - n + in fig. 1c. We can estimate this process if the K M m a t r i x element Vubis fixed. We take tentatively p = 0.14 and ~ = zr/2 following eq. ( 12 ), and then we find that the p r e d i c t e d branching ratio changes to the degree o f 21% (mt = 30 G e V ) - 2 3 % ( m, = 200 G e V ) at k = 3 GeV. I f p << 0.14 the branching ratio is hardly changed. Therefore we can safely neglect the Born process o f fig. lc in the decay B° ~ K - r c +. We have calculated the penguin d i a g r a m in a lowest o r d e r a p p r o x i m a t i o n . The leading logarithmic Q C D corrections o f the penguin coefficients have been done by Ponce [ 15 ]. The i m p r o v e d r e n o r m a l i z a t i o n group 485
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calculation does n o t change in a significant way the b r a n c h i n g ratio, b u t m a y reduce o u r p r e d i c t e d b r a n c h i n g ratios by 10%-20%. We s u m m a r i z e o u r result as follows. We have calculated the processes B ° ~ K - n + a n d B ° ~ K ° 0 in the framework of the s t a n d a r d model. T h e b r a n c h i n g ratios o f these processes are p r e d i c t e d d e p e n d i n g o n mr, k 2, A~--~, F B+~( 0 ) , F B+K( 0 ) a n d Vcb. T h e p r e d i c t e d v a l u e s are s m a l l e r t h a n the p r e v i o u s estimates in refs. [ 5 , 6 ] . T h e mrd e p e n d e n c e is f o u n d to be m i l d in the region o f rn, > 50 GeV, a n d the k 2 - d e p e n d e n c e to be rather strong. It will be necessary to d e t e r m i n e the value o f k, which is fixed in o u r analysis to be in the region 2 . 5 - 3 . 5 GeV. It is f o u n d that the time-like p e n g u i n a m p l i t u d e d o m i n a t e s the processes B ° - - , K - n + a n d B ° ~I~°O. T h e space-like p e n g u i n c o n t r i b u t i o n s in the b r a n c h i n g ratios are a b o u t 25% a n d 2% in the processes B ° - - , K - n + a n d B °--, I~°~, respectively. T h e precise e s t i m a t e o f the f o r m factors a n d the r e n o r m a l i z a t i o n group calculation o f the p e n g u i n coefficients will i m p r o v e the reliability o f the theoretical predictions. A n e x p e r i m e n t a l test o f the s t a n d a r d m o d e l will be possible in the n e a r future, p r o b a b l y f r o m the d a t a f r o m A R G U S a n d CLEO. T h e a u t h o r w o u l d like to t h a n k Professor M. M a t s u d a for helpful discussions a n d careful r e a d i n g o f the manuscript.
References [ 1 ] P.J. O'Donnell, Phys. LeU. B 175 (1986) 369; N.G. Deshpande et al., Phys. Rev. Lett. 57 (1986) 1106; N.G. Deshpande et al., Phys. Rev. LeU. 59 ( 1987 ) 183; S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59 ( 1987 ) 180; J. Ellis and P.J. Franzini, CERN report CERN-TH. 4952/88 (1988). [2]J. Ellis, J.S. Hagelin and S. Rudaz, Phys. Len. B 192 (1987) 201; X-G. He and S. Pakvasa, Phys. Lett. B 194 (1987) 132; I.I. Bigi and A.I. Sanda, Phys. Lett. B 194 (1987) 307; V. Barger, T. Hart and D.V. Nanopoulos, Phys. Lett. B 194 (1987) 312. [3] ARGUS Collab., H. Albrecht et al., Phys. Lett. B 192 (1987) 245; CLEO Collab., R. Fulton et al., Report at XX1Vth Intern. Conf. on High energy physics (Munich, August 1988), report No. 0708B. [4] A.I. Vainshtein, V.I. Zakharov and M.A. Shifman, Soy. Phys. JETP 45 (1987) 670. [5] M.B. Gavela et al., Phys. Lett. B 254 (1985) 425. [ 6] L-L. Chau and H-Y. Cheng, Phys. Rev. Len. 53 (1984) 1037; 59 (1987) 958; Phys. Lett. B 165 ( 1985 ) 429. [7 ] ARGUS Collab., A. Golutvin, talk at XXIVtb Intern. Conf. on High energy physics (Munich, August 1988); CLEO Collab., A. Jawahery, talk at XX1Vth Intern. Conf. on High energy physics (Munich, August 1988). [8] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. [9] M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 29 (1985) 637; C 34 (1987) 103. [ 10] G.P. Lepage and S.J. Brodsky, Phys. Lett. B 87 (1979) 359. [ 11 ] W.J. Marciano, BNL report BNL-41489 (1988). [ 12] J.M. G6rard and W-S. Hou, Max-Planck report MPI-PAE/PTh 26/88 (1988). [ 13 ] Particle Data Group, G.P. Yost et al., Review of particle properties, Phys. Lett. B 204 ( 1988 ) 16. [ 14] K. Kleinknecht, talk at XXIVth Intern. Conf. on High energy physics (Munich, August 1988). [ 15 ] W.A. Ponce, Phys. Rev. D 23 ( 1981 ) 1134.
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