# Structure of 46,47Ca from the $\beta^{-}$ decay of 46,47K in the framework
of nuclear shell model

Priyanka Choudhary  Department of Physics, Indian Institute of Technology
Roorkee, Roorkee 247667, India  Anil Kumar  Department of Physics, Indian
Institute of Technology Roorkee, Roorkee 247667, India  Praveen C. Srivastava
<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Roorkee, Roorkee 247667, India  Toshio Suzuki  Department of
Physics, College of Humanities and Sciences, Nihon University, Sakurajosui 3,
Setagaya-ku, Tokyo 156-8550, Japan

###### Abstract

In the present work, we report a comprehensive theoretical study of the $\log
ft$ values for the allowed and forbidden $\beta^{-}$ decay transitions of 46K
and 47K corresponding to recently available experimental data from the GRIFFIN
spectrometer at TRIUMF-ISAC [Phys. Rev. C 100, 054327 (2019); Phys. Rev. C
102, 054314 (2020)]. We perform the nuclear shell-model calculation in
$sdpf$-valence space, with the SDPF-MU interaction, to calculate low-lying
energy spectra of 46,47K and 46,47Ca. For further investigation, we also
calculate spectroscopic properties of 46,47K, 46,47Ca and compare the results
with the experimental data. We suggest spin-parity of several levels, which
were previously tentative, in both the calcium isotopes. Based on the energy
and $\log ft$ values, we conclude that the state at 3.984 MeV in 46Ca could be
either $3^{-}$ or $2^{+}$ and a state at 4.432 MeV might be $3^{-}$. The spin-
parity of the states at 2.875, 3.951 and 4.453 MeV are predicted to be
$1/2^{-}$, $3/2^{-}$ and $1/2^{-}$ or $1/2^{+}$, respectively, in 47Ca. We
also give the confirmation to spin-parity of 2.850, 3.889 and 4.606 MeV states
for 47Ca. Our results of level schemes of 46,47K are in a reasonable agreement
with the experimental data, while the agreement of calculated $\log ft$ values
for the $\beta^{-}$ decay with the experimental data is limited.

###### pacs:

21.60.Cs, 23.40.-s

##  I Introduction

Understanding the formation of shell structure across an isotopic chain (from
proton drip line to neutron drip line) is a key challenge in nuclear physics.
New experiments have revealed the appearance of new shell closures N = 16, 32,
34 A.Gade ; F.Wienholtz ; D.Steppenbeck ; D.SteppenbeckPRL ; M.Rosenbusch ;
H.N.Liu ; A.Ozawa ; R.Kanungo  and the disappearance of the magic numbers N =
8, 20, 28 A.Navin ; T.Motobayashi ; B.Bastin . Evolution of shell structure in
calcium isotopes has been a crucial area of investigation in nuclear structure
over the past few decades and is still in progress from both experimental and
theoretical perspectives. Accurate experimental data for level energies, spins
and parities, is of utmost importance to obtain complete spectroscopic
information of the calcium isotopes apart from helping us to determine the
nature of interaction among the nucleons therein and set a benchmark for our
theoretical calculations.

The study of $\beta$ decay is a very important tool to determine the structure
of atomic nuclei. In $\beta$ decay, transitions are classified into two
categories jouni : allowed and forbidden, based on the value of orbital
angular momentum ($l$) of the emitted leptons. Transitions with $l=0$
correspond to allowed and those with $l>0$ to forbidden transitions. Allowed
transitions are characterised by a change in total angular momentum $\Delta J$
= 0,1 and no change in parity ($\Delta\pi$ = +1). Forbidden $\beta$ decays are
further classified into forbidden unique (FU) and forbidden non-unique (FNU)
transitions. For the $K^{th}$-FU transition, $\Delta J$ = $K+1$, while $\Delta
J$ $\leq$ $K$ for the $K^{th}$-FNU transition and $\Delta\pi$ = $(-1)^{K}$ in
both the cases. For both allowed and forbidden transitions, $\log ft$ values,
shape factors and electron spectra have been calculated with the effective
value of axial-vector coupling constant gA, obtained corresponding to their
mass region, earlier in $\beta$ decay studies in Refs. mika2016 ; mika2017 ;
mst2006 ; joel12017 ; joel22017 ; VikashK . Study of second-FNU $\beta^{-}$
decay of 24Na($4^{+}$) $\rightarrow$ 24Mg($2^{+}$) and 36Cl($2^{+}$)
$\rightarrow$ 36Ar($0^{+}$) has been reported recently in Ref. Anil .

Several experiments have been performed to investigate the excited states of
46,47Ca, of which, some are nuclear reaction based experiments W.W.Daehnick ;
D.C.Williams ; G.M.Crawley ; M.E.Williams-Norton ; P.Martin  while others are
$\beta$ decay measurements B.Parsa ; P.Kunz ; E.K.Warburton ; D.E.Alburger ;
T.Kuroyanagi . Although several excited states have been identified in these
experiments, experimental assignments of spin-parity to all the states could
not be made. In some instances, tentative assignments are given and in others,
no assignment could be made.

Recently, an experiment has been carried out at TRIUMF in which high-statistic
data from $\beta^{-}$ decay of ground state (g.s.) $J^{\pi}=2^{-}$ of 46K into
46Ca has been measured with GRIFFIN spectrometer 46Ca , where, 42 excited
states were populated and $\beta$ feeding branching ratios into excited states
of 46Ca were also observed. From analysis of these data, it was possible to
assign spin-parity of excited states. From the same experiment, $\beta^{-}$
decay of g.s. $J^{\pi}=1/2^{+}$ of 47K into 47Ca was studied in which spin-
parity of 8 excited states were assigned along precise measurement of half-
life of 47K 47Ca . These experiments measure simultaneously the probability of
electromagnetic transitions among the states of a nucleus and $\beta$ decay
feeding of these states from neighbouring nuclei. Studying them together is a
real stringent test for the nuclear shell-model.

Study of nuclear structures up to $Z=20$ and beyond using first principles or
ab initio approaches has gained momentum over the past few years nocore_review
; P.Choudhary ; arch2 ; G.HagenPRL ; T.D.Morris ; Holtdrip ; Anil2020 . In
Ref. G.Hagen , a coupled-cluster (CC) method with two-nucleon (NN) and three-
nucleon (3N) interaction from chiral effective field theory (EFT) RMP ; EFT1 ;
EFT2 ; EFT3  has been used to treat the many-body system of calcium isotopes.
Their results G.Hagen  confirm a shell closure in 48Ca, a subshell closure in
52Ca and a weak subshell closure in 54Ca. It has been shown that 3$N$ forces
play an important role in explaining the magic number $N=28$ in calcium
isotopes J.D.Holt . In Ref. PRC90 , Holt el. al have calculated energy spectra
of neutron-rich calcium isotopes using many-body perturbation theory (MBPT)
with NN and NN+3N forces for $pf$ and $pf$g9/2 valence spaces. They have also
performed shell-model calculations using the phenomenological $pf$-shell
interactions like GXPF1A M.Honma  and KB3G A.Poves  and compared results with
the experimental data. For 46Ca, their results are in good agreement with the
experimental data, however, $4^{+}$ and $6^{+}$ states are $\sim$ 1 MeV higher
for NN+3N interaction in $pf$g9/2 model space than the experimental values. A
significant spread in the spectra was obtained, indicating the need to
incorporate cross-shell degrees of freedom. Further, they are unable to
reproduce negative parity states and the $0^{+}_{2}$ state, which was expected
to be dominated by $sd$-shell. Thus $sdpf$-model space is crucial for the
study of these states. For 47Ca, they concluded that $pf$-valence space gives
compressed spectra and the best results are obtained with $NN+3N$ forces in
$pf$g9/2 space. This establishes that the extended space $pf$g9/2 with $3N$
forces improves the energy spectra.

In Ref. 46Ca , authors have extended the work of Holt el. al wherein they have
used non-standard valence space for 46Ca in which protons occupy
$s_{1/2}d_{3/2}f_{7/2}p_{3/2}$ orbitals and neutrons occupy
$s_{1/2}d_{3/2}f_{7/2}p_{3/2}p_{1/2}$ orbitals above a 28Si core. They
reproduced correctly the $0^{+}_{2}$ state and also the negative parity states
but obtained a significantly compressed spectra and their results showed that
proton excitations from $sd$-shell are dominant. These calculations reveal
that the protons are no longer inactive despite it having a robust shell-
closure with the $Z=20$.

Other ab initio methods like in-medium similarity renormalization group (IM-
SRG) targeted for a particular nucleus S.R.Stroberg  and self-consistent
Green’s function (SCGF) V.Soma  theory have also been applied to study the
calcium isotopes. Bhoy el. al have used the shell-model with GXPF1Br + VMU
interaction T.Togashi  for calculating energy spectra of 47-58Ca B.Bhoy  where
they have used $pf$, $pf$g9/2 and $pf$g${}_{9/2}d_{5/2}$ model spaces to show
the dependence of states on the inclusion of the higher orbitals.

Nuclear charge radius is an important observable to describe the shell
evolution in nuclei. An experiment at ISOLDE-CERN based on laser spectroscopy
has been performed to measure charge radius for 52Ca Nature , which was found
to be much larger than that expected for a doubly magic nucleus. From ab
initio CC theory Nature , these results are well reproduced. This reflects a
breaking of magicity at $N=32$ in calcium isotopes. A recent shell-model study
L.Coraggio  for calcium isotopes has been reported to find the location of the
drip line of calcium isotopes and showed that the calcium isotope chain is
bound up to 70Ca.

In the present work we have calculated the $\log ft$ values for allowed and
forbidden $\beta^{-}$ decay of 46K and 47K corresponding to the available
experimental data of TRIUMF experiment 46Ca ; 47Ca , using the shell-model
with $sdpf$-model space. For these calculations, we have also extracted the
effective value of gA using chi-square fitting method. We have also studied
the energy spectra, quadrupole moment, magnetic moment and transition
strengths of 46,47K and 46,47Ca isotopes. A comparison of our theoretically
predicted results with the experimental ones is also presented. Wave functions
of various states in both calcium isotopes have also been studied to determine
the contribution from each configuration and also deduce the role of
$sd$-model space.

Our work is the first theoretical description of energy spectra of 47Ca
corresponding to recently available experimental data 47Ca , which refined the
knowledge regarding the level scheme of 47Ca. Our study is the first shell-
model calculation of the $\log ft$ values for allowed and forbidden
$\beta^{-}$ decay of 46,47K 46Ca ; 47Ca .

This paper is organized as follows. In Sec. II, we explain the shell-model
Hamiltonian with adopted model space and theoretical formalism for allowed and
forbidden nuclear $\beta$-decay. Also, we extract the effective value of gA
using chi-square fitting method for $\beta$ decay study. In Sec. III, we
present theoretically calculated level structures and spectroscopic properties
of 46,47K and 46,47Ca. Also we have reported the $\log ft$ results of
$\beta^{-}$ decay of 46K and 47K. In Sec. IV, conclusions are drawn.

##  II Formalism

###  II.1 Shell-model Hamiltonian

The nuclear shell-model Hamiltonian contains a single-particle energy and a
two-nucleon interaction terms. The shell-model Hamiltonian can be written as
follows

$H=T+V=\sum_{\alpha}{\epsilon}_{\alpha}c^{\dagger}_{\alpha}c_{\alpha}+\frac{1}{4}\sum_{\alpha\beta\gamma\delta}v_{\alpha\beta\gamma\delta}c^{\dagger}_{\alpha}c^{\dagger}_{\beta}c_{\delta}c_{\gamma},$
(1)

where $\alpha=\\{n,l,j,t\\}$ is a single-particle state and the corresponding
single particle energy is $\epsilon_{\alpha}$. $c^{\dagger}_{\alpha}$ and
$c_{\alpha}$ are the creation and annihilation operators.
$v_{\alpha\beta\gamma\delta}=\langle\alpha\beta|V|\gamma\delta\rangle$ are the
antisymmetrized two-body matrix elements.

We have used the nuclear shell-model to determine the nuclear structure
properties of 46,47K and 46,47Ca. It has been found that including cross-shell
excitations in the valence space is necessary in order to correctly reproduce
the energy spectra of 46,47K and 46,47Ca 46Ca ; PRC90 . So, we have used a
phenomenological SDPF-MU interaction SDPFMU  for $sdpf$-model space in the
shell-model calculations with 16O as an inert core. In this interaction, there
are seven orbitals ($d_{5/2}$, $s_{1/2}$, $d_{3/2}$, $f_{7/2}$, $p_{3/2}$,
$f_{5/2}$ and $p_{1/2}$) corresponding to two major shells, available for the
valence nucleons to occupy. This allows us to calculate both the positive and
negative parity states of 46,47K and 46,47Ca. For unnatural parity states, we
have allowed either one proton or one neutron excitation from $sd$ to $pf$
shell, known as the 1p-1h excitation.

###  II.2 $\beta$ decay theory for allowed and forbidden transitions

A detailed theoretical formalism of nuclear $\beta$ decay has been presented
in Refs. behrens1982 ; hfs1966 . A relatively simple approach in describing
$\beta$ decay involves the impulse approximation, according to which, the
decaying nucleon experiences just the weak interaction at the moment of decay
and does not interact strongly with the remaining $A-1$ nucleons. As such, the
$A-1$ nucleons are referred to as spectator nucleons with respect to the weak
interaction. In the initial and final nuclear many-body states, however, the
active nucleon does participate in strong interactions jouni . The nuclear
$\beta^{-}$ decay process is described with an effective point-like
interaction vertex with an effective coupling constant $G_{\text{F}}$, known
as Fermi coupling constant. The probability of the emitted electron for the
$\beta^{-}$ decay process to lie in an energy range $W_{e}$ to $W_{e}+dW_{e}$
is given by

$\begin{split}P(W_{e})dW_{e}&=\frac{G_{\text{F}}^{2}}{(\hbar{c})^{6}}\frac{1}{2\pi^{3}\hbar}C(W_{e})p_{e}cW_{e}(W_{0}-W_{e})^{2}\\\
&\times{F_{0}(Z,W_{e})dW_{e}},\end{split}$ (2)

where, $W_{0}$ is the endpoint energy of the $\beta$ spectrum which
corresponds to the maximum energy taken by the electron in a transition, the
factor $F_{0}(Z,W_{e})$ is the Fermi function, and $Z$ is the proton number of
the daughter nucleus. The $p_{e}$ and $W_{e}$ are the momentum and energy of
the emitted electron, respectively and $C(W_{e})$ is the shape factor that
contains information about the nuclear structure. The partial half-life of the
decay process is related to the transition probability as

$\displaystyle
t_{1/2}=\frac{\text{ln}(2)}{\int_{m_{e}c^{2}}^{W_{0}}{P(W_{e})dW_{e}}},$ (3)

where $m_{e}$ is the rest mass of the electron and the integral in the
denominator is called the decay rate or transition probability per unit time.
To simplify this integration, we introduce the dimensionless quantities
$w_{0}=W_{0}/m_{e}c^{2}$, $w_{e}=W_{e}/m_{e}c^{2}$, and
$p=p_{e}c/m_{e}c^{2}=\sqrt{(w_{e}^{2}-1)}$, which are kinematical quantities
divided by the electron rest mass. Now, the partial half-life is related to
shape function as

$ft_{1/2}={\kappa},$ (4)

where, $f$ is the dimensionless integrated shape function and the value of the
constant $\kappa$ Patrignani  is

$\displaystyle\kappa=\frac{2\pi^{3}\hbar^{7}\text{ln(2)}}{m_{e}^{5}c^{4}(G_{\text{F}}\text{cos}\theta_{\text{C}})^{2}}=6289~{}\mathrm{s},$
(5)

where, $\theta_{\text{C}}$ is the Cabibbo angle.

The dimensionless integrated shape function $f$ can be written as

$\displaystyle
f=\int_{1}^{w_{0}}C(w_{e})pw_{e}(w_{0}-w_{e})^{2}F_{0}(Z,w_{e})dw_{e}.$ (6)

For allowed (Gamow-Teller) transition, shape factor $C(w_{e})$ can be
expressed as

$C(w_{e})=B(GT)=\frac{g_{\rm A}^{2}}{2J_{i}+1}|\mathcal{M}_{\rm GT}|^{2},$ (7)

where, $\mathcal{M}_{\rm GT}$ is the Gamow-Teller nuclear matrix element (NME)
of $\sigma\tau$ and $B(GT)$ is reduced transition probability (details of
which can be found in Ref. jouni ), gA is the axial-vector coupling constant
and $J_{i}$ is the angular momentum of initial state. For such a transition,
the reduced half-life is calculated by multiplying the half-life with the
phase-space factor (Fermi integral) given by

$f_{0}=\int_{1}^{w_{0}}pw_{e}(w_{0}-w_{e})^{2}F_{0}(Z,w_{e})dw_{e}.$ (8)

We then take the logarithm of the ft value also known as the reduced half-life
or comparative half-life, as the ft value is generally a large quantity.

$\mbox{log}ft\equiv\mbox{log}(f_{0}t_{1/2}~{}s).$

The shape factor $C(w_{e})$ for forbidden transition has the form of

$\displaystyle\begin{split}C(w_{e})&=\sum_{k_{e},k_{\nu},K}\lambda_{k_{e}}\Big{[}M_{K}(k_{e},k_{\nu})^{2}+m_{K}(k_{e},k_{\nu})^{2}\\\
&-\frac{2\gamma_{k_{e}}}{k_{e}w_{e}}M_{K}(k_{e},k_{\nu})m_{K}(k_{e},k_{\nu})\Big{]},\end{split}$
(9)

where, $k_{e}$ and $k_{\nu}$ (= 1, 2, 3,…) are related to the partial-wave
expansion of the leptonic wave functions, and $K$ is the order of forbiddeness
of the transition. The entities $M_{K}(k_{e},k_{\nu})$ and
$m_{K}(k_{e},k_{\nu})$ are written in terms of NME and leptonic phase-space
factors behrens1982 ; mika2017 . The quantities

$\displaystyle\gamma_{k_{e}}$ $\displaystyle=\sqrt{k_{e}^{2}-(\alpha{Z})^{2}}$
$\displaystyle\mbox{and}\,\,\,\,\lambda_{k_{e}}$
$\displaystyle={F_{k_{e}-1}(Z,w_{e})}/{F_{0}(Z,w_{e})},$

where, $\lambda_{k_{e}}$ is the Coulomb function, in which
${F_{k_{e}-1}(Z,w_{e})}$ is the generalized Fermi function mst2006 ; mika2017
. The $\log ft$ values for forbidden transitions are calculated in a similar
manner as that in the case of allowed transitions by using the phase-space
factor corresponding to the concerned transitions (more details can be found
in Ref. behrens1982 ). The NME contain all the information about nuclear-
structure and can be expressed as

$\displaystyle\begin{split}{}^{\rm
V/A}\mathcal{M}_{KLS}^{(N)}(pn)(k_{e},m,n,\rho)&\\\
=\frac{\sqrt{4\pi}}{\widehat{J}_{i}}\sum_{pn}\,^{\rm
V/A}m_{KLS}^{(N)}(pn)(&k_{e},m,n,\rho)(\Psi_{f}||[c_{p}^{\dagger}\tilde{c}_{n}]_{K}||\Psi_{i}),\end{split}$
(10)

where, the summation, in Eq. 10, runs over the proton $(p)$ and neutron $(n)$
single-particle states with ${\widehat{J}_{i}}=\sqrt{2J_{i}+1}$. The quantity
m is the total power of ($m_{e}R/\hbar$), ($W_{e}R/\hbar$) and $\alpha$Z; n is
the total power of ($W_{e}R/\hbar$) and $\alpha$Z; $\rho$ is the power of
$\alpha$Z. The functions ${{}^{V/A}m_{KLS}^{(N)}}(pn)(k_{e},m,n,\rho)$ are the
single-particle matrix elements (SPMEs) that does not depend on the nuclear
models and $(\Psi_{f}||[c_{p}^{\dagger}\tilde{c}_{n}]_{K}||\Psi_{i})$ are the
one-body transition densities (OBTDs) between the initial $(\Psi_{i})$ and
final $(\Psi_{f})$ states. SPMEs expressions, used in this work, are taken
from mika2017 ; mst2006  and OBTDs are computed from the shell-model code
NUSHELLX nushellx .

###  II.3 Quenching factor

Table 1: Comparison of computed $\log ft$ values and average shape factors for Gamow-Teller and first-forbidden transitions with experimental data 46Ca ; 47Ca . Only confirmed transitions for 46K $\rightarrow$ 46Ca and 47K $\rightarrow$ 47Ca are presented. Calculated values are obtained by using quenching factor q = 1.00 and enhancement factor $\epsilon_{\text{MEC}}$ = 1.6 for $0^{-}$ transitions. Transition |  $\log ft$ |  ${\overline{(C(w_{e}))}}^{1/2}$
---|---|---
Initial | Final | Expt. | Theory | Expt. | Theory
46K$(2^{-})$ |  46Ca$(2^{+}_{1})$ | 6.92 | 6.439 | 10.62 | 18.47
|  46Ca$(0^{+}_{2})$ | 10.27 | 9.525 | 0.22 | 0.53
|  46Ca$(4^{+}_{1})$ | 9.70 | 7.405 | 0.43 | 6.075
|  46Ca$(2^{+}_{2})$ | 8.01 | 8.298 | 3.03 | 2.17
|  46Ca$(3^{-}_{1})$ | 8.04 | 5.755 | 0.0076 | 0.105
|  46Ca$(2^{+}_{3})$ | 8.13 | 6.879 | 2.64 | 11.135
|  46Ca$(4^{+}_{2})$ |  $>$11.5 | 8.868 | 0.054 | 1.13
47K$(1/2^{+})$ |  47Ca$(3/2^{-}_{1})$ |  $>$6.5 | 7.831 | 17.22 | 3.72
|  47Ca$(3/2^{+}_{1})$ | 5.457 | 5.690 | 0.1482 | 0.11
|  47Ca$(1/2^{+}_{1})$ | 4.807 | 4.386 | 0.3132 | 0.51

The bare values of the two coupling constants, gV = 1.0 (vector coupling
constant) and gA = 1.27 (axial-vector coupling constant), associated with the
$\beta$ decay theory are determined from conserved vector current (CVC) and
partial conserved axial-vector current (PCAC), respectively. In the nuclear
shell-model calculations, the renormalizations due to the truncation of model
space and possible meson-exchange current contributions are taken into account
by using effective values of the coupling constants. In this paper, we have
calculated the quenching factor for gA. The averaged shape factor behrens1982
is defined as

$\overline{C(w_{e})}=f/f_{0}\,.$ (11)

The expressions of $f$ and $f_{0}$ are written in Eq. 6 and 8. For Gamow-
Teller transitions, $\overline{C(w_{e})}$ is $B(GT)$ and that for first-
forbidden transitions is enhancement ; PRC85

$\overline{C(w_{e})}(fm^{2})=\frac{6289\lambdabar_{\text{Ce}}^{2}}{ft}=\frac{9378\times
10^{5}}{ft},$ (12)

where, $\lambdabar_{\text{Ce}}$ is the reduced Compton wavelength of electron.
For first-forbidden $\beta$ decays, there are six operators for NME, given by

$\displaystyle\begin{split}O(0^{-})&:\text{g}_{\rm
A}(\bm{\sigma}\cdot\bm{p_{e}}),\,\,\,\text{g}_{\rm
A}(\bm{\sigma}\cdot\bm{r})\\\ O(1^{-})&:\text{g}_{\rm
V}{\bm{p_{e}}},\,\,\,\text{g}_{\rm
A}(\bm{\sigma}\,\times\,\bm{r}),\,\,\text{g}_{\rm V}\bm{r}\\\
O(2^{-})&:\text{g}_{\rm A}[\bm{\sigma r}]_{2}\,,\end{split}$ (13)

where, $r$ and $p_{e}$ are the radial coordinate and electron momentum,
respectively. The operators $O(0^{-})$, $O(1^{-})$ and $O(2^{-})$ correspond
to $\Delta J$ = 0, 1 and 2 transitions, respectively. The operator $\sigma$
$\cdot$ $p_{e}$ is the axial-charge nuclear matrix element $\gamma_{5}$. For
first-forbidden non-unique $J^{+}\leftrightarrow J^{-}$ transitions,
$\gamma_{5}$ NME is enhanced due to nuclear medium effects in the form of
meson-exchange currents. This enhancement factor in $\gamma_{5}$ is
represented by $\epsilon_{\text{MEC}}$, defined as the ratio of the axial-
charge matrix element to its impulse-approximation value. The one-body impulse
NME $M_{0}^{T}$ = $<$ Jf $||$ -gA $\frac{1}{M}\vec{\sigma}\cdot\vec{p}_{e}$
$t_{-}$ $||$ Ji $>$, where M is the nucleon mass and $t_{-}|n>=|p>$, is
multiplied by an effective coupling constant, $\epsilon_{\text{MEC}}$, to take
account of the contributions from the meson-exchange currents Towner ; WTB .
We have taken the value of $\epsilon_{\text{MEC}}$ as 1.6 which is consistent
with Ref. enhancement ; meson .

Figure 1: Comparison of calculated and experimental average shape factors for
Gamow-Teller and first-forbidden transitions to obtain the quenching factor.
For extraction of the quenching factor, we have used only confirmed
transitions of 46,47Ca.

In Table 1, we have shown theoretical and experimental $\log ft$ values and
averaged shape factors corresponding to transitions 46K $\rightarrow$ 46Ca and
47K $\rightarrow$ 47Ca. These calculations are done by using free coupling
constants (quenching factor q = 1.0). We have also included mesonic
enhancement factor $\epsilon_{\text{MEC}}$ = 1.6 for the matrix element
$M_{0}^{T}$ in 0- transitions. In the present calculations, we have also
included next-to-leading-order terms of $\beta$ decay shape function, similar
to mika2017 ; mika2016 . In Fig. 1, theoretical and experimental
${\overline{C(w_{e})}}^{1/2}$ values are plotted. We have extracted the
effective value of gA = q $\times$ g$\rm{}_{A}^{\,\text{free}}$ = 0.72
corresponding to quenching factor q = 0.57 from the chi-square fitting method.

##  III Results and Discussion

We have performed shell-model calculations to study nuclear structures of
46,47K and 46,47Ca in $sdpf$-model space. Low-lying energy spectra of 46K and
46Ca obtained from the shell-model are compared with the experimental data in
Fig. 2. The experimental spin-parity of g.s. is tentative for 46K. Our
theoretical calculations predict the g.s. as $2^{-}$. The g.s. of 46K has the
configuration of $\pi(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{3})$ $\otimes$
$\nu(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{4}f_{7/2}^{7})$ with 70.24% probability.
In Fig. 3, low-lying energy spectra corresponding to 47K and 47Ca are shown
and also compared with the experimental data. The shell-model calculations
with SDPF-MU interaction correctly reproduce the g.s. spin-parity for 47K. The
g.s. configuration of 47K is $\pi(d_{5/2}^{6}s_{1/2}^{1}d_{3/2}^{4})$
$\otimes$ $\nu(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{4}f_{7/2}^{8})$ with probability
of 71.21%. The calculated energy spectra of 47K is in good agreement with the
experimental one.

Table 2: Comparison of theoretical and experimental reduced electric quadrupole transition probabilities from excited states to g.s. in 46,47K and 46,47Ca. Effective charges are taken as $e_{p}=1.35e$ and $e_{n}=0.35e$. Experimental values are taken from Refs. NNDC ; Qandmag . |  |  $B(E2)(W.u.)$
---|---|---
Nuclei | Transition | Expt. | SDPF-MU
46K | $4^{-}_{1}\rightarrow 2^{-}_{1}$ | NA | 2.70
46Ca | $2^{+}_{1}\rightarrow 0^{+}_{1}$ | 3.63 | 0.40
47K | $5/2^{+}_{1}\rightarrow 1/2^{+}_{1}$ | NA | 2.5
47Ca | $3/2^{-}_{1}\rightarrow 7/2^{-}_{1}$ | $<0.28$ | 0.2
| $3/2^{-}_{2}\rightarrow 7/2^{-}_{1}$ | NA | 0.30

We have computed electromagnetic transition strengths $B(E2)$ for transitions
from excited states to g.s. in 46,47K and 46,47Ca, which are shown in Table 2.
In case of 46Ca, $B(E2)$ value for $2^{+}_{1}\rightarrow 0^{+}_{1}$ transition
is 0.4 W.u. while experimental value is 3.63 W.u., which is quite far from
theory but $B(E2$ $3/2^{-}_{1}\rightarrow 7/2^{-}_{1}$) in 47Ca is 0.2 W.u.
which agrees with the experimental value ($<0.28$ W.u.). We have also
predicted experimentally unknown transition strength
$B(E2;3/2^{-}_{2}\rightarrow 7/2^{-}_{1})$ for 47Ca.

In Table 3, we have reported quadrupole and magnetic moments obtained from the
shell-model with SDPF-MU interaction for selected states of 46,47K and
46,47Ca. We have then compared these values with the experimental ones NNDC ;
Qandmag  and found that the magnetic moments of $2^{+}$, $1/2^{+}$ and
$7/2^{-}$ states are consistent with experimental values for 46Ca, 47K and
47Ca, respectively, for $g_{s}^{eff}$ = 0.85 $g_{s}^{free}$. Also, we obtain
magnetic moment of g.s. $2^{-}$ for 46K as -0.478 (-0.585) for $g_{s}^{eff}$ =
0.85 (0.7)$g_{s}^{free}$. As the quenching factor for $g_{s}$ is found to be
0.9 in $fp$-shell for the GXPF1 interaction Honma  and 0.92-0.93 in $sd$-shell
for the USDB interaction Richter , the value of 0.85 is a reasonable one for
SDPF-MU. We have predicted quadrupole and magnetic moments for several states
of 46,47Ca from the shell-model which are experimentally not yet measured.

So, our energy spectra and magnetic moments corresponding to the adopted
interaction are in a reasonable agreement with the experimental data. Thus,
SDPF-MU interaction is a suitable interaction for calculating nuclear
properties of these nuclei. It means that we can use the wave functions,
obtained from SDPF-MU interaction, to compute the NME needed for the $\log ft$
calculations.

Table 3: Comparison of theoretical and experimental quadrupole and magnetic moments of selected states for 46,47K and 46,47Ca. The effective charges are taken as $e_{p}=1.35e$ and $e_{n}=0.35e$. For the magnetic moment calculations two sets of $g_{s}^{eff}$ are employed: $g_{s}^{eff}$ = 0.7 $g_{s}^{free}$ and $g_{s}^{eff}$ = 0.85 $g_{s}^{free}$, separated by a solidus (/). Experimental data are taken from Refs. NNDC ; Qandmag . |  | $\mu(\mu_{N})$ | $Q(eb)$
---|---|---|---
Nuclei | $J^{\pi}_{f}$ | Expt. | SDPF-MU | Expt. | SDPF-MU
46K | $2^{-}_{g.s.}$ | -1.051(6) | -0.585 / -0.478 | NA | 0.061
46Ca | $2_{1}^{+}$ | -0.52(12) | -0.425 / -0.517 | NA | -0.043
| $3_{1}^{+}$ | NA | -1.009 / -1.225 | NA | 0.002
| $4_{1}^{+}$ | NA | -1.176 / -1.429 | NA | -0.015
47K | $1/2^{+}_{g.s.}$ | 1.933(9) | 1.583 / 1.953 | - | \-
47Ca | $7/2_{1}^{-}$ | -1.38(3) | -1.026 / -1.246 | 0.021(4) | 0.051
| $5/2_{1}^{-}$ | NA | -0.734 / -0.892 | NA | -0.014
| $3/2_{1}^{-}$ | NA | -1.060 / -1.287 | NA | -0.026

(a)  (b)

Figure 2: Comparison of theoretical and experimental energy spectra of 46K and 46Ca. Experimental data are taken from Refs. NNDC ; 47Ca . Table 4:  Computed $\log ft$ values for the allowed, FU and FNU $\beta^{-}$ decay from 46K (2-) $\rightarrow$ 46Ca (Jf). The results are obtained by using the quenching factor of q=0.57 for gA. Experimental and theoretically predicted spin-parities are also mentioned. Experimental excitation energies ($E$), $\beta^{-}$ branching ratios and $\log ft$ values are taken from Ref. 46Ca . |  |  |  |  |  $\log ft$
---|---|---|---|---|---
$J^{\pi}_{f}$ (Expt.) |  $J^{\pi}_{f}$ (SM) |  $E$ (keV) | Mode of decay |  $~{}~{}~{}\beta^{-}$ branching | Expt. | Theory
2+ | 2${}^{+}_{1}$ | 1346.00(14) | 1st FNU | 27.4(21) | 6.92(4) | 6.862
0+ | 0${}^{+}_{2}$ | 2421.98(22) | 1st FU | 0.38(1) | 10.27(2) | 10.017
4+ | 4${}^{+}_{1}$ | 2574.53(16) | 1st FU | 1.15(14) | 9.70(6) | 7.898
2+ | 2${}^{+}_{2}$ | 3020.67(14) | 1st FNU | 0.55(20) | 8.01(16) | 8.772
3- | 3${}^{-}_{1}$ | 3610.51(15) | Allowed | 0.28(14) | 8.04(22) | 6.248
2+ | 2${}^{+}_{3}$ | 3638.38(15) | 1st FNU | 0.22(2) | 8.13(4) | 7.370
4+ | 4${}^{+}_{2}$ | 3857.51(17) | 1st FU |  $<$0.003 |  $>$11.5 | 9.361
(2+,3) | 2${}^{+}_{4}\left.\middle/\,\text{3}^{-}_{2}\right.$ | 3984.07(16) | 1st FNU(2${}^{+}_{4})\left.\middle/\,\text{Allowed}(3^{-}_{2})\right.$ | 0.47(3) | 7.63(3) |  $\left.9.080\middle/6.009\right.$
3+ | 3${}^{+}_{1}$ | 4257.26(15) | 1st FNU | 0.092(9) | 8.20(5) | 10.512
2+ | 2${}^{+}_{5}\left.\middle/\,\text{2}^{+}_{4}\right.$ | 4385.91(15) | 1st FNU | 4.39(16) | 6.44(2) |  $\left.9.132\middle/9.137\right.$
3- | 3${}^{-}_{2}\left.\middle/\,\text{3}^{-}_{3}\right.$ | 4404.48(15) | Allowed | 5.09(22) | 6.37(2) |  $\left.6.009\middle/6.040\right.$
2+ | 2${}^{+}_{6}\left.\middle/\,\text{2}^{+}_{5}\right.$ | 4428.02(14) | 1st FNU | 0.017(14) | 8.80(40) |  $\left.9.941\middle/9.133\right.$
(3,4+) | 3${}^{-}_{3}\left.\middle/\,\text{3}^{-}_{4}\right.$ | 4432.05(16) | Allowed | 2.59(7) | 6.65(2) |  $\left.6.040\middle/7.779\right.$
2+ | 2${}^{+}_{7}\left.\middle/\,\text{2}^{+}_{6}\right.$ | 4487.05(16) | 1st FNU | 0.46(2) | 7.36(2) |  $\left.9.173\middle/9.947\right.$

(a)  (b)

Figure 3: Comparison of theoretical and experimental energy spectra of 46K and 46Ca. Experimental data are taken from Refs. NNDC ; 47Ca . Table 5: Computed $\log ft$ values for the allowed, FU and FNU $\beta^{-}$ decay from 47K (1/2+) $\rightarrow$ 47Ca (Jf). The results are obtained by using the quenching factor of q=0.57 for gA. Experimental and theoretically predicted spin-parities are also mentioned. Experimental excitation energies ($E$), $\beta^{-}$ branching ratios and $\log ft$ values are taken from Ref. 47Ca .  |  |  |  |  |  $\log ft$
---|---|---|---|---|---
$J^{\pi}_{f}$ (Expt.) |  $J^{\pi}_{f}$ (SM) |  $E$ (keV) | Mode of Decay |  $\beta^{-}$ branching | Expt. | Theory
$3/2^{-}$ | $3/2^{-}_{1}$ | 2013.4(3) | 1st FNU | $<2.9$ |  $>$6.5 | 7.956
$3/2^{+}$ | $3/2^{+}_{1}$ | 2578.1(3) | Allowed | 18.4(3) | 5.457(8) | 6.183
$1/2^{+}$ | $1/2^{+}_{1}$ | 2599.2(3) | Allowed | 80(2) | 4.807(9) | 4.879
$(3/2)^{-}$ | $3/2^{-}_{2}$ | 2849.9(3) | 1st FNU | 0.013(1) | 8.46(4) | 8.201
$(1/2^{-},3/2^{-})$ | 1/2${}^{-}_{1}$ | 2874.7(3) | 1st FNU | 0.015(3) | 8.40(9) | 8.487
$5/2^{-}$ | $5/2^{-}_{1}$ | 3266.3(3) | 1st FU | 0.0038(2) | 10.28(2) | 8.485
$(5/2^{-})$ | $5/2^{-}_{2}$ | 3889.0(3) | 1st FU | $<0.0017$ |  $>$10.1 | 10.045
$(3/2^{-},5/2^{-})$ | 3/2${}^{-}_{3}$ | 3951.2(3) | 1st FNU | 0.010(5) | 7.9(2) or 9.3(2) | 9.896
$(1/2,3/2)$ | 1/2${}^{-}_{2}$ | 4453.1(4) | 1st FNU | 0.036(5) | 6.99(6) | 8.159
| 1/2${}^{+}_{2}$ | 4453.1(4) | Allowed | 0.036(5) | 6.99(6) | 5.811
$3/2^{+}$ | $3/2^{+}_{2}$ | 4528.3(3) | Allowed | 0.38(1) | 5.91(1) | 5.454
$(5/2^{+})$ | $5/2^{+}_{1}$ | 4606.1(3) | 2nd FNU | $<0.0027$ |  $>$8.0 | 11.751
$3/2^{+}$ | $3/2^{+}_{3}$ | 4989.5(3) | Allowed | 0.246(6) | 5.64(1) | 5.233
$3/2^{+}$ | $3/2^{+}_{4}$ | 5305.9(3) | Allowed | 0.291(7) | 5.20(1) | 5.534
$3/2^{+}$ | $3/2^{+}_{5}$ | 5456.1(3) | Allowed | 0.38(1) | 4.87(1) | 5.668
$3/2^{+}$ | $3/2^{+}_{6}$ | 6064.2(4) | Allowed | 0.0049(2) | 5.56(2) | 5.382

In Table 4, we have presented the calculated $\log ft$ values for $\beta^{-}$
decay from 2- of 46K to various excited states of 46Ca. There are two possible
spin-parities: 2${}^{+}_{4}$ and 3${}^{-}_{2}$, corresponding to the state at
3984.07(16) keV. Accordingly, the spin-parity and $\log ft$ values of the
subsequent states are determined. For instance, if the state at 3984.07(16)
keV is 2${}^{+}_{4}$, then the spin-parity and $\log ft$ value of the state at
4385.91(15) keV are 2${}^{+}_{5}$ and 9.132, respectively. On the other hand,
if the state at 3984.07(16) keV is considered to be 3${}^{-}_{2}$, then, the
4385.91(15) keV state would be 2${}^{+}_{4}$ with a $\log ft$ value of 9.137.
We have also listed the experimentally obtained spin-parity $J_{f}^{\pi}$,
excitation energy, branching ratio and $\log ft$ values of final states.
Further, we have also presented the corresponding modes of decay. Transition
from initial state to final state could be either allowed or FU or FNU, based
on the selection rules of $\beta$ decay.

In computing the $\log ft$, we have used the effective value of gA = 0.72.
Experimental values are used for the $\beta$ decay (ground-state-to-ground-
state) Q values for 46K and 47K as well as for the excitation energies in 46Ca
and 47Ca for the evaluations of the log $ft$ values. The Q values of 7.716 and
6.632 MeV are used for 46K and 47K, respectively.

The g.s. spin-parity of 46Ca is well reproduced with the SDPF-MU interaction.
The energy difference between experimental and theoretical $2_{1}^{+}$ is
0.513 MeV, as can be seen in Fig. 2b and the calculated $\log ft$ value is
6.862 which is very close to the experimental value of 6.92(4). Transition
from $2^{-}$ of 46K to $0^{+}_{2}$ of 46Ca is of the 1st FU type for which
calculated $\log ft$ value is 10.017, which agrees well with the experimental
value of 10.27(2). The difference between the theoretically and experimentally
obtained energies of $3^{-}_{1}$ is 0.022 MeV which is very small. Spin-parity
corresponding to the state with energy 3.984 MeV is unconfirmed. For this
state, experimentally, it is found that the spin-parity based on the $\log ft$
value is constrained to be either $2^{+}$ or $3^{+}$ or $3^{-}$. If we assume
this state to be $2^{+}$, then the $\log ft$ value corresponding to the
$\beta^{-}$ decay to this state is 9.080. By assuming this state to be
$3^{+}$, we obtain $\log ft$ = 10.457. If allowed transition is assumed then
this state would be $3^{-}$ along with $\log ft$ value of 6.009. The energy
difference between calculated and experimental value for this state is 47 keV,
as shown in Fig. 2b. Thus, shell-model calculation with the SDPF-MU
interaction establish this state to be either $3^{-}$ or $2^{+}$. Spin-parity
of the state at 4.432 MeV is likewise unconfirmed from experiments. So, we
have taken all possibilities into account and calculated corresponding $\log
ft$ values. Theoretically, we predict this state to be $3^{-}$ on the basis of
$\log ft$ value.

In Table 5, we have reported calculated $\log ft$ results for the allowed, FU
and FNU $\beta^{-}$ decay from 1/2+ of 47K to various excited states of 47Ca.
The $7/2^{-}$ state is the g.s. of 47Ca which is correctly reproduced with
SDPF-MU interaction. The order of experimental $3/2^{-}_{1}$ and $3/2^{+}_{1}$
states is reversed in theoretical prediction by shell-model, as shown in Fig.
3b. The calculated $\log ft$ for the $3/2^{-}_{2}$ state is 8.201 which is
very near to experimental $\log ft$ of 8.46(4) for $(3/2)^{-}$ state. The
experimental $(3/2)^{-}$ state is at 2.850 MeV while theoretical value of
energy for this state is 3.030 MeV which means shell-model confirm this state
to be $3/2^{-}$. Next unconfirmed state is at 2.875 MeV, which is constrained
to be $(1/2^{-},3/2^{-})$, experimentally. If this state is considered to be
$3/2^{-}$ then calculated $\log ft$ is 9.846, while for $1/2^{-}$, $\log ft$
is 8.487 which is very close to experimental $\log ft$ i.e. 8.40(9). Also
excitation energy for $1/2^{-}$ is 3.237 MeV which is quite close to the
experimental value. So, we predict a state at 2.875 MeV to be $1/2^{-}$.
Experimental first $5/2^{-}$ with $\log ft$ value of 10.28(2) can be assigned
as shell-model $5/2_{1}^{-}$ state. We also get a confirmation between theory
and experiment for $(5/2)^{-}$ state at 3.889 MeV. For this state calculated
$\log ft$ is 10.045, while experimental value is $>$ 10.1.

For a state at 3.951 MeV in 47Ca, there are two possibilities, see in Fig. 3b.
It could be either $3/2^{-}$ or $5/2^{-}$. If this state is assumed as
$5/2^{-}$ then there occurs 1st FU transition and calculated $\log ft$ is
13.170 which is far from experimental $\log ft$ value of 9.3(2). If we
consider this state as $3/2^{-}$ then transition is to be 1st FNU type and
calculated $\log ft$ value is 9.896, while the experimental value is 7.9(2).
The state is more likely to be $3/2^{-}$. There are four possibilities of
spin-parity for a state at 4.453 MeV excitation energy. This could be either
$1/2^{-}$, $1/2^{+}$, $3/2^{-}$ or $3/2^{+}$. The calculated $\log ft$ values
are 8.159, 5.811, 10.815 and 5.454 for $1/2^{-}$, $1/2^{+}$, $3/2^{-}$ and
$3/2^{+}$ states, respectively. Also calculated excitation energies with SDPF-
MU interaction for these states are 4.58, 6.202, 6.725 and 5.290 MeV,
respectively. So, the $\log ft$ values favour this state as either $1/2^{-}$
or $1/2^{+}$. The experimental $(5/2)^{+}$ state is at 4.606 MeV, while the
calculated value of $\log ft$ for this state is 11.751 (experimental one is
$>$ 8.0) and calculated energy is 4.404 MeV. Thus, shell-model results support
this states as $5/2^{+}$.

Figure 4: Comparison of excitation energies of low-lying excited states in
46Ca obtained from SDPF-MU interaction with previously obtained results in
$sd_{3/2}+f_{7/2}p_{3/2}$, $pf$ and $pf$g9/2 valence spaces using NN+3N forces
46Ca ,PRC90 . Experimental data are also shown in first column for comparison.

In Figs. 4-5, we have shown a comparison of the energy levels obtained from
our present calculations with those of previous works, where the level scheme
was obtained in $sd_{3/2}+f_{7/2}p_{3/2}$ 46Ca , $pf$ 46Ca  and $pf$g9/2 PRC90
valence spaces using NN+3N interactions. As can be seen from Fig. 4, although
the lower excited states are well reproduced by previous calculation in
$sd_{3/2}+f_{7/2}p_{3/2}$ model space 46Ca , the higher states, however, are
better reproduced by SDPF-MU interaction. For instance, our shell-model
results with SDPF-MU interaction reported in this paper for $3^{-}$ state of
46Ca is in excellent agreement with the experimental value. Negative parity
states were not obtained below 5 MeV for $pf$g9/2 shell-model space.

For 47Ca, previous studies were performed in $pf$ and $pf$g9/2 valence spaces
using NN+3N forces PRC90 , where only negative parity states were calculated.
In the present work, we have calculated both positive and negative parity
states. From Fig. 5, it is seen, for instance, that previous work with $pf$
and $pf$g9/2 valence spaces give $\sim$ 1.8 MeV and $\sim$ 1 MeV energy
difference between theory and experiment for $3/2^{-}_{1}$ state in 47Ca,
respectively, while the difference is reduced to $\sim$ 600 keV in this work.
This indicates that the $sdpf$-model space is a suitable valence space for
47Ca. For other states also, we obtain improved results with SDPF-MU
interaction as compared to the previous study.

Figure 5: Comparison of excitation energies of low-lying excited states in
47Ca obtained from SDPF-MU interaction with previously obtained results in
$pf$ and $pf$g9/2 valence spaces using NN+3N forces PRC90 . Experimental data
are also shown in first column for comparison.

We have analyzed configuration for low-lying positive and negative parity
states of 46,47Ca corresponding to 1p-1h excitation. For natural parity
states, there is no excitation from $sd$-shell to $pf$-shell. For 47Ca,
$7/2^{-}_{1}$ and $3/2^{-}_{1}$ states are generated from the configurations
$\nu f_{7/2}^{-1}$ with probability of 94.41% and $\nu p_{3/2}^{1}$ with
probability of 92.22%, respectively. These states correspond to single
particle states. In the case of unnatural parity states, either one proton or
one neutron is excited into $pf$-shell. It is expected that for calcium
isotopes, which have a proton shell closure at $Z=20$, protons might not be
excited from $sd$-shell to $pf$-shell. But for these states, we have seen that
not only neutrons but also protons are excited from $sd$-shell to $pf$-shell.
In fact, major contribution comes from proton excitation despite $Z=20$ is
magic number. The $3^{-}_{1}$ state in 46Ca originated from the configuration
$\pi(d_{5/2}^{6}s_{1/2}^{1}d_{3/2}^{4}f_{7/2}^{1})$ $\otimes$
$\nu(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{4}f_{7/2}^{6})$, which has 22.45%
contribution. Dominant configuration for 5/2${}^{+}_{1}$ in 47Ca is
$\pi(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{3}f_{7/2}^{1})$ $\otimes$
$\nu(d_{5/2}^{6}s_{1/2}^{2}d_{3/2}^{4}f_{7/2}^{7})$ with probability of
24.06%. So, protons are not treated as inert in 46,47Ca. This effect is also
seen in the case of 2p-2h excitations. Even for natural parity states, protons
are excited from $sd$-shell to $pf$-shell.

##  IV Conclusion

In the present work, we have performed a systematic nuclear shell-model
calculation for recently available experimental data, measured with the
GRIFFIN spectrometer at TRIUMF-ISAC facility from the $\beta^{-}$ decay of 46K
and 47K, to reproduce the level schemes of 46Ca and 47Ca. We have also
calculated low-lying energy spectra for the parent nuclei 46K and 47K. In our
shell-model calculations, the SDPF-MU interaction has been used to calculate
spectroscopic properties of 46,47K and 46,47Ca. For the first time, we have
calculated the $\log ft$ values corresponding to new data for the $\beta^{-}$
decay from 46,47K to 46,47Ca. Present shell-model study helps us to identify
spin-parity of unconfirmed states for which the $\log ft$ values are reported
in these experimental works. Based on calculated $\log ft$ values, we have
predicted a state with 3.984 MeV to be either $3^{-}$ or $2^{+}$ with $\log
ft$ value 6.009 and 9.080 for 46Ca. A tentative state at 4.432 MeV is also
assigned to be $3^{-}$ in 46Ca. In the case of 47Ca, we have also given spin
confirmation to the tentative states at 2.850, 3.889 and 4.606 MeV. The states
at 2.875, 3.951 and 4.453 MeV could be $1/2^{-}$, $3/2^{-}$ and $1/2^{-}$ or
$1/2^{+}$ in 47Ca, respectively. The agreement of our calculated $\log ft$
values with the experimental data is rather limited. In some cases, the
calculated $\log ft$ values are close to the experimental ones; the difference
in $\log ft$ is within 0.8 for about half of the states measured in 46Ca and
47Ca. The difference, however, amounts to be 1.1-2.7 for other cases. Present
nuclear shell-model study have added more information to the recent
experimental works 46Ca ; 47Ca .

## Acknowledgement

P.C. acknowledges financial support from

MHRD (Government of India) for her Ph.D. thesis work. P.C.S. acknowledges a
research grant from the Science and Engineering Research Board (SERB) of
India, CRG/2019/000556. T. S. acknowledges JSPS KAKENHI Grant JP19K03855. We
would like to thank J.L. Pore for useful discussions during this work.

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# Study of the $B_{c}$ ${\to}$ $B_{s}{\pi}$ decay with the perturbative QCD
approach

SUN Junfeng  Institute of Particle and Nuclear Physics, Henan Normal
University, Xinxiang 453007, China  YANG Yueling<EMAIL_ADDRESS>Institute
of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007,
China  LU Gongru  Institute of Particle and Nuclear Physics, Henan Normal
University, Xinxiang 453007, China

###### Abstract

The $B_{c}$ ${\to}$ $B_{s}{\pi}$ decay is studied with the perturbative QCD
approach. Three types of wave functions for $B_{s}$ meson are considered. The
transition form factor $F_{0}^{B_{c}{\to}B_{s}}(0)$ and the branching ratio
${\cal B}r(B_{c}{\to}B_{s}{\pi})$ are sensitive to the model of the $B_{s}$
meson wave functions. With appropriate inputs, our estimate on ${\cal
B}r(B_{c}{\to}B_{s}{\pi})$ is comparable with the recent LHCb measurement. A
clear signal of $B_{c}$ ${\to}$ $B_{s}{\pi}$ decay should be easily observed
at the Large Hadron Collider.

Keywords: $B_{c}$ meson, weak decay, the perturbative QCD approach, branching
ratio.

###### pacs:

12.38.Bx 12.39.St 13.25.Hw

##  I introduction

The $B_{c}$ meson, the heaviest of the ground pseudoscalar meson with explicit
both bottom and charm quantum numbers, was observed for the first time via the
decay $B_{c}$ ${\to}$ $J/{\psi}{\ell}{\nu}$ in $1.8$ TeV $p{\bar{p}}$
collisions using the CDF detector at the Fermilab Tevatron in 1998 CDF.1998 .
The mass is currently measured at the ${\cal O}(10^{-4})$ level prl.100.182002
; prl.109.232001 . However, there is about $10\%$ uncertainty in the present
$B_{c}$ lifetime measurement prl.102.092001 ; prl.97.012002 .

The $B_{c}$ meson, laying below $BD$ threshold, is stable for both the strong
and electromagnetic annihilation interactions. It can decay only via the weak
interaction. The decay modes can be divided into three classes (taking
$B_{c}^{+}$ as an illustration) zpc.51.549 ; prd.49.3399 : (1) the $c$ quark
decays while the $\bar{b}$ quark as a spectator, that is, $c$ ${\to}$ $W^{+}$
$+$ $s$ (or $d$); (2) the $\bar{b}$ quark decays while the $c$ quark as a
spectator, that is, $\bar{b}$ ${\to}$ $W^{+}$ $+$ $\bar{c}$ (or $\bar{u}$);
(3) the annihilation channel, that is, $c$ $+$ $\bar{b}$ ${\to}$ $W^{+}$;
where the virtual $W^{+}$ boson materializes either into a pair of leptons
${\ell}^{+}{\nu}$, or into a pair of quarks which then hadronizes. Both the
$c$ and $\bar{b}$ quarks in $B_{c}^{+}$ meson can decay individually,
resulting in that the $B_{c}$ lifetime is about three times less than other
ground pseudoscalar $b$-flavored mesons, that is, ${\tau}_{B_{c}}$ $<$
$\frac{1}{3}{\tau}_{B_{u,d,s}}$ pdg2012 . Enhanced by the hierarchy of the
Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix elements ${|}V_{cs}{|}$
$>$ ${|}V_{cb}{|}$, the class (1) decay contributes ${\sim}$ $70\%$ to the
$B_{c}$ decay width prd.53.4991 , which shows the critical contribution of the
charm quark to the $B_{c}$ lifetime. In this paper, we study the $B_{c}$
${\to}$ $B_{s}{\pi}$ decay which belongs to class (1). Our motivations are
listed below.

Firstly, from an experimental consideration. It is estimated pan.67.1559  that
with large production cross section ${\sigma}(B_{c})$ ${\sim}$ $1{\mu}{\rm b}$
(taking into account the cascade decays of the excited states $B_{c}^{\ast}$
below $BD$ threshold) and high luminosity of ${\cal L}$ $=$ $10^{34}{\rm
cm}^{-2}{\rm s}^{-1}$, one could expect $5{\times}10^{10}$ of $B_{c}$ meson
per year at LHC. In addition, LHCb, a dedicated $b$-physics precision
experiment at LHC, allows proper time resolution of ${\sim}$ $50$ fs 1105.5330
which is one order of magnitude shorter than lifetime of the $B_{c,s}$ meson.
This property facilitates the separation between primary and decay vertices.
The rate for the charged pion identification is larger than $90\%$ 1105.5330 .
There seems to be a golden opportunity and a real possibility to investigate
the $B_{c}$ weak decay. The reconstructed events of the $B_{c}$ ${\to}$
$B_{s}{\pi}$ decay is estimated to be ${\sim}$ $10^{3}$ per year pan.67.1559 .
Recently, the LHCb collaboration has presented research that the $B_{c}$
${\to}$ $B_{s}{\pi}$ decay is observed with significance in excess of five
standard deviations based on 3 fb-1 data sample 1308.4544 .

Secondly, from phenomenological considerations. With abundant measurements, we
can carefully test various theoretical models. Accordingly, accurate
theoretical prediction of $B_{c}$ decay is essential, although the $B_{c}$
weak decay is complicated because of strong interaction effects. Based on an
expansion in ${\alpha}_{s}/{\pi}$ and ${\Lambda}_{\rm QCD}/m_{Q}$ (where
${\alpha}_{s}$, ${\Lambda}_{\rm QCD}$ and $m_{Q}$ are the strong coupling
constant, QCD characteristic scale and mass of heavy quark $Q$, respectively),
several attractive methods have been proposed to evaluate the hadronic matrix
elements, such as the QCD factorization qcdf , perturbative QCD method (pQCD)
pqcd , and soft and collinear effective theory scet . These approaches have
been applied to $B_{u,d}$ hadronic decays with reasonable explanation for
measurements. Whether or not these approach is suitable for $B_{c}$ ${\to}$
$B_{s}{\pi}$ decay needs to be examined. Herein we investigate $B_{c}$ ${\to}$
$B_{s}{\pi}$ decay with the pQCD approach to give an estimate of the
associated branching ratio.

##  II effective Hamiltonian

For hadronic decays, typically the effective Hamiltonian calculations with
operator product expansion scheme is used. The effective Hamiltonian for
$B_{c}$ ${\to}$ $B_{s}{\pi}$ decay can be written as rmp68p1125 :

${\cal H}_{\rm eff}=\frac{G_{\rm
F}}{\sqrt{2}}V_{ud}V_{cs}^{\ast}\Big{\\{}C_{1}({\mu})(\bar{s}_{\alpha}c_{\alpha})_{V-A}(\bar{u}_{\beta}d_{\beta})_{V-A}+C_{2}({\mu})(\bar{s}_{\alpha}c_{\beta})_{V-A}(\bar{u}_{\alpha}d_{\beta})_{V-A}\Big{\\}}+{\rm
H.c.},$ (1)

where $G_{\rm F}$ is Fermi coupling constant. The CKM factor
$V_{ud}V_{cs}^{\ast}$ ${\sim}$ ${\cal O}(1)$. ${\alpha}$ and ${\beta}$ are
color indices. $(\bar{q}q^{\prime})_{V-A}$ $=$
$\bar{q}{\gamma}_{\mu}(1-{\gamma}_{5})q^{\prime}$. The Wilson coefficients
$C_{1,2}({\mu})$, incorporating the physics contributions from heavy particles
such as $W$ and $Z$ bosons, top quark, and scales higher than ${\mu}$, have
been calculated to the next-to-leading order with perturbation theory
rmp68p1125 . There most difficult problem remaining in calculation is how to
evaluate the hadronic matrix elements of the local operators properly and
accurately.

##  III Hadronic matrix elements

Using Brodsky-Lepage approach prd22p2157 , the hadronic matrix element is
commonly written as a convolution of hard-scattering kernels and hadron wave
functions (WFs). It is shown that qcdf  due to appearance of the endpoint
divergences, the QCDF formalisms based on the collinear approximation fail to
give the contributions of the spectator and annihilation interactions
satisfactorily. The pQCD approach advocates that pqcd  the endpoint
singularity in collinear approximation could be smeared by retaining the
transverse momentum $k_{T}$ of quarks and by introducing the Sudakov factor
$e^{-S}$. Using the pQCD formalisms, one decay amplitude is factorized into
three factors: the “harder” effects incorporated into the Wilson coefficients
$C$, the heavy quark decay subamplitude $H$, and the universal hadron WFs
${\Phi}$, such that:

${\int}{\bf d}k\,C(t)H(k,t){\Phi}(k)e^{-S}$ (2)

where $k$ and $t$ are the corresponding kinematic variables and characteristic
scale, respectively.

###  III.1 Kinematic variables

In the terms of the light cone coordinate, the momenta of the valence quarks
and hadrons in the rest frame of the $B_{c}$ meson are defined as:

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\frac{m_{1}}{\sqrt{2}}(1,1,\vec{0}_{T}),$ (3) $\displaystyle
p_{2}$ $\displaystyle=$
$\displaystyle\frac{m_{1}}{\sqrt{2}}(r_{B_{s}}^{2},1,\vec{0}_{T}),$ (4)
$\displaystyle p_{3}$ $\displaystyle=$
$\displaystyle\frac{m_{1}}{\sqrt{2}}(1-r_{B_{s}}^{2},0,\vec{0}_{T}),$ (5)
$\displaystyle k_{i}$ $\displaystyle=$ $\displaystyle
x_{i}p_{i}+(0,0,\vec{k}_{iT}),$ (6)

where the subscript $i$ $=$ $1$, $2$, $3$ refer to $B_{c}$, $B_{s}$, ${\pi}$
meson, respectively. Variables $k_{i}$, $\vec{k}_{iT}$, $x_{i}$ are the four-
dimensional momentum, transverse momentum and longitudinal momentum fraction
of light quark, respectively. $r_{B_{s}}$ $=$ $m_{B_{s}}/m_{B_{c}}$.

###  III.2 Wave functions

Hadron wave functions are basic input in Eq.(2). Adopting the notation in
npb592p3 ; plb523p111 ; npb625p239 ; prd55p272 ; prd65p014007 ; prd74p014027 ;
epjc28p515 ; jhep9901p010 ; prd.83.094031 ; jhep0804p061 , the two-valence-
particle WFs for double-light pion, heavy-light $B_{s}$ meson and double-heavy
$B_{c}$ meson are decomposed into :

$\displaystyle{\langle}{\pi}(p_{3}){|}\bar{u}_{\alpha}(z)d_{\beta}(0){|}0{\rangle}$
$\displaystyle=$ $\displaystyle\frac{-i}{\sqrt{2N_{c}}}{\int}{\bf
d}^{4}k_{3}\,e^{+ik_{3}{\cdot}z}\Big{\\{}{\gamma}_{5}\Big{[}\\!\\!\not{\\!p}_{3}{\phi}_{\pi}^{a}\\!+\\!{\mu}_{\pi}{\phi}_{\pi}^{p}\\!+\\!{\mu}_{\pi}\Big{(}\\!\\!\not{\\!n}_{+}\\!\\!\not{\\!n}_{-}\\!\\!-\\!1\Big{)}{\phi}_{\pi}^{t}\Big{]}\Big{\\}}_{{\beta}{\alpha}},$
(7)
$\displaystyle{\langle}B_{s}(p_{2}){|}\bar{s}_{\alpha}(z)b_{\beta}(0){|}0{\rangle}$
$\displaystyle=$ $\displaystyle\frac{-i}{\sqrt{2N_{c}}}{\int}{\bf
d}^{4}k_{2}\,e^{+ik_{2}{\cdot}z}\Big{\\{}{\gamma}_{5}\Big{(}\\!\\!\not{\\!p}_{2}\\!+\\!m_{B_{s}}\\!\Big{)}\Big{[}\frac{\not{\\!n}_{+}}{\sqrt{2}}{\phi}_{B_{s}}^{-}\\!+\frac{\not{\\!n}_{-}}{\sqrt{2}}{\phi}_{B_{s}}^{+}\Big{]}\Big{\\}}_{{\beta}{\alpha}},$
(8)
$\displaystyle{\langle}0{|}\bar{b}_{\alpha}(0)c_{\beta}(z){|}B_{c}(p_{1}){\rangle}$
$\displaystyle=$ $\displaystyle\frac{-i}{\sqrt{2N_{c}}}{\int}{\bf
d}^{4}k_{1}\,e^{-ik_{1}{\cdot}z}\Big{\\{}\Big{[}\frac{\not{\\!n}_{+}}{\sqrt{2}}{\phi}_{B_{c}}^{+}\\!+\frac{\not{\\!n}_{-}}{\sqrt{2}}{\phi}_{B_{c}}^{-}\Big{]}\Big{(}\\!\\!\not{\\!p}_{1}\\!+\\!m_{B_{c}}\\!\Big{)}{\gamma}_{5}\Big{\\}}_{{\beta}{\alpha}},$
(9)

where $N_{c}$ $=$ $3$ is the color number. $n_{-}$ and $n_{+}$ are the minus
and plus null vectors, respectively. $n_{+}{\cdot}n_{-}$ $=$ $1$.

For the heavy-light $B_{s}$ meson, there are two scalar WFs
${\phi}^{+}_{B_{s}}$ and ${\phi}^{-}_{B_{s}}$. The equation of motion for WFs
${\phi}_{B_{s}}^{\pm}$ is npb592p3 ; plb523p111

${\phi}_{B_{s}}^{+}(x)+x\,{\phi}_{B_{s}}^{-{\prime}}(x)=0.$ (10)

The relationship is helpful in constraining models for the $B_{s}$ WFs, which
leads to ${\phi}_{B_{s}}^{+}(x)$ vanished at the endpoint and
${\phi}_{B_{s}}^{-}(x)$ $=$ ${\cal O}(1)$ for $x$ ${\to}$ $0$ npb625p239 .
Here we will investigate three candidates of WFs for $B_{s}$ meson.

The first candidate is the exponential type (GN) suggested in prd55p272 ,

$\displaystyle{\phi}_{B_{s}}^{+}(x,b)$ $\displaystyle=$
$\displaystyle\frac{f_{B_{s}}}{2\sqrt{2N_{c}}}N_{\rm
GN}^{+}\,x{\exp}\Big{[}-\frac{x\,m_{B_{s}}}{{\omega}_{\rm
GN}}\Big{]}\frac{1}{1+(b\;{\omega}_{\rm GN})^{2}},$ (11)
$\displaystyle{\phi}_{B_{s}}^{-}(x,b)$ $\displaystyle=$
$\displaystyle\frac{f_{B_{s}}}{2\sqrt{2N_{c}}}N_{\rm
GN}^{-}{\exp}\Big{[}-\frac{x\,m_{B_{s}}}{{\omega}_{\rm
GN}}\Big{]}\frac{1}{1+(b\;{\omega}_{\rm GN})^{2}}.$ (12)

The second candidate is the Gaussian type (KLS) proposed in prd65p014007 ;
prd74p014027  whereby

$\displaystyle{\phi}_{B_{s}}^{+}(x,b)$ $\displaystyle=$
$\displaystyle\frac{f_{B_{s}}}{2\sqrt{2N_{c}}}N_{\rm
KLS}^{+}x^{2}\bar{x}^{2}\,{\exp}\Big{[}-\frac{1}{2}\Big{(}\frac{x\,m_{B_{s}}}{{\omega}_{\rm
KLS}}\Big{)}^{2}-\frac{1}{2}{\omega}_{\rm KLS}^{2}b^{2}\Big{]},$ (13)
$\displaystyle{\phi}_{B_{s}}^{-}(x,b)$ $\displaystyle=$
$\displaystyle\frac{f_{B_{s}}}{2\sqrt{2N_{c}}}N_{\rm
KLS}^{-}{\exp}\Big{[}-\frac{1}{2}{\omega}_{\rm
KLS}^{2}b^{2}\Big{]}\Big{\\{}{\exp}\Big{[}-\frac{1}{2}\Big{(}\frac{x\,m_{B_{s}}}{{\omega}_{\rm
KLS}}\Big{)}^{2}\Big{]}\Big{(}m_{B_{s}}^{2}\bar{x}^{2}+2{\omega}_{\rm
KLS}^{2}\Big{)}$ (14)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\sqrt{2{\pi}}m_{B_{s}}{\omega}_{\rm
KLS}{\rm Erf}\Big{(}\frac{x\,m_{B_{s}}}{\sqrt{2}{\omega}_{\rm
KLS}}\Big{)}+C_{\rm KLS}\Big{\\}},$

where $\bar{x}$ $=$ $1$ $-$ $x$, and the constant $C_{\rm KLS}$ is chosen so
that ${\phi}_{B_{s}}^{-}(1,b)$ $=$ $0$.

The third candidate is the KKQT type derived form the QCD equations of motion
and heavy-quark symmetry constraint epjc28p515 ; plb523p111 ,

$\displaystyle{\phi}_{B_{s}}^{+}(x,b)$ $\displaystyle=$
$\displaystyle\frac{f_{B_{s}}}{2\sqrt{2N_{c}}}\frac{2x}{{\omega}_{\rm
KKQT}^{2}}{\theta}(y)J_{0}\Big{(}m_{B_{s}}b\sqrt{xy}\Big{)},$ (15)
$\displaystyle{\phi}_{B_{s}}^{-}(x,b)$ $\displaystyle=$
$\displaystyle\frac{f_{B_{s}}}{2\sqrt{2N_{c}}}\frac{2y}{{\omega}_{\rm
KKQT}^{2}}{\theta}(y)J_{0}\Big{(}m_{B_{s}}b\sqrt{xy}\Big{)},$ (16)

where $y$ $=$ ${\omega}_{\rm KKQT}$ $-$ $x$.

In the above equations Eq.(11—16), $b$ denotes the conjugate variables of the
transverse momentum of $s$ quark in $B_{s}$ meson. There is only one parameter
${\omega}$ for each kind of WFs candidate. The normalization constants
$N^{\pm}$ is related to the decay constant $f_{B_{s}}$ through the relation :

${\int}_{0}^{1}{\phi}_{B_{s}}^{\pm}(x,0){\bf
d}x=\frac{f_{B_{s}}}{2\sqrt{2N_{c}}}.$ (17)

For the double-light pion, the expression of ${\phi}_{\pi}^{a,p,t}$ can be
found in prd65p014007 ; prd74p014027 ; epjc28p515 ; jhep9901p010 ;
prd.83.094031 . For the double-heavy $B_{c}$ meson, it can be described
approximatively by nonrelativistic dynamics. At tree level and in leading
order of the expansion in the relative velocities, $b$ and $c$ quarks in the
$B_{c}$ meson just share the total momentum according to their masses
jhep0804p061 ,

${\phi}_{{B}_{c}}^{\pm}(x)=\frac{f_{B_{c}}}{2\sqrt{2N_{c}}}\delta(x-r_{c}),$
(18)

where $f_{B_{c}}$ is the decay constant, and $r_{c}$ $=$ $m_{c}/m_{B_{c}}$.

##  IV $B_{c}$ ${\to}$ $B_{s}$ transition form factors

The $B_{c}$ ${\to}$ $B_{s}$ transition form factors are defined as follows
zpc29p637 :

$\displaystyle{\langle}B_{s}(p_{2}){|}\bar{s}{\gamma}^{\mu}(1-{\gamma}_{5})c{|}B_{c}(p_{1}){\rangle}$
(19) $\displaystyle=$
$\displaystyle\Big{[}(p_{1}+p_{2})^{\mu}-\frac{m_{B_{c}}^{2}-m_{B_{s}}^{2}}{q^{2}}q^{\mu}\Big{]}F_{1}^{B_{c}{\to}B_{s}}(q^{2})$
$\displaystyle+$
$\displaystyle\frac{m_{B_{c}}^{2}-m_{B_{s}}^{2}}{q^{2}}q^{\mu}F_{0}^{B_{c}{\to}B_{s}}(q^{2}),$

where $q$ $=$ $p_{1}$ $-$ $p_{2}$ is the momentum transfer. Usually, the
longitudinal form factor $F_{0}(q^{2})$ is compulsorily equal to the
transverse form factor $F_{1}(q^{2})$ in the largest recoil limit to cancel
singularities appearing at the pole $q^{2}$ $=$ $0$, i.e., $F_{0}(0)$ $=$
$F_{1}(0)$.

Within the pQCD framework, the one-gluon-exchange diagrams contributing to the
$B_{c}$ ${\to}$ $B_{s}$ transition form factors are displayed in Fig.1. It has
been shown that the pQCD approach works ideally in the large recoil region
pqcd ; epjc28p515 ; prd74p014027 ; npb625p239 . The expression of form factors
is :

$\displaystyle F_{0}^{B_{c}{\to}B_{s}}(0)\,=\,F_{1}^{B_{c}{\to}B_{s}}(0)$ (20)
$\displaystyle=$ $\displaystyle 8{\pi}m_{B_{c}}^{2}C_{F}{\int}_{0}^{1}{\bf
d}x_{1}{\int}_{0}^{1}{\bf d}x_{2}{\int}_{0}^{\infty}\\!b_{2}{\bf d}b_{2}$
$\displaystyle{\times}$
$\displaystyle\Big{\\{}E_{a}(t_{a}){\alpha}_{s}(t_{a})H_{a}({\alpha},{\beta}_{a},b_{2})\,r_{B_{s}}{\phi}_{B_{s}}^{+}(x_{2})$
$\displaystyle{\times}\Big{[}{\phi}_{B_{c}}^{+}(x_{1})\Big{(}x_{2}r_{B_{s}}+r_{c}-x_{2}\Big{)}$
$\displaystyle~{}+{\phi}_{B_{c}}^{-}(x_{1})\Big{(}x_{2}r_{B_{s}}+r_{c}-r_{B_{s}}r_{c}\Big{)}\Big{]}$
$\displaystyle+E_{b}(t_{b}){\alpha}_{s}(t_{b})H_{b}({\alpha},{\beta}_{b},b_{2})\,x_{1}{\phi}_{B_{c}}^{-}(x_{1})$
$\displaystyle{\times}\Big{[}{\phi}_{B_{s}}^{-}(x_{2})\Big{(}1-r_{B_{s}}\Big{)}+{\phi}_{B_{s}}^{+}(x_{2})r_{B_{s}}^{2}\Big{]}\Big{\\}},$

where $C_{F}$ $=$ $4/3$ is color factor. $E_{i}(t_{i})$ $=$
$e^{-S_{B_{s}}(t_{i})}$ is the Sudakov factor.

$S_{B_{s}}(t)=s(x_{2}p_{2}^{-},b_{2})+2{\int}_{1/b_{2}}^{t}\frac{{\bf
d}{\mu}}{\mu}{\gamma}_{q},$ (21)

where ${\gamma}_{q}$ $=$ $-{\alpha}_{s}/{\pi}$ is the quark anomalous
dimension. The expression of $s(Q,b)$ can be found in prd52p3958 ; npb642p263
. The hard-scattering kernel function $H_{a,b}$ in Eq.(20) is defined as :

$\displaystyle H_{a}({\alpha},{\beta}_{a},b_{2})$ $\displaystyle=$
$\displaystyle\frac{K_{0}(\sqrt{{\alpha}}b_{2})-K_{0}(\sqrt{{\beta}}b_{2})}{{\beta}_{a}-{\alpha}},$
(22) $\displaystyle H_{b}({\alpha},{\beta}_{b},b_{2})$ $\displaystyle=$
$\displaystyle\frac{K_{0}(\sqrt{{\alpha}}b_{2})}{{\beta}_{b}},$ (23)

where $K_{0}$ and $I_{0}$ are the modified Bessel functions; ${\alpha}$ and
${\beta}$ are the virtualities of internal gluons and quarks, respectively.

$\displaystyle{\alpha}$ $\displaystyle=$ $\displaystyle-
m_{B_{c}}^{2}\Big{[}\bar{x}_{1}^{2}+r_{B_{s}}^{2}\bar{x}_{2}^{2}-\bar{x}_{1}\bar{x}_{2}\Big{(}1+r_{B_{s}}^{2}\Big{)}\Big{]},$
(24) $\displaystyle{\beta}_{a}$ $\displaystyle=$ $\displaystyle-
m_{B_{c}}^{2}\Big{[}1+r_{B_{s}}^{2}\bar{x}_{2}^{2}-\bar{x}_{2}\Big{(}1+r_{B_{s}}^{2}\Big{)}-r_{c}^{2}\Big{]},$
(25) $\displaystyle{\beta}_{b}$ $\displaystyle=$ $\displaystyle-
m_{B_{c}}^{2}\Big{[}r_{B_{s}}^{2}+\bar{x}_{1}^{2}-\bar{x}_{1}\Big{(}1+r_{B_{s}}^{2}\Big{)}\Big{]},$
(26) $\displaystyle t_{i}$ $\displaystyle=$
$\displaystyle{\max}(\sqrt{{|}{\alpha}{|}},\sqrt{{|}{\beta}_{i}{|}},1/b_{2}).$
(27)

##  V decay amplitude

Within the pQCD framework, the Feynman diagrams for $B_{c}$ ${\to}$
$B_{s}{\pi}$ decay are shown in Fig.2, where (a) and (b) are factorizable
topology, (c) and (d) are nonfactorizable topology. After a straightforward
calculation, the decay amplitudes is written as :

${\cal
A}(B_{c}{\to}B_{s}{\pi})=\frac{G_{F}}{\sqrt{2}}V_{ud}V_{cs}^{\ast}\sum\limits_{i=a,b,c,d}{\cal
A}_{i},$ (28)

where the explicit expressions of ${\cal A}_{i}$ are collected in APPENDIX A.
From these expressions, we see that only the twist-2 light-cone distribution
amplitude (LCDA) ${\phi}_{\pi}^{a}$ of pion contributes to nonfactorizable
decay amplitude. The expression of the twist-2 pion LCDA is defined in
Gegenbauer polynomials jhep9901p010  as:

${\phi}_{\pi}^{a}(x,{\mu})=\frac{f_{\pi}}{2\sqrt{2N_{c}}}6x\bar{x}\Big{\\{}1+a_{2}^{\pi}({\mu})C_{2}^{3/2}(x-\bar{x})+a_{4}^{\pi}({\mu})C_{4}^{3/2}(x-\bar{x})\Big{\\}},$
(29)

where the nonperturbative parameter $a_{i}^{\pi}({\mu})$ is the Gegenbauer
moment.

The branching ratio in the $B_{c}$ meson rest frame can be written as:

${\cal
B}r(B_{c}{\to}B_{s}{\pi})=\frac{{\tau}_{B_{c}}}{8{\pi}}\frac{p}{m_{B_{c}}^{2}}{|}{\cal
A}(B_{c}{\to}B_{s}{\pi}){|}^{2},$ (30)

where the common momentum $p$ $=$ $(m_{B_{c}}^{2}-m_{B_{s}}^{2})/2m_{B_{c}}$.

##  VI Numerical results and discussion

The input parameters in our numerical calculation are collected in Table 1. If
not specified explicitly, we shall take their central values as the default
input.

Our results on form factor $F_{0}^{B_{c}{\to}B_{s}}$(0) and branching ratio
${\cal B}r(B_{c}{\to}B_{s}{\pi})$ are listed in Table 2, where the first
uncertainty comes from the mass of charm quark; the second uncertainty comes
from the shape parameter of $B_{s}$ meson WFs, that is, ${\omega}_{\rm GN}$
$=$ $0.65{\pm}0.10$ GeV in Eq.(11-12), ${\omega}_{\rm KLS}$ $=$
$0.75{\pm}0.10$ GeV in Eq.(13-14) and ${\omega}_{\rm KKQT}$ $=$
$0.35{\pm}0.10$ in Eq.(15-16); the third uncertainty comes from the choice of
hard scales $(1{\pm}0.1)t_{i}$ in Eq.(27). We can see that the large
uncertainty come from the $B_{s}$ WFs. In addition, the decay constants
$f_{B_{c}}$ and $f_{B_{s}}$ will bring some $2.4\%$ uncertainty to the form
factor.

For the form factor $F_{0}^{B_{c}{\to}B_{s}}(0)$, our study show that :
Firstly, the interference between Fig.1(a) and (b) is constructive. Compared
with Fig.1(b), the contribution of Fig.1(a) is ${\lesssim}$ $30\%$; Secondly,
both ${\phi}_{B_{c}}^{+}$ and ${\phi}_{B_{c}}^{-}$ contribute to the form
factor, and their interference is constructive. Compared with
${\phi}_{B_{c}}^{-}$, the contribution of ${\phi}_{B_{c}}^{+}$ is ${\lesssim}$
$20\%$; Thirdly, the interference between ${\phi}_{B_{s}}^{+}$ and
${\phi}_{B_{s}}^{-}$ is constructive. Compared with ${\phi}_{B_{s}}^{+}$, the
contribution of ${\phi}_{B_{s}}^{-}$ is $<$ $20\%$ for both GN and KLS type;
Fourthly, the form factor $F_{0}^{B_{c}{\to}B_{s}}(0)$ is sensitive to the
shape parameter of $B_{s}$ WFs; Fifthly, By keeping the parton transverse
momentum $k_{T}$, and employing the Sudakov factors to suppress the long
distance contribution in large $b$ region pqcd , the form factor
$F_{0}^{B_{c}{\to}B_{s}}(0)$ is perturbatively calculable with the pQCD
approach. The contribution to form factor comes completely from
${\alpha}_{s}/{\pi}$ $<$ $0.3$ region with the scale of Eq.(27); Lastly,
considering the uncertainties, our results is reasonable agreement with the
previous results listed in Table 3.

For the branching ratio ${\cal B}r(B_{c}{\to}B_{s}{\pi})$, our study show that
: Firstly, the dominated contribution comes from the factorizable topology
Fig.2(a,b). The interference between nonfactorizable diagrams Fig.2(c) and (d)
is destructive; Secondly, the main uncertainty is from $B_{s}$ WFs.
Considering the uncertainties, our results is basically comparable with the
previous results listed in Table 3, and is also agreement with recent LHCb
estimate ${\cal B}r(B_{c}{\to}B_{s}{\pi})$ ${\sim}$ 10% 1308.4544 .

##  VII Summary

Herein we consider three models (exponential, Gaussian, and KKQT type) of
$B_{s}$ meson WFs, and study the $B_{c}$ ${\to}$ $B_{s}$ transition form
factor and the branching ratio for the $B_{c}$ ${\to}$ $B_{s}{\pi}$ decay with
the pQCD approach. We find that (1) By keeping the parton transverse momentum
$k_{T}$ and employing the Sudakov factors, $F_{0}^{B_{c}{\to}B_{s}}(0)$ and
${\cal B}r(B_{c}{\to}B_{s}{\pi})$ are perturbatively calculable within the
pQCD framework. The contribution to form factor comes completely from
${\alpha}_{s}/{\pi}$ $<$ $0.3$ region. (2) $F_{0}^{B_{c}{\to}B_{s}}(0)$ and
${\cal B}r(B_{c}{\to}B_{s}{\pi})$ are sensitive to the model and the shape
parameter ${\omega}$ of the $B_{s}$ meson wave functions. (3) With appropriate
inputs, our estimates on $F_{0}^{B_{c}{\to}B_{s}}(0)$ and ${\cal
B}r(B_{c}{\to}B_{s}{\pi})$ are comparable with the previous results. The
branching ratio for the $B_{c}$ ${\to}$ $B_{s}{\pi}$ decay is about a few per
cent, which is agreement with the recent LHCb measurement.

With the running of LHC, the more data will be accumulated at LHCb, the more
precise branching ratio for the $B_{c}$ ${\to}$ $B_{s}{\pi}$ decay will be
obtained, and various models for hadron wave function and factorization
treatments for hadron matrix element will be more stringently tested.

## Acknowledgments

This work was supported by National Natural Science Foundation of China under
Grant Nos. 11147008, U1232101 and 11275057). We thank Prof. Hsiangnan Li,
Prof. Caidian lü, Prof. Zhenjun Xiao for their helpful discussion.

##  Appendix A the decay amplitudes

There are four diagrams contributing to the $B_{c}$ ${\to}$ $B_{s}{\pi}$ decay
which are shown in Fig.2. The expression of the decay amplitudes in Eq.(28)
are :

$\displaystyle{\cal A}_{a}$ $\displaystyle=$
$\displaystyle-i8{\pi}C_{F}f_{\pi}m_{B_{c}}^{4}\Big{(}1-r_{B_{s}}^{2}\Big{)}r_{B_{s}}{\int}_{0}^{1}{\bf
d}x_{1}{\bf d}x_{2}{\int}_{0}^{\infty}b_{2}{\bf d}b_{2}$ (31)
$\displaystyle{\times}$ $\displaystyle
E_{a}(t_{a}){\alpha}_{s}(t_{a})H_{a}({\alpha},{\beta}_{a},b_{2})\Big{[}C_{1}(t_{a})+\frac{1}{N_{c}}C_{2}(t_{a})\Big{]}{\phi}_{B_{s}}^{+}(x_{2})$
$\displaystyle{\times}$
$\displaystyle\Big{\\{}{\phi}_{B_{c}}^{+}(x_{1})\Big{[}x_{2}r_{B_{s}}+r_{c}-x_{2}\Big{]}+{\phi}_{B_{c}}^{-}(x_{1})\Big{[}x_{2}r_{B_{s}}+r_{c}-r_{B_{s}}r_{c}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{b}$ $\displaystyle=$
$\displaystyle-i8{\pi}C_{F}f_{\pi}m_{B_{c}}^{4}\Big{(}1-r_{B_{s}}^{2}\Big{)}{\int}_{0}^{1}{\bf
d}x_{1}{\bf d}x_{2}{\int}_{0}^{\infty}b_{2}{\bf d}b_{2}$ (32)
$\displaystyle{\times}$ $\displaystyle
E_{b}(t_{b}){\alpha}_{s}(t_{b})H_{b}({\alpha},{\beta}_{b},b_{2})\Big{[}C_{1}(t_{b})+\frac{1}{N_{c}}C_{2}(t_{b})\Big{]}x_{1}{\phi}_{B_{c}}^{-}(x_{1})$
$\displaystyle{\times}$
$\displaystyle\Big{\\{}{\phi}_{B_{s}}^{-}(x_{2})\Big{[}1-r_{B_{s}}\Big{]}+{\phi}_{B_{s}}^{+}(x_{2})r_{B_{s}}^{2}\Big{\\}},$
$\displaystyle{\cal A}_{c}$ $\displaystyle=$
$\displaystyle\frac{-i32{\pi}C_{F}m_{B_{c}}^{4}}{\sqrt{2N_{c}}}\Big{(}1-r_{B_{s}}^{2}\Big{)}r_{B_{s}}{\int}_{0}^{1}{\bf
d}x_{1}{\bf d}x_{2}{\bf d}x_{3}{\int}_{0}^{\infty}b_{2}{\bf d}b_{2}b_{3}{\bf
d}b_{3}$ (33) $\displaystyle{\times}$ $\displaystyle
E_{c}(t_{c}){\alpha}_{s}(t_{c})H_{c}({\alpha},{\beta}_{c},b_{2},b_{3})C_{2}(t_{c}){\phi}_{B_{s}}^{+}(x_{2}){\phi}_{\pi}^{a}(x_{3})$
$\displaystyle{\times}$
$\displaystyle\Big{\\{}{\phi}_{B_{c}}^{+}(x_{1})\Big{(}1-r_{B_{s}}\Big{)}\Big{(}x_{2}-x_{1}\Big{)}+{\phi}_{B_{c}}^{-}(x_{1})\Big{[}r_{B_{s}}^{2}\Big{(}x_{2}-x_{3}\Big{)}+\Big{(}x_{3}-x_{1}\Big{)}\Big{]}\Big{\\}},$
$\displaystyle{\cal A}_{d}$ $\displaystyle=$
$\displaystyle\frac{-i32{\pi}C_{F}m_{B_{c}}^{4}}{\sqrt{2N_{c}}}\Big{(}1-r_{B_{s}}^{2}\Big{)}{\int}_{0}^{1}{\bf
d}x_{1}{\bf d}x_{2}{\bf d}x_{3}{\int}_{0}^{\infty}b_{2}{\bf d}b_{2}b_{3}{\bf
d}b_{3}$ (34) $\displaystyle{\times}$ $\displaystyle
E_{d}(t_{d}){\alpha}_{s}(t_{d})H_{d}({\alpha},{\beta}_{d},b_{2},b_{3})C_{2}(t_{d}){\phi}_{B_{c}}^{-}(x_{1}){\phi}_{\pi}^{a}(x_{3})$
$\displaystyle{\times}$
$\displaystyle\Big{\\{}{\phi}_{B_{s}}^{+}(x_{2})r_{B_{s}}\Big{[}\Big{(}x_{1}-\bar{x}_{3}\Big{)}-r_{B_{s}}^{2}\Big{(}x_{2}-\bar{x}_{3}\Big{)}\Big{]}-{\phi}_{B_{s}}^{-}(x_{2})\Big{(}1-r_{B_{s}}\Big{)}\Big{(}x_{1}-x_{2}\Big{)}\Big{\\}},$

where the evolution factor $E_{a,b}$ and the kernel function $H_{a,b}$ are the
same as those for the $B_{c}$ ${\to}$ $B_{s}$ transition form factors given in
Section IV. The evolution factor $E_{c,d}$ $=$ $e^{-S_{B_{s}}-S_{\pi}}$. The
Sudakov factors $S_{B_{s}}$ are given in Eq.(21). And the Sudakov factors
$S_{\pi}$ is defined as :

$S_{\pi}(t)=s(x_{3}p_{3}^{+},b_{3})+s(\bar{x}_{3}p_{3}^{+},b_{3})+2{\int}_{1/b_{3}}^{t}\frac{{\bf
d}{\mu}}{\mu}{\gamma}_{q}.$ (35)

The hard-scattering kernel function $H_{c,d}$ is defined as :

$H_{i}({\alpha},{\beta}_{i},b_{2},b_{3})=K_{0}(\sqrt{{\beta}_{i}}b_{3})\Big{\\{}{\theta}\Big{(}b_{2}-b_{3}\Big{)}K_{0}(b_{2}\sqrt{\alpha})I_{0}(b_{3}\sqrt{\alpha})+b_{2}{\leftrightarrow}b_{3}\Big{\\}},$
(36)

where ${\alpha}$ and ${\beta}_{i}$ are the virtualities of internal gluons and
quarks in Fig.2, respectively; $t_{i}$ is the maximal (either longitudinal or
transverse) virtuality of propagators:

$\displaystyle{\beta}_{c}$ $\displaystyle=$ $\displaystyle-
m_{1}^{2}\Big{(}x_{1}-x_{2}\Big{)}\Big{[}x_{1}-x_{3}-r_{B_{s}}^{2}\Big{(}x_{2}-x_{3}\Big{)}\Big{]},$
(37) $\displaystyle{\beta}_{d}$ $\displaystyle=$ $\displaystyle-
m_{1}^{2}\Big{(}x_{1}-x_{2}\Big{)}\Big{[}x_{1}-\bar{x}_{3}-r_{B_{s}}^{2}\Big{(}x_{2}-\bar{x}_{3}\Big{)}\Big{]},$
(38) $\displaystyle t_{i}$ $\displaystyle=$
$\displaystyle{\max}(\sqrt{{|}{\alpha}{|}},\sqrt{{|}{\beta}_{i}{|}},1/b_{2},1/b_{3})~{}~{}~{}{\rm
for}~{}~{}~{}i=c,d.$ (39)

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Table 1: input parameters for $B_{c}$ ${\to}$ $B_{s}{\pi}$ decay parameter | numerical value | Ref.
---|---|---
mass of $B_{c}$ |  $m_{B_{c}}$ $=$ $6.277{\pm}0.006$ GeV | pdg2012
mass of $B_{s}$ |  $m_{B_{s}}$ $=$ $5366.7{\pm}0.4$ MeV | pdg2012
mass of $c$ quark |  $m_{c}$ $=$ $1.275{\pm}0.025$ GeV | pdg2012
lifetime of $B_{c}$ |  ${\tau}_{B_{c}}$ $=$ $0.453{\pm}0.041$ ps | pdg2012
decay constant of ${\pi}$ |  $f_{\pi}$ $=$ $130.41{\pm}0.03{\pm}0.20$ MeV | pdg2012
decay constant of $B_{c}$ |  $f_{B_{c}}$ $=$ $489{\pm}4{\pm}3$ MeV | plb651p171
decay constant of $B_{s}$ |  $f_{B_{s}}$ $=$ $227.6{\pm}5.0$ MeV | lattice
Gegenbauer moment | $a^{\pi}_{2}(1{\rm GeV})=0.16{\pm}0.01$ | ijmpa25p513
| $a^{\pi}_{4}(1{\rm GeV})=0.04{\pm}0.01$ | ijmpa25p513
Table 2: Form factor $F_{0}^{B_{c}{\to}B_{s}}(0)$ and branching ratio ${\cal B}r(B_{c}{\to}B_{s}{\pi})$. | GN | KLS | KKQT
---|---|---|---
$F_{0}$ | $0.84^{+0.06+0.28+0.09}_{-0.05-0.20-0.05}$ | $0.87^{+0.08+0.36+0.09}_{-0.07-0.25-0.05}$ | $0.62^{+0.06+1.08+0.06}_{-0.05-0.37-0.04}$
${\cal B}r{\times}10^{2}$ | $6.20^{+0.90+4.78+1.83}_{-0.75-2.60-0.90}$ | $6.57^{+1.19+6.57+1.90}_{-0.98-3.26-0.94}$ | $3.40^{+0.65+22.32+0.79}_{-0.52-~{}2.84-0.48}$
Table 3: Form factor $F_{0}^{B_{c}{\to}B_{s}}(0)$ and branching ratio ${\cal B}r(B_{c}{\to}B_{s}{\pi})$ (in the unit of %) in the previous literature. Ref. | $F_{0}$ | Ref. | $F_{0}$ | ${\cal B}r{\times}10^{2}$ | Ref. | ${\cal B}r{\times}10^{2}$
---|---|---|---|---|---|---
Phys.Rev.D82.094014.2010  | 0.926111It is estimated with the Isgur-Scora-Grinstein-Wise Phys.Rev.D39.799.1989  quark model. | Phys.Rev.D86.094028.2012  | 1.03 | 12.01 [10.9]222It is estimated with the relativistic quark model, where $a_{1}$ $=$ 1.26 [1.2]. | Phys.Rev.D77.114004.2008  | 5.31333It is estimated with the QCD factorization approach qcdf  at the leading order.
Phys.Rev.D80.074005.2009  | 1.021444It is estimated with a relativistic quark model based on a confining potential in the equally mixed scalar-vector harmonic form. | Phys.Rev.D80.114003.2009  | 0.573 [0.571]555It is estimated with the light-front quark model, where the interaction is Coulomb plus linear [harmonic oscillator] confining potentials. | 3.723 [3.697]555It is estimated with the light-front quark model, where the interaction is Coulomb plus linear [harmonic oscillator] confining potentials. | Phys.Rev.D73.054024.2006  | 3.9666It is estimated with a relativistic quark model.
Phys.Rev.D79.054012.2009  |  $0.73^{+0.03+0.03}_{-0.04-0.03}$777It is estimated with covariant light-front quark model, where the uncertainties are from the decay constant of $B_{c}$ and $B_{s}$ mesons. | Phys.Rev.D79.034004.2009  |  $0.55{\pm}0.03$888It is estimated with the BSW zpc29p637  model. |  $4.8{\pm}0.5$888It is estimated with the BSW zpc29p637  model. | Phys.Rev.D62.014019.2000  | 1.57999It is estimated with a relativistic quark model based on the Bethe-Salpeter equation, where $a_{1}$ $=$ 1.2.
Int.J.Mod.Phys.A23.3237.2008  | 1.02101010It is estimated with the light-cone QCD sum rules. | Phys.Rev.D74.074008.2006  |  $0.58^{+0.01}_{-0.02}$111111It is estimated with the framework of a nonrelativistic constituent quark model. |  $3.51^{+0.19}_{-0.06}$111111It is estimated with the framework of a nonrelativistic constituent quark model. | Phys.Rev.D56.4133.1997  | 4.30999It is estimated with a relativistic quark model based on the Bethe-Salpeter equation, where $a_{1}$ $=$ 1.2.
Mod.Phys.Lett.A16.1439.2001  | 0.297111It is estimated with the Isgur-Scora-Grinstein-Wise Phys.Rev.D39.799.1989  quark model. | Eur.Phys.J.C32.29.2003  | 0.50666It is estimated with a relativistic quark model. | 2.52666It is estimated with a relativistic quark model. | 9803433  | 5.5121212It is estimated with covariant quark model.
Phys.Rev.D63.074010.2001  |  $0.61$666It is estimated with a relativistic quark model. | Phys.Atom.Nucl.67.1559.2004  | 1.3131313It is estimated with QCD sum rules. | 16.4131313It is estimated with QCD sum rules. | prd.49.3399  | 5.93
9810339  | 0.403${\sim}$0.617141414It is estimated with the BSW zpc29p637  formalism, where uncertainty is from mass of $b$, $c$, $s$ quark and shape parameter ${\omega}$ of wave functions. | Nucl.Phys.B585.353.2000  | 1.3131313It is estimated with QCD sum rules. | 17.5131313It is estimated with QCD sum rules. | zpc.51.549  | 2.19888It is estimated with the BSW zpc29p637  model.
Z.Phys.C64.57.1994  |  $0.60{\pm}0.12$151515It is estimated by vertex sum rules. | Phys.Rev.D61.034012.2000  | 0.66161616It is estimated by overlap integral of the meson wave functions. | 4.0161616It is estimated by overlap integral of the meson wave functions. | zpc.51.549  | 3.10111It is estimated with the Isgur-Scora-Grinstein-Wise Phys.Rev.D39.799.1989  quark model.
Z.Phys.C57.43.1993  |  $0.30{\pm}0.05$131313It is estimated with QCD sum rules. | Phys.Usp.38.1.1995  | 0.61 | 3.28888It is estimated with the BSW zpc29p637  model. | Phys.Usp.38.1.1995  | 4.66111It is estimated with the Isgur-Scora-Grinstein-Wise Phys.Rev.D39.799.1989  quark model.
J.Phys.G26.1079.2000  | 0.5917666It is estimated with a relativistic quark model. | Phys.Rev.D39.1342.1989  | 0.340${\sim}$0.925888It is estimated with the BSW zpc29p637  model. | 1.23${\sim}$9.09888It is estimated with the BSW zpc29p637  model. |  |
|  | Phys.Atom.Nucl.62.1739.1999  | 0.564111It is estimated with the Isgur-Scora-Grinstein-Wise Phys.Rev.D39.799.1989  quark model. | 4.37111It is estimated with the Isgur-Scora-Grinstein-Wise Phys.Rev.D39.799.1989  quark model. |  |
Fig. 1: Diagrams contributing to the $B_{c}$ ${\to}$ $B_{s}$ transition form
factor, where the box represents the lowest order hard-scattering kernel $H$,
and the dot denotes an appropriate Dirac matrix. Fig. 2: Feynman diagrams for
$B_{c}$ ${\to}$ $B_{s}{\pi}$ decay within the pQCD framework, where the dots
deonte an appropriate Dirac matrix.