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rpp2010-rev-dalitz-analysis-formalism - 1 DALITZ PLOT...

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– 1– DALITZ PLOT ANALYSIS FORMALISM Written January 2006 by D. Asner (Pacific Northwest National Laboratory) Introduction: Weak nonleptonic decays of D and B mesons are expected to proceed dominantly through resonant two-body decays [1]; see Ref. [2] for a review of resonance phenomenology. The amplitudes are typically calculated with the Dalitz-plot analysis technique [3], which uses the minimum number of independent observable quantities. For three-body decays of a spin-0 particle to all pseudo-scalar final states, D or B abc , the decay rate [4] is Γ = 1 (2 π ) 3 32 s 3 |M| 2 dm 2 ab dm 2 bc , (1) where m ij is the invariant mass of particles i and j . The coefficient of the amplitude includes all kinematic factors, and |M| 2 contains the dynamics. The scatter plot in m 2 ab versus m 2 bc is the Dalitz plot. If |M| 2 is constant, the kinematically allowed region of the plot will be populated uniformly with events. Any variation in the population over the Dalitz plot is due to dynamical rather than kinematical effects. It is straightforward to extend the formalism beyond three-body final states. For N -body final states with only spin-0 particles, phase space has dimension 3 N 7. Other decays of interest include one vector particle or a fermion/anti-fermion pair ( e.g. , B D ππ , B Λ c , B K ) in the final state. For the first case, phase space has dimension 3 N 5, and for the latter two the dimension is 3 N 4. Formalism: The amplitude for the process, R rc, r ab where R is a D or B , r is an intermediate resonance, and a , b , c are pseudo-scalars, is given by M r ( J, L, l, m ab , m bc ) = λ ab | r λ T r ( m ab ) cr λ | R J (2) = Z ( J, L, l, p, q ) B R L ( | p | ) B r L ( | q | ) T r ( m ab ) . The sum is over the helicity states λ of r , J is the total angular momentum of R (for D and B decays, J=0), L is the orbital CITATION: K. Nakamura et al. (Particle Data Group), JPG 37 , 075021 (2010) (URL: http://pdg.lbl.gov) July 30, 2010 14:34
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– 2– angular momentum between r and c , l is the orbital angular momentum between a and b (the spin of r ), p and q are the momenta of c and of a in the r rest frame, Z describes the angular distribution of the final-state particles, B R L and B r L are the barrier factors for the production of rc and of ab , and T r is the dynamical function describing the resonance r . The amplitude for modeling the Dalitz plot is a phenomenological object. Differences in the parametrizations of Z , B L , and T r , as well as in the set of resonances r , complicate the comparison of results from different experiments. Usually the resonances are modeled with a Breit-Wigner form, although some more recent analyses use a K -matrix for- malism [5,6,7] with the P -vector approximation [8] to describe the ππ S-wave. The nonresonant (NR) contribution to D abc is parametrized as constant (S-wave) with no variation in magni- tude or phase across the Dalitz plot. The available phase space is much greater for B decays, and the nonresonant contribu- tion to B abc requires a more sophisticated parametrization.
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