# Possibility of searching for $B_{c}^{\ast}$ ${\to}$ $B_{u,d,s}V$,
$B_{u,d,s}P$ decays

Junfeng Sun  Institute of Particle and Nuclear Physics, Henan Normal
University, Xinxiang 453007, China  Yueling Yang  Institute of Particle and
Nuclear Physics, Henan Normal University, Xinxiang 453007, China  Na Wang
Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang
453007, China  Qin Chang  Institute of Particle and Nuclear Physics, Henan
Normal University, Xinxiang 453007, China  Gongru Lu  Institute of Particle
and Nuclear Physics, Henan Normal University, Xinxiang 453007, China

###### Abstract

The $B_{c}^{\ast}$ ${\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$ decays are investigated
with the QCD factorization approach, where $V$ and $P$ denote the ground
$SU(3)$ vector and pseudoscalar mesons, respectively. The $B_{c}^{\ast}$
${\to}$ $B_{u,d,s}$ transition form factors are calculated with the Wirbel-
Stech-Bauer model. It is found that branching ratios for the color-favored and
Cabibbo-favored $B_{c}^{\ast}$ ${\to}$ $B_{s}{\rho}$, $B_{s}{\pi}$ decays can
reach up to ${\cal O}(10^{-7})$, which might be measurable in the future LHC
experiments.

###### pacs:

12.15.Ji 12.39.St 13.25.Hw 14.40.Nd

##  I Introduction

The vector $B_{c}^{{\ast}}$ meson, a spin-triplet ground state, consists of
two heavy quarks with different flavor numbers $B$ $=$ $C$ $=$ ${\pm}1$, i.e.,
$\bar{b}c$ for $B_{c}^{{\ast}+}$ meson and $b\bar{c}$ for $B_{c}^{{\ast}-}$
meson. With nonzero bottom and charm numbers, the bottom and charm quarks of
the $B_{c}^{\ast}$ meson cannot annihilate into gluons and photons via the
strong and electromagnetic interactions, respectively, unlike the decay modes
of the unflavored $J/{\psi}(1S)$ and ${\Upsilon}(1S)$ mesons. The
$B_{c}^{\ast}$ meson serves as a unique object in studying the heavy quark
dynamics, which is inaccessible through both charmonium and bottomonium.

The $B_{c}^{\ast}$ meson lies below the $B_{q}D_{q}$ ($q$ $=$ $u$, $d$, $s$)
meson pair threshold. And the mass splitting $m_{B_{c}^{\ast}}$ $-$
$m_{B_{c}}$ ${\approx}$ 50 MeV prd86.094510  is less than the pion mass.
Hence, the $B_{c}^{\ast}$ meson decays via the strong interaction are strictly
forbidden. The electromagnetic transition process, $B_{c}^{\ast}$ ${\to}$
$B_{c}{\gamma}$, dominates the $B_{c}^{\ast}$ meson decays, but suffers
seriously from a compact phase space suppression, which results in a lifetime
of ${\tau}_{B_{c}^{\ast}}$ ${\sim}$ ${\cal O}(10^{-17}\,{\rm s})$ epja52.90 .
Besides, the $B_{c}^{\ast}$ meson decays via the weak interaction, although
with very small decay rates, are allowable within the standard model.

The $B_{c}^{\ast}$ meson has a relatively large mass. In addition, both
constituent quarks $b$ and $c$ of the $B_{c}^{\ast}$ meson can decay
individually. Therefore, the $B_{c}^{\ast}$ meson has rich weak decay
channels. The $B_{c}^{\ast}$ meson weak decays, similar to the pseudoscalar
$B_{c}$ meson weak decays zpc51 ; prd49 ; usp38 ; prd77.074013 ; prd89.114019
; ahep2015.104378 ; qwg , can be divided into three classes: (1) the $c$ quark
decay with the spectator $b$ quark, (2) the $b$ quark decay with the $c$ quark
as a spectator, and (3) the $b$ and $c$ quarks annihilation into a virtual
$W^{\pm}$ boson. This property makes the $B_{c}^{\ast}$ meson another
potentially fruitful laboratory for studying the weak decay mechanism of heavy
flavor hadrons.

The study of $B_{c}^{\ast}$ weak decays might be interesting, but has not
really started yet. One of the major reasons is the extraordinary difficulty
of producing the $B_{c}^{\ast}$ meson. The production cross section for the
$B_{c}^{\ast}$ meson in hadronic collisions via the dominant process of $g$
$+$ $g$ ${\to}$ $B_{c}^{\ast}$ $+$ $b$ $+$ $\bar{c}$ qwg ; plb355 ; plb364 ;
prd54.4344 ; epjc38.267 ; prd72.114009  is at least at the order of
${\alpha}_{s}^{4}$. The nature of QCD’s asymptotic freedom implies a much
small possibility of creating two heavy quark pairs ($b\bar{b}$ and
$c\bar{c}$) from the vacuum at the ultrahigh energy. Fortunately, the high
luminosities of the running LHC and the future Super proton proton Collider
(S$pp$C, which is still under discussion today) will promisingly improve this
situation. It is expected that a huge amount of the $B_{c}^{\ast}$ data
samples would be accumulated, and offer a valuable opportunity to investigate
the $B_{c}^{\ast}$ weak decays.

As is well known, there exist some hierarchical structures among the Cabibbo-
Kobayashi-Maskawa (CKM) matrix elements. The CKM coupling strength for the
bottom quark weak decay is proportional to ${|}V_{cb}{|}$ ${\sim}$ ${\cal
O}({\lambda}^{2})$ or ${|}V_{ub}{|}$ ${\sim}$ ${\cal O}({\lambda}^{3})$, while
the CKM coupling strength for the charm quark weak decay is proportional to
${|}V_{cs}{|}$ ${\sim}$ ${\cal O}(1)$ or ${|}V_{cd}{|}$ ${\sim}$ ${\cal
O}({\lambda})$, with the Wolfenstein parameter ${\lambda}$ ${\approx}$ $0.2$
pdg . The $B_{q}$ ($q$ $=$ $u$, $d$, $s$) weak decays are induced dominantly
by the bottom quark decay with the phenomenological spectator scheme. The
$B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays are actually induced by the
charm quark weak decay, where $V$ and $P$ denote respectively the lightest
9-pelts $SU(3)$ vector and pseudoscalar mesons. With respect to the $B_{q}$
weak decays, the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays are favored
by the CKM matrix elements. In this paper, we will study the $B_{c}^{\ast}$
${\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$ weak decays with the QCD factorization
(QCDF) approach prl83.1914 ; npb591.313 ; npb606.245 ; plb488.46 ; plb509.263
; prd64.014036 ; npb774.64 ; npb832.109 ; plb750.348 , in order to provide an
available reference for the future experimental investigation. There is a more
than $2.0\,{\sigma}$ discrepancy between the value for CKM matrix element
${|}V_{cs}{|}$ obtained from semileptonic $D$ decays and that from leptonic
$D_{s}$ decays111The value for CKM matrix element ${|}V_{cs}{|}$ is
${|}V_{cs}{|}$ $=$ $0.953{\pm}0.008{\pm}0.024$ from semileptonic $D$ decays,
and ${|}V_{cs}{|}$ $=$ $1.008{\pm}0.021$ from leptonic $D_{s}$ decays pdg .
pdg . The $B_{c}^{(\ast)}$ ${\to}$ $B_{s}V$, $B_{s}P$ decays, together with
semileptonic $D$ decays and leptonic $D_{s}$ decays, will provide
${|}V_{cs}{|}$ with more stringent constraints. In addition, some of the
$B_{c}$ weak decays, for example, the $B_{c}$ ${\to}$ $B_{s}{\pi}$ decay
prl.111 , have been measured now. One possible background might come from the
$B_{c}^{\ast}$ decays, due to a slightly larger production cross section
${\sigma}(B_{c}^{\ast})$ than ${\sigma}(B_{c})$ in hadronic collisions plb364
; prd54.4344 ; epjc38.267 ; prd72.114009 , and the nearly equal mass
$m_{B_{c}^{\ast}}$ ${\simeq}$ $m_{B_{c}}$ prd86.094510 . Hence, the study of
the $B_{c}^{\ast}$ ${\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$ decays will be helpful
to the experimental analysis on the $B_{c}$ ${\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$
decays.

This paper is organized as follows. The theoretical framework and decay
amplitudes will be presented in Section II. Section III is the numerical
results and discussion. The last section is a summary.

##  II theoretical framework

###  II.1 The effective Hamiltonian

Using the operator product expansion and the renormalization group (RG)
method, the low-energy effective weak Hamiltonian describing the
$B_{c}^{\ast}$ ${\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$ decays has the following
general structure 9512380 ,

${\cal H}_{\rm
eff}\,=\,\frac{G_{F}}{\sqrt{2}}\sum\limits_{q,q^{\prime}=s,d}V_{cq}^{\ast}V_{uq^{\prime}}\Big{\\{}C_{1}({\mu})\,Q_{1}({\mu})\,+C_{2}({\mu})\,Q_{1}({\mu})\Big{\\}}+{\rm
h.c.},$ (1)

where the Fermi coupling constant $G_{F}$ ${\simeq}$
$1.166{\times}10^{-5}\,{\rm GeV}^{-2}$ pdg ; $V_{cq}^{\ast}V_{uq^{\prime}}$ is
a product of the CKM matrix elements. Using the Wolfenstein parameterization,
there are pdg

$V_{cs}^{\ast}V_{ud}\ =\
1-{\lambda}^{2}-\frac{1}{2}\,A^{2}\,{\lambda}^{4}+\frac{1}{2}\,A^{2}\,{\lambda}^{6}\,\big{\\{}1-{\rho}^{2}-{\eta}^{2}-2\,({\rho}-i\,{\eta})\big{\\}}+{\cal
O}({\lambda}^{8}),$ (2) $V_{cs}^{\ast}V_{us}\ =\
{\lambda}-\frac{{\lambda}^{3}}{2}-\frac{{\lambda}^{5}}{8}-\frac{{\lambda}^{7}}{16}-\frac{1}{2}\,A^{2}\,{\lambda}^{5}+\frac{1}{2}\,A^{2}\,{\lambda}^{7}\,\big{\\{}\frac{1}{2}-{\rho}^{2}-{\eta}^{2}-2\,({\rho}-i\,{\eta})\big{\\}}+{\cal
O}({\lambda}^{8}),$ (3) $V_{cd}^{\ast}V_{ud}\ =\
-V_{cs}^{\ast}V_{us}-A^{2}\,{\lambda}^{5}\,({\rho}-i\,{\eta})+{\cal
O}({\lambda}^{8}),$ (4) $V_{cd}^{\ast}V_{us}\ =\
-{\lambda}^{2}+\frac{1}{2}\,A^{2}\,{\lambda}^{6}\,\big{\\{}1-2\,({\rho}-i\,{\eta})\big{\\}}+{\cal
O}({\lambda}^{8}),$ (5)

where the values for these Wolfenstein parameters $A$, ${\lambda}$, ${\rho}$
and ${\eta}$ are given in Table 3.

The renormalization scale ${\mu}$ separates the physical contributions into
two parts. The hard contributions above the scale ${\mu}$ are summarized into
the Wilson coefficients $C_{i}(\mu)$. With the RG equation for $C_{i}(\mu)$,
the Wilson coefficients at an appropriate scale ${\mu}_{c}$ ${\sim}$ ${\cal
O}(m_{c})$ for the charm quark decay are given by 9512380

$\vec{C}({\mu}_{c})\,=\,U_{4}({\mu}_{c},m_{b})\,U_{5}(m_{b},m_{W})\,\vec{C}(m_{W}),$
(6)

where $m_{W}$, $m_{b}$ and $m_{c}$ are the mass of the $W$ boson, $b$ quark
and $c$ quark, respectively. Here $U_{f}(m_{2},m_{1})$ denotes the RG
evolution matrix for $f$ active flavors. The initial values for the Wilson
coefficients $\vec{C}(m_{W})$ at scale ${\mu}_{W}$ $=$ $m_{W}$ to a desired
order in ${\alpha}_{s}$ can be calculated with perturbation theory. The
expressions for the RG evolution matrix $U_{f}(m_{2},m_{1})$ and Wilson
coefficients $\vec{C}(m_{W})$, including both leading order (LO) and next-to-
leading order (NLO) corrections, have been presented in Ref.9512380 . The
contributions below the scale ${\mu}$ are included in the hadronic matrix
elements (HME) where the local four-quark operators $Q_{i}$ are sandwiched
between the initial and final states. The expressions for the four-quark
operators in question are

$Q_{1}\,=\,\big{[}\bar{q}_{\alpha}\,{\gamma}_{\mu}\,(1-{\gamma}_{5})\,c_{\alpha}\big{]}\,\big{[}\bar{u}_{\beta}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,q^{\prime}_{\beta}\big{]},$
(7)
$Q_{2}\,=\,\big{[}\bar{q}_{\alpha}\,{\gamma}_{\mu}\,(1-{\gamma}_{5})\,c_{\beta}\big{]}\,\big{[}\bar{u}_{\beta}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,q^{\prime}_{\alpha}\big{]},$
(8)

where the subscripts ${\alpha}$ and ${\beta}$ are color indices. It should be
pointed out that (1) because the contributions from the penguin operators and
annihilation topologies are proportional to the CKM factor
$V_{cb}^{\ast}V_{ub}$ ${\sim}$ ${\cal O}({\lambda}^{5})$ and therefore
negligible in the actual calculation of branching ratio prd89.114019 , only
the contributions of tree operators are considered here. (2) The participation
of the strong interaction, especially, the nonperturbative QCD effects, makes
the theoretical treatment of HME very complicated. The main problem at this
stage is how to effectively factorize HME into hard and soft parts, and how to
evaluate HME properly.

###  II.2 Hadronic matrix elements

Hadronic matrix elements might be the most intricate part in the calculation
of heavy flavor weak decay, due to the entanglement of perturbative and
nonperturbative contributions. Phenomenologically, one has to turn to some
approximation and assumption, which bring uncertainties and model dependence
to theoretical predictions. A simple approximation is the naive factorization
ansatz (NF) according to Bjorken’s color transparency argument, which says
that the colorless energetic hadron has flown away from the weak interaction
point during the formation time of the emission hadron npb11.325 . With the NF
approach, HME is parameterized as a product of decay constants and hadron
transition form factors plb73.418 ; npb133.315 ; zpc29.637 ; zpc34.103 . A
major flaw of the NF approach is the disappearance of scale dependence and
strong phases from HME, which results directly in a scale-sensitive
nonphysical prediction and none of $CP$ violation for nonleptonic meson weak
decays. In order to overcome these shortcomings of the NF approach,
nonfactorizable contributions to HME should be carefully considered, as
commonly recognized. Some QCD-inspired models, such as, the QCDF approach
prl83.1914 ; npb591.313 ; npb606.245 ; plb488.46 ; plb509.263 ; prd64.014036 ;
npb774.64 ; npb832.109 ; plb750.348 , the soft and collinear effective theory
prd63.014006 ; prd63.114020 ; plb516.134 ; prd65.054022 ; prd66.014017 ;
npb643.431 ; plb553.267 ; npb685.249 , the perturbative QCD approach pqcd1 ;
pqcd2 ; pqcd3 , and so on, have been developed recently, based on the Lepage-
Brodsky treatment on exclusive processes prd22  and some power counting rules
in the expansion in ${\alpha}_{s}$ and ${\Lambda}_{\rm QCD}/m_{Q}$, where
${\alpha}_{s}$ is the strong coupling, ${\Lambda}_{\rm QCD}$ is the QCD
characteristic scale, and $m_{Q}$ is the mass of a heavy quark. In these QCD-
inspired models, HME is generally written as a convolution integral of
hadron’s distribution amplitudes (DAs) and hard rescattering kernels. A virtue
of the QCDF approach is that the NF’s result can be reproduced, if both the
nonfactorizable contributions and the power suppressed contributions are
neglected prl83.1914 ; npb591.313 ; npb606.245 ; plb488.46 ; plb509.263 ;
prd64.014036 .

For the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays ($q$ $=$ $u$, $d$,
$s$), the spectator quark is a heavy quark — the bottom quark. It is generally
assumed that the bottom quark in both the $B_{c}^{\ast}$ and $B_{q}$ mesons is
nearly on shell, and that the gluon exchanged between the heavy spectator
quark and other quarks is soft. The virtuality of emission gluon from the
spectator quark is of order ${\Lambda}_{\rm QCD}^{2}$. The contributions of
spectator scattering are power suppressed relative to the leading order
contributions npb591.313 . In addition, it is supposed that the recoiled
$B_{q}$ meson should move slowly in the rest frame of the $B_{c}^{\ast}$
meson. There should be a large overlap between the $B_{c}^{\ast}$ and $B_{q}$
mesons. The recoiled $B_{q}$ meson cannot be clearly factorized from the
$B_{c}^{\ast}B_{q}$ system due to the soft and nonperturbative contributions.
The $B_{c}^{\ast}B_{q}$ system should be parameterized by some physical from
factors. Hence, with the QCDF approach, up to leading power corrections of
order ${\Lambda}_{\rm QCD}/m_{Q}$, hadronic matrix elements have the following
structure npb591.313 ,

${\langle}B_{q}M{|}Q_{i}{|}B_{c}^{\ast}{\rangle}\,=\,f_{M}\sum_{j}F^{B_{c}^{\ast}{\to}B_{q}}_{j}{\int}dx\,{\cal
H}_{ij}(x)\,{\phi}(x)\,=\,f_{M}\sum_{j}F^{B_{c}^{\ast}{\to}B_{q}}_{j}\,\Big{\\{}1+{\alpha}_{s}\,r_{j}+{\cdots}\Big{\\}},$
(9)

where $f_{M}$ is the decay constant for the light final $M$ (${\equiv}$ $V$
and $P$) meson; $F^{B_{c}^{\ast}{\to}B_{q}}_{j}$ is a transition form factor;
${\cal H}_{ij}(x)$ is a hard rescattering kernel; ${\phi}(x)$ is a DA of
parton momentum fraction $x$. For the light pseudoscalar $P$ and
longitudinally polarized vector $V$ mesons, the leading twist DAs are expanded
in terms of the Gegenbauer polynomials jhep.0605.004 ; jhep.0703.069

${\phi}_{P}(x)\ =\
6\,x\,\bar{x}\,\Big{\\{}1+\sum\limits_{n=1}a_{n}^{P}\,C_{n}^{3/2}(x-\bar{x})\Big{\\}},$
(10)
${\phi}_{V}(x)\,=\,6\,x\,\bar{x}\,\Big{\\{}1+\sum\limits_{n=1}a_{n}^{V}\,C_{n}^{3/2}(x-\bar{x})\Big{\\}},$
(11)

where $\bar{x}$ $=$ $1$ $-$ $x$; $a_{n}^{P,V}$ is a nonperturbative parameter,
also called the Gegenbauer moment. The expressions for the Gegenbauer
polynomials $C_{n}^{3/2}(z)$ are

$C_{1}^{3/2}(z)=3\,z,\quad
C_{2}^{3/2}(z)=\frac{3}{2}\,(5\,z^{2}-1),\quad{\cdots}$ (12)

Figure 1: Feynman diagrams for $B_{c}^{{\ast}}$ ${\to}$ $B_{s}{\pi}$ decay
within the QCDF framework, where (a) denotes the factorizable contributions,
and (b,c,d,e) correspond to the nonfactorizable vertex corrections at the
order of ${\alpha}_{s}$.

###  II.3 Decay amplitudes

The typical Feynman diagrams for the $B_{c}^{{\ast}}$ ${\to}$ $B_{s}{\pi}$
decay within the QCDF framework are shown in Fig.1, where no hard gluons are
exchanged between the spectator quark and other partons. There is no gluon
exchange in factorizable topology of Fig.1(a), so the emitted hadron matrix
element is entirely separated from that of the $B_{c}^{{\ast}}B_{s}$ system.
In this approximation, the hard rescattering kernel ${\cal H}_{ij}$ $=$ $1$
and the integral in Eq.(9) reduces to the normalization condition for
distribution amplitude. According to the QCDF power counting rules, the
leading order contributions come from the factorizable topology of Fig.1(a),
and recover the NF’s results at the order of ${\alpha}_{s}^{0}$. For the
radiative correction diagrams in Fig.1(b-e), hard gluons are exchanged between
the emission meson and the $B_{c}^{{\ast}}B_{s}$ system. The hard rescattering
kernel ${\cal H}_{ij}$ and $x$-integral in Eq.(9) are nontrivial. It has
already been shown npb591.313 ; npb606.245 ; plb488.46 ; plb509.263 ;
prd64.014036  that although both collinear and soft divergences exist for each
of diagrams in Fig.1(b-e), infrared divergences cancel after summing up the
vertex corrections. The strong phases could then come from HME. The
renormalization scale ${\mu}$ dependence of HME is recuperated from the
nonfactorizable contributions, which will reduce partly the ${\mu}$-dependence
of Wilson coefficients.

After a straightforward calculation using the QCDF master formula Eq.(9), the
amplitudes for the $B_{c}^{{\ast}}$ ${\to}$ $B_{q}M$ decays ($q$ $=$ $u$, $d$,
$s$) are written as

${\cal A}(B_{c}^{\ast}{\to}B_{q}M)\,=\,{\langle}B_{q}M{|}{\cal H}_{\rm
eff}{|}B_{c}^{\ast}{\rangle}\,=\,\frac{G_{F}}{\sqrt{2}}\,V_{cq}^{\ast}V_{uq^{\prime}}\,a_{i}\,{\langle}M{|}j^{\mu}{|}0{\rangle}\,{\langle}B_{q}{|}j_{\mu}{|}B_{c}^{\ast}{\rangle}.$
(13)

With the naive dimensional regularization scheme, the effective coefficients
are npb591.313 ; npb606.245 ; plb488.46 ; plb509.263 ; prd64.014036 :

$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle C_{1}^{\rm
NLO}+\frac{1}{N_{c}}\,C_{2}^{\rm
NLO}+\frac{{\alpha}_{s}}{4{\pi}}\,\frac{C_{F}}{N_{c}}\,C_{2}^{\rm LO}\,{\cal
V},$ (14) $\displaystyle a_{2}$ $\displaystyle=$ $\displaystyle C_{2}^{\rm
NLO}+\frac{1}{N_{c}}\,C_{1}^{\rm
NLO}+\frac{{\alpha}_{s}}{4{\pi}}\,\frac{C_{F}}{N_{c}}\,C_{1}^{\rm LO}\,{\cal
V},$ (15) ${\cal
V}=6\,{\log}\Big{(}\frac{m_{c}^{2}}{{\mu}^{2}}\Big{)}-18-\Big{(}\frac{1}{2}+i\,3\,{\pi}\Big{)}+\Big{(}\frac{11}{2}-i\,3\,{\pi}\Big{)}\,a_{1}^{M}-\frac{21}{20}\,a_{2}^{M}+{\cdots},$
(16)

where $N_{c}$ $=$ $3$ and $C_{F}$ $=$ $4/3$; $C_{1,2}^{\rm NLO,LO}$ are Wilson
coefficients containing NLO or LO contributions; $a_{i}^{M}$ is a Gegenbauer
moment. For the transversely polarized vector meson, the vertex factor ${\cal
V}$ $=$ $0$ beyond the leading twist DAs. For convenience, the numerical
values for $a_{1,2}$ of the $B_{c}^{\ast}$ ${\to}$ $B_{q}{\pi}$ decay are
listed in Table 1.

There are some comments on the coefficients $a_{1,2}$. (1) The first two terms
on the right hand side of Eq.(14) and Eq.(15) correspond to the leading order
contributions. The third terms correspond to nonfactorizable contributions.
The NF scenario follows when one neglects the nonfactorizable contributions,
i.e., ${\cal V}$ $=$ $0$. (2) Nonfactorizable vertex corrections to HME are of
order ${\alpha}_{s}$. They include the dependence on the renormalization
scale. It is shown prd64.014036  that with the RG equations for the Wilson
coefficients at leading order logarithm approximation, one can obtain
$\displaystyle{\mu}\frac{{\rm d}}{{\rm d}{\mu}}a_{1,2}\,=\,0$. In principle,
the residual scale dependence could be compensated by higher order corrections
to HME. (3) Compared with the LO contributions, nonfactorizable contributions
are generally suppressed by ${\alpha}_{s}$ and the factor $1/N_{c}$ (see
Eq.(14) and Eq.(15)). Because the LO contributions of $a_{2}$ are color-
suppressed, vertex corrections multiplied by the large Wilson coefficient
$C_{1}^{\rm LO}$ could be sizable to branching rates of the $a_{2}$-dominated
heavy flavor decays. The coefficients $a_{1,2}$ contain strong phases via the
imaginary parts of vertex corrections. Correspondingly, strong scattering
phase of $a_{1}$ ($a_{2}$) is small (large). This argument is also confirmed
by the numerical results for $a_{1,2}$ in Table 1. (4) With the QCDF approach,
nonfactorizable radiative corrections to HME occur first at order
${\alpha}_{s}$ as well as the leading strong phases at order ${\alpha}_{s}$.
In addition, it should be pointed out that nonfactorizable power corrections
beyond leading order are neglected here. For the charm quark decay, power
${\Lambda}_{\rm QCD}/m_{c}$ is comparable to ${\alpha}_{s}$. The strong phases
due to soft (hard) interactions are of order ${\Lambda}_{\rm QCD}/m_{c}$
(${\alpha}_{s}$). One should not expect these phases to have great precision,
as stated in Ref.npb591.313 . (5) With the QCDF approach, the values for
$a_{1,2}$ are close to those for the charm quark decay npb268.16 ; plb252.690
; ijmpa14.937 ; epjc55.607 ; prd81.074021 , ${|}a_{1,2}{|}$ ${\approx}$
${|}C_{1,2}{|}$, and basically consistent with those of the large-$N_{c}$
approach npb268.16 .

Table 1: The numerical values for the Wilson coefficients and $a_{1,2}$ for the $B_{c}^{\ast}$ ${\to}$ $B_{q}{\pi}$ decay. scale | LO | NLO | NF | QCDF
---|---|---|---|---
${\mu}$ | $C_{1}$ | $C_{2}$ | $C_{1}$ | $C_{2}$ | $a_{1}$ | $a_{2}$ | $a_{1}$ | $a_{2}$
$0.8\,m_{c}$ | $1.310$ | $-0.553$ | $1.253$ | $-0.473$ | $1.096$ | $-0.055$ | $1.235+i\,0.081$ | $-0.384-i\,0.192$
$m_{c}$ | $1.259$ | $-0.479$ | $1.209$ | $-0.404$ | $1.074$ | $-0.001$ | $1.193+i\,0.060$ | $-0.313-i\,0.157$
$1.2\,m_{c}$ | $1.227$ | $-0.430$ | $1.180$ | $-0.358$ | $1.061$ | $~{}0.035$ | $1.167+i\,0.048$ | $-0.269-i\,0.137$

The hadronic matrix elements of diquark current operators are defined as
zpc29.637 :

${\langle}V({\epsilon},p){|}\bar{q}_{1}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,q_{2}{|}0{\rangle}\,=\,f_{V}\,m_{V}\,{\epsilon}^{{\ast}{\mu}},$
(17)
${\langle}P(p){|}\bar{q}_{1}\,{\gamma}^{\mu}\,(1-{\gamma}_{5})\,q_{2}{|}0{\rangle}\,=\,-i\,f_{P}\,p^{\mu},$
(18)
$\displaystyle{\langle}B_{q}(p_{2}){|}\bar{q}\,{\gamma}_{\mu}\,(1-{\gamma}_{5})\,c{|}B_{c}^{\ast}(p_{1},{\epsilon}){\rangle}$
(19) $\displaystyle=$
$\displaystyle-{\varepsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,{\epsilon}^{{\nu}}\,q^{\alpha}\,(p_{1}+p_{2})^{\beta}\,\frac{V(q^{2})}{m_{B_{c}^{\ast}}+m_{B_{q}}}-i\,2\,m_{B_{c}^{\ast}}\,\frac{{\epsilon}{\cdot}q}{q^{2}}\,q_{\mu}\,A_{0}(q^{2})$
$\displaystyle-i\,{\epsilon}_{\mu}\,(m_{B_{c}^{\ast}}+m_{B_{q}})\,A_{1}(q^{2})-i\,\frac{{\epsilon}{\cdot}q}{m_{B_{c}^{\ast}}+m_{B_{q}}}\,(p_{1}+p_{2})_{\mu}\,A_{2}(q^{2})$
$\displaystyle+i\,2\,m_{B_{c}^{\ast}}\,\frac{{\epsilon}{\cdot}q}{q^{2}}\,q_{\mu}\,A_{3}(q^{2}),$

where $f_{V}$ and $f_{P}$ are the decay constants of vector $V$ and
pseudoscalar $P$ mesons, respectively; $q$ $=$ $p_{1}$ $-$ $p_{2}$;
${\epsilon}$ is the polarization vector of vector mesons; $V(q^{2})$ and
$A_{0,1,2,3}(q^{2})$ are the $B_{c}^{\ast}$ ${\to}$ $B_{q}$ transition form
factors. To eliminate singularities at the pole of $q^{2}$ $=$ $0$, a
relation, $A_{0}(0)$ $=$ $A_{3}(0)$, is required, with $A_{3}(q^{2})$ given by
zpc29.637 :

$2\,m_{B_{c}^{\ast}}\,A_{3}(q^{2})\,=\,(m_{B_{c}^{\ast}}+m_{B_{q}})\,A_{1}(q^{2})+(m_{B_{c}^{\ast}}-m_{B_{q}})\,A_{2}(q^{2}).$
(20)

In the bottom conservation transition $B_{c}^{\ast}$ ${\to}$ $B_{q}$, both the
initial and final mesons contain a heavy bottom quark. After a sudden kick,
the $B_{q}$ meson would move slowly, even remain nearly intact, with respect
to the $B_{c}^{\ast}$ meson. Therefore, the zero-recoil configuration ($q^{2}$
$=$ $0$) would be a good approximation. Simultaneously, the emission meson
would take up most of the energy available and fly rapidly away from the
interaction point. This fact not only reproduces the NF scenario (Fig.1(a))
but also requires the exchanged gluon in vertex corrections (Fig.1(b-e)) to be
hard. Due to the large virtuality of gluon exchanged between the emitted light
meson and the $B_{c}^{\ast}B_{q}$ system, perturbative calculation of
nonfactorizable vertex corrections with the QCDF approach should be applicable
and reliable.

With the form factors given above, the decay amplitudes are expressed as

$\displaystyle{\cal A}(B_{c}^{\ast}{\to}B_{q}V)$ $\displaystyle=$
$\displaystyle-i\,\frac{G_{F}}{\sqrt{2}}\,m_{V}\,f_{V}\,V_{cq}^{\ast}V_{uq^{\prime}}\,\big{\\{}a_{1}\,{\delta}_{B_{q},B_{d,s}}+a_{2}\,{\delta}_{B_{q},B_{u}}\big{\\}}\,\big{\\{}({\epsilon}_{B_{c}^{\ast}}{\cdot}{\epsilon}_{V}^{\ast})\,(m_{B_{c}^{\ast}}+m_{B_{q}})\,A_{1}$
(21)
$\displaystyle+\,({\epsilon}_{B_{c}^{\ast}}{\cdot}p_{V})\,(p_{B_{c}^{\ast}}{\cdot}{\epsilon}_{V}^{\ast})\,\frac{2\,A_{2}}{m_{B_{c}^{\ast}}+m_{B_{q}}}+i\,{\epsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,{\epsilon}_{B_{c}^{\ast}}^{\mu}\,{\epsilon}_{V}^{{\ast}{\nu}}\,p_{B_{c}^{\ast}}^{\alpha}\,p_{V}^{\beta}\,\frac{2\,V}{m_{B_{c}^{\ast}}+m_{B_{q}}}\big{\\}},$
${\cal
A}(B_{c}^{\ast}{\to}B_{q}P)\,=\,\sqrt{2}\,G_{F}\,m_{B_{c}^{\ast}}\,({\epsilon}_{B_{c}^{\ast}}{\cdot}p_{B_{q}})\,f_{P}\,A_{0}\,V_{cq}^{\ast}V_{uq^{\prime}}\,\big{\\{}a_{1}\,{\delta}_{B_{q},B_{d,s}}+a_{2}\,{\delta}_{B_{q},B_{u}}\big{\\}}.$
(22)

The $B_{c}^{\ast}$ ${\to}$ $B_{q}V$ decay amplitude is a sum of $S$-, $P$-,
$D$-wave amplitudes prd39.3339 ; prd45.193 , i.e.,

${\cal
A}(B_{c}^{\ast}{\to}B_{q}V)=a\,({\epsilon}_{B_{c}^{\ast}}{\cdot}{\epsilon}_{V}^{\ast})+\frac{b}{m_{B_{c}^{\ast}}\,m_{V}}({\epsilon}_{B_{c}^{\ast}}{\cdot}p_{V})\,(p_{B_{c}^{\ast}}{\cdot}{\epsilon}_{V}^{\ast})+\frac{i\,c}{m_{B_{c}^{\ast}}\,m_{V}}{\epsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,{\epsilon}_{B_{c}^{\ast}}^{\mu}\,{\epsilon}_{V}^{{\ast}{\nu}}\,p_{B_{c}^{\ast}}^{\alpha}\,p_{V}^{\beta},$
(23)

with $a$, $b$, $c$, the $S$-, $D$\- and $P$-wave amplitudes respectively, in
the notation of prd45.193 ,

$a\,=\,{\cal F}\,(m_{B_{c}^{\ast}}+m_{B_{q}})\,A_{1},$ (24) $b\,=\,{\cal
F}\,\frac{2\,m_{B_{c}^{\ast}}\,m_{V}}{m_{B_{c}^{\ast}}+m_{B_{q}}}\,A_{2},$
(25) $c\,=\,{\cal
F}\,\frac{2\,m_{B_{c}^{\ast}}\,m_{V}}{m_{B_{c}^{\ast}}+m_{B_{q}}}\,V,$ (26)
${\cal
F}\,=\,-i\,\frac{G_{F}}{\sqrt{2}}\,m_{V}\,f_{V}\,V_{cq}^{\ast}V_{uq^{\prime}}\,\big{\\{}a_{1}\,{\delta}_{B_{q},B_{d,s}}+a_{2}\,{\delta}_{B_{q},B_{u}}\big{\\}}.$
(27)

From the above expressions, one can find that the $P$\- and $D$-wave
amplitudes are suppressed by a factor of
$\frac{2\,m_{B_{c}^{\ast}}\,m_{V}}{(m_{B_{c}^{\ast}}+m_{B_{q}})^{2}}$ relative
to the $S$-wave amplitude. The relations among the helicity amplitudes and the
$S$-, $P$-, $D$-wave amplitudes are prd45.193

$H_{\pm}\ =\ a\,{\pm}\,c\,\sqrt{y^{2}-1},$ (28) $H_{0}\ =\
-a\,y-b\,(y^{2}-1),$ (29) $y\ =\
\frac{p_{B_{c}^{\ast}}{\cdot}p_{V}}{m_{B_{c}^{\ast}}\,m_{V}}\ =\
\frac{m_{B_{c}^{\ast}}^{2}-m_{B_{q}}^{2}+m_{V}^{2}}{2\,m_{B_{c}^{\ast}}\,m_{V}},$
(30) $p_{\rm cm}\ =\
\frac{\sqrt{[m_{B_{c}^{\ast}}^{2}-(m_{B_{q}}+m_{V})^{2}]\,[m_{B_{c}^{\ast}}^{2}-(m_{B_{q}}-m_{V})^{2}]}}{2\,m_{B_{c}^{\ast}}},$
(31) $p_{\rm cm}^{2}\ =\ m_{V}^{2}\,(y^{2}-1),$ (32)

where $p_{\rm cm}$ is the common momentum of final states in the rest frame of
the $B_{c}^{\ast}$ meson.

We assume that the vector mesons are ideally mixed in the singlet-octet basis,
i.e., ${\phi}$ $=$ $s\bar{s}$ and ${\omega}$ $=$
$(u\bar{u}+d\bar{d})/\sqrt{2}$. As for the pseudoscalar ${\eta}$ and
${\eta}^{\prime}$ mesons, they are usually written as a linear superposition
of states in either flavor basis or the singlet-octet basis. Here, we adopt
the quark flavor basis description proposed in Ref. prd.58.114006 , i.e.,

$\left(\begin{array}[]{c}{\eta}\\\
{\eta}^{\prime}\end{array}\right)=\left(\begin{array}[]{cc}{\cos}{\phi}&-{\sin}{\phi}\\\
{\sin}{\phi}&{\cos}{\phi}\end{array}\right)\left(\begin{array}[]{c}{\eta}_{q}\\\
{\eta}_{s}\end{array}\right),$ (33)

where ${\eta}_{q}$ $=$ $(u\bar{u}+d\bar{d})/{\sqrt{2}}$ and ${\eta}_{s}$ $=$
$s\bar{s}$; the mixing angle ${\phi}$ ${\approx}$ $(39.3{\pm}1.0)^{\circ}$
prd.58.114006 . Due to the symmetric flavor configurations of both
${\eta}_{q}$ and ${\eta}_{s}$ states, we assume that DAs for ${\eta}_{q}$ and
${\eta}_{s}$ states are similar to DAs for pion. It should be pointed out that
the contributions from possible $c\bar{c}$ and gluonium compositions are not
considered in our calculation for the moment, because (1) the final states
with $B_{q}$ meson and $c\bar{c}$ or gluonium states lie above the
$B_{c}^{\ast}$ meson mass; (2) the fraction of gluonium components in ${\eta}$
and ${\eta}^{\prime}$ is rather tiny jhep0705.006 . Thus, the amplitudes for
the $B_{c}^{\ast}$ ${\to}$ $B_{u}{\eta}$, $B_{u}{\eta}^{\prime}$ decays are
written as

$\displaystyle{\cal A}(B_{c}^{\ast}{\to}B_{u}{\eta})$ $\displaystyle=$
$\displaystyle{\cos}{\phi}\,{\cal
A}(B_{c}^{\ast}{\to}B_{u}{\eta}_{q})-{\sin}{\phi}\,{\cal
A}(B_{c}^{\ast}{\to}B_{u}{\eta}_{s}),$ (34) $\displaystyle{\cal
A}(B_{c}^{\ast}{\to}B_{u}{\eta}^{\prime})$ $\displaystyle=$
$\displaystyle{\sin}{\phi}\,{\cal
A}(B_{c}^{\ast}{\to}B_{u}{\eta}_{q})+{\cos}{\phi}\,{\cal
A}(B_{c}^{\ast}{\to}B_{u}{\eta}_{s}).$ (35)

###  II.4 Form factors

The hadron transition form factors are the basic input parameters for decay
amplitudes [see Eq.(21) and Eq.(22)]. It is assumed npb591.313  that form
factors come mainly from soft contributions, and form factors are generally
regarded as nonperturbative parameters in the QCDF master formula of Eq.(9).
Fortunately, form factors are universal. Form factors determined by other
means or extracted from data can be employed here to make predictions.
Phenomenologically, form factors are written as overlap integrals of wave
functions.

Here, we will employ the Wirbel-Stech-Bauer model zpc29.637  for evaluating
the form factors. With a factorization of spin and spatial motion, wave
function is written as

${\phi}^{(j,j_{z})}(\vec{k}_{\perp},x)\,=\,{\phi}(\vec{k}_{\perp},x)\,{|}s,s_{z};s_{1},s_{2}{\rangle},$
(36)

where $\vec{k}_{\perp}$ and $x$ are the transverse momentum and longitudinal
momentum fraction, respectively; $j$ ($s$) is the total angular momentum
(spin); $j_{z}$ ($s_{z}$) is the magnetic quantum number; $s_{1}$ and $s_{2}$
are spins of valence quarks. $j$ $=$ $s$ $=$ $1$ for the ground vector
$B_{c}^{\ast}$ meson, and $0$ for the ground pseudoscalar $B_{u,d,s}$ meson.
The spatial wave function of a relativistic scalar harmonic oscillator
potential is given by zpc29.637

${\phi}(\vec{k}_{\perp},x)\,=\,N_{m}\,\sqrt{x\,(1-x)}\,{\exp}\Big{\\{}-\frac{\vec{k}_{\perp}^{2}}{2\,{\omega}^{2}}\Big{\\}}\,{\exp}\Big{\\{}-\frac{m^{2}}{2\,{\omega}^{2}}\,\Big{(}x-\frac{1}{2}-\frac{m_{q_{1}}^{2}-m_{q_{2}}^{2}}{2\,m^{2}}\Big{)}^{2}\Big{\\}},$
(37)

where parameter ${\omega}$ determines the average transverse momentum of
partons, i.e., ${\langle}\vec{k}_{\perp}^{2}{\rangle}$ $=$ ${\omega}^{2}$; $m$
is the mass of the concerned meson; $m_{q_{1}}$ ($m_{q_{2}}$) is the
constituent mass of the decaying (spectator) quark carrying a gluon cloud;
$N_{m}$ is a normalization factor determined by

${\int}d^{2}k_{\perp}{\int}_{0}^{1}dx\,{|}{\phi}(\vec{k}_{\perp},x){|}^{2}\,=\,1.$
(38)

Figure 2: The shapes of the normalized wave functions for the $B_{c}^{\ast}$
and $B_{u,s}$ meson, with parameter ${\omega}$ $=$ $0.4$ GeV (a), $0.6$ GeV
(b), and $m\,{\alpha}_{s}$ (c), respectively.

The form factors at zero momentum transfer are given by zpc29.637

$A_{0}(0)\,=\,A_{3}(0)\,=\,{\int}d^{2}k_{\perp}{\int}_{0}^{1}dx\,{\phi}_{B_{c}^{\ast}}^{(1,0)}(\vec{k}_{\perp},x)\,{\sigma}_{z}^{(1)}\,{\phi}_{B}(\vec{k}_{\perp},x),$
(39)
$J\,=\,\sqrt{2}\,{\int}d^{2}k_{\perp}{\int}_{0}^{1}\frac{dx}{x}\,{\phi}_{B_{c}^{\ast}}^{(1,-1)}(\vec{k}_{\perp},x)\,i\,{\sigma}_{y}^{(1)}\,{\phi}_{B}(\vec{k}_{\perp},x),$
(40) $V(0)\,=\,\frac{m_{c}-m_{q}}{m_{B_{c}^{\ast}}-m_{B_{q}}}\,J,$ (41)
$A_{1}(0)\,=\,\frac{m_{c}+m_{q}}{m_{B_{c}^{\ast}}+m_{B_{q}}}\,J,$ (42)

where ${\sigma}_{z,y}^{(1)}$ are Pauli matrixes acting on the spin indices of
the decaying quark $q_{1}$.

It has been shown zpc29.637  that the form factors are sensitive to the choice
of parameter ${\omega}$. And it is argued zpc29.637  that parameter ${\omega}$
is not expected to be largely different for various mesons due to the flavor
independence of the QCD interactions. Thus the same ${\omega}$ might be
applied to all mesons with the same spectator quark. The motion of the
spectator (bottom) quark is nearly nonrelativistic in the $B_{c}^{\ast}$
${\to}$ $B_{q}$ transition. Thus, nonrelativistic QCD (NRQCD) effective theory
prd46.4052 ; prd51.1125 ; rmp77.1423  could be used to deal with both
$B_{c}^{\ast}$ and $B_{q}$ mesons. According to the NRQCD power counting
rules, the average transverse momentum is the order of ${\omega}$ ${\approx}$
$m\,{\alpha}_{s}$. In order to see the parameter ${\omega}$ effects on the
form factors, we explore two scenarios. One is the same parameter ${\omega}$
for both the $B_{c}^{\ast}$ and $B_{q}$ mesons, and the other is ${\omega}$
$=$ $m\,{\alpha}_{s}$, i.e., ${\omega}$ ${\approx}$ $1.24$ GeV for the
$B_{c}^{\ast}$ meson, $1.10$ GeV for the $B_{s}$ meson, and $1.09$ GeV for the
$B_{u,d}$ mesons. The numerical results for form factors are shown in Table 2.

There are some comments on the form factors. (1) From the expressions in
Eq.(41) and Eq.(42), it is seen that due to the factor
$\frac{m_{c}-m_{q}}{m_{B_{c}^{\ast}}-m_{B_{q}}}$ ${\approx}$ $1$ and
$\frac{m_{c}+m_{q}}{m_{B_{c}^{\ast}}+m_{B_{q}}}$ ${\ll}$ $1$, one can obtain a
relation, $A_{1}(0)$ $<$ $V(0)$. (2) Compared with the integrand in Eq.(39),
there is a factor $1/x$ for the integrand in Eq.(40) with longitudinal
momentum fraction $0$ $<$ $x$ $<$ $1$. Thus, it is expected to have in general
$A_{0,3}(0)$ $<$ $V(0)$. (3) With the relation of form factors in Eq.(20),
$A_{2}(0)$ is significantly enhanced by a factor of
$\frac{2\,m_{B_{c}^{\ast}}}{m_{B_{c}^{\ast}}-m_{B_{q}}}$ (or
$\frac{m_{B_{c}^{\ast}}+m_{B_{q}}}{m_{B_{c}^{\ast}}-m_{B_{q}}}$) relative to
$A_{3}(0)$ (or $A_{1}(0)$). These relations are comprehensively verified by
the numerical results for form factors in Table 2.

In addition, from the numbers in Table 2, it is seen that (1) the form factors
increase as parameter ${\omega}$ increases, due to the fact that the overlap
between wave functions of $B_{c}^{\ast}$ and $B_{q}$ mesons increases as
parameter ${\omega}$ increases, as shown in Fig.2. (2) The flavor symmetry
breaking effects on form factors are small, but the isospin symmetry is
basically held. (3) The values for $A_{2}(0)$ ($V(0)$) are about ten (five)
times as large as those for $A_{1}(0)$, as explained above. The large values
for $A_{2}$ and $V$ would enhance the contributions from the $D$\- and
$P$-wave amplitudes (see Eq.(25) and Eq.(26)).

Table 2: The numerical values for the form factors in the $B_{c}^{\ast}$ ${\to}$ $B_{q}$ transition, where the uncertainties come from both $m_{c}$ and $m_{b}$. transition | ${\omega}$ | $A_{0}(0)$ | $A_{1}(0)$ | $A_{2}(0)$ | $V(0)$
---|---|---|---|---|---
|  $0.4$ GeV | $0.540^{+0.015}_{-0.015}$ | $0.291^{+0.002}_{-0.002}$ | $3.286^{+0.210}_{-0.209}$ | $1.953^{+0.038}_{-0.040}$
$B_{c}^{\ast}$ ${\to}$ $B_{u}$ |  $0.6$ GeV | $0.784^{+0.008}_{-0.008}$ | $0.429^{+0.011}_{-0.011}$ | $4.694^{+0.219}_{-0.222}$ | $2.877^{+0.111}_{-0.109}$
| $m\,{\alpha}_{s}$ | $0.944^{+0.002}_{-0.002}$ | $0.539^{+0.017}_{-0.017}$ | $5.403^{+0.208}_{-0.213}$ | $3.613^{+0.160}_{-0.156}$
|  $0.4$ GeV | $0.540^{+0.015}_{-0.015}$ | $0.291^{+0.002}_{-0.002}$ | $3.288^{+0.210}_{-0.209}$ | $1.954^{+0.038}_{-0.040}$
$B_{c}^{\ast}$ ${\to}$ $B_{d}$ |  $0.6$ GeV | $0.784^{+0.008}_{-0.008}$ | $0.429^{+0.011}_{-0.011}$ | $4.696^{+0.219}_{-0.222}$ | $2.878^{+0.111}_{-0.109}$
| $m\,{\alpha}_{s}$ | $0.944^{+0.002}_{-0.002}$ | $0.539^{+0.017}_{-0.017}$ | $5.405^{+0.208}_{-0.213}$ | $3.614^{+0.160}_{-0.156}$
|  $0.4$ GeV | $0.609^{+0.015}_{-0.015}$ | $0.361^{+0.003}_{-0.003}$ | $3.618^{+0.234}_{-0.234}$ | $1.867^{+0.059}_{-0.061}$
$B_{c}^{\ast}$ ${\to}$ $B_{s}$ |  $0.6$ GeV | $0.821^{+0.007}_{-0.007}$ | $0.494^{+0.012}_{-0.012}$ | $4.785^{+0.242}_{-0.247}$ | $2.554^{+0.122}_{-0.120}$
| $m\,{\alpha}_{s}$ | $0.954^{+0.002}_{-0.002}$ | $0.598^{+0.018}_{-0.017}$ | $5.268^{+0.232}_{-0.238}$ | $3.097^{+0.163}_{-0.159}$

##  III Numerical results and discussion

In the rest frame of the $B_{c}^{\ast}$ meson, branching ratios are defined as

${\cal B}r(B_{c}^{\ast}{\to}BV)\,=\,\frac{1}{24{\pi}}\,\frac{p_{\rm
cm}}{m_{B_{c}^{\ast}}^{2}{\Gamma}_{B_{c}^{\ast}}}\,\big{\\{}{|}H_{+}{|}^{2}+{|}H_{0}{|}^{2}+{|}H_{-}{|}^{2}\big{\\}},$
(43) ${\cal B}r(B_{c}^{\ast}{\to}BP)\,=\,\frac{1}{24{\pi}}\,\frac{p_{\rm
cm}}{m_{B_{c}^{\ast}}^{2}{\Gamma}_{B_{c}^{\ast}}}\,{|}{\cal
A}(B_{c}^{\ast}{\to}BP){|}^{2},$ (44)

where ${\Gamma}_{B_{c}^{\ast}}$ is the full width of the $B_{c}^{\ast}$ meson.

Because the electromagnetic radiation process $B_{c}^{\ast}$ ${\to}$
$B_{c}{\gamma}$ dominates the $B_{c}^{\ast}$ meson decay, to a good
approximation, ${\Gamma}_{B_{c}^{\ast}}$ ${\simeq}$
${\Gamma}(B_{c}^{\ast}{\to}B_{c}{\gamma})$. However, there is still no
experimental information about the partial width
${\Gamma}(B_{c}^{\ast}{\to}B_{c}{\gamma})$ now, because the photon from the
$B_{c}^{\ast}$ ${\to}$ $B_{c}{\gamma}$ process is too soft to be easily
identified. The information on ${\Gamma}(B_{c}^{\ast}{\to}B_{c}{\gamma})$
comes mainly from theoretical estimation on the magnetic dipole (M1)
transition, i.e., epja52.90

${\Gamma}(B_{c}^{\ast}{\to}B_{c}{\gamma})\ =\ \frac{4}{3}\,{\alpha}_{\rm
em}\,k_{\gamma}^{3}\,{\mu}^{2}_{h},$ (45)

where ${\alpha}_{\rm em}$ is the fine-structure constant of electromagnetic
interaction; $k_{\gamma}$ is the photon momentum in the rest frame of initial
state; ${\mu}_{h}$ is the M1 moment of $B_{c}^{\ast}$ meson. There are plenty
of theoretical predictions on ${\Gamma}(B_{c}^{\ast}{\to}B_{c}{\gamma})$, for
example, the numbers in Tables 3 and 6 in Ref.epja52.90 . However, these
estimations still suffer from large uncertainties due to our lack of a precise
value for ${\mu}_{h}$. To give a quantitative evaluation,
${\Gamma}_{B_{c}^{\ast}}$ $=$ $50$ eV will be fixed in our calculation for the
moment. The value of $50$ eV seems reasonable since it is close to the value
given by the potential model (PM) which produces good agreement with
experiment for the measured $J/{\psi}$ ${\to}$ ${\eta}_{c}{\gamma}$ decay
rate. The value for the charm quark magnetic moment ${\mu}_{c}$ obtained from
the charmonium M1 decay width can now be used to predict the $B_{c}^{\ast}$
${\to}$ $B_{c}{\gamma}$ decay width, with a very small $b$ quark magnetic
moment ${\mu}_{b}$ $=$ $-0.06\,{\mu}_{N}$ given in Ref.epja52.90 .

The numerical values for other input parameters are listed in Table 3. Unless
otherwise stated, their central values will be fixed as the default inputs.
Our numerical results are presented in Table 4. The following are some
comments.

Table 3: The numerical values for input parameters. Wolfenstein parameters pdg
---
${\lambda}$ $=$ $0.22506{\pm}0.00050$, $A$ $=$ $0.811{\pm}0.026$, $\bar{\rho}$
$=$ $0.124^{+0.019}_{-0.018}$, $\bar{\eta}$ $=$ $0.356{\pm}0.011$;
Mass of particles and QCD characteristic scale
$m_{B_{c}^{\ast}}$ $=$ $6332{\pm}9$ MeV111Other predictions of the $B_{c}^{\ast}$ meson mass with different models can be found in Table II of Ref.prd93.074010 . prd86.094510 , |  $m_{{\pi}^{+}}$ $=$ $139.57$ MeV pdg , |  $m_{{\pi}^{0}}$ $=$ $134.98$ MeV pdg ,
$m_{B_{u}}$ $=$ $5279.31{\pm}0.15$ MeV pdg , |  $m_{K^{+}}$ $=$ $493.677{\pm}0.016$ MeV pdg , |  $m_{K^{0}}$ $=$ $497.611{\pm}0.013$ MeV pdg ,
$m_{B_{d}}$ $=$ $5279.62{\pm}0.15$ MeV pdg , |  $m_{\eta}$ $=$ $547.862{\pm}0.017$ MeV pdg , |  $m_{{\eta}^{\prime}}$ $=$ $957.78{\pm}0.06$ MeV pdg ,
$m_{B_{s}}$ $=$ $5366.82{\pm}0.22$ MeV pdg , |  $m_{K^{{\ast}+}}$ $=$ $891.66{\pm}0.26$ MeV pdg , |  $m_{K^{{\ast}0}}$ $=$ $895.81{\pm}0.19$ MeV pdg ,
$m_{\rho}$ $=$ $775.26{\pm}0.25$ MeV pdg , |  $m_{\omega}$ $=$ $782.62{\pm}0.12$ MeV pdg , |  $m_{\phi}$ $=$ $1019.461{\pm}0.019$ MeV pdg ,
$m_{b}$ $=$ $4.18^{+0.04}_{-0.03}$ GeV pdg , |  $m_{c}$ $=$ $1.27{\pm}0.03$ GeV pdg , |  ${\Lambda}^{(5)}_{\rm QCD}$ $=$ $210{\pm}14$ MeV pdg ,
$m_{s}$ $=$ $0.51$ GeV uds , |  $m_{u,d}$ $=$ $0.31$ GeV uds , |  ${\Lambda}^{(4)}_{\rm QCD}$ $=$ $292{\pm}16$ MeV pdg ,
Decay constants
$f_{\pi}$ $=$ $130.2{\pm}1.7$ MeV pdg , |  $f_{K}$ $=$ $155.6{\pm}0.4$ MeV pdg , |  $f_{K^{\ast}}$ $=$ $220{\pm}5$ MeV jhep.0703.069 ,
$f_{\rho}$ $=$ $216{\pm}3$ MeV jhep.0703.069 , |  $f_{\omega}$ $=$ $187{\pm}5$ MeV jhep.0703.069 , |  $f_{\phi}$ $=$ $215{\pm}5$ MeV jhep.0703.069 ,
$f_{q}$ $=$ $(1.07{\pm}0.02)\,f_{\pi}$ prd.58.114006 , |  $f_{s}$ $=$ $(1.34{\pm}0.06)\,f_{\pi}$ prd.58.114006 , |
Gegenbauer moments at the sacle of ${\mu}$ $=$ 1 GeV
$a_{1}^{\pi}$ $=$ $a_{1}^{{\eta}_{q}}$ $=$ $a_{1}^{{\eta}_{s}}$ $=$ $0$ jhep.0605.004 , |  $a_{1}^{K}$ $=$ $0.06{\pm}0.03$ jhep.0605.004 , |  $a_{1}^{\rho}$ $=$ $a_{1}^{\omega}$ $=$ $a_{1}^{\phi}$ $=$ $0$ jhep.0703.069 ,
$a_{2}^{\pi}$ $=$ $a_{2}^{{\eta}_{q}}$ $=$ $a_{2}^{{\eta}_{s}}$ $=$ $0.25{\pm}0.15$ jhep.0605.004 , |  $a_{2}^{K}$ $=$ $0.25{\pm}0.15$ jhep.0605.004 , |  $a_{2}^{\rho}$ $=$ $a_{2}^{\omega}$ $=$ $0.15{\pm}0.07$ jhep.0703.069 ,
$a_{1}^{K^{\ast}}$ $=$ $0.03{\pm}0.02$ jhep.0703.069 , |  $a_{2}^{K^{\ast}}$ $=$ $0.11{\pm}0.09$ jhep.0703.069 , |  $a_{2}^{\phi}$ $=$ $0.18{\pm}0.08$ jhep.0703.069 .
Table 4: Branching ratios for the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays calculated with the scale of ${\mu}$ $=$ $m_{c}$, where the parameters in the “CKM” (“$a_{i}$”) column give the CKM factors (coefficients) of the decay amplitude; the uncertainties come from mass $m_{c}$ and $m_{b}$. final | parameters |  | branching ratio
---|---|---|---
state | CKM | $a_{i}$ | case |  ${\omega}$ $=$ $0.4$ GeV |  ${\omega}$ $=$ $0.6$ GeV |  ${\omega}$ $=$ $m\,{\alpha}_{s}$ | unit
$B_{s}^{0}{\rho}^{+}$ |  $V_{cs}^{\ast}V_{ud}$ ${\sim}$ ${\cal O}(1)$ | $a_{1}$ | 1-I | $6.29^{+0.02}_{-0.02}$ | $11.67^{+0.36}_{-0.35}$ | $16.77^{+0.72}_{-0.68}$ | $10^{-7}$
$B_{s}^{0}{\pi}^{+}$ |  $V_{cs}^{\ast}V_{ud}$ ${\sim}$ ${\cal O}(1)$ | $a_{1}$ | 1-I | $4.00^{+0.21}_{-0.20}$ | $7.26^{+0.15}_{-0.15}$ | $9.82^{+0.06}_{-0.06}$ | $10^{-7}$
$B_{s}^{0}K^{{\ast}+}$ |  $V_{cs}^{\ast}V_{us}$ ${\sim}$ ${\cal O}({\lambda})$ | $a_{1}$ | 1-II | $2.21^{+0.02}_{-0.03}$ | $4.12^{+0.18}_{-0.17}$ | $6.01^{+0.32}_{-0.30}$ | $10^{-8}$
$B_{s}^{0}K^{+}$ |  $V_{cs}^{\ast}V_{us}$ ${\sim}$ ${\cal O}({\lambda})$ | $a_{1}$ | 1-II | $1.98^{+0.10}_{-0.10}$ | $3.60^{+0.08}_{-0.08}$ | $4.87^{+0.03}_{-0.03}$ | $10^{-8}$
$B_{d}^{0}{\rho}^{+}$ |  $V_{cd}^{\ast}V_{ud}$ ${\sim}$ ${\cal O}({\lambda})$ | $a_{1}$ | 1-II | $3.51^{+0.05}_{-0.05}$ | $7.51^{+0.17}_{-0.17}$ | $11.47^{+0.44}_{-0.41}$ | $10^{-8}$
$B_{d}^{0}{\pi}^{+}$ |  $V_{cd}^{\ast}V_{ud}$ ${\sim}$ ${\cal O}({\lambda})$ | $a_{1}$ | 1-II | $2.14^{+0.13}_{-0.13}$ | $4.50^{+0.11}_{-0.11}$ | $6.53^{+0.05}_{-0.05}$ | $10^{-8}$
$B_{d}^{0}K^{{\ast}+}$ |  $V_{cd}^{\ast}V_{us}$ ${\sim}$ ${\cal O}({\lambda}^{2})$ | $a_{1}$ | 1-III | $1.45^{+0.01}_{-0.01}$ | $3.13^{+0.11}_{-0.11}$ | $4.84^{+0.24}_{-0.23}$ | $10^{-9}$
$B_{d}^{0}K^{+}$ |  $V_{cd}^{\ast}V_{us}$ ${\sim}$ ${\cal O}({\lambda}^{2})$ | $a_{1}$ | 1-III | $1.15^{+0.07}_{-0.07}$ | $2.41^{+0.06}_{-0.06}$ | $3.50^{+0.02}_{-0.02}$ | $10^{-9}$
$B_{u}^{+}\overline{K}^{{\ast}0}$ |  $V_{cs}^{\ast}V_{ud}$ ${\sim}$ ${\cal O}(1)$ | $a_{2}$ | 2-I | $5.59^{+0.03}_{-0.03}$ | $12.08^{+0.39}_{-0.36}$ | $18.80^{+0.85}_{-0.80}$ | $10^{-8}$
$B_{u}^{+}\overline{K}^{0}$ |  $V_{cs}^{\ast}V_{ud}$ ${\sim}$ ${\cal O}(1)$ | $a_{2}$ | 2-I | $3.48^{+0.29}_{-0.27}$ | $7.32^{+0.34}_{-0.32}$ | $10.60^{+0.31}_{-0.30}$ | $10^{-8}$
$B_{u}^{+}K^{{\ast}0}$ |  $V_{cd}^{\ast}V_{us}$ ${\sim}$ ${\cal O}({\lambda}^{2})$ | $a_{2}$ | 2-III | $1.59^{+0.01}_{-0.01}$ | $3.44^{+0.11}_{-0.10}$ | $5.34^{+0.24}_{-0.23}$ | $10^{-10}$
$B_{u}^{+}K^{0}$ |  $V_{cd}^{\ast}V_{us}$ ${\sim}$ ${\cal O}({\lambda}^{2})$ | $a_{2}$ | 2-III | $9.82^{+0.81}_{-0.77}$ | $20.67^{+0.94}_{-0.90}$ | $29.95^{+0.85}_{-0.82}$ | $10^{-11}$
$B_{u}^{+}{\rho}^{0}$ |  $V_{cd}^{\ast}V_{ud}$ ${\sim}$ ${\cal O}({\lambda})$ | $a_{2}$ | 2-II | $1.84^{+0.04}_{-0.03}$ | $3.95^{+0.07}_{-0.07}$ | $6.08^{+0.20}_{-0.19}$ | $10^{-9}$
$B_{u}^{+}{\omega}$ |  $V_{cd}^{\ast}V_{ud}$ ${\sim}$ ${\cal O}({\lambda})$ | $a_{2}$ | 2-II | $1.36^{+0.02}_{-0.02}$ | $2.93^{+0.06}_{-0.05}$ | $4.51^{+0.15}_{-0.14}$ | $10^{-9}$
$B_{u}^{+}{\phi}$ |  $V_{cs}^{\ast}V_{us}$ ${\sim}$ ${\cal O}({\lambda})$ | $a_{2}$ | 2-II | $1.36^{+0.01}_{-0.01}$ | $2.95^{+0.13}_{-0.12}$ | $4.64^{+0.26}_{-0.25}$ | $10^{-9}$
$B_{u}^{+}{\pi}^{0}$ |  $V_{cd}^{\ast}V_{ud}$ ${\sim}$ ${\cal O}({\lambda})$ | $a_{2}$ | 2-II | $9.24^{+0.77}_{-0.72}$ | $19.44^{+0.89}_{-0.85}$ | $28.17^{+0.81}_{-0.78}$ | $10^{-10}$
$B_{u}^{+}{\eta}$ |  $V_{cd}^{\ast}V_{ud}$, $V_{cs}^{\ast}V_{us}$ | $a_{2}$ | 2-II | $2.37^{+0.20}_{-0.19}$ | $4.98^{+0.23}_{-0.22}$ | $7.22^{+0.21}_{-0.20}$ | $10^{-9}$
$B_{u}^{+}{\eta}^{\prime}$ |  $V_{cd}^{\ast}V_{ud}$, $V_{cs}^{\ast}V_{us}$ | $a_{2}$ |  | $6.62^{+0.55}_{-0.52}$ | $13.93^{+0.64}_{-0.61}$ | $20.18^{+0.58}_{-0.56}$ | $10^{-11}$

(1) According to the relative sizes of coefficients $a_{1,2}$ and CKM factors,
the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays could be classified into
six cases (see Table 4). There is a clear hierarchical relation among
branching ratios, i.e., ${\cal B}r(\text{case 1-I})$ ${\sim}$ ${\cal
O}(10^{-7})$, ${\cal B}r(\text{case 1-II})$ ${\sim}$ ${\cal O}(10^{-8})$,
${\cal B}r(\text{case 1-III})$ ${\sim}$ ${\cal O}(10^{-9})$, and ${\cal
B}r(\text{case 2-I})$ ${\sim}$ ${\cal O}(10^{-8})$, ${\cal B}r(\text{case
2-II})$ ${\sim}$ ${\cal O}(10^{-9})$, ${\cal B}r(\text{case 2-III})$ ${\sim}$
${\cal O}(10^{-10})$.

(2) Branching ratios for the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$ decays are
generally larger than those for the $B_{c}^{\ast}$ ${\to}$ $B_{q}P$ decays
with the same final $B_{q}$ meson, where $V$ and $P$ have the same quark
components. There are two reasons for this. One is the decay constant relation
$f_{V}$ $>$ $f_{P}$, and the other is three partial wave contributions to the
$B_{c}^{\ast}$ ${\to}$ $B_{q}V$ decays rather than only the $P$-wave
contributions to the $B_{c}^{\ast}$ ${\to}$ $B_{q}P$ decays.

It should be pointed out that although the $P$\- and $D$-wave amplitudes for
the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$ decays are enhanced by large values for
the form factors $V$ and $A_{2}$, they are suppressed by a factor of
$\frac{2\,m_{B_{c}^{\ast}}\,m_{V}}{(m_{B_{c}^{\ast}}+m_{B_{q}})^{2}}$ relative
to the $S$-wave amplitude, as discussed above. In addition, the $P$\- and
$D$-wave contributions to helicity amplitudes $H_{\pm}$ in Eq.(28) and $H_{0}$
in Eq.(29) are future suppressed respectively by factors of $\sqrt{y^{2}-1}$
and $(y^{2}-1)/y$ relative to the $S$-wave contribution. Take the
$B_{c}^{\ast}$ ${\to}$ $B_{s}{\rho}$ decay for example,
$\frac{2\,m_{B_{c}^{\ast}}\,m_{\rho}}{(m_{B_{c}^{\ast}}+m_{B_{s}})^{2}}$
${\approx}$ 7%, $\sqrt{y^{2}-1}$ ${\approx}$ $0.7$ and $(y^{2}-1)/y$
${\approx}$ $0.4$, resulting in the polarization fractions $f_{0}$ ${\approx}$
60%, $f_{+}$ ${\approx}$ 30% and $f_{-}$ ${\approx}$ 10% with $f_{0,+,-}$
${\equiv}$
$\frac{{|}H_{0,+,-}{|}^{2}}{{|}H_{0}{|}^{2}+{|}H_{+}{|}^{2}+{|}H_{-}{|}^{2}}$.

(3) The branching ratios for the $B_{c}^{\ast}$ ${\to}$ $B_{s}{\rho}$,
$B_{s}{\pi}$ decays can reach up to ${\cal O}(10^{-7})$. With the estimated
production cross section of the $B_{c}^{\ast}$ meson ${\sim}$ $30$ $nb$ at
LHCprd72.114009 , it is expected to have more than $10^{10}$ $B_{c}^{\ast}$
mesons per ${\rm ab}^{-1}$ data at LHC, corresponding to more than $10^{3}$
events of the $B_{c}^{\ast}$ ${\to}$ $B_{s}{\rho}$, $B_{s}{\pi}$ decays.
Therefore, even with the identification efficiency, the $B_{c}^{\ast}$ ${\to}$
$B_{s}{\rho}$, $B_{s}{\pi}$ decays might be measurable in the future.

(4) Branching ratios for the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays
are several orders of magnitude smaller, especially for the $a_{1}$ dominant
decays, than those for the $B_{c}$ ${\to}$ $B_{q}P$, $B_{q}V$ decays
ahep2015.104378 . This fact might imply that possible background from the
$B_{c}^{\ast}$ ${\to}$ $BV$, $BP$ decays could be safely neglected for an
analysis of the $B_{c}$ ${\to}$ $B_{q}P$, $B_{q}V$ decays, but not vice versa,
i.e., one of main pollution for the $B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$
decays would likely come from the $B_{c}$ decays.

(5) It is seen clearly that the numbers in Table 4 are very sensitive to the
choice of the parameter ${\omega}$. In addition, with a different value for
${\Gamma}_{B_{c}^{\ast}}$, branching ratios in Table 4 should be multiplied by
a factor of ${50\,{\rm eV}}/{{\Gamma}_{B_{c}^{\ast}}}$. Of course, many
factors, such as the choice of scale ${\mu}$, higher order corrections to HME,
$q^{2}$-dependence of form factors, final state interactions, etc., are not
carefully considered in detail here, but have effects on the estimation and
deserve more dedicated study in the future.

##  IV Summary

With the running and upgrading of the LHC, there are certainly huge amounts of
the $B_{c}^{\ast}$ mesons. This would provide us with a possibility of
searching for the $B_{c}^{\ast}$ weak decays in the future. In this paper, the
$B_{c}^{\ast}$ ${\to}$ $B_{q}V$, $B_{q}P$ decays ($q$ $=$ $u$, $d$ and $s$),
induced by the charm quark weak decay, are studied phenomenologically with the
QCDF approach. The form factors for the $B_{c}^{\ast}$ ${\to}$ $B$ transitions
are calculated using the Wirbel-Stech-Bauer model. The nonfactorizable
contributions from the vertex radiative corrections are considered at the
order of ${\alpha}_{s}$. It is found that (1) form factors and branching
ratios are sensitive to models of wave functions; (2) the color-favored and
CKM-allowed $B_{c}^{\ast}$ ${\to}$ $B_{s}{\rho}$, $B_{s}{\pi}$ decays have
large branching ratios of ${\cal O}(10^{-7})$, and might be accessible in the
future LHC experiments.

## Acknowledgments

The work is supported by the National Natural Science Foundation of China
(Grant Nos. U1632109, 11547014 and 11475055). We thank the referees for their
constructive suggestions, and Ms. Nan Li (HNU) for polishing this manuscript.

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# The origin of chirality induced spin selectivity in photo-induced electron
transfer

Thomas P. Fay<EMAIL_ADDRESS>Department of Chemistry, University
of California, Berkeley, CA 94720, USA  David T. Limmer  Department of
Chemistry, University of California, Berkeley, CA 94720, USA  Kavli Energy
Nanoscience Institute at Berkeley, Berkeley, CA 94720, USA  Chemical Sciences
Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley,
CA 94720, USA

###### Abstract

Here we propose a mechanism by which spin polarization can be generated
dynamically in chiral molecular systems undergoing photo-induced electron
transfer. The proposed mechanism explains how spin polarization emerges in
systems where charge transport is dominated by incoherent hopping, mediated by
spin orbit and electronic exchange couplings through an intermediate charge
transfer state. We derive a simple expression for the spin polarization that
predicts a non-monotonic temperature dependence consistent with recent
experiments. We validate this theory using approximate quantum master
equations and the numerically exact hierarchical equations of motion. The
proposed mechanism of chirality induced spin selectivity should apply to many
chiral systems, and the ideas presented here have implications for the study
of spin transport at temperatures relevant to biology, and provide simple
principles for the molecular control of spins in fluctuating environments.

Chirality induced spin selectivity (CISS), in which molecular chirality
controls spin polarization of electrons, has been observed in a wide variety
of systems,[1, 2, 3] including in photo-induced electron transfer in
artificial systems,[4] such as quantum dots,[5] and biological molecules, such
as photosystem I[6] and DNA.[7, 8] Based on this, it has been suggested that
CISS may serve some biological functions[9] and that it could be exploited in
nano- and molecular-scale spintronics technologies.[1, 10] Here we focus on
the CISS effect in photo-induced molecular electron transfer.[6] This effect
is particularly puzzling, as it goes against the conventional notion that the
initial singlet spin state of the system is preserved in this process.[11]

Theoretical descriptions of CISS have primarily been confined to either the
coherent regime,[12, 13, 14, 15, 16] where electron transport is treated as a
coherent tunneling process through a chiral molecular junction between leads,
or static spin polarizations produced by spin-orbit coupling in the ground
state of chiral molecules.[17, 18, 19] However photo-induced electron transfer
in molecules like photosystem I, where electrons incoherently hop between a
discrete set of donor and acceptor sites, cannot be described by either of
these theories.[20] In such systems coupling of the electron transfers to
molecular vibrations is strong,[21] and electronic exchange interactions can
also be significant.[11] This necessitates the development of a theory of CISS
that is appropriate for this incoherent regime of electron transport.
Fortunately such techniques have been developed, primarily for understanding
rates and dynamics of electron transfer reactions,[22, 23] and through the
tools of quantum master equations such theories can be adapted to describe
spin transport.[24, 25] These techniques have recently been used to show that
spin coherence, but no spin polarization, arises in a single electron transfer
step in a chiral environment mediated by spin-orbit coupling.[25] Here we
expand on this theory, and show how the interplay of chirality induced spin
coherence, electron hopping, and electronic exchange coupling can dynamically
produce large spin polarizations in electron transfer reactions. The theory
presented here will be shown to explain the observed temperature dependence of
CISS in photosystem I, and it also provides a framework for engineering
molecules to maximize the CISS effect.

Let us start by reviewing how photo-initiated charge transfer, between a donor
D and acceptor A, in chiral molecules generates spin coherence in charge
transfer states, as described in Ref. 25. This process generally proceeds via
an electron transfer reaction from a bright locally excited singlet state, S_1
= D^*-A, to a charge transfer state, CT = D^$\bullet$+-A^$\bullet$-, which can
exist in either a singlet or triplet spin state,

$\displaystyle\ce{S_{1}->CT}.$

As discussed in Ref. 25 direct diabatic coupling generates the CT state in a
singlet
$\ket{\mathrm{S}}=(\ket{\uparrow_{\mathrm{D}}\downarrow_{\mathrm{A}}}-\ket{\downarrow_{\mathrm{D}}\uparrow_{\mathrm{A}}})/\sqrt{2}$
spin state, whereas spin-orbit coupling (SOC) generates the CT state in the
$i\ket{\mathrm{T}_{0}}=i(\ket{\uparrow_{\mathrm{D}}\downarrow_{\mathrm{A}}}+\ket{\downarrow_{\mathrm{D}}\uparrow_{\mathrm{A}}})/\sqrt{2}$
(taking the $z$ axis to be defined by the spin-orbit coupling vector).[26] So
overall this electron transfer, mediated by spin-orbit coupling in a chiral
molecule, generates the CT state in a coherent superposition of singlet and
$\ket{\mathrm{T}_{0}}$ triplet spin states, and its initial spin density
operator, $\hat{\sigma}_{\ce{CT}}(t)$, is given by

$\displaystyle\hat{\sigma}_{\mathrm{CT}}(0)=\outerproduct{\psi_{\theta}}{\psi_{\theta}},$
(1)

where
$\ket{\psi_{\theta}}=\cos\theta\ket{\mathrm{S}}+i\sin\theta\ket{\mathrm{T}_{0}}$.
The mixing angle $\theta$, is determined by the relative strength of the spin-
preserving diabatic coupling $V_{\mathrm{DA}}$ and the spin-orbit coupling
$\Lambda_{\mathrm{DA}}$, as
$\theta=\arctan(\Lambda_{\mathrm{DA}}/(2V_{\mathrm{DA}}))$, and because the
sign of $\Lambda_{\mathrm{DA}}$ depends on the chirality of the molecule, the
sign of $\theta$ also depends on chirality. The spin polarization of the CT
state is defined as the difference in the $z$ component of the spins between D
and A. This is given by the expectation value of
$\Delta\hat{P}_{z}=\hat{S}_{\mathrm{D}z}-\hat{S}_{\mathrm{A}z}=\outerproduct{\mathrm{S}}{\mathrm{T}_{0}}+\outerproduct{\mathrm{T}_{0}}{\mathrm{S}}$,
so the spin polarization is proportional to the real part of the singlet-
triplet coherence,
$\matrixelement*{\mathrm{S}}{\hat{\sigma}_{\ce{CT}}}{\mathrm{T}_{0}}$. However
the CT state is generated with purely imaginary singlet-triplet coherence, so
no spin polarization is formed directly by the electron transfer.[25]

Now let us consider how the spin state $\hat{\sigma}_{\mathrm{CT}}$ evolves
when there exists an exchange coupling between the donor and acceptor electron
spins in the CT state. In this case the spin Hamiltonian for the CT state can
be taken to be $\hat{H}_{\mathrm{CT}}=-2J\hat{P}_{\mathrm{T}}$
($\hat{P}_{\mathrm{T}}$ is a triplet spin projection operator),[27] and the
time-evolution of the spin state is given by

$\displaystyle
e^{-i\hat{H}_{\mathrm{CT}}t/\hbar}\ket{\psi_{\theta}}=\cos\theta\ket{\mathrm{S}}+i\sin\theta
e^{2iJt/\hbar}\ket{\mathrm{T}_{0}}.$ (2)

We see that exchange coupling generates a complex-valued oscillating singlet-
triplet coherence in the CT spin density operator, and thus the CT state now
has an oscillating real part of its singlet-triplet coherence. This means that
the exchange coupling produces an oscillating spin polarization in the CT
state which is given by,

$\displaystyle\expectationvalue{\Delta
P_{z}(t)}=-\sin(2\theta)\sin(\frac{2Jt}{\hbar}).$ (3)

Figure 1: Schematic of the photo-induced charge transfer mechanism by which
spin polarization can be generated. $\ce{S_{1}}=\ce{D^{*}-A_{1}-A_{2}}$,
$\ce{CT_{1}}=\ce{D^{\bullet+}-A^{\bullet-}_{1}-A_{2}}$ and
$\ce{CT_{2}}=\ce{D^{\bullet+}-A_{1}-A^{\bullet-}}_{2}$. Singlet states are
shown as solid lines and triplet states as dashed lines. Note the non-
negligible S-T gap in CT_1, and the negligible S-T gap in CT_2.

Overall the presence of exchange coupling effects a chirality-dependent
transient oscillating spin polarization in the CT state. However this spin
polarization clearly averages over time to zero. This naturally raises the
question of how a static spin polarization can be generated in a photo-induced
charge transfer reaction. The answer lies in the fact that many photo-
generated CT states are not formed by a simple direct electron transfer, but
rather the final CT state is often formed via a sequence of downhill electron
transfers via intermediate CT states.[11, 21, 20] The final CT state is often
a well-separated radical ion pair state. In this state the exchange coupling
between donor and acceptor spins is very weak, but the intermediate CT states
often have a significantly larger exchange coupling, due to the closer
proximity of the unpaired electrons.[11] In this way, imaginary-valued
singlet-triplet coherence created through spin-orbit coupled charge transfer
can be transiently converted into a net spin polarization between the donor
and acceptor electrons in an intermediate CT state with a large exchange
coupling. This spin polarization can be transferred to the final CT state via
a subsequent incoherent spin state preserving electron transfer. The spin
polarization transferred to the final CT no longer oscillates due to the weak
exchange coupling in this state, thus a net static spin polarization is
generated. This is the basic mechanism by which static spin polarization can
be generated in multi-step electron transfer reactions in chiral molecules.

We will now explore this further, by considering the simplest multi-state
model including only one intermediate CT state in the chiral system (i.e. a
three state model),

$\displaystyle\ce{S_{1}->[$k_{\mathrm{ET}1}$]CT_{1}->[$k_{\mathrm{ET}2}$]CT_{2}}.$

We take the precursor state to be an excited donor state,
$\ce{S_{1}}=\ce{D^{*}-A_{1}-A_{2}}$, the intermediate charge transfer state
corresponds to an electron transfer from D to a primary acceptor A_1, so
$\ce{CT_{1}}=\ce{D^{\bullet+}-A^{\bullet-}_{1}-A_{2}}$, and the final charge
transfer state CT_2 is taken as the state formed by electron transfer from the
primary acceptor A_1 to the secondary acceptor A_2, so
$\ce{CT_{2}}=\ce{D^{\bullet+}-A_{1}-A^{\bullet-}}_{2}$. The CT states can
exist in either singlet or triplet spin states but the S_1 state only exists
in the singlet spin state. A schematic of this system is shown in Fig. 1. To
start with, for simplicity we will assume these downhill electron transfers
occur irreversibly, and we will also assume that only the first electron
transfer is spin-dependent, and occurs via chiral spin-orbit coupled charge
transfer.

In this model, the spin density operators for each state obey the following
set of equations,[24, 25]

$\displaystyle\derivative{t}\hat{\sigma}_{\ce{S_{1}}}(t)$
$\displaystyle=-k_{\mathrm{ET1}}\hat{\sigma}_{\ce{S_{1}}}(t)$ (4a)
$\displaystyle\begin{split}\derivative{t}\hat{\sigma}_{\ce{CT_{1}}}(t)&=-\frac{i}{\hbar}[\hat{H}_{\ce{CT_{1}}}+{\delta\epsilon}\outerproduct{\psi_{\theta}}{\psi_{\theta}},\hat{\sigma}_{\ce{CT_{1}}}(t)]\\\
+k_{\mathrm{ET1}}&\outerproduct{\psi_{\theta}}{\mathrm{S}}\hat{\sigma}_{\ce{S_{1}}}(t)\outerproduct{\mathrm{S}}{\psi_{\theta}}-k_{\mathrm{ET2}}\hat{\sigma}_{\ce{CT_{1}}}(t)\end{split}$
(4b)
$\displaystyle\begin{split}\derivative{t}\hat{\sigma}_{\ce{CT_{2}}}(t)&=-\frac{i}{\hbar}[\hat{H}_{\ce{CT_{2}}},\hat{\sigma}_{\ce{CT_{2}}}(t)]+k_{\mathrm{ET2}}\hat{\sigma}_{\ce{CT_{1}}}(t)\end{split}$
(4c)

Here $\hat{H}_{\ce{CT}_{1}}$ and $\hat{H}_{\ce{CT}_{2}}$ are the spin
Hamiltonians for the $\ce{CT_{1}}$ and $\ce{CT_{2}}$ states, describing all
the spin interactions in these states. Here the spin Hamiltonians are
simplified to just include the exchange couplings, so
$\hat{H}_{\ce{CT_{1}}}=-2J\hat{P}_{\mathrm{T}}$ and $\hat{H}_{\ce{CT_{2}}}=0$
(the energy differences between different CT states enter into the master
equations via the rate constants). This set of coupled equations can be
obtained straightforwardly using perturbative Nakajima-Zwanzig theory,[28, 29]
as has been described previously (a summary of the approximations in Refs. 24
and 25 will be discussed shortly). The shift term $\delta\epsilon$ naturally
emerges from the master equation theory, and its value will depend on the
details of the electron transfer process. For a large downhill driving force,
$\Delta G_{\mathrm{ET1}}$, it can be estimated as
$\delta\epsilon\approx-(V_{\mathrm{DA_{1}}}^{2}+(\Lambda_{\mathrm{DA_{1}}}/2)^{2})/(\lambda_{\mathrm{ET1}}-\Delta
G_{\mathrm{ET1}})$, where $\lambda_{\mathrm{ET1}}$ is the reorganization
energy for the first electron transfer. We can also understand the
$\delta\epsilon$ term as a net chiral superexchange spin-orbit coupling term,
which couples $\mathrm{S}$ and $\mathrm{T}_{0}$ states via the S_1 state.

One can then solve these Eq. (4) analytically for an initial state where only
S_1 is populated, with
$\hat{\sigma}_{\ce{S_{1}}}(0)=\outerproduct{\mathrm{S}}{\mathrm{S}}$, from
which the final spin polarization in CT_2, given by $\expectationvalue{\Delta
P_{z}}=\lim_{t\to\infty}\tr[(\hat{S}_{\mathrm{D}z}-\hat{S}_{\mathrm{A}z})\hat{\sigma}_{\ce{CT_{2}}}(t)]$,
is found to be

$\displaystyle\expectationvalue{\Delta
P_{z}}=-\frac{k_{\mathrm{ET2}}(2J/\hbar)\sin(2\theta)}{k_{\mathrm{ET2}}^{2}+\Omega^{2}},$
(5)

where
$\hbar\Omega=\sqrt{4J^{2}+{\delta\epsilon}^{2}+4J{\delta\epsilon}\cos(2\theta)}$.
Simultaneously a zero quantum coherence (which depends on the imaginary part
of the $\mathrm{S}$-$\mathrm{T}_{0}$ coherence) is produced

$\displaystyle\expectationvalue{S_{\mathrm{D}x}S_{\mathrm{A}y}\\!-\\!S_{\mathrm{D}y}S_{\mathrm{A}x}}\\!=\\!\sin(2\theta)\\!\left(1\\!-\frac{2(J/\hbar)^{2}}{k_{\mathrm{ET2}}^{2}+\Omega^{2}}\\!-\\!\frac{J{\delta\epsilon}\cos(2\theta)}{(\hbar\Omega)^{2}}\\!\right)\\!.$
(6)

This zero quantum coherence is produced directly by the initial SOC mediated
electron transfer from S_1, and modified by the spin dynamics in CT_1.[25]
Lastly, we find the final singlet population is given by

$\displaystyle\expectationvalue{\rho_{\mathrm{S}}}=\cos^{2}\theta\left(1+\frac{2J\delta\epsilon/\hbar^{2}}{k_{\mathrm{ET2}}^{2}+\Omega^{2}}-\frac{2J\delta\epsilon\cos(2\theta)}{(\hbar\Omega)^{2}}\right)$
(7)

and the $\mathrm{T}_{0}$ population is given by
$\expectationvalue*{\rho_{\mathrm{T}_{0}}}=1-\expectationvalue*{\rho_{\mathrm{S}}}$.
We see the final spin density operator for the CT_2 is a combination of the
phenomenological theory in Ref. 30 and the theory obtained from the one-step
transfer model in Ref. 25.

Let us now discuss a few points of interest on the expression for the spin
polarization in Eq. (5). Firstly, this expression is an odd function of
$\theta=\arctan({\Lambda}_{\mathrm{DA_{1}}}/2V_{\mathrm{DA_{1}}})$, so on
changing the chirality of the system the sign of this polarization is
reversed, because the sign of $\Lambda_{\mathrm{DA_{1}}}$ changes, which
confirms that this corresponds to a CISS effect. Secondly, in the absence of
the shift term $\delta\epsilon$ the spin polarization is bounded by
$|\expectationvalue*{\Delta P_{z}}|\leq|\sin(2\theta)|$ as
$k_{\mathrm{ET1}}/(2J)$ is varied. In the limit of weak spin-orbit coupling,
this bound is simply $|\Lambda_{\mathrm{DA_{1}}}/V_{\mathrm{DA_{1}}}|$, which
is the ratio of the spin orbit coupling to the diabatic coupling for the first
electron transfer. However when the shift term is included it is possible for
$\Omega^{2}<(2J)^{2}$. For example if $J$ and $\delta\epsilon$ have opposite
signs and $\cos(2\theta)>0$, the polarization can be enhanced significantly
beyond the maximum that one predicts in the absence of $\delta\epsilon$.

Figure 2: Population and spin polarization dynamics for a three CT state model
of electron transfer calculated with HEOM (solid lines) and the second order
QME (dashed lines). Left column: total populations of S_1 (grey), CT_1
(orange) and CT_2 (purple) as a function of time. Right column: spin
polarization in CT_1 (orange) and CT_2 (purple) as a function of time, with
the value predicted by Eq. (5) indicated by the black dashed line. From top to
bottom the value of the exchange coupling in the CT_1 state is varied with
$J/g_{\mathrm{e}}\mu_{\mathrm{B}}=$ 100 mT (top), 10 mT (middle) and 1 mT
(bottom), where $\mu_{\mathrm{B}}$ is the Bohr magneton and $g_{\mathrm{e}}$
is the $g$ factor for the free electron spin.

As a final point, we note that the magnitude of the spin polarization
$|\expectationvalue*{P_{z}}|$ goes through a maximum when
$k_{\mathrm{ET2}}=\Omega$. This can be understood simply as follows. When
$k_{\mathrm{ET2}}$ is very high compared to the frequency at which
polarization in CT_1 oscillates, $\Omega$, then there is no time for spin
polarization to build up in CT_1 before this state decays to CT_2. Conversely
when $k_{\mathrm{ET2}}$ is very low compared to $\Omega$, the spin
polarization is averaged to zero by the oscillations as it is transferred, so
no net spin polarization in CT_2 is observed. This observation has important
implications for the temperature dependence of $\expectationvalue{\Delta
P_{z}}$, as we will discuss shortly.

In order to further explore this theory, and to demonstrate the validity of
the perturbative quantum master equations (QMEs) and additional approximations
used to obtain the expressions above, we performed simulations of the three
state condensed phase electron transfer process illustrated in Fig. 1. We
employ both numerically exact quantum dynamics and the approximate QMEs, with
the nuclear degrees of freedom treated as a harmonic bath. The total
Hamiltonian in this model is a sum of nuclear (bath) kinetic energy, potential
energy terms for the different states, and a diabatic/SOC term (further
details are given in the SI, Eq. (S.1)). Using the Hierarchical Equations of
Motion (HEOM), we can compute the exact dynamics when the potentials are
harmonic.[31]

For comparison, we also perform simulations of this model with the full
perturbative QMEs, Eq. (4), modified to include back reaction terms.[25] To
briefly summarise the approximations used in Refs. 24 and 25 to obtain these,
in these QMEs we assume the coherences between different electron transfer
states, e.g. $\outerproduct{\ce{S}_{1}}{\ce{CT_{1}},\mathrm{S}}$, are short-
lived and small (due to the large differences between potential energy
surfaces), so these coherences are projected out using Nakajima-Zwanzig theory
(but spin coherences are retained). The Nakajima-Zwanzig kernel is then
expanded at lowest order in the diabatic couplings, with spin Hamiltonian
terms in the kernel ignored (due to the large separation in energy scale
between the nuclear motion and spin dynamics), and the population transfer
dynamics are treated as Markovian. For the harmonic bath model given above,
the second order QME parameters (rate constants and $\delta\epsilon$) can be
computed exactly from the spectral density,[24] details of which are given in
the SI. In the high temperature limit, these rates can be well approximated
with Marcus theory,
$k_{\mathrm{ET}}=(1/\hbar)\Gamma_{\mathrm{DA}}^{2}(\pi/\lambda_{\mathrm{ET}}k_{\mathrm{B}}T)^{1/2}e^{-(\Delta
G_{\mathrm{ET}}+\lambda_{\mathrm{ET}})^{2}/4\lambda_{\mathrm{ET}}k_{\mathrm{B}}T}$,
where
$\Gamma_{\mathrm{DA}}=(V_{\mathrm{DA}}^{2}+\Lambda_{\mathrm{DA}}^{2}/4)^{1/2}$
is the combined diabatic/SOC coupling for the electron transfer.[22] Although
we only consider harmonic potential energy surfaces here, the QMEs apply to
general potential energy surfaces for the electron transfer states.

We set the free energy changes of the electron transfers to be $\Delta
G_{\mathrm{ET1}}=-0.1\text{ eV}$ and $\Delta G_{\mathrm{ET2}}=-0.25\text{
eV}$, with reorganisation energies of $\lambda_{\mathrm{ET1}}=0.1\text{ eV}$
and $\lambda_{\mathrm{ET2}}=0.2\text{ eV}$. Diabatic state couplings are taken
as
$\Gamma_{\mathrm{DA_{1}}}=(V_{\mathrm{DA}_{1}}^{2}+\Lambda_{\mathrm{DA_{1}}}^{2}/4)^{1/2}=0.5\text{
meV}$ (S_1-CT_1 coupling) and
$\Gamma_{\mathrm{A_{1}A_{2}}}=V_{\mathrm{A_{1}A_{2}}}=0.25\text{ meV}$
(CT_1-CT_2 coupling), and $\theta=\pi/16$, which corresponds to $\sim\\!4\%$
of CT_1 molecules being formed in the triplet state.[32, 33, 34] In these
models we treat the intramolecular vibrational modes and solvent bath as a
single harmonic bath with a Debye spectral density,[31]
$\mathcal{J}(\omega)=2\lambda_{\mathrm{D}}\gamma_{\mathrm{D}}\omega/(\gamma_{\mathrm{D}}^{2}+\omega^{2})$,with
a cut-off frequency of $\hbar\gamma_{\mathrm{D}}=0.05k_{\mathrm{B}}T$, at
$T=300\text{ K}$, and with $\lambda_{\mathrm{D}}=\lambda_{\mathrm{ET1}}$.
These parameters are chosen to be representative of electron transfers in
typical organic donor-acceptor systems,[21, 35] whilst being computationally
tractable with the HEOM method.

Figure 2 shows the results of these simulations for a range of values of the
$J$ coupling (decreasing from top to bottom) in the intermediate state for
populations (left) and spin polarizations (right). The population dynamics
clearly show that hopping dominates the ET process in these examples, with a
significant transient population of the CT_1 appearing on a time-scale of
$\sim 0.25\text{ ns}$ and decaying on a time-scale of $\sim 1\text{ ns}$. We
see the final spin polarization in CT_2 increases and then decreases with
decreasing $J$ (note the different $\expectationvalue{\Delta P_{z}(t)}$ scales
from top to bottom), as predicted by the simple theory in Eq. (5). Large spin
polarizations, on the order of 40%, can be observed even with a small fraction
of triplet states formed in ET1. The second order QMEs in this example are
very accurate for both population and spin polarization dynamics, which
validates the approximations of Markovian dynamics and spin-independent rate
constants invoked to derive these. The prediction from Eq. (5) for the final
spin polarization (shown as a black dashed line in the left panels of Fig. 2),
provides an accurate estimate in these examples, with small deviations due to
back reactions.

Figure 3: HEOM dynamics of the three state model with larger diabatic state
couplings. Populations (top panel), and spin polarization $O={\Delta P_{z}}$
and zero quantum coherences
$O={S_{\mathrm{D}x}S_{\mathrm{A}y}-S_{\mathrm{D}y}S_{\mathrm{A}x}}$ (bottom
panel) with the same colour scheme as Fig. 2. Other ET parameters are the same
as in Fig. 2.

In the above examples we have only considered a regime where second order
perturbation theory is valid for the rate constants and $\delta\epsilon$. The
general form of the master equation presented above is only slightly altered
on including higher order effects in the electronic coupling[24, 25];
specifically we should add a decoherence term of the form
$-(k_{\mathrm{D}}/2)[\outerproduct{\psi_{\theta}}{\psi_{\theta}},[\outerproduct{\psi_{\theta}}{\psi_{\theta}},\hat{\sigma}_{\ce{CT_{1}}}(t)]]$
to the equation for $\hat{\sigma}_{\ce{CT_{1}}}(t)$ .[25] This causes the spin
polarization in the intermediate state to decay, but for sufficiently large
$2J$ significant spin polarization can still emerge and subsequently be
transferred to the final CT state. This is demonstrated in Fig. 3 where we
show the population dynamics (top panel), spin polarization (bottom panel,
solid lines), and zero quantum coherence (bottom panel, dashed lines) for the
three state model with $\Gamma_{\mathrm{DA_{1}}}=10\text{ meV}$,
$V_{\mathrm{A_{1}A_{2}}}=5\text{ meV}$ and $J/g_{e}\mu_{\mathrm{B}}=2\text{
T}$. In this limit the second order rate constant for the first electron
transfers are approximately a factor of 5 too large compared to those obtained
by fitting the population dynamics, so this example is clearly outside of the
limits of second order perturbation theory. However, we still clearly see the
emergence of a large static spin polarization ($\sim 30\%$) in the CT_2 state,
as well as significant zero quantum coherence. This demonstrates that this
mechanism can produce significant spin polarizations even outside the second
order limit.

Figure 4: Experimental data from Ref. 6 for the corrected spin-polarization in
photosystem I (left), and a model based on Eq. (5) for the spin polarization
relative to its maximum value (right), $\expectationvalue{\Delta
P_{z}}/\expectationvalue{\Delta
P_{z}}_{\mathrm{max}}=2aT^{-1/2}e^{-E_{a}/k_{\mathrm{B}}T}[1+(aT^{-1/2}e^{-E_{a}/k_{\mathrm{B}}T})^{2}]^{-1}$.
The fitted parameters are $a=8.346\times 10^{6}\text{ K}^{1/2}$ and
$E_{a}=0.32\text{ eV}$.

Having validated the general theory on a model condensed phase electron
transfer, let us consider how this hopping mechanism predicts a temperature
dependent spin polarization in chiral systems. Returning to Eq. (5) for
$\expectationvalue{\Delta P_{z}}$, we can see how this temperature dependence
emerges. Typically the exchange coupling, and energy shift terms are, to a
good approximation, independent of temperature,[24] whereas if the electron
transfer is activated, $k_{\mathrm{ET2}}$ will have an exponential dependence
on inverse temperature.[21] Therefore increasing $T$ will generally increase
the ratio $k_{\mathrm{ET2}}/\Omega$, so the spin polarization will go through
a maximum at a particular temperature, where $k_{\mathrm{ET2}}/\Omega=1$.
Interestingly this temperature dependence of the spin polarization has been
observed in the experiments of Carmeli et al. on CISS in electron transfer in
photosystem I,[6] which is known to proceed via several electron hopping
steps.[20] We have fitted the experimental temperature dependent spin
polarization, relative to its maximum value, for photosystem I to a model
based on Eq. (5) assuming Marcus theory[36] can be applied to describe the
temperature dependence of the rate constants. The experimental data and model
are shown in Fig. 4, where we see this model is consistent with the available
experimental data, and the fitted activation energy of $E_{a}=0.32\text{ eV}$
is in line with the activation energies of the electron transfers in
photosystem I.[35] It should be noted that the spin-polarization reported in
these experiments exceeds the theoretical maximum predicted by Eq. (5) of 50%.
We believe this is due to the error in the correction procedure employed in
Ref. 6 for obtaining the spin polarizations, which, as discussed by the
authors, introduces a large error in the corrected spin polarization. The
“uncorrected” experimental spin polarizations, which have a maximum value of
¡10%, still have a non-monotonic dependence on temperature,[6] which is still
consistent with our proposed mechanism of CISS.

To conclude, in this work we have shown how photo-induced spin-orbit coupled
charge transport in chiral systems, together with multistep charge transport
via CT states with significant exchange couplings, generates the CISS spin
polarization effect. We also see that a zero quantum coherence in the final CT
state emerges. The CISS effect described here depends strongly on electron
transfer parameters, in particular the rate constant for downhill electron
transfer for intermediate CT states. This predicts a specific form for the
temperature dependence of the spin polarization, which is consistent with the
experiments of Carmeli et al. in Ref. 6.

This theory of the CISS effect in molecular systems opens the door to many
potential avenues of research. With a theory for the ZQCs and spin
polarizations generated in chiral molecules, well-established principles for
synthetically tuning electronic properties of donor-acceptor systems[11, 37]
could be applied to engineer systems displaying large CISS effects, which can
be probed using a variety of possible experiments.[4, 3, 25, 30, 38] In fact
the first examples of such molecules have already been synthesised.[39] These
systems could find an array of uses in single molecule-scale quantum computing
and quantum information applications.[37, 1] It is also known that electron
hopping occurs in many important biological electron transfer systems,[21]
such as in photosynthesis in photosystem I[20] and in cryptochromes, which are
involved in signalling and potentially magnetoreception.[40] What role role
could molecular CISS play in these systems?[9] For example, could biological
systems have evolved to inhibit singlet-selective electron recombination
reactions using CISS? Equipped with a microscopic theory of chirality induced
spin effects in electron transfer reactions, it may be possible to answer such
questions.

## Acknowledgements

We would like to thank Anthony Poggioli for useful comments on this
manuscript. We are also thankful to Peter Hore for pointing out the connection
between this work and the CIDEP effect in the EPR literature. T.P.F and D.T.L.
were supported by the U.S. Department of Energy, Office of Science, Basic
Energy Sciences, CPIMS Program Early Career Research Program under Award No.
DE-FOA0002019.

## Supporting Information

Further details of the HEOM and QME calculations, a brief analysis of the
effects of competing back reactions and singlet-triplet dephasing on spin
polarization, and a discussion of the connection between SOC mediated CISS and
chemically induced dynamic electron polarization (CIDEP).

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