Finite top-mass effects in gluon-induced Higgs production with a jet-veto at NNLO

Effects from a finite top quark mass on the H+n-jet cross section through gluon fusion are studied for $n=0/n\ge 1$ at NNLO/NLO QCD. For this purpose, sub-leading terms in $1/m_t$ are calculated. We show that the asymptotic expansion of the jet-vetoed cross section at NNLO is very well behaved and that the heavy-top approximation is valid at the five permille level up to jet-veto cuts of 300 GeV. For the inclusive Higgs+jet rate, we introduce a matching procedure that allows for a reliable prediction of the top-mass effects using the expansion in $1/m_t$. The quality of the effective field theory to evaluate differential K-factors for the distribution of the hardest jet is found to be better than 1-2% as long as the transverse momentum of the jet is integrated out or remains below about 150 GeV.


Introduction
The discovery of a scalar particle [1,2] whose properties are compatible with the particle causing the electro-weak symmetry breaking predicted by the Standard Model (SM), i. e. the Higgs boson, was the first observation of a new elementary particle at the Large Hadron Collider (LHC). Initially, the discovery was based on the combination of various experimental search channels. By now, sufficient significance has been reached to claim an observation alone in the two channels H → γγ [3,4] and H → ZZ * → 4l [3,5]. Some of the experimental signatures rely heavily on the analysis of particular phase-space regions of the final state particles to reduce the contamination from the background processes. In particular, in the search for H → W W * → lνlν [6,7] the huge QCD background is reduced using a veto cut (p jet T < p jet T,veto ) on jets with a large transverse momentum (p T ). The so-called jet-vetoed cross section is used to lower specifically the tt and tW background, where the top quark mainly decays to high-p T bottom quarks.
In the SM, Higgs production proceeds predominantly through gluon fusion. 1 The jet-vetoed cross section in that case has been known up to NNLO for a while [11]. The residual uncertainties associated with this observable have been subject to recent discussion [12], where the resummation of logarithms in p jet T,veto finally allowed to control these uncertainties [13][14][15][16][17][18][19][20]. Another uncertainty, which is very specific to hadronic Higgs production through gluon fusion, is induced by employing an effective theory approach, where the top quark is assumed to be infinitely heavy, to determine higher order corrections to the jet-vetoed rate. Recently, the full top-(m t ) and bottom-mass (m b ) dependence at NLO has been added to the resummed NNLO+NNLL jet-veto efficiencies [21]. 2 At NNLO, finite top-mass effects have been studied in case of the total cross section [25][26][27] so far, which have been found to be below ∼ 1% [28][29][30][31]. Differential studies on the validity of the effective field theory approach at this order in the strong coupling constant (α s ) have been considered only for Higgs quantities [32], but not for jet observables. 3 They were found to be below 3% as long as the transverse momentum of the Higgs is integrated out or is below ∼ 150 GeV.
The goal of this paper is to validate the heavy-top approximation for the Higgs production cross section with a jet-veto at NNLO. For this purpose, we determine the expansion with respect to 1/m k t , where the leading term of this series (k = 0) corresponds to the effective field theory. Additionally, we take into account the first and second non-trivial sub-leading term in the 1/m k t expansion (k = 2/4). We supplement our analysis by further jet-related quantities at NLO such as the inclusive one-jet rate and kinematical distributions of the hardest jet. For the jet-vetoed rate, we find that finite top-quark effects for realistic experimental values of the jet-veto cut (p jet T,veto ∼ 30 GeV) are numerically negligible (about five permille). Even for jet-veto cuts up to 600 GeV they remain below two percent. Therefore, we conclude that the use of the effective field theory approach for the jet-vetoed rate is fully justified. This paper is organized as follows: In Section 2, we define the jet-vetoed cross section and set-up the main ingredients of our calculation. Section 3 contains our results, including our default choices of the input parameters, some considerations at lower order and our analysis of finite top-mass effects on the Higgs+n-jet cross section for n = 0/n ≥ 1 at NNLO/NLO as well as the NLO p T and rapidity (y) distribution of the hardest jet. We conclude in Section 4.

Outline of the calculation
Considering the jet-vetoed (or 0-jet) rate for Higgs production through gluon fusion at NNLO, various contributions have to be taken into account. At LO, the cross section is identical to the total rate, since the only partonic process gg → H has no final state jets 4 , see Fig. 1 (a). Fig. 1 Fig. 3, the corresponding processes gg → ggH, gg → qqH, gq → gqH, qq → qqH, qq → ggH, qq → qqH, qq → qq H andqq →qq H (q = q) enter the calculation of the jet-vetoed cross section at NNLO. It is understood that the charge conjugated processes must be included as well.
The most complicated Feynman diagrams are of the two-loop box-type and three-looptriangle-type with massless and massive (mass m t ) internal and one massive external line (mass m H ) 6 , see Figs. 2 (e) and Figs. 1 (c), for example. Although not out of reach, the complexity of the corresponding integrals is too high for an efficient numerical evaluation. Thus, the NNLO corrections are known only in the effective theory approach with an infinitely heavy top quark (heavy-top limit). Deploying this approximation the corresponding Feynman diagrams simplify to one and two-loop level without internal masses and with an effective Higgs-gluon vertex, multiplied by a Wilson coefficient which can be evaluated perturbatively [37][38][39][40][41].
In this paper, we go beyond the heavy-top approximation and study the effects of a finite 4 Since we do not include any parton showering or hadronization, "jet" denotes a cluster of the outgoing partons throughout this paper. 5 Note that already the LO process is loop-induced. Thus, the single real emission diagrams contain one loop as well. 6 mH denotes the mass of the Higgs. (e-f) mixed real-virtual. The graphical notation for the lines is: thick straight= top quark; thin straight= light quark q ∈ {u, d, c, s, b}; spiraled= gluon; dashed = Higgs boson.
top-quark mass on the jet-vetoed rate. Therefore, we consider the expansion of the cross section with respect to 1/m t , whose leading term is given by the effective theory approach. We use the amplitudes which were calculated in Ref. [28] by applying automated asymptotic expansions [42][43][44].
In practice, we obtain the jet-vetoed Higgs cross section by removing all jet contributions σ ≥1-jet from the total rate σ tot . At NNLO this reads where we use the prime-notation of Ref. [45] to distinguish σ NLO ≥1-jet calculated with NNLO parton density functions (PDFs) from the proper NLO quantity. For the total rate we deploy the program ggh@nnlo [25,28,29,46] including the asymptotic expansion of the amplitudes in 1/m t k up to k = 6. 7 The calculation of the one-jet inclusive cross section σ ≥1-jet was carried out using the program described in Ref. [32], where we implemented the anti-k T jet-algorithm [47] to identify QCD jets. 8 Furthermore, we extended its capabilities to include sub-leading top-mass effects up to 1/m 4 t . Of course, our setup allows to calculate the exclusive Higgs+n-jet rates for n = 1 and n = 2 as well, where we work at NLO and (a) A number of checks have been performed on our results. While the p T distribution of the Higgs in the heavy-top limit was checked [32] against the fixed order part of the program HqT [48][49][50], we used the program HNNLO [24,51,52] for a numerical comparison of the jet-vetoed rate. The agreement was found to be better than one percent. At each order in the 1/m t expansion, we explicitly verified the independence of the 0-jet rate with respect to the so-called α-parameter [53,54], which allows to restrict the phase space of the Catani-Seymour dipoles [55]. The asymptotic expansion of the amplitudes as well as the program ggh@nnlo have been validated previously by the agreement of the inclusive cross section between Refs. [28] and [30].
As observed in Refs. [29,31,32], the 1/m t expansion provides no valid description for the purely quark-induced channels. Therefore, they constitute a solid, though rather minor limitation of the effective field theory, since their contribution is more than two orders of magnitude smaller than the sum of all channels. We can therefore safely disregard them from our considerations.

Input parameters
We study finite top-mass effects on Higgs+n-jet cross sections for n = 0/n ≥ 1 in the gluon fusion process at the LHC with 13 TeV center-of-mass energy. Our choice of the central factorization and renormalization scale is µ F = µ R = m H . All numbers are produced with the MSTW2008 68%CL PDFs [56] which implies that the numerical value for the strong coupling constant is taken as α s (m Z ) = 0.13939 at LO, α s (m Z ) = 0.12018 at NLO, and α s (m Z ) = 0.11707 at NNLO. We set the on-shell top quark mass to m t = 173.5 GeV.
Jets are defined using the anti-k T algorithm [47] with jet radius: R = 0.5. Unless stated otherwise, a jet is required to have transverse momentum of p jet T > 30 GeV, while we apply no cuts on the Higgs momentum. 9

Notation
To deal with the additional expansion of the cross section with respect to 1/m t we introduce the following notation: The truncation of the cross section is defined by where X denotes the order of perturbation theory and k the order at which the 1/m k t expansion of the cross section is truncated. If the index 1/m k t and the brackets are absent, it means that the cross section is not truncated and, consequently, dσ X denotes the cross section with exact top-mass dependence. Here and in what follows we imply that all cross sections are reweighted by the exact top-mass dependence at LO: where dσ denotes the unweighted cross section and σ LO the Born-level cross section for gg → H.
In order to analyse the perturbative corrections to the cross section, we define the K-factor On the right hand side of this definition, it is understood that dσ(b) is integrated over all kinematical variables except the set b, where we consider b = {p jet T,1 } and b = {y jet 1 } (i. e., transverse momentum and rapidity distributions of the hardest jet). For example, K NLO 0 (p H T ) is the NLO K-factor in the heavy-top limit of the p T distribution of the Higgs which has been found to be valid at the 2-3% level for p H T 150 GeV [32]. 10 Using the 1/m t expansion, we will study whether this observation can be expected to carry over also to jet quantities. Fig. 4 (a) shows the NLO jet-vetoed cross section as a function of the Higgs mass. We applied a veto of p jet T,veto = 30 GeV on the jet transverse momenta. At this order, the exact dependence on the top-quark mass is known (solid curve). 11 Comparing it to the 9 We checked that our results directly generalize to experimentally applied jet definitions for this process which usually imply p jet T > 25-30 GeV and a rapidity cut [6,7]. 10 Note that in Appendix A we extend the analysis of the transverse momentum distribution in Ref. [32] by considering an additional term in the 1/m k t expansion (k = 4). 11 To obtain the NLO total cross section with exact top-mass dependence we employed the code SusHi [57].   expansion of the cross section up to 1/m k t for k = 0 (dotted curve), k = 2 (dashed curve) and k = 4 (dash-dotted curve), we can assess the quality of the 1/m t expansion. Clearly, its convergence starts deteriorating once the Higgs mass exceeds the top-quark mass.

Lower order results
In general, the aim of our analysis is to obtain accurate predictions including mass effects for the various jet observable considered in this paper and to use them to estimate the mass corrections with respect to the effective field theory (EFT). The deviation of the higher orders in the asymptotic expansion from the leading term indicates the validity of the EFT to approximate the cross section in the full theory. For this purpose, we normalize all curves to the 1/m 0 t approximation in Fig. 4 (b). For small values of m H , the mass effects are at the percent level. While the expansion up to 1/m 2 t remains extremely close to the full result over the whole mass range, the 1/m 4 t corrections reduce the cross section significantly towards larger values of m H . Assuming the exact cross section was not known, which is the case at NNLO, we would therefore estimate the uncertainty of the mass corrections on the EFT result to be below 5% for m H 200 GeV. Fortunately, all orders of the 1/m t expansion coincide to a very good accuracy at m H 125 GeV.
In Fig. 5 (a), we study the top-mass corrections to the NLO cross section as a function of the jet-veto cut for m H = 125.6 GeV. The horizontal lines denote the total inclusive cross sections, which correspond to p jet T,veto → ∞. The agreement between the curves is remarkable. While the differences are at the permille-level for small jet-veto cuts they remain below 2.5% even at p jet T,veto = 600 GeV. Again, the asymptotic expansion leads to a proper estimation of the mass effects, not underestimating the uncertainty induced by   the heavy-top approximation with respect to exact one. Therefore, the 1/m t terms can be expected to yield a conservative validation of the EFT as well at NNLO. The reason that the 1/m t expansion of the jet-vetoed rate is well behaved even beyond the 2 m t threshold is the phase-space suppression, which strongly reduces contributions from hard jets. However, the 1/m 4 t term receives unjustified large contribution from p jet T 400 GeV. In that region, σ NLO veto , [σ NLO veto ] 1/m 0 t as well as [σ NLO veto ] 1/m 2 t develop a flat behavior, which is expected from phase-space suppression, while [σ NLO veto ] 1/m 4 t grows almost linearly. This reveals that the convergence of the amplitudes at 1/m 4 t in the large-p T tail is broken. The previous observations are in direct analogy to the total cross section. In this case, the bulk of the cross section originates from the region √ s 2 m t , in which the asymptotic expansion is well behaved [29]. Nevertheless, the 1/m 4 t term receives huge contributions as √ s 2 m t [29], since the convergence of the amplitudes is spoiled at large energies. In fact, looking at the total cross sections in Fig. 5 (a), it is obvious that the leading and first sub-leading term in the asymptotic expansion compare better to the exact result than when including the 1/m 4 t terms. 12 To obtain the inclusive Higgs+jet cross section a cut p jet T > p jet T,min is applied, which removes the bulk of the well behaved soft jets and, therefore, enhances the contribution from the problematic large-p T region. Fig. 6 (a)  the deviation between the curves relative to 1/m 0 t is quite large (∼ 27%), convergence of the asymptotic expansion is completely lost at large values of p jet T,min . Thus, we cannot use the ordinary 1/m t expansion to determine a sensible estimate of the mass effects on the inclusive Higgs+jet rate.
However, the same problematic effects contribute to the total inclusive cross section σ tot , as we have seen before. In this case, a matching to the high-energy limit was performed as described in Ref. [29] to control the region √ s > 2 m t . Similarly, a matching of the inclusive Higgs+jet cross section to the p T → ∞ limit would temper unjustified effects from high-p T jets. Let us assume this matched cross section was known and call it σ ≥1-jet, matched . Given the fact that the total cross section can be viewed as the integral over the p T distribution and the asymptotic expansion in the small-p T region works almost perfectly, the following relation should be valid up to a very good precision as long as p jet T,min remains at moderate values: 13 where the primed LO quantity is calculated with NLO parton distributions, as defined in Section 2. This equation allows us to determine the matched inclusive Higgs+jet cross  section by using LO PDFs for all quantities:

Top−Expansion
where we defined the starred NLO cross section to be evaluated with LO PDFs. Fig. 6 (b) shows the matched cross section as defined in Eq. (6). It is very impressive how close all curves are to the exact result with respect to the unmatched case in Fig. 6 (a).
In Fig. 7, the matched predictions of Fig. 6 (b) are normalized to unmatched cross section in the heavy-top limit (dotted curve in Fig. 6 (a)). Comparing first the matched cross sections to the exact curve, their overall agreement is remarkable ( 5% for p jet T,min ≤ 150 GeV). In that region, they are successively closer to the exact result, as k increases. The deviation of the EFT result from the matched curves on the other hand allows its validation at the 3 − 10% level for p jet T,veto ∈ [30, 100] GeV. Thus, with the definition of the matched cross section we recovered the ability to validate the heavy-top limit for the inclusive Higgs+jet rate. This will prove useful at NLO, where the exact result is not available.
There are cases in our analysis where the reliability of the 1/m t expansion appears to be exceptionally good. This happens when the 1/m 4 t corrections become negligible and, consequently, the expansions up to 1/m 2 t and up to 1/m 4 t almost coincide. We already observed this twice: In Fig. 4 (b) around m H = 125 GeV and in Fig. 7 for p jet Overall, our observations so far are encouraging to study the behavior of the 1/m t expansion at higher orders to estimate the range of applicability of the heavy-top limit for jet observables.

Jet-veto at NNLO
We are now ready to analyze the mass effects on the jet-vetoed rate at NNLO, which is the central observable of our study. Fig. 8 (a) shows the truncation of the cross section with a jet-veto at 1/m k t for k = 0 (dotted), k = 2 (dashed) and k = 4 (dash-dotted) as a function of the jet-veto cut. At small values of p jet T,veto , we observe an excellent convergence of the asymptotic expansion, i. e. the cross section is almost independent of the order of expansion in 1/m t . For example, the spread of the curves is about 0.5% at p jet T,veto = 30 GeV, see Fig. 8 (b), where all curves are normalized to the EFT (k = 0). In fact, [σ NNLO veto ] 1/m k t behaves even better with increasing k than the total inclusive cross section, where a matching to the high-energy limit is required [29] to alleviate the unjustified large effects from hard jets. These effects do not appear in case of the jet-vetoed cross section. More precisely, they explicitly cancel between σ NNLO tot and σ NLO ≥1-jet in Eq. (1). At larger values of the jet-veto cut, the deviation between the curves in Fig. 8 Figure 9: Higgs+0-jet cross section at NNLO including terms up to 1/m k t as a function of m H normalized to heavy-top limit (k = 0) for p jet T,veto = 30 GeV. Dotted/dashed/dash-dotted: k = 0/2/4. we found at NLO, the asymptotic expansion of the cross section is well behaved even for a jet-veto beyond the 2 m t threshold, because contributions from large-p T jets are suppressed by phase-space. 14 In a large number of beyond standard model (BSM) theories, additional scalar particles are predicted, e. g. a second (heavier) CP-even Higgs boson. Therefore, we investigate the quality of the m t → ∞ approximation for more general Higgs masses. Fig. 9 shows the 1/m k t expansion (k = 0/2/4) of the jet-vetoed NNLO cross section normalized to the EFT result (k = 0) as a function of m H . Indeed, the effective field theory yields a valid approximation at the one-percent level for m H 150 GeV. At larger Higgs masses the top-mass effects become sizable and the uncertainty induced by the heavy-top limit increases to ∼ 6 (25)% at m H = 200 (300) GeV In summary, for a SM Higgs boson of mass 125.6 GeV it is fully justified to trust the effective field theory approach to determine radiative corrections to the jet-vetoed cross section at NNLO. It is advisable though to account for the full mass dependence at LO through reweighting, as it is common practice and done in our analysis. Furthermore, our results should directly generalize to the resummed jet-vetoed cross section at NNLO+NNLL [16] evaluated in the EFT, since the resummation of Sudakov logarithms from soft-gluon emissions is predominantly described by process independent QCD effects.

Inclusive Higgs+jet rate at NLO
For the LO Higgs+jet cross section, the 1/m t expansion provides no proper approximation of the top-mass effects, as we have seen in Section 3.3. The reason for this are unjustified large contributions from high-p T jets at higher orders in 1/m t . In order to obtain a reliable estimate of the mass effects on the LO Higgs+jet rate, we defined the matched cross section in Eq. (6). Moving to α 4 s , we encounter the same problems, which can be seen from the dash-dotted curve (expansion up to 1/m 4 t ) in Fig. 8 (a) at p jet T 400 GeV, for example. Consequently, not only the Higgs+jet cross section at LO is affected, but also at NLO. This is why we define the matched inclusive Higgs+jet rate at NLO accordingly: where the starred NNLO cross section is calculated with NLO PDFs.
The matched cross section expanded up to different orders in 1/m k t is shown in Fig. 10 (a) (k = 0/2/4). All three curves are very close, extending the validity of the asymptotic expansion to significantly larger values of p jet T,min than in the unmatched case, see Fig. 10 (b). Fig. 11 shows the improved matched predictions of Fig. 10 (a) normalized to unmatched cross section in the heavy-top limit (dotted curve of Fig. 10 (b)   up to 1/m 2 t ) and dashed-dotted curve (expansion up to 1/m 4 t ) to approximate the exact mass effects to better than one percent. Therefore, as long as the minimum jet-p T cut remains at moderate values (p jet T,min 100 GeV) the definition of the matched cross section in Eq. (6) and Eq. (7) allows us to determine a reliable prediction of the inclusive Higgs+jet rate at LO and NLO, respectively. Furthermore, comparing the matched curve at 1/m 4 t to the unmatched EFT result, we validate the heavy-top approximation at the level of 1-2% for p jet T,min ≤ 100 GeV. This result shows that the EFT, in fact, works better in the problematic high-p T region than the corresponding sub-leading 1/m t terms, which are far apart in the unmatched case, see Fig. 10 (b). This is very similar to what was found for the total cross section [29], where it was argued that in the heavy-top limit (k = 0) problematic terms ( √ s/m t ) k vanish, which spoil the convergence of the asymptotic expansion (k > 0) in the high-energy region. Also in this case the matching to the high-energy limit revealed that the unmatched EFT result is valid at the percent level.
However, at larger values (p jet T,min > 100 GeV), the asymptotic expansion starts deteriorating significantly already for the matched cross section in Fig. 11. Therefore, the uncertainty of the EFT due to mass effects in that region is quite sizable. For comparison, the deviation of the EFT from the exact curve is 12(30)% for p jet T,min = 100(200) GeV at LO, see Fig. 7.

Distributions of the hardest jet
Finally, let us consider kinematical distributions of the hardest jet. Fig. 12 shows the p T -dependent K-factors K NLO k ≡ K NLO k (p jet T,1 ) of the cross section up to 1/m k t as defined in Eq. (4) with variable scales In the gg-channel, all three K-factors are almost identical. However, the QCD corrections to the subleading mass terms in the qg-channel behave quite differently to the EFT result once p jet T,1 100 GeV. In the sum of both channels though, the difference remains below ∼ 1.5% for p jet T,1 < 150 GeV, and reaches 6% at p jet T,1 = 300 GeV. Therefore, our results turn out to be quite similar to what was already found for the p T distribution of the Higgs K NLO k (p H T ) [32], yet the asymptotic behavior is slightly improved for the hardest jet. For comparison, we give an updated result for K NLO k (p H T ) up to 1/m 4 t in Appendix A, which shows that K NLO 4 behaves quite differently in the two cases at high p T . to approximate the exact top-mass effects is better that one percent in that region.
The situation for the rapidity distribution of the hardest jet is more involved. The problem is that in the central region the 1/m 4 t term receives unjustified large effects from hard jets, which spoil the convergence of the asymptotic series. Unfortunately, it is not possible to determine a matched cross section in this case, similarly to what we do for the inclusive Higgs+jet cross section. Instead, we introduce a cut p jet T < p jet T,max which simply removes the problematic high-p T jets. This cut is of course arbitrary, therefore, we choose three different values: p jet T,max = 200, 400, 600 GeV.
In fact, the contribution to the y jet 1 distribution from jets with p jet T > 600 GeV should be completely negligible due to phase-space suppression. This is what we observe for the EFT result, but not for the subleading terms in the 1/m t expansion, see Fig. 13, which shows K NLO k ≡ K NLO k (y jet 1 ) for p jet T,max = 200, 400, 600 GeV and without cut. Clearly, the asymptotic behavior in the central region is broken without a cut. It works pretty well though once we apply an upper cut on the jets. The EFT result is almost identical (< 0.5%) in the lower two plots and receives no unjustified large effects from high-p T jets. Therefore, it is legitimate to estimate the quality of the EFT without a cut from the results for p jet T,max = 600 GeV, which we deduce to be better than 2% in the central region and even below one percent in the forward region (y jet 1 > 2.5). In conclusion, the behavior of the K-factors of the hardest jet distributions suggest that the QCD corrections can be safely calculated in the heavy-top approximation. The accuracy remains within 1.5% (6%) below p jet T,1 = 150 GeV (p jet T,1 = 300 GeV) and for p T -integrated quantities at the percent level.

Conclusions
Finite top-mass effects in the gluon fusion process have been studied. The quality of the effective field theory to describe the exact cross section was estimated using subleading terms in 1/m t . They have been evaluated for various jet quantities, namely, the NNLO cross section with a jet veto, the inclusive Higgs+jet rate at NLO and the NLO K-factors of jet distributions.
The corrections of a finite top-mass to the jet-vetoed rate are negligible and the quality of the effective field theory to describe this quantity even at large values of the jet-veto cut is remarkable. Unjustified large contribution from hard jets were found to spoil the convergence of the asymptotic expansion in case of the inclusive Higgs+jet cross section. Only a matching procedure involving the total inclusive cross section allowed for a reliable prediction of this quantity and the estimation of the mass effects from the 1/m t expansion. The EFT was then found to be valid even without the matching at the 1-2% level for jet definitions with a minimal transverse momentum cut lower than 100 GeV. For large values though, the asymptotic expansion of the matched result becomes unreliable. Therefore, also the uncertainty induced by the EFT is large, deviating by 30% from the exact result already at LO for a minimal jet cut of 200 GeV.
Also the perturbative corrections to distributions of the hardest jet turned out to have a rather mild top-mass dependence. For the transverse momentum distribution, the procedure of correcting the LO prediction including the exact top-mass dependence by the K-factor evaluated in the EFT provides an excellent approximation to the full NLO result, valid to better than 1.5(5)% for p jet T < 150(300) GeV. The K-factor of the rapidity distribution determined in the heavy-top limit was validated at the 1-2% level.
We have checked that our results hold also for different machine energies at the LHC. The accuracy of the effective field theory approach is better than the uncertainty on the cross section induced by the PDFs and missing higher order QCD corrections.
Acknowledgements. We would like to thank Pier Francesco Monni for useful comments on the manuscript. We are indebted to Robert Harlander for fruitful discussions, his comments on the manuscript and the private version of his code ggh@nnlo that he provided for our study. The work of TN was supported by BMBF contract 05H12PXE. MW was supported by the European Commission through the FP7 Marie Curie Initial Training Network "LHCPhenoNet" (PITN-GA-2010-264564).
Additionally to Ref. [32], we determine the K-factor expanded up to 1/m 4 t . Our result perfectly confirms the conclusions drawn in that paper, since K 4 lies just right between K 0 and K 2 for most transverse momenta.