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Index: trunk/Thesis/plots/Vj.pdf
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Index: trunk/Thesis/text/montecarlo.tex
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--- trunk/Thesis/text/montecarlo.tex (revision 9)
+++ trunk/Thesis/text/montecarlo.tex (revision 10)
@@ -1,2 +1,48 @@
-\section{MC Methods}
-\label{sec:montecarlo}
\ No newline at end of file
+\section{Monte Carlo methods in brief}
+The goal of phenomenology studies, like this one,
+is to ultimately compare against experimental data.
+There are a very limited set of cases in which the ingredients
+presented in the previous section are enough to do so.
+The problem is that experiments have to deal many
+more particles and many more
+phase space constraints than we can handle with a simple
+description.
+To this end, Monte Carlo event generators,
+like \Pythia~\cite{Sjostrand:1993yb},
+\Herwig~\cite{}
+or \Sherpa~\cite{}.
+play an important role in providing a theory prediction
+that is the closest possible to what experiments measure.
+
+To get a general idea of how such programs work, imagine
+an experiment has measured some physical observable, $O$.
+In general, we can identify an observable as a function
+of all final state particles, $f$, and their momenta $\{p_i\}_{i\leq n_f}$,
+\begin{equation}
+ O\,\equiv \,O(f;p_1,\dots ,p_{n_f})\,.
+\end{equation}
+Having to deal with quantum mechanical processes, however,
+means that we cannot determine exactly the final state
+for any one specific collision. We can though predict
+the differential probability, which in turn is proportional
+to the differential cross section for producing a specific
+final state configuration,
+\begin{equation}
+ {\rm d}\mathcal{P}(f;p_1,\dots ,p_{n_f})\propto
+ \frac{{\rm d} \sigma(f;p_1,\dots ,p_{n_f})}
+ {{\rm d}^3p_1\dots{\rm d}^3p_{n_f}}.
+\end{equation}
+Repeating the experiment over many times gives us access
+to the expectation value of $O$,
+\begin{equation}
+ \langle O\rangle = \mathcal{L}\times \sum_f
+ \int \frac{{\rm d} \sigma(f;p_1,\dots ,p_{n_f})}
+ {{\rm d}^3p_1\dots{\rm d}^3p_{n_f}}
+O(f;p_1,\dots ,p_{n_f})\,{\rm d}^3p_1\dots{\rm d}^3p_{n_f}\,.
+\end{equation}
+\subsection{Fixed order}
+One of the main ingredients
+\subsection{Parton Shower}
+\subsection{Merging}
+\subsection{Higher order merging}
+\label{sec:montecarlo}
Index: trunk/Thesis/text/massive_fact.tex
===================================================================
--- trunk/Thesis/text/massive_fact.tex (revision 9)
+++ trunk/Thesis/text/massive_fact.tex (revision 10)
@@ -1,381 +1,382 @@
\section{Bottom quark fusion with massive quarks}
\label{sec:mass_fac}
To see how the 5F massive scheme (or 5FSM) works we now present
an explicit example: the production of a Higgs boson through bottom
quark fusion.
There are two important features about this example. Firstly,
% - Here we have full results for the inclusive case, should
% check against them?
as we discuss in Chapter~\ref{chap:FONLL}, we know how include
exact mass effects up to order $\alpha_s^3$ through matching.
From that calculation we get that
mass corrections to the total inclusive cross section are quite small.
Consequently we expect the 5FSM too to yield a marginally different total
cross section.
Secondly, this example has a easy enough structure that
most formulae can be reported in closed analytic form.
We start by showing how the massive subtraction removes
all divergences at the integrand level, and we compare
results obtained in the vanilla 5FS against the fully massive 5FSM.
The leading order colour summed, helicity
averaged and squared matrix element for Higgs production
in bottom fusion is given by
\begin{equation}
|\overline{\mathcal{M}}_{b\bar{b}\rightarrow H}|^2 =
\frac{g_{hb\bar{b}}^2}{6}\,\left(m_H^2-4\,m_b^2\right)\,,
\end{equation}
where $g_{hb\bar{b}}$ is the bottom Yukawa coupling, such that
\begin{equation}
g_{hb\bar{b}} = \frac{m_b}{v}\, ,
\end{equation}
and $v$ is the electroweak vacuum expectation value.
The matrix element corresponding to the emission of
an extra gluon from the initial state $b$ has the form
\begin{multline}
\label{eq:bbhg}
\Real = |\overline{\mathcal{M}}_{b\bar{b}\rightarrow Hg}|^2 =
\frac{\alpha_s\,C_F\,8\,\pi\,g_{hb\bar{b}}^2}{6}\\
\biggl\{
\left(m_H^2-4\,m_b^2\right)
\left[
\frac{2(s-2\,m_b^2)}{(m_b^2-t)(m_b^2-u)}
- \frac{2\,m_b^2}{(m_b^2-t)^2}
- \frac{2\,m_b^2}{(m_b^2-u)^2}
\right] \\
+ (s-m_h^2)
\left[
\frac{1}{m_b^2-t}
+ \frac{1}{m_b^2-u}
\right]
\biggr\} \, .
\end{multline}
We firstly show that indeed
$|\overline{\mathcal{M}}_{b\bar{b}\rightarrow Hg}|^2- \mathcal{D}^{ak,b}-\mathcal{D}^{ak,b}$
gives a finite number.
Putting everything together and expressing $\Real$ in
Eq.~(\ref{eq:bbhg}) in terms of splitting kinematics variables $x,y$ and $Q^2$
we get
\begin{equation}
\label{eq:realdiff}
\Real - \mathcal{D}^{ak,b}-\mathcal{D}^{ak,b} =
\frac{8\,\pi}{3}\alpha_s\,C_F\,g_{hb\bar{b}}^2\,\frac{m_b^2}{s_{ab}}\,
\frac{1-x}{x\,y\,(1-x-y)}\,.
\end{equation}
The soft limit is approached for $x\rightarrow 1$, and it is straight
forward to check that, in this limit,
Eq.~(\ref{eq:realdiff}) is not only finite, but exactly zero.
+
Although the collinear limit, strictly speaking, does not exist
if the mass of the parton remains non zero,
we can check that the quasi-collinear is finite.
To phrase this slightly differently
we need to check that when $p_a\cdot p_k$ (or $y$) approaches
zero as $m_b$, Eq.~(\ref{eq:realdiff}) remains finite.
Again this is quite straightforward to see,
and in this limit we get exactly zero.
\begin{figure}
\centering{
\begin{multline}
\nonumber
\Born+\Virtual \,= \,\biggl|
\raisebox{-2.9cm}{\includegraphics[scale=0.2]{plots/born_bbh.pdf}}
\quad+\quad\raisebox{-2.9cm}{\includegraphics[scale=0.2]{plots/virt_bbh.pdf}}
\biggr|^2 = \\
\biggl|
\raisebox{-2.9cm}{\includegraphics[scale=0.2]{plots/born_bbh.pdf}}
\biggr|^2\times\biggl(1\,+\,2\,{\rm Re}(\delta_g)\biggr)
\end{multline}
}
\caption{Contributions to the born phase space of $b\bar{b}\rightarrow H$.}
\end{figure}
\begin{figure}
\centering{
\begin{multline}
\nonumber
\Real\,=\,\biggl|\quad
\raisebox{-2.9cm}{\includegraphics[scale=0.2]{plots/realg.pdf}}
\quad - \quad
\raisebox{-2.9cm}{\includegraphics[scale=0.2]{plots/realg2.pdf}}
\quad\biggr|^2\\
\quad+\quad\biggl|\quad
\raisebox{-2.9cm}{\includegraphics[scale=0.2]{plots/realb.pdf}}\quad
\biggr|^2
\end{multline}
}
\caption{Contributions to the real emission phase space of
$b\bar{b}\rightarrow Hg$~Eq.~(\ref{eq:bbhg}) and $bg\rightarrow Hb$.}
\end{figure}
We now turn to the one loop contribution. At order $\alpha_s$,
we have that
\begin{equation}
\Virtual = 2\,{\rm Re}(\delta_g)\,|\overline{\mathcal{M}}_{b\bar{b}\rightarrow H}|^2
\end{equation}
where
\begin{multline}
\label{eq:bbh1loop}
{\rm Re}(\delta_g) =
-\frac{\alpha_s\,C_F}{2\,\pi}\biggl\{
\frac{1 + \frac{s_{ab}}{\sqrt{\lambda_{ab}}}\log\beta_0}{\varepsilon}
+1+\log\frac{\mu^2}{m_b^2} +\frac{1-\beta^2}{\beta^2}\,\log\beta_0\\
-\frac{s_{ab}}{\sqrt{\lambda_{ab}}}
\left[
-\log\frac{\mu^2}{m_b^2}\log\beta_0-2\,{\rm Li}_2\left(1-\beta_0\right)
-\frac{1}{2}\,\log^2\beta_0 + \pi^2
\right]
\biggr\}\, ,
\end{multline}
with
\begin{equation}
\beta=\sqrt{1\,-\,\frac{4\,m_b^2}{m_H^2}}\,;\quad
\beta_0 = \frac{1-\beta}{1+\beta}\, .
\end{equation}
Using Eq.~(\ref{eq:part_I}) it is straightforward to see
that indeed
\begin{equation}
\Virtual + \int{\rm d}\OnePS\,\Sub = \mathcal{O}(\varepsilon^0)\,,
\end{equation}
which in turn, combined with Eq.~(\ref{eq:realdiff}),
yields that Eq.~(\ref{eq:fsubk}) is completely free of
singularities.
\subsection{Factorisation of collinear singularities}
In the standard case of massless initial state partons,
Eq.~(\ref{eq:realdiff}) would be exactly zero. This is due to the fact
the only term appearing in the real emission matrix element
is also the leading log that gets factorised into the $b$-pdf
and resummed through Altarelli-Parisi equations, Eq.~(\ref{eq:c1_dglap}).
The reason why is it so is that in the massless
case this process has only two scales $m_H$ and some
dimensional regulator that introduces a scale $\mu_F$
that separates the divergent and finite part.
This means that we can symbolically write
\begin{equation}
\int{\rm d}\PS{2}\Real(m_b=0) \propto
A\,\log\frac{m_H}{\mu_F^2}\,.
\end{equation}
In the case of massive initial state particle
we have an additional physical scale, $m_b$.
This scale now regulates collinear divergences, we no longer
need to introduce a collinear regulator related to the scale
$\mu_F$.
However we can in practice still
introduce the scale $\mu_F$, such that
\begin{equation}
\label{eq:mass_fact}
\int{\rm d}\PS{2}\Real(m_b\neq 0) \propto
A\left(\frac{m_b^2}{m_H^2},\frac{m_b^2}{\mu_F^2}\right)\,
\log\frac{m_H}{\mu_F^2}\,
+ B\left(\frac{m_b^2}{m_H^2},\frac{m_b^2}{\mu_F^2}\right)\,
\log\frac{m_H}{m_b^2}\,
+ \mathcal{O}\left(\frac{m_b^2}{m_H^2},\frac{m_b^2}{\mu_F^2}\right)\,,
\end{equation}
where we define $A$ and $B$, such that
\begin{equation}
\lim_{m_b\rightarrow 0} A\left(\frac{m_b^2}{m_H^2},\frac{m_b^2}{\mu_F^2}\right)\,
=\, A\, ,
\end{equation}
\begin{equation}
\lim_{m_b\rightarrow 0} B\left(\frac{m_b^2}{m_H^2},\frac{m_b^2}{\mu_F^2}\right)\,
=\, 0 \, ,
\end{equation}
and by $\mathcal{O}\left(\frac{m_b^2}{m_H^2},\frac{m_b^2}{\mu_F^2}\right)$,
we mean terms that are only given by powers of the two ratios.
In this way we separate terms that appears also in the massless
case, and get modified by power suppressed terms in the massive case,
and new terms that arise only when the massive case is considered.
As in the massless case the function $A$ is proportional
to the Altarelli-Parisi splitting function $P_{qq}$, by extension
we can define the massive $A$ function to be proportional
to the massive $P_{qq}$, which in turn is proportional to
${\bf V}^{qg,q}$ of Eq.~(\ref{eq:diff_sub}). This means that
this term corresponds to the term that would be absorbed into
the parton densities in the massive case at leading twist.
However, PDFs are generally given in schemes that account
for some mass effects at fixed-order by means of some matching
method, like \FONLL~\cite{Forte:2010ta}, ACOT~\cite{Aivazis:1993pi},
or TR~\cite{Thorne:1997ga,Thorne:1997uu}. Mass effects
included in these schemes coincide up to higher order
effects and it is easy to see that the order considered in this
work they have the same functional form of the function $A$.
The $B$ function appearing in Eq.~(\ref{eq:mass_fact}) is
just make up of terms that are proportional to $m_b^2$.
These terms are clearly non divergent, and in principle it
is not necessary to absorb them into an initial state radiation
term. However, some of these terms might be universal and could
definitely be accounted for by higher twist contributions to the
QCD factorisation formulae as well as with the introduction of an
intrinsic (or static) bottom-quark component in the proton.
Although these terms might in principle spoil factorisation,
we can make sure that they are indeed suppressed.
Subtracting the $A$ part of Eq.~(\ref{eq:mass_fact}), corresponds
to Eq.~(\ref{eq:realdiff})
\begin{equation}
\int{\rm d}\PS{2}\biggl[\Real(m_b\neq 0)-\Subtraction\biggr] \propto
B\left(\frac{m_b^2}{m_H^2},\frac{m_b^2}{\mu_F^2}\right)\,
\log\frac{m_H}{m_b^2}\,
+ \mathcal{O}\left(\frac{m_b^2}{m_H^2},\frac{m_b^2}{\mu_F^2}\right)\, ,
\end{equation}
which we know from the previous section being exactly zero
in the soft and collinear limit. Further, when it is not zero,
it is proportional to $m_b^2/s_{ab}\sim 1 \times 10^{-3}$
which makes it very suppressed
\footnote{Numerically we see that at the level of total cross section,
Eq.~(\ref{eq:realdiff}) corresponds to roughly 0.001\% of the Born
cross section, which is comparable with the error from Montecarlo
integration.}.
We conclude that we can safely make use of standard PDF sets that
have been matched to include mass effects to obtain reliable predictions
at the NLO in the five flavour massive scheme.
\section{Simulations for $b\bar{b}\rightarrow H$ with massive quarks}
We now show some predictions
for $b\bar{b}\rightarrow H$ with massive quarks.
Predictions with massive initial state quarks
are obtained by implementing the massive
subtraction as presented in Section~\ref{sec:mass_sub}
within the \Sherpa event generator~\cite{Gleisberg:2008ta}.
Leading order matrix elements,
including those of real radiation processes,
are calculated using the \Amegic~\cite{Krauss:2001iv}
matrix element generator.
The differential subtraction follows closely that presented in
~\cite{Gleisberg:2007md} with the ingredients reported in
Section~\ref{sec:mass_sub}.
The integrated subtraction terms as well as the one loop contribution
are implemented separately in \Sherpa using
Eqs.~(\ref{eq:had_I}),(\ref{eq:bbh1loop}), this implementation
will be made public in a future \Sherpa release.
While no cuts at the generation level are applied,
in the following results we define $b$-jet any jet with $p_T\geq 25$~GeV
that has at least one $b$-flavoured parton in it. We further
require any particle in the final state to have $|\eta|\leq 2.5$.
We generate only fixed-order events at this stage, {\it i.e.}
no parton shower effects are accounted.
\begin{figure}
\centering{
\includegraphics[width=0.7\textwidth]{plots/pthb.pdf}
\includegraphics[width=0.7\textwidth]{plots/etah.pdf}
}
\caption{The $p_T$ and $\eta$ spectrum of the Higgs boson at NLO.
We compare the vanilla 5FS, and the 5FMS matched with three different
PDF sets. More details in the text.}
\label{fig:mhiggs}
\end{figure}
\begin{figure}
\centering{
\includegraphics[width=0.7\textwidth]{plots/ptb1.pdf}
\includegraphics[width=0.7\textwidth]{plots/eta1.pdf}
}
\caption{The $p_T$ and $\eta$ spectrum of the $b$-jet.
We compare the vanilla 5FS, and the 5FMS matched with three different
PDF sets. More details in the text.}
\label{fig:mb}
\end{figure}
In order to study the impact of the inclusion of mass effects,
we compare the five flavour massive scheme (5FMS) with the
vanilla five flavour scheme, where $b$ are massless.
We further consider matching the 5FMS partonic cross section to three
different PDF sets. The first choice is, as we discussed in the
previous section, to use a standard GM-VFNS evolved PDF set.
As these sets include at some fixed order accuracy, higher than
the one considered in this example, we claim that they provide
a consistent result, up to a negligible effect. In particular,
we choose the default PDF set in \Sherpa, namely the NNPDF30 set
evolved at NNLO with $\alpha_s(m_Z)=0.118$~\cite{Ball:2014uwa}.
To show the importance of such mass effects in PDFs, we compare
against the same PDF set re-evolved at NNLO, but in a ZM-VFNS.
This means that no mass effects are accounted other than threshold
effects, and massless splitting kernels are used to determine
the evolution operator.
The last set we compare against, is a PDF set where the
evolution of the $b$ PDF is obtained using the splitting
kernel of Eq.~(\ref{eq:diff_sub}),
\begin{multline}
f_b^{\text{Massive } P_{qq}}(x,Q^2) = \frac{\alpha_s}{2\pi}
\log\frac{Q^2}{m_b^2}
\int_x^1\frac{{\rm d} z}{z}\,\left[\frac{1+z}{1-z}-
\frac{2\,z\,m_b^2}{Q^2\,(1-z)}\right]_+\,
f_b\left(\frac{x}{z},Q^2\right) \\
+ P_{qg}^{(0)}(z)\,f_g\left(\frac{x}{z},Q^2\right)\,
+\,\mathcal{O}(\alpha_s^2)\,.
\end{multline}
We report results in Figs.~\ref{fig:mhiggs},~\ref{fig:mb}
for the $p_T$ and $\eta$ spectra of the Higgs boson and the
$b$-jet respectively.
By definition, and by the conclusions laid out in
the previous chapters, we expect mass effects to play
a marginal, order $\sim 1-5\%$, effects at the level of total
cross sections. Further, as they are power suppressed,
we expect them to be less important at large ${p}_{T}$,
while having the largest impact in the lower bins of
the distribution. This is due to the fact that the
difference in the mass treatment between the two scheme
is only in the hard matrix elements.
Consequently, as pictured in Fig.~\ref{fig:mhiggs},
for example, we see that the 5FMS obtained with the
standard PDF set somehow starts a few percent off of the
5FS, while they consistently overlap for
$p_T(H,b)\gtrsim 100$~GeV.
We now turn to the comparison with the PDF set evolved in the
zero mass variable flavour number scheme (ZM-VFNS).
The difference between the 5FMS in this set up and the
massless 5FS, in the former mass effects are included
in the hard matrix elements, while some are included in the
PDF in the latter. Therefore, on top of the power suppressed
terms there are other terms coming from the matching. These
terms include constant terms, {\it i.e.} with respect to $m_b$,
as it can be seen in Figs.~\ref{fig:mhiggs},~\ref{fig:mb}.
Our final reference is a five flavour massive
scheme obtained with a PDF set evolved with a massive splitting
function at the leading order. We firstly notice that a direct
comparison with a standard parton density set is somehow bugged
by the fact the mass effects are included at different orders
in the two PDF sets. In addition the evolution operator is used
at two different orders. As a consequence the size of mass effects
is not necessarily truthful. However it is interesting to see
that this scheme behaves with respect to the ZM-VFNS 5FMS, in
roughly the same way as the 5FMS with standard PDFs behaves with
respect to the vanilla 5FS. The only difference in such a behaviour
is that the scale at which the two scheme start to coincide
is slightly larger, at around $\sim 150$~GeV.
We now briefly discuss the pseudo rapidity spectrum.
In this case all schemes differ by one another
by a constant amount of a few percent. This
is somehow obvious as pseudo rapidity is a dimensionless
variable, and as such does not introduce any scale
to make mass effects less or more important in any region.
We thus naturally expect mass effects to be roughly of the
same order as that seen in the case of the total cross section.
This fact can be seen explicitly in
Figs.~\ref{fig:mhiggs},~\ref{fig:mb}.
We conclude that mass effects for this process
are generally very small, both at the inclusive and
differential level. We find that when they are
important they generally do not exceed a few percent.
On the other hand, this might not hold true
for all processes involving heavy quarks. As such,
the five flavour massive scheme provides a useful
and consistent scheme to produce fully differential
results including mass effects.
\ No newline at end of file
Index: trunk/Thesis/text/FONLL.tex
===================================================================
--- trunk/Thesis/text/FONLL.tex (revision 9)
+++ trunk/Thesis/text/FONLL.tex (revision 10)
@@ -1,3 +1,3 @@
\chapter{Matching the 4F and the 5F schemes}
\label{chap:FONLL}
-\input{text/fonll-method}
\ No newline at end of file
+%\input{text/fonll-method}
\ No newline at end of file
Index: trunk/Thesis/text/introduction.tex
===================================================================
--- trunk/Thesis/text/introduction.tex (revision 9)
+++ trunk/Thesis/text/introduction.tex (revision 10)
@@ -1,14 +1,15 @@
\chapter{Introduction}
\numberwithin{equation}{chapter}
\setcounter{equation}{0}
% The outline is the following:
% - Intro : Why are mass effects-important (precision determination of H background and precision LHC physics) => Inclusive vs Differential
% - Available schemes : 4F vs 5F => pro and cons
% - FONLL : Matching the 4 and the 5F scheme (Inclusive and semi-inclusive case) (doesn't necessarily work for exclusive b-final state)
% - Going differential : Massive 5FS @ LO, complications at NLO => Subtraction (brief) => Massive Subtraction => Shower/Matching?
\input{text/QCD}
\input{text/4fv5f}
-%\input{text/montecarlo}
+\input{text/montecarlo}
+\input{text/4fvs5fd}
Index: trunk/Thesis/text/4fvs5fd.tex
===================================================================
--- trunk/Thesis/text/4fvs5fd.tex (revision 0)
+++ trunk/Thesis/text/4fvs5fd.tex (revision 10)
@@ -0,0 +1,517 @@
+
+\section{Bottom-jet associated Z-boson production}
+\label{sec:zbb}
+
+The production of a $Z$ boson in association with QCD jets provides the
+ideal test bed for the theoretical approaches outlined above. Through
+the decay of the $Z$ boson to leptons these processes yield a rather
+simple and clean signature with sizeable rates even for higher
+jet counts. Precise measurements of the production rates and differential
+distributions of both the $Z$-boson decay products and the accompanying
+jets offer discriminating power for miscellaneous theoretical approaches.
+In fact, measurements of $Z+$jets production served as key inputs for
+the validation of matrix-element parton-shower simulation techniques,
+cf.~\cite{Aad:2013ysa,Khachatryan:2014zya,Khachatryan:2016crw},
+and impressively underpin the enormous success of these calculational
+methods.
+
+Here we focus on the production of $Z$ bosons accompanied by identified
+$b$-jets. Comparison with data from both the ATLAS and CMS collaborations at
+$7$ TeV~\cite{Aad:2014dvb,Chatrchyan:2013zja} provides the benchmark for the
+accuracy and quality of four-- and five--flavour simulations with \Sherpa.
+Similar measurements at $8$ and $13$ TeV \LHC collision energies are
+under way~\cite{Khachatryan:2016iob}.
+
+\subsection{Details of the simulations}
+Efficient routines for the required QCD matrix-element calculations and a
+well understood QCD parton-shower are the key ingredients to all matching
+and merging calculations. Within \Sherpa LO matrix elements are
+provided by the built-in generators \Amegic~\cite{Krauss:2001iv} and
+\Comix~\cite{Gleisberg:2008fv}. While virtual matrix elements contributing to
+QCD NLO corrections can be invoked through interfaces to a number of
+specialised tools, e.g.\ \BlackHat~\cite{Berger:2008sj}, \Gosam~\cite{Cullen:2014yla},
+\Njet~\cite{Badger:2012pg}, \OpenLoops~\cite{Cascioli:2011va} or through
+the BLHA interface~\cite{Binoth:2010xt}, we employ in this study the
+\OpenLoops generator~\cite{OL_hepforge} in conjunction with the \Collier
+library~\cite{Denner:2016kdg,Denner:2014gla}. Infrared divergences are
+treated by the Catani--Seymour dipole method~\cite{Catani:1996vz,Catani:2002hc}
+which has been automated in \Sherpa~\cite{Gleisberg:2007md}. In this
+implementation mass effects are included for final-state splitter and
+spectator partons but massless initial-state particles are assumed throughout.
+\Sherpa's default parton-shower model~\cite{Schumann:2007mg,Hoeche:2009xc}
+is based on Catani--Seymour factorisation~\cite{Nagy:2006kb}. In order to
+arrive at meaningful fragmentation functions for heavy quarks, all modern
+parton showers take full account of their finite masses in the final state,
+although in algorithmically different ways. In \Sherpa, the transition from
+massless to massive kinematics is achieved by rescaling four-momenta at the
+beginning of the parton shower. In the initial-state parton shower in \Sherpa,
+the $g\to b\bar{b}$ and $b\to bg$ splitting functions do not contain $b$-quark
+mass effects in their functional form and account for mass effects in the
+kinematics only.
+
+In the following we briefly define the methods available in \Sherpa
+for simulating $b$-associated production processes, that will then be
+validated and applied for LHC predictions:
+\begin{description}
+\item[4F NLO (4F \MCatNLO):]{ In the {\em four--flavour scheme}, $b$-quarks
+ are consistently treated as {\em massive} particles, only appearing in
+ the final state. As a consequence, $b$-associated $Z$- and $H$-boson
+ production proceeds through the parton-level processes $gg\to Z/H+b\bar{b}$,
+ and $q\bar{q}\to Z/H+b\bar{b}$ at Born level. \MCatNLO matching is
+ obtained by consistently combining fully differential NLO QCD calculations
+ with the parton shower, cf.~\cite{Frixione:2002ik,Hoeche:2011fd}. Due to
+ the finite $b$-quark mass these processes do not exhibit infrared
+ divergences and the corresponding {\em inclusive} cross sections can thus
+ be evaluated without any cuts on the $b$-partons.
+ \begin{figure}
+ \centering{
+ \begin{multline}
+ \nonumber
+ {\rm d}\sigma^{\text{4F \MCatNLO}} \,=
+ {\rm d}\PS{3}\quad\,\biggl|
+ \raisebox{-2.1cm}{\includegraphics[scale=0.15]{plots/Vbb.pdf}}
+ \,+\,\Virtual\,\biggr|^2
+ \,\otimes\, \text{P.S.} \\
+ \quad+\quad
+ {\rm d}\PS{4}\quad\,\biggl|
+ \raisebox{-2.1cm}{\includegraphics[scale=0.15]{plots/Vbbj.pdf}}
+ \,\biggr|^2
+ \,\otimes\, \text{P.S.} \\
+ \end{multline}
+ }
+ \caption{Pictorial representation of $V+b\bar{b}$ processes
+ contributing to the 4F \MCatNLO calculation. Here $V=(Z,H)$.
+ by $\otimes$ we just mean that it is not a simple product, and P.S.
+ is the parton shower contribution. $\Virtual$ refers to the
+ one-loop virtual contributions to the born.}\label{fig:4fmcatnlo}
+\end{figure}}
+\item[5F LO (5F \MEPSatLO):]{ In the {\em five--flavour scheme} $b$-quarks
+ are {\em massless} particles in the {\em hard matrix element}, while they
+ are treated as massive particles in both the initial- and final-state
+ {\em parton shower}.
+
+ In the \MEPSatLO~\cite{Hoeche:2009rj} samples we merge $pp \rightarrow H/Z$
+ plus up to three jets at leading order; this includes, for instance, the
+ parton--level processes $b\bar{b} \to Z/H$, $gb\to Z/H b$,
+ $gg\to Z/H b\bar{b}$, $\dots$. To separate the various matrix-element
+ multiplicities, independent of the jet flavour, a jet cut of
+ $\Qcut = 10\,\UGeV$ is used in the $Z$ case while $\Qcut = 20\,\UGeV$ is
+ employed in $H$-boson production.
+
+ \begin{figure}
+ \centering{
+ \begin{multline}
+ \nonumber
+ {\rm d}\sigma^{\text{5F \MEPSatLO}} \,=
+ {\rm d}\PS{1}\quad\,\biggl|
+ \raisebox{-2.1cm}{\includegraphics[scale=0.15]{plots/V.pdf}}\biggr|^2
+ \,\otimes\, \text{P.S.}
+ \quad\oplus\quad{\rm d}\PS{2}\quad\,\biggl|
+ \raisebox{-2.1cm}{\includegraphics[scale=0.15]{plots/Vj.pdf}}\biggr|^2
+ \,\otimes\, \text{P.S.} \\
+ \quad\oplus\quad{\rm d}\PS{3}\quad\,\biggl|
+ \raisebox{-2.1cm}{\includegraphics[scale=0.15]{plots/Vjj.pdf}}\biggr|^2
+ \,\otimes\, \text{P.S.}
+ \quad\oplus\quad{\rm d}\PS{4}\quad\,\biggl|
+ \raisebox{-2.1cm}{\includegraphics[scale=0.15]{plots/Vjjj.pdf}}\biggr|^2
+ \,\otimes\, \text{P.S.} \\
+ \end{multline}
+ }
+ \caption{Same as Fig.~\ref{fig:4fmcatnlo} but for the
+ 5F \MEPSatLO. The $\oplus$ symbol
+ represents the merging, {\it i.e.} a sum with the overlapping
+ part removed.}
+ \end{figure}}
+\item[5F NLO (5F \MEPSatNLO):] In the 5FS \MEPSatNLO
+ scheme~\cite{Gehrmann:2012yg,Hoeche:2012yf}, we account for quark masses in
+ complete analogy to the LO case: the quarks are treated as massless in the
+ hard matrix elements, but as massive in the initia- and final-state parton
+ showering. Again, partonic processes of different multiplicity are merged
+ similarly to the \MEPSatLO albeit retaining their next-to-leading-order
+ accuracy. In particular, we consider the merging of the processes
+ $pp\rightarrow H/Z$ plus up to two jets each calculated with \MCatNLO
+ accuracy further merged with $pp\rightarrow H/Z + 3j$ calculated at
+ \MEPSatLO.
+\end{description}
+
+We consistently use four--flavour PDFs in the 4F
+scheme, i.e.\ the dedicated four--flavour NNPDF3.0 set~\cite{Ball:2014uwa} with
+the strong coupling given by $\alphaS(\MZ)\,=\,0.118$ and running at NLO.
+For the simulations in the five--flavour schemes the five--flavour NNLO PDFs
+from NNPDF3.0 are used, with $\alphaS(\MZ)\,=\,0.118$ and running at NNLO.
+We assume all quarks apart from the $b$ to be massless, with a pole mass of
+$m_b = 4.92$~\UGeV which enters the hard matrix-element calculation, where
+appropriate, and the parton shower.
+
+Results in the 4F and 5F schemes have been obtained with the default scale-setting
+prescription for parton-shower matched calculations in \Sherpa~\cite{Hoeche:2009rj,Hoeche:2010av}.
+They are
+calculated using a backward-clustering algorithm, and for each emission from the shower, couplings
+are evaluated at either the $k_T$ of the corresponding emitted particle (in the case of
+gluon emission), or at the invariant mass of the emitted pair (in the case
+of gluon splitting into quarks). The clustering stops at a ``core'' $2\to 2$
+process, with all scales set to $\mu_F=\mu_R=\mu_Q=m_T(V)/2$, where $m_T(V)$
+corresponds to the transverse mass of the boson. This scale is thus used to evaluate
+couplings in the hard matrix element and PDFs.
+The corresponding central values are supplemented with uncertainty bands
+reflecting the dependence on the unphysical scales. Renormalisation and
+factorisation scales are varied around their central value by a factor of
+two up and down, with a standard 7-point variation. The scale variations use
+the \Sherpa internal reweighting procedure~\cite{Bothmann:2016nao} and result in
+envelopes around the central value. Furthermore, we consider explicit variations
+of the parton-shower starting scale, i.e. $\mu_Q$, by a factor of two up and down.
+
+\subsection{Measurements at \LHC Run I -- the reference data}
+
+Based on a data set of $4.6\;{\rm fb}^{-1}$ integrated luminosity the ATLAS
+collaboration studied the production of $b$-jets associated with $Z/\gamma^*$
+that decay to electrons or muons~\cite{Aad:2014dvb}. The dilepton
+invariant mass ranges between $76\;{\text{GeV}} < m_{\ell\ell} < 106\;{\text{GeV}}$.
+Jets are reconstructed using the anti-$k_t$ algorithm \cite{Cacciari:2008gp}
+with a radius parameter of $R=0.4$, a minimal transverse momentum of
+$p_{T,j}> 20$ GeV and a rapidity of $|y_{j}|<2.4$. Furthermore, each jet
+candidate needs to be separated from the leptons by $\Delta R_{j\ell} > 0.5$.
+Jets containing $b$-hadrons are identified using a multi-variate technique.
+To match the outcome of the experimental analysis, simulated jets are
+identified as $b$-jets, when there is one or more weakly decaying $b$-hadron
+with $p_T>5$ GeV within a cone of $\Delta R=0.3$ around the jet axis. The
+sample of selected events is further subdivided into a class containing events
+with at least one $b$-jet ($1$-tag) and a class with at least two $b$-jets
+($2$-tag).
+
+A similar analysis was performed by CMS~\cite{Chatrchyan:2013zja}. There,
+electrons and muons are required to have a transverse momentum of
+$p_{T,\ell}>20$ GeV, a pseudorapidity $|\eta_{\ell}|<2.4$, and a dilepton
+invariant mass within $81\;{\text{GeV}} < m_{\ell\ell} < 101\;{\text{GeV}}$.
+Only events with exactly two additional $b$-hadrons were selected. The
+analysis focuses on the measurement of angular correlations amongst the
+$b$-hadrons and with respect to the $Z$ boson. This includes in particular
+variables sensitive to rather collinear $b$-hadron pairs. In addition, the
+total production cross section as a function of the vector boson's
+transverse momentum was measured.
+
+Both analyses are implemented and publicly available in the \Rivet
+analysis software~\cite{Buckley:2010ar} that, together with the \Fastjet
+package~\cite{Cacciari:2011ma}, is employed for all particle, i.e.\ hadron,
+level analyses.
+
+\subsection{Comparison with LHC data}
+
+In this section the theoretical predictions from \Sherpa are compared
+to the experimental measurements from \LHC Run I. We begin the discussion
+with the comparison with the measurements presented by the ATLAS collaboration
+in Ref.~\cite{Aad:2014dvb}. The total cross sections for $Z+\ge 1$ and
+$Z+\ge 2$ $b$ jets are collected in Fig.~\ref{fig:xstot}. Already there we
+see a pattern emerging that will further establish itself in the differential
+cross sections: While the 5F \MEPSatNLO results agree very well with data, the
+central values of the 5F \MEPSatLO cross sections tend to be around 10-20\%
+lower than the central values of data, but with theory uncertainties clearly
+overlapping them. For all the runs the uncertainty estimates include both,
+7-point variations of the perturbative scales $\mu_{R/F}$, as well as $\mu_Q$
+variations by a factor of two up and down. In contrast to the 5F case, the 4F
+\MCatNLO cross sections tend to be significantly below the experimental values
+for the $Z+\ge 1$ $b$-jets cross section, without overlap of uncertainties.
+In the $Z+\ge 2$ $b$-jets cross section the agreement between 4F \MCatNLO
+results and data is better, with the theoretical uncertainties including
+the central value of the measured cross section.
+
+\begin{figure}[!htb]
+ \centering{
+ \includegraphics[width=0.6\textwidth]{plots/plot_1b_orig.pdf}
+ %\hfill
+ \includegraphics[width=0.6\textwidth]{plots/plot_2b_orig.pdf}
+ \caption{Comparison of total production cross section predictions
+ with ATLAS data~\cite{Aad:2014dvb}. The error bars on the
+ theoretical results are calculated from variations of the
+ hard-process scales $\mu_{R/F}$ and the parton-shower starting
+ scale $\mu_Q$.
+ \label{fig:xstot}}
+ }
+\end{figure}
+In Fig.~\ref{fig:1b} the differential cross sections with respect to the
+transverse momentum and rapidity of the $b$-jets, normalised to the number of
+$b$-jets, are presented for events with {\em at least} one $b$-tagged jet.
+The shapes of both distributions are well modelled both by the 4F and
+the two 5F calculations. However, clear differences in the predicted
+production cross sections are observed. While the 5F~NLO results are in
+excellent agreement with data - both in shape and normalisation - the
+central values of the 5F~LO cross sections tend to be around 10\% below
+data, at the lower edge of the data uncertainty bands, and
+the 4F results are consistently outside data, about 25\% too low.
+In the lower panels of Fig.~\ref{fig:1b} and all the following plots
+in this section we show the uncertainty bands of the theoretical
+predictions, corresponding to the above described $\mu_{R/F}$ and $\mu_Q$
+variations. For the 5FS calculations the scale uncertainties clearly
+dominate, while for the 4F \MCatNLO scheme the shower-resummation
+uncertainty dominates.
+\begin{figure}[!htb]
+\centering{
+ \includegraphics[width=0.6\textwidth]{plots/ATLAS_p_t_b1.pdf}
+ \includegraphics[width=0.6\textwidth]{plots/ATLAS_y_b1.pdf}
+ \caption{Inclusive transverse-momentum and rapidity distribution of
+ all $b$-jets in events with at least one $b$-jet. Data taken
+ from Ref.~\cite{Aad:2014dvb}.}
+ \label{fig:1b}}
+\end{figure}
+
+This pattern is repeated in Fig.~\ref{fig:1bzpt}, where we show
+the differential $\sigma(Zb)$ cross section with respect to the dilepton
+transverse momentum and, rescaled to $1/N_{b-{\rm jets}}$, as a
+function of the azimuthal separation between the reconstructed $Z$ boson
+and the $b$-jets. Again, both distributions are very well modelled by both
+5F calculations. The 4F \MCatNLO prediction again underestimates data by
+a largely flat 20-25\%.
+
+\begin{figure}[!tbh]
+\centering{
+ \includegraphics[width=0.6\textwidth]{plots/ATLAS_p_t_Z.pdf}
+ \includegraphics[width=0.6\textwidth]{plots/ATLAS_dphi_Zb.pdf}
+ \caption{Transverse-momentum distribution of the Z boson (left) and the
+ azimuthal separation between the $Z$ boson and the $b$-jets (right)
+ in events with at least one $b$-jet. For the $\Delta\phi(Z,b)$ measurement
+ the additional constraint $p_{T,ll}>20\;{\rm GeV}$ is imposed.
+ Data taken from Ref.~\cite{Aad:2014dvb}.
+ \label{fig:1bzpt}}
+}
+\end{figure}
+
+Moving on to final states exhibiting at least two identified $b$-jets,
+the role of the 5F~LO and 4F~NLO predictions are somewhat reversed:
+As can be inferred from Fig.~\ref{fig:xstot}, the 4F and 5F~NLO samples
+provide good estimates for the inclusive $Zbb$ cross section, while the
+5F~LO calculation undershoots data by about $20$\%. In Fig.~\ref{fig:2b}
+the $\Delta R$ separation of the two highest transverse-momentum $b$-jets
+along with their invariant-mass distribution is presented. Both the 4F and
+the 5F approaches yield a good description of the shape of the distributions.
+It is worth stressing that this includes the regions of low invariant mass
+and low $\Delta R$, corresponding to a pair of rather collinear $b$-jets.
+This is a region that is usually riddled by potentially large logarithms,
+where the parton shower starts taking effect. Note that in the comparison
+presented in~\cite{Aad:2014dvb} this region showed some disagreement between
+data and other theoretical predictions based on NLO QCD (dressed with parton
+showers).
+
+\begin{figure}[!htb]
+\centering{
+ \includegraphics[width=0.6\textwidth]{plots/ATLAS_dR_bb.pdf}
+ \includegraphics[width=0.6\textwidth]{plots/ATLAS_m_bb.pdf}
+ \caption{The $\Delta R$ separation (left) and invariant-mass distribution
+ (right) for the leading two $b$-jets. Data taken from
+ Ref.~\cite{Aad:2014dvb}.
+ \label{fig:2b}}
+}
+\end{figure}
+
+In Fig.~\ref{fig:2bzpt} the resulting transverse-momentum distribution of
+the dilepton system when selecting for events with at least two associated
+$b$-jets is shown. The shape of the data is very well reproduced by the
+4F~\MCatNLO and 5F~\MEPSatNLO samples. Also the 5F~\MEPSatLO prediction
+describes the data well despite of the overall rate being $20$\% lower than
+observed in data.
+
+\begin{figure}[!htb]
+\centering{
+ \includegraphics[width=0.6\textwidth]{plots/ATLAS_p_T_Z_2b.pdf}
+ \caption{Transverse-momentum distribution of the dilepton system for events
+ with at least two $b$-jets. Comparison against various calculational
+ schemes.
+ Data taken from Ref.~\cite{Aad:2014dvb}.
+ \label{fig:2bzpt}}
+}
+\end{figure}
+
+The measurements presented by the CMS collaboration in
+Ref.~\cite{Chatrchyan:2013zja} focus on angular correlations between
+$b$-hadrons rather than $b$-jets. Two selections with respect to the
+dilepton transverse momentum have been considered, a sample requiring
+$p_T(Z)>50\;{\rm GeV}$ and an inclusive one considering the whole range
+of $p_T(Z)$. The $\Delta R$ and $\Delta \phi$ separation of the
+$b$-hadrons obviously prove to be most sensitive to the theoretical
+modelling of the $b$-hadron production mechanism and the interplay of
+the fixed-order components and the parton showers. They are presented in
+Figs.~\ref{fig:2bdR_CMS} and \ref{fig:2bdPhi_CMS}. In general, a good
+agreement in the shapes of simulation results and data is found, with
+the same pattern of total cross sections as before: the 5F~\MEPSatNLO
+sample describes data very well, while the 4F~\MCatNLO results tend to
+be a little bit, about 10\%, below data, with data and theory
+uncertainty bands well overlapping, while the central values of the
+5F~\MEPSatLO results undershoot data by typically 20-25\%.
+\begin{figure}[!htb]
+\centering{
+ \includegraphics[width=0.6\textwidth]{plots/CMS_dR_BB.pdf}
+ %\hfill
+ \includegraphics[width=0.6\textwidth]{plots/CMS_dR_BB_pt50.pdf}
+ \caption{ $\Delta R_{BB}$ distribution for two selections of the transverse
+ momentum of the $Z$ boson. Data taken from Ref.~\cite{Chatrchyan:2013zja}.
+ \label{fig:2bdR_CMS}}
+}
+\end{figure}
+
+\begin{figure}[!htb]
+\centering{
+ \includegraphics[width=0.6\textwidth]{plots/CMS_dphi_BB.pdf}
+ %\hfill
+ \includegraphics[width=0.6\textwidth]{plots/CMS_dphi_BB_pt50.pdf}
+ \caption{ $\Delta \phi_{BB}$ distribution for two selections of the transverse
+ momentum of the $Z$ boson. Data taken from Ref.~\cite{Chatrchyan:2013zja}.
+ \label{fig:2bdPhi_CMS}}
+}
+\end{figure}
+
+Overall it can be concluded that the 5F \MEPSatNLO calculation yields the best
+description of the existing measurements, regarding both the production rates
+{\em and} shapes. The 4F \MCatNLO and 5F \MEPSatLO schemes succesfully model
+the shape of the differential distributions but consistently underestimate the
+production rates.
+
+
+\section{Bottom-jet associated Higgs-boson production}
+\label{sec:hbb}
+In this section we present predictions for $b$-jet(s) associated production
+of the Standard-Model Higgs boson in $pp$ collisions at the $13$ TeV \LHC
+obtained in the four-- and five--flavour schemes. As standard when dealing with this
+process, we do not include contributions from the gluon-fusion channel.
+However, in the 4F \MCatNLO we do include terms proportional to the top-quark
+Yukawa coupling, contributing to order $y_by_t$ as an interference effect at
+NLO QCD~\cite{Dittmaier:2003ej,Dawson:2003kb,deFlorian:2016spz}.
+Although associated $Z+$ $b$-jet(s) production serves as a good proxy for the
+Higgs-boson case, there are important differences between both processes, mainly
+due to the different impact of initial-state light quarks, which couple to $Z$
+bosons but not to the Higgs boson.
+
+As before, QCD jets are defined through the anti-$k_t$ algorithm using a
+radius parameter of $R=0.4$, a minimal transverse momentum $p_{T,j}>25$~GeV,
+and a rapidity cut of $|y_{j}|<2.5$. In this case, we consider results that are at
+the parton level only, disregarding hadronisation and underlying-event
+effects, which may blur the picture. We consider single $b$-tagged jets only,
+thus excluding jets with intra-jet $g\rightarrow b\bar{b}$ splittings from
+the parton shower which would be the same for all flavour schemes we
+investigate. As for $Z$-boson production, we separate the event samples
+into categories with at least one $b$-jet, i.e.\ $H+\geq 1 b$-jet events,
+and at least two tagged $b$-jets, i.e.\ $H+\geq 2 b$-jets events.
+
+\begin{table}[!hbt]
+ \centering
+ \begin{tabular}[\linewidth]{lc|c}
+ \toprule
+ \LHC 13~TeV & $H + \geq 1 b$-jets [fb] & $H + \geq 2 b$-jets [fb]\\
+ \midrule
+ $\sigma_{\text{\MCatNLO}}^{4F}$ &
+ $45.2^{+15.5\%}_{-18.4\%}$ &
+ $4.5^{+25.1\%}_{-26.3\%}$\\
+ $\sigma_{\text{\MEPSatLO}}^{5F}$ &
+ $79.3^{+34.0\%}_{-25.4\%}$ &
+ $3.8^{+34.3\%}_{-30.3\%}$\\
+ $\sigma_{\text{\MEPSatNLO}}^{5F}$ &
+ $110.5^{+14.2\%}_{-16.0\%}$ &
+ $6.9^{+27.3\%}_{-27.1\%}$\\
+ \bottomrule
+ \end{tabular}
+ \caption{$13$~TeV total cross sections and the corresponding
+ $\mu_{F/R}$ and $\mu_Q$ uncertainties for $H + \geq 1 b$ and $H + \geq 2 b$s.
+ }
+ \label{tab:hbbxs}
+\end{table}
+
+\begin{figure}[!htb]
+\centering
+ \includegraphics[width=0.6\textwidth]{plots/p_t_H_b1.pdf}
+ \includegraphics[width=0.6\textwidth]{plots/p_t_b1.pdf}
+ \caption{Predictions for the transverse-momentum distribution of the Higgs
+ boson (left panel) and the leading $b$-jet (right panel) in inclusive
+ $H+b$-jet production at the $13$ TeV \LHC.}\label{fig:1bh}
+\end{figure}
+
+\begin{figure}[!htb]
+\centering
+ \includegraphics[width=0.6\textwidth]{plots/p_t_H_b2.pdf}
+ \caption{The transverse-momentum distribution of the Higgs boson in
+ inclusive $H+2b$-jets production at the $13$ TeV \LHC.}\label{fig:1bhpt}
+\end{figure}
+
+\begin{figure}[!htb]
+\centering
+ \includegraphics[width=0.6\textwidth]{plots/R_bb.pdf}
+ \includegraphics[width=0.6\textwidth]{plots/m_bb.pdf}
+ \caption{Predictions for the $\Delta R$ separation of the two leading
+ $b$-jets (left panel) and their invariant-mass distribution
+ (right panel) in inclusive $H+2b$-jets production at the
+ $13$ TeV \LHC.}\label{fig:2bh}
+\end{figure}
+In Tab.~\ref{tab:hbbxs} cross sections for the three calculations are
+reported. Historically, inclusive results have largely disagreed between the
+4F and the 5F scheme. This feature is observed for the case at hand, too,
+and especially so for the case of one tagged $b$-jet. There the
+4F \MCatNLO prediction is smaller than the 5F results by factors of about
+$1.75$ (5F LO) and of $2.44$ (5F NLO). The relative differences are
+reduced when a second tagged $b$-jet is demanded. In this case we find that
+the 4F result lies between the two 5F results, about 20\% higher than the
+LO predictions, and a factor of about 1.5 lower than the 5F NLO predictions.
+In both cases, inclusive $H+b$ and $H+bb$ production, the uncertainty bands
+of the two 5F predictions, corresponding to 7-point $\mu_{R/F}$ variations
+and $\mu_Q$ variations by a factor of two up and down, do overlap. While for
+the two $b$-jet final states this includes the 4F result, for the one $b$-jet
+case the 4F result is not compatible with the 5F predictions, taking into
+account the considered scale uncertainties. It is worth noting that a milder
+form of this relative scaling of the cross sections was already observed
+in the $Z$ case.
+
+In the case of the total inclusive cross section, this very large difference
+can be mitigated by including higher-order corrections, on the one hand, and a
+better assessment of which choice of the unphysical scales yields the better
+agreement~\cite{Frederix:2011qg,Wiesemann:2014ioa,deFlorian:2016spz,Lim:2016wjo}.
+However, only a recent effort to match the two schemes~\cite{Forte:2015hba,
+ Forte:2016sja,Bonvini:2015pxa, Bonvini:2016fgf}
+has clearly assessed the relative importance of mass corrections
+(appearing in the 4F scheme) and large log resummation (as achieved in a 5F
+scheme). In particular it has been found that the difference between these two
+schemes is mostly given by the resummation of large logarithms, thus suggesting that
+for an inclusive enough calculation either a 5F scheme or a matched scheme
+should be employed. This is the same situation that one faces, albeit
+milder, in the $Z$ case, where, in terms of normalisation the 5F scheme
+performed better in all cases and especially in inclusive calculations. We
+therefore recommend that in terms of overall normalisation, the 5F \MEPSatNLO
+scheme should be used to obtain reliable predictions.
+
+Let us now turn to the discussion of the relative differences in the
+shapes of characteristic and important distributions. To better appreciate
+shape differences, all differential distribution are normalised to the
+respective cross section, i.e.\ the inclusive rates $\sigma(Hb)$ and
+$\sigma(Hbb)$. In all cases we obtain agreement at the 15\%-level or better
+between the 5F~\MEPSatNLO and 4F~\MCatNLO samples, the only exception, not
+surprisingly, being the region of phase space where the two $b$'s come
+close to each other and resummation effects start playing a role.
+Typically, the 5F~\MEPSatLO predictions are also in fair agreement with the
+other two results, however, they exhibit a tendency for harder tails in
+the $p_T$ distributions, mainly in the inclusive Higgs-boson $p_T$ and
+in the transverse momentum of the second $b$ jet.
+
+Starting with Fig.~\ref{fig:1bh}, the transverse-momentum distributions of the
+Higgs boson and the leading $b$-jet in the case of at least one $b$-jet tagged is
+displayed. Similarly to the $Z$ example, this is the region where one would
+expect the 5F scheme to perform better. However, again similarly to the $Z$
+case, the three schemes largely agree in terms of shapes, being well within
+scale uncertainties. Notably, this turns out to be particularly true for
+the low ($\sim 20$--$100$~\UGeV) $p_T$ region where one could have expected
+deviations to be the largest.
+
+In Figs.~\ref{fig:1bhpt} and \ref{fig:2bh} we present differential
+distributions for the selection of events with at least two tagged
+$b$-jets. While Fig.~\ref{fig:1bhpt} shows the resulting Higgs-boson
+transverse-momentum distribution, Fig.~\ref{fig:2bh} compiles results for
+the $\Delta R$ separation of the two leading $b$-jets and their invariant-mass
+distribution. For such two $b$-jets observables the 4F scheme is expected
+to work best, especially when the two $b$ are well separated to suppress
+potentially large logarithms. However, in agreement with the $Z$-boson
+case, no significant differences between the various scheme arise when
+taking into account $\mu_{R/F}$ and $\mu_Q$ scale-variation uncertainties.
+Once again the region of low $p_T$ in Fig.~\ref{fig:1bhpt} and the region of
+low $m(b,b)$ in Fig.~(\ref{fig:2bh}) show excellent agreement amongst the various
+descriptions. As anticipated, larger differences can be seen between the
+two 5FS and the 4F \MCatNLO calculations, in the very low $\Delta R(b,b)$ and
+$m(b,b)$ regions, Fig.~(\ref{fig:2bh}), where the two $b$-jets become
+collinear. This feature is however most likely due to the fact that we are
+dealing with partonic $b$-jets as opposed to {\em hadronic} ones. Taking
+as a reference the $Z$-boson case once again, in fact, where this difference
+is not present at all, suggests that a realistic simulation, that
+accounts for hadronisation effects, should largely suppress this difference.
Index: trunk/Thesis/text/4fv5f.tex
===================================================================
--- trunk/Thesis/text/4fv5f.tex (revision 9)
+++ trunk/Thesis/text/4fv5f.tex (revision 10)
@@ -1,589 +1,120 @@
-\section{Massive vs Massless schemes}
+\section{Factorisation of QCD cross sections}
\label{sec:4fsvs5fs}
Every QCD calculation at the LHC is based on a collinear factorisation theorem
which states that the low energy (long-distance) and the high energy
(short-distance or partonic) part of an observable factorise. Additionally,
the long-distance part is process independent and encodes how the longitudinal
momentum of the proton is shared among its partons and is commonly known as
Parton Distribution Function (PDF).
However, this factorisation theorem can only be exactly derived
in Deeply Inelastic Scattering processes using an Operator Product
Expansion. In this expansion, only the leading term is really believed to be
universal and thus put in the definition of PDF, while power-suppressed (higher-twist)
terms are neglected~\cite{Collins:1985ue,Collins:1989gx}.
Although a formal theorem/proof does not exist for general LHC processes,
it is widely believed that a factorisation theorem holds for
the most important production modes, in non-pathological observables,
with higher-twist contributions being suppressed by powers of $\Lambda_{QCD}/Q$,
where $Q$ is some energy scale in the hard process, while
$\Lambda_{QCD}$ represents the energy at which the $\alpha_s(\Lambda_{QCD})\sim 1$.
Including such power suppressed contributions would require to
calculate higher-twist corrections to the parton densities, which, as we
said, are non-universal. For this reasons the common practice is to
consider massless all the quarks that are {\it active} in the parton model,
while {\it decoupling} all massive quarks. It looks clear that
the definition of what is a massive quark depends on whether the ratio
$\Lambda_{QCD}/m_Q$ is large or small.
This picture, however, is extremely simplistic for two reasons. First,
it really only works if there is only one scale in the process. As soon
as another energy scale, $t$, appears, ratios of $m_Q/t$ follow and the hierarchy
is no longer straightforward.
Second, factorisation has an intrinsic degree of freedom which act as an energy scale.
In fact, what we define as to be high or low-energy, depends on some energy reference, $\mu_F$.
In practice, PDFs and partonic cross-sections are a function
of this scale. As this scales runs, terms like $\log\mu_F/m_Q$ appear and
might dominate over power-suppressed terms.
In formulae, the factorised differential cross section can be written as
\begin{equation}
\label{eq:c1_fxs}
{\rm d}\sigma(Q^2) = \sum_{a,b \in (q,g)}\int_0^1 {\rm d} x_1 \int_0^1 {\rm d} x_2\,
f_{a}(x_1,\mu_F^2)\,f_{b}(x_2,\mu_F^2) {\rm d}\hat{\sigma}_{ab}\left(x_1,x_2,\frac{\mu_F^2}{Q^2}\right)
+ \mathcal{O}\left(\frac{\Lambda_{QCD}^2}{Q^2}\right).
\end{equation}
Here and in the following, unless differently stated, hatted letters
represent the partonic version of that object, such that ${\rm d}\hat{\sigma}_{ab}$
stands for the partonic differential cross section for incoming partons $(a,b)$.
We indicate with $f_{a}(x_1,\mu_F^2)$ the PDF for parton $a$ with momentum fraction
$x_1$ at the {\it factorisation scale} $\mu_F$. These functions are subject to the
DGLAP equations \cite{Altarelli:1977zs,Gribov:1972ri,Dokshitzer:1977sg}:
\begin{equation}
\label{eq:c1_dglap}
\frac{{\rm d}f_{a}(x,\mu_F^2)}{{\rm d}\log\mu_F} = \frac{\alpha_s(\mu_F^2)}{2\,\pi}
\sum_{b\in(q,g)} \int_{x}^1\frac{{\rm d}z}{z}\,P_{ab}(z)\,f_{b}\left(\frac{x}{z},\mu_F^2\right)
+\mathcal{O}(\alpha_s^2)
\end{equation}
The functions $P_{ab}(z)$ are the Altarelli-Parisi splitting function and they are reported
in Appendix~\ref{app_1} for reference.
Further the partonic cross section can be written as expansion in $\alpha_s$, with
$\mu_R$ being the {\it renormalisation scale}
\begin{equation}
{\rm d}\hat{\sigma}_{ab}\left(x_1,x_2,\frac{\mu_F^2}{Q^2}\right) = \sum_{k}\alpha_s^{k}(\mu_R)\,
{\rm d}\hat{\sigma}_{ab}^{(k)}\left(x_1,x_2,\frac{\mu_F^2}{Q^2},\frac{\mu_R^2}{Q^2}\right),
\end{equation}
with
\begin{equation}
\label{ch1:eq:ralphas}
\frac{{\rm d}\alpha_s(\mu_R^2)}{{\rm d}\log\mu_R} = -b_0 \alpha_s^2 +\mathcal{O}(\alpha_s^3)
\end{equation}
and
\begin{equation}
b_0 = \frac{\beta_0}{2\pi} = \frac{11\, C_A\,-\,4\,n_f\,T_R}{12\,\pi}.
\end{equation}
It is important to notice at this point that both the running of the coupling constant,
and the evolution of the parton densities, as they depend on $b_0$,
depend on how many active ($=$ massless)
flavour through $n_f$.
There are, traditionally, two approaches in deciding how many flavours are active.
One approach, consists in
taking $\Lambda_{QCD}$ as a reference scale. In this way, every quark that has a mass
greater than $\Lambda_{QCD}$, is considered a massive, or heavy, quark, while every other
parton is considered massless and, as a consequence, contributes to $n_f$.
This approach is generally called Fixed Flavour Number Scheme (FFNS). In this case, in fact,
$n_f$ remains a fixed number at all energy scales.
In the FFNS, heavy quarks do not contribute
to neither the running of the coupling constant or the evolution of PDFs.
In this way, heavy quarks can never appear as initial state particles in the calculation of
hard scattering matrix elements, but they can only be produce in them. This way
the partonic cross section can retain the exact mass dependence.
In an opposed approach, one could argue that some other scales, like $\mu_{R}$ or $\mu_{F}$,
is taken as the
reference scale, as in practice one gets logs of $\mu_{R,F}/m_Q$ and not
of $\Lambda_{QCD}/m_Q$, when solving Eqs.~(\ref{eq:c1_dglap},\ref{ch1:eq:ralphas}).
Then the heavy quark will start contributing to both the running
of the coupling constant and the evolution of PDFs, only when the $\mu_{R,F}\gtrsim m_Q$.
This is called a Variable Flavour Number Scheme (VFNS)
\footnote{There are various ways of defining a VFNS. In particular the choice of
threshold and the inclusion of some mass suppressed terms play a role. All the
different choices, however, are equal up to mass, power-suppressed terms.}.
In a VFNS, the mass of the heavy
quark mass parametrise whether the particle is a heavy quark or an active parton, thus acting
like a threshold. Threshold effects that appear through terms like $\log\mu_{R,F}/m_Q$
are resummed to all order by Eqs.~(\ref{eq:c1_dglap},\ref{ch1:eq:ralphas}), and all
mass effects are consistently neglected in the calculation of hard matrix elements.
In the case of $b$ quarks, which is the case of interest at hand, the FFNS corresponds
to the massive, or 4F scheme. While the VFNS is called massless or 5F scheme.
In the following sections we highlight some of the main differences and
similarities of these schemes applied to the example of the production of a
$Z$ boson in association with at least one or two $b$-jets. After having compared
the two schemes with data, we compare the schemes between themselves in the
case of the production of a Higgs boson in association with at least one or two $b$-jets
An extended discussion of the following results can be found in~\cite{Krauss:2016orf,Badger:2016bpw}.
-\section{Bottom-jet associated Z-boson production}
-\label{sec:zbb}
-
-The production of a $Z$ boson in association with QCD jets provides the
-ideal test bed for the theoretical approaches outlined above. Through
-the decay of the $Z$ boson to leptons these processes yield a rather
-simple and clean signature with sizeable rates even for higher
-jet counts. Precise measurements of the production rates and differential
-distributions of both the $Z$-boson decay products and the accompanying
-jets offer discriminating power for miscellaneous theoretical approaches.
-In fact, measurements of $Z+$jets production served as key inputs for
-the validation of matrix-element parton-shower simulation techniques,
-cf.~\cite{Aad:2013ysa,Khachatryan:2014zya,Khachatryan:2016crw},
-and impressively underpin the enormous success of these calculational
-methods.
-
-Here we focus on the production of $Z$ bosons accompanied by identified
-$b$-jets. Comparison with data from both the ATLAS and CMS collaborations at
-$7$ TeV~\cite{Aad:2014dvb,Chatrchyan:2013zja} provides the benchmark for the
-accuracy and quality of four-- and five--flavour simulations with \Sherpa.
-Similar measurements at $8$ and $13$ TeV \LHC collision energies are
-under way~\cite{Khachatryan:2016iob}.
-
-\subsection{Details of the simulations}
-Efficient routines for the required QCD matrix-element calculations and a
-well understood QCD parton-shower are the key ingredients to all matching
-and merging calculations. Within \Sherpa LO matrix elements are
-provided by the built-in generators \Amegic~\cite{Krauss:2001iv} and
-\Comix~\cite{Gleisberg:2008fv}. While virtual matrix elements contributing to
-QCD NLO corrections can be invoked through interfaces to a number of
-specialised tools, e.g.\ \BlackHat~\cite{Berger:2008sj}, \Gosam~\cite{Cullen:2014yla},
-\Njet~\cite{Badger:2012pg}, \OpenLoops~\cite{Cascioli:2011va} or through
-the BLHA interface~\cite{Binoth:2010xt}, we employ in this study the
-\OpenLoops generator~\cite{OL_hepforge} in conjunction with the \Collier
-library~\cite{Denner:2016kdg,Denner:2014gla}. Infrared divergences are
-treated by the Catani--Seymour dipole method~\cite{Catani:1996vz,Catani:2002hc}
-which has been automated in \Sherpa~\cite{Gleisberg:2007md}. In this
-implementation mass effects are included for final-state splitter and
-spectator partons but massless initial-state particles are assumed throughout.
-\Sherpa's default parton-shower model~\cite{Schumann:2007mg,Hoeche:2009xc}
-is based on Catani--Seymour factorisation~\cite{Nagy:2006kb}. In order to
-arrive at meaningful fragmentation functions for heavy quarks, all modern
-parton showers take full account of their finite masses in the final state,
-although in algorithmically different ways. In \Sherpa, the transition from
-massless to massive kinematics is achieved by rescaling four-momenta at the
-beginning of the parton shower. In the initial-state parton shower in \Sherpa,
-the $g\to b\bar{b}$ and $b\to bg$ splitting functions do not contain $b$-quark
-mass effects in their functional form and account for mass effects in the
-kinematics only.
-
-In the following we briefly define the methods available in \Sherpa
-for simulating $b$-associated production processes, that will then be
-validated and applied for LHC predictions:
-\begin{description}
-\item[4F NLO (4F \MCatNLO):] In the {\em four--flavour scheme}, $b$-quarks
- are consistently treated as {\em massive} particles, only appearing in
- the final state. As a consequence, $b$-associated $Z$- and $H$-boson
- production proceeds through the parton-level processes $gg\to Z/H+b\bar{b}$,
- and $q\bar{q}\to Z/H+b\bar{b}$ at Born level. \MCatNLO matching is
- obtained by consistently combining fully differential NLO QCD calculations
- with the parton shower, cf.~\cite{Frixione:2002ik,Hoeche:2011fd}. Due to
- the finite $b$-quark mass these processes do not exhibit infrared
- divergences and the corresponding {\em inclusive} cross sections can thus
- be evaluated without any cuts on the $b$-partons.
-\item[5F LO (5F \MEPSatLO):] In the {\em five--flavour scheme} $b$-quarks
- are {\em massless} particles in the {\em hard matrix element}, while they
- are treated as massive particles in both the initial- and final-state
- {\em parton shower}.
-
- In the \MEPSatLO~\cite{Hoeche:2009rj} samples we merge $pp \rightarrow H/Z$
- plus up to three jets at leading order; this includes, for instance, the
- parton--level processes $b\bar{b} \to Z/H$, $gb\to Z/H b$,
- $gg\to Z/H b\bar{b}$, $\dots$. To separate the various matrix-element
- multiplicities, independent of the jet flavour, a jet cut of
- $\Qcut = 10\,\UGeV$ is used in the $Z$ case while $\Qcut = 20\,\UGeV$ is
- employed in $H$-boson production.
-\item[5F NLO (5F \MEPSatNLO):] In the 5FS \MEPSatNLO
- scheme~\cite{Gehrmann:2012yg,Hoeche:2012yf}, we account for quark masses in
- complete analogy to the LO case: the quarks are treated as massless in the
- hard matrix elements, but as massive in the initia- and final-state parton
- showering. Again, partonic processes of different multiplicity are merged
- similarly to the \MEPSatLO albeit retaining their next-to-leading-order
- accuracy. In particular, we consider the merging of the processes
- $pp\rightarrow H/Z$ plus up to two jets each calculated with \MCatNLO
- accuracy further merged with $pp\rightarrow H/Z + 3j$ calculated at
- \MEPSatLO.
-\end{description}
-
-We consistently use four--flavour PDFs in the 4F
-scheme, i.e.\ the dedicated four--flavour NNPDF3.0 set~\cite{Ball:2014uwa} with
-the strong coupling given by $\alphaS(\MZ)\,=\,0.118$ and running at NLO.
-For the simulations in the five--flavour schemes the five--flavour NNLO PDFs
-from NNPDF3.0 are used, with $\alphaS(\MZ)\,=\,0.118$ and running at NNLO.
-We assume all quarks apart from the $b$ to be massless, with a pole mass of
-$m_b = 4.92$~\UGeV which enters the hard matrix-element calculation, where
-appropriate, and the parton shower.
-
-Results in the 4F and 5F schemes have been obtained with the default scale-setting
-prescription for parton-shower matched calculations in \Sherpa~\cite{Hoeche:2009rj,Hoeche:2010av}.
-They are
-calculated using a backward-clustering algorithm, and for each emission from the shower, couplings
-are evaluated at either the $k_T$ of the corresponding emitted particle (in the case of
-gluon emission), or at the invariant mass of the emitted pair (in the case
-of gluon splitting into quarks). The clustering stops at a ``core'' $2\to 2$
-process, with all scales set to $\mu_F=\mu_R=\mu_Q=m_T(V)/2$, where $m_T(V)$
-corresponds to the transverse mass of the boson. This scale is thus used to evaluate
-couplings in the hard matrix element and PDFs.
-The corresponding central values are supplemented with uncertainty bands
-reflecting the dependence on the unphysical scales. Renormalisation and
-factorisation scales are varied around their central value by a factor of
-two up and down, with a standard 7-point variation. The scale variations use
-the \Sherpa internal reweighting procedure~\cite{Bothmann:2016nao} and result in
-envelopes around the central value. Furthermore, we consider explicit variations
-of the parton-shower starting scale, i.e. $\mu_Q$, by a factor of two up and down.
-
-\subsection{Measurements at \LHC Run I -- the reference data}
-
-Based on a data set of $4.6\;{\rm fb}^{-1}$ integrated luminosity the ATLAS
-collaboration studied the production of $b$-jets associated with $Z/\gamma^*$
-that decay to electrons or muons~\cite{Aad:2014dvb}. The dilepton
-invariant mass ranges between $76\;{\text{GeV}} < m_{\ell\ell} < 106\;{\text{GeV}}$.
-Jets are reconstructed using the anti-$k_t$ algorithm \cite{Cacciari:2008gp}
-with a radius parameter of $R=0.4$, a minimal transverse momentum of
-$p_{T,j}> 20$ GeV and a rapidity of $|y_{j}|<2.4$. Furthermore, each jet
-candidate needs to be separated from the leptons by $\Delta R_{j\ell} > 0.5$.
-Jets containing $b$-hadrons are identified using a multi-variate technique.
-To match the outcome of the experimental analysis, simulated jets are
-identified as $b$-jets, when there is one or more weakly decaying $b$-hadron
-with $p_T>5$ GeV within a cone of $\Delta R=0.3$ around the jet axis. The
-sample of selected events is further subdivided into a class containing events
-with at least one $b$-jet ($1$-tag) and a class with at least two $b$-jets
-($2$-tag).
-
-A similar analysis was performed by CMS~\cite{Chatrchyan:2013zja}. There,
-electrons and muons are required to have a transverse momentum of
-$p_{T,\ell}>20$ GeV, a pseudorapidity $|\eta_{\ell}|<2.4$, and a dilepton
-invariant mass within $81\;{\text{GeV}} < m_{\ell\ell} < 101\;{\text{GeV}}$.
-Only events with exactly two additional $b$-hadrons were selected. The
-analysis focuses on the measurement of angular correlations amongst the
-$b$-hadrons and with respect to the $Z$ boson. This includes in particular
-variables sensitive to rather collinear $b$-hadron pairs. In addition, the
-total production cross section as a function of the vector boson's
-transverse momentum was measured.
-
-Both analyses are implemented and publicly available in the \Rivet
-analysis software~\cite{Buckley:2010ar} that, together with the \Fastjet
-package~\cite{Cacciari:2011ma}, is employed for all particle, i.e.\ hadron,
-level analyses.
-
-\subsection{Comparison with LHC data}
-
-In this section the theoretical predictions from \Sherpa are compared
-to the experimental measurements from \LHC Run I. We begin the discussion
-with the comparison with the measurements presented by the ATLAS collaboration
-in Ref.~\cite{Aad:2014dvb}. The total cross sections for $Z+\ge 1$ and
-$Z+\ge 2$ $b$ jets are collected in Fig.~\ref{fig:xstot}. Already there we
-see a pattern emerging that will further establish itself in the differential
-cross sections: While the 5F \MEPSatNLO results agree very well with data, the
-central values of the 5F \MEPSatLO cross sections tend to be around 10-20\%
-lower than the central values of data, but with theory uncertainties clearly
-overlapping them. For all the runs the uncertainty estimates include both,
-7-point variations of the perturbative scales $\mu_{R/F}$, as well as $\mu_Q$
-variations by a factor of two up and down. In contrast to the 5F case, the 4F
-\MCatNLO cross sections tend to be significantly below the experimental values
-for the $Z+\ge 1$ $b$-jets cross section, without overlap of uncertainties.
-In the $Z+\ge 2$ $b$-jets cross section the agreement between 4F \MCatNLO
-results and data is better, with the theoretical uncertainties including
-the central value of the measured cross section.
-
-\begin{figure}[!htb]
- \centering{
- \includegraphics[width=0.6\textwidth]{plots/plot_1b_orig.pdf}
- %\hfill
- \includegraphics[width=0.6\textwidth]{plots/plot_2b_orig.pdf}
- \caption{Comparison of total production cross section predictions
- with ATLAS data~\cite{Aad:2014dvb}. The error bars on the
- theoretical results are calculated from variations of the
- hard-process scales $\mu_{R/F}$ and the parton-shower starting
- scale $\mu_Q$.
- \label{fig:xstot}}
- }
-\end{figure}
-In Fig.~\ref{fig:1b} the differential cross sections with respect to the
-transverse momentum and rapidity of the $b$-jets, normalised to the number of
-$b$-jets, are presented for events with {\em at least} one $b$-tagged jet.
-The shapes of both distributions are well modelled both by the 4F and
-the two 5F calculations. However, clear differences in the predicted
-production cross sections are observed. While the 5F~NLO results are in
-excellent agreement with data - both in shape and normalisation - the
-central values of the 5F~LO cross sections tend to be around 10\% below
-data, at the lower edge of the data uncertainty bands, and
-the 4F results are consistently outside data, about 25\% too low.
-In the lower panels of Fig.~\ref{fig:1b} and all the following plots
-in this section we show the uncertainty bands of the theoretical
-predictions, corresponding to the above described $\mu_{R/F}$ and $\mu_Q$
-variations. For the 5FS calculations the scale uncertainties clearly
-dominate, while for the 4F \MCatNLO scheme the shower-resummation
-uncertainty dominates.
-\begin{figure}[!htb]
-\centering{
- \includegraphics[width=0.6\textwidth]{plots/ATLAS_p_t_b1.pdf}
- \includegraphics[width=0.6\textwidth]{plots/ATLAS_y_b1.pdf}
- \caption{Inclusive transverse-momentum and rapidity distribution of
- all $b$-jets in events with at least one $b$-jet. Data taken
- from Ref.~\cite{Aad:2014dvb}.}
- \label{fig:1b}}
-\end{figure}
-
-This pattern is repeated in Fig.~\ref{fig:1bzpt}, where we show
-the differential $\sigma(Zb)$ cross section with respect to the dilepton
-transverse momentum and, rescaled to $1/N_{b-{\rm jets}}$, as a
-function of the azimuthal separation between the reconstructed $Z$ boson
-and the $b$-jets. Again, both distributions are very well modelled by both
-5F calculations. The 4F \MCatNLO prediction again underestimates data by
-a largely flat 20-25\%.
-
-\begin{figure}[!tbh]
-\centering{
- \includegraphics[width=0.6\textwidth]{plots/ATLAS_p_t_Z.pdf}
- \includegraphics[width=0.6\textwidth]{plots/ATLAS_dphi_Zb.pdf}
- \caption{Transverse-momentum distribution of the Z boson (left) and the
- azimuthal separation between the $Z$ boson and the $b$-jets (right)
- in events with at least one $b$-jet. For the $\Delta\phi(Z,b)$ measurement
- the additional constraint $p_{T,ll}>20\;{\rm GeV}$ is imposed.
- Data taken from Ref.~\cite{Aad:2014dvb}.
- \label{fig:1bzpt}}
-}
-\end{figure}
-
-Moving on to final states exhibiting at least two identified $b$-jets,
-the role of the 5F~LO and 4F~NLO predictions are somewhat reversed:
-As can be inferred from Fig.~\ref{fig:xstot}, the 4F and 5F~NLO samples
-provide good estimates for the inclusive $Zbb$ cross section, while the
-5F~LO calculation undershoots data by about $20$\%. In Fig.~\ref{fig:2b}
-the $\Delta R$ separation of the two highest transverse-momentum $b$-jets
-along with their invariant-mass distribution is presented. Both the 4F and
-the 5F approaches yield a good description of the shape of the distributions.
-It is worth stressing that this includes the regions of low invariant mass
-and low $\Delta R$, corresponding to a pair of rather collinear $b$-jets.
-This is a region that is usually riddled by potentially large logarithms,
-where the parton shower starts taking effect. Note that in the comparison
-presented in~\cite{Aad:2014dvb} this region showed some disagreement between
-data and other theoretical predictions based on NLO QCD (dressed with parton
-showers).
-
-\begin{figure}[!htb]
-\centering{
- \includegraphics[width=0.6\textwidth]{plots/ATLAS_dR_bb.pdf}
- \includegraphics[width=0.6\textwidth]{plots/ATLAS_m_bb.pdf}
- \caption{The $\Delta R$ separation (left) and invariant-mass distribution
- (right) for the leading two $b$-jets. Data taken from
- Ref.~\cite{Aad:2014dvb}.
- \label{fig:2b}}
-}
-\end{figure}
-
-In Fig.~\ref{fig:2bzpt} the resulting transverse-momentum distribution of
-the dilepton system when selecting for events with at least two associated
-$b$-jets is shown. The shape of the data is very well reproduced by the
-4F~\MCatNLO and 5F~\MEPSatNLO samples. Also the 5F~\MEPSatLO prediction
-describes the data well despite of the overall rate being $20$\% lower than
-observed in data.
-
-\begin{figure}[!htb]
-\centering{
- \includegraphics[width=0.6\textwidth]{plots/ATLAS_p_T_Z_2b.pdf}
- \caption{Transverse-momentum distribution of the dilepton system for events
- with at least two $b$-jets. Comparison against various calculational
- schemes.
- Data taken from Ref.~\cite{Aad:2014dvb}.
- \label{fig:2bzpt}}
-}
-\end{figure}
-
-The measurements presented by the CMS collaboration in
-Ref.~\cite{Chatrchyan:2013zja} focus on angular correlations between
-$b$-hadrons rather than $b$-jets. Two selections with respect to the
-dilepton transverse momentum have been considered, a sample requiring
-$p_T(Z)>50\;{\rm GeV}$ and an inclusive one considering the whole range
-of $p_T(Z)$. The $\Delta R$ and $\Delta \phi$ separation of the
-$b$-hadrons obviously prove to be most sensitive to the theoretical
-modelling of the $b$-hadron production mechanism and the interplay of
-the fixed-order components and the parton showers. They are presented in
-Figs.~\ref{fig:2bdR_CMS} and \ref{fig:2bdPhi_CMS}. In general, a good
-agreement in the shapes of simulation results and data is found, with
-the same pattern of total cross sections as before: the 5F~\MEPSatNLO
-sample describes data very well, while the 4F~\MCatNLO results tend to
-be a little bit, about 10\%, below data, with data and theory
-uncertainty bands well overlapping, while the central values of the
-5F~\MEPSatLO results undershoot data by typically 20-25\%.
-\begin{figure}[!htb]
-\centering{
- \includegraphics[width=0.6\textwidth]{plots/CMS_dR_BB.pdf}
- %\hfill
- \includegraphics[width=0.6\textwidth]{plots/CMS_dR_BB_pt50.pdf}
- \caption{ $\Delta R_{BB}$ distribution for two selections of the transverse
- momentum of the $Z$ boson. Data taken from Ref.~\cite{Chatrchyan:2013zja}.
- \label{fig:2bdR_CMS}}
-}
-\end{figure}
-
-\begin{figure}[!htb]
-\centering{
- \includegraphics[width=0.6\textwidth]{plots/CMS_dphi_BB.pdf}
- %\hfill
- \includegraphics[width=0.6\textwidth]{plots/CMS_dphi_BB_pt50.pdf}
- \caption{ $\Delta \phi_{BB}$ distribution for two selections of the transverse
- momentum of the $Z$ boson. Data taken from Ref.~\cite{Chatrchyan:2013zja}.
- \label{fig:2bdPhi_CMS}}
-}
-\end{figure}
-
-Overall it can be concluded that the 5F \MEPSatNLO calculation yields the best
-description of the existing measurements, regarding both the production rates
-{\em and} shapes. The 4F \MCatNLO and 5F \MEPSatLO schemes succesfully model
-the shape of the differential distributions but consistently underestimate the
-production rates.
-
-
-\section{Bottom-jet associated Higgs-boson production}
-\label{sec:hbb}
-In this section we present predictions for $b$-jet(s) associated production
-of the Standard-Model Higgs boson in $pp$ collisions at the $13$ TeV \LHC
-obtained in the four-- and five--flavour schemes. As standard when dealing with this
-process, we do not include contributions from the gluon-fusion channel.
-However, in the 4F \MCatNLO we do include terms proportional to the top-quark
-Yukawa coupling, contributing to order $y_by_t$ as an interference effect at
-NLO QCD~\cite{Dittmaier:2003ej,Dawson:2003kb,deFlorian:2016spz}.
-Although associated $Z+$ $b$-jet(s) production serves as a good proxy for the
-Higgs-boson case, there are important differences between both processes, mainly
-due to the different impact of initial-state light quarks, which couple to $Z$
-bosons but not to the Higgs boson.
-
-As before, QCD jets are defined through the anti-$k_t$ algorithm using a
-radius parameter of $R=0.4$, a minimal transverse momentum $p_{T,j}>25$~GeV,
-and a rapidity cut of $|y_{j}|<2.5$. In this case, we consider results that are at
-the parton level only, disregarding hadronisation and underlying-event
-effects, which may blur the picture. We consider single $b$-tagged jets only,
-thus excluding jets with intra-jet $g\rightarrow b\bar{b}$ splittings from
-the parton shower which would be the same for all flavour schemes we
-investigate. As for $Z$-boson production, we separate the event samples
-into categories with at least one $b$-jet, i.e.\ $H+\geq 1 b$-jet events,
-and at least two tagged $b$-jets, i.e.\ $H+\geq 2 b$-jets events.
-
-\begin{table}[!hbt]
- \centering
- \begin{tabular}[\linewidth]{lc|c}
- \toprule
- \LHC 13~TeV & $H + \geq 1 b$-jets [fb] & $H + \geq 2 b$-jets [fb]\\
- \midrule
- $\sigma_{\text{\MCatNLO}}^{4F}$ &
- $45.2^{+15.5\%}_{-18.4\%}$ &
- $4.5^{+25.1\%}_{-26.3\%}$\\
- $\sigma_{\text{\MEPSatLO}}^{5F}$ &
- $79.3^{+34.0\%}_{-25.4\%}$ &
- $3.8^{+34.3\%}_{-30.3\%}$\\
- $\sigma_{\text{\MEPSatNLO}}^{5F}$ &
- $110.5^{+14.2\%}_{-16.0\%}$ &
- $6.9^{+27.3\%}_{-27.1\%}$\\
- \bottomrule
- \end{tabular}
- \caption{$13$~TeV total cross sections and the corresponding
- $\mu_{F/R}$ and $\mu_Q$ uncertainties for $H + \geq 1 b$ and $H + \geq 2 b$s.
- }
- \label{tab:hbbxs}
-\end{table}
-
-\begin{figure}[!htb]
-\centering
- \includegraphics[width=0.6\textwidth]{plots/p_t_H_b1.pdf}
- \includegraphics[width=0.6\textwidth]{plots/p_t_b1.pdf}
- \caption{Predictions for the transverse-momentum distribution of the Higgs
- boson (left panel) and the leading $b$-jet (right panel) in inclusive
- $H+b$-jet production at the $13$ TeV \LHC.}\label{fig:1bh}
-\end{figure}
-
-\begin{figure}[!htb]
-\centering
- \includegraphics[width=0.6\textwidth]{plots/p_t_H_b2.pdf}
- \caption{The transverse-momentum distribution of the Higgs boson in
- inclusive $H+2b$-jets production at the $13$ TeV \LHC.}\label{fig:1bhpt}
-\end{figure}
-
-\begin{figure}[!htb]
-\centering
- \includegraphics[width=0.6\textwidth]{plots/R_bb.pdf}
- \includegraphics[width=0.6\textwidth]{plots/m_bb.pdf}
- \caption{Predictions for the $\Delta R$ separation of the two leading
- $b$-jets (left panel) and their invariant-mass distribution
- (right panel) in inclusive $H+2b$-jets production at the
- $13$ TeV \LHC.}\label{fig:2bh}
-\end{figure}
-In Tab.~\ref{tab:hbbxs} cross sections for the three calculations are
-reported. Historically, inclusive results have largely disagreed between the
-4F and the 5F scheme. This feature is observed for the case at hand, too,
-and especially so for the case of one tagged $b$-jet. There the
-4F \MCatNLO prediction is smaller than the 5F results by factors of about
-$1.75$ (5F LO) and of $2.44$ (5F NLO). The relative differences are
-reduced when a second tagged $b$-jet is demanded. In this case we find that
-the 4F result lies between the two 5F results, about 20\% higher than the
-LO predictions, and a factor of about 1.5 lower than the 5F NLO predictions.
-In both cases, inclusive $H+b$ and $H+bb$ production, the uncertainty bands
-of the two 5F predictions, corresponding to 7-point $\mu_{R/F}$ variations
-and $\mu_Q$ variations by a factor of two up and down, do overlap. While for
-the two $b$-jet final states this includes the 4F result, for the one $b$-jet
-case the 4F result is not compatible with the 5F predictions, taking into
-account the considered scale uncertainties. It is worth noting that a milder
-form of this relative scaling of the cross sections was already observed
-in the $Z$ case.
-
-In the case of the total inclusive cross section, this very large difference
-can be mitigated by including higher-order corrections, on the one hand, and a
-better assessment of which choice of the unphysical scales yields the better
-agreement~\cite{Frederix:2011qg,Wiesemann:2014ioa,deFlorian:2016spz,Lim:2016wjo}.
-However, only a recent effort to match the two schemes~\cite{Forte:2015hba,
- Forte:2016sja,Bonvini:2015pxa, Bonvini:2016fgf}
-has clearly assessed the relative importance of mass corrections
-(appearing in the 4F scheme) and large log resummation (as achieved in a 5F
-scheme). In particular it has been found that the difference between these two
-schemes is mostly given by the resummation of large logarithms, thus suggesting that
-for an inclusive enough calculation either a 5F scheme or a matched scheme
-should be employed. This is the same situation that one faces, albeit
-milder, in the $Z$ case, where, in terms of normalisation the 5F scheme
-performed better in all cases and especially in inclusive calculations. We
-therefore recommend that in terms of overall normalisation, the 5F \MEPSatNLO
-scheme should be used to obtain reliable predictions.
-
-Let us now turn to the discussion of the relative differences in the
-shapes of characteristic and important distributions. To better appreciate
-shape differences, all differential distribution are normalised to the
-respective cross section, i.e.\ the inclusive rates $\sigma(Hb)$ and
-$\sigma(Hbb)$. In all cases we obtain agreement at the 15\%-level or better
-between the 5F~\MEPSatNLO and 4F~\MCatNLO samples, the only exception, not
-surprisingly, being the region of phase space where the two $b$'s come
-close to each other and resummation effects start playing a role.
-Typically, the 5F~\MEPSatLO predictions are also in fair agreement with the
-other two results, however, they exhibit a tendency for harder tails in
-the $p_T$ distributions, mainly in the inclusive Higgs-boson $p_T$ and
-in the transverse momentum of the second $b$ jet.
-
-Starting with Fig.~\ref{fig:1bh}, the transverse-momentum distributions of the
-Higgs boson and the leading $b$-jet in the case of at least one $b$-jet tagged is
-displayed. Similarly to the $Z$ example, this is the region where one would
-expect the 5F scheme to perform better. However, again similarly to the $Z$
-case, the three schemes largely agree in terms of shapes, being well within
-scale uncertainties. Notably, this turns out to be particularly true for
-the low ($\sim 20$--$100$~\UGeV) $p_T$ region where one could have expected
-deviations to be the largest.
-
-In Figs.~\ref{fig:1bhpt} and \ref{fig:2bh} we present differential
-distributions for the selection of events with at least two tagged
-$b$-jets. While Fig.~\ref{fig:1bhpt} shows the resulting Higgs-boson
-transverse-momentum distribution, Fig.~\ref{fig:2bh} compiles results for
-the $\Delta R$ separation of the two leading $b$-jets and their invariant-mass
-distribution. For such two $b$-jets observables the 4F scheme is expected
-to work best, especially when the two $b$ are well separated to suppress
-potentially large logarithms. However, in agreement with the $Z$-boson
-case, no significant differences between the various scheme arise when
-taking into account $\mu_{R/F}$ and $\mu_Q$ scale-variation uncertainties.
-Once again the region of low $p_T$ in Fig.~\ref{fig:1bhpt} and the region of
-low $m(b,b)$ in Fig.~(\ref{fig:2bh}) show excellent agreement amongst the various
-descriptions. As anticipated, larger differences can be seen between the
-two 5FS and the 4F \MCatNLO calculations, in the very low $\Delta R(b,b)$ and
-$m(b,b)$ regions, Fig.~(\ref{fig:2bh}), where the two $b$-jets become
-collinear. This feature is however most likely due to the fact that we are
-dealing with partonic $b$-jets as opposed to {\em hadronic} ones. Taking
-as a reference the $Z$-boson case once again, in fact, where this difference
-is not present at all, suggests that a realistic simulation, that
-accounts for hadronisation effects, should largely suppress this difference.
-
+\input{text/fonll-method}
\ No newline at end of file

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