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\begin{document}

\title{\boldmath Updated NNLO QCD predictions for the weak radiative $B$-meson decays}

\author{M.~Misiak}
\affiliation{Institute of Theoretical Physics, University of Warsaw, PL-02-093 Warsaw, Poland}
\author{H.~M.~Asatrian}
\affiliation{Yerevan Physics Institute, 0036 Yerevan, Armenia}
\author{R.~Boughezal}
\affiliation{High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA}
\author{M.~Czakon}
\affiliation{Institut f\"ur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, D-52056 Aachen, Germany}
\author{T.~Ewerth}
\affiliation{Institut f\"ur Theoretische Teilchenphysik, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany}
\author{A.~Ferroglia}
\affiliation{New York City College of Technology, CUNY, Brooklyn, NY 11201, USA}
\affiliation{The Graduate School and University Center, CUNY, New York, NY 10016, USA}
\author{P.~Fiedler}
\affiliation{Institut f\"ur Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, D-52056 Aachen, Germany}
\author{P.~Gambino}
\affiliation{Dipartimento di Fisica, Universit\`a di Torino \&  INFN, Torino, I-10125 Torino, Italy}
\author{C.~Greub}
\affiliation{Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, CH-3012 Bern, Switzerland}
\author{U.~Haisch}
\affiliation{Rudolf Peierls Centre for Theoretical Physics,
    University of Oxford, OX1 3PN Oxford, United Kingdom}
\affiliation{CERN, Theory Division, CH-1211 Geneva 23, Switzerland}
\author{T.~Huber}
\affiliation{Theoretische Physik 1, Naturwissenschaftlich-Technische Fakult\"at, Universit\"at Siegen, D-57068 Siegen, Germany}
\author{M.~Kami\'nski}
\affiliation{Institute of Theoretical Physics, University of Warsaw, PL-02-093 Warsaw, Poland}
\author{G.~Ossola}
\affiliation{New York City College of Technology, CUNY, Brooklyn, NY 11201, USA}
\affiliation{The Graduate School and University Center, CUNY, New York, NY 10016, USA}
\author{M.~Poradzi\'nski}
\affiliation{Institute of Theoretical Physics, University of Warsaw, PL-02-093 Warsaw, Poland}
\affiliation{Theoretische Physik 1, Naturwissenschaftlich-Technische Fakult\"at, Universit\"at Siegen, D-57068 Siegen, Germany}
\author{A.~Rehman}
\affiliation{Institute of Theoretical Physics, University of Warsaw, PL-02-093 Warsaw, Poland}
\author{T.~Schutzmeier}
\affiliation{Physics Department, Florida State University, Tallahassee, FL, 32306-4350, USA}
\author{M.~Steinhauser}
\affiliation{Institut f\"ur Theoretische Teilchenphysik, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany}
\author{J.~Virto}
\affiliation{Theoretische Physik 1, Naturwissenschaftlich-Technische Fakult\"at, Universit\"at Siegen, D-57068 Siegen, Germany}

\begin{abstract}

We perform an updated analysis of the inclusive weak radiative $B$-meson
decays in the standard model, incorporating all our results for
the ${\mathcal O}(\alpha_s^2)$ and lower-order perturbative corrections that
have been calculated after 2006. New estimates of non-perturbative effects are
taken into account, too.  For the CP- and isospin-averaged branching ratios,
we find
%
${\mathcal B}_{s\gamma} = (3.36 \pm 0.23) \times 10^{-4}$
%
and
%
${\mathcal B}_{d\gamma} = \left(1.73^{+0.12}_{-0.22}\right) \times 10^{-5}$,
%
for $E_\gamma > 1.6\,$GeV. These results remain in agreement with the current
experimental averages. Normalizing their sum to the inclusive semileptonic
branching ratio, we obtain
%
$R_\gamma \equiv 
\left({\mathcal B}_{s\gamma}+{\mathcal B}_{d\gamma}\right)/{\mathcal B}_{c\ell\nu}
= (3.31 \pm 0.22) \times 10^{-3}$. 
%
A new bound from ${\mathcal B}_{s\gamma}$ on the charged Higgs boson mass in
the two-Higgs-doublet-model II reads 
%
  $M_{H^{\pm}} > 480\,$GeV at 95\%C.L. 
% $M_{H^{\pm}} > 358\,$GeV at 99\%C.L. 

\end{abstract}

\pacs{12.38.Bx, 13.20.He}
                             
\maketitle

\section{Introduction \label{sec:intro}}

The inclusive decays $\bar B \to X_s\gamma$ and $\bar B \to X_d\gamma$ are
considered among the most interesting flavor changing neutral current
processes.  They contribute in a significant manner to current bounds on
masses and interactions of possible additional Higgs bosons and/or
supersymmetric particles. Measurements of the CP- and isospin-averaged $\bar B
\to X_s\gamma$ branching ratio by CLEO~\cite{Chen:2001fj},
Belle~\cite{Abe:2001hk,Limosani:2009qg} and
BABAR~\cite{Lees:2012ym,Lees:2012ufa,Lees:2012wg,Aubert:2007my} lead to the
combined result~\cite{Amhis:2014hma}
%
\begin{equation}
{\mathcal B}^{\rm exp}_{s\gamma} = (3.43 \pm 0.21 \pm 0.07)\times 10^{-4},
\end{equation}
%
for the photon energy $E_\gamma > E_0 = 1.6\,$GeV in the decaying meson
rest frame. The combination involves an extrapolation from measurements
performed at $E_0 \in [1.7,2.0]\,$GeV. Applying the same extrapolation method
to the available $\bar B \to X_d\gamma$
measurement~\cite{delAmoSanchez:2010ae}, one finds
%
\begin{equation}
{\mathcal B}^{\rm exp}_{d\gamma} = (1.41 \pm 0.57)\times 10^{-5}
\end{equation}
%
at $E_0 = 1.6\,$GeV~\cite{Crivellin:2011ba}. More precise determinations of
${\mathcal B}^{\rm exp}_{q\gamma}$ for $q=s,d$ are expected from
Belle~II~\cite{Aushev:2010bq}.

Theoretical calculations of ${\mathcal B}_{q\gamma}$ have a chance to match
the experimental precision only in a certain range of $E_0$ where the
non-perturbative contribution $\delta \Gamma_{\rm nonp}$ in the relation 
%
\begin{equation}
\Gamma(\bar B \to X_q \gamma) ~=~ \Gamma(b \to X_q^p \gamma) ~+~ \delta \Gamma_{\rm nonp}
\end{equation}
%
remains under control. Here, $\Gamma(b \to X_q^p \gamma)$ denotes the
perturbatively calculable rate of the radiative $b$-quark decay involving only
charmless partons in the final state. Their overall strangeness vanishes for
$X_d^p$ and equals $-1$ for $X_s^p$. The analysis of
Ref.~\cite{Benzke:2010js} implies that unknown contributions to $\delta
\Gamma_{\rm nonp}$ are potentially larger than the so-far determined ones, and
induce around $\pm5\%$ uncertainty in ${\mathcal B}_{s\gamma}$ at $E_0 =
1.6\,$GeV. Non-perturbative uncertainties in ${\mathcal B}_{d\gamma}$ receive
additional sizeable contributions~\cite{Asatrian:2013raa} due to collinear
photon emission in the $b \to du\bar u\gamma$ process whose
Cabibbo-Kobayashi-Maskawa (CKM) factor is only a few times smaller than
the one in the leading term.

Apart from possible future progress in analyzing non-perturbative effects, one
needs to determine $\Gamma(b \to X_q^p \gamma)$ to a few percent accuracy. It
requires evaluating next-to-next-to-leading order (NNLO) QCD
corrections that involve Feynman diagrams up to four loops. The first 
standard model (SM) estimate of the $\bar B \to X_s \gamma$ branching
ratio at this level was presented in Ref.~\cite{Misiak:2006zs} where all the
corrections calculated up to 2006 were taken into account. A part of the
${\mathcal O}(\alpha_s^2)$ contribution was obtained via
interpolation~\cite{Misiak:2006ab} in the charm quark mass between the
large-$m_c$ asymptotic expression~\cite{Misiak:2010sk} and the $m_c=0$
boundary condition that was estimated using the Brodsky-Lepage-Mackenzie (BLM)
approximation~\cite{Brodsky:1982gc}.

In the present paper, we provide an updated prediction for ${\mathcal
B}_{s\gamma}$, including all the contributions and estimates worked out after
2006. They are listed in Sec.~\ref{sec:bsg} where the necessary definitions
are introduced. The interpolation in $m_c$ is still being applied. However,
the $m_c=0$ boundary condition is no longer a BLM-based estimate but rather
comes from an explicit calculation~\cite{Czakon:2015xxx}.

The paper is organized as follows. After discussing ${\mathcal B}_{s\gamma}$
in Sec.~\ref{sec:bsg}, our NNLO analysis is extended to ${\mathcal
B}_{d\gamma}$ in Sec.~\ref{sec:bdg}. Next, in Sec.~\ref{sec:rg}, we consider
%
$R_\gamma \equiv \left({\mathcal B}_{s\gamma}+{\mathcal B}_{d\gamma}\right)/{\mathcal B}_{c\ell\nu}$
%
which may sometimes be more convenient than ${\mathcal B}_{s\gamma}$ for
deriving constraints on new physics.  Sec.~\ref{sec:np} is devoted to
presenting a generic expression for beyond-SM contributions, as well as an
updated bound for the charged Higgs boson mass in the 
two-Higgs-doublet-model II (THDM~II). We conclude
in Sec.~\ref{sec:sum}.

\section{\boldmath ${\mathcal B}_{s\gamma}$ in the SM \label{sec:bsg}}

Radiative $B$-meson decays are most conveniently described in the framework
of an effective theory that arises after decoupling of the $W$ boson and
heavier particles. Flavor-changing weak interactions that are relevant for
$\Gamma(b \to X_q^p \gamma)$ with $q=s,d$ are given by 
%
\begin{equation} \label{Leff}
{\mathcal L}_{\rm eff} ~\sim~ V_{tq}^* V_{tb} \left[ \sum_{i=1}^8 C_i Q_i 
+ \kappa_q \sum_{i=1}^2 C_i (Q_i-Q_i^u)\right].
\end{equation}
%
Explicit expressions for the current-current ($Q_{1,2}$), four-quark penguin
($Q_{3,\ldots,6}$), photonic dipole ($Q_7$) and gluonic dipole ($Q_8$)
operators can be found, e.g., in Eq.~(2.5) of Ref.~\cite{Misiak:2006ab}. The
CKM element ratio $\kappa_q = (V_{uq}^* V_{ub})/(V_{tq}^* V_{tb})$ is small
for $q=s$, and it affects ${\mathcal B}_{s\gamma}$ by less than $0.3\%$. Barring
this effect and the higher-order electroweak ones, $\Gamma(b \to X_s^p
\gamma)$ in the SM is given by a quadratic polynomial in the real Wilson
coefficients $C_i$
%
\begin{equation}
\Gamma(b \to X_s^p \gamma) ~\sim~ \sum_{i,j=1}^8 C_i C_j\; G_{ij}.
\end{equation}
%
A series of contributions to the above expression from our calculations in
Refs.~\cite{Czakon:2015xxx,Czakon:2006ss, Asatrian:2006rq, Ewerth:2008nv,
Boughezal:2007ny, Asatrian:2010rq, Ferroglia:2010xe,Misiak:2010tk,
Kaminski:2012eb, Huber:2014nna} makes the current analysis significantly
improved with respect to the one in Ref.~\cite{Misiak:2006zs}. In particular,
the NNLO Wilson coefficient calculation becomes complete after including the
four-loop anomalous dimensions that describe $Q_{1,\ldots,6} \to Q_8$ mixing
under renormalization~\cite{Czakon:2006ss}. Effects of the charm and bottom
quark masses in loops on the gluon lines in $G_{77}$~\cite{Asatrian:2006rq},
$G_{78}$~\cite{Ewerth:2008nv} and $G_{(1,2)7}$~\cite{Boughezal:2007ny}, as
well as a complete calculation of $G_{78}$~\cite{Asatrian:2010rq} are now
available. Three- and four-body final-state contributions to
$G_{88}$~\cite{Ferroglia:2010xe,Misiak:2010tk} and
$G_{(1,2)8}$~\cite{Misiak:2010tk} are included in the BLM approximation.
Four-body final-state contributions involving the penguin and $Q^u_{1,2}$
operators are taken into account at the leading order
(LO)~\cite{Kaminski:2012eb} and next-to-leading order
(NLO)~\cite{Huber:2014nna}. Last but not least, the complete NNLO
calculation~\cite{Czakon:2015xxx} of $G_{17}$ and $G_{27}$ at $m_c=0$ is used
as a boundary for interpolating their unknown parts in $m_c$.

Following the algorithm described in detail in Ref.~\cite{Czakon:2015xxx},
taking into account new non-perturbative
effects~\cite{Benzke:2010js,Ewerth:2009yr,Alberti:2013kxa}, as well as the
previously omitted parts of the NNLO BLM
corrections~\cite{Ligeti:1999ea}, we arrive at the following SM prediction
%
\begin{equation} \label{bsgsm}
{\mathcal B}^{\rm SM}_{s\gamma} ~=~ (3.36 \pm 0.23) \times 10^{-4}
\hspace{5mm} \mbox{for~}E_0=1.6\,\mbox{GeV}.
\end{equation}
%
Individual contributions to the total uncertainty are of non-perturbative
($\pm 5\%$), higher-order ($\pm 3\%$), interpolation ($\pm 3\%$) and
parametric ($\pm 2\%$) origin. They are combined in quadrature.  The
parametric one gets reduced with respect to Ref.~\cite{Misiak:2006zs}, which
becomes possible thanks to the new semileptonic fits of
Ref.~\cite{Alberti:2014yda}.  Unfortunately, the interpolation uncertainty
cannot be reduced because the interpolated parts of the ${\mathcal
O}(\alpha_s^2)$ non-BLM contributions to $G_{(1,2)7}$ turn out to be
sizeable. Their effect on ${\mathcal B}^{\rm SM}_{s\gamma}$ grows from 0 to
around 5\% when $m_c$ changes from 0 up to the measured value.

\section{\boldmath ${\mathcal B}_{d\gamma}$ in the SM \label{sec:bdg}} 

Extending our NNLO calculation to the ${\mathcal B}_{d\gamma}$ case begins
with inserting the proper CKM factors in Eq.~(\ref{Leff}). Contrary to
$\kappa_s$, the ratio $\kappa_d$ is not numerically small. Using the CKM fits
of Ref.~\cite{Charles:2015gya}, one finds
%
\begin{equation}
\kappa_d ~=~ \left(0.007^{+0.015}_{-0.011}\right) + i \left(-0.404^{+0.012}_{-0.014}\right).
\end{equation}
%
The small real part implies that the effects of $\kappa_d$ on the CP-averaged
${\mathcal B}_{d\gamma}$ are dominated by those proportional to
$|\kappa_d|^2$. In such terms, perturbative two- and three-body
final state contributions arise only at the NNLO and NLO, respectively.
They vanish in the $m_c=m_u$ limit, which effectively makes them
suppressed by $m_c^2/m_b^2 \lsim 0.1$. In consequence, the main $\kappa_d$-effect
comes from $b \to du\bar u\gamma$ at the LO, where phase-space suppression
is partially compensated by the collinear logarithms. 

In the first (rough) approximation, one evaluates the tree-level $b \to du\bar
u\gamma$ diagrams retaining a common light-quark mass $m_q$ inside the
collinear logarithms~\cite{Misiak:2010tk}, and varying $m_b/m_q$ between $10
\sim m_B/m_K$ and $50 \sim m_B/m_\pi$ to estimate the uncertainty. The
considered effect varies then from 2\% to 11\% of ${\mathcal B}_{d\gamma}$.  A
more involved analysis with the help of fragmentation functions gives a very
similar range~\cite{Asatrian:2013raa}. Including this contribution
in our evaluation of the entire $B_{d\gamma}$ from Eq.~(\ref{Leff}), we find
%
\begin{equation} \label{bdgsm}
{\mathcal B}^{\rm SM}_{d\gamma} = \left(1.73^{+0.12}_{-0.22}\right) \times 10^{-5}
\hspace{5mm} \mbox{for~}E_0=1.6\,\mbox{GeV},
\end{equation}
%
where the central value corresponds to $m_b/m_q=50$. Our result is about 12\%
larger than the one given in Ref.~\cite{Crivellin:2011ba} where
the $b \to du\bar u\gamma$ contributions were neglected. The uncertainty
estimate in Eq.~(\ref{bdgsm}) improves with respect to
Ref.~\cite{Crivellin:2011ba} thanks to including the NNLO QCD corrections and
using the updated CKM fit~\cite{Charles:2015gya}. Interestingly, the
parametric uncertainty due to the CKM input amounts to $\pm 2.5\%$ only.

The collinear logarithm problem might seem artificial because isolated photons
are required in the experimental signal sample. Unfortunately, requiring
photon isolation on the perturbative side would necessitate introducing an
infrared cutoff on the gluon energies, e.g., in the NLO corrections to the 
dominant $G_{77}$ term. Without a dedicated analysis (which is beyond
the scope of the present paper), it is hard to verify whether such an approach
would enhance or suppress the uncertainty in ${\mathcal B}_{d\gamma}$.

Another question concerning the $|\kappa_d|^2$-terms is whether the off-shell
light vector meson conversion to photons can be assumed to be included in our
overall $\pm 5\%$ non-perturbative uncertainty. Much smaller effects
found in the vector-meson-dominance analysis of Ref.~\cite{Ricciardi:1995jh}
imply that it is likely to be the case.

\section{\boldmath The ratio $R_\gamma$ \label{sec:rg}}

In the fully inclusive measurements of radiative $B$-meson
decays~\cite{Chen:2001fj,Limosani:2009qg,Lees:2012ym,Lees:2012ufa}, the final
hadronic state strangeness is not verified. The actually measured
quantity is ${\mathcal B}_{s\gamma}+{\mathcal B}_{d\gamma}$. Next, the result
is divided by
%
$\left(1+|(V_{td}^* V_{tb})/(V_{ts}^* V_{tb})|^2\right)$ 
%
to obtain ${\mathcal B}_{s\gamma}$. To avoid such a complication,
we provide here our SM prediction for~ ${\mathcal B}_{s\gamma}+{\mathcal
B}_{d\gamma}$~ with all the correlated uncertainties properly taken into
account. Moreover, we normalize it to the CP- and isospin-averaged inclusive
semileptonic branching ratio ${\mathcal B}_{c\ell\nu}$. In the ${\mathcal
B}_{s\gamma}$ case, such a normalization reduces the parametric uncertainty
from $\pm 2.0\%$ to $\{+1.2,-1.4\}\%$. It may also be useful on the experimental
side because the inclusive semileptonic events can serve for determining the
$B$-meson yield. Proceeding as in the previous sections, we obtain for $E_\gamma
= 1.6\,$GeV
%
\begin{equation}
R^{\rm SM}_\gamma \equiv 
\left({\mathcal B}^{\rm SM}_{s\gamma}+{\mathcal B}^{\rm SM}_{d\gamma}\right)/{\mathcal B}_{c\ell\nu}
= (3.31 \pm 0.22) \times 10^{-3}. 
\end{equation}
%
The relative uncertainties are identical to those in ${\mathcal
B}_{s\gamma}$ (as given below Eq.~(\ref{bsgsm})), except for the
parametric one which amounts to $\{+1.2,-1.7\}\%$ including the effect of
$m_b/m_q$. The gain in the overall theory uncertainty is hardly noticeable,
but this may change with the future progress in determining the perturbative
and non-perturbative corrections.

\section{Beyond-SM effects \label{sec:np}}

In most of the new-physics scenarios considered in the literature, beyond-SM
effects on ${\mathcal B}_{s\gamma}$ are driven by new additive contributions
to the Wilson coefficients of the dipole operators at the matching scale
$\mu_0$ where the heavy particles ($t$, $W$, $Z$, $H^0$, \ldots) are
decoupled. Denoting such contributions by $\Delta C_{7,8}$ and setting $\mu_0$
to $160\,$GeV, we find
%
\begin{eqnarray} 
{\mathcal B}_{s\gamma}\times 10^4 &=& (3.36 \pm 0.23) - 8.22\,\Delta C_7 - 1.99\,\Delta C_8,\nonumber\\[2mm] 
R_\gamma\times 10^3 &=& (3.31 \pm 0.22) - 8.05\,\Delta C_7 - 1.94\,\Delta C_8.\hspace{5mm}
\end{eqnarray}
%
The above expressions are linearized, i.e. it is assumed that the quadratic
terms in $\Delta C_{7,8}$ are negligible when they enter with ${\mathcal O}(1)$
coefficients into the above equations.  If they are not, a detailed analysis of
QCD corrections in the considered beyond-SM scenario is necessary.

Such an analysis is available in the THDM~II~\cite{Abbott:1979dt} for which
the NLO~\cite{Ciuchini:1997xe,Borzumati:1998tg,Borzumati:1998nx} and
NNLO~\cite{Hermann:2012fc} corrections to $\Delta C_{7,8}$ are known. They
are always negative and remain practically independent of the vacuum
expectation value ratio $\tan\beta$ when $\tan\beta \gsim 2$. Sending
$\tan\beta$ to infinity in the expressions for $\Delta C_{7,8}$, we find the
following updated bounds from ${\mathcal B}_{s\gamma}$ on the charged Higgs
boson mass in this model
%
\begin{eqnarray}
M_{H^{\pm}} > 480\,\mbox{GeV at 95\%C.L.}\, ,\nonumber\\[2mm]
M_{H^{\pm}} > 358\,\mbox{GeV at 99\%C.L.}\, . 
\end{eqnarray}
%
For $\tan\beta \lsim 2$ the bounds become considerably stronger, but at the
same time other observables provide competitive
limits~\cite{Eberhardt:2013uba}. In the supersymmetric case, in which the
charged scalar and the neutral pseudoscalar tend to be almost degenerate, the
current direct search bounds~\cite{Khachatryan:2014wca,Aad:2014vgg} exceed
$500\,$GeV for $\tan\beta \gsim 20$.

\section{Summary\label{sec:sum}} 

We presented an updated prediction for ${\mathcal B}_{s\gamma}$ in the SM
taking into account all the perturbative and non-perturbative effects worked
out after the 2006 publication~\cite{Misiak:2006zs} of the first NNLO estimate
for this quantity. Some of the ${\mathcal O}(\alpha_s^2)$ corrections are
still interpolated in $m_c$, but the $m_c=0$ boundary condition now comes from
an explicit calculation. Despite this improvement, the interpolation
uncertainty cannot be reduced because the interpolated correction is sizeable.
Future progress requires extending the calculation of $G_{(1,2)7}$ to
arbitrary $m_c$, which is considered a difficult but manageable task. In
parallel, one should investigate whether non-perturbative uncertainties can be
suppressed by combining lattice inputs with measurements of observables like
the CP- or isospin asymmetries in $\bar B \to X_q\gamma$.

The main outcome of the current update is an upwards shift by around $6.4\%$
in the central value of ${\mathcal B}^{\rm SM}_{s\gamma}$. It originates
mainly from fixing the $m_c=0$ boundary ($+3\%$) and including the 
complete NNLO BLM corrections to the three- and four-body final state
channels ($+2\%$). Since ${\mathcal B}^{\rm SM}_{s\gamma}$ is now closer to
${\mathcal B}^{\rm exp}_{s\gamma}$ (but still ${\mathcal B}^{\rm SM}_{s\gamma}
< {\mathcal B}^{\rm exp}_{s\gamma}$), the bound on $M_{H^{\pm}}$ in the
THDM~II becomes significantly stronger.

We supplemented our analysis with a prediction for ${\mathcal B}_{d\gamma}$ as
well as the ratio
%
$R_\gamma = \left({\mathcal B}_{s\gamma}+{\mathcal B}_{d\gamma}\right)/{\mathcal B}_{c\ell\nu}$
%
where correlated uncertainties are treated in a consistent manner. The
ratio $R_\gamma$ may serve in the future as a more convenient observable for
testing beyond-SM theories with minimal flavor violation.

\begin{acknowledgments} % 17 authors: HA,RB,MC,TE,AF,PF,PG,CG,UH,TH,MM,GO,MP,AR,TS,MS,JV
%
We acknowledge partial support from the 
%
Deutsche Forschungsgemeinschaft (DFG) within research unit FOR 1873 (QFET)                     % Siegen: TH,JV, also MP
%
and within the Sonderforschungsbereich Transregio~9 ``Computational Particle Physics'',        % Aachen: MC,PF,TS, also AR  
%                                                                                              % Karlsruhe: MS,TE, also MP,AR 
from the State Committee of Science of Armenia
  Program No 13-1c153 and Volkswagen Stiftung Program No 86426,                                % Yerevan: HA
%
from the Swiss National Science Foundation,                                                    % Bern: CG          
%
from the National Science Centre (Poland) research project, Decision No. DEC-2014/13/B/ST2/03969, % Warsaw: MM,MP,AR
%
from the US Department of Energy, Division of High Energy Physics, under contract DE-AC02-06CH11357,
                                                                                               % Argonne:  RB
from the US National Science Foundation under Grant No. PHY-1417354,                           % New York: AF, GO
and from MIUR under contract 2010YJ2NYW 006.                                                % Torino:   PG
                                                                                               % Oxford:   HU

\end{acknowledgments}
%
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  %%CITATION = HEP-EX 0108032;%%
%
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%
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%
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%
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%
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%
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%
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%
\end{thebibliography} 
\end{document}
