\chapter{\label{chapter1} Parton distribution functions}
Parton distributions are one of the central pillars of perturbative QCD,
factorising as they do the perturbatively incalculable long distance dynamics
present in calculations involving hadronic initial states. Combined with the
perturbative description of the short-distance cross-section what could seem at
first a hopeless situation is alleviated, and QCD becomes a predictive and
useful theory when applied to hadronic scattering.

In this chapter a brief overview of how parton distribution functions arise in
QCD calculations will be presented.  We shall explore the prototypical example
of the deep inelastic scattering (DIS) of leptons off a hadronic target, first
in the \emph{naive parton model} arising before the advent of QCD and then with
the QCD-improved parton model which allows for an excellent description of DIS
measurements across a wide range of hard scales.

The treatment of heavy quarks in parton distributions is a particularly delicate
issue and therefore will also be discussed in this introductory theory section.
Finally there will be some exploration of the general properties of parton
distributions in order to provide a summary of the available theoretical
constraints upon PDFs.

\section{Partons in deep inelastic scattering}
We shall begin by introducing parton distribution functions as they arise in the
early parton model. The model was originally introduced by Feynman and
Bjorken~\cite{feynman1,Feynmanparton,feynmanparton2, Bjorken:1968dy} in the late
1960's in an effort to understand the scattering behaviour of hadronic states
and successfully describes many properties observed in early deep inelastic
scattering experiments.

In this process, a charged lepton $l$ probes a proton $P$ by the exchange of a
gauge boson. For simplicity we shall describe here the neutral current process
where a photon is exchanged. In the inelastic regime where the momentum transfer
to the target proton is large, the proton does not survive the scattering
process and fragments into an arbitrary hadronic final state $X$. The process
$l(k) + P(p) \to l(k^\prime) + X$ is illustrated at tree level in Figure~\ref{fig:DIS}.

\begin{figure}[ht]
\centering
\includegraphics[scale=0.5]{2-PDFs/figs/DIS.pdf}
\caption{Deep inelastic scattering of a charged lepton with a proton target.}
\label{fig:DIS}
\end{figure}

In this system we can define the standard DIS kinematic variables;  $Q^2$ denotes the momentum transfer from the electron to the target proton, $\nu$ the energy transfer and $y$ the measure of the reaction's inelasticity, or fractional energy transfer. In the rest frame of the proton these are given by
\begin{eqnarray}
 Q^2 &=& -q^2 = -(k - k^{\prime})^2, \\
 \nu &=& M(E- E^\prime), \\
 y &=& (q \cdot p)/(k \cdot p),
\end{eqnarray}
where $M$ refers to the mass of the proton, and the inelasticity ranges between 0 (elastic scattering) and 1. $E$ and $E^\prime$ denote the energies associated with the four-momenta $k$ and $k^\prime$ respectively. Additionally, we may introduce the Bjorken scaling parameter $x$, central to the parton model,
\be x = \frac{Q^2}{2\nu}. \ee
Neglecting spin labels, the amplitude for this diagram in the Feynman gauge is given by
\be\mathcal{M} = ie^2\bar{u}(k^\prime)\gamma^\mu u(k)\left( i\frac{g_{\mu\nu}}{Q^2} \right)\left<X\right|J_h^\nu\left|P\right>,  \label{eq:DISme}\ee
where $J_h^\nu$ represents the hadronic current.
The fundamental difficulty in attempting to compute the cross section for this process is our ignorance of the wavefunction for the hadronic states $\left|X\right>$ and $\left|P\right>$. To isolate the problem, we are able to factorise the spin averaged square of the amplitude in Equation \ref{eq:DISme} into a leptonic ($L_{\mu\nu}$) and a hadronic ($W^{\mu\nu}$) part
\be |\overline{\mathcal M}|^2 = \frac{1}{Q^2} L_{\mu\nu}W^{\mu\nu}, \ee
where the leptonic tensor is straightforwardly calculable:
\ba L_{\mu\nu} &=& e^2\sum_{spin} \bar{u}(k^\prime)\gamma_\mu u(k) \bar{u}(k)\gamma_\nu u(k^\prime), \\
&=& e^2 \mathrm{ tr}\left[ \slashed{k}^\prime\gamma_\mu\slashed{k}\gamma_\nu \right], \\
&=& 4e^2[k_\mu k^\prime_\nu + k_\nu k^\prime_\mu - g_{\mu\nu}k\cdot k^\prime],  \ea
where here we have neglected the fermion masses. The hadronic part of the calculation is considerably more difficult to evaluate, and indeed impossible to compute from first principles in perturbation theory as it is sensitive to the low-scale, and therefore strongly coupled dynamics of the proton target:
\ba
W^{\mu\nu} &\sim& \sum_X \left<P(p)\right| {J_h^\mu}^{\dagger} \left|X\right>\left<X\right| J_h^\nu \left| P(p)\right>, \\
 &\sim& \left<P(p)\right| {J_h^\mu}^{\dagger} J_h^\nu \left| P(p)\right>. \ea

However, we can gain some insight into its structure by noting that the tensor must obey the conservation requirements of the hadronic current $q_\mu W^{\mu\nu}=0$ and $q_\nu W^{\mu\nu}=0$. The tensor may therefore be parametrised without loss of generality by the following structure:
\be W_{\mu\nu} = -\left( g_{\mu\nu} - \frac{q_\mu q_\nu}{q^2}\right) F_1(x,Q^2) +\left(p_\mu -q_\mu \frac{p \cdot q}{q^2}\right)\left(p_\nu -q_\nu \frac{p \cdot q}{q^2}\right)\frac{1}{\nu}F_2(x,Q^2).\label{eq:htensor}\ee
Here we have introduced the parameters in our tensor $F_i$ which are known as the electromagnetic structure functions. For interactions involving parity-violating currents, there is a third contribution to the hadronic tensor arising through the $F_3$ structure function. Here the only possible functional dependence for the structure functions is upon the quantities $Q^2$ and $x$.

It is convenient now to define a projection vector $n$ with the properties $ p \cdot n = 1$, $ n \cdot q = 0$, and $n^2 = p^2 = 0$, where the assumption of negligible proton mass has been made. Any vector can now be written as a combination of $n$, $p$ and a component transverse to the proton momentum as a \emph{Sudakov decomposition}. Using this projection vector we may obtain the structure functions from the hadronic tensor as so:
\begin{eqnarray}
 F_2 &=& \nu n^\mu n^\nu W_{\mu\nu},  \label{eq:proj1} \\
 F_L &=& F_2 - 2xF_1 = \frac{Q^4}{\nu^3}  p^\mu p^\nu W_{\mu\nu}, \label{eq:proj2}
\end{eqnarray}
where the quantity in the second equation is known as the longitudinal structure function. So far, few assumptions have been made about the form of the EM hadronic tensor $W_{\mu\nu}$, we have simply parametrised it in terms of a Lorentz invariant tensor structure and structure functions. Feynman's parton model allows us to describe more of the hadronic tensor with perturbation theory by proposing a composite proton formed as a bound state of fundamental, spin-$1/2$ constituents: the \emph{partons}.

The parton model approximation states that for a sufficiently hard interaction, the virtual photon only interacts with a single point-like parton inside the target proton and we can treat the partons as approximately free particles. The hadronic tensor then admits a probabilistic expansion in terms of Parton Distributions which encode the probability of the hard photon interacting with a constituent parton carrying a faction $\xi$ of the parent proton's momentum. The probability of interacting with a parton carrying between $\xi$ and $\xi+\delta\xi$ of the proton's momentum being given by $f(\xi)\delta\xi$ where $f(\xi)$ is the interaction probability for a parton with momentum $\xi p$. Diagrammatically we may therefore construct the photon-hadron interaction as a weighted sum of partonic diagrams:
\be \left| \vcenter{\hbox{\includegraphics[height=2cm]{2-PDFs/figs/convolution1.pdf}}} \right|^2= \sum_i^{N_{part}}f_i(\xi,Q^2) \otimes \left| \vcenter{\hbox{\includegraphics[height=2cm]{2-PDFs/figs/convolution2.pdf}}}\right|^2(\xi), \nonumber\ee
where we have introduced the multiplicative convolution
\be (f \otimes g )(x) = \int_0^1\; \frac{d\xi}{\xi}\; f\left(\frac{\xi}{x}\right) g(\xi).\ee
The hadronic tensor is then given in terms of a sum of individual hard scattering partonic tensors, denoted  $\widetilde{W}^i_{\mu\nu}(\xi)$ for a target parton of type $i$. Writing the hadronic tensor as the probabilistic sum over all constituent parton types we obtain
\be W_{\mu\nu} =\int_0^1 \frac{d\xi}{\xi} \sum_i f_i(\xi,Q^2)\; \widetilde{W}^i_{\mu\nu}(\xi,Q^2). \label{eq:htensorexpan}\ee
As the parton level tensors must obey the same conservation relations as the full hadronic tensor, we can once again form a general parameterization of $\widetilde{W}^i_{\mu\nu}$:
\be \widetilde{W}^i_{\mu\nu} = -\left( g_{\mu\nu} - \frac{q_\mu q_\nu}{q^2}\right) \widetilde{F_1^i}(\xi,Q^2) +\xi^2 \left(p_\mu -q_\mu \frac{p \cdot q}{q^2}\right)\left(p_\nu -q_\nu \frac{p \cdot q}{q^2}\right)\widetilde{F_2^i}(\xi,Q^2),\ee
where the factors of $\xi^2$ arise from taking $p^\mu \to \xi p^\mu$. Substituting this form for $\widetilde{W}^i_{\mu\nu}(\xi)$ into Eqn \ref{eq:htensorexpan} and comparing with the form in Eqn \ref{eq:htensor}, we find two expressions for the proton EM structure functions,
\be F_1(x,Q^2) = \int_0^1 \frac{d\xi}{\xi} \sum_i f_i(\xi) \widetilde{F_1^i}(\xi,Q^2),  \label{eqn:f1}\ee
\be F_2(x,Q^2) = \int_0^1 \xi d\xi \sum_i f_i(\xi) \widetilde{F_2^i}(\xi,Q^2). \label{eqn:f2}\ee
The naive parton level structure functions $\widetilde{F_1^i}(\xi,Q^2)$ describe the hard scattering subprocess involving a parton of species $i$ and may be computed by considering the parton level squared amplitude for the subprocess, $\gamma^*(q) + q(\xi p) \to q(l)$ and projecting out the desired quantities with the operators defined previously. At leading order, using the parton level version of the projector Eqn \ref{eq:proj1}:
\be { \mathcal M }_\mu = -i e_{q^i}\bar{u}(l)\gamma^\mu u(\xi p),\ee
\be \frac{n^\mu n^\nu}{\xi^2} \widetilde{W}^i_{\mu\nu}  = \frac{n^\mu n^\nu}{\xi^2}\overline{\sum} \left| \mathcal{M} \right|^2_{\mu\nu} = 4e_{q^i}^2,\ee
where we have made the approximation that momenta transverse to the beam axis vanish. Including the phase space for the final state quark in the CM frame we obtain:
\be \widetilde{F^i_2} =  2 e_{q^i}^2 \delta(l^2),\ee
where the delta function can be rewritten in terms of $\xi p$ and $q$:
\be \delta (l^2) = \delta ((\xi p + q )^2 ) = \delta (2\xi \nu - Q^2) = \delta (2\nu (\xi - x)).\ee
This is an interesting result of the analysis at leading order, the kinematical variable $x$ actually describes the momentum fraction of the interacting parton. The parton level structure function
$\widetilde{F^i_2}$ is therefore given by:
\be \widetilde{F^i_2} = 2 e_{q^i}^2 \delta (\xi-x).\ee
The parton level longitudinal structure function is also straightforwardly projected out of the same amplitude,
\be \widetilde{F}^i_L = \frac{Q^4}{\xi \nu^3}p^\mu p^\nu \widetilde{W}^i_{\mu\nu} =  \widetilde{F}^i_2 - \frac{2x}{\xi^2} \widetilde{F}^i_1. \ee
At leading order this projection, and therefore the longitudinal structure function, are exactly zero, consequently
\be \widetilde{F}^i_1 = \frac{\xi^2}{2x} \widetilde{F}^i_2 = e_{q^i}^2\frac{ \xi^2 }{x} \delta (\xi - x).\ee
We may therefore write the full EM proton structure functions in the naive parton model as
\be F_1(x,Q^2) =  \int_0^1 d\xi \sum_i f_i(\xi) e_{q^i}^2\frac{ \xi }{x} \delta (\xi - x) =  \sum_if_i(x)e^2_{q^i},\ee
\be F_2(x,Q^2) = 2 \int_0^1 \xi d\xi \sum_i  f_i(\xi) e_{q^i}^2 \delta (\xi-x) = 2 x \sum_if_i(x)e^2_{q^i}.\ee
These results have a number of important features. Firstly in this model the structure functions have no dependence upon the resolution parameter $Q^2$, a phenomenon known as Bjorken scaling~\cite{Bjorken:1968dy}. This scaling effect was an important achievement of the original parton model, as it was able to describe contemporary experimental results rather well. The lack of any scale dependence in the structure functions is a consequence of the model's assumptions treating interactions with the proton's constituent partons as point like, and consequently having no characteristic length scale.

Secondly we note that $F_2(x) = 2xF_1(x)$, which is known as the Callan-Gross relation~\cite{callangross}. It illustrates a fundamental property of spin-1/2 particles, that they are unable to absorb a longitudinally polarised photon~\cite{pQCDhandbook}.
%
\section{QCD and the parton model}
The naive parton model was able to provide a good phenomenological description of early DIS measurements. Its success also provided great support for QCD as the correct description of the strong interaction. The phenomenon of Bjorken scaling placed substantial constraints upon the theory governing the internal dynamics of the proton. The asymptotic freedom of QCD allows for a consistent description of Bjorken-scaling, where the constituents of the hadron can be viewed as independent, non-interacting point like particles at high  values of the resolution parameter $Q^2$. The partons in Feynman's model were therefore quickly associated with the quarks and gluons of QCD.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{2-PDFs/figs/d96-039f11.pdf}
\caption[Scaling violations in the proton structure function $F_2$]{Scaling violations in the proton structure function $F_2$. Here each curve in $x$ is scaled by a function $C(x)= 0.6(i- 0.4)$ for presentation purposes, where $i$ denotes the bin in $x$. Figure from~\cite{Aid:1996au}.}
\label{fig:F2H1}
\end{figure}

Despite the `snapshot' picture of non-interacting partons at leading order in QCD, we cannot neglect the higher order corrections to the point vertex calculated in the previous section. These corrections introduce logarithms of $Q^2$ which break the naive Bjorken scaling of the structure functions. Indeed, the measurement of such scaling violations provided one of the most powerful experimental verifications of QCD. Such violations are demonstrated in measurements of $F_2$ in Figure \ref{fig:F2H1}. In this section we shall perform an overview of the extension of the parton model to $\mathcal{O}(\alpha_s)$ in QCD.

At one loop order, there are three diagrams that contribute to the $qq\gamma$ vertex studied in the previous section; the real emission of a gluon from the initial (a) or final state (b) quarks, and the virtual correction diagram (c). Additionally at one loop order in QCD there arises a diagram initiated by a gluon splitting into a $q\bar{q}$ pair (d).

\begin{figure}[ht]
\centering
\includegraphics[scale=0.6]{2-PDFs/figs/1loopDIS.pdf}
\end{figure}

All four of these diagrams are separately divergent. When appropriately regularised however, the divergences in the final state real emission and virtual correction diagrams cancel explicitly as a consequence of the IR safety of QCD, yielding a finite contribution to the cross section. However the divergences present in the real emission diagrams from the initial state partons are not subject to the same cancellations, as they modify the momenta at the interaction vertex.

Like the real emission diagram of a gluon from an initial state quark, the initial state gluon diagram (d) suffers from an equivalent divergence mediated by a perturbatively calculable $g\to q\bar{q}$ splitting function $P_{gq}$. Including all of the finite contributions from the other contributing diagrams as the coefficient $W(x)$, the parton level structure function at next to leading order in QCD is given by
\ba
 \widetilde{F_2^i}(\xi,Q^2) &=& 2 e_i^2\left[ \delta(\xi-x) \right. \nonumber\\
 				 &+& \frac{\alpha_S}{2\pi}\sum_j\left(P_{ij}(\xi)\log\frac{Q^2}{\kappa^2} + W_{ij}(\xi)\right) \nonumber\\
				 &+&  \left. \mathcal{O}(\alpha_S^2) \right]. \label{eq:f2plnlo}
\ea
Here the $i$ once again refers to the partonic species at the interaction vertex, and we have introduced an infrared cutoff $\kappa$ to regulate the parton splitting. The sum over splitting functions arises from the multiple contributions from partonic species $j$ splitting to $i$:
\begin{figure}[ht]
\centering
\includegraphics[scale=0.6]{2-PDFs/figs/qgq.pdf}
\end{figure}

The splitting functions $P_{ij}$  were known for some time at leading and next-to-leading accuracy \cite{Gross:1973ju,Georgi:1951sr,Floratos:1977au,Altarelli:1977zs,GonzalezArroyo:1979df,Floratos:1978ny,Furmanski:1980cm,Curci:1980uw,GonzalezArroyo:1979he,Floratos:1981hs,Hamberg:1991qt}, and more recently extended to next-next-to-leading order accuracy \cite{Moch:2004pa,Vogt:2004mw}. After convoluting the parton level functions with the PDFs, we obtain the full structure function
\ba
 F_2(x,Q^2) &=& \sum_i xe_i^2\left[\; f_i(x) \right.  \nonumber\\
 				 &+& \frac{\alpha_S}{2\pi}\int_0^1 \frac{d\xi}{\xi}\sum_j\left(P_{ij}\left(\frac{x}{\xi}\right)\log\frac{Q^2}{\kappa^2} + W_{ij}(x)\right)\; f_j(\xi) \nonumber \\
				 &+&  \left. \mathcal{O}(\alpha_S^2) \right]. \label{eq:f2nlo}
\ea
Our expression for the parton level structure function still suffers from the IR divergence when we take the limit $\kappa\to 0$. This issue may be resolved by concluding that the singularity arises from a breakdown of the ability of perturbation theory to describe physics in the strongly-coupled infrared. We may therefore attempt to factorise out the long distance behaviour of the structure functions into some bare parameters of the theory; analogously to the treatment of ultraviolet divergences by renormalisation of the strong coupling. In this instance we shall absorb the divergences present in the parton level structure functions into our parton distribution functions by replacing the bare quantities $f(x)$ with a physically accessible quantity measured at the \emph{factorisation scale} $\mu_f$. We can express these in terms of an expansion in the bare PDFs as
\be
f_i(x,\mu_F^2) = f_i(x) + \frac{\alpha_S}{2\pi}\int_0^1 \frac{d\xi}{\xi} \Delta^{(1)}_{ij}\left(\frac{x}{\xi}, \frac{\mu_F}{\kappa}\right)\; f_j(\xi) + \mathcal{O}(\alpha_S^2),
\ee
where the counter terms $\Delta^{(n)}_{ij}$ are formed as a sum of a regular part $\Delta^{(n)}_{r,ij}$ and a singular part $\Delta^{(n)}_{s,ij}$, and the sum over the dummy index $j$ is implicit. The singular part of these counterterms is uniquely specified by having to remove the divergence present in the structure functions due to the collinearly divergent parton splitting. Comparing to Eqn.~\ref{eq:f2nlo}, this divergence may be subtracted by setting
\be
\Delta^{(1)}_{s,ij} = P_{ij}\left(\frac{x}{\xi}\right)\log\frac{\mu_F^2}{\kappa^2}.
\ee
Unlike the divergent part, the regular part of the counter-term is not uniquely defined by the factorisation procedure. The choice of a specific regular counter-term is known as a \emph{factorisation scheme}; a choice consisting of shuffling terms between the regular part of the PDF definition and the coefficients present in the calculation. For example one may make a process-specific choice where all of the regular coefficients are absorbed into the PDF definition. In our example case of $F_2$ this is known as the DIS scheme~\cite{Altarelli:1978id}, $\Delta^{(1)}_{r,ij} = W_{ij}(x)$, in terms of which the form of the calculation becomes particularly simple:
\be
 F_2(x,Q^2) = 2 \int_0^1 \xi d\xi \sum_i  f^{\mathrm{DIS}}_i(\xi) e_{i}^2.
\ee
In practice this scheme choice is often rather unhelpful, as it does not permit a consistent definition of PDFs across multiple processes. With this in mind, the most common choice is the \emph{Modified Minimal Subtraction} or $\overline{\mathrm{MS}}$ scheme where the only regular counterterms are a process independent $\Delta^{(1)}_{r,ij} = \log 4\pi - \gamma_E$. In the $\overline{\mathrm{MS}}$ scheme therefore our factorised PDFs are given by
\be
f_i(x,\mu_F^2) = f_i(x) + \frac{\alpha_S}{2\pi} \sum_j  \left[\left(P_{ij}\left(x\right)\log\frac{\mu_F^2}{\kappa^2} + \log 4\pi - \gamma_E \right) \right] \otimes f_j(x) + \mathcal{O}(\alpha_S^2), \label{eq:renormpdf}
\ee
and the expression for $F_2$ becomes
\be
F_2(x,Q^2) = x \sum_i e_i^2 \left\{ f_i(x,\mu_F^2) +  \frac{\alpha_S}{2\pi}\int_x^1 \frac{d\xi}{\xi} f_i(\xi,\mu_F^2)\;\widetilde{W}_i\left(\frac{x}{\xi},\frac{Q^2}{\mu_F^2},\alpha_S\right) \right\},
 \ee
where the $\widetilde{W}_i$ are the finite contributions remaining after factorisation. While the relationship between the PDFs at the factorisation scale and the bare distributions is now divergent, the renormalised quantities may be measured at some scale and used in subsequent calculations, thus making the theory predictive. In general, under a universal factorisation scheme such as $\overline{\mathrm{MS}}$, structure functions may be calculated as
\be F(x,Q^2) = \sum_i \int_x^1 \frac{d\xi}{\xi} C_i\left(\frac{x}{\xi},\frac{Q^2}{\mu_F^2}, \alpha_S \right) f_i(\xi,\mu_F^2), \label{eq:DISsf} \ee
where the $C_i$ are the finite Wilson coefficients determined perturbatively and the PDFs $f_i$ encode the non-perturbative structure of the calculation. This differs from the naive parton model in that the Bjorken-scaling is now broken by logarithms of the hard scale $Q^2$, and the sum over parton species not only runs over spin-$1/2$ partons (the quarks of QCD), but also contains a contribution from an initial state gluon splitting into a quark-antiquark pair.
\subsection{DGLAP and PDF evolution} \label{sec:DGLAP} As a measurable quantity, the structure function itself clearly must be independent of the unphysical factorisation scheme and scale choices. The requirement of scheme independence is of course met when the factorisation scheme is followed consistently for the definition of PDFs and Wilson coefficients in all subsequent calculations. The requirement of factorisation scale independence leads to a renormalisation group equation (RGE) for the structure function
\be \mu_F \frac{d}{d\mu_F} F(x,Q^2) = 0,\ee
and consequently RGEs for the parton distributions and Wilson coefficients, once again in terms of the Altarelli-Parisi splitting functions $P_{ij}$
\be \mu_F \frac{d}{d\mu_F}f_i(y,\mu_F^2) = \sum_j \int_y^1 \frac{dz}{z} P_{ij}\left(\frac{y}{z},\alpha_S \right) f_j(z,\mu_F^2), \label{eq:DGLAP}\ee
\be \mu_F \frac{d}{d\mu_F}C_i\left(x,\frac{Q^2}{\mu_F^2}, \alpha_S \right) = -\sum_i \int_x^1 \frac{dy}{y} C_j\left(y,\frac{Q^2}{\mu_F^2}, \alpha_S \right) P_{ij}\left(\frac{x}{y},\alpha_S \right).\ee
These are known as the Altarelli-Parisi equations~\cite{AP} or the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations~\cite{dokshitzer,gribovlipatov,lipatov}, and they describe how PDFs change, or \emph{evolve} with the factorisation scale. Identically as the RGE for the running of the strong coupling performs a resummation of contributions arising from self energy diagrams, the DGLAP equation resums scale logarithms arising from collinear parton splittings.

The equations may be greatly simplified by moving to a PDF basis that largely diagonalises the matrix of splitting functions $P_{ij}$. For example we may construct a basis of \emph{non-singlet} PDFs, e.g the valence distributions
\be V_i = q_i - \bar{q_i}, \ee
and differences between quark sea distributions $q_s = q + \bar{q}$
\begin{eqnarray}
T_3 &=& u_s - d_s, \\
T_8 &=& u_s + d_s - 2s_s,  \\
T_{15} &=& u_s + d_s +s_s - 3c_s, \\
T_{24} &=&  u_s + d_s +s_s + c_s - 4b_s, \\
T_{35} &=&  u_s + d_s +s_s + c_s + b_s - 5t_s. \label{eq:evolbasis2}
\end{eqnarray}
As QCD is flavour blind, the gluon contribution to the evolution of these PDFs cancels, therefore diagonalising the matrix of splitting functions in this basis. For the nonsinglet distributions the DGLAP equation reduces to
\be \mu_F \frac{d}{d\mu_F}f^{\mathrm{NS}}_i(y,\mu_F^2) =\int_z^1 \frac{dz}{z} P^{\mathrm{NS}}_{i}\left(\frac{y}{z},\alpha_S \right) f^{\mathrm{NS}}_i(z,\mu_F^2).\label{eq:NSDGLAP}\ee
Completing this basis are the gluon and the flavour singlet $\Sigma = \sum_i (q_i +\bar{q}_i)$ PDFs. These remain coupled leading to a $2\times 2$ matrix of integro-differential equations for their evolution:
\be
 \mu_F \frac{d}{d\mu_F}
 \begin{pmatrix} g(x,\mu_F) \\  \Sigma(x,\mu_F) \end{pmatrix}  =
\int_z^1 \frac{dz}{z}
  \begin{pmatrix} P_{gg} & P_{g\Sigma} \\  P_{\Sigma g} & P_{\Sigma\Sigma} \end{pmatrix}
   \begin{pmatrix} g(z,\mu_F) \\  \Sigma(z,\mu_F) \end{pmatrix}. \label{eq:gSDGLAP}\ee
These equations may be solved for a PDF at some scale $Q^2$ evolved from an initial scale $Q_0^2$. Solutions typically follow one of two procedures; arguably the most direct consists of solving the equations iteratively through numerical methods in $x$-space. This method is followed in codes such as HOPPET~\cite{Salam:2008qg}, QCDNUM~\cite{Botje:2010ay} and APFEL~\cite{Bertone:2013vaa} which employ interpolation techniques to improve the speed of the solution. Alternatively the DGLAP equations may be solved by making use of the Mellin convolution theorem
\be \mathcal{M}\left\{f \otimes g\right\} = \mathcal{M}\left\{f\right\}\cdot \mathcal{M}\left\{g\right\}, \ee
whereby the multiplicative convolution present in equations \ref{eq:NSDGLAP}, \ref{eq:gSDGLAP} is reduced to a product in Mellin space; the method employed by QCD-Pegasus~\cite{Vogt:2004ns}. In the Mellin space approach, the emphasis largely lies on a fast numerical implementation of the Mellin inversion integral.

Through either method, the solution of the DGLAP equations provides a perturbative description of the behaviour of parton distributions as they vary in scale. However we remain short of a full description of the distributions having not determined their dependence upon the momentum fraction $x$. Furthermore the precise behaviour of the PDF and structure function renormalisation may be complicated in the attempt to overcome some of the approximations we have made so far regarding the masses of quarks contributing to our parton model, which we shall address here before discussing how the $x$ behaviour of the PDFs may be elucidated.
\clearpage
\section{Treatment of heavy quarks}
So far in our discussion of the QCD parton model we have made the assumption that all the quarks contributing in the theory are massless, an approximation that becomes increasingly untenable when investigating scattering processes with a hard scale approaching a quark's physical mass. A careful treatment of terms depending on quark masses is therefore vital for making theoretical predictions to a dataset that spans heavy quark mass thresholds.

Dealing with heavy quark mass effects is a delicate issue in that different treatments generally have different regions of applicability. The specific combination of approaches to quark masses used when confronting a dataset with a broad reach in hard scale is known as a heavy quark \emph{scheme}, although not necessarily in the spirit of factorisation or renormalisation schemes as the choice often lies in the particulars of the approximation rather than in some arbitrary shuffling of parameters. A heavy quark scheme choice can therefore potentially lead to differences with alternative calculations that do not in principle vanish in the limit of an all-orders calculation.

The space of heavy quark renormalisation schemes is bounded by two regimes where the treatment is fairly simple, the fixed flavour number scheme (FFNS) and the zero-mass variable flavour number scheme (ZM-VFNS). The remaining schemes, known as general-mass variable flavour number schemes (GM-VFNS) aim to interpolate between the FFNS and ZM-VFNS, reducing to the simpler calculations in certain kinematic limits. Motivated by observations suggesting that a more careful treatment of quark mass effects is phenomenologically relevant at the LHC~\cite{Tung:2006tb}, a number of such schemes have arisen in an attempt to better describe experimental data. These typically differ by sub-leading terms in the method of interpolation between the two limiting regimes. We shall now outline some of the available choices and their potential impact in the case of a deep-inelastic scattering analysis. For simplicity we shall discuss a theory with $n_l$ light quarks, and attempt to introduce a single massive quark $h$ with mass $m_h$.

\subsection{The FFN and ZM-VFN schemes}
We consider first the kinematical regime where the hard scale of our scattering problem is of similar order or smaller than our heavy quark mass; $Q^2 \lesssim m_h^2$. Making the assumption that the initial state proton has no intrinsic heavy quark component it is reasonable to treat the heavy quark as a purely final state particle, and the only partons in the theory are the $n_l$ light quark flavours and the gluon. In this instance, setting the factorisation and renormalisation scales $\mu_F^2=\mu_R^2 = \mu^2$; the calculation of a structure function in Eqn.~\ref{eq:DISsf} takes the form
\be F(n_l, Q^2, m_h^2) = \sum_i^{n_l}  C_i\left(n_l, \frac{Q^2}{m_h^2}, \frac{\mu^2}{m_h^2}, \frac{Q^2}{\mu^2} \right) \otimes f_i(n_l, \mu^2), \label{eq:FFN} \ee
where the sum is over light quark flavours only and the full mass dependence of the heavy quark is intact in the calculation. The structure function can be separated into a contribution where only light flavours are present, $F^{L}$, and a contribution including the heavy flavour $F^{H}$ as,
\be F(n_l, Q^2, m_h^2) = F^{L}(n_l, Q^2) + F^{H}(n_l, Q^2,m_h^2), \ee
where
\ba
F^{L}(n_l, Q^2) &=& \sum_i^{n_l}  L_i\left(n_l, \frac{Q^2}{\mu^2} \right) \otimes f_i(n_l, \mu^2),\\
F^{H}(n_l, Q^2,m_h^2) &=& \sum_o^{n_l} H_i\left(n_l, \frac{Q^2}{m_h^2}, \frac{\mu^2}{m_h^2}, \frac{Q^2}{\mu^2} \right) \otimes f_i(n_l, \mu^2).
\ea
Here $L$ denotes the Wilson coefficients that do not contain heavy quark lines, and $H$ includes only the diagrams that do. In this instance the heavy quark structure function first contributes at $\mathcal{O}(\alpha_S)$ via the splitting of an initial state gluon into a $h\bar{h}$ pair:
\begin{figure}[ht]
\centering
\includegraphics[scale=0.6]{2-PDFs/figs/FFNS.pdf}
\end{figure}

This approach is known as the \emph{decoupling} or FFN scheme where the only quarks treated as partons are the $n_l$ light quarks. The expression in Eqn.~\ref{eq:FFN} is unique up to terms of order $m_l^2/Q^2$ in the light quark masses which are typically treated as part of the factorisation level corrections of $\mathcal{O}(\Lambda^2_{\mathrm{QCD}}/Q^2)$. While accurate in the quark mass threshold region and below, this scheme suffers from unresummed logarithms of the ratio $Q^2/m_h^2$ contained in the Wilson coefficients which can become large and damage the convergence of the perturbative series at scales much larger than the heavy quark mass.

These problems may be resolved in a scheme which treats the heavy quark as a massless parton above its mass threshold with the introduction of an associated heavy quark PDF. The subsequent renormalisation of the PDF resums the logarithmic contributions due to parton splitting via solution of the DGLAP equation, removing a significant disadvantage present in the FFN treatment. As this scheme is identical to the zero mass scheme discussed previously, but with an additional partonic flavour, this procedure is known as the Zero-Mass Variable Flavour Number (ZM-VFN) scheme. In the ZM-VFN a structure function calculation is simply

\be F(n_l+1, x,Q^2) = \sum_i^{n_l+1} C_i\left(n_l+1,\frac{Q^2}{\mu^2} \right) \otimes f_i(n_l+1,\mu^2). \label{eq:ZMVFN} \ee
In this instance the heavy quark contribution to the structure function first arises now at leading order via diagrams of the type:

\begin{figure}[h]
\centering
\includegraphics[scale=0.6]{2-PDFs/figs/ZMVFNS.pdf}
\end{figure}

In the ZM-VFNS the heavy quark PDFs are set to zero below mass threshold and evolved as a massless parton according to the DGLAP equations for scales greater than the heavy quark mass. While this method alleviates the difficulties present in the FFN scheme at large scales, its treatment of heavy quarks only in terms of massless partons completely ignores the massive contributions to the Wilson coefficients and is therefore no longer exact. The reliability of the ZM scheme is therefore particularly reduced in the region where powers of $m_h^2/Q^2$ are significant.
\subsection{General mass schemes}
Analyses of QCD measurements are often performed by making a choice between using a suitable FFN scheme at scales in the region of heavy quark mass thresholds or a ZM scheme at high scales where the associated powers of $m_h^2/Q^2$ can be safely neglected. In either case the treatment of heavy quarks is at least unambiguous, with the ZM approach yielding a simpler procedure as there is no requirement to calculate coefficient functions with the heavy quark masses intact.

For analyses of a large dataset, potentially spanning several heavy quark thresholds and extending to very high scales, the desire to improve the perturbative reliability of the calculations has led to the development of a number of hybrid or \emph{general mass} schemes. In such schemes the treatments generally reduce to the FFN regime at low scales and the ZM treatment at high scales, with the intermediate regime handled via some interpolation between the two. Generally in a Variable Flavour Number (VFN) scheme one requires that
\be F^{L}(n_l, Q^2) + \lim_{Q^2 \gg m_h^2} \left[ F^{H}(n_l, Q^2,m_h^2)\right] =  F(n_l+1, x,Q^2), \label{eq:VFN}\ee
i.e. that the ZM-VFN and FFN calculations coincide at large scales, where the heavy quark mass dependance of the FFN Wilson coefficients can be neglected. The constraint in Eqn. \ref{eq:VFN} means that parton distributions in the two schemes may be related by a perturbatively calculable transformation.

\be f_i(n_l+1,\mu^2) = \sum_j^{n_l}A_{ij}\left(n_l, \frac{\mu^2}{m_h^2}\right)\otimes f_j(n_l, \mu^2), \label{eq:FLVreln}\ee
where the $A$ are determined to NNLO in $\alpha_S$ in Refs.~\cite{Buza:1995ie,Buza:1996wv}. It should be noted that the $A$ are not square matrices, with $i$ running over the $n_l+1$ partons in the zero mass scheme, and the $j$ running over the $n_l$ partons in the FFN.

In general a GM-VFN operates as a tower of FFN-type schemes with increasing $n_l$ as the scale increases over each quark mass threshold. In constructing a GM-VFN, the guiding principle is that physical observables should be continuous across these thresholds and therefore continuous across the $n_l$ and $n_l+1$ regimes.
Taking the heavy quark mass itself as the matching point between the two regimes, we demand that the GM-VFN structure function $F^{\text{GM}}$ obeys
\ba
F^{\text{GM}}(m_h^2) &=& \sum_j^{n_l} C^{\text{GM}}_j \left(n_l, m_h^2 \right) \otimes f_j(n_l) \nonumber \\
&=&   \sum_i^{n_l+1} C^{\text{GM}}_i\left( n_l+1, m_h^2 \right) \otimes f_i(n_l+1). \label{eq:GMmatching}
\ea
where the dependance upon the perturbative scales has been omitted for notational simplicity, and the GM superscripts refer to the coefficients in a general mass scheme. Using the relation in Eqn. \ref{eq:FLVreln} we can express the $n_l+1$ expression in the matching Eqn. \ref{eq:GMmatching} in terms of the $n_l$ scheme PDFs, therefore obtaining the relation
\ba && \sum_j^{n_l} C^{\text{GM}}_j \left(n_l, m_h^2 \right) \otimes f_j(n_l) \\
=  && \sum_i^{n_l+1} \sum_j^{n_l } C^{\text{GM}}_i\left( n_l+1, m_h^2 \right) \otimes A_{ij}\left(n_l, m_h^2\right)\otimes f_j(n_l). \ea
Subsequently, we may make the identification
\be  C^{\text{GM}}_j \left(n_l, m_h^2 \right)
=   \sum_i^{n_l+1} C^{\text{GM}}_i\left( n_l+1,m_h^2 \right) \otimes A_{ij}\left(n_l, m_h^2\right),\label{eq:minimalGM}\ee
which provides the minimal description for the construction of a GM-VFN scheme~\cite{Thorne:2008xf}. Ensuring that Eqn.~\ref{eq:minimalGM} is satisfied order by order in $\alpha_S$, we can construct the expression for the GM-VFN scheme coefficient functions above the heavy quark mass threshold. Taking the simplistic example case of Ref.~\cite{Kramer:2000hn} with a theory including only a gluon and a single heavy quark ($h=\bar{h}$), the GM-VFN coefficients may be constructed to order $\alpha_S$ as
\ba
C_g^{\text{LO}}(n_l+1,m_h) &=& C_g^{\text{LO}}(n_l,m_h), \\
C_g^{\text{NLO}}(n_l+1,m_h) &=& C_g^{\text{NLO}}(n_l,m_h) \nonumber \\
&-& C_h^{\text{LO}}(n_l+1,m_h)\otimes A^{\text{LO}}_{hg}(n_l, m_h^2). \label{eq:ACOT}
\ea
where the GM superscript has been omitted, the new superscript specifying the order of the term in the perturbative expansions of the quantities $C$ and $A$. Here the rightmost term in the $\mathcal{O}(\alpha_S)$ expression Eqn.~\ref{eq:ACOT} is known as the \emph{subtraction term} which ensures the cancellation of the IR-unsafe scale logs present in the FFN calculation. The ambiguity in the definition of a GM-VFNS arises upon noticing that terms proportional to powers of $m_h/Q$ may be interchanged between the Wilson coefficients in Eqn.~\ref{eq:ACOT} without changing the final value of the structure function. In this respect changing the distribution of terms between the gluon and heavy quark initiated diagrams in Eqn.~\ref{eq:ACOT} provides the opportunity to perform a \emph{scheme choice}, a freedom which has been exploited by several different GM-VFN scheme implementations.

The earliest complete description of a GM-VFNS was provided by the ACOT method~\cite{Collins:1978wz} which ensures the continuity of physical quantities through Eqn.~\ref{eq:minimalGM}, but does not attempt to take advantage of the degeneracy in the GM-VFN procedure. An important result was achieved with the Simplified-ACOT or S-ACOT  scheme~\cite{Collins:1998rz,Kramer:2000hn} which was able to exploit this ambiguity to considerably simplify the calculation of physical observables. In the S-ACOT scheme it was noted that shifts of the Wilson coefficients by their zero-mass limits may be absorbed into a redefinition of the GM-VFNS. That is, terms such as
\be C_h(n_l+1, m_h) - C_h(n_l+1, 0),\ee
vanish in the limit $Q^2 \gg m_h^2$, and therefore do not spoil the interpolation between the FFN and ZM schemes. This leads to the option of shifting to a simpler scheme where the massive heavy quark initiated coefficients may instead be evaluated with the heavy quark mass set to zero. Other options for the scheme definition were explored by Thorne and Roberts in the TR type schemes~\cite{Thorne:1997uu,Thorne:1997ga}, with the additional constraint that scale derivatives of heavy flavour structure functions should also be continuous at the matching scale.
%
\subsubsection{The FONLL approach}
%
A more recent approach was developed by examining methods previously used to combine fixed order calculations with next-to-leading log resummation via the FONLL method~\cite{Cacciari:1998it}. The method was extended from the original application of studying the \pt spectrum in heavy flavour hadroproduction to the treatment of heavy quarks in DIS by Forte \emph{et al.}~\cite{Forte:2010ta}. The procedure begins by inverting the relationship in Eqn.~\ref{eq:FLVreln} so as to express an $n_l$ flavour structure function in terms of $n_l+1$ flavour PDFs,
\be F(n_l,Q^2) = \sum_i^{n_l} B_i\left(\frac{Q^2}{m_h^2}\right)\otimes f_i(n_l+1,Q^2), \label{eq:FONLLmassive}\ee
where it is important to note that the sum over flavours does not include the heavy flavour PDF, and the full heavy quark mass dependence is present in the coefficients $B$. To perform a matching with the massless scheme, the ZM result in Eqn.~\ref{eq:ZMVFN} can be expressed in terms of light flavour PDFs only, given the assumption that the heavy flavour PDF is generated perturbatively. In this case, Eqn.~\ref{eq:ZMVFN} can be written
\be F(n_l+1, Q^2) = \sum_i^{n_l} \widetilde{C}_i\left(n_l+1,\frac{Q^2}{m_h^2} \right) \otimes f_i(n_l+1,\mu^2). \label{eq:FONLLmassless}\ee
where once again, the sum runs over only light flavours, this time with the heavy flavour contribution being generated via DGLAP evolution included into the modified coefficient function $\widetilde{C}$. To understand which terms are common in the two descriptions, the massive coefficient functions may be decomposed into terms logarithmically dependant upon the heavy quark mass, and terms suppressed by powers of $m_h/Q$:
\be B_i\left(\frac{Q^2}{m_h^2}\right) = \overline{B}_i\left(\frac{Q^2}{m_h^2}\right) + \mathcal{O}\left(\frac{m_h}{Q}\right).\ee
As only the power suppressed terms vanish in the limit of $Q^2 \gg m_h^2$, the terms remaining must be common to both the ZM and massive scheme calculations. We can therefore express the massive structure function in a `\emph{massless}' limit, having dropped those terms in the coefficient functions that are suppressed by powers of $m_h/Q$:
\be \overline{F}(n_l,Q^2) = \sum_i^{n_l} \overline{B}_i\left(\frac{Q^2}{m_h^2}\right)\otimes f_i(n_l+1,Q^2).\label{eq:FONLLdoublecount}\ee
The FONLL result for the structure function is given by the sum of the massive calculation in Eqn.~\ref{eq:FONLLmassive}, and the massless calculation in Eqn.~\ref{eq:FONLLmassless} with the asymptotic limit of the massive calculation in Eqn.~\ref{eq:FONLLdoublecount} subtracted.
\be
F^{\mathrm{FONLL}}(Q^2) =\left[ F(n_l,Q^2) + F(n_l+1, Q^2) \right] - \overline{F}(n_l,Q^2).
\ee
With the subtraction ensuring the cancellation of terms which are double counted between the massive and massless calculations. Therefore in this expression the mass-suppressed terms present in the FFN calculation are fully accounted for in the GM scheme, with the duplicate terms subtracted. The simplicity of this approach helped to elucidate many of the differences between general mass schemes.

It should be noted that while general mass schemes suffer from an ambiguity in their definition compared to the simpler fixed-flavour and zero mass schemes, the differences between them are always of higher order compared to the calculation at hand, as is the case in any true scheme choice. Indeed, a well-defined GM-VFNS will always reduce to the decoupled result at low scales and the zero-mass result at scales much higher than the quark mass, behaving effectively as a tower of fixed flavour schemes with increasing number of partonic quarks. The general-mass schemes therefore do not suffer from a significant loss of predictive power, and are able to provide considerable improvement over the simpler schemes when dealing with datasets spanning quark mass thresholds.
%
\section{General features of parton distributions}
%
While we have now described how the parton distributions functions at an experimental scale $Q^2$ may be found by evolving parton distributions from an initial scale, and discussed briefly how the renormalisation of heavy quark distributions may be accomplished, the issue of determining the functional dependence of the parton distributions upon the momentum fraction $x$ at some initial scale $f_i(x,Q_0^2)$ remains.

The number of independent PDFs to be determined is dependent upon the choice of initial scale, as quark distributions that can be considered \emph{heavy} with respect to $Q^2_0$ may be generated perturbatively through the DGLAP procedure outlined previously. The typical choice is to determine the parton distributions at some scale $ m_s^2< Q^2_0 \le m_c^2$ such that the flavours $c$, $b$, $t$ are produced by evolution. These scale choices minimise the number of distributions to be determined while remaining perturbatively reliable.

As the remaining seven distributions\footnote{The gluon, the $u$, $d$, $s$ quarks and their antiquarks.} are fundamentally a parametrisation of the nonperturbative dynamics of the proton, they are by definition out of reach of a perturbative analysis. There are however some general statements that may be made of their $x$-dependence that are independent of the hard scale. The most important of which are the parton distribution \emph{sum rules} which constrain the relative normalisation of PDFs.

Firstly, the \emph{momentum sum rule} (MSR) ensures that the parton distributions' fractional momenta sum to the momentum of the parent proton
\be  \int_0^1 dx \left[  x\Sigma(x,Q^2) + xg(x,Q^2) \right]= 1, \label{eq:MSR} \ee
where $\Sigma$ is the singlet distribution defined previously. Following this are the quark valence sum rules. These fix the quark distributions such that the resulting proton has the appropriate quantum numbers,
\begin{subequations}
\label{eq:VSR}
\ba \text{up-valence:}& \: &\int_0^1 dx\left( f_u(x,Q^2) - f_{\bar{u}}(x,Q^2) \right) = 2, \label{eq:UVSR} \\
      \text{down-valence:}&  \: & \int_0^1 dx\left( f_d(x,Q^2) - f_{\bar{d}}(x,Q^2) \right) = 1, \label{eq:DVSR}\\
      \text{strange-valence:}&  \: & \int_0^1 dx\left( f_s(x,Q^2) - f_{\bar{s}}(x,Q^2) \right) = 0. \label{eq:SVSR}
\ea
\end{subequations}

From these rules we may infer additional constraints upon individual PDFs. The MSR suggests a form for the large-$x$ behaviour of the distributions, in that they should parametrically tend to zero as $x\to1$. The number sum rules in Eqns. \ref{eq:VSR} require the valence-type distributions to be integrable over the whole $x$-range. While there is no requirement for the singlet and gluon distributions to be integrable, their first moments must be, as required by the MSR. Combining these three constraints we may parametrise the large and small-$x$ behaviour of both valence-like and gluon or singlet-like distributions as:
\ba f_V(x,Q_0^2) &= N_V\;x^{\alpha_V}(1-x)^{\beta_V}\,r_V(x), \nonumber \\
 f_\Sigma(x,Q_0^2) &= N_\Sigma\;x^{\alpha_\Sigma}(1-x)^{\beta_\Sigma}\,r_\Sigma(x). \label{eq:pdflimits}\ea
In these expressions, the parameters $\alpha$ and $\beta$ control the small and large-$x$ PDF behaviour respectively. The $\beta$ should be such that the PDFs tend to zero smoothly at large-$x$, and the $\alpha$ such that the valence distributions are integrable, and the first moment of the gluon and singlet are integrable. The overall PDF normalisations $N$ being constrained via the appropriate sum rules.

Finally, what remains in the determination of the distributions are the remainder terms $r(x)$ which describe the PDFs between the two $x$-limits. Their determination is considerably more complex and is a ongoing source of research. Much of this thesis will be dedicated to discussing the determination of these remainder functions.


\documentclass[10pt,fleqn]{article}
\usepackage{amsmath}
\usepackage{cancel}
\usepackage[letterpaper, landscape, margin=1in]{geometry}

\setlength{\parindent}{0pt}

\begin{document}

\section{Pseudo- visco- acoustic VTI variable density coupled second order self adjoint system}
John Washbourne, Ken Bube\\
September 20, 2013

\section{Introduction}
This note shows the derivation of \textit{time update equations} for the pseudo- visco-
acoustic vertical transverse isotropy (VTI) variable density coupled second order self-adjoint
system. We implement attenuation with a monochromatic approximation to Maxwell bodies, and use
this attenuation model to implement zero outgoing absorbing boundary conditions on the exterior
of the modeling domain.
\vspace{10pt}

The time update equations are used to advance solutions in time, expressing the quasi-p
pressure wavefield $P_{(t + \Delta)}$ and quasi-s pressure wavefield $M_{(t + \Delta)}$
at the next time step $(t + \Delta)$ as functions of $P_{(t - \Delta)}, P_{(t)}$ and $M_{(t - \Delta)}, P_{(t)}$

\subsection{Symbols}
\begin{center}
\begin{tabular}{ll} \\[-10pt]
$\partial_t,\ \partial_x,\ \partial_y,\ \partial_z$ & $
\displaystyle \frac{\partial }{\partial t},\
\displaystyle \frac{\partial }{\partial x},\
\displaystyle \frac{\partial }{\partial y},\
\displaystyle \frac{\partial }{\partial z} $ \\[15pt]
$\Delta$ & Temporal sampling \\[15pt]
$\omega, Q$ & reference frequency for attentuation, attenuation at frequency $\omega$ \\[10pt]
$P, M$ & quasi-P, quasi-S wavefields\\[10pt]
$s_p(x,y,z,t),\ s_m(x,y,z,t)$ & quasi-P, quasi-S source terms\\[10pt]
$f$ & $ 1 - \displaystyle \frac{V_p^2}{V_s^2} $\\[10pt]
$ \widehat{\eta} $ & $ \sqrt{ \displaystyle \frac{2 \left(\epsilon - \delta\right) }{f + 2 \epsilon } } $\\[10pt]
$b$ & buoyancy = $\displaystyle 1/\rho $ (reciprocal density) \\[10pt]
$ \{\ V_p,\ \epsilon,\ \widehat{\eta}\ \} $ & Material parameters \\[10pt]
\end{tabular}
\end{center}

\newpage
\subsection{Coupled second order modeling system}
Equation \ref{eq:system} shows the modeling system with absorbing boundaries
implemented using amplitude only (dissipation only, no dispersion) Q.
\vspace{10pt}

We apply the time derivatives in the terms $\displaystyle \frac{\omega}{Q} \partial_t p$
and $\displaystyle \frac{\omega}{Q} \partial_t m$ using a backward one-sided numerical
difference. We tested both forward one-sided and centered difference alternatives and
found them to be less stable.

\begin{equation}
\begin{aligned}
&\frac{b}{V_p^2} \left( \partial_t^2(P) + \frac{\omega}{Q} \partial_t(P) \right) = \
\partial_x \left( b (1 + 2 \epsilon) \partial_x(P) \right) +
\partial_y \left( b (1 + 2 \epsilon) \partial_y(P) \right) +
\partial_z \left( b (1 - f \widehat{\eta}^2) \partial_z(P) \right) +
\partial_z \left( b f \widehat{\eta} \sqrt{1 - \widehat{\eta}^2} \partial_z(M) \right) + s_p \\[10pt]
&\frac{b}{V_p^2} \left( \partial_t^2(M) + \frac{\omega}{Q} \partial_t(M) \right) = \
\partial_x \left( b (1 - f) \partial_x(M) \right) +
\partial_y \left( b (1 - f) \partial_y(M) \right) +
\partial_z \left( b (1 - f + f \widehat{\eta}^2) \partial_z(M) \right) +
\partial_z \left( b f \widehat{\eta} \sqrt{1 - \widehat{\eta}^2} \partial_z(P) \right) + s_m \\[10pt]
\end{aligned}
\label{eq:system}
\end{equation}

\section{Time update equations}

\subsection{Time update numerical difference formulas, first and second order}
\begin{equation}
\partial_t p = \frac{1}{\Delta} \left[ p_{(t)} - p_{(t - \Delta)} \right]
\label{eq:diff1}
\end{equation}

\begin{equation}
\begin{aligned}
\partial_t^2 p &= \frac{1}{\Delta^2} \left[ p_{(t+\Delta)} - 2 p_{(t)} + p_{(t - \Delta)} \right] \\[10pt]
p_{(t+\Delta)} &= \Delta^2 \partial_t^2 p + 2 p_{(t)} - p_{(t - \Delta)}
\end{aligned}
\label{eq:diff2}
\end{equation}

\subsection{Rearrange terms for $\partial_t^2(P)$ and $\partial_t^2(M)$ }
\begin{equation}
\begin{aligned}
\partial_t^2(P) = \
& \frac{V_p^2}{b} \left[
\partial_x \left( b (1 + 2 \epsilon) \partial_x(P) \right) +
\partial_y \left( b (1 + 2 \epsilon) \partial_y(P) \right) +
\partial_z \left( b (1 - f \widehat{\eta}^2) \partial_z(P) \right) +
\partial_z \left( b f \widehat{\eta} \sqrt{1 - \widehat{\eta}^2} \partial_z(M) \right) + s_p
\right]
- \frac{\omega}{Q} \partial_t(P) \\[10pt]
\partial_t^2(M) = \
& \frac{V_p^2}{b} \left[
\partial_x \left( b (1 - f) \partial_x(M) \right) +
\partial_y \left( b (1 - f) \partial_y(M) \right) +
\partial_z \left( b (1 - f + f \widehat{\eta}^2) \partial_z(M) \right) +
\partial_z \left( b f \widehat{\eta} \sqrt{1 - \widehat{\eta}^2} \partial_z(P) \right) + s_m
\right]
- \frac{\omega}{Q} \partial_t(M) \\[10pt]
\end{aligned}
\nonumber
\end{equation}

\subsection{Apply equations \ref{eq:diff1} and \ref{eq:diff2}, and rearrange}
\begin{equation}
\begin{aligned}
P^{t+\Delta} = \
& \Delta^2\ \frac{V_p^2}{b} \left[
\partial_x b \left( (1 + 2 \epsilon) \partial_x(P) \right) +
\partial_y b \left( (1 + 2 \epsilon) \partial_y(P) \right) +
\partial_z b \left( (1 - f \widehat{\eta}^2) \partial_z(P) \right) +
\partial_z b \left( f \widehat{\eta} \sqrt{1 - \widehat{\eta}^2} \partial_z(M) \right) + s_p \right] \\[10pt]
& \quad - \Delta\ \frac{\omega}{Q} \left( P^{t} - P^{t-\Delta} \right)
+ 2 P^t - P^{t - \Delta}
\\[10pt]
M^{t+\Delta} = \
& \Delta^2\ \frac{V_p^2}{b} \left[
\partial_x b \left( (1 - f) \partial_x(M) \right) +
\partial_y b \left( (1 - f) \partial_y(M) \right) +
\partial_z b \left( (1 - f + f \widehat{\eta}^2) \partial_z(M) \right) +
\partial_z b \left( f \widehat{\eta} \sqrt{1 - \widehat{\eta}^2} \partial_z(P) \right) + s_m \right] \\[10pt]
& \quad - \Delta\ \frac{\omega}{Q} \left( M^{t} - M^{t-\Delta} \right)
+ 2 M^t - M^{t - \Delta}
\end{aligned}
\nonumber
\end{equation}

\end{document}


\subsection{Results by Model Families}
\label{fams}

\begin{center}
\captionof{table}{Ranking of benchmark and horizontal models
                  ($1~\text{km}^2$ pixel size, 60-minute time steps):
                  the table shows the ranks for cases with $2.5 < ADD < 25$
                  (and $25 < ADD < \infty$ in parentheses if they differ)}
\label{t:hori}
\begin{tabular}{|c|ccc|cccccccc|}
\hline
\multirow{2}{*}{\rotatebox{90}{\thead{\scriptsize{Training}}}}
    & \multicolumn{3}{c|}{\thead{Benchmarks}}
    & \multicolumn{8}{c|}{\thead{Horizontal (whole-day-ahead)}} \\
\cline{2-12}
~ & \textit{naive}     & \textit{fnaive}   & \textit{paive}
  & \textit{harima}    & \textit{hcroston} & \textit{hets} & \textit{hholt}
  & \textit{hhwinters} & \textit{hses}     & \textit{hsma} & \textit{htheta} \\
\hline \hline
3 & 11      &  7 (2) &  8 (5) & 5 (7) & 4     & 3
  &  9 (10) & 10 (9) &  2 (6) & 1     & 6 (8) \\
4 & 11      &  7 (2) &  8 (3) & 5 (6) & 4 (5) & 3 (1)
  &  9 (10) & 10 (9) &  2 (7) & 1 (4) & 6 (8) \\
5 & 11      &  7 (2) &  8 (4) & 5 (3) & 4 (9) & 3 (1)
  &  9 (10) & 10 (5) &  2 (8) & 1 (6) & 6 (7) \\
6 & 11      &  8 (5) &  9 (6) & 5 (4) & 4 (7) & 2 (1)
  & 10      &  7 (2) &  3 (8) & 1 (9) & 6 (3)  \\
7 & 11      &  8 (5) & 10 (6) & 5 (4) & 4 (7) & 2 (1)
  &  9 (10) &  7 (2) &  3 (8) & 1 (9) & 6 (3) \\
8 & 11      &  9 (5) & 10 (6) & 5 (4) & 4 (7) & 2 (1)
  &  8 (10) &  7 (2) &  3 (8) & 1 (9) & 6 (3) \\
\hline
\end{tabular}
\end{center}
\

Besides the overall results, we provide an in-depth comparison of models
    within a family.
Instead of reporting the MASE per model, we rank the models holding the
    training horizon fixed to make comparison easier.
Table \ref{t:hori} presents the models trained on horizontal time series.
In addition to \textit{naive}, we include \textit{fnaive} and \textit{pnaive}
    already here as more competitive benchmarks.
The tables in this section report two rankings simultaneously:
The first number is the rank resulting from lumping the low and medium
    clusters together, which yields almost the same rankings when analyzed
    individually.
The ranks from only high demand pixels are in parentheses if they differ.

A first insight is that \textit{fnaive} is the best benchmark in all
    scenarios:
Decomposing flexibly by tuning the $ns$ parameter is worth the computational
    cost.
Further, if one is limited in the number of non-na\"{i}ve methods,
    \textit{hets} is the best compromise and works well across all demand
    levels.
It is also the best model independent of the training horizon for high demand.
With low or medium demand, \textit{hsma} is the clear overall winner; yet,
    with high demand, models with a seasonal fit (i.e., \textit{harima},
    \textit{hets}, and \textit{hhwinters}) are more accurate, in particular,
    for longer training horizons.
This is due to demand patterns in the weekdays becoming stronger with higher
    overall demand.

\begin{center}
\captionof{table}{Ranking of classical models on vertical time series
                  ($1~\text{km}^2$ pixel size, 60-minute time steps):
                  the table shows the ranks for cases with $2.5 < ADD < 25$
                  (and $25 < ADD < \infty$ in parentheses if they differ)}
\label{t:vert}
\begin{tabular}{|c|cc|ccccc|ccccc|}
\hline
\multirow{2}{*}{\rotatebox{90}{\thead{\scriptsize{Training}}}}
    & \multicolumn{2}{c|}{\thead{Benchmarks}}
    & \multicolumn{5}{c|}{\thead{Vertical (whole-day-ahead)}}
    & \multicolumn{5}{c|}{\thead{Vertical (real-time)}} \\
\cline{2-13}
~ & \textit{hets}  & \textit{hsma}   & \textit{varima} & \textit{vets}
  & \textit{vholt} & \textit{vses}   & \textit{vtheta} & \textit{rtarima}
  & \textit{rtets} & \textit{rtholt} & \textit{rtses}  & \textit{rttheta} \\
\hline \hline
3 &  2 (10) &  1  (7) & 6 (4) & 8 (6) & 10 (9)
  &  7  (5) & 11 (12) & 4 (1) & 5 (3) &  9 (8) & 3 (2) & 12 (11) \\
4 &  2  (8) &  1 (10) & 6 (4) & 8 (6) & 10 (9)
  &  7  (5) & 12 (11) & 3 (1) & 5 (3) &  9 (7) & 4 (2) & 11 (12) \\
5 &  2  (3) &  1 (10) & 7 (5) & 8 (7) & 10 (9)
  &  6      & 11      & 4 (1) & 5 (4) &  9 (8) & 3 (2) & 12 \\
6 &  2  (1) &  1 (10) & 6 (5) & 8 (7) & 10 (9)
  &  7  (6) & 11 (12) & 3 (2) & 5 (4) &  9 (8) & 4 (3) & 12 (11) \\
7 &  2  (1) &  1 (10) & 8 (5) & 7     & 10 (9)
  &  6      & 11 (12) &	5 (2) & 4     &  9 (8) & 3     & 12 (11) \\
8 &  2  (1) &  1  (9) & 8 (5) & 7 (6) & 10 (8)
  &  6      & 12 (10) & 5 (2) & 4     &  9 (7) & 3     & 11 \\
\hline
\end{tabular}
\end{center}
\

Table \ref{t:vert} extends the previous analysis to classical models trained
    on vertical time series.
Now, the winners from before, \textit{hets} and \textit{hsma}, serve as
    benchmarks.
Whereas for low and medium demand, no improvements can be obtained,
    \textit{rtarima} and \textit{rtses} are the most accurate with high demand
    and short training horizons.
For six or more training weeks, \textit{hets} is still optimal.
Independent of retraining and the demand level, the models' relative
    performances are consistent:
The \textit{*arima} and \textit{*ses} models are best, followed by
    \textit{*ets}, \textit{*holt}, and \textit{*theta}.
Thus, models that can deal with auto-correlations and short-term forecasting
    errors, as expressed by moving averages, and that cannot be distracted by
    trend terms are optimal for vertical series.

Finally, Table \ref{t:ml} compares the two ML-based models against the
    best-performing classical models and answers \textbf{Q2}:
With low and medium demand, no improvements can be obtained again; however,
    with high demand, \textit{vrfr} has the edge over \textit{rtarima} for
    training horizons up to six weeks.
We conjecture that \textit{vrfr} fits auto-correlations better than
    \textit{varima} and is not distracted by short-term noise as
    \textit{rtarima} may be due to the retraining.
With seven or eight training weeks, \textit{hets} remains the overall winner.
Interestingly, \textit{vsvr} is more accurate than \textit{vrfr} for low and
    medium demand.
We assume that \textit{vrfr} performs well only with strong auto-correlations,
    which are not present with low and medium demand.

\begin{center}
\captionof{table}{Ranking of ML models on vertical time series
                  ($1~\text{km}^2$ pixel size, 60-minute time steps):
                  the table shows the ranks for cases with $2.5 < ADD < 25$
                  (and $25 < ADD < \infty$ in parentheses if they differ)}
\label{t:ml}
\begin{tabular}{|c|cccc|cc|}
\hline
\multirow{2}{*}{\rotatebox{90}{\thead{\scriptsize{Training}}}}
    & \multicolumn{4}{c|}{\thead{Benchmarks}}
    & \multicolumn{2}{c|}{\thead{ML}} \\
\cline{2-7}
~ & \textit{fnaive}  & \textit{hets} & \textit{hsma}
  & \textit{rtarima} & \textit{vrfr} & \textit{vsvr} \\
\hline \hline
3 & 6     & 2 (5) & 1 (4) & 3 (1) & 5 (2) & 4 (3) \\
4 & 6 (5) & 2 (4) & 1 (6) & 3 (2) & 5 (1) & 4 (3) \\
5 & 6 (5) & 2 (4) & 1 (6) & 4 (2) & 5 (1) & 3 \\
6 & 6 (5) & 2     & 1 (6) & 4     & 5 (1) & 3 \\
7 & 6 (5) & 2 (1) & 1 (6) & 4     & 5 (2) & 3 \\
8 & 6 (5) & 2 (1) & 1 (6) & 4     & 5 (2) & 3 \\
\hline
\end{tabular}
\end{center}
\

Analogously, we created tables like Table \ref{t:hori} to \ref{t:ml} for the
    forecasts with time steps of 90 and 120 minutes and find that the relative
    rankings do not change significantly.
The same holds true for the rankings with changing pixel sizes.
For conciseness reasons, we do not include these additional tables in this
    article.
In summary, the relative performances exhibited by certain model families
    are shown to be rather stable in this case study.


\section{Brown Dwarf vs Giant Planet}
\label{sec:BDvsGP}
\subsection{Hot Star vs Cold Star}
\label{sec:hot_vs_cold}
\cite{Marley2007} suggested that young giant planets formed by core accretion in a protoplanetary disk should
have a much lower entropy content at young ages than objects of same mass and age formed by gravitational collapse,
either from a pre-stellar core or by disk instability.
Consequently, the first type of objects should be significantly smaller, cooler and (about 100 times) fainter at young
ages than the second ones. This gave rise to the so-called ?cold start? observational signature, characteristic of young
giant planets formed by core accretion, vs the ?hot start? one, typical of young objects, GP?s or BD?s, formed by collapse
to distinguish these objects from their distinct formation mechanisms.

This suggestion, however, directly relies on two assumptions.
First of all, the assumption that GP?s formed by core accretion have a low entropy content implies
that all the energy of the accretion shock through which most of the planetary mass is processed is radiated
away, leaving the internal energy content of the nascent planet unaffected. This is characteristic of a so-called
supercritical shock. In that case the radiative losses at the accretion shock act as a sink of entropy. This assumption,
however, has never been verified. In fact, a proper treatment of the accretion shock at the onset of a pre-stellar core
formation, the so-called second Larson?s core, shows that the shock in that case is sub-critical, with essentially all the
energy from the infalling material been absorbed by the stellar embryo \citep{Vaytet2013, Tomida2013, Bate2014}.
Although the two formation conditions differ in several ways, they share enough common processes to
at least question the assumption of a supercritical shock for planet formation.
The second underlying assumption about this scenario is the assumption that BD?s form with a high entropy content.
Again, such an assumption is not necessarily correct. Initial conditions for brown dwarf formation are rather uncertain.
Figure \ref{fig:evolution} (see also \cite{Mordasini2012a, Mordasini2013, Spiegel2012}) compares the early
evolution of the luminosity for a 5 $M_{Jup}$ object under several assumptions.
The blue solid and dot-dash lines represent giant planet early evolution.
assuming that either
the accretion energy is entirely converted to radiation, i.e. a supercritical
shock condition as in \cite{Marley2007}, or that this energy is absorbed by the planet, i.e. a sub-critical shock
with no radiative loss at the shock.
The former case yields a low entropy content thus a low luminosity at the end of
the accretion shock while in the second case the luminosity slowly decreases from its value at the end of the shock,
two orders of magnitude brighter, for several Myr?s. The red long-dash and short-dash lines correspond to the early
evolution of a brown dwarf \citep{Baraffe2003} assuming an initial radius $R_i \sim 8 R_{Jup} (\simeq 0.8 R_{Jup})$ (short-dash) or
$R_i \sim 1.6~ R_{Jup}$ (long dash). This corresponds to specific entropies $\tilde{S?} \simeq 1.1 ? 10^9$ and
$\simeq 8.0 ? 10^8 erg~ g^{?1} K^{ ?1}$ , respectively.

\begin{figure}[!t]
\vspace{0cm}
\centerline{\includegraphics[trim=0.5cm 5.5cm 0.5cm 4.5cm,clip, width=0.6\textwidth]{fig_hot_cold_start_L2.pdf}}
\caption{Early evolution of a 5 Jupiter-mass object according to
different formation scenarios. Blue: object formed by core accretion assuming
either a supercritical shock (solid line) or a subcritical shock (dash-dot line)
at the end of the accretion process, after \cite{Mordasini2012a}.
Red: object formed by gravitational collapse for two arbitrary initial conditions (see text),
after \cite{Baraffe2003}}
\label{fig:evolution}
\end{figure}

As seen in the figure, depending on the outcome of the accretion shock episode in one case and on the initial
radius (thus entropy content) in the second case, and given the
long Kelvin-Helmholtz timescale for such low-mass objects (several Myrs) young planets formed by core accretion can
be as bright and even brighter than young BD?s for several Myrs. (Magneto)Radiation-hydrodynamics calculations of
the collapse of prestellar cores \citep{Tomida2013, Vaytet2013} seem to exclude the above rather extreme
?cold start? initial condition for a proto-BD, even though calculations have not been performed yet for such low masses.
They suggest initial entropy contents and radii closer to the hot-start case, the outer region of the protostellar core been
heated up by the shock and attaining a higher entropy. Hot-start conditions for core-accretion GP?s, probably in between
the two above extreme cases, however, are presently not excluded. Interestingly enough, measured temperatures
and luminosities of some directly-imaged exoplanets, notably $\beta$ Pic b and $\kappa$ And b, are consistent with hot-start like
conditions and seem, so far, to exclude the coldest range of initial conditions for these objects \citep{Currie2013,
Marleau2013}. Therefore, at least in the absence of a better knowledge of the accretion shock
condition at the end of the core accretion process, the cold start - hot start argument does not provide a reliable diagnostic
to distinguish core accretion from gravitational instability formed objects.
For sure there is no one-to-one correspondence between cold start vs. hot
start conditions and core accretion vs disk or core collapse.

\subsection {Deuterium Burning}
The distinction between brown dwarfs and giant planets is a topic of intense debate.
In 2003, the IAU has adopted the deuterium-burning (DB) minimum mass, $\sim 10~ M_{Jup}$ , as the official distinction between the
two types of objects.
Deuterium burning has no impact on star formation and a negligible impact on stellar/BD evolution \citep{Chabrier2000c}.
This is in stark contrast with the lifetime impact of hydrogen-burning, making H-burning a genuine physical mechanism
distinguishing objects in nuclear equilibrium for most of their lifetime, defined as stars, from objects which lack
significant support against gravitational contraction and keep contracting for ever since their birth, defined as brown dwarfs.
One of the strongest arguments against deuterium burning to distinguish planets from brown dwarfs is
2M 1207 b \citep{Chauvin2005}, which is a $\sim 4~ M_{Jup}$ companion to a $\sim 20~ M_{Jup}$ brown dwarf. Thus, the companion is
firmly in the non deuterium burning mass regime, but with a system mass ratio of 20\%, the couple appears to be best
described as an extension of the brown dwarf binary population rather than a planetary system. In contrast, it is
presently not excluded that genuine planets formed by core accretion, characterized by a significant heavy element
enrichment, reach masses above the deuterium burning limit and thus ignite D-burning in their core \citep{Baraffe2008, Molliere2012, Bodenheimer2013}.

\subsection {The Brown Dwarf/Planet Overlapping Mass Regime}
There is now ample evidence for the existence of free floating brown dwarfs with
masses of the order of a few Jupiter masses in (low extinction) young clusters and in the field, see \cite[e.g.][]{Caballero2007},
with a mass distribution consistent with the extension of the stellar IMF into the BD regime. The brown
dwarf and planet mass domains thus clearly overlap, arguing against a clear mass separation. The fundamentally
different mass distribution of exoplanets detected by radial velocity surveys, with the mass function rising below
$\sim 30~ M_{Jup}$ \citep{Mayor2011}, in stark contrast with the BD mass distribution clearly suggests two
distinct populations, with different origins.
Of particularly noticeable interest at this stage are the transiting objects Hat-P-2b, with a mass of $9~ M_{Jup}$
(Bakos et al., 2007b) and Hat-P-20b, with a mass of $7.2 M_{Jup}$ \citep{Bakos2011}. Both objects are too dense to be brown
dwarfs. Assuming that the observational error bars on the radius are reliable, and given the age inferred for theses
system, the observed mass-radius determinations imply significant enrichment in heavy material, revealing their planetary
nature \citep{Leconte2009, Leconte2011}. This shows that planets at least 9 times more massive than Jupiter, close to the
DB limit, can form according to the core-accretion scenario, possibly from the merging of lower mass planet embryos.
Note that while massive objects like HAT-P-20 b approach the upper limit of the mass distribution predicted by the core
accretion scenario \citep{Mordasini2012b}, its large metal enrichment ($M_Z \sim 340 M_{\oplus}$, \cite{Leconte2011})
certainly excludes formation by gravitational collapse.
According to the arguments given above, the present IAU definition, based on a
clear-cut mass limit between BD?s and planets, is clearly incorrect and confusing and should be abandoned.

\section {Conclusion}
Even though it is probably still premature to reach definitive conclusions about brown dwarf and giant planet formation
and we must remain open to all possibilities, the confrontation of the various theories with observational constraints
suggests some reasonably sound conclusions. The ability of BD?s to form in isolation
and in wide binaries; the similarity of BD number-density in low and high-density environments, indicating
no significant dependence of BD abundances upon stellar density; the emerging observations of isolated proto-BD?s
and pre-BD cores; the many observational properties shared by young BD?s and young stars; the observed close
similarity between the pre-stellar/BD core mass function and the final stellar/BD initial mass function;
the BD IMF been consistent with the natural extension of the same stellar IMF down to the nearly bottom of the BD domain.
All these points provide evidence that dynamical interactions and dense cluster environments, disk fragmentation or photoionizing
radiation are not required for BD formation. In contrast, all these properties are consistent with BD formation being a
natural scaled down version of star formation by the turbulence induced fragmentation of molecular clumps, leading
to the formation of pre-stellar and pre-BD cores. It is not excluded, however, that, under some specific circumstances
(very dense environment, very massive disks), the other aforementioned mechanisms might play some role but, in
the absence of clear observational evidence for this so far, they seem unlikely to be the dominant mechanisms for star
and BD formation.
Conversely, the overwhelming majority of planet discoveries are consistent with planet formation by core accretion.
As examined in section 5.3, this scenario might also explain planet formation at large orbital distances, not mentioning
the possibility to explain such objects by planet scattering or outward migration. Here again, alternative scenarios like
disk fragmentation might occur in some places, notably in massive circumbinary disks, and thus explain some fraction of the planet population (possibly $\sim$ 10\% or so, \cite[e.g.][]{Vorobyov2013}), but they can hardly be considered as
dominant scenarios for planet formation. Hybrid scenarios invoking both gravitational fragmentation at large orbital
distances followed by inward migration, invoking even in some cases evaporation seem
to raise even more problems than they bring solutions, as migration can only exacerbate problems with the gravitational
instability. Not only one needs to form planets by GI but one needs to prevent them to migrate rapidly all the way
into the star (Vorobyov and Basu, 2006b; Machida et al.,2011; Baruteau et al., 2011; Vorobyov, 2013). Migration,
however, is very likely and fast given that, in that case, the disk mass must be very high, a requirement for GI to occur.
If planets indeed form this way, this requires very finetuning conditions, making the branching ratio for this route
very small.

Finally, as discussed in ?\ref{sec:BDvsGP}, there is ample evidence that the planet and brown dwarf domains overlap and that deuterium
burning plays no particular role in the formation process. There is now growing evidence, possibly including the
WISE survey for the field population, for the existence of non-deuterium burning free floating brown dwarfs
and, conversely, no physical arguments against the possibility for genuine planets to ignite D-burning in their core.
This again shows that the IAU definition has no scientific justification and only brings scientific and mediatic
confusion. This also argues against the use of specific appellations for free-floating objects below the D-burning limit, these
latter being simply non D-burning brown dwarfs.

Given the arguments examined along this review, it seems rather secure to argue that BD?s and GP?s represent
two distinct populations of astrophysical bodies which arise dominantly from two different formation mechanisms.
While BD?s appear to form preferentially like stars, from the gravoturbulent fragmentation of a parent (possibly
filamentary) molecular clump, GP?s arguably form essentially by core accretion in a protoplanetary disk, i.e. from the
growth of solids (planetesimals, pebbles) yielding eventually the accretion of a surrounding gas rich H/He envelope.
So, the very definition of a brown dwarf or a giant planet is intrinsically, tightly linked to its formation mechanism.
%As briefly discussed below and in ?3.2, this latter should leave imprints which might be observationally detectable.
As examined in ?\ref{sec:hot_vs_cold}, the luminosity or effective temperature at early ages, however, cannot be used as a diagnostic
to distinguish between these two populations. On the other hand, we argue in this review that planets are necessarily
companions of a central, significantly more massive object.
Consequently, free floating objects down to a few $M_{Jup}$ can rather unambiguously be identified as genuine (non D-burning) brown dwarfs,
and one should stop giving them different names, which simply adds to the confusion.
This is the most direct observational distinction between
the two types of objects. Ejected planets probably exist and weaken this statement but they are unlikely to represent
a significant fraction of the population. The diagnostic is less clear for wide companions to stars, except if the mass
ratio can safely exclude one of the two possibilities. According to the distinct formation scenarios, planets should
have a substantial enrichment in heavy elements compared with their parent star, as observed for our own solar giant
planets, whereas BD?s of the same mass should have the same composition as their parent cloud. As suggested in
\cite{Chabrier2007} and \citep{Fortney2008}, giant planets should bear the signature of
this enhanced metallicity in their atmosphere. Spectroscopy or even photometry of atmospheric chemical abundances,
notably metal-dominated compounds like e.g. $CO$ or $CO_2$ may provide clues about the formation mechanism and thus
help identifying the nature of the object. In the absence (so far) of clear observational diagnostics (mean density,
atmospheric abundances, oblateness, etc.) to get clues about these formation conditions, the very nature of some of these
objects might remain uncertain. As frustrating as this may sound, we will have to admit such present uncertainties, as
ignorance is sometimes part of science.


\subsection{Policies}

A policy maps the state onto the action

\(a_t=\pi (s_t)\)

The policy does not need to change over time, as discounting is constant. That is, if the policy should be different in future, it should be different now.

The policy affects the transition model, and so we have \(P_\pi \).

\subsubsection{Optimal policy}

There exists a policy that is better than any other policy, under any starting state.

There is no closed form solution to finding the optimal policy.

There are instead iterative methods.