Transaction B: Mechanical Engineering Vol. 16, No. 2, pp. 140{148 c Sharif University of Technology, April 2009
Spectator Model in D Meson Decays
H. Mehrban1
Abstract. In this research, the e ective Hamiltonian theory is described and applied to the calculation of current-current (Q1;2) and QCD penguin (Q3; ;6) decay rates. The channels of charm quark decay in the quark levels are: c ! dud, c ! dus, c ! sud and c ! sus where the channel c ! sud is dominant. The total decay rates of the hadronic of charm quark in the e ective Hamiltonian theory are calculated. The decay rates of D meson decays according to Spectator Quark Model (SQM) are investigated for the calculation of D meson decays. It is intended to make the transition from decay rates at the quark level to D meson decay rates for two body hadronic decays, D ! h1h2. By means of that, the modes of nonleptonic D ! PV , D ! PP , D ! VV decays where V and P are light vector with J P = 0 and pseudoscalar with J P = 1 mesons are analyzed, respectively. So, the total decay rates of the hadronic of charm quark in the e ective Hamiltonian theory, according to Colour Favoured (C-F) and Colour Suppressed (C-S) are obtained. Then the amplitude of the Colour Favoured and Colour Suppressed (F-S) processes are added and their decay rates are obtained. Using the spectator model, the branching ratio of some D meson decays are derived as well.
Keywords: E ective Hamilton; c quark; D meson; Spectator model; Hadronic; Colour favoured; Colour suppressed.
INTRODUCTION: EFFECTIVE like the standard model. A well known example is the HAMILTONIAN THEORY four-Fermi interaction, where the W -boson propagator 2 2 is made local for MW >> q (q denotes the momentum As a weak decay in the presence of strong interaction, transfer through the W ): D meson decays require special techniques. The main tool to calculate such D meson decays is the e ective i(g )=(q2 M 2 ) ! ig [(1=M 2 ) Hamiltonian theory. It is a two step program, starting W W with an Operator Product Expansion (OPE) followed + (q2=M 4 ) + ]; (1) by a Renormalization Group Equation (RGE) analysis. W The necessary machinery has been developed over the past years. The derivation starts as follows: If the where the ellipsis denotes terms of a higher order in kinematics of the decay are of the kind where the 1=MW . Apart from the t quark the basic framework for masses of the internal particle, Mi, are much larger 2 2 weak decays quarks is the e ective eld theory relevant than the external momenta, P , Mi >> p , then, the heavy particle can be integrated out. This con- for scales MW , MZ , Mt >> [1]. This framework, cept takes concrete form with the functional integral as seen above, brings in local operators, which govern formalism. It means that the heavy particles are \e ectively" the transition in question. removed as dynamical degrees of freedom from the It is well known that the decay amplitude is the theory. Hence, their elds no longer appear in the product of two di erent parts the phases of which are e ective Lagrangian. Their residual e ect lies in the made of a weak (Cabbibo-Kobayashi-Maskawa) and a generated e ective vertices. In this way, an e ective strong ( nal state interaction) contribution. The weak low energy theory can be constructed from a full theory contributions to the phases change sign when going to the CP-conjugate process while the strong ones do not. Indeed, the simplest e ective Hamiltonian without 1. Department of Physics, Semnan University, Semnan, P.O. QCD e ects (c ! sus) is: Box 35195-363, Iran. E-mail: [email protected] p Received 13 January 2007; received in revised form 10 November 0 2007; accepted 12 April 2008 He = 2 2GF VscVsuQ1; (2) Spectator Model in D Meson Decays 141
where GF is the Fermi constant, Vij are the relevant The partial decay rate in the c rest frame is (see CKM factors and: Appendix A):
Q1 = (s c )V A(u s )V A; (3) d2 =dp dp = (G2 =3)p p E f (p :p =E E ) Q1; ;Q6 i k F i k j 1 i k i k is a (V A). (V A) is the current-current local + (p :p =E E ) operator. 2 i j i j This simple tree amplitude introduces a new + 3(mkmj=EkEj)g: (9) operator, Q2, and is modi ed by the QCD e ect to: p After the change of variable to x and y, the decay rate He = 2 2GF VscVsu(C1Q1 + C2Q2); (4) is given by:
Q2 = (s c )V A(u s )V A; (5) 2 EH : (10) d Q1; ;Q6 =dxdy = 0cIps where C1 and C2 are Wilson coecients. The situation in the standard model is, however, more complicated SPECTATOR MODEL because of the presence of additional interactions in In the spectator model [3], the spectator quark is given particular penguins, which e ectively generate new a non-zero momentum, having in this work a Gaussian operators. These are, in particular, the gluon, photon distribution represented by a free (but adjustable) and Z0-boson exchanges and penguin c quark contri- parameter, : butions, as seen before.
Consequently, the relevant e ective Hamiltonian 2 3=2 3 (p2=2) for D-meson decays generally involves several oper- P (jpsj ) = (1= )e s : (11) ators, Q , with various colour and Dirac structures, i The probability distribution of a three momentum for which are di erent from Q . The operators can be 1 spectator quarks is given by: grouped into two categories [2]: i = 1; 2-current-current operators; i = 3; ; 6-gluonic penguin operators. 2 3 2 2 Moreover, each operator is multiplied by a calculable dP (ps) = P (jpsj )d ps = P (jpsj )d psdps: (12)
Wilson coecient, Ci(): 2 And P (jpsj ) is normalized according to: 6 p X Z 1 He = 2 2GF di()Qi(); (6) 2 2 4 P (jpsj )psdps = 1: (13) i=1 0 where scale is discussed below, di() = VCKMCi() Here, however, a tentative spectator model is consid- and VCKM denotes the relevant CKM factors that are: ered based upon the idea of duality between quark and hadron physics at the high energies of c quark and D d1;2 = VicVjkC1;2; meson decays. The decays at the quark level, even including the penguins, are basically short distance d3; ;6 = VtcVtkC3; ;6: (7) processes. In our proposed spectator quark model, the long distance hadronization is largely a matter of incoherently assigning regions of the nal quark phase EFFECTIVE HAMILTONIAN DECAY space to the di erent mesonic systems. For example, RATES consider a D meson cu or cd to be a heavy stationary
The e ective C = 1 Hamiltonian at scale = O(mc) c quark accompanied by a light spectator constituent for tree plus penguin term is given by: antiquark, which has a spherically symmetric normal- ized momentum distribution, P (jp j2)d3p . The total p s s C=1 s s meson decay rate through a particular mode is then He = 2 2GF f[d1s()Q1() + d2s()Q2()] assumed to be: d d + [d1d()Q1() + d2d()Q2()] Z 2 d 2 3 total = P (jpsj )d psdpidpk; (14) X6 dpidpk d ()Q ()g: (8) i i which is equal to the initiating decay rate. Ignoring, i=3 for the moment, any constraints due to quark colour, it Here, d1; ;6 are de ned by Equation 7, d1;2s;d = is supposed the spectator antiquark, qs, combines with d1;2(i = j = s; d) and index k refers to d or s quarks. the quark, q (q = qi or qk), to form the meson system. 142 H. Mehrban
, is assigned to For example, if q = qi, a mass, Mqiqs It is also possible that the spectator antiquark the system such that: combines with the quark qk for which we get:
2 2 2 Z M = (pi + ps):(pi + ps) = m + m 2 is i s d 2MksMij Ekps = 2 dMksdMij Mc pk + 2(EiEs pips cos is): (15) d2 Constraining p and p to have mass M , it can be 2 i s is P (jpsj )dpsdpi: (22) inferred from Equation 14 that: dpidpk
Z 2 This process Colour Suppressed (C-S). In some meson d d 2 2 2 2 = 2Mis P (jpsj )(M m m decays, for example, D+(cd) ! (dd) + (du) is initiated dM dp dp is i s is i k by the quark decay, c ! dud, and the spectator, d, could have combined with the d or the u. In this case, 2(EiEs pips cos is)) results are shown if the processes produce incoherence 2 or assume coherence. In sum, the decay rates of B 2p dpsd(cos is)dpidpk: (16) s mesons for process (C-F) and process (C-S) are: Hence: m mZcut is Zcut kj Z 2 2 d psdps d 2 d = 2Mis P (jpsj )dpidpk; (17) (C-F) = dMisdMkj; dMis pi dpidpk dMisdMkj mmin is mmin kj where the integration region is restricted by the condi- m tion j cos isj 1. We also assign a mass, Mkj, to the mZcut ks Zcut ij second quark-antiquark system such that: d2 (C-S) = dMksdMij: (23) dMksdMij 2 2 2 m m M = (p + p ):(p + p ) = m + m min ks min ij kj k j k j k j where m = (m + m ), m = M and + 2(E E p p cos ); (18) min is qi qs cut is qiqs k j k j kj so on. Turning now to the colour factors, what may or: be regarded as two extreme possibilities are examined. In the rst, here called model (A), Equations 9, 20 M 2 = (p + p ):(p + p ) = (p p ):(p p ) and 22 are taken at face value, that is, no attempt is kj k j k j b i c i made to follow the ow of colour and it is assumed 2 2 that all colour ow is looked after by the gluon elds = m + m 2mcEi: (19) c i in the meson system. In the second, here called model (B), consider the possibility that the lowest mass meson The variable Ei or pi determines the mass, Mkj. Taking this mass to be the independent variable, we states are only formed if the quark-antiquark pairs are have: in a colour singlet state. That is, it is assumed that the colour distribution caused by a quark-antiquark pair 2 Z 2 d 2MisMkj Eips d in an octet state will result in more complex meson = P (jp j2)dp dp : dM dM m p2 dp dp s s k systems than the lowest mass 0 and 1 states. is kj c i i k (20) Projecting out the colour singlet states results only in a modi cation of coecients 1, 2 and 3 of The integration range is restricted by j cos kjj 1, Equation A3. Physical hadrons are singlets in colour 2 2 2 space, and are termed colourless or white. Colour cos = (m + m M + 2EkE )=2pkp : (21) kj k j kj j j symmetry is absolute and weak quark currents are, as This mode of quark and antiquark combination pro- hadrons, white. This means, for example, that qsqi is, cess is called (C-F) (Colour Favoured). Finally, by in fact, a sum of three terms: integration, the partial decay rates, (Mis;Mkj), are computed into quark systems with masses less than Mis qsqi = qs (h)qi (h) = qs(1)qi(1) + qs(2)qi(2) and Mkj. With suitable binding, these partial decay rates are equated with corresponding rates into mesons. + qs(3)qi(3); (24) Of particular interest are the quark antiquark systems forming the lowest mass 0 and 1 meson states, as where it suces for 1, 2 and 3 to stand for yellow, blue data exists for charmed quark systems that can be used and red, respectively. The spectator is a colour singlet to test the spectator quark model. with the c quark. Hence, the colour factors for Q1, Spectator Model in D Meson Decays 143
Q2Q3, Q4, Q5 and Q6 are given by: DECAY RATES OF PROCESSES C-F PLUS p Q ;Q ;Q ! process (C-F) 3; C-S (F + S) OF EFFECTIVE 1 4 6 HAMILTONIAN p process (C-S) 1= 3; Now, we want to calculate the decay rates of the e ec- p tive Hamiltonian (Q1; ;Q6) for F + S at quark-level Q2;Q3;Q5 ! process (C-F) 1= 3; and the spectator model. The e ective Hamiltonian for p F + S, is given by: process (C-S) 3: A+B b!ikj b!ijk He = He + He ; (30) Consequently, factors 1, 2 and 3 of Equation A3 are di erent for process (C-F) and process (C-S) of where Hc!ikj is de ned by Equation 8 and we can model (B). For process (C-F), represented by Equa- e c!ijk tion 27, they become: obtain He (see Appendix C). The decay rates of 2 current-current plus penguin for F + S are given by 1 = 3 jd1 + (d2=3) + (d3=3) + d4j ; (see Appendix C): 2 2 = 3 j(d5=3) + d6j : (25) 2 F +S F +S d EH =dxdy = 0cIpc : (31) While for process (C-S), represented by Equation 58, they are: NUMERICAL RESULTS 2 1 = 3 j(d1=3) + d2 + d3 + (d4=3)j ; As an example of the use of the formalism above, we use the standard Particle Data Group [4] parameterization = 3 jd + (d =3)j2 : (26) 2 5 6 of the CKM matrix with the central values,
EFFECTIVE HAMILTONIAN SPECTATOR 12 = 0:221; 13 = 0:0035; 23 = 0:041; MODEL and choose the CKM phase, 13, to be =2. Following We try to calculate the spectator model for the general Ali and Greub [2], internal quark masses are treated in case, which we call the e ective Hamiltonian spectator tree-level loops with the values (GeV) mb = 4:88, ms = model. According to Equation 10, the total decay 0:2, md = 0:01, mu = 0:005, mc = 1:5, me = 0:0005, rates for current-current plus penguin operators in the m = 0:1, m = 1:777 and me = m = m = 0. E ective Hamiltonian are given by: Following G. Buccella [5], the e ective Wilson e = IEH : (27) coecients, Ci , is chosen for the various c ! q Q1; ;Q6 0c ps transitions. The di erential decay rates for the two boson system in the spectator quark model for current-current plus a) The total decay rate and branching ratios hadronic penguin operators in the E ective Hamiltonian are modes, according to the e ective Hamiltonian the- given by (see Appendix B): ory (see Equation 10), are shown in Table 1. We see p that mode c ! sud is dominant. The total c-quark 2 2 d 8qsiq (2m =M )2 +x2 Q1; ;Q6 = p kj i c decay rate of the e ective Hamiltonian is given by: 0c 2 d(qsi=Mc)d(qkj=Mc) Mc x EH total(c ! anything) Z 1 Z 1 e 2z2 dy dzps(q;z)ze : = (c!s anything)+ (c!d anything) 0 0 (28) Now, we can integrate over the two mass cuts (two = 9:261 10 13 + 0:606 10 13 GeV boson systems), and obtain the hadronic decay rates, as follows: = 9:867 10 13 GeV: Z m Z m0 cut cut d2 0 = Q1; ;Q6 dm dm0 ; Q1; ;Q6 cut cut Table 1. Decay rates ( ) and Branching Ratio (BR) of min min0 d(qsi=Mc)d(q =Mc) kj e ective Hamiltonian of c-quark. p Z m Z m0 2 Process 10 15 BR 10 3 cut cut8qsiq (2m =M )2 +x2 EH EH = p kj i c 0c 2 c ! dud 31.689 32.12 min min0 Mc x c ! dus 1.0785 1.093 Z 1 Z 1 e 2z2 0 c ! sud 409.44 414.95 dy dzps(q;z)ze dmcutdmcut: 0 0 (29) c ! sus 23.836 24.157 144 H. Mehrban b) Now the mean lives of the charm quark (D meson) the e ective Hamiltonian spectator model is given are obtained theoretically and compared with the by: 0 + experimental mean life of D , D and Ds , so: BREH (c ! dud)
EH EH Mean life theory(D) = ~= total = EH (c!dud)cut1;cut2= total EH (c!anything)
12 = 1:067 10 sec; (32) = 1:2502 10 14=9:867 10 13 and: = 1:2671 10 2: + 12 Mean life exp(D )=(1:0400:007)10 sec; The masses of some mesons and the masses of cut are shown in Table 2. The results are presented in 0 12 Mean life exp(D )=(0:4100:001)10 sec; Table 3 and compared, where data is available, with the sum of the branching ratios into mesons with + 12 masses less than the above cuto masses. Also, all Mean life exp(Ds )=(0:4610:015)10 sec: (33) the experimental and theoretical D meson decays in the spectator model are classi ed and given in Also, we can compare the branching ratio of the Table 3. semileptonic theoretically and experimentally, so: d) The decay rates of c quark for F + S (shown in Table 4), and the total decay rates of F + S are + + BR(c ! e anything)theory = BR(c ! se e) given by:
+ 3 (c ! dud) D+ ! (0; ; 0;!); (+; +); + BR(c ! de e) = 147:96 10
F +S 2 + 8:702 10 3 = 15:67 10 2; (34) BREH = 2:1023 10 ;
0 + and: (c ! sud) D+ ! (+; +); (K ; K );
+ + 2 F +S 2 BRexp(D ! e anything) = (17:2 1:9) 10 ; BREH = 51:2871 10 ;
+ 0 + + 0 + 2 (c ! sus) D ! ( ; ); (K ;K ); BRexp(D ! e anything) = (6:87 0:8) 10 ; F +S 2 + + 2 BREH = 3:8671 10 : BRexp(Ds ! e anything) < 20 10 : (35)
It is observed that the theoretical and experimental Table 2. The masses of some mesons and the masses of results are close. cut for D meson decay processes. c) In the spectator quark model, the value = 0:6 System of Mass Cuto Mass Particle GeV [6] is used. For the maximum mass of the Quark (GeV) (GeV) quark-antiquark systems, (m ), a value midway cut K 0.494 between the lowest mass 1 state and the next most massive meson is taken. Thus, we take, for (su) or su, sd K 0.892 1.081 K 1.270 (sd), mcut(ud) = 0:877 GeV between (0:770) and 0.140 a0(0:984); for (uu) and (dd), mcut(uu) = mcut(dd) = 0:870 GeV between !(0:782) and 0(0:958). ud 0.770 0.877
For example, it is possible to calculate the a0 0.984 Branching Ratios of mode c ! dud in the tree- 0 0.140 level and in the e ective Hamiltonian spectator 0.547 model. The mode, c ! dud, is for decays D+ ! 0 0+, D+ ! +, D+ ! 0+, D+ ! !+, uu, dd 0.770 0.870 D+ ! 0+, D+ ! +, D+ ! 0+ and ! 0.782 D+ ! !+. In this case, we have got a two- 0 0.958 boson system and, therefore, two masses of cut for 0 0.958 the boson system. Masses of cut for a two boson ss 1.020 1.150 system, mcut1 = 0:870=Mc and mcut2 = 0:877=Mc are chosen. Theoretically, the branching ratio of 1.680 Spectator Model in D Meson Decays 145
Table 3. Experimental and theoretical processes of spectator model of branching ratios of hadronic for D meson decays. 1- Decay of c quark 2- Decay of D meson, process (C-F) 3- Decay of D meson, process (C-S) 4- Experimental branching ratio, process (C-F) 5- Total experimental branching ratio, process (C-F) 6- Experimental branching ratio, process (C-S) 7- Total experimental branching ratio, process (B)
8- mcut1 Mc GeV, mcut2 Mc GeV, process (C-F)
9- mcut1 Mc GeV, mcut2 Mc GeV, process (C-S) 10- E ective Hamiltonian branching ratio, model (A), process (C-F) 11- E ective Hamiltonian branching ratio, model (A), process (C-S) 1- c ! dud + 0 0 + + 0 + + + 0 0 + + 2- D ! ( ; ; ;!), ( ; ), D ! ( ; ), ( ; ), Ds ! (K ;K ), ( ; ) + 0 0 + + 0 0 0 0 0 + 0 0 + + 3- D ! ( ; ; ;!), ( ; ), D ! ( ; ; ;!), ( ; ; ;!), Ds ! ( ; ; ;!), (K ;K ) 4- 0+; (2:5 0:7)E 3 +; (1:25 0:11)E 3 K0+; < 8:0E 3 0+; < 1:4E 3 K(892)0+; (6:5 2:8)E 3 +; (7:5 2:5)E 3 !+; < 7:0E 3 +; 1:2E 2 5- < (3:04 0:25)E 2 (1:25 0:11)E 3 (14:5 2:8)E 3 6- 0+; (2:5 0:7)E 3 00; (8:4 2:2)E 4 K+0; < 2:9E 3 0+; < 1:4E 3 +; (7:5 2:5)E 3 !+; < 7:0E 3 +; 1:2E 2 7- < (3:04 0:25)E 2 (8:4 2:2)E 4 < 2:9E 3 8- 0.870, 0.877 0.877, 0.877 1.081, 0.877 9- 0.870, 0.877 0.870, 0.870 0.870, 1.081 10- 1:2671E 2(F + S 2:1023 E 2) 1:2834E 2 1:2723E 2 11- 1:2931E 2 1:2543E 2 1:2398E 2 1- c ! sus + 0 0 + + 0 + + + 0 + + 2- D ! (K ; K ); (K K ), D ! (K ;K ); (K ;K ), Ds ! ( ; ); (K ;K ) + 0 + + 0 0 0 0 + 0 + + 3- D ! ( ; ); ( ; ), D ! ( ; ); ( ; ; ;!), Ds ! ( ; ); (K ;K ) 4- K0K+; (7:2 1:2)E 3 K+K ; (4:33 0:27)E 3 K+; < 5:0E 4 K(892)0K+; (4:2 0:5)E 3 K(892)+K ; (3:5 0:8)E 3 K0K(892)+; (3:0 1:4)E 2 K+K(892) ; (1:8 1:0)E 3 K(892)0K(892)+; (2:6 1:1)E 2 5- (6:74 2:62)E 2 (9:63 2:07)E 3 < 5:0E 4 6- 0(958); < 9:0E 3 0; < 1:4E 3 K+; < 5:0E 4 0(958)+; < 1:5E 2 ; < 2:8E 3 +; (6:1 0:6)E 3 !; < 2:1E 3 +; < 1:5E 2 0; (1:07 0:29)E 3 7- (4:51 0:1)E 2 (7:37 0:29)E 3 < 5:0E 4 8- 1.081, 1.081 1.081, 1.081 1.510, 1.081 9- 1.150, 0.877 1.150, 0.870 1.150, 1.081 10- 1:9543 E 2 1:9378 E 2 1:9859 E 2 (F + S3:8671E 2) 11- 1:9213 E 2 1:8975 E 2 1:9453 E 2 146 H. Mehrban
Table 4. Decay rates and branching ratio of F + S of colour favoured plus colour suppressed is less than e ective Hamiltonian. 2:1023 E 2. We see that the experimental and 15 2 Process HE 10 BREH 10 theoretical values are close. c ! dud 35.611 31.262 c ! dus 1.4608 1.2824 REFERENCES c ! sud 554.45 486.74 1. Buras, A.J. \Theoretical review of B-physics", Nucl. c ! sus 26.927 23.638 Instrum Meth. A, 368, pp. 1-20 (1995). 2. Ali, A. and Greub, C. \An analysis of two-body non- leptonic B decays involving light mesons in the standard CONCLUSIONS model", Phys. Rev. D, 57, pp. 2996-3016 (1998). In the present paper, the e ective Hamiltonian theory 3. Altarelli, G., Cabibbo, N., Corbo, G., Maiani, L. and and spectator quark model for c quark are used and Martinelli, G. \Decay constants and semileptonic decays the hadronic decays of D mesons are calculated. In of heavy mesons", Nucl. Phys. B, 208, pp. 365-377 (1982); Ali, A. and Greub, C. \ Branching fraction this model, decays of the channel hadronic decays of and photon energy spectrum for b ! s ", Phys. Lett. D mesons are added. For colour favoured and colour B, 259, pp. 182-189 (1991); Atwood, D. and Soni, A. + 0 + suppressed, channel c ! dud (e.g. D ! ) is \B > eta-prime +X and the QCD anomaly", Phys. considered and theoretical values very close to exper- Lett. B, 405, pp. 150-156 (1997) and Kim, C.S., Kim, imental ones are achieved. Finally, cases have been Y.G. and Lee, K.Y. \The bound state corrections to the shown in which the theoretical values are better than semileptonic decays of the heavy baryons", Phys. Rev. the amplitude of all the decay rates calculated. Here, D, 57, pp. 4002-4016 (1998). the total decay rates of the hadronic of charm quark 4. Particle Data Group, European Physical J. C, 46(3) in the e ective Hamiltonian are obtained according to (2006). colour favoured and colour suppressed, and then the 5. Buchalla, G. \QCD corrections to charm quark decay amplitude of colour favoured and colour suppressed in dimensional regularization", Nucl. Phys., B, 391, pp. processes are added to them to obtain their decay rates. 501-514 (1993). Also, using the spectator model, the branching ratio of 6. Cottingham, W.N., Mehrban, H. and Whittingham, some D meson decays was obtained. I.B. \Hadronic B decays: Supersymmetric enhancement According to Tables 1 and 4, it is possible to and a simple spectator model", Phys. Rev. D, 60, pp. compare the decay rates of processes c ! dud, c ! dus 114029-114042 (1999). and c ! sud, c ! sus. The channels of c ! dus and c ! sud have tree-level decay rates and the channels APPENDIX A of c ! dud and c ! sus have tree-level, plus Penguin, decay rates. According to Tables 1 and 4, we saw that Partial Decay Rate the decay rates of channels c ! dud and c ! sus of Squaring spin average term Q1; ;Q6 is given by: colour favoured plus colour suppressed were more than [(~)(~ ) + (~)( ) ]2 the decay rates of the e ective Hamiltonian. Hence, LL LR sp av the branching ratio of colour favoured plus colour suppressed was less than the e ective Hamiltonian. = 1(1=16)(1+vi)(1+vk)(1+vj)[1 cos(k i)] When we put the values of channels c ! dud and c ! sus into the theoretical model, it is observed that + 2(1=16)(1 vi)(1+vk)(1 vj)[1+cos(k j)] the theoretical and experimental values are close. q q 2 2 In Table 3, it is seen that the e ective Hamil- + 3(1=16) 1 vi (1+vk) 1 vj [1+cos(j i) tonian branching ratio of colour favoured plus colour suppressed is better than the e ective Hamiltonian cos(k j) cos(k i)]: (A1) branching ratio of colour favoured or colour suppressed alone. The experimental value of the branching ratio We must obtain the eight terms of the helicity states + 0 + + of the above equation and then add them up. So: of channel c ! sus (e.g. Ds ! K , Ds ! K+) is less than 5:0E 4 and the theoretical value 2 [(~ )(~)LL + (~ )()LR]sp av of the branching ratio of colour favoured plus colour suppressed is less than 3:8671 E 2. It is observed that = ( 1=2)[1 vivk cos(k i)] the experimental and theoretical values are close. Also, + 0 + + 0 + for channel c ! dud (e.g. D ! , D ! , + ( 2=2)[1 + vkvj cos(j k)] D+ ! !+, D+ ! +), the experimental value of q q the branching ratio is less than (3:04 0:25) E 2 + ( =2) 1 v2 1 v2: (A2) and the theoretical value of the branching ratio of 3 i j Spectator Model in D Meson Decays 147
2 2 2 1=2 After adding all colour combinations, 1, 2 and 3 hxa = [1 (x =(x + a ))] ; give: 2 2 2 1=2 2 2 hyc = [1 (y =(y + c ))] ; 1 = jd1 + d2 + d3 + d4j + 2 jd1 + d4j
2 h = [1 (x2=(x2 + b2))]1=2; + 2 jd2 + d3j ; xb
2 2 2 2 2 2 1=2 2 = jd5 + d6j + 2 jd5j + 2 jd6j ; hyb = [1 (y =(y + b ))] : (A8)
3 = Ref(3d1 + d2 + d3 + 3d4)d6 Also:
+ (d1 + 3d2 + 3d3 + d4)d5g: (A3) 2 2 2 1=2 hxa = [1 (x =(x + a ))] ; The angle between the particle velocities must be 2 2 2 1=2 physical, 1 cos(k i) 1 and 1 cos(j hyc = [1 (y =(y + c ))] : (A9) k) 1. So, we should take variable pi and pk, or x and y as: APPENDIX B p = xM =2; p = yM =2: (A4) i c k c Two Boson System Spectator Model
After the change of variable to x and y, the decay rate The di erential decay rates for a two boson system is given by: in the spectator quark model for current-current plus penguin operators in the e ective Hamiltonian was d2 =dxdy = IEH ; (A5) Q1; ;Q6 0c ps obtained, where: where: e e e e = 1 + 2 3 : (B1) EH 1 2 3 ps(q;z) 1 2 3 Ips = 1Ips + 2Ips + 3Ips; (A6) The integration region is restricted by condition where: cos is 1, thus: 1 Ips = 6xy:fab:(1 habc); e e e 1 ; 2 ; 3 2 Ips = 6xy:fbc:(1 + hbca); ( e e e 2 3 1ps; 2ps; 3ps if (fsi(z)) 1 I = 6xy:fac:hxa:hyc: (A7) = (B2) ps 0 otherwise fab, fbc, fac, habc and hbca are de ned by: where: 2 5 3 0c = G M =192 ; F c 2 2 p p fsi(z) = [(mi + ms)=Mc] (qsi=Mc) f = 2 x2 + a2 y2 + b2; ab p 2 2 + (1=Mc) m + ( z) 2 2 2 2 s (fab) (c + x + y ) habc = p p ; p 2 x2 + a2 y2 + b2 2 2 (2mi=Mc) + x =( xz=Mc): p p 2 2 2 2 fbc = 2 x + b y + c ; Therefore, using Equation A7, the phase space param- eters will be de ned by: 2 2 2 2 (fbc) (a + x + y ) hbca = p p ; 2 2 2 2 2 x + b y + c e 1ps = 6xy:fab:(1 habc); p p 2 2 2 2 fac = 2 x + a y + c ; e 2ps = 6xy:fbc:(1 + hbca); 2 2 2 2 (fac) (b + x + y ) hacb = p p ; 2 2 2 2 e 2 x + a y + c 3ps = 6xy:fac:hxa:hyc: (B3) 148 H. Mehrban
APPENDIX C Here A1, A2 and A3 refer to the colour and the helicity factor, Processes C-F Plus C-S (F + S) p p p p Adding the amplitudes of two terms for b quark spin A1 = ( 1=4) 1 + vi 1 + vk 1 + vj; 1/2 and -1/2 gives: p p p p A2 = ( 2=4) 1 vi 1 + vk 1 vj; [(~)(~ )]F +S = A [sin(( )=2) 1=2 1 k j i p p p p A3 = ( 2=4) 1 vi 1 vk 1 + vj: (C2)
+ sin((k + j i)=2)] We must square these terms and, by averaging over the b quark spin 1/2 and -1/2, we have: A2[sin((j k i)=2) F +S 2 ([(~ )(~)]sp av) = [(3=2) 1 + 2 + sin(( )=2)] k j i q q 2 2 3 1 vi 1 vk 1vivk cos(k i)] A1[sin((j k i)=2) q q + sin(( + )=2)] 2 2 j k i [ 3 1 vi 1 vj + 1vivj cos(j i)]
A [sin(( )=2) q q 3 k j i 2 2 + [ 1 1 vk 1 vj
+ sin((j k i)=2)]: + (1=2)( 1 2 2)vkvj cos(k j)]; (C3) F +S [(~ )(~)] 1=2 = A1[cos((k j i)=2) where 1, 2 and 3 are de ned by Equation A3. The decay rates of current-current plus penguin cos(( + )=2)] k j i for F + S was obtained such that:
A2[cos((j k i)=2) F +S Ipc = I1ps + I2ps + I3ps; (C4)
+ cos((k j i)=2)] where:
A1[cos((j k i)=2) I1ps = 6xy:fab:[ 1((3=2) habc) + 2 3hxahyb];
cos((j + k i)=2)] I2ps = 6xy:fac:[ 1hacb + 3hxahyc];
A3[cos((k j i)=2) I3ps = 6xy:fbc:[( 1=2)hbca + 2(hxbhyc hbca)]: (C5) + cos((j k i)=2)]: (C1)