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Semileptonic Decays of Charmed Particles in Terms of Helicity Amplitudes

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Semileptonic Decays of Charmed Particles in Terms of Helicity Amplitudes THE SEHILEPTONIC DECAYS OF CHARMED PARTICLES R.P. NABAVI and X.Y. PHAM Division de Physique Théorique*, Institut de Physique Nucléaire, Paris and Laboratoire de Physique Théorique des Particules Elémentaires, Paris H.N. COTTINGHAM K * H. Hills Fhys ics Laborat ory, University of Bristol Bristol, BS8 1TL - Sogland ABSTRACT We develop a general formalism for analysing the semileptonic decays of charmed particles in terms of helicity amplitudes. We apply this to the !» and F mesons, and propose a simp!'! experimental test for the V ± A coupling of the charmed quark. IPNO.TH-77.-08 ^Laboratoire associé au C.M.R.S. '^Postal address ! Université Pierre et Marie Curie, four 16 - El », Place Jussieu, 75230 Paris Cedex 05 - France The discovery of charmed ^articles « very massive and long-lived,opens up a new field in the physics of weak interactions. The large region of final-state phase space, spanning many hadronic resonances, will make the scudy of these decays both rich in pheno­ menology and potentially useful in providing sensitive tests of ha- dron structure, tests that will bear especially on the relationship between the charmed and uncha/med physics. It is the purpose of this paper to examine the kinematics of the seraileptonic decays of charmed particles, particularly the pseudoscalar mesons, and.to investigate the implications of the quark model for decays to single hadron resonant states and to the more complicated systems described by structure functions. I - DECAYS TO SINGLE RESOKAMT STATES We consider in particular semileptonic decays of the char­ med spin-zero meson D to a general hadron H, either : D —> H + 1+ + |> (1) or D —» H + l" + ^ (2) We will also apply our calculations to the F meson. Although we dis­ cuss only pseudoscalar mesons the helicity formalism we develop here is convenient for generalization to states of higher spin. The decay is assumed to proceed through an intermediate charged vector boson which is assumed to have tbe conventional cou- ' 2 ) " '*" pling to the leptons1 » and an unspecified vector, axial-vector coupling to hadrons. He do not, in the first part of the paper, as­ sume that the "charmed" quark is necessarily that predicted in the GIH scheme . Semileptonic decays of charmed particles have also (4.) been considered in references The general hadronic weak current ma+rix element between n and H has four independent helicity amplitudes, which we define as ft» - <wijA^>^55 T # where g is the weak interaction coupling constant : cf. GF _ ± 4.ai( ,liS JD^is the initial meson in its rest frame. | H If,h} is ^ne recoiling hadron K which ha-î velocity tanhf along the z-axis and helicity A= +/" Ot. D . W = + £ (J« ±CV (5) Jm{0) is the time-like component of the weak current density, and all are evaluated in the rest frame of D at the point in space-time where the decay occurs. ïhe energy and momentum of the recoiling hadron H are given by : and the square of the momentum transfer t is related to f by : t = ^*K -4n,*MJceiti v <7> As is the case w:th electromagnetic decays » the for­ mulae for the partial decay widths are most succinctly expressed in terms of the current matrix elements evaluated in the Lorentz frame where the three momenta of the D and the H are the same. This is the rest frame of the lepton pair. The hadron matrix cle- ments are related to those of equations (3) and <f) by a Lorente transformation along the z-axis with rapidity *R , the rapidity of the lepton pair in the D rest frame. Explicitly, we define : F4 -- F± (8) (10) -ft sink <tL = ynHsinkf = PH (11) The partial decay rate when the lepton spirs are not ob­ served, and in the limit of large intermediate vector boson mass, is then : 4HJ>-»H Ab Juiu + |F («f'JW4 0 (12) where x^ is the lepton (electron or muon) mass, $• is the angle between the charged lepton momentum and the recoil momentum of H in the lepton-pair centre-of-mass system. The upper sign on the terms in eos0 refers to the reaction of equation (1), and the lower sign to equatior (2). For the decays of the charmed mesons, the factor fau is generally very small (less than 5$) even for the rauon, and will hereafter be neglected- We then have : •JF.ttJjWfl I (13) The observation of the angular distribution of the char­ ged lepton will then permit the experimental determination of the three space-like helicity amplitudes F+ , F_ and F (in the lepton pair centre of mass system) Summing over all lepton angles gives : »4 m ,m m 1 ir- kià{ '' '' 'i (I *) II - INCLUSIVE DECAYS AND STRUCTURE FUNCTIONS Lquations (13) and (1*0 are formally true independent of H being a particular resonant state j they are valid if H is any hadronic system of fixed mass, and the F+» F_ and F functions re­ present the decays into states of helicity +1, -1, and 0 respec­ tively. Summing over al]. final hadronic states of this mass for each helicitv immediately gives the partial semileptonic decay ra­ tes in terms of structure functions. Neglecting the lepton mass, we thus have simply : JJTO> Aï Itii + Wft, %*j[*hrtj / (15) where f» is the recoil momentum of the hadron system: <HI J W4 M ) = -^e ^ I * ' *>f * V« 'M ci. ) These structure functions are independent of the direction of Q, being functions only of tctl' CUJL wjj = MJ'+t -*%<£ (18) The matrix elements are here evaluated in the rest frame of the D although equation (17) again takes a somewhat simpler form in the lepton-pair rest frame. The structure functions defined here are related to the structure functions for the scattering of neutrinos from D mesons, the only difference being that the momentum transfer Q is space like for scattering, time like for decay. Defining the covariant tensor ; ri*» (19) "an . (20) M* * 4M? J JH< < c where C( s I'R-|I = ^»+k» . then the invariant functions W. are the direct generalisation from the nucléon to the D mason of those defined by Llewellyn Smith ' 1 -JM, i (21) w0 = \ + il w, The functions W,,» W5, and Wg do not contribute to the decay rate when the lepton mass is neglected. Integrating over angles in Equation (15) we have for the inclusive decay : 4V _ G* k P. (22) (23) III » KINEMATIC SINGULARITY-FREE FORM FACTORS FOR SIMGLE-HADROB MATRIX ELEMENTS The method of Cottingham and Pollard for explicitly exhibiting the angular momentum barrier penetration factors in the electromagnetic current helicity n:atrix elements can be generali­ zed to the weak current to give in the case of the D meson decay to a spin-zero hadron ii : F4(t) =o (2») %») = /,(*) Ihit (25) fft; - Ji(0 (26> and for the case of the D decay to a hadron H with spin s^ 1 (i %H -- (^)'% ^^)^r * ffijo*)'] (ï7) F.!*) - (^J^««*^)(^«JW (26) T(i) Z %(*) [Kvktf (29) IV - A SUM RULE AND QUARK MODEL ESTIMATES From Equations (11-, 16, 17) the semîleptonïc decay width of the D to a lepton pair and hadrons may also be written (30) = MA bJ^<^^><^lTl/^>f -<^HI*><H/JBIJ?> (3D t = Mean value of t in the sum. The term in the weak charge which is not explicitly co- variant has to be evaluated in the frame where the D and H have the same threa momentum. The sum is over all hadron states which have an invariant mass : WH < % (32) and energy H; < + K (33) £M *«, which is a 1 irge region of phase space because of the high mass of charmed particles. Sums like that in equation (31), but over complete sets of states, have the property of being independent of any particu­ lar complete set of states, and could for example be replaced by a sum over a complete set of quark states. The sum in equation (31) is not complete, but the energy span is very large and co­ vers much final-state resonance structure and a large number of many-body final states. In similar sums which occur in the formu­ la for the total e e annihilation cross-section, it seems very likely that such energy averages equal the free quark values, and so, if the same is true here, we have the formula : r _ observed "P (3M) ~ - free ouark This means, as was noted very early by Gaillard, Lee (7) and Rosner, > that total semileptonic rates of charmed parti­ cles may be estimated by calculating the decay rate of a free quark c into a strange quark and leptons. Furthermore, it enables us to raake predictions about the helicities of final states. Assu­ ming from now on the 6IH form of the charmed weak current, we have : J^):««»,^HJÏW (35) The decay rate of a free charmed quark of rest mass m to a free strange quark mass m with momentum p , helicity Àâ, and a lepton-neutrino pair is given by : 41h _ il ±h_ y I F *- I* (36) where (37) and is to be evaluated in the rest frarn? uf the lepton pair. ". * and ii : as the uorrespon- ding spacelike and timelike matrix elements in the charmed quark rest frame, we have : ^ ~, ~ -(*£) F . : -/T F' - -i{4^i C»s{ €. = Œ TL", «J *k - * (38) (39) * 4 * "Ï •» * Performing a Lopenrz transformation to the lepton-pair rest frame, we obtain : c P, , = F (ll0) FH * fill eu 4 t 6 ÇJ = ,| M,AV - f/A Jt4«rt (.2) * A F .
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