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Index: trunk/papers/bbz/bbz_fonll.tex
===================================================================
--- trunk/papers/bbz/bbz_fonll.tex (revision 57)
+++ trunk/papers/bbz/bbz_fonll.tex (revision 58)
@@ -1,700 +1,700 @@
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\begin{document}
\begin{flushright}
TIF-UNIMI-2018-3\\
DAMTP-2018-xx
\end{flushright}
\vspace*{.2cm}
\begin{center}
{\Large \bf{$Z$ boson production in bottom-quark fusion:\\
a study of $b$-mass effects beyond leading order}}
\end{center}
\vspace*{.7cm}
\begin{center}
Stefano Forte$^{1}$, Davide Napoletano$^2$ and Maria Ubiali$^{3}$
\vspace*{.2cm}
\noindent
{\it
$^1$ Tif Lab, Dipartimento di Fisica, Universit\`a di Milano and\\
INFN, Sezione di Milano,
Via Celoria 16, I-20133 Milano, Italy\\
$^2$ IPhT, CEA Saclay, CNRS UMR 3681,\\ F-91191, Gif-Sur-Yvette, France\\
$^3$ DAMTP, University of Cambridge,\\ Wilberforce Road, Cambridge, CB3 0WA, UK\\}
\vspace*{3cm}
%\begin{center}
{\bf Abstract}
\end{center}
\noindent
We compute the total cross-section for $Z$ boson production in
bottom-quark fusion, applying to this case the method we previously
used for Higgs production in bottom fusion. Namely, we match, through
the FONLL procedure,
the next-to-next-to-leading-log five-flavor
scheme result, in which the $b$~quark is
treated as a massless parton, with the next-to-leading-order
$\order{\alpha_s^3}$
four-flavor scheme computation in which bottom is treated as a massive
final-state particle. Our computation provides a test-case for
the discussion of issues of scale dependence and treatment of heavy
quarks, which we discuss in light of our results.
\pagebreak
%\tableofcontents
The production of a $Z$ boson is one of the main standard candles at the LHC,
and is now measured at the sub-percent level. The main production mode
is through quark-anti-quark fusion, of which the bottom-initiated contribution
accounts to ${\cal O}(4\%)$ of the total cross section.
This process is thus an ideal test case for matched computations,
recently applied to Higgs production
in bottom quark
fusion~\cite{Forte:2015hba,Forte:2016sja,Bonvini:2015pxa,Bonvini:2016fgf}. As
we shall show here, it provides a theoretically transparent setting
for the discussion of issues of choice of scheme and scale in the
treatment of heavy quark contributions.
Like any process involving bottom quarks at the matrix-element level,
the bottom-initiated $Z$ production process
may be computed using two different factorization schemes, which we
refer to, as usual, as four- and five-flavor schemes for short. In the
four-flavor scheme (4FS), the $b$~quark is treated as a massive
object, which
decouples from QCD perturbative evolution. Calculations in this scheme
are thus performed by only including
the four lightest flavor together with the gluon in evolution
equations for parton distributions (PDFs), and in the running of
$\alpha_s$,
so $n_f=4$ in the QCD $\beta$ function.
In the five-flavor scheme (5FS), instead, the $b$~ quark is treated on
the same footing as other light quark flavors, there is a $b$~PDF, and
-$n_f=5$ in evolution equations for PDFs and the strong coupling.
+$n_f=5$ in evolution equations for PDFs and in the QCD $\beta$ function.
In matched calculations, both scheme are combined, in such a way that
the result differs by that of each of the two schemes by terms which
are subleading with respect to the accuracy of either of them. Whereas
several matching schemes have been proposed and used in the past in
the context of deep-inelastic (DIS) or hadronic processes, the FONLL scheme,
first proposed for heavy quark production in hadronic
collisions~\cite{Cacciari:1998it} has the advantage of being
universally applicable; also, it allows for the matching of
four- and five-flavor computations performed at any combination of
individual perturbative orders. It has been extended to deep-inelastic
scattering in Ref.~\cite{Forte:2010ta} (also
including~\cite{Ball:2015tna,Ball:2015dpa} the case in which the heavy
quark PDF is independently parametrized) and, as mentioned, it has
been
used in
Refs.~\cite{Forte:2015hba,Forte:2016sja} for the computation of the
total cross-section for Higgs production in bottom
quark fusion.
Here, the methodology of
Refs.~\cite{Forte:2015hba,Forte:2016sja} is applied to $Z$ production.
In the 5FS, the $Z$-production cross section has been known up to
next-to-next-to leading order
(NNLO) (i.e. $\order{\alpha_s^2}$) for almost three decades~\cite{Hamberg:1990np} and
the heavy-quark initiated contribution has been specifically discussed
in
several papers~\cite{Rijken:1995gi,Stelzer:1997ns,Maltoni:2005wd}.
%
The next-to-leading order (NLO) ($\order{\alpha_s^3}$) four-flavor scheme
$Zb\bar{b}$ production cross section was originally computed
in Ref.~\cite{Campbell:2000bg}
for exclusive 2-jet final states, neglecting the $b$~quark mass.
-The $b$~quark mass was subsequently fully included in
+The $b$-quark mass was subsequently fully included in
Refs.~\cite{FebresCordero:2008ci,Cordero:2009kv}.
In this work we combine these two results following the procedure we
presented in Refs.~\cite{Forte:2015hba,Forte:2016sja} for the closely
related case of Higgs production: indeed, the counting of perturbative
orders for these two processes is the same, and many of the Feynman
diagrams are identical, with the only replacement of Higgs Yukawa
couplings with gauge couplings. Following the nomenclature introduced
in Ref.~\cite{Forte:2015hba,Forte:2016sja} (and originally in
Ref.~\cite{Forte:2010ta} for DIS) we have constructed an FONLL-A
result, which combines the NNLO 5FS with the LO $\order{\alpha_s^2}$
4FS fully massive computation, and an FONLL-B, where instead the NNLO
5FS is matched to the full NLO $\order{\alpha_s^3}$ massive
results.
Our construction is essentially identical to that of
Refs.~\cite{Forte:2015hba,Forte:2016sja}, to which we refer for the
details: it can be obtained from it by simply replacing the matrix elements
for Higgs production with those for gauge boson
production. Specifically, we have computed the 5FS NNLO
cross-section using the
code of Ref,~\cite{Maltoni:2005wd}, which we cross-checked both
at LO and NLO against MG5\_aMC@NLO~\cite{Alwall:2014hca}. For the
massive 4FS LO and NLO we have also used MG5\_aMC@NLO.
The construction of the FONLL matched results requires the computation
of the massless limit of the massive result: we have implemented this
in the public code~\cite{code} used in~\cite{Forte:2016sja}, in an
updated version soon to be made public.
All predictions are obtained using the NNLO NNPDF3.1 PDF
set~\cite{Ball:2017nwa}.
In order to be consistent with the PDF set used we take, in the 4F scheme,
the $b$ pole mass to be $m_b=4.92$~GeV, while the strong coupling
is run at NNLO, with $\alpha_s(m_Z) = 0.118$.
New in comparison to Refs.~\cite{Forte:2015hba,Forte:2016sja}, we have
now implemented the possibility of varying the scale $\mu_b$ at which
the 4FS and 5FS schemes are matched. This scale was taken to coincide
with the bottom mass $\mu_b=m_b$ in previous FONLL implementations,
but there is no fundamental reason for this choice. The reason why
results depend on a matching scale is that in
the 5FS the $b$~PDF is not independently
parametrized. Rather, it is assumed that it is radiatively generated
by the gluon. The matching scale is then the scale at which the $b$~PDF
is determined from the gluon.
The matching condition itself depends on
the matching scale in such a way that, at any given order, results
are independent of it up to subleading corrections. This dependence
persists in the FONLL matched results, but it is alleviated at
scales which are not too far from the bottom production threshold,
where the FONLL results almost reduces to the exact mass-dependent result in
which the physical threshold is implemented exactly. It reappears at
high-enough scales, where the FONLL result reduces to the 5FS, and it
only goes away when computing the matching condition to increasingly
high perturbative order, or by independently parametrizing the heavy
quark PDF (indeed, this is the main motivation for independently
parametrizing charm~\cite{Ball:2016neh,Ball:2017nwa}).
The generalization of the FONLL matching formulae of
Refs.~\cite{Forte:2015hba,Forte:2016sja} for a generic choice of
matching scale is given in the Appendix.
Dependence on this
matching scale for Higgs in bottom fusion was studied explicitly in
Ref.~\cite{Bonvini:2015pxa,Bonvini:2016fgf}. The matching scheme of
Ref.~\cite{Bonvini:2015pxa,Bonvini:2016fgf}, based on a EFT approach,
was benchmarked in Ref.~\cite{deFlorian:2016spz} to that of
Refs.~\cite{Forte:2015hba,Forte:2016sja} and found to agree with it at
the percent level, hence a very similar dependence is expected for
FONLL.
Our results are summarized in
Figs.~\ref{fig:muR_var}-\ref{fig:m_mu_var}, where
matched results in the FONLL-A and FONLL-B scheme are compared to each
other and to the 4FS and 5FS scheme computations. In the three plots
we study respectively the renormalization, factorization, and matching
scale dependence of the results.
In each
case, renormalization and factorization scales are fixed, and then
varied about, either a high value $\mu=m_Z$, or a low value
$\mu_{R,F}=\frac{m_Z+2m_b}{3}$. While the higher scale choice is standard in
inclusive $W$ and $Z$ production, the lower choice was advocated in
Refs.~\cite{Maltoni:2012pa,Lim:2016wjo} based on arguments
that it is closer to the physical hard scale of the process and thus leads
to faster perturbative convergence.
Furthermore, in the case of our
preferred FONLL-B result, scale variation is performed in each case in
two different ways which differ by subleading terms, and the result is
shown as a band between these two choices with the central prediction
constructed as the mid-point. The two possibilities correspond to the
observation (see e.g.~\cite{Altarelli:2008aj})
that scale variation by a factor $k$
of a quantity $F(\mu)$ which is scale-independent up to NLO but has a
NNLO scale dependence can be performed by either letting
\begin{equation}\label{eq:scalup}
F(\mu_0;k)=F(\mu_0+\ln k)- \ln k \frac{d}{d\ln\mu}
F(\mu)\Big|_{\mu_0=\mu_0+\ln k},
\end{equation}
or
\begin{equation}\label{eq:scaldown}
F(\mu_0;k)=F(\mu_0+\ln k)- \ln k \frac{d}{d\ln\mu} F(\mu)\Big|_{\mu=\mu_0},
\end{equation}
where the first term on the r.h.s. is computed up to NLO, while the
second term may be computed up to LO, and thus the two expressions
differ by NNLO terms (and similarly for higher orders). The two
options define the two extremes of the band. The width of the band can
be viewed as the ambiguity, i.e. the scale uncertainty,
on the scale uncertainty itself.
Finally, matching scale variation is performed by varying it between the
default $\mu_b=m_b$ and $\mu_b=2m_b$.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{muR_mh_var.pdf}
\includegraphics[width=0.7\textwidth]{muR_var.pdf}
\caption{\label{fig:muR_var} Comparison of the FONLL-A and FONLL-B
matched results to each other, to the 4FS LO ($\order{\alpha_s^2}$)
- and NLO ($\order{\alpha_s^3}$), and to the 5FS NNLO.
- the and and the 4F and 5F scheme. Results are shown as a function of the
+ and NLO ($\order{\alpha_s^3}$), and to the 5FS NNLO. Results are shown as a function of the
renormalization scale, with the factorization scale fixed at a high value
$\mu_F=m_Z$ (top) or a low value $\mu_F=\frac{(m_Z+2m_b)}{3}$
(bottom). The band about the FONLL-B result is obtained from two
different implementations of NLO scale variation that differ by NNLO
terms (see text) and is thus an estimate on the ambiguity of the
scale variation itself.
}
\end{center}
\end{figure}
%
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{muF_mh_var.pdf}
\includegraphics[width=0.7\textwidth]{muF_var.pdf}
\caption{\label{fig:muF_var} Same as Fig.~\ref{fig:muR_var}, but now
with the factorization scale varied with the renormalization scale
kept fixed at a high value
$\mu_R=m_Z$ (top) or a low value $\mu_R=\frac{(m_Z+2m_b)}{3}$ (bottom) .}
\end{center}
\end{figure}
We first describe and comment our
results, then discuss their interpretation, also in view of various
approximations which have been suggested in the literature. A first observation is that comparison of
Figs.~\ref{fig:muR_var}-\ref{fig:muF_var} to the corresponding
plots for Higgs production in bottom fusion (Figs.~2-3 of
Ref.~\cite{Forte:2016sja}) show that they are qualitatively almost
indistinguishable: this is not unexpected given the similarity between
Higgs and $Z$ production which we already repeatedly emphasized.
Coming now to these qualitative features we note that:
\begin{itemize}
\item The factorization scale dependence is generally very slight,
while the renormalization scale dependence is, instead, stronger.
\item The scale dependence is quite large in the 4FS scheme, even at
NLO though it is reduced in comparison to the LO case. It is much
weaker in the 5FS and FONLL cases which all have similar and
similarly weak scale dependence, except for very low values
$\mu_R\sim \frac{m_Z}{10}$ where however the ambiguity on the scale
uncertainty blows up.
\item The perturbative expansion is very
unstable in the 4FS, with the LO and NLO results differing by a
factor two or more. This instability is completely removed when the
4FS is matched to the 5FS: indeed, the FONLL-A and FONLL-B are quite
close to each other.
\item The 4FS and 5FS results are quite far from each other, with
the 4FS NLO significantly
closer to the 5FS than the LO. The FONLL results are in turn quite close to the 5FS.
\item The perturbative expansion is indeed more stable for a lower
choice of factorization and renormalization scale. For very low
scales $\mu\sim\frac{m_Z}{10}$ the 4FS and 5FS results become
similar, but the scale dependence and its uncertainty (i.e. the
slope of the curve, and the band about it) become very large.
\item A change of matching scale has essentially the same effect on
the 5FS and the FONLL results, and it has the effect of moving both
towards the 45S, though by a moderate amount.
\end{itemize}
These qualitative features have a simple theoretical
interpretation. To this purpose, note that the cross-section for
this process contains collinear logarithms regulated by the heavy
quark mass, i.e. powers of $\ln\frac{\mu_Z}{m_b^2}$, one at each
perturbative order. These logs
arise from a transverse
momentum integration, whose the upper limit is the maximum value of the
transverse momentum, i.e. the hard scale of the process,
which is proportional to but not equal to $m_Z$, and the lower limit
is the physical production threshold, which is proportional to but not
equal to $m_b$. Of course, one can always rewrite the ensuing
logarithm as $\ln\frac{\mu_Z^2}{m_b^2}$, plus constants (i.e. terms
which only depend on the dimensionless ratio $\tau=\frac{\mu_z^2}{s}$),
and mass corrections (i.e. terms suppressed by powers of
$\frac{\mu^2_b}{m_Z^2}$).
In the 4FS the result is exact, so whatever is not included in the log is included in
the constants or in the mass corrections; on the other hand at
N$^{k}$LO only the first $k+1$ logs are included. In the 5FS the logs are
rewritten as
$\ln\frac{\mu_Z^2}{m_b^2}=\ln\frac{\mu_Z^2}{\mu_F^2}+\ln\frac{\mu_F^2}{\mu_b^2}+\ln\frac{\mu_b^2}{m_b^2}
$, where $\mu_F$ is the factorization scale and
$\mu_b$ is the matching scale. The logs of the factorization scale
$\ln\frac{\mu_F^2}{\mu_b^2}$ are then resummed to all orders into
the evolution of the PDF, while
the logs of the hard scale $\ln\frac{\mu_Z^2}{\mu_F^2}$ are included to
finite order in the hard partonic cross-section and
logs the matching scale, $\ln\frac{\mu_b^2}{m_b^2}$, are
included to finite order in the matching condition, which
expresses the initial $b$~PDF in terms of the gluon (they would be
implicitly included in the initial PDF
if the $b$~PDF were independently parametrized).
When varying the factorization scale, logs at the upper end
of the evolution are reshuffled between the resummed PDF and the
fixed-order but exact hard cross-section. When varying the matching
scale, logs at the bottom end
of the evolution are reshuffled between the resummed PDF and the
fixed-order but exact hard matching condition.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{m_muR_var.pdf}
\includegraphics[width=0.7\textwidth]{m_muF_var.pdf}
\caption{\label{fig:m_mu_var}
Comparison of the FONLL-B and of the 5FS NNLO results for two different
values of the matching scale $\mu_b=m_b$ (same as in
Figs.~\ref{fig:muR_var}-\ref{fig:muF_var}) to each other and to the
4FS NLO. Results are shown as a function of the renormalization scale
for fixed factorization scale (top) or as a function of the
factorization scale for fixed renormalization scale (bottom), in each
case with the fixed scale chosen as $\mu=\frac{(m_Z+2m_b)}{3}$.}
\end{center}
\end{figure}
Note that both the
hard coefficient and the
matching condition contains logs and constants, but not
mass-suppressed terms: so in the 5FS constants and logs of the
matching scale, as well as constants and logs in the hard
coefficient,
are treated exactly but to fixed order, while logs of the
factorization scale are resummed to all
orders, but neglecting constants.
When the 5FS and the 4FS are matched into FONLL, also
mass-suppressed terms, on top of constants
and logs of the matching scale, are treated
exactly.
-The fact that the 4FS is perturbatively unstable while th 5FS is not
+The fact that the 4FS is perturbatively unstable while the 5FS is not
then is easily explained as a manifestation of the fact that the 4FS
contains large logs which are resummed in the 5FS. This is confirmed
by the fact that the large difference between the 4FS LO and NLO is of
the same order of the scale variation of the LO: indeed the scale
variation by construction captures the size of logarithmic
contribution. So the sizable difference which persists between the 5FS
and NLO 4FS results is explained as being due to the higher order
(NNLO and beyond) logs
-which are missing in the 4FS NLO. This is confirmed by the observation
+which are missing in the 4FS NLO, their size being quantitatively estimated
+in ~\cite{Lim:2016wjo}. This is confirmed by the observation
that the FONLL-A and FONLL-B include both the large log resummation,
and the full constants and mass suppressed terms, up to LO and NLO
respectively. The difference between the FONLL-A and FONLL-B is thus
the size of the constant and mass-suppressed contributions to the
difference between the 4FS LO and NLO. This is seen to be much smaller
than the total difference between 4FS LO and NLO, which must therefore
be due to the log.
In order to further disentangle, within this small contribution,
the constant from mass-suppressed term,
one would have to vary the hard scale, i.e. the $Z$ mass. This was
done in Ref.~\cite{Forte:2016sja} for Higgs production: variation of
the Higgs mass left the difference between FONLL-A and FONLL-B
essentially unchanged, thus showing that mass corrections are
negligible and the bulk of the difference between FONLL-A and FONLL-B
is due to a constant. Given the similarity between the two processes
we expect the same to be the case here. Given the small size of this
contribution the issue is largely academic anyway.
The qualitative form of the renormalization scale dependence of the 4FS
result is also easy to understand: as the scale is decreased, the
value of $\alpha_s$ multiplying the large collinear log increases, and
both the LO and NLO predictions grow; this growth is only partly
reduced by the higher-order compensating term, at least down to
scales $\mu_R\sim 0.2\mu_Z$ where the ambiguity on the scale variation
itself becomes very large. The fact that the 5FS (and FONLL) result
have almost no renormalization scale dependence shows that this scale
dependence is coming from the $b$~quark term which is treated differently
between 4FS and 5FS.
The factorization scale dependence is particularly intriguing. The
fact that this dependence is very slight in the 4FS is again consistent
with the observation that scale dependence is driven by the heavy
quark terms: in this scheme, in the absence of a $b$~PDF, the factorization scale
dependence is related to perturbative evolution of the light
quarks and gluons, which is moderate at NNLO.
On the other hand, in the 5FS (and in FONLL) collinear logs are
resummed in the evolution of the $b$-PDF up to $\mu_F$, and then
expanded out in the partonic cross-section from $\mu_F$ to the
physical hard scale of the process. We therefore expect the
factorization scale dependence in this scheme to be approximately
stationary around this physical hard scale, very slight above it (where
$\alpha_s$ is small) and to only become significant when $\mu_F$ is
lower than the physical hard scale itself. This behaviour is
clearly seen in Fig.~\ref{fig:muF_var}, with the stationary point
close to the low scale advocated in Ref.~\cite{Maltoni:2012pa,Lim:2016wjo}
that indeed this scale of the hard process, and it nicely explains the
very weak factorization scale dependence also seen in the 5FS unless
$\mu_F\lesssim0.2m_Z$ or so.
Finally, the fact that when increasing the matching scale $\mu_b$ the
5FS and FONLL-B result decrease and get closer to the 4FS is
understood as a consequence of the fact that the resummed log becomes
smaller: clearly, if one were to choose $\mu_b=m_Z\sim 10 m_b$ the 5FS would
reduce to the 4FS one because the resummed logs then vanish.
With the choice $\mu_b= 2 m_b$ the FONLL and 5FS move towards the
the 4FS by an accordingly smaller amount.
We can finally discuss, in light of all this, the two related issues
of choosing the various scales, $\mu_F$, $\mu_R$ and $\mu_b$, and of
the validity of various approximations. As discussed, the scale
dependence of this process is driven by the collinear logs in the
$b$~ quark contribution, and thus the bulk of it comes from the
choice of argument in these logs.
In a fully massive 4FS calculation,
these collinear logs are treated exactly, so the scale dependence
comes purely from the choice of argument in the strong coupling. It
then turns out that reducing the renormalization scale increases the
4FS unresummed results up to the point where it agrees with the 5FS
resummed one. This is however accidental: the lack of resummation is made up
by artificially increasing $\alpha_s$, and indeed at low scale the
scale dependence of the 4FS result is not improved: if anything, it
increases. Hence, the 4FS appears to be a poor approximation to this
process and its improvement by lowering the renormalization scale is
unreliable.
In a 5FS calculation, instead, as mentioned, the exact upper and lower
limits of the transverse momentum integration are replaced by $\mu_F$
and $\mu_b$, respectively. As also mentioned, it has been
argued~\cite{Maltoni:2012pa,Lim:2016wjo}
that the exact, kinematics-dependent upper limit of integration is
on average close to a scale $\frac{m_Z+2m_b}{3}\sim 0.35m_Z$. This is
borne out by our results: for all $\mu_F\gtrsim0.3m_Z$ the
factorization scale dependence of the 5FS result is flat, and with
this choice of $\mu_R$ the 5FS scale dependence is visibly
flatter. Given the smallness of mass
corrections, in practice a 5FS with low factorization and
renormalization scales appears to be a good approximation of the full
FONLL result.
On the other hand, it has been recently argued~\cite{Bertone:2017djs}
that a higher choice of matching scale may provide a better
approximation. Clearly, this is a process-dependent statement that
should be checked on a case-by-case basis: as discussed raising the
matching scale improves the accuracy of the starting, dynamically
generated PDF, as it matches it at a scale where perturbation theory
is more reliable, but it reduces the size of the logs which are
resummed. In the present case, the resummed logs are a large effect
and the constants a small correction, so raising the matching scale
does not appear to be advantageous: indeed, the renormalization and
factorization scale dependence is the same with $\mu_b=m_b$ or
$\mu_b=2m_b$, with no obvious improvement.
In fact, when raising $\mu_b$
the 5FS result decreases, and moves towards the low 4FS, but with no
improvement in perturbative stability of the latter. This is to be
contrasted to the case in which $\mu_R$ and $\mu_F$ are lowered, which
also brings the 4FS and the 5FS closer but now towards a high value,
and with a visible increase in perturbative stability. In fact, the
FONLL result shows that exact inclusion of the mass corrections (most
likely the constant) increases the pure 5FS, by a small
amount. On the contrary, raising the matching scale lowers it: this
means that the deterioration of the log resummation is a larger effect
than the improvement made by starting the PDF at a scale at which
perturbation theory is more reliable.
So a 5FS with
large $\mu_b$ does not appear to be a better approximation in our case:
it is likely to be a worse approximation if $\mu_b$ is raised by a
moderate amount,
and it definitely appears to be a poor approximation if $\mu_b$ is
raised up to the point at which the 5FS result reduces to the 4FS
one. On the other hand, a variation of $\mu_b$ by perhaps a factor
two, as shown in Fig.~\ref{fig:m_mu_var}, might well be a reasonable
estimate of the uncertainty due to the use of a fixed-order matching
condition and should be included in the theoretical uncertainty, as was
done in Refs.~\cite{Bonvini:2016fgf,deFlorian:2016spz}. This theoretical
uncertainty can only be removed by parametrizing the $b$~PDF, in which
case it is traded for a PDF uncertainty.
%
In summary, we have determined the total cross-section for $Z$
production in bottom quark fusion at the highest available accuracy in
a matched FONLL scheme, and we have used our results as a test case for the
discussion of issues of scale dependence and heavy quark treatment, by
generalizing our previous results for Higgs production, and studying
not only renormalization and factorization scale, but also matching
scale dependence.
Our main phenomenological conclusion is that,
similarly to the case of Higgs production, mass effects are small,
but non-negligible in comparison to the high experimental accuracy to
which this process is measured. However, the contribution due to the
resummation of collinear logs of the heavy quark is sizable, thereby
making a five-flavor scheme in which the $b$~quark is endowed with a PDF
a better approximation to the full FONLL result than the fixed-order
4FS calculation with massive b, which falls short of the full
prediction and displays large scale uncertainties.
A low choice of
renormalization and factorization scale reduces the scale dependence
of both the full FONLL and pure 5FS result and is likely to improve
their accuracy, though in practice this makes little difference as the scale
dependence of both these results is very slight. However, it
does suggest that the hard physical scale for this process is
lower than the final-state mass, as previously advocated.
All in all, our results support the conclusion that a fully matched
treatment of heavy quarks with a proper inclusion of mass effects is
necessary for LHC phenomenology at the percent level, either through
its direct use, or as a guide to construct efficient and accurate
approximations.
\bigskip
\bigskip
\begin{center}
\rule{5cm}{.1pt}
\end{center}
\bigskip
\bigskip
A public implementation of our NNLL+NLO FONLL-B matched computation
will be added to our code for Higgs
production~\cite{Forte:2016sja}, publicly available from
\begin{center}
\url{http://bbhfonll.hepforge.org/}.
\end{center}
\section*{Acknowledgments}
We thank Fabio Maltoni for providing the code for the calculation of the 5F NNLO
cross section and Valerio Bertone for helping provided in generating
a PDF set with modified threshold using the APFEL code.
D.N. is supported by the French Agence Nationale de la Recherche, under
grant ANR-15-CE31-0016. S.F. is supported by the European Research Council
under the European Union’s Horizon 2020 research and innovation
Programme (grant agreement n◦ 740006).
%%%%%%%%
\begin{appendix}
\section{FONLL expressions with $\mu_b$ different from $m_b$}
\numberwithin{equation}{section}
\setcounter{equation}{0}
We give for completeness the FONLL expressions by using $m_b$ different from $\mu_b$.
Note that the only difference with respect to the formulae presented in~\cite{Forte:2016sja},
is in the logarithm obtained from the expansion of the $b$~PDF, where $L=\log(Q^2/m_b^2)$,
becomes $L=\log(Q^2/\mu_b^2)$.
With this modification in place we get, for the 4F scheme, $B$ coefficients:
\begin{align}
B_{gg}^{(2)}\left(y,\frac{Q^2}{m^2_b}\right) & = \hat{\sigma}_{gg}^{(2)}\left(y,\frac{Q^2}{m_b^2}\right) \\
B_{q\bar{q}}^{(2)}\left(y,\frac{Q^2}{m^2_b}\right) & = \hat{\sigma}_{q\bar{q}}^{(2)}\left(y,\frac{Q^2}{m_b^2}\right)
\end{align}
while at $\order{\alpha_s^3}$ the redefinition of $\alpha_s$ contributes:
\begin{align}
B_{gg}^{(3)}\left(y,\frac{Q^2}{m^2_b},\frac{\mu_R^2}{\mu_b^2},\frac{\mu_F^2}{\mu_b^2}\right) & = \hat{\sigma}_{gg}^{(3)}\left(y,\frac{Q^2}{m_b^2}\right) - \frac{2 T_R}{3\pi} \ln{\frac{\mu_R^2}{\mu_F^2}}\hat{\sigma}_{gg}^{(2)}\left(y,\frac{Q^2}{m_b^2}\right)\\
B_{q\bar{q}}^{(3)}\left(y,\frac{Q^2}{m^2_b},\frac{\mu_R^2}{\mu_b^2},\frac{\mu_F^2}{\mu_b^2}\right) & = \hat{\sigma}_{q\bar{q}}^{(3)}\left(y,\frac{Q^2}{m_b^2}\right)- \frac{2 T_R}{3\pi} \ln{\frac{\mu_R^2}{\mu_b^2}}\hat{\sigma}_{q\bar{q}}^{(2)}\left(y,\frac{Q^2}{m_b^2}\right) \\
B_{gq}^{(3)}\left(y,\frac{Q^2}{m^2_b}\right) & = \hat{\sigma}_{gq}^{(3)}\left(y,\frac{Q^2}{m_b^2}\right) \\
B_{qg}^{(3)}\left(y,\frac{Q^2}{m^2_b}\right) & = \hat{\sigma}_{qg}^{(3)}\left(y,\frac{Q^2}{m_b^2}\right).
\end{align}
The massless limit of the 4F scheme coefficients, $B^{(0)}$, are, in this case, given by
\begin{equation}
\label{eq:massless_lim}
\sigma^{(4),(0)}\left(\alpha_s(Q^2),L\right)=\int_{\tau_H}^{1} \frac{dx}{x}\int_{\frac{\tau_H}{x}}^{1} \frac{dy}
{y^2}\sum_{ij=q,g}f_{i}(x,Q^2)f_j\left(\frac{\tau_H}{x y},Q^2\right)B_{ij}^{(0)}\left(y,L,\alpha_s(Q^2)\right),
\end{equation}
with
\begin{equation}
B_{ij}^{(0)}\left(y,L,\alpha_s(Q^2)\right) = \sum_{p=2}^N\left(\alpha_s(Q^2)\right)^pB_{ij}^{(0),(p)}\left(y,L\right),
\end{equation}
and
\begin{align}
B_{gg}^{(0)(2)} (y,L) & = y\int_y^1\frac{dz}{z}\left[2\mathcal{A}_{gb}^{(1)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right) + 4\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(1)}(z)\right] + \hat{\sigma}_{gg}^{(2)}(y), \\
B_{q\bar{q}}^{(0)(2)} (y,L) &= \hat{\sigma}_{q\bar{q}}^{(2)}(y);
\end{align}
while the new contributions to $\order{\alpha_s^3}$ are
\begin{align}\label{eq:subtrexp}
B_{gg}^{(0)(3)} (y,L) & = y\int_y^1\frac{dz}{z}\left[4\mathcal{A}_{gb}^{(2)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right) + 2\mathcal{A}_{gb}^{(1)}\left(z,L\right)\mathcal{A}_{gb}^{(2)}\left(\frac{y}{z},L\right)\hat{\sigma}_{b\bar{b}}^{(1)}(z) \right.\nonumber \\
& \left.\phantom{asdfdy\int_y^1\frac{dz}{z}4\mathcal{A}_{gb}^{(2)}\left(z,L\right)}+ 4\mathcal{A}_{gb}^{(2)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(1)}(z) + 4\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(2)}(z)\right], \\
B_{gq}^{(0)(3)} (y,L) & = y\int_y^1\frac{dz}{z}\left[2\mathcal{A}_{\Sigma b}^{(2)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right) + 2\mathcal{A}_{\Sigma b}^{(2)}\left(\frac{y}{z},L\right)\hat{\sigma}_{gb}^{(1)}(z) \right.\nonumber \\
& \left.\phantom{asdfdy\int_y^1\frac{dz}{z} 4 \mathcal{A}_{gb}^{(2)}\left(z,L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)}+ 2\mathcal{A}_{gb}^{(1)}\left(\frac{y}{z},L\right)\hat{\sigma}_{qb}^{(2)}(z)\right],
\end{align}
which completes our result in the case in which $\mu_b\neq m_b$.
\end{appendix}
%%%%%%%%%%%%%%%%
\renewcommand{\em}{}
\bibliographystyle{UTPstyle}
\bibliography{bbz_fonll}
%\input{bbH_FONLL.bbl}
\end{document}
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