$gg\to HH$: Combined Uncertainties

In this note we discuss the combination of the usual renormalization and factorization scale uncertainties of Higgs-pair production via gluon fusion with the novel uncertainties originating from the scheme and scale choice of the virtual top mass. Moreover, we address the uncertainties related to the top-mass definition for different values of the trilinear Higgs coupling and their combination with the other uncertainties.


Introduction
Higgs-boson pair production will allow for the first time to probe the trilinear Higgs selfcoupling directly and thus to determine the first part of the Higgs potential as the origin of electroweak symmetry breaking. The dominant Higgs pair production mode is gluon fusion gg → HH that is loop-induced at leading order (LO), mediated by top and to a much lesser extent bottom loops [1]. The total gluon-fusion cross section is about three orders of magnitude smaller than the corresponding single-Higgs production cross section [2]. The dependence of the gluon-fusion cross section on the trilinear Higgs self-coupling λ around the Standard-Model (SM) value is approximately given by ∆σ/σ ∼ −∆λ/λ so that the uncertainties of the cross section are immediately translated into the uncertainty of the extracted trilinear self-coupling. In order to reduce the uncertainties of the cross section higher-order corrections are required. The next-to-leading-order (NLO) QCD corrections have first been obtained in the heavy-top limit (HTL) [3] supplemented by a large top-mass expansion [4] and the inclusion of the full real corrections [5]. Meanwhile, the full NLO calculation including the full top-mass dependence has become available [6,7,8] showing a 15%-difference to the result obtained in the HTL for the total cross section. For the distributions the differences can reach 20-30% for large invariant Higgs pair masses. The full NLO results have been confirmed by suitable expansion methods [9]. Within the HTL the next-to-NLO (NNLO) [10] and next-to-NNLO (N 3 LO) [11] QCD corrections have been derived and raise the cross section by a moderate amount of 20-30% in total. The complete QCD corrections increase the cross section by more than a factor of two. Quite recently, the full NLO result and the NNLO corrections in the HTL have been combined in a fully exclusive Monte Carlo program [12] (including the mass effects of the one-loop double-real contributions at NNLO) that is publicly available 1 . Moreover, the matching of the full NLO results to parton showers has been performed [13] so that there are complete NLO event generators.

Uncertainties
The usual renormalization and factorization scale uncertainties at NLO amount to about 10-15% [6,8] where s denotes the squared center-of-mass energy and σ tot the total cross section. The numbers in brackets are the numerical integration errors and the upper and lower percentage entries denote the combined renormalization and factorization scale uncertainties.
with K 2 ≈ 10.9 and K 3 ≈ 107.11. The scale dependence of the MS mass is treated at next-to-next-to-next-to-leading logarithmic level (N 3 LL), with the coefficient function [15] c This introduces a new scale µ t , the dependence on which induces an additional uncertainty.
For large values of the invariant Higgs-pair mass, the high-energy expansion of the virtual form factors clearly favors the dynamical scale choice µ t ∼ M HH [8,16]. The scale dependence of the total and differential Higgs-pair production cross section on µ t drops by roughly a factor of two from LO to NLO as explicitly described in Ref. [8]. The procedure to obtain the associated uncertainties is to take the envelope of the different predictions with the top pole mass and the MS mass m t (µ t ) at the scale µ t = m t and varying it between M HH /4 and M HH (i.e. a factor of 2 around the central renormalization and factorization scale µ R = µ F = M HH /2) for each M HH bin and integrating the maxima/minima eventually. At NLO we are left with the residual uncertainties related to the top-mass scheme and scale choice [7,8], A further reduction of these uncertainties can only be achieved by the determination of the full mass effects at NNLO which is beyond the state of the art 2 . Since these uncertainties are sizeable, the question arises of how to combine them with the other renormalization and factorization scale uncertainties of Eq. (1). The interplay of the different uncertainties of Eqs. (1,5) at NLO is very simple, i.e. defining the envelope of all uncertainties leads to a linear addition of the renormalization and factorization scale uncertainties of Eq. (1) and the top-mass scheme and scale uncertainties of Eq. (5), since the latter turn out to be (nearly) independent of the renormalization and factorization scale choices. This statement has been evaluated up to NLO explicitly.
The presently recommended predictions and uncertainties are based on the work of Ref. [12]. This work includes the NNLO QCD corrections in the HTL combined with the full mass effects of the LO and NLO predictions. Moreover, the work includes the full mass dependence of the one-loop double-real corrections at NNLO. The central values and residual renormalization and factorization scale uncertainties of this approach are given by [ These uncertainties will be further reduced by consistently including the novel N 3 LO corrections in the HTL [11].

Combination of Uncertainties
In order to find a proper scheme to combine the renormalization and factorization scale uncertainties of Eq. (6) and the uncertainties originating from the top-mass scheme and scale choice of Eq. (5) we have to consider the systematics of these uncertainties in more detail. Each perturbative order of the total (and differential) cross section in QCD can be decomposed in two different pieces of the corrections, where dσ n denotes the n'th-order-corrected differential cross section, dσ (i) the i'th-order correction, K SV the soft+virtual part and K (n) rem the remainder of the n'th-order corrections relative to the previous order of the cross section. The (top-mass independent) part K (i) SV is dominant for the first few orders, while the moderate (top-mass dependent) remainder K (i) rem only adds 10-15% to the bulk of the corrections of ∼ 100%. The soft+virtual corrections K (i) SV are basically the same for the (subleading) mass-effects at all orders, too. Since these pieces are part of the HTL at all perturbative orders the Born-improved [3] and FTapprox [5] approaches provide a reasonable approximation of the total cross section within 10-15% at NLO. The mass effects at a given order are thus multiplied by the same universal corrections factors, too. In the same way, the uncertainties originating from the mass effects are scaling with this dominant part of the QCD corrections. This statement is explicitly corroborated by the fact that the (Born-improved) HTL approximates the NLO cross section within about 15%, while the QCD corrections modify the cross section by close to 100%. Hence, at the state of the art, i.e. full NLO and NNLO 3 within the HTL with massive refinements, the best procedure to combine the relative uncertainties of Eqs. (5) and Eq. (6) is linearly. This will be not only the most conservative approach, but close to the final numbers in a sophisticated combined calculation of the NNLO results in the HTL with the full NLO mass effects, i.e. with a negligible mismatch of the envelope from the linear combination.

Uncertainties for different Higgs self-interactions
A variation of the trilinear Higgs coupling λ modifies the interplay between the LO box and triangle contributions that interfere destructively for the SM case. One of the basic questions is what will happen to the uncertainties for different values of λ. This can be traced back to the approximately aligned uncertainties of the triangle and box diagrams [8,18]. The renormalization and factorization scale uncertainties change by up to about 6% at NLO for large and small values of λ [17] such that the change with respect to the central uncertainties of the SM value of ∼ 10-15% is of moderate size. In a similar way the uncertainties originating from the scheme and scale choice of the top mass depend only mildly on the trilinear coupling λ. Eq. (9) shows the central NNLO F T approx predictions for the total cross section for various choices of κ λ = λ/λ SM for √ s = 13 TeV. The per-cent uncertainties display the usual factorization and renormalization scale uncertainties [19].

Conclusions
We have analyzed the combination of the usual renormalization and factorization scale uncertainties of Higgs-pair production via gluon fusion with the uncertainties originating from the scheme and scale choice of the virtual top mass in the Yukawa coupling and the propagators. Due to the observation that the latter relative uncertainties are nearly independent of the renormalization and factorization scale choices, the proper combination of the relative uncertainties is provided by a linear addition. In a second step we derived the dependence of the uncertainties related to the top-mass scheme and scale choice on a variation of the trilinear Higgs self-coupling λ. The relative uncertainties are again observed to develop only a small dependence on λ. We combined all the uncertainties for √ s = 13 TeV with the ones of the present recommendation of the LHC HXSWG, obtaining state-of-the-art predictions for Higgs pair production cross sections at the LHC including both renormalization/factorization scale and top-quark scale and scheme uncertainties.