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Spectator Model in D Meson Decays

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Spectator Model in D Meson Decays 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.
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