$\Lambda_c$ Semileptonic Decays in a Quark Model

TL;DR

This paper calculates the semileptonic decay form factors of the $ ext{Lambda}_c$ baryon using a nonrelativistic quark model, comparing results with heavy quark effective theory and experimental data.

Contribution

It provides a comprehensive calculation of $ ext{Lambda}_c$ semileptonic decay form factors and branching fractions, including excited states, within a nonrelativistic quark model framework.

Findings

Form factors satisfy HQET relationships.

Decays through $ ext{Lambda}(1405)$ are significant.

Calculated branching fractions agree with experimental data.

Abstract

Hadronic form factors for semileptonic decay of the $\Lambda_c$ are calculated in a nonrelativistic quark model. The full quark model wave functions are employed to numerically calculate the form factors to all relevant orders in ($1/m_c$, $1/m_s$). The form factors obtained satisfy relationships expected from the heavy quark effective theory (HQET). The differential decay rates and branching fractions are calculated for transitions to the ground state and a number of excited states of $\Lambda$. The branching fraction of the semileptonic decay width to the total width of $\Lambda_c$ has been calculated and compared with other theoretical estimates and experimental results. The branching fractions for $\Lambda_c\to\Lambda^{*} l^{+}\nu_l\to\Sigma\pi l^{+}\nu_l$ and $\Lambda_c\to\Lambda^{*} l^{+}\nu_l\to N\bar{K} l^{+}\nu_l$ are also calculated. Apart from decays to the ground state $\Lambda$(1115), it is found that decays through the $\Lambda(1405)$ provide a significant portion of the branching fraction $\Lambda_c\to X_s l\nu_l$. A new estimate for $f=B(\Lambda_c^{+}\to\Lambda l^{+}\nu_l)/B(\Lambda_c^{+}\to X_s l^{+}\nu_l)$ is obtained.

Tables

Table 1. Table 1: Quark masses used in (W. Roberts, 2008 ) .

$m_q$	$m_s$	$m_c$
GeV	GeV	GeV
0.2848	0.5553	1.8182

Table 2. Table 2: Baryon masses and wave function size parameters, α λ subscript 𝛼 𝜆 \alpha_{\lambda} and α ρ subscript 𝛼 𝜌 \alpha_{\rho} obtained from (W. Roberts, 2008 ) . All values are in GeV.

	Mass (GeV)		Size parameters (GeV)
State, $J^P$	Experiment	Model	${\alpha }_{\lambda }$	${\alpha }_{\rho }$
${\Lambda }_c\left(2286\right){\frac{1}{2}}^{+}$	2.29	2.27	0.424	0.393
$\Lambda \left(1115\right){\frac{1}{2}}^{+}$	1.12	1.10	0.387	0.372
$\Lambda \left(1600\right){\frac{1}{2}}_1^{+}$	1.60	1.71	0.387	0.372
$\Lambda \left(1405\right){\frac{1}{2}}^{-}$	1.41	1.48	0.333	0.320
$\Lambda \left(1520\right){\frac{3}{2}}^{-}$	1.52	1.53	0.333	0.308
$\Lambda \left(1890\right){\frac{3}{2}}^{+}$	1.89	1.81	0.325	0.303
$\Lambda \left(1820\right){\frac{5}{2}}^{+}$	1.82	1.81	0.325	0.303

Table 3. Table 3: Coefficients in parametrization of the form factors, from eqn. 53 .

Transition	$a_n$(${\text{GeV}}^{-n}$)	$F_1$	$F_2$	$F_3$	$F_4$	$G_1$	$G_2$	$G_3$	$G_4$
${\Lambda }_c\to \Lambda \left(1115\right)$	$a_0$	1.382	-0.235	-0.146	$-$	0.868	-0.440	0.203	$-$
	$a_2$	-0.073	0.022	-0.003	$-$	0.013	-0.116	-0.009	$-$
	$a_4$	0.000	0.006	-0.001	$-$	0.004	0.003	0.000	$-$
${\Lambda }_c\to \Lambda \left(1600\right)$	$a_0$	0.172	0.036	-0.015	$-$	0.144	-0.002	0.021	$-$
	$a_2$	-0.257	0.121	0.020	$-$	-0.102	0.160	-0.040	$-$
	$a_4$	0.025	-0.008	-0.001	$-$	0.005	-0.026	0.004	$-$
${\Lambda }_c\to \Lambda \left(1405\right)$	$a_0$	0.300	-0.797	0.162	$-$	0.881	-0.516	0.027	$-$
	$a_2$	-0.126	0.028	-0.010	$-$	-0.058	-0.066	0.025	$-$
	$a_4$	0.008	-0.003	-0.000	$-$	0.002	0.009	-0.001	$-$
${\Lambda }_c\to \Lambda \left(1520\right)$	$a_0$	1.496	-0.530	-0.172	0.094	0.613	-0.810	0.351	-0.170
	$a_2$	-0.080	0.019	-0.005	0.001	0.005	0.122	-0.010	0.008
	$a_4$	0.002	0.003	-0.001	0.000	0.002	-0.001	0.000	0.000
${\Lambda }_c\to \Lambda \left(1890\right)$	$a_0$	0.251	-0.358	0.165	-0.090	0.625	-0.257	0.016	0.040
	$a_2$	-0.079	-0.107	-0.006	0.004	-0.030	-0.041	0.021	-0.011
	$a_4$	0.005	0.950	0.000	-0.000	0.001	0.006	-0.001	0.001
${\Lambda }_c\to \Lambda \left(1820\right)$	$a_0$	1.148	-0.441	-0.177	0.139	0.322	-0.677	0.381	-0.325
	$a_2$	-0.059	0.008	-0.005	0.001	0.002	0.089	-0.008	0.013
	$a_4$	0.002	0.001	-0.001	0.000	0.000	-0.002	0.000	0.000

Table 4. Table 4: The leading order HQET form factors in terms of the general form factors. The second and third columns show the expressions in terms of the vector form factors, while the fifth and sixth columns show them in terms of the axial-vector form factors. The fourth and seventh columns show the ratios ξ 2 / ξ 1 ​ ( ζ 2 / ζ 1 ) subscript 𝜉 2 subscript 𝜉 1 subscript 𝜁 2 subscript 𝜁 1 \xi_{2}/\xi_{1}\,\,(\zeta_{2}/\zeta_{1}) calculated at the non-recoil point.

State, $J^P$	Vector			Axial Vector
	${\xi }_1$ (${\zeta }_1$)	${\xi }_2$ (${\zeta }_2$)	${\xi }_2/{\xi }_1$ (${\zeta }_2/{\zeta }_1$)	${\xi }_1$ (${\zeta }_1$)	${\xi }_2$ (${\zeta }_2$)	${\xi }_2/{\xi }_1$ (${\zeta }_2/{\zeta }_1$)
$\Lambda \left(1115\right),{\frac{1}{2}}^{+}$	$F_1+F_2/2$	$F_2/2$	-0.093	$G_1-G_2/2$	$G_2/2$	-0.202
$\Lambda \left(1600\right),{\frac{1}{2}}^{+}$			0.095			-0.007
$\Lambda \left(1405\right),{\frac{1}{2}}^{-}$	$-F_1+F_2/2$	$-F_2/2$	-0.571	$-G_1-G_2/2$	$-G_2/2$	-0.414
$\Lambda \left(1520\right),{\frac{3}{2}}^{-}$	$F_1+F_2/2$	$F_2/2$	-0.215	$G_1-G_2/2$	$-G_2/2$	-0.398
$\Lambda \left(1890\right),{\frac{3}{2}}^{+}$	$-F_1+F_2/2$	$-F_2/2$	-0.416	$-G_1-G_2/2$	$-G_2/2$	-0.259
$\Lambda \left(1820\right),{\frac{5}{2}}^{+}$	$F_1+F_2/2$	$F_2/2$	-0.238	$G_1-G_2/2$	$G_2/2$	-0.512

Table 5. Table 5: Integrated total decay widths for Λ c + → Λ ( ∗ ) ​ l + ​ ν l → superscript subscript Λ 𝑐 superscript Λ superscript 𝑙 subscript 𝜈 𝑙 \Lambda_{c}^{+}\to\Lambda^{(*)}l^{+}\nu_{l} in units of 10 11 ​ s − 1 superscript 10 11 superscript 𝑠 1 10^{11}s^{-1} , for the states we consider in this work. Also shown are the results obtained in PRCI. The last row shows the branching fraction of the elastic decay channel, where Γ Total subscript Γ Total \Gamma_{\text{Total}} is the total semileptonic decay width assuming the decays shown in the table saturate the semileptonic decays.

Spin	Mass (GeV)	Model estimates
		This Work		PRCI (M. Pervin and Capstick, 2005)
		${\Lambda }_c^{+}\to {\Lambda }^{\ast }{e}^{+}{\nu }_e$	${\Lambda }_c^{+}\to {\Lambda }^{\ast }{\mu }^{+}{\nu }_{\mu }$
${\frac{1}{2}}^{+}$	1.115	$1.92$	1.86	$2.10$
${\frac{1}{2_1}}^{+}$	1.600	$0.63\times {10}^{-2}$	$0.55\times {10}^{-2}$	$2.00\times {10}^{-2}$
${\frac{1}{2}}^{-}$	1.405	$0.12$	0.11	$0.19$
${\frac{3}{2}}^{-}$	1.519	$2.97\times {10}^{-2}$	$2.6\times {10}^{-2}$	$5.00\times {10}^{-2}$
${\frac{3}{2}}^{+}$	1.890	$1.58\times {10}^{-4}$	$1.01\times {10}^{-4}$	$--$
${\frac{5}{2}}^{+}$	1.820	$0.66\times {10}^{-4}$	$0.42\times {10}^{-4}$	$--$
Total		$2.08$	2.00	$2.36$
${\Gamma }_{{\Lambda }_c^{+}\to \Lambda l^{+}{\nu }_l}/{\Gamma }_{\text{total}}$		0.92	0.93	0.89

Table 6. Table 6: Branching fractions of the semileptonic decay Λ c + → Λ ​ ( 1115 ) ​ l + ​ ν l → superscript subscript Λ 𝑐 Λ 1115 superscript 𝑙 subscript 𝜈 𝑙 \Lambda_{c}^{+}\to\Lambda(1115)l^{+}\nu_{l} , compared with other theoretical estimates and experimental results. In the table, CQM refers to the covariant quark model of reference 𝑒𝑡 . 𝑎𝑙 . formulae-sequence 𝑒𝑡 𝑎𝑙 \it{et.al.} . ( 2016 ) , while LCSR refers to the light cone sum rules of reference Yong-lu Liu ( 2009 ) .

Branching	Model estimates ($\%$)				Experimental results($\%$)
fraction	This work	PRCI (M. Pervin and Capstick, 2005)	CQM $\mathrm{𝑒𝑡}.\mathrm{𝑎𝑙}.$. (2016)	LCSR Yong-lu Liu (2009)	Belle $\mathrm{𝑒𝑡}.\mathrm{𝑎𝑙}.$ (Belle Collaboration)	BESIII $\mathrm{𝑒𝑡}.\mathrm{𝑎𝑙}.$ (BESIII Collaboration); $\mathrm{𝑒𝑡}.\mathrm{𝑎𝑙}.$ (BESIII Collaboration)
${\Gamma }_{{\Lambda }_c^{+}\to \Lambda e^{+}{\nu }_e}/{\Gamma }_{{\Lambda }_c^{+}}$	3.84	4.2	2.78	$3.05\pm 0.27$	$2.9\pm 0.5$	$3.63\pm 0.38\pm 0.20$
${\Gamma }_{{\Lambda }_c^{+}\to \Lambda {\mu }^{+}{\nu }_{\mu }}/{\Gamma }_{{\Lambda }_c^{+}}$	3.72		2.69	$1.96\pm 0.32$	$2.7\pm 0.6$	$3.49\pm 0.46\pm 0.27$

Table 7. Table 7: Parameters of the excited Λ Λ \Lambda states that we use in our study. Shown are their total decay widths, and their partial decay widths and the strong couplings for the decays Λ ∗ → Σ ​ π → superscript Λ Σ 𝜋 \Lambda^{*}\to\Sigma\pi and Λ ∗ → N ​ K ¯ → superscript Λ 𝑁 ¯ 𝐾 \Lambda^{*}\to N\bar{K} .

Spin of ${\Lambda }^{(\ast )}$	Mass(GeV)	Total width	Partial width (MeV)		Strong coupling constant
		(MeV)	${\Gamma }_{{\Lambda }^{\ast }\to \Sigma \pi }$	${\Gamma }_{{\Lambda }^{\ast }\to N\overline{K}}$	$g_{\Lambda \Sigma \pi }$	$g_{\Lambda N\overline{K}}$
${\frac{1}{2}}^{+}$	1.115	-	-	-	15.73	14.03
${\frac{1}{2}}_1^{+}$	1.600	150	52.5	33.8	8.21	5.76
${\frac{1}{2}}^{-}$	1.405	50.5	50.5	-	1.57	1.90
${\frac{3}{2}}^{-}$	1.519	15.7	6.6	7.1	3.64	15.38
${\frac{3}{2}}^{+}$	1.890	100	6.5	27.5	0.14	1.05
${\frac{5}{2}}^{+}$	1.820	80.0	8.8	48.0	0.40	8.45

Table 8. Table 8: Integrated decay widths for Λ c + → Λ ( ∗ ) ​ l + ​ ν l → Σ ​ π ​ l + ​ ν l → superscript subscript Λ 𝑐 superscript Λ superscript 𝑙 subscript 𝜈 𝑙 → Σ 𝜋 superscript 𝑙 subscript 𝜈 𝑙 \Lambda_{c}^{+}\to\Lambda^{(*)}l^{+}\nu_{l}\to\Sigma\pi l^{+}\nu_{l} and Λ c + → Λ ( ∗ ) ​ l + ​ ν l → N ​ K ¯ ​ l + ​ ν l → superscript subscript Λ 𝑐 superscript Λ superscript 𝑙 subscript 𝜈 𝑙 → 𝑁 ¯ 𝐾 superscript 𝑙 subscript 𝜈 𝑙 \Lambda_{c}^{+}\to\Lambda^{(*)}l^{+}\nu_{l}\to N\bar{K}l^{+}\nu_{l} in units of 10 11 ​ s − 1 superscript 10 11 superscript 𝑠 1 10^{11}s^{-1} , for individual Λ ( ∗ ) superscript Λ \Lambda^{(*)} states for both l + = e + superscript 𝑙 superscript 𝑒 l^{+}=e^{+} and l + = μ + superscript 𝑙 superscript 𝜇 l^{+}=\mu^{+} . The last row shows the coherent totals for the four-body decays Λ c + → Σ ​ π ​ l ​ ν l → superscript subscript Λ 𝑐 Σ 𝜋 𝑙 subscript 𝜈 𝑙 \Lambda_{c}^{+}\to\Sigma\pi l\nu_{l} and Λ c + → N ​ K ​ l ​ ν l → superscript subscript Λ 𝑐 𝑁 𝐾 𝑙 subscript 𝜈 𝑙 \Lambda_{c}^{+}\to NKl\nu_{l} .

Spin of ${\Lambda }^{(\ast )}$	Mass(GeV)	${\Gamma }_{{\Lambda }_c\to {\Lambda }^{\ast }{e}^{+}{\nu }_e\to \Sigma \pi e^{+}{\nu }_e}$	${\Gamma }_{{\Lambda }_c\to {\Lambda }^{\ast }{\mu }^{+}{\nu }_{\mu }\to \Sigma \pi {\mu }^{+}{\nu }_{\mu }}$	${\Gamma }_{{\Lambda }_c\to {\Lambda }^{\ast }{e}^{+}{\nu }_e\to N\overline{K}{e}^{+}{\nu }_e}$	${\Gamma }_{{\Lambda }_c\to {\Lambda }^{\ast }{\mu }^{+}{\nu }_{\mu }\to N\overline{K}{\mu }^{+}{\nu }_{\mu }}$
${\frac{1}{2}}^{+}$	1.115	$4.18\times {10}^{-2}$	$3.81\times {10}^{-2}$	$1.86\times {10}^{-2}$	$1.65\times {10}^{-2}$
${\frac{1}{2}}_1^{+}$	1.600	$1.66\times {10}^{-3}$	$1.44\times {10}^{-4}$	$9.79\times {10}^{-4}$	$8.42\times {10}^{-4}$
${\frac{1}{2}}^{-}$	1.405	$9.54\times {10}^{-2}$	$8.78\times {10}^{-2}$	$1.82\times {10}^{-2}$	$1.65\times {10}^{-2}$
${\frac{3}{2}}^{-}$	1.519	$2.21\times {10}^{-2}$	$1.98\times {10}^{-2}$	$2.54\times {10}^{-2}$	$2.25\times {10}^{-2}$
${\frac{3}{2}}^{+}$	1.890	$2.11\times {10}^{-5}$	$1.66\times {10}^{-5}$	$1.12\times {10}^{-4}$	$9.00\times {10}^{-5}$
${\frac{5}{2}}^{+}$	1.820	$2.23\times {10}^{-5}$	$1.74\times {10}^{-5}$	$1.81\times {10}^{-4}$	$1.43\times {10}^{-4}$
Total		$18.31\times {10}^{-2}$	$16.59\times {10}^{-2}$	$9.33\times {10}^{-2}$	$8.23\times {10}^{-2}$

Equations

$$\mathcal{B}({\Lambda }_c^{+}\to p{K}^{-}{\pi }^{+})=RfF\frac{B(D\to X{l}^{+}{\nu }_l)}{1+|\frac{V_{cd}}{V_{cs}}{|}^2}\tau ({\Lambda }_c^{+}),$$

$$R$$

$$f$$

$$F$$

$$f=B({\Lambda }_c^{+}\to \Lambda l^{+}{\nu }_l)/B({\Lambda }_c^{+}\to X_s{l}^{+}{\nu }_l)=0.87{}_{-0.17}^{+0.13},$$

$$\mathcal{M}=\frac{G_F}{\sqrt{2}}{V}_{cs}{L}^{\mu }⟨\Lambda (p^{\prime },s^{\prime })|J_{\mu }|{\Lambda }_c(p,s)⟩,$$

$$H_{\mu }=⟨{\Lambda }^{(\ast )}|J_{\mu }|{\Lambda }_c⟩.$$

$$⟨\Lambda (p^{\prime },s^{\prime })|V_{\mu }|{\Lambda }_c(p,s)⟩$$

$$⟨\Lambda (p^{\prime },s^{\prime })|A_{\mu }|{\Lambda }_c(p,s)⟩$$

$$⟨\Lambda (p^{\prime },s^{\prime })|V_{\mu }|{\Lambda }_c(p,s)⟩$$

$$⟨\Lambda (p^{\prime },s^{\prime })|A_{\mu }|{\Lambda }_c(p,s)⟩$$

$$p_{\alpha }^{\prime }{\bar{u}}^{\alpha }(p^{\prime },s^{\prime })=0,{\bar{u}}^{\alpha }(p^{\prime },s^{\prime }){\gamma }_{\alpha }=0,{\bar{u}}^{\alpha }(p^{\prime },s^{\prime })/p^{\prime }=m_{{\Lambda }^{3/2}}{\bar{u}}^{\alpha }(p^{\prime },s^{\prime }).$$

$$⟨\Lambda (p^{\prime },s^{\prime })|V_{\mu }|{\Lambda }_c(p,s)⟩$$

$$⟨{\Lambda }_c(p^{\prime },s^{\prime })|A_{\mu }|{\Lambda }_c(p,s)⟩$$

$$p_{\alpha }^{\prime }{\bar{u}}^{\alpha \beta }(p^{\prime },s^{\prime })=p_{\beta }^{\prime }{\bar{u}}^{\alpha \beta }(p^{\prime },s^{\prime })=0,{\bar{u}}^{\alpha \beta }(p^{\prime },s^{\prime }){\gamma }_{\alpha }$$

$${\bar{u}}^{\alpha \beta }(p^{\prime },s^{\prime })/p^{\prime }=m_{{\Lambda }^{5/2}}{\bar{u}}^{\alpha \beta }(p^{\prime },s^{\prime }),{\bar{u}}^{\alpha \beta }(p^{\prime },s^{\prime })g_{\alpha \beta }$$

$$d\Gamma =\frac{1}{2m_{{\Lambda }_c}}\overline{|\mathcal{M}{|}^2}\frac{d^3{p}_l{d}^3{p}_{{\nu }_l}{d}^3{p}^{\prime }}{2E_{l}2E_{{\nu }_l}2E^{\prime }}\frac{(2\pi {)}^4{\delta }^4(p-p^{\prime }-p_l-p_{{\nu }_l})}{(2\pi {)}^3(2\pi {)}^3(2\pi {)}^3},$$

$$\overline{|\mathcal{M}{|}^2}$$

$$=\frac{G_F^2}{4}|V_{cs}{|}^2{H}_{\mu \nu }{L}^{\mu \nu }.$$

$$H_{\mu \nu }$$

$$L^{\mu \nu }=8[p_l^{\mu }{p}_{{\nu }_l}^{\nu }+p_{{\nu }_l}^{\mu }{p}_l^{\nu }-g^{\mu \nu }(p_l.p_{{\nu }_l})+i{ϵ}^{\mu \nu \alpha \beta }{p}_{l\alpha }{p}_{{\nu }_l\beta }].$$

$$\int \frac{d^3{p}_l{d}^3{p}_{{\nu }_l}}{(2\pi {)}^3(2\pi {)}^{3}2E_{l}2E_{{\nu }_l}}{L}^{\mu \nu }=\int d{\Omega }_l(A{g}^{\mu \nu }+A^{\prime }{L}^{\mu }{L}^{\nu }),$$

$$A=-\frac{(q^2-m_l^2{)}^2(2q^2+m_l^2)}{384{\pi }^6{q}^4},A^{\prime }=\frac{(q^2-m_l^2{)}^2(q^2+2m_l^2)}{192{\pi }^6{q}^6}.$$

$$\frac{d\Gamma }{d{q}^2}=\frac{|V_{cs}{|}^2}{192}\frac{G_F^2}{{\pi }^3{m}_{{\Lambda }_c}^3}{\lambda }^{1/2}(m_{{\Lambda }_c}^2,m_{\Lambda }^2,q^2)\frac{(q^2-m_l^2{)}^2}{4q^4}$$

$$\displaystyle+\Big{[}\beta_{LP}(P\cdot L)+\beta_{PL}(P\cdot L)+\beta_{LL}q^{2}\Big{]}3m_{l}^{2}\bigg{)},$$

$$P\equiv p_{\Sigma }+p_{\pi },Q\equiv p_{\Sigma }-p_{\pi },L\equiv p_l+p_{\nu },$$

$$P$$

$$L$$

$$p_{\pi }$$

$$p_{\Sigma }$$

$$p_l$$

$$p_{\nu }$$

$$⟨(B(p_B)M(p_M){)}_i|J_{\mu }|{\Lambda }_c(p_{{\Lambda }_c})⟩=\bar{u}(p_B){{\rm Y}}^{s}R(P){\mathcal{J}}_{\mu }^{i}u(P+L).$$

$$P\equiv p_B+p_M,Q\equiv p_B-p_M,L\equiv p_l+p_{\nu }.$$

$$⟨B(p_B)M(p_M)|V_{\mu }^i|{\Lambda }_c(P+L)⟩$$

$$⟨B(p_B)M(p_M)|A_{\mu }^i|{\Lambda }_c(P+L)⟩$$

Full text

I Introduction and Motivation

Semileptonic decays of hadrons are the main sources for precise knowledge on Cabibo-Kobayashi-Maskawa (CKM) matrix elements Glashow (1961). The form factors that parametrize the non-perturbative QCD effects in these transitions play a crucial role in the extraction of CKM matrix elements and the precision depends on how well the form factors are calculated.

A great deal of work has been done on semileptonic decay processes to calculate and improve the modeling of the form factors. For example, monopole type form factors were used to study semileptonic decay of heavy mesons by Wirbel, Stech and Bauer M. Wirbel and Bauer (1985). Isgur, Scora, Grinstein and Wise caculated the semileptonic $B$ and $D$ meson decays in a non-relativistic quark model (N. Isgur and Wise, 1989). Lattice QCD calculations of semileptonic decay form factors have been done in ref ($\it{et.al.}$., 2004). These are a very few out of a huge number of articles. More work has been done on semileptonic meson decays than baryon decays. Pervin, Roberts and Capstick worked on semileptonic baryon decays of $\Lambda_{Q}$ (M. Pervin and Capstick, 2005) and $\Omega_{Q}$ (M. Pervin and Capstick, 2006)) in a constituent quark model. Some baryon decays have also been addressed in QCD sum rules (Huang and Wang, 2004), perturbative lattice QCD ($\it{et.al.}$ [UKQCD Collboration], 1998) and a number of other approaches ($\it{et.al.}$, 1999).

The description of the weak decays of heavy hadrons are somewhat simplified because of the so-called heavy quark symmetry. This was first pointed out by Isgur and Wise (N.Isgur and Wise, 1989). Hadrons containing one heavy quark $Q$ (with $m_{Q}>>\Lambda_{\text{QCD}}$) possess this symmetry, which has been formalized into the heavy quark effective theory (HQET). In HQET the properties of the hadrons are governed by the light degrees of freedom and are independent of the heavy quark degrees of freedom. For semileptonic decays of heavy hadrons, HQET reduces the number of independent form factors needed to describe the decays.

In this paper, we examine the semileptonic decays of the $\Lambda_{c}^{+}$ to a number of $\Lambda$s, including the ground state. Because it is the lightest charmed baryon, $\Lambda_{c}^{+}$ plays an important role in understanding charm and bottom baryons. The lowest-lying bottom baryon is most often detected through its weak decay to $\Lambda_{c}^{+}$. In addition, the study of all of the $\Lambda_{c}^{+}$-type and $\Sigma_{c}$-type baryons are directly linked to the understanding of the ground state of $\Lambda_{c}^{+}$, as these baryons eventually decay into a $\Lambda_{c}^{+}$.

Among the branching fractions of the $\Lambda_{c}$, $\mathcal{B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})$ is used to normalize most of its other branching fractions. The Particle Data Group (PDG), in their previous version ($\it{et.al.}$, Particle Data Group) reported that there was no model independent measurement of $\mathcal{B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})$. Two model-dependent measurements were reported, with two different results obtained from different assumptions. The model that calculated branching fractions $\mathcal{B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})$ from semileptonic decays, estimated that

[TABLE]

where,

[TABLE]

They estimated $B(\Lambda_{c}^{+}\to pK^{-}\pi^{+})=(7.3\pm 1.4)\%$ with the theoretical estimate of $f=F=1.0$ with significant uncertainties.

However, in their most recent release, PDG ($\it{et.al.}$, Particle Data Group) reports a model independent measurement of $\mathcal{B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})$. A. Zupanc et al. (Belle Collaboration) ($\it{et.al.}$, Belle Collaboration) measured it to be $6.84{}^{+0.32}_{-0.40}\%$, while M. Ablikim et al. (BESIII Collaboration) ($\it{et.al.}$, BESIII Collaboration) measured it to be $5.84\pm 0.27\pm 0.23\%$. The PDG fit is $6.35\pm 0.33\%$ that leads to a new estimate of

[TABLE]

with the assumption of $F=1.0$. Pervin, Roberts and Capstick (PRCI) (M. Pervin and Capstick, 2005) estimated the value of $f$ to be $0.85\pm 0.04$. Mott and Roberts (Mott and Roberts, 2012) later estimated the rare decay branching fractions of the $\Lambda_{b}$ using two different methods. Their results indicated that the results were sensitive to the precision with which the form factors were estimated, and this further implied that $f$ could be even smaller than $0.85$. The semileptonic branching fraction, $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda l^{+}\nu_{l})$ is reported to be $2.8\pm 0.5\%$ with the assumption that the $\Lambda_{c}^{+}$ decays only to the ground state $\Lambda(1115)$. No semileptonic decays to excited $\Lambda$ have been reported. This provides the motivation for our work.

There have been a number of theoretical articles on the semileptonic decay of $\Lambda_{c}^{+}$ in recent years. Gutsche et al. used a covariant quark model to estimate the branching fraction for $\Lambda_{c}\to\Lambda l^{+}\nu_{l}$ ($\it{et.al.}$., 2016). Liu et al. used QCD light cone sum rules to examine this decay (Yong-lu Liu, 2009), while Ikeno and Oset have examined the semileptonic decay to the $\Lambda(1405)$, treating that state as a dynamically generated molecular state (N. Ikeno, 2016).

In the work presented herein, we work in the framework of a constituent quark model. Such models have been quite successful in explaining the main features of hadron phenomenology. In computing the form factors for $\Lambda_{c}\to\Lambda^{*}$, we have deployed two approximations. In the first approximation, single component wave functions are used to compute the analytic form factors for $\Lambda_{c}\to\Lambda^{*}$ transitions. As in PRCI (M. Pervin and Capstick, 2005) a variational diagonalization of a quark model Hamiltonian was used to extract the single component wave functions and the quark operators were reduced to their non-relativistic Pauli form. In the second method we keep the full relativistic form of the quark spinors and use the full quark model wave functions. We believe that this second method provides more reliable numerical values of the form factors as it uses fewer approximations.

We calculate the decay widths and branching fractions for decays to ground state and a number of excited $\Lambda^{(*)}$. We also study the decay widths and branching fractions of two other decay channels, namely $\Lambda_{c}^{+}\to\Sigma\pi l^{+}\nu_{l}$ and $\Lambda_{c}^{+}\to N\bar{K}l^{+}\nu_{l}$, via a set of $\Lambda$ resonances.

The rest of this paper is organized as follows: in section II, we discuss the hadronic matrix elements and decay rates. Section III presents a concise overview of HQET and the relationships predicted by HQET among the form factors for the transitions we study. In section IV we describe the model we employ to calculate the form factors. Section V is devoted to discussing the numerical results such as form factors, decay rates and branching fractions. Section VI presents our conclusions and outlook. A number of details of the calculation are shown in the appendices.

II Matrix Elements and Decay Widths

II.1 Semileptonic decay ($\Lambda_{c}^{+}\to\Lambda l^{+}\nu_{l}$)

II.1.1 Matrix Elements

Fig.1 depicts the semileptonic decay $\Lambda_{c}^{+}\to\Lambda^{(*)}l^{+}\nu_{l}$. We work in the rest frame of the parent $\Lambda_{c}$. The transition matrix element for the decay is

[TABLE]

where $V_{cs}$ is the CKM matrix element, $L^{\mu}=\bar{u}_{\nu_{l}}\gamma^{\mu}(1-\gamma_{5})v_{l}$ is the lepton current and $J_{\mu}=\bar{s}\gamma_{\mu}(1-\gamma_{5})c$ is the hadronic current. The momenta of the $\Lambda_{c}$, $\Lambda$, $l$, $\nu_{l}$ are labeled as $p$, $p^{\prime}$, $p_{l}$ and $p_{\nu_{l}}$, respectively. The hadronic matrix element is defined as

[TABLE]

The hadronic matrix elements are parametrized in terms of a number of form factors. For transitions from the ground state $\Lambda_{c}$ ($J^{P}=\frac{1}{2}^{+}$) to the ground state $\Lambda$ ($J^{P}=\frac{1}{2}^{+}$), the matrix elements for the vector ($V_{\mu}$) and axial-vector ($A_{\mu}$) currents are, respectively,

[TABLE]

where the $F_{i}$’s and $G_{i}$’s are the form factors and $s(s^{\prime})$ is the spin of $\Lambda_{c}(\Lambda)$. The matrix elements for transitions to a daughter baryon with $J^{P}=\frac{3}{2}^{-}$ are

[TABLE]

The Rarita-Schwinger spinor $\bar{u}^{\alpha}$ satisfies the conditions

[TABLE]

The corresponding matrix elements for transitions to a daughter baryon with $J^{p}=\frac{5}{2}^{+}$ are

[TABLE]

The spinor $\bar{u}^{\alpha\beta}$ satisfies the conditions

[TABLE]

Here we have shown the hadronic transition matrix elements for the decays to daughter baryons with natural parity. For decays to states with unnatural parity, the matrix elements are constructed by switching $\gamma_{5}$ from the equations defining the $G_{i}$ to the equations defining the $F_{i}$.

II.1.2 Decay Width

The differential decay rate for the transition $\Lambda_{c}\to\Lambda^{(*)}l^{+}\nu_{l}$ is

[TABLE]

where

[TABLE]

$\overline{|\mathcal{M}|^{2}}$ is the squared amplitude averaged over the initial spins (the factor of $\frac{1}{2}$) and summed over the final spins.

The most general Lorentz form of the hadronic tensor can be written as

[TABLE]

where we have defined $P=p^{\prime}$ and $L=p-p^{\prime}$. The lepton tensor is

[TABLE]

Integrating over the lepton momenta allows us to write the lepton tensor as

[TABLE]

where $q^{2}=(p-p^{\prime})^{2}$ and

[TABLE]

The complete expression for the differential decay rate becomes

[TABLE]

where $P\cdot L=\frac{1}{2}(m_{\Lambda_{c}}^{2}-m_{\Lambda}^{2}-q^{2})$ and $\lambda^{1/2}(x,y,z)=(x^{2}+y^{2}+z^{2}-2xy-2yz-2zx)^{1/2}$. When contracted with the lepton tensor, all of the $\beta$s (except $\beta_{PP}$) are proportional to powers of the lepton mass $m_{l}$ and thus give small contributions to the decay rate. The complete form of $\beta_{PP}$ is given in appendix E.1.

II.2 $\Lambda_{c}\to\Lambda^{*}l^{+}\nu_{l}\to\Sigma\pi l^{+}\nu_{l}/N\bar{K}l^{+}\nu_{l}$

We include six $\Lambda^{(*)}$ in this calculation. We denote these as $\Lambda_{i},\,\,i=1\dots 6$. In this notation, $\Lambda_{1}=\Lambda(1115)\,\,1/2^{+}$; $\Lambda_{2}=\Lambda(1600)\,\,1/2^{+}$; $\Lambda_{3}=\Lambda(1405)\,\,1/2^{-}$; $\Lambda_{4}=\Lambda(1520)\,\,3/2^{-}$; $\Lambda_{5}=\Lambda(1890)\,\,3/2^{+}$; $\Lambda_{6}=\Lambda(1820)\,\,5/2^{+}$. With the exception of $\Lambda_{1}$, these excited $\Lambda_{i}$ are not stable particles and will decay strongly to $\Sigma\pi$ or $N\bar{K}$. Thus we study the four-body decays, $\Lambda_{c}\to\Lambda_{i}l^{+}\nu_{l}\to\Sigma\pi l^{+}\nu_{l}$ and $\Lambda_{c}\to\Lambda_{i}l^{+}\nu_{l}\to N\bar{K}l^{+}\nu_{l}$ as shown in Fig. 2(a, b). There are other contributions to each of these four-body final states, two of which are shown in Fig. 2 (c, d). However, in each case, the intermediate resonance is very heavy and very far from the mass shell. Thus, we expect these contributions to be small.

II.2.1 Kinematics

Fig 3 shows the kinematic diagram for the four-particle decay $\Lambda_{c}\to\Sigma\pi l^{+}\nu_{l}$. We define

[TABLE]

so that $p_{\Lambda_{c}}=P+L$. In the rest frame of the $\Lambda_{c}$, the back-to back momenta $\vec{P}$ and $\vec{L}$ define a common $z$-axis. In the rest frame of the daughter hadrons, $\theta_{h}^{*}$ is the polar angle between the pion momentum and $\vec{P}$. Similarly, in the rest frame of the lepton pair, $\theta_{l}^{*}$ is the polar angle between the lepton momentum and $\vec{L}$. $\phi^{*}$ is then the angle between the lepton and hadron planes.

In the overall rest frame of $\Lambda_{c}$, the momenta $P$ and $L$ are

[TABLE]

In the rest frame of the daughter hadrons, the momenta $p_{\Sigma}$ and $p_{\pi}$ are

[TABLE]

In the rest frame of the lepton pair, the lepton momenta are

[TABLE]

II.2.2 Matrix Elements

The hadron matrix elements for the decays $\Lambda_{c}\to\Lambda_{i}l\nu_{l}\to BMl\nu_{l}$, where $B$ is a baryon with $J^{P}=1/2^{+}$ and $M$ is a pseudoscalar meson, can be written as

[TABLE]

In this expression, $\Upsilon^{s}$ represents the strong decay vertex, $p_{B}$ and $p_{M}$ are the momenta of the daughter baryon $B$ and meson $M$, respectively, $R(P)$ is the propagator with momentum $P$. $J_{\mu}$ is the weak current leading to the weak decay, while ${\cal J}_{\mu}^{i}$ is the matrix element for the semileptonic decay $\Lambda_{c}\to\Lambda_{i}$, written in terms of the form factors of section II.1.1. In this notation, the momenta of eqn. 16 are more generally written as

[TABLE]

When the intermediate baryon has $J^{P}=1/2^{+}$, the hadron matrix elements are

[TABLE]

where $\nabla^{i}=1/(P^{2}-{M^{i}_{\Gamma}}^{2})$ and $M^{i}_{\Gamma}=m_{\Lambda_{i}}-i\Gamma_{i}/2$, with $m_{\Lambda_{i}}$ and $\Gamma_{i}$ the mass and total decay width of the $\Lambda_{i}$, respectively. $g_{\Lambda_{i}BM}$ is the strong coupling constant for the decay $\Lambda_{i}\to BM$.

For an intermediate state with $J^{P}=3/2^{-}$, the hadron matrix elements are

[TABLE]

where $R^{\alpha\beta}(P)$ is the Rarita-Schwinger tensor for a massive spin $3/2$ propagator, which takes the form

[TABLE]

For an intermediate state with $J^{P}=5/2^{+}$, the hadronic matrix elements are

[TABLE]

where $R^{\alpha\beta}_{\alpha^{\prime}\beta^{\prime}}(P)$ is the Rarita-Schwinger propagator tensor for a massive particle with total angular momentum $5/2$ (V. Shklyar, 2010).

We need to cast the matrix elements from the previous three equations into a more general form that makes it easier to organize the calculation. The most general form of the contribution of the $i$th state to the matrix element for the four-body decay $\Lambda_{c}^{+}\to\Lambda_{i}l^{+}\nu_{l}\to BMl^{+}\nu_{l}$ can be written

[TABLE]

where the Lorentz-Dirac operators ${\cal O}_{i}$ are

[TABLE]

Because of the forms of the propagators in eqns. II.2.2 - II.2.2, there are no terms containing $\not{L}$ or $\not{Q}$ among the ${\cal O}_{i}$. For the cases of eqns. II.2.2 and II.2.2, the meson momentum $p_{M}$ can be replaced by $P-p_{B}$, and any factors of $\not{p}_{B}$ can be commuted leftward until they are adjacent to the spinor $\bar{u}(p_{B})$. The Dirac equation can then be used to write this as the scalar $m_{B}$.

The $c^{i}_{j}$ can be written

[TABLE]

where $k$ runs from $1$ to $3$ for spin $\frac{1}{2}$ states and from $1$ to $4$ for states with higher spin.

In the above, we have shown the forms for the $\Lambda_{i}$ states with natural parity. For the states with unnatural parity, the weak and strong vertices each acquire an extra multiplicative factor of $\gamma_{5}$.

II.2.3 Decay Width

The differential decay rate for the decay $\Lambda_{c}^{+}\to BMl^{+}\nu_{l}$ is,

[TABLE]

The hadron tensor that arises from each intermediate state $i$ can be written as,

[TABLE]

In this expression,

[TABLE]

with similar forms for all of the other coefficients. The terms in $\gamma_{i}$ do not contribute to the decay rates that we consider, due to the symmetry of the lepton tensor.

For the process $\Lambda_{c}\to BMl\nu_{l}$ we examine the contribution from each $\Lambda_{i}$ individually, as well as the coherent contribution of all the $\Lambda_{i}$. For the coherent sum, we write

[TABLE]

which ultimately leads to

[TABLE]

Integrating over the lepton momenta, and making use of eqn. (13) leads to

[TABLE]

where

[TABLE]

III Heavy Quark Effective Theory

The heavy quark effective theory (HQET) has been a very useful tool in the study of the electroweak decays of hadrons containing one heavy quark. In this effective theory, the matrix elements are expanded in increasing orders of $1/m_{Q}$, where $m_{Q}$ is the mass of the heavy quark. This expansion has facilitated the extraction of CKM matrix elements with decreasing model dependence.

Hadrons containing a single charm or beauty quark are considered to be heavy hadrons as the mass $m_{Q}>>\Lambda_{\text{QCD}}$. For such hadrons, HQET reduces the number of independent form factors required to describe the transitions mediated by electroweak transitions that change a heavy quark of one flavor into a heavy quark of different flavor. At leading order in the $1/m_{Q}$ expansion, such heavy to heavy transitions require a single form factor, the so-called Isgur-Wise function. This is the case independent of the total angular momentum of the daughter hadron (we assume that the parent hadron is a ground-state hadron), integer (meson) or half-integer (baryon). For transitions between a ground-state heavy hadron and a light one, HQET is not as powerful. However, for transitions between a heavy baryon (ground state) and a light one, HQET indicates that a pair of form factors is all that is needed to describe the transition, independent of the angular momentum of the daughter baryon.

The semileptonic decays $\Lambda_{c}\to\Lambda^{*}$ fall into this second category, and are therefore described by two independent form factors. We may represent one of these light baryons of angular momentum $J$ by a generalized Rarita-Schwinger field $u^{\mu_{1}\dots\mu_{n}}(p)$ where $n=J-1/2$. This field is symmetric under exchange of any pair of its Lorentz indices, and satisfies the conditions

[TABLE]

The matrix element we are interested in is

[TABLE]

where $\Gamma=\gamma^{\mu}\text{ or }\gamma^{\mu}\gamma_{5}$ defines vector or axial vector current and $\mathit{M}_{\mu_{1}\mu_{2}...\mu_{n}}$ is a tensor. The most general tensor can be constructed as

[TABLE]

where $\mathit{A_{n}}$ is the most general Lorentz scalar that can be constructed. This takes the form

[TABLE]

where $v=p/m_{\Lambda_{c}}$ is the velocity of the parent baryon. For the transitions to daughter baryons with unnatural parity, $M_{\mu_{1}\mu_{2}...\mu_{n}}$ must be a pseudo-tensor. This is easily constructed by including a factor of $\gamma_{5}$, so that

[TABLE]

III.1 Form Factors

The matrix elements can be written in terms of six general form factors for spin $\frac{1}{2}^{\pm}$, or eight general form factors for spin $\frac{3}{2}^{\pm}$ and $\frac{5}{2}^{+}$, as shown in Section II.1.1. Comparing the predictions of HQET with the most general form of the matrix elements leads to a number of relations among the general form factors $F_{i}/G_{i}$ and the HQET form factors $\xi_{i}/\zeta_{i}$.

For spin $\frac{1}{2}^{+}$, these relationships are

[TABLE]

For spin $\frac{1}{2}^{-}$, they are

[TABLE]

For spin $\frac{3}{2}^{-}$, they are

[TABLE]

For spin $\frac{3}{2}^{+}$, the relationships are

[TABLE]

For spin $\frac{5}{2}^{+}$, they are

[TABLE]

III.2 Decay Width

At leading order in HQET, the differential decay rates take simple forms for all the excited states we discuss. This general form is

[TABLE]

where $\Phi^{J}$ is a dimensionless quantity that depends on the angular momentum of the daughter baryon. $X,A_{i},B_{i},C_{i}$ are, respectively,

[TABLE]

The decay width for states with total spin $J$ does not depend on parity. The $\Phi^{J}$ for states with angular momentum $J$ are

[TABLE]

IV The Model

IV.1 Wave Function Components

In our model, a baryon state has the form

[TABLE]

where $\Lambda_{Q}$ is a flavored baryon ($\Lambda_{c}^{+}$ or $\Lambda$) having a flavored quark ($c$ or $s$) $Q$, which may or may not be considered heavy. $q_{i}(\vec{p}_{i},s_{i})$ is the creation operator for quark $q_{i}$ with momentum $\vec{p}_{i}$ and spin $s_{i}$. $|q_{1}(\vec{p}_{1},s_{1})q_{2}(\vec{p}_{2},s_{2})q_{3}(\vec{p}_{3},s_{3})\rangle$ is the three quark state with quarks $q_{i}$ having momenta and spins $(\vec{p}_{i},s_{i})$. $\vec{p}_{\rho}=\frac{1}{\sqrt{2}}(\vec{p}_{1}-\vec{p}_{2})$ and $\vec{p}_{\lambda}=\frac{1}{\sqrt{6}}(\vec{p}_{1}+\vec{p}_{2}-2\vec{p}_{3})$ are the Jacobi momenta. $C^{A}$ is the antisymmetric color wave function and $\Psi_{\Lambda_{Q}}^{S}=\phi_{\Lambda_{Q}}\psi_{\Lambda_{Q}}\chi_{\Lambda_{Q}}$ is a symmetric combination of flavor, momentum and spin wave functions. For $\Lambda_{Q}$ the flavor wave function is

[TABLE]

This is antisymmetric under the exchange of the first two quarks, so the spin-space wave function must also be antisymmetric under such exchange.

The total spin of a system of three spin-$\frac{1}{2}$ particles can be either $\frac{3}{2}$ or $\frac{1}{2}$. The maximally stretched spin states are

[TABLE]

where the superscript $S$ indicates that the state is totally symmetric under the exchange of any pair of quarks, while $\rho$, $\lambda$ denote the mixed-symmetric states that are antisymmetric and symmetric under the exchange of first two spins, respectively.

The momentum-space wave function $\psi_{\Lambda_{Q}}$ can be constructed from the Clebsch-Gordan sum of the product of wave functions of the two jacobi momenta $p_{\rho}$, $p_{\lambda}$ with total angular momentum $\vec{L}=\vec{l}_{\rho}+\vec{l}_{\lambda}$,

[TABLE]

This wave function is then coupled to the spin wave function $\chi_{\Lambda_{Q}}$ to give a spin-momentum wave function of total spin $J$ and parity $(-)^{(l_{\rho}+l_{\lambda})}$,

[TABLE]

The full wave function is then constructed as

[TABLE]

The $\eta_{i}$ are the coefficients determined by diagonalizing the Hamiltonian in the basis of the states $\Psi_{JM}^{i}$ (M. Pervin and Capstick, 2005). In this model the expansion is restricted to $N\leq 2$, where $N=2(n_{\rho}+n_{\lambda})+l_{\rho}+l_{\lambda}$.

In the notation introduced above, the wave functions for states with $J^{P}=\frac{1}{2}^{+}$ are written as

[TABLE]

where we have used $[\psi_{LM_{L}n_{\rho}l_{\rho}n_{\lambda}l_{\lambda}}(\vec{p}_{\rho},\vec{p}_{\lambda})\chi_{S}(M-M_{L})]_{J,M}$ as a shorthand notation for the Clebsch-Gordan sum $\sum_{M_{L}}C^{JM}_{LM_{L},SM-M_{L}}\psi_{LM_{L}n_{\rho}l_{\rho}n_{\lambda}l_{\lambda}}(\vec{p}_{\rho},\vec{p}_{\lambda})\chi_{S}(M-M_{L})$.

In our analytic calculation of the form factors we have used the following single component representation of the $\Lambda$ states with different $J^{P}$:

[TABLE]

For details of the construction of the wave functions, see appendix D.

$\psi_{nlm}$ is expanded in the harmonic oscillator basis, whose wave functions in momentum space are,

[TABLE]

where, $L_{n}^{\beta}(x)$ are the generalized Laguerre polynomials with $p=|\vec{p}|$ and $\mathcal{Y}_{lm}(\vec{p})$ are the solid harmonics.

IV.2 Extraction of Form Factors

The hadron matrix elements for any arbitrary current $\bar{s}\Gamma c$ take the form

[TABLE]

where $\langle q_{1}^{\prime}q_{2}^{\prime}s|\bar{s}\Gamma c|q_{1}q_{2}c\rangle=\langle q_{1}^{\prime}q_{2}^{\prime}|q_{1}q_{2}\rangle\langle s|\bar{s}\Gamma c|c\rangle$. In our spectator approximation $\langle q_{1}^{\prime}q_{2}^{\prime}|q_{1}q_{2}\rangle$ gives delta functions in spin, momentum and flavor.

The analytic expressions for the form factors shown in appendix C are obtained using the single-component wave functions of eq. IV.1. We also calculate the form factors numerically using the full multi-component wave functions extracted from the diagonalization of the Hamiltonian. For this, we adapted the semi-analytic approach used by Mott and Roberts (Mott and Roberts, 2012) in their calculation of the rare dileptonic decay of $\Lambda_{b}$. In this method some of the calculation is done analytically, leaving a couple of integrations to be done numerically.

In the rest frame of the parent $\Lambda_{c}$, we write the initial quark momenta in terms of the Jacobi momenta as

[TABLE]

We use the spectator approximation in which the first two quarks are unaffected by the transition. This allows us to integrate over the Jacobi momenta separately and write the matrix element as

[TABLE]

where the coefficients $h_{b(b^{\prime})}$ are the products of the normalization of the baryon states, the expansion coefficients $\eta_{i}$, and the various Clebsch-Gordan coefficients that appear in the parent (daughter) baryon wave function. The indices $b(b^{\prime})$ contain all the relevant quantum numbers being summed over for the parent (daughter) baryon state. $B_{n_{\rho}l_{\rho}m_{\rho}}^{n_{\rho^{\prime}}l_{\rho^{\prime}}m_{\rho^{\prime}}}$ is the spectator overlap,

[TABLE]

This integral can be done analytically and is given in appendix B.

The interaction overlap $D_{\Gamma;n_{\lambda}l_{\lambda}m_{\lambda}s_{q}}^{n_{\lambda^{\prime}}l_{\lambda^{\prime}}m_{\lambda^{\prime}}s_{q^{\prime}}}$ is

[TABLE]

where $\beta^{(\prime)}=\sqrt{2/3}\alpha_{\lambda}^{(\prime)}$ is the reduced length parameter for the parent (daughter) baryon, $p^{\prime}=(2m_{\sigma}/\tilde{m}_{\Lambda})\vec{p}_{\Lambda}+\vec{p}$, $\tilde{m}_{\Lambda}=m_{s}+2m_{\sigma}$, and $m_{s}$ and $m_{\sigma}$ are the masses of the strange quark and each light quark, respectively. In terms of the generalized Laguerre polynomials,

[TABLE]

The angular dependence in the exponential is eliminated by using the substitutions

[TABLE]

where

[TABLE]

and $\alpha_{\lambda\lambda^{\prime}}=\sqrt{(\alpha_{\lambda}^{2}+\alpha_{\lambda^{\prime}}^{2})/2}$. The interaction overlap then takes the form

[TABLE]

where $\alpha^{2}=\frac{\alpha_{\lambda}^{2}+\alpha^{2}_{\lambda^{\prime}}}{2\alpha^{2}_{\lambda}\alpha^{2}_{\lambda^{\prime}}}$. The Laguerre polynomials and the solid harmonics are functions of $k$ and $p_{\Lambda}$. The details of the semi-analytic calculations are given in appendix A.

V Numerical Results

V.1 Form Factors

The form factors in this work are calculated using the parameters for the quark model wave functions taken from (W. Roberts, 2008). The quark masses relevant for this calculation are shown in Tables 1, while the wave function size parameters are shown in table 2. The calculated form factors are parametrized to have the simple form

[TABLE]

where $q^{2}$ is the momentum transfer $(p_{\Lambda_{c}}-p_{\Lambda})^{2}$. $p_{\Lambda}$ is calculated in the rest frame of the parent $\Lambda_{c}$, and takes the form

[TABLE]

The parameters for the form factors we obtain are given in table 3.

Figure 4 shows the form factors for the transitions to the ground state and the excited states that we consider. In the language of HQET, the form factors ($F_{1}$, $G_{1}$) associated with leading order in the $1/m_{c}$ expansion are dominant, while all of the others are smaller. With the exception of transitions to the $\Lambda(1600){1/2}^{+}$, all of the form factors have their largest absolute values at their respective non-recoil points.

V.2 Comparison with HQET

In Section III.1, we obtained expressions for the general transition form factors in terms of the leading order HQET form factors. Those expressions can be inverted to write the HQET form factors in terms of the general ones. Since the pair of leading order HQET form factors are valid for both the vector and axial-vector hadronic matrix elements, we can extract them from both sets of general form factors. The expressions for $\xi_{1}$ and $\xi_{2}$ are shown in table 4, and the curves are shown in Fig. 5.

The leading order HQET expectation is that the extraction of $\xi_{1}$ and $\xi_{2}$ should be independent of whether they are extracted from axial or vector form factors. However, the curves we obtain indicate that there is some sensitivity to which set of form factors is used. This sensitivity can be attributed to the fact that our form factors include effects that arise in all orders of $1/m_{c}$, while the relationships between the $\xi_{i}\,\,(\zeta_{i})$ and the $F_{i}$ and $G_{i}$ are obtained at leading order. Higher order terms in the $1/m_{c}$ expansion will modify the expressions shown in eqns. 32 - 36, and hence the inverted relationships.

V.3 Decay Widths

V.3.1 $\Lambda_{c}^{+}\to\Lambda^{(*)}l^{+}\nu_{l}$

The differential decay rates, $d\Gamma/dq^{2}$ (in ${\rm s}^{-1}{\rm GeV}^{-2}$), for the semileptonic decays $\Lambda_{c}^{+}\to\Lambda^{(*)}l\nu_{l}$ are shown in figure 6. Fig 6(a) shows the decay rates for the transition to the elastic channel (the ground state) as well as to the excited states that we consider. The elastic channel is dominant but the decay rates for the decays to $1/2^{-}$ and $3/2^{-}$ are significant. Fig 6(b) shows an enlarged version of the decay rates to the excited states. This figure shows that the rates for decays to the radially excited ${1/2}^{+}$, the $3/2^{+}$ and the $5/2^{+}$ states are small compared to the rates for the $1/2^{-}$ and $3/2^{-}$ states.

The integrated total decay widths that we obtain for $\Lambda_{c}^{+}\to\Lambda^{(*)}l^{+}\nu_{l}$ are shown in table 5. Also shown are the results presented in PRCI. The calculated total decay widths to the elastic channel are $1.92\times 10^{11}s^{-1}$ for $l=e$, and $1.86\times 10^{11}s^{-1}$ for $l=\mu$. The branching fractions calculated are $\frac{\Gamma(\Lambda_{c}^{+}\to\Lambda l^{+}\nu_{l})}{\Gamma_{\Lambda_{c}}}=3.84\%$ for the electron channel, and 3.72% for the muon channel. $\Gamma_{\Lambda_{c}}$ is the total decay width of the $\Lambda_{c}$. Table 6 compares our results with other theoretical estimates ($\it{et.al.}$., 2016; Yong-lu Liu, 2009; N. Ikeno, 2016) and the experimental results from the Belle ($\it{et.al.}$, Belle Collaboration) and BESIII ($\it{et.al.}$, BESIII Collaboration; $\it{et.al.}$, BESIII Collaboration) collaborations. Our results are in very good agreement with the most recent experimental result from BESIII.

From table 5, it is evident that the elastic channel dominates the semileptonic decay rate of the $\Lambda_{c}$ but does not saturate it. We find that the branching fraction to the $\Lambda^{*}(1405)$ state with $J^{P}=1/2^{-}$ is $6\%$ of the total semileptonic decay, while the branching fraction to $\Lambda^{*}(1520)$ is $1\%$ of the total. Decays to the other states we consider are significantly smaller.

V.3.2 $\Lambda_{c}^{+}\to\Sigma\pi l^{+}\nu_{l}$ and $\Lambda_{c}^{+}\to N\bar{K}l^{+}\nu_{l}$

Table 7 lists the states that we have included in this study, their total and partial widths in the $\Sigma\pi$ and $N\bar{K}$ channels, and the corresponding strong coupling constants, ($g_{\Lambda\Sigma\pi}$, $g_{\Lambda N\bar{K}}$). The $\Lambda(1405)$ lies just below the $N\bar{K}$ threshold, so its coupling to this channel must be estimated by other means. We use the value estimated by Schat, Scoccola and Gobbi (C.L. Schat, 1994), but also explore the effects on the decay rate of allowing departures from their value.

Figures 7 and 8 show the differential decay rates $d\Gamma/dq^{2}$ and $d\Gamma/dS_{\Sigma\pi}$, respectively, for the decays $\Lambda^{+}_{c}\to\Lambda^{(*)}l^{+}\nu_{l}\to\Sigma\pi l^{+}\nu_{l}$. The dominant contribution to this total decay width is through the $\Lambda(1405)$ resonance. Transitions through the $\Lambda(1115)$ and $\Lambda(1520)$ also provide a significant contribution to the total decay rate for $\Lambda_{c}^{+}\to\Sigma\pi l^{+}\nu_{l}$. The contributions from the transitions through the $\Lambda(1600)$, $\Lambda(1890)$ and $\Lambda(1820)$ are small.

The differential decay rates $d\Gamma/dq^{2}$ and $d\Gamma/dS_{N\bar{K}}$ for the decay $\Lambda^{+}_{c}\to\Lambda^{(*)}l^{+}\nu_{l}\to N\bar{K}l^{+}\nu_{l}$ are shown in figs 9 and 10, respectively. In the total decay width, the transition through the $\Lambda(1520)$ is dominant, and the transition through the sub-threshold $\Lambda(1405)$ is still large. The contributions to the total rate, from transitions through the $\Lambda(1600)$, $\Lambda(1890)$ and $\Lambda(1820)$, are small.

The integrated total decay widths are shown in table 8. In this calculation, we assume $\Lambda_{c}^{+}\to\Lambda^{*}\to\Sigma\pi$ and $\Lambda_{c}^{+}\to\Lambda^{*}\to N\bar{K}$ are the dominant decay modes, and that these two decay modes are saturated by contributions from the states we consider. We also assume that other semileptonic decay modes of the $\Lambda_{c}$ are suppressed. The total decay width $\Gamma_{\Lambda_{c}^{+}\to\Lambda^{(*)}l^{+}\nu_{l}\to\Sigma\pi l^{+}\nu_{l}}$ and $\Gamma_{\Lambda_{c}^{+}\to\Lambda^{(*)}l^{+}\nu_{l}\to N\bar{K}l^{+}\nu_{l}}$ are calculated to be $\sim 0.18\times 10^{11}s^{-1}$, and $\sim 0.1\times 10^{11}s^{-1}$ respectively. We calculate the semileptonic branching fraction $\frac{\Gamma_{\Lambda_{c}^{+}\to\Lambda^{(*)}l^{+}\nu_{l}\to\Sigma\pi l^{+}\nu_{l}}}{\Gamma_{\Lambda_{c}}}$ to be $0.08$, while the branching fraction $\frac{\Gamma_{\Lambda_{c}^{+}\to\Lambda^{(*)}l^{+}\nu_{l}\to N\bar{K}l^{+}\nu_{l}}}{\Gamma_{\Lambda_{c}}}$ is $0.04$. Our calculation contradicts the assumption of the CLEO collaboration ($\it{et.al.}$, Particle Data Group) that the elastic channel saturates the semileptonic decay of $\Lambda_{c}$. In our model, we find the branching fractions for the multi-particle final states are $12\%$ of the total semileptonic decay. This suggests that the semileptonic decay of $\Lambda_{c}^{+}$ is not saturated to decay to elastic channel and further investigation is needed to see evidence of the channels we have discussed here.

We have treated the $\Lambda(1405)$ as a three-quark state that is the lightest excitation of the $\Lambda$, with $J^{P}=1/2^{-}$. In fig. 7, this state contributes a clear resonant structure at $\sqrt{S_{\Sigma\pi}}\approx 1405$. This would suggest that examination of the decay channel $\Lambda_{c}^{+}\to\Sigma\pi l^{+}\nu_{l}$ would provide confirmation of this state as a three-quark state. If no evidence is found for this resonance then it may well be that this state is not a simple three-quark state.

There are a number of other conjectures regarding the structure of the $\Lambda(1405)$. It has been suggested that it could be a dynamically generated molecular state of $\bar{K}N$ and $\Sigma\pi$ (Jennings, 1986; Dalitz and Tuan, 1960; R. H. Dalitz and Rajasekaran, 1967; A. Ramos, 2005; N. Ikeno, 2016), or a multi-quark state (Choe, 1998). Recently, Roca and Oset (Roca and Oset, 2013) explained it as a molecular state of $\bar{K}N$. Hall et al. ($\it{et.al.}$., 2015) drew the same conclusion based on a lattice simulation. Ikeno and Oset (N. Ikeno, 2016) have estimated the semileptonic decay rate of the $\Lambda_{c}$ to this state, assuming that it is a dynamically-generated molecular state. They obtained a value of $2\times 10^{-5}$ for the branching fraction $\mathcal{B}(\Lambda_{c}^{+}\to\Lambda(1405)l^{+}\nu_{l})=2\times 10^{-5}$. For $\Lambda_{c}^{+}\to\Lambda(1405)l^{+}\nu_{l}\to\Sigma\pi l^{+}\nu_{l}$ our branching fraction/ratio is $\sim 2.0\times 10^{-3}$, while for $\Lambda_{c}^{+}\to\Lambda(1405)l^{+}\nu_{l}\to N\bar{K}l^{+}\nu_{l}$ it is $\sim 0.4\times 10^{-3}$. Our values are therefore about 20 times larger than the prediction by Ikeno and Oset.

VI Conclusions and Outlook

In this work, semileptonic decays of the $\Lambda_{c}^{+}$ have been studied using a constituent quark model to calculate the required form factors. These form factors for the $\Lambda_{c}\to\Lambda^{(*)}$ transitions have been obtained both analytically and numerically, using the harmonic oscillator basis to describe the baryon wave functions. The form factors obtained in this model are compared with the HQET expectations at leading order, and are seen to be largely consistent with those expectations. The decay rates of $\Lambda_{c}^{+}$ to the ground state and a number of excited $\Lambda$ states have been evaluated.

The original motivation for this work was that there was no model independent calculation for $\mathcal{B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})$ reported in the previous edition of PDG ($\it{et.al.}$, Particle Data Group). PDG estimated $B(\Lambda_{c}^{+}\to pK^{-}\pi^{+})=RfF\frac{B(D\to Xl^{+}\nu_{l})}{1+|\frac{V_{cd}}{V_{cs}}|^{2}}\tau(\Lambda_{c}^{+})$, based on the measurements by the ARGUS ($\it{et.al.}$., ARGUS Collaboration) and CLEO ($\it{et.al.}$, CLEO Collaboration) Collaborations, using the semileptonic decays of the $\Lambda_{c}$. They assumed that $f=\frac{B(\Lambda_{c}^{+}\to\Lambda l^{+}\nu_{l})}{B(\Lambda_{c}^{+}\to X_{s}l^{+}\nu_{l})}=1.0$ and $F=\frac{B(\Lambda_{c}^{+}\to X_{s}l^{+}\nu_{l})}{B(D\to X_{s}l^{+}\nu_{l})}=1.0$. The latest edition reports a model independent measurement that makes the old estimate obsolete. A. Zupanc et al. (Belle Collaboration) ($\it{et.al.}$, Belle Collaboration) and M. Ablikim (BESIII Collaboration) ($\it{et.al.}$, BESIII Collaboration) measured $B(\Lambda_{c}^{+}\to pK^{-}\pi^{+})$ to be $6.84^{+0.32}_{-0.40}$ and $5.84\pm 0.27\pm 0.23\%$ respectively. PDG reports their fit for $B(\Lambda_{c}^{+}\to pK^{-}\pi^{+})$ to be $6.35\pm 0.33\%$. This result lets us estimate the branching fraction $f=0.87^{+0.13}_{-0.17}$, still assuming that $F=1.0$.

We have calculated branching fractions of the semileptonic decays and they are in a good agreement with the calculations done by Pervin et al. in PRCI (M. Pervin and Capstick, 2005). The branching fraction of the decay to the elastic channel has been calculated to be $3.84\%$ (for $l=e$) and $3.72\%$ (for $l=\mu$). Our prediction is in agreement with the recent results from BESIII ($\it{et.al.}$, BESIII Collaboration; $\it{et.al.}$, BESIII Collaboration) that measured it to be $3.63\pm 0.38\pm 0.20\%$ (for $l=e$) and $3.49\pm 0.46\pm 0.27$ (for $l=\mu$).

We have used the form factors obtained to examine the semileptonic decays to two four-particle final states, namely $\Sigma\pi l^{+}\nu_{l}$ and $NKl^{+}\nu_{l}$. We find that the branching fraction for these two channels totals $12\%$ of the inclusive semileptonic decay $\Lambda_{c}\to X_{s}l^{+}\nu_{l}$. We estimate $f=0.88$, in disagreement with the CLEO ($\it{et.al.}$, CLEO collaboration) assumption that the decay to the ground state $\Lambda(1115)$ saturates the semileptonic decays of the $\Lambda_{c}$.

The two lowest-lying $\Lambda$ resonances, the $\Lambda(1405)1/2^{-}$ and $\Lambda(1520)3/2^{-}$ are seen to be important in both the rate and the shape of the spectrum. The $\Lambda(1405)$ produces a sharp resonant structure in the $S_{\Sigma\pi}$ spectrum, suggesting that this state may be detectable in the $\Lambda_{c}^{+}\to\Sigma\pi l^{+}\nu_{l}$ transition. The $\Lambda(1520)$ also generates sharp resonant structures in both the $\Sigma\pi$ and $N\bar{K}$ decay spectra. This state may therefore also be detectable in these channels. This can have some impact on baryon phenomenology, as it would confirm these states as orbital excitations of the ground states $\Lambda$. The broader resonances that were included in the study are less likely to be identifiable in the decay spectra.

In this calculation we have assumed that the states we include saturate the resonant decays of the $\Lambda_{c}$. The available phase space limits the number of excited states that can contribute significantly to the semileptonic decay rate. There is ample phase space to produce some of the lighter excitations, such as the $\Lambda(1670)1/2^{-}$ and $\Lambda(1690)3/2^{-}$, but the very small wave function overlap with that of the $\Lambda_{c}$ means that the form factors are tiny, so that the decays are very effectively suppressed.

The work presented in this manuscript can be extended in a number of directions. The form factors calculated here may be used to study any of the polarization observables that can arise in these semileptonic decays. With a suitable parametrization of the factorization assumption, they can also be used to examine a number of nonleptonic decays of the $\Lambda_{c}$. The form factors were evaluated using the harmonic oscillator basis, and this leads to form factors that have exponential dependence on $q^{2}$. One possible extension of the project would be to use a different basis, such as the sturmian basis, to extract the form factors. This basis leads to form factors with multipole dependence on $q^{2}$, closer to popular expectations. The semianalytic method we have developed for use with the harmonic oscillator basis can easily be adapted for the sturmian basis. The semileptonic decays of the $\Lambda_{b}$ to both charmed and charmless final states may also be re-examined.

Acknowledgements

We gratefully acknowledge the support of the Department of Physics, the College of Arts and Sciences, and the Office of Research at Florida State University. This research is supported by the U.S. Department of Energy under contracts DE-SC0002615.

Appendix A Semi Analytic Treatment of Hadronic Matrix Elements

The hadronic matrix element can be written in the form,

[TABLE]

where the coefficients $h^{A}_{b(b^{\prime})}$ are the products of the normalization of the baryon state $A$, the expansion coefficients $\eta_{i}$, and the various Clebsch-Gordan coefficients that appear in the parent (daughter) baryon wave function. The indices $b(b^{\prime})$ contain all the relevant quantum numbers being summed over for the parent (daughter) baryon state. $B_{n_{\rho}l_{\rho}m_{\rho}}^{n_{\rho^{\prime}}l_{\rho^{\prime}}m_{\rho^{\prime}}}$ is the spectator overlap,

[TABLE]

where $b^{2}=\frac{\alpha_{\rho}^{2}+\alpha_{\rho^{\prime}}^{2}}{2\alpha_{\rho}^{2}\alpha_{\rho^{\prime}}^{2}}$. The results for the overlap integrals that appear in this calcuation are shown in appendix B.

The interaction overlap $D_{\Gamma;n_{\lambda}l_{\lambda}m_{\lambda}s_{q}}^{n_{\lambda^{\prime}}l_{\lambda^{\prime}}m_{\lambda^{\prime}}s_{q^{\prime}}}$ is

[TABLE]

where, $\beta^{(\prime)}=\sqrt{2/3}\alpha_{\lambda}^{(\prime)}$, $\alpha^{2}=\frac{\alpha_{\lambda}^{2}+\alpha^{2}_{\lambda^{\prime}}}{2\alpha^{2}_{\lambda}\alpha^{2}_{\lambda^{\prime}}}$ and $\alpha_{\lambda\lambda^{\prime}}=\sqrt{(\alpha_{\lambda}^{2}+\alpha_{\lambda^{\prime}}^{2})/2}$.

Using the changes in variable $\vec{p}=\vec{k}+a\vec{p}_{\Lambda}$, and $\vec{p}^{\prime}=\vec{k}+a^{\prime}\vec{p}_{\Lambda}$, where $a=-\frac{m_{\sigma}}{\tilde{m}_{\Lambda}}\frac{\alpha_{\lambda}^{2}}{2\alpha_{\lambda\lambda^{\prime}}^{2}}$ and $a^{\prime}=\frac{m_{\sigma}}{\tilde{m}_{\Lambda}}\frac{\alpha_{\lambda^{\prime}}^{2}}{2\alpha_{\lambda\lambda^{\prime}}^{2}}$, the solid harmonics take the form $\mathcal{Y}_{lm}(a\vec{p}_{1}+b\vec{p}_{2})$, and can be decomposed using the addition theorem as,

[TABLE]

where

[TABLE]

Equation A then takes the form

[TABLE]

where

[TABLE]

The quark current $\langle s(\vec{p}_{\Lambda}+\vec{p},s_{q^{\prime}})|\bar{s}\Gamma c|c(\vec{p},s_{q})\rangle$ can be written in its most general form as

[TABLE]

where $\xi_{\Gamma;lm_{l}}^{s_{q^{\prime}}s_{q}}(\vec{k},\vec{p}_{\Lambda})$ can be expanded in Legendre polynomials as

[TABLE]

Here, $x=\hat{k}\cdot\hat{p}_{\Lambda}$. The coefficients $\zeta_{\Gamma;lm_{l}l^{\prime}}^{s_{q^{\prime}}s_{q}}(k,\vec{p}_{\Lambda})$ are obtained as

[TABLE]

and the integral on the right hand side is evaluated numerically. In practice, the sum in eq. 59 includes a finite number of terms, determined by the values of $l_{\lambda}$, $n_{\lambda}$, $l_{\lambda^{\prime}}$, $n_{\lambda^{\prime}}$ and the maximum value of $l$ in eq. 58. The Legendre polynomial $P_{l_{0}}(x)$ can be written as

[TABLE]

Thus, $\xi_{\Gamma;lm_{l}}^{s_{q^{\prime}}s_{q}}(\vec{k},\vec{p}_{\Lambda})$ takes the form

[TABLE]

The product of the Laguerre polynomials $\mathcal{L}_{n_{\lambda^{\prime}}}^{l_{\lambda^{\prime}}+\frac{1}{2}*}(p^{\prime 2}/\beta^{\prime 2})\times\mathcal{L}_{n_{\lambda}}^{l_{\lambda}+\frac{1}{2}}(p^{2}/\beta^{2})$ can be written as

[TABLE]

where $(\vec{k}\cdot\vec{p}_{\Lambda})$ can be expanded as

[TABLE]

where $P_{l_{\sigma}}(\hat{k}\cdot\hat{p}_{\Lambda})$ is the Legendre polynomial and $C_{l_{\sigma}}^{\prime\sigma}$ is defined as

[TABLE]

Eqn. A therefore becomes

[TABLE]

The angular integrations in eq. 63 are evaluated using the properties of the spherical and solid harmonics. The remaining integral is done numerically.

Appendix B Spectator Overlap

The spectator overlaps for the set of quantum numbers ($n_{\rho}l_{\rho}m_{\rho}$,$n_{\rho^{\prime}}l_{\rho^{\prime}}m_{\rho^{\prime}}$) used in our calculation are listed in this appendix. We define $\alpha_{\rho\rho^{\prime}}=\sqrt{\alpha_{\rho}^{2}+\alpha_{\rho^{\prime}}^{2}/2}$.

[TABLE]

Appendix C Analytic Expressions For The Form Factors

The analytical expressions for the form factors for transition to $\Lambda^{*}$ states with the $J^{P}$ are shown. We obtained these form factors using the single component wave-functions in the harmonic oscillator basis.

C.1 $1/2^{+}$

[TABLE]

where

[TABLE]

C.2 $1/2_{1}^{+}$

[TABLE]

where

[TABLE]

C.3 $1/2^{-}$

[TABLE]

where

[TABLE]

C.4 $3/2^{-}$

[TABLE]

where

[TABLE]

C.5 $3/2^{+}$

[TABLE]

where

[TABLE]

C.6 $5/2^{+}$

[TABLE]

where

[TABLE]

Appendix D Wave Functions

The baryon wave functions are expanded in the harmonic oscillator basis. For $\Lambda$ states with spin-parity $J^{P}=\frac{1}{2}^{+}$, the wave function expansion is

[TABLE]

where the $\eta_{i}$’s are the expansion coefficients and ${[\psi_{LM_{L}n_{\rho}m_{\rho}n_{\lambda}m_{\lambda}}\chi_{S}(M-M_{L})]}_{J,M}$ is a short-hand notation for the Clebsch-Gordan sum $\sum_{M_{L}}C^{JM}_{LM_{L},SM-M_{L}}$.

For $J^{P}=\frac{1}{2}^{-}$ and $J^{P}=\frac{3}{2}^{-}$, the wave function expansion is

[TABLE]

For $J^{P}=\frac{3}{2}^{+}$, the wave function is

[TABLE]

For $J^{P}=\frac{5}{2}^{+}$, the wave function is

[TABLE]

No other states are expected to have significant overlap with the decaying ground state $\Lambda_{c}$ in the spectator approximation.

Appendix E Hadron Tensors

E.1 Hadron Tensor in $\Lambda_{c}^{+}\to\Lambda^{*}l^{+}\nu_{l}$ transitions

E.1.1 $1/2^{+}$

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

where $X\equiv[(m_{\Lambda_{c}}+m_{\Lambda})^{2}-q^{2}]$ and $Y\equiv[(m_{\Lambda_{c}}-m_{\Lambda})^{2}-q^{2}]$.

E.1.2 $1/2^{-}$

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

E.1.3 $3/2^{-}$

[TABLE]

where $Z=3m_{\Lambda_{c}}^{2}m_{\Lambda}^{2}$ and the non vanishing coefficients are

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

E.1.4 $3/2^{+}$

[TABLE]

where $Z=3m_{\Lambda_{c}}^{2}m_{\Lambda}^{2}$ and the non vanishing coefficients are

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

E.1.5 $5/2^{+}$

[TABLE]

where $Z^{\prime}=20m_{\Lambda_{c}}^{4}m_{\Lambda}^{4}$ and the non vanishing coefficients are

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\lambda(m_{\Lambda_{c}}^{2},m_{\Lambda}^{2},q^{2})=(m_{\Lambda_{c}}^{4}+m_{\Lambda}^{4}+q^{4}-2m_{\Lambda_{c}}^{2}m_{\Lambda}^{2}-2m_{\Lambda_{c}}^{2}q^{2}-2m_{\Lambda}^{2}q^{2})$.

[TABLE]

[TABLE]

E.2 Hadron tensor in $\Lambda_{c}^{+}\to\Lambda^{*}l^{+}\nu_{l}\to\Sigma\pi l^{+}\nu_{l}/NKl^{+}\nu_{l}$ transitions

The most general form of the contribution of the $i$th state to the matrix element for the four-body decay $\Lambda_{c}^{+}\to\Lambda_{i}l^{+}\nu_{l}\to BMl^{+}\nu_{l}$ can be written

[TABLE]

where the Lorentz-Dirac operators ${\cal O}_{i}$ are

[TABLE]

Thus, there are sixteen independent Lorentz-Dirac structures in the amplitude. The $c^{i}_{j}$ can be written

[TABLE]

where $k$ runs from $1$ to $3$ for spin $\frac{1}{2}$ states and from $1$ to $4$ for states with higher spin.

The hadron tensor arising from a single intermediate $\Lambda_{i}$ can be written

[TABLE]

The terms in $\gamma^{i}$ do not contribute to the decay width. Because they are proportional to at least one power of the lepton mass, contributions from $\beta^{i}_{PL}$, $\beta^{i}_{QL}$, $\beta^{i}_{LL}$, $\beta^{i}_{LP}$, $\beta^{i}_{LQ}$ are small. The $\alpha^{i}$ from each intermediate state considered takes the form

[TABLE]

Similarly, for the $\beta^{i}_{P_{1}P_{2}}$ ($P_{1}$ and $P_{2}$ denotes $P$, $Q$ or $L$),

[TABLE]

When we treat the coherent sum of the contributions from all the states we consider, we write

[TABLE]

which ultimately leads to

[TABLE]

In this case, the hadron tensor takes the same form as in eq. 80 with the superscripts $i$ removed. The coefficients contributing to the differential decay widths we consider are then

[TABLE]

For each intermediate $\Lambda_{i}$ we consider, the $c^{i}_{j}$ can be written

[TABLE]

where $k$ runs from $1$ to $3$ for spin $\frac{1}{2}$ states and from $1$ to $4$ for states with higher spin. Here, $g_{\Lambda\Sigma\pi}$ is the strong coupling constant for the decay $\Lambda^{i}\to BM$.

For future convenience, we define

[TABLE]

The nonzero coefficients $a_{jk}$s and $b_{jk}$s for $\alpha$ and $\beta$s are listed in the next few subsections.

E.2.1 $\alpha$

[TABLE]

[TABLE]

[TABLE]

E.2.2 $\beta_{PP}$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

E.2.3 $\beta_{PQ}$, $\beta_{QP}$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

E.2.4 $\beta_{QQ}$

[TABLE]

[TABLE]

E.2.5 $\beta_{QL}$, $\beta_{LQ}$

[TABLE]

[TABLE]

E.2.6 $\beta_{LL}$

[TABLE]

[TABLE]

E.2.7 $\beta_{PL}$, $\beta_{LP}$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The nonzero terms in the coefficients $c^{i}_{j}$ are listed in the following subsections.

E.2.8 $\Lambda_{1}=\Lambda\frac{1}{2}^{+}(1115)$, $\Lambda_{2}=\Lambda\frac{1}{2}^{+}(1600)$

[TABLE]

[TABLE]

[TABLE]

E.2.9 $\Lambda_{3}=\Lambda\frac{1}{2}^{-}(1405)$

[TABLE]

[TABLE]

[TABLE]

E.2.10 $\Lambda_{4}=\Lambda\frac{3}{2}^{-}(1520)$

[TABLE]

E.2.11 $\Lambda_{5}=\Lambda\frac{3}{2}^{+}(1890)$

[TABLE]

E.2.12 $\Lambda_{6}=\Lambda\frac{5}{2}^{+}(1820)$

The coefficients take the form

[TABLE]

[TABLE]