\(\Lambda _b\) decays into \(\Lambda _c^*\ell \bar{\nu }_\ell \) and \(\Lambda _c^*\pi ^\) \([\Lambda _c^*=\Lambda _c(2595)\) and \(\Lambda _c(2625)]\) and heavy quark spin symmetry
Abstract
We study the implications for \(\Lambda _b \rightarrow \Lambda _c^*\ell \bar{\nu }_\ell \) and \(\Lambda _b \rightarrow \Lambda _c^*\pi ^\) \([\Lambda _c^*=\Lambda _c(2595)\) and \(\Lambda _c(2625)]\) decays that can be deduced from heavy quark spin symmetry (HQSS). Identifying the odd parity \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) resonances as HQSS partners, with total angular momentum–parity \(j_q^P=1^\) for the light degrees of freedom, we find that the ratios \(\Gamma (\Lambda _b\rightarrow \Lambda _c(2595)\pi ^)/\Gamma (\Lambda _b\rightarrow \Lambda _c(2625)\pi ^)\) and \(\Gamma (\Lambda _b\rightarrow \Lambda _c(2595) \ell \bar{\nu }_\ell )/ \Gamma (\Lambda _b\rightarrow \Lambda _c(2625) \ell \bar{\nu }_\ell )\) agree, within errors, with the experimental values given in the Review of Particle Physics. We discuss how future, and more precise, measurements of the above branching fractions could be used to shed light into the inner HQSS structure of the narrow \(\Lambda _c(2595)\) oddparity resonance. Namely, we show that such studies would constrain the existence of a sizable \(j^P_q=0^\) component in its wavefunction, and/or of a twopole pattern, in analogy to the case of the similar \(\Lambda (1405)\) resonance in the strange sector, as suggested by most of the approaches that describe the \(\Lambda _c(2595)\) as a hadron molecule. We also investigate the lepton flavor universality ratios \(R[\Lambda _c^*] = \mathcal{B}(\Lambda _b \rightarrow \Lambda _c^* \tau \,\bar{\nu }_\tau )/\mathcal{B}(\Lambda _b \rightarrow \Lambda _c^* \mu \,\bar{\nu }_\mu )\), and discuss how \(R[\Lambda _c(2595)]\) may be affected by a new source of potentially large systematic errors if there are two \(\Lambda _c(2595)\) poles.
1 Introduction
Masses and widths of the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) resonances (MeV units). Thresholds (MeV) of some possible \(S\)wave decay channels are also given. In addition, the thresholds of the threebody channels, after the \(P\)wave decay of the \(\Sigma _c^{(*)}\) resonances, are \(M(\Lambda _c+\pi ^++\pi ^)= 2565.60 \pm 0.14\) MeV and \(M(\Lambda _c+\pi ^0+\pi ^0)= 2556.41 \pm 0.14\) MeV. Data taken from the Review of Particle Physics (RPP) [12]
M  \(\Gamma \)  \(M(\Sigma _c^{(*)+}+\pi ^0)\)  \(M(\Sigma _c^{(*)0}+\pi ^+)\)  \(M(\Sigma _c^{(*)++}+\pi ^)\)  

\(\Lambda _c(2595)\)  \(2592.25\pm 0.28\)  \(2.6\pm 0.6\)  \(2587.9 \pm 0.4\)  \(2593.32 \pm 0.14\)  \(2593.54 \pm 0.14\) 
\(\Lambda _c(2625)\)  \(2628.11\pm 0.19\)  \(< 0.97\)  \(2652.5 \pm 2.3\)  \(2658.05 \pm 0.20\)  \(2657.98 \pm 0.20\) 
Constituent quark models (CQMs) predict a nearly degenerate pair of \(P\)wave \(\Lambda _c^*\) excited states, with spin–parity \(J^P=1/2^\) and \(3/2^\), whose masses are similar to those of the isoscalar oddparity \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) resonances [4, 5, 6, 7, 8]. In the most recent of these CQM studies [8], two different types of excitationmodes are considered: The first one, \(\lambda \)mode, accounts for excitations between the heavy quark and the brown muck as a whole, while the second one, \(\rho \)mode, considers excitations inside the brown muck. When all quark masses are equal, \(\lambda \) and \(\rho \)modes are degenerate [8]. However for singlyheavy baryons, the typical excitation energies of the \(\lambda \)mode are smaller than those of the \(\rho \)mode. This is because for singly charm or bottom baryons, the interactions between the heavy quark and the brown muck are more suppressed than between the light quarks [8, 9]. Thus, one should expect the \(\lambda \) excitation modes to become dominant for lowlying states of singly heavyquark baryons. Within this picture, the \(\Lambda ^\mathrm{CQM}_c(2595)\) and \(\Lambda ^\mathrm{CQM}_c(2625)\) resonances would correspond to the members of the HQSS–doublet associated to \((\ell _\lambda =1,\ell _\rho =0)\), with total spin \(S_q=0\) for the light degrees of freedom (ldof), leading to a spinflavorspatial symmetric wavefunction for the light isoscalar diquark subsystem inside of the \(\Lambda _c^*\) baryon. The total spins of these states are the result of coupling the orbitalangular momentum \(\ell _\lambda \) of the brown muck – with respect to the heavy quark – with the spin (\(S_Q\)) of the latter. Thus both \(\Lambda ^\mathrm{CQM}_c(2595)\) and \(\Lambda ^\mathrm{CQM}_c(2625)\) states are connected by a simple rotation of the heavyquark spin, and these resonances will be degenerate in the heavyquark limit.^{1}
Since the total angular momentum and parity of the ldof in the \(S\)wave \(\pi \Sigma _c\) and \(\pi \Sigma ^*_c\) pairs are \(1^\), as in the CQM \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) resonances, the \(\Lambda ^\mathrm{CQM}_c(2595) \rightarrow \pi \Sigma _c \rightarrow \pi \pi \Lambda _c\) and \(\Lambda ^\mathrm{CQM}_c(2625) \rightarrow \pi \Sigma ^*_c \rightarrow \pi \pi \Lambda _c \) decays respect HQSS, and hence one should expect sizable widths for these resonances, unless these transitions are kinematically suppressed. This scenario seems plausible, as can be inferred from the masses and thresholds compiled in Table 1. Indeed, the recent works of Refs. [10, 11] find widths for the CQM \((\ell _\lambda =1,\ell _\rho =0)\) states (\(j_q^P=1^\)) predicted in [8] consistent with data.
A different mechanism to explain the small width of the \(\Lambda _c (2595) \) would be that its wavefunction had a large \(j_q^P=0^\) ldof component.^{2} This is because the transition of this \(j_q^P=0^\) term of the \(\Lambda _c(2595)\) to the final \(\pi \Sigma _c\) state will be suppressed by HQSS. This new mechanism will act in addition to any possible kinematical suppression. As we will see in the next section, it turns out that some of the approaches that describe the \( \Lambda _c(2595)\) as a hadronmolecule predict precisely a significant \(j_q^P=0^\) component for the inner HQSS structure of this resonance. These models generate also the existence of a second, broad, resonance in the region of the \(\Lambda _c(2595)\), with a large \(j_q^P=1^\) ldof component, that could be naturally identified to the HQSS partner of the \(\Lambda _c(2625)\), since both states will have the same brown muck configuration in the heavyquark limit.^{3}
In this work, we will derive HQSS relations between the \(\Lambda _b\) decays into \(\Lambda _c^*\pi ^\) and \(\Lambda _c^*\ell \bar{\nu }_\ell \) \([\Lambda _c^*=\Lambda _c(2595)\) and \(\Lambda _c(2625)]\), supposing firstly that the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) form the lowestlying \(j_q^P=1^\) HQSS doublet. We will also discuss how measurements of the ratio of branching fractions \(\Gamma [\Lambda _b\rightarrow \Lambda _c(2595)]/\Gamma [\Lambda _b\rightarrow \Lambda _c(2625)]\) can be used to constrain the existence of a sizable \(j^P_q=0^\) ldof component in the \(\Lambda _c(2595)\) wavefunction, and/or of a second pole, in analogy to the case of the similar \(\Lambda (1405)\) resonance.
Exclusive semileptonic \(\Lambda _b\) decays into excited charmed \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) baryons have been studied using heavy quark effective theory (HQET), including order \(\Lambda _{\text {QCD}}/m_Q\) corrections [13, 14], and nonrelativistic and semirelativistic CQMs [15], always assuming a single pole structure for the first of these resonances and a dominant \(j^P_q=1^\) configuration. Recently, it has also been suggested that measurements of these decays by LHCb could be used to perform precise lepton flavor universality (LFU) tests [16, 17], comparing branching fractions with \(\tau \) or \(\mu \)leptons in the final state. The analyses of Refs. [16, 17] assumed that both excited charmed baryons form a doublet under HQSS, and therefore it neither contemplated the possibility that the narrow \(\Lambda _c(2595)\) might not be the HQSS partner of the \(\Lambda _c(2625)\), nor that it could contain a nonnegligible \(j_q^P=0^\) component, as it occurs in most of the molecular descriptions of this resonance. It is therefore timely and of the utmost interest to test the HQSS doublet assumption for the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) with the available data.
A first step in that direction was given in Refs. [18, 19]. In these two works, the semileptonic \(\Lambda _b \rightarrow \Lambda _c^*\) transitions, together with the \(\Lambda _b\) decays into \(\Lambda _c^*\pi ^\) and \(\Lambda _c^*D_s^\) were studied. It was found that the ratios of the rates obtained for \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) final states are very sensitive to the couplings of these resonances to the \(D^*N\) channel, which also becomes essential to obtain agreement with the available data. Following the claims of Refs. [18, 19], these results seem to give strong support to the molecular picture of the two \(\Lambda _c^*\) states, and the important role of the \(D^*N\) component in their dynamics.^{4} As we will discuss in the next section, the \(\Lambda _c(2595) D^*N\) and \(\Lambda _c(2625) D^*N\) couplings, together with those to the DN and \(\pi \Sigma _c^{(*)}\) pairs, can also be used to obtain valuable information on the inner HQSS structure of these resonances.
Within a manifest Lorentz and HQSS invariant formalism [21, 22, 23], we will reexamine here some of the results obtained in Refs. [18, 19], and will connect the findings of these two works with the quantum numbers of the ldof in the \(\Lambda _c(2595)\) wave function. Specifically, we will discuss how future accurate measurements of the different ratios of branching fractions proposed in [18, 19] may be used to constrain or discard (i) a sizable \(j^P_q=0^\) component in the \(\Lambda _c(2595)\) wavefunction, and (ii) the existence of a second pole, analog to the second (broad) \(\Lambda (1405)\) resonance [12]. The study will also shed some light on the validity of some of the most popular hadronmolecular interpretations of the oddparity lowestlying \(\Lambda _c^*\) states.
This work is structured as follows. After this introduction, in Sect. 2 we critically review different molecular descriptions of the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) baryons, and discuss in detail the main features of those models that predict a twopole pattern for the \(\Lambda _c(2595)\). Next in Sect. 3, we study the semileptonic \(\Lambda _b \rightarrow \Lambda _c^*\ell \bar{\nu }_\ell \) decays and the constrains imposed by HQSS to these processes. We derive a scheme that preserves spinsymmetry in the \(b\)quark sector and that leads to simple and accurate expressions for the differential widths, including \(\mathcal{O}(1/m_c)\) corrections and full finitelepton mass contributions that are necessary for testing LFU. Semileptonic decays to molecular \(\Lambda _c^\mathrm{MOL}\) states are addressed in Sect. 3.3, and the pion mode is examined in Sect. 4. The numerical results of this work are presented in Sect. 5. First in Sect. 5.1, we discuss the semileptonic (\(\mu ^ \bar{\nu }_\mu \) or \(e^ \bar{\nu }_e\)) and pion \(\Lambda _b\rightarrow \Lambda _c^*\) decays, and present \(m_Q\rightarrow \infty \), \(\mathcal{O}(1/m_Q)\) HQET and molecularmodel predictions for the ratios of branching fractions studied in [18, 19]. Next in Sect. 5.2, we show results for \(\Lambda _b\) semileptonic decays with a \(\tau \) lepton in the final state that can be of interest for LFU tests. Finally, we outline the main conclusions of this work in Sect. 6.
2 HQSS structure of the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) states in hadronmolecular approaches
In this section, we will discuss the most important common features and results obtained from approaches where the \( \Lambda _c(2595)\) and \(\Lambda _c (2625)\) are described as hadronmolecules. These studies are motivated by the appealing similitude of these resonances to the \(\Lambda (1405) \) and \( \Lambda (1520) \) in the strange sector. In particular the two isoscalar Swave \(\Lambda (1405)\) and \(\Lambda _c(2595)\) resonances have several features in common. The mass of the former lies in between the \(\pi \Sigma \) and \(\bar{K} N\) channel thresholds, to which it couples strongly [24, 25, 26]. In turn, the \(\Lambda _c(2595)\) lies below the DN and just slightly above the \(\pi \Sigma _c\) thresholds, and substituting the c quark by a s quark, one might expect the interaction of DN to play a role in the dynamics of the \(\Lambda _c(2595)\) similar to that played by \(\bar{K}N\) in the strange sector.
The hadronic molecular interpretation of the \(\Lambda (1405)\) provides a good description of its properties. Actually, the dynamics of this resonance is mostly governed by the leading order (LO) SU(3) chiral Weinberg–Tomozawa (WT) meson–baryon interaction. The resonance is dynamically generated from the interaction of the mesons of the \(0^\) octet (Goldstone bosons) with the \(1/2^+\) octet of ground state baryons [27, 28, 29, 30, 31, 32, 33, 34] (see also the most recent works of Refs. [35, 36] and references therein for further details and other related studies on the \(\Lambda (1405)\)). One of the distinctive features of this resonance is its twopole structure [29, 31, 32, 33, 34, 35, 36], that have found experimental confirmation [37, 38] as discussed in Ref. [39]. This twopole pattern^{5} is by now widely accepted by the community (see f.i. the mini review on this issue in the RPP by the Particle Data Group [12]).
On the other hand, many works have been also devoted to the study of dynamically generated \(J^P=3/2^\) states in the SU(3) sector [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. Early works considered only the chiral interaction of pseudoscalar \(0^\) mesons with the baryons of the \(3/2^+\) decuplet, but more recently, vectormesons degrees of freedom have also been incorporated in the coupledchannel approach, using different schemes (see for instance the discussion in [51]). In these approaches, the \(\Lambda (1520)\) is dynamically generated mostly from the \(S\)wave \(\pi \Sigma ^*\bar{K}^* N\) coupledchannels dynamics, appearing it slightly above the \(\pi \Sigma ^*\) threshold. It has a nonvanishing width, since the \(\pi \Sigma ^*\) channel is open. In clear analogy, one might naturally think of a similar mechanism to generate the \(\Lambda _c(2625)\) from the \(\pi \Sigma ^*_cD^* N\) dynamics, though the major difference is that the charmresonance is located around 3025 MeV below the \(\pi \Sigma ^*_c\) threshold.
2.1 Molecular models
The general scheme consists of taking some \(S\)wave interactions as kernel of a BetheSalpeter equation (BSE), conveniently ultraviolet (UV) renormalized, and whose solutions fulfill exact elastic unitarity in coupledchannels. In this context, bound and resonant states appear as poles in the appropriate Riemannsheets,^{6} and the residues provide the coupling of the dynamically generated states to the different channels considered in the approach.
The resemblance of the physics in the oddparity charm \(C=1\) baryon sector to the phenomenology seen in \(\bar{K} N\pi \Sigma \) dynamics was first exploited in the works of Refs. [52, 53]. These first two works had some clear limitations. In the first one, the \(J^P= 1/2^\) sector is studied using the scattering of Goldstone bosons off \(1/2^+\) heavylight baryon resonances. Despite the interactions were fully consistent with chiral symmetry, neither the DN, nor the \(D^*N\) channels were considered [52]. The work of Ref. [53] also studied the \(\Lambda _c(2595)\) and there, the interactions were obtained from chirally motivated Lagrangians upon replacing the s quark by the c quark. Though in this way, the DN channel was accounted for, the HQSS counterpart \(D^*N\) was not considered.
The subsequent works of Refs. [54, 55, 56] for the \(J^P=3/2^\) sector, introduced some improvements on the schemes of Refs. [52, 53]. Namely, the BSE interaction kernels were obtained from tchannel exchange of vector mesons between pseudoscalar mesons and baryons, in such a way that chiral symmetry is preserved in the light meson sector. Besides, the universal vector meson coupling hypothesis [Kawarabayashi–Suzuki–Fayyazuddin–Riazudden (KSFR) [57, 58]] was modified to take into account the reduction of the interaction strength provoked by the mass of the \(t\)channel exchanged meson. In this way, some SU(4) flavorsymmetry breaking corrections, additional to those induced by the use of the physical masses, were considered. Similar qualitative findings were obtained in the work of Ref. [59], where some finite range effects were explored.
A detailed treatment of the interactions between the groundstate singly charmed and bottomed baryons and the pseudoNambuGoldstone bosons, discussing also the effects of the nexttoleadingorder chiral potentials, was carried out in [60]. However, channels not involving Goldstone bosons, like DN or \(D^*N\), were again not considered. In this reference, several aspects related to the renormalization procedure were also critically discussed.^{7}
In all cases, the \(\Lambda _c(2595)\), or the \(\Lambda _c(2625)\) if studied, could be dynamically generated after a convenient tuning of the renormalization constants. However, none of these works were consistent with HQSS since none of them considered the \(D^*N\) [66]. Heavy pseudoscalar and vector mesons should be treated on equal footing, since they are degenerated in the heavy quark limit, and are connected by a spinrotation of the heavy quark that leaves unaltered the QCD Hamiltonian in that limit. This is to say the D and \(D^*\) mesons form a HQSSdoublet.
The first molecular description of the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) resonances, using interactions fully consistent with HQSS, was derived in Refs. [66, 67]. In these works a consistent \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) extension of the WT \(\pi N\) Lagrangian –where “lsf” stands for lightspinflavor symmetry–, is implemented, although the adopted renormalization scheme (RS) [54, 56] might not respect HQSS (see the discussion below). Within such scheme, two states are dynamically generated in the region of 2595 MeV. The first one, identified with the \(\Lambda _c(2595)\) resonance, is narrow and it strongly couples to DN and especially to \(D^*N\), with a small coupling to the open \(\pi \Sigma _c\) channel. The second state is quite broad since it has a sizable coupling to this latter channel. On the other hand, a \(J^P=3/2^\) state is generated mainly by the \((D^*N\pi \Sigma _c^*)\) coupledchannel dynamics. It would be the charm counterpart of the \(\Lambda (1520)\), and could be identified with the \(\Lambda _c(2625)\) resonance. The same \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) scheme also dynamically generates the \(\Lambda _b(5912)\) and \(\Lambda _b(5920)\) narrow resonances, discovered by LHCb in 2012 [68], which turn out to be HQSS partners, naturally explaining in this way their approximate mass degeneracy [69]. The extension of the model to the hidden charm sector was carried out in [70], and more recently, it was shown [71] that some (probably at least three) of the narrow \(\Omega _c^*\) states recently observed by LHCb [72] in the \(\Xi _c^+K^\) spectrum in pp collisions can be also dynamically generated within the same scheme.
Several \(\Lambda _c^*\) poles were also obtained in the approach followed in Ref. [73]. There, the interaction of DN and \(D^*N\) states, together with their coupled channels are considered by using an extension to four flavours of the SU(3) local hidden gauge formalism from the light meson sector [74, 75, 76]. The scheme also respects LO HQSS constraints [77] and, as in Refs. [66, 67], a twopole structure for the \(\Lambda _c(2595)\) was also found, with the \(D^*N\) channel playing a crucial role in its dynamics. This is a notable difference to the situation in the strange sector, where the analog \(\bar{K}^*N\) channel is not even considered in most of the studies of the \(\Lambda (1405)\), because of the large \(\bar{K}^*\bar{K}\) mass splitting. (See also the discussion carried out in Ref. [78]).
The beauty \(\Lambda _b(5912)\) and \(\Lambda _b(5920)\) states were also studied in the extended local hidden gauge (ELHG) approach in Ref. [79], while the the predictions of this scheme referred to the LHCb \(\Omega _c^*\) states can be found in [80]. These latter states were also addressed in Ref. [81] using a model constructed out of the SU(4)flavor tchannel exchange of vector mesons. There, the original model of Ref. [54] is revisited, and after taking an appropriate regularization scheme with physically sound parameters, two of the LHCb \(\Omega _c^*\) resonances could be accommodated.
2.2 HQSS structure of the \(\Lambda _c(2595)\) and \(\Lambda _c (2625)\) hadronmolecules
2.2.1 \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\)
The RS adopted in Refs. [66, 67], proposed in [54, 56], plays an important role in enhancing the influence of the \(D^*N\) channel in the dynamics of the narrow \(\Lambda _{c\, (n)}^\mathrm{MOL}(2595)\) state. Furthermore, this RS also produces a reduction in the mass of the resonance of around 200 MeV, which thus appears in the region of 2.6 GeV, instead of in the vicinity of the DN threshold. The RS establishes that all loop functions are set to zero at a common point [\(\mu = \sqrt{m_\mathrm{th}^2+M_\mathrm{th}^2}\), where \((m_\mathrm{th}+M_\mathrm{th})\) is the mass of the lightest hadronic channel], regardless of the total angular moment J of the sector. However, we should point out that such RS might not be fully consistent with HQSS.
In addition, there appears a second \(J^P=1/2^\) pole [\(\Lambda _{c\, (b)}^\mathrm{MOL}(2595)\)] in the 2.6 GeV region [66, 67]. Although it is placed relatively close to the \(\pi \Sigma _c\) threshold, this resonance is broad (\(\Gamma \sim 7090\) MeV) thanks to its sizable coupling to this open channel, which in this case is larger than those to DN and \(D^*N\). The study of Ref. [67] associates this isoscalar resonance to a \(\mathbf{15}\) \(\mathrm SU(6)_{ lsf}\) irrep, where the ldof effectively behave as an isoscalar spintriplet diquark (antisymmetric spinflavor configuration). Thus, it is quite reasonable to assign a dominant \(j_q^P=1^\) configuration to the ldof in this second pole. However, the ratios of \(\pi \Sigma _c, DN\) and \(D^*N\) couplings of this second resonance do not follow the pattern inferred from \(v_1^\mathrm{atr}^2 _{1/2^}\) in Eq. (10) as precisely as in the case of the narrow state. Actually, the couplings of this broad state to the DN and \(D^*N\) pairs, though smaller, turn out to be comparable (absolute value) in magnitude to the \(\pi \Sigma _c\) one (1.6, 1.4 and 2.3, respectively [67]). This points to the possibility that this second pole might also have a sizable component of the \(1^\) repulsive configuration, for which we should expect DN and \(D^*N\) couplings much larger than the \(\pi \Sigma _c\) one (likely in proportion 9 to 1 for the squares of the absolute values, just opposite to what is expected from the \(1^\) attractive eigenvector in Eq. (10)). Indeed, the fact that the \(\Lambda _{c\, (b)}^\mathrm{MOL}(2595)\) is located above the \(\pi \Sigma _c\) threshold reinforces this picture, where there would be a significant mixing among the attractive and repulsive \(1^\) configurations, provoked by the flavor breaking corrections incorporated in the model of Refs. [66, 67]. These symmetry breaking terms affect the kernel f(s) of the BSE, the meson–baryon loops and the renormalization of the UV behaviour of the latter to render finite the unitarized amplitudes. The large difference between the actual \(\pi \Sigma _c\) and \(D^{(*)}N\) thresholds, which are supposed to be degenerate to obtain the results of Eq. (10), should certainly play an important role. The mass breaking effects were less relevant for the narrow \(\Lambda _{c\, (n)}^\mathrm{MOL}(2595)\) resonance, because in that case i) the \(\pi \Sigma _c\) channel had little influence in the dynamics of the state, and ii) the dominant DN and \(D^*N\) thresholds turn out to be relatively close, thanks to HQSS. In addition, other higher channels like \(\eta \Lambda _c\), \(K \Xi ^{(\prime )}_c \), \(D_s \Lambda \), \(\rho \Sigma _c, \ldots \) which are considered in [66, 67], have not been included here in the simplified analysis that leads to the results of Eq. (10). Finally, one should neither discard a small \(0^\) ldof component in the \(\Lambda _{c\, (b)}^\mathrm{MOL}(2595)\) wavefunction that will also change the couplings of this broad state to the different channels.
Note that the total angular momentum and parity of the ldof are neither really conserved in the \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) model, nor in the real physical world because the charm quark mass is finite. Hence, both the narrow and broad \(\Lambda _{c\, (n,b)}^\mathrm{MOL}(2595)\) resonances reported in [66, 67] will have an admixture of the \(0^\) and \(1^\) configurations^{10} in their inner structure. More importantly, the physical \(\Lambda _c(2595)\) and the second resonance, if it exists, will also contain both type of ldof in their wavefunction. As stressed in the Introduction, a nonnegligible \(0^\) component in the \(\Lambda _c(2595)\) or a doublepole structure have not been considered in the theoretical analyses of the exclusive semileptonic \(\Lambda _b\) decays into \(\Lambda _c(2595)\) carried out in Refs. [13, 14, 16]. One of the main objectives of this work is precisely the study of how these nonstandard features affect the \(\Lambda _b\rightarrow \Lambda _c^*\) transitions.
Finally, the lowestlying \(J^P=3/2^\) isoscalar resonance found in Refs. [66, 67] is clearly the HQSS partner of the broad \(J^P=1/2^\) \(\Lambda _{c\, (b)}^\mathrm{MOL}(2595)\) state, with quantum number \(j_q^P=1^\) for the ldof. It is located above the \(\pi \Sigma ^*_c\) threshold, with a width of around 40–50 MeV, and placed in the \(\mathbf{15}\) \(\mathrm SU(6)_{ lsf}\) irrep [67], as the broad \(\Lambda _{c\, (b)}^\mathrm{MOL}(2595)\) resonance. Moreover, the complex coupling of this \(J^P=3/2^\) pole to the \(\pi \Sigma _c^*\) channel is essentially identical to that of the \(\Lambda _{c\, (b)}^\mathrm{MOL}(2595)\) to \(\pi \Sigma _c\). In turn, the square of the absolute value of its coupling to \(D^*N\) compares reasonably well with the sum of the squares of the couplings of the \(\Lambda _{c\, (b)}^\mathrm{MOL}(2595)\) to DN and \(D^*N\), as one would expect from Eqs. (10) and (11). This \(J^P=3/2^\) isoscalar resonance is identified with the \(D\)wave \(\Lambda _c(2625)\) in Refs. [66, 67]. In these works, it is argued that a small change in the renormalization subtraction constant could easily move the resonance down by 40 MeV to the nominal position of the physical state, and that in addition, this change of the mass would considerably reduce the width, since its position would get much closer to the threshold of the only open channel \(\pi \Sigma _c^*\).
Thus, within the \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) model, the \(\Lambda _c(2625)\) turns out to be the HQSS partner of the second broad \(\Lambda _{c\, (b)}^\mathrm{MOL}(2595)\) pole instead of the narrow \(\Lambda _{c\, (n)}^\mathrm{MOL}(2595)\) resonance, as commonly assumed in the theoretical analyses of the exclusive semileptonic \(\Lambda _b\) decays into \(\Lambda _c(2595)\). This picture clearly contradicts the predictions of the CQMs where first, there is no a second 2595 pole, and second, the \(\Lambda _c(2625)\) and the narrow \(\Lambda _c(2595)\) are HQSS siblings, produced by a \(\lambda \)mode excitation of the ground \(1/2^+\) \(\Lambda _c\) baryon.
2.2.2 Extended local hidden gauge (ELHG)

First, \(DN\rightarrow \pi \Sigma _c\) transitions in the \(J^P=1/2^\) sector, which would give rise to \(\quad \langle 1  V^\mathrm{HG}  2 \rangle _{1^} = \sqrt{2}f(s)/4\). The factor 1 / 4 roughly accounts for the ratio \((m_\rho /m_{D^*})^2\), which one would expect to suppress the diagrams induced by the \(t\)channel exchange of charmed vector mesons compared to those mediated by members of the light \(\rho \)octet [55]. This assumes a universal KSFR vectormeson coupling. However, the effects due to \(\quad \langle 1  V^\mathrm{HG}  2 \rangle _{1^}\ne 0\) are, inconsistently with HQSS, not considered in the \(J^P=3/2^\) sector, and thus \(D^*N\) and \(\pi \Sigma _c^*\) channels are not connected^{11} in the formalism of Ref. [73]. Actually, the isoscalar \(\pi \Sigma _c^*\) pair is separately treated as a single channel. We will come back to this point below.

Second, \(D^{(*)}N\rightarrow D N\) transitions in the \(J^P=1/2^\) sector obtained from box diagrams, which also generate contributions to the \(DN\rightarrow DN\) and \(D^*N\rightarrow D^*N\) diagonal interactionterms. In the \(J^P=3/2^\) sector, modifications of the \(D^*N\rightarrow D^*N\) potential induced by boxdiagrams constructed out, in this case, of the anomalous \(D^*D^*\pi \) coupling are also taken into account in [73].
On the other hand, in the \(J^P=3/2^\) sector the isoscalar \(\pi \Sigma _c^*\) is treated as a single channel in [73]. It gives rise to a further broad state (\(\Gamma \sim \) 100 MeV) in the region of 2675 MeV, which is not related to the \(\Lambda _c(2625)\) in that reference.
Finally, we should mention that the boxdiagrams interaction terms evaluated in this ELHG model break HQSS at the charm scale, and it becomes difficult to identify any HQSS resonance doublet among the results reported in [73].
2.2.3 SU(4) flavor tchannel exchange of vector mesons
As already mentioned in this kind of models [54, 55], the BSE potentials are calculated from the zerorange limit of \(t\)channel exchange of vector mesons between pseudoscalar mesons and baryons. Chiral symmetry is preserved in the light meson sector, while the interaction is still of the WT type. Thus, the \(J=1/2\) lowestlying oddparity \(\Lambda _c^*\) resonances are mostly generated from \(DN, \pi \Sigma _c\) coupledchannels dynamics. SU(4) flavor symmetry is used to determine the \(DN\rightarrow DN\) and \(DN\rightarrow \pi \Sigma _c\) interactions, which could be also derived assuming that the KSFR coupling relation holds also when charm hadrons are involved. The flavor symmetry is broken by the physical hadron masses, and in particular the large mass of the \(D^*\) suppresses the off diagonal matrix element \(DN\rightarrow \pi \Sigma _c\), as compared to the diagonal ones that are driven by \(\rho \)meson exchange (see also the discussion in the previous subsection about the factor 1/4 included in the ELHG approach of Ref. [73]). These approaches do not include the \(D^*N\rightarrow D^*N, DN, \pi \Sigma _c\) transitions, and therefore are not consistent with HQSS. Nevertheless, a \(J^P = 1/2^\) narrow resonance close to the \(\pi \Sigma _c\) threshold, which can be readily identified with the \(\Lambda _c(2595)\), is generated. It couples strongly to DN, and its nature is therefore very different from those obtained in the \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) and in the ELHG models, for which the \(D^*N\) channel plays a crucial role. The reason why these SU(4) models can generate the \(\Lambda _c(2595)\) is that the lack of the \(D^*N\) in the \(J^P=1/2^\) sector is compensated by the enhanced strength in the DN channel. For instance, the DN coupling in the approaches of Refs. [54, 55] turned out to be of the same magnitude as that of the narrow \(\Lambda _{c\, (n)}^\mathrm{MOL}(2595)\) to \(D^*N\) in the \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) model of Refs. [66, 67]. On the other hand, the \(\pi \Sigma _c\) coupling, though still small, was found twice larger in Refs. [54, 55]. By construction, the resonance described in [54, 55] will mix \(j_q^P=0^ \) and \(1^\) ldof configurations. The gross features of this dynamically generated state are similar to those of the the resonance reported in Ref. [53], where the similarity between the DN and \(\bar{K} N\) systems, once the strange quark in the later is replaced by a charm quark, was exploited.
In addition, the models based on the tchannel exchange of vector mesons, when the unitarized amplitudes are renormalized as suggested in [54, 56], produce also a second \(J^P=1/2^\) broad resonance (\(\Gamma \sim 100\) MeV) above 2600 MeV, with \(\pi \Sigma _c\) (largest) and DN couplings similar to those found in the \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) and in the ELHG approaches (see Table XIV of Ref. [66] and the related discussion for an update of the results of the model used in [55]). Therefore, this type of molecular models might also predict a double pole structure for the \(\Lambda _c(2595)\), in analogy with what happens in the unitary chiral descriptions of the \(\Lambda (1405)\). We should, however, note that this second broad state is not generated when a RS based on an UV hardcutoff is used [55, 59].
2.2.4 Chiral isoscalar \(\pi \Sigma ^{(*)}_c\) molecules
The chiral interactions between the groundstate singly charmed baryons and the Goldstone bosons lead to scenarios [52, 56, 60] where \(\pi \Sigma _c\) and \(\pi \Sigma ^*_c\) isoscalar molecules naturally emerge in the \(J^P=1/2^\) and \(J^P=3/2^\) sectors, respectively. These states will form a \(1^\) HQSS doublet, whose masses and widths depend on the details of the used RS. The works of Refs. [52, 56] found \(J^P=1/2^, 3/2^\) resonances of around 50 MeV of width and masses in the 2660 MeV region using a RS, inspired in the success of Refs. [30, 34, 40] to describe the chiral SU(3) meson–baryon \(J^P=1/2^\) and \(J^P=3/2^\) sectors, later also employed in the \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) model of Refs. [66, 67].^{12} The \(\pi \Sigma _c^*\) pole found in the ELHG scheme followed in [73] clearly matches the results of Ref. [56], though it was not identified with the \(\Lambda _c(2625)\) in the work of Ref. [73].
In sharp contrast, subtraction constants or UV cutoffs were finetuned in Ref. [60] in such a way that the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) experimental masses were reproduced, leading to weakly \(\pi \Sigma _c\) and \(\pi \Sigma _c^*\) bound states. Thus, the needed UV cutoffs turned out to be slightly higher than expected, 1.35 and 2.13 GeV, respectively. This could indicate some degrees of freedom that are not considered in the approach, such that CQM states or \(D^{(*)}N\) components, and that could play a certain role, being their effects effectively accounted for the fitted real parts of the unitarity loops [82, 83].
2.3 Weinberg compositeness condition
In recent years, the compositeness condition, first proposed by Weinberg to explain the deuteron as a neutronproton bound state [84, 85], has been advocated as a model independent way to determine the relevance of hadronhadron components in a molecular state. With renewed interests in hadron spectroscopy, this method has been extended to more deeply bound states, resonances, and higher partial waves [86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97]. However, we should mention that the compositeness analysis proposed by Weinberg [84, 85] is only valid for bound states. For resonances, it involves complex numbers and, therefore, a strict probabilistic interpretation is lost as pointed out in Ref. [91].
For the particular case of the \(\Lambda _c(2595)\), the situation is a bit unclear. For instance, it was shown in Ref. [98] that the \(\Lambda _c(2595)\) is not predominantly a \(\pi \Sigma _c\) molecular state using the effective range expansion. A similar conclusion was reached in Ref. [99], using a generalized effective range expansion including CastillejoDalitzDyson pole contributions. In this latter work, the effects of isospin breaking corrections are also taken into account and the extended compositeness condition for resonances developed in Ref. [100] was applied to calculate the component coefficients. Furthermore, although in the unitary approaches, the \(\Lambda _c(2595)\) is found to be of molecular nature [52, 53, 54, 55, 60, 66, 67, 73], there is no general agreement on its dominant meson–baryon components yet.
In general, one can conclude that the compositeness of the \(\Lambda _c(2595)\) depends on the number of considered coupled channels, and on the particular regularization scheme adopted in the unitary approaches and, therefore, would be model dependent [61].
3 Semileptonic \(\Lambda _b \rightarrow \Lambda _c^*\ell \bar{\nu }_\ell \) decays
3.1 Infinite heavy quark mass limit
The single heavy baryon and heavy quark velocities are equal in the \(m_Q\rightarrow \infty \) limit. The heavy baryon can be viewed as a freely propagating pointlike color source (the heavy quark), dressed by strongly interacting brown muck bearing appropriate color, flavor, baryon number, energy, angular momentum and parity to make up the observed physical state. Since an infinitely massive heavy quark does not recoil from the emission and absorption of soft \((E\sim \Lambda _\mathrm{QCD})\) gluons, and since chromomagnetic interactions of such a quark are suppressed as \(1/m_Q\), neither its mass (flavor) nor its spin affect the state of the light degrees of freedom. This results in a remarkable simplification of the description of transitions in which a hadron containing a heavy quark, with velocity \(v^\mu \), decays into another hadron containing a heavy quark of a different flavor. To the heavy quark, this looks like a free decay (up to pertubative QCD corrections), in which the light dressing plays no role. The brown muck, on the other hand, knows only that its pointlike source of color is now recoiling at a new velocity \(v'^{\mu }\), and it must rearrange itself about it in some configuration [23]. Hence, in the \(m_Q\rightarrow \infty \) limit, the weak matrix elements must become invariant under independent spin rotations of the c and b quarks. This is easily shown in the brick wall frame (\(\vec {v}=\vec {v}\,'\), \(v^0=v'^0\)) by quantizing the angular momentum of the ldof (brown muck) about the spatial axis defined by \(\vec {v}\). It follows that neither the initial and final heavy baryons, nor the c and b quarks have orbital angular momentum about this decay axis. Thus, in the IsgurWise (IW) limit, the spins of c and b are decoupled from the light quanta, and the component of the ldof total angular momentum along the decay axis is conserved [23, 101].
Under a Lorentz transformation, \(\Lambda \), and b and c quark spin transformations \(\widehat{S}_b\) and \(\widehat{S}_c\), the above spinor wave functions transform as \(S(\Lambda )\,u_b,\, \Lambda ^\mu _\nu S(\Lambda )\, \mathcal{U}^\nu _c\) and \(\widehat{S}_b\, u_b\) and \(\widehat{S}_c \,\mathcal{U}^\mu _c\), respectively, with \(S(\Lambda )=\exp \{i\sigma _{\mu \nu }S^{\mu \nu }/4\}\), the usual spinor representation. Note that \(\widehat{S}_b\) and \(\widehat{S}_c\) are also of the form \(S(\hat{\Lambda })\), but with \(\hat{\Lambda }\) restricted to spatial rotations and affecting only to the heavy quark spinor.
On the other hand, for the groundstate \(\Lambda _b\) transition to the \(J^P=1/2^\) charmed baryon with \(j_Q^P=0^\) ldof, one can use for the latter a spinor \(u_c(v')\), but the formfactors must be pseudoscalar and therefore involve a LeviCivita tensor [22]. At leading order in the \(1/m_Q\) expansion, there are not enough vectors available to contract with the indices of the epsilon tensor so these unnatural^{14} parity matrix elements vanish [13, 14].
A different way to understand why the \(\Lambda _b[1/2^+, j_q^P=0^+] \rightarrow \Lambda _c^*[1/2^, j_q^P=0^]\) is forbidden in the IW limit is adopting the picture introduced in Refs. [18, 19]. In the heavyquark limit, the weak transition occurs on the b quark, which turns into a c quark and a \(W^\) boson, as shown in the left panel of Fig. 1. Since we will have a \(1/2^\) or \(3/2^\) state at the end, and the u, d quarks are spectators, remaining in a \(0^+\) spinparity configuration, the final charm quark must carry negative parity and hence must be in an \(L = 1\) level. This corresponds to an orbital angular momentum excitation between the heavy quark and the isoscalar u, d diquark as a whole, which maintains the same spinparity quantum numbers, \(0^+\), as in the initial \(\Lambda _b\), leading to a nonzero ldof wavefunction overlap. Within this picture, the total angular momentum and parity of the light subsystem will be \(j_q^P=1^\,[= 0^+\otimes (L=1)]\), and the transition will be described by the matrix element in Eq. (21), that will go through \(P\)wave, giving rise to the \((\omega ^21)\) factor in Eq. (23). In sharp contrast, the \((j_q^P=0^,J^P=1/2^)\) final baryon contains a \(P\)wave excitation inside the brown muck and a realignment of the light quarks spins to construct a spin triplet state. That requires going beyond the spectator approximation of Fig. 1, involving dynamical changes in the QCD dressing of the heavy baryon during the transition, which are \(1/m_Q\) suppressed. Thus in the heavy quark limit, the initial and final ldof overlap for the unnatural \(0^+\rightarrow 0^\) transition vanishes. It would be parametrized by a pseudoscalar formfactor, involving the LeviCivita tensor. As mentioned above, at leading order in the \(1/m_Q\) expansion, there are not enough vectors available to contract with the indices of the epsilon tensor.
3.2 \(\mathcal{O}(\Lambda _\mathrm{QCD}/m_c)\) corrections
Corrections of order \(1/m_Q\) to \(d\Gamma (\Lambda _b\rightarrow \Lambda _c^{*3/2}[j^P_q=1^])/d\omega \) and \(d\Gamma (\Lambda _b\rightarrow \Lambda _c^{*1/2}[j^P_q=1^])/d\omega \) distributions were studied in [14] and shown to be quite large, specially in the \(J^P=1/2^\) case (see Fig. 1 of that reference).
If \(\Delta _1= \Omega _1= \sigma \) and \(\Delta _2=\Omega _2=0\), the IW limit of Eq. (23) is recovered for transitions to \(\Lambda _c^{*3/2^,\, 1/2^}\,[j_q^P=1^]\) states.^{15}
 \(j_q^P= 1^\): Neglecting \(1/m_b\) corrections and QCD shortrange logarithms [14],$$\begin{aligned} \Delta _1(\omega )= & {} \sigma (\omega ) + \frac{1}{2m_c}\left( \phi _\mathrm{kin}^{(c)}(\omega )  2 \phi _\mathrm{mag}^{(c)}(\omega )\right) ,\quad \nonumber \\ \Delta _2(\omega )= & {} \frac{1}{2m_c} \left( 3(\omega \bar{\Lambda }'\bar{\Lambda })\sigma (\omega )\right. \nonumber \\&\left. +2(1\omega ^2)\sigma _1(\omega ) \right) \end{aligned}$$(29)with \(m_c\sim 1.4\) GeV, the charm quark mass, and \(\bar{\Lambda }\sim 0.8\) GeV [\(\bar{\Lambda }'\sim (1 \pm 0.1\) GeV)] the energy of the ldof in the \(m_Q\rightarrow \infty \) limit in the \(\Lambda _b\) [\(\Lambda _c^*\,(j_q^P=1^)\)] baryon. The \(\sigma _1(\omega )\) formfactor determines, together with \(\bar{\Lambda }\) and \(\bar{\Lambda }'\), the \(1/m_c\) corrections stemming from the matching of the QCD and effective theory currents. This subleading IW function is unknown and in Ref. [14], it was varied in the range \(\pm 1.2 \left[ 11.6(\omega 1)\right] \) GeV. In addition, \(\phi _\mathrm{kin}^{(c)}\) and \(\phi _\mathrm{mag}^{(c)}\) account for the time ordered product of the dimensionfive kinetic energy and chromomagnetic operators in the effective Lagrangian. The chromomagnetic term is neglected in [14], because it is argued that it should be small relative to \(\Lambda _\mathrm{QCD}\). In addition, the kinetic energy correction is estimated in the large \(N_c\) limit, \(\phi _\mathrm{kin}^{(c)} = \frac{\bar{\Lambda }}{8}\sqrt{\frac{\bar{\Lambda }^3}{\kappa }}\left( \omega ^21\right) \sigma (\omega )\), with \(\kappa =(0.411\,\mathrm{GeV})^3\) [14].$$\begin{aligned} \Omega _1(\omega )= & {} \sigma (\omega ) + \frac{1}{2m_c}\left( \phi _\mathrm{kin}^{(c)}(\omega ) + \phi _\mathrm{mag}^{(c)}(\omega )\right) ,\quad \nonumber \\ \Omega _2(\omega )= & {} \frac{\sigma _1(\omega )}{2m_c} \end{aligned}$$(30)The Eqs. (29) and (30) can be rederived fromwhere the \(\mathcal{O}(1/m_c)\) \(\beta _2\) and \(\beta _3\) formfactors and the subleading term of \(\beta _1\) depend on J. Thus, we have$$\begin{aligned}&\big \langle \Lambda _c^{*1/2^,\,3/2^}; j_q^P=1^\left \, J_{bc}^\mu (0)\, \right \Lambda _b\big \rangle \nonumber \\&= \bar{\mathcal{U}}^\lambda _c(v')\, \left\{ v_\lambda [\beta _1+(\omega \not v)\beta _2]+ \gamma _\lambda \beta _3/3\right\} \nonumber \\&\qquad \times \gamma ^\mu (1\gamma _5) u_b(v) + \mathcal{O}(1/m_{b})\,, \end{aligned}$$(31)with \(c_{J=1/2}= 2\) and \(c_{J=3/2}= 1\), which correspond to the eigenvalues of the operator \(2\, \vec {S}_c \cdot \vec {j}_q\)$$\begin{aligned} \beta _1(\omega )_J= & {} \sigma (\omega ) + \frac{1}{2m_c}\left( \phi _\mathrm{kin}^{(c)}(\omega ) +c_J \phi _\mathrm{mag}^{(c)}(\omega )\right) \nonumber \\ \beta _2(\omega )_J= & {} \frac{c_J}{2m_c}\, \sigma _1(\omega )\, ,\quad \nonumber \\ \beta _3(\omega )= & {} 3\,\frac{(\omega \bar{\Lambda }'\bar{\Lambda })}{2m_c}\,\sigma (\omega ) \end{aligned}$$(32)for \(j_q= 1\) and \(S_c=1/2\), and$$\begin{aligned} c_J= J\,(J+1)\frac{1}{2}\,\left( \frac{1}{2}+1\right) 1\,(1+1)\,, \end{aligned}$$(33)The \(1/m_b\) contributions, not taken into account, are much smaller than the theoretical uncertainties induced by the errors on \((\bar{\Lambda }\bar{\Lambda }')\) and the \(\sigma _1(\omega )\) formfactor. Hence, the formfactors of Eqs. (29) and (30) provide an excellent approximation to the results reported in Ref. [14].$$\begin{aligned}&\Omega _1 = \beta _1(\omega )_{J=3/2}, \quad \Delta _1 = \beta _1(\omega )_{J=1/2}, \quad \nonumber \\&\Omega _2 = \beta _2(\omega )_{J=3/2}, \nonumber \\&\Delta _2 = \beta _3(\omega )+\beta _2(\omega )_{J=1/2}. \end{aligned}$$(34)
Two final remarks to conclude this discussion: (i) The \((\omega \bar{\Lambda }' \bar{\Lambda })\) difference in \(\Delta _2\) [\(\gamma _\lambda \) formfactor in Eq. (31)] provides a \(S\)wave \(W^\Lambda _c^* (1/2^)\) term that should scale as \(\sqrt{\omega ^21}\), and hence should dominate this differential rate at zero recoil. ii) The kinetic operator correction is the only \(1/m_c\) term that does not break HQSS.

\(j_q^P= 0^\): For the case of this unnatural transition, the matrix elements of the \(1/m_Q\) current and kinetic energy operator corrections are zero for the same reason that the leading form factor vanished [14]. The time ordered products involving the chromomagnetic operator lead to nonzero contributions, which however vanish at zero recoil [14] and can be cast in a \(\Delta _1\)type form factor. At order \(1/m_Q\) the corresponding \(\Delta _2\) formfactor is zero.
From the above results, we conclude that the \(\Lambda _b\) semileptonic decay to a \(J^P=1/2^\)daughter charm excited baryon with a \(j_q^P= 0^\) ldof–configuration can be visible only if HQSS is severely broken and higher \(\left( 1/m_Q\right) ^n\) corrections are sizable.
3.3 Decays to molecular \(\Lambda _c^\mathrm{MOL}\) states
Following the spectator image of Fig. 1, the c quark created in the weak transition must carry negative parity and hence must be in a relative \(P\)wave. The parity and total angular momentum of the final resonance are those of the intermediate system before hadronization. Since the molecular \(\Lambda _c^\mathrm{MOL}\) states come from meson–baryon interaction in our picture, we must hadronize the final state including a \(q\bar{q}\) pair with the quantum numbers of the vacuum (\(^{2S+1}L_J=^3P_0\)). This is done following the work of Refs. [18, 19], and thus we include \(u\bar{u}+d\bar{d} + s\bar{s}\) as in the right panel of Fig. 1. The c quark must be involved in the hadronization, because it is originally in an \(L = 1\) state, but after the hadronization produces the \(D^{(*)}N\) state, and the c quark in the \(D^{(*)}\) meson is in an \(L = 0\) state. Neglecting hiddenstrange contributions, the hadronization results in isoscalar \(S\)wave DN and \(D^*N\) pairs, but does not produce \(\pi \Sigma _c^{(*)}\) states [18, 19].
The function \(\varphi (\omega )\) accounts for some \(\omega \) dependences induced by the hadronization process and by the matrix element between the initial \(S\)wave \(b\)quark, the outgoing \(W\)plane wave and the \(P\)wave \(c\)quark created in the intermediate hadronic state. This latter factor should scale like \(\vec {q}\, \propto \sqrt{\omega ^21}\) close to zero recoil [18, 19]. In the heavy quark limit assumed in the mechanism depicted in Fig. 1, one expects \(\varphi (\omega )\) to be independent of the angular momentum, J, of the final resonance.
4 \(\Lambda _b \rightarrow \Lambda _c^* \pi ^\) decay
5 Results
5.1 Semileptonic (\(\mu ^ \bar{\nu }_\mu \) or \(e^ \bar{\nu }_e\)) and pion \(\Lambda _b\rightarrow \Lambda _c^*\) decays
Ratios of semileptonic (\(\mu ^ \bar{\nu }_\mu \) or \(e^ \bar{\nu }_e\)) and pion \(\Lambda _b\) decays into odd parity \(J=1/2\) and 3 / 2 charm baryons. We show predictions obtained from the molecular schemes of Refs. [66, 67] (\(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\)) and [73] (ELHG), together with the \(m_Q\rightarrow \infty \) limit (IW\(_\infty \)) ratios, and those found including the subleading corrections [IW\(_{\mathcal{O}(1/m_Q)}\)] derived in Ref. [14] for the case of a \(j_q^P=1^\) HQSS doublet. The ELHG results for the narrow \(\Lambda _c(2595)\) are taken from Refs. [18, 19]. In the case of molecular approaches, the gG factors that enter in Eq. (37) are compiled in Table 3. The ranges quoted for the \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) pionmode ratios account for the factor of two introduced at the end of Sect. 4, suggested by the findings of Ref. [19]. We also show in the last column experimental estimates for these ratios obtained from branching fractions given in the RPP [12]. See the text for more details
IW\(_\infty \)  IW\(_{\mathcal{O}(1/m_Q)}\)  \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\)  ELHG  RPP  

\(\Gamma _\mathrm{sl}[\Lambda _b \rightarrow \Lambda _{c\, (n)}(2595)]/\Gamma _\mathrm{sl}[\Lambda _b \rightarrow \Lambda _c(2625)]\)  0.5  \(1.4^{+1.7}_{1.0}\)  0.14  0.39 – 0.48  \(0.6^{+0.4}_{0.3}\) 
\(\Gamma _\mathrm{sl}[\Lambda _b \rightarrow \Lambda _{c\, (b)}(2595)]/\Gamma _\mathrm{sl}[\Lambda _b \rightarrow \Lambda _c(2625)]\)  −  −  0.39  \(\sim 0.02\)  − 
\(\Gamma _\mathrm{\pi }[\Lambda _b \rightarrow \Lambda _{c\, (n)}(2595)]/\Gamma _\mathrm{\pi }[\Lambda _b \rightarrow \Lambda _c(2625)]\)  0.5  \(1.4^{+3.3}_{1.1}\)  0.14–0.28  0.76 – 0.91  \(1.0\pm 0.6\) 
\(\Gamma _\mathrm{\pi }[\Lambda _b \rightarrow \Lambda _{c\, (b)}(2595)]/\Gamma _\mathrm{\pi }[\Lambda _b \rightarrow \Lambda _c(2625)]\)  −  −  0.39–0.78  \(\sim 0.02\)  − 
Values (MeV) of the factors \(g_{R_J}^{D^{(*)}N}\, G_{D^{(*)}N}\) from Ref. [66] (\(\mathrm{SU(6)}_\mathrm{lsf} \times \) \( \mathrm{SU(2)}_\mathrm{HQSS}\)) and [73] (ELHG). The signs of \(g_{R_J=1/2}^{D^*N}\, G_{D^*N}\) are changed with respect to Ref. [73], as discussed in [19]. The values quoted for the \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) \(g_{R_J=1/2}^{D^*N}\,G_{D^*N}\) take into account the order meson–baryon used in this work to couple the spins (see footnote 8)
\(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\)  ELHG  

\(g_{R_J}^{DN}\, G_{DN}\)  \(g_{R_J}^{D^*N}\, G_{D^*N}\)  \(g_{R_J}^{DN}\, G_{DN}\)  \(g_{R_J}^{D^*N}\, G_{D^*N}\)  
\(\Lambda _{c\, (n)}^\mathrm{MOL}(2595)\)  \(10.54 + 0.02\,i\)  \(11.65  0.42\,i\)  \(\phantom {}13.881.06\,i\)  \( 26.51 + 2.10\,i \) 
\(\Lambda _{c\, (b)}^\mathrm{MOL}(2595)\)  \(3.16  3.45\,i\)  \(4.14 + 0.17\,i\)  \(0.68 + 3.13\,i\)  \(4.66 + 3.42\, i\) 
\(\Lambda _{c\,}^\mathrm{MOL}(2625)\)  −  \(5.82 + 2.58\,i \)  −  29.10 
In Table 2, we show results for the ratios of semileptonic (\(\mu ^ \bar{\nu }_\mu \) or \(e^ \bar{\nu }_e\)) and pion \(\Lambda _b\) decays into odd parity \(J=1/2\) and 3 / 2 charm baryons, obtained within the molecular schemes of Refs. [66, 67] (\(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\)) and [73] (ELHG). As commented in Sect. 2.2, a double pole structure for the \(\Lambda _c(2595)\) is found in these approaches, with clear similarities to the situation for the \(\Lambda (1405)\), and hence we give results for both, the narrow (n) and (b) broad \(\Lambda _c^\mathrm{MOL}(2595)\) states. In Table 2, we also show experimental estimates for these ratios deduced from branching fractions given in the RPP [12]. We have considered that the reconstructed \(\Lambda _c(2595)\) resonance observed in the decays corresponds to the molecular narrow resonance. In addition, \(m_Q\rightarrow \infty \) limit results (IW\(_\infty \)) and predictions obtained incorporating the subleading corrections (IW\(_{\mathcal{O}(1/m_Q)}\)) discussed in Ref. [14] are also shown in Table 2. In this latter work, it is assumed that the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) form the lowestlying \(j_q^P=1^\) HQSS doublet, and the values quoted in the table follow mostly from Eqs. (2.26) and (2.28) of that reference. To the error budget deduced from these equations, we have added in quadrature the effects due to the uncertainty (\(\pm 0.1 \) GeV) on the \(\bar{\Lambda }'\) parameter in Eq. (29), which produces variations in the ratios of about 25%–30% [14]. The errors on the IW\(_{\mathcal{O}(1/m_Q)}\) ratios are largely dominated by the uncertainties on the subleading \(\sigma _1\) formfactor. It leads to opposite effects for \(\Lambda _c(2595)\) or \(\Lambda _c(2625)\) final states [14], as can be inferred here from Eqs. (32) and (33). The biggest (smallest) \(\Gamma ^{\Lambda _{1/2}}_\mathrm{sl,\pi }/ \Gamma ^{\Lambda _{3/2}}_\mathrm{sl,\pi }\) values correspond to \(\sigma _1(1)=1.2\) (+1.2) GeV, while the central values are obtained for \(\sigma _1(\omega )=0\). The \(\Gamma ^{\Lambda _{1/2}}_\mathrm{sl}\) rate, depending on \(\sigma _1\), could be significantly enhanced (around a factor 2.5 for \(\sigma _1=0\)) compared to the infinite mass prediction (\(\sim 0.020\,\Gamma _0)\), while \(1/m_Q \) effects are much smaller for \(\Gamma ^{\Lambda _{3/2}}_\mathrm{sl}\). Predictions for the pion decay widths depend on \(d\Gamma _\mathrm{sl}/d\omega \) at \(q^2=m_\pi ^2\), and turn out to be quite uncertain due to \(\sigma _1\). We see that IW\(_{\mathcal{O}(1/m_Q)}\) predictions and experimental estimates for the \(\Gamma ^{\Lambda _{1/2}}/ \Gamma ^{\Lambda _{3/2}}\) ratios agree, within errors, for both semileptonic and pion \(\Lambda _b\) decay modes. A certain tendency is observed in the central values, for which the theoretical estimations are greater than the experimental ones, in particular in the semileptonic mode. However, it would not be really significant due to the great uncertainties.
In what respects to the ELHG ratios for the narrow molecular \(\Lambda _c(2595)\) state, we give in Table 2 the ranges quoted in the original works of Refs. [18, 19]. The lowest ratios can be found using the gG coefficients compiled in Table 3, while the highest values account for corrections due to the contribution of hiddenstrange (\(D^{(*)}_s\Lambda \)) channels in the hadronization. Within the ELHG scheme the broad \(\Lambda _c(2595)\) ratios are negligible. This is because in this approach, the \(J^P=3/2^\) \(\Lambda _c(2625)\) is a quasibound \(D^*N\) state with a large coupling to this channel, whose absolute value is around five times bigger than that of the broad \(\Lambda _c(2595)\) resonance to \(D^*N\) or DN [19]. The narrow ELHG \(\Lambda _c(2595)\) molecule has DN and \(D^*N\) couplings (in absolute value) similar to \(g_{\Lambda _c(2625)}^{D^*N}\), and its \(\Gamma ^{\Lambda _{1/2}}_\mathrm{sl,\pi }/ \Gamma ^{\Lambda _{3/2}}_\mathrm{sl,\pi }\) ratios are larger and about 0.4 and 0.8, respectively, compatible within errors with the experimental expectations. It should be also noted that after renormalization, the DN loop function is almost a factor of two smaller than the \(D^*N\) one, which produces a significant source of HQSS breaking in the ELHG approach of Ref. [19].
Finally, we see that the \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) ratios for the narrow molecular \(\Lambda _c(2595)\) resonance, though small (\(0.140.28\)), are neither negligible, nor totally discarded by the available data. As we expected, they are suppressed because within this approach this state has a large \(j_q^P=0^\) ldof component. Semileptonic decays into the broad \(\Lambda _c(2595)\) resonance are about a factor of three larger, but the \(\Gamma ^{1/2 (b)}_{sl,\,\pi }/ \Gamma ^{3/2}_{sl,\,\pi }\) ratios are still below 1/2, the \(m_Q\rightarrow \infty \) prediction, and well below the IW\(_{\mathcal{O}(1/m_Q)}\) central values obtained in [14] (see Fig. 2). Both sets of results point to important \(\left( 1/m_Q\right) ^n\) corrections, induced by the meson–baryon interactions that generate the molecular states. On the other hand, we do not expect large variations from the consideration of hidden strange channels as intermediate states. From the couplings reported in Refs. [66, 67], only \(\Lambda \, D_s\) and \(\Lambda \, D_s^*\) might be important through their coupling to the narrow \(\Lambda _c(2595)\) state, but the respective thresholds are located (around 3.1 and 3.2 GeV) well above the resonance position, and it is not reasonable to claim for large effects produced by these high energy physics contributions. Actually, we have checked that the ratios given in Table 2 for the \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) model hardly change if the large number of coupledchannels used in Refs. [66, 67] is reduced only to \(D^{(*)}N\) and \(\pi \Sigma _c^{(*)}\).
The predictions for the ratios in molecular schemes are very sensitive to the interference and relative weights of the DN and \(D^*N\) contributions [18, 19], and thus future accurate measurements of these ratios will shed light on the nature of the \(\Lambda _c(2595)\), allowing us to address issues as the existence of two poles or the importance of the \(D^*N\) channel in the formation of the resonance(s). Such studies will also help to understand the interplay between CQM and hadronscattering degrees of freedom [107, 108, 109, 110, 111, 112] in the dynamics of the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\).
Semileptonic decay widths \(\Gamma [\Lambda _b \rightarrow \Lambda _c^*\,\ell \,\bar{\nu }_\ell \)] (in \(\Gamma _0/100\) units) for \(\mu \) and \(\tau \) modes. The rates are calculated using Eq. (26), with formfactors given in Eqs. (29) and (30) or equivalently in Eq. (32). They contain the subleading \(\mathcal{O}(1/m_c)\) corrections derived in Ref. [14], assuming that the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) form the lowestlying \(j_q^P=1^\) HQSS doublet. We also show the \(\tau /\mu \)semileptonic ratios for both final baryon states, and \(\Gamma _\mathrm{sl}[\Lambda _b \rightarrow \Lambda _{c}(2595)]/\Gamma _\mathrm{sl}[\Lambda _b \rightarrow \Lambda _c(2625)]\) for the \(\tau \,\bar{\nu }_\tau \) semileptonic mode. Errors are derived from the uncertainties on the \(\sigma _1\) formfactor and the \(\bar{\Lambda }'\) parameter, and are added in quadrature for the ratios shown in the last three columns. Central values are obtained for \(\sigma _1(\omega )=0\) in all cases. Results from Ref. [17] are taken from TABLE II
\(\Gamma [\Lambda _b \rightarrow \Lambda _c(2595)\tau \,\bar{\nu }_\tau \)]  \(\Gamma [\Lambda _b \rightarrow \Lambda _c(2595)\mu \,\bar{\nu }_\mu \)]  \(\Gamma [\Lambda _b \rightarrow \Lambda _c(2625)\tau \,\bar{\nu }_\tau ]\)  \(\Gamma [\Lambda _b \rightarrow \Lambda _c(2625)\mu \,\bar{\nu }_\mu ]\)  

\(0.55_{0.18}^{+0.23}(\sigma _1)^{+0.19}_{0.15} (\bar{\Lambda }')\)  \(4.8\pm 2.4 (\sigma _1)^{+1.3}_{1.1} (\bar{\Lambda }')\)  \(0.38_{0.08}^{+0.09}(\sigma _1)\)  \(3.5_{1.2}^{+1.3}(\sigma _1)\)  
\(R[\Lambda _c(2595)]\)  \(R[\Lambda _c(2625)]\)  \(\Gamma ^{1/2}_\mathrm{sl;\, \tau }/ \Gamma ^{3/2}_\mathrm{sl;\, \tau }\)  \(R[\Lambda _c(2595)]\) [17]  \(R[\Lambda _c(2625)]\) [17] 
\(0.11_{0.01}^{+0.04}\)  \(0.11_{0.01}^{+0.02}\)  \(1.5^{+1.2}_{0.8}\)  \(0.13\pm 0.03\)  \(0.11\pm 0.02\) 
5.2 \(\Lambda _b \rightarrow \Lambda _c(2595)\tau \,\bar{\nu }_\tau \) and \(\Lambda _b \rightarrow \Lambda _c(2625)\tau \,\bar{\nu }_\tau \) decays
We have used Eq. (26) to compute \(\Gamma [\Lambda _b \rightarrow \Lambda _c(2595)\tau \,\bar{\nu }_\tau ]\) and \(\Gamma [\Lambda _b \rightarrow \Lambda _c(2625)\tau \,\bar{\nu }_\tau ]\), assuming that the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) form the lowestlying \(j_q^P=1^\) HQSS doublet, and have taken the \(\mathcal{O}(1/m_c)\) improved form factors given in Eq. (32). Therefore, spin symmetry in the \(b\)quark sector is conserved, which implies neglecting terms of order \(\Lambda _\mathrm{QCD}/m_b\). This is an excellent approximation, and we reproduce within a 5% the \(\Lambda _c(2595)\) differential and integrated rates reported in Ref. [14]. The approximation works even better for the \(\Lambda _c(2625)\), and moreover it leads to simple expressions for the \(\omega \)differential widths, including full finitelepton mass contributions that are necessary for testing LFU. Note that the calculations of Ref. [14] were made in the \(m_\ell \rightarrow 0\) limit.
Predictions for semileptonic \(\tau \)decays are relatively stable against the uncertainties on the \(\mathcal{O}(1/m_c)\) corrections, because in this case \(\omega _\mathrm{max}\sim 1.2\), and the largest contributions to the integrated width come from regions relatively close to zero recoil (see blue solid line of Fig. 2). However, there are still some uncertainties associated with the lack of information about the form factor \(\sigma _1(\omega ) \), although they are significantly smaller than those shown in Table 2 for the case of massless leptons. The \(\sigma _1\) term produces, also for \(\tau \)decays, opposite effects for \(\Lambda _c(2595)\) or \(\Lambda _c(2625)\) final states (see Eqs. (32) and (33)). Uncertainties partially cancel in the \(R[\Lambda _c(2595)]\) and \(R[\Lambda _c(2625)]\) ratios, which are predicted in Table 4 with moderate errors. We expect these ratios to be comprised in the intervals [0.10, 0.15] and [0.10, 0.13], respectively. These estimates compare rather well with those obtained in the covariant confined quark model employed in Ref. [17].
Next we discuss the \(\Gamma ^{1/2}_\mathrm{sl;\, \tau }/ \Gamma ^{3/2}_\mathrm{sl;\, \tau }\) ratio, for which theoretical errors are larger. The central value of this ratio compares rather well with that quoted in Table 2 for light leptons (\(\mu \) or e), though its errors for the \(\tau \) mode are slightly smaller.
The \(\Gamma ^{1/2}_\mathrm{sl;\, \tau }/ \Gamma ^{3/2}_\mathrm{sl;\, \tau }\) ratio would drastically change if the final charmed baryons turned out to be predominantly hadronic molecules. In that situation, we would obtain the same values as in Table 2 from the gG factors compiled in Table 3. We should point out that because the available phase space is smaller for the \( \tau \) mode, the decay most likely occurs near the zerorecoil point where the approximations that lead to the quotient of gG factors in Eq. (37) are more precise. The predicted ratios would depend on the molecular scheme, and on the member of the double pole structure of the \(\Lambda _c(2595)\) involved in the decay. However, in all cases, we would obtain values below 0.5, at least onesigma away from the predictions collected in Table 4, based on the hypothesis that the \(\Lambda _c(2595)\) and \(\Lambda _c(2595)\) form the lowestlying \(1^\) HQSS multiplet of excited charmbaryons. This latter picture also discards the existence of a second \(J^P=1/2^\) (broad) resonance in the 2.6 GeV region.
It is not clear how the \(R[\Lambda _c(2595)]\) and \(R[\Lambda _c(2625)]\) ratios would be affected if any of the resonances has a large molecular component, since this will also affect the decay widths into light leptons that appear in the denominators of these ratios. Therefore, one might think that they would not be significantly modified with respect to the values given in Table 4, that mostly account for the reduction of phase space. Nevertheless, it is difficult to be more quantitative. However, \(R[\Lambda _c(2595)]\) may be affected by a new source of potentially large systematic errors, if in the \(\tau \) and \(\mu \) or e modes, the same \(\Lambda _c(2595)\) molecular state is not observed. This confusion would produce large numerical variations that would suggest false violations of LFU.
Finally, in Table 5 we collect several predictions [17, 118, 119, 120, 121, 122] of the LFU ratios for the \(\Lambda _b\) semileptonic decay into the groundstate \(\Lambda _c\) \((1/2^+)\). Comparing the ratios of Tables 4 and 5, we see that \(R[\Lambda _c]\) is predicted to be significantly larger than \(R[\Lambda _c^*]\). (Note, however, that the result of Ref. [122] is considerably smaller than those given by the other authors.)
6 Summary
In this work, we have studied the \(\Lambda _b \rightarrow \Lambda _c^*\ell \bar{\nu }_\ell \) and \(\Lambda _b \rightarrow \Lambda _c^*\pi ^\) \([\Lambda _c^*=\Lambda _c(2595)\) and \(\Lambda _c(2625)]\) decays, paying special attention to the implications that can be derived from HQSS. We have critically reviewed different molecular descriptions of these charm excited baryons, and have discussed in detail the main features of those schemes that predict a twopole pattern for the \(\Lambda _c(2595)\), in analogy to the case of the similar \(\Lambda (1405)\) resonance in the strange sector.
We have calculated the ratios \(\Gamma (\Lambda _b\rightarrow \Lambda _c(2595)\pi ^)/\Gamma (\Lambda _b\rightarrow \Lambda _c(2625)\pi ^)\) and \(\Gamma (\Lambda _b\rightarrow \Lambda _c(2595)\, \ell \,\bar{\nu }_\ell )/ \Gamma (\Lambda _b\rightarrow \Lambda _c(2625)\, \ell \,\bar{\nu }_\ell )\), and have shown that molecular schemes are very sensitive to the interference and relative weights of the DN and \(D^*N\) contributions, as firstly pointed out in Refs. [18, 19]. Actually, we have rederived some of the results of these latter works using a manifest Lorentz and HQSS invariant formalism. In this context, we have argued that future accurate measurements of the above ratios will shed light on the nature of the \(\Lambda _c(2595)\), allowing us to address issues as the existence of two poles or the importance of the \(D^*N\) channel in the formation of the resonance(s).
We have also investigated the LFU ratios \(R[\Lambda _c^*] = \mathcal{B}(\Lambda _b \rightarrow \Lambda _c^* \tau \,\bar{\nu }_\tau )/\mathcal{B}(\Lambda _b \rightarrow \Lambda _c^* \mu \,\bar{\nu }_\mu )\). We have computed \(\Gamma [\Lambda _b \rightarrow \Lambda _c(2595)\tau \,\bar{\nu }_\tau ]\) and \(\Gamma [\Lambda _b \rightarrow \Lambda _c(2625)\tau \,\bar{\nu }_\tau ]\) assuming that the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) form the lowestlying \(j_q^P=1^\) HQSS doublet, and have taken \(\mathcal{O}(1/m_c)\) improved form factors [14]. We have used a scheme that preserves spinsymmetry in the \(b\)quark sector, which implies neglecting corrections of order \(\Lambda _\mathrm{QCD}/m_b\). This is an excellent approximation that leads to simple expressions for the \(\omega \)differential widths, including full finitelepton mass contributions that are necessary for testing LFU.
Finally, we have pointed out that the \(R[\Lambda _c(2595)]\) ratio may be affected by a new source of potentially large systematic errors if there are two \(\Lambda _c(2595)\) poles.
At the LHC, a large number \(\Lambda _b\) baryons are produced, and the LHCb collaboration has reported large samples of \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) baryons in its semileptonic decays. Hence, there are good prospects that LHCb can measure in the near future some of the ratios discussed in this work.
Footnotes
 1.
The lowestlying \(\rho \)mode, \((\ell _\lambda =0,\ell _\rho =1)\) gives rise to two \(\frac{1}{2}^\) and also two \(\frac{3}{2}^\) multiplets of \(\Lambda ^*_c\)’s, together with an additional \(\frac{5}{2}^\) \(\Lambda _c\)excited state, significantly higher in the spectrum [8]. Note that the isoscalar light diquark could have \(0^\), \(1^\) and \(2^\) quantumnumbers, resulting from the coupling of the spin, \(S_q=1\), and the orbitalangular momentum, \(\ell _\rho =1\), of the light quarks. In the heavy quark limit all the baryons with the same light diquark \(j_q^P\) configuration will be degenerate [9].
 2.
Note that, in principle, both \(j_q=0^\) and \(j_q=1^\) configurations can couple with the spin (\(S_Q=\frac{1}{2}\)) of the charm quark to give a total \(J^P = \frac{1}{2}^\) for the \(\Lambda _c(2595)\).
 3.
Since the spinparity of the \(\Lambda _c(2625)\) is \(\frac{3}{2}^\) and it is the lowestlying state with these quantum numbers, one should expect the total angular momentum and parity of the ldof in the \(\Lambda _c(2625)\) to be \(1^\).
 4.
The same type of ideas were extended in Ref. [20] to the semileptonic and one pion decays of the \(\Xi _b^\) baryons into \(\Xi _c^*\) resonances, analogs of the \(\Lambda _c(2595)\) and \(\Lambda _c(2625)\) states in the charmstrange sector.
 5.
One narrow state situated below the \(\bar{K} N\) threshold and with a small coupling to the open \(\pi \Sigma \) channel, and a second state much wider because its large coupling to the open \(\pi \Sigma \) channel.
 6.
This is in gross features also the scheme used in the previous works on the \(\Lambda (1405)\) and \(\Lambda (1520)\), and in most of the studies leading to hadronmolecular interpretations of many other resonances.
 7.
It is also worth mentioning Ref. [61], where the properties of the \(\Lambda _c(2595)\) are discussed in the limit of large number of colors (\(N_c\)), within several schemes. The \(N_c\gg 3\) behaviour of the resonance properties (mass, width, couplings, etc.) puts constrains on its possible dynamical origin, since the importance of the unitary loops involving Goldstone bosons decreases as \(N_c\) grows [62, 63, 64, 65].
 8.
Note that the order baryonmeson, instead of meson–baryon, is used in Refs. [66, 67]. This induces a minus sign for off diagonal elements involving the \(D^*N\) pair in the \(J=1/2\) sector. In addition, there exists a minus sign of difference between the conventions of [66, 67] and those adopted here for the \(\Sigma _c^*\).
 9.
In the approach of Refs. [66, 67] sizable flavor symmetry breaking terms are included. Actually, the symmetrypattern exhibited by the reduced matrix elements in Eq. (7) is modified, by computing the function f(s) using physical hadron masses and decay constants (see for instance, Eq. (7) of Ref. [67]). This induces mostly SU(4)flavor breaking corrections, since the charmedhadrons masses and decay constants follow in good approximation the HQSSpredictions, which do not significantly alter the discussion that follows.
 10.
However, the previous discussion has allowed us to reasonably identify the dominant one in each case. The existence of a certain mixing is out of doubt, thus for instance, the narrow state can decay into \(\pi \Sigma _c\) through its \(1^\) small component.
 11.
We should also point out that the \(D^*N\rightarrow \pi \Sigma _c\) transition in the \(J^P=1/2^\) sector is also set to zero in [73]. This is also inconsistent with HQSS, since this symmetry relates this off diagonal term of the interaction with the \( DN \rightarrow \pi \Sigma _c \) one (a factor \( 1/\sqrt{3}\)).
 12.
 13.
The sum over the initial and final polarizations in the definition of the hadronic tensor in Eq. (15) can be written as trace in the Dirac space, with the help of the spin 1/2 and 3/2 projectors. These latter operators are \(u(v)\,\bar{u}(v) = (1+\not v)/2\) and \(u^\rho (v)\,\bar{u}^\lambda (v) =\left( g^{\rho \lambda }+v^\rho \,v^\lambda + (\gamma ^\rho +v^\rho )(\gamma ^\lambda v^\lambda )/3 \right) (1+\not v)/2\).
 14.
A semileptonic baryonic transition is unnatural if it involves transitions between tensor (\(0^+, 1^, 2^+, \dots \)) to pseudotensor (\(0^, 1^+, 2^, \dots \)), or viceversa, \(j_q^P\) ldof quantum numbers.
 15.
Note that for the \(1/2^\) member of the \(j_q^P=1^\) multiplet, we have \(v_\lambda \left[ (\gamma ^\lambda + v'^\lambda \right) \gamma _5 u^{1/2}_c(v')]^\dagger \gamma ^0 = \bar{u}_c^{1/2}(v')\left( \not v\omega \right) \gamma _5\) and \(\gamma _5\gamma ^\mu (1\gamma _5)=\gamma ^\mu (1\gamma _5) \).
 16.
We are assuming that \(G_{D^*N}\) is the same both for \(J=1/2\) and \(J=3/2\). This is correct as long as the renormalization of the UV divergences of this loop function does not depend on the angular momentum, as in the \(\mathrm{SU(6)}_\mathrm{lsf} \times \mathrm{SU(2)}_\mathrm{HQSS}\) and ELHG models of Refs. [66, 67, 73], respectively.
Notes
Acknowledgements
We warmly thank E. Oset for useful discussions. R.P. Pavao wishes to thank the program Santiago Grisolia of the Generalitat Valenciana. This research has been supported by the Spanish Ministerio de Ciencia, Innovación y Universidades and European FEDER funds under Contracts FIS201784038C21P and SEV20140398. S. Sakai acknowledges the support by NSFC and DFG through funds provided to the SinoGerman CRC110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No. 11621131001), by the NSFC (Grant Nos. 11747601 and 11835015), by the CAS Key Research Program of Frontier Sciences (Grant No. QYZDBSSWSYS013), by the CAS Key Research Program (Grant No. XDPB09) and by the CAS President’s International Fellowship Initiative (PIFI) (No. 2019PM0108).
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