# Heavy quark expansion of $\Lambda_{b}\to\Lambda^{*}(1520)$ form factors
beyond leading order

Marzia Bordone<EMAIL_ADDRESS>Dipartimento di Fisica, Università di
Torino & INFN, Sezione di Torino, I-10125 Torino, Italy

###### Abstract

I review the parametrisation of the full set of
$\Lambda_{b}\to\Lambda^{*}(1520)$ form factors in the framework of Heavy Quark
Expansion, including next-to-leading-order $\mathcal{O}(\alpha_{s})$ and, for
the first time, next-to-leading-power $\mathcal{O}(1/m_{b})$ corrections. The
unknown hadronic parameters are obtained by performing a fit to recent lattice
QCD calculations. I investigate the compatibility of the Heavy Quark Expansion
and the current lattice data, finding tension between these two approaches in
the case of tensor and pseudo-tensor form factors, whose origin could come
from an underestimation of the current lattice QCD uncertainties and higher
order terms in the Heavy Quark Expansion.

##  1 Introduction

The flavour changing neutral current (FCNC)-mediated $b\to s\ell^{+}\ell^{-}$
transition plays an important role in the search for physics beyond the
Standard Model (SM). Its potential has been extensively studied through the
$B\to K^{(*)}\ell^{+}\ell^{-}$ decays. Interestingly, the LHCb experiment
found some discrepancies with respect to the SM predictions in a few
observables: $R_{K}$ and $R_{K^{*}}$, which test universality between the muon
and electron final states and the angular coefficient $P_{5}^{\prime}$ in the
$B\to K^{*}\mu^{+}\mu^{-}$ angular distribution [1, 2, 3, 4, 5, 6]. These
hints, together with all available $b\to s\ell^{+}\ell^{-}$ data, form a
coherent pattern of discrepancies. They can be addressed by introducing New
Physics (NP) effects. Low-energy fits point toward a breaking of lepton
flavour universality with a combined significance for the NP hypothesis higher
than $5\sigma$ [7, 8, 9, 10]. Whether these data show without doubt first
signs of NP is not clear yet. Only further measurements with higher statistics
or measurements of new processes able to corroborate these data will give a
final answer.

A possibility to better understand these data is studying further $b\to
s\ell^{+}\ell^{-}$-mediated decays, among which baryon decays are promising
candidates. The decay channel involving ground-state baryons
$\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ has already received attention both from
the experimental and theoretical point of view. The LHCb experiment measured
the $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ angular distribution [11, 12],
finding good agreement between the measured values of the angular observables
and thieir SM predictions [13, 14, 15] for the angular observables. Even
though this result might be discouraging for NP searches, Refs. [14, 15]
showed that this is still consistent with the NP hypothesis. In fact, angular
distributions describing baryon decays are very different from those in the
meson cases, and NP affects them differently.
Another possibility is studying excited $\Lambda^{*}$ states. The LHCb
experiment used the decay chain $\Lambda_{b}\to\Lambda^{*}(\to
pK^{-})\ell^{+}\ell^{-}$ to measure $R(pK)$, the universality ratio between
muons and electrons, finding results consistent with both the SM expectation
and the measured values of $R_{K^{(*)}}$ [16]. In this analysis, the various
$\Lambda^{*}$ resonances below a certain mass threshold are not distinguished.
However, in Ref. [17] it is shown that the $\Lambda^{*}(1520)$ is expected to
be the most frequent among the $\Lambda^{*}$ resonances, with quite narrow
mass distribution. Therefore, it is motivated to study in detail the
$\Lambda_{b}\to\Lambda^{*}(1520)\ell^{+}\ell^{-}$ decay from both experimental
and theoretical point of view, the latter being the main focus of this paper.
The determination of the form factors of the $\Lambda_{b}\to\Lambda^{*}(1520)$
decays had already been object of study in the literature [18, 19, 20, 21,
22]. The Heavy Quark Expansion (HQE) of the form factors up to next-to-leading
(NLO) order in $\alpha_{s}$ and leading power in $1/m_{b}$ has been employed
[20, 21], using Quark Models to constrain the unknown hadronic parameters
[19]. Recently, a lattice QCD determination of the full base of form factors
has become available [22], but constrained to the low-recoil region.
In this work, I investigate the compatibility between the HQE form factors and
the recent lattice QCD determination. At this scope, I perform a HQE of form
factors including NLO $\alpha_{s}$ and next-to-leading power (NLP) $1/m_{b}$
corrections, the latter being not known so far in the literature. The results
are then matched onto the lattice QCD calculation. In the HQE the number of
independent, hadronic parameters is reduced compared to the lattice QCD case,
introducing strict correlations among the form factors. I perform a fit to the
lattice QCD results to obtain the central values, uncertainties, and
correlations among the HQE parameters. The comparison between lattice QCD
results and HQE predictions shows a tension between the two methods for tensor
and pseudo-tensor form factors, whose origin is not yet completely determined.
This paper is organised as follows: in Sect. 2 I present the HQE of the form
factors; in Sect. 3 I discuss the fit to lattice data; in Sect. 4 I conclude.
Appendix A and Appendix B report details on the calculation of the form
factors and Appendix C contains the covariance matrix for the fitted values of
the HQE parameters.

##  2 Setup

In the following, I investigate the form factors mediating the transition
$\Lambda_{b}(p,s_{b})\to\Lambda^{*}(1520)(k,\eta(\lambda_{\Lambda}),s_{\Lambda})$,
where $p$ and $k$ are the momenta of the initial and final states,
respectively, $s_{b}$ and $s_{\Lambda}$ are the rest-frame helicities of the
two baryons and $\eta(\lambda_{\Lambda})$ is the polarisation vector of the
$\Lambda^{*}$ for each polarisation state $\lambda_{\Lambda}$. Since in this
work I refer to the $\Lambda^{*}(1520)$, only, I denote this state as
$\Lambda^{*}$ in the following. It is worth noticing that the $\Lambda^{*}$ is
considered stable in this discussion. The subsequent $\Lambda^{*}$ decay has
to be taken into account when comparing to experimental data. In Refs. [20,
21] the $\Lambda^{*}\to N\bar{K}$111The $\Lambda^{*}\to N\bar{K}$ is the
$\Lambda^{*}$ decay mode with larger branching fraction and the one that will
be employed by the LHCb collaboration to reconstruct the $\Lambda^{*}$ in
future experimental analysis. decay is discussed, and the four-dimensional
differential decay width of the decay chain $\Lambda_{b}\to\Lambda^{*}(\to
N\bar{K})\ell^{+}\ell^{-}$ is presented.

I define the helicity form factors for
$\Lambda_{b}(p,s_{b})\to\Lambda^{*}(k,\eta(\lambda_{\Lambda}),s_{\Lambda})$ as

$\displaystyle\bra{\Lambda^{*}(k,\eta(\lambda_{\Lambda}),s_{\Lambda})}\bar{s}\gamma^{\mu}b\ket{\Lambda_{b}(p,s_{b})}$
$\displaystyle=+\bar{u}_{\alpha}(k,\eta(\lambda_{\Lambda}),s_{\Lambda})\left[\sum_{i}F_{i}(q^{2})\Gamma^{\alpha\mu}_{V,i}\right]u(p,s_{b})\,,$
(2.1)
$\displaystyle\bra{\Lambda^{*}(k,\eta(\lambda_{\Lambda}),s_{\Lambda})}\bar{s}\gamma^{\mu}\gamma_{5}b\ket{\Lambda_{b}(p,s_{b})}$
$\displaystyle=-\bar{u}_{\alpha}(k,\eta(\lambda_{\Lambda}),s_{\Lambda})\left[\sum_{i}G_{i}(q^{2})\gamma_{5}\Gamma^{\alpha\mu}_{A,i}\right]u(p,s_{b})\,,$
$\displaystyle\bra{\Lambda^{*}(k,\eta(\lambda_{\Lambda}),s_{\Lambda})}\bar{s}i\sigma^{\mu\nu}q_{\nu}b\ket{\Lambda_{b}(p,s_{b})}$
$\displaystyle=-\bar{u}_{\alpha}(k,\eta(\lambda_{\Lambda}),s_{\Lambda})\left[\sum_{i}T_{i}(q^{2})\Gamma^{\alpha\mu}_{T,i}\right]u(p,s_{b})\,,$
$\displaystyle\bra{\Lambda^{*}(k,\eta(\lambda_{\Lambda}),s_{\Lambda})}\bar{s}i\sigma^{\mu\nu}q_{\nu}\gamma_{5}b\ket{\Lambda_{b}(p,s_{b})}$
$\displaystyle=-\bar{u}_{\alpha}(k,\eta(\lambda_{\Lambda}),s_{\Lambda})\left[\sum_{i}T^{5}_{i}(q^{2})\gamma_{5}\Gamma^{\alpha\mu}_{T5,i}\right]u(p,s_{b})\,,$

where $\bar{u}_{\alpha}$ is the spin $3/2$ projector of a Rarita-Schwinger
object [23]. The Dirac structures $\Gamma^{\alpha\mu}_{L,i}$, with
$L=V,A,T,T5$ are given in Appendix A. In the cases $L=V,A$ I adopt the
parametrisation in Ref. [24], while for $L=T,T5$ I follow and adapt the
parametrisation in Refs. [20, 22]. In the following I use the convention
$\sigma^{\mu\nu}=\frac{i}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})$.

###  2.1 The Heavy Quark Expansion

In the low-recoil limit, a HQE of the $\Lambda_{b}\to\Lambda^{*}$ form factors
can be performed. At leading power in $1/m_{b}$ and leading order in
$\alpha_{s}$, the hadronic matrix element for $\Lambda_{b}\to\Lambda^{*}$
transitions reads:

$\langle\Lambda^{*}(k,\eta,s_{\Lambda})|\bar{s}\Gamma^{\mu}b|\Lambda_{b}(p,s_{b})\rangle=\sqrt{4}\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\zeta^{\alpha}(k,\eta,s_{\Lambda})\Gamma^{\mu}u(m_{\Lambda_{b}}v,s_{b})\,,$
(2.2)

where $v$ is the velocity of the initial state and $\Gamma^{\mu}$ denotes a
Dirac structure. In the following, I focus on the cases
$\Gamma^{\mu}=\gamma^{\mu},\,\gamma^{\mu}\gamma_{5},\,i\sigma^{\mu\nu}q_{\nu},\,i\sigma^{\mu\nu}q_{\nu}\gamma_{5}$.
The most general decomposition for the leading-order and leading-power
contribution $\zeta^{\alpha}$ reads

$\zeta^{\alpha}=v^{\alpha}[\zeta_{1}+\zeta_{2}\not{v}]\,,$ (2.3)

where $\zeta_{1}$ and $\zeta_{2}$ are the leading Isgur-Wise (IW) functions.
The discussion of $1/m_{b}$ and $\alpha_{s}$ corrections closely follows Refs.
[25, 26, 27]. In this spirit, I replace the (axial-)vector and (pseudo-)tensor
currents with:

$\displaystyle\bar{s}\gamma^{\mu}b\mapsto\bar{s}J^{\mu}_{V(A)}h_{v}=\,$
$\displaystyle(1+C_{0}^{(v)})\bar{s}\gamma^{\mu}(\gamma_{5})h_{v}\pm
C_{1}^{(v)}v^{\mu}\bar{s}(\gamma_{5})h_{v}+\frac{1}{2m_{b}}\bar{s}\Delta
J^{\mu}_{V(A)}h_{v}\,,$ (2.4)
$\displaystyle\bar{s}(\gamma_{5})i\sigma^{\mu\nu}q_{\nu}b\mapsto\bar{s}J^{\mu}_{T(5)}h_{v}=\,$
$\displaystyle(1+C_{0}^{(t)})\bar{s}i\sigma^{\mu\nu}q_{\nu}h_{v}+\frac{1}{2m_{b}}\bar{s}\Delta
J^{\mu}_{T(5)}h_{v}\,.$ (2.5)

The matching coefficients at NLO read [25, 26]

$\displaystyle C_{0}^{(v)}(\mu)=$
$\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\left[3\log\left(\frac{\mu}{m_{b}}\right)+4\right]+\mathcal{O}(\alpha_{s}^{2})\,,$
(2.6) $\displaystyle C_{1}^{(v)}(\mu)=$
$\displaystyle+\frac{\alpha_{s}C_{F}}{2\pi}+\mathcal{O}(\alpha_{s}^{2})\,,$
$\displaystyle C_{0}^{(t)}(\mu)=$
$\displaystyle-\frac{\alpha_{s}C_{F}}{4\pi}\left[5\log\left(\frac{\mu}{m_{b}}\right)+4\right]+\mathcal{O}(\alpha_{s}^{2})\,.$

For numerical purposes, the scale of the Wilson coefficients is set to
$\mu\sim 2\,\text{GeV}$. The NLP $1/m_{b}$ corrections due to the expansion of
the current are parametrised as

$\langle\Lambda^{*}(k,\eta,s_{\Lambda})|\bar{s}\Delta
J^{\mu}_{V(A,T,T5)}b|\Lambda_{b}(p,s_{b})\rangle=\sqrt{4}\sum_{i}\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\zeta^{\alpha\beta}[\mathcal{O}_{i}^{V(A,T,T5)}]^{\mu}_{\beta}u(m_{\Lambda_{b}}v,s_{b})\,,$
(2.7)

where

$\zeta^{\alpha\beta}=g^{\alpha\beta}\left[\zeta^{\text{SL}}_{1}+\not{v}\zeta^{\text{SL}}_{2}\right]+v^{\alpha}v^{\beta}\left[\zeta^{\text{SL}}_{3}+\not{v}\zeta^{\text{SL}}_{4}\right]+v^{\alpha}\gamma^{\beta}\left[\zeta_{5}^{\text{SL}}+\not{v}\zeta^{\text{SL}}_{6}\right]\,.$
(2.8)

The functions $\zeta^{\text{SL}}_{1\dots 6}$ are the subleading Isgur-Wise
functions, and they correspond to all the independent Dirac structures that
can appear in $\zeta^{\alpha\beta}$. The possible operators
$\mathcal{O}^{\mu\beta}_{i}$ are listed in Ref. [25]. Out of the possible six
of them, only the operator $[\mathcal{O}_{1}^{\Gamma}]^{\mu}_{\beta}$ arises
at order $1/m_{b}$, while the others are suppressed by
$\mathcal{O}(\alpha_{s}/m_{b})$ and therefore are beyond the precision here
required. Therefore, the only contributions that I consider for this analysis
come from:

$\displaystyle\left[\mathcal{O}_{1}^{V}\right]^{\mu}_{\beta}=$
$\displaystyle+\gamma^{\mu}\gamma_{\beta}\,,$
$\displaystyle\left[\mathcal{O}_{1}^{A}\right]^{\mu}_{\beta}=$
$\displaystyle-\gamma_{5}\gamma^{\mu}\gamma_{\beta}\,,$ (2.9)
$\displaystyle\left[\mathcal{O}_{1}^{T}\right]^{\mu}_{\beta}=$
$\displaystyle+i\sigma^{\mu\nu}q_{\nu}\gamma_{\beta}\,,$
$\displaystyle\left[\mathcal{O}_{1}^{T5}\right]^{\mu}_{\beta}=$
$\displaystyle+i\gamma_{5}\sigma^{\mu\nu}q_{\nu}\gamma_{\beta}\,,$

By means of Dirac algebra, and using the properties of Rarita-Schwinger
objects in Ref. [23], inserting Eq. (2.9) in Eq. (2.8) yields

$\displaystyle\langle\Lambda^{*}(k,\eta,s_{\Lambda})|\bar{s}\Delta
J^{\mu}_{V}b|\Lambda_{b}(p,s_{b})\rangle=2\bigg{\\{}$ $\displaystyle
2\bar{u}_{\alpha}(k,\eta,s_{\Lambda})u(m_{\Lambda_{b}}v,s_{b})g^{\alpha\mu}(\zeta^{\text{SL}}_{1}+\zeta^{\text{SL}}_{2})$
$\displaystyle+\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\gamma^{\mu}(m_{\Lambda_{b}}v,s_{b})v^{\alpha}(\zeta^{\text{SL}}_{3}-\zeta^{\text{SL}}_{4}-2\zeta^{\text{SL}}_{2}-2\zeta^{\text{SL}}_{5})$
$\displaystyle+2\bar{u}_{\alpha}(k,\eta,s_{\Lambda})u(m_{\Lambda_{b}}v,s_{b})v^{\alpha}v^{\mu}(\zeta^{\text{SL}}_{4}+2\zeta^{\text{SL}}_{6})\bigg{\\}}\,,$
(2.10) $\displaystyle\langle\Lambda^{*}(k,\eta,s_{\Lambda})|\bar{s}\Delta
J^{\mu}_{A}b|\Lambda_{b}(p,s_{b})\rangle=2\bigg{\\{}$ $\displaystyle
2\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\gamma_{5}u(m_{\Lambda_{b}}v,s_{b})g^{\alpha\mu}(-\zeta^{\text{SL}}_{1}+\zeta^{\text{SL}}_{2})$
$\displaystyle-\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\gamma_{5}\gamma^{\mu}(m_{\Lambda_{b}}v,s_{b})v^{\alpha}(\zeta^{\text{SL}}_{3}+\zeta^{\text{SL}}_{4}+2\zeta^{\text{SL}}_{2}+2\zeta^{\text{SL}}_{5})$
$\displaystyle+2\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\gamma_{5}u(m_{\Lambda_{b}}v,s_{b})v^{\alpha}v^{\mu}(\zeta^{\text{SL}}_{4}-2\zeta^{\text{SL}}_{6})\bigg{\\}}\,,$
(2.11) $\displaystyle\langle\Lambda^{*}(k,\eta,s_{\Lambda})|\bar{s}\Delta
J^{\mu}_{T}b|\Lambda_{b}(p,s_{b})\rangle=2\bigg{\\{}$
$\displaystyle-\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\gamma^{\mu}u(m_{\Lambda_{b}}v,s_{b})v^{\alpha}\bigg{[}2m_{\Lambda_{b}}\zeta_{1}^{\text{SL}}+2m_{\Lambda^{*}}\zeta_{2}^{\text{SL}}$
$\displaystyle+(m_{\Lambda_{b}}+m_{\Lambda^{*}})(\zeta_{3}^{\text{SL}}+2\zeta_{6}^{\text{SL}})+\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda_{b}}m_{\Lambda^{*}}-q^{2}}{m_{\Lambda_{b}}}\zeta_{4}^{\text{SL}}\bigg{]}$
$\displaystyle+2\bar{u}_{\mu}(k,\eta,s_{\Lambda})u(m_{\Lambda_{b}}v,s_{b})\bigg{[}(m_{\Lambda_{b}}-m_{\Lambda^{*}})\zeta_{1}^{\text{SL}}-\frac{m_{\Lambda^{*}}^{2}-m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2}}{m_{\Lambda_{b}}}\zeta_{2}^{\text{SL}}\bigg{]}$
$\displaystyle+2\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\gamma^{\mu}u(m_{\Lambda_{b}}v,s_{b})k^{\alpha}(\zeta_{1}^{\text{SL}}-\zeta_{2}^{\text{SL}})$
$\displaystyle+\bar{u}_{\alpha}(k,\eta,s_{\Lambda})u(m_{\Lambda_{b}}v,s_{b})v^{\alpha}k^{\mu}(\zeta_{3}^{\text{SL}}+2\zeta_{2}^{\text{SL}}+\zeta_{4}^{\text{SL}}+2\zeta_{6}^{\text{SL}})$
$\displaystyle+4\bar{u}_{\alpha}(k,\eta,s_{\Lambda})u(m_{\Lambda_{b}}v,s_{b})v^{\mu}k^{\alpha}\zeta_{2}^{\text{SL}}$
$\displaystyle+\bar{u}_{\alpha}(k,\eta,s_{\Lambda})u(m_{\Lambda_{b}}v,s_{b})v^{\alpha}v^{\mu}$
$\displaystyle\times\big{[}m_{\Lambda_{b}}\zeta_{3}^{\text{SL}}-2m_{\Lambda_{b}}\zeta_{2}^{\text{SL}}+(2m_{\Lambda^{*}}-m_{\Lambda_{b}})\zeta_{4}^{\text{SL}}+2m_{\Lambda_{b}}\zeta_{6}^{\text{SL}}\big{]}\bigg{\\}}\,,$
(2.12) $\displaystyle\langle\Lambda^{*}(k,\eta,s_{\Lambda})|\bar{s}\Delta
J^{\mu}_{T5}b|\Lambda_{b}(p,s_{b})\rangle=2\bigg{\\{}$
$\displaystyle-\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\gamma_{5}\gamma^{\mu}u(m_{\Lambda_{b}}v,s_{b})v^{\alpha}\bigg{[}m_{\Lambda_{b}}\zeta_{1}^{\text{SL}}+2m_{\Lambda^{*}}\zeta_{2}^{\text{SL}}$
$\displaystyle+(m_{\Lambda_{b}}-m_{\Lambda^{*}})(\zeta_{3}^{\text{SL}}+2\zeta_{6}^{\text{SL}})-\frac{m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}m_{\Lambda^{*}}-q^{2}}{m_{\Lambda_{b}}}\zeta_{4}^{\text{SL}}\bigg{]}$
$\displaystyle+2\bar{u}_{\mu}(k,\eta,s_{\Lambda})\gamma_{5}u(m_{\Lambda_{b}}v,s_{b})\bigg{[}(m_{\Lambda_{b}}+m_{\Lambda^{*}})\zeta_{1}^{\text{SL}}+\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2}}{m_{\Lambda_{b}}}\zeta_{2}^{\text{SL}}\bigg{]}$
$\displaystyle+2\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\gamma_{5}\gamma^{\mu}u(m_{\Lambda_{b}}v,s_{b})k^{\alpha}(\zeta_{1}^{\text{SL}}+\zeta_{2}^{\text{SL}})$
$\displaystyle+\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\gamma_{5}u(m_{\Lambda_{b}}v,s_{b})v^{\alpha}k^{\mu}(+\zeta_{3}^{\text{SL}}-2\zeta_{2}^{\text{SL}}-\zeta_{4}^{\text{SL}}+2\zeta_{6}^{\text{SL}})$
$\displaystyle-4\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\gamma_{5}u(m_{\Lambda_{b}}v,s_{b})v^{\mu}k^{\alpha}\zeta_{2}^{\text{SL}}$
$\displaystyle+\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\gamma_{5}u(m_{\Lambda_{b}}v,s_{b})v^{\alpha}v^{\mu}$
$\displaystyle\times\big{[}m_{\Lambda_{b}}\zeta_{3}^{\text{SL}}+2m_{\Lambda_{b}}\zeta_{2}^{\text{SL}}+(2m_{\Lambda^{*}}+m_{\Lambda_{b}})\zeta_{4}^{\text{SL}}+2m_{\Lambda_{b}}\zeta_{6}^{\text{SL}}\big{]}\bigg{\\}}\,.$
(2.13)

The number of independent subleading IW functions can be reduced by using
equations of motions. In particular, for this decay, the relation
$v_{\beta}\zeta^{\alpha\beta}=0$ gives the following conditions:

$\displaystyle\zeta_{1}^{\text{SL}}+\zeta_{3}^{\text{SL}}+\zeta_{6}^{\text{SL}}=0\,,$
(2.14)
$\displaystyle\zeta_{2}^{\text{SL}}+\zeta_{4}^{\text{SL}}+\zeta_{5}^{\text{SL}}=0\,.$
(2.15)

I choose to retain as independent quantities the subleading IW functions
$\zeta_{1}^{\text{SL}}$, $\zeta_{2}^{\text{SL}}$, $\zeta_{3}^{\text{SL}}$ and
$\zeta_{4}^{\text{SL}}$.
Corrections to the form factors arise also by of non-local insertions of the
heavy quark Lagrangian at order $1/m_{b}$. Following the discussions in
Refs.[28, 29, 25], non-local insertion of the kinetic operator yields a
universal shift proportional to the tree-level matrix elements. Hence I
reabsorb such shift in a redefinition of the leading order IW function
$\zeta_{1}$. Non-local insertion of the chromomagnetic operator can be
parametrised as [30]

$R_{\mu\nu}^{\alpha}\bar{u}_{\alpha}(k,\eta,s_{\Lambda})\Gamma\frac{1+\not{v}}{2}i\sigma^{\mu\nu}u(m_{\Lambda_{b}}v,s_{b})\,,$
(2.16)

where the object $R_{\mu\nu}^{\alpha}$ is an antisymmetric tensor in the
indices $\mu$ and $\nu$ and contains the velocity $v$. $\Gamma$ symbolises all
the possible Dirac structures which mediate $\Lambda_{b}\to\Lambda^{*}$
transitions. Using equations of motion, it can be shown that no possible form
of $R_{\mu\nu}^{\alpha}$ gives a non-zero contribution for the chromomagnetic
operator.

The expressions of the form factors in terms of the leading and subleading IW
functions are obtained by matching the helicity amplitudes with their HQE
expansion. For the vector current, by comparing Eqs. (A.17)–(A.20) to Eqs.
(A.33)–(A.36), I get

$\displaystyle F_{1/2,0}=$
$\displaystyle\frac{\sqrt{s_{+}}}{2m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{3/2}}\bigg{\\{}\zeta_{1}s_{-}\left[(1+C_{0}^{(v)})+C_{1}^{(v)}\frac{s_{+}}{2m_{\Lambda_{b}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})}\right]$
$\displaystyle+$
$\displaystyle\zeta_{2}s_{-}\left[(1+C_{0}^{(v)})\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2}}{m_{\Lambda_{b}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})}+C_{1}^{(v)}\frac{s_{+}}{2m_{\Lambda_{b}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})}\right]$
$\displaystyle-$
$\displaystyle\zeta_{1}^{\text{SL}}\frac{\lambda+m_{\Lambda_{b}}^{2}(m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}^{2}+q^{2})}{m_{\Lambda_{b}}^{2}(m_{\Lambda^{*}}+m_{\Lambda_{b}})}-\zeta_{2}^{\text{SL}}\frac{(m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}^{2}+q^{2})}{(m_{\Lambda^{*}}+m_{\Lambda_{b}})}$
$\displaystyle-$ $\displaystyle
s_{-}\left[\zeta_{3}^{\text{SL}}\frac{(2m_{\Lambda^{*}}^{2}+3m_{\Lambda^{*}}m_{\Lambda_{b}}+m_{\Lambda_{b}}^{2}-2q^{2})}{2m_{\Lambda_{b}}^{2}(m_{\Lambda^{*}}+m_{\Lambda_{b}})}-\zeta_{4}^{\text{SL}}\frac{(m_{\Lambda^{*}}^{2}+3m_{\Lambda^{*}}m_{\Lambda_{b}}+2m_{\Lambda_{b}}^{2}-q^{2})}{2m_{\Lambda_{b}}^{2}(m_{\Lambda^{*}}+m_{\Lambda_{b}})}\right]\bigg{\\}}\,,$
(2.17) $\displaystyle F_{1/2,t}=$
$\displaystyle\frac{\sqrt{s_{-}}s_{+}}{2m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{3/2}}\bigg{\\{}\zeta_{1}\left[(1+C_{0}^{(v)})-C_{1}^{(v)}\frac{m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}^{2}-q^{2}}{2m_{\Lambda_{b}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}\right]$
$\displaystyle-$
$\displaystyle\zeta_{2}\frac{1}{2m_{\Lambda_{b}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}\left[2(1+C_{0}^{(v)})(m_{\Lambda^{*}}^{2}-m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2})+C_{1}^{(v)}(m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}^{2}-q^{2})\right]$
$\displaystyle-$
$\displaystyle\frac{1}{m_{\Lambda_{b}}-m_{\Lambda^{*}}}\left[\frac{s-m_{\Lambda^{*}}^{2}}{m_{\Lambda_{b}}^{2}}\zeta_{1}^{\text{SL}}-\zeta_{2}^{\text{SL}}\right]$
$\displaystyle+$
$\displaystyle\left[\frac{2m_{\Lambda^{*}}^{2}-m_{\Lambda^{*}}m_{\Lambda_{b}}-m_{\Lambda_{b}}^{2}-2q^{2}}{2m_{\Lambda_{b}}^{2}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}\zeta_{3}^{\text{SL}}-\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-2m_{\Lambda_{b}}^{2}-q^{2}}{2m_{\Lambda_{b}}^{2}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}\zeta_{4}^{\text{SL}}\right]\,,$
(2.18) $\displaystyle F_{1/2,\perp}=$
$\displaystyle\frac{\sqrt{s_{+}}}{2m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{3/2}}\bigg{\\{}s_{-}(1+C_{0}^{(v)})(\zeta_{1}-\zeta_{2})-m_{\Lambda^{*}}(\zeta_{1}^{\text{SL}}+\zeta_{2}^{\text{SL}})+\frac{s_{-}}{2m_{\Lambda_{b}}}(\zeta_{3}^{\text{SL}}+\zeta_{4}^{\text{SL}})\bigg{\\}}\,,$
(2.19) $\displaystyle F_{3/2,\perp}=$
$\displaystyle-\frac{\sqrt{s_{+}}}{2m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{1/2}}\bigg{\\{}\zeta_{1}^{\text{SL}}+\zeta_{2}^{\text{SL}}\bigg{\\}}\,,$
(2.20)

and for the axial vector current, by matching Eqs. (A.21)–(A.24) to Eqs.
(A.37)–(A.40) I obtain

$\displaystyle G_{1/2,0}=$
$\displaystyle\frac{\sqrt{s_{-}}}{2m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{3/2}}\bigg{\\{}\zeta_{1}s_{+}\left[(1+C_{0}^{(v)})+C_{1}^{(v)}\frac{s_{-}}{2m_{\Lambda_{b}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}\right]$
$\displaystyle-$
$\displaystyle\zeta_{2}s_{+}\left[(1+C_{0}^{(v)})\frac{m_{\Lambda^{*}}^{2}-m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2}}{m_{\Lambda_{b}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}-C_{1}^{(v)}\frac{s_{-}}{2m_{\Lambda_{b}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}\right]$
$\displaystyle-$
$\displaystyle\zeta_{1}^{\text{SL}}\frac{\lambda+m_{\Lambda_{b}}^{2}(m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}^{2}+q^{2})}{m_{\Lambda_{b}}^{2}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}+\zeta_{2}^{\text{SL}}\frac{(m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}^{2}+q^{2})}{(m_{\Lambda_{b}}-m_{\Lambda^{*}})}$
$\displaystyle-$ $\displaystyle
s_{+}\left[\zeta_{3}^{\text{SL}}\frac{(2m_{\Lambda^{*}}^{2}-3m_{\Lambda^{*}}m_{\Lambda_{b}}+m_{\Lambda_{b}}^{2}-2q^{2})}{2m_{\Lambda_{b}}^{2}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}+\zeta_{4}^{\text{SL}}\frac{(m_{\Lambda^{*}}^{2}-3m_{\Lambda^{*}}m_{\Lambda_{b}}+2m_{\Lambda_{b}}^{2}-q^{2})}{2m_{\Lambda_{b}}^{2}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}\right]\bigg{\\}}\,,$
(2.21) $\displaystyle G_{1/2,t}=$
$\displaystyle\frac{\sqrt{s_{+}}s_{-}}{2m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{3/2}}\bigg{\\{}\zeta_{1}\left[(1+C_{0}^{(v)})-C_{1}^{(v)}\frac{m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}^{2}-q^{2}}{2m_{\Lambda_{b}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})}\right]$
$\displaystyle+$
$\displaystyle\zeta_{2}\frac{1}{2m_{\Lambda_{b}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})}\left[2(1+C_{0}^{(v)})(m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2})-C_{1}^{(v)}(m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}^{2}-q^{2})\right]$
$\displaystyle-$
$\displaystyle\frac{1}{m_{\Lambda_{b}}+m_{\Lambda^{*}}}\left[\frac{s-m_{\Lambda^{*}}^{2}}{m_{\Lambda_{b}}^{2}}\zeta_{1}^{\text{SL}}+\zeta_{2}^{\text{SL}}\right]$
$\displaystyle+$
$\displaystyle\left[\frac{2m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-m_{\Lambda_{b}}^{2}-2q^{2}}{2m_{\Lambda_{b}}^{2}(m_{\Lambda_{b}}+m_{\Lambda^{*}})}\zeta_{3}^{\text{SL}}+\frac{m_{\Lambda^{*}}^{2}-m_{\Lambda^{*}}m_{\Lambda_{b}}-2m_{\Lambda_{b}}^{2}-q^{2}}{2m_{\Lambda_{b}}^{2}(m_{\Lambda_{b}}+m_{\Lambda^{*}})}\zeta_{4}^{\text{SL}}\right]\,,$
(2.22) $\displaystyle G_{1/2,\perp}=$
$\displaystyle\frac{\sqrt{s_{-}}}{2m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{3/2}}\bigg{\\{}s_{+}\left[(1+C_{0}^{(v)})(\zeta_{1}+\zeta_{2})\right]+m_{\Lambda^{*}}(\zeta_{1}^{\text{SL}}-\zeta_{2}^{\text{SL}})+\frac{s_{+}}{2m_{\Lambda_{b}}}(\zeta_{3}^{\text{SL}}-\zeta_{4}^{\text{SL}})\bigg{\\}}\,,$
(2.23) $\displaystyle G_{3/2,\perp}=$
$\displaystyle-\frac{\sqrt{s_{-}}}{2m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{1/2}}\bigg{\\{}\zeta_{1}^{\text{SL}}-\zeta_{2}^{\text{SL}}\bigg{\\}}\,.$
(2.24)

For the tensor current, the comparison between Eqs. (A.25)–(A.27) and Eqs.
(A.41)–(A.43) yields

$\displaystyle T_{1/2,0}=$
$\displaystyle\frac{\sqrt{s_{+}}}{m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{1/2}}\bigg{\\{}(1+C_{0}^{(t)})(\zeta_{1}-\zeta_{2})s_{-}-\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda_{b}}^{2}-q^{2}}{m_{\Lambda_{b}}}(\zeta_{1}^{\text{SL}}-\zeta_{2}^{\text{SL}})$
$\displaystyle-\frac{s_{-}}{2m_{\Lambda_{b}}}(\zeta_{4}^{\text{SL}}+\zeta_{3}^{\text{SL}})\bigg{\\}}\,,$
(2.25) $\displaystyle T_{1/2,\perp}=$
$\displaystyle\frac{\sqrt{s_{+}}}{m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{1/2}}\bigg{\\{}(1+C_{0}^{(t)})\left[\zeta_{1}+\frac{m_{\Lambda^{*}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})-q^{2}}{m_{\Lambda_{b}}(m_{\Lambda^{*}}+m_{\Lambda_{b}})}\zeta_{2}\right]s_{-}$
$\displaystyle+$ $\displaystyle
m_{\Lambda^{*}}\frac{m_{\Lambda^{*}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})+q^{2}}{m_{\Lambda_{b}}(m_{\Lambda^{*}}+m_{\Lambda_{b}})}\zeta_{1}^{\text{SL}}+\frac{m_{\Lambda^{*}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}{m_{\Lambda^{*}}+m_{\Lambda_{b}}}\zeta_{2}^{\text{SL}}$
$\displaystyle+$
$\displaystyle\frac{s_{-}}{2m_{\Lambda_{b}}}\left[-\zeta_{3}^{\text{SL}}+\frac{m_{\Lambda^{*}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})-q^{2}}{m_{\Lambda_{b}}(m_{\Lambda^{*}}+m_{\Lambda_{b}})}\zeta_{4}^{\text{SL}}\right]\bigg{\\}}\,,$
(2.26) $\displaystyle T_{3/2,\perp}=$
$\displaystyle+\frac{\sqrt{s_{+}}}{m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{1/2}}\bigg{\\{}-\zeta_{1}^{\text{SL}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})+\frac{m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}m_{\Lambda^{*}}-q^{2}}{m_{\Lambda_{b}}}\zeta_{2}^{\text{SL}}\bigg{\\}}\,,$
(2.27)

while for the axial-tensor form factors the comparison between Eqs.
(A.29)–(A.31) and Eqs. (A.44)–(A.46) gives

$\displaystyle T_{1/2,0}^{5}=$
$\displaystyle\frac{\sqrt{s_{-}}}{m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{1/2}}\bigg{\\{}(1+C_{0}^{(t)})(\zeta_{2}+\zeta_{1})s_{+}-\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda_{b}}^{2}-q^{2}}{m_{\Lambda_{b}}}(\zeta_{1}^{\text{SL}}-\zeta_{2}^{\text{SL}})$
$\displaystyle-$
$\displaystyle\frac{s_{+}}{2m_{\Lambda_{b}}}(\zeta_{3}^{\text{SL}}-\zeta_{4}^{\text{SL}})\bigg{\\}}\,,$
(2.28) $\displaystyle T_{1/2,\perp}^{5}=$
$\displaystyle\frac{\sqrt{s_{-}}}{m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{1/2}}\bigg{\\{}(1+C_{0}^{(t)})\left[\zeta_{1}-\frac{m_{\Lambda^{*}}(-m_{\Lambda_{b}}+m_{\Lambda^{*}})-q^{2}}{m_{\Lambda_{b}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}\zeta_{2}\right]s_{+}$
$\displaystyle+$ $\displaystyle
m_{\Lambda^{*}}\frac{m_{\Lambda^{*}}(+m_{\Lambda_{b}}+m_{\Lambda^{*}})-q^{2}}{m_{\Lambda_{b}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}\zeta_{1}^{\text{SL}}+\frac{m_{\Lambda^{*}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})}{m_{\Lambda_{b}}-m_{\Lambda^{*}}}\zeta_{2}^{\text{SL}}$
$\displaystyle-$
$\displaystyle\frac{s_{+}}{2m_{\Lambda_{b}}}\left[\zeta_{3}^{\text{SL}}+\frac{m_{\Lambda^{*}}(-m_{\Lambda_{b}}+m_{\Lambda^{*}})-q^{2}}{m_{\Lambda_{b}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})}\zeta_{4}^{\text{SL}}\right]\bigg{\\}}\,,$
(2.29) $\displaystyle T_{3/2,\perp}^{5}=$
$\displaystyle-\frac{\sqrt{s_{-}}}{m_{\Lambda_{b}}^{3/2}m_{\Lambda^{*}}^{1/2}}\bigg{\\{}\zeta_{1}^{\text{SL}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})+\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda_{b}}m_{\Lambda^{*}}-q^{2}}{m_{\Lambda_{b}}}\zeta_{2}^{\text{SL}}\bigg{\\}}\,.$
(2.30)

The expressions in Eqs. (2.17)–(2.30) have been checked against the results in
Ref. [21], where the HQE for the form factors including NLO $\alpha_{s}$
corrections is presented. I find agreement with the expressions reported in
Ref. [21], apart from a sign flip in the term proportional to
$C_{1}^{(v)}\zeta_{2}$ in $G_{1/2,t}$. This misalignment is not expected to
invalidate the analysis in Ref. [21] since it appears in a term that is
$\alpha_{s}$ suppressed. In the following analysis, I stick to my findings and
adopt the signs in Eq. (2.22).

##  3 Form Factors relations and comparison with lattice QCD results

The first lattice QCD calculation for the full basis of
$\Lambda_{b}\to\Lambda^{*}$ form factors is presented in Ref. [22]. The
calculation is performed in the low-recoil region, very close to the zero-
recoil point $q^{2}_{\text{max}}=(m_{\Lambda_{b}}-m_{\Lambda^{*}})^{2}$. Two
lattice points per form factor are obtained, allowing to determine the
normalisation and the slope of each form factor. In the kinematical limit
where the lattice QCD computation is valid, it is more convenient to
substitute the variable $q^{2}$ with the adimensional variable $w=p\cdot
k/(2m_{\Lambda_{b}}m_{\Lambda^{*}})=(m_{\Lambda_{b}}^{2}+m_{\Lambda^{*}}^{2}-q^{2})/(2m_{\Lambda_{b}}m_{\Lambda^{*}})$,
where the zero-recoil point corresponds to $w=1$. The continuum extrapolation
of the results in Ref. [22] is performed using the following functional form
for each of the form factor $f_{i}$:

$f_{i}=F_{i}+A_{i}(w-1)\,.$ (3.1)

Values for the coefficients $F_{i}$ and $A_{i}$ and their covariance matrix
are given in ancillary files of Ref. [22].
Given that both the results in Ref. [22] and the parametrisation based on HQE
in Sect. 2 are valid in the low-recoil region, the former is suitable to
extract the unknown, hadronic parameters for the leading and sub-leading IW
functions introduced in Sect. 2. I want to stress that it is not possible to
extrapolate these results to the large-recoil region without any further
information on the form factors valid at low $q^{2}$.
The form factor base employed in Ref. [22] differs from the one presented in
Sect. 2 and the matching between the two is given in Appendix B. In the
following, I denote with capital letters the HQE base and with lower cases the
base of Ref. [22].

###  3.1 Relations in the zero-recoil point

I study the form factors first at the zero-recoil point. At this particular
kinematical configuration, all the axial-vector and pseudo-tensor HQE form
factors become zero because they are weighted by the factor $s_{-}$. The
remainder is further simplified since the terms associated with
$\zeta_{1},\zeta_{2},\zeta_{3}^{\text{SL}},\zeta_{4}^{\text{SL}}$, are always
proportional to $s_{-}$, hence vanish, leaving only $\zeta_{1}^{\text{SL}}$
and $\zeta_{2}^{\text{SL}}$ to determine the form factors in the zero-recoil
point. Even more interestingly, from Eqs. (2.17)–(2.30) it can be seen that
only the combination $\zeta_{1}^{\text{SL}}+\zeta_{2}^{\text{SL}}$ appears,
with different weights for each of the form factor. Therefore, HQE provides
predictions for ratios of form factors at zero-recoil independent of the IW
functions. They read:

$\displaystyle\frac{F_{1/2,0}}{F_{1/2,\perp}}=$
$\displaystyle\,\frac{F_{1/2,0}}{F_{3/2,\perp}}=-2\frac{m_{\Lambda_{b}}-m_{\Lambda^{*}}}{m_{\Lambda_{b}}+m_{\Lambda^{*}}}=-1.15\,,$
$\displaystyle\frac{F_{1/2,\perp}}{F_{3/2,\perp}}=$ $\displaystyle\,1\,,$
(3.2) $\displaystyle\frac{T_{1/2,0}}{T_{1/2,\perp}}=$
$\displaystyle-2\frac{m_{\Lambda_{b}}+m_{\Lambda^{*}}}{m_{\Lambda_{b}}-m_{\Lambda^{*}}}=-3.48\,,$
$\displaystyle\frac{T_{1/2,0}}{T_{3/2,\perp}}=$
$\displaystyle\frac{2m_{\Lambda^{*}}}{m_{\Lambda_{b}}-m_{\Lambda^{*}}}=0.74\,,$
$\displaystyle\frac{T_{1/2,\perp}}{T_{3/2,\perp}}=$
$\displaystyle-\frac{m_{\Lambda^{*}}}{m_{\Lambda_{b}}+m_{\Lambda^{*}}}=-0.21\,,$

where errors on the baryon masses have been neglected. These predictions have
to be compared with the results in Ref. [22]. Extracting pseudodata for the
parameters entering the lattice QCD form factors, according to their central
values and covariance matrix, I find (in the HQE basis)

$\displaystyle\frac{F_{1/2,0}^{\text{latt}}}{F_{1/2,\perp}^{\text{latt}}}=$
$\displaystyle-0.63\pm 1.82\,,$
$\displaystyle\frac{F_{1/2,0}^{\text{latt}}}{F_{3/2,\perp}^{\text{latt}}}=$
$\displaystyle-0.94\pm 0.14\,,$
$\displaystyle\frac{F_{1/2,\perp}^{\text{latt}}}{F_{3/2,\perp}^{\text{latt}}}=$
$\displaystyle\,1.48\pm 0.39\,,$ (3.3)
$\displaystyle\frac{T_{1/2,0}^{\text{latt}}}{T_{1/2,\perp}^{\text{latt}}}=$
$\displaystyle-5.62\pm 7.86\,,$
$\displaystyle\frac{T_{1/2,0}^{\text{latt}}}{T_{3/2,\perp}^{\text{latt}}}=$
$\displaystyle+1.03\pm 0.20\,,$
$\displaystyle\frac{T_{1/2,\perp}^{\text{latt}}}{T_{3/2,\perp}^{\text{latt}}}=$
$\displaystyle-0.18\pm 0.05\,.$

The central values showed above are the medians of the distributions and the
errors correspond to the $68\%$ intervals. Concerning the ratios
$F_{1/2,0}^{\text{latt}}/F_{1/2,\perp}^{\text{latt}}$ and
$T_{1/2,0}^{\text{latt}}/T_{1/2,\perp}^{\text{latt}}$, I stress that their
uncertainties are very large to reflect the fact that their distributions are
highly non gaussian. For the other ratios, the gaussian approximation well
describes lattice data. The predictions for the ratios obtained in the HQE
framework and using the lattice QCD results agree within $\sim 1.1\sigma$.

###  3.2 Form factors parametrisation and fit

The HQE of the $\Lambda_{b}\to\Lambda^{*}$ form factors presented in Sect. 2
depends on 6 unknown IW functions. The HQE does not predict the $w$ dependence
of the IW functions, and it has to be inferred from other principles. Since
the formalism of HQE is valid mainly in the low-recoil region for $b\to s$
transitions, the form factors can be expanded around the zero-recoil point.
Substituting $q^{2}$ with $w$, the IW functions $\zeta_{i}$ can be expanded as

$\zeta_{i}=\sum_{n=0}^{N}\frac{\zeta_{i}^{(n)}}{n!}(w-1)^{n}\,.$ (3.4)

The truncation order $N$ of the expansion depends on the precision required
and how far from $w=1$ the form factors are evaluated. The parameters
$\zeta_{i}^{(n)}$ are unknown and have to be fixed using some dynamical
information. Notice that since the $\Lambda^{*}$ is not a ground state baryon,
the HQE does not predict any normalisation for the leading IW functions.
For convenience, I perform a fit to the lattice QCD data using the base in
Ref. [22], using the chiral and continuum extrapolation there provided to
obtain pseudo-points for the form factors. Since the lattice QCD data do not
provide information on the curvature of the form factors, it is useful to
express the HQE form factors as in the form of Eq. (3.1). At this scope, I use
the parametrisation in Eq. (3.4) up to $N=1$ and then re-expand the full form
factors up to the first order in $w-1$. After this procedure, it can be
noticed that i) the parameters $\zeta_{1}^{(1)}$ and $\zeta_{2}^{(1)}$ appear
always in the combination $\zeta_{1}^{(1)}+\zeta_{2}^{(1)}$ and ii) the
parameters $\zeta_{3}^{\text{SL},(1)}$ and $\zeta_{4}^{\text{SL},(1)}$ appear
always in the combination
$\zeta_{4}^{\text{SL},(1)}-\zeta_{3}^{\text{SL},(1)}$. Therefore these
parameters cannot be determined on their own, but only the combinations
$\zeta_{1}^{(1)}+\zeta_{2}^{(1)}$ and
$\zeta_{4}^{\text{SL},(1)}-\zeta_{3}^{\text{SL},(1)}$ are determined. This
makes the number of independent, unknown HQE parameters $10$.
Before discussing the fit results, a couple of technical comments are in
order:

  1. 1.

Given the available information in Ref. [22], it is possible to use two
pseudo-points for each form factor. I choose to evaluate them at $w=1.02$ and
$w=1.04$. Given that the HQE parametrisation depends on fewer parameters than
the lattice QCD one, it is possible to perform a fit to only a subset of the
lattice QCD data. In particular, I choose to employ the data on the vector and
axial-vector form factors and provide predictions for the tensor and pseudo-
tensor form factors based on the fit results. I comment on the consequences of
this choice in the following. I stress that this is a common procedure and has
been already employed in Ref. [31].

  2. 2.

The HQE based form factors are affected by uncertainties due to the unknown
contributions from higher orders of expansion. By naive dimensional arguments
these contributions are expected to be roughly $\mathcal{O}(\text{few}\,\%)$.
Hence, I introduce an uncorrelated $1\%$ uncertainty on all HQE expressions of
the form factors to take these effects into account. Comments on this choice
can be found later in the text.

Parameter | Best fit point
---|---
$\zeta_{1}^{(0)}$ | $0.454\pm 0.070$
$\zeta_{2}^{(0)}$ | $-0.0303\pm 0.0552$
$\zeta_{1}^{(1)}+\zeta_{2}^{(1)}$ | $0.113\pm 0.024$
$\zeta_{1}^{\text{SL},(0)}$ | $0.125\pm 0.038$
$\zeta_{1}^{\text{SL},(1)}$ | $0.0487\pm 0.0614$
$\zeta_{2}^{\text{SL},(0)}$ | $0.0110\pm 0.0363$
$\zeta_{2}^{\text{SL},(1)}$ | $0.00362\pm 0.06184$
$\zeta_{3}^{\text{SL},(0)}$ | $0.228\pm 0.190$
$\zeta_{4}^{\text{SL},(0)}$ | $0.0883\pm 0.185$
$\zeta_{4}^{\text{SL},(1)}-\zeta_{3}^{\text{SL},(1)}$ | $-0.0267\pm 0.0773$
Table 3.1: Best fit points for the HQE parameters.

The fit is performed with a $\chi^{2}$ minimisation, yielding at the minimum
$\chi^{2}_{\text{min}}/\text{d.o.f.}\sim 0.25$. This low value is a direct
consequence of the poor current knowledge of the exact size of the theory
uncertainties and their correlations. If the theory uncertainties were
uncorrelated, the fit procedure would indicate that their natural size is
smaller than the one inferred from HQE. However, such a low value for the
$\chi^{2}_{\text{min}}/\text{d.o.f.}$ could also indicate that strong
correlations among unknown higher-order terms in the HQE have been neglected.
At the current status, it is not possible to obtain a more precise estimation
of theory uncertainties and their correlations. Therefore, I retain the
conservative choice of an uncorrelated $1\%$ uncertainty on all the form
factors in the analysis.

The best-fit point for the hadronic parameters and their uncertainties is
shown in Table 3.1, and the correlation matrix among the parameters is given
in Appendix C. With this result, I compare the HQE form factors to the lattice
QCD ones. The comparison is given in Fig. 3.1, showing excellent agreement
between the two parametrisations.

|
---|---
|
|
|
Figure 3.1: Comparison between the lattice results in Ref. [22] (grey band)
and the fit results for the HQE form factors (red band) for the vector and
axial-vector form factors. The two bands represent the $68\%$ interval.

I use the results of the fit to obtain predictions for the tensor and pseudo-
tensor form factors. The comparison between them and the lattice QCD
computation is shown in Fig. 3.2 222The lattice QCD results employed here for
tensor and pseudo-tensor form factors differ by a global sign with respect to
the first version of Ref.[22]. This sign inconsistency will be fixed in a
forthcoming update of Ref.[22].. The form factors $h_{\perp^{\prime}}$ and
$\tilde{h}_{\perp^{\prime}}$ show a tension between lattice QCD data and HQE
predictions. This tension manifests itself also when introducing the tensor
and pseudo-tensor form factors in the fit. This procedure increases the
$\chi^{2}/\text{d.o.f.}$, making it much higher than 1. This corroborates the
choice of excluding them from the fit procedure.

|
---|---
|
|
Figure 3.2: Comparison between the lattice results in Ref. [22] (grey band)
and the predictions for tensor and pseudo-tensor form factors based on the fir
results in Table 3.1 (red band). The two bands represent the $68\%$ interval.

Sources of these tensions can be looked for in the lattice QCD data and HQE
framework. The hypotheses are mainly two: i) the uncertainties on the lattice
QCD parameters describing tensor and pseudo-tensor form factors are
underestimated, and ii) missing corrections in the HQE cause a shift in the
hadronic parameters. In the case of i), I checked explicitly the result of
inflating lattice QCD uncertainties by $20\%$ for $h_{\perp^{\prime}}$ and
$\tilde{h}_{\perp^{\prime}}$. In Fig. 3.3 the results of this test are shown,
proving that the compatibility slightly improves, even if it is still poor in
the case of $\tilde{h}_{\perp^{\prime}}$. Concerning ii), the most important
corrections in the HQE, beside the ones already discussed in this work, arise
at order $\mathcal{O}(\alpha_{s}/m_{b})$, and could produce a
$\mathcal{O}(\text{few}\,\%)$ shift in the central value of the form factors
parameters. However, assessing the quantitative impact of these corrections
requires understanding how they affect the form factors in a correlated way.
From these estimates, it seems that neither i) nor ii) can explain the tension
on their own. Most likely, a combination of the two effects might help to
reconcile the HQE for the tensor and pseudo-tensor form factors with the
current lattice QCD data.

(a)  (b)

Figure 3.3: Comparison between the lattice QCD results in Ref. [22] (grey
band) and the predictions for tensor and pseudo-tensor form factors based on
the fir results in Table 3.1 (red band) with lattice QCD uncertainties
inflated of $20\%$. The two bands represent the $68\%$ interval.

##  4 Conclusions

I revisit the Heavy Quark Expansion of the $\Lambda_{b}\to\Lambda^{*}(1520)$
form factors including next-to-leading order $\alpha_{s}$ corrections and for
the first time next-to-leading power $1/m_{b}$ corrections. In this framework,
form factors depend on unknown hadronic parameters which are obtained by
fitting the form factor parametrisation here discussed to a recent lattice QCD
computation [22]. I perform the fit using data for vector and axial-vector
form factors, finding good agreement between the lattice QCD calculation and
the Heavy Quark Expansion predictions. The fit results are used to predict
tensor and pseudo-tensor form factors, showing tensions between the Heavy
Quark Expansion-based predictions and the lattice QCD data. I discuss two
possible sources of the tensions: an underestimation of the uncertainties on
the lattice QCD parameters involved in these form factors and missing higher-
order terms in the Heavy Quark Expansion, e.g. at order
$\mathcal{O}(\alpha_{s}/m_{b})$. Most likely, only a combination of these two
effects could reconcile lattice QCD determination and Heavy Quark Expansion
based parametrisation of tensor and pseudo-tensor form factors. Until then, it
is not possible to claim high precision in the Heavy Quark Expansion
parametrisation of $\Lambda_{b}\to\Lambda^{*}(1520)$ form factors.
Besides this, I want to point out the need to extend the calculation of the
$\Lambda_{b}\to\Lambda^{*}(1520)$ form factors to the large-recoil region.
Quark models [19] are available, although without a consistent treatment of
uncertainties. It is therefore needed to perform up-to-date calculations of
the $\Lambda_{b}\to\Lambda^{*}(1520)$ form factors using e.g. sum rules at
$q^{2}\leq 0$. Estimates of this type will allow to extrapolate the form
factors to the large-recoil region and to assess the magnitude of their
curvature. These studies will be crucial for future LHCb analysis of
$\Lambda_{b}\to\Lambda^{*}(1520)\ell^{+}\ell^{-}$ decays.

## Acknowledgments

I thank Stefan Meinel and Gumaro Rendon for insightful discussions on lattice
QCD data. I am also grateful to Bernat Capdevila, Thorsten Feldmann, Paolo
Gambino, Martin Jung and Danny van Dyk for useful discussions. I also
acknowledge useful inputs on the text from Nico Gubernari.
This work is supported by Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under grant 396021762 - TRR 257 “Particle Physics
Phenomenology after the Higgs Discovery” and by the Italian Ministry of
Research (MIUR) under grant PRIN 20172LNEEZ.

##  Appendix A Details on the form factor parametrisation

Concerning the vector and the axial vector currents, we follow the notation of
Ref.[24]. For the vector current I have:

$\displaystyle\Gamma_{V,(1/2,t)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{+}}}\,\frac{2m_{\Lambda^{*}}}{\sqrt{s_{+}s_{-}}}p^{\alpha}\,\frac{m_{\Lambda_{b}}-m_{\Lambda^{*}}}{\sqrt{q^{2}}}\,\frac{q^{\mu}}{\sqrt{q^{2}}}\,,$
(A.1) $\displaystyle\Gamma_{V,(1/2,0)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{-}}}\,\frac{2m_{\Lambda^{*}}}{\sqrt{s_{+}s_{-}}}p^{\alpha}\,\frac{m_{\Lambda_{b}}+m_{\Lambda^{*}}}{s_{+}}\left[(p+k)^{\mu}-\frac{m_{\Lambda_{b}}^{2}-m_{\Lambda^{*}}^{2}}{q^{2}}q^{\mu}\right]\,,$
(A.2) $\displaystyle\Gamma_{V,(1/2,\perp)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{-}}}\,\frac{2m_{\Lambda^{*}}}{\sqrt{s_{+}s_{-}}}p^{\alpha}\,\left[\gamma^{\mu}-\frac{2m_{\Lambda^{*}}}{s_{+}}p^{\mu}-\frac{2m_{\Lambda_{b}}}{s_{+}}k^{\mu}\right]\,,$
(A.3) $\displaystyle\Gamma_{V,(3/2,\perp)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{-}}}\,\frac{-4i\epsilon^{\alpha\mu
pk}}{\sqrt{s_{+}s_{-}}}\gamma_{5}+\Gamma_{V,(1/2,\perp)}\,,$ (A.4)

while for the axial vector current:

$\displaystyle\Gamma_{A,(1/2,t)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{-}}}\,\frac{2m_{\Lambda^{*}}}{\sqrt{s_{+}s_{-}}}p^{\alpha}\,\frac{m_{\Lambda_{b}}+m_{\Lambda^{*}}}{\sqrt{q^{2}}}\,\frac{q^{\mu}}{\sqrt{q^{2}}}\,,$
(A.5) $\displaystyle\Gamma_{A,(1/2,0)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{+}}}\,\frac{2m_{\Lambda^{*}}}{\sqrt{s_{+}s_{-}}}p^{\alpha}\,\frac{m_{\Lambda_{b}}-m_{\Lambda^{*}}}{s_{-}}\left[(p+k)^{\mu}-\frac{m_{\Lambda_{b}}^{2}-m_{\Lambda^{*}}^{2}}{q^{2}}q^{\mu}\right]\,,$
(A.6) $\displaystyle\Gamma_{A,(1/2,\perp)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{+}}}\,\frac{2m_{\Lambda^{*}}}{\sqrt{s_{+}s_{-}}}p^{\alpha}\,\left[\gamma^{\mu}+\frac{2m_{\Lambda^{*}}}{s_{-}}p^{\mu}-\frac{2m_{\Lambda_{b}}}{s_{-}}k^{\mu}\right]\,,$
(A.7) $\displaystyle\Gamma_{A,(3/2,\perp)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{+}}}\,\frac{-4i\epsilon^{\alpha\mu
pk}}{\sqrt{s_{+}s_{-}}}\gamma_{5}-\Gamma_{A,(1/2,\perp)}\,.$ (A.8)

Concerning the tensor currents, we modify the parametrisation in Ref. [20] by
rescaling each structure with suitable factors. We have:

$\displaystyle\Gamma_{T,(1/2,0)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{+}}}\,\frac{q^{2}}{s_{+}s_{-}}p^{\alpha}\left[(p+k)^{\mu}-\frac{m_{\Lambda_{b}}^{2}-m_{\Lambda^{*}}^{2}}{q^{2}}q^{\mu}\right]\,,$
(A.9) $\displaystyle\Gamma_{T,(1/2,\perp)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{+}}}\,\frac{m_{\Lambda_{b}}+m_{\Lambda^{*}}}{s_{-}}p^{\alpha}\,\left[\gamma^{\mu}-2\frac{m_{\Lambda^{*}}}{s_{+}}p^{\mu}-2\frac{m_{\Lambda_{b}}}{s_{+}}k^{\mu}\right]\,,$
(A.10) $\displaystyle\Gamma_{T,(3/2,\perp)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{+}}}\,\left[g^{\alpha\mu}+\frac{m_{\Lambda^{*}}}{s_{-}}p^{\alpha}\left(\gamma^{\mu}-2\frac{1}{m_{\Lambda^{*}}}k^{\mu}+2\frac{m_{\Lambda^{*}}}{s_{+}}p^{\mu}+2\frac{m_{\Lambda_{b}}}{s_{+}}k^{\mu}\right)\right]\,,$
(A.11)

and

$\displaystyle\Gamma_{T5,(1/2,0)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{-}}}\,\frac{q^{2}}{s_{+}s_{-}}p^{\alpha}\left[(p+k)^{\mu}-\frac{m_{\Lambda_{b}}^{2}-m_{\Lambda^{*}}^{2}}{q^{2}}q^{\mu}\right]\,,$
(A.12) $\displaystyle\Gamma_{T5,(1/2,\perp)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{-}}}\,\frac{m_{\Lambda_{b}}-m_{\Lambda^{*}}}{s_{+}}p^{\alpha}\,\left[\gamma^{\mu}+2\frac{m_{\Lambda^{*}}}{s_{-}}p^{\mu}-2\frac{m_{\Lambda_{b}}}{s_{-}}k^{\mu}\right]\,,$
(A.13) $\displaystyle\Gamma_{T5,(3/2,\perp)}^{\alpha\mu}$
$\displaystyle=\frac{\sqrt{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}{\sqrt{s_{-}}}\,\left[g^{\alpha\mu}-\frac{m_{\Lambda^{*}}}{s_{+}}p^{\alpha}\left(\gamma^{\mu}+2\frac{1}{m_{\Lambda^{*}}}k^{\mu}-2\frac{m_{\Lambda^{*}}}{s_{-}}p^{\mu}+2\frac{m_{\Lambda_{b}}}{s_{-}}k^{\mu}\right)\right]\,.$
(A.14)

I define the helicity amplitudes as

$\mathcal{A}_{\Gamma}(s_{b},s_{\Lambda},\lambda_{\Lambda},\lambda_{q})=\bra{\Lambda^{*}(s_{\Lambda},\eta(\lambda_{\Lambda}))}\bar{s}\Gamma^{\mu}\epsilon^{*}_{\mu}(\lambda_{q})b\ket{\Lambda_{b}(s_{b})}\,,$
(A.15)

where $\epsilon^{*}_{\mu}(\lambda_{q})$ are a basis of polarisation vectors
for the virtual W exchange with polarisation states
$\lambda_{q}=\\{t,0,+1,-1\\}$. For details see Ref. [24]. The physical
helicity amplitudes are identified by the following set:

$\displaystyle\mathcal{A}_{\Gamma}(+1/2,+3/2,+1)$
$\displaystyle\equiv\mathcal{A}_{\Gamma}(+1/2,+1/2,+1,+1)\,,$ (A.16)
$\displaystyle\mathcal{A}_{\Gamma}(+1/2,+1/2,0)$
$\displaystyle\equiv\sqrt{\frac{2}{3}}\mathcal{A}_{\Gamma}(+1/2,+1/2,0,0)+\sqrt{\frac{1}{3}}\mathcal{A}^{(3/2)}_{\Gamma}(+1/2,-1/2,+1,0)\,,$
$\displaystyle\mathcal{A}_{\Gamma}(+1/2,+1/2,t)$
$\displaystyle\equiv\sqrt{\frac{2}{3}}\mathcal{A}_{\Gamma}(+1/2,+1/2,0,t)+\sqrt{\frac{1}{3}}\mathcal{A}^{(3/2)}_{\Gamma}(+1/2,-1/2,+1,t)\,,$
$\displaystyle\mathcal{A}_{\Gamma}(+1/2,-1/2,-1)$
$\displaystyle\equiv\sqrt{\frac{2}{3}}\mathcal{A}_{\Gamma}(+1/2,-1/2,0,-1)+\sqrt{\frac{1}{3}}\mathcal{A}^{(3/2)}_{\Gamma}(+1/2,+1/2,-1,-1)\,,$

where $\Gamma$ represents the four possible currents. In the case of the
vector and axial vector current, the helicity amplitudes are already
calculated in Ref. [24]. For convenience, I adapt them to this case and report
them here:

$\displaystyle\mathcal{A}_{V}(+1/2,+3/2,+1)$
$\displaystyle=-4\sqrt{m_{\Lambda_{b}}m_{\Lambda^{*}}}F_{3/2,\perp}\,,$ (A.17)
$\displaystyle\mathcal{A}_{V}(+1/2,+1/2,0)$
$\displaystyle=2\sqrt{\frac{2\,m_{\Lambda_{b}}m_{\Lambda^{*}}}{3\,q^{2}}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})F_{1/2,0}\,,$
(A.18) $\displaystyle\mathcal{A}_{V}(+1/2,+1/2,t)$
$\displaystyle=2\sqrt{\frac{2\,m_{\Lambda_{b}}m_{\Lambda^{*}}}{3\,q^{2}}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})F_{1/2,t}\,,$
(A.19) $\displaystyle\mathcal{A}_{V}(+1/2,-1/2,-1)$
$\displaystyle=-\frac{4}{\sqrt{3}}\sqrt{m_{\Lambda_{b}}m_{\Lambda^{*}}}F_{1/2,\perp}\,,$
(A.20)

and for the axial vector current:

$\displaystyle\mathcal{A}_{A}(+1/2,+3/2,+1)$
$\displaystyle=-4\sqrt{m_{\Lambda_{b}}m_{\Lambda^{*}}}G_{3/2,\perp}\,,$ (A.21)
$\displaystyle\mathcal{A}_{A}(+1/2,+1/2,0)$
$\displaystyle=2\sqrt{\frac{2\,m_{\Lambda_{b}}m_{\Lambda^{*}}}{3\,q^{2}}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})G_{1/2,0}\,,$
(A.22) $\displaystyle\mathcal{A}_{A}(+1/2,+1/2,t)$
$\displaystyle=2\sqrt{\frac{2\,m_{\Lambda_{b}}m_{\Lambda^{*}}}{3\,q^{2}}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})G_{1/2,t}\,,$
(A.23) $\displaystyle\mathcal{A}_{A}(+1/2,-1/2,-1)$
$\displaystyle=\frac{4}{\sqrt{3}}\sqrt{m_{\Lambda_{b}}m_{\Lambda^{*}}}G_{1/2,\perp}\,.$
(A.24)

In the case of tensor currents, with the definitions in Eqs. (A.11)–(A.14), I
find

$\displaystyle\mathcal{A}_{T}(+1/2,+3/2,+1)$
$\displaystyle=-2\sqrt{m_{\Lambda_{b}}m_{\Lambda^{*}}}\,T_{3/2,\perp}\,,$
(A.25) $\displaystyle\mathcal{A}_{T}(+1/2,+1/2,0)$
$\displaystyle=-\sqrt{\frac{2}{3}}\sqrt{\frac{m_{\Lambda_{b}}}{m_{\Lambda^{*}}}q^{2}}\,T_{1/2,0}\,,$
(A.26) $\displaystyle\mathcal{A}_{T}(+1/2,-1/2,-1)$
$\displaystyle=+\frac{2}{\sqrt{3}}\sqrt{\frac{m_{\Lambda_{b}}}{m_{\Lambda^{*}}}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})T_{1/2,\perp}\,,$
(A.27) $\displaystyle\mathcal{A}_{T}(+1/2,+1/2,t)$ $\displaystyle=0\,,$ (A.28)

and

$\displaystyle\mathcal{A}_{T5}(+1/2,+3/2,+1)$
$\displaystyle=+2\sqrt{m_{\Lambda_{b}}m_{\Lambda^{*}}}\,T^{5}_{3/2,\perp}\,,$
(A.29) $\displaystyle\mathcal{A}_{T5}(+1/2,+1/2,0)$
$\displaystyle=+\sqrt{\frac{2}{3}}\sqrt{\frac{m_{\Lambda_{b}}}{m_{\Lambda^{*}}}q^{2}}\,T^{5}_{1/2,0}\,,$
(A.30) $\displaystyle\mathcal{A}_{T5}(+1/2,-1/2,-1)$
$\displaystyle=+\frac{2}{\sqrt{3}}\sqrt{\frac{m_{\Lambda_{b}}}{m_{\Lambda^{*}}}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})T_{1/2,\perp}\,,$
(A.31) $\displaystyle\mathcal{A}_{T5}(+1/2,+1/2,t)$ $\displaystyle=0\,.$
(A.32)

In the heavy quark expansion, the helicity amplitudes concerning the vector
current read:

$\displaystyle\mathcal{A}_{V}(+1/2,+3/2,+1)$
$\displaystyle=2\frac{s_{+}}{m_{\Lambda_{b}}}(\zeta_{1}^{\text{SL}}+\zeta_{2}^{\text{SL}})\,,$
(A.33) $\displaystyle\mathcal{A}_{V}(+1/2,+1/2,0)$
$\displaystyle=\frac{\sqrt{2s_{+}}}{m_{\Lambda_{b}}m_{\Lambda^{*}}\sqrt{3q^{2}}}\bigg{\\{}\frac{s_{-}}{m_{\Lambda_{b}}}\left[(1+C_{0}^{(v)})(m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2})+\frac{1}{2}C_{1}^{(v)}s_{+}\right]\zeta_{2}$
$\displaystyle+$ $\displaystyle
s_{-}\left[(1+C_{0}^{(v)})(m_{\Lambda^{*}}+m_{\Lambda_{b}})+\frac{1}{2m_{\Lambda_{b}}}C_{1}^{(v)}s_{+}\right]\zeta_{1}$
$\displaystyle-$
$\displaystyle\left[m_{\Lambda_{b}}^{2}(m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}^{2}+q^{2})+s_{-}s_{+}\right]\zeta_{1}^{\text{SL}}-(m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}^{2}+q^{2})\zeta_{2}^{\text{SL}}$
$\displaystyle-$
$\displaystyle\frac{s_{-}}{2m_{\Lambda_{b}}^{2}}\left[(2m_{\Lambda^{*}}^{2}+3m_{\Lambda^{*}}m_{\Lambda_{b}}+m_{\Lambda_{b}}^{2}-2q^{2})\zeta_{3}^{\text{SL}}-(m_{\Lambda^{*}}^{2}+3m_{\Lambda^{*}}m_{\Lambda_{b}}+2m_{\Lambda_{b}}^{2}-q^{2})\zeta_{4}^{\text{SL}}\right]\bigg{\\}}\,,$
(A.34) $\displaystyle\mathcal{A}_{V}(+1/2,+1/2,t)$
$\displaystyle=\frac{\sqrt{2s_{-}}s_{+}}{m_{\Lambda_{b}}m_{\Lambda^{*}}\sqrt{3q^{2}}}\bigg{\\{}\frac{1}{m_{\Lambda_{b}}}\left[(1+C_{0}^{(v)})(-m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}+q^{2})+\frac{1}{2}C_{1}^{(v)}(-m_{\Lambda^{*}}^{2}+m_{\Lambda_{b}}^{2}+q^{2})\right]\zeta_{2}$
$\displaystyle+$
$\displaystyle\left[(1+C_{0}^{(v)})(m_{\Lambda_{b}}-m_{\Lambda^{*}})+C_{1}^{(v)}\frac{-m_{\Lambda^{*}}^{2}+m_{\Lambda_{b}}^{2}+q^{2}}{2m_{\Lambda_{b}}}\right]\zeta_{1}+\left[\frac{m_{\Lambda^{*}}^{2}-q^{2}}{m_{\Lambda_{b}}^{2}}\zeta_{1}^{\text{SL}}+\zeta_{2}^{\text{SL}}\right]$
$\displaystyle+$
$\displaystyle\frac{1}{2m_{\Lambda_{b}}^{2}}\left[(2m_{\Lambda^{*}}^{2}-m_{\Lambda^{*}}m_{\Lambda_{b}}-m_{\Lambda_{b}}^{2}-2q^{2})\zeta_{3}^{\text{SL}}-(m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-2m_{\Lambda_{b}}^{2}-q^{2})\zeta_{4}^{\text{SL}}\right]\bigg{\\}}\,,$
(A.35) $\displaystyle\mathcal{A}_{V}(+1/2,-1/2,-1)$
$\displaystyle=\frac{2\sqrt{s_{+}}}{\sqrt{3}m_{\Lambda_{b}}m_{\Lambda^{*}}}\bigg{\\{}s_{-}(1+C_{0}^{(v)})(\zeta_{2}-\zeta_{1})+m_{\Lambda^{*}}(\zeta_{1}^{\text{SL}}+\zeta_{2}^{\text{SL}})-\frac{s_{-}}{2m_{\Lambda_{b}}}(\zeta_{3}^{\text{SL}}+\zeta_{4}^{\text{SL}})\bigg{\\}}\,,$
(A.36)

and for the axial vector current:

$\displaystyle\mathcal{A}_{A}(+1/2,+3/2,+1)$
$\displaystyle=2\frac{s_{-}}{m_{\Lambda_{b}}}(\zeta_{1}^{\text{SL}}-\zeta_{2}^{\text{SL}})\,,$
(A.37) $\displaystyle\mathcal{A}_{A}(+1/2,+1/2,0)$
$\displaystyle=\frac{\sqrt{2s_{-}}}{\sqrt{3q^{2}}m_{\Lambda_{b}}m_{\Lambda^{*}}}\bigg{\\{}\frac{s_{+}}{m_{\Lambda_{b}}}\left[(1+C_{0}^{(v)})(-m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}+q^{2})+\frac{s_{-}}{2}C_{1}^{(v)}\right]\zeta_{2}$
$\displaystyle+s_{+}\left[(m_{\Lambda_{b}}-m_{\Lambda^{*}})(1+C_{0}^{(v)})+\frac{s_{-}}{2m_{\Lambda_{b}}}C_{1}^{(v)}\right]\zeta_{1}$
$\displaystyle+\frac{m_{\Lambda_{b}}^{2}(m_{\Lambda_{b}}^{2}-m_{\Lambda^{*}}^{2}-q^{2})-s_{+}s_{-}}{m_{\Lambda_{b}}^{2}}\zeta_{1}^{\text{SL}}+(m_{\Lambda^{*}}^{2}-m_{\Lambda_{b}}^{2}+q^{2})\zeta_{2}^{\text{SL}}\,,$
$\displaystyle-\frac{s_{+}}{2m_{\Lambda_{b}}^{2}}\left[(2m_{\Lambda^{*}}^{2}-3m_{\Lambda^{*}}m_{\Lambda_{b}}+m_{\Lambda_{b}}^{2}-2q^{2})\zeta_{3}^{\text{SL}}+(m_{\Lambda^{*}}^{2}-3m_{\Lambda^{*}}m_{\Lambda_{b}}+2m_{\Lambda_{b}}^{2}-q^{2})\zeta_{4}^{\text{SL}}\right]\bigg{\\}}$
(A.38) $\displaystyle\mathcal{A}_{A}(+1/2,+1/2,t)$
$\displaystyle=\frac{\sqrt{2s_{+}}s_{-}}{\sqrt{3q^{2}}m_{\Lambda_{b}}m_{\Lambda^{*}}}\bigg{\\{}\frac{1}{m_{\Lambda_{b}}}\left[(m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2})(1+C_{0}^{(v)})+\frac{1}{2}(m_{\Lambda_{b}}^{2}-m_{\Lambda^{*}}^{2}+q^{2})C_{1}^{(v)}\right]\zeta_{2}\,,$
$\displaystyle+\left[(m_{\Lambda_{b}}+m_{\Lambda^{*}})(1+C_{0}^{(v)})+\frac{m_{\Lambda_{b}}^{2}-m_{\Lambda^{*}}^{2}+q^{2}}{2m_{\Lambda_{b}}}C_{1}^{(v)}\right]\zeta_{1}+\frac{m_{\Lambda^{*}}^{2}-q^{2}}{m_{\Lambda_{b}}^{2}}\zeta_{1}^{\text{SL}}-\zeta_{2}^{\text{SL}}$
$\displaystyle+\frac{1}{2m_{\Lambda_{b}}^{2}}\left[(2m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-m_{\Lambda_{b}}^{2}-2q^{2})\zeta_{3}^{\text{SL}}+(m_{\Lambda^{*}}^{2}-m_{\Lambda^{*}}m_{\Lambda_{b}}-2m_{\Lambda_{b}}^{2}-q^{2})\zeta_{4}^{\text{SL}}\right]$
(A.39) $\displaystyle\mathcal{A}_{A}(+1/2,-1/2,-1)$
$\displaystyle=\frac{2\sqrt{s_{-}}}{\sqrt{3}m_{\Lambda_{b}}m_{\Lambda^{*}}}\bigg{\\{}s_{+}(1+C_{0}^{(v)})(\zeta_{1}+\zeta_{2})+m_{\Lambda^{*}}(\zeta_{1}^{\text{SL}}-\zeta_{2}^{\text{SL}})+\frac{s_{+}}{2m_{\Lambda_{b}}}(\zeta_{3}^{\text{SL}}-\zeta_{4}^{\text{SL}})\bigg{\\}}\,.$
(A.40)

In the case of the tensor current, the non-zero helicity amplitudes in the
heavy quark limit are

$\displaystyle\mathcal{A}_{T}(+1/2,+3/2,+1)$
$\displaystyle=\frac{2\sqrt{s_{+}}}{m_{\Lambda_{b}}}\bigg{\\{}(m_{\Lambda_{b}}-m_{\Lambda^{*}})\zeta_{1}^{\text{SL}}+\frac{m_{\Lambda^{*}}m_{\Lambda_{b}}-m_{\Lambda^{*}}^{2}+q^{2}}{m_{\Lambda_{b}}}\zeta_{2}^{\text{SL}}\bigg{\\}}\,,$
(A.41) $\displaystyle\mathcal{A}_{T}(+1/2,+1/2,0)$
$\displaystyle=\frac{\sqrt{2q^{2}s_{+}}}{\sqrt{3}m_{\Lambda_{b}}m_{\Lambda^{*}}}\bigg{\\{}s_{-}(\zeta_{2}-\zeta_{1})+\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda_{b}}^{2}-q^{2}}{m_{\Lambda_{b}}}(\zeta_{2}^{\text{SL}}+\zeta_{1}^{\text{SL}})+\frac{s_{-}}{2m_{\Lambda_{b}}}(\zeta_{3}^{\text{SL}}+\zeta_{4}^{\text{SL}})\bigg{\\}}\,,$
(A.42) $\displaystyle\mathcal{A}_{T}(+1/2,-1/2,-1)$
$\displaystyle=\frac{2\sqrt{s_{+}}}{\sqrt{3}m_{\Lambda_{b}}m_{\Lambda^{*}}}\bigg{\\{}s_{-}\left[(m_{\Lambda^{*}}+m_{\Lambda_{b}})\zeta_{1}+\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2}}{m_{\Lambda_{b}}}\zeta_{2}\right]$
$\displaystyle+\frac{m_{\Lambda^{*}}(m_{\Lambda^{*}}m_{\Lambda_{b}}-m_{\Lambda^{*}}^{2}+q^{2})}{m_{\Lambda_{b}}}\zeta_{1}^{\text{SL}}+m_{\Lambda^{*}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})\zeta_{2}^{\text{SL}}$
$\displaystyle+\frac{s_{-}}{2m_{\Lambda_{b}}}\left[(m_{\Lambda_{b}}+m_{\Lambda^{*}})\zeta_{3}^{\text{SL}}+\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2}}{m_{\Lambda_{b}}}\zeta_{4}^{\text{SL}}\right]\,,$
(A.43)

and for the tensor axial current:

$\displaystyle\mathcal{A}_{T5}(+1/2,+3/2,+1)$
$\displaystyle=-\frac{2\sqrt{s_{-}}}{m_{\Lambda_{b}}}\bigg{\\{}(m_{\Lambda_{b}}+m_{\Lambda^{*}})\zeta_{1}^{\text{SL}}+\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2}}{m_{\Lambda_{b}}}\zeta_{2}^{\text{SL}}\bigg{\\}}\,,$
(A.44) $\displaystyle\mathcal{A}_{T5}(+1/2,+1/2,0)$
$\displaystyle=\frac{\sqrt{2q^{2}s_{-}}}{\sqrt{3}m_{\Lambda_{b}}m_{\Lambda^{*}}}\bigg{\\{}s_{+}(\zeta_{1}+\zeta_{2})+\frac{m_{\Lambda^{*}}^{2}+m_{\Lambda_{b}}^{2}-q^{2}}{m_{\Lambda_{b}}}(-\zeta_{1}^{\text{SL}}+\zeta_{2}^{\text{SL}})+\frac{s_{+}}{2m_{\Lambda_{b}}}(-\zeta_{3}^{\text{SL}}+\zeta_{4}^{\text{SL}})\bigg{\\}}\,,$
(A.45) $\displaystyle\mathcal{A}_{T5}(+1/2,-1/2,-1)$
$\displaystyle=\frac{2\sqrt{s_{-}}}{\sqrt{3}m_{\Lambda_{b}}m_{\Lambda^{*}}}\bigg{\\{}s_{+}\left[(m_{\Lambda_{b}}-m_{\Lambda^{*}})\zeta_{1}+\frac{m_{\Lambda^{*}}m_{\Lambda_{b}}-m_{\Lambda^{*}}^{2}+q^{2}}{m_{\Lambda_{b}}}\zeta_{2}\right]$
$\displaystyle+\frac{m_{\Lambda^{*}}(m_{\Lambda^{*}}^{2}+m_{\Lambda^{*}}m_{\Lambda_{b}}-q^{2})}{m_{\Lambda_{b}}}\zeta_{1}^{\text{SL}}+m_{\Lambda^{*}}(m_{\Lambda^{*}}+m_{\Lambda_{b}})\zeta_{2}^{\text{SL}}$
$\displaystyle+\frac{s_{+}}{2m_{\Lambda_{b}}}\left[-(m_{\Lambda_{b}}-m_{\Lambda^{*}})\zeta_{3}^{\text{SL}}+\frac{m_{\Lambda^{*}}m_{\Lambda_{b}}-m_{\Lambda^{*}}^{2}+q^{2}}{m_{\Lambda_{b}}}\zeta_{4}^{\text{SL}}\right]\,.$
(A.46)

##  Appendix B Relations with lattice form factors

The definitions of the form factors here employed differ from other
conventions in the literature. In particular, the translation with the Lattice
determination in Ref. [22] is needed. I find

$\displaystyle F_{1/2,t}=$
$\displaystyle\,\frac{1}{2}\sqrt{\frac{s_{-}}{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}f_{0}\,,$
$\displaystyle F_{1/2,0}=$
$\displaystyle\,\frac{1}{2}\sqrt{\frac{s_{+}}{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}f_{+}\,,$
(B.1) $\displaystyle F_{1/2,\perp}=$
$\displaystyle\,\frac{1}{2}\sqrt{\frac{s_{+}}{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}f_{\perp}\,,$
$\displaystyle F_{3/2,\perp}=$
$\displaystyle\,-\frac{1}{2}\sqrt{\frac{s_{+}}{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}f_{\perp^{\prime}}\,,$
$\displaystyle G_{1/2,t}=$
$\displaystyle\,\frac{1}{2}\sqrt{\frac{s_{+}}{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}g_{0}\,,$
$\displaystyle F_{1/2,0}=$
$\displaystyle\,\frac{1}{2}\sqrt{\frac{s_{-}}{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}g_{+}\,,$
$\displaystyle G_{1/2,\perp}=$
$\displaystyle\,\frac{1}{2}\sqrt{\frac{s_{-}}{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}g_{\perp}\,,$
$\displaystyle G_{3/2,\perp}=$
$\displaystyle\,\frac{1}{2}\sqrt{\frac{s_{-}}{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}g_{\perp^{\prime}}\,,$
$\displaystyle T_{1/2,0}=$
$\displaystyle\,s_{+}^{1/2}\sqrt{\frac{m_{\Lambda^{*}}}{4m_{\Lambda_{b}}}}h_{+}\,,$
$\displaystyle T_{1/2,\perp}=$
$\displaystyle\,s_{+}^{1/2}\sqrt{\frac{m_{\Lambda^{*}}}{4m_{\Lambda_{b}}}}h_{\perp}\,,$
$\displaystyle T_{3/2,\perp}=$
$\displaystyle\,\sqrt{\frac{s_{+}}{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}(m_{\Lambda_{b}}+m_{\Lambda^{*}})h_{\perp^{\prime}}\,,$
$\displaystyle T^{5}_{1/2,0}=$
$\displaystyle\,s_{-}^{1/2}\sqrt{\frac{m_{\Lambda^{*}}}{4m_{\Lambda_{b}}}}\tilde{h}_{+}\,,$
$\displaystyle T^{5}_{1/2,\perp}=$
$\displaystyle\,s_{-}^{1/2}\sqrt{\frac{m_{\Lambda^{*}}}{4m_{\Lambda_{b}}}}\tilde{h}_{\perp}\,,$
$\displaystyle T^{5}_{3/2,\perp}=$
$\displaystyle\,-\sqrt{\frac{s_{-}}{4m_{\Lambda_{b}}m_{\Lambda^{*}}}}(m_{\Lambda_{b}}-m_{\Lambda^{*}})\tilde{h}_{\perp^{\prime}}\,.$

##  Appendix C Correlations between the fit parameters

The correlation matrix for the HQE parameters is reported in Table C.1. The
order of the various correlation coefficients is the same as in Table 3.1.

$1$ | $-0.879$ | $0.440$ | $0.0458$ | $0.0460$ | $-0.120$ | $-0.0619$ | $0.363$ | $0.337$ | $0.0312$
---|---|---|---|---|---|---|---|---|---
$-0.879$ | $1$ | $-0.160$ | $0.0109$ | $-0.0585$ | $0.130$ | $0.0936$ | $-0.325$ | $-0.343$ | $-0.121$
$0.440$ | $-0.160$ | $1$ | $-0.00723$ | $-0.00712$ | $0.0101$ | $0.0465$ | $0.211$ | $0.139$ | $-0.218$
$0.0458$ | $0.0109$ | $-0.00723$ | $1$ | $0.861$ | $-0.512$ | $-0.382$ | $-0.221$ | $-0.0500$ | $0.611$
$0.0460$ | $-0.0585$ | $-0.00712$ | $0.861$ | $1$ | $-0.406$ | $-0.435$ | $-0.214$ | $-0.0603$ | $0.707$
$-0.120$ | $0.130$ | $0.0101$ | $-0.512$ | $-0.406$ | $1$ | $0.887$ | $0.0941$ | $-0.119$ | $-0.783$
$-0.0619$ | $0.0936$ | $0.0465$ | $-0.382$ | $-0.435$ | $0.887$ | $1$ | $0.160$ | $-0.0287$ | $-0.871$
$0.363$ | $-0.325$ | $0.211$ | $-0.221$ | $-0.214$ | $0.0941$ | $0.160$ | $1$ | $0.966$ | $-0.164$
$0.337$ | $-0.343$ | $0.139$ | $-0.0500$ | $-0.0603$ | $-0.119$ | $-0.0287$ | $0.966$ | $1$ | $0.0606$
$0.0312$ | $-0.121$ | $-0.218$ | $0.611$ | $0.707$ | $-0.783$ | $-0.871$ | $-0.164$ | $0.0606$ | $1$

Table C.1: Correlation matrix for the HQE parameters.

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#  KAER: A Knowledge Augmented Pre-Trained Language Model for Entity
Resolution

Liri Fang1, Lan Li1, Yiren Liu2, Vetle I. Torvik1, Bertram Ludäscher1

###### Abstract

Entity resolution has been an essential and well-studied task in data cleaning
research for decades. Existing work has discussed the feasibility of utilizing
pre-trained language models to perform entity resolution and achieved
promising results. However, few works have discussed injecting domain
knowledge to improve the performance of pre-trained language models on entity
resolution tasks. In this study, we propose Knowledge Augmented Entity
Resolution (KAER), a novel framework named for augmenting pre-trained language
models with external knowledge for entity resolution. We discuss the results
of utilizing different knowledge augmentation and prompting methods to improve
entity resolution performance. Our model improves on Ditto, the existing
state-of-the-art entity resolution method. In particular, 1) KAER performs
more robustly and achieves better results on “dirty data”, and 2) with more
general knowledge injection, KAER outperforms the existing baseline models on
the textual dataset and dataset from the online product domain. 3) KAER
achieves competitive results on highly domain-specific datasets, such as
citation datasets, requiring the injection of expert knowledge in future work.

††Accepted to Workshop on Knowledge Augmented Methods for Natural Language
Processing, in conjunction with AAAI 2023.

## Introduction

Recent studies using _transformer-based Pre-trained Language Models_ (PLMs)
have shown their strong ability to perform various types of NLP tasks (Min et
al. 2021). However, few studies have discussed the application of PLMs in the
domain of data cleaning (Li et al. 2020; Narayan et al. 2022; Vos, Döhmen, and
Schelter 2022). Entity resolution is a common data cleaning task that aims to
identify the entries referring to the same real-world entities within or
across databases (Christen 2012).

Most existing techniques on entity resolution assume the same schema for
records from different sources (Elmagarmid, Ipeirotis, and Verykios 2007).
However, in many situations, raw records are obtained from heterogeneous
sources and use different schema (Enríquez et al. 2017; Li et al. 2021b). In
addition, source data is often from varied domains (e.g., publications, online
products, musicians) and in different formats (e.g., numerical, textual,
geolocations). All of these increase the difficulty for practitioners to
perform entity resolution tasks without prior knowledge of the domain-specific
information about the data. Thus, we hypothesize that enhancing the external
knowledge at the schema and entity level can improve entity resolution tasks.

With transformer-based PLMs, recent studies draw increasing attention to
entity resolution problems (Li et al. 2020; Trabelsi, Heflin, and Cao 2022).
However, current studies show that the performance might not be ideal when
simply inputting the serialized entity pairs into PLMs for classification.
Ditto (Li et al. 2020) injects domain information: pre-defined entity types
(i.e., PRODUCT and NUM), and standardizes the numerical formats to improve the
performance before feeding the serialized entity pairs into PLMs.

We push this idea further by injecting more external knowledge at the schema
and entity level. Knowledge injection at the schema level aims to infer the
fine-grained semantic types (e.g., ALBUM, ARTIST, PUBLISHER) for each column
based on data values. For the entity level, entity mentions are identified
from WikiData and annotated in the initial text with semantic type information
of the linked entities. In addition, different formats used to inject external
knowledge into the initial entity pairs may vary the performance of PLMs.
Thus, three prompting methods are further explored in this study: space,
slash, and constrained tuning of PLMs.

To summarize, starting from state-of-the-art method Ditto (Li et al. 2020), we
propose a framework for Knowledge Augmented Entity Resolution (KAER):

  * •

using Column Semantic Type (CST) inference and Entity Linking (EL) in order to
inject domain-specific information as additional signals to pre-trained
language models.

  * •

leveraging three prompting methods to better augment the acquired knowledge to
PLMs.

  * •

analyzing the effectiveness of different combinations of knowledge injection
and prompting methods on entity resolution tasks from different domains and
data types.

## Related Work

Figure 1: The framework of KAER. First, (a) column semantic type augmentation
and (b) entity semantic type augmentation are used to inject domain knowledge
from both column and entity levels of the input table. Then three options of
prompting types are provided after external knowledge injection: (1) using
space; (2) using “/”; (3) using constrained tuning. With the knowledge
injection and prompting methods, the serialized sequence is fed into the PLMs.

### Pretrained Model for Entity Resolution

A few recent works apply transformer-based PLMs to entity resolution tasks.
Paganelli et al. (2022) discover that simply fine-tuning BERT can benefit
matching/non-matching classification tasks and recognize the input sequence as
a pair of records. Li et al. (2021a) leverage siamese network structure with
PLMs to improve the efficiency of PLMs during the blocking phase. Ditto by (Li
et al. 2020) is the state-of-the-art entity matching system based on PLMs,
i.e., RoBERTa. In addition, Ditto provides a deeper language understanding for
entity resolution by injecting domain knowledge, summarizing the key
information, and augmenting with more difficult examples for training data.

### Knowledge Augmentation

Recent studies on semantic column type augmentation and entity linking can
inject external domain knowledge into PLMs at the schema and entity levels.
Semantic column-type augmentation can inject domain-specific knowledge for
columns with/without existing column names. Existing studies (Hulsebos et al.
2019; Suhara et al. 2022) use deep learning approaches to detect semantic data
types at the column level. Entity linking (Li et al. 2020) refers to linking
entity mentions appearing in natural language text with their corresponding
entities in an external knowledge base, e.g., Wikidata. Ayoola et al. (2022)
introduces an entity linking method by fine-tuning a PLM over Wikidata, which
is used for entity linking in this study.

“prompting” is the method of conditioning the language model Liu et al.
(2021). Kojima et al. (2022) propose Zero-shot-CoT, a single zero-shot prompt
that highlights that the performance of the language model has been affected
by different templates of the prompt. Furthermore, the method to inject the
identified external knowledge into PLMs matters. Liu et al. (2020) propose the
model K-BERT adding soft-position encoding and visible matrix to the augmented
input sequence to reduce knowledge noise injected into the original input and
corresponding models.

## Methodology

### Notation of Entity Resolution Task

The formulation of the entity resolution task in this paper is based on the
current work by Li et al. (2020). The input of the entity resolution task
consists of a set $M\subseteq D_{1}\times D_{2}$, where $D_{1}$ and $D_{2}$
are two sets of data entry collections that contain duplicated entries. For
each data entry, $e\in{(col_{i},val_{i})}_{1\leq i\leq N}$ where $N$ is the
number of columns, and $(e_{1},e_{2})\in M$. The task discussed in this paper
focuses on: for each data entry pair $(e_{1},e_{2})\in M$, determine whether
$e_{1}$ and $e_{2}$ refer to the same data entity.

### Model Overview

This study introduces a new framework for entity resolution: KAER (See Figure
1). KAER uses PLMs for entity resolution and contains three modules for
knowledge augmentation: a) column semantic type augmentation, b) entity
semantic type augmentation, and c) three options of prompting types. The
following sections first describe the PLMs for entity resolution and then each
module.

#### Pre-trained language model for entity resolution.

Following the work by Li et al. (2020), KAER uses RoBERTa as the backbone
model. For each data entry pair, $(e_{1},e_{2})$, the text context of column
names and values of $e_{1}$ and $e_{2}$ are serialized and concatenated as the
input for PLMs. The [CLS] token position is used to classify whether $e_{1}$
and $e_{2}$ refer to the same entity. The loss for optimizing the
classification objective is:

$\ell=-log\ p(y|s(e_{1},e_{2}))$ (1)

where y denotes whether $e_{1}$ and $e_{2}$ refer to the same entity, and
$s(\cdot,\cdot)$ denotes the serialization and transformation of entity pairs
with knowledge injection and prompting methods.

$s(e_{i},e_{j})::=\text{[CLS]}\ serialize(e_{i})\ \text{[SEP]}\
serialize(e_{j})\ \text{[SEP]}$

(2)

where $serialize(\cdot)$ serializes each data entry.

$\begin{split}serialize(e_{i})::=&\text{[COL]}\ f(col_{1},pt)\ \text{[VAL]}\
g(val_{1},pt)\ ...\\\ &\text{[COL]}\ f(col_{N},pt)\ \text{[VAL]}\
g(val_{N},pt)\end{split}$

(3)

where $f(col_{i},pt)$ denotes the semantic column type injection with
prompting method $pt$, and $g(val_{i},pt)$ denotes the entity linking
injection with prompting method $pt$. The following sections describe column
semantic type injection, entity linking, and three prompting methods.

#### Column semantic type augmentation.

Column type prediction aims to predict the semantic type of each column by
considering the intra-column context, so that fine-grained semantic
information can be augmented to the framework. For example, “name” as a column
name can provide more semantic information if injected with the semantic type
“song”. Here, KAER uses Sherlock (Hulsebos et al. 2019) for column semantic
type prediction. In detail, Sherlock is a multi-input deep neural network pre-
trained on 686,765 data columns retrieved from the VizNet corpus by matching
78 semantic data types from DBpedia to column headers (Hulsebos et al. 2019).

| Dirty | Structured | Textual
---|---|---|---
Models |  | DBLP
---
-
GoogleScholar
| iTunes
---
-
Amazon
| iTunes
---
-
Amazon
| Amazon
---
-
Google
| DBLP
---
-
ACM
| DBLP
---
-
GoogleScholar
| Abt
---
-
Buy
RoBERTa | 95.81 | 72.73 | 61.76 | 73.12 | 98.77 | 95.32 | 88.41
Ditto | 95.54 | 60.38 | 47.06 | 74.44 | 98.55 | 95.85 | 88.89
KAER ( Roberta + CST ) | 95.47 | 62.96 | 80.70**++ | 68.70 | 98.77 | 95.05 | 92.27**++
KAER ( Roberta + CST + / ) | 95.74 | 54.90* | 89.66**++ | 74.85 | 98.99 | 95.14 | 92.61**++
KAER ( Roberta + CST + PCT) | 95.20 | 72.00 | 57.14 | 76.25 | 98.09* | 95.62 | 78.24**++
KAER ( Roberta + EL ) | 95.12 | 50.00* | 61.54 | 73.77 | 98.54 | 95.72 | 89.11
KAER ( Roberta + EL + / ) | 95.75 | 81.82 | 86.21**++ | 74.02 | 98.43 | 96.11* | 88.19
KAER ( Roberta + EL + PCT) | 94.80**+ | 77.78+ | 67.69 | 75.63 | 97.04**+ | 95.29 | 73.80**++
KAER ( Roberta + CST + EL ) | 95.87 | 47.06* | 59.26 | 72.28 | 98.43 | 95.65 | 89.66
KAER ( Roberta + CST + EL + / ) | 95.63 | 63.33* | 57.14+ | 75.15 | 98.34 | 95.67 | 89.81
KAER ( Roberta + CST + EL + PCT) | 94.24**++ | 71.11+ | 61.29 | 71.37* | 96.26**++ | 94.93+ | 67.20**++

Table 1: Experimental results using different knowledge injection methods,
measured by F1 score. The two baseline models: RoBERTa (without domain
knowledge injection); and Ditto (RoBERTa and Ditto’s general domain knowledge
injection methods). Other models represent KAER with various knowledge
injection and prompting methods combined with RoBERTa. “+CST” indicates Column
Semantic Type injection.“+EL” indicates knowledge injection with Entity
Linking. “+/” represents prompting with a slash. “+PCT” represents Prompting
with Constrained Tuning, i.e., soft positions and visible matrix. The paired
t-test is processed on between KAER injections and two baseline models, i.e.,
RoBERTa(*) and Ditto(+). ** and ++ represent 99% confidence level, and * and +
represent 95% confidence level.

#### Entity semantic type augmentation.

Entity semantic type augmentation leverages the entity linking method to
identify all entity mentions from a given knowledge base (KB) within the
target text sequence. More specifically, 1) to identify the text span of each
entity mention $m\in M$, and 2) map each mention to the entity set from an
external KB $\mathcal{M}:M\rightarrow E$. Thus, additional knowledge from the
KB is used to augment the input text records. In this study, WikiData is used
as the external KB since it covers a wide range of domains and is suitable for
entity resolution over different datasets. KAER uses the state-of-the-art
method proposed by Ayoola et al. (2022), which uses a RoBERTa model jointly
pre-trained over entity typing and entity description modeling objectives to
inject coarse entity types.

#### Prompting methods

##### Template-based prompting.

After acquiring domain knowledge, we experiment using text-based templates to
combine the knowledge with the initial text input. Two different characters
are used to concatenate the original entity mention/column name, i.e. slash
(“/”) and space. The slash symbol is chosen because it contains a semantic
meaning equivalent to ’or’ in general web text.

##### Prompting with constrained tuning

Inspired by hard-soft prompt hybrid tuning (Han et al. 2021), prompting with
constrained tuning is used to generate the token embedding by controlling the
visible area of the augmented knowledge only to its corresponding entity
mentions or column names. This type of domain knowledge injection modifies the
initial input sequence in two dimensions, including the token position and
token semantics. More specifically, the token from the initial input sequence
has a different absolute position when the template-based prompting is
concatenated. In addition, the injected knowledge for a specific entity
mention is shared with the surrounding tokens in the same sequence. These two
modifications might introduce erroneous knowledge and overload to the
framework.

To limit the erroneous knowledge and overload, we leverage constrained tuning,
i.e., soft-position encoding and visible matrix, as an additional input of our
backbone model, inspired by KBert (Liu et al. 2020). Here, the knowledge-
injected sequence is constructed into a tree structure, where the injected
knowledge subsequences are considered branches. In this way, as shown in
Figure 1, the soft-position encoding assigns the initial tokens the same
position encoding in the knowledge-injected sequence as those in the initial
input sequence, and assigns the injected knowledge tokens the position in the
branches. The visible matrix $V\in\mathcal{R}^{N\times N}$ is a binary matrix,
where the injected knowledge token is only visible to itself and the
corresponding entity mention or column name tokens, and set as 1, denoted as
follows:

$V_{ij}=\begin{cases}1,&\text{if $token_{i}$ \text{and} $token_{j}$ co-
occur}.\\\ 0,&\text{otherwise}.\end{cases}$ (4)

where the co-occurrence of $token_{i}$ and $token_{j}$ indicates that the
tokens co-occur in the initial input sequences or co-occur as head and tail in
domain knowledge injection. Here, the head means the entity mention/column
name, and the tail indicates the injected knowledge text.

## Experiments

### Dataset

KAER is evaluated on the Magellan datasets (Das et al. 2022) across various
domains. The overview of the datasets used in this study is shown in Table 2.
The Magellan datasets contain three types of datasets: dirty dataset,
structured dataset, and textual dataset. The dirty dataset is generated from
the structured dataset by randomly removing attribute values and appending the
initial values to another randomly selected attribute (Li et al. 2020).

Data Type | Dataset | Domain | Size | # Positive | # Attr.
---|---|---|---|---|---
Structured | Amazon-Google | software | 11,460 | 1,167 | 3
| iTunes-Amazon | music | 539 | 132 | 8
| DBLP-ACM | citation | 12,363 | 2,220 | 4
| DBLP-GoogleScholar | citation | 28,707 | 5,347 | 4
Dirty | iTunes-Amazon | music | 539 | 132 | 8
DBLP-ACM | citation | 12,363 | 2,220 | 4
DBLP-GoogleScholar | citation | 28,707 | 5,347 | 4
Textual | Abt-Buy | product | 9,575 | 1,028 | 3

Table 2: Dataset Summary

### Results

The experimental results are presented in Table 1.

Knowledge augmentation performs better on smaller datasets. According to the
experimental results, incorporating external knowledge can improve the
performance on entity resolution tasks. In particular, the proposed knowledge
augmentation methods outperform the two baseline models in a data-scarce
context. For example, KAER (RoBERTa+EL+/) outperforms both baseline models
Roberta and Ditto by 9.09% and 21.44% $\uparrow$ on dataset Dirty/iTunes-
Amazon, but not with statistic significance. Meanwhile, it statistically
significantly outperforms both baseline models 24.45% and 39.15% $\uparrow$ on
the dataset Structured/iTunes-Amazon. The two baseline models perform worse on
smaller datasets because the language model might not be able to learn enough
information to distinguish between different entities when fine tuning, given
fewer training instances correctly. Prompting with PCT achieves better
performance in datasets with smaller text lengths. For example, KAER
(RoBERTa+CST+PCT) reaches the best F1 score (76.25%) without statistic
significance on dataset Structured/Amazon-Google, which contains the smallest
length of serialized data entry. Moreover, KAER (RoBERTa+CST+PCT) on
Structured/Amazon-Google makes a significant difference compared to both
baseline models, at 99% confidence level.

Knowledge augmentation does not perform well on datasets from certain domains.
The datasets used in our experiments span several different domains. Using our
proposed knowledge augmentation methods (CST and EL), records from certain
domains, such as music and online product, benefit from more accurate
retrieved knowledge. For example, KAER (RoBERTa+CST+/) achieves the best
F1-score with statistic significance, i.e., 92.61%, on the textual dataset
Abt-Buy from the online product domain. KAER (RoBERTa+EL+/) achieves the best
F1-score (81.82%) without statistical significance on the musical dataset
Dirty/iTunes-Amazon. One potential reason for this notable improvement is that
the entities mentioned in the datasets from these domains are more general.
However, publication datasets from domains like citation require more domain-
specific or even expert knowledge rather than general commonsense. Our
knowledge injection methods are primarily designed for retrieving general
knowledge. Thus, for datasets, like Structured/DBLP-ACM, KAER (RoBERTa+CST)
performs as well as Roberta and significantly makes a difference with Ditto at
95% confidence level, and KAER (RoBERTa+CST+/) achieves the best F1-score
98.99% without statistic significance. But all the other KAER models perform
worse. For future improvement, additional external expert knowledge from the
citation domain should be incorporated for knowledge injection.

Knowledge augmentation is affected by data quality. KAER, with certain
knowledge augmentation methods, outperforms the baseline models on dirty
datasets. For instance, KAER (RoBERTa+CST+EL) slightly exceeds RoBERTa, and
Ditto on Dirty/DBLP-GoogleScholar. However, the F1 scores of all the other
injections on the Dirty/DBLP-GoogleScholar cannot compete with the results on
the baseline models (RoBERTa and Ditto). One reason might be that the
misleading predictions based on the dirty input by CST result in semantic
noise and propagate to the PLMs. On the other hand, knowledge augmentation
with PCT can advance the robustness of the model dealing with low-quality
injections. For instance, KAER (RoBERTa+CST+PCT) obtains better F1 score than
KAER (RoBERTa+CST) (72.00% vs. 62.96%). Similar improvement applies to KAER
(RoBERTa+EL+PCT) vs. KAER (RoBERTa+EL) and KAER (RoBERTa+CST+EL+PCT) vs. KAER
(RoBERTa+CST+EL).

## Conclusion

In this study, we introduced a novel framework, KAER, for entity resolution
problems by augmenting PLMS with external knowledge using prompting
techniques. The framework is evaluated over a set of datasets from varied
domains and data types (i.e., dirty, structured, and textual) and improves
performance in certain applications over the baseline models. Further analysis
revealed the effectiveness of different domain knowledge injection and
prompting methods on various datasets from different domains and data types.
The results show a promising direction of injecting knowledge to improve
entity resolution performance using PLMs. Future studies can improve upon our
method by introducing more domain-specific knowledge.

## Acknowledgement

The experiments are conducted on the NCSA Delta supercomputing system,
supported by the National Science Foundation under Grant No. OAC-2005572. This
work is supported by the National Institute on Aging of the NIH (Award Number
P01AG039347). The funders had no role in study design, data collection and
analysis, decision to publish, or preparation of the manuscript.

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## APPENDIX

##### A. Hyperparameter Settings.

Baseline and KAER models share the same hyperparameter setting, in which the
batch size equals 64, the max length equals 512, the learning rate is 3e-5,
and the number of training epochs is 20.

##### B. Baseline models.

Dataset | Injection
---|---
dirty/DBLP-Google | Ditto + General
dirty/iTunes-Amazon | Ditto + Product
structured/iTunes-Amazon | Ditto + Product
structured/Amazon-Google | Ditto + Product
structured/DBLP-ACM | Ditto + General
structured/DBLP-Google | Ditto + General
textual/Abt-Buy | Ditto + Product
Table 3: Datasets and Corresponding Ditto default knowledge injection methods

This paper includes two baseline models, i.e., RoBERTa without any knowledge
injection and Ditto with its default knowledge injection methods.

In detail, Table 3 shows the datasets and corresponding injection methods
utilized by Ditto. ”+ General” indicates that Ditto injects seven entity types
into the corresponding dataset. The seven types include ’PERSON’, ’ORG’,
’LOC’, ’PRODUCT’, ’DATE’, ’QUANTITY’, and ’TIME’. ”+ Product” indicates that
Ditto annotates as ’PORODUCT’, any entity mentions in the following types,
i.e., ’NORP’, ’GPE’, ’LOC’, ’PERSON’, and ’PRODUCT’.

##### C. Two-sided Paired T-test.

The two-sided paired T-test is implemented to test whether the predictions of
different models are significantly different. We first compare whether the
predicted label is equal to the ground-truth label. The value is 1 if the
prediction is equal to the ground truth and 0 otherwise. And then, the paired
T-test is conducted.