Muon mass correction in partial wave analyses of charmed meson semi-leptonic decays

We derive the parameterization formula for partial wave analyses of charmed meson semi-leptonic decays with consideration of the effects caused by the lepton mass. As the proposed super-tau-charm factory will reach much enhanced luminosity and BESIII is taking $\psi(3770)\to D\bar{D}$ data, our results are helpful to improve the measurement precision of future partial wave analyses of charmed meson semi-muonic decays.


I. INTRODUCTION
The Standard Model, as the theory of elementary particle interactions, has achieved great success to describe experimental data with various energies.Nevertheless, charmed mesons' ∼ 2 GeV masses make them in the region where perturbative QCD is not applicable, and raise challenges to both theory and experiment.In recent years, testing the Standard Model with high precision measurements become one of the hottest topics in the charm sector.
The semi-leptonic decays of charmed mesons, in which the hadronic and weak currents could be well separated, provide a clean platform to study the mechanism of the c quark to the d(s) quark transition and play an important role to understand the strong and weak interactions.Their partial decay width accesses to the product of the hadronic form factor, which describes the strong-interaction in the hadronic current connecting initial and final hadrons, and the Cabibbo-Kobayashi-Maskawa matrix element |V cs | or |V cd |, which parameterizes the weak interaction between different quark flavors.Partial wave analyses of four-body semi-leptonic decays of charmed meson allow to extract the form factors in the D → V ℓ + ν ℓ and D → Sℓ + ν ℓ transitions (where ℓ = e, µ, and V and S denote vector and scalar mesons, respectively).The K * (892) −(0) resonance have been studied in the D 0(+) → K 0(−) π + e + ν e decay by the CLEO, BABAR, and BESIII collaborations [1][2][3][4].BE-SIII has also studied ρ − resonance in D 0 → π − π 0 e + ν e [5]; ρ 0 , ω, and f 0 (500) resonances in D + → π + π − e + ν e [5].In the near future, more partial wave analyses are expected to be performed with high-statistics datasets, including D (s) semi-muonic decays. 1owever, the mass of leptons is neglected in the parameterization formula of the amplitude analyses for the charmed meson semi-leptonic decays.This should be fine in the case of semi-electronic decays, but will cause bias in semi-muonic decays, which downgrades the advantage of high-statistics from the new data and is against the purpose of precision measurement.In this work, we derive the formula to consider the mass of leptons based on Refs.[6,7].The results will be present in the format used in experimental analyses.
Experiments can easily adopt our results to their analyses.Charge-conjugated decay modes are implied throughout this paper.
First of all, we define kinematic variables and discuss their relations.A four-body semi- 2) is the product meson, and ℓ = e, µ.The momentum four-vectors and invariant masses are denoted by p and m, respectively.For convenience, the independent four-vectors combinations are defined as (5) where β M is the three-momentum modulus of the meson in the center-of-mass frame of the meson-meson system, β L the three-momentum modulus of the lepton in the center-of-mass frame of the lepton-neutrino system, and X an element of phase space, with Comparing to Refs.[6,7], we don't neglect the mass of leptons.
Next, from the effective Hamiltonian at the quark level for D → M 1 M 2 ℓ + ν ℓ , the decay amplitude is given by where G F is the Fermi constant and V q 1 q 2 is the element of the Cabibbo-Kobayashi-Maskawa matrix.The hadronic matrix element can be written in terms of four form factors w ± , r and h that are defined by where the form factors w ± , r, and h are function of s M , s L , and cos θ M .The ǫ µναβ is Levi-Civita symbol.
The differential decay rate takes the form In order to study the structure of the hadron system, i.e. form factors of the M 1 M 2 system, the decay intensity I is decomposed with respect to θ L and φ, written as where I 1,...,9 depend only on s M , s L , and φ.One can further express these I 1,...,9 in terms of form factors: where F 1−4 are the form factors, For purpose of discussing angular momentum of M 1 M 2 , e.g.S-and P -wave, the partial wave expansions in spherical harmonics for the form factors F 1−4 are written as Moreover, the decay of D → M 1 M 2 ℓ + ν ℓ could happen via intermediate states, such as vector or scalar mesons, which provides information about the intermediate resonances [8,9]. where with Here, f ± (s L ) is the D → S form factor; A 0,1,2 are the D → V axial-vector form factors and V 0 is the D → V vector form factor [10,11].Finally, one can obtain F 1−4 in the helicity basis (for S-and P -wave only): with where ) is derived from the propagator for S(V ).In the case of Breit-Wigner lineshapes, [12].The Breit-Wigner lineshape can be replaced by the Flatté formula [13], the Gounaris-Sakurai lineshape [14], etc. for various situations.

III. DISCUSSION AND CONCLUSION
In this work, we have derived the parameterization for a four-body semi-leptonic decay with consideration of the effects caused by the lepton mass, and expressed the differential decay width in the format used in partial wave analyses.One can obtain the parameterization formula used in Refs.[3][4][5] as neglecting the lepton mass (or setting β L = 1).While it does not significantly influence semi-electronic decays, neglecting the lepton mass could lead up to ∼ 1% bias to partial wave analyses for charmed meson semi-muonic decays.
BESIII is accumulating data samples with integrated luminosity of 20 fb −1 at center-ofmass energy 3.773 GeV (for D 0 and D ± mesons) and has collected 7.33 fb −1 data samples at 4.128 − 4.226 GeV (for D + s mesons) [15].In addition, the proposed super-tau-charm factory will be able to reach much enhanced luminosity.These aim for testing the Standard Model with high precision in the charm sector, but a precise theoretical parameterization is also needed.With the correction for the lepton mass presented in this work, partial wave analyses can be performed to study the form factors in D (s) → V µ + ν µ and D (s) → Sµ + ν µ decays more precisely.