Anatomy of $\Lambda_c^+$ semileptonic decays

We present a systematic study of $\Lambda_c^+ \to {\cal B}_q \ell^+ \nu_\ell $ with ${\cal B}_q = (\Lambda, n)$ and $\ell =( e, \mu)$, examining all the possible decay observables based on the homogeneous bag model (HBM) and lattice QCD (LQCD). With the HBM, we find that the branching fractions and polarization asymmetries of the daughter baryon $\Lambda$ are ${\cal B}(\Lambda_c^+ \to \Lambda e^+ \nu_e, \Lambda \mu^+ \nu_\mu, n \ell ^+ \nu_\ell ) = (3.78 \pm 0.25, 3.67\pm 0.23, 0.40\pm 0.04 )\%$ and $\alpha_\Lambda (\Lambda_c^+ \to \Lambda e^+ \nu_e,\Lambda\mu^+ \nu_\mu ) =(-82.6,-82.3)\%$, respectively. From the LQCD, we obtain that $\alpha_{\Lambda}(\Lambda_c^+ \to \Lambda e^+ \nu_e, \mu^+ \nu_\mu ) = (-87.4\pm 1.0,-87.2\pm 1.0)\%$. We also explore the time-reversal asymmetries due to new physics beyond the standard model. All our results are consistent with the current experimental data, while some of them are accessible to the experiments at BESIII and Belle II.

It is reasonable to expect that some angular observables including α Λ can be soon measured at BESIII and Belle II. In the view of the recent experiments of Λ + c at BESIII [3][4][5][6][7][8], the strange charmed baryons at Belle [9][10][11][12], and the doubly charmed baryons at LHCb [13], it is no doubt that the era of the precision measurements in the charm baryons has begun.
In this paper, we concentrate on the decays of Λ + c → B q ℓ + ν with the bag model (BM) and LQCD. We compute all the possible decay observables of Λ + c → B q ℓ + ν in the SM and discuss some of the possible effects from new physics (NP). In particular, we discuss the time-reversal (T) asymmetries. As these asymmetries are contaminated little by the hadronic uncertainties, they provide a reliable way to probe NP. This work is organized as follows. In Sec. II, we give the baryon wave functions and helicity amplitudes from the homogeneous BM (HBM). In Sec. III, we present the angular distributions and extract the physical observables in Λ + c → B q ℓ + ν. In Sec. IV, we study the T asymmetries from NP. Finally, we conclude this study in Sec. V.

II. BARYON WAVE FUNCTIONS AND HELICITY AMPLITUDES
The amplitudes of Λ + c → B q ℓ + ν are given as where V cs = 0.973 and V cd = 0.221 are the Cabibbo-Kobayashi-Maskawa quark mixing matrix elements [34], G F is the Fermi constant, q = (s, d) for B q = (Λ, n), and v and u are the Dirac spinors of ν and ℓ + , respectively. We expand the Minkowski metric g µν by where ε and q are the polarization vector and four-momentum of the off-shell W boson (W * ), respectively, and the subscript in ε denotes the helicity of W * . In the center of frames of W * and Λ + c , we have [35] and respectively. We note that in the calculations, we always choose the 3-momentum of the outgoing fermion toward theẑ direction.
By inserting Eq. (4) into Eq. (3), the three-body problem is reduced to a product of the two-body ones, given by along with describing W * → ℓ + ν and Λ + c → B q W * , respectively.
As L λ W and B λ W are Lorentz scalars, they can be calculated in different Lorentz frames. We adopt the rest frames of W * and Λ + c for L λ W and B λ W , respectively. In the SM, the helicity amplitudes are given as with λ c = λ q − λ W and where N q = (1, 3/2) for q = (s, d) are the spin-flavor overlappings, the subscript of Dirac spinors stands for helicity, s 3 and c 3 act only on the third quarks, J 3 is the angular momentum of the third quark, and v represents the velocity.
The adopted convention of the Dirac spinors can be found in Appendix of Ref. [35].
The helicity amplitudes h ± are calculated as where δ ℓ = 2M 2 ℓ /q 2 and M ℓ corresponds to the charged lepton mass. Due to the lefthanded nature of the weak interaction, we have h + ≫ h − as ℓ + has a positive helicity in the massless limit.
On the other hand, V and A depend on the baryon wave functions and vary in quark models. The relevant baryon wave functions are given as where ǫ αβγ is the totally antisymmetric tensor, the Latin and Greek letters stand for the spinor and color indices, and q † ( x) is the creation operator of a quark at x and t = 0. Without specifying the distributions of Ψ, the wave functions in Eq. (12) are general results of quark models in the instant form. In the MIT BM, the distributions read as [36] Ψ abc where N is the normalization constant, and φ a q ( x) is a bag state centering at with χ ↑ = (1, 0) T and χ ↓ = (0, 1) T representing J z = ±1/2, and j 0,1 the spherical Bessel functions, respectively. Here, the kinematic factors are defined as ω ± q = E k q ± m q with E k q , p q and m q the kinematic energy, 3-momentum and mass of the quark. In turn, p q has to obey the boundary condition with R the bag radius.
Although the MIT BM successfully explains most of the low-lying baryon masses, it is problematic when it applies to decays. It is due to that the baryon wave functions in Eq. (13) are localized at x = 0. According to the Heisenberg uncertainty principle, it can not be a momentum eigenstate, which is a basic requirement in calculating decay widths. To resolve the problem, we take the baryon wave functions as infinite linear superpositions of Eq. (13), given by which clearly distribute homogeneously over the space, and thus are named as the HBM [37]. The overall normalization constants, depending on the quark components, are calculated as with where Λ µ ν is a Lorentz boost toward the −ẑ direction, defined as Λ 3 0 = −γv with γ = 1/(1 − v 2 ). From the baryon wave functions given in Eq. (16), we obtain with where S ±v = a + ± a − γ 0 γ 3 are the Lorentz boost matrix for Dirac spinors toward the ±ẑ direction, and a ± = (γ ± 1)/2. In Eq. (20), Υ (5) describes the quark transition of the (axial) current operator, and D v q are the overlappings of the spectator quarks.
The main uncertainties of the model calculations come from the quark energies. In this work, we take the values [37] with M p the proton mass. We use the capital M to represent hadron masses and the lower case m for quark masses. The adopted bag radius and quark masses are [36,37] By collecting Eqs. (19), (20) and (21), one shall be able to evaluate Eq. (9), which completes the evaluations. See Ref. [37] for calculation details.
Before we end this section, we note that the baryon transition matrix elements are conventionally parameterized as follows: where u q and u c are the Dirac spinors of B q and Λ + c , σ µν = i(γ µ γ ν − γ ν γ µ )/2 , and M q and M c are the masses of B q and Λ + c , respectively. The form factors in Eq. (24) can be numerically extracted by matching Eqs. (9) and (24) once V and A are computed.
The relations between the helicity amplitudes and form factors are with M ± = M c ± M q and Q ± = M 2 ± − q 2 .

III. DECAY OBSERVABLES
In this section, we study physical observables by the means of the angular distributions. For convenience, we take the following abbreviations: with ξ = a, b and t. Here, we do not explicitly write down the q 2 dependencies of the helicity amplitudes.
We start with the partial decay width, given as where p s is the three-momentum of Λ in the rest frame of Λ + c . The angular momentum of a timelike W * in its rest frame is essentially zero, resulting in that ℓ + and ν ℓ have opposite helicities. As t ± attribute to the timelike W * solely, they are always followed by δ ℓ as shown in Eq. (27). Numerically, δ ℓ can be taken as zero.
By adopting a different set of parameter input, Refs. [30,31] show that the LFQM is capable of explaining the experimental data, but the predictive power is questionable in turn. A parameter-independent study of the LFQM is clearly required. One of the great advantage of the BM is that the parameters are fitted from the mass spectra. In particular, the uncertainties are considerably smaller than other quark models. Our central values of the branching fractions are slightly larger than the ones from the experiments but smaller than those of the LQCD.
where P 2 = (3 cos θ ℓ − 1)/2, Φ = φ c + φ ℓ , and the observables X 2-26 are defined in TABLE II. Note that within the SM, ξ ± are real, leading to Re(X ) = X . To obtain the distributions of the charge conjugate processes, one can take the transformation of θ ℓ → π − θ ℓ and α → α so that X i = X i in the absence of NP, where the overline denotes the charge conjugation.
To test the results with the experiments, we define the q 2 averages of the decay observables in TABLE II, as The form factors calculated in the LQCD can be found in Refs. [25,26], where B are also provided. In this work, we calculate the angular observables from both the HBM and LQCD, listed in 64.0(1) 73.9(9) 67.0(8) 11.6(1) 11.6(1) 6.1(19) 6.4(17) 78.0(5) 77.8(5) 83.1(10) 83.1(10)     We believe that it is somewhat a universal factor in the BM. For instance, the magnetic dipole moments of the octet baryons, proportional to f 2 , are also systematically underestimated by a factor of two third in the MIT BM [36].
To compare our results with the experiments, we calculate the form factors f ∈ {g ⊥ , g + , f ⊥ , f + } adopted by the experiments. The definitions of f can be found explic-   (13) itly in Ref. [26], while their q 2 dependencies are governed by where m f pole is the pole mass, which are extracted by the cascade decays of Λ in the experiments, given as Likewise, the forward-backward asymmetries are Note that we have adopted the shorthand notations of Γ ℓ Ω = ∂Γ ℓ /(Γ ℓ ∂ Ω) .
We compare α Λ with those in the literature in TABLE V. In all the quark models,  As the cascade decays of the neutron can not be observed in the experiments, there are only eight possible decay observables in Λ + c → ne + ν , in which five of them require P b = 0 for the measurements, given as D(q 2 , Ω) ∝ 1 + X 2 P 2 + X 3 cos θ ℓ + P b X 4 cos θ c + X 5 cos θ c P 2 + X 6 cos θ c cos θ ℓ +Re(X 7 e iφ ℓ ) sin θ c sin θ ℓ + Re(X 8 e iφ ℓ ) sin θ c sin θ ℓ cos θ ℓ .
The discussions are parallel to Λ + c → Λℓ + ν ℓ . We list out the branching fractions and decay observables in TABLE. VI and TABLE VII, respectively. The branching fractions in the HBM are compatible with those in the LQCD [26] and Ref. [30], but twice larger than the results in Ref. [29] . On the other hand, the SU(3) F symmetry predicts relatively large branching fractions comparing to the others.
The calculated values of X i (ℓ) show consistencies between the HBM and LQCD.
Nevertheless, X 4-8 require Λ + c to be polarized for measurements, imposing difficulties in the experiments. The form factors of Λ + c → n are given in TABLE. VIII and FIG. 4. We see that once again the HBM underestimates f 2 by a factors of two third, whereas the others are compatible with the LQCD.

IV. TIME REVERSAL ASYMMETRIES AND NEW PHYSICS
In general, contaminated by the hadron uncertainties, it is often difficult to draw a sharp conclusion on whether NP is needed to explain the experimental data. Nevertheless, NP can generate clear signals of T asymmetries, which vanish in the SM and are equivalent to the CP asymmetries based on the CPT theorem. A great advantage of T violation is that a strong phase is not necessary. The simplest T violating observables are studied in Ref. [35], and we highlight some the results here.
Two of the simplest T asymmetries in Λ + c → Λ(→ pπ − )ℓ + ν ℓ are defined as HBM LQCD which are closely related to the azimuthal angles. The reason is that they correspond to the triple product asymmetries of the three-momenta in the final states. Keeping in mind ξ ± are real in the SM, it is easy to see that nonvanishing T ℓ,s require NP beyond the SM.
As an illustration, we explore NP with the effective Hamiltonian as To the first order, we find that [35] with where C L,R are the Wilson coefficients from NP, and ξ ± in Eq. (40) are calculated within the SM. In practice, C L can be absorbed by redefining V cs , so their effects vanish in the first order. We only considered NP with the left-handed neutrinos and leptons, as the right-handed ones are suppressed by the lepton masses [35].
The values of Y ℓ,s can be viewed as the sensitivity coefficients of NP. We take Im(C R ) = (0.1, 0.2) as examples in TABLE IX. We see that T ℓ can be as large as 10%, but they require Λ + c to be polarized for an experimental measurement. On the other hand, though T c can be probed with unpolarized Λ + c , their values are twice smaller than T ℓ , making them hard to be observed. Finally, we emphasize that the T asymmetry does not require a comparison with the charge conjugate as the strong phase is irrelevant.

V. SUMMARY
We have given a systematical study on all the possible observables in Λ + c semileptonic decays, including the effects of NP. The model independent angular distributions with polarized Λ + c have been presented. We have found that the BM underestimate f 2 by a factor of two third, where the same underestimations were also found in the magnetic dipole moments of the octet baryons. The branching fractions and polarization asymmetries have been found to be B(Λ + c → Λe + ν e , Λµ + ν µ , nℓ + ν ℓ ) = , and the ones with I = 1 read as , The spin-flavor-antisymmetric spatial distribution Ψ A is defined in Eq. (16), and the symmetric one is given as where Similarly, the heavy baryons with a single heavy quark are given as for the antitriplet baryons, and for the sextet baryons. The bottom baryons are obtained directly by substituting bottom quarks for the charmed quarks.
On the other hand, the low-lying spin 3/2 baryons are constructed by where S B is the number of the identical quark in the baryon, J z the angular momentum inẑ direction, and Ψ abc The baryon wave functions with negative angular momenta can be obtained by flipping the spin directions in both sides of Eq. (A7).