4. QCD interpolation grids
4.7. Future steps towards beyond fixed-order interpolation grids
The lack of multi-order or all-order support inQCDinterpolation grids puts limits on what data is usable forPDFfits, because these fits rely on the speed of the grid reweighting ap-proach, at least beyondLO. This means that data points have to be removed from the fit, when they require resummation effects to be taken into account, or if they are better
de-10−6
Figure 4.7.:A comparison of uncertainty bands for thepWT in a Sherpa NLOPS W production.
The left-hand plot features bands that are generated by reweighting an APPLgrid filled by Sherpa via MCgrid. In the right-hand plots, these bands are compared individually to ones that are generated with the internal reweighting of Sherpa, where the reweighting of parton-shower emissions is disabled.
scribed as a combined result of calculations at different jet multiplicities. All Monte-Carlo results that rely on parton showers or multi-jet merging fall into this category. Resumma-tion effects play a role in the small-pWT/Zbins, as we have seen, which is why e.g. in the NNPDF3.0 fit the corresponding data points forpWT are not used [99]. Another example is the small-angle region of the dijet azimuthal decorrelation observable, which is dominated by soft gluon emissions [168,169]. Both observables,pWT/Zand the azimuthal decorrelation, are also examples for observables that profit from a multi-jet merged calculation. They feature overlapping regions (at largepT and large angles, respectively), where each one is best described by a certain number of hard jets that provide the recoil.
4.7.1. Multi-jet merging via stacked interpolation grids
How easy is it to modify interpolation grids to support multi-jet merged calculations? We have seen inSection 3.7that multi-jet merged calculations feature several aspects that re-quire a reweighting. There are the different matrix elements that can be handled individually
as fixed-order calculations, safe for the simplePDFfactors, which are replaced with several ratios ofPDFs determined by the clustering, cf.Eq. (3.43). These ratios are accompanied for NLOmatrix elements with appropriate counter-terms, given inEq. (3.50), which subtract theirαSexpansion to retain theNLOaccuracy. The clustering also determines shower-like αSscales, one perQCDcluster node, which are combined to determine an overall renormal-isation scale, seeEq. (3.44). Morover, there are the Sudakov rejection weights,Eq. (3.46), for which truncated showers are sampled for emissions above the merging scale, which leads to the scale-/parameter-dependent veto. And finally,MENLOPScalculations come withKfactors, that reweight as compositions of matrix-elements, involving products and divisions, seeEq. (3.53).
Leaving aside the all-order parts for the moment, i.e. the parton shower and the re-lated Sudakov veto, there are still elements in the remaining fixed-order calculations that can not be filled into existing interpolation grids. Matrix elements with varying multi-plicities themselves are easily supported, either by implementing the support for addi-tional sub-grids per multiplicity withinAPPLgrid/fastNLO, or by operating with several APPLgrid/fastNLO instances managed byMCgrid. Also the dynamic renormalisa-tion scale determinarenormalisa-tion by itself is not an issue, because the variarenormalisa-tion of the combined scale is correct toNLOaccuracy. However, interpolation grids assumeµ2R=µ2F, or at least µ2R = µ2R(µ2F), which is broken by the different prescription for theµF scale (which is de-termined by the lowest invariant mass or negative virtuality in the core process). Also the PDFratios of the clustering can not be filled into interpolation grids, as they requirePDFs to enter as simple prefactor pairs. On the other hand, the counter-terms only feature a singlePDFfactor, and the current grid implementations do not allow a mixture of single and pairedPDFprefactors. TheKfactor provides furtherPDFratios, via its dependence on a B/B term.
A perfect representation of these complications by a modified interpolation grid tech-nology is a question of scalability. It is feasible to add a one grid per multiplicity and one per parton (e.g. to fill the single-PDFcounter-terms), but the storage size and convolution times will increase by orders of magnitude when we begin to provide grids for each possible combination ofPDFratios, and the question of how to encode the individual cluster scales and longitudinal momentum fractions would remain.
However, one could study with a simple modification of the internal reweighting of Sherpa, how accurate it would be to neglect those complications, i.e. to only reweight the matrix-element contributions per multiplicity, and to restrict thePDFreweighting to the external incoming partons (i.e. before clustering), and to setµ2F =µ2R. At first,NLOmatrix elements could be omitted to factor out the additional problems by the counter-terms and theKfactor.
4.7.2. Approximate approaches to include parton-shower emissions
The problem of a variable number ofPDFratios discussed in the previous section is also encountered when we attempt to reweight parton-shower emissions with interpolation
grids, with the number of partonic combinations growing too large to get by with some variation of the usual approach.
One idea is motivated by our finding that some observables, such as the pWT, are only sensitive to the dependences of the first two or three hardest emissions. However, even then the number ofrejectedemissions still varies and will be larger than the number of accepted ones. Also, the dependence structure is different between the two, which prevents a unified handling, effectively doubling the number of grids again. Hence, the problem is even more complex than with the clusterPDFratios of the merging.
Therefore, a second idea is to not track the dependences of the parton shower at all, and instead have a number of grid replica, each one filled by events from a matrix-element plus parton-shower generation, where some scale or parameter used is varied. Then the respective parton-shower dependence is encoded in the difference between the predictions of those grid replica, and one could interpolate between those predictions. Unfortunately, even if this can be done for a single-valued parameter likeαS(m2Z), there is no universal ordering between differentPDFs. The approach would only be feasible, if aPDFcan be parametrised by a small number of values. If we look at theMMHT2014parametrisation, there are about 40 independent parameters entering its fit. So if one wants three grid replicas perPDFparameter—a central one along with an up and a down variation, a total of 120 grids is needed. An increase of the number of grids by two orders of magnitude seems impractical in terms of storage andCPUtime requirements.
In conclusion, a direct encoding of emission dependences is made infeasible by the large number ofPDFratio combinations and the duplication by the differing dependence of re-jected trial emissions, and an indirect and approximate encoding by grid replicas is difficult due to the large number of degrees of freedom inPDFdeterminations.
4.8. Discussion
In this chapter we have discussed new developments for pQCDreweighting using interpo-lation grids, and validated them with closure tests.
Typical applications for interpolation grids are parameter fits that use an iterative ap-proach, such asPDFfits. In these, the exact variations are not known beforehand, and the associated predictions must be calculable within milliseconds due to the large number of iterations needed. Currently, interpolation grids allow for variations of thePDFs,αS(m2Z), and bothµF andµR, for fixed-orderLO,NLOand evenNNLOcalculations.
We have presented and validated new features in theMCgrid interface, which led to its 2.0 release. First, fastNLOinterpolation grids are now supported, such that the two main interpolation grid implementations can be used. Through this common interface, they can be compared in a reliable manner. Also, their feature sets are not equal, such that it can be advantageous to use one or another. For example, fastNLOin principle supportsNNLO cross sections, and allows for two additional sub-grids for encoding kinematic variables for a more flexible scale variation based on an arbitrary function of these two variables [53].
MCgrid does not yet support these two features, but especially the addition ofNNLO calculations will certainly be relevant for future applications, and would be a natural next step of the development. A second addition toMCgrid we presented is the support for dedicatedAPPLgrids for the scale logarithm coefficients, which allow for scale variations that go beyond simple scale factors. Thirdly, we discussed the added support for theO(αS) expansion of S-MC@NLOcalculations, by filling their(DA−DS)contribution.
This last feature is a first step beyond encoding fixed-order calculations in interpolation grids. However, proper resummation effects by the inclusion of parton-shower emissions are still absent. A precondition for adding these is a precise understanding of their de-pendences, which we achieved in the previous chapter with our detailed account on this subject in the context of the internal reweighting. However, there are still major difficulties to overcome, such as variable numbers ofPDFratios. These need to be solved, possibly by an approximate account, before a reweighting of parton-shower emissions in interpolation grids is in reach. For now we suggest to use the on-the-fly reweighting as a fast exploratory method before the actual grid production, to assess the validity of doing fixed-order vari-ations only. Similarly, one could assess with a modification of the on-the-fly reweighting the viability of a future implementation for an approximate handling of multi-jet merged calculations within interpolation grids. Hence, internal-reweighting methods can play a guiding role in future interpolation-grid developments.