# Heavy quark symmetry in quark models

Jiabao<EMAIL_ADDRESS>Xiang-Nan
<EMAIL_ADDRESS>Chia-Wei<EMAIL_ADDRESS>and
Chao-Qiang<EMAIL_ADDRESS>School of Fundamental Physics and
Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China
University of Chinese Academy of Sciences, 100190 Beijing, China

###### Abstract

We study the heavy quark symmetry with the homogeneous bag model (HBM) and
light-front quark model (LFQM) based on the decays of
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\ell^{-}\overline{\nu}_{\ell}~{}(\ell=e,\mu,\tau)$.
In particular, we calculate various parameters in the heavy quark expansions,
including the Isgur-Wise functions and their first order corrections. The
parameters in the HBM are fitted from the mass spectra, while the ones in the
LFQM are tightly constrained by the heavy quark symmetry, granting the
predictive power of our results. We explicitly obtain that ${\cal
B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}e^{-}\overline{\nu}_{e})=(5.69\pm
0.58,5.35\pm 0.50)$, ${\cal
B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\mu^{-}\overline{\nu}_{\mu})=(5.67\pm
0.58$, $5.33\pm 0.49)$, and
$\Gamma(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\tau^{-}\overline{\nu}_{\tau})/\Gamma(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\mu^{-}\overline{\nu}_{\mu})=(0.3243\pm
0.0126$, $0.3506\pm 0.0046)$ for the numerical values of (HBM, LFQM). Our
results of the branching fractions in both models agree well with the
experimental data and lattice QCD calculations. In addition, we find that the
hard gluon corrections decrease the branching fractions around $10\%$.

##  I Introductions

Testing the lepton universality via beauty quark decays has raised great
interest in both theories and experiments LHCb:2021trn ; LHCb:2022wrs ;
Iguro:2022yzr ; Choudhury:2022ktg ; Belle:2021dgc ; Sheng:2020drc ;
Das:2019cpt ; Zhang:2019xdm . Recently, the LHCb collaboration has reported
the ratio of $R_{\Lambda_{c}}={\cal
B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\tau^{-}\overline{\nu}_{\tau})/{\cal
B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\mu^{-}\overline{\nu}_{\mu})$, given by
LHCb:2022piu ; LHCb:2017vhq

$R_{\Lambda_{c}}=0.242\pm 0.026\pm 0.040\pm 0.059,$ (1)

where the first and second uncertainties are systematic and statistical in
${\cal B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\tau^{-}\overline{\nu}_{\tau})$,
and the third one comes from ${\cal
B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\mu^{-}\overline{\nu}_{\mu})$,
respectively. On the other hand, the lattice quantum chromodynamics (LQCD)
gives a slighly larger ratio Detmold:2015aaa ; Bernlochner:2018kxh . The
results along with the meson versions are summarized in FIG. 1 with
$R_{D^{(\ast)}}=\Gamma(B\to
D^{(\ast)}\tau^{-}\overline{\nu}_{\tau})/\Gamma(B\to
D^{(\ast)}e^{-}\overline{\nu}_{e})$ HFLAV:2022pwe ; Martinelli:2022fgg .
Notice that the experimental values of $R_{D^{(\ast)}}$ are larger than the
theorical ones in contrast to $R_{\Lambda_{c}}$.

Figure 1: The ratios of $R_{D},R_{D^{*}}$ and $R_{\Lambda_{c}}$ in the
experiments and LQCD.

On the theoretical side, the form factors of
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}$ provide ideal playgrounds for quark
models. The main reason is that at the massless limit of $(u,d)$, there are
only two energy scales in $\Lambda_{Q}$ with $Q=(b,c)$, given by

$\overline{\Lambda}=\lim_{m_{Q}\to\infty}\left(M_{Q}-m_{Q}\right)\,,~{}~{}~{}\varepsilon_{Q}=\overline{\Lambda}/2m_{Q}\ll
1\,,$ (2)

where $M_{Q}$ and $m_{Q}$ are the heavy baryon and quark masses, respectively.
As $\varepsilon_{Q}$ are tiny, the physical quantities are Taylor-expanded
regarding to $\varepsilon_{Q}$, namely the heavy quark expansion (HQE).

We take the Isgur-Wise function as an illustration, given as

$\overline{\xi}(\omega)=\xi(\omega)+(\varepsilon_{b}+\varepsilon_{c})\xi_{ke}(\omega)+{\cal
O}(\varepsilon_{c}^{2})\,,$ (3)

where $\xi(\omega)$ is the Isgur-Wise function, $\xi_{ke}(\omega)$ is the
first order correction, and $\omega=v_{b}\cdot v_{c}$ with $v_{Q}$ the
4-velocities of $\Lambda_{Q}$. As we will see in the next section,
$\overline{\xi}(\omega)$ governs the recoil effects of the form factors.
Although the experiments can probe $\overline{\xi}(\omega)$ only, Eq. (3)
allows us to compare $\xi(\omega)$ and $\xi_{ke}(\omega)$ separately among the
quark models, without the dependence of the quark masses.

In this work, we take the homogeneous bag model (HBM) and light-front quark
model (LFQM) to illustrate the heavy quark symmetry. These models are
unimpressive but interesting in different aspects. On the one hand, the HBM is
a relativistic quark model, in which $u$ and $d$ quarks can be safely taken as
massless, and the parameters are fitted from the mass spectra. It is reliable
at the zero recoil point $(\omega=1)$, but unequal time commutators are needed
for a boosted state, causing several uncertainties as $\omega$ goes up. On the
other, the LFQM describes the bound state in a frame-independent way, in which
only the relative motions between the constituents are required
Schlumpf:1992vq ; Cheng:1996if ; Cheng:2004cc ; Ke:2007tg ; Ke:2012wa ;
Wang:2017mqp ; Geng:2022xpn ; Zhao:2022vfr ; Chen:2021ywv ; Chang:2018zjq ;
Chua:2019yqh . The Lorentz boosts in the front form are generated by the
kinematical operators, which leave the $x^{+}=0$ plane invariant Dirac:1949cp
; Chiu:2017ycx , so the unequal time commutation relations are not needed.
However, the LFQM suffers the uncertainties from the parameters input. In
addition the Z-graph contributions forbid us from computing the form factors
in the time-like region ODonnell:1995dio ; Simula:1996pk ; Cardarelli:1997sx ;
Demchuk:1998gv ; Choi:1998nf ; Choi:2012zzb ; Choi:2011xm . We will show that
these aforementioned difficulties are resolved by the heavy quark symmetry.

This paper is organized as follows. We present the formalism of the HBM and
LFQM in Sec. II. The numerical results are given in Sec. III. Sec. IV is the
conclusion.

##  II Formalism

We briefly review some of the results of the HQE, where the details can be
found in Ref. Bernlochner:2018kxh . In the heavy quark system, there are two
popular parametrizations for the form factors, given as

$\displaystyle\begin{aligned}
\langle\Lambda_{c}^{+}|\overline{c}\gamma^{\mu}(1-\gamma_{5})b|\Lambda_{b}^{0}\rangle=\overline{u}_{c}&\left[(F^{V}_{1}(\omega){\gamma^{\mu}}+F^{V}_{2}(\omega)v_{b}^{\mu}+F^{V}_{3}(\omega)v_{c}^{\mu})\right.\\\
&~{}-\left.(F^{A}_{1}(\omega){\gamma^{\mu}}+F^{A}_{2}(\omega)v_{b}^{\mu}+F^{A}_{3}(\omega)v_{c}^{\mu})\gamma_{5}\right]u_{b}\\\
=\overline{u}_{c}&[(f_{1}(q^{2})\gamma^{\mu}-f_{2}(q^{2})\frac{i\sigma^{\mu\nu}q_{\nu}}{M_{b}}+f_{3}(q^{2})\frac{q^{\mu}}{M_{b}})\\\
&~{}-(g_{1}(q^{2})\gamma^{\mu}-g_{2}(q^{2})\frac{i\sigma^{\mu\nu}q_{\nu}}{M_{b}}+g_{3}(q^{2})\frac{q^{\mu}}{M_{b}})\gamma_{5}]u_{b},\\\
\end{aligned}$ (4)

where $q=p_{b}-p_{c}$, and $p_{b}$ and $p_{c}$ are the four momenta of
$\Lambda_{b}^{0}$ and $\Lambda_{c}^{+}$, respectively. From
$p_{Q}=m_{Q}v_{Q}$, it is straightforward to show that
$q^{2}=M_{b}^{2}+M_{c}^{2}-2M_{b}\,M_{c}\,\omega$. Two sets of
parametrizations are related as

$\displaystyle
f_{1}=F_{1}^{V}+\frac{M_{+}}{2M_{b}}F_{2}^{V}+\frac{M_{+}}{2M_{c}}F_{3}^{V},~{}$
$\displaystyle f_{2,3}=\mp\frac{1}{2}F_{2}^{V}-\frac{M_{b}}{2M_{c}}F_{3}^{V},$
(5) $\displaystyle
g_{1}=F_{1}^{A}-\frac{M_{-}}{2M_{b}}F_{2}^{A}-\frac{M_{-}}{2M_{c}}F_{3}^{A},~{}$
$\displaystyle g_{2,3}=\mp\frac{1}{2}F_{2}^{A}-\frac{M_{b}}{2M_{c}}F_{3}^{A},$

with $M_{\pm}\equiv M_{b}\pm M_{c}$.

At the zero-recoil point of $\omega=1$, the form factors are simply written as
Falk:1992ws

$\displaystyle
F^{V}_{1}(1)=1+\varepsilon_{b}+\varepsilon_{c}+\varepsilon_{c}^{2}(b_{1}-b_{2})\,,$
$\displaystyle F_{1}^{A}=1+\varepsilon_{c}^{2}b_{1}\,,$ (6) $\displaystyle
F_{2}^{V}(1)=F_{2}^{A}(1)=-\varepsilon_{c}+b_{2}\varepsilon^{2}_{c},$
$\displaystyle F_{3}^{V,A}(1)=\mp\varepsilon_{b}\,,$

to the precision of ${\cal O}(\varepsilon_{b}\varepsilon_{c})$, providing that
hard gluon corrections are absent. A great advantage in the heavy quark system
is that the recoil effects are taken account by a single function of
$\overline{\xi}(\omega)$, $i.e.$

$F_{1,2,3}^{V,A}(\omega)=\overline{\xi}(\omega)F_{1,2,3}^{V,A}(1)\,.$ (7)

After including the hard gluon corrections shown in FIG. 2, the form factors
receive several corrections, given by

$\displaystyle\frac{F_{1}^{V}(\omega)}{\overline{\xi}(\omega)}=1+\hat{\alpha}_{s}C_{V_{1}}+\varepsilon_{c}+\varepsilon_{b}+\hat{\alpha}_{s}\left[C_{V_{1}}+2(\omega-1)C_{V_{1}}^{\prime}\right]\left(\varepsilon_{c}+\varepsilon_{b}\right)+\varepsilon_{c}^{2}\left(b_{1}-b_{2}\right)\,,$
$\displaystyle\frac{F_{2}^{V}(\omega)}{\overline{\xi}(\omega)}=\hat{\alpha}_{s}C_{V_{2}}-\frac{2\varepsilon_{c}}{\omega+1}+\hat{\alpha}_{s}\left\\{C_{V_{2}}\frac{3\omega-1}{\omega+1}\varepsilon_{b}-\left[2C_{V_{1}}-(\omega-1)C_{V_{2}}+2C_{V_{3}}\right]\frac{\varepsilon_{c}}{\omega+1}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+2(\omega-1)C_{V_{2}}^{\prime}\left(\varepsilon_{c}+\varepsilon_{b}\right)\bigg{\\}}+\varepsilon_{c}^{2}b_{2}\,,$
$\displaystyle\frac{F_{3}^{V}(\omega)}{\overline{\xi}(\omega)}=\hat{\alpha}_{s}C_{V_{3}}-\frac{2\varepsilon_{b}}{\omega+1}+\hat{\alpha}_{s}\left\\{C_{V_{3}}\frac{3\omega-1}{\omega+1}\varepsilon_{c}-\left[2C_{V_{1}}+2C_{V_{2}}-(\omega-1)C_{V_{3}}\right]\frac{\varepsilon_{b}}{\omega+1}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+2(\omega-1)C_{V_{3}}^{\prime}\left(\varepsilon_{c}+\varepsilon_{b}\right)\bigg{\\}}\,,$
(8)
$\displaystyle\frac{F_{1}^{A}(\omega)}{\overline{\xi}(\omega)}=1+\hat{\alpha}_{s}C_{A_{1}}+\left(\varepsilon_{c}+\varepsilon_{b}\right)\frac{\omega-1}{\omega+1}+\hat{\alpha}_{s}\left[C_{A_{1}}\frac{\omega-1}{\omega+1}+2(\omega-1)C_{A_{1}}^{\prime}\right]\left(\varepsilon_{c}+\varepsilon_{b}\right)+\varepsilon_{c}^{2}b_{1}\,,$
$\displaystyle\frac{F_{2}^{A}(\omega)}{\overline{\xi}(\omega)}=\hat{\alpha}_{s}C_{A_{2}}-\frac{2\varepsilon_{c}}{w+1}+\hat{\alpha}_{s}\left\\{C_{A_{2}}\frac{3w+1}{w+1}\varepsilon_{b}-\left[2C_{A_{1}}-(w+1)C_{A_{2}}+2C_{A_{3}}\right]\frac{\varepsilon_{c}}{w+1}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+2(w-1)C_{A_{2}}^{\prime}\left(\varepsilon_{c}+\varepsilon_{b}\right)\bigg{\\}}+\varepsilon_{c}^{2}b_{2}\,,$
$\displaystyle\frac{F_{3}^{A}(\omega)}{\overline{\xi}(\omega)}=\hat{\alpha}_{s}C_{A_{3}}+\frac{2\varepsilon_{b}}{w+1}+\hat{\alpha}_{s}\left\\{C_{A_{3}}\frac{3w+1}{w+1}\varepsilon_{c}+\left[2C_{A_{1}}-2C_{A_{2}}+(w+1)C_{A_{3}}\right]\frac{\varepsilon_{b}}{w+1}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+2(w-1)C_{A_{3}}^{\prime}\left(\varepsilon_{c}+\varepsilon_{b}\right)\bigg{\\}}\,,$

where $\hat{\alpha}_{s}$ is the strong coupling constant and the definitions
of $C^{(\prime)}_{A,V_{1,2,3}}$ can be found in Ref. Bernlochner:2018kxh . To
include both soft and hard gluon corrections, we extract the relevant
parameters of $\overline{\xi}(\omega)$, $\overline{\Lambda}$ and $b_{1,2}$
from the quark models, where the soft gluon (non-perturbative) effects are
taken account by the wave functions. After that, we plug the computed
parameters into Eq. (II) to contain the hard gluon corrections.

We note that the heavy baryon masses can be expanded as

$M_{Q}=m_{Q}+\overline{\Lambda}+\frac{\lambda_{1}}{2m_{Q}}+{\cal
O}(\varepsilon_{Q}^{2}).$ (9)

It is convenient to rewrite the binding energy as
$\overline{\Lambda}=CE_{\text{di}}$, where $E_{\text{di}}$ stands for the
energy of the diquark system, and $C$ describes the correction of it in the
presence of the heavy quark with an infinite mass.

Figure 2: The hard gluon corrections of the current operators.

###  II.1 Homogeneous bag model

We begin our study with the MIT bag model, of which the quark wave functions
of the baryons are confined in a finite region, given as bag

$\displaystyle\phi_{q\updownarrow}(\vec{x})=\left(\begin{array}[]{c}j_{0}(p_{q}r)\chi_{\updownarrow}\\\
i\frac{p_{q}}{E_{q}^{k}+p_{q}}j_{1}(p_{q}r)\hat{r}\cdot\vec{\sigma}\chi_{\updownarrow}\\\
\end{array}\right)\,,~{}~{}~{}~{}$ $\displaystyle\text{for}~{}r\leq R\,,$ (10)
$\displaystyle\phi_{q\updownarrow}(\vec{x})=0\,,$
$\displaystyle\text{for}~{}r>R\,,$

with $r=|\vec{x}|$ and $R$ the bag radius. Here, $j_{0,1}$ are the spherical
Bessel functions, $E_{q}^{k}=\sqrt{p_{q}^{2}+m_{q}^{2}}$ with $m_{q}$ and
$p_{q}$ the quark mass and 3-momentum, and $\chi_{\uparrow}=(1,0)^{T}$ and
$\chi_{\downarrow}=(0,1)^{T}$ represent a spin-up and a spin-down quarks,
respectively. From the boundary condition, $p_{q}$ must satisfy bag

$\tan(p_{q}R)=\frac{p_{q}R}{1-m_{q}R+E_{q}^{k}R}\,.$ (11)

In particular, we have

$\lim_{m_{u,d}\to 0}p_{u,d}=2.043/R\,,$ (12)

and

$p_{Q}=\frac{\pi}{R}\left(1-\
\frac{1}{2m_{Q}R}\right)\,,~{}~{}~{}E_{Q}^{k}=m_{Q}+\frac{\pi^{2}}{2m_{Q}R^{2}}\,,$
(13)

to the precision of ${\cal O}(1/m_{Q}^{2}R^{2})$. It is interesting to point
out that the zeroth order corrections are absent in $E_{Q}^{k}$, and the first
order corrections correspond to the kinematic energies of the heavy quarks in
the nonrelativistic limit. Plugging Eq. (13) into the bag wave functions, we
are led to

$\phi_{Q}(\vec{x})=\left(\begin{array}[]{c}\left(j_{0}(\pi r/R)-\frac{\pi
r}{2R^{2}m_{Q}}j_{1}(\pi r/R)\right)\chi\\\ i\frac{\pi}{2m_{Q}R}j_{1}(\pi
r/R)\hat{r}\cdot\vec{\sigma}\chi\\\ \end{array}\right)\,.$ (14)

On the other hand, the masses of the baryons are given as

$M_{Q}=Z_{0}/R+\frac{4\pi}{3}R^{3}B_{0}^{4}+E_{I}+\sum_{q}E_{q}^{k}\,,$ (15)

where $Z_{0}$ and $B_{0}$ are associated with the zero-point and bag volume
energies, respectively, $E_{I}$ is the interaction energies, and $R$
minimalizes the baryon masses, given by

$\frac{\partial M_{Q}}{\partial R}=0\,.$ (16)

In calculating the Isgur-Wise function, we do not include the interaction
corrections, so we set $E_{I}=0$ for consistency, which is the major source of
errors.

Combining Eqs. (12) and (13), we arrive at

$\frac{\partial M_{Q}}{\partial R}=4\pi
R^{2}B_{0}^{4}-(Z_{0}+4.086)/R^{2}=0\,.$ (17)

There are two sets of bag parameters in Ref. bag , given as

$(Z_{0},B_{0})=(1.84,0.145~{}\text{GeV}),(1.95,0.125~{}\text{GeV})\,,$ (18)

resulting in that

$(\overline{\Lambda},R^{-1})=(0.665,0.223),(0.554,0.195)~{}\text{GeV},$ (19)

respectively.

The twist is that although the MIT bag model successfully explains most of the
low-lying baryon masses, it is difficult to be applied in decays, especially
in taking account the recoil effects ($\omega$ dependencies). The problem can
be traced back to that bag states are localized. According to the the
Heisenberg uncertainty principle, it can not be an eigenstate of 3-momenta,
which is referred to as the center-of-mass motion (CMM) problem. The
difficulties were tackled a few years ago in Ref. Geng:2020ofy , where
localized bags are replaced by linear superpositions of infinite ones,
distributed homogeneously all over the space.

In the HBM, the baryon states at rest are given as

$\displaystyle|\Lambda_{Q},\uparrow\rangle=\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}d_{a\alpha}^{\dagger}(\vec{x}_{d})u_{b\beta}^{\dagger}(\vec{x}_{u})Q_{c\gamma}^{\dagger}(\vec{x}_{Q})\Psi^{abc}_{A_{\uparrow}(duQ)}(\vec{x}_{d},\vec{x}_{u},\vec{x}_{Q})[d^{3}\vec{x}]|0\rangle\,,$
(20)

where the Greek (Latin) letters represent the color (Dirac spinor) indices,
$\epsilon$ stands for the totally anti-symmetric tensor,
$[d^{3}\vec{x}]=d^{3}\vec{x}_{d}d^{3}\vec{x}_{u}d^{3}\vec{x}_{Q}$,
$q^{\dagger}$ is the creation operator of the quark, satisfying

$\\{q^{\dagger}_{a\alpha}(\vec{x}),q_{b\beta}(\vec{x}\,^{\prime})\\}=\delta_{ab}\delta_{\alpha\beta}\delta^{3}\left(\vec{x}-\vec{x}\,^{\prime}\right)\,,$
(21)

with $q\in\\{u,d,Q\\}$, and the spatial distributions are described by

$\displaystyle\Psi_{A_{\uparrow}(duQ)}^{abc}(\vec{x}_{d},\vec{x}_{u},\vec{x}_{Q})=$
$\displaystyle\frac{{\cal
N}_{Q}}{\sqrt{2}}\int\left[\phi^{a}_{d\uparrow}(\vec{x}_{d}-\vec{x}_{\Delta})\phi^{b}_{u\downarrow}(\vec{x}_{u}-\vec{x}_{\Delta})\right.$
(22)
$\displaystyle\left.-\phi^{a}_{d\downarrow}(\vec{x}_{d}-\vec{x}_{\Delta})\phi^{b}_{u\uparrow}(\vec{x}_{u}-\vec{x}_{\Delta})\right]\phi^{c}_{Q\uparrow}(\vec{x}_{Q}-\vec{x}_{\Delta})d^{3}\vec{x}_{\Delta},$

where ${\cal N}_{Q}$ are the normalization constants. If not stated otherwise,
$q^{\dagger}(\vec{x})$ is evaluated at $t=0$ in this work.

In Eq. (22), $\phi_{q\updownarrow}(\vec{x}_{q}-\vec{x}_{\Delta})$ represents
the quark state in the static bag centering at $\vec{x}_{\Delta}$. By carrying
out the integral of $d^{3}\vec{x}_{\Delta}$, it is straightforward to see that
the baryons are distributed uniformly all over the 3-dimensional space.
Therefore, the unwanted CMM is removed. We note that the formula are reduced
to the ones of the MIT bag model when the integral of $d^{3}\vec{x}_{\Delta}$
is eliminated.

To get a baryon state with a nonzero momentum, we apply the Lorentz boost to
Eq. (20). The transformation rule of the creation operators reads

$U_{v}q^{\dagger}_{\alpha
a}(x^{\mu})U_{v}^{-1}=\left(S_{v}\right)_{ab}q^{\dagger}_{\alpha
b}\left(\left(\Lambda_{v}^{-1}\right)^{\mu}\,_{\nu}x^{\nu}\right)\,,$ (23)

where $U_{v}$, $S_{v}$, and $\Lambda_{v}$ are the Lorentz boost operators of
states, Dirac spinors, and coordinates, respectively. Without lost of
generality, we take the Lorentz boost toward the $\hat{z}$ direction,
resulting in

$S_{\pm v}=\left(\begin{array}[]{cc}a_{+}&\pm a_{-}\sigma_{3}\\\ \pm
a_{-}\sigma_{3}&a_{+}\end{array}\right)\,,~{}~{}~{}S_{\pm
v}^{2}=\left(\begin{array}[]{cc}\gamma&\pm\gamma v\sigma_{3}\\\ \pm\gamma
v\sigma_{3}&\gamma\end{array}\right)\,,$ (24)

where $a_{\pm}=\sqrt{\gamma\pm 1}$ and $\gamma=1/\sqrt{1-v^{2}}$.

From Eq. (23), we see that even if we start with $t=0$, the $t$-dependencies
of quark states are nevertheless still required. It is due to that the $t=0$
plane is not invariant under Lorentz boosts. Thus, for the state having a
Lorentz boost, we have to evaluate the anticommutation relations with unequal
times, which can not be treated perturbatively. To overcome the problem, we
utilize that the quark states are energy eigenstates in the bag model, and
make the substitution in Eq. (22)

$\displaystyle
q^{\dagger}_{a\alpha}(\vec{x})=e^{-iE_{q}t}q^{\dagger}_{a\alpha}(t,\vec{x})\,,$
(25)

where $E_{q}$ are the energies of the quarks. It is clear that we do not
consider the interaction energies here, as the quark energies are independent
to each other. The reasonable range of $E_{q}$ is

$\frac{1}{3}M_{N}<E_{q}<E_{q}^{k}+\frac{1}{3}(E_{0}+E_{v})\,,$ (26)

where $M_{N}$ is the neutron mass, $E_{0}$ and $E_{v}$ are the zero-point and
vacuum energies arisen from the bag, respectively, which are allocated evenly
among the quarks. In the HBM, Eq. (26) serves as the major source of the
uncertainties.

After some algebra, we arrive at Liu:2022pdk

$\Psi^{abc}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3})\overset{v}{\to}(S_{v})_{aa^{\prime}}(S_{v})_{bb^{\prime}}(S_{v})_{cc^{\prime}}\Psi^{a^{\prime}b^{\prime}c^{\prime}}(\vec{x}_{1}^{v},\vec{x}_{2}^{v},\vec{x}_{3}^{v}),$
(27)

where $\vec{x}^{v}=(x,y,\gamma z)$. From the normalization condition

$\langle\Lambda_{Q},\vec{p}\,^{\prime}|\Lambda_{Q}\vec{p}\,\rangle=u^{\dagger}u(2\pi)^{3}\delta^{3}(\vec{p}-\vec{p}^{\prime})\,,$
(28)

we derive that

$\frac{\overline{u}u}{{\cal N}^{2}}=\int
d^{3}\vec{x}_{\Delta}\prod_{q}\int\phi_{q}^{\dagger}\left(\vec{x}_{q}^{\,+}\right)\phi_{q}\left(\vec{x}_{q}^{\,-}\right)d^{3}\vec{x}_{q}\,,$
(29)

where $\vec{x}^{\pm}_{q}=\vec{x}_{q}\pm\vec{x}_{\Delta}/2$, and
$\sum_{q}E_{q}=M_{Q}$ has been used.

We adopt the Briet frame so that $\Lambda_{b}^{0}$ and $\Lambda_{c}^{+}$ have
opposite velocities. Collecting Eqs. (20), (21) and (27), we find that

$\displaystyle\langle\Lambda_{c}^{+}(\vec{v}\,)|c^{\dagger}\Gamma
b(0)|\Lambda_{b}^{0}(-\vec{v}\,)\rangle={\cal N}_{c}\,{\cal N}_{b}\int
d^{3}\vec{x}_{\Delta}\Gamma_{cb}(\vec{x}_{\Delta})\prod_{l}D^{v}_{l}(\vec{x}_{\Delta})\,,$
(30)

along with

$\displaystyle\Gamma_{cb}(\vec{x}_{\Delta})=\int
d^{3}\vec{x}\phi_{c}^{\dagger}\left(\vec{x}+\frac{1}{2}\vec{x}_{\Delta}\right)S_{v}\Gamma
S_{-v}\phi_{b}\left(\vec{x}-\frac{1}{2}\vec{x}_{\Delta}\right)e^{2iE_{\text{di}}\vec{v}\cdot\vec{x}}\,,$
$\displaystyle D^{v}_{l}(\vec{x}_{\Delta})=\frac{1}{\gamma}\int
d^{3}\vec{x}\phi_{l}^{\dagger}\left(\vec{x}+\frac{1}{2}\vec{x}_{\Delta}\right)\phi_{l}\left(\vec{x}-\frac{1}{2}\vec{x}_{\Delta}\right)e^{-2iE_{l}\vec{v}\cdot\vec{x}}\,,~{}~{}~{}l=u,d\,,$
(31)

where $\Gamma$ is an arbitrary Dirac matrix, $E_{\text{di}}=E_{u}+E_{d}$, and
$\omega=\gamma^{2}(1+v^{2})$. In general, $\Gamma_{cb}$ and $D_{l}^{v}$ would
be some complicated functions of the quark masses and bag radius. To examine
the model, we take $\vec{v}\to 0$ and Taylor-expand the formulae regarding to
$M_{Q}$, leading to 555In the calculation of the form factors, we have taken
the normalization of $\overline{u}_{Q}u_{Q}=1$.

$\frac{1}{{\cal
N}_{\infty}^{2}}=16\pi^{2}\int\left(D^{0}_{l}(r_{\Delta})\right)^{2}r_{\Delta}^{2}dr_{\Delta}\int_{0}^{\sqrt{R^{2}-r_{\Delta}^{2}/4}}d\rho\rho\int_{0}^{\sqrt{R^{2}-r_{\Delta}^{2}/4}-r_{\Delta}/2}dzj_{0}^{+}j_{0}^{-}\,,$
(32)

with the abbreviations of $j_{0,1}^{\pm}=j_{0,1}(\pi r_{\pm}/R)$,
$r_{\pm}=|\vec{x}\pm\vec{x}_{\Delta}/2|$ and $r_{\Delta}=|\vec{x}_{\Delta}|$.
In addition, with $m_{u,d}\to 0$, we are led to

$D^{0}_{l}(r_{\Delta})=4\pi\int_{0}^{\sqrt{R^{2}-r_{\Delta}^{2}/4}}d\rho\rho\int_{0}^{\sqrt{R^{2}-r_{\Delta}^{2}/4}-r_{\Delta}/2}dz\left[l_{0}^{+}l_{0}^{-}+\left(r^{2}-r_{\Delta}^{2}/4\right)l_{1}^{+}l_{1}^{-}\right]\,,$
(33)

where $l_{0,1}^{\pm}=j_{0,1}(2.043r_{\pm}/R)$.

The form factors of $F_{1}^{V}$ and $F_{1}^{A}$ are extracted by

$\displaystyle F_{1}^{V}(1)=\lim_{v\to
0}\frac{1}{v}\langle\Lambda_{c}^{+}(\vec{v}\,)|\left(\overline{c}\,\gamma^{1}b\right)(0)|\Lambda_{b}^{0}(-\vec{v}\,)\rangle\,,$
$\displaystyle F_{1}^{A}(1)=\lim_{v\to
0}\langle\Lambda_{c}^{+}(\vec{v}\,)|\left(\overline{c}\,\gamma^{1}\gamma_{5}b\right)(0)|\Lambda_{b}^{0}(-\vec{v}\,)\rangle,$
(34)

resulting in that

$\displaystyle
F_{1}^{V}(1)=1+\left(\frac{1}{2m_{c}}+\frac{1}{2m_{c}}\right)E_{\text{di}}{\cal
N}_{\infty}^{2}4\pi\int r_{\Delta}^{2}dr_{\Delta}{\cal
C}(r_{\Delta})\left(D^{0}_{l}(r_{\Delta})\right)^{2}\,,$ $\displaystyle
F_{1}^{A}(1)=1\,,$ (35)

where

$\displaystyle{\cal
C}(r_{\Delta})=\frac{4\pi^{2}}{3R}\int_{0}^{\sqrt{R^{2}-r_{\Delta}^{2}/4}}d\rho\rho\int_{0}^{\sqrt{R^{2}-r_{\Delta}^{2}/4}-r_{\Delta}/2}dz\Big{[}\left(j_{0}^{+}j_{1}^{-}\frac{1}{r_{-}}+j_{1}^{+}j_{0}^{-}\frac{1}{r_{+}}\right)r^{2}$
$\displaystyle+\left(j_{0}^{-}j_{1}^{+}\frac{r_{\Delta}z}{2r_{+}}-j_{1}^{-}j_{0}^{+}\frac{r_{\Delta}z}{2r_{-}}\right)\Big{]}\,,$
(36)

to the precision of ${\cal O}(1/m_{c}^{2}R^{2})$. By matching it with Eq. (6)
, we find that the first order correction of $F_{1}^{A}$ indeed vanishes, and

$C={\cal N}_{\infty}^{2}4\pi\int r_{\Delta}^{2}dr_{\Delta}{\cal
C}(r_{\Delta})\left(D^{0}_{l}(r_{\Delta})\right)^{2}=1\,,$ (37)

which holds exactly by an actual calculation. It is a sensible result, since
the zeroth order corrections of $E_{Q}^{k}$ are absent as shown in Eq. (13),
and the interactions among quarks are omitted. We conclude that the diquark
energies are unaffected by the presence of the heavy quark at the zeroth
order, namely $C=1$. However, Eq. (37) itself is a nontrivial result, and it
indicates that our treatments for the CMM are self-consistent.

We emphasize that Eq. (37) is a parameter-independent result, which can be
shown explicitly by changing the variables

$(\rho,z,\vec{x}_{\Delta})\to\left(\frac{\rho}{R},\frac{z}{R},\frac{\vec{x}_{\Delta}}{R}\right)\,,$
(38)

in Eqs. (II.1) and (37). To take account the recoil effects, we use the
following trick

$\displaystyle\xi(\omega)=\left.\sum_{i=1,2,3}F_{i}^{V}(\omega)\right|_{\varepsilon_{Q}=0}\,,$
$\displaystyle\xi_{ke}(\omega)=\left.\frac{\partial}{\partial\varepsilon_{c}}\left(\sum_{i=1,2,3}F_{i}^{V}(\omega)\right)\right|_{\varepsilon_{Q}=0}\,,$
(39)

where $\sum_{i}F_{i}^{V}$ is evaluated by taking $\Gamma=1$ in Eq. (II.1).
Consequently, we get

$\displaystyle\xi(\omega)$ $\displaystyle=$ $\displaystyle{\cal
N}_{\infty}^{2}\int
d^{3}\vec{x}_{\Delta}d^{3}\vec{x}\left(j_{0}^{+}j_{0}^{-}\right)e^{2iE_{\text{di}}\vec{v}\cdot\vec{x}}\left(D^{0}_{l}(\vec{x}_{\Delta})\right)^{2}\,,$
(40) $\displaystyle\xi_{ke}(\omega)$ $\displaystyle=$ $\displaystyle
A(\vec{v}\,)\overline{\Lambda}-A(0)\overline{\Lambda}\,,$ $\displaystyle
A(\vec{v})$ $\displaystyle\equiv$ $\displaystyle\frac{\pi{\cal
N}_{\infty}^{2}}{R^{2}}\int
d^{3}\vec{x}_{\Delta}d^{3}\vec{x}\left(r_{-}j_{0}^{+}j_{1}^{-}+r_{+}j_{1}^{+}j_{0}^{-}\right)e^{2iE_{\text{di}}\vec{v}\cdot\vec{x}}\left(D^{0}_{l}(\vec{x}_{\Delta})\right)^{2}\,.$

By taking $\vec{v}=0$, we get that $\xi(1)=1$ and $\xi_{ke}(1)=0$ as demanded
by the Luke’s theorem.

Similarly, the first order corrections of $b_{2}$ and $b_{2}$ are obtained,
given by

$\displaystyle\left.\frac{\partial^{2}F_{1}^{V}(1)}{\partial\varepsilon_{c}^{2}}\right|_{\varepsilon_{Q}=0}=(b_{1}-b_{2})\,,$
$\displaystyle\left.\frac{\partial^{2}F_{1}^{A}(1)}{\partial\varepsilon_{c}^{2}}\right|_{\varepsilon_{Q}=0}=b_{1}\,,\quad\left.\frac{\partial^{2}F_{2}^{V,A}(1)}{\partial\varepsilon_{c}^{2}}\right|_{\varepsilon_{Q}=0}=b_{2}\,.$
(41)

The operations can be easily done by a computer program. However, their
expressions are much more complicated and lengthy as well, so we do not list
them out here. As a cross check, we compute

$\left.\frac{\partial^{2}}{\partial\varepsilon_{c}^{2}}\left(F_{1}^{V}(\omega)-F_{1}^{A}(\omega)+F_{2}^{V,A}(\omega)\right)\right|_{\varepsilon_{Q}=0}\,,$
(42)

which is indeed found to be zero, consistent with Eq. (6).

We note that $\xi$, $\xi_{ke}$ and $b_{1,2}$ depend only on $E_{\text{di}}R$,
which is the only dimensionless parameter. To be specific, they are invariant
under the transformation of $(R,E_{\text{di}})\to(a^{-1}R,aE_{\text{di}})$,
where $a$ is an arbitrary constant. For a practical purpose, we can fix $R$
and vary $E_{\text{di}}$ solely to cover the model uncertainties on
$\overline{\Lambda}$, $\xi$, $\xi_{ke}$ and $b_{1,2}$.

###  II.2 Light-front quark model

In the LFQM, the baryon states are expressed as

$\displaystyle|\Lambda_{Q},\uparrow\rangle=$
$\displaystyle\int\frac{1}{\sqrt{6}}\epsilon^{\alpha\beta\gamma}u^{\dagger}_{\alpha}(\tilde{p}_{u},\lambda_{u})d^{\dagger}_{\beta}(\tilde{p}_{d},\lambda_{d})Q^{\dagger}_{\gamma}(\tilde{p}_{Q},\lambda_{Q})\Psi^{[\lambda]}_{Q}(\tilde{p}_{u},\tilde{p}_{d},\tilde{p}_{Q})[d^{3}\tilde{p}]|0\rangle,$
(43)

where $\Psi^{[\lambda]}_{Q}(\tilde{p}_{u},\tilde{p}_{d},\tilde{p}_{Q})$
represent the vertex functions between the $\Lambda_{Q}$ and $udQ$,
$[\lambda]$ and $[d^{3}\tilde{p}]$ stand for the light-front helicities
$(\lambda_{u},\lambda_{d},\lambda_{Q})$ and light-front 3-momentum integrals
$d^{3}\tilde{p}_{u}d^{3}\tilde{p}_{d}d^{3}\tilde{p}_{Q}$, respectively, and

$p_{q}=(p_{q}^{-},\tilde{p}_{q})=(p_{q}^{-},p_{q}^{+},p_{q\perp}),~{}d^{3}\tilde{p}_{q}\equiv\frac{dp_{q}^{+}d^{2}p_{q\perp}}{2(2\pi)^{3}},$
(44)

with $p_{q}^{\pm}=p_{i}^{0}\pm p_{q}^{3}$ and
$p_{q\perp}=(p_{q}^{1}\,,p_{q}^{2})\,$. At the equal light-front time
($x^{+}=0$), the commutation relationships for
$q^{\dagger}_{\alpha}(\tilde{p}_{q},\lambda_{q})$ are

$\\{q^{\dagger}_{\alpha}(\tilde{p},\lambda),q_{\beta}(\tilde{p}^{\prime},\lambda^{\prime})\\}=\delta_{\lambda\lambda^{\prime}}\delta_{\alpha\beta}\delta^{3}(\tilde{p}-\tilde{p}^{\prime}).$
(45)

Notice that the wave functions are made of position eigenstates in the HBM
shown in Eq. (20), whereas they are built out of momentum eigenstates here in
contrast.

The vertex functions are further decomposed as

$\Psi^{[\lambda]}_{Q}=2(2\pi)^{3}\frac{1}{\sqrt{P^{+}}}\delta^{3}\left(\tilde{P}-\tilde{p}_{u}-\tilde{p}_{d}-\tilde{p}_{Q}\right)\Phi(\tilde{p}_{u},\tilde{p}_{d},\tilde{p}_{Q})\Xi^{1/2,\uparrow}(\lambda_{u},\lambda_{d},\lambda_{Q}),$
(46)

where $\Phi$ are the momentum distribution functions of the quarks,
$\Xi^{1/2,\uparrow}$ stands for the helicity wave function

$\Xi^{1/2,\uparrow}(\lambda_{u},\lambda_{d},\lambda_{Q})=\sum_{s_{u},s_{d},s_{Q}}\prod_{q}\langle\lambda_{q}|R_{q}^{\dagger}|s_{q}\rangle\langle\frac{1}{2}s_{u},\frac{1}{2}s_{d},\frac{1}{2}s_{Q}\mid
SS_{z}\rangle,$ (47)

and $R_{q}$ is the Melosh matrix, which brings the quark from its spin state
at rest to a light-front helicity state with momentum $\tilde{p}_{q}$
Chiu:2017ycx .

In this work, we consider the two-particle forces between the quarks, which
are effectively described by the harmonic oscillator potentials

$V=\beta^{4}r^{2}/(2M),$ (48)

where $r$ are the distances of the quarks, and $M$ and $\beta$ are the reduced
masses and shape parameters, respectively, to be specified later. We take the
quarks of $u$ and $d$ to form a diquark cluster denoted as $[ud]$, as they are
(iso)spin singlet. By integrating out the delta function of
$\delta(\tilde{P}-\tilde{p}_{u}-\tilde{p}_{d}-\tilde{p}_{Q})$, Eqs. (43) and
(46) can be rewritten as

$[d^{3}\tilde{p}]\to
d^{3}\vec{q}\,d^{3}\vec{Q}\,,~{}~{}~{}\Phi(\tilde{p}_{u},\tilde{p}_{d},\tilde{p}_{Q})\to\Phi(\vec{q},\vec{Q})$
(49)

where

$\Phi(\vec{q},\vec{Q})=\phi_{ud}\,\phi_{Q[ud]}=(\pi\beta_{ud}\beta_{Q[ud]})^{-3/2}\exp(-\frac{\vec{q}\,^{2}}{2\beta_{ud}^{2}}-\frac{\vec{Q}^{2}}{2\beta_{Q[ud]}^{2}}),$
(50)

$\phi_{ud}$ and $\phi_{Q[ud]}$ are the wave functions of $(u,d)$ and
$(Q,[ud])$, and the shape parameters of $\beta_{ud}$ and $\beta_{Q[ud]}$ are
the typical relative 3-momenta, respectively. Note that we have projected the
light-front 3-momenta to the instant forms by defining

$\displaystyle
x_{Q}\equiv\frac{p_{Q}^{+}}{P^{+}}=\frac{E_{Q}-Q_{z}}{E_{ud}+E_{Q}},~{}$
$\displaystyle Q_{\perp}=x_{Q}(p_{u\perp}+p_{d\perp})-(1-x_{Q})p_{Q\perp}$ ,
(51) $\displaystyle
y\equiv\frac{p_{u}^{+}}{p_{u}^{+}+p_{d}^{+}}=\frac{E_{d}-q_{z}}{E_{u}+E_{d}},~{}$
$\displaystyle q_{\perp}=yp_{u\perp}-(1-y)p_{d\perp}$ ,

where $E_{u,d}=\sqrt{\vec{q}\,^{2}+m^{2}}$,
$E_{Q}=\sqrt{\vec{Q}^{2}+m_{Q}^{2}}$,
$E_{ud}=\sqrt{\vec{Q}^{2}+(E_{u}+E_{d})^{2}}$, and $m$ stand for the
constituent quark masses of $u$ and $d$ in the LFQM.

By utilizing that the potentials are independent of quark masses, we find

$\displaystyle\beta_{Q[ud]}^{4}M_{Q[ud]}^{-1}=\beta_{d[ud]}^{4}M_{d[ud]}^{-1}=\beta_{ud}^{4}(m/2)^{-1}$
(52)

where $M_{q[ud]}$ is the reduced effective mass of $q$ and $[ud]$, given as

$M_{q[ud]}^{-1}=m_{q}^{-1}+E_{\text{di}}^{-1}.$ (53)

The second equality in Eq. (52) comes from the Isgur-Karl model for the equal
mass scenario Isgur:1978xj . By considering the well-measured ratio of
$g_{A}/g_{V}=-1.275$ from the neutron beta decay, we get

$\beta_{d[ud]}=0.952m\,.$ (54)

To the precision of ${\cal O}(m_{Q}^{-2})$, we have

$\beta_{Q[ud]}=\left(1-\frac{E_{\text{di}}}{4m_{Q}}+{\cal
O}(\frac{1}{m_{Q}^{2}})\right)\beta_{\infty},~{}~{}~{}\beta_{\infty}=(1+E_{\text{di}}/m)^{1/4}\beta_{d[ud]}.$
(55)

The existence of the heavy quark limit for $\beta_{\infty}$ has already been
discussed and put by hand in Refs. Cheng:2004cc ; Ke:2007tg . Here, we provide
a clear theoretical background to justify their assumptions.

Now, we are ready to compute $\overline{\Lambda},b_{1,2},\xi(\omega)$, and
$\xi_{ke}(\omega)$. We choose the frame of $q^{+}=0$ to avoid the Z-graph
contribution Choi:1998nf . As $q^{2}=q^{+}q^{-}-q_{\perp}^{2}$, the LFQM is
unable to calculate the region of $q^{2}>0$. We set $m_{b}=m_{c}/x=m_{Q}$. At
$q^{2}=0$, $\omega$ corresponds to

$\omega(x)=\frac{M_{Q}^{2}+M_{Qx}^{2}}{2M_{Q}M_{Qx}},$ (56)

with

$M_{Q{(x)}}=(x)m_{Q}+\overline{\Lambda}+\frac{\lambda_{1}}{(x)2m_{Q}}$ (57)

from Eq. (9), leading to 666The form factors in the $q^{+}=0$ frame are
extracted as Eq. (24) in Ref. Ke:2007tg .

$\displaystyle\xi(\omega(x))$
$\displaystyle=f_{1}(q^{2}=0,x,\varepsilon_{Q}\to 0),$ (58)
$\displaystyle\left(\frac{1}{x}+1\right)\xi_{ke}(\omega(x))$
$\displaystyle=\frac{\partial}{\partial\varepsilon_{Q}}f_{1}(q^{2}=0,x,\varepsilon_{Q}\to
0).$

In reality, we have that $x\approx 3$ and $\lambda_{1}=-0.5$ GeV2,
corresponding to $\omega=1.43$. The trick is that $\xi(\omega)$ and
$\xi_{ke}(\omega)$ are independent of $m_{b,c}$ at a certain value of
$\omega$. In the computation, we vary $x$ to compute $\xi(\omega)$ and
$\xi_{ke}(\omega)$ at $q^{2}=0$ and get the desired $\omega(x)$.

For $\overline{\Lambda}$ and $b_{1,2}$, we again use the trick that they are
independent of $m_{b,c}$. We compute them with $x=1$. Accordingly, $g_{1}$ and
$f_{2}$ are expanded as

$g_{1}=1+\frac{g_{1}^{(2)}}{m_{Q}^{2}}+{\cal
O}(1/m_{Q}^{3}),~{}f_{2}=\frac{f_{2}^{(1)}}{m_{Q}}+\frac{f_{2}^{(2)}}{m_{Q}^{2}}+{\cal
O}(1/m_{Q}^{3}),$ (59)

where

$\displaystyle g_{1}^{(2)}=-\frac{\beta_{\infty}^{2}}{2},\quad\quad
f_{2}^{(1)}=\int
d^{3}\vec{q}\,d^{3}\vec{Q}\,|\Phi_{\infty}|^{2}\frac{E_{ud}}{2},$ (60)
$\displaystyle f_{2}^{(2)}=\int
d^{3}\vec{q}\,d^{3}\vec{Q}\,|\Phi_{\infty}|^{2}\frac{E_{ud}}{8\beta_{\infty}^{2}}\left[7E_{\text{di}}\beta_{\infty}^{2}-4E_{ud}\beta_{\infty}^{2}-2E_{\text{di}}Q^{2}\right],$

and $\Phi_{\infty}$ is the momentum wave function under the heavy quark limit,
given as

$\Phi_{\infty}(\vec{q},\vec{Q})=(\pi\beta_{ud}\beta_{\infty})^{-3/2}\exp(-\frac{\vec{q}\,^{2}}{2\beta_{ud}^{2}}-\frac{\vec{Q}^{2}}{2\beta_{\infty}^{2}}).$
(61)

Note that the zeroth order of $g_{1}$ is consistent with the heavy quark
symmetry, which is a nontrivial result.

To match Eqs. (5) and (6) to Eqs. (59) and (60) with $x=1$, there would be
extra crossing terms at the order of $\varepsilon_{c}\varepsilon_{b}$ in Eq.
(6), which introduce two additional free parameters. To eliminate them, we
assume that $F^{V,A}_{1}(\omega)$ are factorized as

$F^{V(A)}_{1}(\varepsilon_{b},\varepsilon_{c},\omega)=f^{V(A)}(\varepsilon_{c},\omega)f^{V(A)}(\varepsilon_{b},\omega),$
(62)

with $f^{V,A}(\varepsilon_{Q},\omega)$ to be determined. Now we have

$\overline{\Lambda}=2f_{2}^{(1)},~{}\overline{b}_{1}=2g_{1}^{(2)},~{}\overline{b}_{2}=2(f_{2}^{(1)})^{2}-4f_{2}^{(2)}+2g_{1}^{(2)},$
(63)

where $\overline{b}_{1,2}\equiv b_{1,2}\overline{\Lambda}^{2}$.

From Eqs. (52), (60) and (61), we see that $f_{2}^{(1)}$ depends only on
$E_{\text{di}}$ and $m$. We arrive at

$f_{2}^{(1)}(E_{\text{di}},m)=E_{\text{di}}{\cal
F}\left(E_{\text{di}},m\right)=E_{\text{di}}{\cal
F}\left(\frac{E_{\text{di}}}{m}\right)\,,$ (64)

where ${\cal F}$ is a function to be determined, and the second equality comes
from that ${\cal F}$ can only depends on the dimensionless quantities, i.e.
$E_{\text{di}}/m$ in this case. By demanding the heavy quark limit of
$\overline{\Lambda}=E_{\text{di}}$ or equivalently ${\cal F}=1/2$, we find

$E_{\text{di}}=3.293m\,.$ (65)

Collecting Eqs. (52), (60), (61) and (65), we see that only $m$ remains
unfixed.

The twist is that $m$ is canonically dimensional-one, whereas except for
$\overline{\Lambda}$, all the other HQE parameters are dimensionless. It turns
out that $\xi$, $\xi_{ke}$ and $b_{1,2}$ can not depend on $m$, so they are
parameter-independent results from the LFQM. Explicitly, we have

$b_{1}=-0.173,~{}b_{2}=0.518,$ (66)

while $\xi$ and $\xi_{ke}$ are shown in FIG. 3

Figure 3: The Isgure-Wise function and its first order correction from the
LFQM.

##  III Numerical results

In both of the models, $m_{b,c}$ are taken as the pole quark masses with
$m_{b,c}=(4.78,1.655)$ GeV, and the adopted value of the Cabibbo-Kobayashi-
Maskawa (CKM) matrix element is $|V_{cb}|=(42.2\pm 0.7)\times 10^{-3}$ from
the inclusive semileptonic $B$ decays pdg . The formalism of the decay widths
can be found in Ref. Geng:2022hmf .

For the HBM, we fix $R=4.8$ GeV-1 without lost of generality as explained in
the end of Sec. II.1. From Eq. (26), the reasonable range of $E_{\text{di}}$
is then given as

$0.585~{}\text{GeV}<E_{\text{di}}<0.686~{}\text{GeV}\,,$ (67)

which is consistent with Eq. (19). For the LFQM, we take $m=0.30\pm 0.08$ GeV,
which is consistent with the experiments of the nucleon magnetic moments and
${\cal B}(\Lambda_{b}^{0}\to\Lambda\gamma)=(7.1\pm 1.7)\times 10^{-6}$ pdg ;
LHCb:2019wwi . To be conservative, we allow $m$ to vary in a wide range, which
shall cover all the reasonable values.

The HQE parameters are computed with the formula given in Sec. II. The Isgur-
Wise function and its first order correction of the HBM are given in FIG. 4.
Notice that the region of $\omega>1.43$ is equivalent to $q^{2}<0$ and thus it
is irrelevant to
$\Lambda_{b}^{0}\to\Lambda_{c}^{+}\ell^{-}\overline{\nu}_{\ell}$.
Nevertheless, we have plotted them in the figures to see the dependencies in
the high $\omega$ region, as they are physical in the scattering processes.

To compare the results with those in the literature, we expand the Isgur-Wise
function regarding to $\omega$ to the fourth order, given as

$\overline{\xi}(\omega)=1+\overline{\xi}^{(1)}(\omega-1)+\frac{1}{2}\overline{\xi}^{(2)}(\omega-1)^{2}+\frac{1}{3!}\overline{\xi}^{(3)}(\omega-1)^{3}+\frac{1}{4!}\overline{\xi}^{(4)}(\omega-1)^{4}.$
(68)

The numerical results along with those in the literature are collected in
TABLE 1. In the literature, Ref. Bernlochner:2018kxh  fits the HQE parameters
from the experimental data LHCb:2017vhq  and LQCD calculations Detmold:2015aaa
, Ref. Ebert:2006rp  adopts the relativistic quark model (RQM), Refs.
Miao:2022bga ; Duan:2022uzm  employ the light-cone sum rule (LCSR), Ref.
Li:2021qod  also considers the LFQM with parameters fitted from the mass
spectra, and Ref. Huang:2005mea  utilizes the QCD sum rule (QCDSR).

Figure 4: The Isgure-Wise function and its first order correction from the HBM. Table 1: The HQE parameters with $\overline{\Lambda}$ and $\overline{b}_{1,2}$ in units of GeV and GeV2, respectively. | HBM | LFQM | LHCb LHCb:2017vhq ; Bernlochner:2018kxh  | LQCD+LHCb Bernlochner:2018kxh  | RQM Ebert:2006rp  | QCDSR Huang:2005mea
---|---|---|---|---|---|---
$\overline{\xi}^{(1)}$ | $-1.94(26)$ | $-2.35(10)$ | $-2.17(26)$ | $-2.04(8)$ | $-1.51$ | $-1.35(13)$
$\overline{\xi}^{(2)}$ | $4.78(93)$ | $5.75(55)$ | $4.10(105)$ | $3.16(38)$ | $4.06$ |
$\overline{\xi}^{(3)}$ | $-10.8(25)$ | $-10.5(21)$ |  |  |  |
$\overline{\xi}^{(4)}$ | $13.1(32)$ | $10.1(38)$ |  |  |  |
$\overline{\xi}(1.43)$ | $0.470(43)$ | $0.392(15)$ | $0.446(4)$ | $0.415(1)$ |  |
$\overline{\Lambda}$ | $0.681(55)$ | $0.988(263)$ |  $0.81(5)$777Extracted from $M_{\Lambda_{Q}}$, where $0.81(5)$ means $0.81\pm 0.05$ | $0.81(5)^{a}$ | $0.764$ |
$\overline{b}_{1}$ | $-0.141(1)$ | $-0.181(90)$ | $0.24(192)$ | $-0.46(15)$ |  |
$\overline{b}_{2}$ | $0.351(35)$ | $0.541(270)$ | $0.45(188)$ | $-0.39(39)$ |  |

Surprisingly, the results in the HBM and LFQM agree well with each other, even
though they are two very different quark models. Our values of
$\overline{\xi}^{(1)}$ also agree well with those from the LHCb and LQCD, but
significantly larger than the results from the RQM and QCDSR. Note that in
Ref. Bernlochner:2018kxh , $\overline{\xi}^{(3)}$ and $\overline{\xi}^{(4)}$
have been omitted. On the contrary, we find that they are sizable. In
particular, $\overline{\xi}^{(3)}$ is opposite in sign and twice larger in
comparison to $\overline{\xi}^{(2)}$. Although our results of
$\overline{\xi}^{(2)}$ are larger than the values from the LHCb and LQCD,
those of $\xi(\omega=1.43)$ are consistent with them due to the presences of
$\overline{\xi}^{(3)}$.

Note that our values of $\overline{b}_{1,2}$ from the LFQM are contaminated by
the uncertainties of $\overline{\Lambda}$, but their signs are not.
Explicitly, both models give $\overline{b}_{1}<0$ and $\overline{b}_{2}>0$.
The predicted sign of $\overline{b}_{2}$ is opposite to the one of LHCb+LQCD.
As the HBM and LFQM show well consistence in all the HQE parameters, we are
confident on these results.

Table 2: The branching fractions of $\Lambda_{b}^{0}\to\Lambda_{c}^{+}\to\ell^{-}\overline{\nu}_{\ell}$ in units of $\%$, where the lower and upper columns in the HBM and LFQM for each lepton pair correspond to the values with and without the hard gluon corrections, respectively. lepton pair |  HBM |  LFQM |  Exp pdg  | RQM Faustov:2016pal  | LFQM Li:2021qod  | LCSR Miao:2022bga  | LCSR Duan:2022uzm
---|---|---|---|---|---|---|---
$e^{-}\overline{\nu}$ | $6.23(58)$ | $5.53(77)$ |  $6.2^{+1.4}_{-1.3}$ | $6.48$ | $6.47(96)$ | $5.81$ | $5.71(98)$
$5.69(58)$ | $5.35(50)$
$\mu^{-}\overline{\nu}$ | $6.21(57)$ | $5.52(77)$ |  $6.2^{+1.4}_{-1.3}$ | $6.46$ | $6.45(95)$ | $5.78$ | $5.69(98)$
$5.67(58)$ | $5.33(49)$
$\tau^{-}\overline{\nu}$ | $1.95(11)$ | $1.91(23)$ |  $1.5(4)$ | $2.03$ | $1.97(29)$ | $1.55$ | $1.66(26)$
$1.83(12)$ | $1.87(15)$

To see the hard gluon effects, we adopt two different schemes for the form
factors. In the first one, we calculate them directly from the quark models.
In the second one, we plug the values of $\overline{\Lambda}$, $b_{1}$,
$b_{2}$ and $\overline{\xi}(\omega)$ from TABLE 1 into Eq. (II). The branching
fractions along with those in the literature are collected in TABLE 2, where
the lower and upper columns in the HBM and LFQM for each lepton pair are the
values with and without the hard gluon corrections, respectively, which
decrease the branching fractions about $10\%$. The branching fractions in
Table 2 are consistent with the experimental data and the those in literature,
where the uncertainties of our results are smaller than those of other
approaches with uncertainties provided.

To further test the lepton universality, we compute the ratio of
$R_{\Lambda_{c}}$, given in TABLE 3 and FIG. 5, where
$R_{\Lambda_{c}}(\text{LQCD})$ come from two different works, in which the
upper column is purely from the LQCD calculations Detmold:2015aaa , while the
lower column corresponds to the one which the heavy quark symmetry is imposed
to lower the uncertainties Bernlochner:2018kxh . The predictions of the HBM
fit well with the experimental data, but the ones of the LFQM disagree with
the LHCb results in contrast.

Table 3: Comparisons of $R_{\Lambda_{c}}$ in different approaches. HBM | LFQM | LQCD Bernlochner:2018kxh ; Detmold:2015aaa  |  LHCb LHCb:2017vhq  | LFQM Li:2021qod  | LCSR Duan:2022uzm
---|---|---|---|---|---
$0.3154(109)$ | $0.3457(70)$ | $0.3328(102)$ | $0.242(76)$ | $0.30(9)$ | $0.292$
$0.3243(126)$ | $0.3506(46)$ | $0.3237(36)$
Figure 5: The comparison of $R_{\Lambda_{c}}$.

##  IV Conclusion

We have examined the heavy quark symmetry in the HBM and LFQM. The inputs of
the HBM are fixed from the mass spectra, while the parameters in the LFQM are
tightly constrained by the heavy quark symmetry. We have demonstrated that the
two models all respect the heavy quark symmetry.

The HQE parameters of $\overline{\Lambda},b_{1,2}$, $\xi(\omega)$ and
$\xi_{ke}(\omega)$ have been computed. Our results of $\overline{\Lambda}$,
$\overline{\xi}(\omega)$ and $b_{1}$ are compatible with those in the
literature, but the sign of $b_{2}$ is opposite comparing to the LHCb+LQCD
results. Explicitly, we have shown that $(b_{1},b_{2})=(-0.173,0.518)$ from
the LFQM, and $(\overline{b}_{1},\overline{b}_{2})=(-0.141\pm 0.001,0.351\pm
0.035)$ from the HBM in units of GeV2.

We have also calculated ${\cal
B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\ell^{-}\overline{\nu}_{\ell})$ and
$R_{\Lambda_{c}}$ with and without the hard gluon corrections. We have found
that the hard gluon corrections decrease the branching fractions about $10\%$.
Explicitly, we have obtained that ${\cal
B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}e^{-}\overline{\nu}_{e})=(5.69\pm
0.58,5.35\pm 0.50)$, ${\cal
B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}\mu^{-}\overline{\nu}_{\mu})=(5.67\pm
0.58$, $5.33\pm 0.49)$, and $R_{\Lambda_{c}}=(0.3243\pm 0.0126$, $0.3506\pm
0.0046)$, for the results of (HBM, LFQM), respectively. Our predicted values
of the branching fractions show good consistencies with the experimental data
of results ${\cal
B}(\Lambda_{b}^{0}\to\Lambda_{c}^{+}e^{-}\overline{\nu}_{e},\tau^{-}\overline{\nu}_{\tau})=(6.2^{+1.4}_{-1.3}\,,1.5\pm
0.4)\%$.

## Acknowledgments

This work is supported in part by the National Key Research and Development
Program of China under Grant No. 2020YFC2201501 and the National Natural
Science Foundation of China (NSFC) under Grant No. 12147103.

## References

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# Conceptual Design and Analysis of No-Insulation High-Temperature
Superconductor Tubular Wave Energy Converter

Kyoungmo Koo, Wonseok Jang, Jeonghwan Park, Jaemyung Cha and Seungyong Hahn K.
M. Koo, C. Im, G.Kim, J. Kim and S. Hahn are with the Department of Electrical
and Computer Engineering, Seoul National University, Seoul 08826, Korea
(e-mail: hahnsy@snu.ac.kr).

###### Abstract

So far, a number of wave energy converters (WEC) have been proposed to
increase efficiency and economic feasibility. Particularly, tubular WEC with
permanent magnets and coil winding packs is mostly used to convert the wave
energy. Due to the demand for high magnetic flux density in WEC, research has
been conducted on high-temperature superconductors (HTS) WEC. In this paper,
the conceptual design of no-insulation (NI) HTS tubular WEC and its
optimization process are proposed. Using NI technology, it has become possible
to design WEC with high volumetric efficiency and cost-effectiveness.
Furthermore, the design is analyzed in the aspect of electromagnetism,
mechanical force, and cryogen. The performance of the proposed WEC is
evaluated as a response to various waveforms and their amplitudes. A
rectifying circuit of WEC connected in parallel with load resistance is used
for the output power study.

###### Index Terms:

material cost, NI technology, optimization, output power measurement,
stability

##  I Introduction

The high demand for electricity stemming from the expansion of the electric
mobility industry implies an upcoming electric power shortage problem in the
future. Also, there has been growing concern about global warming due to
fossil fuel usage. These factors cast light on the importance of renewable
energy harvest. Among renewable energies, wave energy has been emphasized as a
growing energy resource. Wave energy around the world has been increasing
annually by 0.47 % from 1948 to 2008 [1]. (Southern Ocean 0.58 %, Pacific 0.35
%, Atlantic and India 0.26 % per year) Besides, wave energy is one of the most
affluent energy resources around the world. Total wave power around the
coastlines of the world is about 2.11 TW. Among the total wave, the power that
can be harvested is about 96.6 GW [2]. A large amount of energy has not been
harvested due to technical difficulties, compared to inland generator systems,
such as wind turbine and solar power system. A number of countries are trying
to build wave energy converters (WECs) to avoid future energy shortages and to
lower their dependencies on fossil fuels. For instance, in 2021, China had
been building a 1 MW WEC in Wanshan. Plus, in the same year, India built a 0.9
MW WEC that uses a wave-powered navigated buoy in Chennai, Tamil Nadu [3].
Meanwhile, the range of NI HTS applications on electric machinery has been
broadening, recently. By eliminating the turn-to-turn insulation of the HTS
pancake coil, much higher operational reliability is achieved, compared to
conventional HTS magnets [4]. Due to its strength, research regarding the NI
technique has been conducted actively [5, 6, 7, 8, 9]. Also, NI HTS technology
has begun to be adopted in the application area where a high magnetic field is
needed, including particle accelerators, magnetic resonance imaging (MRI), and
electric motors [10, 11, 12, 13]. Considering that existing WEC is having a
tradeoff problem regarding magnetic flux density and volume of a magnet, NI
HTS will be a valuable solution. Previously, several studies have been
conducted in the field of NI HTS rotating machines, including wind power
generators [14, 15, 16]. Meanwhile, the range of NI HTS applications on
electric machinery has been broadening, recently. By eliminating the turn-to-
turn insulation of the HTS pancake coil, much higher operational reliability
is achieved, compared to conventional HTS magnets [4]. Due to its strength,
research regarding the NI technique has been conducted actively [5, 6, 7, 8,
9]. Also, NI HTS technology has begun to be adopted in the application area
where a high magnetic field is needed, including particle accelerators,
magnetic resonance imaging (MRI), and electric motors [10, 11, 12, 13].
Considering that existing WEC is having a tradeoff problem regarding magnetic
flux density and volume of a magnet, NI HTS will be a valuable solution.
Previously, several studies have been conducted in the field of NI HTS
rotating machines, including wind power generators [14, 15, 17, 16].

This paper focuses on the design and analysis of WEC whose field winding part
is substituted with NI HTS magnets. In such a process, magnet design for the
minimum material cost was found through the parameter sweep analysis. Also,
multiphysics design approaches including electromagnetics, solid mechanics,
and cryogenic engineering for the design of NI HTS tubular WEC are conducted.
Last but not least, operational analysis has been conducted for several ocean
waves with different amplitudes. Output power for the design was calculated
through a three-phase bridge rectifier circuit including its load resistance.

##  II SYSTEM CONFIGURATION

Point absorber is chosen among various WEC designs, which uses oscillation of
buoy to extract wave energy and drive generator [18]. A point absorber is a
typical WEC structure since it can generate power from any directional wave
flow. Fig. 1 illustrates how the buoy moves according to the wave and moves
the actuator to generate electricity.

As the buoy fluctuates according to an ocean wave, the actuator part of the
generator that is connected to the buoy also moves and the voltage is induced
in the coil by Faraday’s law. It should be noted that the figure is not a
exactly scaled version of the actual system. Connecting part between the buoy
is about 30 m, which means that its actual length is much longer than the
figure. The basic structure of WEC is mainly derived from [19]. Cryocooler for
HTS cooling and auxiliary battery which is used for constant current flow in
HTS is placed on a buoy. The wire for those components is connected to HTS
through the structure between the buoy and the generator. The power that the
cryocooler consumes come from a generator and the power is rectified by a
converter that will be discussed in 5.1.

Figure 1: Conceptual design of WEC system.

LH2 is used as the coolant of the generator. Since LH2 initially enters the
actuator, it will flow through its path until it arrives at the lowest HTS
magnet among DP (double pancake). Dunking is used as a cooling method, which
fully immerses magnets into the coolant. Fig. 2 shows the specific structure
of the generator and cooling mechanism for two HTS magnets. Coolant follows
the main path and enters each magnet. Super insulation surrounds each HTS
magnet and protects it from transferred heat. The actuator frame and stator
frame are both made of fiber-reinforced plastic (FRP) to reduce total mass,
except for the outer yoke of the stator frame, which is iron. By using NI HTS
technology, the high magnetic flux density can be increased effectively in
terms of volume. The mass of the magnet is also greatly reduced, leading to
the enhancement of mechanical stability. As typical generators, an air gap
between the actuator and stator will be maintained using bearings. Every
neighboring HTS magnet will have an opposite magnetic pole for the dramatic
rate of magnetic flux change per second. The phase of each coil is designed to
be apart from each other, composing a three-phase circuit. A rectangular form
coil is used for the armature part and SuNAM superconducting tape is used for
HTS magnets.

Figure 2: Stator and actuator of WEC generator.

Table I shows key parameters of the design of WEC and ocean waveform during
operation.

TABLE I: Key Operational Parameters of WEC Parameters | Units | Value
---|---|---
Buoy radius | [m] | 0.510
Buoy height | [m] | 0.300
Number of magnets | [-] | 4
Number of slots | [-] | 12
Bobbins per phase | [-] | 4
Magnet height | [mm] | 119.80
Magnet width | [mm] | 22.50
Air gap | [mm] | 10
Cryo-vessel height | [mm] | 131.80
Cryo-vessel width | [mm] | 34.50
Armature coil height | [mm] | 50
Armature coil width | [mm] | 220
Actuator height | [mm] | 1150
Actuator radius | [mm] | 100
Total height | [mm] | 1150
Total radius | [mm] | 400
Operating current | [A] | 179
Operating temperature | [K] | 20
Coolant | [-] | LH2
Total HTS usage 4 mm; 6 mm | [km] | 3.57; 2.38
Waveform | [-] | Sinusoidal
Wave amplitude | [m] | 1.25
Frequency | [Hz] | 0.167

##  III DESIGN APPROACH

###  III-A Overall design approach and considering ocean wave condition

A variety of parameters had to be considered during the design. The number of
slots per pole is simply designed as three since the THD (Total Harmonic
Distortion) and performance remained the same while multiplying the number.
THD is calculated as the sum of the energy of all frequencies except
fundamental frequency, divided by that of fundamental frequency [20]. There
have been some significant changes compared to previous works. Compared to the
design of [19], the cross-sectional surface of HTS has been reduced by 40 %.
The air gap has increased by 25 % for more mechanical safety. Buoy diameter is
determined regarding resonance. Assuming the wavelength of the ocean wave is
longer than four times of buoy diameter, buoy diameter is 1.5% of wavelength
and absorbs 99 100 % of maximum energy, which is why the buoy radius is
designed as 0.510 m, assuming the wavelength of 34 m [21]. The buoy height is
designed at 0.300 m, with a 50 % margin from being completely submerged. As
illustrated in [22], which conducted HTS WEC experiment based on Chinese sea
conditions, an actuator is assumed to move at 0.4 m/s. Furthermore, because
the average coastline ocean wave period is less than 8 seconds [23], the
period is supposed to be 6 seconds and the amplitude to be 1.25 m.

###  III-B Optimization: Parameter sweep and multi-width winding method

The system consists of 4 sets of a three-phase circuit, which is composed of
12 coils and 4 HTS magnets in total. A generator is designed to produce target
power when connected to the rectifying circuit. For magnet width and height,
parameters that produce the highest voltage using the same total HTS length
are determined by the parameter sweep method. As a result, as demonstrated in
Fig. 3, magnet height of 119.8 mm, and magnet width of 22.4 mm turned out to
produce the highest output power, 14.68 kW.

Figure 3: Stator and actuator of WEC generator.

Additionally, the multi-width method is used to increase critical current, and
check whether the output power increases, compared to the single-width model
[24]. Fig. 4 shows the initial design of HTS that is composed solely of 4 mm
tape and the multi-width design is made of 6 mm and 4 mm tape. Using the
multi-width method, the critical current turned out to be 215 A, which is
about 43.3% higher than the initial design. Also, the multi-width design
produced an output power of 14.10 kW, which is about 27.5% higher than that of
the initial design.

Figure 4: Stator and actuator of WEC generator.

Fig. 5a and Fig. 5b shows the magnetic flux density norm of each model. As
shown in the figure, the multi-width model has a higher maximum magnetic flux
density and higher center flux density (4.45 T, 3.14 T), compared to the
single-width model (3.46 T, 2.75 T), in the condition of the same conduction
current. The result shows that the multi-width method certainly enhanced the
magnetic field in the design. Since the performance significantly improved,
the design is based on the multi-width model with 4 mm and 6 mm DP.

(a) single width

(b) multi width

Figure 5: Distribution of magnetic flux density in each design: (a) single
width, (b) multi width

###  III-C Specific design of HTS magnet

The design of the HTS magnet is as follows. The magnet is composed of a
winding pack, insulation plate, and coolant for maintaining an operating
temperature of 20 K. HTS magnet is composed of 13 layers as shown in Fig. 6.
All layers are composed of one 4 mm DP bulk or one 6 mm DP bulk. Additionally,
0.5 mm insulation plates are placed between two DPs. Each DP is 22.4 mm in
width, letting the coolant flow on both sides. Also, the upper path, which has
a width of 6mm enables effective cooling of the whole magnet. As shown in the
figure, coolant flows from the main path, which starts from the top of the
actuator. SI (Super Insulation) is designed to have a 6 mm width on each side,
making a total magnet height of 131.8 mm and width of 34.5 mm.

Figure 6:  Cross-sectional view of HTS magnet and insulation

##  IV Analysis

###  IV-A Electromagnetic analysis: magnetic field and critical current of HTS
magnet

Fig. 7 shows the distribution of magnetic flux density of the system. As shown
in the figure, the maximum magnetic flux density turns out to be 4.5 T in the
operating parameters, and the direction of magnetic flux density is mainly the
moving direction of the actuator. The left figure shows the distribution.

Figure 7: Distribution of magnetic field in the generator

Fig. 8a, 8b, 8c shows the distribution of critical current and magnetic flux
density in a single HTS magnet. As expected, since the magnetic flux density
increases as the point of measurement moves toward the center, the critical
current turns out to have a tendency of decreasing accordingly. But it is not
always the case because the direction of magnetic flux greatly affects the
value of critical current at the measured point. Maximum and minimum critical
current turned out to be 951.0 A, and 248.9 A at the operating current.

(a) magnetic flux density

(b) critical current of 6 mm DP

(c) critical current of 4 mm DP

Figure 8: Distribution of critical current and magnetic flux density in HTS
magnet ( a)magnetic flux density, b) critical current of 6 mm DP, c) critical
current of 4 mm DP)

Fig. 9 shows the load line graph of critical current. As illustrated in the
figure, the system has a critical current value of 215 A, so the current
margin obtained by the operating current, 179 A, is calculated as 36 A. The
current margin is 20 % of the operating current, which means that the system
is stable in the view of electromagnetism.

Figure 9: Load line graph of the system

Also, to avoid the saturation of the magnetic field that could lead to
degradation of system performance, it should be confirmed that the magnetic
flux density of the stator yoke should not exceed 1.5 T. It is because the
stator yoke is composed of an electrical steel sheet. Fig. 7 shows the
magnetic flux density of the cut plane of the stator and the maximum flux
density turns out to be 1.22 T, which means that the stator is free from
saturation issues.

(a) Cut plane of stator

(b) Magnetic flux density of stator

Figure 10: Distribution of critical current and magnetic flux density in HTS
magnet ( a) magnetic flux density, b) critical current of 6 mm DP, c) critical
current of 4 mm DP)

###  IV-B Mechanical stability: Lorentz force

The following figure is the analysis regarding magnetic force load, due to
Lorentz force inside HTS magnets. Fig. 11a, 11b shows the distribution of
radial and hoop stress, which is created by the inner Lorentz force of the
magnet, among four HTS magnets and the maximum value is 24.2 MPa. The
distribution of each magnet turned out to be identical because of the symmetry
of the design. The design is stable compared to conventional generators, due
to the compactness of magnets.

(a) radial stress of HTS magnet

(b) hoop stress of HTS magnet

Figure 11: Distribution of stress of each HTS magnet: (a) radial (b) hoop

###  IV-C Cryogenic analysis

Since the design use dunk mode for cooling the magnets, the initially needed
amount of coolant is important for safely operating a generator. The coolant
needed to force the temperature of HTS magnets to 20 K is calculated as eq.
(1).

$\displaystyle\frac{M_{H}}{H_{mg}}$
$\displaystyle=\frac{{h_{Cu}}(300K)-{h_{Cu}}(20K)}{h_{H}},$ (1)

MH is the mass of needed liquid hydrogen for initiating the generator. Mmgis
the total mass of the magnet, hCu is the specific heat of copper, and hH is
the specific heat of hydrogen. The value is calculated as 8.23 kg.
Additionally, the armature coil part should not flow a current of more than 6
A/mm2, in the case of the water- cooled type of design. This will be
specifically discussed in 5.2. Lastly, the critical current of the
superconducting wire according to the temperature can be calculated with
linear approximation applied as eq. (2), considering the critical temperature
of 0 A is 92 K. IC is a critical current at a certain temperature T. TC is a
critical temperature. IC0 is critical current at operating temperature, Top.

$\displaystyle I_{C}(T)$
$\displaystyle=I_{C0}\frac{T_{C}-T}{T_{C}-T_{op}},T_{op}<T<T_{C}$ (2)

Using the formula, the stability of the coolant system can be determined by
temperature margin. T-IC graph of SuNAM REBCO tape can be drawn as Fig. 12.

Figure 12: Critical current - temperature graph of HTS magnets

The stability margin of the magnet in terms of enthalpy is also calculated by
heat capacity per unit volume, C(T), integrated over temperature margin.

$\displaystyle E$ $\displaystyle=\int_{T_{C}S}^{T_{o}p}C(T)\,dT\,,$ (3)

Using the upper formula, 8.31 GJ/m3 of energy is tolerable in the perspective
of cryostability. TABLE II shows various parameters that summarize what is
discussed in this section.

TABLE II: Multiphysics Design Parameters of WEC Parameters | Units | Value
---|---|---
Maximum magnetic flux density | [T] | 4.45
Average magnetic flux density | [T] | 1.94
Critical current | [A] | 215
Current margin | [A] | 36
Maximum stress | [MPa] | 24.2
Average stress | [MPa] | 11.9
Young’s modulus of coil | [GPa] | 160
Mass of LN2 | [kg] | 10.17
Critical temperature | [K] | 32
Stability margin | [GJ/m3] | 8.31

##  V OPERATION ANALYSIS

###  V-A Modeling of the power grid

Three phase rectifier is connected to the three-phase voltage sources. 0.15 H
of an inductor is connected in series with source and resistance. A resistor
is optimized to 4.2 $\Omega$ so as to produce maximum output power. 2.7 H
inductor is connected in series with resistance to smooth out the output
voltage graph and reduce the distortion rate. Fig. 13 shows how the input
voltage is rectified through a circuit. The total distortion ratio has greatly
decreased after the rectifier from 2.99 to 1.12. The fundamental frequency of
the input voltage source turned out to be 0.35 Hz, and that of the output
voltage turned out to be 2.01 Hz.

Figure 13: Circuit diagram of the whole system

###  V-B Voltage and current analysis of each phase

Fig. 14 shows the voltage results of the design. The voltage shows three
cycles of output power waves with changing amplitude as time pasts. The
voltage of each phase had a maximum voltage of 589.4 V, 635.0 V, and 646.5 V.
Vrms (Root Mean Square) of each voltage source turned out to be 267.1 V, 267.2
V, and 264.8 V.

Figure 14: Output voltage of each phase

Fig. 15 shows the current that flows in each armature coil during a
simulation. As stated in 4.3, the current density in the armature coil should
be maintained under 6 A/mm2 most time. It turns out that the current in each
phase stays under 80 A most time except for several peaks, which are in the
range of 80-85 A. Therefore, the system is proven stable considering the
cross-sectional surface of the rectangular form coil, considering that the
critical current of the armature coil is 140 A.

Figure 15: Armature coil current of each phase

###  V-C Output power of various conditions

Various simulations are conducted with different conduction currents and wave
amplitudes. Fig. 16 and TABLE III show the results of each case. Obviously,
voltage wave amplitude was proportional to the conduction current of magnet
and wave amplitude, aligning with the result of [25]. The power, Vrms of input
voltage and output voltage, loss, and efficiency of each case is stated in the
following table. Since the maximum instant current that flows through the
armature coil is maintained under 6 A/mm2 for the 2nd, 3rd, and 4th cases, the
armature coil is stable considering the cooling method.

Figure 16: Output voltage of various conditions TABLE III: Simulation results of various conditions Wave amplitude, conduction current |  1.25 m, 179 A |  1.25 m, 215 A |  1.75 m, 179 A |  1.75 m, 215 A
---|---|---|---|---
Vrms of input voltage (V) |  284.6 |  347.3 |  462.3 |  555.2
Vrms of output voltage (V) |  245.8 |  297.6 |  321.5 |  387.5
Power (kW) |  14.1 |  21.1 |  24.6 |  35.7
Joule loss (kW) |  8.4 |  12.2 |  14.6 |  21.3
Efficiency (%) |  63.0 |  63.4 |  62.7 |  62.7
Power factor |  0.84 |  0.84 |  0.84 |  0.84

The wave can be also modeled in triangular form. Fig. 17 compares the output
voltage result of triangular form wave and sinusoidal wave. It turned out to
be that the case a triangular wave can generate power of 10.63 kW, which is
24.77 % lower than that of the sinusoidal wave, assuming the same wave
amplitude and frequency. The voltage waveform has much less noise and has the
form of a sinusoidal wave with a DC offset.

Figure 17: Output voltage of the triangular wave

##  VI CONCLUSION

The conceptual design of a no-insulation HTS tubular wave energy generator is
proposed in this paper. A multiphysics design approach including
electromagnetics, solid mechanics, and cryogenic engineering is conducted for
the design. The magnet operates at 20 K, 179 A with the maximum magnetic flux
density of 4.45 T. Based on the proposed design, the voltage results of
various waveforms have been obtained. The whole system is modeled as an
equivalent circuit and analyzed in terms of the current that flows through
armature coils. Using the circuit model, the output power has been calculated
for different cases. Lastly, in the condition of a sinusoidal form ocean wave
with an amplitude of 1.25 m and 6 Hz, the output power was calculated as 14.10
kW. The performance turns out to increase as the current inside the magnet and
amplitude of the ocean wave increase. The design shows high volumetric
efficiency of electricity production, by applying NI HTS technology. Also,
although the paper only discusses the generator production of 10-50 kW, due to
the simplicity of the design, production can be increased to a 100 kW-1 MW
scale. However, for the scaled-up version, other cooling methods may be needed
for the armature part such as forced water cooling, which allows 12 A/mm2
[26], since a much higher current will flow.

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