# The $\Lambda_{c}\to\Lambda\,\ell^{+}\nu_{\ell}$ weak decay including new
physics

Fernando Alvarado<EMAIL_ADDRESS>Instituto de Física
Corpuscular (IFIC) and Departamento de Física Teórica,
Universidad de Valencia and Consejo Superior de Investigaciones Científicas
(CSIC)
E-46980 Paterna, Valencia, Spain  Luis Alvarez-Ruso  Instituto de Física
Corpuscular (IFIC),
Consejo Superior de Investigaciones Científicas (CSIC) and Universidad de
Valencia
E-46980 Paterna, Valencia, Spain  Eliecer Hernández  Departamento de Física
Fundamental and IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain
Juan Nieves  Instituto de Física Corpuscular (IFIC),
Consejo Superior de Investigaciones Científicas (CSIC) and Universidad de
Valencia
E-46980 Paterna, Valencia, Spain  Neus Penalva  Institut de Ciencies del
Cosmos (ICC), Facultat de Física
Universitat de Barcelona, E-08028 Barcelona, Spain

###### Abstract

We investigate the $\Lambda_{c}\to\Lambda\ell^{+}\nu_{\ell}$ decay with a
focus on potential new physics (NP) effects in the $\ell=\mu$ channel. We
employ an effective Hamiltonian within the framework of the Standard Model
Effective Field Theory (SMEFT) to consider generalized dimension-6
semileptonic $c\to s$ operators of scalar, pseudoscalar, vector, axial-vector
and tensor types. We rely on Lattice QCD (LQCD) for the hadronic transition
form factors, using heavy quark spin symmetry (HQSS) to determine those that
have not yet been obtained on the lattice. Uncertainties due to the truncation
of the NP Hamiltonian and different implementations of HQSS are taken into
account. As a result, we unravel the NP discovery potential of the
$\Lambda_{c}\to\Lambda$ semileptonic decay in different observables. Our
findings indicate high sensitivity to NP in lepton flavour universality
ratios, probing multi-TeV scales in some cases. On the theoretical side, we
identify LQCD uncertainties in axial and vector form factors as critical for
improving NP sensitivity, alongside better SMEFT uncertainty estimations.

##  I  Introduction

Hadron weak decays probe the flavour structure of matter, where unknown Beyond
Standard Model (BSM) effects would manifest as differences between the SM
prediction and precise measurements. The reader can consult Refs. [1, 2, 3]
for comprehensive reviews in both light and heavy quark sectors. In recent
years, experimental hints to lepton flavour universality (LFU) violation
observed in $b\to c\ell\bar{\nu}_{\ell}$ with different hadrons in the initial
and final states have received considerable attention. Theoretical studies and
experimental searches have also been extended to the weak decays of $c$ and
$s$ quarks. Although, for some quantities, the deviations from the SM have
become smaller (a recent update can be found in Ref. [4]), the theoretical
strategies developed to identify and predict promising observables are
valuable tools to keep putting the SM under scrutiny.

Baryon decays are complementary to meson ones, providing a different, richer,
sensitivity to new physics (NP), particularly when the study of decays rates
is generalized to consider angular distributions and asymmetries. The
$\Lambda_{b}\to\Lambda_{c}\,\ell^{-}\bar{\nu}_{\ell}$ decay has been
extensively investigated in the SM and beyond [5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Some
studies considered $\Lambda_{b}\to p\,\ell^{-}\bar{\nu}_{\ell}$ as well [5,
10, 32]. In most cases, lattice QCD (LQCD) input has been used but quark model
[33, 34, 35, 36, 37], heavy quark effective theory [38] and sum rule [39, 40,
41, 42] calculations have also been performed. The $\Lambda_{b}$ inclusive
semileptonic decays were examined in Ref. [43]. The transitions from
$\Lambda_{b}$ to $\Lambda_{c}^{*}$ excited states were considered in Refs.
[44, 45, 46, 47, 48, 49, 50]. The $\Lambda_{c}\to\Lambda\,\ell^{+}\nu_{\ell}$
decay was also discussed in Ref. [35, 25]. In the light-quark sector, the NP
discovery potential of $\Lambda$, $\Sigma$ and $\Xi$ semileptonic decays has
been explored in Ref. [51]. From the perspective of BSM searches, rare
$\Sigma\to p\ell\ell$ decays are also interesting [52, 53]. Asymmetries in the
angular distributions of $\Sigma^{0}\to\Lambda\gamma$ compared to the
corresponding ones with antiparticles would give access to new sources of CP
violation [54].

New experimental results are the driving force for these theory developments.
The study of hadron decays, including hyperon ones, proceeds at BESIII, Belle
II and LHCb. The experimental program of LHCb and Belle II encompasses a
variety of $b\to s$ and $b\to c$ processes. BESIII, operating in a regime
where $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pairs are produced in the
exclusive two-body channel with no additional particles, has recently
performed updated measurements of
$\Lambda_{c}\to\Lambda\,\ell^{+}\,\nu_{\ell}$ decays with $\ell=e,\,\mu$ [55,
56].

We study the impact of NP contributions to the
$\Lambda_{c}\to\Lambda\,\ell^{+}\,\nu_{\ell}$ decay, relying on the general
framework of Ref. [21] for general $q\to q^{\prime}$ charged-current
transitions involving left-handed neutrinos. We consider scalar, pseudo-scalar
and right-handed tensor NP interactions, as well as left- and right-handed
corrections to the SM, with interaction strengths given by Wilson
coefficients. The effective Hamiltonian is a low-energy realization of the
Standard Model Effective Field Theory (SMEFT) [57, 58].

The QCD input can be parametrized in terms of hadronic structure functions
built from matrix elements of the operators involved. These structure
functions are written in terms of transition form factors, which depend on the
masses of the initial and final particles and on the invariant mass of the
outgoing lepton pair. These form factors are directly taken from lattice QCD
determinations, when possible. Otherwise we resort to approximations to
express the unknown form factors in terms of known ones.

In line with the standard hypothesis of flavour hierarchy of NP, we assume
that $\ell=e$ is well described by the SM and focus on the $\ell=\mu$ channel.
The fact that (pseudo)scalar and tensor NP contributions are
$\mathcal{O}(m_{\ell})$ and therefore suppressed in the $e$ channel further
reinforces the assumption.

In Sec. II we first present the effective Hamiltonian, define the observables
under consideration, the QCD input, and present the NP contributions and their
constrains. The sensitivity to BSM effects is explored in Sec. III for LFU
ratios and in Ref. IV for the coefficients in the angular decomposition of the
decay rate. The conclusions of our study are summarised in Sec. V.

##  II  The $\Lambda_{c}\to\Lambda\,\ell^{+}\nu_{\ell}$ weak decay including
new physics

###  II.1 Effective Hamiltonian

To investigate the $\Lambda_{c}\to\Lambda$ semileptonic decay we rely on the
most general low-energy dimension-six effective Hamiltonian for the $c\to
s\,\ell^{+}\,\nu_{\ell}$ transition, considering only left-handed neutrinos
and lepton-flavour conserving transitions (see for instance Ref. [59])

$\displaystyle H_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{4G_{F}}{\sqrt{2}}V_{cs}\Big{[}(1+\epsilon^{\ell}_{L})\,(\bar{s}\gamma_{\alpha}P_{L}c)\
(\bar{\nu}_{\ell}P_{R}\gamma^{\alpha}\ell)+\epsilon^{\ell}_{R}\,(\bar{s}\gamma_{\alpha}P_{R}c)\
(\bar{\nu}_{\ell}P_{R}\gamma^{\alpha}\ell)$ (1) $\displaystyle\hskip
56.9055pt+\frac{1}{2}\,\epsilon^{\ell}_{S}(\bar{s}c)\
(\bar{\nu}_{\ell}P_{R}\ell)+\frac{1}{2}\,\epsilon^{\ell}_{P}(\bar{s}\gamma_{5}c)\
(\bar{\nu}_{\ell}P_{R}\ell)+\epsilon^{\ell}_{T}(\bar{s}\sigma_{\alpha\beta}P_{R}c)\
(\bar{\nu}_{\ell}\sigma^{\alpha\beta}P_{R}\ell)\Big{]}\,,$

with $P_{L,R}=(1\mp\gamma_{5})/2$ denoting the projectors onto left and right
chiralities, respectively. $G_{F}$ is the Fermi decay constant and $V_{cs}$
the pertinent Cabibbo-Kobayashi-Maskawa (CKM) matrix element. In the domain of
applicability of SMEFT, where the NP scale $\Lambda$ is much higher than the
electroweak one, the strength of NP terms is determined by flavour ($\ell$)
dependent Wilson coefficients $\epsilon^{\ell}_{X}$, which are suppressed by
$v^{2}/\Lambda^{2}$, where $v$ is the Higgs vacuum expectation value [57].
Although these Wilson coefficients are in general complex, we adopt the CP-
conserving limit and treat them as real in the following. Sub-index $X=L$,
$R$, $S$, $P$, $T$ denotes the left-handed, right-handed, scalar, pseudoscalar
and tensor structure of the corresponding $c\to s$ transition
operators111Notice that the Wilson coefficient for the term proportional to
the left-handed quark current is denoted $\epsilon^{\ell}_{L}$ differing with
the notation of Ref. [59] where it is represented as $\epsilon^{\ell}_{V}$..
Note that, for left-handed neutrinos, a tensor operator with quark chirality
$P_{L}$ instead of $P_{R}$ vanishes identically, as can be shown with the help
of Dirac algebra.

###  II.2 Observables

For the
$\Lambda_{c}^{+}(p)\rightarrow\Lambda(p^{\prime})\,\ell^{+}(k^{\prime})\,\nu_{l}(k)$
decay, the double-differential branching ratio
$\mathcal{B}\equiv\Gamma/\Gamma_{\text{tot}}$, in terms of
$q^{2}=(k+k^{\prime})^{2}$ and the angle $\theta$ between the $\Lambda$ and
$\ell^{+}$ three-momenta in the leptonic rest frame, where
$q^{\mu}=(\sqrt{q^{2}},\vec{0})$, can be cast as [60, 59, 21]

$\frac{{\rm d}^{2}\mathcal{B}}{{\rm d}q^{2}{\rm
d}\cos\theta}=|V_{cs}|^{2}\left[c^{\ell}_{0}(q^{2})+c^{\ell}_{1}(q^{2})\cos\theta+c^{\ell}_{2}(q^{2})\cos^{2}\theta\right]\,,$
(2)

where $|V_{cs}|^{2}$ has been factorised for convenience. The full
$\Lambda_{c}$ width $\Gamma_{\text{tot}}=(3.252\pm 0.050)\times 10^{-12}$ GeV
has been obtained from its mean life $\tau=(202.4\pm 3.1)\times 10^{-15}$ s
[61]. The functions $c_{i}(q^{2})$ entirely characterise the decay and can be
extracted from experimentally measurable quantities. For example, as shown in
Eqs. (30)-(32), they can be obtained from the single-differential decay width
${\rm d}\Gamma/{\rm d}q^{2}$, the forward-backward asymmetry,
$\mathcal{A}_{\rm{FB}}$, and the convexity, $\mathcal{A}_{\pi/3}$. Our
analysis of NP contributions is performed in terms of
$c^{\ell}_{0,1,2}(q^{2})$. We take advantage of the general framework
developed in Refs. [21, 23] for the study of hadron semileptonic decays in the
presence of NP. Appendix A discloses how the formalism derived in these
references is adapted to the present case.

In order to reduce the theoretical and experimental uncertainties, it is
useful to consider LFU ratios. In our case

$R_{i}\equiv\frac{c_{i}(m_{\mu},q^{2},\epsilon^{\mu})}{c_{i}(m_{e},q^{2},\epsilon^{e})}=\frac{c_{i}^{\mu}(q^{2},\epsilon^{\mu})}{c_{i}^{e\rm{SM}}(q^{2})}+\mathcal{O}(\epsilon^{e})\underbrace{\simeq}_{\text{if
}\epsilon^{e}\ll\epsilon^{\mu}}1+\mathcal{O}(m_{\mu}^{2}/x^{2})+\mathcal{O}(\epsilon^{\mu})\
,$ (3)

with $x^{2}=q^{2},m_{\Lambda}^{2},m_{\Lambda_{c}}^{2}$. The common assumption
$\epsilon^{e}\ll\epsilon^{\mu}$ arises, for example, in many NP models which
explain the fermion mass hierarchy [62, 63]. In these ratios, not only $V_{\rm
CKM}$ cancels out but also an important advantage resides in the fact that in
the SM the deviation from unity enters at $\mathcal{O}(m_{\mu}^{2}/x^{2})$ and
so does the SM uncertainty. By using ratios a significant improvement in the
sensitivity to NP is hence achieved, as compared to $c_{i}$ values themselves,
where the SM uncertainties enter without any suppression.

###  II.3 QCD input

The functions $c_{0,1,2}(q^{2})$ depend on the NP Wilson coefficients
introduced above and on purely hadronic structure functions which are given in
terms of transition form factors (see Appendix E of Ref. [21]). For a
$J^{P}=1/2^{+}\to 1/2^{+}$ transition, like the $\Lambda_{c}\to\Lambda$ one,
there are 12 real $q^{2}$ dependent form factors: scalar and pseudoscalar
$F_{S,P}$, vector $F_{1,2,3}$, axial-vector $G_{1,2,3}$ and tensor
$T_{1,2,3,4}$. Adopting the notation of Ref. [21] and references therein

$\displaystyle\langle\Lambda(p^{\prime})|\bar{s}\left(1-\gamma_{5}\right)c|\Lambda_{c}(p)\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{\Lambda}(p^{\prime})\left(F_{S}-\gamma_{5}F_{P}\right)u_{\Lambda_{c}}(p),$
$\displaystyle\langle\Lambda(p^{\prime})|\bar{s}\gamma^{\alpha}\left(1-\gamma_{5}\right)c|\Lambda_{c}(p)\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{\Lambda}(p^{\prime})\left\\{\gamma^{\alpha}\left(F_{1}-\gamma_{5}G_{1}\right)+\frac{p^{\alpha}}{M}\left(F_{2}-\gamma_{5}G_{2}\right)+\frac{p^{\prime\alpha}}{m}\left(F_{3}-\gamma_{5}G_{3}\right)\right\\}u_{\Lambda_{c}}(p)\,,$
(4)
$\displaystyle\langle\Lambda(p^{\prime})|\bar{s}\sigma^{\alpha\beta}c|\Lambda_{c}(p)\rangle$
$\displaystyle=$
$\displaystyle\bar{u}_{\Lambda}(p^{\prime})\left\\{\frac{i}{M^{2}}\left(p^{\alpha}p^{\prime\beta}-p^{\beta}p^{\prime\alpha}\right)T_{1}+\frac{i}{M}\left(\gamma^{\alpha}p^{\beta}-\gamma^{\beta}p^{\alpha}\right)T_{2}\right.$
$\displaystyle\hskip
31.2982pt\left.+\frac{i}{M}\left(\gamma^{\alpha}p^{\prime\beta}-\gamma^{\beta}p^{\prime\alpha}\right)T_{3}+\sigma^{\alpha\beta}T_{4}\right\\}u_{\Lambda_{c}}(p)\,,$

where $M(m)$ denotes the $\Lambda_{c}$($\Lambda$) mass. The vector and axial
form factors have been calculated within LQCD in Ref. [64] using gauge field
configurations generated by the RBC and UKQCD collaborations with $2+1$
flavours of dynamical domain-wall fermions222A recent extraction of these form
factors in the framework of QCD sum rules is also of interest [65].. Two
different lattice spacings are analysed and the physical pion mass is reached
for one ensemble. The translation of the form factors from the helicity basis
adopted in Ref. [64] to the one adopted here is given by Eq. (E1) of Ref. [21]
once the baryon masses are changed accordingly. Using these form factors, in
agreement with Ref. [64] and propagating the statistical uncertainties
according to the correlation matrices provided in Ref. [64] we obtain:

$\frac{\Gamma(\Lambda_{c}\to\Lambda
e^{+}\nu_{e})}{\absolutevalue{V_{cs}}^{2}}=0.2008(71)^{\rm LQCD}_{\rm{stat}}\
\rm{ps}^{-1},\quad\frac{\Gamma(\Lambda_{c}\to\Lambda\mu^{+}\nu_{\mu})}{\absolutevalue{V_{cs}}^{2}}=0.1945(69)^{\rm
LQCD}_{\rm{stat}}\ \rm{ps}^{-1},\ $ (5)

yielding the LFU ratio

$R_{\mathrm{SM}}=\frac{\Gamma(\Lambda_{c}\to\Lambda\mu^{+}\nu_{\mu})}{\Gamma(\Lambda_{c}\to\Lambda
e^{+}\nu_{e})}=0.96884(61)^{\rm LQCD}_{\rm{stat}}\ .$ (6)

We notice the difference between this SM ratio and the value of $0.974(1)$
reported in Ref. [25] although both are consistent with the experimental
result of $R^{\rm exp}=0.98(5)_{\rm stat}(3)_{\rm syst}$ [56] owing to its
large error.

Ideally one would also take the rest of form factors from LQCD, relying in
this way on an accurate parametrisation of the QCD structure. However this
input is not yet available. Therefore, we express the scalar, pseudoscalar and
tensor form factors in terms of the vector and axial ones employing
approximations based on the fact that the charm quark mass is considerably
larger than the masses of the light ones: $m_{c}\gg\Lambda_{\rm{QCD}}\gg
m_{s,u,d}$. There is some freedom in the way one exploits this condition. The
resulting differences are used to estimate the systematic uncertainty of the
approximations. As shown in Sec. (2.10) of Ref. [66], HQSS for a heavy-light
quark transition implies that at leading order in $1/m_{c}$ all possible form
factors can be given in terms of only two independent functions. For this
purpose we choose either $F_{1,2}$ or $G_{1,2}$ ($F_{3}=G_{3}=0$ in the HQSS
limit). We refer to these prescriptions as HQSSV and HQSSA respectively. A
less stringent choice arises from the assumption that the $c$ quark is on-
shell. Then, in the $m_{c}/M\to 1$ limit, one obtains a consistent set of
equations which allow to relate all (pseudo)scalar and tensor form factors to
$F_{1,2,3}$ and $G_{1,2,3}$. This solution, labelled OSHQ (for On-Shell Heavy
Quark), is consistent with HQSS in the $F_{3}=G_{3}=0$ limit. The three
prescriptions are listed below. Details of the derivations can be found in
Appendix B.

  1. 1.

OSHQ:

$\displaystyle F_{S}$ $\displaystyle=F_{1}+F_{2}+\omega F_{3}\ ,\qquad\qquad$
$\displaystyle F_{P}=G_{1}-G_{2}-\omega G_{3}\ ,$ $\displaystyle T_{1}$
$\displaystyle=-\frac{M}{m}\left(F_{3}+G_{3}\right)\ ,\qquad\qquad$
$\displaystyle T_{2}=F_{1}-G_{1}-G_{3}\ ,$ $\displaystyle T_{3}$
$\displaystyle=\frac{M}{m}G_{3}\ ,\qquad\qquad\qquad\qquad$ $\displaystyle
T_{4}=G_{1}-\left(\omega-1\right)G_{3}\ ,$ (7)

where $\omega=(M^{2}+m^{2}-q^{2})/(2Mm)$ is the scalar product of the four-
velocities of the initial and final hadrons.

  2. 2.

HQSSV:

$F_{S}=F_{1}+F_{2}\ ,\quad F_{P}=F_{1}\ ,\quad T_{1}=0\ ,\quad T_{2}=-F_{2}\
,\quad T_{3}=0\ ,\quad T_{4}=F_{1}+F_{2}\ .$ (8)

  3. 3.

HQSSA:

$F_{S}=G_{1}\ ,\quad F_{P}=G_{1}-G_{2}\ ,\quad T_{1}=0\ ,\quad T_{2}=-G_{2}\
,\quad T_{3}=0\ ,\quad\ T_{4}=G_{1}\ .$ (9)

(a)

(b)

(c)

Figure 1: Scalar, pseudoscalar and tensor form factors obtained from the
vector and axial ones using different approximations. The continuous, dashed
and dotted lines are obtained with OSHQ, HQSSV and HQSSA prescriptions,
respectively. Notice that $T_{1}=T_{3}=0$ within HQSSV and HQSSA.

Figure 1 displays $F_{P,S}$ and $T_{1-4}$ in terms of $F_{1-3}$ and $G_{1-3}$
from LQCD [64] in the $q^{2}$ range of relevance for the $\Lambda_{c}$ decay
using the three prescriptions. In the following, our theoretical error bands
for each of these form factors cover the results of all three prescriptions
including the LQCD errors given by the bands in Fig. 1. This defines a
conservative estimate of the QCD uncertainty.

###  II.4  New physics contributions

In order to analyse individual NP contributions to $c^{\ell}_{0,1,2}$ we
introduce

$c^{\ell}_{i}(q^{2},\epsilon^{\ell}_{X})\equiv
c^{\ell}_{i}(q^{2},\epsilon^{\ell}_{X},\epsilon^{\ell}_{Y\neq X}=0)\,,\qquad
X,Y=L,R,S,P,T\,,\,i=0,1,2$ (10)

which can be cast as (lepton index and $q^{2}$ dependence are implicit)

$\displaystyle c_{i}(\epsilon_{X})$ $\displaystyle=$ $\displaystyle
c_{i}^{\text{SM}}+\epsilon_{X}c_{i}^{X\rm{l}}+\epsilon_{X}^{2}c_{i}^{X\rm{q}}=c_{i}^{\text{SM}}+\epsilon_{X}\left(c_{i}^{X\rm{l}}+\epsilon_{X}c_{i}^{X\rm{q}}\right)\equiv
c_{i}^{\text{SM}}+\epsilon_{X}c^{X}_{i}(\epsilon_{X})\ .$ (11)

The NP part has terms linear and quadratic in $\epsilon_{X}$. The leading
linear part $\epsilon_{X}c_{i}^{X\rm{l}}$ arises from the interference between
SM and NP terms in the decay amplitude squared. On the other hand, one has to
keep in mind that the sub-leading $\epsilon^{2}_{X}c_{i}^{X\rm{q}}$ terms
provide an incomplete account of the $\mathcal{O}(\epsilon^{2})$ corrections,
which would also receive contributions from the interference between the SM
and terms from a higher order $(\epsilon^{2})$ SMEFT Hamiltonian, controlled
by additional Wilson coefficients [67].

(a)

(b)

(c)

Figure 2: $q^{2}$ dependence of the NP individual perturbations
$c^{X}_{i}=c_{i}^{X\text{l}}+\epsilon_{X}c_{i}^{X\text{q}}$, Eq. (11)
($i=0,1,2$ from left to right). Each colour denotes a type of interaction:
$X=L$ (blue), $R$ (red), $S$ (black), $P$ (brown) and $T$ (green). Wilson
coefficients are set to their central values in Ref. [60] (see text). The
dashed lines originate from the linear terms $c_{i}^{X\rm{l}}$. The bands
represent the QCD uncertainty from the transition form factors.

The structure of the NP perturbation of the SM values is therefore encoded in
$c^{X}_{i}(\epsilon_{X})$, which are depicted in Fig. 2 for $\ell=\mu$.
Propagated QCD uncertainties, from LQCD and HQSS prescriptions discussed in
Sec. II.3, give rise to the bands. The dashed lines denote the linear
$c_{i}^{X\rm{l}}$ terms that contribute at $\mathcal{O}(1/\Lambda^{2})$.
Although for $X=S,P,T$ these interference terms are suppressed by the muon
mass, the tensor contribution turns out to be comparable or larger than the
$X=L,R$ ones for $c_{0}$ and $c_{1}$ observables. To account for the residual
$\epsilon_{X}$ dependence of $c^{X}_{i}(\epsilon_{X})$, we choose as
indicative the central $\epsilon_{X}$ values extracted in Ref. [60] from
leptonic and semileptonic decays of pseudoscalar mesons. Under the assumption
of real Wilson coefficients, the relations between the definitions in Ref.
[60] ($g$) and the ones in the present work ($\epsilon$) are

$\epsilon_{L,R}=\frac{g_{V}\mp
g_{A}}{2}=g_{V_{L,R}},\quad\epsilon_{S,P,T}=g_{S,P,T}\,.$ (12)

We therefore obtain the following ranges: $\epsilon_{L}=(1.2\pm 2.0)\times
10^{-2}$, $\epsilon_{R}=(-0.9\pm 2.0)\times 10^{-2}$, $\epsilon_{S}=(-1\pm
2)\times 10^{-2}$, $\epsilon_{P}=(0.7\pm 1.4)\times 10^{-3}$,
$\epsilon_{T}=(1.2\pm 1.8)\times 10^{-2}$.333These values differ from those in
Ref. [59] although they are obtained from the same source.

The smallness of the quadratic terms is apparent from the difference between
the dashed and the central solid lines, which can hardly be discerned except
for $c_{0,2}^{T}$. Indeed, $\epsilon_{T}c_{0}^{T\text{q}}$ is relatively large
for a typical value of $\epsilon_{T}=1.2\times 10^{-2}$. The tensor
contribution to $c_{2}$ is purely quadratic ($c_{2}^{T\text{l}}=0$) and quite
large. $c_{0}$ is the only observable where the quadratic, $c_{0}^{S\rm{q}}$
and $c_{0}^{P\rm{q}}$, scalar and pseudoscalar contributions are exactly zero.

The plots of Fig. 2 show the magnitude and shape of the individual
contributions. They allow us to identify the NP perturbations that are in
principle more easily revealed in an experiment. The sensitivity to these NP
contributions is carefully studied in Sec. III. To assess the results of Fig.
2 from this perspective it is useful to bear in mind that
$c_{i}^{\text{SM}}=c_{i}^{L\rm{l}}/2$. For $c_{0}$ [Fig. 2 (a)], tensor and
left-handed contributions prevail, while the right-handed one is also sizeable
and changes sign. In $c_{1}$ [Fig. 2 (b)] the tensor interaction is the
largest, peaking in the $q^{2}$ region where the SM has a local minimum. Since
the SM relative error is quite constant in $q^{2}$, a minimum in the SM
magnitude corresponds to a minimum in the SM error (see the width of the
$c_{1}^{L}$ band). Therefore one can expect a good sensitivity to tensor NP at
low $q^{2}$ in $c_{1}$. Finally, in $c_{2}$ [Fig 2 (c)] the left-handed term
dominates, having a maximum in its absolute value at low $q^{2}$. The
suppression of all contributions with respect to the left-handed one implies
that the latter is the only potentially relevant NP mechanism in $c_{2}$. It
is also worth noting that the scalar and pseudoscalar contributions linear in
$\epsilon_{S,P}$ are small in $c_{0,1}$ and exactly zero in $c_{2}$.

###  II.5 Phenomenological constraints

In the first row of Fig. 3 functions $c_{i}(q^{2})$ are plotted in the SM and
also adding NP according to the aforementioned phenomenological constraints to
the Wilson coefficients from Ref. [60] in the $\mu$ channel. We account
simultaneously for all NP effects, considering linear terms as well as
quadratic ones, which include interference
$\mathcal{O}(\epsilon_{X}\epsilon_{Y}$) contributions. We have varied the
$\epsilon_{X}$ values in the given ranges, approximating their probability
distributions by Gaussians and neglecting correlations. The LQCD errors of the
vector and axial form factors have been propagated according to the
correlation matrices given in Ref. [64]. All three prescriptions of the HQSS
form factors in $c_{i}^{S,P,T}$ are encompassed by a single band.444In this
case the HQSSV choice practically covers the bands of the other two
prescriptions. Given that the Wilson coefficients under study are compatible
with zero, the SM and NP bands overlap even in the absence of QCD
uncertainties. Nevertheless, one can already see where the possible NP
contribution is more prominent. Figure 3 (b) reflects that the SM local
minimum is a good place to look for a NP (tensor) signal. Indeed, it is
apparent that at this local minimum the SM and NP central values are more
separated than anywhere else.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: $q^{2}$ dependence of $c^{\mu}_{i}$ and $\mu/e$ LFU ratios $R_{i}$
in the SM and in the presence of NP (with the empirical values obtained in
Ref. [60]). The SM bands (grey) include errors in the LQCD vector form
factors. The NP bands (red) arise from both the uncertainties in all form
factors and in the Wilson coefficients.

Under the assumption of the standard hierarchy,
$\epsilon_{X}^{\mu}\gg\epsilon_{X}^{e}$, one would expect to observe
deviations from the SM only in the muon channel. However, the comparison of
the SM differential decay width, evaluated with LQCD input, to the BESIII data
of Refs. [55, 56] shown in Fig. 4 reveals tensions not only in the $\ell=\mu$
mode but also in the $\ell=e$ one. Moreover, the SM fits performed in Refs.
[55] produce form factors that clearly differ from those of Ref. [64], as can
be seen in Fig. 3 of Ref. [55]. We have also obtained that this tension with
experimental data cannot be accommodated by natural $\epsilon^{e}_{X}\neq 0$.

Therefore, the extraction of allowed ranges on $\epsilon^{\mu}_{X}$ under the
assumption of $\epsilon^{e}\ll\epsilon^{\mu}$ directly from the present data
on $\Lambda_{c}\to\Lambda\mu^{+}\nu_{\mu}$ [56] would be undermined by these
discrepancies in the $\ell=e$ channel that cast doubts about the QCD input.

Figure 4: Differential decay width for $\Lambda_{c}^{+}\to\Lambda
e^{+}\nu_{e}$ (left) and $\Lambda_{c}^{+}\to\Lambda\mu^{+}\nu_{\mu}$ (right)
decay modes. The grey band is the SM prediction with 1$\sigma$ errors from the
LQCD form factors (neglecting 1% error in $V_{cs}$). The blue points are the
experimental data from BESIII [55, 56].

Turning now to the LFU violation ratios, Eq. (3), setting $\epsilon^{e}_{X}=0$
one gets

$R_{i}(\epsilon^{\mu}_{X})=\frac{c_{i}(\Lambda_{c}\to\Lambda\mu^{+}\nu_{\mu},\epsilon^{\mu}_{X})}{c_{i}^{\rm
SM}(\Lambda_{c}\to\Lambda e^{+}\nu_{e})}\ ,$ (13)

which depend on all $\epsilon^{\mu}_{X}$, and implicitly on $q^{2}$. They are
shown in the second row of Fig. 3 with $\epsilon^{\mu}_{X}$ values set to the
range of the phenomenological constraints from Ref. [60]. As anticipated, the
reduction of the SM uncertainty is evident: the SM error bands become
negligible compared with the bands generated by the $\epsilon^{\mu}_{X}$
ranges. This provides a much higher sensitivity to NP.

##  III Sensitivity to new physics in the $\mu$ channel: LFU ratios

We have posed ourselves the question of how large should a new physics
contribution be in order to be experimentally identifiable, given the QCD
uncertainties. We focus on the $\mu$ channel, under the assumption
$\epsilon^{\mu}\gg\epsilon^{e}\simeq 0$, and study ratios to reduce the
uncertainties. We have defined the signals to which we are sensitive as the
ones that do not overlap with the SM prediction within $1\sigma$ errors. In
other words, if a Wilson coefficient has a given true value, $\epsilon_{X}$,
we would be sensitive to it at a certain $q^{2}$ only if its magnitude is
large enough to completely separate the $1\sigma$ NP band from the SM one.
This condition has some degree of arbitrariness and would still be
insufficient to claim a discovery in an experiment but serves the purpose of
getting an idea of the NP discovery potential of each of the LFU ratios,
$R_{i}$, as functions of $q^{2}$. We define $\epsilon_{X}^{\rm{sens}}$ as the
minimum, in magnitude, $\epsilon_{X}$ value which satisfies this condition. We
evaluate them one by one, setting the rest of the Wilson coefficients to zero.
Recalling the notation of Eq. (13) we define $\epsilon_{X}^{\rm{sens}}$ as the
one that satisfies

$\absolutevalue{R_{i}(\epsilon_{X})-R_{i}^{\rm{SM}}}>\Delta
R_{i}(\epsilon_{X})+\Delta R_{i}^{\rm{SM}}$ (14)

with a minimal $|\epsilon_{X}|$. Here $\Delta R_{i}^{SM}$ and $\Delta
R_{i}(\epsilon_{X})$ denote the $1\sigma$ errors of $R_{i}^{\rm{SM}}$ and
$R_{i}(\epsilon_{X})$, which includes NP. On the one hand, the larger the
$\epsilon_{X}$, the more promising the observable would be to detect a given
NP signal. On the other hand, the SMEFT calculation has a worse convergence
for larger $\epsilon_{X}$. We take this into account by introducing a SMEFT
truncation error within $\Delta R_{i}(\epsilon_{X})$, which is defined below.

We explore the condition of Eq. (14) for positive and negative $\epsilon_{X}$
separately. In general we do not obtain the same absolute value in both cases
due to quadratic corrections. Quadratic terms in SMEFT are briefly discussed
in Ref. [68], while the theoretical uncertainty of SMEFT is addressed in Sect.
2.7.6 of Ref. [69]. According to this reference, to properly account for the
truncation error one should calculate to one-loop order. It is also argued
that an estimate can be given without such a computation, but including Wilson
coefficients of the dimension-eight Lagrangian (Eq. (2.79) therein). Such
studies are beyond the scope of the present work. We content ourselves with a
simpler approach, relying on a common truncation error estimate given by the
difference between linearization and quadratization of the matrix element
squared, and define the following uncertainty for $c_{i}$, Eq. (11)

$\Delta
c_{i}^{\text{SMEFT}}(\epsilon_{X})=\epsilon_{X}^{2}\max\\{|c_{i}^{\rm{SM}}|,|c^{X\rm{l}}_{i}|,|c^{X\rm{q}}_{i}|\\}\,.$
(15)

It estimates the size of the unaccounted $\mathcal{O}(\epsilon^{2})$
contributions, only with information from $\mathcal{L}_{\rm{SM}}$ and the NP
dimension-six Lagrangian $\mathcal{L}_{6}$. To simplify the notation, index
$\ell=\mu$ has been omitted throughout this section. Correspondingly,

$\Delta R_{i}^{\text{SMEFT}}(\epsilon_{X})=\frac{\Delta
c_{i}^{\text{SMEFT}}(\epsilon_{X})}{c_{i}^{e{\rm SM}}}\ .$ (16)

We consider this a conservative estimation of the truncation error.

In order to obtain $\epsilon_{X}^{\rm{sens}}$ from Eq. (14) we sum in
quadratures the error propagated from the LQCD vector and axial form factors,
$\Delta R_{i}^{\rm{LQCD}}$, with the truncation one, yielding

$\left(\Delta R_{i}(\epsilon_{X})\right)^{2}=\left(\Delta
R_{i}^{\rm{LQCD}}(\epsilon_{X})\right)^{2}+\left(\Delta
R_{i}^{\text{SMEFT}}(\epsilon_{X})\right)^{2}\ .$ (17)

In addition, the HQSS uncertainty in $c_{i}^{S,P,T}$ is taken into account by
covering all three versions of the form factors introduced in Sec. II.5. In
this case, this reduces to taking the least favourable choice, namely the HQSS
prescription associated with the minimum perturbation to the SM, maximising
therefore $\absolutevalue{\epsilon_{X}^{\rm{sens}}}$.

First of all, we study the ratio of integrated widths, i.e. we introduce NP
contributions in the numerator of Eq. (6). In this observable,
$\absolutevalue{\epsilon_{L}^{\rm{sens}}}=6.3\times 10^{-4}$,
$\absolutevalue{\epsilon_{R}^{\rm{sens}}}=2.1\times 10^{-3}$,
$\absolutevalue{\epsilon_{T}^{\rm{sens}}}=8.0\times 10^{-4}$,
$\absolutevalue{\epsilon_{S}^{\rm{sens}}}=8.9\times 10^{-3}$, but we are not
sensitive to $\epsilon_{P}$. This is a very good sensitivity, beyond the
multi-TeV scale. Indeed, the $\epsilon$ value corresponding approximately to a
NP scale $\Lambda$ of 1 TeV is estimated by a naive power counting (see Sec.
2.1 of Ref. [69]), to be $\absolutevalue{\epsilon}=v^{2}/\Lambda^{2}\simeq
0.0625$, with $v\approx 250$ GeV.

Next, we analyse the LFU ratios defined in Eq. (13). The first thing to notice
is that the SM $R_{2}$ is independent of the form factors. Therefore $\Delta
R_{2}^{\rm SM}=0$. Applying the criterion of Eq. (14), we are sensitive to
arbitrarily small NP $\epsilon_{L,R,T}$, which are the ones that contribute to
$c_{2}$. This has to be understood as a result of neglecting
$\mathcal{O}(\epsilon^{2})$ (and disregarding the experimental uncertainty).
Nevertheless, this result clearly indicates that $R_{2}$ is a very sensitive
observable to NP contributions. The drawback is that it might be difficult to
measure, because $c_{2}$ is related to the angular asymmetry
$\mathcal{A}_{\pi/3}$, Eq. (32).

(a)

(b)

(c)

(d)

Figure 5: The sensitivity regions to $\epsilon_{X}$ in $R_{0}$ as a function
of $q^{2}$ are depicted in light blue. The solid blue line represents
$\epsilon_{X}^{\rm{sens}}$ defined in Eq. (14). Currents of chirality $L$,
$T$, $R$ and $S$ are studied in (a), (b), (c) and (d), respectively. Notice
the different scale of the ordinate axis in upper and lower panels. The red
solid (dashed) lines represent the Wilson coefficient value for a $\Lambda=10$
TeV ($\Lambda=2$ TeV) in the naive power counting. Black vertical lines
delimit the phase space. $R_{0}$ is sensitive to Wilson coefficients smaller
than the central values obtained in Ref. [60].

We have also observed that $R_{0}$, which is dominated by the ratio of decay
widths, is more sensitive to NP than $R_{1}$. In consequence we put the focus
on $R_{0}$ in the following. In Fig. 5 we study the sensitivity of $R_{0}$ to
NP signals of different chirality, $\epsilon_{X}$, as a function of $q^{2}$.
The $\epsilon_{X}^{\rm{sens}}(q^{2})$ are reported; they set the boundaries of
the regions of sensitivity. Red lines denote the Wilson coefficient values
corresponding to a NP scale of 2 (dashed) and 10 TeV (solid). The plots reveal
an excellent sensitivity to $\epsilon_{L}$ and $\epsilon_{T}$ (upper panels).
For $\epsilon_{R}$, the sensitivity is higher in the region of medium to high
$q^{2}$ (lower left panel). We are also sensitive to quite small
$\epsilon_{S}$ (lower right panel). The only current to which this ratio is
not sensitive is the pseudoscalar one. In this case the $\epsilon_{P}$ signal
is too small to be discerned from the SM, taking into account the
$\mathcal{O}(\epsilon_{P}^{2})$ theoretical error.

Finally, from our analysis we conclude that the main obstacle to resolve a NP
signal from the theoretical point of view is the LQCD error of the
(axial)vector form factors, prevailing over the HQSS errors that accompany
some NP contributions, and the SMEFT uncertainty, which can be sizeable for
relatively large $\epsilon_{X}$.

One may also wonder about the experimental uncertainty and therefore the real
sensitivity in practice. At present, the experimental error is much larger
than the theoretical one, at least in observables where SM uncertainties
cancel. To illustrate this, we consider the integrated ratio $R$ (whose SM
value is given in Eq. (6)), setting all the $\epsilon_{X\neq L}=0$ and
accounting for the theoretical uncertainties. For instance, to be sensitive to
$\epsilon_{L}=1.2\times 10^{-2}$ as the central value extracted in Ref. [60],
one would need the experimental error to be reduced to 10% of its current
size555Recall that we set all the $\epsilon_{X\neq L}=0$ and account for the
theoretical uncertainties.. In addition, as discussed in Sec. II.5, the
tensions that arise in the description of $\Lambda_{c}\to\Lambda e^{+}\nu_{e}$
with SM LQCD input compromise the extraction of $\epsilon_{X}^{\mu}$ from
$\Lambda_{c}\to\Lambda\mu^{+}\nu_{\mu}$ data alone. On the other hand,
recalling the aforementioned $\mathcal{O}(m_{\mu}^{2})$ suppression of the
form factor uncertainty in the SM contribution to LFU ratios, we find that
these ratios are more reliable in order to look for NP deviations, because
systematic differences in the SM form factor largely cancel out. In Fig. 6 we
report the LFU ratio for the differential decay width, defined in analogy with
Eq. (13) as

$R_{\Gamma}(\epsilon^{\mu}_{X})=\frac{d\Gamma/dq^{2}(\Lambda_{c}\to\Lambda\mu^{+}\nu_{\mu},\epsilon^{\mu}_{X})}{d\Gamma/dq^{2\,{\rm
SM}}(\Lambda_{c}\to\Lambda e^{+}\nu_{e})}\ ,$ (18)

compared to the corresponding experimental result from BESIII [56]. The
theoretical result includes a black 1$\sigma$ SM band from LQCD and a red one
accounting for the allowed range in the Wilson coefficients from Ref. [60]. In
line with the previous comments about the ratio of integrated widths, the
present experimental uncertainty is still insufficient to put competitive
bounds on the $\epsilon_{\mu}$ Wilson coefficients.

Figure 6: $\mu/e$ LFU ratio for the differential decay width ${\rm
d}\Gamma/{\rm d}q^{2}$ as a function of $q^{2}$. The black band stands for the
SM prediction from 1$\sigma$ errors in the LQCD form factors. The red band
corresponds to the inclusion of NP with the empirical bounds obtained in Ref.
[60]. The blue points are the experimental data from BESIII [56].

##  IV Sensitivity to new physics in the $\mu$ channel: $c^{\mu}_{i}$
functions

Here we report an analysis similar to the one performed for LFU ratios but at
the level of the $c^{\mu}_{i}(q^{2})$ functions. It is free of assumptions on
the flavour hierarchy of Wilson coefficients. Furthermore,
$c^{\mu}_{i}(q^{2})$ would not receive any contribution from the statistical
experimental error in the $\Lambda_{c}\to\Lambda e^{+}\nu_{e}$ decay.
Unfortunately, the sensitivity is considerably reduced in comparison with the
one expected for the LFU ratios explored in the previous section. We present
only the cases where the NP perturbation was found to be relatively large in
Sec. II.4 (see Fig. 2). In consequence, we do not take into account
(pseudo)scalar terms.

The left-handed contributions to $c_{0,1,2}$ are studied in Fig. 7 (a)-(c). In
this case the truncation error is governed by
$\epsilon_{L}^{2}c_{i}^{L\rm{l}}=2\epsilon_{L}^{2}c_{i}^{\rm{SM}}$ and the
convergence is good, at least from the perspective of the uncertainty that we
have defined. The error is dominated by the one of the LQCD vector form
factors, even for high values of $\epsilon_{L}$. The plots show that the study
of $c_{0,1,2}$ for the present process is mostly insensitive to the
$\epsilon_{L}$ range constrained by pseudoscalar meson decays in Ref. [60].

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: The sensitivity regions to $\epsilon_{X}$ in selected $c_{i}$ is
represented by the light blue regions. The solid blue line denotes
$\epsilon_{X}^{\rm{sens}}$ defined in Eq. (14). Panels (a), (b) and (c) show
the sensitivity to $\epsilon_{L}$ in $c_{0}$, $c_{1}$ and $c_{2}$,
respectively. The (d) panel reports the poor sensitivity to the right-handed
contribution in $c_{0}$ (note the different scale). The (e) and (f) panels
display the sensitivity to $\epsilon_{T}$ in $c_{0}$ and $c_{1}$ respectively.
The dashed lines encompass the allowed region obtained in Ref. [60] from
pseudoscalar meson decays. The red lines represent the values of Wilson
coefficients for a $\Lambda=1$ TeV in the naive power counting. Black vertical
lines delimit the phase space.

The right-handed contributions are in general modest, being more prominent in
$c_{0}$ than in $c_{1}$ and $c_{2}$. We show the sensitivity to $\epsilon_{R}$
in $c_{0}$ in Fig. 7 (d). It is much poorer than in the other displayed cases.
Indeed this observable is only sensitive to quite low $\Lambda$ (below the
TeV) in the naive power counting estimation666Approaching the electroweak
scale, less NP models survive that have not been ruled out empirically.. At
low $q^{2}$, $c_{0}$ is not sensitive to $\epsilon_{R}$ at all. In other
words, the condition in Eq. (14) is not satisfied. Actually, due to the small
magnitude of $c_{0}^{R}$ at low $q^{2}$, $\epsilon_{R}$ needs to be large to
be distinguishable. However, at such $\epsilon_{R}$ values, the truncation
error becomes large enough to make Eq. (14) impossible to satisfy. The
difference between positive and negative $\epsilon_{R}^{\rm{sens}}$ is caused
by the $\epsilon_{R}^{2}c_{0}^{R\rm{q}}$ term, which is important at high
$\epsilon_{R}$. A similar scenario is encountered for the tensor contribution
to $c_{0}$ studied in Fig. 7 (e). Now the decay under study is not sensitive
to negative $\epsilon_{T}$. This is due to the twofold impact of the quadratic
term in Eq. (14), which is relatively large: not only does it notably increase
$\Delta c_{0}(\epsilon_{T})$, it also conspires to cancel the linear
contribution to $c_{0}(\epsilon_{T})$, having the opposite sign. As argued in
the previous section, the quadratic contribution is not complete. Hence, the
fact that $\epsilon_{R,T}^{2}$ are large implies that our sensitivity
estimates are less solid in these cases. This is partially accounted by the
truncation error. In all the other $c_{i}^{X}$ analysed in Fig. 7, $\Delta
c_{i}^{\rm{SMEFT}}$ is negligible for low values of $\epsilon_{X}$
($\epsilon_{X}\leq 0.1$).

The most promising analysis turns out to be the study of $\epsilon_{T}$ in
$c_{1}$, Fig. 7 (f). As anticipated in Sec. II.4, the maximum in the tensor
contribution coincides with a SM minimum (Fig. 2 (b)). This leads to a good
sensitivity in the low-medium $q^{2}$ region as can be seen in Fig. 7 (f). The
quadratic coefficient $c_{1}^{T\rm{q}}$ is small and the truncation error is
negligible (it is given by $\epsilon_{T}^{2}c_{1}^{T\rm{l}}$). The critical
uncertainty in this scenario is actually the one of the LQCD vector and axial
form factors $\Delta c_{1}^{\rm{LQCD}}$. In this case, the
$\Lambda_{c}\to\Lambda\mu^{+}\nu_{\mu}$ decay is sensitive to a part of the
band allowed by pseudoscalar meson decays [60] without resorting to ratios.
Finally, the tensor contribution to $c_{2}$ is not discussed because there is
no linear term in $\epsilon_{T}$.

As a final remark, we stress that the key uncertainties to asses in order to
gain sensitivity to NP with the $\Lambda_{c}$ semileptonic decay are the ones
of the LQCD vector and axial form factors because they appear in the leading
SM term. In the case of the tensor contribution to $c_{0}$, this error
competes with the truncation one. In this sense, a more precise estimate of
the SMEFT uncertainty, taking into account all terms at
$\mathcal{O}(1/\Lambda^{4})$ would be very useful as well. In addition, it is
clear that a reliable determination of the truncation error would provide more
consistency to the analysis in general, including the left and right-handed
contributions. In the tensor term the HQSS uncertainty is not so critical,
since it is part of the NP correction (hence it is multiplied by
$\epsilon_{T}$) and is therefore small.

##  V Conclusions and outlook

We have studied the $\Lambda_{c}\to\Lambda\ell^{+}\nu_{\ell}$ decay,
investigating the impact of NP currents in the $\ell=\mu$ channel. To
accomplish this task, an effective Hamiltonian for $c\to s\ell^{+}\nu_{\ell}$,
based on SMEFT has been employed. We have also relied on the LQCD results for
the axial and vector $\Lambda_{c}\to\Lambda$ form factors obtained in Ref.
[64]. The form factors required to describe the $\Lambda_{c}\to\Lambda$
transitions driven by scalar, pseudoscalar and tensor operators have been
expressed in terms of the SM axial and vector ones by taking advantage of the
large $c$-quark mass. Differences arising from alternative approximations to
implement this connection among form factors are treated as a theoretical
uncertainty.

The NP analyses are performed in terms of the $c_{i}(q^{2})$ functions that
drive the dependence on the angle between the $\Lambda$ and the $\mu^{+}$ in
the leptonic rest frame. We have first obtained the $q^{2}$ shapes of the NP
contributions to $c_{i}$, with their QCD uncertainty bands. Lepton flavour
universality ratios received special attention. In the first place, owing to
the LQCD form factor input, a SM ratio
$\Gamma(\Lambda_{c}\to\Lambda\mu^{+}\nu_{\mu})/\Gamma(\Lambda_{c}\to\Lambda
e^{+}\nu_{e})=0.96884(61)^{\rm LQCD}_{\rm{stat}}$ is obtained. LFU ratios are
particularly advantageous in the search for NP due to the
$\mathcal{O}(m_{\mu}^{2})$ suppression of the SM uncertainty. This is apparent
in the reported $R_{i}=c_{i}^{\mu}/c_{i}^{e}$, $i=0,1,2$, ratios.

We have then studied the theoretical sensitivity to a potential NP current of
a certain chirality in the $\mu$ channel, accounting also for the theoretical
uncertainty in the SMEFT truncation. Assuming a
$\epsilon^{\mu}\gg\epsilon^{e}$ hierarchy, we have analysed different LFU
ratios. In the ratio of integrated widths the sensitivity is already very
high, probing the multi-TeV scale, except for the pseudoscalar interaction.
However, to realise this theoretical expectation, the experimental uncertainty
should be reduced as detailed in the work. In addition, we have shown that the
ratios $R_{0}$ and $R_{2}$ have an excellent sensitivity to NP of any type
except the pseudoscalar one. Finally, an analogous sensitivity analysis has
been performed on the $c_{i}^{\mu}$ observables themselves, which are free of
the flavour hierarchy assumption. In this case the sensitivity varies notably
depending on the NP chirality. In particular, the NP tensor contribution to
$c_{1}$ displays a promising discovery potential.

We have identified the LQCD uncertainty of the axial and vector form factors
as the most important one when it comes to improving the sensitivity to NP.
Besides, we have also seen that a better estimation of the SMEFT uncertainty
would substantially improve the precision of this kind of analysis.

As a future development, the inclusion of right-handed neutrinos in the
analysis would be interesting, since they are present in a large variety of NP
models and can be implemented in a straightforward way within the framework
established in Refs. [21] and [23].

## Acknowledgements

This work has been partially supported by the Ministry of Science and
Innovation of Spain under grants PID2020-112777GB-I00 and
PID2022-141910NB-I00, funded by MCIN/AEI/10.13039/501100011033, by the EU
STRONG-2020 project under the program H2020-INFRAIA-2018-1, grant agreement
no. 82409 and by Generalitat Valencia’s CIDEGENT program, Ref.
CIDEGENT/2019/015.

##  Appendix A General framework and observables

With the aim of taking advantage of the general analytic results derived in
Refs. [21, 23], we rewrite the Hamiltonian of Eq. (1) as

$\displaystyle H_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{4G_{F}}{\sqrt{2}}V_{cs}\left[C^{V}_{LR}{\cal
O}^{V}_{LR}+C^{V}_{RR}{\cal O}^{V}_{RR}+C^{S}_{LR}{\cal
O}^{S}_{LR}+C^{S}_{RR}{\cal O}^{S}_{RR}+C^{T}_{RR}{\cal O}^{T}_{RR}\right],$
(19)

where

${\cal
O}^{V}_{(L,R)R}=(\bar{s}\gamma_{\alpha}P_{L,R}\,c)(\bar{\ell}^{c}\gamma^{\alpha}P_{R}\nu^{c}_{\ell}),\,{\cal
O}^{S}_{(L,R)R}=(\bar{s}\,P_{L,R}\,c)(\bar{\ell}^{c}\,P_{R}\nu^{c}_{\ell}),\,{\cal
O}^{T}_{RR}=(\bar{s}\,\sigma_{\alpha\beta}P_{R}\,c)(\bar{\ell}^{c}\sigma^{\alpha\beta}P_{R}\nu^{c}_{\ell})\,;$
(20)

$\ell^{c},\nu_{\ell}^{c}$ are the charge-conjugated lepton fields. Taking into
account that

$\displaystyle\bar{\nu}_{\ell}P_{R}\Gamma\ell\equiv-\bar{\ell}^{c}C\Gamma^{T}C^{\dagger}P_{R}\nu^{c}_{\ell},$
(21)

and

$\displaystyle C\Gamma^{T}C^{\dagger}=(-1)^{n}\Gamma\ ;\
\left\\{\begin{array}[]{ll}n=0&,\,\Gamma=I\\\
n=1&,\,\Gamma=\gamma_{\alpha},\sigma_{\alpha\beta}\end{array}\right.,$ (24)

where $C=i\gamma^{2}\gamma^{0}$ denotes the charge conjugation matrix, we
identify

$C^{V}_{LR}=1+\epsilon^{\ell}_{V},\ C^{V}_{RR}=\epsilon^{\ell}_{R},\
C^{S}_{LR}=-\frac{\epsilon^{\ell}_{S}-\epsilon^{\ell}_{P}}{2},\
C^{S}_{RR}=-\frac{\epsilon^{\ell}_{S}+\epsilon^{\ell}_{P}}{2},\
C^{T}_{RR}=\epsilon^{\ell}_{T}\,.$ (25)

Since the charge conjugated fields play for antiparticles the same role as the
original fields for particles, all the expressions derived in Ref. [23] for
$b\rightarrow c\ell^{-}\bar{\nu}_{\ell}$ are directly applicable (see Appendix
E of that work).

In particular, one can write the differential decay width as

$\displaystyle\frac{{\rm d}^{2}\Gamma}{{\rm d}\omega{\rm d}s_{13}}$
$\displaystyle=$
$\displaystyle\frac{G^{2}_{F}|V_{cs}|^{2}Mm^{2}}{16\pi^{3}}\Big{[}{\cal
A}(\omega)+{\cal B}(\omega)\frac{k\cdot p}{M^{2}}+{\cal
C}(\omega)\frac{(k\cdot p)^{2}}{M^{4}}\Big{]}\,.$ (26)

$M$ and $m$ are the masses of the initial and final hadrons, respectively;
$s_{13}\equiv(p-k)^{2}=M^{2}-2k\cdot p$; $\omega\equiv p\cdot p^{\prime}/(Mm)$
is the product of the hadron four-velocities, which is related to
$q^{2}=M^{2}+m^{2}-2Mm\omega$. The functions ${\cal A}(\omega),\,{\cal
B}(\omega)$ and ${\cal C}(\omega)$ are given in Appendix D of Ref. [23] as a
combination of Wilson coefficients777It is useful to bear in mind that from
the definitions of Appendix A of Ref. [23] and Eq. (25) follows that
$C^{V}_{R}=1+\epsilon^{\ell}_{V}+\epsilon^{\ell}_{R},\
C^{A}_{R}=-(1+\epsilon^{\ell}_{V}-\epsilon^{\ell}_{R}),\
C^{S}_{R}=-\epsilon^{\ell}_{S},\ C^{P}_{R}=-\epsilon^{\ell}_{P},\
C^{T}_{R}=\epsilon^{\ell}_{T}$. and purely hadronic structure functions. The
latter are given in terms of transition form factors: in the case of
$1/2^{+}\to 1/2^{+}$ decays, as the one under study here, explicit expressions
can be found in Ref. [21].

The $k\cdot p$ product can be written as

$k\cdot
p=\frac{M}{2}\Big{(}1-\frac{m^{2}_{\ell}}{q^{2}}\Big{)}(M_{\omega}+m\sqrt{\omega^{2}-1}\cos\theta)$
(27)

with $M_{\omega}=M-m\omega$. Accordingly,

$\frac{{\rm d}^{2}\Gamma}{{\rm d}\omega{\rm
d}\cos\theta}=\frac{G^{2}_{F}|V_{cs}|^{2}M^{2}m^{3}}{16\pi^{3}}\sqrt{\omega^{2}-1}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}[a_{0}(\omega)+a_{1}(\omega)\cos\theta+a_{2}(\omega)\cos^{2}\theta]\,,$
(28)

where $a_{0,1,2}(\omega)$ are linear combinations of ${\cal A}(\omega),\,{\cal
B}(\omega)$ and ${\cal C}(\omega)$ (see Eq. (18) of Ref. [21]). Finally, the
$c_{i}$ functions introduced in Eq. (2) correspond to $a_{i}$ up to a
normalization:

$c_{i}=\frac{G^{2}_{F}Mm^{2}}{32\pi^{3}}\sqrt{\omega^{2}-1}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}\frac{a_{i}}{\Gamma_{\rm{tot}}}.$
(29)

The $c_{i}$ functions can be extracted from measurable quantities: the
differential width $d\Gamma/dq^{2}$, the forward-backward asymmetry
$\mathcal{A}_{\rm{FB}}$, and the convexity $\mathcal{A}_{\pi/3}$. They are
related as follows [60, 59, 21],

$\frac{1}{\Gamma_{\rm tot}}\frac{{\rm d}\Gamma}{{\rm
d}q^{2}}=\absolutevalue{V_{cs}}^{2}\,2(c_{0}+c_{2}/3)\ ,$ (30)
$\displaystyle\mathcal{A}_{\rm{FB}}$ $\displaystyle\equiv$
$\displaystyle\left(\frac{{\rm d}\Gamma}{{\rm
d}q^{2}}\right)^{-1}\left[\int_{0}^{1}{\rm d}\cos\theta\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}{\rm d}\cos\theta}-\int_{-1}^{0}{\rm
d}\cos\theta\frac{{\rm d}^{2}\Gamma}{{\rm d}q^{2}{\rm d}\cos\theta}\right]$
(31) $\displaystyle=$ $\displaystyle\frac{c_{1}}{2(c_{0}+c_{2}/3)}$
$\displaystyle=$ $\displaystyle
c_{1}\,\absolutevalue{V_{cs}}^{2}\left(\frac{{\rm d}\mathcal{B}}{{\rm
d}q^{2}}\right)^{-1}\ ,$ $\displaystyle\mathcal{A}_{\pi/3}$
$\displaystyle\equiv$ $\displaystyle\left(\frac{{\rm d}\Gamma}{{\rm
d}q^{2}}\right)^{-1}\Bigg{[}\int_{1/2}^{1}{\rm d}\cos\theta\frac{{\rm
d}^{2}\Gamma}{{\rm d}q^{2}{\rm d}\cos\theta}-\int_{-1/2}^{1/2}{\rm
d}\cos\theta\frac{{\rm d}^{2}\Gamma}{{\rm d}q^{2}{\rm
d}\cos\theta}+\int_{-1}^{-1/2}{\rm d}\cos\theta\frac{{\rm d}^{2}\Gamma}{{\rm
d}q^{2}{\rm d}\cos\theta}\Bigg{]}$ (32) $\displaystyle=$
$\displaystyle\frac{c_{2}}{4(c_{0}+c_{2}/3)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,c_{2}\,\absolutevalue{V_{cs}}^{2}\left(\frac{{\rm
d}\mathcal{B}}{{\rm d}q^{2}}\right)^{-1}\ .$

##  Appendix B  Relations among form factors

Here we describe the approximations and prescriptions adopted to express
scalar, pseudoscalar and tensor $\Lambda_{c}\to\Lambda$ transition form
factors in terms of vector and axial ones, which have been determined using
LQCD [64]. All of them rely on the fact that $m_{c}/\Lambda_{\rm QCD}>>1$.

###  B.1 OSHQ approximation

Let us consider the transition between a heavy ($H$) and a light ($L$) spin
$1/2$ baryons induced by vector current
$J^{\mu}_{V}(x)=\bar{l}(x)\gamma^{\mu}h(x)$, in terms of heavy and light quark
fields, $h$ and $l$. We assume that

  * •

$h$ can be treated as a free field obeying the Dirac equation
$(i\not{\partial}-m_{h})h(x)=0$, with $m_{h}$ the heavy quark mass.

  * •

$\ket{H(s,p)}$ can be factorized as
$\ket{H(s,p)}=\ket{h(\tilde{s},\tilde{p})}\ket{\lambda}$, where
$\ket{\lambda}$ is a residual light system. $\tilde{p}$ stands for the
fraction of the $H$ momentum $p$ carried by the heavy quark, satisfying the
on-shell condition $\tilde{p}^{2}=m_{h}^{2}$; $\tilde{s}$ is the corresponding
fraction of the spin projection $s$ of $H$.

Relying on the standard second quantization decomposition of $h(x)$, these two
premises lead to

$\bra{L(s^{\prime},p^{\prime})}J_{V}^{\mu}(x)\ket{H(s,p)}=\bra{L}\bar{l}(x)\gamma^{\mu}u(\tilde{s},\tilde{p})e^{-i\tilde{p}\cdot
x}\ket{0}_{h}\ket{\lambda}$ (33)

for the matrix element of the vector current. Proceeding analogously for the
scalar current $J_{S}=\bar{l}(x)h(x)$ one gets

$\bra{L}J_{S}(x)\ket{H}=\bra{L}\bar{l}(x)u(\tilde{s},\tilde{p})e^{-i\tilde{p}\cdot
x}\ket{0}_{h}\ket{\lambda}$ (34)

We further assume that

  * •

$\tilde{p}\approx p$ or, more specifically, that the heavy quark spinor,
$u(\tilde{s},\tilde{p})$, satisfies
$\not{p}u(\tilde{s},\tilde{p})=m_{h}u(\tilde{s},\tilde{p})$.

We can therefore contract

$\displaystyle
p_{\mu}\bra{L}J_{V}^{\mu}(x)\ket{H}=\bra{L}\bar{l}(x)\not{p}u(\tilde{s},\tilde{p})e^{-i\tilde{p}\cdot
x}\ket{0}_{h}\ket{\lambda}=m_{h}\bra{L}\bar{l}(x)u(\tilde{s},\tilde{p})e^{-i\tilde{p}\cdot
x}\ket{0}_{h}\ket{\lambda}\ .$ (35)

This result, together with Eqs. (33,34), provides the following connection
between the matrix elements of the vector and scalar currents

$p_{\mu}\bra{L}J_{V}^{\mu}(x)\ket{H}=m_{h}\bra{L}J_{S}(x)\ket{H}\,.$ (36)

Then, identifying $H=\Lambda_{c}$, $L=\Lambda$, $h=c$, $l=s$ one finds that

$F_{S}\bar{u}_{\Lambda}(p^{\prime})u_{\Lambda_{c}}(p)=\frac{1}{m_{c}}\bar{u}_{\Lambda}(p^{\prime})p_{\mu}\Gamma^{\mu}_{V}u_{\Lambda_{c}(p)}\,,$
(37)

where

$\Gamma^{\mu}_{V}=\gamma^{\mu}F_{1}+\frac{p^{\mu}}{M}F_{2}+\frac{p^{\prime\mu}}{m}F_{3}\,,$
(38)

[see Eq. (II.3)]. With the help of the Dirac equations for baryons, Eq. (37)
allows to write $F_{S}$ in terms of $F_{1-3}$. This relation is given in Eq.
(1) in the $m_{c}/M\to 1$ limit.

Replacing $J_{V}(x)$ by the axial current
$J^{\mu}_{A}(x)=\bar{l}(x)\gamma^{\mu}\gamma_{5}h(x)$ and $J_{S}(x)$ by the
pseudoscalar one $J_{P}=\bar{l}(x)\gamma_{5}h(x)$ one obtains

$F_{P}\bar{u}_{\Lambda}(p^{\prime})\gamma_{5}u_{\Lambda_{c}}(p)=-\frac{1}{m_{c}}\bar{u}_{\Lambda}(p^{\prime})p_{\mu}\Gamma^{\mu}_{A}u_{\Lambda_{c}(p)}\,,$
(39)

with

$\Gamma^{\mu}_{A}=\left(\gamma^{\mu}G_{1}+\frac{p^{\mu}}{M}G_{2}+\frac{p^{\prime\mu}}{m}G_{3}\right)\gamma_{5}\,.$
(40)

The resulting expression for $F_{P}$ in terms of axial form factors $G_{1-3}$
is also given in Eq. (1).

Next we repeat the procedure with tensor currents
$J_{T}^{\mu\nu}(x)=\bar{l}(x)\sigma^{\mu\nu}h(x)$ and
$J_{TA}^{\mu\nu}(x)=\bar{l}(x)\sigma^{\mu\nu}\gamma_{5}h(x)$ and obtain

$\displaystyle\bar{u}_{\Lambda}(p^{\prime})\left[(p_{\alpha}\Gamma_{V}^{\alpha})p^{\mu}-m_{c}^{2}\Gamma_{V}^{\mu}\right]u_{\Lambda_{c}}(p)$
$\displaystyle=$ $\displaystyle
im_{c}\bar{u}_{\Lambda}(p^{\prime})\Gamma^{\mu\nu}p_{\nu}u_{\Lambda_{c}}(p)$
(41)
$\displaystyle\bar{u}_{\Lambda}(p^{\prime})\left[(p_{\alpha}\Gamma_{A}^{\alpha})p^{\mu}-m_{c}^{2}\Gamma_{A}^{\mu}\right]u_{\Lambda_{c}}(p)$
$\displaystyle=$ $\displaystyle-
im_{c}\bar{u}_{\Lambda}(p^{\prime})\Gamma_{A}^{\mu\nu}p_{\nu}u_{\Lambda_{c}}(p)$
(42)

where ($\epsilon_{0123}=+1$)

$\Gamma_{A}^{\mu\nu}\equiv-\frac{i}{2}\epsilon^{\mu\nu\alpha\beta}\Gamma_{\alpha\beta}\
,$ (43)

thanks to the relation

$\sigma^{\mu\nu}\gamma_{5}=-\frac{i}{2}\epsilon^{\mu\nu\alpha\beta}\sigma_{\alpha\beta}\
,$ (44)

and

$\Gamma^{\alpha\beta}=\frac{i}{M^{2}}\left(p^{\alpha}p^{\prime\beta}-p^{\beta}p^{\prime\alpha}\right)T_{1}+\frac{i}{M}\left(\gamma^{\alpha}p^{\beta}-\gamma^{\beta}p^{\alpha}\right)T_{2}+\frac{i}{M}\left(\gamma^{\alpha}p^{\prime\beta}-\gamma^{\beta}p^{\prime\alpha}\right)T_{3}+\sigma^{\alpha\beta}T_{4}\,.$
(45)

In the $m_{c}/M\to 1$ limit, Eqs. (41,42) lead to a consistent and independent
set of four equations for $T_{1-4}$ in terms of $F_{1,3}$ and $G_{1,3}$,
completing the OSHQ prescription of Eq. (1).

###  B.2 HQSS relations

Following Sec. 2.10 of Ref. [66], at leading order in $1/m_{c}$, one can write
that

$\langle\Lambda(p^{\prime})|\bar{s}\gamma
c|\Lambda_{c}(p)\rangle\to\langle\Lambda(p^{\prime})|\bar{s}\gamma
c_{v}|\Lambda_{c}(v)\rangle=\bar{u}_{\Lambda}(p^{\prime})\Gamma
u_{\Lambda_{c}}(v)\,$ (46)

where $\not{v}u_{\Lambda_{c}}(v)=u_{\Lambda_{c}}(v)$ and $v=p/M$;
$\gamma=\mathds{1},\,\gamma_{5},\,\gamma^{\mu},\,\gamma^{\mu}\gamma_{5},\,\sigma^{\mu\nu}$
while $\Gamma$ denotes the corresponding representation in terms of form
factors given by Eq. (II.3). On the other hand, in general,

$\langle\Lambda(p^{\prime})|\bar{s}\gamma
c_{v}|\Lambda_{c}(v)\rangle=\bar{u}_{\Lambda}(p^{\prime})(\tilde{F}_{1}+\tilde{F}_{2}\not{v})\gamma
u_{\Lambda_{c}}(v)\,.$ (47)

Therefore, in the HQSS limit, all form factors introduced in Eq. (II.3) can be
expressed in terms of two, $\tilde{F}_{1,2}$. In fact, straightforward Dirac
algebra manipulations lead to

$\gamma=\mathds{1}$

$F_{S}=\tilde{F}_{1}+\tilde{F}_{2}$

$\gamma=\gamma_{5}$

$F_{P}=\tilde{F}_{1}-\tilde{F}_{2}$

$\gamma=\gamma^{\mu}$

$F_{1}=\tilde{F}_{1}-\tilde{F}_{2}$, $F_{2}=2\tilde{F}_{2}$, $F_{3}=0$

$\gamma=\gamma^{\mu}\gamma_{5}$

$G_{1}=\tilde{F}_{1}+\tilde{F}_{2}$, $G_{2}=2\tilde{F}_{2}$, $G_{3}=0$

$\gamma=\sigma^{\mu\nu}$

$T_{1}=T_{3}=0$, $T_{2}=-2\tilde{F}_{2}$, $T_{4}=\tilde{F}_{1}+\tilde{F}_{2}$.

These equations allow to write scalar, pseudoscalar and tensor form factors as
a function of $F_{1,2}$ or $G_{1,2}$, giving rise to the models denoted by
HQSSV [Eq. (8)] or HQSSA [Eq. (9)], respectively.

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