# Is the $\Lambda_{c}(2625)^{+}$ the heavy quark spin symmetry partner of the
$\Lambda_{c}(2595)^{+}$?

Meng-Lin Du<EMAIL_ADDRESS>Instituto de Física Corpuscular (centro
mixto CSIC-UV),
Institutos de Investigación de Paterna, Apartado 22085, 46071, Valencia, Spain
Eliecer Hernández<EMAIL_ADDRESS>Departamento de Física Fundamental e
IUFFyM, Universidad de Salamanca,
Plaza de la Merced s/n, E-37008 Salamanca, Spain  Juan Nieves
<EMAIL_ADDRESS>Instituto de Física Corpuscular (centro mixto CSIC-UV),
Institutos de Investigación de Paterna, Apartado 22085, 46071, Valencia, Spain

###### Abstract

We use a $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ heavy quark effective
theory scheme, where only $\mathcal{O}(\Lambda_{\text{QCD}}/m_{b})$ and
perturbative QCD short distance corrections are neglected, to study the matrix
elements of the scalar, pseudoscalar, vector, axial-vector and tensor currents
between the $\Lambda_{b}$ ground state and the odd parity charm
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ resonances. We show that
in the near-zero recoil regime, the scheme describes reasonably well, taking
into account uncertainties, the results for the 24 form-factors obtained in
lattice QCD (LQCD) just in terms of only 4 Isgur-Wise (IW) functions. We also
find some support for the possibility that the $\Lambda_{c}(2595)^{+}$ and
$\Lambda_{c}(2625)^{+}$ resonances might form a heavy-quark spin symmetry
(HQSS) doublet. However, we argue that the available LQCD description of these
two resonances is not accurate enough to disentangle the possible effects of
the $\Sigma_{c}\pi$ and $\Sigma_{c}^{*}\pi$ thresholds, located only a few MeV
above their position, and that it cannot be ruled out that these states are
not HQSS partners. Finally, we study the ratio
$\frac{d\Gamma[\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}\ell\bar{\nu}_{\ell}]/dq^{2}}{d\Gamma[\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}\ell\bar{\nu}_{\ell}]/dq^{2}}$
of the Standard Model differential semileptonic decay widths, with $q^{\mu}$
the momentum transferred between the initial and final hadrons. We provide a
natural explanation for the existence of large deviations, near the zero
recoil, of this ratio from 1/2 (value predicted in the infinite heavy quark
mass limit, assuming that the $\Lambda_{c,1/2^{-}}^{*}$ and
$\Lambda_{c,3/2^{-}}^{*}$ are the two members of a HQSS doublet) based on
S-wave contributions to the $\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}$ decay
amplitude driven by a sub-leading IW function.

##  I Introduction

Heavy quark symmetry plays an important role in our understanding of low-
energy strong interactions and the classification of the heavy-light hadronic
spectrum. In the infinite heavy quark mass limit ($m_{Q}\to\infty$), the
degrees of freedom of the infinitely massive heavy quark decouple from the
light quark ones, and hence the heavy and light degrees of freedom are
separately conserved. The dynamics of hadrons containing a heavy quark is
blind to the flavour and spin of the heavy quark, with the latter exhibiting
an SU(2)Q pattern known as the heavy-quark spin symmetry. Light degrees of
freedom (ldof) with spin-parity $j_{\rm ldof}^{P}$ yield to a degenerate
doublet under rotations of the heavy quark spin $s_{Q}$. The quantum numbers
of this doublet are $J^{P}=\left[j_{\rm ldof}\pm\frac{1}{2}\right]^{P}$ since
the spin-parity of the heavy quark is $\frac{1}{2}^{+}$. Note here that the
ldof quantum numbers contain both the orbital angular-momentum and spin parts.

In the real world, HQSS is broken due to the large
($m_{Q}\gg\Lambda_{\text{QCD}}$), but finite mass of the heavy quark, e.g. $c$
and $b$ quarks. These doublets are not exactly degenerate with a hyperfine-
splitting of order $\Lambda_{\text{QCD}}/m_{Q}$. The similar masses of the
isoscalar odd parity resonances $\Lambda_{c}(2595)^{+}$ and
$\Lambda_{c}(2625)^{+}$, with $J^{P}=\frac{1}{2}^{-}$ and $\frac{3}{2}^{-}$
respectively, makes them promising candidates for the lightest charmed baryon
doublet with $j_{\rm ldof}^{P}=1^{-}$. This assignment has been widely adopted
in the literature, see e.g. Refs. [1, 2, 3, 4, 5, 6, 7]. Various constituent
quark models (CQMs) predict a nearly degenerate pair of P-wave charmed
isoscalar baryons $\Lambda_{c}^{*}$ with $J^{P}=\frac{1}{2}^{-}$ and
$\frac{3}{2}^{-}$, respectively, which masses are close to those of the
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ [8, 9, 10, 3]. Two
different orbital excitations are considered in Ref. [3], driven by the so-
called $\lambda$\- and $\rho$-degrees of freedom. While the former accounts
for the excitation between the heavy quark and the light quark subsystem, the
latter considers the excitation between the two light quarks. Owing to the big
difference between the heavy- and light-quarks masses, the low-lying states
are dominated by the lowest $\lambda$-mode excitation and the mixture with
$\rho$ excitations is small [3]. In this picture, the $\Lambda_{c}(2595)^{+}$
and $\Lambda_{c}(2625)^{+}$ correspond to the HQSS doublet associated to
($\ell_{\lambda}=1$, $\ell_{\rho}=0$) with total ldof spin $s_{\rm ldof}=0$,
leading to $j_{\rm ldof}^{P}=1^{-}$. The predicted decay widths of these
resonances within this scheme are found to be consistent with data [4, 11].

The $\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ have also been
described in hadronic-molecular models, as the counterparts of the
$\Lambda(1405)$ and $\Lambda(1520)$ with the strange quark replaced by the
charm one [12, 13, 14, 15, 16, 17, 18, 19, 20]. Since the $\Sigma_{c}$ and
$\Sigma_{c}^{*}$ form the ground $j_{\rm ldof}^{P}=1^{+}$ HQSS doublet, the
quantum numbers of the ldof of the S-wave $\Sigma_{c}\pi$ and
$\Sigma_{c}^{*}\pi$ pairs are $1^{-}$ for both cases. In Ref. [12], the
$\Lambda_{c}(2595)^{+}$ is dynamically generated from the S-wave scattering of
the Goldstone bosons off the $J^{P}=\frac{1}{2}^{+}$ charmed baryon octet. A
similar treatment is performed in Ref. [19], where the masses of the
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ are reproduced as
$\Sigma_{c}\pi$ and $\Sigma_{c}^{*}\pi$ bound states by fine-tuned parameters.
A coupled-channel approach including the $DN$ is studied in Ref. [13] for the
$\Lambda_{c}(2595)^{+}$, and it is extended to the $\Lambda_{c}(2625)^{+}$
sector subsequently in Refs. [14, 15]. HQSS, however, is not respected since
the $D^{*}N$ channel is not considered and the $D$ and $D^{*}$ mesons form the
ground $j_{\rm ldof}^{P}=1/2^{-}$ HQSS doublet in the meson sector. The first
molecular description of the $\Lambda_{c}(2595)^{+}$ and
$\Lambda_{c}(2625)^{+}$ respecting HQSS was provided in Refs. [17, 18], where
the SU(3) Weinberg-Tomozawa chiral Lagrangian is extended to SU(6)${}_{\rm
lsf}\times$SU(2)Q, with lsf standing here for the spin-flavor symmetry in the
light sector. The model is supplemented by a pattern of symmetry breaking
corrections and using the particular renormalization scheme proposed in Refs.
[14, 16]. One $J^{P}=\frac{3}{2}^{-}$ state is dynamically generated by the
$\Sigma_{c}^{*}\pi$-$D^{*}N$ coupled-channel dynamics, which is identified in
[17] with the $\Lambda_{c}(2625)^{+}$, even though its mass is about 40 MeV
larger and it is significantly wider than the physical resonance. Thus within
this scheme, the $\Lambda_{c}(2625)^{+}$ would be the counterpart of the
$\Lambda(1520)$ in the charm sector. Interestingly, the model of Refs. [17,
18] produce two $J^{P}=\frac{1}{2}^{-}$ states generated near the nominal
position of the $\Lambda_{c}(2595)^{+}$. One is narrow and it strongly couples
to $DN$ and especially to $D^{*}N$, with a small mixing with $\Sigma_{c}\pi$.
It is identified with the $\Lambda_{c}(2595)^{+}$ resonance in Refs. [17, 18].
The other $J^{P}=\frac{1}{2}^{-}$ molecular state, however, is quite broad
because of the sizable coupling to the open channel $\Sigma_{c}\pi$. The two
states would be analogous to those forming the two-pole structure of the
$\Lambda(1405)$ [21, 22, 23, 24], where the states couple to $\Sigma\pi$ and
$\bar{K}N$, respectively [24]. A big difference, between the charm and strange
sectors, is that while $D^{*}N$ plays a crucial role in the charm sector
within the scheme of Refs. [17, 18], the $\bar{K}^{*}N$ channel is not
considered in the chiral unitary approaches [21, 22, 23, 24] because of the
large $\bar{K}^{*}$-$\bar{K}$ mass gap.

In Ref. [25], it is stressed that the narrow $J^{P}=\frac{1}{2}^{-}$
$\Lambda_{c}(2595)^{+}$ found in [17, 18] is mostly generated from the
$DN$-$D^{*}N$ coupled-channel dynamics with a dominant $j_{\rm
ldof}^{P}=0^{-}$ configuration. The small coupling of this state to the
$\Sigma_{c}\pi$ channel is then a consequence of HQSS due to its small $j_{\rm
ldof}=1^{-}$ component. The coupling of the broad $J^{P}=\frac{1}{2}^{-}$
state to the $\Sigma_{c}\pi$, however, is larger than those to $DN$ and
$D^{*}N$ and thus dominated by the $j_{\rm ldof}^{P}=1^{-}$ configuration. It
means that the isoscalar $J^{P}=\frac{3}{2}^{-}$ $\Lambda_{c}(2625)^{+}$ state
found in Refs. [17, 18] would be the HQSS partner of the broad
$J^{P}=\frac{1}{2}^{-}$ state with $j_{\rm ldof}^{P}=1^{-}$ [25], instead of
the observed $\Lambda_{c}(2595)^{+}$. A similar two-pole structure for the
$J^{P}=\frac{1}{2}^{-}$ sector is found in Ref. [26] by making use of a SU(4)
flavor extension of the local hidden gauge formalism. In that work, an
additional broad state around 2675 MeV is found in the isoscalar
$J^{P}=\frac{3}{2}^{-}$ sector with the single-channel $\Sigma_{c}^{*}\pi$
Weinberg-Tomozawa interaction, which would not be related to the
$\Lambda_{c}(2625)^{+}$.

The interplay between the $\Sigma_{c}^{(*)}\pi$-$D^{(*)}N$ baryon-meson pairs
and the bare P-wave CQM states has been recently studied, in the framework of
an effective field theory, respecting heavy quark spin and chiral symmetries
[27]. It is shown that the $\Lambda_{c}(2625)^{+}$ should be viewed mostly as
a dressed three-quark state, that is originated from a bare CQM state. The
$\Lambda_{c}(2595)^{+}$, however, should be either dynamically generated by
the chiral $\Sigma_{c}\pi$ interaction, or the result of the $DN$-$D^{*}N$
coupled-channel dynamics with a $j_{\rm ldof}^{P}=0^{-}$ configuration,
depending on the employed renormalization procedure. In any case, these two
resonances would not be HQSS partners [27]. This is because the bare
$J^{P}=\frac{3}{2}^{-}$ CQM state and the $\Sigma_{c}\pi$ threshold are
located extremely close to the $\Lambda_{c}(2625)^{+}$ and
$\Lambda_{c}(2595)^{+}$, respectively, and thus play different roles in each
sector.

HQSS also puts constraints on the form factors of currents containing heavy
quarks, based on the observation that the current $\bar{Q}\Gamma q$
($\bar{Q}\Gamma Q^{\prime}$) transforms as a spinor under SU(2)Q
(SU(2)${}_{Q^{\prime}}$ as well) [28]. In particular in the heavy quark limit,
the semileptonic decay of a $\Lambda_{b}$ into the lowest HQSS-doublet with
$j_{\rm ldof}^{P}=1^{-}$ can be described by a universal leading order Isgur-
Wise (IW) function ($\sigma$) [29]. Besides, at zero recoil, the weak-current
matrix elements between a $\Lambda_{b}$ and any excited charmed baryon vanish
[1]. The form factors for the exclusive semileptonic $\Lambda_{b}$ decays to
the excited $\Lambda_{c}$ baryons were first obtained to order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ in heavy quark effective theory
(HQET) in Ref. [30], and were improved to order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{b})$ in Refs. [1, 5], where the
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ were regarded as the
lowest-lying $j_{\rm ldof}^{P}=1^{-}$ doublet. At order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{Q})$ (here $Q=c$ and $b$), there appear
five additional independent functions ($\sigma_{1}^{(c)}$,
$\phi_{\text{kin}}^{(c)}$, $\phi_{\text{kin}}^{(b)}$,
$\phi_{\text{mag}}^{(c)}$ and $\phi_{\text{mag}}^{(b)}$) and two low energy
constants (LECs) $\bar{\Lambda}$ and ${\bar{\Lambda}}^{\prime}$, following the
notation introduced in Ref. [1]. The LECs (${\bar{\Lambda}}^{(\prime)}$)
denote the energy in the hadron of the light degrees of freedom in the
$m_{Q}\to\infty$ limit. The functions $\phi_{\text{kin}}^{(c)}$ and
$\phi_{\text{kin}}^{(b)}$ can be re-absorbed, neglecting
$\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{Q}^{2})$, in the leading order IW
function $\sigma$, common for both decays. Therefore, to describe the various
form factors of the currents between the $\Lambda_{b}$ and the
$\Lambda_{c,1/2^{-}}^{*}$ and $\Lambda_{c,3/2^{-}}^{*}$ states at order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{Q})$, only three of these sub-leading
functions and the two LECs $\bar{\Lambda}$ and ${\bar{\Lambda}}^{\prime}$ are
needed [30, 25]. Furthermore, if the $\mathcal{O}(\Lambda_{\text{QCD}}/m_{b})$
corrections are neglected, and one only considers up to order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$, all the form factors for the
$\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*},\Lambda_{c,3/2^{-}}^{*}$ transitions
are given in terms of only three independent functions,
$\sigma(\omega),\phi_{\text{kin}}^{(c)}$ and $\sigma_{1}^{(c)}$, and the
$\bar{\Lambda}$ and ${\bar{\Lambda}}^{\prime}$ LECs (see for instance also
Eqs. (31)-(33) of Ref. [25]). The $\Lambda_{\text{QCD}}/m_{b}$ or
$\Lambda_{\text{QCD}}^{2}/m_{c}^{2}$ contributions, not taken into account in
this limit, are expected to be smaller than the theoretical uncertainties
induced by the errors on $({\bar{\Lambda}}-{\bar{\Lambda}}^{\prime})$.

Moreover, the semileptonic form factors between the $\Lambda_{b}$ and a final
$J^{P}=\frac{1}{2}^{-}$ charm-baryon, but with $j_{\rm ldof}^{P}=0^{-}$,
denoted as $\Lambda_{c,1/2^{-}}^{\prime*}$, vanish in the $m_{Q}\to\infty$
limit. The unique non-vanishing correction at order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ comes from the chromomagnetic
operator, which can be described by a universal function with a structure of
the type $\epsilon_{\mu\nu\rho\tau}\sigma^{\mu\nu}v^{\rho}v^{\prime\tau}$ with
$v$ and $v^{\prime}$ the four-velocities of the $\Lambda_{b}$ and
$\Lambda_{c,1/2^{-}}^{\prime*}$, respectively [1]. Thus, the different HQSS
pattern of form factors for the $\Lambda_{b}$ semileptonic decay into a charm
$j_{\rm ldof}^{P}=1^{-}$ doublet or a $j_{\rm ldof}^{P}=0^{-}$ singlet might
be used to test whether the ldof configuration in the $\Lambda_{c}(2595)^{+}$
corresponds mainly to any of these quantum numbers, provided that the decays
are measured.

Given the lack of data for the form-factors of the
$\Lambda_{b}\to\Lambda_{c}(2595)^{+}$ and
$\Lambda_{b}\to\Lambda_{c}(2625)^{+}$ semileptonic decays, the lattice QCD
(LQCD) simulations carried out in Refs. [6, 31] provide valuable information
from first principles111The form factors for the
$\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}$ and
$\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}$ transitions have also been
investigated within the framework of constituent quark-models, see e.g., in
Refs. [2, 32, 33].. (In the second of the references, the exact zero-recoil
rotational symmetry relations among the different form factors are imposed to
ensure the correct behavior of the angular observables). Using LQCD form
factors [34, 35, 36, 37], the line shape of the
$\Lambda_{b}\to\Lambda_{c}\mu^{-}\bar{\nu}_{\mu}$ decay, with $\Lambda_{c}$
the $J^{P}=1/2^{+}$ ground state charm baryon for which $j_{\rm
ldof}^{P}=0^{+}$, has been found to be consistent with the LHCb measurement
[38]. This gives strong support to the LQCD calculation of semileptonic form
factors. Note, however, that the form factors for the
$\Lambda_{b}\to\Lambda_{c}$ and those for the
$\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*},\Lambda_{c,3/2^{-}}^{*}$ transitions
are not related at all, since the ldof in the final charm baryon have
different configurations. In Ref. [1], branching fractions and heavy quark sum
rules for $\Lambda_{b}$ decays to $\Lambda_{c}(2595)^{+}$ and
$\Lambda_{c}(2625)^{+}$ (identified as the HQSS-doublet of $j_{\rm
ldof}^{P}=1^{-}$) within the Standard Model (SM) were evaluated in the large
$N_{c}$ limit of QCD, using the bound state soliton picture.

The relevant expressions for the matrix elements, up to order $\alpha_{s}$ and
$\Lambda_{\text{QCD}}/m_{Q}$ in HQET, to test lepton flavour universality are
provided in Ref. [5]. However in that work, only rough estimates of the form-
factors, obtained from the zero recoil sum rule [39, 40], could be used. New
physics signatures for various $b\to c\ell\bar{\nu}_{\ell}$ four Fermi
interactions in the baryon sector are also investigated in Ref. [7], using a
HQET-based parametrization of the form factors to existing quark model results
[2]. As previously noted also in Ref. [6], the results of Ref. [7] show a
tension between LQCD data and HQET predictions to order $\alpha_{s}$ and
$\Lambda_{\text{QCD}}/m_{Q}$, and points out to the existence of unexpectedly
large HQET-violating terms, potentially large $1/m_{c}^{2}$ corrections near
zero recoil, in the LQCD form-factors. Further studies of this issue are thus
needed, since it is still an open question how well HQSS works for the
semileptonic $\Lambda_{b}\to\Lambda_{c}(2595)^{+}$ and
$\Lambda_{b}\to\Lambda_{c}(2625)^{+}$ transitions, and if the
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ form a HQSS doublet [27].

In this work, we will try to answer some of these questions using the LQCD
form factors recently obtained in Refs. [6, 31]. In Sec. II, we will provide
the explicit definition of the $\Lambda_{b}\to\Lambda_{c}^{*}$ semileptonic
form factors used in this work, and the relations between them and those
obtained in Refs. [6, 31]. We briefly review the form factors in HQET, up to
order $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$, in Sec. III. We test how well
HQSS describes these form factors and perform a detailed numerical analysis in
Sec. IV. Section V includes a brief summary of the main results of this work.
Finally in the Appendices A and B, we provide relations between the different
sets of form-factors considered in this work.

##  II Form factors for $\Lambda_{b}\to\Lambda_{c}^{*}$ semileptonic
transitions

The form factors for the
$\Lambda_{b}\to\Lambda_{c}(2595)^{+}\ell\bar{\nu}_{\ell}$
[$\Lambda_{b}\to\Lambda_{c}(2625)^{+}\ell\bar{\nu}_{\ell}$] semileptonic
transitions parameterize the matrix elements of the local currents
$[\bar{c}(0)\Gamma b(0)]$ between the $\Lambda_{b}$ and the
$\Lambda_{c}(2595)^{+}$ [$\Lambda_{c}(2625)^{+}$] states. Following the
notation of Ref. [1] for the vector and axial-vector currents and extending it
to the scalar, pseudoscalar, and tensor currents, we have for the
$\Lambda_{b}\to\Lambda_{c}(2595)^{+}$

$\displaystyle\langle\Lambda_{c}(2595)^{+}(p^{\prime},s^{\prime})|\bar{c}b|\Lambda_{b}(p,s)\rangle$
$\displaystyle=$ $\displaystyle
d_{S}\bar{u}_{c}(p^{\prime},s^{\prime})\gamma_{5}u_{b}(p,s),$
$\displaystyle\langle\Lambda_{c}(2595)^{+}(p^{\prime},s^{\prime})|\bar{c}\gamma_{5}b|\Lambda_{b}(p,s)\rangle$
$\displaystyle=$ $\displaystyle
d_{P}\bar{u}_{c}(p^{\prime},s^{\prime})u_{b}(p,s),$
$\displaystyle\langle\Lambda_{c}(2595)^{+}(p^{\prime},s^{\prime})|\bar{c}\gamma^{\mu}b|\Lambda_{b}(p,s)\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{c}(p^{\prime},s^{\prime})\left[d_{V_{1}}\gamma^{\mu}+d_{V_{2}}\frac{p^{\mu}}{m_{\Lambda_{b}}}+d_{V_{3}}\frac{p^{\prime\mu}}{m_{\Lambda_{c,1/2}^{*}}}\right]\gamma_{5}u_{b}(p,s),$
$\displaystyle\langle\Lambda_{c}(2595)^{+}(p^{\prime},s^{\prime})|\bar{c}\gamma^{\mu}\gamma_{5}b|\Lambda_{b}(p,s)\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{c}(p^{\prime},s^{\prime})\left[d_{A_{1}}\gamma^{\mu}+d_{A_{2}}\frac{p^{\mu}}{m_{\Lambda_{b}}}+d_{A_{3}}\frac{p^{\prime\mu}}{m_{\Lambda_{c,1/2}^{*}}}\right]u_{b}(p,s),$
$\displaystyle\langle\Lambda_{c}(2595)^{+}(p^{\prime},s^{\prime})|\bar{c}\sigma^{\mu\nu}\gamma_{5}b|\Lambda_{b}(p,s)\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{c}(p^{\prime},s^{\prime})\Big{[}i\frac{d_{T_{1}}}{m_{\Lambda_{b}}^{2}}(p^{\mu}p^{\prime\nu}-p^{\nu}p^{\prime\mu})+i\frac{d_{T_{2}}}{m_{\Lambda_{b}}}(\gamma^{\mu}p^{\nu}-\gamma^{\nu}p^{\mu})$
(1)
$\displaystyle+i\frac{d_{T_{3}}}{m_{\Lambda_{b}}}(\gamma^{\mu}p^{\prime\nu}-\gamma^{\nu}p^{\prime\mu})+d_{T_{4}}\sigma^{\mu\nu}\Big{]}u_{b}(p,s),$

where $m_{\Lambda_{c,1/2}^{*}}$ is the mass of $\Lambda_{c}(2595)^{+}$, and
$u_{b}(p,s)$ and ${u}_{c}(p^{\prime},s^{\prime})$ are the spinors of
$\Lambda_{b}$ and $\Lambda_{c}(2595)^{+}$ baryons, respectively ($p$ and
$p^{\prime}$ are four-momenta, while $s$ and $s^{\prime}$ are spin indices).
The form-factors $d_{S,P,V_{i},A_{i},T_{i}}$ are scalar functions of
$q^{2}=(p-p^{\prime})^{2}$, or equivalently
$\omega=(m_{\Lambda_{b}}^{2}+m_{\Lambda_{c,1/2}^{*}}^{2}-q^{2})/(2m_{\Lambda_{b}}m_{\Lambda_{c,1/2}^{*}})$.

The form factor decomposition for the
$\langle\Lambda_{c}(2595)^{+}(p^{\prime},s^{\prime})|\bar{c}\sigma^{\mu\nu}b|\Lambda_{b}(p,s)\rangle$
matrix element can be straightforwardly obtained from that of the
$\sigma^{\mu\nu}\gamma_{5}$ operator by making use of
$\sigma^{\mu\nu}\gamma_{5}=-\frac{i}{2}\epsilon^{\mu\nu\lambda\rho}\sigma_{\lambda\rho}$,
with the convention $\epsilon_{0123}=+1$.

Likewise, the $\Lambda_{b}\to\Lambda_{c}(2625)^{+}$ matrix elements can be
parameterized as

$\displaystyle\langle\Lambda_{c}(2625)^{+}(p^{\prime},s^{\prime})|\bar{c}b|\Lambda_{b}(p,s)\rangle$
$\displaystyle=$ $\displaystyle
l_{S}\frac{p^{\lambda}}{m_{\Lambda_{b}}}\bar{u}_{c,\lambda}(p^{\prime},s^{\prime})u_{b}(p,s),$
$\displaystyle\langle\Lambda_{c}(2625)^{+}(p^{\prime},s^{\prime})|\bar{c}\gamma_{5}b|\Lambda_{b}(p,s)\rangle$
$\displaystyle=$ $\displaystyle
l_{P}\frac{p^{\lambda}}{m_{\Lambda_{b}}}\bar{u}_{c,\lambda}(p^{\prime},s^{\prime})\gamma_{5}u_{b}(p,s),$
$\displaystyle\langle\Lambda_{c}(2625)^{+}(p^{\prime},s^{\prime})|\bar{c}\gamma^{\mu}b|\Lambda_{b}(p,s)\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{c,\lambda}(p^{\prime},s^{\prime})\Bigg{[}\frac{p^{\lambda}}{m_{\Lambda_{b}}}\left(l_{V_{1}}\gamma^{\mu}+l_{V_{2}}\frac{p^{\mu}}{m_{\Lambda_{b}}}+l_{V_{3}}\frac{p^{\prime\mu}}{m_{\Lambda_{c,3/2}^{*}}}\right)$
$\displaystyle+l_{V_{4}}g^{\lambda\mu}\Bigg{]}u_{b}(p,s),$
$\displaystyle\langle\Lambda_{c}(2625)^{+}(p^{\prime},s^{\prime})|\bar{c}\gamma^{\mu}\gamma_{5}b|\Lambda_{b}(p,s)\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{c,\lambda}(p^{\prime},s^{\prime})\Bigg{[}\frac{p^{\lambda}}{m_{\Lambda_{b}}}\left(l_{A_{1}}\gamma^{\mu}+l_{A_{2}}\frac{p^{\mu}}{m_{\Lambda_{b}}}+l_{A_{3}}\frac{p^{\prime\mu}}{m_{\Lambda_{c,3/2}^{*}}}\right)$
$\displaystyle+l_{A_{4}}g^{\lambda\mu}\Bigg{]}\gamma_{5}u_{b}(p,s),$
$\displaystyle\langle\Lambda_{c}(2625)^{+}(p^{\prime},s^{\prime})|\bar{c}\sigma^{\mu\nu}b|\Lambda_{b}(p,s)\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{c,\lambda}(p^{\prime},s^{\prime})\bigg{[}\frac{p^{\lambda}}{m_{\Lambda_{b}}}\Big{(}i\frac{l_{T_{1}}}{m_{\Lambda_{b}}^{2}}(p^{\mu}p^{\prime\nu}-p^{\nu}p^{\prime\mu})+i\frac{l_{T_{2}}}{m_{\Lambda_{b}}}(\gamma^{\mu}p^{\nu}-\gamma^{\nu}p^{\mu})$
(2)
$\displaystyle+i\frac{l_{T_{3}}}{m_{\Lambda_{b}}}(\gamma^{\mu}p^{\prime\nu}-\gamma^{\nu}p^{\prime\mu})+l_{T_{4}}\sigma^{\mu\nu}\Big{)}+il_{T_{5}}(g^{\lambda\mu}\gamma^{\nu}-g^{\lambda\nu}\gamma^{\mu})$
$\displaystyle+i\frac{l_{T_{6}}}{m_{\Lambda_{b}}}(g^{\lambda\mu}p^{\nu}-g^{\lambda\nu}p^{\mu})\bigg{]}u_{b}(p,s),$

where $m_{\Lambda_{c,3/2}^{*}}$ is now the mass of $\Lambda_{c}(2625)^{+}$,
$u_{c,\lambda}(p^{\prime},s^{\prime})$ is the Rarita-Schwinger spinor for the
spin-$\frac{3}{2}$ $\Lambda_{c}(2625)^{+}$, which satisfies
$p^{\prime\lambda}u_{c,\lambda}=\gamma^{\lambda}u_{c,\lambda}=0$, and the
form-factors $l_{S,P,V_{i},A_{i},T_{i}}$ are scalar functions of
$q^{2}=(p-p^{\prime})^{2}$, or $\omega$, which is now expressed in terms of
$m_{\Lambda_{c,3/2}^{*}}$. Note there is also one more independent structure
$i\frac{l_{T_{7}}}{m_{\Lambda_{b}}}(g^{\lambda\mu}p^{\prime\nu}-g^{\lambda\nu}p^{\prime\mu})$
in the form factor decomposition of the tensor operator, which can not be
eliminated simply by the equation of motion or transversality conditions.
However, it is shown in Ref. [7] that the combination of operators

$\mathcal{K}_{\mu\nu}=\bar{u}_{c}^{\lambda}\Big{[}v_{\lambda}\Big{(}\sigma_{\mu\nu}-i(v^{\prime}_{\mu}\gamma_{\nu}-v^{\prime}_{\nu}\gamma_{\mu})\Big{)}+i\Big{(}g_{\lambda\mu}\big{[}(\omega+1)\gamma_{\nu}-{v_{\nu}}-v^{\prime}_{\nu}\big{]}-g_{\lambda\nu}\big{[}(\omega+1)\gamma_{\mu}-{v_{\mu}}-v^{\prime}_{\mu}\big{]}\Big{)}\Big{]}u_{b}$
(3)

does not contribute to physical amplitudes. Here we have introduced the
notations $v_{\mu}=p_{\mu}/m_{\Lambda_{b}}$,
$v_{\mu}^{\prime}=p_{\mu}^{\prime}/m_{\Lambda_{c,3/2}^{*}}$ and $\omega=v\cdot
v^{\prime}$, determined by $q^{2}$. Thus, we can set to zero the contribution
of $l_{T_{7}}$ to the physical amplitudes by redefining $l_{T_{i}}$ for
$i=3,4,5,6$.

On the other hand, as in the case of the $\Lambda_{c}(2595)^{+}$, the
$\langle\Lambda_{c}(2625)^{+}(p^{\prime},s^{\prime})|\bar{c}\sigma^{\mu\nu}\gamma_{5}b|\Lambda_{b}(p,s)\rangle$
form-factors can be obtained from the tensor ones introduced in the
decomposition of the matrix element of the $\sigma^{\mu\nu}$ operator.

The helicity form factors for $\Lambda_{b}\to\Lambda_{c}(2595)^{+}$ and
$\Lambda_{b}\to\Lambda_{c}(2625)^{+}$ determined in the LQCD simulation
carried out in Refs. [31, 6] are linear combinations of those introduced
above, and the relation between both sets of form-factors is given in Appendix
A.

##  III $\Lambda_{b}\to\Lambda_{c}^{*}$ form factors and heavy-quark effective
theory

In this section, we will provide the form factors for the semileptonic decay
of the $\Lambda_{b}$ into the members of the $j_{\rm ldof}^{P}=1^{-}$ HQSS
doublet $\Lambda_{c,1/2^{-}}^{*}$ and $\Lambda_{c,3/2^{-}}^{*}$ in HQET up to
order $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ and neglecting QCD short range
logarithms [7].

In the infinite heavy quark limit ($m_{Q}\to\infty$), the form factors $d_{i}$
and $l_{i}$ in Eqs. (II) and (II) for the $\Lambda_{b}$ decay into the two
members ($\Lambda_{c}^{*}$) of the $j_{\rm ldof}^{P}=1^{-}$ HQSS-doublet can
be described by a universal IW function [29]. In this limit, the ground state
$\Lambda_{b}$ is a HQSS-singlet with $j_{\rm ldof}^{P}=0^{+}$, and thus can be
described by a Dirac spinor $u_{b}(v)$, with $v$ the velocity of the
$\Lambda_{b}$, satisfying the condition $\not{v}u_{b}(v)=u_{b}(v)$ [41]. For
the $j_{\rm ldof}^{P}=1^{-}$ doublet with velocity $v^{\prime}$, the ldof are
represented by a vector $A^{\mu}$ subject to the transversality condition
$v^{\prime}\cdot A=0$. Then, the $\Lambda_{c}^{*}$ doublet can be introduced
by the multiplet spinor

$\displaystyle\mathcal{U}^{\mu}_{c}(v^{\prime})=A^{\mu}u_{h}(v^{\prime}),$ (4)

where the spinor $u_{h}$ describes the heavy quark $c$ obeying
$\not{v}^{\prime}u_{h}=u_{h}$. Note that $\mathcal{U}^{\mu}_{c}(v^{\prime})$
is not an irreducible representation under the Lorentz group, instead it
contains both spin $3/2=(1+1/2)$ and $1/2=(1-1/2)$ components,

$\displaystyle\mathcal{U}^{\mu}_{c}(v^{\prime})=u_{c}^{\mu}(v^{\prime})+\frac{1}{\sqrt{3}}(\gamma^{\mu}+v^{\prime\mu})\gamma_{5}u_{c}(v^{\prime}),$
(5)

where $u_{c}^{\mu}(v^{\prime})$ and $u_{c}(v^{\prime})$ are the Rarita-
Schwinger and Dirac spinors for $J^{P}=\frac{3}{2}^{-}$ and $\frac{1}{2}^{-}$
states, respectively. Note that
$v^{\prime}\cdot\mathcal{U}_{c}(v^{\prime})=0$,
$\not{v}^{\prime}\mathcal{U}_{c}(v^{\prime})=\mathcal{U}_{c}(v^{\prime})$, and
$\gamma_{\mu}u_{c}^{\mu}(v^{\prime})=0$. Then one can easily obtain that in
the $m_{Q}\to\infty$ limit, the most general form for the matrix element
respecting HQSS is (i.e., being invariant under arbitrary separate rotations
of the $b$\- and $c$-quarks spins [25])

$\displaystyle\langle\Lambda_{c}^{*};j_{\rm
ldof}^{P}=1^{-}|\bar{h}_{v^{\prime}}^{(c)}\Gamma
h_{v}^{(b)}|\Lambda_{b}\rangle=\sigma(\omega)v_{\lambda}\bar{\mathcal{U}}_{c}^{\lambda}(v^{\prime})\Gamma
u_{b}(v).$ (6)

where $h_{v}^{(Q)}$ is the heavy quark field in HQET, $\Gamma$ is a Dirac
matrix, and $\sigma(\omega)$ is the dimensionless leading IW function. Using
Eqs. (5) and (6), one finds

$\displaystyle\langle\Lambda_{c,1/2^{-}}^{*}|\bar{h}_{v^{\prime}}^{(c)}\Gamma
h_{v}^{(b)}|\Lambda_{b}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\sigma(\omega)\bar{u}_{c}(\not{v}-\omega)\gamma_{5}\Gamma
u_{b},$
$\displaystyle\langle\Lambda_{c,3/2^{-}}^{*}|\bar{h}_{v^{\prime}}^{(c)}\Gamma
h_{v}^{(b)}|\Lambda_{b}\rangle$ $\displaystyle=$
$\displaystyle\sigma(\omega)v_{\lambda}\bar{u}_{c}^{\lambda}\Gamma u_{b}.$ (7)

and thus, it is straightforward to obtain [25] the $d_{i}$ and $l_{i}$ form-
factors in terms of $\sigma(\omega)$.

At order $\mathcal{O}(\Lambda_{\text{QCD}}/m_{Q})$ , there are corrections
originating from the matching of the $b\to c$ flavor changing current onto the
effective theory (HQET) and from order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{Q})$ effective Lagrangian [1].
Considering for simplicity only $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$
contributions, this is to say keeping invariance under rotations of the spin
of the $b$ quark, we have first for the current corrections [42]

$\displaystyle\bar{c}\Gamma
b=\bar{h}_{v^{\prime}}^{(c)}\left(\Gamma-\frac{i}{2m_{c}}\overleftarrow{\not{D}}\Gamma\right)h_{v}^{(b)},$
(8)

where $D$ is the gauge covariant derivative. The charm quark next-leading
order effective Lagrangian contains the kinetic energy and the chromomagnetic
terms [42]

$\displaystyle\delta\mathcal{L}_{v}^{\prime(c)}=\frac{1}{2m_{c}}\left(\bar{h}_{v^{\prime}}^{(c)}(iD_{\perp})^{2}+\frac{g_{s}}{2}\sigma\cdot
G\right)h_{v^{\prime}}^{(c)}.$ (9)

The operator that appears in the correction of Eq. (8) can be parameterized as
[30]

$\displaystyle\bar{h}_{v^{\prime}}^{(c)}i\overleftarrow{D}_{\lambda}\Gamma
h_{v}^{(b)}=\left[\sigma_{1}^{(c)}(\omega)v_{\mu}v_{\lambda}+\sigma_{2}^{(c)}(\omega)v_{\mu}v^{\prime}_{\lambda}+\sigma_{3}^{(c)}(\omega)g_{\mu\lambda}\right]\bar{\Psi}_{v^{\prime}}^{\mu\,(c)}\Gamma\Psi_{v}^{(b)}.$
(10)

with $\Psi_{v^{\prime}}^{\mu\,(c)}$ and $\Psi_{v}^{(b)}$ the HQET fields which
destroy the spin 1/2 and spin 3/2 members of the charm $j_{\rm
ldof}^{P}=1^{-}$ doublet and the ground state $\Lambda_{b}$, respectively.
Multiplying Eq. (10) by $v^{\prime\lambda}$ and making use of the equation of
motion $(v^{\prime}\cdot D)h_{v^{\prime}}^{(c)}=0$, one obtains
$\sigma_{2}^{(c)}(\omega)=-\omega\sigma_{1}^{(c)}(\omega)$. Additionally,
translational invariance allows to write [1]

$\sigma_{3}^{(c)}(\omega)=({\bar{\Lambda}}-\omega{\bar{\Lambda}}^{\prime})\sigma(\omega)+(\omega^{2}-1)\sigma_{1}^{(c)}(\omega)$
(11)

Then one has

$\displaystyle\langle\Lambda_{c}^{*};j_{\rm
ldof}^{P}=1^{-}|\bar{h}_{v^{\prime}}^{(c)}i\overleftarrow{\not{D}}\Gamma
h_{v}^{(b)}|\Lambda_{b}\rangle=\bar{\mathcal{U}}_{c}^{\mu}(v^{\prime})\left[v_{\mu}\left(\not{v}-\omega\right)\sigma_{1}^{(c)}(\omega)+\gamma_{\mu}\sigma_{3}^{(c)}(\omega)\right]{\Gamma}u_{b}(v),$
(12)

and thus

$\displaystyle\langle\Lambda_{c,1/2^{-}}^{*}|\bar{h}_{v^{\prime}}^{(c)}i\overleftarrow{\not{D}}\Gamma
h_{v}^{(b)}|\Lambda_{b}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\bar{u}_{c}\left[(\omega^{2}-1)\sigma_{1}^{(c)}(\omega)-3\sigma_{3}^{(c)}(\omega)\right]\gamma_{5}\Gamma
u_{b},$
$\displaystyle\langle\Lambda_{c,3/2^{-}}^{*}|\bar{h}_{v^{\prime}}^{(c)}i\overleftarrow{\not{D}}\Gamma
h_{v}^{(b)}|\Lambda_{b}\rangle$ $\displaystyle=$
$\displaystyle\sigma_{1}^{(c)}(\omega)v_{\lambda}\bar{u}_{c}^{\lambda}(\not{v}-\omega)\Gamma
u_{b}.$ (13)

In addition, the correction $\phi_{\text{kin}}^{(c)}(\omega)$ from the charm
kinetic energy operator
$\bar{h}_{v^{\prime}}^{(c)}(iD_{\perp})^{2}h_{v^{\prime}}^{(c)}$ in Eq. (9)
respects HQSS and hence it enters in the same way as the leading IW function
$\sigma(\omega)$, and it simply re-normalizes the latter [30, 1]

$\sigma(\omega)\to\widetilde{\sigma}(\omega)=\sigma(\omega)+\frac{\phi_{\text{kin}}^{(c)}(\omega)}{2m_{c}}$
(14)

However, the chromomagnetic operator
$\bar{h}_{v^{\prime}}^{(c)}g_{s}\sigma\cdot Gh_{v^{\prime}}^{(c)}/2$ breaks
HQSS and it leads to a contribution to the matrix elements of
$\bar{h}_{v^{\prime}}^{(c)}\Gamma h_{v}^{(b)}$ [30, 1]

$\displaystyle
i\phi_{\text{mag}}^{(c)}\,g_{\mu\alpha}\,v_{\nu}\,\bar{\mathcal{U}}_{c}^{\alpha}(v^{\prime})\sigma^{\mu\nu}\frac{1+\not{v}^{\prime}}{2}\Gamma
u_{b}(v)=\phi_{\text{mag}}^{(c)}v_{\lambda}\bar{u}_{c}^{\lambda}(v^{\prime})\Gamma
u_{b}(v)-\frac{2\phi_{\text{mag}}^{(c)}}{\sqrt{3}}\bar{u}_{c}(v^{\prime})\left(\not{v}-\omega\right)\gamma_{5}\Gamma
u_{b}(v)$ (15)

Combining the leading order (Eq. (III)), and the
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ (Eqs. (III)-(15)) contributions, we
obtain that the general form of the semileptonic matrix elements, keeping the
invariance under spin rotations of the quark $b$, reads [25],

$\displaystyle\langle\Lambda_{c,1/2^{-}}^{*}|\bar{c}\Gamma
b|\Lambda_{b}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\bar{u}_{c}\left[(\not{v}-\omega)\Delta_{1}(\omega)-\Delta_{2}(\omega)\right]\gamma_{5}\Gamma
u_{b},$ $\displaystyle\langle\Lambda_{c,3/2^{-}}^{*}|\bar{c}\Gamma
b|\Lambda_{b}\rangle$ $\displaystyle=$ $\displaystyle
v_{\lambda}\bar{u}_{c}^{\lambda}\left[\Omega_{1}(\omega)-(\not{v}-\omega)\Omega_{2}(\omega)\right]\Gamma
u_{b},$ (16)

where we have introduced the scalar form-factors

$\displaystyle\Delta_{1}(\omega)$ $\displaystyle=$
$\displaystyle\sigma(\omega)+\frac{1}{2m_{c}}\left[\phi_{\text{kin}}^{(c)}(\omega)-2\phi_{\text{mag}}^{(c)}(\omega)\right]=\widetilde{\sigma}(\omega)-\frac{\phi_{\text{mag}}^{(c)}(\omega)}{m_{c}},$
$\displaystyle\Delta_{2}(\omega)$ $\displaystyle=$
$\displaystyle\frac{(\omega^{2}-1)\sigma_{1}^{(c)}(\omega)-3\sigma_{3}^{(c)}(\omega)}{2m_{c}}=-\frac{2(\omega^{2}-1)\sigma_{1}^{(c)}(\omega)+3({\bar{\Lambda}}-\omega{\bar{\Lambda}}^{\prime})\sigma(\omega)}{2m_{c}},$
$\displaystyle\Omega_{1}(\omega)$ $\displaystyle=$
$\displaystyle\sigma(\omega)+\frac{1}{2m_{c}}\left[\phi_{\text{kin}}^{c}(\omega)+\phi_{\text{mag}}^{(c)}(\omega)\right]=\widetilde{\sigma}(\omega)+\frac{\phi_{\text{mag}}^{(c)}(\omega)}{2m_{c}},\qquad\Omega_{2}(\omega)=\frac{\sigma_{1}^{(c)}(\omega)}{2m_{c}}.$
(17)

which are determined by the leading IW function $\sigma$ and the sub-leading
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ correction discussed in Ref. [1].

It is worth stressing that the $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$
decomposition of Eq. (III) is valid for any $\Lambda_{b}$ transition to
$J^{P}=\frac{1}{2}^{-}$ and $J^{P}=\frac{3}{2}^{-}$ baryons, regardless of the
ldof quantum numbers in the final charm states. However the expressions of the
$\Delta_{1,2}$ and $\Omega_{1,2}$ form-factors in terms of the leading and
sub-leading IW functions are specific to $\Lambda_{b}$ decays into the two
members of the $j_{\rm ldof}^{P}=1^{-}$ HQSS doublet. Otherwise, the relations
of Eq. (III) are lost and thus, for instance, $\Delta_{1}(\omega)$ does not
need to approach $\Omega_{1}(\omega)$ in the heavy charm quark mass limit.
This might be of special interest, since as we argued in the introduction, the
$\Lambda_{c}(2595)^{+}$ [$J^{P}=\frac{1}{2}^{-}$] could contain, in addition
to $j_{\rm ldof}^{P}=1^{-}$, ldof components with $j_{\rm ldof}^{P}=0^{-}$
quantum numbers. For the case of this latter 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 [1]. The time
ordered products involving the chromomagnetic operator lead to non-zero
contributions, which however vanish at zero recoil [1] and can be cast in a
$\Delta_{1}$-type form factor. To order $1/m_{Q}$, the corresponding
$\Delta_{2}$ form-factor is therefore zero. These different HQSS patterns for
the $\Lambda_{b}$ semileptonic decay into a charm $j_{\rm ldof}^{P}=1^{-}$
doublet or a $j_{\rm ldof}^{P}=0^{-}$ singlet might shed light into the ldof
configuration in the $\Lambda_{c}(2595)^{+}$ resonance. Similarly, the
$J^{P}=\frac{3}{2}^{-}$ $\Lambda_{c}(2625)^{+}$ resonance could have, in
addition to $j_{\rm ldof}^{P}=1^{-}$, a $j_{\rm ldof}^{P}=2^{-}$ component.
For this also unnatural $0^{+}\to 2^{-}$ transition, in contrast to what we
saw for the $j^{P}_{\rm ldof}=1^{-}$ component, the leading IW function and
the kinetic energy operator correction vanish, as it occurs for the $0^{+}\to
0^{-}$ case [30, 1]. Different to the latter case, the matrix element of the
$1/m_{Q}$ current term does not vanish for the transition to $j_{\rm
ldof}^{P}=2^{-}$, and it takes a form that can be absorbed into the
$\Omega_{2}(\omega)$ form factor. The chromomagnetic correction provides a
$1/m_{c}-$suppressed contribution to $\Omega_{1}(\omega)$. Both corrections,
however, vanish at zero recoil.

By employing Eq. (III), it is straightforward to obtain for
$\Lambda_{c,1/2^{-}}^{*}$

$\displaystyle\langle\Lambda_{c,1/2^{-}}^{*}|\bar{c}b|\Lambda_{b}\rangle$
$\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{3}}\bar{u}_{c}\left[(\omega+1)\Delta_{1}+\Delta_{2}\right]\gamma_{5}u_{b},$
$\displaystyle\langle\Lambda_{c,1/2^{-}}^{*}|\bar{c}\gamma_{5}b|\Lambda_{b}\rangle$
$\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{3}}\bar{u}_{c}\left[(\omega-1)\Delta_{1}+\Delta_{2}\right]u_{b},$
$\displaystyle\langle\Lambda_{c,1/2^{-}}^{*}|\bar{c}\gamma_{\mu}b|\Lambda_{b}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\bar{u}_{c}\big{(}[(\omega-1)\Delta_{1}+\Delta_{2}]\gamma_{\mu}-2\Delta_{1}v_{\mu}\big{)}\gamma_{5}u_{b},$
$\displaystyle\langle\Lambda_{c,1/2^{-}}^{*}|\bar{c}\gamma_{\mu}\gamma_{5}b|\Lambda_{b}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\bar{u}_{c}\big{(}[(\omega+1)\Delta_{1}+\Delta_{2}]\gamma_{\mu}-2\Delta_{1}v_{\mu}\big{)}u_{b},$
$\displaystyle\langle\Lambda_{c,1/2^{-}}^{*}|\bar{c}\sigma_{\mu\nu}\gamma_{5}b|\Lambda_{b}\rangle$
$\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{3}}\bar{u}_{c}\big{(}[\Delta_{1}(\omega-1)+\Delta_{2}]\sigma_{\mu\nu}+2i\Delta_{1}(\gamma_{\mu}v_{\nu}-\gamma_{\nu}v_{\mu})\big{)}u_{b},$
(18)

and for $\Lambda_{c,3/2^{-}}^{*}$,

$\displaystyle\langle\Lambda_{c,3/2^{-}}^{*}|\bar{c}b|\Lambda_{b}\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{c}^{\lambda}v_{\lambda}[\Omega_{1}+(\omega-1)\Omega_{2}]u_{b},$
$\displaystyle\langle\Lambda_{c,3/2^{-}}^{*}|\bar{c}\gamma_{5}b|\Lambda_{b}\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{c}^{\lambda}v_{\lambda}[\Omega_{1}+(\omega+1)\Omega_{2}]u_{b},$
$\displaystyle\langle\Lambda_{c,3/2^{-}}^{*}|\bar{c}\gamma_{\mu}b|\Lambda_{b}\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{c}^{\lambda}v_{\lambda}\big{(}[\Omega_{1}+(\omega+1)\Omega_{2}]\gamma_{\mu}-2\Omega_{2}v_{\mu}\big{)}u_{b},$
$\displaystyle\langle\Lambda_{c,3/2^{-}}^{*}|\bar{c}\gamma_{\mu}\gamma_{5}b|\Lambda_{b}\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{c}^{\lambda}v_{\lambda}\big{(}[\Omega_{1}+(\omega-1)\Omega_{2}]\gamma_{\mu}-2\Omega_{2}v_{\mu}\big{)}\gamma_{5}u_{b},$
$\displaystyle\langle\Lambda_{c,3/2^{-}}^{*}|\bar{c}\sigma_{\mu\nu}b|\Lambda_{b}\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{c}^{\lambda}v_{\lambda}\big{(}[\Omega_{1}+(\omega-1)\Omega_{2}]\sigma_{\mu\nu}+2i\Omega_{2}(\gamma_{\mu}v_{\nu}-\gamma_{\nu}v_{\mu})\big{)}u_{b}.$
(19)

The matrix elements above can be used to express the form factors $d_{i}$ and
$l_{i}$ in terms of $\Delta_{1,2}$ and $\Omega_{1,2}$. The relations are
collected in Appendix B.

The $\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}\ell\bar{\nu}_{\ell}$ and
$\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}\ell\bar{\nu}_{\ell}$ SM differential
decay widths deduced from the vector and axial matrix elements in Eqs. (III)
and (III) are given by [25]

$\displaystyle\frac{d\Gamma[\Lambda_{b}\to\Lambda_{c,J^{P}}^{*}]}{d\omega}$
$\displaystyle=$
$\displaystyle\left(2J+1\right)\frac{8\,\Gamma_{0}}{3}\left(\frac{m_{\Lambda_{c}^{*}}}{m_{\Lambda_{b}}}\right)^{3}\left(1-\frac{m^{2}_{\ell}}{q^{2}}\right)^{2}(\omega^{2}-1)^{J}\,\left\\{\alpha_{J}^{2}\Bigg{[}3\omega\frac{q^{2}+m_{\ell}^{2}}{m_{\Lambda_{b}}^{2}}\right.$
(20)
$\displaystyle\left.+2\frac{m_{\Lambda_{c}^{*}}}{m_{\Lambda_{b}}}(\omega^{2}-1)\left(1+\frac{2m^{2}_{\ell}}{q^{2}}\right)\Bigg{]}+2\,(\omega^{2}-1)\,\left[\alpha_{1}(\omega)\alpha_{2}(\omega)\right]_{J}\times\right.$
$\displaystyle\times\left.\left[\frac{2q^{2}+m_{\ell}^{2}}{m_{\Lambda_{b}}^{2}}+\left(1-\frac{m^{2}_{\Lambda_{c}^{*}}}{m_{\Lambda_{b}}^{2}}\right)\left(1+\frac{2m^{2}_{\ell}}{q^{2}}\right)\right]+\mathcal{O}\left(\frac{\Lambda_{\text{QCD}}}{m_{b}}\right)\right\\}$

with $m_{\ell}$ the final charged lepton mass,
$\Gamma_{0}=|V_{cb}|^{2}G_{F}^{\,2}m_{\Lambda_{b}}^{5}/(192\pi^{3})$, where
$|V_{cb}|$ is the modulus of the Cabibbo–Kobayashi–Maskawa matrix element for
the $b\to c$ transition and $G_{F}=1.16638\times\ 10^{-11}$ MeV-2 is the Fermi
decay constant, and

$\displaystyle\alpha_{J=1/2}^{2}(\omega)$ $\displaystyle=$
$\displaystyle\Delta_{2}^{2}(\omega)+(\omega^{2}-1)\Delta_{1}^{2}(\omega),\qquad\left.\alpha_{1}(\omega)\,\alpha_{2}(\omega)\,\right|_{J=1/2}=\Delta_{1}(\omega)\,\,\Delta_{2}(\omega)\,$
$\displaystyle\alpha_{J=3/2}^{2}(\omega)$ $\displaystyle=$
$\displaystyle\Omega_{1}^{2}(\omega)+(\omega^{2}-1)\Omega_{2}^{2}(\omega),\qquad\left.\alpha_{1}(\omega)\,\alpha_{2}(\omega)\,\right|_{J=3/2}=\Omega_{1}(\omega)\,\,\Omega_{2}(\omega)$
(21)

In the strict $m_{c}\to\infty$ limit,

$\displaystyle\lim_{m_{c}\to\infty}\left(\frac{d\Gamma[\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}]/d\omega}{d\Gamma[\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}]/d\omega}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,\left(\frac{\lim_{m_{c}\to\infty}\Delta_{1}(\omega)}{\lim_{m_{c}\to\infty}\Omega_{1}(\omega)}\right)^{2}$
(22)

and if $\Lambda_{c,1/2^{-}}^{*}$ and $\Lambda_{c,3/2^{-}}^{*}$ are the two
members of the $j_{\rm ldof}^{P}=1^{-}$ HQSS doublet, the above ratio will be
1/2, since the relations of Eq. (III) will be satisfied and
$\Delta_{1}(\omega)\sim\Omega_{1}(\omega)\sim\sigma(\omega)$. However in the
vicinity of zero recoil ($\omega\leq 1.05$), one might find large deviations,
because in this kinematic region, the sub-leading contribution
$\Delta_{2}^{2}(\omega)$ to $\alpha_{J=1/2}^{2}(\omega)$ could be comparable
or larger than the other one, $(\omega^{2}-1)\Delta_{1}^{2}(\omega)$, which is
used to obtain222Note that $(\omega^{2}-1)$ multiplying
$\Delta_{1}^{2}(\omega)$ compensates the overall extra $(\omega^{2}-1)$ factor
which appears for the $J^{P}=3/2^{-}$ decay width in Eq. (20).  Eq. (22).
Actually, the $\Delta_{2}(\omega)$ form-factor accounts for a S-wave term to
the $\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}$ decay amplitude proportional to
$3({\bar{\Lambda}}-{\bar{\Lambda}}^{\prime})\sigma(1)/(2m_{c})\sim(0.2-0.3)\sigma(1)$
[25]. This follows from Eq. (III), using $m_{c}\sim 1.4$ GeV for the charm
quark mass, $\bar{\Lambda}\sim 0.8$ GeV and $\bar{\Lambda}^{\prime}\sim 1\pm
0.1$ GeV for the energies of the ldof, in the $m_{Q}\to\infty$ limit, in the
$\Lambda_{b}$ and the P-wave $\Lambda_{c}^{*}\,(j_{q}^{P}=1^{-})$ baryon.
Since the $\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}$ semileptonic decay proceeds
necessarily in P-wave, it appears an extra factor $(\omega^{2}-1)$, and the
ratio
$\frac{d\Gamma[\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}]/d\omega}{d\Gamma[\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}]/d\omega}$
could differ significantly from $1/2$ when $\omega$ is close to 1.

##  IV Numerical analysis

In the last two sections, we have presented a general parameterization and
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET form factors for the
$\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}/\Lambda_{c,3/2^{-}}^{*}$ semileptonic
transitions. Although in recent years, the LHC has collected large numbers of
$\Lambda_{b}\to\Lambda_{c}(2595)^{+}\mu^{-}\bar{\nu}_{\mu}$ and
$\Lambda_{b}\to\Lambda_{c}(2625)^{+}\mu^{-}\bar{\nu}_{\mu}$ samples [38], the
extraction of the form factors is, however, not available yet. Without
experimental input for the form factors, LQCD provides a valuable framework to
test how well the $j_{\rm ldof}^{P}=1^{-}$ HQSS predictions, with corrections
of order $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$, works to describe the
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ form factors measured on
the lattice, and whether there is a possibility of a sizable $j_{\rm
ldof}^{P}=0^{-}$ configuration for the $\Lambda_{c}(2595)^{+}$ resonance.

###  IV.1 LQCD and $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET form-factors

Figure 1: Comparison between the LQCD [31] (purple uncertainty bands and solid
curves) and the $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET (Eqs. (24) and
(30)) form factors for the $\Lambda_{b}\to\Lambda_{c}(2595)^{+}$ transition,
as a function of $\omega$. The HQET predictions are displayed by blue bands
and dotted lines. The LQCD results are obtained using the relations given in
Appendix A between the form-factors displayed here and the helicity ones
computed in Ref. [31]. Figure 2: Comparison of the
$\Lambda_{b}\to\Lambda_{c}(2595)^{+}$ LQCD (purple uncertainty bands and solid
curves) and $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET results for the
helicity form factors, which are those directly computed in Refs. [6, 31], as
function of $\omega$. The HQET predictions are obtained by inverting the
relations given in Appendix A, and using for the $d_{i}$ form-factors, the
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ expressions of Eq. (30). The HQET
predictions are displayed by blue bands and dotted lines.

The first LQCD calculation of the form factors for these transitions was
carried out in Ref. [6], using three different ensembles of gauge-field
configurations with $2+1$ flavors of domain-wall fermions generated by the RBC
and UKQCD collaborations [43, 44], and three-quark interpolating fields to
excite $\Lambda_{c}^{*}$ states. The form-factors in Ref. [6] are defined
through a helicity decomposition of the amplitudes and are evaluated at three
unphysical pion masses, $m_{\pi}=0.4312(13)$, $0.3400(11)$ and $0.3030(12)$
GeV. Finally, the results were extrapolated to the physical point (continuum
limit and physical pion mass), where the LQCD form factors are parameterized
as [6, 31],

$\displaystyle f(\omega)=F^{f}+A^{f}(\omega-1),$ (23)

which corresponds to a Taylor expansion of the form factors around zero recoil
$\omega=1$. Because only two different close kinematics ($\omega=1.01$ and
$\omega=1.03$) are available at the physical point in the LQCD calculation of
Ref. [6], the parameterization of Eq. (23) is expected to be only reliable for
small ($\omega-1$). Systematic uncertainties are estimated from the variation
of results obtained from two different extrapolations of the lattice results
to the physical limit. The helicity form factors, which parameterize the
matrix elements of the vector, axial-vector and tensor currents, for the two
available values of $\omega$ are treated in Ref. [6] as independent
quantities. Thus, a total of 48 parameters were fitted to the lattice data. In
the improved analysis carried out by the same authors in Ref. [31], relations
among the different form factors at zero recoil, which follow from rotational
symmetry, are imposed to ensure the correct behavior of the angular
observables near the endpoint, and thus the number of free parameters is
reduced to 39. In this work, we will employ the updated results of Ref. [31].
The form factors $d_{i}$ and $l_{i}$ introduced in Eqs. (II) and (II) are
computed from the helicity form factors determined in Refs. [6, 31] with the
help of Eqs. (A) and (A), respectively.

As discussed in Sec. III, there are only two independent functions
$\Delta_{1,2}(\omega)$ and $\Omega_{1,2}(\omega)$ for the
$\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}$ and
$\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}$ decays, respectively, including up to
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ contributions. This is to say,
neglecting $\mathcal{O}(\Lambda_{\text{QCD}}/m_{b})$ and perturbative QCD
short distance corrections. In order to test if the HQSS describes the
semileptonic $\Lambda_{b}\to\Lambda_{c}(2595)^{+}$ and
$\Lambda_{b}\to\Lambda_{c}(2625)^{+}$ decays at this order, we determine the
$\Delta_{1,2}(\omega)$ and $\Omega_{1,2}(\omega)$ functions from two of the
form factors in each of the transitions, and then predict the rest of the form
factors using the relations collected in Appendix B. We take,

$\displaystyle\Delta_{1}(\omega)=\dfrac{\sqrt{3}}{2}\left[d_{A_{1}}(\omega)-d_{V_{1}}(\omega)\right],\quad$
$\displaystyle\Delta_{2}(\omega)=\frac{\sqrt{3}}{2}\left[(1+\omega)d_{V_{1}}(\omega)+(1-\omega)d_{A_{1}}(\omega)\right],$
(24)
$\displaystyle\Omega_{1}(\omega)=\dfrac{1}{2}\left[(1-\omega)l_{V_{1}}(\omega)+(1+\omega)l_{A_{1}}(\omega)\right],\quad$
$\displaystyle\Omega_{2}(\omega)=\dfrac{1}{2}\left[l_{V_{1}}(\omega)-l_{A_{1}}(\omega)\right].$

Here there is a subtle point. We note that all type of contributions
suppressed by any power of the charm quark mass should be included in the
unknown $\Delta_{1,2}$ and $\Omega_{1,2}$ functions empirically determined
from the LQCD form-factors. Thus, we expect only perturbative QCD short
distance [7] and $\mathcal{O}(\Lambda_{\text{QCD}}/m_{b})$ corrections, from
the breaking of $b$-quark spin rotational invariance, to the matrix elements
of the scalar, pseudoescalar, vector, axial-vector and tensor currents between
the $\Lambda_{b}$ ground state and the final odd parity charm
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ resonances, for the whole
range of $\omega$ values accessible in the decay. However, far from zero
recoil, the energy scale of the heavy quark symmetry breaking corrections will
not necessarily be $\Lambda_{\rm QCD}\sim(200-300)$ MeV anymore. For instance,
the momentum of the outgoing hadron in the $\Lambda_{b}$-rest frame
($|\vec{q}\,|=m_{\Lambda_{c}^{*}}\sqrt{\omega^{2}-1}\,$) could be much larger
than $\Lambda_{\rm QCD}$. For $\omega=1.05$, we have already
$|\vec{q}\,|/m_{c}>1/2$ and the $(\Delta_{1},\Delta_{2})$ or
$(\Omega_{1},\Omega_{2})$ hierarchy relations inferred from Eqs. (III) clearly
break down. Even corrections suppressed by the bottom quark mass might become
relevant for $\omega\geq 1.1$, where $|\vec{q}\,|\geq 1.2$ GeV. Note that
$\omega$ reaches values of about $\sim 1.2\,(1.3)$ at the end-point of the
spectrum for the tau (muon) semileptonic mode.

The $\Lambda_{b}\to\Lambda_{c}(2595)^{+}$ and
$\Lambda_{b}\to\Lambda_{c}(2625)^{+}$ form factors $d_{i}$ and $l_{i}$
evaluated using Eq. (30) are compared with the LQCD results in Figs. 1
($1/2^{+}\to 1/2^{-})$, 3 (vector and axial-vector $1/2^{+}\to 3/2^{-}$) and 5
(tensor $1/2^{+}\to 3/2^{-}$). There, the purple solid and blue dotted curves
correspond to the LQCD results and the HQET predictions, obtained using the
$\Delta_{1,2}$ or $\Omega_{1,2}$ functions determined as specified in Eq.
(24), respectively. Statistic and systematic uncertainties, given in Refs. [6,
31], are added in quadrature and shown in the plots. We do not show the scalar
and pseudoscalar form factors ($d_{S,P}$ and $l_{S,P}$), since they were not
computed in the LQCD simulation of Refs. [6, 31]. They are given in Appendix A
using their relation to the vector and axial-vector form factors obtained from
the heavy quarks equations of motion.

Figure 3: Same as Fig. 1 for the $\Lambda_{b}\to\Lambda_{c}(2625)^{+}$ vector
and axial-vector form-factors.  Figure 4: Same as Fig. 2 for the
$\Lambda_{b}\to\Lambda_{c}(2625)^{+}$ scalar, pseudoscalar,vector and axial-
vector helicity form-factors. Figure 5: Same as Fig. 1 for the
$\Lambda_{b}\to\Lambda_{c}(2625)^{+}$ tensor form-factors.  Figure 6: Same as
Fig. 2 for the $\Lambda_{b}\to\Lambda_{c}(2625)^{+}$ tensor helicity form-
factors.

The comparison of the LQCD and $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET
results for the helicity form factors, which are those directly obtained in
the lattice calculation [6, 31], are presented in Fig. 2 for
$\Lambda_{b}\to\Lambda_{c}(2595)^{+}$, and Figs. 4 and 6 for
$\Lambda_{b}\to\Lambda_{c}(2625)^{+}$. The comparison for the scalar and
pseudoscalar form factors ($d_{S,P}$ and $l_{S,P}$) can be inferred from the
results shown in these figures for $f_{0}^{(J^{P})}$ and $g_{0}^{(J^{P})}$,
cf. Eqs. (A) and (A).

For the $\Lambda_{b}\to\Lambda_{c}(2595)^{+}$ transition, we see that LQCD and
HQET agree in general within uncertainties. The largest discrepancies are
found for the vector $d_{V_{3}}$ and tensor $d_{T_{1}}$ and $d_{T_{3}}$ form-
factors, which vanish at this order within the HQET scheme, and moreover they
are poorly determined on the lattice. In fact, the large uncertainties
affecting these form factors make these discrepancies of little significance.
Note that the axial $d_{A_{3}}$ form-factor, which also vanishes at this order
of the HQET expansion, is accurately determined in Ref. [31], and it turns out
to be compatible with zero in the whole $\omega$ interval shown in the figure.
The results in Fig. 2, where we pay attention to the helicity form-factors
directly determined in the LQCD simulation, confirm the quite reasonable
comparison between LQCD and HQET predictions. The most significant discrepancy
is found now for the axial $g_{0}^{(\frac{1}{2}^{-})}$ helicity form-factor,
which contributes to the axial $d_{A_{2}}$ and $d_{A_{3}}$, where however LQCD
and HQET predictions agree within errors. On the other hand, we do not
appreciate significant discrepancies between both sets of predictions for the
helicity form-factors involved in the vector $d_{V_{3}}$ and tensor
$d_{T_{1}}$ and $d_{T_{3}}$. Therefore, the origin of the differences noted
above must be sought in the large cancellations responsible for these form-
factors being zero at order $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$.

At first sight, the comparison of LQCD and
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET form-factors for the
$\Lambda_{b}\to\Lambda_{c}(2625)^{+}$ decay does not look as satisfactory as
that described above for $\Lambda_{b}\to\Lambda_{c}(2595)^{+}$, as one might
infer from Figs. 3 and 5. In particular, the discrepancies are clearly visible
in $l_{V_{3}}$, $l_{V_{4}}$, $l_{A_{4}}$, $l_{T_{5}}$ and $l_{T_{6}}$, which
are predicted to be zero at this order of the HQET expansion. We however note,
the LQCD predictions of Ref. [31] for these form factors are compatible with
typical values of order $\mathcal{O}(\Lambda_{\text{QCD}}/m_{b})\sim 0.1$,
except for the case of $|l_{V_{3}}|$, which takes notably higher values. As in
the case of $d_{V_{3}}$ for the $\Lambda_{b}\to\Lambda_{c}(2595)^{+}$ mode, we
also believe that the origin of the differences for $l_{V_{3}}$ stems from
some inaccuracies in the required large cancellations, among the
$f_{0}^{(\frac{3}{2}^{-})},f_{+}^{(\frac{3}{2}^{-})},f_{\perp}^{(\frac{3}{2}^{-})}$
and $f_{\perp^{\prime}}^{(\frac{3}{2}^{-})}$ helicity form-factors, to make
this form-factor vanish at order $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$.
The same helicity form-factors, but in a different linear combination, appear
also in $l_{V_{2}}$, for which some disagreement, near zero recoil, between
LQCD and $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET predictions can be
appreciated in Fig. 3. Paying now attention to the results in Figs. 4 and 6,
we observe there are five dominant helicity transitions, those associated to
$f_{0}^{(\frac{3}{2}^{-})},g_{+}^{(\frac{3}{2}^{-})}$ and
$g_{\perp}^{(\frac{3}{2}^{-})}$ for vector and axial-vector currents, and
$\widetilde{h}_{+}^{(\frac{3}{2}^{-})}$ and
$\widetilde{h}_{\perp}^{(\frac{3}{2}^{-})}$ for the tensor ones. Importantly,
the $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET scheme provides a good
reproduction of the LQCD results for these leading form-factors. The most
significant differences appear for $f_{0}^{(\frac{3}{2}^{-})}$, where the
central values of both schemes are separated by one sigma, which might explain
the discrepancies pointed out above for $l_{V_{2}}$ and $l_{V_{3}}$. The rest
of the helicity form-factors are very small, around a factor 20-40 smaller (in
absolute value) than the five dominant ones mentioned above. Order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{b})$ and/or of short physics corrections
to the HQET results or further systematic errors affecting the LQCD ones might
significantly change the predictions displayed in Figs. 4 and 6, and improve
the apparent disagreement exhibited there for these sub-leading form-factors.

Therefore, we conclude that $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET
scheme describes reasonably well the LQCD results, taking into account that
the neglected HQET sub-leading corrections or LQCD systematic uncertainties
might be important in those cases where discrepancies are more apparent on a
naive visual inspection of Figs. 3 and 5. The results of the next subsection
will give further support to this general conclusion.

###  IV.2 HQSS form-factors and differential decay rates

Figure 7: Comparison between the leading and sub-leading $\Delta_{1,2}$ and
$\Omega_{1,2}$ functions, determined as specified in Eq. (24). Note that it
would be necessary to include a relative phase between the states
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ to obtain $\Delta_{1}$ and
$\Omega_{1}$ with the same sign.

In this subsection, we try first to assess how well the
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ are described by a $j_{\rm
ldof}^{P}=1^{-}$ configuration for the ldof. We discussed in Sec. III that
$\Delta_{2}(\omega)=\Omega_{2}(\omega)=0$ in the heavy charm quark limit, and
that in addition, $\Delta_{1}(\omega)=\Omega_{1}(\omega)=\sigma(\omega)$ if
the $\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ are the members of the
lowest-lying HQSS $j_{\rm ldof}^{P}=1^{-}$ doublet. Thus, the difference
between the $\Delta_{1}(\omega)$ and $\Omega_{1}(\omega)$ should be of order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$, while $\Delta_{2}(\omega)$ and
$\Omega_{2}(\omega)$ are both of order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$, and hence suppressed compared to
the leading $\Delta_{1}(\omega)$ and $\Omega_{1}(\omega)$ functions, cf.
(III). Note that the overall sign of the form factors for each decay mode
depends on the phase conventions of the $\Lambda_{c}^{*}$ states [6], and only
the relative signs among the form factors for a specific mode are well
determined. The comparisons between the $\Delta_{1,2}$ and $\Omega_{1,2}$ IW
functions, obtained from the LQCD results for the
$d_{V_{1}},d_{A_{1}},l_{V_{1}}$ and $l_{A_{2}}$ form-factors (Eq. (24)) are
shown in Fig. 7. We see that the differences between $\Delta_{1}$ and
$-\Omega_{1}$ are small compared to any, $\Delta_{1}$ or $\Omega_{1}$, of
these leading IW functions, and can be naturally attributed to
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ contributions. This gives some
support, or at least does not contradict, that the $\Lambda_{c}(2595)^{+}$ and
$\Lambda_{c}(2625)^{+}$ resonances might form the lowest-lying HQSS $j_{\rm
ldof}^{P}=1^{-}$ doublet. Nevertheless, we would remind that in the LQCD
simulation carried out in Refs. [6, 31], the $\Lambda_{c}^{*}$ states were
described using three-quark interpolating fields. This should not create any
bias for an unquenched simulation, since these operators should capture any
baryon-meson ($\Sigma_{c}^{(*)}\pi$) component in the QCD state, assuming a
sufficiently large evolution time. The chiral-continuum extrapolations of the
masses of the states excited on the lattice are [6]

$m_{\Lambda_{c,1/2}^{*}}=\left(2693\pm 43_{\rm stat}\pm 95_{\rm
sys}\right)\,\,{\rm MeV},\qquad m_{\Lambda_{c,3/2}^{*}}=\left(2742\pm 43_{\rm
stat}\pm 96_{\rm sys}\right)\,\,{\rm MeV}$ (25)

which, though are consistent with the experimental values of
$m_{\Lambda_{c,1/2}^{*}}=2592.25(28)$ MeV and
$m_{\Lambda_{c,3/2}^{*}}=2628.11(19)$ MeV [45], are not accurate enough to
disentangle the possible effects of the $\Sigma_{c}\pi$ and
$\Sigma_{c}^{*}\pi$ thresholds located only a few MeV above the position of
the physical resonances, especially in the case of the $\Lambda_{c}(2595)^{+}$
and $\Sigma_{c}\pi$ [27]. Therefore, the claim made in this latter reference
that the $\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ resonances are
not HQSS partners cannot be discarded using the LQCD data studied here. In
Ref. [27], the $J^{P}=3/2^{-}$ state is described mostly as a dressed three-
quark state, whose origin is determined by a bare state [3], predicted to lie
very close to the mass of the resonance. The $J^{P}=1/2^{-}$ resonance seemed
to have, however, a predominant molecular structure. This is, depending on the
renormalization scheme, either because the $\Lambda_{c}(2595)^{+}$ is the
result of the chiral $\Sigma_{c}\pi$ interaction [27, 19], whose threshold is
located much closer than the mass of the bare three-quark state, or because
the ldof in its inner structure are coupled to the unnatural $j_{\rm
ldof}^{P}=0^{-}$ quantum numbers which gives rise to a double-pole pattern for
this resonance analog to that established for the $\Lambda(1405)$ [17, 18]. To
shed light into this problem, it would require new LQCD simulations giving
rise to more accurate determinations of the masses of the
$m_{\Lambda_{c,1/2}^{*}}$ and $m_{\Lambda_{c,3/2}^{*}}$, with precision better
than the mass difference with the $\Sigma_{c}\pi$ and $\Sigma_{c}^{*}\pi$
thresholds, and using not only three-quark interpolating operators, but also
other ones with larger overlaps to hadron-molecular degrees of freedom.

Figure 8: SM
$d\Gamma[\Lambda_{b}\to\Lambda_{c}(2595)^{+}\tau\bar{\nu}_{\tau}]/d\omega$ and
$d\Gamma[\Lambda_{b}\to\Lambda_{c}(2625)^{+}\tau\bar{\nu}_{\tau}]/d\omega$
differential decay widths, as function of $\omega$. We show distributions
obtained using the full set of LQCD form-factors computed in Ref. [31] (LQCD)
and the expressions (HQET) of Eqs. (20)-(21), employing the $\Delta_{1,2}$ and
$\Omega_{1,2}$ functions determined in Eq. (24). For the latter scheme, in the
right-bottom panel, we compare the full
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET results with those obtained
after setting to zero the sub-leading $\Delta_{2}$ and $\Omega_{2}$ functions.

Coming back to Fig. 7, we observe that both $\Delta_{2}$ and $\Omega_{2}$ are
also smaller (absolute value) than $\Delta_{1}$ and $\Omega_{1}$,
respectively, near zero recoil. Actually, the ratios $\Delta_{2}/\Delta_{1}$
and $\Omega_{2}/\Omega_{1}$ (right-bottom plot) have a typical size of order
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$, which could be already large for
$\omega=1.05$, as we mentioned in the discussion of Eq. (24). This supports
the compatibility of the LQCD and $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$
HQET predictions for the form-factors, discussed in the previous subsection.
Actually in the two top plots and the left-bottom one of Fig. 8, we compare
the SM
$d\Gamma[\Lambda_{b}\to\Lambda_{c,J^{P}}^{*}\tau\bar{\nu}_{\tau}]/d\omega$
distributions obtained using the full set of LQCD form-factors computed in
Ref. [31] and the $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET expressions
of Eqs. (20)-(21), employing the $\Delta_{1,2}$ and $\Omega_{1,2}$ functions
determined in Eq. (24). We find a good agreement, not only up to $\omega=1.05$
(region of validity of the analysis of Ref. [31]), but also for the whole
phase-space available in the semileptonic decay in the tau mode. This latter
result confirms that the $\Delta_{1,2}$ and $\Omega_{1,2}$ functions,
empirically determined from the LQCD form-factors, contain all type of
contributions suppressed by any power of the charm quark mass. Note also that
in the left-bottom plot of Fig. 8, some discrepancies between LQCD and HQET
predictions can be observed for $\omega>1.1$, which should be expected, since
in that kinematical region, the $\mathcal{O}(|\vec{q}\,|/m_{b})$ corrections
could start to become sizable. Nevertheless, we should stress that predictions
above $\omega=1.05$ might not be correct, because this region is beyond the
limit of reliability of the LQCD form-factors obtained in Refs. [31, 6].

On the other hand, we can see in Fig. 7 that in the vicinity of zero recoil,
$\Delta_{2}/\Delta_{1}$ is higher than $\Omega_{2}/\Omega_{1}$, reaching the
former ratio values around 0.3-0.35. Moreover, since $\Delta_{2}$ accounts for
a S-wave term to the $\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}$ decay amplitude
(see discussion of Eq. (22)), which becomes dominant close to $\omega=1$, we
see in the right-bottom panel of Fig. 8 that this sub-leading IW function has
an enormous numerical impact in the $\Lambda_{b}\to\Lambda_{c}(2595)^{+}$
differential decay distribution, and its contribution becomes totally
dominant. This result confirms the findings of Refs. [1, 25], obtained using
the soliton model derived in Ref. [1], where the large size of these finite
charm quark mass HQSS breaking terms on the $\Lambda_{c}(2595)^{+}$
differential decay distribution was firstly pointed out (see for instance Fig.
2 of Ref. [25]). In sharp contrast, the effects on the $\Lambda_{c}(2625)^{+}$
distribution of the sub-leading $\Omega_{2}$ function are small below
$\omega=1.05$ (right-bottom panel of Fig. 8).

From the above discussion, we naturally find an explanation for large
deviations of the
$\frac{d\Gamma[\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}]/d\omega}{d\Gamma[\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}]/d\omega}$
ratio from the value of 1/2 predicted in the $m_{Q}\to\infty$ limit assuming
that the $\Lambda_{c,1/2^{-}}^{*}$ and $\Lambda_{c,3/2^{-}}^{*}$ are the two
members of the $j_{\rm ldof}^{P}=1^{-}$ HQSS doublet. Actually, this ratio at
$\omega=1.05$ takes values around 2.5 and much larger ones in the near-zero
recoil regime. This tension between LQCD data and the HQET predictions
triggered the claim in Refs. [7, 31] of unexpectedly large HQSS-violating
terms, with potentially large $1/m_{c}^{2}$ corrections, near zero
recoil333The authors of Refs. [7, 31] pointed out that such large HQSS-
violating terms cannot persist uniformly over the full recoil spectrum, as
they would be incompatible with the measurement of the ratio of total decay
rates in the muon mode [46]
$\frac{\Gamma[\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}\mu\bar{\nu}_{\mu}]}{\Gamma[\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}\mu\bar{\nu}_{\mu}]}=0.6\pm
0.2^{+0.5}_{-0.3}$ (26) We note, however, that at the end of the spectrum in
the semileptonic muon mode $|\vec{q}\,|\sim 2.2$ GeV and, thus talking about
corrections suppressed by the charm quark mass is meaningless.. We assign here
these huge HQSS breaking corrections, not necessarily just to big values of
the $\Delta_{2}(\omega)/\Delta_{1}(\omega)$ ratio of leading to sub-leading IW
functions, but also to the fact that the $\Delta_{2}(\omega)$ form-factor
accounts for a S-wave term to the $\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}$
decay amplitude, while the $\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}$
semileptonic decay proceeds necessarily in P-wave, and it is therefore
suppressed by a factor $(\omega^{2}-1)^{3}/2$ at zero recoil444We remind here
that the in the $\Lambda_{b}$-rest frame the three-momentum of the outgoing
hadron ($\Lambda^{*}_{c}$) in the semileptonic decay is proportional to
$\sqrt{\omega^{2}-1}$..

Below $\omega\sim 1.05$, when $\Delta_{2}$ and $\Omega_{2}$ are set to zero,
we recover within errors the value of 1/2 for the ratio of differential decay
widths, see right-bottom plot of Fig. 8. This is so because, taking into
account uncertainties, $\Delta_{1}(\omega)\sim-\Omega_{1}(\omega)$ even in the
presence of $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ corrections to these
leading IW functions (see top left panel of Fig. 7). This is in accordance
with the $\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ being HQSS
partners.

##  V Summary

We have used the $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET scheme derived
in Ref. [25], in which invariance under rotations of the spin of the $b$ quark
is preserved, to study the matrix elements of the scalar, pseudoescalar,
vector, axial-vector and tensor currents between the $\Lambda_{b}$ ground
state and the odd parity charm $\Lambda_{c}(2595)^{+}$ and
$\Lambda_{c}(2625)^{+}$ resonances. Only
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{b})$ and perturbative QCD short distance
corrections are neglected. There are only four independent functions
$\Delta_{1,2}(\omega)$ and $\Omega_{1,2}(\omega)$ for the
$\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}$ and
$\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}$ semileptonic decays, respectively,
which are determined from a recent LQCD computation of the corresponding
helicity form-factors [6, 31]. We have shown that in the near-zero recoil
regime, this $\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ HQET scheme describes
reasonably well, taking into account systematic uncertainties, the results for
the total of 24 form-factors obtained in the LQCD studies of Refs. [6, 31].

We have found that the differences between $\Delta_{1}$ and $-\Omega_{1}$ are
small compared to any, $\Delta_{1}$ or $\Omega_{1}$, of these leading IW
functions, and can be naturally attributed to
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ contributions. This gives some
support, or at least does not contradict, to the scenario in which the
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ resonances might form the
lowest-lying HQSS $j_{\rm ldof}^{P}=1^{-}$ doublet. However, we have argued
that the available LQCD description of these two resonances is not accurate
enough to disentangle the possible effects of the $\Sigma_{c}\pi$ and
$\Sigma_{c}^{*}\pi$ thresholds, located only few MeV above the position of
these excited states, especially in the case of the $\Lambda_{c}(2595)^{+}$
and $\Sigma_{c}\pi$ [27]. The claim made in this latter reference that the
$\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ resonances are not HQSS
partners cannot be discarded using the LQCD data studied here. To clarify this
problem, new and more precise LQCD simulations capable of elucidating the role
played by the $\Sigma_{c}\pi$ and $\Sigma_{c}^{*}\pi$ channels and that make
use, not only of three-quark interpolating fields, but also of other operators
with larger overlaps to hadron-molecular degrees of freedom will be required.

Finally, we have naturally given an explanation to the large deviations, near
zero recoil, of the
$\frac{d\Gamma[\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}]/d\omega}{d\Gamma[\Lambda_{b}\to\Lambda_{c,3/2^{-}}^{*}]/d\omega}$
ratio from 1/2, value predicted in the $m_{Q}\to\infty$ limit assuming that
the $\Lambda_{c,1/2^{-}}^{*}$ and $\Lambda_{c,3/2^{-}}^{*}$ are the two
members of the $j_{\rm ldof}^{P}=1^{-}$ HQSS doublet. We have related these
huge HQSS breaking corrections, not necessarily to just big values of the
$\Delta_{2}(\omega)/\Delta_{1}(\omega)$ ratio of sub-leading to leading IW
functions, but also to the S-wave character of the contributions to the
$\Lambda_{b}\to\Lambda_{c,1/2^{-}}^{*}$ decay amplitude driven by
$\Delta_{2}(\omega)$.

## Acknowledgements

This research has been supported by the Spanish Ministerio de Ciencia e
Innovación (MICINN) and the European Regional Development Fund (ERDF) under
contracts PID2020-112777GB-I00 and PID2019-105439GB-C22, the EU STRONG-2020
project under the program H2020-INFRAIA-2018-1, grant agreement no. 824093 and
by Generalitat Valenciana under contract PROMETEO/2020/023.

##  Appendix A LQCD form-factors

The relations between the form factors in Eqs. (II) and (II) and the LQCD ones
computed in Ref. [31] are for $\Lambda_{b}\to\Lambda_{c}(2595)^{+}$,

$\displaystyle d_{S}$ $\displaystyle=$ $\displaystyle
f_{0}^{(\frac{1}{2}^{-})}\frac{m_{\Lambda_{b}}+m_{\Lambda_{c,1/2}^{*}}}{m_{b}-m_{c}},$
$\displaystyle d_{P}$ $\displaystyle=$ $\displaystyle
g_{0}^{(\frac{1}{2}^{-})}\frac{m_{\Lambda_{b}}-m_{\Lambda_{c,1/2}^{*}}}{m_{b}+m_{c}},$
$\displaystyle d_{V_{1}}$ $\displaystyle=$ $\displaystyle-
f_{\perp}^{(\frac{1}{2}^{-})},$ $\displaystyle d_{V_{2}}$ $\displaystyle=$
$\displaystyle
m_{\Lambda_{b}}\left[\frac{m_{\Lambda_{b}}+m_{\Lambda_{c,1/2}^{*}}}{q^{2}}f_{0}^{(\frac{1}{2}^{-})}+\frac{m_{\Lambda_{b}}-m_{\Lambda_{c,1/2}^{*}}}{s_{-}}\left(1-\frac{m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,1/2}^{*}}^{2}}{q^{2}}\right)f_{+}^{(\frac{1}{2}^{-})}+\frac{2m_{\Lambda_{c,1/2}^{*}}}{s_{-}}f_{\perp}^{(\frac{1}{2}^{-})}\right],$
$\displaystyle d_{V_{3}}$ $\displaystyle=$ $\displaystyle
m_{\Lambda_{c,1/2}^{*}}\left[-\frac{m_{\Lambda_{b}}+m_{\Lambda_{c,1/2}^{*}}}{q^{2}}f_{0}^{(\frac{1}{2}^{-})}+\frac{m_{\Lambda_{b}}-m_{\Lambda_{c,1/2}^{*}}}{s_{-}}\left(1+\frac{m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,1/2}^{*}}^{2}}{q^{2}}\right)f_{+}^{(\frac{1}{2}^{-})}-\frac{2m_{\Lambda_{b}}}{s_{-}}f_{\perp}^{(\frac{1}{2}^{-})}\right],$
$\displaystyle d_{A_{1}}$ $\displaystyle=$ $\displaystyle-
g_{\perp}^{(\frac{1}{2}^{-})},$ $\displaystyle d_{A_{2}}$ $\displaystyle=$
$\displaystyle-
m_{\Lambda_{b}}\left[\frac{m_{\Lambda_{b}}-m_{\Lambda_{c,1/2}^{*}}}{q^{2}}g_{0}^{(\frac{1}{2}^{-})}+\frac{m_{\Lambda_{b}}+m_{\Lambda_{c,1/2}^{*}}}{s_{+}}\left(1-\frac{m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,1/2}^{*}}^{2}}{q^{2}}\right)g_{+}^{(\frac{1}{2}^{-})}-\frac{2m_{\Lambda_{c,1/2}^{*}}}{s_{+}}g_{\perp}^{(\frac{1}{2}^{-})}\right],$
$\displaystyle d_{A_{3}}$ $\displaystyle=$ $\displaystyle
m_{\Lambda_{c,1/2}^{*}}\left[\frac{m_{\Lambda_{b}}-m_{\Lambda_{c,1/2}^{*}}}{q^{2}}g_{0}^{(\frac{1}{2}^{-})}-\frac{m_{\Lambda_{b}}+m_{\Lambda_{c,1/2}^{*}}}{s_{+}}\left(1+\frac{m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,1/2}^{*}}^{2}}{q^{2}}\right)g_{+}^{(\frac{1}{2}^{-})}+\frac{2m_{\Lambda_{b}}}{s_{+}}g_{\perp}^{(\frac{1}{2}^{-})}\right],$
$\displaystyle d_{T_{1}}$ $\displaystyle=$ $\displaystyle
2m_{\Lambda_{b}}^{2}\left(\frac{1}{s_{-}}h_{+}^{(\frac{1}{2}^{-})}-\frac{(m_{\Lambda_{b}}-m_{\Lambda_{c,1/2}^{*}})^{2}}{q^{2}s_{-}}h_{\perp}^{(\frac{1}{2}^{-})}-\frac{1}{s_{+}}\widetilde{h}_{+}^{(\frac{1}{2}^{-})}+\frac{(m_{\Lambda_{b}}+m_{\Lambda_{c,1/2}^{*}})^{2}}{q^{2}s_{+}}\widetilde{h}_{\perp}^{(\frac{1}{2}^{-})}\right),$
$\displaystyle d_{T_{2}}$ $\displaystyle=$ $\displaystyle
2m_{\Lambda_{b}}\left(\frac{m_{\Lambda_{c,1/2}^{*}}}{s_{-}}h_{+}^{(\frac{1}{2}^{-})}-\frac{(m_{\Lambda_{b}}-m_{\Lambda_{c,1/2}^{*}})(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,1/2}^{*}}^{2}-q^{2})}{2q^{2}s_{-}}h_{\perp}^{(\frac{1}{2}^{-})}+\frac{m_{\Lambda_{b}}+m_{\Lambda_{c,1/2}^{*}}}{2q^{2}}\widetilde{h}_{\perp}^{(\frac{1}{2}^{-})}\right),$
$\displaystyle d_{T_{3}}$ $\displaystyle=$ $\displaystyle
2m_{\Lambda_{b}}\left(-\frac{m_{\Lambda_{b}}}{s_{-}}h_{+}^{(\frac{1}{2}^{-})}+\frac{(m_{\Lambda_{b}}-m_{\Lambda_{c,1/2}^{*}})(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,1/2}^{*}}^{2}+q^{2})}{2q^{2}s_{-}}h_{\perp}^{(\frac{1}{2}^{-})}-\frac{m_{\Lambda_{b}}+m_{\Lambda_{c,1/2}^{*}}}{2q^{2}}\widetilde{h}_{\perp}^{(\frac{1}{2}^{-})}\right),$
$\displaystyle d_{T_{4}}$ $\displaystyle=$ $\displaystyle
h_{+}^{(\frac{1}{2}^{-})},$ (27)

and for $\Lambda_{b}\to\Lambda_{c}(2625)^{+}$,

$\displaystyle l_{S}$ $\displaystyle=$ $\displaystyle
f_{0}^{(\frac{3}{2}^{-})}\frac{m_{\Lambda_{b}}m_{\Lambda_{c,3/2}^{*}}}{s_{+}}\frac{m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}}}{m_{b}-m_{c}},$
$\displaystyle l_{P}$ $\displaystyle=$ $\displaystyle
g_{0}^{(\frac{3}{2}^{-})}\frac{m_{\Lambda_{b}}m_{\Lambda_{c,3/2}^{*}}}{s_{-}}\frac{m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}}}{m_{b}+m_{c}},$
$\displaystyle l_{V_{1}}$ $\displaystyle=$
$\displaystyle\frac{m_{\Lambda_{b}}m_{\Lambda_{c,3/2}^{*}}}{s_{-}}(f_{\perp}^{(\frac{3}{2}^{-})}+f_{\perp^{\prime}}^{(\frac{3}{2}^{-})}),$
$\displaystyle l_{V_{2}}$ $\displaystyle=$
$\displaystyle\frac{m_{\Lambda_{b}}^{2}m_{\Lambda_{c,3/2}^{*}}}{s_{+}s_{-}}\bigg{[}f_{0}^{(\frac{3}{2}^{-})}\frac{(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})s_{-}}{q^{2}}+f_{+}^{(\frac{3}{2}^{-})}(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})\left(1-\frac{m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2}}{q^{2}}\right)$
$\displaystyle-2m_{\Lambda_{c,3/2}^{*}}f_{\perp}^{(\frac{3}{2}^{-})}+2m_{\Lambda_{c,3/2}^{*}}f_{\perp^{\prime}}^{(\frac{3}{2}^{-})}\bigg{]},$
$\displaystyle l_{V_{3}}$ $\displaystyle=$
$\displaystyle\frac{m_{\Lambda_{b}}m_{\Lambda_{c,3/2}^{*}}^{2}}{s_{+}s_{-}}\bigg{[}-f_{0}^{(\frac{3}{2}^{-})}\frac{(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})s_{-}}{q^{2}}+f_{+}^{(\frac{3}{2}^{-})}(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})\left(1+\frac{m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2}}{q^{2}}\right)$
$\displaystyle-2m_{\Lambda_{b}}f_{\perp}^{(\frac{3}{2}^{-})}+2(m_{\Lambda_{b}}-\frac{s_{+}}{m_{\Lambda_{c,3/2}^{*}}})f_{\perp^{\prime}}^{(\frac{3}{2}^{-})}\bigg{]},$
$\displaystyle l_{V_{4}}$ $\displaystyle=$ $\displaystyle
f_{\perp^{\prime}}^{(\frac{3}{2}^{-})},$ $\displaystyle l_{A_{1}}$
$\displaystyle=$
$\displaystyle\frac{m_{\Lambda_{b}}m_{\Lambda_{c,3/2}^{*}}}{s_{+}}(g_{\perp}^{(\frac{3}{2}^{-})}+g_{\perp^{\prime}}^{(\frac{3}{2}^{-})}),$
$\displaystyle l_{A_{2}}$ $\displaystyle=$
$\displaystyle\frac{m_{\Lambda_{b}}^{2}m_{\Lambda_{c,3/2}^{*}}}{s_{+}s_{-}}\bigg{[}-\frac{(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})s_{+}}{q^{2}}g_{0}^{(\frac{3}{2}^{-})}+\frac{(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2}-q^{2})}{q^{2}}g_{+}^{(\frac{3}{2}^{-})}$
$\displaystyle-2m_{\Lambda_{c,3/2}^{*}}g_{\perp}^{(\frac{3}{2}^{-})}+2m_{\Lambda_{c,3/2}^{*}}g_{\perp^{\prime}}^{(\frac{3}{2}^{-})}\bigg{]},$
$\displaystyle l_{A_{3}}$ $\displaystyle=$
$\displaystyle\frac{m_{\Lambda_{b}}m_{\Lambda_{c,3/2}^{*}}^{2}}{s_{+}s_{-}}\bigg{[}\frac{(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})s_{+}}{q^{2}}g_{0}^{(\frac{3}{2}^{-})}-\frac{(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2}+q^{2})}{q^{2}}g_{+}^{(\frac{3}{2}^{-})}$
$\displaystyle+2m_{\Lambda_{b}}g_{\perp}^{(\frac{3}{2}^{-})}-2(m_{\Lambda_{b}}+\frac{s_{-}}{m_{\Lambda_{c,3/2}^{*}}})g_{\perp^{\prime}}^{(\frac{3}{2}^{-})}\bigg{]},$
$\displaystyle l_{A_{4}}$ $\displaystyle=$ $\displaystyle
g_{\perp^{\prime}}^{(\frac{3}{2}^{-})},$ $\displaystyle l_{T_{1}}$
$\displaystyle=$
$\displaystyle\frac{2m_{\Lambda_{b}}^{3}}{q^{2}s_{-}s_{+}}\bigg{(}m_{\Lambda_{c,3/2}^{*}}\left[(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})^{2}h_{\perp}^{(\frac{3}{2}^{-})}+q^{2}(\widetilde{h}_{+}^{(\frac{3}{2}^{-})}-h_{+}^{(\frac{3}{2}^{-})})-(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})^{2}\widetilde{h}_{\perp}^{(\frac{3}{2}^{-})}\right]$
$\displaystyle+h_{\perp^{\prime}}^{(\frac{3}{2}^{-})}(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})(m_{\Lambda_{b}}^{2}+m_{\Lambda_{b}}m_{\Lambda_{c,3/2}^{*}}-q^{2})+(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})(m_{\Lambda_{b}}^{2}-m_{\Lambda_{b}}m_{\Lambda_{c,3/2}^{*}}-q^{2})\widetilde{h}_{\perp^{\prime}}^{(\frac{3}{2}^{-})}\bigg{)},$
$\displaystyle l_{T_{2}}$ $\displaystyle=$
$\displaystyle\frac{m_{\Lambda_{b}}^{2}m_{\Lambda_{c,3/2}^{*}}}{q^{2}s_{-}s_{+}}\bigg{(}s_{+}(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})(h_{\perp}^{(\frac{3}{2}^{-})}+h_{\perp^{\prime}}^{(\frac{3}{2}^{-})})$
(28)
$\displaystyle+(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2}-q^{2})(\widetilde{h}_{\perp^{\prime}}^{(\frac{3}{2}^{-})}-\widetilde{h}_{\perp}^{(\frac{3}{2}^{-})})+2m_{\Lambda_{c,3/2}^{*}}q^{2}\widetilde{h}_{+}^{(\frac{3}{2}^{-})}\bigg{)},$
$\displaystyle l_{T_{3}}$ $\displaystyle=$
$\displaystyle-\frac{m_{\Lambda_{b}}}{q^{2}s_{-}s_{+}}\bigg{(}s_{+}\left[m_{\Lambda_{b}}^{3}-q^{2}(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})+m_{\Lambda_{c,3/2}^{*}}^{3}\right]h_{\perp^{\prime}}^{(\frac{3}{2}^{-})}$
$\displaystyle+m_{\Lambda_{b}}m_{\Lambda_{c,3/2}^{*}}\left[s_{+}(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})h_{\perp}^{(\frac{3}{2}^{-})}-(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2}+q^{2})\widetilde{h}_{\perp}^{(\frac{3}{2}^{-})}+2m_{\Lambda_{b}}q^{2}\widetilde{h}_{+}^{(\frac{3}{2}^{-})}\right]$
$\displaystyle+(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})(m_{\Lambda_{b}}^{2}-m_{\Lambda_{b}}m_{\Lambda_{c,3/2}^{*}}+m_{\Lambda_{c,3/2}^{*}}^{2}-q^{2})(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2}+q^{2})\widetilde{h}_{\perp^{\prime}}^{(\frac{3}{2}^{-})}\bigg{)},$
$\displaystyle l_{T_{4}}$ $\displaystyle=$
$\displaystyle-\frac{m_{\Lambda_{c,3/2}^{*}}}{q^{2}s_{+}}\bigg{(}s_{+}(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})h_{\perp^{\prime}}^{(\frac{3}{2}^{-})}+(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2}+q^{2})\widetilde{h}_{\perp^{\prime}}^{(\frac{3}{2}^{-})}-m_{\Lambda_{b}}q^{2}\widetilde{h}_{+}^{(\frac{3}{2}^{-})}\bigg{)}$
$\displaystyle l_{T_{5}}$ $\displaystyle=$
$\displaystyle-\frac{s_{+}h_{\perp^{\prime}}^{(\frac{3}{2}^{-})}(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})+(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})(m_{\Lambda_{b}}^{2}-m_{\Lambda_{c,3/2}^{*}}^{2}+q^{2})\widetilde{h}_{\perp^{\prime}}^{(\frac{3}{2}^{-})}}{2m_{\Lambda_{b}}q^{2}},$
$\displaystyle l_{T_{6}}$ $\displaystyle=$
$\displaystyle\frac{h_{\perp^{\prime}}^{(\frac{3}{2}^{-})}(m_{\Lambda_{b}}+m_{\Lambda_{c,3/2}^{*}})^{2}+(m_{\Lambda_{b}}-m_{\Lambda_{c,3/2}^{*}})^{2}\widetilde{h}_{\perp^{\prime}}^{(\frac{3}{2}^{-})}}{q^{2}},$
(29)

where $q=p-p^{\prime}$, $s_{\pm}=(m_{\Lambda_{b}}\pm
m_{\Lambda_{c}^{*}})^{2}-q^{2}$ and $m_{b}$($m_{c}$) the mass of $b$($c$)
quark. The equations of motion of the heavy quarks ($b$ and $c$ quark) have
been used to derive the scalar and pseudoscalar form factors.

##  Appendix B HQET predictions for the form factors

The non-vanishing form factors in HQET up to
$\mathcal{O}(\Lambda_{\text{QCD}}/m_{c})$ read

$\displaystyle
d_{S}=-\dfrac{1}{\sqrt{3}}[(1+\omega)\Delta_{1}+\Delta_{2}],\qquad$
$\displaystyle d_{P}=-\frac{1}{\sqrt{3}}[(\omega-1)\Delta_{1}+\Delta_{2}],$
(30) $\displaystyle
d_{V_{1}}=\dfrac{1}{\sqrt{3}}[(\omega-1)\Delta_{1}+\Delta_{2}],\qquad$
$\displaystyle d_{V_{2}}=-\frac{2}{\sqrt{3}}\Delta_{1},$ $\displaystyle
d_{A_{1}}=\dfrac{1}{\sqrt{3}}[(\omega+1)\Delta_{1}+\Delta_{2}],\qquad$
$\displaystyle d_{A_{2}}=-\frac{2}{\sqrt{3}}\Delta_{1},$ $\displaystyle
d_{T_{2}}={-}\dfrac{2}{\sqrt{3}}\Delta_{1},\qquad\qquad\qquad\quad$
$\displaystyle
d_{T_{4}}=-\frac{1}{\sqrt{3}}[(\omega-1)\Delta_{1}+\Delta_{2}],$
$\displaystyle l_{S}=\Omega_{1}+(\omega-1)\Omega_{2},\qquad\qquad$
$\displaystyle l_{P}=\Omega_{1}+(\omega+1)\Omega_{2},$ $\displaystyle
l_{V_{1}}=\Omega_{1}+(\omega+1)\Omega_{2},\qquad\qquad$ $\displaystyle
l_{V_{2}}=-2\Omega_{2},$ $\displaystyle
l_{A_{1}}=\Omega_{1}+(\omega-1)\Omega_{2},\qquad\qquad$ $\displaystyle
l_{A_{2}}=-2\Omega_{2},$ $\displaystyle
l_{T_{2}}=2\Omega_{2},\qquad\qquad\qquad\quad\qquad$ $\displaystyle
l_{T_{4}}=\Omega_{1}+(\omega-1)\Omega_{2}.$

The form factors not listed above, i.e. $d_{V_{3}}$, $d_{A_{3}}$, $d_{T_{1}}$,
$d_{T_{3}}$, $l_{V_{3}}$, $l_{V_{4}}$, $l_{A_{3}}$, $l_{A_{4}}$, $l_{T_{1}}$,
$l_{T_{3}}$, $l_{T_{5}}$, $l_{T_{6}}$, vanish at this order.

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