The precision measurements of SM parameters, such as the
mass of the $W$ boson $M_W$ or the running of the strong coupling
constant $\alpha_S(Q)$, represent powerful ways of constraining
BSM dynamics at the LHC.
%
For instance, following the discovery of the Higgs boson,
in the absence of new physics the standard model is an over-constrained
theory: one can use a set of input parameters, such as the
Higgs mass $m_h$ and the top quark mass $m_t$, in the
context of the global electroweak precision fit~\cite{Baak:2014ora},
to predict other parameters, such as $M_W$.
%
By comparing these indirect predictions of the $W$ mass with
direct experimental measurements, one can provide a stress-test
of the SM, where any tension might indicate hints for BSM dynamics
at scales higher than those that are currently directly accessible.
%
Tests of this type have already resulted in the famous $g_{\mu}-2$ anomaly,
where a persistent $3$ to $4$-sigma discrepancy is found
between the theoretical predictions of the muon anomalous magnetic
moment~\cite{Benayoun:2015gxa}
and the corresponding experimental measurements.

In order to make these comparison between indirect predictions
and direct measurements as stringent as possible, it is
important to improve the precision of the latter.
%
And for many SM parameters, PDF uncertainties represent one of the
dominant theoretical
uncertainties in their determination, providing
another motivation for the need of improved PDFs.
%
Focusing on the case of the $W$ mass measurements, the role
of PDF uncertainties has been quantified in detail
in a number of studies, both from the
phenomenological~\cite{Bozzi:2015hha,Bozzi:2015zja,Bozzi:2011ww}
and from the experimental point of view.
%
In the latter case, the first direct measurement of $M_W$
at the LHC has recently been presented by the
ATLAS collaboration~\cite{Aaboud:2017svj}, yielding
a total uncertainty of only 19 MeV, of which around half of
it (9 MeV) is estimated to come from PDF uncertainties.
%
In Fig.~\ref{fig:mw} we show a comparison
 between the direct
  measurements of $m_W$, $m_t$, and $m_h$ from ATLAS
  with the predictions from the global electroweak fit,
  from~\cite{Aaboud:2017svj}.
  %
  We observe that there is good agreement between the direct measurements
and the indirect predictions, providing a highly non--trivial
validation test of the SM.
%
Future measurements of $m_W$ and $m_t$, as well as their
combination with other experiments, should be able
to reduce the uncertainties in this comparison.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.48]{8-pheno/plots/mw.pdf}
  \caption{\small Left: comparison between the direct
  measurements of $m_W$, $m_t$, and $m_h$ from ATLAS
  with the predictions from the global electroweak fit,
  from~\cite{Aaboud:2017svj}.
  %
  Right: estimate of the PDF uncertainties in the $m_W$
  determination using different PDF sets and collider scenarios,
  from~\cite{Bozzi:2015hha}. 
  %
  This estimate has been obtained from template fits to the
  $p_T^l$ distribution, imposing the constraint that $p_T^W\le 15$ GeV.
    \label{fig:mw}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In Fig.~\ref{fig:mw} we also show a phenomenological estimate
of the PDF uncertainties associated to the $m_W$
  measurements using different PDF sets and collider scenarios,
  from~\cite{Bozzi:2015hha}. 
  %
  This estimate has been obtained from template fits to the
  $p_T^l$ distribution, imposing the additional
  constraint that $p_T^W\le 15$ GeV.
  %
  A number of NNLO PDF sets are used in this comparison,
  in order to achieve a robust estimate of the
  PDF uncertainties.
  %
  In general one finds that there is good agreement within
  PDF uncertainties, although in some cases this
  agreement is only marginal, as in the case of NNPDF3.0
  and CT10 at the LHC 13 TeV for the $W^-$ case.
  %
  From this study, one estimates that at the LHC 7 TeV
  PDF uncertainties using state-of-the-art
  sets are around 6 MeV, a number close to the one estimated
  in the ATLAS measurement~\cite{Aaboud:2017svj}.

Another SM parameter that could potentially provide
indirect information
on BSM dynamics is the QCD coupling $\alpha_s(Q)$, and
specifically of its running at the TeV scale.
%
It is well known that the presence of new colored heavy degrees of freedom
will modify the QCD $\beta$-function and therefore lead to a different
running with $Q$ as compared to the corresponding SM prediction.
%
This fact is for example at the basis of the improved
agreement at high scales between the strong, weak and electromagnetic
couplings in the case of low-scale supersymmetry,
which suggest the unification of the three forces
at a GUT scale of around $\Lambda_{\rm GUT}\sim 10^{16}$
GeV~\cite{Dimopoulos:1981yj}.

If  new BSM heavy particles are present at the TeV scale,
the difference induced in the QCD coupling running
could be accessible at the LHC, see {\it e.g.}~\cite{Becciolini:2014lya}
and Fig.~\ref{fig:alphasrunning}, where we show the
modifications in the running of $\alpha_s(Q)$ induced
  by a new heavy colored fermion of mass $m=0.5$ TeV for
  various representations of its color gauge group.
%
With this motivation,
as well as to compare with other measurements
of $\alpha_s$ at lower energies, 
the ATLAS and CMS
collaborations have presented a number of measurements
of both $\alpha_s(m_Z)$ and of $\alpha_s(Q)$ for restricted
$Q$ ranges, mostly from jet
production~\cite{Khachatryan:2014waa,Aaboud:2017fml,
Chatrchyan:2013txa} but also
from top-quark pair production~\cite{Chatrchyan:2013haa}
(see also~\cite{Rojo:2014kta} for a review, and Fig.~\ref{fig:alphasrunning}
for a graphical overview).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.48]{8-pheno/plots/alphasrunning.pdf}
  \caption{\small Left: the modifications
  in the running of $\alpha_s(Q)$ induced
  by a new heavy colored fermion of mass $m=0.5$ TeV as compared to the SM
  expectation, for various representations of its color gauge group,
  from~\cite{Becciolini:2014lya}.
  %
  Right: comparison of recent direct determinations of $\alpha_s(Q)$ at the Tevatron
  and the LHC as a function of $Q$, together with the PDG 2016 world
  average and with the results of the
  ATLAS TEEC 2012 extraction~\cite{Aaboud:2017fml}.
    \label{fig:alphasrunning}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In these collider-based
determinations of the strong coupling, PDF uncertainties, which are
significant
at the TeV scale (see Sect.~\ref{sec:LHCpheno.BSM}),
represent an important source of theoretical uncertainties.
%
For instance, in the recent ATLAS determination of $\alpha_s(m_Z)$ from
transverse energy-energy correlations (TEEC) at 8 TeV~\cite{Aaboud:2017fml}, the PDF uncertainty is
$\delta_{\rm pdf}=0.0018$, almost a factor 2 larger than the experimental
uncertainty of $\delta_{\rm exp}=0.0011$.
%
While in this analysis PDF uncertainties are sub-dominant with respect
to the scale uncertainties, $\delta_{\rm scale}\simeq 0.006$, the latter were
computed using NLO theory and can be reduced significantly by exploiting
the NNLO calculation.
%
Likewise, in the CMS measurement at 7 TeV based on a QCD
analysis of the inclusive jet cross-sections~\cite{Khachatryan:2014waa}, one finds
that the PDF uncertainties $\delta_{\rm pdf}=0.0028$ are larger than
the experimental uncertainties $\delta_{\rm exp}=0.0019$, though still
sub-dominant with respect to the large scale variations of the
NLO calculation $\delta_{\rm scale}=^{+0053}_{-0.0024}$.
%
In both cases, it is manifest that if one is able to reduce PDF uncertainties,
and exploit the reduction of scale errors arising from the NNLO calculation,
it should be possible to achieve a rather
competitive determination of $\alpha_s(m_Z)$ and 
also perform stringent tests of its running in the TeV region.

As a related topic, we would like to mention
that there are also proposals to measure the running
of the electroweak running couplings at the LHC~\cite{Alves:2014cda}
and use these to impose model-independent constraints
on new particles with electroweak quantum numbers without any assumptions
about their decay properties.
%
Again, here PDFs are one of the dominant sources of theoretical
uncertainty, for instance in the high-mass tail of $W$ and $Z$
production at the LHC due to the quark-antiquark luminosity.
%
Improving our knowledge of large-$x$ anti-quarks will therefore
be helpful in providing such indirect constraints
of new heavy electroweak sectors.


