Defiducialization: Providing Experimental Measurements for Accurate Fixed-Order Predictions

An experimental procedure is proposed to perform measurements of differential cross sections which can be compared to fixed-order QCD predictions with improved accuracy. The procedure can be applied to the Drell-Yan cross-section measurements which are differential in the boson transverse momentum. An example analysis is performed using the ATLAS measurement of the $Z$-boson production cross section at center-of-mass energy of $8$ TeV. The resulting full phase space measurement of the cross section differential in the boson rapidity is compared to theoretical predictions computed with next-to-next-to leading-order accuracy in QCD.


I. INTRODUCTION
Accurate knowledge of parton distribution functions of the proton (PDFs) is essential for the physics program at the LHC. PDF uncertainties are the leading source of systematics for precision measurements of the W -boson mass and effective weak mixing angle, sin 2 θ W [1,2].
Reduction of the PDF uncertainties is important in particular for the interpretation of high statistics measurements for the future LHC data.
PDFs can be constrained using W -and Z-boson production in the charged and neutral current Drell-Yan processes. These processes can be measured with sub-percent experimental accuracy and thus provide a valuable input for the PDF determination. An example of an accurate measument is the ATLAS result on γ * /Z-and W -boson production at the center-of-mass energy of √ s = 7 TeV [3]. For the Z boson, the measurement is perfromed in bins of the invariant mass of the lepton pair, m , and of the lepton pair rapidity, y .
For the W boson, the results are reported as a function of the lepton pseudorapidity, η .
Excluding the global normalisation uncertainty, the measurement reaches better than 0.5% experimental uncertainty.
The cross sections differential in y , m and η are known at next-to-next-to leadingorder (NNLO) accuracy in perturbative QCD [4][5][6][7][8]. For these observables, the corresponding computations are inclusive in the boson transverse momentum p T and thus insenstive to ln p T /m divergences, providing a robust input for determination of collinear PDFs. However the ATLAS measurement is performed in a fiducial volume with experimental cuts on the lepton transverse momentum and lepton pseudorapidity which are required due to the detector acceptance. These selection criteria introduce dependence on p T modelling thereby spoiling accuracy of the fixed-order predictions. The study performed in Ref. [3] shows that predictions become unstable with respect to small variations of the cuts and differ as much as 1% for different subtraction methods which is significant compared to the experimental accuracy.
The impact of the fiducial cuts on fixed-order predictions is under investigation since some time. It is known to be large when transverse momentum of individual leptons (or jets) approaches half of the invariant mass of the lepton pair (or jet pair). It has been proposed in Ref. [9] to introduce an asymmetry for the cut values for the leading and sub-leading objects.
This procedure stabilizes the computation of the fixed-order predictions, however it does not elliminate the uncertainty arising from the logarithmic corrections [10]. Computation of higher order corrections and/or inclusion of the ln p T /m resummation should reduce the uncertainty arising from the fiducial cuts. However the full next-to-next-to-next-to leadingorder (N 3 LO) corrections for Drell-Yan processes are not availible yet, while resummation corrections may bring additional uncertainties from the recoil prescirption [11].
Experimentally, it is sometimes possible to isolate regions in the phase space which are not affected by fiducial cuts. For example, in Ref. [12] the fiducial acceptance for the triple differential measurement of dσ dy d cos θdm , where cos θ is the lepton polar angular variable, is above 99% for | cos θ| < 0.4, y < 1 and m > 66 GeV [13]. This region can be safely compared to fixed-order predictions. Other experimental methods include explicit correction to the full phase space during the data analysis, as in e.g. Ref. [14].
In this paper, another approach is proposed. It is based on an observation that an acceptance correction from the fiducial to full phase space can be determined accurately using fixed-order calculations for the vector boson plus jet process for measurements given in bins of p T . It is demonstrated in the following using the ATLAS measurement of Zboson production differential in y and p T performed using data collected at √ s = 8 TeV [15]. The acceptance correction is computed at NLO for the Z plus jet process (O(α 2 S ), where α S is the strong coupling constant), using the MCFM v6.8 program [16], interfaced to APPLGRID [17]. The correction is used to determine the full phase space measurement differential in y and p T . The result is integrated in p T and the inclusive dσ/dy differential cross-section measurement is compared to the NNLO (O(α 2 S )) computation for inclusive Zboson production obtained using the MCFM v9 program [18,19]. The paper concludes with a discussion of the results and possible applications of the method to other measurements.

II. FORMALISM
The fully differential Z-boson production cross section can be expressed as d 5 σ dp T dy dm d cos θdφ = 3 16π Here φ is the lepton azimuthal angular variable, P i (cos θ, φ) are the nine harmonic polynomials, and σ U +L is the unpolarised cross section. The harmonic polynomials depend on eight angular coefficients A i (p T , y , m ) which define fractions of helicity cross sections with respect to the unpolarised case. Equation 1 relies on factorization of the Z boson production and decay processes. In essence, it represents production of a spin one particle and its following decay. It is violated by electroweak corrections which contain interaction of the initial quarks and final state leptons. It is also modified for the γγ → + − scattering process.
These corrections are however small at the Z-pole region and neglected in the following. For the measurements insensitive to the azimuthal angle φ and forward-backward asymmetry in cos θ, the most important coefficient is A 0 . The coefficient vanishes for p T → 0 and saturates The coefficients A i (p T , y , m ) can be calculated using fixed-order predictions. It is possible that the A 2 coefficient is sensitive to non-perturbative effects at low p T [20], however this should have a small impact on the acceptance. It was demonstrated in Ref. [21] which compared predictions for the A i coefficients to the data obtained by the ATLAS and CMS collaborations [22,23]. The calculations were performed at LO (O(α S )), NLO (O(α 2 S )) and NNLO (O(α 3 S )) order for the Z-boson plus jet process, as provided by the NNLOJET program [24]. For the coefficient A 0 , good perturbative convergence is observed with already LO predictions being in reasonable agreement with the data.
Given the p T , y , m values and the A i coefficients, the kinematics of the final state leptons and thereby the fraction of events passing fiducial cuts is fully determined. Any fixedorder prediction for the Z boson plus jet process must obey decomposition of equation 1. Therefore, the fiducial acceptance for each p T , y , m bin can be determined at fixed order, provided it is narrow enough to neglect the p T dependence inside the bin. The residual theoretical uncertainties arising in this procedure can be estimated the usual way, by PDF and scale variations.
The corrected to the full phase space cross section can be integrated in p T to provide the full phase space measurement as a function of y and m to be compared to corresponding predictions. The main advantage of this procedure compared to the standard approach that it eliminates dependence on the p T distribution in predictions which is poorly modeled at fixed order: the p T dependence as measured in data is used instead.

III. RESULTS
The correction procedure is tested using the ATLAS γ * /Z data at √ s = 8 TeV [15].
The cross sections in this analysis are measured differentialy in p T and y for the 66 < m < 115 GeV invariant mass range. There are 20 variable-size bins in p T , starting with narrow 2 GeV bins for low p T and ending with a wide 700 GeV bin from 200 to 900 GeV. The fiducial acceptance is estimated using the MCFM v6.8 program interfaced to APPLGRID, for fast evaluation of theoretical uncertainties. The Z plus jet process is computed at LO and NLO yielding acceptance estimates A LO and A NLO , respectively. It is also possible to use NNLO calculations for the Z plus jet process which became available recently [27], however they are not used in the present analysis. Note that in order of α S , the NLO calculations for the Z plus jet process match NNLO calculations for inclusive Z-boson production.
The renormalization and factorization scales are set to m and the CT14NNLO PDF set [28] is used for the calculations. The jet transverse momentum is required to be above 1 GeV. The resulting fiducial acceptance is shown in figure 1.   are evaluated using the CT18ANNLO set [29,30]     other sets for y < 2. For the highest rapidity bin, the acceptance based on ATLASepwz16 deviates by as much as −0.2% for low p T and up to +0.35% for p T around m Z . No additional uncertainty is introduced due to this deviation.
The size of the theoretical uncertainties on the acceptance corrections suggests that it is possible to proceed with applying them to the data. Figure 4 shows the ratio of the dσ/dp T distributions corrected to the full phase space for a given y bin to the most central 0 < y < 0.4 bin. The ratio is close to one for small p T decreasing almost linearly for large p T and large y . For p T > 35 GeV, the data are compared to fixed-order predictions at NLO for the Z-boson plus jet process computed using the same APPLGRID as for the acceptance correction which is convoluted with the CT14NNLO PDF set. The data and predictions are found to be in reasonable agreement. 116 GeV obtained for the CT14NNLO PDF set is 1114.9(1) pb, which is 0.4% larger than the value of 1110(1) pb obtained in Ref. [26] using the DYTURBO program [34].
The comparison between the full phase space measurement of dσ/dy at timate (overestimate) the data. Given the significant global normalization uncertainty the difference is however not significant.
More quantitatively, the comparison can be performed by taking ratios of the predictions to the data. It is also interesting to compare the ratios of the full cross sections to the ratios of the fiducial cross sections, to see the impact of the p T -dependent acceptance corrections.
The fiducial cross section in data is computed by simple integration over the differential cross section. It is verified that the total fiducial cross section agrees with the one reported in the ATLAS publication. The NNLO predictions are obtained using the MCFM v9 program with fiducial cuts enabled. The resulting full and fiducial cross-section ratios for the two PDF sets are shown in figure 7. For the CT14NNLO based prediction, the ratio of the full phase space cross sections is closer to the unity compared to the ratio of the fiducial cross sections.
The dependence of the ratio on y is also reduced significantly. For the ATLASepWZ16 based prediction, the fiducial ratio is closer to unity, which is not too surprising since the set is fitted to the fiducial ATLAS measurement at three bins the fiducial ratio is lower than the full one, by as much as 2%, for the two lowest y bins. For bins with 1.2 < y < 2.0, they roughly agree. For the highest rapidity bin, the behavior is different depending on the PDF set. For the CT14NNLO based prediction the two ratios agree while for the ATLASepWZ16 based prediction the fiducial ratio is below the full one by almost 2%.
For the full phase space comparison based on the CT14NNLO and ATLASepWZ16 sets, the ratios of σ theory /σ data show opposite trend as a function of y . Given that the strangequark distribution affects low rapidity region more, based on this observation, it is possible to make a conjecture that the ATLAS data may prefer somewhat larger strangeness content than in the CT14NNLO set and somewhat smaller than in the ATLASepWZ16 set. The sets [29,30]  The strange-quark distribution in CT18NNLO is similar to CT14NNLO. The CT18ANNLO set in addition uses a charm-quark mass that is increased compared to other PDF sets.
This leads to suppression of the charm-quark contribution, compensated by an increase of the light-quark sea, yielding in turn an increase of the Drell-Yan cross sections at the LHC, as it was demonstrated in Ref. [35]. The predictions based on the CT18ANNLO and CT18ZNNLO sets show better agreement with the ATLAS data compared to those based on the CT18NNLO set: the ratio shows no dependence on y and the overall normalisation of the ratio is closer to unity.

IV. DISCUSSION
In the discussion above, it is argued that for comparison of fixed-order predictions with data, it is better to use full σ full,theory = dσ theory dp T dp T vs σ full,data = dσ data dp T instead of fiducial cross sections σ fidu,theory = dσ theory dp T A(p T )dp T vs σ fidu,data = dσ data dp T dp T .
Both formula require accurate prediction for the acceptance A(p T ), however equation 2 does not require accurate modeling of the p T distribution by the prediction. Moreover, the acceptance A(p T ) is well-known for fixed-order predictions with the residual theoretical uncertainties being significantly smaller than the current experimental errors. For future, more accurate experimental measurements, the scale uncertainty may be reduced by using existing higher order calculations. If needed, the PDF uncertainty can be reduced further too, by using the same PDF in the calculation of σ full,theory and A(p T ). This could be relevant for PDF fits in particular and can be arranged without difficulty since the PDF-dependent acceptance A(p T ) can be computed using exact and fast methods such as APPLGRID.
The method is applicable to the measurements differential in the boson transverse momentum p T . It is natural for Z-boson production, but can be extended for the W boson too.
Given that A(p T ) has only mild variation at low p T (see figure 1), it is probably possible to use coarser p T binning, which is required by the experimental resolution. Since the rapidity of the W boson can not be reconstructed, this may lead to increased PDF dependence of the acceptance correction factor, requiring its update when performing comparisons for predictions based on different PDFs. Studies of the defiducialization method for W -boson production are however beyond the scope of the current analysis and they await for experimental measurements, such as of the lepton pseudorapidity distribution, reported in bins of the W -boson transverse momentum.

V. SUMMARY
In summary, a method is proposed to perform comparisons of experimental data on Drell-Yan production, performed in fiducial phase space, and fixed-order QCD predictions with improved accuracy. The method requires the data to be measured differentially in the boson transverse momentum p T in addition to other variables of interest such as y and m . It relies on the ability to calculate the fiducial acceptance correction for a given p T value at fixed order.
The method is applied to the ATLAS Z-boson production data at a center-of-mass energy of 8 TeV. The uncertainty on the acceptance correction is shown to be small compared to the current level of experimental uncertainties. The data are integrated in p T and compared to the NNLO predictions. To check impact of the new approach, the comparison is also performed using fiducial cross sections. A significant, compared to experimental uncertainties, improvement in the data description by the predictions based on CT14NNLO set is observed when performing comparison in the full phase space.

VI. ACKNOWLEDGEMENTS
The author would like to thank Simone Amoroso, Ludovica Aperio Bella, Maarten Boonenkamp, Stefano Camarda, and Jan Kretzschmar for discussions and comments to the paper draft.