text:
\section{Introduction}

Elastic $\pi\pi$ scattering is among the simplest processes governed by Quantum Chromodynamics (QCD) in its nonperturbative regime, yet it contains a rich amount of interesting physics ranging from  spontaneous chiral symmetry breaking to low-lying meson resonances. First-principle studies of the scattering process thus play a critical role in refining our understanding of the properties of strong interactions at low energies. In particular, there has been a continuous effort since the 1990s in the simulation of the $\pi\pi$ scattering using lattice QCD that covers all isospin channels~\cite{Sharpe:1992pp,Kuramashi:1993ka,Gupta:1993rn,Fukugita:1994ve,
	Aoki:2002in,Du:2004ib,Chen:2005ab,Li:2007ey,Beane:2007xs,Aoki:2007rd,Feng:2009ij,Feng:2010es,Dudek:2010ew,Lang:2011mn,Aoki:2011yj,Beane:2011sc,Dudek:2012gj,Pelissier:2012pi,
	Dudek:2012xn,Fu:2013ffa,Feng:2014gba,Bai:2015nea,Helmes:2015gla,Wilson:2015dqa,Bali:2015gji,Bulava:2016mks,Guo:2016zos,Briceno:2016mjc,
	Liu:2016cba,Alexandrou:2017mpi}.

Studies of  $\pi\pi$ scattering in lattice QCD involve the computation of four-point correlation functions. Quantized energy levels are fixed from the time-dependence of these correlation functions, and the scattering observables are extracted from these volume-dependent energies using
L\"uscher's method~\cite{Luscher:1986pf,Luscher:1990ux}. Pion-pion scattering amplitudes for total isospin $I=1$ and $I=2$  are relatively easy to compute.
However, it is well-known that the lattice computation of $I=0$ $\pi\pi$ scattering is significantly more challenging due to the existence of quark contraction diagrams in which the quark propagators begin and end at the same temporal point on the lattice: These diagrams are known as ``disconnected diagrams" for historical reasons. In particular, there are the so-called vacuum-type contractions, in which the quark lines in the source and those in the sink are separated from each other. Such contractions possess a low signal-to-noise ratio, and usually require special algorithms (such as the stochastic LapH quark smearing method~\cite{Peardon:2009gh,Morningstar:2011ka} that provides all-to-all propagators) that are computationally demanding to achieve good statistical precision. For instance, in Ref.~\cite{Liu:2016cba} it is clear that the disconnected correlation functions suffer from larger noise than the connected ones, while Ref.~\cite{Briceno:2016mjc} achieved better precision using a larger basis of interpolating fields but their computation is significantly more expensive.

In the meantime, corresponding theoretical tools have been developed in an attempt to model the properties of disconnected diagrams and eventually to serve as a gauge for any possible future improvement of relevant lattice computational techniques. They generally involve extensions of the flavor sector in QCD such that the contraction diagrams of interest can be singled out by choosing appropriately the quark flavors in the external scattering states. The first attempt of such a kind was taken in Ref.~\cite{Guo:2013nja} that made use of an SU(4) chiral perturbation theory (ChPT) to describe the separation of different contraction diagrams at leading order (LO) in the chiral expansion. Follow-up works were carried out in Ref.~\cite{Acharya:2017zje} based on the SU$(4|2)$ partially-quenched chiral perturbation theory (PQChPT)~\cite{Bernard:1993sv,Sharpe:1999kj,Sharpe:2000bc,Sharpe:2001fh,Bernard:2013kwa,Sharpe:2006pu,Golterman:2009kw} that fully preserves the internal dynamics of the ordinary two-flavor ChPT. Scattering amplitudes corresponding to each individual contraction were computed to the next-to-leading order (NLO), and analytic expressions for static quantities such as scattering lengths were derived. The problem with these calculations, however, is that the results were obtained in a field theory in the infinite volume, whereas what one obtains on the lattice are always discrete energy levels because of the  finite volumes. It is therefore unclear how the energy levels extracted from the lattice correlation functions of definite contractions are related to the quantities (such as scattering lengths) derived in Ref.~\cite{Acharya:2017zje}.

\begin{figure}[tb]
	\begin{centering}
		\includegraphics[scale=0.2]{D}\hfill
		\includegraphics[scale=0.2]{C}\hfill
		\includegraphics[scale=0.2]{R}\hfill
		\includegraphics[scale=0.2]{V}
		\par\end{centering}
	\caption{\label{fig:contraction}(From left to right) direct (D), crossed (C), rectangular (R) and vacuum (V) contractions in a generic $\pi\pi$ scattering amplitude.}
\end{figure}
We provide in this paper a satisfactory solution to the above-mentioned problem by combining PQChPT with the single-channel L\"{u}scher's formula~\cite{Luscher:1986pf,Luscher:1990ux}. First, we introduce auxiliary valence flavors, which are degenerate with the sea up and down quarks (we consider the isospin symmetric case), using the graded group SU$(4|2)$. This enables us to have enough channels to separate various types of contractions, which include direct (D), crossed (C), rectangular (R) and vacuum (V) types, see Fig.~\ref{fig:contraction}. Because of the degeneracy of all the sea and valence quarks in the setting of our theory, by diagonalizing a matrix for multi-channel $S$-wave scattering amplitudes in SU$(4|2)$, we obtain four linear combinations of scattering amplitudes that are effectively single-channel. Two of them consist of only the connected contractions (D and C) and the other two involve the R and V contractions as well. The single-channel L\"{u}scher's formula then provides the relation between the discrete energy level of each combination in a finite volume and the corresponding low-energy scattering observables, such as the scattering length and effective range, in the infinite volume.  The latter are also calculated in PQChPT up to NLO and are expressed in terms of the low-energy constants (LECs) of the theory. Since the discrete energies involving only the connected contractions are easily obtainable on the lattice, they can be used to fix the poorly-known ``unphysical" LECs in the theory, which have so far only been fitted to the meson masses and decay constants in the lattice calculations of Ref.~\cite{Boyle:2015exm}. After doing so, the two remaining energy levels involving disconnected diagrams can be predicted in PQChPT. The comparison between the energy levels extracted from lattice correlation functions and the theory predictions then serves as a powerful gauge for the accuracy of the former, because the latter are model-independent results that must be obeyed (up to corrections from higher orders in the chiral power counting and from finite lattice spacing) by any lattice setup that satisfies unitarity.

This paper is organized as follows. In Section~\ref{sec:Luescher}, we review the single-channel L\"{u}scher's formula that relates the energy levels in a finite volume to the scattering observables in the infinite volume. In Section~\ref{sec:effective}, we derive four linear combinations of the $S$-wave $\pi\pi$ amplitudes that comprise effective single-channel scattering amplitudes in SU$(4|2)$. The PQChPT results of their respective scattering length and effective range are summarized in Section~\ref{sec:ere}. In Section~\ref{sec:numerics}, we make use of existing lattice data on connected diagrams in $\pi\pi$ scattering to fit a subset of the so-called ``unphysical" LECs in SU$(4|2)$, and thus give our predictions of the two remaining energy levels that depend on the disconnected diagrams. We sum up with conclusions in Section~\ref{sec:conclusion}.

\section{Single-Channel L\"{u}scher's Formula}
\label{sec:Luescher}

We start by reviewing  L\"{u}scher's formula for single-channel partial-wave amplitudes. A single-channel scattering amplitude $T(s,t,u)$ between two non-identical particles in the center of mass (CM) frame can be expanded in terms of its partial-wave amplitudes $\{T_l(E)\}$,
\begin{equation}
T(s,t,u)=\sum_{l=0}^{\infty}(2l+1)T_l(E)P_l(\cos\theta), \label{eq:PWE}
\end{equation}
where the $P_l(\cos\theta)$ are Legendre polynomials, $E=\sqrt{s}$ is the CM energy, and $\theta$ is the scattering angle. The partial-wave amplitudes satisfy the following unitarity relation:
\begin{equation}
\mathrm{Im}T_l(E)=\Theta(E-m_1-m_2)\frac{p}{8\pi E}|T_l(E)|^2,\label{eq:unitarity}
\end{equation}
with $p$ the CM momentum. This allows us to parameterize the partial-wave amplitudes in terms of their phase shift $\delta_l$ as
\begin{equation}
T_l(E)=\frac{8\pi E}{p\cot\delta_l(E)-ip}.\label{eq:phaseshift}
\end{equation}
From now on we shall restrict ourselves to the special case where $m_1=m_2=M_\pi$, and thus $p=\sqrt{E^2-4M_\pi^2}/2$.

Lattice studies of scattering processes are performed in finite spatial volumes characterized by the lattice size $L$. Four-point correlation functions are computed, and the measured outcomes are discrete energy levels which correspond to the poles of the finite-volume $T$-matrix.
L\"{u}scher \cite{Luscher:1986pf,Luscher:1990ux} derived a relation between the volume dependence of two-particle energies in the finite volume and the partial-wave amplitudes in the infinite volume. We consider here the $S$-wave case. The relation is recast into an equivalent but more understandable form in Ref.~\cite{Doring:2011vk} which we shall adopt here\footnote{Notice that the right-hand side of our Eq. \eqref{eq:Luscher} has a sign difference with that in Ref.~\cite{Doring:2011vk} because our partial-wave amplitudes are defined to have a positive imaginary part above threshold.},
\begin{equation}
T_0^{-1}(E_0)=-\Delta G(E_0),\label{eq:Luscher}
\end{equation}
which holds for the eigenenergies of the two-particle system in a finite volume, denoted by $E_0$. Here $\Delta G(E)=\tilde{G}(E)-G(E)$, where
\begin{equation}
G(E)=\int_{|\vec{q}|<\Lambda}\frac{d^3q}{(2\pi)^3\omega_\pi}\frac{1}{E^2-4\omega_\pi^2+i\varepsilon},\qquad
\tilde{G}(E)=\frac{1}{L^3}\sum_{|\vec{q}|<\Lambda}\frac{1}{\omega_\pi}\frac{1}{E^2-4\omega_\pi^2}
\end{equation}
with $\omega_\pi=\sqrt{M_\pi^2+\vec{q}^2}$, are the infinite-volume and finite-volume versions of the $s$-channel 2-point scalar loop function, both regularized at the ultraviolet (UV) end by a hard cutoff $\Lambda$. Notice that the discrete loop momentum $\vec{q}$ in $\tilde{G}(E)$ is given by $\vec{q}=(2\pi/L)\vec{n}$ with $\vec{n}\in \mathbb{Z}^3$. Also, the function $\Delta G(E)$ is UV finite and insensitive to the cutoff $\Lambda$ as long as $\Lambda\gg M_\pi$, as the UV part gets cancelled between $G$ and $\tilde G$.

When $p$ is small, one could perform the following effective-range expansion to $p\cot\delta_0$,\footnote{The scattering length defined here has a sign difference to that in Ref.~\cite{Helmes:2015gla}.}
\begin{equation}
p\cot\delta_0=-\frac{1}{a_0}+\frac{1}{2}r_0p^2+\mathcal{O}(p^4).\label{eq:EFexpand}
\end{equation}
The right-hand side is valid both above and below threshold, for the latter case we simply have $p^2<0$. The quantities $a_0$ and $r_0$ are known as the ($S$-wave) scattering length and effective range, respectively. Substituting Eqs.~\eqref{eq:phaseshift} and \eqref{eq:EFexpand} into Eq.~\eqref{eq:Luscher} yields
\begin{equation}
-\frac{1}{a_0}+\frac{1}{2}r_0p^2+...=-8\pi E_0\Delta G(E_0)+ip\,,\label{eq:central}
\end{equation}
which is the starting point of our finite-volume analysis. Notice that the right side is always real: Above threshold, the imaginary part of $-8\pi E_0\Delta G(E_0)$ cancels with $ip$, whereas below threshold one performs the analytic continuation $p=i\sqrt{4M_\pi^2-E_0^2}/2$ \cite{Doring:2011vk}. This equation serves two purposes: (1) If $E_0$ could be extracted from lattice for several values of $L$, then one could use them to fit the scattering length and effective range $\{a_0,r_0\}$ to determine the unknown theoretical parameters (LECs) within these quantities, and (2) if the values of $\{a_0,r_0\}$ could be calculated from theory, then one could use Eq.~\eqref{eq:central} to predict the discrete energy $E_0$, which can be used to test the accuracy of the corresponding lattice calculation.

\section{Effective Single-Channel $S$-wave Amplitudes}
\label{sec:effective}

The $I=0$ $\pi\pi$-scattering amplitude consists of four types of contraction diagrams as depicted in Fig.~\ref{fig:contraction}: D, C, R and V as well as their $t\leftrightarrow u$ crossings. As described in the Introduction, D and C are connected diagrams that could be obtained rather easily on the lattice, whereas R needs more efforts and V is the most challenging. Each contraction can be rigorously defined in terms of ``physical" scattering amplitudes between pseudo-Nambu--Goldstone bosons (pNGBs) in a degenerate SU$(4|2)$ partially-quenched QCD (PQQCD), for which we refer the readers to Ref.~\cite{Acharya:2017zje} for details,
\begin{eqnarray}
T^D(s,t,u)&\equiv &T_{(u\bar{d})(j\bar{k})\rightarrow (u\bar{d})(j\bar{k})}(s,t,u) ,\nonumber\\
T^C(s,t,u)&\equiv &T_{(u\bar{d})(j\bar{k})\rightarrow (u\bar{k})(j\bar{d})}(s,t,u) ,\nonumber\\
T^R(s,t,u)&\equiv &T_{(j\bar{u})(u\bar{k})\rightarrow (j\bar{d})(d\bar{k})}(s,t,u) ,\nonumber\\
T^V(s,t,u)&\equiv &T_{(u\bar{d})(d\bar{u})\rightarrow (j\bar{k})(k\bar{j})}(s,t,u),
\end{eqnarray}
where $u,d,j$ and $k$ refer to the quark flavors in the SU$(4|2)$ theory, and $(u\bar d)$ and so on refer to the corresponding pNGBs.

None of the amplitudes above describes a single-channel scattering in SU$(4|2)$. However, it is straightforward to construct linear combinations of amplitudes that are effectively single-channel. To demonstrate the mechanism, let us consider a multi-channel scattering process that involves only the D, C and R contractions. This can be done by including only five channels as follows: $(j\bar{u})(u\bar{k})$, $(j\bar{d})(d\bar{k})$, $\pi^0(j\bar{k})$, $\tilde{\eta}(j\bar{k})$ and $\tilde{\phi}(j\bar{k})$, where
\begin{equation}
\pi^0=\frac{1}{\sqrt{2}}\left(u\bar{u}-d\bar{d}\right),\:\:\:
\tilde{\eta}=\frac{1}{\sqrt{6}}\left(u\bar{u}+d\bar{d}-2j\bar{j}\right),\:\:\:
\tilde{\phi}=\frac{1}{\sqrt{12}}\left(u\bar{u}+d\bar{d}+j\bar{j}-3k\bar{k}\right)
\end{equation}
in the flavor space. The $S$-wave amplitudes in this coupled-channel system are then a $5\times 5$ symmetric matrix,
\begin{equation}
T_{0}=\left(\begin{array}{ccccc}
T_{0}^{D}+T_{0}^{R} & T_{0}^{R} & \frac{1}{\sqrt{2}}T_{0}^{C} & \frac{1}{\sqrt{6}}\left(T_{0}^{C}-2T_{0}^{R}\right) & \frac{1}{2\sqrt{3}}\left(T_{0}^{C}-2T_{0}^{R}\right)\\
T_{0}^{R} & T_{0}^{D}+T_{0}^{R} & -\frac{1}{\sqrt{2}}T_{0}^{C} & \frac{1}{\sqrt{6}}\left(T_{0}^{C}-2T_{0}^{R}\right) & \frac{1}{2\sqrt{3}}\left(T_{0}^{C}-2T_{0}^{R}\right)\\
\frac{1}{\sqrt{2}}T_{0}^{C} & -\frac{1}{\sqrt{2}}T_{0}^{C} & T_{0}^{D} & 0 & 0\\
\frac{1}{\sqrt{6}}\left(T_{0}^{C}-2T_{0}^{R}\right) & \frac{1}{\sqrt{6}}\left(T_{0}^{C}-2T_{0}^{R}\right) & 0 & T_{0}^{D}+\frac{2}{3}\left(T_{0}^{C}+T_{0}^{R}\right) & \frac{1}{3\sqrt{2}}\left(2T_{0}^{R}-T_{0}^{C}\right)\\
\frac{1}{2\sqrt{3}}\left(T_{0}^{C}-2T_{0}^{R}\right) & \frac{1}{2\sqrt{3}}\left(T_{0}^{C}-2T_{0}^{R}\right) & 0 & \frac{1}{3\sqrt{2}}\left(2T_{0}^{R}-T_{0}^{C}\right) & T_{0}^{D}+\frac{5}{6}T_{0}^{C}+\frac{1}{3}T_{0}^{R}
\end{array}\right),
\end{equation}
which can be diagonalized through an orthogonal transformation to give
\begin{equation}
T'_0=UT_0U^T=\mathrm{diag}\left(T_0^D+T_0^C,T_0^D+T_0^C,T_0^D+T_0^C,T_0^D-T_0^C,T_0^D-\frac{1}{2}T_0^C+3T_0^R\right).
\end{equation}
Furthermore, since all pNGBs are degenerate, each of the diagonal entries of $T'_0$ (among which only three of them are independent) is by itself an effective single-channel $S$-wave amplitude that satisfies the unitarity relation in Eq.~\eqref{eq:unitarity}, and can be parameterized according to
Eq.~\eqref{eq:phaseshift}.

The treatment above can be generalized to include the vacuum contraction, but then it will involve the diagonalization of a $12\times 12$ matrix. In fact, such a generalization is not necessary because we already know the fourth linear combination. It is just the $I=0$ $S$-wave amplitude,\footnote{While the lattice community usually adopts the opposite sign convention for the crossed and rectangular contractions: $T_0^{I=0}=2T_0^D+T_0^C-6T_0^R+3T_0^V$, our sign convention is more natural in terms of the interpretation of different contractions as physical scattering amplitudes in SU$(4|2)$.}
\begin{equation}
T_0^{I=0}=2T_0^D-T_0^C+6T_0^R+3T_0^V.
\end{equation}
The only complication is that we need to rescale it by a factor 1/2 to match the normalization in Eq. \eqref{eq:unitarity}, due to the fact that $T_0^{I=0}$ involves identical particles. Up to this point, we have successfully constructed four linear combinations of D, C, R and V that give rise to effective single-channel $S$-wave amplitudes in SU$(4|2)$:
\begin{eqnarray}
T_0^\alpha&=&T_0^D+T_0^C,\nonumber\\
T_0^\beta&=&T_0^D-T_0^C,\nonumber\\
T_0^\gamma&=&T_0^D-\frac{1}{2}T_0^C+3T_0^R,\nonumber\\
T_0^\delta&=&T_0^D-\frac{1}{2}T_0^C+3T_0^R+\frac{3}{2}T_0^V,
\end{eqnarray}
among which $T_0^\alpha$ and $T_0^\delta$ correspond (up to a constant factor of 2) to the $I=2$ and $I=0$ $\pi\pi$ scattering amplitude, respectively. The implications of this construction on lattice are as follows. Suppose $\tilde{C}^{I=0}(\tau)=2\tilde{C}^D(\tau)-\tilde{C}^C(\tau)+6\tilde{C}^R(\tau)+3\tilde{C}^V(\tau)$ is the lattice correlation function used to extract the discrete energy levels in the $I=0$ $\pi\pi$ scattering, then the following linear combinations of $\tilde{C}^X(\tau)$ ($X=D,C,R,V$) will decay as a single exponential function at large Euclidean time $\tau$:
\begin{eqnarray}
\tilde{C}^\alpha(\tau)\equiv 2\tilde{C}^D(\tau)+2\tilde{C}^C(\tau)&\sim &\exp\{-E_0^\alpha\tau\},\nonumber\\
\tilde{C}^\beta(\tau)\equiv 2\tilde{C}^D(\tau)-2\tilde{C}^C(\tau)&\sim &\exp\{-E_0^\beta\tau\},\nonumber\\
\tilde{C}^\gamma(\tau)\equiv 2\tilde{C}^D(\tau)-\tilde{C}^C(\tau)+6\tilde{C}^R(\tau)&\sim &\exp\{-E_0^\gamma\tau\},\nonumber\\
\tilde{C}^\delta(\tau)\equiv 2\tilde{C}^D(\tau)-\tilde{C}^C(\tau)+6\tilde{C}^R(\tau)+3\tilde{C}^V(\tau)&\sim & \exp\{-E_0^\delta\tau\},
\label{eq:Edef}
\end{eqnarray}
where the energy levels $E^i_0$ ($i=\alpha,\beta,\gamma,\delta$) correspond to the poles of the finite-volume $T_0^i(E)$, and thus satisfy the single-channel L\"{u}scher formula in Eq.~\eqref{eq:Luscher}. We wish also to point out the group theoretical origin of the four linear combinations. They are nothing but elements of the four irreducible representations resulting from the symmetric product of two 15-plets (that represent the two pNGBs) in SU(4)~\cite{Sharpe:1992pp},
\begin{equation}
(\textbf{15}\otimes\textbf{15})_\mathrm{symm}=\textbf{1}\oplus\textbf{15}\oplus\textbf{20}\oplus\textbf{84}~.
\end{equation}

\section{PQChPT Predictions of Scattering Lengths and Effective Ranges}
\label{sec:ere}

The expansion of $\mathrm{Re}\left[\left(T_0^i(E)\right)^{-1}\right]$ around the two-pion threshold gives the scattering length $a_0^i$ and the effective range $r_0^i$ for $i=\alpha,\beta,\gamma,\delta$. The PQChPT predictions of these quantities up to NLO can be obtained through simple manipulations of the results in Ref.~\cite{Acharya:2017zje}. The outcomes are \footnote{We shall take this opportunity to point out that the analytic expression for the $I=0$ effective range given in Appendix A of Ref.~\cite{Liu:2016cba} is incorrect.}
\begin{eqnarray}
a_0^\alpha&=&\frac{M_\pi}{16\pi F_\pi^2}\left[1+\frac{M_\pi^2}{\pi^2F_\pi^2}\left(-\frac{\bar{l}_1}{12}-\frac{\bar{l}_2}{6}+\frac{\bar{l}_3}{32}+\frac{\bar{l}_4}{8}-\frac{1}{32}\right)\right],\nonumber\\
r_0^\alpha&=&\frac{48\pi F_\pi^2}{M_\pi^3}\left[1+\frac{M_\pi^2}{\pi^2F_\pi^2}\left(\frac{\bar{l}_1}{12}+\frac{\bar{l}_2}{18}-\frac{7\bar{l}_3}{96}-\frac{\bar{l}_4}{8}+\frac{31}{288}\right)\right],\label{eq:aralpha}
\end{eqnarray}
\begin{eqnarray}
a_0^\beta&=&-\frac{M_\pi}{16\pi F_\pi^2}\left[1-4\mu_\pi+\frac{M_\pi^2}{F_\pi^2}\left(\frac{\bar{l}_1}{12\pi^2}+\frac{\bar{l}_2}{6\pi^2}-\frac{\bar{l}_3}{32\pi^2}-\frac{\bar{l}_4}{8\pi^2}-96L_0^\mathrm{PQ,r}\right.\right.\nonumber\\
&&\left.\left.-32L_3^\mathrm{PQ,r}+32L_5^\mathrm{PQ,r}-32L_8^\mathrm{PQ,r}-\frac{3}{32\pi^2}\right)\right],\nonumber\\
r_0^\beta&=&-\frac{48\pi F_\pi^2}{M_\pi^3}\left[1+\frac{100}{9}\mu_\pi+\frac{M_\pi^2}{F_\pi^2}\left(-\frac{\bar{l}_1}{12\pi^2}-\frac{\bar{l}_2}{18\pi^2}+\frac{7\bar{l}_3}{96\pi^2}+\frac{\bar{l}_4}{8\pi^2}+\frac{160}{3}L_0^\mathrm{PQ,r}\right.\right.\nonumber\\
&&\left.\left.+32L_3^\mathrm{PQ,r}-\frac{160}{3}L_5^\mathrm{PQ,r}+\frac{224}{3}L_8^\mathrm{PQ,r}+\frac{61}{288\pi^2}\right)\right],
\end{eqnarray}
\begin{eqnarray}
a_0^\gamma&=&-\frac{7M_\pi}{32\pi F_\pi^2}\left[1-\frac{18}{7}\mu_\pi+\frac{M_\pi^2}{F_\pi^2}\left(\frac{\bar{l}_1}{42\pi^2}+\frac{\bar{l}_2}{21\pi^2}-\frac{\bar{l}_3}{112\pi^2}-\frac{\bar{l}_4}{28\pi^2}+\frac{144}{7}L_0^\mathrm{PQ,r}\right.\right.\nonumber\\
&&\left.\left.+\frac{48}{7}L_3^\mathrm{PQ,r}+\frac{48}{7}L_5^\mathrm{PQ,r}+\frac{48}{7}L_8^\mathrm{PQ,r}+\frac{5}{56\pi^2}\right)\right],\nonumber\\
r_0^\gamma&=&-\frac{288\pi F_\pi^2}{49M_\pi^3}\left[1-\frac{110}{63}\mu_\pi+\frac{M_\pi^2}{F_\pi^2}\left(\frac{\bar{l}_1}{126\pi^2}+\frac{17\bar{l}_2}{189\pi^2}+\frac{25\bar{l}_3}{1008\pi^2}-\frac{\bar{l}_4}{84\pi^2}\right.\right.\nonumber\\
&&\left.\left.+\frac{592}{21}L_0^\mathrm{PQ,r}+\frac{496}{21}L_3^\mathrm{PQ,r}+\frac{16}{7}L_5^\mathrm{PQ,r}-\frac{400}{21}L_8^\mathrm{PQ,r}-\frac{25}{189\pi^2}\right)\right],\label{eq:argamma}
\end{eqnarray}
\begin{eqnarray}
a_0^\delta&=&-\frac{7M_\pi}{32\pi F_\pi^2}\left[1+\frac{M_\pi^2}{\pi^2F_\pi^2}\left(\frac{5\bar{l}_1}{84}+\frac{5\bar{l}_2}{42}-\frac{5\bar{l}_3}{224}+\frac{\bar{l}_4}{8}+\frac{5}{32}\right)\right],\nonumber\\
r_0^\delta&=&-\frac{288\pi F_\pi^2}{49M_\pi^3}\left[1+\frac{M_\pi^2}{\pi^2F_\pi^2}\left(\frac{11\bar{l}_1}{84}+\frac{43\bar{l}_2}{378}+\frac{125\bar{l}_3}{2016}-\frac{\bar{l}_4}{8}-\frac{479}{864}\right)\right],
\label{eq:ardelta}\end{eqnarray}
where $\mu_\pi=\left(M_\pi^2/(32\pi^2 F_\pi^2)\right)\ln(M_\pi^2/\mu^2)$, with $\mu$ the renormalization scale and $M_\pi,F_\pi$ the scale-independent physical pion mass and decay constant ($F_\pi^{\mathrm{phy}} = 92.1$ MeV). The constants $\{\bar{l}_1\}$ and $\{L_i^\mathrm{PQ,r}\}$ are known as the scale-independent physical and scale-dependent unphysical LECs in SU$(4|2)$ PQChPT, respectively.
The former ones enter  processes in the real world, but their values are $M_\pi$-dependent: $\bar{l}_i(M_\pi)=\bar{l}_i^\mathrm{phy}-\ln(M_\pi^2/M_{\pi,\mathrm{phy}}^2)$ where $M_{\pi,\mathrm{phy}}\approx 138$ MeV is the physical pion mass. The latter ones are $M_\pi$-independent, but they can only show up either when unphysical quark flavors appear in the external states, or when the valence and sea quark masses in an ordinary QCD are intentionally made different. In the second case, $\{L_{5,8}^\mathrm{PQ,r}\}$ affect the pion mass and decay constant at NLO, which allows for a fairly precise determination of their values on the lattice as shown in Ref.~\cite{Boyle:2015exm}. On the other hand, the effect of $\{L_{0,3}^\mathrm{PQ,r}\}$ on the pion mass and decay constant appears only at NNLO so that their values are poorly constrained. Finally, we note that the $\mu$-dependence in $\mu_\pi$ and $L_i^\mathrm{PQ,r}$ always cancel each other to make the final expressions of $\{a_0^i,r_0^i\}$ scale-independent.

Until now we have presented all theoretical ingredients needed in this work.
The usefulness of the results above is as follows. First, the energy levels $E_0^{\alpha,\beta}$ that could be obtained rather precisely from the D and C correlation functions on lattice can be used to determine a number of physical and unphysical LECs in PQChPT through the fitting of $\{a_0^{\alpha,\beta},r_0^{\alpha,\beta}\}$. In particular, $a_0^\beta$ and $r_0^\beta$ depend on different linear combinations of the two poorly-constrained unphysical LECs $L_0^\mathrm{PQ,r}$ and $L_3^\mathrm{PQ,r}$, so that the lattice calculation of $E_0^\beta$ with several values of $L$ will help pinning down these two LECs more precisely. After doing so, the energy level $E_0^\gamma$ can be predicted from
Eq.~\eqref{eq:central} using the PQChPT values of $\{a_0^\gamma,r_0^\gamma\}$. The comparison between the theory prediction and the lattice-extracted value of $E_0^\gamma$ then serves as a concrete accuracy check of the lattice calculation of the more difficult rectangular correlation function $\tilde{C}^R(\tau)$. The last energy level $E_0^\delta$ can be deduced in the same way, but it is just the ordinary $I=0$ energy level that does not depend on any unphysical LECs and can be worked out in the ordinary SU(2) ChPT.

\section{Numerical Analysis}
\label{sec:numerics}

In this section we shall carry out explicitly the procedures outlined in the last paragraph to demonstrate their feasibility. Any lattice working group that has $I=2$ $\pi\pi$ scattering data off the shelf may, however, easily repeat the whole program and even improve upon the precision by performing a more sophisticated error analysis.

The basis of our analysis below is the lattice data of connected $\pi\pi$ correlation functions $\tilde{C}^{D,C}(\tau)$ kindly provided to us by the \textit{European Twisted Mass} (ETM) Collaboration with $N_f=2+1+1$ and lattice spacing $a\approx 0.086$ fm \cite{Baron:2010bv,Helmes:2015gla}. To study the volume-dependence, we acquire data from two ensembles that share the same bare QCD parameters but with different volumes: A40.32 and A40.24 which correspond to $(L/a)^3\times T/a=32^3\times 64$ and $24^3\times 48$, respectively. The first step is to extract the energy levels $E_0^{\alpha,\beta}$ from the connected $\pi\pi$ correlation functions. In order to minimize pollutions from thermal states, we adopt the technique introduced in Ref.~\cite{Feng:2009ij} and define the function
\begin{equation}
R_i(\tau+a/2)\equiv\frac{\tilde{C}^i(\tau)-\tilde{C}^i(\tau+a)}{C_\pi^2(\tau)-C_\pi^2(\tau+a)}, \label{eq:R_function}
\end{equation}
($i=\alpha,\beta$), where $C_\pi(\tau)$ is the single pion two-point correlation function. We then fit it to the functional form~\cite{Feng:2009ij}
\begin{equation}
R_i(\tau+a/2)=A\left(\cosh(\delta E_0^i \tau')+\sinh(\delta E_0^i\tau')\coth(2M_\pi \tau')\right),
\end{equation}
where $\tau'=\tau+a/2-T/2$, to extract the energy shift $\delta E_0^i=E_0^i-2M_\pi$. The pion masses at finite volume are taken directly from
Ref.~\cite{Helmes:2015gla}: $aM_\pi=0.1415(2)$ and $0.1446(3)$ for the ensemble A40.32 and A40.24, respectively.

The error of the fitted energy levels are estimated using the bootstrap method. The distribution of mean values for 1500 bootstrap samples is plotted in Fig.~\ref{fig:fit}, and the error is taken as the standard deviation of the distribution. One extra complication is that the result of fitting depends on the fit range, and in principle one should average over all the results obtained from a large number of different fit ranges
with a carefully-assigned weight as done in Ref.~\cite{Helmes:2015gla}. In this work, however, we take a shortcut by performing only a single fit for each ensemble over a specially chosen fit range which reproduces best the $I=2$ energy shifts in Ref. ~\cite{Helmes:2015gla}: $16a\leq \tau\leq 31a$ for A40.32 and $16a\leq \tau\leq 23a$ for A40.24. The best-fit curves are displayed in Fig.~\ref{fig:fit} for illustration. The fitted energy levels and their implied threshold parameters are summarized in Table~\ref{tab:fitresult}.
One observes that the scattering lengths are quite well-determined whereas the effective ranges bear a large uncertainty, which is expected because the second term in the effective range expansion, Eq.~\eqref{eq:EFexpand}, has smaller effects on the discrete energy levels, as also observed in Ref.~\cite{Helmes:2015gla}. As a consistency check, we also include in the same table the $I=2$ threshold parameters predicted in the NLO ChPT using Eq.~\eqref{eq:aralpha} (with the values of physical LECs in Table~\ref{tab:LECs}).\footnote{A technical detail: the $M_\pi$ and $F_\pi$ appear in the ChPT expressions refer to the values in the infinite-volume limit; they can be obtained from their finite-volume values divided by the ``finite-volume correction factor" in Table~7 of Ref.~\cite{Helmes:2015gla}. After combining the A40.24 and A40.32 values we obtain $aM_\pi=0.14102(42)$ in lattice units or around 323 MeV, and $M_\pi/F_\pi=2.876(14)$ at $L\rightarrow\infty$.} We observe a slight discrepancy between the NLO ChPT-prediction of the $I=2$ scattering length and the lattice fit at the level of 2.4~sigma,
which could originate from systematic uncertainties not accounted for in the lattice fitting of $\bar{l}_1^\mathrm{phys}$ and $\bar{l}_2^\mathrm{phys}$ \cite{Baron:2010bv}, as they do not enter the pion mass and decay constant at NLO in the infinite-volume ChPT and can only be determined through finite-size effects.

\begin{figure}[tb]
	\begin{centering}
		\includegraphics[width=0.49\textwidth]{RDm2X}\hfill
		\includegraphics[width=0.49\textwidth]{RDp2X}
		\par\end{centering}
	\caption{\label{fig:fit}Fit to the function $R_i(\tau+a/2)$ defined in Eq.~\eqref{eq:R_function} for $i=\alpha$ (left) and $\beta$ (right). Blue and red dots represent the data from the ensemble A40.24 and A40.32 of Ref.~\cite{Helmes:2015gla}, respectively. }
\end{figure}

\begin{table}
		\begin{ruledtabular}
		\begin{tabular}{|c |c c c c| }
		 Correlation function& $a\, \delta E_0 $ (A40.32) & $a\, \delta E_0 $ (A40.24) &  { $a_0/a$} & $r_0 /a$ \\ \hline
		$\tilde{C}^{I=2}$~\cite{Helmes:2015gla} & $\phantom{-}$0.0033(1) & $\phantom{-}$0.0082(3) & $\phantom{-}${ 1.09(6)} & 53(107) \\
		 $\tilde{C}^{I=2}$, SU(2) ChPT & & & 1.300(19) & 83(2)\\
        $\tilde{C}^\alpha$ & $\phantom{-}$0.0034(1) & $\phantom{-}$0.0083(3) & $\phantom{-}${ 1.124(54)} &  41(97) \\
        $\tilde{C}^\beta$ & $-0.0036(1)$ & $-0.0086(3)$ &  $-1.429(77)$ &  140(85) \\
		\end{tabular}
		\end{ruledtabular}
	\caption{The fitted energy shifts, the extracted inverse scattering lengths and effective ranges (with a pion mass in  infinite volume of about 323~MeV) obtained using the connected $\pi\pi$
		correlation functions. Notice that $\delta E_0^\alpha$ is just the $I=2$ $\pi\pi$ energy shift. For comparison, we list the $I=2$ values with the corresponding statistical errors in the original lattice paper~\cite{Helmes:2015gla} in the second row, and include the NLO ChPT predictions of the $I=2$ threshold parameters in the third row. \label{tab:fitresult} }

\end{table}

\begin{table}
	\begin{centering}
		\begin{tabular}{|c|c|c|}
			\hline
			Parameters & Previous & This Work\\
			\hline
		    $F_0$ [MeV] &  85.58(38)& \\
			$\bar{l}_1^\mathrm{phy}$ &  $-0.309(139)$& \\
			$\bar{l}_2^\mathrm{phy}$ & 4.325(10)&\\
			$\bar{l}_3^\mathrm{phy}$ & 3.537(47)&\\
			$\bar{l}_4^\mathrm{phy}$ & 4.735(17)&\\
			$10^3L_{5}^{\mathrm{PQ},r}$ & 0.501(43)& \\
			$10^3L_{8}^{\mathrm{PQ},r}$ &  0.581(22)&\\
			$10^3(3L_{0}^{\mathrm{PQ},r}+L_{3}^{\mathrm{PQ},r})$ &$-0.6(1.4)$ & $-0.70(18)$ \\
			$10^3L_{0}^{\mathrm{PQ},r}$ & 1.0(1.1)& $5.7(1.9)$  \\
			\hline
			\hline
		\end{tabular}
		\par\end{centering}
	\caption{\label{tab:LECs}Parameters in SU$(4|2)$ PQChPT relevant to this work. In the second column, values for the pion decay constant $F_0$ in the chiral limit and the physical LECs
		$\{\bar{l}_i^\mathrm{phy}\}$ are taken from Ref.~\cite{Baron:2010bv}, whereas
		$\{L^{\mathrm{PQ,r}}_5,L^{\mathrm{PQ,r}}_8\}$ and $\{3L^{\mathrm{PQ,r}}_0+L_{3}^{\mathrm{PQ},r},L^{\mathrm{PQ,r}}_0\}$ are taken from Ref.~\cite{Boyle:2015exm} through NLO and NNLO fits respectively. In the third column we present our fit of $\{3L^{\mathrm{PQ,r}}_0+L_{3}^{\mathrm{PQ},r},L^{\mathrm{PQ,r}}_0\}$. The renormalization scale of the unphysical LECs is 1 GeV.}
\end{table}

After evaluating $a_0^\beta$ and $r_0^\beta$, one may proceed to fit the unphysical LECs $\{L_{0,3}^{\mathrm{PQ,r}}\}$ through the following equations:
\begin{eqnarray}
3L_0^\mathrm{PQ,r}+L_3^\mathrm{PQ,r}&=&-\frac{1}{512\pi M_\pi a_0^\beta}-\frac{F_\pi^2}{32M_\pi^2}\left(1+4\mu_\pi\right)+\frac{\bar{l}_1}{384\pi^2}+\frac{\bar{l}_2}{192\pi^2}-\frac{\bar{l}_3}{1024\pi^2}-\frac{\bar{l}_4}{256\pi^2}\nonumber\\
&&+L_5^\mathrm{PQ,r}-L_8^\mathrm{PQ,r}-\frac{3}{1024\pi^2}, \nonumber\\
L_0^\mathrm{PQ,r}&=&\frac{1}{2048\pi}\left(-\frac{3}{M_\pi a_0^\beta}+M_\pi r_0^\beta\right)+ \frac{F_\pi^2 \mu_\pi}{6M_\pi^2}+\frac{\bar{l}_2}{384\pi^2}+\frac{\bar{l}_3}{1024\pi^2}-\frac{1}{2}L_5^\mathrm{PQ,r}\nonumber\\
&&+L_8^\mathrm{PQ,r}+\frac{17}{6144\pi^2}.
\end{eqnarray}
Special attention is paid to the combination $3L_0^\mathrm{PQ,r}+L_3^\mathrm{PQ,r}$ because it is the combination that enters the scattering lengths $\{a_0^\beta,a_0^\gamma\}$, and it is free from the large uncertainty in the effective range $r_0^\beta$. The results are summarized in Table~\ref{tab:LECs}.
One observes that for the quantity $3L_0^\mathrm{PQ,r}+L_0^\mathrm{PQ,r}$, our extraction is in a perfect agreement with the NNLO fit to different quantities by
Ref.~\cite{Boyle:2015exm}, but with the uncertainty reduced by one order of magnitude. This represents a significant triumph in the PQChPT analysis of $\pi\pi$ scattering. At the same time, a disagreement is found between our extracted $L_0^\mathrm{PQ,r}$ and that obtained in
Ref.~\cite{Boyle:2015exm} at the level of 1.6 sigma. Here we would like to stress that our analysis is consistently performed at NLO, while the NNLO fits in Ref. \cite{Boyle:2015exm} are known to be more unstable and the results depend quite sensitively on the fitting scheme. We thus believe that our result represents a more realistic estimation of $L_0^\mathrm{PQ,r}$.

Finally, we present the most important prediction of this paper, namely the energy shifts $\{\delta E_0^{\gamma,\delta}\}$ as a function of the spatial lattice size $L$ (to remind the readers, the $\delta$-type correlation function is simply the $I=0$ $\pi\pi$ correlation function, while the $\gamma$-type is exactly the same quantity but with vacuum-diagram contributions taken away). To do that, one first calculates the threshold parameters using Eqs.~\eqref{eq:argamma},\eqref{eq:ardelta} and the updated LECs in
Table~\ref{tab:LECs}.\footnote{The quantity $F_\pi$ in the analytical expressions refers to the physical pion decay constant which is a function of the pion mass. It should be in principle determined directly from lattice, but here we obtain a nice approximated value by using the NLO ChPT expression:  $F_\pi=F_0\left[1+\bar{l}_4\,M_\pi^2/(16\pi^2F_0^2)\right]$.}
Notice that $a_0^\gamma$ depends only on the combination $3L_0^\mathrm{PQ,r}+L_3^\mathrm{PQ,r}$ which is tightly constrained, so its value is determined by satisfactory precision; on the other hand, $r_0^\gamma$ is exposed to the large uncertainty of $L_0^\mathrm{PQ,r}$ especially for the case of a large pion mass which enhances the counterterm effects. Meanwhile, $\{a_0^\delta,r_0^\delta\}$ depend only on physical LECs and are hence tightly constrained. After that, the energy shift is deduced by solving Eq.~\eqref{eq:central} numerically, and the result is plotted in
Fig.~\ref{fig:Egamma} for $3<LM_\pi<5$ at three pion masses:  $M_\pi=$138, 236 and 330~MeV.

Here we briefly discuss the qualitative features of the result. First, both the $\gamma$ and $\delta$ channels are attractive so that the energy shifts are negative, and their magnitudes increase with increasing the pion mass and decreasing the lattice size.
The uncertainty of $\delta E_0^\gamma$ comes primarily from $r_0^\gamma$, whose role is important only when $\delta E_0^\gamma$ is large (in other words, when the effective range expansion in Eq.~\eqref{eq:central} starts breaking down). Therefore, one can see that even at a relatively large $M_\pi$ of 330~MeV, the error bar in $\delta E_0^\gamma$ already starts to diminish at $LM_\pi>4$ and one arrives at a rather controlled prediction of the energy shift. Predictions as in Fig.~\ref{fig:Egamma} are useful because they provide extra consistency checks to the lattice calculations of disconnected contractions in $\pi\pi$ scattering.  As we stress before, our formalism is based on the single-channel L\"{u}scher formula in  partially-quenched QCD which follows from the $S$-matrix unitarity. Thus, its results, bearing higher-order PQChPT corrections, must be obeyed by any lattice setup that satisfies unitarity.

\begin{figure}[tb]
	\begin{centering}
		\includegraphics[width=0.49\textwidth]{deltaE}\hfill
		\includegraphics[width=0.49\textwidth]{deltaEdelta}
		\par\end{centering}
	\caption{\label{fig:Egamma}Prediction of the energy shift $\delta E_0^\gamma$ (left) and $\delta E_0^\delta$ (right) as a function of the lattice size $L$ at $M_\pi=$~138 MeV (blue), 236~MeV (red) and 330~MeV (purple). {\color{black} The definitions of $\delta E_0^\gamma$ and $\delta E_0^\delta$ can be found in Eq.~\eqref{eq:Edef}: $\gamma$ corresponds to the channel involving D, C and R types of contractions, and $\delta$ is the isoscalar channel which has all of the D, C, R and V types of contractions. } }
\end{figure}

\section{Conclusions}
\label{sec:conclusion}

This work serves as an important follow-up to Refs.~\cite{Guo:2013nja,Acharya:2017zje} which represent the initial attempts in the study of generic features of various contraction diagrams in lattice simulations of the $\pi\pi$ scattering using PQQCD. The general motivation is to provide useful insights from the theory side to the noisy ``disconnected diagrams" that usually require special treatments.  A missing ingredient in those works is the discussion of the relation between quantities predictable in PQChPT and the discrete energy levels directly measured on lattice. This work fills up the gap and presents, for the first time, practically-useful theory constraints on the disconnected contributions to two-pion correlation functions that may now be checked by lattice community working in this direction.

Let us briefly summarize our approach. First, each contraction in $\pi\pi$ scattering can be represented by a physical scattering process between two pNGBs in a SU$(4|2)$ PQQCD. The multi-channel scattering matrix can be diagonalized to obtain four effective single-channel scattering amplitudes, two of which involve only connected contractions and the others contain disconnected pieces. With that, we can immediately relate the threshold parameters of each single-channel scattering amplitude with the corresponding discrete energy levels in a finite volume through the usual single-channel L\"uscher formula. The threshold parameters can be expressed in terms of physical and unphysical LECs in the SU$(4|2)$ PQChPT, which can be fitted to discrete energy levels extracted from the connected $\pi\pi$ correlation functions. In particular, we achieve an order-of-magnitude improvement over the result in the existing literature in the determination of the LECs combination $3L_0^{\mathrm{PQ},r}+L_3^{\mathrm{PQ},r}$. Performing lattice computations using more volumes, one can also improve the precision of the worst known $L_0^{\text{PQ},r}$. With these fitted LECs we are able to predict the discrete energy shifts $\delta E_0^{\gamma,\delta}$, which involve disconnected contractions, as functions of the lattice size.

The method can be extended to other sectors, such as the low-energy pion-nucleon scattering, for which lattice calculations are still scarce. Knowledge of the PQChPT LECs extracted from the connected contributions, together with known physical LECs, would be helpful to quantify the disconnected contribution in these more difficult cases, and thus reduce the related systematic uncertainties.

\bigskip

\noindent {\bf Acknowledgements}

We thank Xu Feng and Akaki Rusetsky for many useful discussions, and are particularly indebted to Carsten Urbach, Liuming Liu, Markus Werner and Martin Ueding for providing the lattice data for $\pi\pi$ scattering of the ETM Collaboration and all the patient explanations.
This work is supported in part by  the DFG (Grant No. TRR110)
and the NSFC (Grant No. 11621131001) through the funds provided
to the Sino-German CRC 110 ``Symmetries and the Emergence of
Structure in QCD", by the NSFC under Grant No. 11747601 and No. 11835015, by the CAS under Grant No. QYZDB-SSW-SYS013 and No. XDPB09, by
the CAS Center for Excellence in Particle Physics (CCEPP),  by the Alexander von Humboldt Foundation through the Humboldt Research Fellowship, by the VolkswagenStiftung (Grant No. 93562), and by  the  CAS  President's  International
Fellowship  Initiative  (PIFI)  (Grant  No. 2018DM0034).

\begin{appendix}

\end{appendix}

redpajama_set_name: RedPajamaArXiv


text:
SUNDANCE NOW TO EXPAND ITS OFFERINGS WITH THE LAUNCH OF ORIGINAL SCRIPTED PODCAST "EXETER" STARRING JEANNE TRIPPLEHORN AND RAY MCKINNON
The SVOD Service's First Scripted Podcast Features All-New Dynamic Captioning
LINK TO SCENE W/ DYNAMIC CAPTIONS
"Exeter" to Premiere on Sundance Now in September 2018
Los Angeles, CA – July 28, 2018 – Sundance Now, AMC Networks' direct-to-consumer SVOD service curated with high quality dramas and crime thrillers, announced today from the Television Critics Association (TCA) Press Tour that it is expanding its content offerings to include podcasts, beginning with its first-ever scripted podcast, "Exeter." Created by indie writer/director Ronnie Gunter (Lighter) and starring Jeanne Tripplehorn ("Big Love," "Grey Gardens") and Ray McKinnon ("Rectify," "Sons of Anarchy," "Deadwood"), "Exeter" will premiere on the service in September 2018.
The six-episode podcast series, produced by Insurrection Media, will be further brought to life with dynamic captioning — also known as 'kinetic typography' — a process by which words and phrases in the character's dialogue are animated to underscore emotion and heighten tension. This marks the first time this process has been utilized for a scripted dramatic podcast.
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"Sundance Now could not be more excited to be entering the scripted podcast space with such a riveting story, brilliantly brought to life by two outstanding actors — the indomitable Jeanne Tripplehorn and Ray McKinnon, whom we are thrilled to have back in the Sundance fold," said Jan Diedrichsen, General Manager, Sundance Now. "'Exeter' embodies the best of the Sundance brand – it's high-quality, engaging storytelling that we know our audience responds to, with an innovative twist in the form of dynamic captioning. We are particularly excited about this process, as it will provide unprecedented access to audio content for our deaf audience."
"Exeter," produced by Insurrection Media, is executive produced by Ronnie Gunter, Kiliaen Van Rensselaer, Jordanna Fraiberg and Deborah Henderson and directed by Gunter. George Ducker and Ronnie Gunter co-wrote all six episodes. Icemen Audio (Evan Menak, Michael Capuano) is responsible for the sound design and mixing.
About Sundance Now:
Sundance Now is AMC Networks' premium streaming service, offering original and exclusive dramas, comedies, and true crime series, in addition to award-winning movies from every genre, including foreign-language and documentary features – all streaming commercial-free. Built on the Sundance legacy and curated by acclaimed filmmakers and cultural icons, Sundance Now is proud to present an extensive array of entertainment for a passionate and intellectually curious audience.
To stream all Sundance Now content via web, iOS, Android, Roku, Apple TV, and Chromecast, visit SundanceNow.com and sign up for a free trial.
About Insurrection Media
Insurrection Media is an independent talent-friendly digital TV studio developing and producing high quality scripted shows for over-the-top video platforms as well as broadcast and cable networks internationally. The Company is headquartered in Beverly Hills with a division in London that is identifying and developing television shows for traditional and streaming networks in Europe.
SundanceTV and Sundance Now
Sarah Chaikin
Sarah.Chaikin@amcnetworks.com

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nChain, the global leader in research and development of blockchain technologies, announces the creation of Bitcoin SV, a new full node implementation of the original Bitcoin protocol now restored in the form of Bitcoin Cash (BCH). Reflecting its mission to fulfil the vision of Bitcoin's founder Satoshi Nakamoto, the project name represents the "Satoshi Vision" or SV. Created at the request of leading BCH mining enterprise CoinGeek and other miners, Bitcoin SV is intended to provide a clear BCH implementation choice for miners who support Bitcoin's original vision, over implementations that seek to make unnecessary changes to the original Bitcoin protocol.
"Because miners should drive the roadmap in the Bitcoin space, CoinGeek and other miners asked nChain to create a professionally-driven implementation of the Bitcoin full node software (for use on BCH) that restores the original Bitcoin protocol. CoinGeek is sponsoring the project and intends to mine with Bitcoin SV. We invite other BCH miners to join us in using Bitcoin SV to voice their support for the Satoshi Vision." nChain Group CEO Jimmy Nguyen comments.
A comprehensive roadmap is in development, but the first goal is to have the initial Bitcoin SV release ready for testing by first week of September 2018. The code will be based off Bitcoin ABC v0.17.2. The initial release will contain as minimal a changeset as possible to support the November 15, 2018 BCH protocol upgrade because the first priority for Bitcoin SV is to establish security and QA best practices.
Once ready, code and supporting information will be available at the new Bitcoin SV repository on Github. Bitcoin SV will be made available for open source usage under the MIT License.
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The Bitcoin SV team has been constructed with a view to realizing industry best practices, in order to deliver and maintain a full node implementation with an unprecedented commitment to quality assurance and stability.
The Lead Developer will be Daniel Connolly, who joined nChain after 20 years in enterprise systems and IT positions for United Nations agencies. Daniel contributed anonymously to Bitcoin for several years, and has contributed to the Electron Cash project and is a primary contributor to the BitcoinJ-Cash project. nChain's Steve Shadders will act as Technical Director, providing project oversight and liaison with sponsors. Steve began contributing to Bitcoin in 2011, authoring one of the first open source mining pool engines and was one of the earliest contributors to BitcoinJ. Additionally, the team will begin with a pool of 5 C++ developers with over 95 collective years of development experience, a part time Dev Ops resource, a full time QA engineer and range of business support personnel.
The project team is looking to hire additional C++ developers with either direct Bitcoind experience or infrastructure experience. Applicants are invited to submit their expression of interest to careers@nchain.com.
CoinGeek will provide Bitcoin SV with resources to support this professionalization of BCH development.
The Bitcoin SV project will reinforce its commitment to stringent quality with several measures.
Complete a full security audit of the Bitcoin SV code base scheduled to begin in mid- October. This audit will cover not only the code itself but also development practices and processes to assist in building the most robust node development team in the industry.
Provide ongoing review and threat analysis of all code changes as and when they are submitted as candidates.
Using such an outside security audit firm requires significant expense, but CoinGeek and nChain are willing to contribute such costs for the good of the entire BCH ecosystem.
In order to demonstrate its commitment to rewarding responsible bug disclosures, the Bitcoin SV team will begin by offering the highest tier reward of USD $100,000 (sponsored by CoinGeek and of course, payable in BCH) to Cory Fields, a security researcher who found and responsibly disclosed a potential BCH chain-splitting bug in May of this year. The result of this anonymous disclosure was a rapid and professional response that ensured the vulnerability was patched quickly and the network secured. Through its bug bounty program, the Bitcoin SV project wants to encourage more professional disclosures like that to help strengthen BCH.
More node implementations and more competition are good for Bitcoin. Therefore, the Bitcoin SV project welcomes engagement from other implementation developer teams that choose to support the consensus changes proposed by Bitcoin SV. The Bitcoin SV team invites collaboration with those teams and encourages cooperation on testing and consensus code sharing.


Hyundai i20 Boot Space, Size, Seats | What Car?
TThe i20 is one of the roomiest in the class, with enough leg and head room to accommodate tall drivers with ease. It has a noticeably wider interior than most rivals', too.
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From Premium Nav trim upwards, there is a central cubby that doubles up as an armrest and a phone dock. If you don't have the phone dock, there's a shallow cubby in front of the gearlever that's an ideal storage space, given the proximity to the USB and 12V ports, although it's not rubberised.
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From SE trim and upwards, a height-adjustable boot floor can create a hidden storage area. When the floor is raised, it sits flush with the folded rear seats for a smooth, uninterrupted load bay.
Two bag hooks in the boot are also a practical touch. There is, however, a big drop down to the floor over the i20's load lip.


text:
India crushes New Zealand to go 1-0 up
India are not No1 but most improved
Airtel Super Singer western round – Day 1
Western round began last night with 7 singers still left in the competition. This round threw up some interesting possibilities as Prasanna, Ajeesh and Vijay escaped elimination last week. This week contestants might not be that lucky. All of them have to perform well to get through and there will surely be elimination this round (At least that's what I hope).
My guess this week was that it will be tough for Renu and Ajeesh as rest of the pack will pull it off with comfort. Okay now coming back to last night's show, I knew that there will only be 3 singers performing and the night started with Ranjani.
Ranjani began the western round with "Jillunnu oru Kadhal" song. This was a Jazz composition by AR Rahman, sung by Tanvi in the original. Ranjani did a fair job. Even though there were few errors here and there overall Ranjani's attitude impressed the judges. She was rightly spot selected.
Sometimes during this season Vijay TV seems to go out of the way to extend or prolong super singer 2008. It's really very irritating as the viewers have been watching it for almost a year and all of us would want to know who the super singer is. I did not get the idea of making the singers sing the song which they have prepared if they were not spot selected. This was another cheap trick to make sure that there are only three singers performing on the first evening on the show.
Anyways my frustrations aside, the second singer of the night was Vijay. As I mentioned earlier this is a bread and butter round for Vijay "Adams" (As he is called by the judges). Vijay is very comfortable with western songs which he has constantly proven in the previous rounds. Vijay sang "Uyirin Uyire" from "Kaaka kaaka", the original sung by one of my favorite singers Kay kay. Vijay did a fantastic job. His attitude and improvisations were brilliant and as Srinivas rightly said Vijay knows how to perform on stage. He always displays right energy and attitude required for a western number. Vijay was spot selected and was requested to sing "Hotel California" song on the final night of the competition. That was the song he would have sung if did not get spot selected.
So that makes it 2 in 2. The final singer of the night was Ravi (The classical Guru of the group). Ravi sang "Oh Mama Mama" from the movie Minnale, another Harris Jeyraj song. He did a good job and with the hood pulled over his head he gave a rap song effect to the performance. There were few mistakes or probably I should say some improvements pointed out by Unni but overall everyone liked his performance. He was spot selected too. He was asked to sing his other song choice again and he sang a Linkin Park song.
So three singers perform on the first evening and all three get spot selected. This leaves Prasanna, Raghini, Renu and Ajeesh to perform on the second night. It will be very interesting to see how Renu performs. We have seen her struggle with breath while singing fast numbers in the past and let's see how she goes. I don't think that Raghini and Prasanna would have any issues in this round.
That's all about last night's show and just before I end this piece just wanted to mention that Malini Yugendran at last gave Yugendran chance to talk last night and hopefully this trend continues. I am just fed up of the bad pronunciations and horrible anchoring by Malini.
Tags: Airtel Super Singer, Ajeesh, Harris jeyraj, Prasanna, Raghini, Ranjani, ravi, Renu, Srinivas, Sujatha, Unni, Vijay, Western Round
What sort of pitches are these? Are we practising for Tests? Even though good to see bowlers getting help. Too much… twitter.com/i/web/status/1… 1 hour ago


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text:
❯Liverpool ❯ Deal Agreed: Liverpool set to sell player after agreeing bid?
Deal Agreed: Liverpool set to sell player after agreeing bid?
12 Jan 2015 3 min read
For a majority of footballers making a switch to the Premier League represents an upgrade in their respective careers. Over the past we have seen relative unknowns have gone on to prosper in the highest echelons of English Football. However, for some its a complete reality check when they fail to make their mark.
Liverpool's Oussama Assaidi is one such example. Signed in the summer of 2012 when Brendan Rodgers took charge in 2012 the Moroccan was a relative unknown and no one saw the recruitment coming. Assaidi arrived from SC Heerenveen for a fee worth s little as £2.4m.
Majority of the Liverpool fans after watching Youtube videos of the player heaped praise on what they believed was a money ball transfer. Rodgers joining the bandwagon had said, "We're delighted with the signing, he's a young player who has done very well, in particular in the last couple of seasons, in Holland. He fits our style perfectly, he's an exciting player who is very good in one versus one situations and I think he's a player that will excite the crowd."
The winger consequently failed to muster a starting eleven spot and made just 12 appearances for the Anfield side. Assadi joined Stoke City on loan on the final day of the summer transfer window to resurrect his career in England but he has had very few chances to come by.
Now according to reports from the Telegraph, Assaidi has been recalled from his loan sell and Liverpool are set to sell him to to Middle East side Al-Ahli for £4.75m. Confirming the termination of Assaidi's temporary deal Stoke boss Mark Hughes said: "Assaidi has been frustrated this season because I haven`t been able to give him as much game time as he would have liked. He was a player we liked but he has found it difficult to force his way into the team over the past couple of months."
It frees up a loan opportunity for us and obviously we will look to utilise that before the end of the month."
Can't imagine conversation lasted long when Ali-Ahli offered £4.7m for Assaidi. Somehow Liverpool getting double what they paid for him. — James Pearce (@JamesPearceEcho) January 11, 2015
This certainly represents fantastic business to get double the price on what they had spent to buy the player. Maybe Liverpool are raising funds to bring Divock Origi on a permanent basis. The Reds had bought the Belgium striker in the summer but had sent him on loan to Lille till the end of the season. However with Mario Balotelli drawing blanks, Rickie Lambert hardly matching expectations and Daniel Sturridge's long term injury, Liverpool are in desperate need of a striker who can fire them in the top four.
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In reconstructing the development of the sport wheelchair the project will bring together the history of science and technology, the history of Paralympic sport and the history of disability politics and activism. It will, in particular, look at the role of disabled athletes in this story and their agency and actions both in developing new innovations and incorporating new technologies into the wheelchair and in challenging those who sought to resist this development. What will make this account so powerful is that it is a success story for a group who have historically been excluded from design processes.
The project will draw on archival and documentary sources and on oral histories. In reconstructing the history the student will interview wheelchair users, wheelchair athletes, wheelchair designers, representatives from wheelchair manufacturers, medical and health professionals and government officials and civil servants. These oral histories will be supplemented by an extensive search of archival resources of a range of relevant publications including publications by disability groups such as; Magic Carpet (the journal of the Invalid Tricycle Association, ITA), Paraplegia News (the journal of the Paralysed Veteran's of America, PVA) Sports'n'Spokes (a journal also published by the PVA focusing on wheelchair sports). They will also review technical-scientific publications, such as Paraplegia, Archives of Physical Medicine and Rehabilitation and the British Journal of Biomedical Science. Other archives will include the UK's Public Records Office and the National Register of Archives and the National Archives in the US.
•	Applicants should also have a Masters-level degree that satisfies AHRC eligibility requirements for advanced research training, or equivalent professional/occupational experience.
•	We may accept relevant work experience in lieu of a Masters.
Please also see additional eligibility on the postgraduate research funding website.


text:
La serie A2 1991-92 fu il 58º campionato di seconda divisione di rugby a 15 in Italia.

 e fu vinto dal ; insieme alla squadra veneta fu promosso il ; entrambe le squadre disputarono inoltre i playoff scudetto della serie A1 1991-92 ma furono eliminate al primo turno.

Le squadre dal terzo al sesto posto di A2 spareggiarono contro quelle dal settimo al decimo posto di A1 per tentare di salire in categoria superiore, mentre per la retrocessione furono le ultime due della classifica a scendere in serie B, nell'occasione  e l'aquilana .

Squadre partecipanti

Prima fase

Risultati

Classifica

Play-off A1/A2

Verdetti
   e : ammesse ai playoff scudetto serie A1 1991-92
   e : promosse direttamente in serie A1 1992-93
   e : retrocesse in serie B 1992-93

Note

Bibliografia

Voci correlate
 Serie A1 1991-1992 (rugby a 15)

redpajama_set_name: RedPajamaWikipedia


text:
Governor Newsom Issues New Executive Order on Climate Change and Biodiversity
By BARC Staff | Date: October 7, 2020
Governor Newsom today issued a new Executive Order seeking to address climate change and preserve California's biodiversity through a number of measures related to lands and coastal waters. The Executive Order includes a goal for the state to conserve 30% of land and coastal waters by 2030, and directs a number of state agencies to take actions to scale up restoration and conservation efforts to conserve biodiversity.
Executive Order N-82-20, Office of Governor Gavin Newsom
People keep moving to the worst places for climate risk


text:
Coinfomania: Where blockchain and cryptos live.
On-Chain Data
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By PR Desk
Decentraland Announces Second LAND Auction
Decentraland, a virtual LAND ownership platform built on the Ethereum Blockchain which allows users to create, experience, discover, and monetize 3D content and applications, is inviting members of the crypto communities to participate in the upcoming second LAND auction, according to a document shared with Coinfomania.
Decentraland's first LAND auction was held in December 2017 and concluded in January 2018, with over 3,000 participants and over 160 million MANA (approximately $30 million during that period) spent on the acquisition of 34,356 LAND parcels.
However, at the end of the last auction, 9,331 parcels of LAND were not claimed. This has led to the idea of holding a second LAND auction in the Q4 of 2018 in order to allocate the unclaimed LAND properties to members of the community.
Importance of a Second LAND Auction
The second auction will offer several opportunities to both existing landowners and new users who could not purchase a parcel of LAND during the first auction. These benefits include, but not limited to
Acquiring several adjacent parcels of LAND in order to create new Estates. Purchasing multiple parcels, however, could be difficult in the current market situation.
Expanding current LAND property holdings.
Buying affordable plots of LAND where users can build, share, and monetize their 3D content and experiences.
Process and Participation
The second LAND auction will begin on December 10, 2018, and will run for two weeks, adopting a Dutch Auction format. The auctioning of the parcels of LAND will have the same starting rate, and the price will gradually decrease over the course of the auction.
Users are expected to place an order for the LAND property they wish to acquire once it gets to their desired price. This method eliminates bidding.
Interested participants are required to have enough MANA tokens in their Ledger, Trezor, MetaMask or Mist wallet in order to purchase their desired LAND parcel(s).
At the end of the auction, the total MANA spent for the acquisition of parcels of Land and Estate will be removed or burned from circulation.
Official website: https://decentraland.org
Twitter: https://twitter.com/decentraland
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Follow us on Twitter, Facebook, and Telegram to receive timely updates. Subscribe to our weekly Newsletter.
Decentraland MANA Press Release
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Virtual Assets Unleashes Retail Cash Reload Innovation for Buying Cryptocurrencies
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Life is full of stresses, and what now is called "post traumatic stress syndrome" is actually quite widespread in the general population.
Abused children, battered wives, those living through life threatening illnesses or accidents all may suffer variations of the syndrome.
. As many as 70% of adults in the United States have experienced at least one major trauma in their lives, and many of them have suffered from the emotional reactions that are called PTSD. It is estimated that 5% of the population currently have PTSD and that 8% have had PTSD at some point in their lives.
Most of my experience with the "syndrome" has been with abused women. But often the women don't present with abuse: they present with fatigue, or stomach pain, or headaches and often have signs and symptoms of depression on examination.
The ability to tolorate stress, including the stress of combat, is partly genetic, and partly depends on how bad the stress is. And there are few stresses as bad as being a soldier in a war zone.
Their faces were expressionless, their eyes blank and unseeing, and they tended to go to sleep wherever they were. The sick, injured, lightly wounded, and psychiatric cases were usually indistinguishable on the basis of their appearance. Even casual observation made it evident that these men were fatigued to the point of exhaustion….. In spite of this lack of sleep he must undergo long periods of severe exertion, more often than not on a diet that is at best deficient in calories. Often the food is there for him, but he either cannot carry enough of it with him or is too frightened to eat the proper amount. Sometimes the type available has become distasteful through monotony.
If the stress is severe enough, people will break. And some will remain with psychiatric problems.
Yet most of the soldiers described recovered, simply by removing them from the combat situation for a rest.
The Manual then goes on to discuss how combat stress often causes somatic (regular illness) type symptoms, especially symptoms of peptic ulcer or dyspepsia, and stresses that soldiers should remain on limited duty while they recovered. One paradoxical lesson learned early in World War II was that hospitalizing milder cases resulted in what is now called "survivor guilt", resulting in a worse disability than if they were allowed time to recover on their own and regain their self esteem.
But wars are about as high stress a situation as anything humankind suffers, so the problem remains.
It was during World War II that researchers began compiling lots of data on troop stress and its effects. It was discovered that most troops were likely to develop debilitating PTSD after about 200 days of combat (that is, the stress of having your life threatened by enemy fire). But today there are other factors.
In Iraq, army combat troops often get 200 days of combat in one 12 month tour, which is more than their grandfathers got during all of World War II….. A recent survey of troops who had served in Iraq and Afghanistan, found half of them still had some mental or physical health problems six months after returning from overseas.
Again, like what was observed in World War II, simply having a safe place to rest (including decent beds and airconditioning), and contact with family (nowadays that means email) are doing a lot toward keeping morale high in Iraq.
One complicating problem in Iraq: IED's.
Body Armour has proven life saving, especially in combat, but the head trauma due to explosions have many suffering from a poorly defined post concussion syndrome, whose symptoms overlap with that of stress related illnesses.
So what happens when the soldiers go home? This is especially worrisome, since many are National Guards troops who might see their local doctors rather than a military health care worker or a VA physician.
So the military has encouraged physicians and the families to learn about these problems.
For families whose family members might suffer from problems, a good place for information would be this link, which includes a film for family members.
For those who might be involved in treating cases, the core module for diagnosis and treatment can be found HERE. And there are PDF and palmpocket downloads of the core module HERE.
No one hates war more than soldiers, but too many politicians are spinning the war for political gain, and one worries that, like Vietnam veterans, the soldiers will be made a scapegoat. Of course, one advantage here is that, unlike VietNam, the war in Iraq has been successful. But those who "hate war" on principal or because of political beliefs, need to know that the Army is aware of the problem, and trying to address it, by screening, by treatment, including counseling and medications, and by trying to lower stress during rotations in Afghanistan or Iraq.
Of course, like any government bureaucracy, these things take forever, and are enmeshed in bureaucracy, but that's a government systematic problem, which is inherent in any federal hospital.
Behavior: irritability, avoidence, anger, non compliance, self risk.
Evokes aversion or fear in health care provider.
Change in ability to function.
Increased arousal: Difficulty sleeping, Anger outbursts, Difficulty concentrating, hypervigilance, exaggerated Startle Response.
Nancy Reyes is a physician living in the rural Philippines. She writes medical essays at HeyDoc Xanga Blog.
1 user commented in " The Military and Post Traumatic Stress Syndrome, Part II "
Being the wife of a World War 11 Marine, shot in the head and leg with a vitallium plate in his scull who survived Iwojima, Post Ttaumatic stress is not new to me. Although it surfaced after 40 years with nightmares and sweats there were many indicators before that. I feel guilty that I did not encourage him to speak about his war experiences.
Com pensating for his neurological deficiencies, aphasia, numbness all down his right side, loss of smell and taste plus his PSTD has been just too much. He is my hero. With my help found a way to build a business and together we raised a family.


It was a family tradition during my childhood to leave out a mince pie, sherry and carrots for Father Christmas. I'm fascinated by the way traditions and Christmas is celebrated around the world when I speak with friends based in different countries. I decided to record this quickly on Christmas Eve to share with them, as from childhood, I still like to read the poem to myself on a night and think back fondly.


text:
Yinka @ Large
Stop wishing 'Baba Sala' dead – Son
Son of the popular comedian, Moses Adejumo, 'aka Baba Sala' Emmanuel, has called on the public to ignore the rumours of his father's death, while also urging those he described as 'rumour mongers' to stop spreading the false rumours.
There were unsubstantiated reports of the death of the retired comedian, on the internet late Tuesday, with one of them claiming that the octogenarian died at the University College Hospital, Ibadan, where he was rushed to for treatment.
Some months ago, the comedian was taken to the UCH for treatment on ailment related to old age and was discharged after regaining fitness. He returned to his home in Osun State immediately.
rumour was posted. People wanted to know the true position of things. Some of the callers were his (Baba Sala) fans living abroad. I told them he is alive. But how many people will I explain that to, considering the damage that the spread of that rumour had created?
"People should stop toying with others' emotion and psychological positions. Every rumour will definitely have a deeply negative effect on people concerned. When I called my father, he was eating his favourite food and his wife is with him at home. He is in good condition and in high spirit. He is an old man with good health.
"Like every person of his age, he fell ill sometimes ago and UCH gave him the best treatment he could get and he returned home. That was several months ago and since then, he has been in good health. The Adejumo family is not happy that some people feel happy to feed on rumour just to create excitement or sensation for themselves. It is an anti-human action and those behind it should stop," added Adejumo junior.
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About Yinka Adedehinbo
M. Adeyinka Adedehinbo was born on 21st December in Hadeija in the present Jigawa state, Nigeria, West Africa. He had his secondary education in Sokoto state and got B. Sc. (Microbiology) from University of Lagos.
He also has MBA (management) from Usman Dan Fodio University, Sokoto. He has many years work experience (mostly in marketing) in several renowned organizations.
An acute observer and critic of politics and public affairs. He is married and blessed with twin girls.