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Final-state-interaction and SU(3)-breaking effects in D0→ππ, KK̄
Physics Letters B 280 (1992) 281-286 North-Holland
PHYSICS LETTERS B
Final-state-interaction and SU ( 3 )-breaking effects in
rcg, KI(
Ling-Lie Chau Department of Physics, University of California, Davis, CA 95616, USA
and Hai-Yang Cheng Institute of Physics, Academia Sinica, TaipeL Taiwan 11529 Received 6 January 1992
Based on the quark-diagram scheme and the recent measurement of D °--,~%°, we analyze the decays D°--*~+ ~-, ~t°n°, K +K-, and K°I(°. To fit the data, we find that there must be both final-state-interaction and SU(3)-breaking effects: of the former, nonzero elastic phase shifts in both gg and KI~ are determined and no inelasticity in phase shifts is required; the SU(3 ) breaking effects are at the maximal level of 30% and of particular interest are those in the spectator W-emission and the non-spectator Wexchange diagrams.
The ratio F ( D °--, K + K - ) / F ( D ° - + ~ + n - ) presents a long outstanding problem: While the theoretical expectation lies in the range o f 0.9-1.4 [ 1,2 ] experimentally it deviates from unity substantially. The previous world average o f this ratio [ 3 ] was 3.95 + 1.24, b u t the recent i m p r o v e d m e a s u r e m e n t s by A R G U S [ 4 ], CLEO [ 5,6 ] a n d E691 [ 7 ] e x p e r i m e n t s gave smaller values 2.0-2.5. Nevertheless, the discrepancy between experimental results a n d theoretical expectations still remains. It has long been p o i n t e d out that the d e v i a t i o n o f this ratio from unity is caused b y final-state interactions ( F S I s ) or SU ( 3 ) - b r e a k i n g effects [ 8,1,2 ] or both. In the quarkd i a g r a m scheme [ 8,9 ] ~ such effects can be systematically studied. In o r d e r to resolve this puzzle, d a t a o f all four decay channels D ° - + K + K - , K ° I ( °, n+Tt - , ~o~o are needed. W i t h the recent m e a s u r e m e n t o f D°--,x°n ° by CLEO [ 6 ], we can finally m a k e a meaningful analysis. In this paper, we present our analysis a n d report the FSIs a n d SU ( 3 ) - b r e a k i n g effects we have f o u n d in these decays. Let us first s u m m a r i z e in table 1 the recent e x p e r i m e n t a l m e a s u r e m e n t s o f the branching ratios o f D°-+KI(, 7t~ by A R G U S [ 4 ], CLEO [ 5,6 ] a n d E691 [ 7 ], where the branching ratio [ 10 ] BR ( D O~ K - x + ) = (4.2 + 0.6 ) % has been used for normalization. In the below we shall take the average value (0.46 + 0.05)% for the branching ratio o f D ° - - , K + K - a n d (0.21 + 0.03)% for D ° - + n + x - . Two quantities o f interest are #~ Some misprints in the quark-diagram amplitudes of D°--,K+K- and K°I(° are corrected in eq. (3) of this paper. Table 1 Recent experimental measurements of the branching ratios D °--,KR, xft (in units of 10- 2). Collaboration
BR(D°-.K+K - )
BR (D°-~K°I(° )
BR(D°-~Tr+g -)
BR(DO-~% ° )
ARGUS CLEO E691
0.42 + 0.08 0.49+0.08 0.48 + 0.09
0.13+0.07
0.17 __0.04 0.21 __.0.05 0.25 + 0.06
0.09+0.03
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F ( D ° ~ K + K - ) + F ( D ° ~ K°I~°) F(D°~K+K - ) F(DO~n+n _) +F(DO~nOnO ) , R+_ = F(DO~n+n _) .
(1)
The experimental results of table 1 give R=2.0+0.4,
R+
=2.2+0.4.
(2)
In terms of the quark-diagram amplitudes, the decay amplitudes of the above four decay modes are [ 9 ] A ( D ° ~ K + K - ) = ( s , c , ) { d + ~¢+ ( 5 8 + 2 8 ~ v ) - ½[ d + ~g- ( S f f - 88) ] [1 - e x p ( - i ~ )
1} exp(iSo~),
A ( D ° ~ K°I(°) = (s, c, ){ (5 c¢_ 58) + ( 5 8 + 2 8 ~ ) + ½[ d + ~¢- (5 ~¢- 58) ] [ 1 - e x p ( - i A r . ~ ) ] } exp (i8~J~) , A ( D ° ~ n + x - ) = - (sic,){M+ i f - ( 8 8 + 28~v) - ~ ( M + ~ ) [ 1 - e x p ( - iA~) 1} exp ( i O U ) , A ( D ° ~ x ° n °) = (½V/2) (s, c, ) { ~ -
if+ ( 8 8 + 2 8 ~ ) - ~ ( d + ~ ) [1 - e x p ( - i A ~ )
]} e x p ( i g U ) ,
(3)
where s l - sin 0, ~ I V~I ~ I W~l, cl = cos 0~ ~ I W~l ~ I V~ol, AK~ ----( go -- 51 )Kilo and A~ = ( g o - ~2).~. (The g;s are phase shifts of isospin I. ) In eq. (3) ~t is the external W-emission amplitude, ~ is the internal W-emission amplitude, (g is the W-exchange amplitude (the underline ~ denotes the W-exchange graph involving the creation of gs), ~ is the horizontal W-loop amplitude with the loop-quark i. (The one-gluon-exchange approximation of the 8 graph is the so-called "penguin diagram".) ~ is the vertical W-loop amplitude. These quarkdiagram amplitudes have well-defined meaning with all QCD strong-interaction effects included. In eq. (3), 8 ~- ~-(g, 58-8~-do, 8~-~-~; they characterize potential SU(3)-violating effects in these quark-diagram amplitudes. The FSIs are expressed by the phase shifts, the O's, which in general have both real and imaginary parts; the real parts are related to the elastic scattering effects while the imaginary parts indicate effects of inelasticity [ 9 ] (see also footnote 1 ). In the absence of FSIs and SU (3) breaking, it is evident from eq. (3) that A (D°-~K+K - )/A (D°~Tr+~ - ) = 1 . Fro m our previous analysis of D + ~ I~°x ÷, D O~ K - x +, I~°~°, we had obtained [ 9 ] [~¢+:~[=(I.48_+0.11)×10-6GeV,
~¢+(g - - - 1
~-~ - -
=2.02_+0.22,
(4)
and A ~ = 79 ° _+ 11 °. Given these values of (~¢ + ~g) and (~¢ + ~ ), the independent parameters in all four decays are ( 8 8 + 2 8 ~ r ) , (5 ~g-Sg), A~, Av,g, 8 ~ and gg~i~ [see eq. (3) ]. It seems that the data should be easily fitted since there are more free parameters than the number of decay rates to be fitted. Surprisingly, we find that we cannot fit the data with these free parameters, including the introduction of the imaginary parts in gg~ and goI~. We need to introduce SU ( 3 )-breaking effects in the amplitudes ~¢ and ~gso that ~trat # ~1,~, ~gr,x # %~. From eq. (3) we see that in the absence of FSIs in the x+~ - and xo~o modes, the only free parameter in the decay amplitudes is the SU (3)-breaking term ( 5 8 + 2 8 ~ ) and it cannot fit the measured rates of both 7r+n and 7r°x°. (The calculated branching ratio of D°--,n°~ ° would be too small by a factor of 3. ) From this very general observation, it is evident that the FSIs must be in D°--,~r+~r -, n°Tr° decays. Allowing the FSIs, the two parameters A~ and ( 5 8 + 28~:) obey the equations F ( D ° ~ n + n - ) + F ( D ° ~ n°~r°) = (s~ c~ )2 (~¢+ (g)z. 9 ( 1 - ~ x + x ~) [ exp (igg ~) [ 2 (p.s.) , F(D°--, x + ~ - ) - F ( D ° - * ~°n °) (S1Cl)2(~¢.4. ~ Z . l r23
5X'4"3X2"4-IN//2 ( 5 - 6 x ) cos d ~ ] [exp(ig~ ~) [2(p.s.)
(5)
where x = (8 8 + 25 ~ ) / ( ~ + ff ), (p. s. ) is the phase-space factor, and the value ( ~¢ + (g) / ( ~¢ + ~ ) ~ 2 from eq. (4) has been used. Fitting eq. (5) to the observed rates of D ° ~ g without assuming inelasticity, i.e. I exp(ig~ ~ )1 = 1, yields (88+28~)/(~+ 282
cg)~O.05,
'~ - g z' ~ ~~ 8 3 ° , A,~,~=go
(6)
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where only the central values are given. We see that the elastic FSIs alone can fit the D o . n + g - , ~ogo rates. In the future, the value of I~¢+ 81 of D - - . ~ can be independently determined from experiment by measuring the Cabibbo-suppressed decay mode D + ~ x + ~ °, whose quark-diagram amplitude is given by A ( D + ~n+~o) = ( 1/x/~) (s~c~) (~¢+ 8 ) exp (ig~ ~ ). From A (D + ~I(°~ + ) = (S~Cl) (~¢+ 8 ) exp (i8~/"2), one can easily obtain the following long outstanding prediction [ 8 ]
r(D+--,n+n°) /r(D+--,K°n +)=½1 V¢a/V¢slZ(1.07) ,
(7)
provided that there are no SU (3)-breaking effects in the amplitude (~¢ + 8 ), i.e., (A + B ),~ = (A + B ) ~ . (We do not expect inelasticity in both channels since they are exotic. ) From the known value of I V~a/Vcs]2, we can predict from eq. (7) the branching ratio of D + ~ + ~ ° to be 0.08%. It then can be compared with the value obtained from D-~ I(~, eq. ( 4 ). If future experimental measurements give (A + B ),~ substantially different from (A + B ) i~=given by eq. (4), we shall then need to modify our analysis presented here. We now turn to the decay modes D ° ~ K + K - , K°I~°. As in the D ° - ~ case, we also need FSIs for the same reason that without them the decay amplitudes will have only one free parameter (8 c g - 8 g ) and it certainly cannot fit the two D--.KI( decay rates simultaneously. When FSIs are put in, the two free parameters Ara~ and (~ cg_~g) satisfy the equations F ( D O ~ K + K - ) + F(DO~KOi~O) = ( s I el )2(d2~j I_ ( ~ ) 2 [ ( 1 - l - x ) 2-]- ( x - ~ y ) 2 ] Iexp (i60K~')I2(p.s.) ,
F(D°~K+K-)-F(D°~K°f~°)=(SiCl)2(~t+ ~)2[(l+x)2-(x+y)2]lexp(iSo~)lZ(p.s.)cosAr~,
(8)
with y = (8 ~¢- 8 8 ) / (~¢ + if). However, fitting eq. ( 8 ) to the data given in table 1 we cannot find a solution for z/v,~ and (8 c¢_ 8 ¢ ). Allowing inelastic effects in 8or'x does not help solving the difficulty because the imaginary component ofS~¢~ will suppress the decay rate of D ° ~ K K . Next, we ask if we can put in inelastic effects in 8gn so that there is one more free parameter, Im ~ , in fitting D ° ~ n ~ decays and the value of ( 8 g + 2 8 ~ ) can be increased to help to fit D ~ KI( rates. Having inelastic effects in the I = 0 amplitude of the ng channel is reasonable since there are two known 0 + resonances fo (975 ) and fo (1400), which couple to both K I ( and n~ channels. It is plausible to assume 8~n ~ 0 owing to the absence of higher isospin resonances. However, we find that this alternative does not work. To compensate the decreasing effect from inelasticity in 8~n, we need constructive contribution from (8 g + 2 8 ~ ) to D ~ n ~ in order to accommodate data. Since this term then contributes destructively to D--.KI( decays [el. eq. (3) ], so it does not help to fit D°--.KI( decays. Therefore, no inelastic effects are called for and elastic scattering phase shifts are adequate. This means that we finally need to retreat to allow SU (3) violation in the amplitudes J and cg. It has been advocated that the penguin contribution in the W-loop amplitude may help to explain the ratio R+ _ [ 11 ]. Our analysis indicates that the W-loop amplitudes are small [cf. eq. (6) ]. In fact, a model calculation of the penguin contribution gives [ 12] 8g/d=
- ~ols[ (rnZ~-m2)lm2]{1 + 2m2/[(mu + ms) ( m e - m s ) l } ,
(9)
with ors being the strong-interaction coupling constant. It is clear that the penguin effect is rather small, 8 g / ~¢ ~ 1%. Another empirical argument against a significant 8g is as follows [ 13 ]. From the measured decay rates of D + ~ I ( ° n + , I(°K + and D + ~ I ( ° K +, we obtain I d + S I ~ 1.5× 10 -6 GeV, I d - ~ + S g l ~ 3 . 5 × 10 -6 GeV, and 18 + ~ l ~ 2 × 10 -6 GeV. Obviously, they are consistent with each other only if 8 g is small compared to other quark-diagram amplitudes. Similarly we also expect the vertical W-loop contribution 6 ~ to be small. Furthermore, we have shown that given the amplitude (~¢+ cg) from D--.I(g decays and ( 8 g + 2 8 ~ ) from D ~ n ~ , no solutions for AK~ and (8 c¢- 88) can be found to fit the observed rates o f D ° ~ K + K -, K°I( °. We thus conclude that the SU (3)-breaking effects solely from the W-loop amplitudes cannot explain the puzzle of R + _. 283
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To estimate the amount of m a x i m u m S U ( 3 ) violation required in the amplitudes d and (g [i.e. ( a ¢ + ~ ¢ ) r . ~ ( ~ + ~¢)~n ], we consider the extreme case of 8 ~ ¢ = 8 ~ = 8 ~ = 0 . It follows from eq. (8) and table 1 that 32 (d+
ff)K~:/(~¢+ ff)~n-----1.50,
AK~_--__56° ,
(10)
where we have only quoted the central values. The degree of SU (3) breaking in the spectator amplitude d can be calculated. In the factorization approach, ~¢r.~ ( K + I~u(1 - ~ s ) u l 0 ) ( K - I ~ U ( 1 - y s ) s l D ° ) ~¢~ - ( n + IdYu( 1 -Ys)U[ 0 ) ( n - [ ~ u ( 1 - r s ) d l D°> "
( 11 )
Using the form factor decomposition [ 15 ] (p (K-IVuID°)
=
D--PK
2 2 ) q2 q
mD--mK
FDK(q2) +
/1
2 -2- m K q2
mD
quFDK(q2)'
(12)
where qu= (PD--PK)u,Fl (0) = F o ( 0 ) , and the pole dominance for the q2 dependence of form factors, we get
alKr~ fK (m2--m 2) FDK(0) (FD~(0) .~-I d,~ - f~ (m2D--m~) 1--(m~/m~) 1--(m~/m~)] '
(13)
where the 0 +-pole masses are m~ = 2.60 GeV, m2=2.47 GeV, and form factors at q 2 = 0 are given by [ 15 ] FDK(O) =0.762, Ft)~(0) =0.692. With fidf~ = 1.22 [3], it follows that A I ~ = 1.30A~n.
(14)
Therefore, the SU (3) breaking in the spectator amplitude is about 30%. To determine the magnitude of SU (3) breaking in the W-exchange amplitude, we need to know cg~. As we pointed out before [ 13 ], the magnitude and relative signs of the six quark-diagram amplitudes ~¢, M, ¢g, ~ , ~ , _~ ( ~ and ~ being W-annihilation amplitudes) can be determined from the measured decay rates of D + --,I~°n +, I(°K +, D ° - ) K - x +, I(°n °, I(°rl, I(°H ' , and D~ --)I(°K +, n+n, n+rl ' . Using updated experimental data we find ~ d ~ 3 . 8 X 10-6 GeV,
cg~ - 1.1X 10-6 G e V ,
(15)
and hence ( cg/~d)~ g - 0 . 2 7 . It follows from eqs. (10) and (14) that ~¢Kg:~0.75 Cg~ ~ --0.75 × 1.1 × 10 -6 G e V ,
(16)
where we have set
(17)
(sl+ cg)Z [(l_x:)+~(5_6x)(cosA,~,_l) ] ,2 When 8 ~¢= 8 # = 0, the isospin phase-shift difference Av.~obeys the equation F(D 0 --*K0I(0 ) = tan 2 ( ½Av.x) F(D 0 --,K + K- ). This equation was first obtained by Lipkin [ 14 ]. '3 In contradiction to some previous observations that D°~K°I(° proceeds solely through SU (3)-breaking effects in the W-exchange diagram ~iff and the vertical W-loop diagram 6~, our solution indicates that D °--,K°K° occurs essentially via the final-state interactions. A previous theoretical consideration along this line was given by Pham [ 16 ]. 284
Volume 280, number 3,4
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where the factor o f 0.86 comes from the phase-space differences for the KI~ and ng modes. Let us neglect the FSIs for the moment. We see that if all non-spectator contributions ( if, 8 if, 8g, 6 ~ ) are neglected, we will have R +_ = 1.45. This means that the effect o f SU (3) breaking in the spectator amplitude alone is not adequate to explain the experimental observation. However, when we put in the W-exchange mechanism, eq. ( 15 ), R + _ is enhanced from 1.45 to 1.70 due to the destructive interference of ff with ~¢. Hence, the presence of W-exchange helps to solve the puzzle for the ratios R+_ andR. As we further take into account the SU (3)-breaking effect in the W-exchange amplitude, namely ~¢r,K~ ~¢~, we obtain R + _ = 1.94, to be compared with the observed value 2.2 + 0.4. Finally, as we include the FSIs o f A~, Ar,~ given in eqs. (6) and (10) we further improve the agreement with measured R + _ . In conclusion, we have analyzed the effects o f S U ( 3 ) breaking and final-state interactions in the decays D°~Tr~, KI~ based on the quark-diagram scheme. To fit the D ° ~ n ~ decays, we find that the elastic phase shift must be introduced and are sufficient; no inelasticity or SU ( 3 )-breaking effects are called for. To fit the D o__.KI~ decays, besides the necessary elastic phase shift, SU (3)-breaking effects are definitely needed, especially those in the amplitudes (~¢+ ~); those solely from (6 ~ - 6 g ) are not adequate. Introducing inelasticity in D°--.Tr~ does not help in reducing the SU (3)-breaking effects in D °--,KI~ It is obvious that introducing inelasticity in D ° - , K K will only make fitting the data more difficult and increase the need o f SU (3)-breaking effects. We finally arrived at the following solution to fit D°--,Trg, KI~: The elastic phase shifts in the final-state interactions are ~0l t ~ - ~27tg ~ 83 ° and ~or'x - ~ r ~ ~ 56 °; the SU (3)-breaking effects in the spectator W-emission and the nonspectator W-exchange amplitudes are definitely required, however at a maximal level o f 30%, which can be reduced (but not to zero) by introducing the SU(3)-breaking effects from ( 6 g + 2 f i ~ ) and (6 i f - 6 g ) . Future more accurate measurements o f all the decay modes, D--,rcg, KK, nK, in particular D ÷ ~ n + ~ °, will improve our analysis. One o f us (H.Y.C.) would like to thank the Physics Department o f Brookhaven National Laboratory and University o f California at Davis for their hospitality. This work was supported in part by the US Department o f Energy ( D O E ) and the National Science Council o f Taiwan under the contract number NSC80-0208-M00181.
References
[ 1] V. Barger and S. Pakvasa, Phys. Rev. Lett. 43 (1979) 812; H. Fritszch and P. Minkowski, Nucl. Phys. B 171 (1980) 413. [ 2 ] R. Kingsley, S. Treiman, F. Wilczek and A. Zee, Phys. Rev. D 11 ( 1975) 1919; G. Altarelli, N. Cabibbo and L. Maiani, Nucl. Phys. B 88 ( 1975) 285; J.F. Donoghue and B.R. Holstein, Phys. Rev. D 12 (1975) 1454; M.B. Einhorn and C. Quigg, Phys. Rev. D 12 (1975 ) 2015; M.B. Voloshin, V.I. Zakharov and L.B. Okun, JETP Lett. 21 (1975) 183; L.L. Chau Wang and F. Wilczek, Phys. Rev. Lett. 43.(1979) 816; M. Suzuki, Phys. Lett. B 85 (1979) 91; Phys. Rev. Lett. 43 (1980) 818; A.N. Kamal and E.D. Cooper, Z. Phys. C 8 ( 1981 ) 67; A.N. Kamal and R.C. Verma, Phys. Rev. D 35 (1987) 3515. [ 3 ] Panicle Data Group, J.J. Hermtndez et al., Review of particle properties, Phys. Lctt. B 239 (1990) S 1. [4] ARGUS Collab., H. Albrecht et al., Z. Phys. C 46 (1990) 9. [5] CLEO Collab., J. Alexander et al., Phys. Rev. Lett. 65 (1990) 1184. [6] R. Poling, talk presented at the Lepton-photon Conf. (Geneva, July 1991); R. Ammar, talk presented at Hadron 1991 (MD, August 1991); B. Gittleman, talk presented in the 1991 DPF Meeting (Vancouver, August 1991). [7] E691 CoUab., J.C. Anjos et al., preprint FERMILAB-Pub-91/147 ( 1991). [8 ] L.L. Chau, in: Proc. Guangzhou Conf. on Theoretical panicle physics (Guangzhou, China, 1980) (Science Press, Beijing, China/ Van Nostrand Reinhold, New York, 1980); Phys. Rep. 95 (1983) 1. 285
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[9] L.L. Chau and H.Y. Cheng, Phys. Rev. D 36 (1987)
PHYSICS
LETTERS B
137; D 39 (1989)
2788; Phys. Rev. Lett. 56 (1986)
[lo] D. Hitlin, Nucl. Phys. B (Proc. Suppl.) 3 (1988) 179. [ 111 M. Ghick, Phys. Lett. B 88 ( 1979) 145;
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G. Eilam and J.P. Leveille, Phys. Rev. Lett. 44 (1980) 1648; J. Finjord, Nucl. Phys. B 18 I ( 198 1) 74; M.J. Savage, Phys. Lett. B 251 (1991) 414. L.L. Chau and H.Y. Cheng, Phys. Lett. B 165 (1985) 429. L.L. Chau and H.Y. Cheng, Phys. Lett. B 222 (1989) 285; Phys. Rev. D 42 ( 1990) 1837; L.L. Chau, H.Y. Cheng and T. Huang, Z. Phys. C., to be published. H.J. Lipkin, Phys. Rev. Lett. 44 (1980) 710. M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29 (1985) 637. X.Y. Pham, Phys. Lett. B 193 (1987) 331.
30 April 1992 1655.
PHYSICS LETTERS B
Final-state-interaction and SU ( 3 )-breaking effects in
rcg, KI(
Ling-Lie Chau Department of Physics, University of California, Davis, CA 95616, USA
and Hai-Yang Cheng Institute of Physics, Academia Sinica, TaipeL Taiwan 11529 Received 6 January 1992
Based on the quark-diagram scheme and the recent measurement of D °--,~%°, we analyze the decays D°--*~+ ~-, ~t°n°, K +K-, and K°I(°. To fit the data, we find that there must be both final-state-interaction and SU(3)-breaking effects: of the former, nonzero elastic phase shifts in both gg and KI~ are determined and no inelasticity in phase shifts is required; the SU(3 ) breaking effects are at the maximal level of 30% and of particular interest are those in the spectator W-emission and the non-spectator Wexchange diagrams.
The ratio F ( D °--, K + K - ) / F ( D ° - + ~ + n - ) presents a long outstanding problem: While the theoretical expectation lies in the range o f 0.9-1.4 [ 1,2 ] experimentally it deviates from unity substantially. The previous world average o f this ratio [ 3 ] was 3.95 + 1.24, b u t the recent i m p r o v e d m e a s u r e m e n t s by A R G U S [ 4 ], CLEO [ 5,6 ] a n d E691 [ 7 ] e x p e r i m e n t s gave smaller values 2.0-2.5. Nevertheless, the discrepancy between experimental results a n d theoretical expectations still remains. It has long been p o i n t e d out that the d e v i a t i o n o f this ratio from unity is caused b y final-state interactions ( F S I s ) or SU ( 3 ) - b r e a k i n g effects [ 8,1,2 ] or both. In the quarkd i a g r a m scheme [ 8,9 ] ~ such effects can be systematically studied. In o r d e r to resolve this puzzle, d a t a o f all four decay channels D ° - + K + K - , K ° I ( °, n+Tt - , ~o~o are needed. W i t h the recent m e a s u r e m e n t o f D°--,x°n ° by CLEO [ 6 ], we can finally m a k e a meaningful analysis. In this paper, we present our analysis a n d report the FSIs a n d SU ( 3 ) - b r e a k i n g effects we have f o u n d in these decays. Let us first s u m m a r i z e in table 1 the recent e x p e r i m e n t a l m e a s u r e m e n t s o f the branching ratios o f D°-+KI(, 7t~ by A R G U S [ 4 ], CLEO [ 5,6 ] a n d E691 [ 7 ], where the branching ratio [ 10 ] BR ( D O~ K - x + ) = (4.2 + 0.6 ) % has been used for normalization. In the below we shall take the average value (0.46 + 0.05)% for the branching ratio o f D ° - - , K + K - a n d (0.21 + 0.03)% for D ° - + n + x - . Two quantities o f interest are #~ Some misprints in the quark-diagram amplitudes of D°--,K+K- and K°I(° are corrected in eq. (3) of this paper. Table 1 Recent experimental measurements of the branching ratios D °--,KR, xft (in units of 10- 2). Collaboration
BR(D°-.K+K - )
BR (D°-~K°I(° )
BR(D°-~Tr+g -)
BR(DO-~% ° )
ARGUS CLEO E691
0.42 + 0.08 0.49+0.08 0.48 + 0.09
0.13+0.07
0.17 __0.04 0.21 __.0.05 0.25 + 0.06
0.09+0.03
Elsevier Science Publishers B.V.
281
Volume 280, number 3,4
R=
PHYSICS LETTERS B
30 April 1992
F ( D ° ~ K + K - ) + F ( D ° ~ K°I~°) F(D°~K+K - ) F(DO~n+n _) +F(DO~nOnO ) , R+_ = F(DO~n+n _) .
(1)
The experimental results of table 1 give R=2.0+0.4,
R+
=2.2+0.4.
(2)
In terms of the quark-diagram amplitudes, the decay amplitudes of the above four decay modes are [ 9 ] A ( D ° ~ K + K - ) = ( s , c , ) { d + ~¢+ ( 5 8 + 2 8 ~ v ) - ½[ d + ~g- ( S f f - 88) ] [1 - e x p ( - i ~ )
1} exp(iSo~),
A ( D ° ~ K°I(°) = (s, c, ){ (5 c¢_ 58) + ( 5 8 + 2 8 ~ ) + ½[ d + ~¢- (5 ~¢- 58) ] [ 1 - e x p ( - i A r . ~ ) ] } exp (i8~J~) , A ( D ° ~ n + x - ) = - (sic,){M+ i f - ( 8 8 + 28~v) - ~ ( M + ~ ) [ 1 - e x p ( - iA~) 1} exp ( i O U ) , A ( D ° ~ x ° n °) = (½V/2) (s, c, ) { ~ -
if+ ( 8 8 + 2 8 ~ ) - ~ ( d + ~ ) [1 - e x p ( - i A ~ )
]} e x p ( i g U ) ,
(3)
where s l - sin 0, ~ I V~I ~ I W~l, cl = cos 0~ ~ I W~l ~ I V~ol, AK~ ----( go -- 51 )Kilo and A~ = ( g o - ~2).~. (The g;s are phase shifts of isospin I. ) In eq. (3) ~t is the external W-emission amplitude, ~ is the internal W-emission amplitude, (g is the W-exchange amplitude (the underline ~ denotes the W-exchange graph involving the creation of gs), ~ is the horizontal W-loop amplitude with the loop-quark i. (The one-gluon-exchange approximation of the 8 graph is the so-called "penguin diagram".) ~ is the vertical W-loop amplitude. These quarkdiagram amplitudes have well-defined meaning with all QCD strong-interaction effects included. In eq. (3), 8 ~- ~-(g, 58-8~-do, 8~-~-~; they characterize potential SU(3)-violating effects in these quark-diagram amplitudes. The FSIs are expressed by the phase shifts, the O's, which in general have both real and imaginary parts; the real parts are related to the elastic scattering effects while the imaginary parts indicate effects of inelasticity [ 9 ] (see also footnote 1 ). In the absence of FSIs and SU (3) breaking, it is evident from eq. (3) that A (D°-~K+K - )/A (D°~Tr+~ - ) = 1 . Fro m our previous analysis of D + ~ I~°x ÷, D O~ K - x +, I~°~°, we had obtained [ 9 ] [~¢+:~[=(I.48_+0.11)×10-6GeV,
~¢+(g - - - 1
~-~ - -
=2.02_+0.22,
(4)
and A ~ = 79 ° _+ 11 °. Given these values of (~¢ + ~g) and (~¢ + ~ ), the independent parameters in all four decays are ( 8 8 + 2 8 ~ r ) , (5 ~g-Sg), A~, Av,g, 8 ~ and gg~i~ [see eq. (3) ]. It seems that the data should be easily fitted since there are more free parameters than the number of decay rates to be fitted. Surprisingly, we find that we cannot fit the data with these free parameters, including the introduction of the imaginary parts in gg~ and goI~. We need to introduce SU ( 3 )-breaking effects in the amplitudes ~¢ and ~gso that ~trat # ~1,~, ~gr,x # %~. From eq. (3) we see that in the absence of FSIs in the x+~ - and xo~o modes, the only free parameter in the decay amplitudes is the SU (3)-breaking term ( 5 8 + 2 8 ~ ) and it cannot fit the measured rates of both 7r+n and 7r°x°. (The calculated branching ratio of D°--,n°~ ° would be too small by a factor of 3. ) From this very general observation, it is evident that the FSIs must be in D°--,~r+~r -, n°Tr° decays. Allowing the FSIs, the two parameters A~ and ( 5 8 + 28~:) obey the equations F ( D ° ~ n + n - ) + F ( D ° ~ n°~r°) = (s~ c~ )2 (~¢+ (g)z. 9 ( 1 - ~ x + x ~) [ exp (igg ~) [ 2 (p.s.) , F(D°--, x + ~ - ) - F ( D ° - * ~°n °) (S1Cl)2(~¢.4. ~ Z . l r23
5X'4"3X2"4-IN//2 ( 5 - 6 x ) cos d ~ ] [exp(ig~ ~) [2(p.s.)
(5)
where x = (8 8 + 25 ~ ) / ( ~ + ff ), (p. s. ) is the phase-space factor, and the value ( ~¢ + (g) / ( ~¢ + ~ ) ~ 2 from eq. (4) has been used. Fitting eq. (5) to the observed rates of D ° ~ g without assuming inelasticity, i.e. I exp(ig~ ~ )1 = 1, yields (88+28~)/(~+ 282
cg)~O.05,
'~ - g z' ~ ~~ 8 3 ° , A,~,~=go
(6)
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where only the central values are given. We see that the elastic FSIs alone can fit the D o . n + g - , ~ogo rates. In the future, the value of I~¢+ 81 of D - - . ~ can be independently determined from experiment by measuring the Cabibbo-suppressed decay mode D + ~ x + ~ °, whose quark-diagram amplitude is given by A ( D + ~n+~o) = ( 1/x/~) (s~c~) (~¢+ 8 ) exp (ig~ ~ ). From A (D + ~I(°~ + ) = (S~Cl) (~¢+ 8 ) exp (i8~/"2), one can easily obtain the following long outstanding prediction [ 8 ]
r(D+--,n+n°) /r(D+--,K°n +)=½1 V¢a/V¢slZ(1.07) ,
(7)
provided that there are no SU (3)-breaking effects in the amplitude (~¢ + 8 ), i.e., (A + B ),~ = (A + B ) ~ . (We do not expect inelasticity in both channels since they are exotic. ) From the known value of I V~a/Vcs]2, we can predict from eq. (7) the branching ratio of D + ~ + ~ ° to be 0.08%. It then can be compared with the value obtained from D-~ I(~, eq. ( 4 ). If future experimental measurements give (A + B ),~ substantially different from (A + B ) i~=given by eq. (4), we shall then need to modify our analysis presented here. We now turn to the decay modes D ° ~ K + K - , K°I~°. As in the D ° - ~ case, we also need FSIs for the same reason that without them the decay amplitudes will have only one free parameter (8 c g - 8 g ) and it certainly cannot fit the two D--.KI( decay rates simultaneously. When FSIs are put in, the two free parameters Ara~ and (~ cg_~g) satisfy the equations F ( D O ~ K + K - ) + F(DO~KOi~O) = ( s I el )2(d2~j I_ ( ~ ) 2 [ ( 1 - l - x ) 2-]- ( x - ~ y ) 2 ] Iexp (i60K~')I2(p.s.) ,
F(D°~K+K-)-F(D°~K°f~°)=(SiCl)2(~t+ ~)2[(l+x)2-(x+y)2]lexp(iSo~)lZ(p.s.)cosAr~,
(8)
with y = (8 ~¢- 8 8 ) / (~¢ + if). However, fitting eq. ( 8 ) to the data given in table 1 we cannot find a solution for z/v,~ and (8 c¢_ 8 ¢ ). Allowing inelastic effects in 8or'x does not help solving the difficulty because the imaginary component ofS~¢~ will suppress the decay rate of D ° ~ K K . Next, we ask if we can put in inelastic effects in 8gn so that there is one more free parameter, Im ~ , in fitting D ° ~ n ~ decays and the value of ( 8 g + 2 8 ~ ) can be increased to help to fit D ~ KI( rates. Having inelastic effects in the I = 0 amplitude of the ng channel is reasonable since there are two known 0 + resonances fo (975 ) and fo (1400), which couple to both K I ( and n~ channels. It is plausible to assume 8~n ~ 0 owing to the absence of higher isospin resonances. However, we find that this alternative does not work. To compensate the decreasing effect from inelasticity in 8~n, we need constructive contribution from (8 g + 2 8 ~ ) to D ~ n ~ in order to accommodate data. Since this term then contributes destructively to D--.KI( decays [el. eq. (3) ], so it does not help to fit D°--.KI( decays. Therefore, no inelastic effects are called for and elastic scattering phase shifts are adequate. This means that we finally need to retreat to allow SU (3) violation in the amplitudes J and cg. It has been advocated that the penguin contribution in the W-loop amplitude may help to explain the ratio R+ _ [ 11 ]. Our analysis indicates that the W-loop amplitudes are small [cf. eq. (6) ]. In fact, a model calculation of the penguin contribution gives [ 12] 8g/d=
- ~ols[ (rnZ~-m2)lm2]{1 + 2m2/[(mu + ms) ( m e - m s ) l } ,
(9)
with ors being the strong-interaction coupling constant. It is clear that the penguin effect is rather small, 8 g / ~¢ ~ 1%. Another empirical argument against a significant 8g is as follows [ 13 ]. From the measured decay rates of D + ~ I ( ° n + , I(°K + and D + ~ I ( ° K +, we obtain I d + S I ~ 1.5× 10 -6 GeV, I d - ~ + S g l ~ 3 . 5 × 10 -6 GeV, and 18 + ~ l ~ 2 × 10 -6 GeV. Obviously, they are consistent with each other only if 8 g is small compared to other quark-diagram amplitudes. Similarly we also expect the vertical W-loop contribution 6 ~ to be small. Furthermore, we have shown that given the amplitude (~¢+ cg) from D--.I(g decays and ( 8 g + 2 8 ~ ) from D ~ n ~ , no solutions for AK~ and (8 c¢- 88) can be found to fit the observed rates o f D ° ~ K + K -, K°I( °. We thus conclude that the SU (3)-breaking effects solely from the W-loop amplitudes cannot explain the puzzle of R + _. 283
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To estimate the amount of m a x i m u m S U ( 3 ) violation required in the amplitudes d and (g [i.e. ( a ¢ + ~ ¢ ) r . ~ ( ~ + ~¢)~n ], we consider the extreme case of 8 ~ ¢ = 8 ~ = 8 ~ = 0 . It follows from eq. (8) and table 1 that 32 (d+
ff)K~:/(~¢+ ff)~n-----1.50,
AK~_--__56° ,
(10)
where we have only quoted the central values. The degree of SU (3) breaking in the spectator amplitude d can be calculated. In the factorization approach, ~¢r.~ ( K + I~u(1 - ~ s ) u l 0 ) ( K - I ~ U ( 1 - y s ) s l D ° ) ~¢~ - ( n + IdYu( 1 -Ys)U[ 0 ) ( n - [ ~ u ( 1 - r s ) d l D°> "
( 11 )
Using the form factor decomposition [ 15 ] (p (K-IVuID°)
=
D--PK
2 2 ) q2 q
mD--mK
FDK(q2) +
/1
2 -2- m K q2
mD
quFDK(q2)'
(12)
where qu= (PD--PK)u,Fl (0) = F o ( 0 ) , and the pole dominance for the q2 dependence of form factors, we get
alKr~ fK (m2--m 2) FDK(0) (FD~(0) .~-I d,~ - f~ (m2D--m~) 1--(m~/m~) 1--(m~/m~)] '
(13)
where the 0 +-pole masses are m~ = 2.60 GeV, m2=2.47 GeV, and form factors at q 2 = 0 are given by [ 15 ] FDK(O) =0.762, Ft)~(0) =0.692. With fidf~ = 1.22 [3], it follows that A I ~ = 1.30A~n.
(14)
Therefore, the SU (3) breaking in the spectator amplitude is about 30%. To determine the magnitude of SU (3) breaking in the W-exchange amplitude, we need to know cg~. As we pointed out before [ 13 ], the magnitude and relative signs of the six quark-diagram amplitudes ~¢, M, ¢g, ~ , ~ , _~ ( ~ and ~ being W-annihilation amplitudes) can be determined from the measured decay rates of D + --,I~°n +, I(°K +, D ° - ) K - x +, I(°n °, I(°rl, I(°H ' , and D~ --)I(°K +, n+n, n+rl ' . Using updated experimental data we find ~ d ~ 3 . 8 X 10-6 GeV,
cg~ - 1.1X 10-6 G e V ,
(15)
and hence ( cg/~d)~ g - 0 . 2 7 . It follows from eqs. (10) and (14) that ~¢Kg:~0.75 Cg~ ~ --0.75 × 1.1 × 10 -6 G e V ,
(16)
where we have set
(17)
(sl+ cg)Z [(l_x:)+~(5_6x)(cosA,~,_l) ] ,2 When 8 ~¢= 8 # = 0, the isospin phase-shift difference Av.~obeys the equation F(D 0 --*K0I(0 ) = tan 2 ( ½Av.x) F(D 0 --,K + K- ). This equation was first obtained by Lipkin [ 14 ]. '3 In contradiction to some previous observations that D°~K°I(° proceeds solely through SU (3)-breaking effects in the W-exchange diagram ~iff and the vertical W-loop diagram 6~, our solution indicates that D °--,K°K° occurs essentially via the final-state interactions. A previous theoretical consideration along this line was given by Pham [ 16 ]. 284
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where the factor o f 0.86 comes from the phase-space differences for the KI~ and ng modes. Let us neglect the FSIs for the moment. We see that if all non-spectator contributions ( if, 8 if, 8g, 6 ~ ) are neglected, we will have R +_ = 1.45. This means that the effect o f SU (3) breaking in the spectator amplitude alone is not adequate to explain the experimental observation. However, when we put in the W-exchange mechanism, eq. ( 15 ), R + _ is enhanced from 1.45 to 1.70 due to the destructive interference of ff with ~¢. Hence, the presence of W-exchange helps to solve the puzzle for the ratios R+_ andR. As we further take into account the SU (3)-breaking effect in the W-exchange amplitude, namely ~¢r,K~ ~¢~, we obtain R + _ = 1.94, to be compared with the observed value 2.2 + 0.4. Finally, as we include the FSIs o f A~, Ar,~ given in eqs. (6) and (10) we further improve the agreement with measured R + _ . In conclusion, we have analyzed the effects o f S U ( 3 ) breaking and final-state interactions in the decays D°~Tr~, KI~ based on the quark-diagram scheme. To fit the D ° ~ n ~ decays, we find that the elastic phase shift must be introduced and are sufficient; no inelasticity or SU ( 3 )-breaking effects are called for. To fit the D o__.KI~ decays, besides the necessary elastic phase shift, SU (3)-breaking effects are definitely needed, especially those in the amplitudes (~¢+ ~); those solely from (6 ~ - 6 g ) are not adequate. Introducing inelasticity in D°--.Tr~ does not help in reducing the SU (3)-breaking effects in D °--,KI~ It is obvious that introducing inelasticity in D ° - , K K will only make fitting the data more difficult and increase the need o f SU (3)-breaking effects. We finally arrived at the following solution to fit D°--,Trg, KI~: The elastic phase shifts in the final-state interactions are ~0l t ~ - ~27tg ~ 83 ° and ~or'x - ~ r ~ ~ 56 °; the SU (3)-breaking effects in the spectator W-emission and the nonspectator W-exchange amplitudes are definitely required, however at a maximal level o f 30%, which can be reduced (but not to zero) by introducing the SU(3)-breaking effects from ( 6 g + 2 f i ~ ) and (6 i f - 6 g ) . Future more accurate measurements o f all the decay modes, D--,rcg, KK, nK, in particular D ÷ ~ n + ~ °, will improve our analysis. One o f us (H.Y.C.) would like to thank the Physics Department o f Brookhaven National Laboratory and University o f California at Davis for their hospitality. This work was supported in part by the US Department o f Energy ( D O E ) and the National Science Council o f Taiwan under the contract number NSC80-0208-M00181.
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