and INFN, Sezione di Roma I, Roma, Italy and INFN, Sezione di Napoli, Napoli, Italy ABSTRACT

Transcription

1 ROMA Prerint n NAPOLI DSF-T-56/94 he-h/ Nonletonic weak decays of charmed mesons F. Buccella (a), M. Lusignoli (b;), G. Miele (c), A. Pugliese (b;) and P. Santorelli (c) (a) Diartimento di Scienze Fisiche, Universita di Naoli, Naoli, Italy (b) Diartimento di Fisica, Universita \La Saienza", Roma, and INFN, Sezione di Roma I, Roma, Italy (c) Diartimento di Scienze Fisiche, Universita di Naoli, and INFN, Sezione di Naoli, Naoli, Italy ABSTRACT HEP-PH A revious analysis of two-body Cabibbo allowed nonletonic decays of D 0 mesons and of Cabibbo allowed and rst-forbidden decays of D + and D s + has been adjourned using more recent exerimental data and extended to the Cabibbo forbidden decays of D 0. Annihilation and W-exchange contributions as well as nal state interaction eects (assumed to be dominated by nearby resonances) have been included and are in fact crucial to obtain a reasonable agreement with the exerimental data, which show large avour SU(3) violations. New tting arameters are necessary to describe rescattering eects for Cabibbo forbidden D 0 decays, given the lack of exerimental informations on isoscalar resonances. We kee their number to a minimum - three - using henomenologically based considerations. We also discuss CP violating asymmetries. () artially suorted by the Euroean Community under the Human Caital and Mobility Programme, contract CHRX-CT

2 This aer relaces the revious one with the same title submitted to the ICHEP94 conference (ref. gls0658), also circulated as ROMA rerint n / NAPOLI DSF-T- 18/94. The 1994 edition of the Particle Data (Phys. Rev. D 50 (August 1994) art I) did in fact show noticeable changes in the exerimental branching ratios for D decays. We therefore made a new analysis, that we reort here. 1. Introduction A theoretical descrition of exclusive nonletonic decays of charmed hadrons based on general rinciles is not yet ossible. Even if the short distance eects due to hard gluon exchange can be resummed and an eective hamiltonian has been constructed (recently, at next-to-leading order [1] ), the evaluation of its matrix elements requires nonerturbative techniques. Waiting for future rogress in lattice QCD calculations one has to rely on aroximate methods and/or models. The largely dierent lifetimes of charmed hadrons make it clear that the innitely heavy quark limit is quite far from the actual situation. Therefore, the exansion in inverse owers of the heavy quark mass characteristic of heavy quark eective theory [] is resumably not a useful tool in this case. Moreover, the methods of HQET are not obviously extended to coe with exclusive hadronic decays. On the other hand, the simle factorized ansatz for the matrix elements is known not to describe roerly Cabibbo allowed D 0 decays. The color-suression of some contributions seems in fact to be stronger than the factor 1/3 exected from QCD [3] and the data exhibit large hase dierences between amlitudes with denite isosin. We are thus forced, still using the factorization aroximation as a starting oint of the matrix element evaluation, to include imortant corrections due to rescattering eects in the nal states. This we do assuming the dominance of nearby resonances and taking from exeriment - when ossible - their masses and widths. We also include W-exchange and annihilation contributions that turn out to be larger than generally believed. The resence of nearby resonances may well have the eect of increasing these terms relative to their naive PCAC estimates. In two revious aers the Cabibbo allowed [4] two-body decays of charmed mesons were described in the framework discussed above and the model was alied to the analysis of Cabibbo rst-forbidden decays of the charged mesons [5] D + and D s + and to their CP violating asymmetries. The considerable success of that analysis romts us to extend it to the Cabibbo forbidden two-body decays of D 0. The recent exerimental determination of the branching ratio B.R.(D +! + 0 ) = % [6] [7] - that agrees with our rediction [5] - allows

3 to erform an amlitude analysis on the comlex of D! decays, that shows a large (' 90 ) hase dierence between I = 0 and I = amlitudes. Moreover, a comarison of the I = amlitude with the I = 3 from D+! + K 0 shows a considerable violation of avour SU(3) in the direction of larger amlitudes; on the other hand, it is known since a long time and recently conrmed [7] that the ratio of D 0 decay branching fractions to K + K and to + is much larger than the SU(3) rediction (i.e. 1), showing an oosite attern of SU(3) breaking in exotic and nonexotic channels. Another striking signal of the imortance of SU(3) violations is given by the value, quite similar to other Cabibbo forbidden decays, of the B.R.(D 0! K 0 K 0 ) that should be vanishing in the symmetric limit. Our model describes satisfactorily the exerimental situation. For what concerns SU(3) breaking in exotic channels it is the combination of several small eects that yields a large result. These eects would not be enough to exlain the large K + K to + ratio and to roduce a nonvanishing decay rate to K 0 K 0 : in the nonexotic channels the rescattering eects are essential. In order to introduce these rescattering eects we need to know masses, widths and coulings of yet unobserved sinless, isoscalar resonances with ositive and negative arity and masses around 1.9 GeV. One exects for each arity two resonances of this tye, a SU(3) singlet and a member of an octet, that generally mix among themselves. Such a large number of new arameters to t eight new data (or limits) for branching ratios is obviously unaealing, unless some arguments can be given to reduce it. In the following we will show that reasonable henomenological assumtions may reduce the number of new arameters to three. We have to determine these by a t to the data. Before doing that we reeated the t to all Cabibbo allowed and to charged meson rst-forbidden decay branching ratios, which in the meantime have got lower error bars and in some cases have also changed. The model is therefore assing a more demanding test.. Decay amlitudes in the factorized aroximation The eective weak hamiltonian for Cabibbo allowed nonletonic decays of charmed articles is given by (U ij are elements of the CKM matrix) H C=S eff = G F U ud Ucs C s (1 5 ) c u (1 5 ) d + (:1) C 1 u (1 5 ) c s (1 5 ) d +h:c:; 3

4 while for C = S rocesses the hamiltonian is obtained from the same equation with the substitution s $ d. The eective weak hamiltonian for Cabibbo rst-forbidden nonletonic decays is H C=1;S=0 eff = G F U ud Ucd C1 Q d 1 + C Q d + C1 Q s 1 + C Q s + G F U us U cs G F U ub U cb 6X i=3 C i Q i + h:c:; (:) In eq. (.) the oerators are [8]: Q d 1 =u (1 5 ) d d (1 5 ) c ; Q d =u (1 5 ) d d (1 5 ) c ; X Q 3 =u (1 5 ) c q (1 5 ) q ; Q 4 =u (1 q X 5 ) c q (1 5 ) q ; Q 5 =u (1 q X 5 ) c q (1 + 5 ) q ; Q 6 =u (1 q X 5 ) c q (1 + 5 ) q : q (:3) The oerator Q s 1 (Q s ) in eq. (.) is obtained from Q d 1 (Q d ) with the substitution (d! s). In eqs. (.1) and (.3) and are colour indices (that we will omit in the following formulae) and in the \enguin" oerators q (q) is to be summed over all active avours (u, d, s). If we neglect mixing with the third generation (U ub = 0 and U us Ucs = U ud Ucd = sin C cos C ) then the three eective hamiltonians H C=S eff cos C ; H S=0 eff sin C cos C ; H C= S eff sin C (:4) form a U-sin trilet. Therefore, in the limit of exact avour SU(3) symmetry a number of relations between decay amlitudes should hold. We shall discuss some of them in Section 4 and we will see that they are often violated rather strongly. Wehaveevaluated the coecients C i at the scale 1.5 GeV using the two-loo anomalous dimension matrices recently calculated by A. Buras and collaborators [1], assuming 4

5 MS 4 = 300 MeV. This value, which corresonds to the best agreement between the exerimental data and the theoretical results on the exclusive decay channels of D mesons, is comatible with the exerimental determination from LEP measurements [6]. The coef- cients at next-to-leading order are renormalization scheme deendent: we assume in the following the values obtained using the "scheme indeendent rescrition" of Buras et al., namely C 1 = 0:68, C =1:347, C 3 =0:07, C 4 = 0:057, C 5 =0:015, C 6 = 0:070. In the factorized aroximation the matrix elements of H eff are written in terms of matrix elements of currents, (V q q 0 ) =q 0 qand (A q q 0 ) =q 0 5 q. We recall the denitions of the decay constants for seudoscalar (, K:::) and vector (, K :::) mesons <P i ()ja j0>= if Pi ; <V i (; )j V j0 >=M i f Vi () ; (:5) and the usual denitions [9] for the matrix elements of the currents: <P i jv jp j >=( i + j M j M i q q )f + (q )+ + M j M i q q f 0 (q ); <V i ja jp j >=i(m j +M i )A 1 (q )( q q q ) ia (q ) q ( i M j +M + j i + i M i A 0 (q ) q q q ; <V i jv jp j >=V(q ) " j i M j + M i : M j M i q q )+ (:6) To avoid the resence of a surious singularity atq = 0, one has to require that f + (0) = f 0 (0) v qq0 ; (:7) A 0 (0) = M i + M j M i A 1 (0) + M i M j M i A (0) a qq0 : (:8) The semiletonic decay rate for D 0! K e + [10] indicates a value v cs ' 0:79; assuming SU(3) symmetry for the weak charges, we will set v cs = v cd = v cu =0:79 (:9) 5

6 Using the denition (.8) and the data on the semiletonic decay D 0! K e + [11] we obtain a cs ' :54 :13 (E653 Collaboration) or a cs ' :71 :16 (E691 Collaboration). Dierent lattice QCD calculations [1] give similar results: in average a cs ' :74 0:15. The limited statistics for the data on D 0! e + decay does not allow an analysis of the dierent form factors, but within large errors the measured branching fraction is larger than theoretical redictions based on quark models [13]. Lattice results and results obtained in the framework of QCD sum rules by using (.8) on SU(3) breaking are inconclusive because of large errors on the A form factor. Since the data on D meson decays show large SU(3) breaking eects and since the axial charges are not rotected by the so-called Ademollo-Gatto theorem, in our t we allowed a cs and a cd = a cu to vary between.5 and 1 indeendently. The values chosen by the t are a cs =0:59, consistent with exerimental data, and a cd = 1 (in fact, an even better t would be obtained allowing larger values for a cd ). We note that these values do not agree with the direct QCD sum rule calculation of A 0 (q ) erformed in [14], where the authors conclude that SU(3) breaking eects should be small (a cs =a cu =1:10 0:05) and the q deendence of the form factors comatible with a olar deendence dominated by the 0 ole. The q deendence of the form factors is assumed to be dominated by the nearest singularity. This entails for the c-quark decay terms the usual simle-ole form factors v cu(s;d) f + (q )= 1 q =MD ; (s) (1 ) v cu(s;d) f 0 (q )= 1 q =MD ; (s) (0+ ) (:10) and analogous exressions, with the mass of the lightest article with aroriate quantum numbers, for the other form factors. In the W-exchange and annihilation terms, however, the large and time-like q values needed, together with the suggested existence of resonances with masses near to the D-meson mass, make a rediction based on the lightest mass singularity unjustied. These terms deend on the matrix elements of current divergences between the vacuum and two-meson states. We write them, with the hel of the equations of motion, in the 6

7 way indicated in the following examles: <K + j@ (V d s) j0>=i(m s m d ) <K + jsd j0 > i (m s m d ) M D f D W PP ; <K + j@ (A d s ) j0>=i(m s +m d ) <K + js 5 d j0 > (:11) (m s +m d ) M f D K W PV : We assume SU(3) symmetry for the matrix elements of scalar and seudoscalar densities, and exress all of them in terms of W PP, W PV. In our aroach the W i 's are free arameters of the t. Their magnitude turns out to be considerably larger than what one would obtain assuming form factors dominated by the ole of the lightest scalar or seudoscalar meson, i.e. K0 (1430) or K(497). We note that to obtain the amlitudes for Cabibbo rst-forbidden decays one has to evaluate matrix elements like <j d 5 dj0>or, for enguin oerators, <j d 5 dj0>.to get the correct result it is necessary to take into account the anomaly of the singlet axial current; we have followed the method discussed in [15]. In this scheme the angle, 0, results to be equal to 0 mixing 10. Remarkably, this value which is consistent with the Gell-Mann Okubo mass formula, is also erfectly comatible with the exerimental value of (! ) obtained by two-hoton roduction exeriments [16]. Therefore the 0 mixing angle is not a arameter of the t, as it was in our revious analyses. If the nal K meson in Cabibbo allowed decays is neutral, it has been observed as a short-lived neutral K, K S. There is therefore an interference between Cabibbo allowed (D! K 0 + X) and doubly suressed (D! K 0 + X) decay amlitudes. We have tted the exerimental data ex [6] using the denition (D! K S + X) 1 ex(d! K 0 + X) ; (:1) and included FSI modications also for the doubly Cabibbo suressed art of the amlitude. The correction due to this eect is not negligible and it hels in obtaining a better t to the exerimental data. We write now a few examles of amlitudes for Cabibbo allowed and rst forbidden 7

8 decays in the factorized aroximation: A w (D 0! K + )= = G F U ud U cs = G F U ud U cs (C + C 1 ) <K j(v c s) jd 0 >< + j(a d u) j0>+ +(C 1 +C ) <K + j(vs d) j0><0j(a c u ) jd 0 > = (C + C 1 )f <K j@ (Vs c) jd 0 >+ +(C 1 +C )f D <K + j@ (V d s) j0> : (:13) In (.13) the two terms corresond to c-quark decay and W-exchange, resectively. A w (D +! K S + )= = G F U ud U cs + G F U us U cd = G F U ud U cs + G F U us U cd (C + C 1 ) <K 0 j(v c s) jd + >< + j(v d u) j0>+ +(C 1 +C ) <K 0 j(a d s ) j0>< + j(a c u ) jd + > + (C + C 1 ) <0j(A c d ) jd + ><K 0 + j(a s u ) j0>+ +(C 1 +C ) <K 0 j(a s d) j0>< + j(a c u) jd + > = (C + C 1 )f M ( + ) <K 0 j(vs) c jd + >+ +(C 1 +C )f K < + j@ (A c u ) jd + > + (C + C 1 )f D <K 0 + j@ (A s u ) j0> + +(C 1 +C )f K < + j@ (A c u) jd + > : (:14) In (.14) we have given an examle of an amlitude with K 0 and K 0 interfering contributions, that also contains an annihilation term. A w (D +! 0 + )= = G F U ud U cd = G F U ud U cd (C + C 1 ) < 0 j(v c d) jd + >< + j(a d u) j0>+ +(C 1 +C ) < + j(vu) c jd + >< 0 j(a d d) j0> = (C + C 1 )f < 0 j@ (Vd c) jd + >+ (C 1 +C ) f < + j@ (V c u) jd + > = (:15) =+ G F U ud U cd (C 1 + C )(1+)f < + j@ (V c u) jd + > : 8

9 A w (D 0! + )= = G F U ud U cd (C + C 1 ) < j(v c d ) jd 0 >< + j(a d u ) j0>+ + G F U ub U cb (C4 + C 3 ) < j(v c d ) jd 0 >< + j(a d u ) j0>+ (C 6 +C 5 ) < + juuj0><0ju 5 cjd 0 > + +(C 6 +C 5 ) < j dcjd 0 >< + ju 5 dj0> = = G F U ud U cd (C + C 1 )f < j@ (V c d) jd 0 > + (:16) + G F U ub U cb (C4 + C 3 )f < j@ (V c d) jd 0 > + +i(c 6 +C 5 ) < + juuj0 > f D M D m u + m c + i (C 6 + C 5 ) < j dcjd 0 > f M m u + m d : In (.15) and (.16) we give examles of Cabibbo forbidden decay amlitudes, to the second of which enguin oerators contribute. The arameter aearing in the above equations should be equal to 1 in QCD. 3 Since however other colour suressed, nonfactorizable contributions have been neglected we will consider it as a free arameter to be tted to the exerimental data, following [3]. The result of the t favours a value ' Final state interaction eects We make the assumtion that FSI are dominated by resonant contributions, and we neglect the hase-shifts in exotic channels. In the mass region of seudoscalar charmed articles there is evidence, albeit not very strong [6], for a J P =0 K(1830) (with = 50 MeV and an observed decay tok [17]) and a J P =0 (1770) with = 310 MeV [18]. The couling of an octet of 0 e P resonance to 0 +1 (PV)channels is determined from charge conjugation and SU(3) symmetry to be: Equation (3.1) imlies: hf abc (@ e Pa ) V b P c : (3:1) ( e K! PV) (e! PV) =0:8' ex ( e K) ex(e) ; (3:) consistent with the assumtion that the e P resonances decay redominantly into the lowest lying P and V mesons, which we shall make for simlicity. 9

10 For Cabibbo allowed and doubly forbidden D 0 decays and for Cabibbo rst and doubly forbidden D + decays, the FSI eect modies the amlitudes in the following way: A(D! V h P k )=A w (D!V h P k )+c hk [ex(i 8 ) 1] X h0k0 c h0k0 A w (D! V h0 P k0 ) ; (3:3) In (3.3) c hk are the normalized ( P c hk = 1) coulings e PPV and sin 8 ex(i 8 )= (M ep ( e P ) M D ) i ( e P ) ; (3:4) where e P is the resonance aroriate to the decay channel considered (e or e K). Equation (3.4) determines 8 u to a 180 ambiguity. The choice can be made according to the number of resonances and bound states [19] that are resent in the channel at lower energies, each one of them increasing the hase-shift by. In this case the e P resonance may be assumed as the third resonance in the channel, with 8 close to 1 (in our recent rerint a dierent choice was made for the PV channels). For Cabibbo forbidden D 0! PV decays one exects e and e 0 resonances to take also art to rescattering. Due to charge conjugation invariance, the singlet comonents have vanishing couling and the combined eect of the two exected isoscalar resonances may be described by a hase attached to the isosinglet octet art of the decay amlitudes. This hase is the only added arameter for these channels to be tted. The tted value is 43, corresonding to two resonances with masses one below and the other above the D 0 mass. Coming now to FSI eects for arity violating D! PP decays, we note that some evidence exists for a J P =0 + resonance K0 (with mass MeV, width MeV and 514% branching ratio in K [0]). No a 0 isovector resonance has been observed u to now in the interesting mass region. In [4] we assumed its existence and we estimated its mass from the equisacing formula M a 0 = M K 0 M K + M : (3:5) In the t we allowed the mass and the width of K0 to vary within the exerimental bounds. The best t values are 198 MeV and 300 MeV, resectively. From Eq. (3.5) we get M a0 = 1869 MeV. The SU(3) and C invariant couling of a scalar S resonance to two seudoscalar mesons is: r 3 g 888 d abc P a P b S c +g 818 ( P a P 0 + P 0 P a ) S a + +g 881 P a P a S 0 +g 111 P 0 P 0 S 0 : (3:6) 10

11 In (3.6) a; b; c =1:::8 are SU(3) indices, P 0 and S 0 are SU(3) singlets. Assuming that the S resonances decay dominantly to a air of mesons belonging to the lowest mass nonet, one obtains from (3.6) the branching ratio for (K0! K)as a function of the ratio of couling constants r = g 818 =g 888. The further assumtion of nonet symmetry would imly r = 1. The exerimental data allow two ossible values for r, one ositive (and consistent with 1) and another negative and close to 1. The best t value of r is 0:84, corresonding to a branching ratio for the decay ofthek0resonance in K of about 64%. The descrition of rescattering eects for Cabibbo forbidden D 0 decays is comlicated by the resence of yet unobserved f 0 and f0 0 isoscalar resonances, which should be singlet-octet mixtures. Denoting by jf 0 i the lower mass state, we dene jf 0 i = sin jf 8 i + cos jf 1 i jf 0 0i = cos jf 8 i + sin jf 1 i: (3:7) The results will also deend on the arameters a = g 881 =g 888 and c = g 111 =g 888, see (3.6). To reduce the number of new arameters, we assume that these scalar resonances behave similarly to the tensor mesons f (170) and f 0 (155): the f 0 is very weakly couled to, and the f has in turn a small couling to KK. In order to forbid the f 0 0! decay, we required a and the mixing angle to be related by a = 1 tan : The value tan = would then imly a vanishing branching ratio for the decay f 0! KK. The best t value is tan =1:14, not very far from 1:41. The t has therefore two new free arameters, = Mf 0 0 Mf 0 and. For any air ( ;) there are two ossible values for c, that are solutions of a quadratic condition coming from the requirement of unitarity of the rescattering transformation. 4. Comarison with exerimental data on Branching Ratios Starting from the weak amlitudes A w dened in Section and modifying them with FSI eects as exlained in the revious Section, we evaluated the rates for all Cabibbo allowed two-body decays and for Cabibbo forbidden D + and D s + decays as functions of the arameters of the t. These are, the arameters W PP and W PV of the annihilation contributions, the axial charges a cu = a cd and a cs, the mass and width of the scalar K0 resonance. We use the values (m u ;m d ;m s ;m c ) = (4.5, 7.4, 150, 1500 MeV) for 11

12 the quark masses. The decay constants are (f ;f K ; f = f K ;f! ;f ) = (133, 160, 16, 156, 33 MeV). The ole masses in the form factors (.10) corresonding to yet undetected charmed articles have been taken to be M D (0 + ) = 470 MeV and M D s (0 + ) = 600 MeV. Other arameters relevant for decay to nal states containing ( 0 )have been xed following [15]. The results are resented in Tables 1,, 3. The best-t results are reorted in column three of the Tables. The total is The 5 data oints for Cabibbo allowed decays contribute 61.8 and the 1 data oints for Cabibbo rst forbidden decays The best t arameters are = 0:07, r = 0:84, W PP = 0:9, W PV =+0:9, a cu = a cd =1:0, a cs =0:59, M K 0 = 198 MeV and K 0 = 300 MeV. A searate t to Cabibbo allowed data alone gives quite similar values for the arameters. In the Tables we have also reorted the theoretical redictions for the decays to nal states containing K L, in order to show the imortance of interference eects with doubly Cabibbo-suressed amlitudes. decays [6] We note that the relatively large SU(3) violation resent in the data for exotic D + U cd R + = U cs (D +! + 0 ) (D +! K S + ) ' 3:55; (4:1) (instead of R + =1)iswell reroduced by the tted data, as it also haened in [5]. This oint has been discussed in detail in [1] and more recently in []. The reason is that several SU(3) breaking eects, each one rather small, contribute coherently to enhance R +. We have also evaluated, not including them in the t, two recently measured doubly Cabibbo forbidden branching ratios. The exerimental data are [3], [4] R K 0 = B:R:(D0! K + ) B:R:(D 0! + K ) =(0:77 0:5 0:5) 10 (4:) R 0 + = B:R:(D+! K + ) B:R:(D +! + ) =(5:8+3: :6 0:7) 10 ; (4:3) and the theoretical redictions using the best-t arameters are R K 0 =0:89 10 and R 0 + =0: In the SU(3) limit the U-sin roerties of the hamiltonian should give R K 0 = tan 4 C =0:6 10, also discussed in []. Our model redicts R K 0 to be 3.4 times larger than this value. This is due mainly to the W-exchange contributions, that should vanish in the symmetric limit and have oosite signs in the two amlitudes, and also to rescattering eects. 1

13 On the other hand the theoretical value for R 0 + is much smaller than the exerimental datum, that however diers from zero by only.5 standard deviations. We note that in a factorized model the decay D +! K + may only roceed through annihilation and rescattering. It is therefore dicult to reroduce the resent very large value for R 0 +. We consider now D 0 decay rocesses, and in articular Cabibbo rst-forbidden decays. The D 0 meson is a U-sin singlet and it should only decay tou-sin trilet states, if avour SU(3) is a good symmetry. Therefore in that limit several relations among decay amlitudes hold. For the arity violating D 0! PP decays they are: A(D 0! K 0 K 0 )= 0; (4:4) A(D 0! K + K )= A(D 0! + )= = A(D0!K + ) tan C = tan C A(D 0! K + ) (4:5) (the last equality corresonds to the symmetry rediction discussed after (4.)), 1 3 A(D 0! K S 0 ) = cos 0A(D 0! K S ) + sin 0A(D 0! K S 0 ) ; (4:6) 1 3 A(D 0! 0 0 ) = cos 0A(D 0! 0 ) + sin 0A(D 0! 0 0 ) ; (4:7) A(D 0! 0 0 )= (1 tan 0)A(D 0! ) + tan 0A(D 0! 0 ) ; (4:8) A(D 0! 0 ) = tan 0 A(D 0! )+ + 3 sin 0A(D 0! 0 ) cos 0A(D 0! 0 0 ) : (4:9) Comaring the above formulae to the exerimental data [6], [7], [3], we note that (4.4) is denitely not true, the moduli of the amlitudes in (4.5) are in the ratios (1:0.59:1.15:0.67) instead of being equal, relation (4.6) is comatible with the data, but only with large hases, and nally no data exist for (4.7), (4.8), (4.9). In the factorization scheme the amlitude A(D 0! K 0 K 0 )vanishes, and we may only obtain a nonzero rate in that channel through rescattering from the other decay channels. Analogously, the small SU(3) breaking eects (f 6= f K, MD M >M D M K,:::) are not enough to reroduce the large ratio of K + K to + rates. In Table 4 we show the results of our best t to the Cabibbo rst-forbidden decay rates. As exlained in Section 3, to the arameters reviously determined we added two more free arameters, and, and chose the value of another one, c, between the two solutions of a quadratic consistency condition. The t we obtain is good ( =1:7 with 4 data oints). The best t results for 13

14 q the arameters are = Mf 0 0 Mf 0 = 105 MeV, =48:7 and c ' :69. The corresonding masses and widths of the scalar and isoscalar resonances are M f0 = 1778 MeV, f 0 = 361 MeV, M f 0 0 = 148 MeV and f 0 0 = 389 MeV. For arity conserving decays, the relations obtained from avour symmetry are less restrictive in view of the few existing data. For instance, the relation corresonding to (4.4), namely A(D 0! K 0 K 0 )+A(D 0!K 0 K 0 )= 0; (4:10) is valid in the factorization aroximation and rescattering does not soil its validity in our model. Only uer limits exist exerimentally. Similarly, other SU(3) redictions are A(D 0! K + K )= A(D 0! + ) A(D 0!K K + )= A(D 0! + ): (4:11) The W-exchange terms strongly violate these relations and our redictions are therefore at variance with them. Data for nal states are unfortunately still missing. The exerimental data and the tted values for arity conserving Cabibbo rstforbidden D 0! PV decays are also reorted in Table 4. The only arameter added to those determined in the revious ts is the hase-shift of the isosinglet octet art of the decay amlitudes. This arameter turns out to be 43 and the quality of the t is reasonably good ( =7:7 for 4 data oints). 5. CP violation It is well known that CP violating eects show u in a decay rocess only if the decay amlitude is the sum of two dierent arts, whose hases are made of a weak (CKM) and a strong (nal state interaction) contribution. The weak contributions to the hases change sign when going to the CP-conjugate rocess, while the strong ones do not. Let us denote a generic decay amlitude of this tye by and the corresonding CP conjugate amlitude by A = Ae i 1 + Be i (5:1) A = A e i 1 + B e i : (5:) The CP violating asymmetry in the decay rates will be therefore a CP jaj j Aj jaj + j Aj = =(AB ) sin( 1 ) jaj + jbj +<(AB ) cos( 1 ) : (5:3) 14

15 Both factors in the numerator of eq. (5.3) should be nonvanishing to have a nonzero eect. Moreover, to have a sizeable asymmetry the moduli of the two amlitudes A and B should not dier too much. In Cabibbo rst-forbidden D decays, the enguin terms in the eective hamiltonian (.) rovide the dierent hases of the weak amlitudes A and B. Having obtained a reasonable descrition of the decay rocesses, including a model for their strong hases, we may envisage the more ambitious goal to derive CP violating asymmetries using our model for the hase-shifts. The asymmetries resulting are around 10 3, somewhat larger than reviously exected. We stress however that the actual numbers may vary areciably for arameter variations that still give reasonable ts to the decay rates. For D + (and D + s ) decays the total charge allows to directly measure the rates to be combined in the asymmetry (5.3). In the neutral D decays, however, the need of tagging the decaying article to tell its charm and the ossibility ofd Dmixing make (5.3), as it stands, not directly measurable. Let us consider the articular case of an e + e at the 00, assume to tag D 0 by semiletonic decays and dene a eff: factory CP = ja(f i;e )j ja(f i ;e + )j (5:4) ja(f i ;e )j +ja(f i ;e + )j: Limiting our considerations to the simlest case, D decays to a CP self-conjugate nal state (f i = f i ), and considering the time-integrated asymmetry, one obtains 1 a eff: CP = q 1+ qa + A 1 y 1+x 1+ q 1 qa A 1+ q 1+ qa + A 1 y 1+x 1 q 1 qa : (5:5) A In (5.5) A = A(D 0! f i ) is the decay amlitude, the mass matrix eigenstates are dened as and the mixing arameters are jd S;L i/jd 0 i q jd0 i x = (M S M L ) S+ L ; y= S L S+ L : The mixing for charmed mesons is exerimentally known to be small (jxj < 0:083, y<0:085) [6] and the theoretical calculations of the short-distance contributions give very small redictions. A reliable calculation of long-distance terms is roblematic, even more so for CP violation in the mass matrix and the ratio =q. We did not consider at all the 15

16 time-deendence in the asymmetries, since these deend on the hase of =q. We note anyhow that the smallness of x and y revents the develoment in time of areciable asymmetries even if the hases would allow this. We exect moreover that the modulus j=qj will dier from one by a small amount, O(10 3 ). Therefore we can exand (5.5) in the small quantities a CP = 1 1 A A and < = q ; with the result a eff: CP ' a CP (1 ) < + ::: (5:6) In (5.6) we have used the denition = 1 (1 y ) = (1 + x ). Exerimentally <0:014. Therefore, if a CP is not much smaller than <, the measured asymmetry will be a eff: CP ' a CP. We reort in Table 5 the values of a CP that we obtain in our model for several decay rocesses. We chose to give only the results that corresond to a good t for the branching ratios, even if the redictions for some other decay channels are also large. For arity conserving D 0 decays the nal states are not CP eigenstates, and the corresonding formula for a eff: CP is more comlicated; however, we note that for amlitudes of not too dierent absolute values, as it haens in Cabibbo rst-forbidden decays, and given the smallness of the mixing arameters, the result is again a eff: CP ' a CP. If the nal state contains a neutral kaon, normally identied by its decay to +, one has to disentangle the CP violating eects in D and K decays. How to do this for the D + decays has been discussed in [5]. We note that these channels are not very romising candidates to look for CP violating eects in D decays. Weevaluated the central value for a CP choosing for the Maiani { Wolfenstein arameters (,) the values (0:, 0:3), following a recent analysis of CKM arameters [5], and U cb =0:040. We varied (,) in the one-sigma region obtained in [5] for f B = MeV. The error given in Table 5 reects only this uncertainty, which is already quite large. 6. Conclusion We have resented a generally successfull descrition of the comlex of two-body nonletonic decays of charmed mesons. We tted 45 exerimental branching ratios with eleven free arameters, that assume values close to the exected ones at the minimum of the =

17 We note that the large SU(3) breaking eects shown by the data are well reroduced in our results. Rescattering (FSI) eects are articularly imortant in this resect: their arameters are derived from exerimental data on nearby resonances. The masses of these being unequal, avour SU(3) breaking is induced through the dierence in the hase shifts for each isosin channel. W-exchange/annihilation contributions are substantial in many cases. The danger of getting too big decay rates for D s Cabibbo-favoured decays has been avoided in our model imosing chiral symmetry requirements. Moreover, the rather large nal state hase shifts and \enguin" oerator contributions lead us to envisage CP violating asymmetries larger than 10 3 in some decay channels. 17

18 f i B.R. ex (D 0! f i ) B.R. th (D 0! f i ) K + 4:01 0:14 4:03 K S 0 1:0 0:13 0:78 K L 0 0:57 K S 0:34 0:06 0:46 K L 0:34 K S 0 0:83 0:15 0:84 K L 0 0:67 K 0 0 3:0 0:4 3:49 K S 0 0:55 0:09 0:49 K L 0 0:39 K + 4:9 0:6 4:69 K + 10:4 1:3 11:19 K 0 1:9 0:5 0:51 K 0 0 < 0:11 0:005 K S! 1:0 0: 1:1 K L! 1:04 K S 0:415 0:060 0:4 K L 0:48 TABLE 1 Decay branching ratios in ercent for Cabibbo allowed two-body D 0 nonletonic decays. In the rst column the exerimental data are reorted (uer bounds are 90% c.l.) [6], in the second column the theoretical values obtained in the best t. See text for further exlanations. 18

19 f i B.R. ex (D +! f i ) B.R. th (D +! f i ) K S + 1:37 0:15 1:08 K L + 1:43 K 0 + : 0:4 0:64 K S + 3:30 1:5 5:8 K L + 6: :5 0:07 0:17 + 0:75 0:5 0: < 0:9 0:79 K 0 K + 0:78 0:17 0: < 0:14 0: :37 + < 1: 0: < 1:5 0:13! + < 0:7 0: :67 0:08 0:59 K 0 K + 1:70 K 0 K + 0:51 0:10 0:5 TABLE Decay branching ratios in ercent for Cabibbo allowed and rst-forbidden two-body D + nonletonic decays. In the rst column the exerimental data are reorted (uer bounds are 90% c.l.)[6], in the second column the theoretical rates obtained in the best t. See text for further exlanations. 19

20 f i B.R. ex (D s +! f i ) B.R. th (D s +! f i ) K S K + 1:75 0:35 :53 K L K + :6 + 1:90 0:40 1: :7 1:4 5: :0 : 9: :0 3:0 :61 K 0 K + 3:3 0:5 3:86 K S K + :1 0:5 1:44 K L K + 1:93 + 3:5 0:4 :89! + < 1:7 0:0 0 + < 0:8 0: :080 K + 0 0:16 K + 0:7 K + 0 0:5 K 0 + < 0:7 0:43 K + 0 0:09 K + 0 0:4 K + 0:04 K + 0 0:04 K +! 0:07 K + < 0:5 0:015 K 0 + 0:33 K 0 + 1:95 TABLE 3 Same as Table for D s + nonletonic decays. 0

21 f i B.R. ex (D 0! f i ) B.R. th (D 0! f i ) 0 0: :17 0:10 0 0: 0 0 0:088 0:03 0: :159 0:01 0:159 K + K 0:454 0:09 0:456 K 0 K 0 0:11 0:04 0:093! 0 0: : :010! 0:19! 0 0: :11 0:057 K 0 K 0 < 0:08 0:099 K 0 K 0 < 0:15 0:099 K + K 0:34 0:08 0:45 K K + 0:18 0:10 0:8 + 0:8 + 0: :17 TABLE 4 Decay branching ratios in ercent for Cabibbo rst-forbidden two-body D 0 nonletonic decays. See the cation of Table and text for details. 1

22 decay channel 10 3 a CP decay channel 10 3 a CP D +! 0 + 1:17 0:68 D +! K 0 K + 0:51 0:30 D +! :8 0:74 D 0! 0 +1:43 0:83 D 0! K 0 K 0 +0:67 0:39 D 0! 0 0 0:98 0:57 D 0! K 0 K 0 +0:67 0:39 D 0! +0:50 0:9 D 0! K + K 0:038 0:0 D 0! 0 +0:8 0:16 D 0! K K + 0:16 0:09 D 0! 0 0 0:54 0:31 D 0! + 0:37 0: D 0! + +0:0 0:01 D 0! + +0:36 0:1 D 0! K + K +0:13 0:8 D 0! K 0 K 0 0:8 0:16 TABLE 5 CP-violating decay asymmetries for some D + and D 0 Cabibbo forbidden decays. See text for exlanation.

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