Angular distributions for multi-body semileptonic charmed baryon decays

We perform an analysis of angular distributions in semileptonic decays of charmed baryons B1(′)→B2(′)(→B3(′)B4(′))ℓ+νℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_1^{(\prime )}\rightarrow B_2^{(\prime )}(\rightarrow B_3^{(\prime )}B_4^{(\prime )})\ell ^+\nu _{\ell }$$\end{document}, where the B1=(Λc+,Ξc(0,+))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_1{=}(\Lambda _c^+,\Xi _c^{(0,+)})$$\end{document} are the SU(3)-antitriplet baryons and B1′=Ωc-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_1'{=}\Omega _c^-$$\end{document} is an SU(3) sextet. We will firstly derive analytic expressions for angular distributions using the helicity amplitude technique. Based on the lattice quantum chromodynamics (QCD) results for Λc+→Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _c^+\rightarrow \Lambda $$\end{document} and Ξc0→Ξ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi _c^0\rightarrow \Xi ^-$$\end{document} form factors and model calculation of the Ωc0→Ω-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _c^0\rightarrow \Omega ^-$$\end{document} transition, we predict the branching fractions: B(Λc+→Λ(→pπ-)e+νe)=2.48(15)%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(\Lambda _{c}^{+} \rightarrow \Lambda (\rightarrow p \pi ^{-}) e^{+} \nu _{e})=2.48(15)\%$$\end{document}, B(Λc+→Λ(→pπ-)μ+νμ)=2.50(14)%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(\Lambda _{c}^+\rightarrow \Lambda (\rightarrow p \pi ^{-})\mu ^{+}\nu _{\mu })=2.50(14)\%$$\end{document}, B(Ξc0→Ξ-(→Λπ-)e+νe)=2.40(30)%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(\Xi _{c}^0\rightarrow \Xi ^-(\rightarrow \Lambda \pi ^{-})e^{+}\nu _{e})=2.40(30)\%$$\end{document}, B(Ξc0→Ξ-(→Λπ-)μ+νν)=2.41(30)%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(\Xi _{c}^0\rightarrow \Xi ^-(\rightarrow \Lambda \pi ^{-})\mu ^{+}\nu _{\nu })=2.41(30)\%$$\end{document}, B(Ωc0→Ω-(→ΛK-)e+νe)=0.362(14)%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(\Omega _{c}^0\rightarrow \Omega ^-(\rightarrow \Lambda K^{-})e^{+}\nu _{e})=\!0.362(14)\%$$\end{document}, B(Ωc0→Ω-(→ΛK-)μ+νν)=0.350(14)%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {B}}(\Omega _{c}^0\rightarrow \Omega ^-\!(\rightarrow \Lambda K^{-})\mu ^{+\!}\nu _{\nu })=0.350(14)\%$$\end{document}. We also predict the q2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^2$$\end{document} dependence and angular distributions of these processes, in particular the coefficients for the cosnθℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos n\theta _{\ell }$$\end{document} (cosnθh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos n\theta _{h}$$\end{document}, cosnϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos n\phi $$\end{document}) (n=0,1,2,…)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n=0, 1, 2, \ldots )$$\end{document} terms. This work can provide a theoretical basis for the ongoing experiments at BESIII, LHCb, and BELLE-II.

The rest of this paper is organized as follows. In Sect. 2, we presen the theoretical framework for calculating the helicity amplitudes of charmed baryon decays, including the theoretical results of the Lorentz invariant leptonic and hadronic matrix elements. In Sect. 3, we list the differential decay widths of the three-body and four-body decay formulas. Integrating out the q 2 , we obtain numerical results of partial decay width, as well as the illustration of momentum transfer and angular distributions of the decay width. A brief summary will be presented in the last section.

Formalism
We will focus on semileptonic four-body decays of both SU(3)3 and 6 ground states of singly charmed baryons, denoted as B ( ) The effective weak Hamitonian for the semileptonic decays of charmed baryons can be written as where G F is the Fermi constant, and V cs is the CKM matrix element. Based on the above effective Hamiltonian, we can obtain the decay amplitudes of B ( ) With the decomposition of g μν the above amplitude can be decomposed into the Lorentz invariant hadronic and leptonic matrix elements: where the hadronic part and leptonic part L ν ≡ + ν |ν γ ν (1 − γ 5 ) | 0 , and is the polarization vector of the decomposed W + boson with helicity state λ and t, in which μ (t) ≡ q μ / q 2 . Focusing on the hadronic part and inserting the oneparticle completeness states of the intermediate baryon we find the H μ given as: In the above, p 2 , m 2 , and 2 are the four-momentum, mass, and decay rate of intermediate state B where the subscripts "H " and "I " denote a Heisenberg and interaction representation matrix element, respectively, g H 2 represents the coupling constant of hadronic vertex B ( ) 4 , and factors A and B are constants weighting the contributions from scalar and pesudoscalar density operators [62].
Therefore, the decay amplitude in Eq. (5) can be expressed as a convolution of the Lorentz invariant leptonic part L(s , s ν , s W ) and two hadronic parts, with Averaging the spin of the initial state and summing over the spins of final states, we obtain the squared amplitude as

Kinematics
With the abbreviations the momentum of the initial and final states in the subprocesses can be set as follows.
2 moves along the positive zdirection (θ = 0, φ = 0) and W + boson along the negative z-direction (θ = π, φ = π ), so we have with For the phase space of n-body decays, the four-body one can be generated recursively by the two-body subprocesses. The two-body phase space can be expressed as where √ŝ = p 2 is the center-of-mass energy, and θ is the angle between the two final states. Based on this, the three- + ν can be written as wherem l = m l / q 2 . Similarly, the total four-body phase space is Note that the integration variable p 2 2 is artificially introduced from the insertion of intermediate state B ( ) 2 , and in the narrow-width limit, this integration will be conducted as while where B ( ) are the total width and branching fraction of the subprocess B ( ) 4 , respectively. Combining the four-body phase space in Eq. (25) and squared amplitude in Eq. (15), we can write the differential decay width of the four-body process B ( )

Leptonic part
In this section, we will focus on the Lorentz invariant leptonic part defined in Eq. (12): Combining each component of spinors of leptons and polarization vectors of the W + boson, we can obtain the following nonvanishing matrix elements: where the factor N = i 2 q 2 − m 2 l andm l = m l / q 2 .

Hadronic part H 1 with spin-1/2 intermediate state
If B 1 is a singly charmed baryon antitriplet, the transition matrix elements with weak current (13) can be parameterized as [57,72] with the momentum transfer q μ = p The form factors f i and g i are functions of q 2 , and the relations between these form factors and other parameterizations are collected in the Appendix.
By combining each spin component of B 1 , B 2 , and W + , we give the nonzero terms of H 1 (s 1 , s 2 , s W ) as in which the abbreviation s ± has been defined in Eq. (16).
Reversing the helicities gives the same results for the vector current, but an opposite sign exists for the axial vector current. The total amplitudes are

Hadronic part H 1 with a spin-3/2 intermediate state
The charmed baryons B 1 = 0 c will weakly decay into spin-3/2 baryon decuplets − , and the transition matrix elements for this process can be parametrized as The definition of vectorial spinor U α ( p, S z ) for the spin-3/2 baryon is shown in the Appendix. Therefore, the nonzero terms of H 1 (s 1 , s 2 , s W ) are collected as and the total transition amplitudes give which corresponds to the hadronic coupling constant g H = G F m 2 4 in Eq. (14) with low-energy effective theory. Here, G F is a standard model electroweak coupling constant. A and B are factors weighting the contributions from scalar and pesudoscalar operators [30]. It is convenient to introduce the asymmetry parameter with s = A and r = B × | p 3 |/(E 3 + m 3 ), and follow (s ± r ) 2 = (|s| 2 + |r | 2 )(1 ± α). Therefore, combining each spin component of B 2 and B 3 , we get the following hadronic matrix elements H 2 (s 2 , s 3 ): . Using the two-body phase space and squared matrix elements, we obtain the spinaveraged decay width

Hadronic part H 2 with spin-3/2 intermediate state
The transition for a spin-3/2 baryon − decaying into a spin-1/2 baryon and a spin-0 pseudoscalar meson K − is parametrized as [73] where g h denotes the coupling constant of the hadronic vertex K . This nonperturbative parameter is usually determined from the experimental partial decay width of the relevant process, but here we can absorb it into the spin-averaged decay width in Eq. (73) to avoid the uncertainties and model dependence of nonperturbative methods.

Input
In this section, all parameters used in the calculations will be collected, including the baryon masses and CKM matrix ( Table 1). In addition, the lepton mass m e = 0.005 GeV, m μ = 0.1134 GeV, and Fermi constant G F = 1.166 × 10 −5 GeV −2 . The CKM matrix element V cs = 0.973. The decay widths of the baryons can be obtained from the reciprocal of lifetimes: (MeV) = 1/τ × 6.582 × 10 −22 . In the calculation of the heavy baryon four-body decays, the asymmetry parameters and branch ratios are collected as [29]  and 0 c → − + ν , respectively. In order to access the q 2 distribution, we use the modified z expansions in the physical limits [75], and the fit functions are shown as The expansion variable is defined as • c decays.
We collect the fitted form factor parameters from [74] in Table 2. The resulting SM predictions for the + c → + ν and + c → pπ − + ν decay widths and branching fractions with corresponding error estimates are listed as    [37,38], which are highly consistent with our results in Eqs. (106)-(107). For the processes of four-body decays, based on the form factors, we predict the differential decay widths for + c → pπ − + ν as a function of q 2 in Fig. 3a. Note that the increasing errors in the small-q 2 region come from the uncertainties of form factors at large momentum transfer in the lattice calculations (Fig. 4). In Eqs. (76)(77)(78)(79)(80)(81)(82)(83)(84), we show the theoretical results of θ l -, θ h -and φ-dependence of the total four-body decay width with different final-state leptons, in which the coefficients are functions of q 2 only. After we integrate out q 2 , the angular distributions with cos θ l , cos θ h , and φ are shown as • 0 c decays Based on the recently announced c → form factors calculated by lattice QCD [68], where the z-expansion parameters of helicity-based form factors are collected in Table 3, we perform the predictions for 0 c → − + ν and 0 c → π − + ν decay widths and branching fractions with corresponding uncertainties We also predict the differential decay widths for 0 c → π − + ν as a function of q 2 in Fig. 5; the errors in plots mainly come from the uncertainties of form fac-   tors extracted from lattice QCD, and its behavior is the same as + c → pπ − + ν . The results of the θ l -, θ h -and φ-dependence of differential decay widths with different leptonic final states are listed as follows and the corresponding plots of angular distributions are collected in Fig. 5. • 0 c decays Due to the absence of the lattice QCD calculation, we use 0 c → − transition form factors calculated by the light-front quark model [69]. The form factors can be expressed as the following double-pole form: where F(0) is the value of the form factors at q 2 = 0 with m F = 1.86 GeV, δ ≡ δm c /m c = ±0.04. The numerical results from the light-front quark model are collected in Table 4.
Through the decay width of c in Eq. (88) and integrating out all variables, we can obtain the numerical results of decay widths and branching fractions with errors:  In addition, we give the angular distributions of all the processes with different angular cos θ l ,cos θ h and φ. In the future, we expect to study the more general cases of semileptonic charmed baryon decays by calculating form factors such as c → and so on. This work can provide a theoretical basis for the ongoing experiments at BESIII, LHCb, and BELLE-II.