HZ and BCK were supported in part by the Joint Large-Scale Scientific Facility Fund of the National Natural Science Foundation of China (NSFC) (11875054, 12192263) and the Chinese Academy of Sciences (U2032104); YY was supported in part by the NSFC (11905023, 12047564, 12147102), the Natural Science Foundation of Chongqing, China (cstc2020jcyj-msxmX0555), and the Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202200605, KJQN202200621); EW and HZ were supported in part by the Natural Science Foundation of Henan, China (222300420554, 232300421140), the Project of Youth Backbone Teachers of Colleges and Universities of Henan Province, China (2020GGJS017), the Youth Talent Support Project of Henan Province, China (2021HYTP002), and the Open Project of Guangxi Key Laboratory of Nuclear Physics and Nuclear Technology (NLK2021-08)

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

We derive a parameterization formula for the partial wave analyses of charmed meson semi-leptonic decays while considering the effects of lepton mass. Because the proposed super-tau-charm factory will reach a significantly enhanced luminosity and BESIII is collecting new

Article funded by SCOAP^{3}

The standard model, as a theory of elementary particle interactions, has achieved great success in describing experimental data with various energies. Nevertheless, the

The semi-leptonic decays of charmed mesons, in which hadronic and weak currents can be well separated, provide a clean platform for studying the mechanism of the ^{
①
}.

However, lepton mass is neglected in the parameterization formula used for the amplitude analyses of charmed meson semi-leptonic decays. This should be manageable 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 conflicts with the purpose of precision measurement. In this study, we derive the formula while considering the mass of leptons based on Refs. [

First, we define the kinematic variables and discuss their relations. The four-body semi-leptonic decay

A four-body decay can be uniquely described via kinematic parameterization with five variables (besides spin). The squared masses of the hadronic system

and the following relations can be easily derived:

The other three variables are chosen as the angle between the ^{
②
}. The angles

(color online) Definition of angles

where

with

Compared to Refs. [

Next, from the effective Hamiltonian at the quark level for

where

where the form factors

The differential decay rate takes the form

To study the structure of the hadron system, that is, the form factors of the

where

where

For the purpose of discussing the angular momentum of

Moreover, the decay

where

with

Here,

with

where _{0}(980), it couples to the channels

In this study, we derive the parameterization formula of a four-body semi-leptonic decay while considering the effects of lepton mass and express the differential decay width in the format used in partial wave analyses. We can obtain the parameterization formula used in Refs. [

BESIII is accumulating data samples with an integrated luminosity of 20 fb

At present, there is no partial wave analyses of