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207 lines
9.5 KiB
207 lines
9.5 KiB
\newpage
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\topheading{Z production at N$^3$LO and N$^4$LL}
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\label{n3losec}
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Based on \href{https://arxiv.org/abs/2207.07056}{arXiv:2207.07056} (Neumann, Campbell '22).
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This page describes how to obtain Z-boson predictions at the level of up
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to N$^4$LL+N$^3$LO and at a fixed order of up to N$^3$LO. The
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highest order predictions are then are at the level of $\alpha_s^3$ up
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to missing N$^3$LO PDFs, which both affect the logarithmic accuracy
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and the fixed-order accuracy.
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\textbf{Warning}: Please note that predictions at the level of
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$\alpha_s^3$ are computationally very expensive due to the Z+jet NNLO
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matching corrections calculated with a small (5 GeV) cutoff. Our
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production plots typically run on 128 NERSC Perlmutter nodes for 12
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hours, about 100k CPU hours. If you do not have these resources and are
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mostly interested in the region of small $q_T$ (less than about 40 GeV), the
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matching to fixed order can be performed at the level of $\alpha_s^2$.
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This changes results by about 10\% above 40 GeV (missing
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$\alpha_s^3$/Z+jet NNLO corrections at large $q_T$), but typically
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just at the level of 2\% below 30 GeV, depending on cuts.
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For $Z$ production one can start with the input file
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\texttt{Bin/input\_Z.ini} that has a set of default cuts for $Z$
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production, i.e.~a mass window of the lepton pair around $m_Z$
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(\texttt{m34min} and \texttt{m34max} are set), and lepton minimum
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transverse momenta (\texttt{ptleptmin} and \texttt{ptlept2min}, both the
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same, i.e.~symmetric cuts).
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After choosing a set of PDFs (\texttt{lhapdf\%lhapdfset}), beamfunctions
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grids should be pre-generated by running MCFM with
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\texttt{resummation\%makegrid=.true.}.
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\midheading{N$^4$LL + matching at $\alpha_s^2$ fixed-order (NLO~$Z$+jet)}
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The fully matched result consists of the purely resummed part, the
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fixed-order Z+jet calculation and the fixed-order expansion of the
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resummation to remove overlap. At N$^3$LL$^\prime$+NNLO these three
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parts can be computed together automatically with
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\texttt{general\%part=resNNLOp}, or with \texttt{general\%part=resNNLO}
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at N$^3$LL+NNLO (\texttt{general\%part=resNLO} at NNLL+NLO). At the
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level of N$^4$LL+N$^3$LO the matching is with NNLO Z+jet predictions
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and, due to the computational requirements, these three parts are kept
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separate and have to be assembled manually.
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\bottomheading{Purely resummed N$^4$LL}
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The purely resummed N$^4$LL part can be obtained by running with
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\texttt{part\ =\ resonlyN3LO}. Similarly the N$^3$LL resummation is
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obtained with \texttt{part\ =\ resonlyNNLO} and N$^3$LL$^\prime$
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with \texttt{part\ =\ resonlyNNLOp} (see overview of configuration
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options). Scale variation of hard, low and rapidity scale can be enabled
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with \texttt{scales\%doscalevar\ =\ .true.}.
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The resummation part will be cut off at large transverse momenta through
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a transition function defined in the plotting routine. We recommend to
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use the default transition function with a parameter $(q_T^2/Q^2)=0.4$
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or $0.6$. The default plotting routine generates histograms with both
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choices that allows for estimating a matching uncertainty.
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Since the resummation becomes also invalid and numerically unstable for
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$q_T>m_Z$, we select the resummation integration range between $0$
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and $80$ GeV with \texttt{resummation\%res\_range=0\ 80}.
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\bottomheading{Fixed-order expansion of the resummed result}
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The fixed-order expansion of the resummed result (removing overlap with
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fixed-order Z+jet at NLO) (in the following called resexp) can be
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obtained by running \texttt{part\ =\ resexpNNLO}. We recommend a lower
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cutoff of 1 GeV, setting \texttt{resexp\_range\ =\ 1.0\ 80.0} in the
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\texttt{resummation} section.
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This part makes use of the transition function to ensure that this part
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is turned off at large $q_T$. Therefore the range is also limited to
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80 GeV.
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\bottomheading{Fixed-order Z+jet at NLO}
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The fixed-order $\alpha_s^2$ corrections (in the following called
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resabove) can be obtained by running \texttt{part\ =\ resaboveNNLO}. We
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recommend a cutoff of 1 GeV, setting \texttt{fo\_cutoff\ =\ 1.0} in the
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\texttt{resummation} section. This cutoff disables matching corrections
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below 1 GeV and must agree with the lower value of
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\texttt{resexp\_range}.
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\bottomheading{Combination and scale uncertainties}
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After running all three parts separately, the generated histograms can
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be added manually in a plotting program. The matching corrections
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consist of fixed-order result + fixed-order expansion of the resummed
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result. At $\alpha_s^2$ a manual combination should agree with an
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automatic combination through \texttt{part\ =\ resNNLO}, for example.
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To obtain uncertainties from scale variation the following procedure
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should be followed. The scales in the matching corrections must match,
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i.e.~resexp\_scalevar\_01 should be added to resabove\_scalevar\_01, and
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resexp\_scalevar\_02 should be added to resexp\_scalevar\_02. Note that
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the scale variation histograms only give the difference to the central
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value. So the minimum of the scale varied matching corrections consist
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of:
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\begin{verbatim}
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min(resabove + resabove_scalevar_01 + resexp + resexp_scalevar_01,
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resabove + resabove_scalevar_02 + resexp + resexp_scalevar_02)
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\end{verbatim}
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Similarly the maximum can be taken, both giving an envelope of
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uncertainties. Note that in the resummation and its fixed-order
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expansion we have not decoupled the scale in the PDFs from other scales.
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Therefore when combining resexp with resabove, only the simultaneous
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variation of factorization scale and renormalization scale upwards and
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downwards can be used for the scale variation, corresponding to "\_01"
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and "\_02".
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Finally the scalevar\_maximum and scalevar\_minimum histograms of the
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purely resummed result should be considered as an additional envelope.
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For this part the envelope of all scale variations is taken. The
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variation of the rapidity scale plays an important role and can be
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enabled by setting \texttt{scalevar\_rapidity\ =\ .true.} in the
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\texttt{{[}resummation{]}} section. It gives two important additional
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variations to the 2, 6, or 8-point variation of hard and resummation
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scale in the resummed part.
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\midheading{Adding $\alpha_s^3$ matching corrections (Z+jet NNLO coefficient)}
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To obtain the matching corrections at $\alpha_s^3$ we compute just the
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$\alpha_s^3$ \emph{coefficient} and add it to the previously obtained
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lower order results.
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\bottomheading{Fixed-order Z+jet NNLO coefficient}
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To obtain the fixed-order $\alpha_s^3$ corrections please run with
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\texttt{part\ =\ resaboveN3LO}. We recommend a matching cutoff of 5 GeV,
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setting \texttt{fo\_cutoff\ =\ 5.0} in the \texttt{resummation} section
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and consequently a jettiness cutoff of \texttt{taucut=0.08} in the
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\texttt{nnlo} section. It is possible to run with a larger
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\texttt{fo\_cutoff} keeping the same \texttt{taucut} value, but either a
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smaller \texttt{fo\_cutoff} or a larger \texttt{taucut} value will
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require a new validation of results.
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\bottomheading{Fixed-order Z+jet NNLO coefficient}
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To obtain the fixed-order $\alpha_s^3$ corrections please run the
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$Z$+jet process (\texttt{nproc=41}) with \texttt{part=nnlocoeff} in
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the \texttt{{[}general{]}} section with a fixed $q_T$ cutoff, i.e.~by
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setting \texttt{pt34min\ =\ 5.0} in the \texttt{{[}masscuts{]}} section.
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The Z+jet calculation is based on jettiness slicing, which requires a
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jettiness cutoff. For a $q_T$ cutoff of 5 GeV (for resummation this is
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the matching-corrections cutoff) we recommend a jettiness cutoff of
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\texttt{taucut=0.08} in the \texttt{{[}nnlo{]}} section. It is possible
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to run with a larger $q_T$ cutoff, keeping the same \texttt{taucut}
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value, but either a smaller $q_T$ cutoff or a larger \texttt{taucut}
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value will require a new validation of results. See
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\href{https://arxiv.org/abs/2207.07056}{arXiv:2207.07056} for technical
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details.
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\bottomheading{$\alpha_s^3$ fixed-order expansion coefficient of the resummed result}
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The $\alpha_s^3$ fixed-order expansion coefficient of the resummed
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result (removing overlap with fixed-order Z+jet at NNLO) can be obtained
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by running \texttt{part\ =\ resexpN3LO}. \textbf{\emph{NOTE}} that this
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only returns the N$^3$LO expansion \textbf{\emph{coefficient}}, to
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match with the fixed-order \texttt{nnlocoeff} part. Similarly, to match
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with the fixed-order part, we recommend a cutoff of 5 GeV, setting
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\texttt{resexp\_range\ =\ 5.0\ 80.0} in the \texttt{resummation}
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section.
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\bottomheading{Combination}
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Similary to the lower order, the matching corrections $\alpha_s^3$
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coefficient can be added to lower order $\alpha_s^2$ results.
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\midheading{Fixed order N$^3$LO}
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To compute fixed-order N$^3$LO cross-sections with $q_T$
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subtractions one needs to calculate the fixed-order Z+jet NNLO
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coefficient with a $q_T$ cutoff, as outlined above. The below-cut
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contribution can be obtained via \texttt{part=n3locoeff} in the
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\texttt{{[}general{]}} section for $Z$ production,
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i.e.~\texttt{nproc=31}, where the \texttt{qtcut} value in the
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\texttt{{[}nnlo{]}} section has to match the \texttt{pt34min} value
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chosen for the Z+jet NNLO calculation.
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We recommend to calculate the fixed-order NNLO coefficient first, as it
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is instructional to understand the procedure at N$^3$LO. This proceeds
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by combining NLO Z+jet result with a \texttt{pt34min} value with the
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\texttt{part=nnloVVcoeff} part (below-cut at NNLO), where \texttt{qtcut}
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has to be set to match the \texttt{pt34min} value. The result of this
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manual procedure must agree with the automatic calculation,
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i.e.~calculating Z with \texttt{part=nnlo} or \texttt{part=nnlocoeff}.
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Please pay particular attention to the difference of calculating the
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NNLO ($\alpha_s^2$) and N$^3$LO ($\alpha_s^3$) coefficients and
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the full result.
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