## Chapter 11

Z production at
N${}^{3}$LO and
N${}^{4}$LL

Based on arXiv:2207.07056 (Neumann, Campbell ’22).
This page describes how to obtain Z-boson predictions at the level of up to N${}^{4}$LL+N${}^{3}$LO and at a fixed order of up to N${}^{3}$LO. The highest order predictions are then are at the level of ${\alpha}_{s}^{3}$ up to missing N${}^{3}$LO PDFs, which both affect the logarithmic accuracy and the fixed-order accuracy.

Warning: Please note that predictions at the level of ${\alpha}_{s}^{3}$ are computationally very expensive due to the Z+jet NNLO matching corrections calculated with a small (5 GeV) cutoff. Our production plots typically run on 128 NERSC Perlmutter nodes for 12 hours, about 100k CPU hours. If you do not have these resources and are mostly interested in the region of small ${q}_{T}$ (less than about 40 GeV), the matching to fixed order can be performed at the level of ${\alpha}_{s}^{2}$. This changes results by about 10% above 40 GeV (missing ${\alpha}_{s}^{3}$/Z+jet NNLO corrections at large ${q}_{T}$), but typically just at the level of 2% below 30 GeV, depending on cuts.

For $Z$
production one can start with the input file `Bin/input_Z.ini`

that has a set of default cuts
for $Z$
production, i.e. a mass window of the lepton pair around
${m}_{Z}$
(`m34min`

and `m34max`

are set), and lepton minimum transverse momenta (`ptleptmin`

and
`ptlept2min`

, both the same, i.e. symmetric cuts).

After choosing a set of PDFs (`lhapdf%lhapdfset`

), beamfunctions grids should be
pre-generated by running MCFM with `resummation%makegrid=.true.`

.

### 11.1 N${}^{4}$LL + matching at ${\alpha}_{s}^{2}$ fixed-order (NLO $Z$+jet)

The fully matched result consists of the purely resummed part, the fixed-order Z+jet
calculation and the fixed-order expansion of the resummation to remove overlap. At
N${}^{3}$LL${}^{\prime}$+NNLO
these three parts can be computed together automatically with
`general%part=resNNLOp`

, or with `general%part=resNNLO`

at
N${}^{3}$LL+NNLO
(`general%part=resNLO`

at NNLL+NLO). At the level of
N${}^{4}$LL+N${}^{3}$LO
the matching is with NNLO Z+jet predictions and, due to the computational requirements,
these three parts are kept separate and have to be assembled manually.

#### 11.1.1 Purely resummed N${}^{4}$LL

The purely resummed N${}^{4}$LL
part can be obtained by running with `part = resonlyN3LO`

. Similarly the
N${}^{3}$LL
resummation is obtained with `part = resonlyNNLO`

and
N${}^{3}$LL${}^{\prime}$
with `part = resonlyNNLOp`

(see overview of configuration options). Scale variation of hard,
low and rapidity scale can be enabled with `scales%doscalevar = .true.`

.

The resummation part will be cut off at large transverse momenta through a transition function defined in the plotting routine. We recommend to use the default transition function with a parameter $({q}_{T}^{2}\u2215{Q}^{2})=0.4$ or $0.6$. The default plotting routine generates histograms with both choices that allows for estimating a matching uncertainty.

Since the resummation becomes also invalid and numerically unstable for
${q}_{T}>{m}_{Z}$, we select the resummation
integration range between $0$
and $80$
GeV with `resummation%res_range=0 80`

.

#### 11.1.2 Fixed-order expansion of the resummed result

The fixed-order expansion of the resummed result (removing overlap with fixed-order Z+jet at
NLO) (in the following called resexp) can be obtained by running `part = resexpNNLO`

. We
recommend a lower cutoff of 1 GeV, setting `resexp_range = 1.0 80.0`

in the `resummation`

section.

This part makes use of the transition function to ensure that this part is turned off at large ${q}_{T}$. Therefore the range is also limited to 80 GeV.

#### 11.1.3 Fixed-order Z+jet at NLO

The fixed-order ${\alpha}_{s}^{2}$
corrections (in the following called resabove) can be obtained by running ```
part =
resaboveNNLO
```

. We recommend a cutoff of 1 GeV, setting `fo_cutoff = 1.0`

in the
`resummation`

section. This cutoff disables matching corrections below 1 GeV and must agree
with the lower value of `resexp_range`

.

#### 11.1.4 Combination and scale uncertainties

After running all three parts separately, the generated histograms can be
added manually in a plotting program. The matching corrections consist
of fixed-order result + fixed-order expansion of the resummed result. At
${\alpha}_{s}^{2}$ a
manual combination should agree with an automatic combination through `part = resNNLO`

,
for example.

To obtain uncertainties from scale variation the following procedure should be followed. The scales in the matching corrections must match, i.e. resexp_scalevar_01 should be added to resabove_scalevar_01, and resexp_scalevar_02 should be added to resexp_scalevar_02. Note that the scale variation histograms only give the difference to the central value. So the minimum of the scale varied matching corrections consist of:

min(resabove + resabove_scalevar_01 + resexp + resexp_scalevar_01, resabove + resabove_scalevar_02 + resexp + resexp_scalevar_02)

Similarly the maximum can be taken, both giving an envelope of uncertainties. Note that in the resummation and its fixed-order expansion we have not decoupled the scale in the PDFs from other scales. Therefore when combining resexp with resabove, only the simultaneous variation of factorization scale and renormalization scale upwards and downwards can be used for the scale variation, corresponding to “_01” and “_02”.

Finally the scalevar_maximum and scalevar_minimum histograms of the purely
resummed result should be considered as an additional envelope. For this part the envelope of
all scale variations is taken. The variation of the rapidity scale plays an important role and
can be enabled by setting `scalevar_rapidity = .true.`

in the `[resummation]`

section. It
gives two important additional variations to the 2, 6, or 8-point variation of hard and
resummation scale in the resummed part.

### 11.2 Adding ${\alpha}_{s}^{3}$ matching corrections (Z+jet NNLO coefficient)

To obtain the matching corrections at ${\alpha}_{s}^{3}$ we compute just the ${\alpha}_{s}^{3}$ coefficient and add it to the previously obtained lower order results.

#### 11.2.1 Fixed-order Z+jet NNLO coefficient

To obtain the fixed-order ${\alpha}_{s}^{3}$
corrections please run with `part = resaboveN3LO`

. We recommend a matching cutoff of 5
GeV, setting `fo_cutoff = 5.0`

in the `resummation`

section and consequently a jettiness
cutoff of `taucut=0.08`

in the `nnlo`

section. It is possible to run with a larger `fo_cutoff`

keeping the same `taucut`

value, but either a smaller `fo_cutoff`

or a larger `taucut`

value will
require a new validation of results.

#### 11.2.2 Fixed-order Z+jet NNLO coefficient

To obtain the fixed-order ${\alpha}_{s}^{3}$
corrections please run the $Z$+jet
process (`nproc=41`

) with `part=nnlocoeff`

in the `[general]`

section with a fixed
${q}_{T}$
cutoff, i.e. by setting `pt34min = 5.0`

in the `[masscuts]`

section. The Z+jet
calculation is based on jettiness slicing, which requires a jettiness cutoff. For a
${q}_{T}$ cutoff
of 5 GeV (for resummation this is the matching-corrections cutoff) we recommend a jettiness
cutoff of `taucut=0.08`

in the `[nnlo]`

section. It is possible to run with a larger
${q}_{T}$
cutoff, keeping the same `taucut`

value, but either a smaller
${q}_{T}$ cutoff
or a larger `taucut`

value will require a new validation of results. See arXiv:2207.07056 for
technical details.

#### 11.2.3 ${\alpha}_{s}^{3}$ fixed-order expansion coefficient of the resummed result

The ${\alpha}_{s}^{3}$
fixed-order expansion coefficient of the resummed result (removing overlap with fixed-order Z+jet
at NNLO) can be obtained by running `part = resexpN3LO`

. NOTE that this only returns the
N${}^{3}$LO
expansion coefficient, to match with the fixed-order `nnlocoeff`

part. Similarly, to match
with the fixed-order part, we recommend a cutoff of 5 GeV, setting ```
resexp_range = 5.0
80.0
```

in the `resummation`

section.

#### 11.2.4 Combination

Similary to the lower order, the matching corrections ${\alpha}_{s}^{3}$ coefficient can be added to lower order ${\alpha}_{s}^{2}$ results.

### 11.3 Fixed order N${}^{3}$LO

To compute fixed-order N${}^{3}$LO
cross-sections with ${q}_{T}$
subtractions one needs to calculate the fixed-order Z+jet NNLO coefficient with a
${q}_{T}$ cutoff, as
outlined above. The below-cut contribution can be obtained via `part=n3locoeff`

in the `[general]`

section for $Z$
production, i.e. `nproc=31`

, where the `qtcut`

value in the `[nnlo]`

section has to match the
`pt34min`

value chosen for the Z+jet NNLO calculation.

We recommend to calculate the fixed-order NNLO coefficient first, as it is instructional to understand the
procedure at N${}^{3}$LO.
This proceeds by combining NLO Z+jet result with a `pt34min`

value with the
`part=nnloVVcoeff`

part (below-cut at NNLO), where `qtcut`

has to be set to
match the `pt34min`

value. The result of this manual procedure must agree with
the automatic calculation, i.e. calculating Z with `part=nnlo`

or `part=nnlocoeff`

.
Please pay particular attention to the difference of calculating the NNLO
(${\alpha}_{s}^{2}$) and
N${}^{3}$LO
(${\alpha}_{s}^{3}$)
coefficients and the full result.