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IL NUOVO CIMENTO VOL. 102 A, N. 4 Ottobre 1989
Charmed-Meson Nonleptonic Decays and S U(3) Symmetry.
FENG-ZHI CHEN and PING WANG
Institute of High Energy Physics, P.O. Box 918(4) - Beijing, China
(ricevuto il 16 Settembre 1988)
Summary. - - Based on the analysis of SU(3) symmetry property, we propose that the Cabibbo-favoured nonleptonic decays of charmed mesons should be treated in two different cases: i) octet-octet final states and ii) octet-singlet final states. In case i) the spectator process gives correct predictions, but in case ii) the decay mechanism is different. We give a theoretical lower limit of BR(D~ + ~ ~o~ +) which cannot go through spectator process. This channel, as well as D~ +-~ we § , are recommended to search for in experiments.
PACS 12.90 - Miscellaneous theoretical ideas and models.
Recent ly the observed decay(1) D ~ 1 7 6 has imposed a puzzle to the otherwise highly successful calculation of charged meson decays (2.~). I t has been shown that for nonleptonic decays generally good results are obtained by the factorization approach with only contribution from spectator processes(2). However the channel D ~ ~o~ needs some special t r ea tment C~). On the other
(1) ARGUS COLLABORATION (H. ALBRECHT et al.): Phys. Lett. B, 158, 525 (1985); CLEO COLLABORATION (C. BEBEK et al.): Phys. Rev. Lett., 56, 1893 (1986); MARK III COLLABORATION (R. BALTRUSAITIS et al.): Phys. Rev. Lett., 56, 2136 (1986). (2) M. BAUER, B. STECH and M. WIRBEL: Z. Phys. C, 34, 103 (1987); A. J. BURAS, J. M. GERARD and R. RUCKL: Nucl. Phys. B, 268, 16 (1986). (3) B. YU. BLOK and M. A. SHIFMAN: ITEP preprint; ITEP-9 (1986); ITEP-17 (1986); ITEP-37 (1986). (4) j . F. DONOGHUE: Phys. Rev. D, 33, 1516 (1986); U. BAUR, A. J. BURAS, J. M. GERARD and R. RUCKL: Phys. Lett. B, 175, 377 (1986). (~) I. I. Y. BIGI: International Symposium on the Production and Decay of Heavy Flavors, Stanford, 1987.
977
978 FENG-ZHI CHEN and PING WANG
hand, the charmed-meson nonleptonic decays can also be analysed in the frame of SU(3) flavour symmetry (,9) and the decay amplitudes can be expressed in terms of the reduced matrix elements. In this paper we shall demonstrate that the spectator process dominance corresponds to certain cancellation conditions of the reduced matrix elements and the conditions for octet-octet final states are different from those for octet-singlet final states. Therefore in the point of view of SU(3) symmetry the spectator process gives good predictions only for the octet-octet final states and it cannot be simply extended to octet-singlet final states. Thus we give a mathematical formalism to the observation that the possible solution to D~ ~ 0 puzzle lies on the fact that + has a SU(3) singlet component.
In this work we concentrate on the Cabibbo-favoured two-body VP (vector- pseudoscalar) decay channels, because there is rich experimental information for quantitative estimates.
First consider processes with octet-octet final states. In the constituent quark model the spectator process dominance means the following equalities among different decay amplitudes in the limit of SU(3) symmetry:
(la) A(D ~ ~+ K*-) = - ~ A ( D ~ + ~ r. + Vs), Y~
(lb) V~A(D0__) p0~0) = V~A(DO~ ~o Vs) = A(D~ + ~ ~o K,+),
(lc)
Od)
V~A(DO___) ~o~,o) = ~/-~A(DO___) ~ ~,o) = A(D+___) K § K,O),
3 + + A(D~ K- p+) = - -~]=,A(Ds ~ ~ ).
Here Vs is the vector meson with ]Vs> = (1/V6)]u~ + dd - 2s~>. We decompose and (o into Vs and SU(3) singlet state W1)=(1/V~)]u-~+d-d+s-~) for
convenience of the theoretical analysis. The mixing between ~ and ~' is neglected. To prove eq. (1) one can use either the factorization approach or quark diagram analysis. For example, by the factorization approach the two sides of eq. (la) can be written as
~ / ~ 2 A(D~ ~+ K*-) = GF cos 0c <=+]~d]0)<K*-]~c]D ~
(6) C. QUIGG: Z. Phys. C, 4, 55 (1980). (7) X. Y. LI and S. F. TUAN: DESY preprint; DESY 83-078 (1983), unpublished. (s) A. N. KAMAL and R. C. VERMA: University of Alberta preprint, Alberta Thy-13-86 (1986). (9) M. GORN: Nucl. Phys. B, 191, 269 (1981).
CHARMED-MESON NONLEPTONIC DECAYS AND SU(3) SYMMETRY
and
_ 3 + + = 1 2 , 4 A ( D s - o = V,)(-~/~)'4GFCOSOc<=+l-Ed,O><Vsl-sclD+~>
9 7 9
respectively, and are equal due to SU(3) symmetry. In the quark diagram analysis both D~ + and D+--,Vs~ + are represented by the spectator diagram in fig. la). The other equalities in (1) are represented by quark line graphs in fig. lb), c) and d).
c
)v ~)
~ )v
)p
c)
c
)v b)
c - - , ~ )v
~)
Fig. 1. - The quark line graphs (spectator diagrams) which contribute to the octet-octet final states.
The expressions of these decay amplitudes in terms of reduced matrix elements are found in ref. (7.9). For convenience of the reader we list them in table I with the notations of ref. (9). The reduced matrix elements are denoted by <m, n>, where m stands for the irreducible tensor of the final state, n stands for the representation of the Hamiltonian, number 20 referring to 6 in SU(3), number 84 referring to 1__55.
Inserting the expressions of table I into eq. (1) we find that the necessary and sufficient conditions for all equalities in (1) to be true are:
(2) (8s,84> = <27,84), <8A,20> = <10",20>, <8A,84> = <10,84>.
These conditions mean that the reduced matrix elements only depend on i) the representation of the Hamiltonian and ii) the symmetry property of the final state under the exchange of the two particles.
Using the expressions of table I, it is easy to see that the conditions in (2) also make
A(D+~ ~~ = 0
980 FENG-ZHI CHEN and PING WANG
TABLE I. - Decay amplitudes in terms of SU(3) reduced matrix elements. A factors of (1/4) cos2Oc is to be understood with each amplitude.
D O decays
K*- =+
-~o o
-K,O o
K - a +
K*~
~ o Vs
~o V1
K § K,o
~-o K* +
1 10",20} 1(8a,20 ) 2(27,84)- 1 (8~,20) -~( -~ +~
1 1 (SA, 84} -1(10'S4)3 + (S,,84) -~
~-~(- 1(8~, 20) + 2 (10",20) -~ (8a,20)1 +~3 (27,84) _
1(10, S4)-1(8,,S4} 1(8a,84))
~/~(-1 (S,, 20}-2 {10",20)+6 (SA,20)+53-- (27, S4) +
1(10, 84) --1(8, , 84} 1(8a 84)) +~ +ii '
1 (8A,20) 2 (2v, 84) + 1(8"20) +1(10"20) +6 +5
1 (8a 84) +~1 (10, 84} +1(8~,S4) +g
~ ( -~1 (<, 20) -~1 (8a, 20) + a~ (27, 84) + (10, 84)-
1 (8,,84) 1 (8A 84)) 10 -2 '
~ ( 1 (8~,20) a (27,84)- (10,84)- 1 -�89 +~ +g
1 (8~ 84) 1 ) +~(8A 84) 10 '
1 (_ (8,20) + (8,84)) av~
1 1(8a,20) 2(27,84)+ -~(8~,20)-~(10",20)-~ +g
+3(10,84) +~0 (8~,84)+1(8A,84)
1(8A,20) 2(27,84)_ - l(s~,2o) +~(lo*, 2o) +~ +~
1(10,84)+1(8,,84) 1(8A,84) 3
CHARMED-MESON NONLEPTONIC DECAYS AND SU(3) SYMMETRY
TABLE I (continued).
981
D: decays
o +
~+ ~o
~},z +
n+Vs
~+ V1
1 1 , 1 +1(10,84) 1(8A,84)) ~/~(~ (10 ,20) --~ (8A ,20)
~ ( 1 , 1(8A,20) 1(10,84) l(8A 84)) --~(10 ,20)+~ --~ +~ ,
~/~( 1 (8~'84)) -- (8~,20) -- (10",20) --6 (27,84) -- (10, 84) +~
~--~( 6 (27,84)+ (10, 84)+ 1(8~ 84)) - (8~,20) + (I0",20) - ~
3 l-V3 ((8, 20) + (8, 84))
D + decays
~ o + (10",20) + (27,84) ~.0~+ - (10",20) + (27,84)
and
A(D~ + ~ ~+ n ~ = 0
as predicted by spectator dominance decay mechanism, because these two channels can only go through annihilation processes. The experimental limit (lo) is
BR(D+~ po=+) < 0.09. BR(D + ~ ~ =+)
With eq. (2) all the decay amplitudes of octet-octet final states can be expressed in terms of four reduced matrix elements corresponding to four spectator dia_grams of fig. 1:
A(D~ K*-~ +) = 1((8~, 20) - (BA, 20) + (Bs, 84) - (BA, 84)),
A(D0_o~op0) = 1 (_ (88,20) + (8A,20) + (8~,84) - (8A,84)),
(10) M. WITHERELL: International Symposium on the Production and Decay of Heavy Flavors, Stanford, 1987.
63 - II N u o v o C imen to A .
982 FENG-ZHI CHEN and PING WANG
A(DO _)~o~.o)= 1 (_ (8~,20} - (8A,20} + (8s,84} + (8A,84}),
1 A(D~ K-p +) =~({8~,20} + (8A,20) + (8s,84} + (8A, 84}).
Other decay amplitudes can be related to them by using eq. (1). Next consider octet-singlet states. We have
A(D~ o~K~ = ~/~ A(D~ Vs K~ + ~/~ A(D~ VIK~ ,
A(D~176 - ~/~A(D~ VsK~ + ~A(D~ K~ ,
If the spectator process dominates and the annihilation contribution is negligible, then A(D~176 and A(D~+---)o~+)~0, because these two channels can only go through annihilation process. From (3) that these two amplitudes vanish means
(4) "A(D +--> V8 r: +) = - V~A(D +--> V~ r,+),
A(D ~ Vs ~o) = (I/V~) A(D ~ Vl ~o).
These two relations are justified in quark diagram approach, if only spectator diagrams are considered: they are represented by the quark line graphs in fig. la) and b), respectively. Equation (4) cannot be a consequence of SU(3) symmetry, because it relates a process involving SU(3) octet particles to a process involving a SU(3) singlet particle. Substituting the expressions in table I into (4) we get
(5)
2 - (8~, 10} + (8A,20} + (8~,84} - (8A, 84) = ~ ( - - (8,20} + (8,84)),
[(8~,20} - (8A,20} + (8~,84}-- (8A,84)=3( (8 ,20) + (8,84}).
This is the condition of spectator dominance for octet-singlet final states, which is different from condition (2) for octet-octet final states.
CHARMED-MESON NONLEPTONIC DECAYS AND SU(3) SYMMETRY 983
Experimental ly the decay D ~ ~K ~ has been observed. This means that the spectator process does not give correct predictions for octet-singlet final states. Equivalently it means that eqs. (4) and (5) do not hold. Notice that (5) is totally different from (2). The octet-octet final states may have different decay mechanisms from octet-singlet final states. As it is pointed out that the lat ter may receive contributions from the quark diagrams in fig. 2 (11). The success of spectator dominance for octet-octet final states does not necessarily contradict with its failure for octet-singlet final states.
Fig. 2. - The new quark line graphs (Hairpin diagrams) which may contribute to the octet-singlet final states.
Using experimental data we can estimate the amplitudes and phase of A(D ~ V1 ~o). With A(D ~ V8 ~o) = (l /V3) A(D ~ ~o po), inputing the values of D~ polo and ~ o in table II by Mark III, we fit numerically
A(D~ V1 K~176 ---~ Vs K ~ -~ 5.4 exp [i 51~
which obviously is not consistent with eq. (4).
TABLE II. - In this paper, these data by MARK III (12) are used for numerical estimates.
Channel Branching ratio (%)
DO---) ,z + K- 10.8 + 0.4 + 1.7 D~ ,z~ ~ 0.75 + 0.09 + 0.47 D~ ,~ + K* 5.3 + 0.4 + 1.0 DO___~ ~o~.o 2.6 _+ 0.3 + 0.7 D~ ~oK ~ 3.8+ 1.3+ 1.0 DO__~ r a Q~+o.5+o 31
v.~v-0.41-0.17
In the Ds + decays the channel D~ + --~ ~o~ § which can only go through annihilation process does not have to be vanishing, meanwhile the partial width of D~ + --~ ~,~+ may deviate from the prediction of pure spectator process. Relating A(Ds + --~ Vs ~§ with the experimentally measured channel A(D~ K*- ~*) by eq.
(,1) X. Y. LI, X. Q. LI and P. WANG: Beijing Institute of High Energy Physics preprint, BIHEP-TH-87-1 (1987), unpublished. (,2) D. H. COWARD: Annual Meeting of the Division of Particle and Fields of the APS, Eugene, Oregon, August 1985; D. HITLIN: Charm Physics Workshop, Beijing, 1987.
984 FENG-ZHI CHEN and PING WANG
(2) and using A(D~ + -* ~b~ +) as a variable, we can get a lower limit to A(D7 -o ~o~+). This is done by taking the phase between A(D~ + --. Vs~ +) and A(D~ + ~ V1 ~+) 180 ~ degrees, so that DT--* r is a constructive interference, whereas D~ +-o o~ + is a destructive interference. Because the absolute branching ratios of D~ + decay channels are still poorly determined experimentally, we express the results in terms of ratios of partial widths I'(D+~--,K*~ § and F(D~ +--* ~+)/F(D~ +--* ~=+) for the convenience to compare with the experiments. Here + .0 § Ds--* K K is a decay with octet-octet final states and its amplitude is related with the measured channel D~176 ~ by eq. (2). It has been given experimentally (~,~):
BR(D~ + --. K*o K +)
BR(D~ +--~ r t ~0.75 + 0.12 _+ 0.06 E691,
= 0.85+ 0.23 MARK II,
[1.44 + 0.37 ARGUS.
Corresponding to these three numbers and using the data in table II our theoretical lower limit of F(D~ +--~ ~+) relative to F(D~ +-* ~+) is listed in table III. These predictions are contrary to most of the other calculations (2) which predict it to be negligible, although QCD sum rule gives a small value of 0.15 (9 which is quite close to our lower limit. We strongly recommend that the decay D~+--~ o~= + should be searched by experiments.
TABLE I I I . - The theoretical lower limit of BR(D~ + --* ~o~ +) relative to BR(D~ + --~ Cu+) using BR(D+ __.-~.o K+)/BR(D+ __. r247 as input.
BR(D+ ~ ~.0 K §
BR(D~ +--. r
BR(D~ + ~ ~oT. +)
BR(D~ +--> ~=+)
0.75 0.85 1.44
>0.19 >0.10 >0.08
Finally to study the 0ZI-suppressed annihilation process which is a characteristic of the decays involving SU(3) singlet particles, we suggest that the semi-leptonic channels involving the meson with the ,~wrong quark content~ like D~+-. ~e§ should also be searched in experiments.
To sum up we propose that the theoretical analysis of charmed meson nonleptonic decays should be treated in two different cases: i) octet-octet final states and ii) octet-singlet final states. Only in case i) the spectator process gives correct results. But in case ii) the decay mechanism is different. To further advance our understanding the nonleptonic decays the experimental information about the decay channel DT-* ~o~ § and D~+--* ~oe+~ is highly desirable.
CHARMED-MESON NONLEPTONIC DECAYS AND SU(3) SYMMETRY 985
�9 R I A S S U N T O (*)
Basandosi sull 'analisi della propr ie ta di s immetr ia SU(3), si propone che i decadimenti non leptonici favoriti secondo Cabibbo dei mesoni con charm dovrebbero essere t r a t t a t i in due casi diversi: i) s tat i finali di ot te t to-ot te t to e ii) s tat i finali di ottet to-singoletto. Nel caso i) il processo di spet ta tore da previsioni corret te , ma in caso ii) il meccanismo di decadimento diverso. Si da un limite inferiore teorico di BR(D~ + ~ ~o~. +) che non pub passare a t t raverso il processo di spet tatore. Si consiglia di utilizzare negli esper iment i questo canale, oppure D + ~ ~oe +,~.
(*) Traduzione a cura della Redazione.
HeaenToxnbm pacna~bl oqaponaHnbtx Me3OHOB il SU(3) CFIMMeTpXl;I.
Pe3,o~e (*). - - I/Icno~ib3y~l aHam43 CBOI~ICTB SU(3) c n i i e r p n n , Mbl npeaonaraeM, qTO HeJIeIITOHHbIe pacnaJIbi otIapOBaHHblX Me3OHOB JIOJI~KHbI 6bITb pacCMOTpeHb~ B ;/e~-VX
pa3~IrtqHblX cHyqa~lx: 1) OKTeT-OKTeTHble KOHeqHbIe COCTOIIIHI4$1 H 2) OKTeT-CHr~rYleTIfblC
KOHeqnbIe COCTO~HH~. B cnyqae 1) npot~ecc-Ha6moJIaweab ~IaeT npasnJIbHb~e npe~cKa3aHv6i, HOB cayqae 2) MexaHn3M pacna~Ia ~Ba~eTc~ OTarIqHbIU. MSI iloayqaeM TeOpeTI4qeCKllfI HtlZg~l~Ifi npeilea iln~l Br(Ds+--* o)r~+), KOTOpt, Ifi He MO~eT H~TH qepe3 npor4ecc-sa6nro~IaTem,. ~TOT KaHaJI, a TaK)Ke D + --~ toe% peKoMenllyrorca ~Ina IdCCJIeJIOBaHH~I B :~KCIIepeMeHTaX.
(*) Flepe6ec)eno pec)am4ue~.
