Gluon Fusion

Gluon fusion predictions for the Higgs boson p_T spectrum have been provided by theorists [104, 105] for all combinations of theκcandκbvalues given in Table 6.1. These predictions can be

κ_c: -10 -5 0 1 5 10 κ_b: -2 -1 0 1 2

Table 6.1.|Values forκcandκbfor which gluon fusion predictions have been provided.

used to determine the coefficientsc^qq_ggF and c^qq_ggF⁰,q,q⁰∈ {t,b,c}, introduced in Eq. (6.7): the predictions allow to construct an overconstrained set of equations of the form

σggF=c^tt_ggF·κ²_t +c^tb_ggF·κ_tκ_b+c^tc_ggF·κ_tκ_c+c^bc_ggF·κ_bκ_c

+c^bb_ggF·κ²_b+c^cc_ggF·κ²_c (6.9)

for each bin of the differential p^H_T distribution. These sets of equations were solved using a method of least squares, resulting in the above-mentioned coefficients. The coefficientsc^qq_ggF, which correspond to the cross section contributions from gluon fusion processes with a quark of typeqin the gluon fusion loop, are generally positive. The interference termsc^qq_ggF⁰, on the other hand, can give both positive and negative contributions to the cross section. Numerical values for these coefficients are shown in Figure 6.1. Because thet-quark gluon fusion contribution is much larger than all other contributions, all contributions are scaled up for this plot such that the visibility of their features is improved. In allp_T^Hbins, the dominant contribution to the ggF cross section results from thet-quark loop, i.e. the coefficientc^tt_ggF. The interferences of thet-quark gluon fusion loop with theb- andc-quark loop, denoted by c^tb_ggF and c^tc_ggF, are comparatively small, but not negligible. They are negative at p_T^Hvalues below 80 GeV. The coefficientsc^bb_ggF, c^cc_ggF, andc^bc_ggF, are very small.

The gluon fusion predictions for the individual combinations ofκ_candκ_bare computed using RadISH; see Section 2.6.4. For the renormalization, factorization and resummation scale, a cen-tral value ofm_H/2 is assumed. The PDF set used for these predictions is PDF4LHC15_nnlo_mc.

In order to improve the accuracy of the predictions for the p^H_T spectra, they are multiplied with

[GeV]

pT

0 20 40 60 80 100 120 140

[pb/GeV] Tp∆c/∆

−1

−0.5 0 0.5

1 gg →H

tt bb x 100 cc x 10.000 tb x 10 tc x 10 bc x 100

→H

q q

x 10

→H

b b

x 100

→ H

c c

Figure 6.1. |Coefficients for gluon fusion contributions from different quark types in the gluon fusion loop in Eq. 6.9. Additionally, cross section contributions from thecc¯→H andbb¯ →H Higgs boson production modes, described in Section 6.2.2, are shown. Coefficients with relatively small values are scaled up in order to facilitate a comparison.

a scaling factor

σ_N3LO/σ_RadISH(κ_b=1, κ_c=1)=1.165,

whereσ_RadISH(κ_b=1, κ_c=1) denotes the total cross section of the RadISH prediction at SM parameter values, andσ_N³_LOto the inclusive cross section of the state-of-the-art gluon fusion prediction with N³LO accuracy for the inclusive cross section, see Section 2.6.4.

The predictions are given for inclusive Higgs boson production in the gluon fusion production mode, its decay was not taken into account at that stage. Therefore, the SMH→γγbranching ratio has been multiplied to the predicted cross sections. In order to achieve comparability with the observedp_T^Hspectrum, which was measured in a fiducial phase-space defined by the kinematics and isolation of the photons (see Section 5.3.2), the inclusive gluon fusion predictions need to be corrected for the difference in considered phase-space volume. For this, it is assumed that the correction does not depend on the value ofκ_c andκ_b. Ideally, one would derive such acceptance corrections as a function of κ_c and κ_b, but the calculations necessary for this are lacking. The acceptance corrections are determined using the nominal NNLOPS ggF simulation,

see Section 2.6.4, and are shown as a function of p_T^Hin Figure 6.2.

0 20 40 60 80 100 120 140

[GeV]

pT

0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55

Fiducial Correction

Figure 6.2. |Phase-space corrections to be applied to RadISH predictions as derived from the NNLOPS gluon fusion simulation. The fiducial selection includes requirements on the photon transverse momentum, photon pseudorapidity, and photon isolation. The shown error bars correspond to a combination of PDF uncertainties and uncertainties due to missing higher orders of QCD calculation.

The upper and lower limits of the provided prediction for the p_T^H spectrum are 0.5 GeV and 245 GeV, respectively. Based on the nominal NNLOPS ggH simulation, this has been extrapolated to 0 GeV and 250 GeV, respectively, in order to be consistent with bin boundaries in the measured p^H_T distribution.

Uncertainties

The gluon fusion predictions have uncertainties due to the imperfect knowledge of the proton PDF as well as due to missing higher-orders of the QCD calculation. The phase-space corrections for the gluon fusion prediction are subject to these types of uncertainties as well. Correlation between the uncertainties on the phase-space corrections and the cross section predictions in an inclusive phase-space volume are not known; they are treated as uncorrelated in this measurement.

Perturbative uncertainties for the gluon fusion prediction are estimated through variations of the renormalization scale µ_r, the factorization scaleµ_f, and the resummation scale Qaround the central value ofm_H/2. The correlation scheme between the three scale variations has been chosen such that the resulting uncertainty is maximal, corresponding to a simultaneous variation of the renormalization and factorization scales. The resummation scale is varied independently

of the other two scales. In Figure 6.3a the relative effects of corresponding scale variations by factors of 1/2 and 2 are shown. For each value ofκ_bandκc, the scale variations lead to slightly different relative differences between the nominal cross sections and the cross sections based on varied scales. Among those different values for the relative differences from scale variations at differentκ_b and κ_c values, the maximal value is chosen in order to obtain a conservative uncertainty estimate.

The PDF-related uncertainty of the RadISH gluon fusion prediction is estimated using the NNLOPS ggF simulation, which is warranted because the same PDF set is used in both predic-tions. Uncertainties on the proton PDF are encapsulated in 30 eigenvectors of the covariance matrix of the parameters that describe the PDFs [184, 211]. By repeating the PDF fit on the input data with varied parameters according to the direction in parameter space of these orthogonal eigenvectors, variations of the best-fit PDF set are derived and are accessible in the NNLOPS ggF sample. The relative differences between the differential cross section as computed with the nominal PDF set and the differential cross sections as computed with the varied PDF set serve as input to the overall PDF uncertainty on the measurement ofκ_candκ_b. Not all of these variations, however, lead to significant changes in the differential cross section. The three PDF eigenvectors resulting in the largest deviations from the nominal prediction enter the analysis; see Figure 6.3b for the corresponding relative differences which are used as uncertainties in theκcandκ_bfit.

The correction factors that are used to render the ggF predictions comparable to the measured differential cross section, which are given in a fiducial phase-space volume, are derived from the nominal NNLOPS gluon fusion simulation. They are subject to theoretical uncertainties related to missing higher orders of the QCD calculation and to PDF uncertainties. The combined uncertainties are shown as error bars in Figure 6.2.

Bringing the three considered uncertainty contributions to the gluon fusion differential cross section in a fiducial volume into relation to each other, the QCD-scale-related uncertainties dominate, having relative uncertainties of approximately 20 % at larger values of p_T^Hand approx-imately 10 % at low p_T^H. The PDF uncertainties on the differential cross section in an inclusive phase-space volume and the combined uncertainties on the phase-space corrections are of the orderO(1 %).

0 20 40 60 80 100 120 140 [GeV]

pT

−0.3

−0.2

−0.1

0 0.1 0.2

σ/σ∆ 0.3 down µF

&

µR

F up µ &

µR

down Q

up Q

(a)

0 20 40 60 80 100 120 140

[GeV]

pT

−0.01

0 0.01 0.02 0.03

σ/σ∆ Eigenvector 4

Eigenvector 5 Eigenvector 17

(b)

Figure 6.3.|(a) QCD-scale-related uncertainty contributions to the gluon fusion differential cross section.

The renormalization and factorization scalesµ_randµ_fare varied simultaneously, while the resummation scaleQis varied independently. (b) PDF-related uncertainty contributions to the gluon fusion differential cross section.