We talk about skin a lot – not surprising with the gut-skin connection. After colonics, clean food pre and pro biotics, this is what I use on my skin. I had acne from 13 until I went fully raw vegan in London in 2006. I'd given up on having clear skin by this point, so this was an incredible surprise. Before and since then, I've been honing my skin care routine – this is what works for me at the moment.
I was just planning to write about the clay mask, but hey.
I do no recommend using soap, toner or any other manufactured product as a cleaner if you have problem skin. Each morning I gently exfoliate with a cheap generic hand glove, in clean water, followed by the best moisturiser for my skin type, season and location. I LOVE Ren's Moroccan Rose Otto body oil – a little goes a long way, mixed with a little Pure Potions Intensive hand cream or Natural Siberica. And, I use the same oil to remove makeup. When I make up my own oil blends I add frankincense, a drop of castor oil (always have to hand) and olive oil.
After I've done my dry skin brushing, for a deeper face and body cleanse twice a week, I use the ancient practice of Guasha – simply made from bicarbonate (baking) soda and vodka. Take great care on your face until you gauge how you respond to it. I'll let Tyler explain how to make and use it.
Using a non-metal spoon and bowl: 1 teaspoon of bentonite clay + 1 capsule of activated charcoal (skip the fancy organic bamboo charcoal – it's gritty and doesn't work as well). Add a little water and mix into a paste. It takes awhile to get it smooth. Seal it in a tight pot. These Sistema ones are found in the kitchen sections – I use them for everything.
There you go, hope that helps. Suki Says Skin 101.


text:
\section{Introduction}

Jet algorithms are essential tools for connecting long-distance measurements made on hadrons to short-distance interpretations based on perturbative quarks and gluons.  Since there is a fundamental mismatch between color-singlet hadrons and color-carrying partons, there is no way to define an ideal jet finding procedure.  For this reason, a variety of jet algorithms have been introduced with different underlying philosophies and different practical advantages \cite{Ellis:2007ib,Salam:2009jx}.

In this paper, we expose a surprising connection between three seemingly unrelated approaches to jet finding:  jet function maximization \cite{Georgi:2014zwa,Ge:2014ova,Bai:2014qca}, 1-jettiness minimization \cite{Stewart:2010tn,Thaler:2010tr,Thaler:2011gf,Jouttenus:2013hs}, and stable cone finding \cite{Blazey:2000qt,Ellis:2001aa,Salam:2007xv}, all reviewed in \Sec{sec:review}.  Philosophically, these algorithms are quite different, so one might think that they would yield rather different jets.  Instead, all three algorithms yield (approximately) conical jets of (approximately) fixed radius $R$, with a high degree of correlation between the methods.  As we will show, this correlation is not an accident, since jet function maximization and 1-jettiness minimization can be viewed as descendants of a mother optimization problem, whose solution is a stable cone jet.

This relationship is most transparent in electron-positron collisions, where jets are typically defined in terms of particle energies and angles.  Remarkably, with appropriate definitions, all three algorithms can be made to yield \emph{identical} cone jets.  We will prove this in \Sec{sec:meta} by showing how these three methods can be derived from optimizing a common meta function.  This optimization is equivalent to finding all stable cones and then choosing the one with the largest value of $J$ (defined below).  In \Sec{sec:twoparticle}, we explain this relationship in more detail by performing a two particle case study.

Turning to the Large Hadron Collider (LHC), the intrinsic differences between the three algorithms become apparent.  In proton-proton collisions, jets are typically defined in terms of particle transverse momenta and rapidity-azimuth distances.  As discussed in \Sec{sec:LHC}, the transverse momentum of a jet is not exactly the same as the summed transverse momenta of its constituents.  For this reason, there is now a difference between optimizing a four-vector (as in jet functions), optimizing a light-like axis (as in 1-jettiness), and aligning the jet axis with the jet momentum (as in stable cones).  Despite these differences, though, we show that the algorithms still give similar results when the jet radius $R$ is small.

\section{Review of Jet Algorithms}
\label{sec:review}

To begin, we briefly review these three approaches to jet finding, all of which are infrared and collinear (IRC) safe.  For simplicity, in this paper we only consider finding the single hardest jet in an event, though all of these methods can be adapted to identify multiple jets.  For consistency of notation, we always use $R$ to refer to the adjustable jet radius parameter in each algorithm, which can be identified with the true jet radius only in the small $R$ limit.

\begin{itemize}
\item \textit{Jet function maximization} \cite{Georgi:2014zwa,Ge:2014ova,Bai:2014qca}.  Here, the jet finding strategy is to find the subset of particles in an event that maximizes a jet function $J(P_\mu)$, where $P_\mu$ is the four-vector of the candidate jet.\footnote{The name ``jet function'' and the symbol $J$ should not be confused with earlier usage in the context of factorization and resummation, e.g. \cite{Sterman:1986aj,Catani:1989ne,Korchemsky:1994jb}.}  The original paper \cite{Georgi:2014zwa} introduced a jet function appropriate for electron-positron collisions:
\begin{equation}
\label{eq:Jorig}
J_{\rm orig}(P_\mu) = E - \frac{1}{R^2} \frac{m^2}{E},
\end{equation}
where $E$ and $m$ are the total energy and mass of the candidate jet, and the original paper used the notation $R^2 \to 1/\beta$.  Maximizing this $J_{\rm orig}$ gives quasi-conical jets with nearly fixed radius $\simeq R$.  Since then, this algorithm has been adapted to collisions at the LHC \cite{Bai:2014qca} to yield nearly conical jets in the rapidity-azimuth plane.
\item \textit{1-jettiness minimization} \cite{Stewart:2010tn,Thaler:2010tr,Thaler:2011gf,Jouttenus:2013hs}.  This approach is based on finding a light-like jet axis $n = (1, \hat{n})$ that minimizes 1-jettiness $\mathcal{T}_1(n)$.  For electron-positron collisions, one possible choice of 1-jettiness measure is (see also \cite{Jouttenus:2013hs})
\begin{equation}
\label{eq:tauorig}
\mathcal{T}_1(n) = \sum_{i \in {\rm event}} \min \left\{E_i, \frac{2 \, n\cdot p_i}{R^2} \right\},
\end{equation}
where $p_i = (E_i, \vec{p}_i)$ is the four-vector of particle $i$, and one often sees the notation $R^2 \to 2 \rho$.  The minimum inside of $\mathcal{T}_1$ partitions the event into an unclustered (or ``beam'') region and a jet region, and after minimizing over $\hat{n}$, the jet region is exactly conical (for massless particles) with radius $\simeq R$.  This approach has been generalized to $N$-jettiness jet finding through the \textsc{XCone} jet algorithm \cite{Stewart:2015waa,Thaler:2015xaa}.
\item \textit{Stable cone finding} \cite{Blazey:2000qt,Ellis:2001aa,Salam:2007xv}.   Algorithms like \textsc{SISCone} \cite{Salam:2007xv} search for all conical jet regions of radius $R$ which are stable, meaning that the total jet momentum is aligned with the jet axis.  This algorithm works equally well for opening angle as for rapidity-azimuth distance.  From the set of stable cones, one must then choose the desired jet(s).  By construction, the hardest jet after progressive removal (e.g.\ \textsc{SISCone-PR}  as described in \cite{fjmanual}) is a perfect cone.  While ``hardest'' typically refers to the jet with highest (transverse) momentum, we will find it interesting to consider the jet with largest $J$.   As shown in \cite{Ellis:2001aa}, stable cone finding can be viewed as an optimization problem, which is closely related to 1-jettiness minimization \cite{Thaler:2011gf}.  While historically there were issues with IRC safety of specific iterative cone algorithms (see discussion in \cite{Ellis:2007ib,Salam:2009jx}), those issues can be resolved using seedless methods (like in \textsc{SISCone}) or IRC safe seeds (like in \textsc{XCone}).
\end{itemize}

\section{A Meta Optimization Problem}
\label{sec:meta}

Since the three above approaches yield (nearly) conical jets and all are based on optimization, it is perhaps not surprising that, with suitable modifications, they can yield identical jets.  Here, we show this explicitly for cone jets of fixed opening angle, as relevant for electron-positron collisions.  We discuss obstructions to generalizing this to the LHC in the \Sec{sec:LHC}.

Consider the following meta function which depends both on a candidate jet $P_\mu = (E, \vec{P})$ as well as on an auxiliary light-like axis $n = (1, \hat{n})$:
\begin{equation}
\label{eq:Mdef}
M(P_\mu,n) = E - \frac{2 \, n \cdot P}{R^2},
\end{equation}
where $R$ will soon be identified with the same parameter in \Eq{eq:tauorig}.  Here, we work with massless final-state particles with $E_i = |\vec{p}_i|$; one can adapt this construction to massive particles by performing measurements in, say, the $p$-scheme or $E$-scheme where massive particles are replaced by massless proxies (see e.g.\ \cite{Salam:2001bd,Mateu:2012nk}). We claim that maximizing $M$ over all possible $P_\mu$ and $n$ simultaneously yields a jet function maximum, a 1-jettiness minimum, and a stable cone jet.

First, consider maximizing $M$ with respect to $n$, keeping the candidate jet $P_\mu$ fixed.  Introducing a Lagrange multiplier $\lambda$ to enforce unit norm
\begin{equation}
\label{eq:lagrange}
M(P_\mu,n) \rightarrow M(P_\mu,n) + \lambda (\hat{n}^2 - 1),
\end{equation}
one can show that $M$ is maximized by
\begin{equation}
n^{\rm opt} = \Biggl(1, \frac{\vec{P}}{|\vec{P}|} \Biggr).
\end{equation}
Note that $\hat{n}$ lives on a smooth compact space (i.e.\ the surface of a unit sphere), so there are no edge conditions to check.  Plugging this back into \Eq{eq:Mdef}, we find
\begin{equation}
M(P_\mu,n^{\rm opt}) = E - \frac{2 \bigl(E - |\vec{P}|\bigr)}{R^2} \equiv J(P_\mu).
\end{equation}
Thus, maximizing $M$ over $\{P_\mu,n \}$ is equivalent to maximizing the jet function $J$ over $P_\mu$.  Since
\begin{equation}
2\bigl(E - |\vec{P}|\bigr) \simeq \frac{m^2}{E}
\end{equation}
for $m \ll E$, this jet function is very similar to the original definition in \Eq{eq:Jorig}.\footnote{\Ref{Ge:2014ova} introduced a generalization of \Eq{eq:Jorig} with an extra parameter $n$,
\begin{equation}
J^{(n)} = E^n \left(1 - \frac{n}{R^2}\frac{m^2}{E^2} \right),
\end{equation}
where $R^2 / n \to 1/\beta$ in the original notation.  Because $J^{(n)}$ is not linear in the jet kinematics, we know of no meta function that can directly handle this case.  That said, maximizing $J^{(n)}$ is equivalent to maximizing $\sqrt[n]{J^{(n)}}$, and $\sqrt[n]{J^{(n)}} \simeq J$ in the $m \ll E$ limit.}
    Following the logic of \cite{Georgi:2014zwa}, a non-trivial $J$ maximum is guaranteed to exist since there are a finite number of $P_\mu$ partitions and a single particle has a larger $J$ value than the empty partition.

Next, consider maximizing $M$ over the candidate jet $P_\mu$, keeping $n$ fixed.  Crucially, $M$ is linear in the final state particles, so we can write $M$ as
\begin{equation}
\label{eq:Mlinear}
M(P_\mu,n) = \sum_{i \in \rm jet}  E_i - \frac{2 \, n \cdot p_i}{R^2},
\end{equation}
where ``jet'' refers to the set of particles contributing to $P_\mu$ (i.e.\ $\sum_{i \in \rm jet} E_i = E$, $\sum_{i \in \rm jet} \vec{p}_i = \vec{P}$).  Any final-state particle that contributes positively to $M$ will increase $M$'s value, so for the optimal jet $P^{\rm opt}_\mu$, we must have
\begin{equation}
\label{eq:Msumoverevent}
M(P^{\rm opt}_\mu,n) = \sum_{i \in {\rm event}} \max \left\{E_i - \frac{2 \, n \cdot p_i}{R^2} , 0\right \},
\end{equation}
where now the sum runs over all particles in the event.  Rearranging this formula as
\begin{equation}
E - M(P^{\rm opt}_\mu,n)= \sum_{i \in {\rm event}} \min \left\{E_i, \frac{2 \, n \cdot p_i}{R^2} \right\} \equiv \mathcal{T}_1(n),
\end{equation}
we recover precisely \Eq{eq:tauorig}.  Thus, taking into account the minus sign, maximizing $M$ over $\{P_\mu,n \}$ is equivalent to minimizing 1-jettiness $\mathcal{T}_1$ over $n$.

Finally, we can maximize $M$ with respect to both $n$ and $P_\mu$ by combining these two analyses.  Since $M$ has a maximum, there must be an optimal subset $P^{\rm opt}_\mu$ (perhaps more than one in degenerate phase space configurations) and a corresponding optimal axis $n^{\rm opt}$.  Fixing $P^{\rm opt}_\mu$ and minimizing $\mathcal{T}_1(n)$ with respect to $n$ using the same Lagrange multiplier trick in \Eq{eq:lagrange}, the optimal axis must be
\begin{equation}
\label{eq:stablecone}
n^{\rm opt} = \Biggl(1, \frac{\vec{P}^{\rm opt}}{|\vec{P}^{\rm opt}|} \Biggr).
\end{equation}
Now looking at \Eq{eq:Msumoverevent} with fixed $n^{\rm opt}$, the optimal subset $P_\mu^{\rm opt}$ must consist of all particles with
\begin{equation}
\label{eq:Joptcondition}
\frac{2 \, n^{\rm opt} \cdot p_i}{E_i} < R^2.
\end{equation}
This is precisely the condition for a stable cone.  To see why it is stable, note that \Eq{eq:stablecone} implies that the jet axis $n^{\rm opt}$ is aligned with the jet momentum $\vec{P}^{\rm opt}$.  To see why it is a cone, note that for massless particles with $|\vec{p}_i| = E_i$,
\begin{equation}
\frac{2 \, n \cdot p_i}{E_i} = 2 - 2 \cos \theta_{\hat{n},i},
\end{equation}
where $\theta_{\hat{n},i}$ is the angle to the axis.  Thus, the condition in \Eq{eq:Joptcondition} specifies particles contained in a cone of half opening angle $R_{\rm true}$ around $n^{\rm opt}$ with the identification
\begin{equation}
\label{eq:halfopen}
\sqrt{2 - 2 \cos R_{\rm true}} \equiv R,
\end{equation}
where $R_{\rm true} \simeq R$ at small radii.

We have therefore demonstrated that optimizing $M(P_\mu,n)$ is equivalent to optimizing the following two functions:
\begin{align}
\label{eq:Jnew}
J(P_\mu) &= E - \frac{2 \bigl(E - |\vec{P}|\bigr)}{R^2}, \\
\mathcal{T}_1(n) &= \sum_{i \in {\rm event}} \min \left\{E_i, \frac{2 \, n \cdot p_i}{R^2} \right\}.
\end{align}
Moreover, the optimal jet $P^{\rm opt}_\mu$ is specified by a stable cone centered on the axis $\hat{n}^{\rm opt} \propto \vec{P}^{\rm opt}$.  Of course, actually \emph{finding} the optimal configuration is a challenging computational problem.  In practice, one can make use of the converse statement that among all stable cones, the one that has the largest value of $J(P_\mu)$ is guaranteed to be the one that optimizes $M(P_\mu,n)$.  Therefore, one can run spherical \textsc{SISCone-PR} \cite{Salam:2007xv,fjmanual} to find all stable cone jets, and then simply select the jet with the largest value of $J(P_\mu)$.

\section{Two Particle Case Study}
\label{sec:twoparticle}

To better understand the above analysis, it is instructive to study the simplest case of two massless particles with equal energies $E/2$, separated by opening angle $\theta_{12}$.  For ease of notation, we define
\begin{equation}
\sqrt{2 - 2 \cos \frac{\theta_{12}}{2}} \equiv \frac{R_{12}}{2}, \qquad \sqrt{2 - 2 \cos \theta_{12}} \equiv \widetilde{R}_{12},
\end{equation}
such that $\theta_{12} \ge R_{12} \ge \widetilde{R}_{12}$, but all three agree in the small angle limit.

In the jet function approach using \Eq{eq:Jnew}, there are three candidate jets to consider:  particle 1 alone, particle 2 alone, or particles $1$ and $2$ together.  The corresponding jet function values are
\begin{align}
J(P_\mu^1) = J(P_\mu^2)    &= \frac{E}{2},\\
J(P_\mu^1 + P_\mu^2) &= E \left(1 - \frac{R_{12}^2}{4R^2} \right).
\end{align}
Choosing the maximum $J$ value, the condition for the two particles to be clustered into a single jet is
\begin{equation}
\text{$J$ maximum:} \quad R_{12} < \sqrt{2} R.
\end{equation}

In the 1-jettiness approach, it is straightforward to prove that the axis that minimizes $\mathcal{T}_1(n)$ must lie along the line between particles 1 and 2, such that
\begin{equation}
\theta_{\hat{n},1} + \theta_{\hat{n},2} = \theta_{12}.
\end{equation}
Furthermore, one can show that local minima of $\mathcal{T}_1(n)$ can only appear at three possible points:
\begin{equation}
\label{eq:withmidpoint}
\theta_{\hat{n},1} = 0, \qquad \theta_{\hat{n},2} = 0, \qquad \theta_{\hat{n},1} = \theta_{\hat{n},2} = \frac{\theta_{12}}{2},
\end{equation}
corresponding to the three candidate jet configurations above.  For $\widetilde{R}_{12} < R$ (note the tilde), only $\theta_{\hat{n},1} = \theta_{\hat{n},2}$ is a true local minimum, so the two particles are always clustered together.  For $R_{12} > 2 R$ (no tilde), only $\theta_{\hat{n},1} = 0$ and $\theta_{\hat{n},2} = 0$ are true local minima, so the two particles are never clustered together. For intermediate $R_{12}$ values, all three minima are present with corresponding 1-jettiness values of
\begin{align}
\mathcal{T}_1(\theta_{\hat{n},1} = 0) = \mathcal{T}_1(\theta_{\hat{n},2} = 0) &= \frac{E}{2},\\
\mathcal{T}_1(\theta_{\hat{n},1} = \theta_{\hat{n},2}) & =  \frac{E}{4} \frac{R_{12}^2}{R^2}.
\end{align}
Choosing the global $\mathcal{T}_1$ minimum, the particles will be clustered if
\begin{equation}
\text{global $\mathcal{T}_1$ minimum:} \quad R_{12} < \sqrt{2} R,
\end{equation}
yielding the same result as the jet function approach.   Alternatively, one might be content with only finding a local $\mathcal{T}_1$ minimum.  For example, \textsc{XCone} \cite{Stewart:2015waa,Thaler:2015xaa} searches for local $\mathcal{T}_1$ minima using IRC safe seeds based on $k_T$-style clustering \cite{Catani:1993hr,Ellis:1993tq}, leading to the clustering condition
\begin{equation}
\text{$k_T$-seeded local $\mathcal{T}_1$ minimum:} \quad R_{12} < R.
\end{equation}
With IRC safe seeds, this is a perfectly acceptable jet finding strategy from the point of view of perturbative QCD calculations.\footnote{Crucial to achieving IRC safety for \textsc{XCone}, the number of seeds is always equal to the desired number of jets $N$, independent of the final state particle multiplicity.}

Turning to the stable cone approach, the three local minima in $\mathcal{T}_1$ correspond precisely to the three possible stable cone configurations.  The default behavior of spherical \textsc{SISCone-PR} \cite{Salam:2007xv,fjmanual} is to progressively remove the stable cone with the largest energy, meaning that the two particles will always be clustered if they fall within a common cone:
\begin{equation}
\text{$E$-ordered \textsc{SISCone-PR}:}  \quad R_{12} < 2R.
\end{equation}
To mimic precisely the behavior of $J$ maximization or $\mathcal{T}_1$ minimization, though, one has to progressively remove the jet with the largest value of $J(P_\mu)$:
\begin{equation}
\label{eq:Jordered}
\text{$J$-ordered \textsc{SISCone-PR}:}  \quad R_{12} < \sqrt{2} R.
\end{equation}
Only for this version of stable cone finding does one recover the remarkable equivalence derived in \Sec{sec:meta}.

As a historical note, the presence of local $\mathcal{T}_1$ minima led to many Tevatron-era discussions about cone jet algorithms \cite{Blazey:2000qt,Ellis:2001aa}, where a variant of $\mathcal{T}_1$ was known as the ``Snowmass potential''.  One issue is that of midpoint seeds, since if one only uses the location of the two particles themselves as seeds, then one misses out on a possible stable cone centered on their midpoint, as in \Eq{eq:withmidpoint}.  Since the location of $\mathcal{T}_1$ minima is an IRC safe property, this is the reason why seedless methods are preferred.   A related midpoint issue is that of disappearing minima, since in data, where the two particle directions are smeared out by showering and hadronization effects, the local $\mathcal{T}_1$ minimum at the midpoint disappears for $R_{12} \gtrsim R_{\rm sep} R$, where $R_{\rm sep} \simeq 1.3$  \cite{Ellis:1992qq}.  Disappearing minima do not present an issue for IRC safety, but they do lead to poor convergence of perturbation theory.  In that context, it is interesting to note that $1.3$ is comparable to the $\sqrt{2}$ factor in \Eq{eq:Jordered}.  This suggests that $J$-ordered cone finding may be more robust to showering and hadronization effects than $E$-ordered cone finding, since $J$-ordering aims to find global $\mathcal{T}_1$ minimum instead of just a local one.  We leave a detailed analysis of this possibility to future work.

\section{Differences at the LHC}
\label{sec:LHC}

For LHC applications, it is typically advantageous to work with transverse momenta and rapidity-azimuth distances.  Here, jet function maximization, 1-jettiness minimization, and stable cone finding are indeed inequivalent, which we now show.

Jet functions are based on optimizing the jet four-vector as a whole.  Therefore, we expect the jet function to be a function of the total transverse momentum $P_T$, total mass $m$, or the combination $E_T = \sqrt{P_T^2 + m^2}$.  For example, the chosen jet function in \cite{Bai:2014qca} was
\begin{equation}
\label{eq:Jhad}
J_{\rm had}(P_\mu) = E_T - \frac{1}{R^2} \frac{m^2}{E_T},
\end{equation}
where $R \to 1/ \sqrt{\beta}$ in the original notation.  As explained in \cite{Bai:2014qca}, while the jet region is nearly conical when using $J_{\rm had}$, the jet momentum direction is somewhat offset from the jet center at forward rapidities.

By contrast, 1-jettiness is based on optimizing a light-like axis, using a measure that is additive over the individual particles $i$.  As shown in \cite{Thaler:2011gf} (see also \cite{Stewart:2015waa,Thaler:2015xaa}), a cone algorithm can be defined via
\begin{equation}
\label{eq:tau1had}
\mathcal{T}^{\rm had}_1(n) = \sum_{i \in {\rm event}} p_{Ti} \, \min \left\{1,  \frac{\Delta{R_{i,\hat{n}}}}{R} \right\}^\beta,
\end{equation}
where $R$ is the jet radius and $\beta$ is an angular exponent (not to be confused with $\beta$ in the jet function literature).  For any value of $\beta$, the jet region is perfectly conical and the jet axis is located at the center of the jet region.  For the special value of $\beta = 2$, the jet axis is aligned with the $p_T$-weighted jet momentum (see \Eq{eq:stablecone1} below).

At first glance, one might think that  $J_{\rm had}$ and $\mathcal{T}^{\rm had}_1$ could be massaged into a common meta function $M_{\rm had}$ by just making suitable $E \to p_T$ replacements.  However, there is a fundamental mismatch due to the fact that
\begin{equation}
\label{eq:mismatch}
\sum_i p_{Ti} \not= \Bigl(\sum_i p_i \Bigr)_T.
\end{equation}
For this reason one cannot construct a suitable $M_{\rm had}$ that depends solely on $P_\mu$ and $n$, since one also needs information about the scalar $p_T$ sum within the jet.  Of course, one could find an $M_{\rm had}$ if one relaxes the requirements that $J_{\rm had}$ depends solely on $P_\mu$ or that $\mathcal{T}^{\rm had}_1$ can be expressed as a sum over $i$, though that goes somewhat against the original philosophies of those approaches.

\setlength{\tabcolsep}{0.2cm}
\begin{table*}
\begin{tabular}{rcccc|cc}
\hline \hline
 & $J_{E_T}$ & \textsc{XCone} $N =1$ & \textsc{SISCone-PR} & anti-$k_t$ & C/A &  $k_t$ \\
 \hline
$J_{E_T}$ &--- & 93\% & 96\% & 93\% & 81\% & 74\% \\
\textsc{XCone} $N =1$ &93\% & --- & 94\% & 96\% & 85\% & 77\% \\
\textsc{SISCone-PR} &96\% & 94\% & --- & 94\% & 82\% & 75\% \\
anti-$k_t$ & 93\% & 96\% & 94\% & --- & 84\% & 76\% \\
\hline
C/A & 81\% & 85\% & 82\% & 84\% & --- & 83\% \\
$k_t$ & 75\% & 78\% & 76\% & 76\% & 83\% & --- \\
\hline
\hline
\end{tabular}
\caption{Comparing the hardest jet found in $Z$ plus jets production at the LHC for $R = 0.4$.  Shown is the percentage of jets found by the column algorithm that match the row algorithm at the 3\% level, as measured by \Eq{eq:qualitymetric}.}
\label{fig:compare}
\end{table*}

\setlength{\tabcolsep}{0.2cm}
\begin{table*}
\begin{tabular}{rcccc|cc}
\hline \hline
 & $J_{E_T}$ & \textsc{XCone} $N =1$ & \textsc{SISCone-PR} & anti-$k_t$ & C/A &  $k_t$ \\
 \hline
$J_{E_T}$ &--- & 64\% & 72\% & 64\% & 45\% & 42\% \\
\textsc{XCone} $N=1$ &63\% & --- & 84\% & 91\% & 57\% & 48\% \\
\textsc{SISCone-PR} &71\% & 85\% & --- & 87\% & 53\% & 45\% \\
anti-$k_t$ & 63\% & 91\% & 87\% & --- & 56\% & 47\% \\
\hline
C/A & 44\% & 56\% & 53\% & 56\% & --- & 67\% \\
$k_t$ & 41\% & 48\% & 45\% & 47\% & 67\% & --- \\
\hline
\hline
\end{tabular}
\caption{Same as \Tab{fig:compare}, but for $R = 0.8$ where larger differences are expected.}
\label{fig:compare2}
\end{table*}

For stable cone finding, there are two different definitions in common use.  In \cite{Ellis:2001aa}, a stable cone is defined such that the jet axis aligns with the $E_T$-weighted centroid of its constituents.  For massless constituents, this is the same as minimizing $\mathcal{T}^{\rm had}_1$ in \Eq{eq:tau1had} with $\beta = 2$ \cite{Thaler:2011gf}, yielding a jet axis located at rapidity/azimuth location
\begin{equation}
\label{eq:stablecone1}
y = \frac{\sum_{i \in {\rm jet}} y_i \, p_{Ti} }{\sum_{i \in {\rm jet}} p_{Ti}}, \qquad \phi = \frac{\sum_{i \in {\rm jet}} \phi_i \, p_{Ti} }{\sum_{i \in {\rm jet}} p_{Ti}}.
\end{equation}
Alternatively, one can define a stable cone to have its jet axis aligned with the true jet three-momentum, which is the default behavior in \textsc{SISCone-PR} \cite{Salam:2007xv,fjmanual}.  For small radius jets, the distinction is small, but we know of no 1-jettiness measure (or jet function) that yields exactly stable cones by this second definition.

Despite the above discussion, the mismatch in \Eq{eq:mismatch} is a small effect at small $R$, with corrections that scale  like $R^2$ or $m^2/p_T^2$.  Moreover, these finite $R$ corrections are often subdominant to other effects present in practical jet algorithm implementations, such as the difference between local or global optimization or the treatment of jets at forward rapidities.

In typical situations, maximizing $J_{\rm had}$, minimizing $\mathcal{T}^{\rm had}_1$, and finding stable cones (by either definition) yield rather similar jets, which are also similar to anti-$k_t$ jets \cite{Cacciari:2008gp}.  This is shown in \Tab{fig:compare} for a $Z$ plus jets sample from \textsc{Pythia}~8.209 \cite{Sjostrand:2006za,Sjostrand:2007gs,Sjostrand:2014zea} at the 14 TeV LHC, looking at the hardest jet with $p_T > 50~\text{GeV}$ and $R = 0.4$ for all algorithms.  The jet function approach is implemented with the $J_{E_T}$ algorithm \cite{Bai:2014qca}, which finds jets ordered by \Eq{eq:Jhad} but we take the hardest jet by $p_T$.  The 1-jettiness approach is implemented with \textsc{XCone} \cite{Stewart:2015waa,Thaler:2015xaa}, using \Eq{eq:tau1had} with $\beta = 2$.\footnote{Note that \Eq{eq:tau1had} differs from the default \textsc{XCone} measure based on Lorentz dot products.  We use iterative one-pass minimization, which finds a local $\mathcal{T}_1^{\rm had}$ minimum from IRC safe seeds.}  The stable cone approach is implemented with \textsc{SISCone-PR} \cite{Salam:2007xv,fjmanual}, taking the hardest jet by $p_T$.  The anti-$k_t$ approach is implemented with \textsc{FastJet} \cite{Cacciari:2011ma}.
For comparison, we also show results for the Cambridge/Aachen (C/A) \cite{Dokshitzer:1997in,Wobisch:1998wt,Wobisch:2000dk} and $k_t$ \cite{Catani:1993hr,Ellis:1993tq} algorithms.  As a similarity metric between the row and column jet algorithms in \Tab{fig:compare}, we look at the fraction of the scalar $p_T$ not contained in the shared constituents of the two algorithms:
\begin{equation}
\label{eq:qualitymetric}
\frac{\sum_{i \not\in \text{row} \cap \text{column}} p_{Ti}}{\sum_{i \in \text{row}} p_{Ti}}.
\end{equation}
Around 95\% of the time, the four cone-like algorithms yield similar jet kinematics at the 3\% level, while the differences with C/A and $k_t$ are noticeably larger.  Because all of the algorithms considered are longitudinally boost invariant, the similarities/differences between algorithms are essentially independent of the jet rapidity.  In \Tab{fig:compare2}, we perform the same comparison for $R = 0.8$, where the differences are much larger as expected.

Note that the study in \Tab{fig:compare} involves events that typically only have one hard jet without substructure.  The four algorithms do exhibit larger variations when going to multiple jets, due to their differing treatment of jet overlap regions, though the algorithms still yield similar results when the jets are widely separated.  For jets with substructure, interesting differences can arise for subjets at wide angles (see further discussion in \cite{Bai:2014qca}).  Analogous to the discussion in \Sec{sec:twoparticle}, two particles with equal momentum sharing will be clustered together if:
\begin{equation}
\label{eq:Rsep12}
\Delta R_{12} \lesssim
\begin{cases}
R & \text{\textsc{XCone}, anti-$k_t$}, \\
\sqrt{2} R & \text{$J_{E_T}$}, \\
2 R &  \text{\textsc{SISCone-PR}},
\end{cases}
\end{equation}
where $2 \to R_{\rm sep}$ in the last line if there is sufficient smearing of the subjet directions.\footnote{The fact that the $J_{E_T}$ behavior is intermediate between anti-$k_t$ and \textsc{SISCone-PR} is also reflected in the NLO single jet inclusive cross section, where the $J_{E_T}$ result is very nearly the average of the anti-$k_t$ and \textsc{SISCone}-PR results \cite{Kaufmann:2014nda,Kaufmann:2015hma}.}  As in \Sec{sec:twoparticle}, we can make \textsc{SISCone-PR} behave more like the $J_{E_T}$ algorithm by switching from $p_T$-ordered to $J_{\rm had}$-ordered cones, and we can make \textsc{XCone} behave more like the $J_{E_T}$ algorithm by performing brute force global $\mathcal{T}_1$ minimization.  In cases like this where the four algorithms give different results, the desired behavior depends on the physics of interest.

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

The idea of jet finding as an optimization problem has a long history (see also~\cite{Berger:2002jt,Angelini:2002et,Angelini:2004ac,Grigoriev:2003yc,Grigoriev:2003tn,Chekanov:2005cq,Lai:2008zp,Volobouev:2009rv}), so we find it remarkable that three seemingly different optimization strategies with different underlying philosophies correspond to the same meta optimization problem.  As we saw, the exact correspondence only holds for fixed-angle cone jets, whereas at the LHC, complications arise when projecting four-vectors to the rapidity/azimuth plane.  That said, the differences between the algorithms are small for small radius jets, and this correspondence helps explain why jet function maximization and 1-jettiness minimization yield such similar jets to stable cone jet algorithms.

There are different conclusions one might draw from this analysis.  To the extent that jet function maximization, 1-jettiness minimization, and stable cone finding yield similar jets (which are similar to anti-$k_t$), one should simply choose the jet algorithm with the best computational performance.  Alternatively, to the extent that the three algorithms have different underlying philosophies, one might try to exploit those differences to develop new jet finding techniques with qualitatively different behaviors or with better convergence in perturbative QCD calculations.  Ultimately, one should choose the jet finding approach that best suits the intended physics application.   In that spirit, this paper suggests that jet algorithms should be judged not by the elegance of their underlying philosophies but by the utility of their jet objects.

\begin{acknowledgments}
We thank Gavin Salam and Gregory Soyez for collaborating during the conceptual stages of this work and for subsequent suggestions.  We thank Steve Ellis and Andrew Larkoski for detailed comments on a draft of this manuscript.  We benefitted from conversations with Yang Bai, Yang-Ting Chien, Matteo Cacciari, Howard Georgi, Jakub Scholtz, Matthew Schwartz, Iain Stewart, Werner Vogelsang, and Bryan Webber.  J.T. is supported by the U.S. Department of Energy (DOE) under cooperative research agreement DE-SC-00012567, the DOE Early Career research program DE-SC-0006389, and by a Sloan Research Fellowship from the Alfred P. Sloan Foundation.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
Asp while Breastfeeding
It is recommended to breastfeed exclusively for six months and then while introducing to other food sources extend it to twelve months. In this duration most mothers will need help of some sort of medication, It could be for short term like could and flue or it could be something chronic like Arthritis or Diabetes and here comes the question of safety of medication in use. In this post we will figure out what is Asp and whether its safe to use Asp while nursing or not.
What is Asp used for?
For temporary relief of symptoms of cough with expectoration and dyspepsia, ill effects of melons, water and dried fruits, thick yellow ropy mucous secretions.
Is Asp usage safe while breastfeeding? If a lactating mother is using it can there be any effect on growth or development of infant?
FDA does not regulate Asp. There is no credible study done on safety of Asp while breastfeeding, Same holds truth for almost all homeopathic medicines however homeopathic medicines go through a process called potentisation. In potentisation homeopathic preparation goes through repeated dilution and shaking. Homeopaths state that repeated dilution and shaking helps the body to heal naturally. Due to extreme dilution of active ingredients homeopathic medicines are mostly safe in breastfeeding, Hence we can consider Asp as safe to use while breastfeeding.. Below we have provided analysis of its active ingredients. Safety rating of ingredients holds truth for herbal product but may not apply for homeopathic diluted drugs.
Asp Breastfeeding Analsys
Echinacea angustifolia while Breastfeeding
Plant that is widely used even during pregnancy and breastfeeding. Because a lack of toxicity with an appropriate dose and moderate consumption it should be compatible with breastfeeding. The roots and aerial summits are used. It contains polysaccharides, essential oil, flavonoids, pyrrolizidine alkaloids among others. Unproven effects: immune stimulant, wound healing, anti-inflammatory. Indications are: common cold, bronchitis, skin lesions.Roots and aerial summits are used. It contains polysaccharides, essential oil, flavonoids, pyrrolizidine alkaloids ... Unproven effects: immune stimulant, wound healing, anti-inflammatory. Indications according to Commission E of German Ministry of Health: common cold, bronchitis, skin lesions. Contrary to the European Scientific Cooperative on Phytotherapy (ESCOP), the European Medication Agency does not recommend usage in younger than 12 years (allergy risk). Avoid using for longer than 8 weeks (risk for leukopenia)
Arsenic trioxide while Breastfeeding
Used in the treatment of promyelocitic leukemia in adults.
Bryonia alba root while Breastfeeding
Climbing plant. The female inflorescences or flower tips are used.It contains phloroglucinols, estrogenic, quercetin, kaempferol, tannins, phenolic acids essential oil and flavonoids. One of its components, 8-prenylnaringenin (8-PN) is the most powerful phytoestrogen known. Properties that are attributed: hypnotic, sedative, orexigenic.It is used as a flavoring and stabilizer of the beer.Indications German Commission E Ministry of Health, EMA and ESCOP: insomnia, nervousness, anxiety There is no scientific evidence showing an improvement in milk production.A possible estrogenic effect may be a decrease in milk production.The best galactogogue is a frequent and on-demand breastfeeding along with proper technique. During breastfeeding its consumption should be moderate or occasional.
Colchicum autumnale bulb while Breastfeeding
It is used for treatment of Gout, Familial Mediterranean Fever. As well as for other conditions like Amyloidosis, Behçet's Syndrome, Pericarditis, Primary Biliary Cirrhosis and Pyoderma gangrenosum. It is excreted in breast milk in quantities that differ according to different published studies, from no clinically significant in some, until being important in others, but there have been no problems in infants whose mothers were treated, even after months, with undetectable plasma levels in these infants. The concomitant use of macrolide antibiotics and fruit juices (both the mother and the infant) , especially grape-fruit, is not recommended since they may interfere with bile excretion and greatly increase toxicity. The American Academy of Breastfeeding rates it as compatible with breastfeeding. List of Essential Medicines of WHO (2002): compatible with breastfeeding
Sodium sulfate while Breastfeeding
Osmotic laxatives are often used to cleanse the bowel before medical procedures.They base their action on the very little or nothing they are absorbed and by this means forcing the water to enter the intestine with the result of watery diarrhea. At latest update no published data on excretion into breast milk were found. Another laxative with similar estructure which is Sodium Picosulfate (see specific info) is not excreted into breast milk Be aware of drinking enough fluids to avoid dehydration of the nursing mother who is already loosing water by producing about 1 liter of milk a day.
Silicon dioxide while Breastfeeding
A polymer made out of silicon-oxygen-methyl combination with a high molecular weight, water repellent and low superficial tension. It is used in many ways (dimethicone, simethicone, -see specific items)orally to treat infant colic and flatulence; as pediculicide, in cosmetic creams and lotions and skin protectants as to prevent ulcers and scars; arthroplasties, retinal detachments and reconstruction or cosmetic surgery as injections and implants. Silicone is widely distributed in our environment with several cosmetic and medicinal uses. No evidence of toxicity on human tissues has been shown. A 1994 report on immunological side effects in infants breastfed by mothers with silicone implants, was denied categorically by means of meta-analysis and other work. The absorption by oral or dermal route is negligible. Both a high molecular weight and polymer molecular structure make it practically impossible excretion in the milk and hence a significant amount of intestinal absorption by the infant. Those circumstances make silicone implants safe for lactation even if broken or manufacturing fault (Poly Implant Prothèse, PIP). After extensive analysis of such silicone prosthesis, where lack of health risk was demonstrated, it can be concluded that many of the initial recommendations published lacked scientific validity, including that carriers of such prosthesis should not breastfeed. Silicon levels in blood and milk of women with implants (55 ng / ml) are similar to those of women without implants (51 ng / ml), 13 times lower than that found in cow's milk (709 ng / ml) and 80 times lower than in commercial infant formulas (4403 ng / ml). American Academy of Pediatrics: Product usually compatible with breastfeeding. To view other possible effects on breastfeeding of breast implant unrelated to silicone, see the term 'Augmentation Mammoplasty'. See below the information of these related products:
Asp Breastfeeding Analsys - 2
Echinacea species (Echinacea angustifolia, Echinacea purpurea, Echinacea pallida) contain high molecular weight polysaccharides (e.g., heteroxylan, arabinogalactan) and lower molecular weight compounds (e.g., alkylamides, caffeoyl conjugates such as cichoric acid and echinacosides), but no single chemical is known to be responsible for echinacea's biological activity. Some products have been standardized based on echinacoside, and others on cichoric acid. Echinacea has no specific uses during breastfeeding, but is commonly used orally to treat or prevent upper respiratory infections. It is also used topically to treat skin infections. Excretion of some of the purportedly active alkamides was found in breastmilk in one mother. No data exist on the safety and efficacy of echinacea in nursing mothers or infants. In general, echinacea is well tolerated with gastrointestinal upset, diarrhea and constipation, skin rash and rarely allergic reactions reported. It may also alter the metabolism of some dugs metabolized by the P450 enzyme system. Some sources indicate that echinacea is safe in recommended doses,[1] while others recommend avoiding it during breastfeeding because of the lack of published safety data. Dietary supplements do not require extensive pre-marketing approval from the U.S. Food and Drug Administration. Manufacturers are responsible to ensure the safety, but do not need to the safety and effectiveness of dietary supplements before they are marketed. Dietary supplements may contain multiple ingredients, and differences are often found between labeled and actual ingredients or their amounts. A manufacturer may contract with an independent organization to verify the quality of a product or its ingredients, but that does certify the safety or effectiveness of a product. Because of the above issues, clinical testing results on one product may not be applicable to other products. More detailed information #about dietary supplements# is available elsewhere on the LactMed Web site.
Goldenseal while Breastfeeding
Goldenseal (Hydrastis canadensis) root contains berberine and other isoquinoline alkaloids. Goldenseal has traditionally been used as an anti-infective both systemically and topically, although high-quality studies of its efficacy and safety are lacking. It has also been used to mask illicit drugs in the urine, although it appears to be ineffective with modern laboratory methods. Goldenseal has been used topically by nursing mothers to treat sore nipples.[1] No data exist on the excretion of any components of goldenseal into breastmilk or on the safety and efficacy of goldenseal in nursing mothers. Berberine can displace bilirubin from serum albumin, causing concern about exposure of newborn infants, because bilirubin can build up in the infant's brain, causing brain damage. However, the extent of berberine's passage from the mother to the infant is unknown. Most sources recommend avoiding exposure of neonates to goldenseal via breastfeeding or otherwise.[2][3][4] Dietary supplements do not require extensive pre-marketing approval from the U.S. Food and Drug Administration. Manufacturers are responsible to ensure the safety, but do not need to the safety and effectiveness of dietary supplements before they are marketed. Dietary supplements may contain multiple ingredients, and differences are often found between labeled and actual ingredients or their amounts. A manufacturer may contract with an independent organization to verify the quality of a product or its ingredients, but that does certify the safety or effectiveness of a product. Because of the above issues, clinical testing results on one product may not be applicable to other products. More detailed information #about dietary supplements# is available elsewhere on the LactMed Web site.
Most sources consider breastfeeding to be contraindicated during maternal antineoplastic drug therapy. It might be possible to breastfeed safely during intermittent therapy with an appropriate period of breastfeeding abstinence; the manufacturer recommends an abstinence period of 1 week after the last dose. Chemotherapy may adversely affect the normal microbiome and chemical makeup of breastmilk.[1] Women who receive chemotherapy during pregnancy are more likely to have difficulty nursing their infant.[2]
Long-term prophylactic maternal doses of colchicine up to 1.5 mg daily produce levels in milk that result in the infant receiving less than 10% of the maternal weight-adjusted dosage. The highest milk levels occur 2 to 4 hours after a dose, so avoiding breastfeeding during this time can minimize the infant dose, although some clinicians simply recommend taking the drug after nursing. No adverse effects in breastfed infants have been reported in case series and a case-control study and some authors consider colchicine safe during breastfeeding in women being treated for familial Mediterranean fever or rheumatic conditions.[1][2][3][4]
Pulsatilla vulgaris while Breastfeeding
Pulsatilla (Anemone pulsatilla and other related species) contains ranunculin, protoanemonin, and anemonin as well as triterpene saponins and flavonoids. The fresh plant is extremely irritating to the skin, gastrointestinal tract and mucous membranes. Allergic reactions have been reported to pulsatilla. Homeopathic preparations of pulsatilla are reportedly used for sore nipples and mastitis,[1] to reduce an overabundant milk supply,[2] or to increase milk supply.[3] Galactogogues should never replace evaluation and counseling on modifiable factors that affect milk production.[4] No scientifically valid clinical trials support either of these uses. Because of a lack of information, other agents may be preferred in nursing mothers. Dietary supplements do not require extensive pre-marketing approval from the U.S. Food and Drug Administration. Manufacturers are responsible to ensure the safety, but do not need to the safety and effectiveness of dietary supplements before they are marketed. Dietary supplements may contain multiple ingredients, and differences are often found between labeled and actual ingredients or their amounts. A manufacturer may contract with an independent organization to verify the quality of a product or its ingredients, but that does certify the safety or effectiveness of a product. Because of the above issues, clinical testing results on one product may not be applicable to other products. More detailed
Goldenseal and Breastfeeding
Ledum palustre twig and Breastfeeding
Labrador tea is a plant. The leaves and flowering shoots are used to make medicine. People take Labrador tea for sore throat, chest congestion, coughs, lung infections, and other chest ailments. They also take it for diarrhea, kidney problems, joint and muscle pain (rheumatism), headache, and cancer.
Women use it to cause an abortion or treat �female disorders.�Some people add Labrador tea to bath water or apply it directly to the skin to treat skin problems. The flowers and leaves of the plant are used to prepare tea. However, frequent intake of the same have some side effects like headaches, indigestion, cramps, and vomiting.
Labrador tea has narcotic properties. If taken in concentrations that are too high, it can cause symptoms of intoxication that can lead to paralysis and death. Due to its narcotic properties Its likely unsafe to use Labrador tea if you are breastfeeding. If you are pregnant It might cause an abortion.
Pulsatilla vulgaris and Breastfeeding
Note: Mostly safe in Homeopathic preparations
Sepia officinalis juice and Breastfeeding
SEPIA OFFICINALIS is usually low in mercury and its likely safe in breastfeeding.
What if I already have used Asp?
Due to high dilution of ingredients in homeopathic medicines they do not create much problem for baby. Asp is a homeopathic medicine and if your baby does not have any abnormal symptoms then there is nothing to worry about. Be careful with too much usage of ethanol based homeopathic medicines during breastfeeding.
I am nursing mother and my doctor has suggested me to use Asp, is it safe?
Homeopathic medicines are usually safe in breastfeeding and if Asp has been recommended by doctor then there should be no concern about its usage in breastfeeding.
If I am using Asp, will my baby need extra monitoring?
Who can I talk to if I have questions about usage of Asp in breastfeeding?

redpajama_set_name: RedPajamaCommonCrawl


text:
A glimpse into the minds of the Positioning Pioneers
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