\label{sec:fcc}

Now we turn to discuss the role that PDFs would have in some of the recent
proposals for future colliders involving hadrons in the initial state.
%
There are three main families of potential future high-energy colliders currently under
active discussion.
%
Electron-positron colliders, such as the ILC~\cite{Baer:2013cma},
CLIC~\cite{Linssen:2012hp},
TLEP/FCC-ee~\cite{Gomez-Ceballos:2013zzn}, or CEPC~\cite{CEPC-SPPCStudyGroup:2015csa}, offer the potential
for ultra-high precision measurements of the Higgs, electroweak
and top-quark sectors.
%
On the other hand,
hadron colliders with energy much greater than the LHC would allow
 the continued exploration of the high-energy frontier and significantly
extend the coverage of 
searches for new BSM particles, including Dark Matter candidates,
while at the same time providing  unprecedented opportunities for the study of the Higgs sector, for example the Higgs self-interactions.
%
With this motivation, there is ongoing work towards a circular
collider hosted at the CERN site which would accelerate protons
up to the extreme energies of $\sqrt{s}=100$
TeV~\cite{Contino:2016spe,Mangano:2016jyj}, dubbed FCC-hh,
while a similar machine under study by the Chinese HEP community~\cite{CEPC-SPPCStudyGroup:2015csa}.

Another avenue for future high-energy collisions would be new machines based
on electron-proton collisions, exploiting the successful strategy
adopted at HERA.
%
One of the open proposals is the Large Hadron electron Collider (LHeC)~\cite{AbelleiraFernandez:2012cc}, where the $E_{p}=7$ TeV proton beam from the LHC would collide with an electron/positron
beam with $E_e=60$ GeV coming from a new LinAc, and that would be able
to reach the region down to $x_{\rm min}\simeq 2\cdot 10^{-6}$ at
$Q^2=2$ GeV$^2$.
%
A more extreme incarnation of the same idea corresponds to colliding these
$E_e=60$ GeV electrons
with the $E_p=50$ TeV beam of the FCC-hh.
%
The resulting collider, dubbed FCC-eh, would be able to reach down to
 $x_{\rm min}\simeq 2\cdot 10^{-7}$ at
$Q^2=2$ GeV$^2$.
%
These two machines would therefore extend
the PDFs coverage  the small-$x$ by more than two orders of magnitude
in comparison to HERA.

The Electron Ion Collider (EIC)~\cite{Boer:2011fh}, which
might start construction soon either at the BNL or the
JLAB site, falls under the same category.
%
The EIC would offer
the possibility to polarize both leptons and protons and
to accelerate heavy nuclei, although its $\sqrt{s}$ would
be smaller than that of HERA.
%
In this respect, it would not be able to provide new information
on the unpolarized proton PDFs, though it would be a unique machine
to explore the spin and 3D content of the proton, as well as
map in detail the nuclear modifications of the free-proton PDFs.

In this section, we review the role that PDFs would play
first at a future higher-energy lepton-proton collider
such as the LHeC/FCC-eh, and then at a future
hadron collider with a center-of-mass energy
of $\sqrt{s}=100$ TeV.

\subsubsection{PDFs at high-energy lepton-hadron colliders}

As mentioned above, one of the possibilities for a future high-energy collider
now under active discussion
would be to exploit the LHC/FCC proton beam and collide it with a high energy lepton
beam, which would be delivered by a new LinAc to be built at the CERN site.
%
In the case of using the LHC beams, the
LHeC~\cite{AbelleiraFernandez:2012cc,AbelleiraFernandez:2012ty,Armesto:2013dia} 
would then represent
a scaled-up version of HERA, and as such would offer immense opportunities for
improved determinations of the proton structure down to very low $x$
and up to high-$Q^2$, as well as providing a wealth of information on nuclear PDFs
in a kinematic region where they are currently essentially unconstrained.
%
Several options are now being considered, with some preference now for synchronous
operation during the final years of the HL-LHC upgrade, since then the LHeC
program can be extended to include measurements of the Higgs sector.

In Fig.~\ref{fig:kinplotLHeC} we show
the kinematic coverage in  $(x,Q^2)$
of several existing and future deep-inelastic scattering experiments,
including the EIC, the LHeC, and the FCC-eh.
%
We observe that by starting from the fixed-target experiments and then moving to HERA, the
LHeC and finally the FCC-eh, as the center of mass energy increases,
the kinematic reach extends both towards higher $Q^2$ and smaller $x$ values.
%
At the FCC-eh in particular, it should be able to cover the region down to
$x\simeq 10^{-7}$ without leaving the perturbative region $Q \gsim 1$ GeV.
%
It is important to emphasize that the same coverage would be achieved
for nuclear PDFs, extending the coverage by four or five orders of magnitude in
$x$ as compared to existing measurements.

One of the most important aspects of the LHeC/FCC-eh scientific case
is being able to probe the proton/nuclear PDFs with an unprecedented precision,
not only by means of inclusive structure functions but also with measurements
of the strange, charm, and bottom structure functions, that provide
a direct handle on the heavy flavour PDFs.
%
The LHeC/FCC-eh would also allow a measurement of the strong
coupling constant $\alpha_s(M_Z)$ with per-mile uncertainties,
for instance using jet production~\cite{Currie:2016ytq}, and high-precision
measurements of the electroweak sector parameters.
%
To illustrate these possibilities,
in Fig.~\ref{fig:kinplotLHeC} we also show the
results of an {\tt xFitter} PDF feasibility study that
compares the impact on the gluon PDF of adding either LHeC or FCC-eh (or both)
pseudo-data in addition to the HERA inclusive structure function dataset.
%
The reduction of the PDF uncertainties down to very small-$x$ values reflects
the extended kinematic reach of these future high energy
lepton proton colliders.
%
A similar reduction of the PDF uncertainty is expected for the quark PDFs.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centering
\includegraphics[width=0.93\textwidth]{9-future/plots/lhec.pdf}
\caption{\small Left: kinematic coverage in the $(x,Q^2)$
of several existing and proposed deep-inelastic scattering experiments.
%
Starting from the fixed-target experiments and then moving to HERA, the
LHeC and finally the FCC-eh, as the center of mass energy increases,
the kinematic reach extends both towards higher $Q^2$ and smaller $x$ values.
%
Right: results of an {\tt xFitter} PDF feasibility study that
compares the impact on the gluon PDF of adding either LHeC or FCC-eh (or both)
pseudo-data in addition to the HERA inclusive structure function dataset.
}
\label{fig:kinplotLHeC}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Another important aspect of the interplay between PDFs and the LHeC/FCC-eh
is related to the small-$x$ resummation
framework~\cite{Ciafaloni:2007gf,White:2006yh,Altarelli:2008aj,Bonvini:2016wki,Bonvini:2017ogt}.
%
This framework is based on extending the collinear DGLAP formalism to account for
the all-order resummation of terms of the form $\alpha_s^k\ln^m(1/x)$ in the
splitting functions and the partonic coefficient functions,
as implemented in the BFKL equation~\cite{Lipatov:1976zz,Fadin:1975cb,Kuraev:1976ge}.
%
Using this approach, the reliability of theoretical predictions
for DIS structure functions and collider cross-sections can be extended
down to much smaller values of $x$, as compared to the calculations
based on the collinear DGLAP framework.
%
Recently, a version of the NNPDF3.1 global analysis, called
NNPDF3.1sx, based on
NLO+NLL$x$ and NNLO+NLL$x$ theory, has been
presented~\cite{Rottoli:2017ifw,smallxPDF},
which suggest that accounting for small-$x$ effects leads to an
improved description of the inclusive HERA structure
function data (see also the discussion in~\cite{Caola:2010cy,Caola:2009iy}).
%
One may expect such effects to become even more relevant for higher-energy
lepton-proton colliders (see Fig.~\ref{fig:kinplotLHeC}).

With the motivation of providing a first estimate of the relevance of small-$x$ resummation
for the LHeC/FCC-eh, in
Fig.~\ref{fig:LHeCpredictions} we provide
predictions for the $F_2$ and $F_L$ structure functions using the NNPDF3.1sx NNLO
   and NNLO+NLL$x$ fits at $Q^2=5$ GeV$^2$ for the kinematics of the LHeC
   and the FCC-eh.
   %
   For these calculations,
   we have used {\tt APFEL} to produce  NNLO(+NLL$x$) predictions,
each using as input the corresponding NNPDF3.1sx fits,
for the most updated version of the simulated LHeC/FCC-eh pseudo-data kinematics.
   %
    In the case of $F_2$, we also show the expected total experimental uncertainties
    based on the simulated pseudo-data, assuming the NNLO+NLL$x$ curve as central prediction.
    %
    The total uncertainties of the simulated pseudo-data are at the few percent
    level, and therefore they are rather smaller than the PDF uncertainties in the
    entire kinematic range.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.38]{9-future/plots/F2p_lhec.pdf}
\includegraphics[scale=0.38]{9-future/plots/FLp_lhec.pdf}
\includegraphics[scale=0.38]{9-future/plots/F2p_fcceh.pdf}
\includegraphics[scale=0.38]{9-future/plots/FLp_fcceh.pdf}
\caption{\small The theoretical predictions  for the $F_2(x,Q)$ and $F_L(x,Q)$
structure functions at the
LHeC (upper) and FCC-eh (lower plots) using the NNPDF3.1sx NNLO
   and NNLO+NLL$x$ fits at $Q^2=5$ GeV$^2$.
%
In the case of $F_2(x,Q)$, we also show the expected total experimental uncertainties
   based on the simulated pseudo-data, assuming the NNLO+NLL$x$ values as central prediction.
       \label{fig:LHeCpredictions}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From the comparisons in Fig.~\ref{fig:LHeCpredictions},
we see how the FCC-eh would allow the small-$x$ region to be probed
by about an order of magnitude deeper than the LHeC (which in turn
extends HERA by about the same amount).
%
The differences between NNLO and NNLO+NLL$x$ theory are  small for
$F_2$, in particular taking into account the large PDF uncertainties, implying that 
refitting the pseudo-data is required to first reduce the PDF errors and then discriminate
between the two theoretical scenarios.
%
Given the small experimental uncertainties,
these inclusive $F_2^p$ measurements would
represent a sensitive probe of small-$x$ dynamics.
%
We also see that differences are more marked for $F_L$, with the central values
differing by several times the PDF uncertainty in most
of the accessible kinematic range.
%
This illustrates the sensitivity of $F_L$ measurements to probe
small-$x$ QCD dynamics.
%
We also note that small-$x$ predictions based on non-linear
effects~\cite{Albacete:2012rx,Albacete:2010sy}
(`saturation') have typically the opposite trend to small-$x$ resummation,
suppressing the structure functions as compared to the NNLO fixed-order
calculation.
%
Therefore, the measurements in Fig.~\ref{fig:LHeCpredictions} and related
ones would open a unique window to the novel dynamical regime of QCD at very
small $x$.

\subsubsection{PDFs at a 100 TeV hadron collider}

Next we discuss parton distributions at the FCC-hh,
a proposal for a future hadron collider with a center of mass energy of
$\sqrt{s}=100$ TeV.
%
In order to illustrate the extended kinematic coverage that
would be achieved at a 100 TeV proton-proton collider
as compared to the one at the LHC, in Fig.~\ref{fig:kinplot}, taken from~\cite{Rojo:2016kwu}, 
we compare the $(x,M_X)$ coverage at 100 TeV and 14 TeV, where
the dotted lines indicate the regions of constant rapidity at the
FCC-hh.
%
In addition,  we also indicate the relevant $M_X$ regions for some representative
  processes, from low masses (Drell-Yan, low $p_T$ jets), to
  electroweak scale processes (Higgs, $W,\,Z$, top) and new
  high-mass particles (squarks, $Z'$).
  %
  A significant increase in the kinematic
  coverage as compared to the LHC is clear.
  %
  A particularly interesting aspect is that
 at the FCC-hh even
 electroweak-scale processes
  such as $W,Z$, or $h$ production become sensitive to the small-$x$
  region.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centering
\includegraphics[width=0.85\textwidth]{9-future/plots/kinplot}
\caption{\small Kinematic coverage in the $(x,M_X)$ plane of a
  $\sqrt{s}=100$ TeV 
  hadron collider (solid blue line),
  compared with the corresponding coverage of the LHC at
  $\sqrt{s}=14$ TeV (dot-dashed red line), from~\cite{Rojo:2016kwu}.
}
\label{fig:kinplot}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As discussed in the FCC-hh Yellow Report~\cite{Mangano:2016jyj}, there are two main aspects
of PDF phenomenology relevant for a 100 TeV collider.
%
On the one hand, as for the LHC, at $\sqrt{s}=100$ TeV, parton distributions
are one of the dominant theoretical uncertainties for
several cross-sections.
%
In particular, electroweak scale cross-sections, such as $W$ or $h$ production,
become sensitive to the small-$x$ region where PDF uncertainties
are currently large.
%
To illustrate this point,
in Fig.~\ref{fig:fcc-xsecs} we show the
comparison of cross-sections for different representative
     processes at the FCC with $\sqrt{s}=100$ TeV,
     between the NNPDF3.0 predictions and those of the NNPDF3.0+LHCb
     sets~\cite{Gauld:2016kpd}, see Ref.~\cite{Gauld:2017rbf} for more details.
     %
     The acceptance cuts are different in each process.
     %
     In the left plot we show the results for direct photon
     production, off-peak Drell-Yan cross-sections, and inclusive
     weak boson production.
     %
     In the right plot we show the fiducial cross-sections for
     $c\bar{c}$ and $b\bar{b}$ production.
     %
     In all cases, and specially for heavy quark pair production,
     we can observe the reduction of PDF uncertainties
     that is derived once the NNPDF3.0+LHCb sets are used,  highlighting the
     sensitivity to the small-$x$ region.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.60]{9-future/plots/fcc-xsecs.pdf}
   \caption{\small 
     Comparison of cross-sections for different representative
     processes at the FCC with $\sqrt{s}=100$ TeV,
     between the NNPDF3.0 predictions and those of the NNPDF3.0+LHCb
     sets.
     %
     Note that the acceptance cuts are different in each process.
     %
     In the left plot we show the results for direct photon
     production, off-peak Drell-Yan cross-sections, and inclusive
     weak boson production.
     %
     In the right plot we show the fiducial cross-sections for
     $c\bar{c}$ and $b\bar{b}$ production.
    \label{fig:fcc-xsecs}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The other aspect of PDFs at the FCC-hh is the onset of new  phenomenon that are absent
at the lower energies of the LHC.
%
These include the possibility of treating the top quark as a
massless parton~\cite{Han:2014nja,Dawson:2014pea},
the need for resummation of ``collinear'' weak gauge boson radiation
and the consequent introduction of
electroweak PDFs~\cite{Bauer:2017isx,Ciafaloni:2001mu,Chen:2016wkt}, as well as the increased role
of photon-induced processes~\cite{Bertone:2015lqa,Mangano:2016jyj,Harland-Lang:2016kog}.
%
Moreover, just as in the case of the LHeC/FCC-eh, the role of small-$x$ resummation is expected
to become more important at the FCC-hh than at the LHC, given the sensitivity of even standard
candles such as $W$, $Z$ and Higgs production to the small-$x$ region.

Here we provide two representative illustrations of these new PDF-related phenomena at the FCC-hh.
%
First of all, as mentioned above, at 100 TeV the electroweak gauge bosons become effectively massless,
and thus it is possible to construct electroweak PDFs with the corresponding evolution
equations.
%
In Fig.~\ref{fig:FCCtopEW} we show the PDF of the $W^+$ boson normalized to that of the gluon,
    as a function of $x$ for different scales: $q=10^4$ GeV,
    $10^6$ GeV, $10^8$ GeV, computed using the framework of~\cite{Bauer:2017isx}.
    %
    We observe that the dependence of the $W$ PDF with the energy $q$ is rather mild.
    %
    For most of the range of $x$, the $W$ PDF is at most a few percent of the gluon
    PDF, while for $x\ge 0.1$ it becomes larger, up to 40\% of the gluon PDF.
    %
    This does not necessarily mean that the effects of the $W$ PDF will be phenomenologically
    relevant: this can be assessed only at the cross-section level, comparing calculations
    with massive gauge bosons and those where these are treated as massless (and thus resummed
    into the electroweak PDFs).
    %
    Interestingly, the chiral nature of the weak interactions implies that
    once electroweak effects are accounted for in the DGLAP evolution, the proton
    can become spontaneously polarized, with implications for BSM searches~\cite{Fuks:2014uka}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.44]{9-future/plots/FCCtopEW.pdf}
   \caption{\small 
    Left: the PDF of the $W^+$ boson normalized to that of the gluon,
    as a function of $x$ for different scales: $q=10^4$ GeV,
    $10^6$ GeV, $10^8$ GeV, from Ref.~\cite{Bauer:2017isx}.
    %
    Right: the cross-section of inclusive Higgs production by $t\bar{t}$ associated
    production, comparing the results of the $n_f=5$ scheme ($gg\to h t\bar{t}$),
    the $n_f=6$ scheme ($t\bar{t}\to h$), and of their interpolation by means
    of the ACOT general-mass scheme, from~\cite{Han:2014nja}.
\label{fig:FCCtopEW}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Following a similar line of thought, at 100 TeV it is conceivable to treat the top quark
as massless partons, much in the same way as at the LHC the bottom quark is treated
as massless in most calculations.
%
In Fig.~\ref{fig:FCCtopEW} we show the
cross-section of inclusive heavy scalar (``Higgs'') production by $t\bar{t}$ associated
    production, comparing the results of the $n_f=5$ scheme ($gg\to H^0 t\bar{t}$),
    the $n_f=6$ scheme ($t\bar{t}\to H^0$), and of their interpolation by means
    of the ACOT general-mass scheme~\cite{Han:2014nja}
    (see also Sect.~\ref{sec:pdffitting.heavyq}).
    %
    The comparison is performed as a function of the heavy Higgs boson mass $m_{H^0}$.
    %
    We find that the $n_f=6$ calculation, where the top quark is treated as massless and resummed
    into a top quark PDF $t(x,Q)$, is rather far from the matched calculation up to at least $m_{H^0}=10$ TeV.
    %
    This suggest that the massless top approximation is not suitable even for the extreme
    FCC energies.
    %
    On the other hand, the concept of top PDF is still useful in order to improve fixed
    order calculations, using general-mass schemes such as ACOT or FONLL, but it seems not
    advisable using it without accounting for the top quark mass effects.
