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%%%%%%%%%%%%%%%%% Begin document %%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
%% \linenumbers

\preprint{TTP15-006}

\title{
%======
\boldmath
Penguin contributions to \CP phases in $B_{d,s}$ decays to charmonium
\unboldmath}

\author{Philipp Frings$^{\,a}$,
  Ulrich Nierste$^{\,a}$, and Martin Wiebusch$^{\,c}$
\vspace{0.6cm}}

\affiliation{
\mbox{$^{a}$ Institut f\"ur Theoretische Teilchenphysik,
Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany,}
\mbox{email: philipp.frings@kit.edu, ulrich.nierste@kit.edu}\\
\mbox{$^{c}$  IPPP, Department of Physics, University of Durham, DH1
	      3LE, UK,	{e-mail: martin.wiebusch@durham.ac.uk}}
}
%\date{Draft of \today}


\begin{abstract}
  The precision of the \CP phases $2\beta$ and $2\beta_s$ determined
  from the mixing-induced \CP asymmetries in $B_d\to J/\psi K_S$ and
  $B_s\to J/\psi \phi$, respectively, is limited by the unknown
  long-distance contribution of a penguin diagram involving up
  quarks. We analyze the infrared QCD structure of this contribution and
  find that all soft and collinear divergences either cancel between
  different diagrams or factorize into matrix elements of local
  four-quark operators {up to terms suppressed by $\lqcd/m_{\psi}$,
    where $m_{\psi}$ denotes the $J/\psi$ mass}. Our results allow us to
  calculate the penguin-to-tree ratio $P/T$ in terms of the matrix
  elements of these operators and to constrain the penguin contribution
  to the phase $2\beta$ as $|\Delta \phi_d|\leq {0.68}^\circ$. The
  penguin contribution to $2\beta_s$ is bounded as $|\Delta
  \phi_s^{0}|\leq {1.03}^\circ$, $|\Delta \phi_s^{\parallel}|\leq
  {1.28}^\circ$, and $|\Delta \phi_s^{\perp}|\leq {0.95}^\circ$
  for the {case of} longitudinal, parallel, and perpendicular $\phi$
  and $J/\psi$ polarizations, respectively. {We further place bounds on
    $|\Delta \phi_d|$ for $B_d\to \psi(2S) K_S$ and the polarization
    amplitudes in $B_d\to J/\psi K^*$.}  In our approach it is further
  possible to constrain $P/T$ for decays in which $P/T$ is
  Cabibbo-unsuppressed and we {derive upper limits on} the 
  {penguin contribution to the} mixing-induced \CP asymmetries
  {in} $B_d\to J/\psi \pi^0$, $B_d\to J/\psi \rho^0$, $B_s\to J/\psi
  K_S$, and {$B_s\to J/\psi K^*$. For all studied decay modes we
    also constrain the sizes of the direct \CP asymmetries.}  
\end{abstract}


\pacs{13.25.Hw 	}
%========================================


\maketitle



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The mixing-induced \CP asymmetry in the decay $B_d\to J/\psi K_S$ is the
key quantity to measure the \CP phase of the \bbmd\ amplitude.
Within the Standard Model (SM) this \CP asymmetry
determines the angle $\beta=\arg[-V_{tb}V_{td}^*/(V_{cb}V_{cd}^*)]$ of
the unitarity triangle.  $B_s\to J/\psi \phi$ plays the same role for
the \bbs\ system. Since the unitarity of the Cabibbo-Kobayashi-Maskawa
(CKM) matrix essentially fixes
$\beta_s=\arg[-V_{tb}^*V_{ts}/(V_{cb}V_{cs}^*)]=1.0^\circ$,
\CP studies of $B_s\to J/\psi \phi$ directly probe physics beyond the
SM. The decay amplitude $A_f$ for an $\ov b\to \ov c c \ov s$
decay $B_q \to f$, where $f$ is a \CP eigenstate consisting of
a charmonium state and a light meson,
can be written as
\begin{equation}
A_f = \lambda_c^s T_f + \lambda_u^s P_f \label{eq:atp}
\end{equation}
with $\lambda_{p}^s=V_{pb}^*V_{ps}$, {$p=u,c$,} and
\begin{eqnarray}
 T_f &=& \frac{G_F}{\sqrt{2}} \braOket{f}{
     C_1 Q_1^c +C_2 Q_2^c -\sum_{j} C_j Q_j}{B_q},\\
 P_f &=& \frac{G_F}{\sqrt{2}} \braOket{f}{
     C_1 Q_1^u +C_2 Q_2^u -\sum_{j} C_j Q_j}{B_q}.
\label{eq:tp}
\end{eqnarray}
Here,  $Q_1=\ov s{}^\alpha \gamma_\mu (1-\gamma_5) q^\beta
\ov q{}^\beta \gamma^\mu (1-\gamma_5)b^\alpha$ and $Q_2=\ov s{}^\alpha
\gamma_\mu (1-\gamma_5) q^\alpha
\ov q{}^\beta \gamma^\mu (1-\gamma_5)b^\beta$ are the current-current
operators. The index $j$ labels the penguin operators $Q_j$ which involve the 
CKM elements $\lambda^s_t =-\lambda^s_c-\lambda^s_u$. While the QCD penguin 
operators % Eigentlich finde ich komisch, dass in dem Satz die lambdas auftreten
$Q_{3-6}$ and $Q_{8G}$ are important for this paper (see
Ref.~\cite{bbl} for their definition), electroweak penguin operators
have negligible effects.  The time-dependent \CP asymmetry
$A_{\rm CP}^{B_q\to f}(t)\equiv
[\Gamma(\Bbar_q(t)\to f)-\Gamma(B_q(t)\to f)]/[\Gamma(\Bbar_q(t)\to f)+
\Gamma(B_q(t)\to f)]$ reads
\begin{equation}
A_{\rm CP}^{B_q\to f}(t) =
\frac{S_f \sin(\dm_q t)- C_f \cos(\dm_q t)}{\cosh(\dg_q
  t/2)+A_{\dg_q}\sinh(\dg_q t/2)}.
\end{equation}
Here $\dm_q$ and $\dg_q$ are the mass and width difference,
respectively, between the mass eigenstates of the \bbq\ system.  We
write $S_f{\approx -\eta_f}\sin(\phi_q+ \Delta \phi_q)$, where 
{$\CP\!\ket f=\eta_f\ket f$ and } $\phi_q$
is the \CP phase in the limit $P_f=0$. The SM predictions are
$\phi_d=2\beta$ and $\phi_s=-2\beta_s$. To first order in
$\epsilon=|V_{us}V_{ub}/(V_{cs}V_{cb})|\approx 0.02$ one has
\begin{equation}
    \tan(\Delta \phi)\simeq 2
    \epsilon\sin \gamma \, \real \frac{P_f}{T_f} .
%\qquad
%    \qquad\epsilon \equiv \left|
%    \frac{V_{us}V_{ub}}{V_{cs}V_{cb}}\right|
	\label{eq:Dphi}
\end{equation}
$T_f$ and $P_f$ are non-perturbative multi-scale matrix elements, which
defy calculations from first principles of QCD.

For the prediction of the branching ratio $B(B_d\to J/\psi K_S)$ one
only needs $T_f$, which was addressed with the method of QCD
factorization \cite{qcdf} in Ref.~\cite{Chay:2000xn}: in the limit of
infinite charm and bottom masses $T_f$ can be expressed in terms of the
$J/\psi$ decay constant and the $B_d\to K_S$ form factor. The result of
Ref.~\cite{Chay:2000xn} underestimates $B(B_d\to J/\psi K_S)$ by a
factor of 8. This failure, however, is not surprising, because the
corrections to the infinite-mass limit are of order
$\lqcd/(m_c\alpha_s)$ and therefore numerically unsuppressed for the
actual value of the charm mass \cite{qcdf,Beneke:2002bs}.  The standard
approach to quantify $P_f/T_f$ in $B_d\to J/\psi K_S$ uses the
approximate SU(3)$_{\rm F}$ symmetry of QCD (or its U-spin subgroup)
which relates the decay of interest to $b\to c \ov c d$ modes like
$B_s\to J/\psi K_S$ and $B_d \to J/\psi \pi^0$
{\cite{Fleischer:1999nz,su3}}.  A drawback of this method is our poor
knowledge of the quality of the SU(3)$_{\rm F}$ symmetry in $B_{d,s}\to
J/\psi X$ (with $X=K_S,\pi^0,\ldots$) decays. (Comparisons of branching
ratios essentially test SU(3)$_{\rm F}$ in $T_f$ only, with little
sensitivity to $P_f$.) Furthermore, the $b\to c\ov c d$ control channels
have 20 times smaller statistics than their $b\to c\ov c s$
counterparts.  SU(3)$_{\rm F}$ seemingly fails in $B_s\to J/\psi \phi$,
because the $\phi$ meson cannot be closely approximated by an
SU(3)$_{\rm F}$ eigenstate, but is an equal mixture of octet and
singlet.

In this paper, we present a dynamical calculation of $P_f/T_f$ which does
not assume an approximate SU(3)$_{\rm F}$ symmetry. Our results permit,
for the first time, the prediction of $S_f$ and $C_f$ also for $b\to
c\ov c d$ decays.

\section{Operator product expansion}
For definiteness we first specify the discussion to $B_d\to J/\psi K_S$
and return to $B_s\to J/\psi \phi$ and other modes in the phenomenology
section.  {For} $B(B_d\to J/\psi K_S)$ we only need $T_f$ and can
  neglect the penguin coefficients.  It is useful to express $T_f$ in
terms of the matrix elements of
\begin{eqnarray}
Q_{0V} & \equiv & \ov s{} \gamma_\mu (1-\gamma_5) b \,
		  \ov c{}  \gamma^\mu c ,
		\nn
Q_{0A} & \equiv & \ov s{} \gamma_\mu (1-\gamma_5) b \,
		  \ov c{}  \gamma^\mu \gamma_5 c,
\nn
Q_{8V} & \equiv & \ov s{} \gamma_\mu (1-\gamma_5) T^a b\,
		  \ov c{}  \gamma^\mu T^a c,
		\nn
Q_{8A} & \equiv & \ov s{} \gamma_\mu (1-\gamma_5) T^a b\,
		  \ov c{}  \gamma^\mu \gamma_5 T^a c.
\label{eq:ops}
\end{eqnarray}
Then $T_f$ in \eq{eq:tp} becomes $T_f= \frac{G_F}{\sqrt 2}
\braOket{J/\psi K_S}{C_0(Q_{0V}-Q_{0A}) +C_8 (Q_{8V}-Q_{8A})}{B_d}$ with
$C_0 = C_2/N_c +C_1$ and $C_8=2C_2$, where $N_c=3$ is the number of
colors.  Using next-to-leading order (NLO) Wilson coefficients in the
NDR scheme \cite{bw,bbl} at the scale $\mu=m_{\psi}$ one finds $C_0=0.13 $
and $C_8=2.2$. The smallness of $C_0$ is a well-known numerical accident
entailing that the weak decay produces the $(c,\ov c)$ pair almost in a
color octet state. We normalize the matrix elements (for $j=0,8$) as
\begin{equation}
%  \braOket{J/\psi K_S}{Q_{jV}}{B_d} =
% V_0
% \, v_j \quad
%  \braOket{J/\psi K_S}{Q_{jA}}{B_d} =
% V_0 \, a_j
 \langle Q_{jV} \rangle =
V_0
\, v_j, \qquad
 \langle Q_{jA} \rangle =
V_0 \, a_j 
\label{eq:v0aj}
\end{equation}
to the factorized matrix element $ V_0\equiv \langle Q_{0V} \rangle {}_{\rm fact}
=2 f_{J/\psi} m_{B_d} p_{cm} F_1^{B\to K}(m_{\psi}^2)= (4.26 \pm 0.16)
\gev{}^{3} $.
The uncertainty stems from the form factor $F_1^{B\to
  K}(m_{\psi}^2)=0.586 \pm 0.021$ \cite{ff} and the $J/\psi$ decay
constant $f_{J/\psi}=(0.405\pm 0.005) \gev$. 
% With m_J/psi = 3.09692\pm 0.00001 GeV
% and m_KS = 0.4976 GeV
% I find p_cm = 1.68251 GeV
% From 1409.4557 I find F_1^{B\to K}=0.58623 +/- 0.0206828 including
%			the correlation matrix of the errors
$m_{B_d}=5.28\gev$ and $p_{cm}=1.68 \gev$ are the $B_d$ mass and the
magnitude of the three-momentum of the $J/\psi$ or $K_S$ in the $B_d$
rest frame.  $v_{0,8},a_{0,8}$ depend on $\mu$ in such a way that the
$\mu$-dependence of $C_0,C_8$ cancels from physical quantities. When we
quote numerical values we refer to the choice $\mu=m_{\psi}$.  The
large-$N_c$ counting of our (complex) hadronic parameters is
$v_0=1+{\cal O}(1/N_c^2)$, $v_8,a_8={\cal O}(1/N_c)$, and $a_0={\cal
  O}(1/N_c^2)$. Normalizing the branching ratio to the experimental
value we find
\begin{eqnarray}
  \frac{B(B_d\to J/\psi K_S)}{B(B_d\to J/\psi K_S)_{\rm exp}}
  &=& \nn 
&& \hspace{-15mm} 
  \lt[1\pm 0.08 \rt] \lt| 0.47v_0 +7.8 (v_8-a_8) \rt|^2.
\label{eq:bf}
\end{eqnarray}
\begin{figure*}[t]
% \hrule
%\begin{center}
	\begin{subfigure}{0.23\textwidth}
		\includegraphics[width=\textwidth,clip=true]
 {penguin_lo.eps}
		\caption{LO Penguin \label{fig:1a}}
		
	\end{subfigure}
	\begin{subfigure}{0.2\textwidth}
		\includegraphics[width=\textwidth,clip=true]%
		  {penguin13a.eps}
		  \caption{\label{fig:1b}} %Soft divergence}
		   
	\end{subfigure}
	\begin{subfigure}{0.15\textwidth}
		\includegraphics[width=\textwidth,clip=true]%
		  {penguin13ae.eps}
		  \caption{\label{fig:1c}	} %Soft factorized divergence}
		  	  
	\end{subfigure}
	\begin{subfigure}{0.2\textwidth}
		\includegraphics[width=\textwidth,clip=true]%
		   {penguin12b.eps}
		  \caption{}%Collinear divergence}
		  \label{fig:1d}		  
	\end{subfigure}
	\begin{subfigure}{0.19\textwidth}
		\includegraphics[width=\textwidth,clip=true]%
		      {penguinspectator.eps}
		         \caption{Spectator scattering}
		         \label{fig:1e}
	\end{subfigure}
%\end{center}
  \caption{The soft IR divergence of the
    diagram (b) factorizes with the corresponding diagram 
    of the effective-theory side shown in (c).  
    The diagram (d) is an example of a diagram with a collinear IR 
    divergence.}\label{fig:pengs}
~\\[-3mm]
\hrule
\end{figure*}
Varying the phase of $v_8-a_8$ between $-\pi$ and $\pi$ one finds the
correct branching ratio for $0.07\leq |v_8-a_8|\leq 0.19$ if $v_0$ is
set to 1.  Thus, there is no mystery with the branching ratio and the
hadronic parameters obey the hierarchy expected from $1/N_c$ counting.
The terms involving $a_0$ are negligible in view of other uncertainties
and are omitted throughout this paper.

$P_f$ in \eq{eq:tp} receives contributions from $Q_{1,2}^u$ and the
penguin operators $Q_j$, $j\geq 3$. The matrix elements of the latter
can be trivially expressed in terms of the operators in
\eq{eq:ops}. Therefore, this contribution to $P_f/T_f$ only depends on
$v_8/v_0$ and $a_8/v_0$. Below we will see that the magnitudes
of these ratios are under control thanks to the $1/N_c$ hierarchy of
$v_0,v_8,a_8$ and the information from $B(B_d\to J/\psi K_S)_{\rm exp}$.
By varying the parameters in the allowed ranges we can then find the
maximal contribution of the penguin operators to
$|\Delta\phi|$. 

In order to apply the same strategy to $Q_{1,2}^u$ we
must first express the up-quark penguin depicted in 
\fig{fig:1a} in terms of matrix elements of the local operators in
\eq{eq:ops}. In Ref.~\cite{Bander:1979px} it is argued that a penguin
loop flown through by a hard momentum $q$ (in our case $q^2\sim
m_\psi^2=(3.1\gev)^2$) can be calculated in perturbation theory (``BSS
mechanism'').  In Ref.~\cite{Boos:2004xp} this idea is used to find an
estimate of $\langle Q_2^u\rangle$ which leads to an upper bound on
$|\Delta\phi|$ which is smaller than the values found by SU(3)$_{\rm F}$
arguments \cite{su3}. In this paper, we turn the BSS idea into a rigorous
field-theoretic method by proving an operator product expansion (OPE)
\begin{eqnarray}
\!\!  \braOket{J/\psi K_S}{Q_j^u}{B_d} &=&  \sum_k
     \widetilde{C}_{j,k}
  \braOket{J/\psi K_S}{Q_{k}}{B_{d}}
    % \langle Q_k \rangle
    + \ldots % {\cal O} \lt( \rt)
\label{eq:ope}
\end{eqnarray}
with $k$ running over $k=0V,0A,8V,8A$ and the dots representing terms
suppressed by higher powers of $\lqcd/\sqrt{q^2}$. The Wilson
coefficients $\widetilde{C}_{j,k}=\widetilde{C}_{j,k}^{(0)}+
(\alpha_s(\mu)/(4\pi))\widetilde{C}_{j,k}^{(1)}+\ldots $ are calculated
in perturbation theory to the desired order in $\alpha_s(\mu)$, with the
renormalization scale $\mu={\cal O}(m_{\psi},m_b)$. A similar OPE has
been derived to calculate charm-loop effects in the rare semileptonic
decays $B\to K^{(*)} \ell^+\ell^-$ \cite{rare}. Since leptons carry no
color charges, this application involves no four-quark operators like
those in \eqsand{eq:ops}{eq:ope}. From the LO diagrams (see 
\fig{fig:1a}) one finds $ \widetilde{C}_{j,k}^{(0)}=0$ except for
$\widetilde{C}_{8G,8V}^{(0)}=-\frac{m_b^2}{q^2}\frac{\alpha_s}{\pi}$ and
$\widetilde{C}_{2,8V}^{(0)} =P(q^2) $ with the penguin function
\begin{eqnarray}
P(q^2)& = &\frac23 \frac{\alpha_s}{4 \pi}\lt[ \ln\lt(\frac{q^2}{\mu^2}\rt) -i 
\pi -\frac23 \rt].
\end{eqnarray}


\section{Proof of Factorization}

In order to establish \eq{eq:ope} we must prove that the coefficients $
\widetilde{C}_{j,k}$ are infrared (IR) safe. To this end we analyze {i)}
the soft IR divergences of the two-loop diagrams contributing to
$\langle Q_j^u\rangle$,
{ii)} the collinear IR divergences of these diagrams, {iii)}
spectator scattering diagrams, and {iv)} higher-order diagrams in
which the large momentum bypasses the penguin loop (``long distance
penguins'').

{i)} An example of a diagram with a soft divergence is shown in
\fig{fig:1b}. This soft divergence is reproduced by the corresponding
diagram of the effective-theory side (i.e.\ RHS) of \eq{eq:ope},
depicted in \fig{fig:1c}, so that this divergence
factorizes with $\widetilde{C}_{j,k}^{(0)}$ and does not affect
$\widetilde{C}_{j,k}^{(1)}$. All soft divergences are from diagrams in
which the additional gluon connects two external lines and cancel from
$\widetilde{C}_{j,k}^{(1)}$ in the same way. 

{ii)} Collinear divergences occur in diagrams in which a gluon is
attached to the line with the strange quark, which we treat as
massless. An example is shown in \fig{fig:1d}. If $l$ denotes the loop
momentum flowing through the gluon propagator and $p_s$ is the momentum
of the external strange quark, the collinear divergence corresponds to
the region with $l^2=0$ and $l \propto p_s$. We can then reduce the
problem to the study of one-loop diagrams with an external on-shell
gluon: If we sum over all possibilities to attach this gluon to one of
the lines of the LO diagram in \fig{fig:1a}, the collinear Ward identity
of QCD ensures that this sum vanishes when the open Lorentz index of the
gluon line is contracted with $l^\mu$. This feature ensures that the
collinear divergences of the sum of the two-loop diagrams vanish. (For a
discussion in the context of QCD factorization see
Refs.~\cite{qcdf,Bauer:2001cu,Buchalla:2002pd}.)  It equally holds for
the effective-theory side of the OPE. On both sides of \eq{eq:ope} the
sum over diagrams needed for the cancellation involves the diagram with
a strange-quark self-energy which does not contribute to the truncated
matrix element.  But adding these unphysical diagrams to both sides of
the OPE has no effect, since these diagrams factorize with
$\widetilde{C}_{j,k}^{(0)}$. The cancellation of collinear divergences
is conceptually identical to the situation in typical processes in
collider physics; it is further known to be much simpler (with fewer
diagrams to be discussed) if a physical gauge (with only two propagating
gluon degrees of freedom) is adopted.

{iii)} Next we discuss the spectator scattering contributions: diagrams
in which the gluon connects the $b$ or $s$ line with the spectator quark
line trivially factorize with the corresponding diagrams on the
effective side.  If the gluon {connects} the {spectator} with the gluon
{line or a} charm or up line, we have to take into account that the
squared momentum in the penguin loop is $(q+l)^2$ instead of $q^2$. If
the gluon is soft, $l^\mu\sim\lqcd$, the expansion of the loop function
$P$ around $q^2$ reproduces a term which correctly factorizes with
$\widetilde{C}_{j,k}^{(0)}$ up to term suppressed by
$\lqcd/\sqrt{q^2}$. If the gluon is hard-collinear, with virtuality
$l^2\sim p_{\rm cm} \lqcd$, the situation is more subtle: the LO diagram
is suppressed by $\lqcd/p_{\rm cm}$, because the momentum of the
spectator quark changes from zero to ${\cal O}(p_{\rm cm})$ in the
decay, which is penalized by the light-cone distribution amplitude
(LCDA) of the kaon \cite{qcdf}. The asymptotic form of the kaon LCDA,
$\Phi(x)=6x(1-x)$, where $x$ and $1-x$ are the fractions of the kaon
momentum carried by the $\ov s$ and $d$ quarks, {favors} momentum
configurations in which the kaon momentum is roughly equally shared
between the two valence quarks. While the propagator of the scattered
hard-collinear gluon is suppressed as $\sim 1/(\lqcd p_{\rm cm})$, the
suppression of the LO diagram is lifted, because the spectator momentum
is in the region $x\sim 1/2$ {favored} by the kaon LCDA. To identify
further suppression factors we first discuss the case that the gluon
connect a charm line with the spectator: counting $q^2\sim m_{\psi}^2$
and the energies of $\ov s$ and spectator-$d$ quarks as $p_{\rm cm}/2$,
the penguin loop gives $P((q+l)^2)\simeq P(q^2) + \frac{p_{\rm cm}}
{m_{\psi}}P^\prime(q^2)$.  The non-factorizing piece involving the
derivative $P^\prime(q^2)$ \uli{comes with a factor of $p_{\rm
    cm}/m_{\psi}$.}  The virtuality of the (anti-)charm propagator is
around $p_{\rm cm}$ \uli{entailing a suppression factor of $\lqcd/p_{\rm
    cm}$}.  Thus, in total spectator scattering from the charm lines
obeys \eq{eq:ope} up to terms of order $\lqcd/m_{\psi}$.  Next we
discuss the spectator scattering from the up line, with a sample diagram
depicted in \fig{fig:1e}. We find that these diagrams are
power-suppressed by $\lqcd/m_{\psi}$. In this respect these spectator
diagrams differ from the similar photon penguins calculated in
Ref.~\cite{Beneke:2006mk}, which involve $P(q^2)$ for $q^2\sim 0$ rather
than $q^2\sim m_{\psi}^2$.

{vi)} So far we have assumed
that the underlying hard process is the penguin loop with the hard scale
$\sqrt{q^2}$. But it may also be possible that the hard momentum
transfer to the $J/\psi$ occurs through a hard gluon radiated from the
$b$ or $s$ line, while the penguin loop is {a ``long-distance
  penguin''} governed by soft QCD. Such a situation is exemplified by
the diagram in \fig{fig:1b} with the left gluon having virtuality $\sim
m_{\psi}^2$. These diagrams, in which the whole weak decay process
occurs with small momentum transfers, have a suppression factor
$(\lqcd/\sqrt{q^2})^3$ stemming from the hard gluon propagator and an
off-shell $b$ quark propagator (or $s$ quark propagator).

In our discussion above we have considered the effect of $Q_2^u$.  
The proof equally holds for the contribution from  $Q_{8G}$, which 
contributes to $\widetilde{C}_{8G,8V}^{(0)}$ through a tree-level diagram.

In our power counting {in i)--iv)} we have treated $p_{\rm cm}$ as
an intermediate scale between $\lqcd$ and $m_\psi$ and have found no
non-factorizable non-perturbative effects of order $p_{\rm
  cm}/m_\psi$. While $p_{\rm cm}$ enters two-loop diagrams through
$p_b\cdot p_s \sim m_bp_{\rm cm} $, such terms do not come with IR
divergences and end up in the NLO corrections to the coefficients
$\widetilde{C}_{j,k}$. We find that the counting rule for $p_{\rm cm}$
is irrelevant, one can reproduce our results above as well for the
limiting cases $p_{\rm cm}\sim \lqcd$ and $p_{\rm cm}\sim \sqrt{q^2}$.

{The choice $q^2=m_\psi^2$ for $P(q^2)$ may be altered by adding a 
contribution of order $\lqcd$ to $\sqrt{q^2}$. This shuffles a piece 
proportional to $(\lqcd/m_\psi) P^\prime(m_\psi^2) $ into the
coefficient of the sub-leading operator $\ov s{} \gamma_\mu (1-\gamma_5) T^a b\,
\lt[ \Box-m_\psi^2 \rt] \ov c{}  \gamma^\mu T^a c $, which removes the
ambiguity associated with the choice of $q^2$. At NLO in  $\alpha_s$ 
one generates non-zero coefficients $\widetilde{C}_{j,k}^{(1)}$ also for 
$j=1$ or $k=0A,8A$. 
% and further populates the coefficients of operators 
% describing effects suppressed by $\lqcd/m_\psi$. 
}

\section{Phenomenology}
The penguin amplitude {depends on the Wilson coefficients as}
\begin{equation}
  P_f= V_0 \lt( 2 C_4 +2 C_6 
  +2 C_2 
  \widetilde{C}_{2,8V}^{(0)}+C_{8G} \widetilde{C}_{8G,8V}^{(0)}\rt)v_8
    + \dots 
  \label{eq:24p}
\end{equation}
where the dots represent the terms with $v_0$ and $a_8$ which have
  much smaller coefficients. The dependence of $\widetilde{C}_{2,8V}^{(0)}$
  (calculated from \fig{fig:1a}) on the renormalization scheme cancels
  with the scheme dependence of $C_4+C_6$ in \eq{eq:24p}. In the NDR
  scheme adopted by us these penguin coefficients give a larger 
  contribution to $P_f$ than the $u$-penguin loop contained in
  $\widetilde{C}_{2,8V}^{(0)}$.  

  For the prediction of $P_f/T_f$ we implement the constraint from
  $B(B\to f)$ exemplified for $f={J/\psi K_S}$ in \eq{eq:bf} in the
  following way: adapting a phase convention in which $A_f$ in
  \eq{eq:atp} is {real and} positive, we can determine $a_8$ in terms of
  {$V_0$,} $v_0$, $v_8$, and the measured $B(B\to f)$
  \cite{BranchingRatios}.  Then we use this to eliminate $a_8$ from
  $P_f/T_f$.  For example, we find
\begin{equation}
\frac{P_{J/\psi K_S}}{T_{J/\psi K_S}}= 0.01- 0.02 v_0 - (0.71 +0.33i)v_8 
\label{eq:ptv}
\end{equation}
{for} the central values of $V_0$ {(quoted after \eq{eq:v0aj})}
and $B(B_d \to J/\psi K_S)$. {We vary $v_8$ and $v_0$ in their
  allowed ranges $|v_8|<1/3$ and $|v_0|=1\pm 0.15$ 
  with the constraint that $|a_8|\leq 
  1/3$ 
  must be obeyed. The
  allowed ranges for $\Delta \phi$, $C_f$, and $\Delta S_f \equiv S_f +
  \eta_f \sin \phi_q$ are almost symmetric around zero. We list the
  upper bounds on their magnitudes 
  {for {several} decay modes} in Tabs.~\ref{tab:resultsBd} and
  \ref{tab:resultsBs}. The results include the uncertainties from $V_0$,
  the branching ratios, CKM parameters \cite{Charles:2004jd}, 
  and higher-order terms in our 
  OPE.  For the
  $b\to \ov{c} c d$ decay modes with Cabibbo-unsuppressed $P_f/T_f$ the
  expansion in \eq{eq:Dphi} has been replaced by the exact formula (see
  e.g.\ Ref.~\cite{Fleischer:1999nz,su3}). Our bounds are conservative, as
  the considered ranges for $v_8$ and $a_8$ are wide (permitting
  even sizable cancellations in \eq{eq:bf}).  From
  \eqsand{eq:bf}{eq:ptv} one verifies that any additional information on
  magnitude or phase of one of these parameters
  will substantially reduce the ranges quoted in
  Tabs.~\ref{tab:resultsBd} and \ref{tab:resultsBs}. Our results for
  $B_d\to J/\psi \pi^0$ favor the Belle measurement $C_{J/\psi
    \pi^0}=-0.08\pm0.17$, $S_{J/\psi \pi^0}=-0.65\pm0.22$ \cite{Lee:2007wd} 
  over the BaBar
  result $C_{J/\psi \pi^0}=-0.20\pm0.19$, $S_{J/\psi \pi^0}=-1.23\pm0.21$}
\cite{Aubert:2008bs}. (In the absence of penguin pollution $C_{J/\psi
  \pi^0}=0$ and $S_{J/\psi \pi^0}=-\sin(2\beta)=-0.69\pm 0.02$.)
In the the case of a more precise and non-vanishing measurement of 
$C_{J/\psi \pi^0}$, for example, $C_{J/\psi \pi^0}=-0.10\pm0.01$, which corresponds
to the current world average with a ten times smaller error, we can also
put stronger restrictions on the shift of the mixing-induced 
\CP violation $|\Delta S_{J/\psi \pi^0}|\leq 0.13 $.
A measurement of $C_{J/\psi \pi^0}$ that is consistent with zero,
however, does not improve the bound. This {feature occurs in} all
decay modes {with} Cabibbo-unsuppressed $P_f/T_f$. The recent 
measurements of $S_f$ and $C_f$ for the $B_d\to J/\psi \rho^0$
  polarization amplitudes \cite{Aaij:2014vda} comply with the ranges in 
  Tab.~\ref{tab:resultsBd}. 
 
\begin{table*}
  \caption{The maximal phase shift of $\phi_d$ due to penguin 
    pollution and limits for the $\CP$ violation observables $S_f$ and $C_f$ in 
    various $B_d\to f$ decays. 
    {Decays into two vector mesons involve different polarization
      amplitudes, 
  indicated by $0$, $\parallel$, and $\perp$ \cite{Dighe:1995pd}.} 
% The {superscripts} $0$, $\parallel$, and $\perp$ correspond to the the 
% different polarizations of the respective final state
% \cite{Dighe:1995pd}. 
}

\begin{ruledtabular}
\begin{tabular}{lccccccccc}
Final State \rule{0pt}{2.3ex} & $J/\psi K_S$ & $\psi(2S) K_S$ & 
$J/\psi\pi^0$  & $(J/\psi\rho)^0$&  $(J/\psi\rho)^\parallel$ & 
$(J/\psi\rho)^\perp$& $(J/\psi K^*)^0$&$ (J/\psi K^*)^\parallel$& $(J/\psi 
K^*)^\perp $\\\hline

$\max(|\Delta \phi_d|)$  $[{}^\circ]$&$0.68$&$0.74$ & n. a.  & n. a.  & n. a.  & n. a. &$0.87$&$1.28$&$1.07$ 
\\ 
$\max(|\Delta S_f|)$ $[10^{-2}]$&$0.86$&$0.94$&$18$&$22$&$27$&$22$&$1.11$&$1.64$&$1.38$ \\ 
$\max(|C_f|)$ $[10^{-2}]$&$1.33$&$1.33$&$29$&$35$&$41$&$36$&$1.70$&$2.47$&$2.08$ 

\end{tabular}
\end{ruledtabular}
\label{tab:resultsBd}
\end{table*}

\begin{table*}
\caption{{Same as Tab.~\ref{tab:resultsBd} for  $B_s\to f$ decays.}}
\begin{ruledtabular}
\begin{tabular}{lccccccc}
Final State & $J/\psi K_S$&   $(J/\psi\phi)^0$& 
$(J/\psi\phi)^\parallel$& $\left(J/\psi\phi\right)^\perp$& $(J/\psi K^*)^0$&$ 
(J/\psi K^*)^\parallel$& $(J/\psi K^*)^\perp$  \\\hline

 $\max(|\Delta \phi_s|)$  $[{}^\circ]$ & n. a. &$1.03$&$1.28$&$0.95$ & n. a.  & n. a.  & n. a.  \\ 
 $\max(|\Delta S_f|)$ $[10^{-2}]$&$37$&$1.80$&$2.22$&$1.65$&$39$&$60$&$24$ \\ 
 $\max(|C_f|)$ $[10^{-2}]$&$40$&$2.01$&$2.44$&$1.84$&$42$&$59$&$24$ 
 
 \end{tabular}
\end{ruledtabular}
\label{tab:resultsBs}
\end{table*}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We have established a factorization formula (to leading power in
  $\lqcd/m_{\psi}$) for the penguin
  contribution to the \CP$\!\!$-violating coefficients $S_f$ and $C_f$ in
  $A_{\rm CP}^{B_q\to f}(t)$ for final states $f$ containing
  charmonium. As a crucial result
  the penguin contributions involve the same hadronic matrix elements as
  the tree amplitude. This allows us to constrain $P_f/T_f$ and e.g.\
  find $|\Delta \phi_d |\leq {0.68}^\circ$ for $B_d\to J/\psi K_S$ and
  $|\Delta \phi_s^\perp|\leq {0.95}^\circ $ for $B_d\to J/\psi\phi$,
  representing bounds that were
  thought to be uncalculable from first principles. Novel territory are
  our predictions for $S_f$ and $C_f$ in $b\to c\ov c d$ decays, in
  which $P_f/T_f$ is Cabibbo-unsuppressed. Our results do not depend
  on any properties of the charmonium wave functions. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Acknowledgments}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We thank Gerhard Buchalla, Marco Ciuchini, Enrico Franco, Yuval
Grossman, Sebastian J\"ager, Kirill Melnikov, and Luca Silvestrini for
fruitful discussions .  This work is supported by BMBF under grant
no.~05H12VKF.  P.F.~acknowledges the support by the DFG-funded Doctoral
School \emph{KSETA}.


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  %30 citations counted in INSPIRE as of 13 Feb 2015

We modify the experimental $B(B_s\to J/\psi K_S)$ and $B(B_s \to J/\psi \phi)$ 
branching ratios according to:
  %\cite{DeBruyn:2012wj}
  K.~De Bruyn, R.~Fleischer, R.~Knegjens, P.~Koppenburg, M.~Merk and N.~Tuning,
  %``Branching Ratio Measurements of $B_s$ Decays,''
  Phys.\ Rev.\ D {\bf 86}, 014027 (2012)
  [arXiv:1204.1735 [hep-ph]].
  %%CITATION = ARXIV:1204.1735;%%
  %106 citations counted in INSPIRE as of 29 gen 2015

%CKMfitter  
  %\cite{Charles:2004jd}
\bibitem{Charles:2004jd} 
  J.~Charles {\it et al.}  [CKMfitter Group Collaboration],
  %``CP violation and the CKM matrix: Assessing the impact of the asymmetric $B$ factories,''
  Eur.\ Phys.\ J.\ C {\bf 41}, 1 (2005)
  [hep-ph/0406184]; {update from http://ckmfitter.in2p3.fr.}
  %%CITATION = HEP-PH/0406184;%%
  %1287 citations counted in INSPIRE as of 26 Feb 2015


% Citations B_d \to J/\psi \pi^0  
%\cite{Lee:2007wd}
\bibitem{Lee:2007wd} 
  S.~E.~Lee {\it et al.}  [Belle Collaboration],
  %``Improved measurement of time-dependent CP violation in B0 ---> J/Psi pi0 decays,''
  Phys.\ Rev.\ D {\bf 77}, 071101 (2008)
  [arXiv:0708.0304 [hep-ex]].
  %%CITATION = ARXIV:0708.0304;%%
  %31 citations counted in INSPIRE as of 21 Jan 2015  
  %\cite{Aubert:2008bs}
\bibitem{Aubert:2008bs} 
  B.~Aubert {\it et al.}  [BaBar Collaboration],
  %``Evidence for CP violation in $B^0 \to J/\psi \pi^0$ decays,''
  Phys.\ Rev.\ Lett.\  {\bf 101}, 021801 (2008)
  [arXiv:0804.0896 [hep-ex]].
  %%CITATION = ARXIV:0804.0896;%%
  %30 citations counted in INSPIRE as of 21 Jan 2015

\bibitem{Aaij:2014vda} 
  R.~Aaij {\it et al.}  [LHCb Collaboration],
  %``Measurement of the CP-violating phase $\beta$ in $B^0\rightarrow J/\psi \pi^+\pi^-$ decays and limits on penguin effects,''
  Phys.\ Lett.\ B {\bf 742}, 38 (2015)
  [arXiv:1411.1634 [hep-ex]].
  %%CITATION = ARXIV:1411.1634;%%

% Polarization amplitudes
    %\cite{Dighe:1995pd}
\bibitem{Dighe:1995pd} 
  A.~S.~Dighe, I.~Dunietz, H.~J.~Lipkin and J.~L.~Rosner,
  %``Angular distributions and lifetime differences in $B_s \to J/\psi \phi$ decays,''
  Phys.\ Lett.\ B {\bf 369}, 144 (1996)
  [hep-ph/9511363].
  %%CITATION = HEP-PH/9511363;%%
  %266 citations counted in INSPIRE as of 25 Feb 2015

  


  
\end{thebibliography}


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