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\begin{document}

\preprint{TTP15-020}

\vspace*{-30mm}

\title{Sum Rules of Charm CP Asymmetries beyond the SU(3)$_F$ Limit}
\thanks{dedicated to the memory of Lincoln Wolfenstein (1923-2015)}
\author{Sarah M\"uller$^{\,a,b}$}
\email{sarah.mueller2@kit.edu}
\author{Ulrich Nierste$^{\,b}$}
\email{ulrich.nierste@kit.edu}
\author{Stefan Schacht$^{\,b}$}
\email{stefan.schacht@kit.edu}
\affiliation{
$^{\,a}$ Institut f\"ur Experimentelle Kernphysik, Karlsruher
  Institut f\"ur Technologie, 76021 Karlsruhe, Germany\\
$^{\,b}$ Institut f\"ur Theoretische Teilchenphysik, Karlsruher
  Institut f\"ur Technologie, 76128 Karlsruhe, Germany}

\vspace*{1cm}

\begin{abstract}
We find new sum rules between direct CP asymmetries in $D$
  meson decays with coefficients that can be determined from a
  global fit to branching ratio data. Our sum rules eliminate the
  penguin topologies $P$ and $PA$, which cannot be determined from
  branching ratios.  In this way we can make predictions about direct
CP asymmetries in the Standard Model without ad-hoc assumptions on
the sizes of penguin diagrams. We consistently include first-order
  SU(3)$_F$ breaking in the topological amplitudes extracted from the
  branching ratios. By confronting our sum rules with future precise data
  from LHCb and Belle~II one will identify or constrain new-physics
  contributions to $P$ or $PA$.
The first sum rule correlates the CP asymmetries $a_{CP}^{\mathrm{dir}}$
in $D^0\to K^+K^-$, $D^0\to \pi^+\pi^-$, and $D^0\to \pi^0\pi^0$.  We
study the region of the $a_{CP}^{\mathrm{dir}}(D^0\to
\pi^+\pi^-)$--$a_{CP}^{\mathrm{dir}} (D^0\to \pi^0\pi^0)$ plane allowed
by current data and find that our sum rule excludes more than half of
the allowed region at 95\% CL. Our second sum rule correlates the direct
CP asymmetries in $D^+\rightarrow \bar{K}^0 K^+$, $D_s^+\rightarrow K^0
\pi^+$, and $D_s^+\rightarrow K^+\pi^0$.
\end{abstract}

\maketitle

\section{Introduction}
Decays of charmed mesons are currently the only way to probe flavour
violation in the up-quark sector. A major goal of experimental charm
physics is the discovery of CP violation in nonleptonic charm decays
(see e.g.~\cite{Aaij:2011in, Collaboration:2012qw, Aaij:2014gsa,
  Nisar:2014fkc}). To this end it is promising to study singly
Cabibbo-suppressed (SCS) decays~$d$ whose
direct CP asymmetries may be large enough to be detected in
the near future:
\begin{align}
a_{CP}^{\mathrm{dir}}(d) &amp;\equiv \frac{
	\vert \mathcal{A}^{\mathrm{SCS}}(d)\vert^2 - \vert \overline{\mathcal{A}}^{\mathrm{SCS}}(d)\vert^2
}{
	\vert \mathcal{A}^{\mathrm{SCS}}(d)\vert^2 + \vert \overline{\mathcal{A}}^{\mathrm{SCS}}(d)\vert^2
}\,. \label{eq:defacp}
\end{align}
We decompose the decay amplitude $\mathcal{A}^{\mathrm{SCS}}(d)$ as
$\mathcal{A}^{\mathrm{SCS}}(d) = \mathcal{A}_{sd}^{\mathrm{SCS}}(d) +
\mathcal{A}_b^{\mathrm{SCS}}(d)$ with
\begin{align*}
\mathcal{A}_{sd}^{\mathrm{SCS}}(d) &amp;= \lambda_{sd} \mathcal{A}_{sd}(d)
\,, 
&amp;\mathcal{A}_b^{\mathrm{SCS}}(d) &amp;=   -\frac{\lambda_b}{2} \, \mathcal{A}_b(d),
\end{align*}
and the shorthand notations $\lambda_q \equiv V_{cq}^* V_{uq}$ and
$\lambda_{sd} \equiv (\lambda_s-\lambda_d)/2$ for the elements of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix involved.  Adopting the PDG
convention (with $\lambda_s&gt;0$) one may safely neglect subleading powers
of $\lambda_{b}/\lambda_{sd} \sim 10^{-3}$.
\eq{eq:defacp} reads
\begin{align}
  a_{CP}^{\mathrm{dir}}(d) &amp;=
%
\frac{ \mathrm{Im}\,\lambda_b \; \mathrm{Im}\left[ e^{-i\delta(d)}
  \mathcal{A}_b(d) \right]}{ \vert
  \mathcal{A}^{\mathrm{SCS}}_{sd}(d)\vert } \label{eq:numACP-3}
%
\end{align}
in terms of the strong phase
$\delta(d)\equiv\mathrm{arg}\left[\mathcal{A}_{sd}(d)\right]$. The
smallness of $\mathrm{Im}\, \lambda_b$ renders
$a_{CP}^{\mathrm{dir}}(d)$ highly sensitive to physics beyond the
Standard Model (SM). To establish a ``smoking gun'' signal of new
physics one needs reliable SM predictions for
$a_{CP}^{\mathrm{dir}}(d)$ or at least robust theoretical upper bounds
on $|a_{CP}^{\mathrm{dir}}(d)|$ which cannot be exceeded within the SM.
The difficulty of such theory predictions can be witnessed from $\Delta
a_{CP}^{\mathrm{dir}}\equiv a_{CP}^{\mathrm{dir}}(D^0\rightarrow K^+K^-)
- a_{CP}^{\mathrm{dir}}(D^0\rightarrow\pi^+\pi^-)$: estimates vary
between $\mathcal{O}(0.01\%)$ \cite{Bigi:2011re}, $\mathcal{O}(0.1\%)$
\cite{Buccella:1994nf, Grossman:2006jg, Artuso:2008vf, Petrov:2010gy,
  Li:2012cfa}, $\sim -0.25\%$ \cite{Cheng:2012wr} and $\sim -0.4\%$
\cite{Brod:2011re}, not excluding an enhanced SM value between
$\sim-0.6\%$ and $\sim-0.8\%$ \cite{Brod:2011re,Pirtskhalava:2011va,
  Brod:2012ud,Feldmann:2012js,Hiller:2012xm,Franco:2012ck}.  There are
claims that CP-violating effects in charm physics can be
$\mathcal{O}(1\%)$ \cite{Golden:1989qx}. The situation is not any better
in CP violation induced by \ddm\ \cite{Bobrowski:2010xg}.  The key
problem is our lack of knowledge of the penguin amplitude entering
$\mathcal{A}_b(d)$ in \eq{eq:numACP-3} \cite{Feldmann:2012js,
  Brod:2011re, Brod:2012ud}.

All theoretical analyses of $D\to PP^\prime$ decays, where
$P$,$P^\prime$ denote pseudoscalar mesons, rely on the approximate
SU(3)$_F$ symmetry of the strong interaction \cite{Altarelli:1974sc,
  Kingsley:1975fe, Einhorn:1975fw, Voloshin:1975yx, Quigg:1979ic,
  Wang:1980ac, Chau:1982da, Golden:1989qx, Savage:1991wu, Chau:1991gx,
  Grossman:2006jg, Isidori:2011qw, Brod:2011re, Pirtskhalava:2011va,
  Cheng:2012wr,Bhattacharya:2012ah, Altmannshofer:2012ur,
  Feldmann:2012js, Brod:2012ud, Li:2012cfa, Franco:2012ck,
  Hiller:2012xm, Bhattacharya:2012pc, Bhattacharya:2012kq,
  Atwood:2012ac, Grossman:2012ry, Hiller:2012wf, Buccella:2013tya,
  Gronau:2013xba, Lenz:2013pwa, Gronau:2015rda, Muller:2015lua}.
{In analyses of branching ratios one can include first-order 
SU(3)$_F$ breaking
  \cite{Grossman:2006jg,Li:2012cfa, Cheng:2012wr,Brod:2011re,
    Pirtskhalava:2011va,Feldmann:2012js, Brod:2012ud,
    Hiller:2012xm,Franco:2012ck,Savage:1991wu,
    Chau:1991gx,Bhattacharya:2012ah, Bhattacharya:2012pc,
    Bhattacharya:2012kq,Grossman:2012ry,Buccella:2013tya,Gronau:2013xba,
    Gronau:2015rda,Muller:2015lua}.}  An intuitive way to exploit
SU(3)$_F$ relations involves topological amplitudes (pioneered in
Refs.~\cite{Wang:1980ac,Zeppenfeld:1980ex,Chau:1982da}) which
characterize the flavor-flow in terms of tree ($T$), color-suppressed
tree ($C$), exchange ($E$), annihilation ($A$), penguin ($P_{d,s,b}$),
and penguin annihilation ($PA_{d,s,b}$) diagrams.  This method has been
extended to include linear SU(3)$_F$ breaking in applications to $B$
\cite{Gronau:1995hm} and $D$ \cite{Muller:2015lua} decays.

With our inability to predict individual CP asymmetries it is natural to
study correlations among several asymmetries. There are two sum rules
which hold in the limit of exact SU(3)$_F$ symmetry
\cite{Grossman:2006jg,Grossman:2013lya,Hiller:2012xm}:
\begin{align}
  a_{CP}^{\mathrm{dir}}(D^0\rightarrow K^+K^-) +
  a_{CP}^{\mathrm{dir}}(D^0\rightarrow \pi^+\pi^-) &amp;=
  0\,, \label{eq:su3limit1}\\ 
a_{CP}^{\mathrm{dir}}(D^+\rightarrow
  \bar{K}^0K^+) + a_{CP}^{\mathrm{dir}}(D_s^+\rightarrow K^0\pi^+) &amp;=
  0\,, \label{eq:su3limit2}
\end{align}
{No relations among direct CP asymmetries hold to first order in
  SU(3)$_F$ breaking \cite{Grossman:2013lya}.}
Analyses of branching ratios permit the determination of $\vert
\mathcal{A}^{\mathrm{SCS}}_{sd}(d)\vert$ in \eq{eq:numACP-3} and,
through global fits, also to constrain the phase $\delta(d)$.  The
branching ratios of the decays entering
\eqsand{eq:su3limit1}{eq:su3limit2} exhibit sizable SU(3)$_F$ breaking
which limits the power of these SU(3)$_F$-limit sum rules to test the
SM.  In this letter we derive new sum rules which incorporate SU(3)$_F$
breaking in $\mathcal{A}^{\mathrm{SCS}}_{sd}(d)$ to linear order. To
this end we use the result of our global fit to $D\to PP^\prime$
branching ratios in Ref.~\cite{Muller:2015lua} in two ways: On one hand
we extract $\vert \mathcal{A}^{\mathrm{SCS}}_{sd}(d)\vert$ and
$\delta(d)$ for the decays of interest from the fit to find the
numerical relation between $a_{CP}^{\mathrm{dir}}(d)$ and
$\mathcal{A}_b(d)$ in \eq{eq:numACP-3}.  On the other hand the same fit
also returns the values of the topological amplitudes entering the
fitted branching ratios. However, the desired $\mathcal{A}_b(d)$ of
individual decay modes involve additional topological amplitudes (new
combinations of $P_{d,s,b}$ and $PA_{d,s,b}$) which do not appear in the
branching ratios. Our sum rules are constructed in a way to eliminate
these unknowns. Unlike \eqsand{eq:su3limit1}{eq:su3limit2} these new sum
rules use the SU(3)$_F$ limit for the eliminated $P_{d,s,b}$,
$PA_{d,s,b}$ in $\mathcal{A}_b(d)$ only, while consistently including
SU(3)$_F$ breaking in $T$, $A$, $C$, $E$. 

\section{CP Asymmetry Sum Rules}
Our conventions and the methodology of the global fit can be found in
Ref.~\cite{Muller:2015lua}.
The CKM-subleading part of the SCS
amplitude is expanded in terms of topological amplitudes
$\mathcal{T}_i={C,P_{\rm break}},\ldots$ as
\begin{align}
\mathcal{A}_b (d) = {c_{sd}^d \mathcal{A}_{sd}(d) +}
\sum_i c_i^d \mathcal{T}_i \, , \label{eq:atop}
\end{align}
with the coefficients $c_i^d$ specified in
Table~\ref{tab:topoparametrization-subleading}.  {With the first term
  (with $c_{sd}^d=\pm 1$) we eliminate the numerically large parameters
  $T$ and $E$ from the sum in \eq{eq:atop}. Since
  $e^{-i\delta(d)}\mathcal{A}_{sd}(d)=|\mathcal{A}_{sd}(d)|$ is real,
  $c_{sd}^d \mathcal{A}_{sd}(d)$ does not contribute to
  $a_{CP}^{\mathrm{dir}}(d)$ in \eq{eq:numACP-3}.}  The branching
fractions involve the SU(3)$_F$-breaking penguin topology
$P_{\mathrm{break}}\equiv P_s - P_d $, with $s,d$ denoting the quark
flavour in the penguin loop. The new penguin topologies in
$a_{CP}^{\mathrm{dir}}(d)$ are $P\equiv P_d + P_s - 2 P_b$ and $PA\equiv
PA_d + PA_s - 2P_b$.  Since we consistently neglect subleading terms
{(such as $PA_{\mathrm{break}}\equiv PA_s - PA_d $ which vanishes in the
SU(3) limit)} in $\mathcal{A}_b (d)$, we can use the fitted values in
\eq{eq:atop}.
\begin{table*}
\begin{center}
\begin{tabular}{ccccccccccc}
  \hline \hline Decay ampl. $\mathcal{A}(d)$ &amp; $A_{sd}(d)$ &amp; $A_1^{(1)}$
  &amp; $A_2^{(1)}$ &amp; $A_3^{(1)}$ &amp; $C+A$ &amp; $C_3^{(1)}$ &amp; 
 $P_{\mathrm{break}}$
  &amp; $P{{+}2A}$ &amp; $P{+PA}$ \\\hline
%
  $\mathcal{A}(D^0\rightarrow K^+ K^-)$ &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; $-1$ &amp;
  {0} &amp; 1 \\\hline
%                                                                                                                                                                                                                                         
  $\mathcal{A}(D^0\rightarrow \pi^+ \pi^-)$ &amp; $-1$ &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp;
  1 &amp; {0} &amp; 1 \\\hline
% 
  $\mathcal{A}(D^0 \rightarrow \pi^0 \pi^0)$ &amp; $-1$ &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0
  &amp; $-\frac{1}{\sqrt{2}}$ &amp; {0} &amp;
  $-\frac{1}{\sqrt{2}}$ \\\hline
%                                                                                                                                                                                                                                         
  $\mathcal{A}(D^+ \rightarrow \bar{K}^0 K^+)$ &amp; 1 &amp; 0 &amp; 0 &amp; 2 &amp; 0 &amp; 0
  &amp;$-1$ &amp; 1 &amp; 0 \\\hline
%                                                                                                                                                                                                                                         
  $\mathcal{A}(D_s^+ \rightarrow K^0 \pi^+)$ &amp; $-1$ &amp; 2 &amp; 2 &amp; 0 &amp; 0 &amp; 0
  &amp; 1 &amp; 1 &amp; 0 \\\hline
%                                                                                                                                                                                                                                         
  $\mathcal{A}(D_s^+ \rightarrow K^+ \pi^0)$ &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp;
  $\sqrt{2}$ &amp; $\sqrt{2}$ &amp; $\frac{1}{\sqrt{2}}$ &amp; $-\frac{1}{\sqrt{2}}$
  &amp; 0 \\\hline\hline
%
\end{tabular}
\caption{The coefficients {$c_i^d$}
of the topological decomposition of
  $\mathcal{A}_b(d)$ in \eq{eq:atop}. 
\label{tab:topoparametrization-subleading}}
\end{center}
\end{table*}

%
\begin{table*}
\begin{center}
\begin{tabular}{ccc}
\hline \hline
Decay $d$ &amp;  ${\mathrm{Im}}\left[X(d)\right]$  &amp; $\mathcal{S}(d)$ \\\hline
%
$D^0\rightarrow K^+ K^-$        &amp; $\mathrm{Im}\left[e^{-i \delta\left( D^0\rightarrow K^+ K^-\right)} \left( - P_{\mathrm{break}} \right)\right]$  &amp; $ \left( P + PA \right)- e^{2 i \delta\left( D^0\rightarrow K^+ K^-\right) }\left( P + PA \right)^*$  \\\hline
%
$D^0\rightarrow \pi^+ \pi^- $   &amp; $\mathrm{Im}\left[e^{-i \delta\left( D^0\rightarrow \pi^+ \pi^-\right)} \left( P_{\mathrm{break}} \right)\right]$  &amp; $\left( P + PA \right) -  e^{2 i \delta\left( D^0\rightarrow \pi^+ \pi^-\right) }   \left( P + PA \right)^* $  \\\hline
%
$D^0 \rightarrow \pi^0 \pi^0$   &amp; $\mathrm{Im}\left[e^{-i \delta\left( D^0 \rightarrow \pi^0 \pi^0\right)} \left( -\frac{1}{\sqrt{2}} P_{\mathrm{break}}\right)\right]$   &amp; $\frac{1}{\sqrt{2}}\left(  - P - PA \right)-  e^{2 i \delta\left( D^0 \rightarrow \pi^0 \pi^0\right) }  \frac{1}{\sqrt{2}}\left(  - P - PA \right)^* $  \\\hline
%
$D^+ \rightarrow \bar{K}^0 K^+$ &amp;  $\mathrm{Im}\left[e^{-i \delta\left( D^+ \rightarrow \bar{K}^0 K^+\right)} \left( -2 A^{\mathrm{fac}}_{ D^+ \rightarrow \bar{K}^0 K^+} + 2 \delta_A - P_{\mathrm{break}}\right)\right]$  &amp; $P  -   e^{2 i \delta\left( D^+ \rightarrow \bar{K}^0 K^+\right) }    P^* $  \\\hline
%
$D_s^+ \rightarrow  K^0 \pi^+$    &amp; $\mathrm{Im}\left[e^{-i \delta\left( D_s^+ \rightarrow  K^0 \pi^+\right)} \left( 2 A^{\mathrm{fac}}_{ D_s^+ \rightarrow K^0 \pi^+} + 2 \delta_A + P_{\mathrm{break}} \right)\right]  $   &amp; $P - e^{2 i \delta\left( D_s^+ \rightarrow  K^0 \pi^+\right) } P^* $  \\\hline
%
$D_s^+ \rightarrow  K^+ \pi^0$    &amp; $\mathrm{Im}\left[e^{-i \delta\left( D_s^+ \rightarrow  K^+ \pi^0\right)} \left( \sqrt{2} {C} + \sqrt{2} C_3^{(1)} + \frac{1}{\sqrt{2}} P_{\mathrm{break}}\right)\right]$    &amp; $\frac{1}{\sqrt{2}}\left( - P\right) -    e^{2 i \delta\left( D_s^+ \rightarrow  K^+ \pi^0\right) }  \frac{1}{\sqrt{2}}\left( - P\right)^* $  \\\hline\hline
%
\end{tabular}
\caption{Definitions of $X(d)$ and results for $\mathcal{S}(d)$ as used
  and defined in Eq.~(\ref{eq:defineS}) in case of the SU(3)$_F$ fit
  including $1/N_c$ {counting}.  For the sign conventions of
  $A^{\mathrm{fac}}(d)$ see Ref.~\cite{Muller:2015lua}. Note
  that {$A^{\mathrm{fac}}_{D^+
    \rightarrow \bar{K}^0 K^+} = 0$ by} isospin {symmetry}.
\label{tab:S}}
\end{center}
\end{table*}
Our strategy involves three steps: in step~1, we determine
{$A_i^{(1)}$}, $C$, {$C_3^{(1)}$},
$P_{\mathrm{break}}$, $|\mathcal{A}_{sd}(d)|$, and the phases
$\delta(d)$ in a global fit to branching ratio data as described in
Ref.~\cite{Muller:2015lua}. In step~2 we eliminate all hadronic
parameters but $P$ and $PA$ from $ a_{CP}^{\mathrm{dir}}(d)$. To this
end we define
\begin{align}
\mathcal{S}(d) &amp; \equiv 2i e^{i \delta(d)}
\left[\frac{\vert\mathcal{A}^{\mathrm{SCS}}_{sd}(d)\vert
    a_{CP}^{\mathrm{dir}}(d)}{ \mathrm{Im}\, \lambda_b}   -
  \mathrm{Im}\, X(d) \right] \nn \\
 &amp;=  2i e^{i \delta(d)}  \mathrm{Im}
  \left[ e^{-i \delta(d)}\mathcal{A}_b(d)-X(d)	 \right]
 \,, \label{eq:defineS}
\end{align}
with $\mathrm{Im}\, X(d)$ given in Table~\ref{tab:S}. The right column
of Table~\ref{tab:S} shows the expansion of $\mathcal{S}(d)$ in terms of
$P$, $PA$, their complex conjugates {$P^*$, $PA^*$}, and $\delta(d)$ which
is determined from the fit.  In step 3 we construct sum rules to
eliminate $P$, $PA$, $P^*$, and $PA^*$.  In this last step SU(3)$_F$
breaking in $P$, $PA$ is neglected.  From the expressions in
Table~\ref{tab:S} one can read off the two possible independent linear
combinations, connecting three $a_{CP}^{\mathrm{dir}}(d)$ each:
%
\paragraph{\textbf{CP Asymmetry Sum Rule 1:}}
\begin{align}
&amp;\frac{\mathcal{S}(D^0\rightarrow K^+K^-) - \mathcal{S}(D^0\rightarrow \pi^+\pi^- )}{e^{2 i \delta(D^0\rightarrow K^+K^-)} - e^{2 i \delta(D^0\rightarrow \pi^+\pi^-)}} - \nn\\
&amp;\frac{\mathcal{S}(D^0\rightarrow K^+K^-) + \sqrt{2} \mathcal{S}(D^0\rightarrow \pi^0\pi^0 )}{e^{2 i \delta(D^0\rightarrow K^+K^-)} - e^{2 i \delta(D^0\rightarrow \pi^0\pi^0)}}
= 0 \,. \label{eq:sumrule1}
\end{align}
%
\paragraph{\textbf{CP Asymmetry Sum Rule 2:}}
\begin{align}
&amp;\frac{\mathcal{S}(D^+\rightarrow \bar{K}^0 K^+) - \mathcal{S}(D_s^+\rightarrow K^0 \pi^+)}{e^{2i\delta( D^+\rightarrow \bar{K}^0 K^+) } - e^{2i\delta( D_s^+\rightarrow K^0 \pi^+) } } - \nn\\
&amp;\frac{\mathcal{S}(D^+\rightarrow \bar{K}^0 K^+) + \sqrt{2} \mathcal{S}(D_s^+\rightarrow K^+\pi^0)}{ e^{2i\delta( D^+\rightarrow \bar{K}^0 K^+) } - e^{2i\delta(D_s^+\rightarrow K^+\pi^0) } }
= 0 \,. \label{eq:sumrule2}
%
\end{align}
If some of the phases in the denominators of the sum rules are equal
(covering the SU(3)$_F$ limit as a special case) one finds:
for $e^{2i\delta(D^0\rightarrow K^+K^-)} = e^{2i\delta(D^0\rightarrow \pi^+\pi^-)} $
	\eq{eq:sumrule1} collapses to
	\begin{align}
	\mathcal{S}(D^0\rightarrow K^+K^-) - \mathcal{S}(D^0\rightarrow \pi^+\pi^-) &amp;= 0\,, \label{eq:specialFirst}
	\end{align}
%
while for $e^{2i\delta(D^0\rightarrow K^+K^-)} =
e^{2i\delta(D^0\rightarrow \pi^0\pi^0)} $ the sum rule becomes
	\begin{align}
	  \mathcal{S}(D^0\rightarrow K^+K^-) + \sqrt{2}
	  \mathcal{S}(D^0\rightarrow \pi^0\pi^0 ) &amp;= 0\,.
	    \label{eq:specialMiddle}
	\end{align}
%
If all three phase factors are equal,
\eqsand{eq:specialFirst}{eq:specialMiddle} hold simultaneously.
The special cases of Sum Rule {2} are obtained from those
in Eqs.~(\ref{eq:specialFirst})--(\ref{eq:specialMiddle}) by obvious
replacements.

When linking Sum Rule {2} to experimental quantities, we use
$a_{CP}^{\mathrm{dir}}( D^+\rightarrow K_S K^+ ) =
a_{CP}^{\mathrm{dir}}( D^+\rightarrow \bar{K}^0 K^+ )$ and
$a_{CP}^{\mathrm{dir}}( D_s^+\rightarrow K_S \pi^+) =
a_{CP}^{\mathrm{dir}}( D_s^+\rightarrow K^0\pi^+ )$, meaning that in our
definition of $a_{CP}^{\mathrm{dir}}$ kaon CP violation is properly
subtracted \cite{Grossman:2011zk}.  The two sum rules probe the
$SU(3)_F$ limit in {$P$} and $P+PA$. If future experiments find
deviations of order $30\%$, one will ascribe those to $SU(3)_F$-breaking
hadronic effects. The smallness of $\mathrm{Im}\, \lambda_b$ makes the
sum rules highly sensitive to new physics, which may well violate the
sum rules at a far higher level.


\section{SM Prediction of CP Asymmetries}

\begin{table}
\begin{center}
\begin{tabular}{ccr}
\hline\hline
Observable	      &amp;      Measurement &amp;  References	\\\hline
%
$\Delta a_{CP}^{\mathrm{dir}}$	 &amp; $-0.00253\pm 0.00104$ &amp; \cite{Amhis:2012bh,Aubert:2007if,Aaij:2011in,Staric:2008rx, Collaboration:2012qw, Ko:2012px, LHCb:2013dka, Aaij:2013bra, Aaij:2014gsa} \\
%
$\Sigma a_{CP}^{\mathrm{dir}}$ &amp; $-0.0011\pm 0.0026$ &amp; $^\dagger$\cite{Ko:2012px,Aubert:2007if,Aaij:2011in,Aaltonen:2011se,Collaboration:2012qw}   \\
%
$a_{CP}^{\mathrm{dir}}(D^0 \rightarrow K_S K_S)$ &amp; $-0.23\pm 0.19$ &amp; \cite{Bonvicini:2000qm} \\
%
$a_{CP}^{\mathrm{dir}}(D^0 \rightarrow \pi^0 \pi^0)$   &amp;  $-0.0004\pm 0.0064$ &amp; $^\dagger$\cite{Bonvicini:2000qm, Nisar:2014fkc}  \\
%
$a_{CP}^{\mathrm{dir}}(D^+ \rightarrow \pi^0 \pi^+)$   &amp;  $+0.029\pm 0.029$ &amp; \cite{Mendez:2009aa} \\
%
$a_{CP}^{\mathrm{dir}}(D^+ \rightarrow K_S K^+)$    &amp; $+0.0011\pm 0.0017$ &amp;  $^\dagger$\cite{Ko:2012uh, Lees:2012jv, Mendez:2009aa, Link:2001zj,Aaij:2014qec}	\\
%
$a_{CP}^{\mathrm{dir}}(D_s^+\rightarrow K_S \pi^+)$  &amp; $+0.006 \pm 0.005$  &amp; $^\dagger$\cite{Lees:2012jv,Ko:2010ng,Mendez:2009aa,Aaij:2014qec,Aaij:2013ula}   \\
%
$a_{CP}^{\mathrm{dir}}(D_s^+ \rightarrow K^+ \pi^0)$ &amp; $+0.266\pm 0.228$ &amp;  \cite{Mendez:2009aa}  \\\hline\hline
%
\end{tabular}
\caption{Current data on SCS charm CP asymmetries with subtracted indirect CP violation from kaon and charm mixing \cite{Grossman:2011zk, Cenci:2012ps},
see Appendix~A of Ref.~\cite{Hiller:2012xm}. We use the notation
$\Sigma a_{CP}^{\mathrm{dir}}\equiv a_{CP}^{\mathrm{dir}}(D^0\rightarrow K^+K^-) + a_{CP}^{\mathrm{dir}}(D^0\rightarrow \pi^+\pi^-)$.
No correlations between CP asymmetries are taken into account in the fits.
$^\dagger$Our average. Table adapted from Ref.~\cite{Hiller:2014prep}.
\label{tab:cpasyms} }
\end{center}
\end{table}

We combine the sum rules Eqs.~(\ref{eq:sumrule1})--(\ref{eq:sumrule2})
with the branching ratio fit presented in Ref.~\cite{Muller:2015lua}:
{for each point in the parameter space complying with all measured
  branching fractions (and the strong phase $\delta_{K\pi}$) we
  determine $S(d)$ for the decays entering the sum rules. In the same
  step \eqsand{eq:sumrule1}{eq:sumrule2} are used to predict one CP
  asymmetry in terms of the other two.  In our fit} we demand $\vert
{(C{+}\delta_A)}/T^{\mathrm{fac}}\vert,
\vert(E{+}\delta_A)/T^{\mathrm{fac}}\vert\leq1.3$ to {enforce} {proper}
$1/N_c$ counting.  {($T^{\mathrm{fac}}$ and $A^{\mathrm{fac}}$ are the
      factorized tree and annihilation amplitudes.)}  Apart from the fit
    {with} current data {(see Table~X of Ref.~\cite{Muller:2015lua})} we
    also consider a hypothetical future scenario {with improved}
    branching ratios {by scaling their errors with} a factor
    $1/\sqrt{50}$.
To illustrate the impact of the $1/N_c$ counting for the SM predictions,
we perform an additional fit {without $1/N_c$ input.}  {This plain
  SU(3)$_F$ fit relies on} the topological parameterization of Table~III
in Ref.~\cite{Muller:2015lua} with the SU(3)$_F$ counting described in
Sec.~IIIB of Ref.~\cite{Muller:2015lua}.
The redundancy of the four SU(3)$_F$-limit topologies
\cite{Muller:2015lua} {is removed by absorbing $A$} into $T$, $C$ and
$E$. We {further} demand $\vert A_i^{(1)}/T\vert\leq 50\%$ {to
  respect the SU(3)$_F$ counting.} 

The {experimental values} of {the} CP asymmetries included in the
fit are summarized in Table~\ref{tab:cpasyms}.  Our global fit results
are shown in Fig.~\ref{fig:sumrule}.  {The $\chi^2$ of the global minima
  range from 0.0 to 2.0 in the considered scenarios, i.e.\ with or
  without $1/N_c$ counting and with current or future data, indicating
  an excellent fit.}  The sum rules have nontrivial implications for
direct CP asymmetries, especially when combined with the input from
$1/N_c$ counting.  With current data we see the largest impact of the
sum rules in Fig.~\ref{plot-2d-02-26}: roughly 47\% of the 95\% CL
region allowed by the measurements of the CP asymmetries is excluded by
our global fit result.

Our results show that future improved measurements of branching ratios
{will} play a key role {to sharpen our} predictions: {drastic examples
  are} the prediction of $a_{CP}^{\mathrm{dir}}(D_s^+\rightarrow
K^+\pi^0)$ in Figs.~\ref{rangeplot-0-23} and \ref{rangeplot-2-23} and
the correlation of $a_{CP}^{\mathrm{dir}}(D^0\rightarrow \pi^0\pi^0)$
and $a_{CP}^{\mathrm{dir}}(D^0\rightarrow \pi^+\pi^-)$ in
Fig.~\ref{plot-2d-02-26}, where one of the two {overlapping} ellipses
vanishes in our future-data scenario. In Fig.~\ref{plot-2d-02-24} a
smaller region is excluded than {in} Fig.~\ref{plot-2d-02-26},
because $a_{CP}^{\mathrm{dir}}(D^0\to\pi^0\pi^0)$ is measured less
precisely than $\Delta a_{CP}^{\mathrm{dir}}$ (see
Table~\ref{tab:cpasyms}).  {In general, the correlation of two CP
  asymmetries can be better predicted once improved data for the} third
one appearing in \eqsand{eq:sumrule1}{eq:sumrule2} {becomes}
available.  {In case of the fit {without $1/N_c$ counting} the
  {smaller errors of the} branching ratios in our {future
    scenario} do not {improve} the predictions for CP asymmetries.
  Note that with {current} data the {predicted ranges} {barely}
  depend on the {additional} input from $1/N_c$ counting.}

{In principle one can obtain the quantities {$P$} and $P+PA$, which we
  eliminate through our sum rules, from the individual CP
  asymmetries. If future data challenge our sum rules at a level which
  cannot be explained with SU(3)$_F$ breaking in {$P$} and $P+PA$, this
  will point to new physics which couples differently to $s$ and $d$
  quarks. (For an SU(3)$_F$ analysis of such models see
  e.g.~Ref.~\cite{Hiller:2012xm}.) However, SU(3)$_F$-symmetric new
  physics in {$P$} and $P+PA$ vanishes from the sum rules.  {(A
    similar situation can be found in the isospin sum rules of
    Ref.~\cite{Grossman:2012eb} which are insensitive to new physics in
    $\Delta I=1/2$ amplitudes.)}


\section{Conclusions}

To find {reliable} SM predictions for charm CP asymmetries {we derive
two sum rules which treat
{$T$, $A$, $C$, $E$, $P_{\mathrm{break}}$} correctly to linear order in 
SU(3)$_F$ breaking and eliminate the penguin topologies {$P$} and $P+PA$ 
to leading order in SU(3)$_F$. Thus we treat large tree-level parameters 
at sub-leading order to increase the precision in the extraction of 
loop-induced quantities sensitive to new physics. 
Unlike previously known  SU(3)$_F$-limit 
sum rules our new sum rules correlate 
three CP asymmetries each. }
The interplay of the two sum rules probes both
the quality of SU(3)$_F$ for penguin topologies and new physics.  Future
branching ratio measurements play a key role in order to reduce the
uncertainties of the consequent SM predictions for charm CP asymmetries.

\begin{figure*}
\begin{center}
\subfigure[\, 
    {Fit including
    $1/N_c$ counting in the topological amplitudes with current (future)
    branching ratio data} in blue (green).
  {The black (magenta) line delimits the 95\% CL region found
  in a fit without using $1/N_c$ counting in the topological amplitudes
  for current (future) branching ratio data.} 
  Note that the black and magenta curves lie on
    top of each other.
  \label{plot-2d-02-26}]{
	\includegraphics[width=0.45\textwidth]{plot-2d-02-26.pdf} }
\hfill \subfigure[\, {Color coding as in Fig.~\ref{plot-2d-02-26}.}
The dashed-dotted line denotes the
  SU(3)$_F$-limit sum rule of \eq{eq:su3limit1}.\label{plot-2d-02-24}]{
  \includegraphics[width=0.45\textwidth]{plot-2d-02-24.pdf} } \hfill
\begin{minipage}{0.45\textwidth}
\subfigure[\, Prediction from current branching ratio data. 
The
  dashed, solid, and dashed-dotted lines correspond to 
the $1\sigma$, $2\sigma$, and $3\sigma$ intervals, respectively.
{The blue bars show our fit results for} current branching ratio data. 
{The corresponding result without $1/N_c$ counting (i.e.\ only using 
SU(3)$_F$) is shown in black.}  
\label{rangeplot-0-23}]{
	\includegraphics[width=\textwidth]{rangeplot-0-23.pdf} }
\subfigure[\, Same as in Fig.~(c), but for our future scenario with
  smaller errors of the branching ratios. {The green (magenta) bars 
 correspond to the analysis with (without)
    $1/N_c$ counting.}\label{rangeplot-2-23}]{
  \includegraphics[width=\textwidth]{rangeplot-2-23.pdf} }
\end{minipage}
%
\begin{minipage}{0.45\textwidth}
\subfigure[\, Future scenario assuming
  $a_{CP}^{\mathrm{dir}}(D_s^+\rightarrow K^+\pi^0)=-1\%$ {and branching
    ratios with} current (future) {errors} {including $1/N_c$ counting}
  in blue (green).  The {corresponding allowed area found in a fit
    without $1/N_c$ counting is delimited by the} black (magenta) line. Note
  that the black, magenta, and red curves lie on top of each other. The
  dashed-dotted line denotes the SU(3)$_F$-limit sum rule of
  \eq{eq:su3limit2}.  \label{plot-2d-47-25}]{
    \includegraphics[width=\textwidth]{plot-2d-47-25.pdf} }
\end{minipage}
\end{center}
\caption{SM predictions for CP asymmetries obtained from {our}
  global fit. 
  In Figures (a),(b),(e) the dashed (solid) {red} lines {delimit
    the experimental} 68\% (95\%)~CL regions. {The other dashed (solid) lines are the 68\% (95\%)~CL regions
    of the respective fit scenarios explained in the captions of the subfigures.}
  The generic error of order $\sim 30\%$ from SU(3)$_F$ breaking in {$P$}
  and {$P+PA$} is not shown.  The experimental error ellipses are obtained
  by scaling the errors quoted in Table~\ref{tab:cpasyms} by factors of
  $\sqrt{2.28}$ and $\sqrt{5.99}$ in order to obtain the 
  two-dimensional $68\%$ and $95\%$ CL regions {from the
    corresponding one-dimensional ranges}. The experimental error
  ellipse shown in Fig.~\ref{plot-2d-02-24} is calculated from the
  corresponding one for $\Delta a_{CP}^{\mathrm{dir}}$ and $\Sigma
  a_{CP}^{\mathrm{dir}}$.  
\label{fig:sumrule}
}
\end{figure*}

\section{Acknowledgments}

\begin{acknowledgments}
We thank {Michael Gronau, Yuval Grossman}, Alexander Kagan, Luca
Silvestrini, Gudrun Hiller, Martin Jung, {Dean Robinson} and Jure
Zupan for useful discussions {and Michael Gronau vor a critical
  proofreading of our paper.  UN acknowledges the kind hospitality of
  the \emph{Munich Institute for Astro- and Particle Physics.}}  The
fits are performed using the \texttt{python} version of the software
package \texttt{myFitter}~\cite{Wiebusch:2012en}.  StS thanks Martin
Wiebusch for \texttt{myFitter} support and the provision of
\texttt{myFitter-python-0.2-beta}.  Parts of the computations were
performed on the {NEMO cluster of Baden-W\"urttemberg (framework
  program bwHPC).}  The presented work is supported by BMBF under
contract no.~05H12VKF.
\end{acknowledgments}

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