††thanks: corresponding author

# Study on the ${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ weak decays

Junfeng Sun  Institute of Particle and Nuclear Physics, Henan Normal
University, Xinxiang 453007, China  Lili Chen  Institute of Particle and
Nuclear Physics, Henan Normal University, Xinxiang 453007, China  Na Wang
Institute of Particle and Key Laboratory of Quark and Lepton Physics, Central
China Normal University, Wuhan 430079, China  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  Institute of Particle and Key Laboratory of Quark and Lepton
Physics, Central China Normal University, Wuhan 430079, China  Jinshu Huang
College of Physics and Electronic Engineering, Nanyang Normal University,
Nanyang 473061, China  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

###### Abstract

Motivated by the prospects of the potential ${\Upsilon}(1S)$ particle at high-
luminosity heavy-flavor experiments, we studied the ${\Upsilon}(1S)$ ${\to}$
$B_{c}M$ weak decays, where $M$ $=$ ${\pi}$, ${\rho}$, $K^{(\ast)}$. The
nonfactorizable contributions to hadronic matrix elements are taken into
consideration with the QCDF approach. It is found that the CKM-favored
${\Upsilon}(1S)$ ${\to}$ $B_{c}{\rho}$ decay has branching ratio of ${\cal
O}(10^{-10})$, which might be measured promisingly by the future experiments.

${\Upsilon}$ meson; weak decay; QCD factorization

###### pacs:

13.25.Gv 12.39.St 14.40.Pq

##  I Introduction

The first evidence for upsilons, the bound states of $b\bar{b}$, was observed
in collisions of protons with a stationary nuclear target at Fermilab in 1977
herb ; innes . From that moment on, bottomonia have been a subject of
intensive experimental and theoretical research. Some of the salient features
of upsilons are as follows ann1983 : (1) In the upsilons rest frame, the
relative motion of the $b$ quark is sufficiently slow. Nonrelativistic
Schrödinger equation can be used to describe the spectrum of bottomonia states
and thus one can learn about the interquark binding forces. (2) The
${\Upsilon}(nS)$ particles, with the radial quantum number $n$ $=$ $1$, $2$
and $3$, decay primarily via the annihilation of the $b\bar{b}$ quark pairs
into three gluons. Thus the properties of the invisible gluons and of the
gluon-quark coupling can be gleaned through the study of the upsilons decay.
(3) Compared with the $u$, $d$, $s$ light quarks, the relatively large mass of
the heavy $b$ quark implies a nonnegligible coupling to the Higgs bosons,
making upsilons to be one of the best hunting grounds for light Higgs
particles. By now, our knowledge of the properties of bottomonia comes
primarily from $e^{+}e^{-}$ annihilation.

The ${\Upsilon}(1S)$ particle is the ground state of the vector bottomonia
with quantum number of $I^{G}J^{PC}$ $=$ $0^{-}1^{--}$ pdg . The mass of the
${\Upsilon}(1S)$ particle, $m_{{\Upsilon}(1S)}$ $=$ $9460.30{\pm}0.26$ pdg ,
is about three times heavier than the mass of the $J/{\psi}$ particle (the
ground state of charmonia with the same quantum number of $I^{G}J^{PC}$). On
one hand, compared with the $J/{\psi}$ decay, much richer decay channels could
be accessed by the ${\Upsilon}(1S)$ particle. On the other hand, the coupling
constant ${\alpha}_{s}$ for the ${\Upsilon}(1S)$ decay is smaller than that
for the $J/{\psi}$ decay due to the QCD nature of asymptotic freedom, which
results in hadronic partial width ${\Gamma}({\Upsilon}(1S){\to}ggg)$ $<$
${\Gamma}(J/{\psi}{\to}ggg)$, although the possible phase space in the
${\Upsilon}(1S)$ decay is larger than that in the $J/{\psi}$ decay. In
addition, the squared value of the $b$ quark charge, $Q^{2}_{b}$ $=$ $1/9$, is
less than that of the $c$ quark charge, $Q^{2}_{c}$ $=$ $4/9$, which results
in electromagnetic partial width
${\Gamma}({\Upsilon}(1S){\to}{\gamma}^{\ast})$ $<$
${\Gamma}(J/{\psi}{\to}{\gamma}^{\ast})$. So the full decay width of the
${\Upsilon}(1S)$ particle, ${\Gamma}_{{\Upsilon}(1S)}$ $=$ $54.02{\pm}1.25$
keV, is less than that of the $J/{\psi}$ particle, ${\Gamma}_{J/{\psi}}$ $=$
$92.9{\pm}2.8$ keV pdg . Furthermore, one of the outstanding properties of all
upsilons below $B\bar{B}$ threshold is their narrow decay width of tens of keV
pdg .

The ${\Upsilon}(1S)$ and $J/{\psi}$ particles share the similar decay
mechanism. As is the case for the $J/{\psi}$ particle, strong decays of the
${\Upsilon}(1S)$ particle are suppressed by the phenomenological OZI (Okubo-
Zweig-Iizuka) rules ozi-o ; ozi-z ; ozi-i , so electromagnetic interactions
and radiative transitions become competitive. It is expected that, at the
lowest order approximation, the decay modes of the ${\Upsilon}(1S)$ particle
could be subdivided into four types: (1) The lion’s share of the decay width
is the hadronic decay via the annihilation of the $b\bar{b}$ quark pairs into
three gluons, i.e., some $(81.7{\pm}0.7)\%$ via ${\Upsilon}(1S)$ ${\to}$ $ggg$
pdg . (2) The partial width of the electromagnetic decay via the annihilation
of the $b\bar{b}$ quark pairs into a virtual photon could be written
approximately as $(3+R){\Gamma}_{{\ell}{\ell}}$, where the value of $R$ is the
ratio of inclusive production of hadrons to the ${\mu}^{+}{\mu}^{-}$ pair
production rate at the energy scale of $m_{{\Upsilon}(1S)}$, and
${\Gamma}_{{\ell}{\ell}}$ is the partial width of decay into dileptons. (3)
Branching ratio of the radiative decay is about
$Br({\Upsilon}(1S){\to}{\gamma}gg)$ ${\simeq}$ $(2.2{\pm}0.6)\%$ pdg . The up-
to-date research for light Higgs bosons in the ${\Upsilon}(1S)$ radiative
decay has been performed by CLEO 0807.1427 , Bellepos2013.higgs  and BaBar
1502.06019  Collaborations. (4) The magnetic dipole transition decay,
${\Upsilon}(1S)$ ${\to}$ ${\gamma}{\eta}_{b}(1S)$, is very challenging to
experimental physicist due to the very soft photon and pollution from other
processes, such as ${\Upsilon}(1S)$ ${\to}$ ${\pi}^{0}X$ ${\to}$
${\gamma}{\gamma}X$ 1212.6552 . The experimental signal for ${\Upsilon}(1S)$
${\to}$ ${\gamma}{\eta}_{b}(1S)$ has not been discovered until now. Besides,
the ${\Upsilon}(1S)$ particle could also decay via the weak interactions,
although the branching ratio for a single $b$ or $\bar{b}$ quark decay is
tiny, about $2/{\tau}_{B}{\Gamma}_{{\Upsilon}(1S)}$ ${\sim}$ $10^{-8}$ pdg .
In this paper, we will estimate the branching ratios for the flavor-changing
nonleptonic ${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ weak decays with the QCD
factorization (QCDF) approach qcdf1 ; qcdf2 , where $M$ $=$ ${\pi}$, ${\rho}$,
$K$ and $K^{\ast}$. The motivation is listed as follows.

Table 1: Summary of data samples (in the unit of $10^{6}$) of the ${\Upsilon}(nS)$ particles below $B\bar{B}$ threshold collected by Belle, BaBar and CLEO Collaborations. The data in the 5th column correspond to total number of the ${\Upsilon}(1S)$ particle, including events from hadronic dipion transitions between upsilons ${\Upsilon}(2S,3S)$ ${\to}$ ${\pi}{\pi}{\Upsilon}(1S)$, where branching ratios for the dipion decays ${\Upsilon}(2S)$ ${\to}$ ${\pi}^{+}{\pi}^{-}{\Upsilon}(1S)$, ${\pi}^{0}{\pi}^{0}{\Upsilon}(1S)$ and ${\Upsilon}(3S)$ ${\to}$ ${\pi}^{+}{\pi}^{-}{\Upsilon}(1S)$, ${\pi}^{0}{\pi}^{0}{\Upsilon}(1S)$ are $(17.85{\pm}0.26)\%$, $(8.6{\pm}0.4)\%$, $(4.37{\pm}0.08)\%$ and $(2.20{\pm}0.13)\%$, respectively pdg . | ${\Upsilon}(1S)$ | ${\Upsilon}(2S)$ | ${\Upsilon}(3S)$ | total ${\Upsilon}(1S)$
---|---|---|---|---
Belle 1212.5342  | 102 | 158 | 11 | 145
BaBar 1108.5874  |  | 121.8 | 98.6 | 39
CLEO 0704.2766  | 22.78 | 9.45 | 8.89 | 26

From the experimental point of view, (1) there is plenty of upsilons at the
high-luminosity dedicated bottomonia factories, for example, over $10^{8}$
${\Upsilon}(1S)$ at Belle (see Table 1). Upsilons are also observed by the on-
duty ALICE 1403.3648 , ATLAS 1212.7255 , CMS 1501.07750 , LHCb 1304.6977
experiments at LHC. It is hopefully expected that more upsilons could be
accumulated with great precision at the running upgraded LHC and forthcoming
SuperKEKB. The huge ${\Upsilon}(1S)$ data samples will provide good
opportunities to search for the ${\Upsilon}(1S)$ weak decays which in some
cases might be detectable. Theoretical studies on the ${\Upsilon}(1S)$ weak
decays are just necessary to offer a ready reference. (2) For the two-body
${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ decay, final states with opposite charges
have definite energies and momenta in the center-of-mass frame of the
${\Upsilon}(1S)$ particle. Particularly, identification of a single charged
$B_{c}$ meson in the final state, which is free from inefficiently double
tagging of the bottomed hadron pairs occurring above the $B\bar{B}$ threshold,
would provide an unambiguous signature of the ${\Upsilon}(1S)$ weak decay. Of
course, small branching ratios make the observation of the ${\Upsilon}(1S)$
weak decays extremely difficult, and evidences of an abnormally large
production rate of single $B_{c}$ mesons in the ${\Upsilon}(1S)$ decay might
be a hint of new physics beyond the standard model.

From the theoretical point of view, the bottom-changing upsilon weak decay
could permit overconstraining parameters obtained from $B$ meson decay, but
few studies devoted to the nonleptonic upsilon weak decay in the past. For the
${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ decay, the amplitude is usually treated as
the factorizable product of two independent factors: one describing the
transition between heavy quarkonium ${\Upsilon}(1S)$ and $B_{c}$, and the
other depicting the production of the $M$ state from the vacuum. Previous
works, such as Ref. zpc62.271  based on the spin symmetry and nonrecoil
approximation, Ref. ijma14  based on the heavy quark effective theory, and
Ref. adv2013  based on the Bauer-Stech-Wirbel (BSW) model bsw1 , concentrated
mainly upon the ${\Upsilon}(1S)$ ${\to}$ $B_{c}$ transition form factors which
are related to the space integrals of the meson wave functions. As is well
known, there exist hierarchical scales with nonrelativistic quantum
chromodynamics (NRQCD) prd46 ; prd51 ; rmp77  which is an approach to deal
with the heavy quarkonium, i.e. $M^{2}$ ${\gg}$ $(Mv)^{2}$ ${\gg}$
$(Mv^{2})^{2}$, where $M$ is the mass of heavy quark with typical velocities
$v$ ${\sim}$ ${\alpha}_{s}$ ${\ll}$ $1$. The physical contributions at scales
of $Mv^{2}$ ${\sim}$ ${\Lambda}_{\rm QCD}$ are absorbed into wave functions of
the ${\Upsilon}(1S)$ and $B_{c}$ particles, thus the ${\Upsilon}(1S)$ ${\to}$
$B_{c}$ transition should be dominated by the nonperturbative dynamics. The
nonfactorizable contributions above scales of $Mv^{2}$ has not be taken
seriously in previous works. About 2000, M. Beneke et al. proposed the QCDF
approach qcdf1 ; qcdf2 , where nonfactorizable contributions could be
estimated systematically with the perturbation theory based on collinear
factorization approximation and power countering rules in the heavy quark
limit qcdf2 , and the QCDF approach has been widely applied to nonleptonic $B$
meson decays. So it should be very interesting to study the ${\Upsilon}(1S)$
${\to}$ $B_{c}M$ weak decays by considering nonfactorizable contributions with
the attractive QCDF approach.

This paper is organized as follows. In section II, we will present the
theoretical framework and the amplitudes for the nonleptonic two-body
${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ weak decays with the QCDF approach. Section
III is devoted to numerical results and discussion. The last section is our
summary.

##  II theoretical framework

###  II.1 The effective Hamiltonian

The low energy effective Hamiltonian responsible for the ${\Upsilon}(1S)$
${\to}$ $B_{c}M$ decays is 9512380

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

where the Fermi coupling constant $G_{F}$ $=$ $1.166{\times}10^{-5}\,{\rm
GeV}^{-2}$ pdg ; Using the Wolfenstein parameterization, the Cabibbo-
Kobayashi-Maskawa (CKM) factors can be expanded as a power series in the small
parameter ${\lambda}$ $=$ $0.22537(61)$ pdg ,

$\displaystyle V_{cb}V_{ud}^{\ast}$ $\displaystyle=$ $\displaystyle
A{\lambda}^{2}-\frac{1}{2}A{\lambda}^{4}-\frac{1}{8}A{\lambda}^{6}+{\cal
O}({\lambda}^{8}),$ (2) $\displaystyle V_{cb}V_{us}^{\ast}$ $\displaystyle=$
$\displaystyle A{\lambda}^{3}+{\cal O}({\lambda}^{8}).$ (3)

The Wilson coefficients $C_{1,2}(\mu)$ summarize the physical contributions
above scales of ${\mu}$. The Wilson coefficients $C_{i}$ are calculable with
the perturbation theory and have properly been evaluated to the next-to-
leading order (NLO) with the renormalization group (RG) equation. The
numerical values of the Wilson coefficients $C_{1,2}$ at scales of ${\mu}$
${\sim}$ ${\cal O}(m_{b})$ in naive dimensional regularization scheme are
listed in Table 2. The local tree four-quark operators are defined as follows.

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle[\bar{c}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\alpha}][\bar{q}_{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})u_{\beta}],$
(4) $\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle[\bar{c}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\beta}][\bar{q}_{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})u_{\alpha}],$
(5)

where ${\alpha}$ and ${\beta}$ are color indices and the sum over repeated
indices is understood. To obtain the decay amplitudes, the remaining and the
most intricate works are to calculate accurately hadronic matrix elements of
local operators.

###  II.2 Hadronic matrix elements

Phenomenologically, the simplest treatment with hadronic matrix elements of
four-quark operators is approximated by the product of two current matrix
elements with color transparency ansatz bjorken  and naive factorization (NF)
scheme nf1 ; nf2 , and current matrix elements are further parameterized by
decay constants and transition form factors. For example, previous studies on
the ${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ decay ijma14 ; adv2013  were based on NF
approach.

As is well known, NF’s defect is the disappearance of the renormalization
scale dependence, the strong phases and the nonfactorizatable contributions
from hadronic matrix elements, resulting in nonphysical amplitudes and no $CP$
violating asymmetries. To remedy NF’s deficiencies, M. Beneke et al. proposed
that hadronic matrix elements could be written as the convolution integrals of
hard scattering kernels and light cone distribution amplitudes with the QCDF
approach qcdf1 ; qcdf2 .

For the ${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ decay, the spectator quark is the
heavy bottom (anti)quark. According to the QCDF’s power counting rules qcdf2 ,
contributions from the spectator scattering are power suppressed. With the
QCDF master formula, hadronic matrix elements could be written as :

${\langle}B_{c}M{|}Q_{i}{|}{\Upsilon}(1S){\rangle}=\sum\limits_{i}F_{i}^{{\Upsilon}{\to}B_{c}}{\int}\,dx\,H_{i}(x)\,{\phi}_{M}(x),$
(6)

where transition form factor $F_{i}^{{\Upsilon}{\to}B_{c}}$ and light cone
distribution amplitudes ${\phi}_{M}(x)$ of the emitted meson $M$ are
nonperturbative input parameters, hard scattering kernels $H_{i}(x)$ are
computable order by order with the perturbation theory in principle.

The leading twist two-valence-particle distribution amplitudes of pseudoscalar
and longitudinally polarized vector meson are defined in terms of Gegenbauer
polynomials ballv ; ballp :

${\phi}_{M}(x)=6\,x\bar{x}\sum\limits_{n=0}^{\infty}a_{n}^{M}\,C_{n}^{3/2}(x-\bar{x}),$
(7)

where $\bar{x}$ $=$ $1$ $-$ $x$; $a_{n}^{M}$ is the Gegenbauer moment and
$a_{0}^{M}$ ${\equiv}$ $1$.

After calculation, the decay amplitudes could be written as

${\cal A}({\Upsilon}(1S){\to}B_{c}M)={\langle}B_{c}M{|}{\cal H}_{\rm
eff}{|}{\Upsilon}(1S){\rangle}=\frac{G_{F}}{\sqrt{2}}\,V_{cb}V_{uq}^{\ast}\,a_{1}\,{\langle}M{|}J^{\mu}{|}0{\rangle}{\langle}B_{c}{|}J_{\mu}{|}{\Upsilon}(1S){\rangle}.$
(8)

The coefficient $a_{1}$ in Eq.(8), including nonfactorizable contributions
from QCD radiative vertex corrections, is written as prd77 :

$a_{1}=C_{1}^{\rm NLO}+\frac{1}{N_{c}}\,C_{2}^{\rm
NLO}+\frac{{\alpha}_{s}}{4{\pi}}\,\frac{C_{F}}{N_{c}}\,C_{2}^{\rm LO}\,V.$ (9)

For the transversely polarized vector meson, the factor $V$ is zero beyond
leading twist (twist-2) contributions. For the pseudoscalar and longitudinally
polarized vector meson, with the modified minimal subtraction ($\overline{\rm
MS}$) scheme, the factor $V$ is written as prd77 :

$V=3\,{\log}\Big{(}\frac{m_{b}^{2}}{{\mu}^{2}}\Big{)}+3\,{\log}\Big{(}\frac{m_{c}^{2}}{{\mu}^{2}}\Big{)}-18+{\int}_{0}^{1}dx\,T(x)\,{\phi}_{M}(x),$
(10)

where

$\displaystyle T(x)$ $\displaystyle=$
$\displaystyle\frac{c_{a}}{1-c_{a}}\,{\log}(c_{a})-\frac{4\,c_{b}}{1-c_{b}}\,{\log}(c_{b})$
(11) $\displaystyle+$
$\displaystyle\frac{c_{d}}{1-c_{d}}\,{\log}(c_{d})-\frac{4\,c_{c}}{1-c_{c}}\,{\log}(c_{c})$
$\displaystyle-$ $\displaystyle
r_{c}\,\Big{\\{}\frac{c_{a}}{(1-c_{a})^{2}}\,{\log}(c_{a})+\frac{1}{1-c_{a}}\Big{\\}}$
$\displaystyle-$ $\displaystyle
r_{c}^{-1}\,\Big{\\{}\frac{c_{d}}{(1-c_{d})^{2}}\,{\log}(c_{d})+\frac{1}{1-c_{d}}\Big{\\}}$
$\displaystyle+$ $\displaystyle f(c_{a})-f(c_{b})-f(c_{c})+f(c_{d})$
$\displaystyle+$ $\displaystyle
2\,{\log}(r_{c}^{2})\\{{\log}(c_{a})-{\log}(c_{b})\\},$ $f(c)\ =\ 2\,{\rm
Li}_{2}\Big{(}\frac{c-1}{c}\Big{)}-{\log}^{2}(c)-\frac{2\,c}{1-c}{\log}(c),$
(12)

and the relations

$\displaystyle r_{c}$ $\displaystyle=$ $\displaystyle m_{c}/m_{b},$ (13)
$\displaystyle c_{a}$ $\displaystyle=$ $\displaystyle x\,(1-r_{c}^{2}),$ (14)
$\displaystyle c_{b}$ $\displaystyle=$ $\displaystyle\bar{x}\,(1-r_{c}^{2}),$
(15) $\displaystyle c_{c}$ $\displaystyle=$ $\displaystyle-c_{a}/r_{c}^{2},$
(16) $\displaystyle c_{d}$ $\displaystyle=$ $\displaystyle-c_{b}/r_{c}^{2}.$
(17)

The numerical values of coefficient $a_{1}$ at scales of ${\mu}$ ${\sim}$
${\cal O}(m_{b})$ are listed in Table 2.

Table 2: The numerical values of the Wilson coefficients $C_{1,2}$ and $a_{1}$ for the ${\Upsilon}(1S)$ ${\to}$ $B_{c}{\pi}$ decay at different scales, where $m_{b}$ $=$ 4.78 GeV pdg . | NLO | LO | QCDF
---|---|---|---
${\mu}$ | $C_{1}$ | $C_{2}$ | $C_{1}$ | $C_{2}$ | Re($a_{1}$) | Im($a_{1}$)
$0.5\,m_{b}$ | $1.128$ | $-0.269$ | $1.168$ | $-0.337$ | $1.076$ | $+0.027$
$m_{b}$ | $1.076$ | $-0.173$ | $1.110$ | $-0.235$ | $1.054$ | $+0.015$
$1.5\,m_{b}$ | $1.054$ | $-0.128$ | $1.085$ | $-0.188$ | $1.043$ | $+0.011$
$2.0\,m_{b}$ | $1.041$ | $-0.100$ | $1.070$ | $-0.159$ | $1.036$ | $+0.008$

###  II.3 Decay constants and form factors

The matrix elements of current operators are defined as follows:

$\displaystyle{\langle}P(p){|}A_{\mu}{|}0{\rangle}$ $\displaystyle=$
$\displaystyle-if_{P}\,p_{\mu},$ (18)
$\displaystyle{\langle}V(p,{\epsilon}){|}V_{\mu}{|}0{\rangle}$
$\displaystyle=$ $\displaystyle f_{V}\,m_{V}\,{\epsilon}_{V,{\mu}}^{\ast},$
(19)

where $f_{P}$ and $f_{V}$ are the decay constants of pseudoscalar and vector
mesons, respectively; $m_{V}$ and ${\epsilon}_{V}$ denote the mass and
polarization of vector meson, respectively.

The transition form factors are defined as follows ijma14 ; adv2013 ; bsw1 :

$\displaystyle{\langle}B_{c}(p_{2}){|}V_{\mu}-A_{\mu}{|}{\Upsilon}(p_{1},{\epsilon}){\rangle}$
(20) $\displaystyle=$
$\displaystyle-{\epsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,{\epsilon}_{{\Upsilon}}^{{\nu}}\,q^{\alpha}\,(p_{1}+p_{2})^{\beta}\,\frac{V^{{\Upsilon}{\to}B_{c}}(q^{2})}{m_{{\Upsilon}}+m_{B_{c}}}-i\,\frac{2\,m_{{\Upsilon}}\,{\epsilon}_{{\Upsilon}}{\cdot}q}{q^{2}}\,q_{\mu}\,A_{0}^{{\Upsilon}{\to}B_{c}}(q^{2})$
$\displaystyle-i\,{\epsilon}_{{\Upsilon},{\mu}}\,(m_{{\Upsilon}}+m_{B_{c}})\,A_{1}^{{\Upsilon}{\to}B_{c}}(q^{2})-i\,\frac{{\epsilon}_{{\Upsilon}}{\cdot}q}{m_{{\Upsilon}}+m_{B_{c}}}\,(p_{1}+p_{2})_{\mu}\,A_{2}^{{\Upsilon}{\to}B_{c}}(q^{2})$
$\displaystyle+i\,\frac{2\,m_{{\Upsilon}}\,{\epsilon}_{{\Upsilon}}{\cdot}q}{q^{2}}\,q_{\mu}\,A_{3}^{{\Upsilon}{\to}B_{c}}(q^{2}),$

where $q$ $=$ $p_{1}$ $-$ $p_{2}$; and $A_{0}(0)$ $=$ $A_{3}(0)$ is required
compulsorily to cancel singularities at the pole $q^{2}$ $=$ $0$. There is a
relation among these form factors

$2m_{{\Upsilon}}A_{3}(q^{2})=(m_{{\Upsilon}}+m_{B_{c}})A_{1}(q^{2})+(m_{{\Upsilon}}-m_{B_{c}})A_{2}(q^{2}).$
(21)

It is clearly seen that there are only three independent form factors,
$A_{0,1}(0)$ and $V(0)$, at the pole $q^{2}$ $=$ $0$ for the ${\Upsilon}(1S)$
${\to}$ $B_{c}M$ decays. The form factors at the pole $q^{2}$ $=$ $0$ could be
written as the overlap integrals of wave functions of mesons bsw1 , i.e.,

$A_{0}^{{\Upsilon}{\to}B_{c}}(0)={\int}d\vec{k}_{\perp}{\int}_{0}^{1}dx\,\Big{\\{}{\Phi}_{\Upsilon}(k_{\perp},x,1,0)\,{\sigma}_{z}\,{\Phi}_{B_{c}}(k_{\perp},x,0,0)\Big{\\}},$
(22) $\displaystyle A_{1}^{{\Upsilon}{\to}B_{c}}(0)$ $\displaystyle=$
$\displaystyle\frac{m_{b}+m_{c}}{m_{{\Upsilon}(1S)}+m_{B_{c}}}I^{{\Upsilon}{\to}B_{c}}$
(23) $\displaystyle V^{{\Upsilon}{\to}B_{c}}(0)$ $\displaystyle=$
$\displaystyle\frac{m_{b}-m_{c}}{m_{{\Upsilon}(1S)}-m_{B_{c}}}I^{{\Upsilon}{\to}B_{c}},$
(24)
$I^{{\Upsilon}{\to}B_{c}}=\sqrt{2}{\int}d\vec{k}_{\perp}{\int}_{0}^{1}\frac{dx}{x}\,\Big{\\{}{\Phi}_{\Upsilon}(k_{\perp},x,1,-1)\,i{\sigma}_{y}\,{\Phi}_{B_{c}}(k_{\perp},x,0,0)\Big{\\}},$
(25)

where ${\sigma}_{y,z}$ is a Pauli matrix acting on the spin indices of the
decaying bottom quark; $x$ and $\vec{k}_{\perp}$ denote the fraction of the
longitudinal momentum and the transverse momentum carried by the nonspectator
quark, respectively.

Using the separation of momentum and spin variables, the wave functions of
mesons can be written as

${\Phi}(\vec{k}_{\perp},x,j,j_{z})\,={\phi}(\vec{k}_{\perp},x)\,{|}s,s_{z},s_{1},s_{2}{\rangle},$
(26)

with the normalization condition,

$\sum\limits_{s_{1},s_{2}}{\int}d\vec{k}_{\perp}{\int}_{0}^{1}dx\,{|}{\Phi}(\vec{k}_{\perp},x,j,j_{z}){|}^{2}\,=1,$
(27)

where $s_{1,2}$ denote the spin of valence quark in meson; $\vec{s}$ $=$
$\vec{s}_{1}$ $+$ $\vec{s}_{2}$; $s$ $=$ $1$ and $0$ for the ${\Upsilon}(1S)$
and $B_{c}$ particles, respectively.

For the ground states of heavy quarkonia ${\Upsilon}(1S)$ and $B_{c}$
particles, we will take the solution of the Schödinger equation with
nonrelativistic three-dimensional scalar harmonic oscillator potential,

${\phi}(\vec{k})\ {\sim}\
{\int}d\vec{r}\,{\phi}(\vec{r})\,e^{-i\vec{k}{\cdot}\vec{r}}\ {\sim}\
{\int}d\vec{r}\,e^{-{\beta}^{2}r^{2}/2}\,e^{-i\vec{k}{\cdot}\vec{r}}\ {\sim}\
e^{-\vec{k}^{2}/2{\beta}^{2}},$ (28)

where the parameter ${\beta}$ determines the average transverse quark
momentum, i.e., ${\langle}\vec{k}^{2}_{\perp}{\rangle}$ $=$ ${\beta}^{2}$.
According to the power counting rules of NRQCD prd46 , the characteristic
magnitude of the moment is order of $Mv$ and $v$ ${\sim}$ ${\alpha}_{s}$. So
we will take ${\beta}$ $=$ $M{\alpha}_{s}$ in our calculation. Employing the
substitution ansatz xiao ,

$\vec{k}^{2}\ {\to}\
\frac{1}{4}\Big{\\{}\frac{\vec{k}_{\perp}^{2}+m_{1}^{2}}{x_{1}}+\frac{\vec{k}_{\perp}^{2}+m_{2}^{2}}{x_{2}}+\frac{(m_{1}^{2}-m_{2}^{2})^{2}}{\frac{\vec{k}_{\perp}^{2}+m_{1}^{2}}{x_{1}}+\frac{\vec{k}_{\perp}^{2}+m_{2}^{2}}{x_{2}}}\Big{\\}},$
(29)

where $x_{1}$ $+$ $x_{2}$ $=$ $1$; $m_{1,2}$ is the mass of valence quark.
Setting $x_{1}$ $=$ $x$, we can obtain

${\phi}(\vec{k}_{\perp},x)\
=N{\exp}\Big{\\{}\frac{-1}{8{\beta}^{2}}\Big{[}\frac{\vec{k}_{\perp}^{2}+\bar{x}\,m_{1}^{2}+x\,m_{2}^{2}}{x\,\bar{x}}+\frac{(m_{1}^{2}-m_{2}^{2})^{2}\,x\,\bar{x}}{\vec{k}_{\perp}^{2}+\bar{x}\,m_{1}^{2}+x\,m_{2}^{2}}\Big{]}\Big{\\}},$
(30)

where $N$ is a normalization factor.

Using the aforementioned convention, we get

$\displaystyle
A_{0}^{{\Upsilon}{\to}B_{c}}(0)\,=\,A_{3}^{{\Upsilon}{\to}B_{c}}(0)\,=\,0.81{\pm}0.01,$
(31) $\displaystyle A_{1}^{{\Upsilon}{\to}B_{c}}(0)\,=\,0.83{\pm}0.01,$ (32)
$\displaystyle A_{2}^{{\Upsilon}{\to}B_{c}}(0)\,=\,0.73{\pm}0.08,$ (33)
$\displaystyle V^{{\Upsilon}{\to}B_{c}}(0)\,=\,1.98{\pm}0.06,$ (34)

where the uncertainties come from variation of valence quark mass $m_{b,c}$.
In addition, according to the NRQCD argument, the relativistic corrections and
higher-twist effects might give uncertainties of ${\cal O}(v^{2})$, about
10%${\sim}$30%. Values at $q^{2}$ $\neq$ $0$ could, in principle, be
extrapolated by assuming the form factors dominated by a proper pole which is
unknown, or calculated with other method, such as the approach using the
Bethe-Salpeter wave functions with the help of the nonrelativistic
instantaneous approximation and the potential model based on the Mandelstam
formalism prd49  and so on. Here, we will follow the common practice for
nonleptonic $B$ decays with the QCDF approach. Values of form factors at
$q^{2}$ $=$ $0$ are taken to offer an order of magnitude estimation, because
both ${\Upsilon}$ and $B_{c}$ are heavy quarkonium and the recoil effects
might be not so significant.

###  II.4 Decay amplitudes

With the aforementioned definition of hadronic matrix elements, the decay
amplitudes of ${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ decays can be written as

$\displaystyle{\cal A}({\Upsilon}{\to}B_{c}^{+}{\pi}^{-})$ $\displaystyle=$
$\displaystyle\sqrt{2}\,G_{F}\,V_{cb}\,V_{ud}^{\ast}\,a_{1}\,f_{\pi}\,m_{\Upsilon}\,({\epsilon}_{\Upsilon}{\cdot}p_{\pi})\,A_{0}^{{\Upsilon}{\to}B_{c}},$
(35) $\displaystyle{\cal A}({\Upsilon}{\to}B_{c}^{+}K^{-})$ $\displaystyle=$
$\displaystyle\sqrt{2}\,G_{F}\,V_{cb}\,V_{us}^{\ast}\,a_{1}\,f_{K}\,m_{\Upsilon}\,({\epsilon}_{\Upsilon}{\cdot}p_{K})\,A_{0}^{{\Upsilon}{\to}B_{c}},$
(36) $\displaystyle{\cal A}({\Upsilon}{\to}B_{c}^{+}{\rho}^{-})$
$\displaystyle=$
$\displaystyle-i\,\frac{G_{F}}{\sqrt{2}}\,V_{cb}\,V_{ud}^{\ast}\,a_{1}\,f_{\rho}\,m_{\rho}\,\Big{\\{}({\epsilon}_{\Upsilon}{\cdot}{\epsilon}_{\rho}^{\ast})\,(m_{\Upsilon}+m_{B_{c}})\,A_{1}^{{\Upsilon}{\to}B_{c}}$
(37)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+({\epsilon}_{\Upsilon}{\cdot}p_{\rho})\,({\epsilon}_{\rho}^{\ast}{\cdot}p_{\Upsilon})\,\frac{2\,A_{2}^{{\Upsilon}{\to}B_{c}}}{m_{\Upsilon}+m_{B_{c}}}-i\,{\epsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,{\epsilon}_{\Upsilon}^{\mu}\,{\epsilon}_{\rho}^{{\ast}{\nu}}\,p_{\Upsilon}^{\alpha}\,p_{\rho}^{\beta}\,\frac{2\,V^{{\Upsilon}{\to}B_{c}}}{m_{\Upsilon}+m_{B_{c}}}\Big{\\}},$
$\displaystyle{\cal A}({\Upsilon}{\to}B_{c}^{+}K^{{\ast}-})$ $\displaystyle=$
$\displaystyle-i\,\frac{G_{F}}{\sqrt{2}}\,V_{cb}\,V_{us}^{\ast}\,a_{1}\,f_{K^{\ast}}\,m_{K^{\ast}}\,\Big{\\{}({\epsilon}_{\Upsilon}{\cdot}{\epsilon}_{K^{\ast}}^{\ast})\,(m_{\Upsilon}+m_{B_{c}})\,A_{1}^{{\Upsilon}{\to}B_{c}}$
(38)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+({\epsilon}_{\Upsilon}{\cdot}p_{K^{\ast}})\,({\epsilon}_{K^{\ast}}^{\ast}{\cdot}p_{\Upsilon})\,\frac{2\,A_{2}^{{\Upsilon}{\to}B_{c}}}{m_{\Upsilon}+m_{B_{c}}}-i\,{\epsilon}_{{\mu}{\nu}{\alpha}{\beta}}\,{\epsilon}_{\Upsilon}^{\mu}\,{\epsilon}_{K^{\ast}}^{{\ast}{\nu}}\,p_{\Upsilon}^{\alpha}\,p_{K^{\ast}}^{\beta}\,\frac{2\,V^{{\Upsilon}{\to}B_{c}}}{m_{\Upsilon}+m_{B_{c}}}\Big{\\}}.$

For the ${\Upsilon}(1S)$ ${\to}$ $B_{c}V$ decays, the hadronic matrix elements
in Eq.(8) can also be expressed as vv

$\displaystyle{\cal H}_{\lambda}$ $\displaystyle=$
$\displaystyle{\langle}V{|}J^{\mu}{|}0{\rangle}{\langle}B_{c}{|}J_{\mu}{|}{\Upsilon}(1S){\rangle}$
(39) $\displaystyle=$
$\displaystyle{\epsilon}_{V}^{{\ast}{\mu}}{\epsilon}_{{\Upsilon}}^{\nu}\Big{\\{}a\,g_{{\mu}{\nu}}+\frac{b}{m_{{\Upsilon}}\,m_{V}}(p_{{\Upsilon}}+p_{B_{c}})^{\mu}p_{V}^{\nu}+\frac{i\,c}{m_{{\Upsilon}}\,m_{V}}{\epsilon}_{{\mu}{\nu}{\alpha}{\beta}}p_{V}^{\alpha}(p_{{\Upsilon}}+p_{B_{c}})^{\beta}\Big{\\}}.$

The definition of helicity amplitudes is

$\displaystyle{\cal H}_{0}$ $\displaystyle=$
$\displaystyle-a\,y-2b\,(y^{2}-1),$ (40) $\displaystyle{\cal H}_{\pm}$
$\displaystyle=$ $\displaystyle a\,{\pm}\,2c\,\sqrt{y^{2}-1},$ (41)

where invariant amplitudes $a$, $b$, $c$ and variable $y$ are

$\displaystyle a$ $\displaystyle=$
$\displaystyle-i\,f_{V}\,m_{V}\,(m_{{\Upsilon}}+m_{B_{c}})\,A_{1}^{{\Upsilon}{\to}B_{c}}(q^{2}),$
(42) $\displaystyle b$ $\displaystyle=$
$\displaystyle-i\,f_{V}\,m_{{\Upsilon}}\,m_{V}^{2}\,\frac{A_{2}^{{\Upsilon}{\to}B_{c}}(q^{2})}{m_{{\Upsilon}}+m_{B_{c}}},$
(43) $\displaystyle c$ $\displaystyle=$
$\displaystyle+i\,f_{V}\,m_{{\Upsilon}}\,m_{V}^{2}\,\frac{V^{{\Upsilon}{\to}B_{c}}(q^{2})}{m_{{\Upsilon}}+m_{B_{c}}},$
(44) $\displaystyle y$ $\displaystyle=$
$\displaystyle\frac{p_{{\Upsilon}}{\cdot}p_{V}}{m_{{\Upsilon}}\,m_{V}}\ =\
\frac{m_{{\Upsilon}}^{2}-m_{B_{c}}^{2}+m_{V}^{2}}{2\,m_{{\Upsilon}}\,m_{V}}.$
(45)

The scalar amplitudes $a$, $b$, $c$ describe the $s$, $d$, $p$ wave
contributions, respectively. Clearly, compared with the $s$ wave amplitude,
the $p$ and $d$ wave amplitudes are suppressed by a factor
$m_{V}/m_{{\Upsilon}}$.

##  III Numerical results and discussion

In the rest frame of ${\Upsilon}(1S)$ particle, branching ratio for
nonleptonic ${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ weak decays can be written as

${\cal B}r({\Upsilon}(1S){\to}B_{c}M)\ =\ \frac{1}{12{\pi}}\,\frac{p_{\rm
cm}}{m_{{\Upsilon}}^{2}{\Gamma}_{{\Upsilon}}}\,{|}{\cal
A}({\Upsilon}(1S){\to}B_{c}M){|}^{2},$ (46)

where the decay width ${\Gamma}_{\Upsilon}$ $=$ $54.02{\pm}1.25$ keV pdg ; the
momentum of final states is

$p_{\rm cm}\ =\
\frac{\sqrt{[m_{\Upsilon}^{2}-(m_{B_{c}}+m_{M})^{2}][m_{\Upsilon}^{2}-(m_{B_{c}}-m_{M})^{2}]}}{2\,m_{\Upsilon}}.$
(47)

The input parameters, including the CKM Wolfenstein parameters, masses of $b$
and $c$ quarks, decay constants, and Gegenbauer moments of distribution
amplitudes in Eq.(7), are collected in Table 3. If not specified explicitly,
we will take their central values as the default inputs. Our numerical results
on the $CP$-averaged branching ratios for the ${\Upsilon}(1S)$ ${\to}$
$B_{c}M$ decays are displayed in Table 4, where theoretical uncertainties of
the QCDF results come from the CKM parameters, the renormalization scale
${\mu}$ $=$ $(1{\pm}0.5)m_{b}$, masses of $b$ and $c$ quarks, hadronic
parameters (decay constants and Gegenbauer moments), respectively. For the
sake of comparison, previous results of Refs. ijma14 ; adv2013  are re-
evaluated with coefficient $a_{1}$ $=$ $1.054$, where the scenario of the
flavor dependent parameter ${\omega}$ in Ref. adv2013  is taken. The following
are some comments.

(1) The QCDF’s results fall in between those of Ref. ijma14  and Ref. adv2013
, because the form factors $A_{0,1}^{{\Upsilon}{\to}B_{c}}$ in our calculation
fall in between those of Ref. ijma14  and Ref. adv2013 .

(2) There is a clear hierarchical relationship, ${\cal
B}r({\Upsilon}(1S){\to}B_{c}{\rho})$ $>$ ${\cal
B}r({\Upsilon}(1S){\to}B_{c}{\pi})$ $>$ ${\cal
B}r({\Upsilon}(1S){\to}B_{c}K^{\ast})$ $>$ ${\cal
B}r({\Upsilon}(1S){\to}B_{c}K)$. These are two dynamical reasons. One is that
the CKM factor $V_{cb}V_{us}^{\ast}$ responsible for the ${\Upsilon}(1S)$
${\to}$ $B_{c}K^{(\ast)}$ decay is suppressed by a factor of ${\lambda}$
relative to the CKM factor $V_{cb}V_{ud}^{\ast}$ responsible for the
${\Upsilon}(1S)$ ${\to}$ $B_{c}{\pi}$, $B_{c}{\rho}$ decays. The other is that
the orbital angular momentum $L_{B_{c}P}$ $>$ $L_{B_{c}V}$.

(3) The CKM-favored $a_{1}$ dominated ${\Upsilon}(1S)$ ${\to}$ $B_{c}{\rho}$
decay has the largest branching ratio, ${\sim}$ $10^{-10}$, which should be
sought for with high priority and firstly observed at the running LHC and
forthcoming SuperKEKB. For example, the ${\Upsilon}(1S)$ production cross
section in p-Pb collision can reach up to a few ${\mu}b$ with the ALICE
detector at LHC plb740 . Therefore, per 100 $fb^{-1}$ data collected at ALICE,
over $10^{11}$ ${\Upsilon}(1S)$ particles are in principle available,
corresponding to tens of ${\Upsilon}(1S)$ ${\to}$ $B_{c}{\rho}$ events if with
about 10% reconstruction efficiency.

(4) There are many uncertainties on the QCDF’s results. The first uncertainty,
about 7${\sim}$8%, from the CKM factors could be lessened with the improvement
on the precision of the Wolfenstein parameter $A$. The second uncertainty from
the renormalization scale should, in principle, be reduced by inclusion of
higher order ${\alpha}_{s}$ corrections to hadronic matrix elements. The third
uncertainty is due to the fact that masses of $b$ and $c$ quark affect the
shape lines of wave functions, and hence the magnitude of form factors and
branching ratios. The fourth uncertainty from hadronic parameters is expected
to be reduced with the relative ratio of branching ratios.

(5) Other factors, such as the contributions of higher order corrections to
hadronic matrix elements, relativistic effects, $q^{2}$ dependence of form
factors and so on, which are not considered in this paper, deserve the
dedicated study. Our results just provide an order of magnitude estimation.

Table 3: The values of input parameters. Wolfenstein parameters
---
${\lambda}$ $=$ $0.22537{\pm}0.00061$ pdg  |  $A$ $=$ $0.814^{+0.023}_{-0.024}$ pdg
masses of quarks
$m_{c}$ $=$ $1.67{\pm}0.07$ GeV pdg  |  $m_{b}$ $=$ $4.78{\pm}0.06$ GeV pdg
decay constants
$f_{\pi}$ $=$ $130.41{\pm}0.20$ MeV pdg  |  $f_{K}$ $=$ $156.2{\pm}0.7$ MeV pdg
$f_{\rho}$ $=$ $216{\pm}3$ MeV ballv  |  $f_{K^{\ast}}$ $=$ $220{\pm}5$ MeV ballv
Gegenbauer moments at the scale ${\mu}$ $=$ 1 GeV
$a_{1}^{\rho}$ $=$ $0$ ballv  |  $a_{2}^{\rho}$ $=$ $0.15{\pm}0.07$ ballv
$a_{1}^{K^{\ast}}$ $=$ $-0.03{\pm}0.02$ ballv  |  $a_{2}^{K^{\ast}}$ $=$ $0.11{\pm}0.09$ ballv
$a_{1}^{\pi}$ $=$ $0$ ballp  |  $a_{2}^{\pi}$ $=$ $0.25{\pm}0.15$ ballp
$a_{1}^{K}$ $=$ $-0.06{\pm}0.03$ ballp  |  $a_{2}^{K}$ $=$ $0.25{\pm}0.15$ ballp
Table 4: The $CP$-averaged branching ratios for the ${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ decays. | Ref. ijma14  | Ref. adv2013  | this work
---|---|---|---
$10^{10}{\times}{\cal B}r({\Upsilon}(1S){\to}B_{c}{\rho})$ | $1.84$ | $1.2$ | $1.53^{+0.11+0.10+0.03+0.04}_{-0.10-0.04-0.04-0.04}$
$10^{11}{\times}{\cal B}r({\Upsilon}(1S){\to}B_{c}{\pi})$ | $6.91$ | $2.8$ | $5.03^{+0.36+0.34+0.09+0.02}_{-0.34-0.14-0.11-0.02}$
$10^{12}{\times}{\cal B}r({\Upsilon}(1S){\to}B_{c}K^{\ast})$ | $10.47$ | $6.2$ | $8.75^{+0.68+0.55+0.16+0.40}_{-0.64-0.20-0.22-0.39}$
$10^{12}{\times}{\cal B}r({\Upsilon}(1S){\to}B_{c}K)$ | $5.03$ | $2.3$ | $3.73^{+0.29+0.25+0.07+0.03}_{-0.27-0.10-0.08-0.03}$

##  IV Summary

With the sharp increase of the ${\Upsilon}(1S)$ data sample at high-luminosity
dedicated heavy-flavor factories, the bottom-changing ${\Upsilon}(1S)$ ${\to}$
$B_{c}M$ weak decays are interesting in exploring the underlying mechanism
responsible for transition between heavy quarknoia, investigating perturbative
and nonperturbative effects and overconstraining parameters from $B$ decays.
The ${\Upsilon}(1S)$ weak decays are allowable within the standard model,
though their branching ratios are expected to be tiny in comparison to the
conventional strong and electromagnetic decays. In this paper, we studied the
nonleptonic ${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ weak decays, which are
$a_{1}$-dominated based on the low energy effective theory, and hence should
have large branching ratios among weak decay modes. Considering the
nonfactorizable contributions to hadronic matrix elements with the QCDF
approach, we estimated the branching ratios of the ${\Upsilon}(1S)$ ${\to}$
$B_{c}M$ weak decays, where transition form factors are obtained by the
integrals of wave functions with the nonrelativistic isotropic harmonic
oscillator potential. The prediction on branching ratios for the
${\Upsilon}(1S)$ ${\to}$ $B_{c}M$ decays is the same order as previous works
ijma14 ; adv2013 . The CKM favored ${\Upsilon}(1S)$ ${\to}$ $B_{c}{\rho}$
decay has relatively large branching ratio, ${\sim}10^{-10}$, and might be
detectable in future experiments.

## Acknowledgments

We thank the referees for their helpful comments. The work is supported by the
National Natural Science Foundation of China (Grant Nos. 11475055, 11275057,
U1232101 and U1332103). Dr. Wang thanks for the support from CCNU-QLPL
Innovation Fund (QLPL201411).

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March 15, 2023

# Mean-variance constrained priors have finite maximum Bayes risk in the
normal location model

JIAFENG CHEN<EMAIL_ADDRESS>Department of Economics, Harvard
University, Cambridge, Massachusetts, U.S.A.

###### Abstract

Consider a normal location model
$X\mid\theta\sim\mathcal{N}(\theta,\sigma^{2})$ with known $\sigma^{2}$.
Suppose $\theta\sim G_{0}$, where the prior $G_{0}$ has zero mean and unit
variance. Let $G_{1}$ be a possibly misspecified prior with zero mean and unit
variance. We show that the squared error Bayes risk of the posterior mean
under $G_{1}$ is bounded, uniformly over $G_{0},G_{1},\sigma^{2}>0$.

###### keywords:

Bayes risk; normal location model; Bayesian robustness; moments

††journal: Preprint

##  section Introduction and main result

We consider the estimation problem in the normal location model. For a known
$\sigma^{2}>0$, let $X\mid\theta\sim\mathcal{N}(\theta,\sigma^{2})$ be a
Gaussian signal on some unknown parameter $\theta\in\mathbb{R}$. We assume
that $\theta\sim G_{0}$ is itself random, and we refer to $G_{0}$ as the true
prior. We can view $G_{0}$ either as some latent subjective beliefs or as some
objective sampling process for $\theta$. We additionally assume $G_{0}$ has
zero mean and unit variance. However, suppose the data analyst does not know
$G_{0}$, but they do know that the true prior has zero mean and unit variance.
They instead use some possibly misspecified prior $G_{1}$. Given $G_{1}$, the
analyst’s Bayes decision rule, for estimating $\theta$ under squared error
loss, is given by the posterior mean under $G_{1}$

$\mu_{1}(x)=\frac{\int_{-\infty}^{\infty}\theta\varphi\left({\frac{x-\theta}{\sigma}}\right)\sigma^{-1}G_{1}(d\theta)}{\int_{-\infty}^{\infty}\varphi\left({\frac{x-\theta}{\sigma}}\right)\sigma^{-1}G_{1}(d\theta)},\quad\varphi(t)=\frac{1}{\sqrt{2\pi}}e^{-t^{2}/2}.$

The Bayes risk of $\mu_{1}(x)$, under the true prior $G_{0}$, is

$R(G_{0},G_{1},\sigma^{2})=E[(\mu_{1}(X)-\theta)^{2}],$

where the expectation integrates over $\theta\sim G_{0}$ and
$X\mid\theta\sim\mathcal{N}(\theta,\sigma^{2})$. Let $\mathcal{M}$ denote the
set of distributions on $\mathbb{R}$ with zero mean and unit variance. The
worst-case Bayes risk, over $G_{0},G_{1},\sigma^{2}$, is given by

$\overline{R}=\sup\left\\{{R(G_{0},G_{1},\sigma^{2}):{G_{0}\in\mathcal{M},G_{1}\in\mathcal{M},\sigma>0}}\right\\}.$

This paper proves the following bound on $\overline{R}$.

######  Theorem section.theorem.

Under the preceding setup, $\overline{R}<535$.

##  section Discussion

Gaussian decision theory is a canonical problem in statistics. Our result
upper bounds the Bayes risk of misspecified Bayes decision rules in this
setting. Misspecification refers to the fact that the prior $G_{1}$ used to
derive Bayes decision rules may be different from the prior $G_{0}$ used to
evaluate their Bayes risk. This risk quantity features in some classical
results, for instance, in Lemma 4.8 of Johnstone (2019).

Our result relates to the Bayesian literature on partial prior information
(see chapter 3 in Robert, 2007, and references therein), where information is
available for some features of a decision-maker’s beliefs, but the full prior
is too difficult to elicit. With only partial information on the prior, this
literature is interested in robustness of the posterior distribution with
respect to the prior choice. Viewed through this lens, we think of $G_{0}$ as
some true subjective beliefs that are hard to elicit, and $G_{1}$ as some
convenient approximation of $G_{0}$. In our case, the decision-maker has
access to the first two moments of the prior distribution $G_{0}$, which are
normalized to zero and one without loss of generality. When the prior
distribution admits constraints in terms of moments, the closest result to our
knowledge is Goutis (1994), who shows that the posterior mean is pointwise
finite. That is, in our notation, for all $x$,

$\sup_{G_{1}\in\mathcal{M}}|\mu_{1}(x)|<\infty.$

Compared to Goutis (1994), we restrict to the normal location model, but our
bound for the Bayes risk, which integrates over $x$, is not directly implied
by the pointwise-in-$x$ finiteness of the posterior mean.

Our interest in the result is motivated by empirical Bayes procedures (Efron,
2019). Empirical Bayes in the Gaussian model envisions a setup where
$X_{i}\mid\theta_{i},\sigma_{i}^{2}\sim\mathcal{N}(\theta_{i},\sigma_{i}^{2})$
for $i=1,\ldots,n$ and $\theta_{i}\sim G_{0}$. Since we have repeated
measurements over $i$, empirical Bayesians can estimate the prior distribution
$G_{0}$ and report posterior means relative to an estimate $G_{1}$ of $G_{0}$.
Here, $G_{0}$ no longer denotes a subjective belief, but instead some assumed
physical sampling process for the parameters $\theta_{i}$. Empirical Bayes
procedures face the challenge that the distribution $G_{0}$ itself is often
much harder to estimate than its moments (Dedecker & Michel, 2013; Pensky,
2017). In this context, $\overline{R}$ puts a ceiling on the worst-case
behavior of empirical Bayes procedures corresponding to $G_{1}$, assuming that
$G_{1}$ has the same moments as $G_{0}$. Specifically, Chen (2023) considers
an empirical Bayes model where $(\theta_{i},\sigma_{i})\sim P$, but
$\theta_{i}$ may not be independent with $\sigma_{i}$ under $P$. Chen (2023)
proposes to model $P$ as a location-scale family for
$\theta_{i}\mid\sigma_{i}$:

$P(\theta_{i}\leq
t\mid\sigma_{i})=G_{1}\left({\frac{t-E[\theta_{i}\mid\sigma_{i}]}{\operatorname{var}(\theta_{i}\mid\sigma_{i})^{1/2}}}\right),$

for $G_{1}$ of mean zero and variance one. When this location-scale assumption
fails, empirical Bayes procedures based on this assumption use some common
$G_{1}$ to compute posterior means for
$\tau_{i}=(\theta_{i}-E[\theta_{i}\mid\sigma_{i}])/\operatorname{var}(\theta_{i}\mid\sigma_{i})^{1/2}$,
whose true prior distribution may be different from $G_{1}$, but obeys the
moment restrictions by construction. Our result would then bound the worst-
case behavior of this procedure.

To be clear, if we are solely concerned about worst-case behavior and are free
to choose $G_{1}$, then we do not need Theorem section.theorem. The choice
$G_{1}=\mathcal{N}(0,1)$ minimizes the maximum risk
$\sup\left\\{{R(G_{0},G_{1},\sigma^{2}):G_{0}\in\mathcal{M}}\right\\}$ for any
$\sigma^{2}>0$. This minimax risk is the posterior variance
$\sigma^{2}/(\sigma^{2}+1)$, which is at most 1. However, note that, under
$G_{1}=\mathcal{N}(0,1)$,
$R(G_{0},G_{1},\sigma^{2})=\sigma^{2}/(\sigma^{2}+1)$ is constant over
$G_{0}\in\mathcal{M}$, and so the Gaussian prior $G_{1}$ is pessimistic in the
sense that every underlying $G_{0}$ attains $G_{1}$’s worst-case behavior. In
contrast, using a non-Gaussian $G_{1}$ may be advantageous when $G_{1}$ is
close to $G_{0}$. In this case, Theorem section.theorem is useful in providing
a worst-case assurance.

The specific value of the bound, $535$, is not tight. We suspect our proof
method can be refined in various places to obtain a tighter constant. We
conjecture that the tightest possible upper bound is 2, which is obtained by
the construction used to prove the following lemma. Unfortunately, an early
step in our proof already pushes the upper bound over 2, and it is thus
unlikely that we could achieve this upper bound with our line of argument.

######  Lemma section.theorem.

Under the preceding setup, $\overline{R}\geq 2$.

######  Proof section.theorem.

Suppose $G_{1}$ puts $1/2$ mass each on $\left\\{{-1,+1}\right\\}$. Let
$\epsilon>0$. Suppose $G_{0}$ puts mass $\epsilon^{2}/(1+\epsilon^{2})$ on
$-1/\epsilon$, and mass $1/(1+\epsilon^{2})$ on $\epsilon$. We verify that
$G_{0},G_{1}\in\mathcal{M}$. The posterior distribution under prior $G_{1}$
has mass for $\theta=1$ equal to

$p_{1}(X)=\operatorname{pr}_{G_{1}}(\theta=1\mid
X)=\Lambda\left({\frac{1}{2\sigma^{2}}\left[{\left({-1-X}\right)^{2}-(1-X)^{2}}\right]}\right),\quad\Lambda(t)=\frac{1}{1+e^{-t}}.$

Thus the posterior mean is $2p_{1}(X)-1$. The Bayes risk is

$\displaystyle R(G_{0},G_{1},\sigma^{2})$
$\displaystyle=\frac{\epsilon^{2}}{1+\epsilon^{2}}E\left[{\left({2p_{1}(X)-1+\frac{1}{\epsilon}}\right)^{2}\mid\theta=-\frac{1}{\epsilon}}\right]$
$\displaystyle\quad+\frac{1}{1+\epsilon^{2}}E\left[{\left({2p_{1}(X)-1-\epsilon}\right)^{2}\mid\theta=\epsilon}\right].$

Now, taking $\sigma\to 0$ and then $\epsilon\to 0$ shows that

$\lim_{\epsilon\to 0}\lim_{\sigma\to
0}R(G_{0},G_{1},\sigma^{2})=\lim_{\epsilon\to
0}\frac{\epsilon^{2}}{1+\epsilon^{2}}(1/\epsilon-1)^{2}+\frac{1}{1+\epsilon^{2}}(1-\epsilon)^{2}=2.$

This completes the proof.

There may be reasons to suspect the upper bound is indeed $2$. Note that if we
simply ignored the data and sampled $\hat{\theta}\sim G_{1}$ independently as
our decision, then we would achieve Bayes risk of 2. Conditioning on the data
$X$ should bring the posterior under $G_{1}$ closer to $G_{0}$, and so it
should not hurt the Bayes risk.\arabicfootnote\arabicfootnotefootnoteNote that
this heuristic leads to an incorrect conclusion if we report the prior mean
under $G_{1}$ when we ignore the data, instead of sampling $\hat{\theta}\sim
G_{1}$. In that case, we achieve a risk of 1 but the worst-case risk is at
least 2. Zellner (1988) provides one information-theoretic sense in which this
is true, as the posterior minimizes a weighted average of the Kullback–Leibler
divergence to the prior and the integrated log-likelihood. However, since we
make no support or continuity restrictions on $G_{0},G_{1}$, it is not
immediate how to use these results in terms of information-theoretic
divergences.

## Acknowledgement

The author is particularly grateful to Isaiah Andrews, Xiao-Li Meng, Natesh
Pillai, Neil Shephard, and Elie Tamer for their comments. In early
conversations, Alexander Frankel suggested the configuration in the proof of
Lemma section.theorem.

## References

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  * Dedecker & Michel (2013) Dedecker, J. & Michel, B. (2013).  Minimax rates of convergence for wasserstein deconvolution with supersmooth errors in any dimension.  Journal of Multivariate Analysis 122, 278–291.
  * Efron (2019) Efron, B. (2019).  Bayes, oracle bayes and empirical bayes.  Statistical science 34, 177–201.
  * Goutis (1994) Goutis, C. (1994).  Ranges of posterior measures for some classes of priors with specified moments.  International Statistical Review/Revue Internationale de Statistique , 245–256.
  * Johnstone (2019) Johnstone, I. M. (2019).  Gaussian estimation: sequence and wavelet models.  Tech. rep.
  * Pensky (2017) Pensky, M. (2017).  Minimax theory of estimation of linear functionals of the deconvolution density with or without sparsity .
  * Robert (2007) Robert, C. P. (2007).  The Bayesian choice: from decision-theoretic foundations to computational implementation, vol. 2.  Springer.
  * Zellner (1988) Zellner, A. (1988).  Optimal information processing and bayes’s theorem.  The American Statistician 42, 278–280.

## Proof of Theorem section.theorem

The main idea of the proof is as follows: Using a result of Goutis (1994), any
distribution in $\mathcal{M}$ may be represented by a mixture of discrete
distributions in $\mathcal{M}$ with at most three support points. Jensen’s
inequality then shows that the optimal for $\overline{R}$ is achieved at a
distribution with at most three support points, and so we may assume $G_{1}$
satisfies this restriction. Chebyshev’s inequality yields a bound on the mass
points of $G_{1}$ that are far away from zero, and implies that some mass
point is close to zero. The posterior mass ratio between any mass point far
away from zero and the mass point close to zero is tractable, and the expected
squared error of $\mu_{1}(x)$ can be bounded in terms of the posterior mass
ratio. We obtain an upper bound on $\overline{R}$ through further bounding
various expressions.

######  Proof .theorem (of Theorem section.theorem).

We first recall Lemma 1 in Goutis (1994), which states that $G_{1}$ can be
represented as a mixture over mean-zero, variance-one distributions with at
most three support points. Observe that a mean-zero, variance-one distribution
supported on at most three points is uniquely determined by its support
points, since the probability masses assigned to the support points must
satisfy moment constraints. We let $a$ denote a particular arrangement of the
support points, and $\mathcal{A}$ denote the set of all such arrangements.
Goutis (1994) states that there exists some mixing distribution $\Pi$ over
$\mathcal{A}$ such that, for any Borel set $A\subset\mathbb{R}$,

$\operatorname{pr}_{G_{1}}(A)=\int\operatorname{pr}_{G_{a}}(A)\,d\Pi(a),$

where $G_{a}$ is a mean-zero, variance-one distribution supported on $a$.

For a given arrangement of support points $a$, let $\theta_{k}(a)$ denote the
individual support points, where $k$ ranges from $1$ to either $2$ or $3$.
Then, the posterior mean over $G_{1}$ is an average over the posterior means
for $G_{a}$, which can further be written as an average of $\theta_{k}(a)$:

$\mu_{1}(x)=\int\sum_{k}\operatorname{pr}_{G_{a}}\left({\theta=\theta_{k}(a)\mid
X=x}\right)\theta_{k}(a)\,d\Pi(a).$

By Jensen’s inequality,

$E[(\mu_{1}(X)-\theta)^{2}]\leq\int\sum_{k}E\left[{\operatorname{pr}_{G_{a}}\left({\theta=\theta_{k}(a)\mid
X=x}\right)(\theta_{k}(a)-\theta)^{2}}\right]\,d\Pi(a).$

We can write $X=\theta+\sigma Z$, where $Z\sim\mathcal{N}(0,1)$.
Correspondingly, we can define the posterior mass in terms of $\sigma,\theta,$
and $Z=z$: Let
$\pi_{k,a}(\sigma,\theta,z)=\operatorname{pr}_{G_{a}}(\theta=\theta_{k}(a)\mid
X=\theta+\sigma z).$ We can also define its expectation over $Z$: Let
$\overline{\pi}_{k,a}(\sigma,\theta)=\int_{-\infty}^{\infty}\pi_{k,a}(\sigma,\theta,z)\varphi(z)\,dz.$
Substituting these into (.theorem), we have that

$E[(\mu_{1}(X)-\theta)^{2}]\leq\int\sum_{k}E\left[{\overline{\pi}_{k,a}(\sigma,\theta)(\theta_{k}(a)-\theta)^{2}}\right]\,d\Pi(a)$

where the expectation $E[\cdot]$ integrates solely over $\theta\sim G_{0}$.

It suffices to bound
$\sum_{k}E\left[{\overline{\pi}_{k,a}(\sigma,\theta)(\theta_{k}(a)-\theta)^{2}}\right]$
over choices of $G_{0},\sigma,$ and the support point arrangement $a$. To that
end, consider a fixed but arbitrary $a$ with support points
$\left\\{{\theta_{k}}\right\\}$. Since we have fixed some $a$, we suppress it
from notation.

Case 1: All of $G_{a}$’s support points are within $[-2,2]$. In this case, we
can expand

$\displaystyle\sum_{k}E\left[{\overline{\pi}_{k,a}(\sigma,\theta)(\theta_{k}-\theta)^{2}}\right]$
$\displaystyle=E[\theta^{2}]+\sum_{k}E[\overline{\pi}_{k,a}(\sigma,\theta)]\theta_{k}^{2}-2E\left[{\theta\sum_{k}\overline{\pi}_{k,a}(\sigma,\theta)\theta_{k}}\right]$
$\displaystyle\leq 1+2^{2}+2E[|\theta|\cdot 2]$ $\displaystyle\leq 1+4+4$
$\displaystyle=9.$

Case 2: At least one of $G_{a}$’s support points are outside of $[-2,2]$. In
this case, there are at least one and at most two support points within
$[-2,2]$. By Chebyshev’s inequality, then, there is at least one support point
within $[-2,2]$ with mass exceeding
$\pi_{0}=\frac{1}{2}\left({1-\frac{1}{4}}\right)=\frac{3}{8}$. We can denote
that support point as $\theta_{0}$. For a given $\theta_{k}$, we shall compare
its posterior probability to $\theta_{0}$. As a result, it will be convenient
to translate by $-\theta_{0}$ so that we can normalize this comparison point
to zero.

Precisely speaking, let $H\sim\theta-\theta_{0}$ when $\theta\sim G_{a}$. Let
$H_{0}\sim\theta-\theta_{0}$ when $\theta\sim G_{0}$. By construction,
$\operatorname{pr}_{H}(\left\\{{0}\right\\})\geq\pi_{0}$. Denote the mass
points of $H$ by $\tau_{k}=\theta_{k}-\theta_{0}$. Observe that

$\operatorname{pr}_{H}(\tau=\tau_{k}\mid X=\theta+\sigma
z)=\operatorname{pr}_{G_{a}}(\theta=\theta_{k}\mid X=\theta+\sigma
z)=\pi_{k,a}(\sigma,\theta,z).$

Thus, we define

$\pi_{k}(\sigma,\tau,z)=\pi_{k,a}(\sigma,\tau+\theta_{0},z),\quad\overline{\pi}_{k}(\sigma,\tau)=\int\pi_{k}(\sigma,\tau,z)\varphi(z)\,dz.$

We are interested in
$\sum_{k}E\left[{\overline{\pi}_{k,a}(\sigma,\theta)(\theta_{k}-\theta)^{2}}\right]$,
which is preserved by the translation:

$\sum_{k}E\left[{\overline{\pi}_{k,a}(\sigma,\theta)(\theta_{k}-\theta)^{2}}\right]=\sum_{k}E\left[{\overline{\pi}_{k}(\sigma,\tau)(\tau_{k}-\tau)^{2}}\right]$

where the $E[\cdot]$ on the right-hand side integrates over $\tau\sim H_{0}$.
As a result, we shall bound
$\sum_{k}E\left[{\overline{\pi}_{k}(\sigma,\tau)(\tau_{k}-\tau)^{2}}\right]$.

After the translation, we note that the second moments of $H$ and $H_{0}$ are
now bounded by $5=1+2^{2}$. Since there are at most two non-zero support
points $\tau_{k}\neq 0$, we note that

$\displaystyle\sum_{k}E\left[{\overline{\pi}_{k}(\sigma,\tau)(\tau_{k}-\tau)^{2}}\right]$
$\displaystyle\leq 2\max_{k:\tau_{k}\neq
0}E[(\tau_{k}-\tau)^{2}\overline{\pi}_{k}(\sigma,\tau)]+E[\tau^{2}]$
$\displaystyle=5+2\max_{k:\tau_{k}\neq
0}E[(\tau_{k}-\tau)^{2}\overline{\pi}_{k}(\sigma,\tau)].$ (equation)

Thus, it suffices to bound
$E[(\tau_{k}-\tau)^{2}\overline{\pi}_{k}(\sigma,\tau)]$ for some nonzero
$\tau_{k}$. Without loss of generality, assume $\tau_{k}>0$. Define the
constants $C_{0}=\pi_{0}/5=3/40<1$ and
$s_{0}=\sqrt{1/C_{0}}=2\sqrt{10/3}>\sqrt{5}>2$.

Case 2.1: $\tau_{k}\leq s_{0}$. In this case,

$\displaystyle E[(\tau_{k}-\tau)^{2}\overline{\pi}_{k}(\sigma,\tau)]\leq
E[(\tau_{k}-\tau)^{2}]=E[(\theta_{k}-\theta)^{2}]=1+\theta_{k}^{2}$

Note that $|\theta_{k}|=|\tau_{k}+\theta_{0}|\leq s_{0}+2$. Thus, plugging
into the preceding display, in this case,

$E[(\tau_{k}-\tau)^{2}\overline{\pi}_{k}(\sigma,\tau)]\leq 33,$

and thus

$\sum_{k}E\left[{\overline{\pi}_{k}(\sigma,\tau)(\tau_{k}-\tau)^{2}}\right]\leq
71.$

Case 2.2: $\tau_{k}>s_{0}$. We decompose
$E[(\tau_{k}-\tau)^{2}\overline{\pi}_{k}(\sigma,\tau)]$ on four intervals for
$\tau$, where

$\mathbb{R}=(-\infty,-\tau_{k}]\cup[\tau_{k}/2,\infty)\cup(-\tau_{k},0)\cup(0,\tau_{k}/2).$

Note that if $\tau\in(-\infty,-\tau_{k}]$, then $(\tau_{k}-\tau)^{2}\leq
4\tau^{2}$. If $\tau\in[\tau_{k}/2,\infty)$, then
$(\tau_{k}-\tau)^{2}\leq\tau^{2}$. Thus,

$\displaystyle
E\left[{\overline{\pi}_{k}(\sigma,\tau)(\tau_{k}-\tau)^{2}}\right]\leq\,$
$\displaystyle
E[4\tau^{2}\mathbbm{1}\left({\tau\leq-\tau_{k}}\right)]+E[\tau^{2}\mathbbm{1}\left({\tau\geq\tau_{k}/2}\right)]$
$\displaystyle\quad+\int_{-\tau_{k}}^{0}(\tau_{k}-\tau)^{2}\overline{\pi}_{k}(\sigma,\tau)H_{0}(d\tau)+\int_{0}^{\tau_{k}/2}(\tau_{k}-\tau)^{2}\overline{\pi}_{k}(\sigma,\tau)H_{0}(d\tau).$

The first two terms in the preceding display are bounded by
$E[4\tau^{2}\mathbbm{1}(\tau<-\tau_{k}\cup\tau>\tau_{k}/2)].$ The next two
terms are respectively bounded by the ensuing Lemmas .theorem and .theorem.
The key to doing so is a bound of $\overline{\pi}_{k}(\sigma,\tau)$ in terms
of the posterior ratio (Lemma .theorem). Note that

$\pi_{k}(\sigma,\tau,z)\leq\min\left({\frac{\operatorname{pr}_{H}(\tau=\tau_{k}\mid\sigma
z+\tau)}{\operatorname{pr}_{H}(\tau=0\mid\sigma z+\tau)},1}\right),$

where the right-hand side can be integrated tractably.

Collecting the results in Lemmas .theorem and .theorem, we find that

$E\left[{\overline{\pi}_{k}(\sigma,\tau)(\tau_{k}-\tau)^{2}}\right]\leq\left({4\vee\left({1.5+\frac{3}{\sqrt{C_{0}}}}\right)\vee\frac{2.92}{C_{0}}}\right)E[\tau^{2}]+\frac{2.92}{C_{0}}+\frac{3}{2C_{0}}+\frac{3}{\sqrt{C_{0}}}\leq
264.6$

where $E[\tau^{2}]\leq 5$. Plugging into (equation), we obtain an upper bound

$E\left[{\overline{\pi}_{k}(\sigma,\tau)(\tau_{k}-\tau)^{2}}\right]\leq 535.$

This is worse than the bounds obtained from Case 2.1 and Case 1. Hence,
$\overline{R}\leq 535.$

######  Lemma .theorem.

Let $\Phi(t)=\int_{-\infty}^{t}\varphi(x)\,dx$ be the cumulative distribution
function of a standard Gaussian random variable. Let
$\overline{\Phi}(t)=1-\Phi(t)$ be its complement. If $\tau_{k}\geq\sqrt{5}$,
then

$\displaystyle\overline{\pi}_{k}(\sigma,\tau)$
$\displaystyle\leq\frac{1}{C_{0}\tau_{k}^{2}}\exp\left({\frac{\tau_{k}\tau}{\sigma^{2}}}\right)\overline{\Phi}\left({\frac{\tau_{k}/2+\tau}{\sigma}-\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right)+\overline{\Phi}\left({\frac{\tau_{k}/2-\tau}{\sigma}+\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right)$
(equation)

where we recall $C_{0}=\frac{\pi_{0}}{5}=\frac{3}{40}$.

######  Proof .theorem.

Note that,

$\pi_{k}(\sigma,\tau,z)=\left({\operatorname{pr}_{H}(\tau=0\mid\sigma
z+\tau)\frac{\operatorname{pr}_{H}(\tau=\tau_{k}\mid\sigma
z+\tau)}{\operatorname{pr}_{H}(\tau=0\mid\sigma z+\tau)}}\right)\wedge 1\leq
R(\sigma,\tau,z)\wedge 1.$

where $R=\frac{\operatorname{pr}_{H}(\tau=\tau_{k}\mid\sigma
z+\tau)}{\operatorname{pr}_{H}(\tau=0\mid\sigma z+\tau)}$ is the posterior
mass ratio. Note that, since $\operatorname{pr}(\tau=\tau_{k})\tau_{k}^{2}\leq
E_{H}[\tau^{2}]\leq 5,$ the prior mass is bounded:
$\operatorname{pr}(\tau=\tau_{k})\leq\frac{5}{\tau_{k}^{2}}$. We can thus
compute the posterior mass ratio

$R(\sigma,\tau,z)\leq\frac{1}{C_{0}}\frac{1}{\tau_{k}^{2}}\exp\left({\frac{\tau_{k}}{\sigma}z+\frac{\tau_{k}(\tau-\tau_{k}/2)}{\sigma^{2}}}\right),$

where the inequality stems from bounding the prior mass ratio with
$\frac{5/\tau_{k}^{2}}{\pi_{0}}.$ Define the right-hand side of the above
display as $\overline{R}(\sigma,\tau,z)$. We note that
$\overline{R}(\sigma,\tau,z)\leq 1$ if and only if $z\leq u$ where
$u=\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})+\frac{\tau_{k}/2-\tau}{\sigma}.$
Hence, letting $\overline{\Phi}(t)=1-\Phi(t)$ denote the complementary normal
CDF, we compute that

$\displaystyle\overline{\pi}_{k}(\sigma,\tau)$
$\displaystyle\leq\int_{-\infty}^{u}\overline{R}(\sigma,\tau,z)\varphi(z)\,dz+(1-\Phi(u))$
$\displaystyle=\frac{1}{C_{0}\tau_{k}^{2}}\exp\left({\frac{\tau_{k}\tau}{\sigma^{2}}}\right)\overline{\Phi}\left({\frac{\tau_{k}/2+\tau}{\sigma}-\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right)+\overline{\Phi}\left({\frac{\tau_{k}/2-\tau}{\sigma}+\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right).$

This completes the proof.

######  Lemma .theorem.

Recall that $s_{0}=\sqrt{1/C_{0}}=\sqrt{40/3}$. Define
$K_{(-\tau_{k},0)}=\int_{-\tau_{k}}^{0}(\tau_{k}-\tau)^{2}\overline{\pi}(\sigma,\tau)\,H_{0}(d\tau)$.
For $\tau_{k}\geq s_{0}$,

$K_{(-\tau_{k},0)}\leq\frac{3}{2C_{0}}+\frac{3}{\sqrt{C_{0}}}+\left({\frac{3}{2}+\frac{3}{\sqrt{C_{0}}}}\right)E[\tau^{2}\mathbbm{1}(-\tau_{k}\leq\tau\leq
0)].$

######  Proof .theorem.

Define $c=c(\tau)=\tau/\tau_{k}\in[-1,0]$ when $\tau\in[-\tau_{k},0]$. Then,
plugging in (equation),

$K_{(-\tau_{k},0)}\leq\int_{-\tau_{k}}^{0}\frac{(1-c)^{2}}{C_{0}}+(1-c)^{2}\tau_{k}^{2}\overline{\Phi}\left({(1/2-c)\frac{\tau_{k}}{\sigma}+\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right)\,H(d\tau)$

where we use the fact that $\exp(\tau_{k}\tau/\sigma^{2})\leq 1$ and
$\overline{\Phi}(\cdot)<1$ to simplify the first term in (equation). These are
true since we integrate over $(-\tau_{k},0)$.

Observe that, by Corollary .theorem,

$\overline{\Phi}\left({(1/2-c)\frac{\tau_{k}}{\sigma}+\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right)\leq\frac{1}{2}\exp\left({-(1-2c)\log(C_{0}\tau_{k}^{2})}\right)=\frac{1}{2}(C_{0}\tau_{k}^{2})^{2c-1}=\frac{1}{2}C_{0}^{2c-1}\tau_{k}^{4c-2}.$

Hence,

$\displaystyle K_{(-\tau_{k},0)}$
$\displaystyle\leq\int_{-\tau_{k}}^{0}\frac{(1-c)^{2}}{C_{0}}+\frac{(1-c)^{2}}{2}C_{0}^{2c-1}\tau_{k}^{4c}\,H_{0}(d\tau)$
$\displaystyle=\frac{1}{C_{0}}\int_{-\tau_{k}}^{0}(1-c)^{2}\left[{1+\frac{1}{2}(\sqrt{C_{0}}\tau_{k})^{4c}}\right]\,H_{0}(d\tau)$
$\displaystyle\leq\frac{1}{C_{0}}\int_{-\tau_{k}}^{0}\frac{(\tau_{k}-\tau)^{2}}{\tau_{k}^{2}}\left({1+\frac{1}{2}}\right)\,H_{0}(d\tau)$
(Note that $(\sqrt{C_{0}}\tau_{k})^{4c}\leq 1$)
$\displaystyle=\frac{3}{2}E\left[{\mathbbm{1}(-\tau_{k}\leq\tau\leq
0)\frac{(\tau_{k}-\tau)^{2}}{C_{0}\tau_{k}^{2}}}\right]$
$\displaystyle\leq\frac{3}{2C_{0}}\left({1+\frac{2}{\tau_{k}}E[|\tau|\mathbbm{1}(-\tau_{k}\leq\tau\leq
0)]+\frac{1}{\tau_{k}^{2}}E[\tau^{2}\mathbbm{1}(-\tau_{k}\leq\tau\leq
0)]}\right)$
$\displaystyle\leq\frac{3}{2C_{0}}+\frac{3}{\sqrt{C_{0}}}E[|\tau|\mathbbm{1}(-\tau_{k}\leq\tau\leq
0)]+\frac{3}{2}E[\tau^{2}\mathbbm{1}(-\tau_{k}\leq\tau\leq 0)].$
($\tau_{k}>1/\sqrt{C_{0}}$)

Conclude by noting that

$E[|\tau|\mathbbm{1}(-\tau_{k}\leq\tau\leq 0)]\leq E[(\tau^{2}\vee
1)\mathbbm{1}(-\tau_{k}\leq\tau\leq 0)]\leq
E[\tau^{2}\mathbbm{1}(-\tau_{k}\leq\tau\leq 0)]+1.$

This completes the proof.

######  Lemma .theorem.

Assume that $\tau_{k}>s_{0}$. Define
$K_{(0,\tau_{k}/2)}=\int_{0}^{\tau_{k}/2}(\tau_{k}-\tau)^{2}\overline{\pi}(\sigma,\tau)\,H_{0}(d\tau)$.
Then

$K_{(0,\tau_{k}/2)}\leq\frac{2.92}{C_{0}}+\frac{2.92}{C_{0}}E[\tau^{2}\mathbbm{1}(\tau\in(0,\tau_{k}/2))].$

######  Proof .theorem.

Let $c=\frac{\tau}{\tau_{k}}\in[0,1/2]$. Then we can write (equation),
multiplied by $(\tau_{k}-\tau)^{2}=(1-c)^{2}\tau_{k}^{2}$, as

$\frac{(1-c)^{2}}{C_{0}}\exp\left({\frac{c\tau_{k}^{2}}{\sigma^{2}}}\right)\overline{\Phi}\left({(c+1/2)\frac{\tau_{k}}{\sigma}-\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right)+(1-c)^{2}\tau_{k}^{2}\overline{\Phi}\left({(1/2-c)\frac{\tau_{k}}{\sigma}+\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right)$

This quantity upper bounds the integrand in $K_{(0,\tau_{k}/2)}$. Let us
consider the first term.

First term, case a:  In particular, let us consider the case where

$\tau_{k}^{2}\leq(1/2+c)^{-1}\sigma^{2}\log(C_{0}\tau_{k}^{2}).$

In this case, the first term in (.theorem) is bounded by the following, via
ignoring the $\overline{\Phi}(\cdot)$ term:

$\displaystyle\frac{(1-c)^{2}}{C_{0}}\exp\left({\frac{c}{1/2+c}\log(C_{0}\tau_{k}^{2})}\right)$
$\displaystyle=(1-c)^{2}C_{0}^{-\frac{1}{1+2c}}\tau_{k}^{\frac{4c}{1+2c}}$
$\displaystyle\leq\frac{1}{C_{0}}(1-c)^{2}\tau_{k}^{\frac{4c}{1+2c}}$
$\displaystyle\leq\frac{1}{C_{0}}\tau^{4c/(1+2c)}(1-c)^{2}\left({1/c}\right)^{\frac{4c}{1+2c}}$
$\displaystyle\leq\frac{1.76}{C_{0}}(\tau\vee 1)^{2}.$

The last inequality follows from the bound

$\sup_{c\in[0,1/2]}(1-c)^{2}(1/c)^{\frac{4c}{1+2c}}\leq 1.76.$

and $\tau^{4c/(1+2c)}\leq\tau\vee 1\leq(\tau\vee 1)^{2}$

First term, case b:  Now, assume that

$\tau_{k}^{2}>(1/2+c)^{-1}\sigma^{2}\log(C_{0}\tau_{k}^{2})\iff(1/2+c)\tau_{k}/\sigma>\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})$

so that the argument in $\overline{\Phi}$ in the first term is positive. Now,

$\displaystyle\frac{(1-c)^{2}}{C_{0}}\exp\left({c\tau_{k}^{2}/\sigma^{2}}\right)\overline{\Phi}\left({(c+1/2)\frac{\tau_{k}}{\sigma}-\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right)$
$\displaystyle\leq\frac{1}{2C_{0}}(1-c)^{2}\exp\left[{c\frac{\tau_{k}^{2}}{\sigma^{2}}-\frac{1}{2}\left({(c+1/2)\frac{\tau_{k}}{\sigma}-\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right)^{2}}\right]$
(Lemma .theorem)

We compute that, by expanding and applying the inequality $(a+b)\geq
2\sqrt{ab}$ for nonnegative $a,b$,

$\displaystyle
c\frac{\tau_{k}^{2}}{\sigma^{2}}-\frac{1}{2}\left({(c+1/2)\frac{\tau_{k}}{\sigma}-\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right)^{2}$
$\displaystyle=c\frac{\tau_{k}^{2}}{\sigma^{2}}-\frac{1}{2}(c+1/2)^{2}\frac{\tau_{k}^{2}}{\sigma^{2}}-\frac{\sigma^{2}}{2\tau_{k}^{2}}\log(C_{0}\tau_{k}^{2})^{2}+(c+1/2)\log(C_{0}\tau_{k}^{2})$
$\displaystyle=(c+0.5)\log(C_{0}\tau_{k}^{2})-\left\\{{(0.5(c+0.5)^{2}-c)\frac{\tau_{k}^{2}}{\sigma^{2}}+\frac{1}{2}\log(C_{0}\tau_{k}^{2})^{2}\frac{\sigma^{2}}{\tau_{k}^{2}}}\right\\}$
$\displaystyle\leq-\left\\{{2\sqrt{\frac{1}{2}\left({\frac{1}{2}(c+0.5)^{2}-c}\right)}-(c+0.5)}\right\\}\log(C_{0}\tau_{k}^{2})$
$\displaystyle=2c\log(C_{0}\tau_{k}^{2}).$

Hence, the first term in (.theorem) is bounded by

$\frac{1}{2C_{0}}(1-c)^{2}(C_{0}\tau_{k}^{2})^{2c}=(1-c)^{2}\frac{\tau^{4c}}{2C_{0}}C_{0}^{2c}(1/c)^{4c}\leq\frac{1.16}{C_{0}}(\tau\vee
1)^{2}.$

The last inequality comes from maximizing the expression over $c\in[0,1/2]$.
The two cases imply that the first term is bounded by $1.76C_{0}^{-1}(\tau\vee
1)^{2}$.

Second term:  The second term in (.theorem) is bounded by

$\displaystyle(1-c)^{2}\tau_{k}^{2}\overline{\Phi}\left({(1/2-c)\frac{\tau_{k}}{\sigma}+\frac{\sigma}{\tau_{k}}\log(C_{0}\tau_{k}^{2})}\right)$
$\displaystyle\leq(1-c)^{2}\tau_{k}^{2}\frac{1}{2}\exp\left({(2c-1)\log(C_{0}\tau_{k}^{2})}\right)$
(Corollary .theorem)
$\displaystyle\leq\frac{1}{2C_{0}}\tau^{1+2c}c^{-4c}(1-c)^{2}$
$\displaystyle\leq\frac{1.16}{C_{0}}(\tau\vee 1)^{2}$ (Maximizing over $c$ and
using $\tau^{1+2c}\leq(\tau\vee 1)^{2}$)

Hence, overall, the integrand (.theorem) is bounded by
$\frac{2.92}{C_{0}}(\tau\vee 1)^{2}.$ We use $\tau^{2}\vee 1\leq\tau^{2}+1$ to
conclude the proof.

######  Lemma .theorem (Mill’s ratio bound).

For all $t>0$,

$\overline{\Phi}(t)\leq\frac{\varphi(t)}{t\vee\sqrt{2/\pi}}\leq\sqrt{\pi/2}\varphi(t)=\frac{1}{2}e^{-t^{2}/2}.$

######  Proof .theorem.

It is well known that $\overline{\Phi}(t)\leq\frac{\varphi(t)}{t}$ for all
positive $t$, which is tighter when $t$ is large. At $t=0$, the truncation
$\sqrt{2/\pi}$ is such that both sides are equal to $1/2$. We can
differentiate for $t<\sqrt{2/\pi}$ to verify that
$\overline{\Phi}^{\prime}(t)<\sqrt{\pi/2}\varphi^{\prime}(t)$.

######  Corollary .theorem.

For nonnegative $a,b\in\mathbb{R}$,
$\overline{\Phi}(a+b)\leq\frac{1}{2}\exp\left({-2ab}\right).$

######  Proof .theorem.

Immediate by noting that the arithmetic-mean–geometric-mean inequality implies
$\overline{\Phi}(a+b)\leq\overline{\Phi}(2\sqrt{ab})$ and applying Lemma
.theorem.