text:
\section{Introduction}
\begin{sloppypar}
Despite tremendous progress in simulating Quantum Chromodynamics (QCD) on space-time lattices using physical quark masses, many studies of complicated observables within lattice QCD are still performed with light quarks that are heavier than they are in the real world (see \textit{e.g.}\ Refs.~\cite{Aoki:2016frl,Briceno:2017max} for reviews). To extrapolate such simulations to the physical point, additional theoretical input is required, which should ideally be based on systematically improvable effective field theories.  At low energies, the effective field theory that controls the quark-mass dependence by construction is chiral perturbation theory (ChPT)~\cite{Weinberg:1978kz,Gasser:1983yg,Gasser:1984gg}, which describes the interactions of the pseudo-Goldstone bosons of spontaneous chiral symmetry breaking, the pions (as well as kaons and the $\eta$).

However, the vast majority of states in QCD are resonances, and to perform \textit{chiral extrapolations} for these is less straightforward.  A popular tool in this regard has been to employ unitarized versions of ChPT, such as the inverse amplitude method (IAM)~\cite{Truong:1991gv,Dobado:1992ha,Dobado:1996ps,GomezNicola:2007qj}: a resummation of higher-order effects obeying $S$-matrix unitarity (\textit{i.e.}, probability conservation) allows one to generate poles on unphysical Riemann sheets in the complex-energy plane, the signatures of resonances.  The IAM can be justified using dispersion theory; scattering amplitudes constructed via the IAM match smoothly on the ChPT expansion at low energies.
In this manner, the properties of elastic resonances such as the $f_0(500)$ and the $\rho(770)$ in pion--pion as well as
the $K_0^*(700)$ and $K^*(892)$ in pion--kaon scattering have been investigated with respect to their
quark-mass dependence~\cite{Hanhart:2008mx,Nebreda:2010wv}.

Nevertheless, by far not all hadronic resonances appear in two-body scattering processes.  The lightest resonance that decays only into a three-body final state (in QCD in the isospin limit) is the $\omega(782)$, with its dominant decay $\omega\to3\pi$.  Clearly, the quark-mass dependence of the $\omega$ cannot be assessed within an IAM-type formalism; it could at best be studied within the appropriate partial wave of the $3\pi\to3\pi$ scattering process, and the formalism to study such processes on the lattice is currently under intense investigation~\cite{Polejaeva:2012ut,Briceno:2012rv,Hansen:2014eka,Hansen:2015zga,Hammer:2017uqm,Hammer:2017kms,Mai:2017bge}.
In this article, we suggest an approach to assess the quark-mass dependence of the $\omega\to3\pi$ decay amplitude based on dispersion relations.  We employ the so-called Khuri--Treiman equations~\cite{Khuri:1960zz} that require the pion--pion two-body phase shift as input, which we extract from the known quark-mass-dependent IAM partial wave.  While we still need to rely on effective field theory ideas to describe the variation of the $\omega$ \textit{mass} with the quark masses, the dispersive framework allows us to predict its quark-mass-dependent \textit{width}.  The idea to employ dispersion theory to extend the applicability of IAM-generated phase shifts is not new: it has already been applied to describe the pion vector form factor~\cite{Guo:2008nc}, as well as, in a formalism closely related to what we present here, to the reaction $\gamma\pi\to\pi\pi$~\cite{Niehus:2017}.
\end{sloppypar}

The outline of the present article is as follows.
We recall the description of pion--pion scattering with the IAM formalism in Sect.~\ref{sec:rho}.
The Khuri--Treiman formalism for $\omega\to3\pi$ is described in Sect.~\ref{sec:dispframework}.
Supplementary assumptions to describe the quark-mass dependence of the $\omega$ width
are collected in Sect.~\ref{sec:width}, before we show results in Sect.~\ref{sec:results}.
We summarize our findings in Sect.~\ref{sec:summary}.

\section{\boldmath Pion--pion scattering and the $\rho$ resonance in one-loop unitarized ChPT}\label{sec:rho}
Before starting with the discussion of the quark-mass dependence of the $\omega$ as a three-pion resonance, we briefly summarize the investigation of the quark-mass dependence in $\pi\pi\to\pi\pi$ scattering. The results of this section will be an essential input for the study of $\omega\to3\pi$. We follow the formalism introduced in Refs.~\cite{Hanhart:2008mx,Pelaez:2010fj}. For the later purpose our investigation will focus on the $\rho(770)$ resonance, which appears as a pole in the $P$-wave scattering amplitude. We will treat the $\pi\pi$ rescattering as elastic in this whole section.

The partial-wave decomposition of the $\pi\pi\to\pi\pi$ scattering amplitude $\mathcal{T}_{I}$ of definite isospin $I$ is defined by
\begin{equation}
\mathcal{T}_{I}(s,z_{s})=32\pi\sum_{\ell=0}^{\infty}(2\ell+1)P_{\ell}(z_{s})\,t_{\ell}^{I}(s)\,,
\label{eq_pipi_T}
\end{equation}
with the partial-wave amplitude $t_{\ell}^{I}$ of angular momentum $\ell$, the Legendre polynomial $P_{\ell}$, and the $s$-channel scattering angle $z_{s}=\cos\theta_{s}$. Below any inelastic thresholds the partial-wave amplitude is given in terms of the scattering phase shift $\delta_{\ell}^{I}$ only,
\begin{equation}
t_{\ell}^{I}(s) = \frac{\sin\delta_{\ell}^{I}(s)\,e^{i\delta_{\ell}^{I}(s)}}{\sigma(s)}\,,
\label{eq:amplitude_phase_shift}
\end{equation}
where $\sigma(s)=\sqrt{1-4M_{\pi}^2/s}$. Since we are interested in the $P$-wave only ($I=\ell=1$) we will drop the labels for simplicity from now on.

In ChPT the pion mass is given in terms of the light quark masses as an expansion $M_{\pi}^2 = 2B\hat{m}+\mathcal{O}(m_{q}^{2})$, where the leading term is known as the Gell-Mann--Oakes--Renner (GMOR) relation~\cite{GellMann:1968rz}, with the two light quark masses combined in $\hat{m}=\tfrac{1}{2}(m_{u}+m_{d})$. The constant $B$ is related to the scalar quark condensate in the chiral limit, which measures the strength of spontaneous symmetry breakdown in QCD. We will work in the isospin limit, meaning that $m_{u} = m_{d}$ and $M_{\pi^{\pm}}=M_{\pi^0}\equivM_{\pi}$. A brief discussion of the isospin-breaking effects in $\omega\to3\pi$ can be found in Sect.~\ref{sec:omegaisospinbreaking}. The GMOR relation implies that studying the quark-mass dependence is equivalent to an investigation of the pion-mass dependence. Hence from now on we will refer to the pion-mass dependence instead. Since we are interested in pion-mass-dependent quantities, it turns out to be useful to define parameters at the physical point as \textit{e.g.}\ $\bar{M}_{\pi}\equivM_{\pi}^{\text{phys}}$.

In the ChPT power counting the $P$-wave amplitude for $\pi\pi\rightarrow\pi\pi$ scattering up to next-to-leading order can be written as
\begin{equation}
\label{tChPT1}
t_{\rm ChPT}(s)=t_2(s)+t_4(s)+\mathcal{O}(p^6)\,,
\end{equation}
where $t_{i}$ denotes the contribution of chiral order $p^i$. The $P$-wave projection of the scattering amplitude $\mathcal{T}_{I}$ given in Ref.~\cite{Gasser:1983yg} yields
\begin{align}
\label{t2t4}
t_2(s)&=\frac{s\sigma^2}{96\pi F^2}\,, \notag\\
t_4(s)&=\frac{t_{2}(s)}{48\pi^2F^2}\bigg[s\left(\bar{l}+\frac{1}{3}\right)-\frac{15}{2}M_{\pi}^2\notag\\
&\qquad-\frac{M_{\pi}^4}{2s}\Big(41-2L_\sigma\big(73-25\sigma^2\big)\notag\\
&\qquad\quad +3L_\sigma^2\big(5-32\sigma^2+3\sigma^4\big) \Big)\bigg]+i\sigma\,t_2(s)^2\,,
\end{align}
where we made use of the abbreviation
\begin{equation}
\label{Abbrev_Lsig_sig}
L_\sigma=\frac{1}{\sigma^2}\left(\frac{1}{2\sigma}\log\frac{1+\sigma}{1-\sigma}-1\right)\,.
\end{equation}
The value for pion decay constant in the chiral limit $F$ is taken from the ratio $F_\pi/F=1.064(7)$~\cite{Bazavov:2010hj,Beane:2011zm,Borsanyi:2012zv,Durr:2013goa,Blum:2014tka,Aoki:2016frl}, where $F_\pi = 92.28(9)\,\text{MeV}$~\cite{Tanabashi:2018} is the pion decay constant at the physical point. For our purpose it is beneficial to work with $F$ instead of $F_\pi$, since $F$ is independent of $M_{\pi}$.  We treat the combination of low-energy constants (LECs) $\bar{l}=\bar{l}_2-\bar{l}_1$, which occurs in the ChPT expression at next-to-leading order, as a free parameter that will be fixed in the following. Note that $\bar{l}$ is also independent of $M_{\pi}$, since the individual mass dependences of $\bar{l}_{1}$ and $\bar{l}_{2}$ cancel~\cite{Gasser:1983yg}.

This amplitude however cannot capture the effects of the $\rho$ resonance, which we expect to be the dominant effect in the $P$-wave above the threshold region, since unitarity is only fulfilled perturbatively ($\Im t_{4} = \sigma |t_{2}|^2$). Furthermore $t_{2}$ and $t_{4}$ are polynomials in $s$ (up to cuts encoded in the $\sigma$ dependence), thus the analytic structure of the standard ChPT expression $t_{\text{ChPT}}$ does not allow for any poles on the second Riemann sheet.

In order to include the $\rho$ resonance into our amplitude we will use the IAM. This method allows us to construct an amplitude that fulfills unitarity exactly. Up to next-to-leading order the IAM yields
\begin{equation}
\label{tIAM1}
t_{\text{IAM}}(s)=\frac{t_2(s)^2}{t_2(s)-t_4(s)}\,,
\end{equation}
which is equivalent to $t_{\rm ChPT}$ up to corrections of $\mathcal{O}(p^6)$. Note that  crossing symmetry is now only fulfilled perturbatively.

\subsection{\boldmath Pole position and residue}\label{sec:rhopole}
The characteristic properties of the $\rho$ resonance are encoded in the pole position and residue of the amplitude on the second Riemann sheet. By analytic continuation the amplitude on the second sheet can be expressed in terms of the amplitude on the first sheet~\cite{Gribov:2009zz}. We employ the specific representation~\cite{Moussallam:2011zg}
\begin{align}
t^{\text{II}}(s) &=\frac{t^\text{I}(s)}{1-2\hat{\sigma}(s)\,t^{\text{I}}(s)}\,, \notag\\
\hat{\sigma}(s) &= \sqrt{\frac{4M_{\pi}^2}{s}-1} \,, \quad \hat{\sigma}(s\pm i\epsilon) = \mp i \sigma(s) \,,
\label{eq:amp_2nd_sheet}
\end{align}
where $t^{\text{I}}$ and $t^{\text{II}}$ denote the amplitudes on the first and second sheets, respectively. Thus the pole position of the amplitude is determined by
\begin{equation}
1-2\hat{\sigma}(s_{\text{pole}})\,t^{\text{I}}(s_{\text{pole}})=0\,,
\label{eq:s_pole_def}
\end{equation}
where $s_{\text{pole}}$ corresponds to
\begin{equation}
\label{spole_M_Gamma_rho}
\sqrt{s_{\text{pole}}} = M_{\rho}-\frac{i}{2}\Gamma_{\rho}\,.
\end{equation}
This allows us to identify the mass $M_{\rho}$, as well as the decay width $\Gamma_{\rho}$ of the $\rho$ resonance (that is assumed to be a purely elastic resonance with a single decay channel $\pi\pi$). The location of the $\rho$ pole at the physical pion mass stemming from two studies of $\pi\pi$ scattering with Roy-type dispersion relations~\cite{Colangelo:2001df,GarciaMartin:2011jx} is used to constrain the up to now undetermined LEC. This is done in the following way: we minimize the distance of the $\rho$ pole position at the physical point of the IAM amplitude with respect to the most precise extraction of the pole position from the GKPY analysis\footnote{Note that all three determinations of the $\rho$ pole position in Refs.~\cite{Colangelo:2001df,GarciaMartin:2011jx} lead to compatible results for $\bar{l}$.} of Ref.~\cite{GarciaMartin:2011jx}. This yields $\bar{l}=5.73(8)$.

The pole position of the IAM at the physical point is then given by
\begin{align}
\sqrt{s_{\text{pole}}} &= 0.7620(15)\,\text{GeV}-i0.0778(11)\,\text{GeV}\,,
\end{align}
which is in good agreement with Refs.~\cite{Hanhart:2008mx,Pelaez:2010fj} and the real part of the pole position coincides with Refs.~\cite{Colangelo:2001df,GarciaMartin:2011jx} within the error bars. Nevertheless we observe a tension in the imaginary part, which is $\sim4\,\,\text{MeV}$ ($\sim2-3$ standard deviations) larger than the imaginary parts from the Roy analyses. Thus here we reach the limits of the one-loop IAM description with only one free parameter; for more elaborate studies with an $\mathcal{O}(p^6)$ IAM amplitude containing several LECs, see Refs.~\cite{Pelaez:2010fj,Niehus:2018}. However, for the purposes of our study of $\omega\to3\pi$, we consider the one-loop IAM a sufficiently reasonable description of $\pi\pi$ scattering.

\begin{figure}
	\centering
	\large
	\scalebox{0.68}{\input{plots/rho_pole_trajectory.tex}}
	\caption{Trajectory of the $\rho$ pole position of $t^{\text{II}}$ in the complex $s$ plane as given in Eq.~\eqref{eq:amp_2nd_sheet} for different pion masses. The gray error band is generated by the uncertainty of $F$ and $\bar{l}$. The colored ellipses mark the one-$\sigma$ uncertainty regions of the pole position for the respective pion mass.}
	\label{Fig:poleTrajectory}
\end{figure}

Since the amplitude is not limited to the physical value of the pion mass, we are able to calculate $M_{\rho}$ and $\Gamma_{\rho}$ as a function of $M_{\pi}$. The trajectory of the pole position on the second Riemann sheet is displayed in Fig.~\ref{Fig:poleTrajectory}. As expected from its quark content, the mass of the $\rho$ increases if the pion becomes heavier. This behavior can be described to good approximation as a linear function in $M_{\pi}^2$ given by
\begin{equation}
M_{\rho}(M_{\pi}^2) = M_{\rho}(0) +a \, M_{\pi}^2\,,
\end{equation}
where $M_{\rho}(0)$ and $a$ can be matched to the pion-mass-dependent pole trajectory extracted from Eq.~\eqref{eq:s_pole_def}, which yields
\begin{align}
M_{\rho}(0) = 0.7480(16)\,\text{GeV}\,,\quad a = 0.719(9)\,\text{GeV}^{-1}\,. \label{eq:Mrho-Mpi-num}
\end{align}
Similar observations have been made by investigating chiral symmetry constraints~\cite{Bruns:2017gix}.

\begin{sloppypar}
The available phase space for the decay decreases with growing pion mass, since the $\rho$ mass increases much more slowly than the pion mass~\cite{Hanhart:2008mx}. Thus the width of the $\rho$ becomes smaller for larger values of $M_{\pi}$. The coupling $g_{\rho\pi\pi}$ of the $\rho$ to the $\pi\pi$ system is defined via the residue
\begin{equation}
g_{\rho\pi\pi}^{2} = -48\pi\lim_{s\to s_{\text{pole}}} \frac{s-s_{\text{pole}}}{s-4M_{\pi}^2}\,t^{\text{II}}(s)\,,
\label{eq:g_rhopipi_res}
\end{equation}
where the normalization factors are chosen such that it coincides with the naive expression
\begin{equation}
\Gamma_{\rho} = \frac{|g_{\rho\pi\pi}|^2}{48\piM_{\rho}^2}\big(M_{\rho}^2-4M_{\pi}^2\big)^{3/2} \,,
\label{eq:g_rhopipi_VMD}
\end{equation}
as obtained from a Lagrangian-based narrow-width approximation or a vector-meson-dominance (VMD) model. Equation~\eqref{eq:g_rhopipi_res} yields a numerical value of $|g_{\rho\pi\pi}|=6.12(4)$, which is in fair agreement with other determinations~\cite{Pelaez:2010fj,GarciaMartin:2011jx}. Note that the coupling $g_{\rho\pi\pi}$ extracted from the IAM is pion-mass independent to very good approximation. Thus the pion-mass dependence of $\Gamma_{\rho}$ is driven by the phase space factor only, see Eq.~\eqref{eq:g_rhopipi_VMD}, as confirmed by lattice QCD calculations~\cite{Alexandrou:2017mpi}.
\end{sloppypar}

\subsection{Scattering phase shift}\label{sec:phaseshift}
The $I=1$ $\pi\pi$ system is one of the most widely-studied resonant scattering phase shifts in lattice QCD~\cite{Aoki:2007rd,Feng:2010es,Aoki:2011yj,Lang:2011mn,Dudek:2012xn,Pelissier:2012pi,Bali:2015gji,Wilson:2015dqa,Bulava:2016mks,Alexandrou:2017mpi,Hu:2017wli,Andersen:2018mau}. As defined in Eq.~\eqref{eq:amplitude_phase_shift} the scattering phase shift of the IAM amplitude can be extracted via $\delta(s)\equiv\arg t(s)$. The results for different values of the pion mass are shown in Fig.~\ref{Fig:IAMphase}. As expected, the slope of the phase shift becomes steeper for heavier pions, while the whole curve moves to the right (decreasing width and increasing mass of the $\rho$). This behavior is also observed by various lattice QCD calculations carried out at different pion masses~\cite{Aoki:2007rd,Feng:2010es,Aoki:2011yj,Lang:2011mn,Dudek:2012xn,Pelissier:2012pi,Bali:2015gji,Wilson:2015dqa,Bulava:2016mks,Alexandrou:2017mpi,Hu:2017wli,Andersen:2018mau}. At the physical pion mass the phase shift is in perfect agreement with the Roy analyses of Refs.~\cite{Caprini:2011ky,GarciaMartin:2011cn} in the low-energy regime up to $\sqrt{s}\sim0.8\,\text{GeV}$.

Above this energy the Roy solutions are typically continued to an asymptotic value of $\pi$. The IAM amplitude on the other hand behaves like
\begin{equation}
\lim_{s\to\infty}t_{\text{IAM}}(s) = -\frac{3\pi}{6\bar{l}+2+3\pi i}
\end{equation}
in the high-energy limit, which depends only on the value of the LEC. Thus the phase of the IAM amplitude will not reach $\pi$ for all reasonable values of $\bar{l}$.
Nevertheless the phase shift gives a reasonable parametrization up to the $\rho$ resonance region and thus will be used as a key ingredient for the dispersive representation of the $\omega\to3\pi$ amplitude, as described in Sect.~\ref{sec:dispframework}. In order to test the effect of the discrepancy of the IAM compared to the Roy solution phase shifts, we will also use the parametrization
\begin{equation}
\delta(s) = \pi-\frac{\delta_{a}}{\delta_{b}+s/\Lambda^{2}}
\end{equation}
to account for the correct asymptotic behavior. The parameters $\delta_{a}$ and $\delta_{b}$ are fixed by ensuring continuity of the phase shift and its derivative at some high-energy scale $\Lambda^2\sim1.2\,\text{GeV}^2$, at which we switch from the IAM to the asymptotic phase shift. Note that this high-energy continuation has no relevant influence on the $\omega\to3\pi$ amplitude, since its effects can be fully absorbed by adjusting the unknown normalization of the amplitude, see Eq.~\eqref{eq_basis_functions} below. Similarly, any hypothetical, sub-leading, pion-mass dependence in $\Lambda^2$ would be far too small to be of relevance.

\begin{figure}
	\centering
	\large
	\scalebox{0.68}{\input{plots/iam_phase_shift.tex}}
	\caption{IAM amplitude $\pi\pi$ $P$-wave scattering phase shift for different values of the pion mass: 0 (blue), $\tfrac{1}{2}\bar{M}_{\pi}$ (cyan), $\bar{M}_{\pi}$ (green), $\tfrac{3}{2}\bar{M}_{\pi}$ (yellow), $2\bar{M}_{\pi}$ (orange), and $\tfrac{5}{2}\bar{M}_{\pi}$ (red). The error bands result from the uncertainty of $F$ and $\bar{l}$. The color coding is identical to Fig.~\ref{Fig:poleTrajectory}.}
	\label{Fig:IAMphase}
\end{figure}

\section{Formalism and dispersive representation of the $\omega\to3\pi$ amplitude}
\label{sec:dispframework}
The transition amplitude for the $\omega\to3\pi$ decay is defined as
\begin{align}
&\langle \pi^{+}(p_{1})\pi^{-}(p_{2})\pi^{0}(p_{3})|T|\omega(P)\rangle \notag\\
&\hspace{1cm}= (2\pi)^4\delta^{(4)}(P-p_{1}-p_{2}-p_{3})\,\mathcal{M}(s,t,u)\,.
\end{align}
In our convention the Mandelstam variables are defined according to
\begin{equation}
s=(P-p_{3})^2\,,\quad t=(P-p_{1})^2\,,\quad u=(P-p_{2})^2\,,
\end{equation}
which fulfill the relation
\begin{equation}
s+t+u=M_{\omega}^2+3M_{\pi}^2 =: 3s_{0}.
\end{equation}
In the $s$-channel center-of-mass system, $t$ and $u$ can be expressed as
\begin{equation}
t(s,z_{s}) = u(s,-z_{s}) = \frac{1}{2}\big(3s_{0}-s + z_{s}\kappa(s)\big)\,,
\end{equation}
where $z_s=\cos\theta_{s}$ is the scattering angle
\begin{align}
z_{s}=\cos\theta_{s}=\frac{t-u}{\kappa(s)}\,,\quad\kappa(s) = \sigma(s)\,\lambda^{1/2}(M_{\omega}^2,M_{\pi}^2,s)\,,
\end{align}
with the K\"all\'en function $\lambda(x,y,z) = x^2+y^2+z^2-2(xy+xz+yz)$. Similar expressions hold for the $t$ and $u$-channels, respectively. The physical thresholds in the three channels are
\begin{equation}
s_{\text{thr}}=t_{\text{thr}}=u_{\text{thr}}=4M_{\pi}^2\,.
\end{equation}

Since the transition $\omega\to3\pi$ is of odd intrinsic parity, the amplitude can be further decomposed into a kinematic prefactor and a scalar function $\mathcal{F}(s,t,u)$ containing the dynamical information,
\begin{equation}
\mathcal{M}(s,t,u) = i\epsilon_{\mu\nu\alpha\beta}\,\epsilon^{\mu}(P)\,p_{1}^{\nu}\,p_{2}^{\alpha}\,p_{3}^{\beta}\,\mathcal{F}(s,t,u)\,.
\label{eq_decopm_M_F}
\end{equation}
Here $\epsilon^{\mu}(P)$ denotes the polarization vector of the $\omega$ meson. The modulus of the amplitude is given by
\begin{align}
|\mathcal{M}(s,t,u)|^2 = \frac{1}{4}\Big[stu-M_{\pi}^2\big(M_{\omega}^2-M_{\pi}^2\big)^2\Big]|\mathcal{F}(s,t,u)|^2\,.
\label{eq:kin_prefactor}
\end{align}
The expression in the square brackets is also known as the \textit{Kibble cubic}~\cite{Kibble:1960zz}.

Due to Bose symmetry only odd partial waves are allowed to contribute to the process. Thus the partial-wave decomposition for the scalar function $\mathcal{F}(s,t,u)$ in the $s$-channel reads
\begin{equation}
\mathcal{F}(s,t,u) = \sum_{\ell~\text{odd}}P'_{\ell}(z_s)\,f_{\ell}(s)\,,
\end{equation}
where $P'_{\ell}(z_s)$ denotes the differentiated Legendre polynomials. A particular partial wave can be projected out by making use of
\begin{equation}
f_{\ell}(s) = \frac{1}{2}\int_{-1}^{1}{\rm d} z_{s}\big[P_{\ell-1}(z_{s})-P_{\ell+1}(z_{s})\big]\mathcal{F}(s,t,u)\,. \label{eq_PW}
\end{equation}
As the available phase space in the $\omega\to3\pi$ decay is rather small, the dominant contribution will come from the $\ell=1$ partial wave (see Ref.~\cite{Niecknig:2012sj} for a discussion of potential $F$-wave contributions). Neglecting discontinuities from $F$- and higher partial waves allows us to decompose the scalar function into a sum of single-variable functions~\cite{Niecknig:2012sj,Hoferichter:2012pm,Danilkin:2014cra,Hoferichter:2014vra,Hoferichter:2017ftn,Hoferichter:2018kwz}
\begin{equation}
\mathcal{F}(s,t,u) = \mathcal{F}(s)+\mathcal{F}(t)+\mathcal{F}(u)\,,
\label{eq_reconstruction_thm}
\end{equation}
where $\mathcal{F}(s)$ possesses only a right-hand cut. This kind of decomposition is known as a \textit{reconstruction theorem}~\cite{Stern:1993rg,Knecht:1995tr,Ananthanarayan:2000cp,Zdrahal:2008bd}. The symmetry in this decomposition reflects the fact that the process is invariant under the exchange of the pions.
Combining Eqs.~\eqref{eq_PW} and \eqref{eq_reconstruction_thm} leads to
\begin{align}
f_{1}(s) &=\mathcal{F}(s)+\hat{\mathcal{F}}(s) \,, \quad
\hat{\mathcal{F}}(s) = 3\big\langle(1-z_{s}^2)\mathcal{F}\big\rangle(s)\,, \notag\\
\big\langle z_{s}^{n}\mathcal{F}\big\rangle(s) &= \frac{1}{2}\int_{-1}^{1}{\rm d} z_{s}\,z_{s}^{n}\mathcal{F}\big(t(s,z_{s})\big)\,. \label{eq_inhomogeneity}
\end{align}
The right-hand cut of the partial wave $f_1(s)$ is contained in $\mathcal{F}(s)$, while its left-hand cut contributions
reside entirely in the projection of the crossed-channel single-variable functions $\hat{\mathcal{F}}(s)$.

\begin{sloppypar}
The dispersive framework to describe this decay was already used in previous studies of $\omega\to3\pi$~\cite{Niecknig:2012sj,Danilkin:2014cra} as well as the closely related processes $\gamma^{(*)}\to3\pi$~\cite{Hoferichter:2012pm,Hoferichter:2014vra,Hoferichter:2017ftn,Hoferichter:2018dmo,Hoferichter:2018kwz}.  It uses the formalism of the so-called Khuri--Treiman equations~\cite{Khuri:1960zz,Bronzan:1963mby}, which is based on analytic continuation of a crossed scattering amplitude in the decay mass~\cite{Gribov:1962fu}.  Final-state pion--pion rescattering is assumed to be elastic, and can be described in terms of the phase shift only, see Eq.~\eqref{eq:amplitude_phase_shift}.
The unitarity relation for the partial wave $f_1(s)$ is given by\footnote{Here and in the following relations that involve the discontinuity are always meant to be valid along the right-hand cut only, which starts at the two-pion threshold in the respective channel.}
\begin{equation}
{\rm disc}\, f_1(s) = 2i f_1(s) \sin\delta(s)\,e^{-i\delta(s)}\,,
\label{eq_unitarity_f1}
\end{equation}
where $\delta(s)$ is the $\pi\pi$ $P$-wave phase shift.  Inserting Eq.~\eqref{eq_inhomogeneity} and noting that
${\rm disc}\, f_{1}(s) = {\rm disc}\, \mathcal{F}(s)$ along the right-hand cut, we conclude
\begin{equation}
{\rm disc}\,\mathcal{F}(s) = 2i\big[\mathcal{F}(s)+\hat{\mathcal{F}}(s)\big]\sin\delta(s)\,e^{-i\delta(s)}\,,
\label{eq_unitarity_F}
\end{equation}
which is an inhomogeneous Omn\`es problem for the single-variable function $\mathcal{F}(s)$
with the inhomogeneity $\hat{\mathcal{F}}(s)$.
Assuming the Froissart--Martin bound~\cite{Froissart:1961ux,Martin:1962rt}, a solution of the unitarity relation~\eqref{eq_unitarity_F} can be written in terms of a single subtraction constant $\alpha$~\cite{Niecknig:2012sj},
\begin{align}
\mathcal{F}(s) &= \alpha\mathcal{F}_{\alpha}(s)\,,\qquad\hat{\mathcal{F}}(s)=\alpha\hat{\mathcal{F}}_{\alpha}(s)\,, \notag\\
\mathcal{F}_{\alpha}(s) &= \Omega(s)\bigg\{1+\frac{s}{\pi}\int_{4M_{\pi}^2}^{\infty}\frac{{\rm d} s'}{s'}\frac{\hat{\mathcal{F}}_{\alpha}(s')\,\sin\delta(s')}{|\Omega(s')|(s'-s)}\bigg\}\,,
\label{eq_basis_functions}
\end{align}
where the Omn\`es function $\Omega(s)$ is given by~\cite{Omnes:1958hv}
\begin{equation}
\Omega(s) = \exp\bigg\{\frac{s}{\pi}\int_{4M_{\pi}^2}^{\infty}\frac{{\rm d} s'}{s'}\frac{\delta(s')}{(s'-s)}\bigg\}\,.
\end{equation}
The \textit{basis function} $\mathcal{F}_{\alpha}(s)$ can be constructed independently of the numerical value of the subtraction constant $\alpha$, which therefore can be determined \textit{a posteriori}.  As $\alpha$ serves as an overall normalization of the amplitude, at physical pion masses it is fixed to the total rate $\Gamma(\omega\to3\pi)$, with the energy dependence of the amplitude or the Dalitz plot distribution then being a theoretical prediction~\cite{Niecknig:2012sj}.
\end{sloppypar}

\section{\boldmath Decay width and $M_{\pi}$ dependencies}\label{sec:width}
The process $\omega\to3\pi$ gives the by far dominant contribution to the total $\omega$ decay width, $\mathcal{B}(\omega\to3\pi)=(89.2\pm0.7)\%$. Besides that, the main subleading contributions stem from $\mathcal{B}(\omega\to\pi\gamma)=(8.4\pm0.2)\%$ (electromagnetic) and $\mathcal{B}(\omega\to\pi\pi)=(1.5\pm0.1)\%$ (isospin-breaking). Together these contributions account for $>99\%$ of the decay width at the physical point~\cite{Tanabashi:2018}. As long as we restrict ourselves to the isospin limit and strong contributions only, the decay width is fully driven by $\omega\to3\pi$.

Thus the decay width $\Gamma(\omega\to3\pi) \equiv \Gamma_{\omega}$ is obtained by integrating the squared amplitude over phase space according to
\begin{equation}
\label{Eq_DecWid_LorInvPhaseSpInt}
\Gamma_{\omega}=\frac{1}{256\pi^3M_{\omega}^3}\int{\rm d} s\,{\rm d} t\,|\mathcal{M}(s,t,u)|^2\,.
\end{equation}
This expression has several $M_{\pi}$ dependencies besides the explicit ones (integration boundaries and Mandelstam variables), which will be discussed in the following.

\subsection{Pion-mass dependence of the $\omega$ mass}\label{sec:omegamass}
\begin{sloppypar}
In contrast to the case of the $\rho$, for which we can derive the pion-mass dependence of the complete
pole position in the $\pi\pi\to\pi\pi$ $P$-wave amplitude by means of the IAM, we are not in the position
to do the same for the $\omega$ within some $3\pi\to3\pi$ amplitude of the appropriate quantum numbers.
We will discuss the complicated pion-mass dependence of the \textit{width} of the $\omega$, or the imaginary
part of its pole in the complex plane, which is the
main focus of this study, in the following; for the pion-mass dependence of its \textit{mass}, the
corresponding real part, we have to resort to symmetry arguments based on effective Lagrangians.
These will relate $M_{\omega}(M_{\pi}^2)$ to $M_{\rho}(M_{\pi}^2)$, which we have discussed in Sect.~\ref{sec:rhopole}.
\end{sloppypar}

\begin{sloppypar}
We briefly recapitulate the analysis of the leading symmetry-breaking effects
in the masses of the vector meson nonet~\cite{Jenkins:1995vb,Bijnens:1996nq,Bijnens:1997ni}.
Here, the vector mesons are treated as static matter fields; the effective Lagrangian is organized
in terms of increasing chiral dimension as well as using the expansion in the inverse number of colors
$1/N_c$.  We neglect isospin breaking and electromagnetic effects~\cite{Bijnens:1996nq}, and ignore
deviations from ideal mixing.  In this approximation, the symmetry-breaking part of the
effective Lagrangian can be written as
\begin{align}
\mathcal{L}_{\textrm{SB}} &= \frac{\delta}{2} \langle W_\mu^\dagger\rangle \langle W^\mu \rangle
+ \frac{a}{2}\langle \chi \{W_\mu^\dagger,W^\mu\} \rangle \nonumber\\
&\quad+ \frac{b}{4} \big( \langle \chi W_\mu^\dagger\rangle \langle W^\mu\rangle + \text{h.c.} \big)
+ \frac{c}{2}\langle \chi \rangle \langle W_\mu^\dagger W^\mu \rangle  \nonumber\\
&\quad+ \mathcal{O}(m_q^2,1/N_c^2)\,, \label{eq:Lagr}
\end{align}
where
\begin{equation}
W_\mu = \left( \begin{array}{ccc}
\frac{\rho^0_\mu}{\sqrt{2}} + \frac{\omega_\mu}{\sqrt{2}} & \rho_\mu^+ & K_\mu^{*+} \\
\rho_\mu^- & - \frac{\rho^0_\mu}{\sqrt{2}} + \frac{\omega_\mu}{\sqrt{2}} & K_\mu^{*0} \\
K_\mu^{*-} & \bar K_\mu^{*0} & \phi_\mu
\end{array} \right)
\end{equation}
contains the (nonrelativistic) vector-meson fields, and $\chi = \text{diag}(M_{\pi}^2,M_{\pi}^2,2M_K^2-M_{\pi}^2)$
breaks $SU(3)$ flavor symmetry due to the different quark masses.
Among the terms in Eq.~\eqref{eq:Lagr}, the quark-mass-independent
operator $\propto \delta$ is $1/N_c$ suppressed and
breaks nonet symmetry; the term $\propto a$ is chirally suppressed, but the dominant flavor-breaking
term in the large-$N_c$ limit; and the operators $\propto b$ and $\propto c$ are both chirally and
$1/N_c$ suppressed.  The term $\propto c$ leads to a common shift in all nonet masses and hence
cannot be discerned from the common mass $M_V$ using experimental data only; on account of the fact
that we can show the operator $\propto b$ indeed to be strongly suppressed below, we will neglect
the former in the following.
\end{sloppypar}

Equation~\eqref{eq:Lagr} then leads to the vector-meson masses
\begin{align}
M_{\rho} &= M_V + a \,M_{\pi}^2\,, \nonumber\\
M_{\omega} &= M_V + \delta + (a+b)M_{\pi}^2\,, \nonumber\\
M_\phi &= M_V + \frac{\delta}{2} +  \Big(a+\frac{b}{2}\Big)\big(2M_K^2-M_{\pi}^2\big)\,, \nonumber\\
M_{K^*} &= M_V + a\, M_K^2\,,
\label{Eq:mV_mpi_dependence}
\end{align}
which allows us to extract the coupling constants according to
\begin{align}
a &= \frac{M_{K^*}-M_{\rho}}{M_K^2-M_{\pi}^2}\,,\nonumber\\
b &= \frac{M_\phi-2M_{K^*}+\frac{3}{2}M_{\rho}-\frac{1}{2}M_{\omega}}{M_K^2-M_{\pi}^2}\,, \nonumber\\
\delta &= M_{\omega}-M_{\rho} - b\,M_{\pi}^2\,.
\end{align}
In particular the value for $b$ depends quite sensitively on the precise values inserted for
the masses of the broad $\rho$ and $K^*$ resonances; if we employ the real parts of their
pole positions~\cite{GarciaMartin:2011jx,Pelaez:2016klv}, we find
$a = 0.57(1)\,\text{GeV}^{-1}$, $b = -0.045(20)\,\text{GeV}^{-1}$,  $\delta = 20(2)\,\text{MeV}$.

\begin{sloppypar}
We hence conclude that the formal $1/N_c$ suppression of $b/a$ translates, in fact, into a numerical
suppression by more than an order of magnitude; we will therefore neglect $b$, too.
Furthermore, we observe that the determination of $a$ based on $SU(3)$ symmetry leads to an estimate
that is about 20\% smaller than the value deduced from the one-loop IAM representation,
see Eq.~\eqref{eq:Mrho-Mpi-num}.  As we expect that $SU(3)$ breaking effects ought to affect the
relation between $\rho$ and $\omega$ observables less, we use the arguments above to employ
a pion-mass dependence of $M_{\omega}$ that equals the one of $M_{\rho}$ up to the constant offset $\delta$,
hence
\begin{align}
\label{Eq_kin_Mw_mp}
M_{\omega}(M_{\pi}^2) &=\bar{M}_{\omega}+\frac{0.719(9)}{\unit{GeV}}\left(M_{\pi}^2-\bar{M}_{\pi}^2\right) \,, \notag\\
M_{\omega}(0) &= 0.7686(20)\,\text{GeV} \,.
\end{align}

At higher orders in the chiral expansion, Goldstone-boson loops induce nonanalytic dependencies
of the vector meson masses of the form $\mathcal{O}(m_q^{3/2})$ and $\mathcal{O}(m_q^2\log m_q)$,
which have been studied extensively in the literature~\cite{Jenkins:1995vb,Bijnens:1996nq,Bijnens:1997ni,Bruns:2004tj,Bruns:2013tja,Bavontaweepanya:2018yds}; such terms will obviously break the similarity in
$M_{\rho}(M_{\pi}^2)$ and $M_{\omega}(M_{\pi}^2)$ due to the different coupling of $\rho$ and $\omega$ to pions.
We ignore such terms in the present study solely based on the observation in Sect.~\ref{sec:rhopole}
that a linear dependence of $M_{\rho}$ on $M_{\pi}^2$ is sufficient to describe the behavior of the $\rho$ pole
of the $\mathcal{O}(p^4)$ IAM amplitude.
\end{sloppypar}

\subsection{Subtraction constant}
As derived in Sect.~\ref{sec:dispframework}, we require one subtraction constant in the dispersive representation of the $\omega\to3\pi$ decay amplitude in order to maintain a convergent integral representation. Since this subtraction constant is not fixed by unitarity (and, for the process at hand, cannot be matched to ChPT as for other processes such as $\gamma\pi\to\pi\pi$~\cite{Hoferichter:2012pm,Niehus:2017}), we need to fix its pion-mass dependence in a different way.

In Sect.~\ref{sec:rho} we have recounted that the coupling $g_{\rho\pi\pi}$ at the $\rho\to\pi\pi$ vertex is (essentially) pion-mass independent and in good agreement with a narrow-width formula or a VMD model. In an isobar model of subsequent two-body decays, $\omega\to3\pi$ is typically understood in terms of processes $\omega\to\rho\pi$, followed by $\rho\to\pi\pi$ decays, see Fig.~\ref{fig:VMD_omega_3pi} (\textit{cf.}\ \textit{e.g.}\ Refs.~\cite{GellMann:1962jt,Klingl:1996by,Leupold:2008bp}).  Reducing the dispersive representation Eq.~\eqref{eq_basis_functions} to such a simplified picture, we find the subtraction constant $\alpha$ in one-to-one correspondence with the product of coupling constants $g_{\omega\rho\pi}\times g_{\rho\pi\pi}$.  We therefore conjecture that, by analogy, it is reasonable to assume $g_{\omega\rho\pi}$, and hence $\alpha$, to be also pion-mass independent, and we fix the subtraction constant to the total decay width $\Gamma_\omega$ at the physical point.

\begin{figure}
\centering
\includegraphics[width=0.45\linewidth]{plots/VMD_omega_3pi.eps}
\caption{VMD tree-level diagram for $\omega\to3\pi$ given by a $\omega\to\rho\pi$ and a subsequent $\rho\to\pi\pi$ decay.}
\label{fig:VMD_omega_3pi}
\end{figure}

\subsection{Isospin-breaking effects}\label{sec:omegaisospinbreaking}
Up to now all calculations have been carried out under the assumption of isospin symmetry ($M_{\pi^{\pm}}=M_{\pi^0}$). We now want to briefly discuss the influence of isospin breaking effects in $\omega\to3\pi$; a more detailed discussion can be found in Ref.~\cite{Dax:2017}.  It is well known that in the context of precision analyses of $\eta\to3\pi$, in particular comparing $\eta\to\pi^+\pi^-\pi^0$ and $\eta\to3\pi^0$, taking into account the pion-mass difference for the available phase space at least is mandatory~\cite{Gasser:1984pr,Ditsche:2008cq,Schneider:2010hs,Colangelo:2018jxw}. In this section, we therefore only investigate isospin breaking in the kinematical contribution, ignoring all dynamical effects (\textit{i.e.}\ using $|\mathcal{F}(s,t,u)|^2=const.$), expecting this to be the dominant change.

The pion-mass difference
\begin{equation}
\Delta_{\pi} = \big(M_{\pi^{\pm}}^2-M_{\pi^0}^2\big) = 1.26116(15)\times10^{-3}\,\text{GeV}^{2}
\end{equation}
originates from two sources: electromagnetic effects and the light-quark-mass difference $(m_{u}-m_{d})$.
At leading-order in ChPT (for three flavors), the latter can be evaluated to
\begin{align}
\Delta_{\pi}^{\text{QCD}} = \frac{(m_{u}-m_{d})^2}{8\hat{m}(m_{s}-\hat{m})}M_{\pi}^2 \approx 3 \times10^{-5}\,\text{GeV}^{2} \,,
\end{align}
hence this effect is very small: the pion-mass difference is dominantly caused by electromagnetism, and we neglect the effect of the difference of the light quark masses completely. Consequently, as long as we neglect higher-order corrections of $\mathcal{O}(e^2m_q)$, the pion-mass difference $\Delta_{\pi}$ stays constant when varying $M_{\pi}$. This allows us to relate the neutral pion mass to the charged one according to
\begin{equation}
M_{\pi^0}(M_{\pi^{\pm}}^{2}) = \sqrt{M_{\pi^{\pm}}^2-\Delta_{\pi}}\,.
\label{Eq:mpn_mpc_isobreak}
\end{equation}
Obviously, this relation breaks down at a minimal charged pion mass of $M_{\pi^{\pm}} = \sqrt{\Delta_\pi} \approx 35.5\,\text{MeV}$.

It turns out that isospin breaking in the kinematical dependence of $\Gamma_{\omega}$ as given in Eq.~\eqref{Eq_DecWid_LorInvPhaseSpInt} when tuning the charged pion mass gives only a tiny correction to $\Gamma_{\omega}$ of less than $2\%$.  Due to the connection of the pion masses in Eq.~\eqref{Eq:mpn_mpc_isobreak} it is easy to see that the effect of isospin breaking becomes smaller when increasing the mass of the pions; on the other hand, since the ratio of the $\omega$ mass to the pion masses becomes larger when approaching the chiral limit, the increasing isospin-breaking effects are lifted by the $\omega$ mass. Thus using the isospin limit is entirely justified for our purposes.

\section{Results}\label{sec:results}
\begin{figure*}
	\large
	\centering
	\scalebox{0.665}{\input{plots/KT_omnes_abs_1.tex} \input{plots/KT_omnes_arg_1.tex}}

	\vspace{-0.8cm}

	\scalebox{0.665}{\input{plots/KT_omnes_abs_2.tex} \input{plots/KT_omnes_arg_2.tex}}

	\vspace{-0.8cm}

	\scalebox{0.665}{\input{plots/KT_omnes_abs_3.tex} \input{plots/KT_omnes_arg_3.tex}}

	\vspace{-0.8cm}

	\scalebox{0.665}{\input{plots/KT_omnes_abs_4.tex} \input{plots/KT_omnes_arg_4.tex}}
\caption{Comparison of the absolute value (left column) and phase motion (right column) of the Omn\`es function (red) and single-variable amplitude $\mathcal{F}_{\alpha}$ (blue) for various pion masses: first row $M_{\pi}=0$, second row $M_{\pi}=\bar{M}_{\pi}$, third row $M_{\pi}=\tfrac{3}{2}\bar{M}_{\pi}$, and last row $M_{\pi}=1.96\bar{M}_{\pi}\approx\frac{1}{3}M_{\omega}(M_\pi^2)$. The dashed black lines mark the pion-mass-dependent lower and upper phase-space boundaries for the $\omega\to3\pi$ decay given by $4M_{\pi}^2$ and $(M_{\omega}-M_{\pi})^2$, respectively. The error bands are generated by taking the uncertainties of the IAM phase shift $\delta(s)$ and $M_{\omega}(M_{\pi}^2)$ into account, see Fig.~\ref{Fig:IAMphase} and Eq.~\eqref{Eq_kin_Mw_mp}. The latter only affects $\mathcal{F}_{\alpha}$.}
	\label{fig:KT_omnes_abs_delta}
\end{figure*}

In this section we discuss the final results of our dispersive representation of $\omega\to3\pi$. First of all we want to compare the resulting pion-mass-dependent Omn\`es function or two-body final-state interaction\footnote{Here the third pion will act as spectator, meaning $\hat{\mathcal{F}}_{\alpha}(s)=0$, thus the single-variable amplitude equals the Omn\`es function in this case (see Eq.~\eqref{eq_basis_functions}).} (2-body FSI) to the single-variable amplitude $\mathcal{F}_{\alpha}$ (3-body FSI) to study the dynamical effects generated by the interaction with the third pion as depicted in Fig.~\ref{fig:KT_omnes_abs_delta}.

Besides pion masses close to the chiral limit, the 3-body FSI leads to an enhancement of the modulus of the single-variable amplitude compared to the Omn\`es function. While the peak position in the absolute values (due to the $\rho$ resonance) is essentially identical for 2- and 3-body FSI at a given value of $M_{\pi}$, the phases behave rather differently. The Omn\`es function fulfills Watson's theorem and thus its phase is identical to the $\pi\pi$ scattering phase shift (the IAM phase). Due to the dynamical effects stemming from the interaction with the third pion (and the generation of a three-pion cut), this does not hold for the single-variable amplitude (see Sect.~\ref{sec:dispframework}). Thus the argument of the single-variable function is shifted compared to the input IAM phase. The lower and upper phase-space boundaries are marked by dashed vertical lines, and hence denote the kinematical range directly accessible in the decay.  Already here we want to point out that mainly the tails of the $\rho$ resonance will only contribute to the dynamics when increasing the pion mass (\textit{cf.}~Fig.~\ref{fig:phsp_mpi_dep_rho_bands}).

\begin{figure}
	\large
	\centering
	\scalebox{0.68}{
		\input{plots/omega_width.tex}
	}
	\caption{Pion-mass-dependent decay width for $\omega\to3\pi$ shown for three different cases: kinematic contribution only (black), considering 2-body FSI (red), and full 3-body FSI (blue). The physical point is marked by the black diamond. The error bands are generated by taking the uncertainties of the IAM phase shift $\delta(s)$ and $M_{\omega}(M_{\pi}^2)$ into account, see Fig.~\ref{Fig:IAMphase} and Eq.~\eqref{Eq_kin_Mw_mp}.}
	\label{fig:omega_width}
\end{figure}

In order to study the dynamical effects on the decay width we consider three different scenarios: first we consider only kinematic contributions to the decay width (all pion--pion dynamics are disregarded, $\delta(s)=0$, and thus $|\mathcal{F}(s,t,u)|^2=const.$), secondly we allow for 2-body rescattering effects (meaning $|\mathcal{F}(s,t,u)|^2 \propto |\Omega(s)+\Omega(t)+\Omega(u)|^2$), and third the full 3-body dynamics are taken into account. A comparison of the different cases is displayed in Fig.~\ref{fig:omega_width}.

\begin{sloppypar}
First of all we notice that the $\omega\to3\pi$ width decreases with increasing pion mass. This is not surprising since the mass of the three pions is increasing faster than the mass of the $\omega$, \textit{cf.}~Eq.~\eqref{Eq_kin_Mw_mp}. Hence the phase space shrinks for larger pion masses as depicted in Fig.~\ref{fig:phsp_mpi_dep_rho_bands}. For $M_{\pi}>\tfrac{1}{3}M_{\omega}\approx1.96\bar{M}_{\pi}$ the masses of the three pions exceed the $\omega$ mass, thus the reaction $\omega\to3\pi$ is no longer allowed and the $\omega$ becomes stable with respect to the considered decay channel (\textit{i.e.}, in QCD in the isospin limit).
\end{sloppypar}

\begin{figure}
	\large
	\centering
	\scalebox{0.68}{\input{plots/phsp_mpi_dep_rho_bands.tex}}
	\caption{Pion-mass-dependent phase space boundaries of $\omega\to3\pi$ (solid lines) and position of the ``on-shell'' $\rho$ in the respective $s$-, $t$-, and $u$-channels (dashed lines). The color coding is kept identical to Figs.~\ref{Fig:poleTrajectory} and \ref{Fig:IAMphase} with the following pion masses: 0 (blue), $\tfrac{1}{2}\bar{M}_{\pi}$ (cyan), $\bar{M}_{\pi}$ (green), $\tfrac{3}{2}\bar{M}_{\pi}$ (yellow), $2\bar{M}_{\pi}$ (orange), and $\tfrac{5}{2}\bar{M}_{\pi}$ (red). For the last two the phase space already vanishes, and thus the $\omega$ becomes stable with respect to the investigated decay mode.}
	\label{fig:phsp_mpi_dep_rho_bands}
\end{figure}

\begin{sloppypar}
We now study the effects of the 2- and 3-body dynamics. First of all we notice that the kinematical prefactor given in Eq.~\eqref{eq:kin_prefactor} vanishes at the phase space boundaries in all directions. This leads to a stronger weighting of the inner region compared to the outskirts of the phase space when evaluating the integral Eq.~\eqref{Eq_DecWid_LorInvPhaseSpInt}. Secondly, the dynamics are mainly governed by the $\rho$ resonance in the respective 2-body channels, leading to a three-band structure (one for each channel) in the function $\mathcal{F}(s,t,u)$. The peak position and width of these bands will be determined by the respective 2- and 3-body FSI effects and the information of the scattering phase input from the IAM. We conclude that strong imprints of the $\rho$ as a dynamical effect will only affect the decay width close to the chiral limit, since the $\rho$ is only allowed to go on-shell within the phase space boundaries for $M_{\pi}<0.15\bar{M}_{\pi}$. This effect is even reinforced by the increasing mass of the $\rho$ and its decreasing width with growing pion mass, as investigated in Sect.~\ref{sec:rho}. Thus the main contributions to the $\omega$ width will come from the tail of the $\rho$ resonances at low pion masses, see Fig.~\ref{fig:phsp_mpi_dep_rho_bands}.
\end{sloppypar}

\begin{figure}
	\large
	\centering
	\scalebox{0.68}{\input{plots/omega_pole_trajectory.tex}}
	\caption{Trajectory of the pion-mass-dependent $\omega$ mass and width in the complex-energy plane of $3\pi\to3\pi$ scattering for the three investigated scenarios: pure kinematics (black), 2-body FSI (red), and full 3-body FSI (blue). This plot can be compared with the one for the $\rho$ resonance, see Fig.~\ref{Fig:poleTrajectory}. The physical point is marked by the black diamond. The error bands are generated by taking the uncertainties of the IAM phase shift $\delta(s)$, $M_{\omega}(M_{\pi}^2)$, and $\bar{\Gamma}_{\omega}$ into account, see Fig.~\ref{Fig:IAMphase}, Eq.~\eqref{Eq_kin_Mw_mp}, and Ref.~\cite{Tanabashi:2018}.}
	\label{fig:omega_pole_trajectory}
\end{figure}

\begin{sloppypar}
This is consistent with the result given in Fig.~\ref{fig:omega_width}. We observe that the impact of the 2- and 3-body dynamics on the decay width is very strong close to the chiral limit, where it leads to a reduction of the decay width compared to the one of pure kinematics. The damping due to the 3-body dynamics is four times stronger than the influence of the 2-body effects and results in a width smaller by about one third than the purely kinematic effects. In the chiral limit we find
\begin{align}
\Gamma_{\omega}^{\text{kinematic}}(0) &= 4.02(1)\,\bar{\Gamma}_{\omega} = 30.4(1)\,\text{MeV}\,, \notag\\
\Gamma_{\omega}^{\text{2-body FSI}}(0) &= 3.72(2)\,\bar{\Gamma}_{\omega} = 28.1(1)\,\text{MeV}\,, \notag\\
\Gamma_{\omega}^{\text{3-body FSI}}(0) &= 2.81(2)\,\bar{\Gamma}_{\omega} = 21.3(2)\,\text{MeV}\,,
\end{align}
for the widths of the three discussed scenarios, where we took $\bar{\Gamma}_{\omega}\equiv\bar{\Gamma}(\omega\to3\pi)=7.57(9)\,\text{MeV}$~\cite{Tanabashi:2018}. Since the subtraction constant of all three curves is fixed at the physical point, the difference between them shrinks when approaching this point. Above the behavior is opposite, here the 2- and 3-body dynamics generate a decay width that is larger than the one obtained from pure kinematic effects. Since the phase space at the physical point and above does not allow for on-shell $\rho$ mesons in the respective 2-body systems, the influence of the dynamical effects above this point is very small. Thus the 2- and 3-body FSI curves do not differ strongly from the curve of pure kinematics.

Considering Eq.~\eqref{spole_M_Gamma_rho} we are in the position to predict a pion-mass-dependent trajectory of the $\omega$ pole position in the complex-energy plane of $3\pi\to3\pi$ scattering similar to the $\rho$ resonance in Fig.~\ref{Fig:poleTrajectory}. A plot of this pole trajectory is given in Fig.~\ref{fig:omega_pole_trajectory}. Note that, strictly speaking, our formalism does not determine such a complex pole position: rather, we have discussed mass (via effective Lagrangians) and width (via the Khuri--Treiman formalism) individually. Given the smallness of the width of the $\omega$, we regard the error committed thereby as negligible, although this may not obviously be so in the general case of an arbitrary $3\pi$ resonance.
\end{sloppypar}

\begin{figure}
	\large
	\centering
	\scalebox{0.68}{
		\input{plots/phi_width.tex}
	}
	\caption{Pion-mass-dependent decay width for $\phi\to3\pi$ shown for three different cases: kinematic contribution only (black), considering 2-body FSI (red) and full 3-body FSI (blue). The physical point is marked by the black diamond. The error bands are generated by taking the uncertainty of the IAM phase shift $\delta(s)$ into account, see Fig.~\ref{Fig:IAMphase}.}
	\label{fig:phi_width}
\end{figure}

Since dynamical effects in the investigated process are limited due to the small phase space, we want to emphasize that the dispersive representation derived in Sect.~\ref{sec:dispframework} is valid for general $V\to3\pi$ decays, with $V$ denoting an arbitrary isoscalar vector meson, at least to the extent that the elastic approximation in the pion--pion rescattering is justifiable. Thus we are able to describe $\phi\to3\pi$ by just replacing the decay mass according to $M_{\omega}\mapsto M_{\phi}$ within the same formalism~\cite{Niecknig:2012sj,Danilkin:2014cra}. With a mass of $1019\,\text{MeV}$~\cite{Tanabashi:2018} its decay allows for much richer dynamics due to the larger phase space, in particular the three $\rho$ bands are visible inside the Dalitz plot~\cite{Aloisio:2003ur,Akhmetshin:2006sc}. In the quark-model picture, the $\phi$ can be understood as a pure $s\bar{s}$ state, thus its mass does not depend on the pion mass at leading order. This follows from Eq.~\eqref{Eq:mV_mpi_dependence} when using the GMOR relation to relate $M_{K}^{2}=B(\hat{m}+m_{s})+\mathcal{O}(m_{q}^{2})$. The results for the pion-mass-dependent partial decay width $\Gamma(\phi\to3\pi)$ are displayed in Fig.~\ref{fig:phi_width}. Since the $\rho$ is allowed to go on-shell up to $M_{\pi}\approx1.7\bar{M}_{\pi}$, the dynamical imprints are much stronger compared to the case of the $\omega\to3\pi$ even above the physical point.

Overall, given how subtle rescattering effects beyond two-body rescattering are usually thought to be (and more often than not neglected altogether in experimental Dalitz plot analyses), it is remarkable to see that in both cases studied here, $\omega\to3\pi$ and $\phi\to3\pi$, the 3-body FSI effects tend to affect the quark-mass dependence of the (partial) widths about as strongly as the 2-body FSI.

We wish to add a few caveats concerning the precision of the predictions shown in this section.
Error bands are estimated solely based on the uncertainty within the one-loop IAM phase shift input, as well
as the one in $M_\omega(M_\pi^2)$, which is nonetheless based on the large-$N_c$ expansion and leading-order
symmetry breaking in the quark masses.   We do not attempt to quantify corrections due to higher orders
in either case, and refer to Ref.~\cite{Niehus:2018} for future work concerning an improved pion-mass-dependent
phase shift.  Furthermore, a high-precision measurement of the $\phi\to3\pi$ Dalitz plot~\cite{Aloisio:2003ur}
revealed the need to include a \textit{second} subtraction in the dispersive representation of the decay
amplitude~\cite{Niecknig:2012sj}, which has also not been considered here; experimental data on the $\omega\to3\pi$
Dalitz plot is not conclusive in this respect yet~\cite{Adlarson:2016wkw}.

\section{Summary}\label{sec:summary}
\begin{sloppypar}
We have investigated the pion-mass (and hence the quark-mass) dependence of the $\omega\to3\pi$ decay width,
generalizing previous studies of two-pion resonances based on the inverse amplitude method.
To this end, we have employed a dispersive formalism, based on Khuri--Treiman equations,
that uses inverse-amplitude-method phase shifts as input.  The pion-mass dependence of the $\omega$ mass is estimated
using a symmetry relation based on chiral perturbation theory for vector mesons.
Deviations from phase space behavior alone, induced by the pion-mass-dependent decay amplitude, are clearly
visible, although suppressed for larger-than-physical pion masses due to the smallness of phase space.
We have demonstrated for the decay of the heavier $\phi$ into three pions that this need not be the case
in general.  Remarkably, a simple description of the decays in terms of two-body rescattering alone does
not yield a good approximation to the full pion-mass dependence seen.

The three-pion decays of the lightest isoscalar vector mesons only serve as a paradigm case for the investigation
of three-body resonances; extensions to other, similar decays within the same formalism ought to be tested in the future.
\end{sloppypar}

\begin{acknowledgements}
\begin{sloppypar}
We thank C.~Hanhart, M.~Hoferichter, M.~Niehus, J.~Ruiz de Elvira, M.~Ueding, and M.~Werner for useful discussions,
as well as M.~Hoferichter, S.~Holz, and M.~Niehus for carefully reading this manuscript.
Partial financial support by DFG and NSFC through funds provided to
the Sino--German CRC~110 \enquote{Symmetries and the Emergence of
Structure in QCD} (DFG Grant No.~TRR110 and NSFC Grant No.~11621131001)
and by the Bonn--Cologne Graduate School of Physics and Astronomy is gratefully acknowledged.
\end{sloppypar}
\end{acknowledgements}

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redpajama_set_name: RedPajamaArXiv


Today I had a stall at the We Heart Vintage market in Mount Lawley. I wore a dress from Miss Paisley Vintage (my own stall), I liked it too much to sell!
That dress is gorgeous, you look amazing! I'll have to come down to the market next time.
I can't believe I still haven't been to one of these! I'll have to make sure I don't miss the next one. Great dress!


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Home > Introducing Query Anomaly Detection
Introducing Query Anomaly Detection
By Baron Schwartz
November 11, 2015 | Database
Anomaly detection sure is a hot topic. We've written about it ourselves a number of times, and Preetam Jinka and I just co authored a book for O'Reilly called Anomaly Detection For Monitoring. One of the challenges, as we've discussed so often, is that catch-all, generic anomaly detection is hard to do. In special cases, however, there's often a niche use case that can be done well and is highly beneficial. Query behavior changes are an example of that, and I'm happy to announce that SolarWinds® Database Performance Monitor (DPM) now has advanced statistical algorithms running to detect important changes to your most important queries continually. What does query anomaly detection mean? Good question! In general, a lot of anomaly detection techniques try to compare current behavior to past behavior and determine if we're within ranges of expected behavior. You're probably familiar with various ways to do this, such as Holt-Winters forecasting. HWF includes seasonality so you won't have skewed expectations at 5am based on 5pm's traffic, for example. There are so many ways that system behavior can change, however, that most anomaly detection techniques alarm on lots of false positives. That is, they tell you something's unusual far too often. To avoid this, we've taken a more sophisticated approach that we'll write up in more detail later. As for what we check for anomalous changes, though, we're again taking a very specific view of that. Database Performance Monitor measures many thousands of metrics per server every second (sometimes much more). We're not going to check all of those; as I've discussed previously, it's vital to understand the meaning of the metrics and only check meaningful metrics in sensible ways. In brief, we look for anomalous changes in query metrics first of all:
For each category of queries, we detect changes in frequency, total accumulated time, and latency
We detect important changes in overall error and warning rate, globally (not per-category-of-query)
There are more metrics we're approaching cautiously, including some system metrics, but this is a starting point. As we've said many times, databases are meant to execute queries, so by far the most important thing to monitor about a database is is high-definition query monitoring. We monitor MySQL, PostgreSQL, Redis, and MongoDB queries, and all of these automatically get anomaly detection as a result. We take various steps to avoid false positives, including accounting for multiple kinds of seasonality. We also suppress anomaly detection if a query isn't very important–that is, if it's not a heavy-hitter–relative to the overall workload. The result is dynamically generated, intelligent baselining that is biased towards avoiding false-positive events. It might be more sophisticated than the "baselines" you might be accustomed to in many monitoring products. It detects changes such as the following on one of our Redis servers. Anomaly detection uses our standard Events functionality, creating an event with details of the anomaly in the query metric. As a result, you can easily configure alerts for the event, using email or any of our standard integrations (Slack, HipChat, PagerDuty, VictorOps, and more) to get notified. Of course, you can also inspect the anomalies in context. The events are deep-linked to the query's summary dashboard, where you can not only look at the metrics, but also examine individual samples, use Compare Queries to compare across time ranges, and more. Clearly that query suddenly became drastically slower around this time. The frequency didn't change, but something else did. Armed with that knowledge, you can dig in and figure out what and why. Our query anomaly detection features have been running in production for all of our customers for a while, although we didn't publicize it previously. We wanted to take more time and examine the anomaly detection algorithm's behavior across a variety of customer workloads and ensure that we aren't spamming anyone with false positives. If you'd like help with anything you see in Database Performance Monitor, please chat with us using the in-app support chat at the bottom left of the app. We'll be glad to help investigate or take a look at anything you'd like to know more about.
Baron Schwartz
Baron is a performance and scalability expert who participates in various database, open-source, and distributed systems communities. He has helped build and scale many large,…
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redpajama_set_name: RedPajamaCommonCrawl


Every birthday is a milestone as far was we're concerned and the "Babe-A-Licious" party accessories from the Birthday-licous collection covers all your birthday bases. The neoprene koozie will keep her birthday beverage cool, no matter how hot she is. It's fabu-licious, especially when matched with the birthday tiara, party beads and shot glasses all with the same green leopard print! Measuring 4 1/2 inches tall, it will fit 12 ounce cans and standard bottles. The front reads "Babe-A-licious" and "It took years to get this hot." These party favors combine to make a beautiful birthday gift basket.
Girlfriend Tested. Girlfriend Approved. This is the perfect drink sleeve to use again and again and coordinates with other party supplies from the licious line for a gorgeous gift for her. It is sturdy and thick to last year after year.
Love this. It is a gift. Couldn't find anything like it at my local party store. Great price.


I've had an eventful week.
Another visit to the local hospital curtesy of the blue flashing light brigade, and very nice they were too. I was much more sociable after an arm full of morphine!
The health stuff is boring. I do hate that blooming place. It was particularly hot on this occasion. Coming home wasn't a moment too soon.
I thought I'd recuperate with a trip down memory lane. Does anyone remember those large round macaroons with the whole almond on top? Why don't they do those anymore? When I was small my Nan used to take me shopping and we would usually stop at a bakery (Nan would always say that Grandad deserved a treat but I reckon she liked a treat just as much!). I thought that shopping in a bakery was the height of luxury since Mum did most of her shopping in a supermarket. I got to choose a cake or biscuit and I would always choose the macaroon. Yum!
This is the simplest recipe which uses just ground almonds, sugar and crushed cardomom seeds. I was supposed to roll them in rose watered hands, but c'mon rosewater! Not in my kitchen cupboard at the moment but I'm sure Nigella would have some in hers!
The recipe didn't mention that they don't spread that much. I was hoping for an exact replica according to my childhood memory thank you very much! Next time I will squish them down a lot more, make them almost the shape and size they are supposed to end up.
Have you spotted the rogue pecan? I ran out of almonds! They weren't bad at all. Even Ella approved but she said she'd prefer them without the cardomom. Fair enough.
I have made them, yummy they are too and no I didn't bother with the rose water either!


Chinese scientists have shown that needling active acupoints in migraine patients reduces pain and affects key disease-affected brain regions involving pain.
A secondary analysis of data from the Acupuncture Trialists' Collaboration has shown that acupuncture is significantly superior to control, irrespective of the type of control used.
American authors who have spent the last two decades investigating the neural mechanisms underlying the ability of acupuncture to regulate blood pressure have summarised their work in a review paper.
Swedish authors have reviewed the results of randomised studies on the use of acupuncture for women with polycystic ovary syndrome (PCOS).
An American group has found that the brain's response to electro-acupuncture (EA) correlates with symptom reduction following EA for carpal tunnel syndrome (CTS).
A Danish study has attempted to discern the contribution of specific and non-specific factors to acupuncture analgesia in patients undergoing dental surgery.
Acupuncturists outperform non-acupuncturists (NA) in tests of behavioural skills associated with their profession and this is reflected in structural differences in their brains.
Needling non-acupoints that lie on an acupuncture channel has a stronger effect on blood perfusion (BP) in the surrounding tissue than needling non-acupoints that do not line on a channel.
Needle rotation and sensation play an important role in the effects of acupuncture, Korean investigators report.
Japanese researchers have found evidence that acupuncture points have specific actions on the autonomic functions of particular internal organs.
Is ATP the key to acupuncture's effects?
A British neuroscientist has proposed that a novel extracellular signalling system, based on adenosine 5'-triphosphate (ATP), is the mechanism underlying the effects of acupuncture.
Facial cosmetic acupuncture (FCA) results in improvement of objective measures of facial elasticity, a Korean pilot study suggests.
A study by a team of US researchers has shown that an enhanced response to sham acupuncture (SA) may be present in fibromyalgia patients who are more sensitive to experimental-pressure stimuli.
A small case control study carried out in Canada has found evidence that electrodermal activity at acupuncture points (APs) differs between patients with current pain and pain-free controls.
Relief of pain is rewarding, according to a researchers from the USA. They demonstrated that treatments that relieve the unpleasant feeling of pain also activate reward circuits in the brain and thus reinforce behaviors that result in relief of pain.
Another Chinese study has found that acupuncture needle sensations of deqi and sharp pain are associated with different patterns of activations and deactivations in the brain.
Investigators from South Korea have found that the effect of acupuncture on the autonomic nervous system differs between the wood and metal points of the Pericardium and San Jiao channels.
An experimental study carried out in Korea has found that the skin at acupuncture points contains higher levels of the signaling molecule nitric oxide (NO) than surrounding areas.
An fMRI study from China has found changes in brain activity following electro-acupuncture (EA) at acupoints Yintang N-NH-3 and Baihui DU-20, both of which are used in the treatment of psychiatric disorders.
American scientists have investigated the brain circuitry involved in the antipruritic effects of acupuncture.
Experimental studies carried out in humans have shown that acupuncture triggers local increases in interstitial adenosine concentrations.
Acupuncture is now supported by a broad range of surveys of safety, clinical trials and basic science studies of physiological action.
Acupuncture is a cost-effective treatment modality, according to Korean authors who performed a systematic review of economic evaluations carried out alongside randomised controlled trials.
An Australian systematic review has found preliminary support for the use of acupuncture as a means to enhance exercise performance and post-exercise recovery.
Up until now, the placebo effect has seemed to make no sense…if people are able to heal themselves unaided, why does the body wait for a sugar pill to prompt recovery?
A review by an Italian author has examined the placebo effect in terms of the 'therapeutic ritual' created around the patient and how this plays a crucial role in the therapeutic outcome.
Blood perfusion at the acupoint Hegu L.I.-4 increases when the contralateral Hegu L.I.-4 is stimulated by acupuncture, according to Chinese team.
An international team, consisting of acupuncturists, doctors, patients, researchers and medical insurance company representatives has produced a consensus document aimed at those conducting comparative effectiveness research (CER) trials in acupuncture.
Acupuncture is cost-effective for the treatment of chronic pain, according to a systematic review carried out by UK researchers.


Ray Bradbury said a lot of cool things. Fahrenheit 451 is one of my favorite books, and it grows more relevant by the hour when it comes to the age of the infowar that we currently occupy. Some wise words for your Friday from r/quotesporn.
Skin – Wet the dye stained area of skin with warm water.


Background: Robot-assisted radical prostatectomy (RARP) has been associated with reduced bleeding and shorter hospital stays than open retropubic radical prostatectomy (RRP), but it is unclear whether these differences translate into shorter absence from work. Objective: To investigate short-and long-term rates of work disability following RARP and RRP. Design, setting, and participants: We conducted a nationwide population-based cohort study of 2571 men of working age treated with RARP or RRP between 2007 and 2009 identified in the National Prostate Cancer Register of Sweden. Information about physician-certified sick leave and disability pension was retrieved from the Swedish Social Insurance Agency through 2012. Outcome measurements and statistical analysis: We used Cox regression to calculate time to return to work (RTW, or duration of sick leave) after surgery and used generalised estimating equations to analyse days lost from work (because of sick leave and disability pension) after RTW. Results and limitations: Men treated with RARP returned to work after a median of 35 d, whereas the corresponding time for RRP was 48 d (p < 0.001). The difference was seen early; within the first month, men treated with RARP returned to work nearly four times faster than men treated with RRP (adjusted relative RTW rate 3.76; 95% confidence interval [CI], 3.04-4.66). During a median of 3.6 yr after return to work, men treated with RARP lost fewer days from work per person-year than men treated with RRP-12 d versus 15 d-but the association was not statistically significant (p = 0.10). The adjusted rate ratio was 1.08 (95% CI, 0.82-1.42). One limitation is the nonrandomised design of this study. Conclusions: RARP was associated with a faster RTW compared with RRP, but the surgical method did not influence long-term rates of work disability in terms of days lost from work after RTW. Patient summary: We compared disease-related absence from work between two surgical methods for the removal of the prostate. Robot-assisted surgery was associated with a faster return to work compared with open surgery but did not influence absence from work in a long-term perspective.


San Francisco, CA, July 11, 2018 – VIZ Media, LLC (VIZ Media),a premier company in the fields of publishing, animation distribution, and global entertainment licensing, delivers a memorable weekend of events, creator appearances, and more at Comic-Con International: San Diego 2018.
Comic-Con is one of the nation's largest comic book, science fiction, and pop culture events, and will take place July 19th–22nd at the San Diego Convention Center, in San Diego, CA. VIZ Media will be located in Booth #2813. Complete details on Comic-Con are available at: comic-con.org.
This year, VIZ Media welcomes Kohei Horikoshi, creator of the bestselling manga series MY HERO ACADEMIA. Horikoshi Sensei is scheduled to participate in a variety of events throughout Comic-Con including an official MY HERO ACADEMIA panel and VIZ Media's SHONEN JUMP panel. A pair of autograph sessions will also take place at VIZ Media's booth located in Comic-Con's Main Hall.
Attendees are also invited to catch special in-booth signing sessions from VIZ Media's acclaimed THE ART OF MAGIC: THE GATHERING series featuring lead writer James Wyatt and artist Tyler Jacobson.
VIZ Media will offer a wide range of convention exclusives and prestreet releases available for purchase in the retail area of the booth. Don't miss a variety of special poster giveaways for TOKYO GHOUL:RE and MY HERO ACADEMIA as well as other fun in-booth activities.
Join us as we talk about some of your favorite titles. We'll also be making announcements, showing off exclusive first looks at new titles, and giving away prizes!
Lead Magic: The Gathering writer James Wyatt and artist Tyler Jacobson visit the booth to sign posters for attendees!
Don't miss your last chance to meet writer James Wyatt and artist Tyler Jacobson.
See the creator of the world's most popular superhero manga! Horikoshi Sensei will talk about his manga and do a live drawing. He'll also have voice actors Justin Briner (Deku) and Christopher Sabat (All Might) at his side!
*Tickets will be distributed the morning of each signing session at the VIZ Media booth. Please visit VIZ Media's front desk for details.
VIZ Media's retail area of the booth will offer a variety of must-have Comic-Con exclusives, prestreet items, and more. Each purchase will come with a special VIZ Media 2018 limited edition tote bag offered in two distinct designs: MY HERO ACADEMIA and POKÉMON: SUN AND MOON.


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App profile
Limbo app review: a platform puzzle game with challenging levels
07 Jul 2015 Reviews
Limbo combines the elements of platform gaming and physics-based puzzler that gives mobile gamers a totally new adventure experience. It offers a dark environment, shadow characters, and frustratingly difficult puzzles. The game takes you to limbo where death always lurks in the corner. Developed by Playdead, this game requires the latest models of iPhone, iPad, and iPod Touch.
The best iPhone apps for spot the difference
Get for $3.99
Discover the Dark and Dangerous World of Limbo
Limbo belongs to the genre of puzzle-platformer where you explore a surreal world full of danger. More often than not, you will find your character in an agonizing situation where death is a sure thing. Playdead really recreates death and destruction quite impressively with morbid sound effects and equally horrifying graphics.
The best puzzle apps for iPad
Limbo is set in a dark black and white environment where shadows rule. There is a thin storyline which only sets the mood of the game. It says that the boy enters the world of limbo after failing to ascertain the fate of his sister. That's the storyline and you will now start your adventure.
The game takes you to different platform puzzles. You have to avoid deadly trees, falling objects, razor sharp blades, and many more. The game keeps you on your toes as it throws hazards after hazards on every level.
Unfortunately, the eerie horror-like atmosphere of the game takes a back seat because you will be too busy solving the puzzles. For example, one level takes you in a room with two boxes. Then, the room suddenly spins and the boxes will move threatening to crush you. You have to quickly arrange the boxes so you can avoid them. If you fail in this level, you will find the boy crushed and his blood splattered throughout the room.
There seems to be no end to the sufferings of the boy; but this is just part of the design and appeal of Limbo. The game challenges you to think quickly so you can survive the punishing obstacles.
Best Features of Limbo
Limbo offers impressive 2D graphics set in a dark background. The game is well-designed and can match the quality of other top-end platformers. It uses real physics to depict movements.
Best iphone and ipad apps
This game offers surreal display of gore and doom. For those who are fans of Temple of Doom, then Limbo is a worthy substitute when it comes to graphic display of destruction.
The game offers dozens of challenging levels. Each level forces you to think logically to solve the puzzles. You also need to be quick in order to avoid the hazards. The game progressively develops and it becomes more difficult as you overcome different levels.
Offers stunning 2D graphics.
With eerie soundtrack and realistic sound effects.
Uses real physics to display movements.
Offers dozens of challenging levels.
Offers different scenarios for solving puzzles.
With intuitive game controls.
Responsive gaming screen.
With thin storyline.
Depicts graphic violence and gore.
Limbo is an eerie game that is both challenging and scary. It offers extremely difficult puzzles that will keep you glued to your device for many hours. If you are a fan of physics-based puzzle-platformers, then Limbo is the perfect game for you.
Cherry is a hard-working writer and an aspiring web designer who can't go a day without using her favorite apps. She's always on the lookout for new apps and enjoys discovering ones that others have overlooked.


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Trump pardons Russia inquiry figures, Blackwater guards, Republicans
EFE News Services | Dec 23, 2020 | News | 3
(EFE).- President Donald Trump granted clemency to 20 people on Tuesday, including two convicted over the Russia inquiry, three former Republican congressmen implicated in corruption, and four Blackwater guards convicted in relation to the killing of Iraqi civilians.
The White House announced Trump granted full pardons to 15 people, and commuted all or part of the sentences of another five.
The highest-profile pardon is that of George Papadopoulos, who was a foreign affairs adviser to the Trump campaign in 2016 and who pleaded guilty to making false statements to federal officials during the Robert Mueller investigation into Russian meddling in the 2016 presidential election.
The outgoing president also pardoned Alex van der Zwaan, a lawyer who pleaded guilty to the same charge as Papadopoulos.
GOPUSA Editor's Note: Please keep in mind that this is a mainstream media story with the typical lean to the left. We publish the story for the purpose of informing our readers so don't let their choice of language trigger you. Conservatives are not soft like millennials.
The three former Republican congressmen granted clemency are Duncan Hunter of California, Chris Collins of New York and Steve Stockman of Texas.
Stockman was serving a 10-year prison sentence for fraud and money laundering, but the rest of his sentence was commuted. Collins was serving a 26-month sentence for making false statements to the FBI and conspiring to commit securities fraud, and was granted a full pardon. Hunter was soon to begin serving 11 months for misuse of campaign funds, was also granted full pardon.
To add a little perspective let's remember a far more famous pardon:
Bill Clinton's pardon of fugitive Marc Rich continues to pay big
On Jan. 20, 2001, his last day in office, Bill Clinton issued a pardon for international fugitive Marc Rich. It would become perhaps the most condemned official act of Clinton's political career. A New York Times editorial called it "a shocking abuse of presidential power." The usually Clinton-friendly New Republic noted it "is often mentioned as Exhibit A of Clintonian sliminess."
Of the four Blackwater contractors granted full pardons, Nicholas Slatten had been sentenced to life in prison for his role in the killing of 17 Iraqi civilians in Baghdad in 2007.
Recently, Trump pardoned his former national security adviser Michael Flynn, who had pleaded guilty to lying to the FBI about his Russian contacts.
The New York Times reported that "a tabulation by the Harvard Law School professor Jack Goldsmith found that of the 45 pardons or commutations that Mr. Trump had granted up until Tuesday, 88 percent aided someone with a personal tie to the president or furthered his political aims."
The pardons are an indicator that even though Trump has not acknowledged his November election defeat to Democrat Joe Biden, he is aware that he will have to leave power on Jan. 20. EFE
© 2020 EFE News Services (U.S.) Inc.
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capricorn1	on 7:17 am December 23, 2020 at 7:17 am
out going president? when did he concede?
ltuser	on 3:45 pm December 23, 2020 at 3:45 pm
NO idea. B UT the longer he waits to do something, the MORE IT DOES LOOK like he's given up…
ltuser	on 1:42 am December 24, 2020 at 1:42 am
I am wondering, WHY HE DIDN'T take, to pardoning Rittenhouse!!


The Ministry of Interior with the Centre de Diagnostic et de Recherché en Medecine Moleculaire (CDRMM) one of Africa's leading scientific molecular center in Senegal on Wednesday 19th July signed a Fourty-Eight Million dollar project in revolutionizing the art of fighting crimes in the Gambia at a ceremony held at its office in Bertil Harding Highway.
The project look to ensure proper investigations backed with scientific proofs as well address DNA, balestic, biological investigations, cyber investigation, documents examination that will be conducted at this center.
The Minister of Interior Mai Ahmad Fatty said the Gambia has been struggling for the past fifty years to achieve significance in the fight against crime, adding that half a century after independence this dream has now reached not only Gambia but by the sub region.
The completion of the project he noted will be a greater mile stone in integrating police services, potential to attract international funding for police forces through cooperation with international security forces and organizations to provide efficient crime prevention mechanisms as well, said Fatty.
This project he said is a fulfillment of a promise by the new government to keep the Gambia safe and secure as well as to provide a judicial system that is second to known in Africa, adding that a judicial system that is characterize by free and fair evidential processes and a criminal jurisprudence that would be the envy of Africa.
The project he said went through several processes for months of consultation between the various stakeholders and will be number one in the Africa sub region and one of the best in the world.
The relation between Senegal and the Gambia he said is one of the best in African countries and we are proud that this project is pioneered by a Senegalese showing Africans can do things on their own by protecting its sovereignty and independence.
The future of fighting crimes Fatty noted is in technology and the Gambia he noted has difficulties with proving cases before courts, adding that Police investigation have been fraud with inefficiency and ineffectiveness as a result of not having the tools to produce and provide the evidences necessary to secure conviction before courts of law.
Fatty remarked that having this facility in the Gambia will cement a truthful international security organization such as Interpol, FPI, and CIA and will create direct and indirect employment for the Gambian citizens.
He however extends thanks and appreciation of the government of the Gambia to the sister republic of Senegal for the confidence and support noting that this project will have a positive impact in addressing security and health of the Gambia.
Dr. Cheikh Tidane Sy director general Centre de Diagnostic et de Recherché en Mececine Moleculaire CDRMM said the project will cover a variety of impacts in the Gambia including security as without security no development will be achieve. He said that security now goes with scientific research and technology, adding that security is well represented under this project as it will allow investigations and raise alarm for crimes to serve better justice.
The other impacts of this project he said will be in the medical and scientific field. The same approach that is used in hospitals he revealed will be used to detect early diagnostic diseases and once this project is implemented in the Gambia, no more need for patients to travel outside the country for medical services .Sy thanked the government of the Gambia and the ambassador of Senegal to the Gambia for the collaboration, support and tireless involvement in the project.
The Minister of Tourism and Culture Hamat Bah said the signing of this project marks a history in the political history of the new government of the Gambia as the barrow administration wants to prove a case to the Gambian people, adding that the new government will make sure that Gambians benefit as much as they expected. This project he noted will further deal with crimes but will also address medical tourism matters as it will address the problem of Gambians travelling abroad for medical attention.
Professor Ndiaye Senegal Ambassador to the Gambia said that Gambia and Senegal are two special countries in the region as there is no comparison between the two countries in the world.
He said the relationship between Senegal and the Gambia did not start today, adding that the project will go a long way in increasing the force of the judicial systems and the wellbeing of the medical services in the Gambia. The relationship between the Gambia and Senegal he said is healthy and that the two countries have been benefitting one another.
Ndiaye however urged both citizens to jealously guard the cordial relationship that exist between the two countries.


All about the Trails of the Swiss Alps.
Located in the heart of the Bernese Oberland, where the great arc of the Alps culminates in some of its highest and most spectacular peaks, Kandersteg is a walker's paradise. This area epitomises the 'chocolate-box' Alps: lush green meadows bright with wildflowers, against a backdrop of rugged, wild mountains, topped with glistening glaciers. We stay at the charming Hotel Alfa Soleil, owned by the Seiler family, and discover the best of the region offers, including towering cliffs, a hidden valley headed by a glacier, a sheer gorge carved out by glacial meltwaters and more.
Your holiday begins at the Alfa Soleil hotel in Kandersteg. You should aim to arrive late morning to early afternoon for a pre-dinner briefing from your walking leader.
Day 2 Walk along the Kandertal Valley to the world famous Blausee Lake.
A gentle walk to start our week, this varied and peaceful route takes us along the bottom of the upper Kandertal Valley to the world-famous Blausee – Blue lake. Lunch can be taken at the lake, followed by an optional swim. The afternoon is free to enjoy the hotel's facilities and the wonderful views of the valley.
Day 3 Ride on the Sunnbüel cable car to the start of the walk. Cross the cantonal border into the Valais; hike to the northern end of the Daubensee.
In the morning we take ride on the Sunnbüel cable car to the start of the walk. We then cross the cantonal border into the Valais and walk to the Schwarenbach Hotel, where even Lenin and Picasso have signed the visitors' book! From here we walk to the northern end of the Daubensee (sometimes still partially frozen) to the famous high Gemmi Pass (2322m), a major transalpine route first used by the Celts. The extraordinary panorama of the Valais Alps to the south includes the Matterhorn (4478m) and Monte Rosa on the Italian border.
Day 4 Short ride on the Oeschinensee cable car followed by a circular walk to the Heuberg viewpoint.
Day 6 Ride on the Allmenalp cable car followed by a walk along the paths into the Ueschinen valley; descend to Kandersteg.
It's a gentle start today with a spectacular cable car ride to ascend to the Allmenalp (1725m), a colourful meadow brimming with rural mountain charm. There are stunning views across the valley to the Oeschinensee with its magnificent turquoise-coloured waters and towering cliffs adorned with waterfalls. From the summit the 360° view is nothing less than spectacular, with 12 mountain peaks above 3000m in close proximity, and occasionally eagles frequent the adjoining peaks.
Day 7 Walk through the Gasterntal Valley to the Kanderfirn Glacier; return with local bus.
This is a day full of thrilling experiences, starting with an exciting 20 minute minibus ride along a narrow road literally carved out of the rock wall. The Gasterntal is Kandersteg's 'hidden valley', dozens of snow-fed waterfalls thundering down the sheer rock faces carved out by a gigantic glacier 10,000 years ago. From this romantic nature reserve, and with the mighty rock walls of the Tatlishorn and Atels towering up on the right, we meander through woods and open meadows; look out for lady slippers, fire lilies and more than 25 varieties of orchid before we return to the hotel with a local bus.
End Kandersteg. Following breakfast we say our goodbyes and take our train to the airport.
The Hotel Alfa Soleil is a traditional, cosy and family-run hotel on the edge of the village (only 5 minutes' walk from the centre) with a wonderful view over fields to the mountains behind. It has been owned by the same family for three generations. All rooms are en suite and have a television, telephone, hairdryer and wifi.
Single rooms are available for a supplement – subject to availability.
With such a mixture of language and tradition co-existing in such a small country, it is no wonder that there are so many regional dishes. In the Bernese Oberland, Austrian and German influences predominate; gastronomic local delicacies include geschnetzeltes Kalbsfleisch (veal in a rich cream sauce), Kandersteger Bratwurst au Mad re-Jus (roast sausage with a Madeira sauce), Lammbraten gefüllt mit Kraütern (herb roasted lamb) and smoked pork with sauerkraut, and a selection of colorful salads. Typical desserts include Caramel-Birne mit Rum-Sabayon (caramelized pear with a rum sabayon) and Aprikosen-Symphonie (apricot gateau).