Closure tests for multi-jet merged calculations

The reweighting for multi-jet merged calculations as discussed in the previous sections has been implemented within Sherpa with the CSShower for leading-order matrix elements (MEPS@LO), next-to-leading order matrix elements (MEPS@NLO) and next-to-leading-order matrix elements with additional next-to-leading-order ones on top (MENLOPS). For the validation, we again perform closure tests between reweighted and dedicated predictions for the transverse momentum of the W-boson inFigs. 3.11to3.13. Again, the definitions for the scale uncertainty band are listed inTable 2.1, and the lepton cuts and the jet definition are the same as inSection 3.2. In addition, we employ a merging cut ofQ_cut=20 GeV.

For theMEPS@LOvalidation inFig. 3.11, we combineLOmatrix elements for 0-, 1- and 2-jet multiplicities, obtained from Comix [70]. We can observe that we populate a much larger phase space than for a mereLOPScalculation in terms ofp^W_T. Below the merging cut (i.e.p^W_T ≲20 GeV), the scale uncertainty band is equal to the one of theLOPScalculation.

For higherp^W_T, the scale uncertainty increases corresponding to the larger uncertainty of the higher-multiplicity matrix elements, that contribute renormalisation scale uncertainties.

InFig. 3.12, we consider theMENLOPScase. We combine anNLOmatrix element for the 0-jet multiplicities withLOmatrix elements for the 1- and 2-jet multiplicities. The scale uncertainty for lowp^W_T values now features the reduced scale uncertainty, that we already have seen in theNLOPSvalidation.

The same is true in theMEPS@NLOcase depicted in3.13. A direct comparison of the scale uncertainties to theMENLOPScase is not straightforward though, as we combine NLO matrix elements for the 0- and the 1-jet multiplicity, where the virtual amplitudes are obtained from BlackHat. Hence, the 2-jet multiplicity is described at leading order through the 1-jetH-events. As such, the set-up is not a simple “upgrade” from ourMENLOPS calculation.

ForMEPS@NLO, we have also added the nominal result for the NLOPS calculation for

10⁻⁶

other maximum numbers of reweighted emissionsnPS

Figure 3.11.:The same as inFig. 3.8, but for a multi-jet merged generation with LO matrix ele-ments for 0-, 1- and 2-jet multiplicities. The uncertainty bands are calculated by reweighting the matrix element and a maximum number of emissionsnPSof parton-shower emissions.

In the upper four plots,nPS =3, thus up to three emissions are reweighted. In the lower plots,nPSis varied for comparison. Again, we find a saturation when reproducing dedicated calculations fornPS≥2, with no further improvement whennPSis increased from 2 to 3.

10⁻⁶

other maximum numbers of reweighted emissionsnNLOPS,nPS

Figure 3.12.:The same as inFig. 3.11, but for a multi-jet merged generation with one NLO matrix element for the 0-jet multiplicity, and LO matrix elements for the 1- and 2-jet multiplicities.

The uncertainty bands are calculated by reweighting the matrix element and a maximum number of emissions from the MC@NLO (nNLOPS) and the ordinaryPS(nPS). In the upper four plots,nNLOPS=1 andnPS=2, thus up to three emissions are reweighted. In the lower plots, bothnare varied for comparison. Again, we find a saturation when reproducing dedicated calculations fornNLOPS+nPS≥2, with no further improvement whennPSis increased from 1 to 2.

p^W_⊥[GeV]

other maximum numbers of reweighted emissionsn_NLOPS,n_PS

Figure 3.13.:The same as inFig. 3.12, but for a multi-jet merged generation withNLOmatrix elements for the 0- and 1-jet multiplicities. The uncertainty bands are calculated by reweight-ing the matrix element and a maximum number of emissions from the MC@NLO (n_NLOPS) and the ordinary PS (n_PS). In the upper four plots,n_NLOPS=1 and=n_PS=2, thus up to three emissions are reweighted. In the lower plots, bothnare varied for comparison. Again, we find a saturation when reproducing dedicated calculations forn_NLOPS+n_PS ≥ 2, with no further improvement whenn_PSis increased from 1 to 2.

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dσ/dpT[pb/GeV] Sherpa MEPS@LO,pp →H+0, 1, 2 jets,√s=13 TeV

CT10 variation dedicated

ME-only reweighting

0 10 20 30 40 50

H p_T[GeV]

0.9 1.0 1.1

ratiotoCV

Figure 3.14.:The transverse momentum of the H-boson in H production. A reweighting is performed to generate the CT10 uncertainty band, which is compared to a dedicated calcula-tion. The reweighting is restricted to the matrix elements of the Sherpa multi-jet calculation, merging the 0-, 1- and 2-jet multiplicities and a parton shower.

comparison. BelowQ_cut=20 GeV, it is identical to theMEPS@NLOresult, but it falls more steeply thereafter, as it describes 0 jets toNLOand 1 jet toLO, whereas ourMEPS@NLO calculation describes the 0- and 1-jet multiplicities toNLO, and 2 jets toLOaccuracy, such that large W-boson recoils can be predicted with higher (and more faithful) rates.

In all multi-jet merging validations, we find a similar behaviour with respect to the imprint of including emissions in the reweighting. Forn_NLOPS+n_PS = 2, the dedicated calculations are well reproduced, and no further improvement is found forn_NLOPS+n_PS =3.

It is noteworthy, that for theMENLOPScase we find a worse reproduction forn_NLOPS =1 andn_PS = 0 compared to theNLOPSand theMEPS@NLOcases. This originates in the fact that in the latter two cases, we enable the reweighting of emissions offS-events at all involved multiplicities, whereas in theMENLOPScase only the first of the three multiplicities is affected, because the other two are atLOand therefore do not haveS-events. Thus, the overall importance of theSemission reweighting gets restricted to the region belowQ_cut of the 1-jet configuration in theMENLOPScase.

We close this section by noting that the too small uncertainty for very lowp^W_T for varia-tions of the matrix elements only seems to be a general feature, with the form being inde-pendent of the partonic initial state. InFig. 3.14, we perform aCT10 PDFband closure test for the transverse momentum distribution of the H-boson in H production. The collider set-up is as for the W production, i.e. pp collisions at 13 TeV.LOmatrix elements for H plus 0, 1 or 2 jets are merged. The Higgs is produced via an effective coupling to two incoming gluons via a top-quark loop which is integrated out in the infinite top-mass limit. Hence, our lowest-order initial state is now two gluons instead of a weak quark doublet. However,

the deviation of the matrix-element-only reweighting band from the dedicated one for low transverse momentum is very similar to what we observed in the W-boson case, although less pronounced.