Predictions for Higgs production at the Tevatron and the associated uncertainties

We update the theoretical predictions for the production cross sections of the Standard Model Higgs boson at the Fermilab Tevatron collider, focusing on the two main search channels, the gluon-gluon fusion mechanism $gg \to H$ and the Higgs-strahlung processes $q \bar q \to VH$ with $V=W/Z$, including all relevant higher order QCD and electroweak corrections in perturbation theory. We then estimate the various uncertainties affecting these predictions: the scale uncertainties which are viewed as a measure of the unknown higher order effects, the uncertainties from the parton distribution functions and the related errors on the strong coupling constant, as well as the uncertainties due to the use of an effective theory approach in the determination of the radiative corrections in the $gg \to H$ process at next-to-next-to-leading order. We find that while the cross sections are well under control in the Higgs--strahlung processes, the theoretical uncertainties are rather large in the case of the gluon-gluon fusion channel, possibly shifting the central values of the next-to-next-to-leading order cross sections by more than $\approx 40%$. These uncertainties are thus significantly larger than the $\approx 10%$ error assumed by the CDF and D0 experiments in their recent analysis that has excluded the Higgs mass range $M_H=$162-166 GeV at the 95% confidence level. These exclusion limits should be, therefore, reconsidered in the light of these large theoretical uncertainties.


Introduction
We are approaching the exciting and long awaited times of discovering the "Holy Grail" of nowadays particle physics: the Higgs boson [1,2], the remnant of the mechanism breaking the electroweak gauge symmetry and at the origin of the particle masses. Indeed, the Large Hadron Collider (LHC) has started to have its first collisions [3], although at energies and with instantaneous luminosities yet far from those which would be required for discovery. Most importantly in this context, the CDF and D0 experiments at the Fermilab Tevatron collider have collected enough data to be sensitive to the Higgs particle of the Standard Model. Very recently, the two collaborations performed a combined analysis on the search for this particle and excluded at the 95% confidence level the possibility of a Higgs boson in the mass range between 162 and 166 GeV [4]; this exclusion range is expected to increase to 159 GeV ≤ M H ≤ 168 GeV [5]. We are thus entering a new era in the quest of the Higgs particle as this is the first time that the mass range excluded by the LEP collaborations in the late 1990s, M H ≥ 114.4 GeV [6], is extended.
However, in contrast to the Higgs LEP limit which is rather robust, as the production cross section is mainly sensitive to small electroweak effects that are well under control, the Tevatron exclusion limit critically depends on the theoretical prediction for the Higgs production cross sections which, at hadron colliders, are known to be plagued with various uncertainties. Among these are the contributions of yet uncalculated higher order corrections which can be important as the strong coupling constant α s is rather large, the errors due to the folding of the partonic cross sections with the parton distribution functions (PDFs) to obtain the production rates at the hadronic level, and the errors on some important input parameters such as α s . It is then mandatory to estimate these uncertainties in order to have a reliable theoretical prediction for the production rates, that would allow for a consistent confrontation between theoretical results and experimental measurements or exclusion bounds 1 . The present paper critically addresses this issue.
At the Tevatron, only two production channels are important for the Standard Model Higgs boson 2 . In the moderate to high mass range, 140 GeV < ∼ M H < ∼ 200 GeV, the Higgs boson decays dominantly into W boson pairs (with one W state being possibly off mass-shell) [8] and the main production channel is the gluon-gluon fusion mechanism gg → H [9] which proceeds through heavy (mainly top and, to a lesser extent, bottom) quark triangular loops. The Higgs particle is then detected through the leptonic decays of the W bosons, H → W W ( * ) → ℓ + νℓ −ν with ℓ = e, µ, which exhibits different properties than the pp → W + W − → ℓℓ plus missing energy continuum background [10].
It is well known that the gg → H production process is subject to extremely large QCD radiative corrections [11][12][13][14][15][16][17][18][19]. In contrast, the electroweak radiative corrections are much smaller, being at the level of a few percent [20][21][22], i.e. as in the case of Higgs production at the LEP collider. For the corrections due to the strong interactions, the K-factor defined as the ratio of the higher order (HO) to the lowest order (LO) cross sections, consistently evaluated with the α s value and the PDF sets at the chosen order, is about a factor of two at next-to-leading order (NLO) [11,12] and about a factor of three at the next-to-next-to-leading order (NNLO) [14][15][16]. In fact, this exceptionally large Kfactor is what allows a sensitivity on the Higgs boson at the Tevatron with the presently collected data. Nevertheless, the K-factor is so large that one may question the reliability of the perturbative series, despite of the fact that there seems to be kind of a convergence of the series as the NNLO correction is smaller than the NLO correction 3 .
In the low mass range, M H < ∼ 140 GeV, the main Higgs decay channel is H → bb [8] and the gg fusion mechanism cannot be used anymore as the gg → H → bb signal is swamped by the huge QCD jet background. The Higgs particle has then to be detected through its associated production with a W boson qq → W H [23] which leads to cleaner ℓνbb final states [24]. Additional topologies that can also be considered in this context are qq → W H with H → W W * → ℓℓνν or the twin production process qq → ZH with the subsequent decays H → bb and Z → νν or ℓ + ℓ − . Other production/decay channels are expected to lead to very low rates and/or to be afflicted with too large QCD backgrounds.
At the Tevatron, the Higgs-strahlung processes qq → V H with V = W, Z receive only moderate higher order corrections: the QCD corrections increase the cross sections by about 40% at NLO [25] and 10% at NNLO [26], while the impact of the one-loop electroweak corrections is small, leading to a ≈ 5% decrease of the cross sections [27]. Thus, in contrast to the gluon-gluon fusion process, the production cross sections in the Higgs-strahlung processes should be well under control.
In this paper, we first update the cross sections for these two main Higgs production channels at the Tevatron, including all known and relevant higher order QCD and electroweak corrections and using the latest MSTW2008 set of parton distribution functions [28]. For the the gg → H process, this update has been performed in various recent analyses [18,20] and, for instance, the normalized Higgs production cross sections used by the CDF/D0 collaborations in their combined analysis [5] are taken from these references. Such an update is lacking in the case of the Higgs-strahlung production channels qq → V H and, for instance, the normalised cross sections used by the Tevatron experiments [5] are those given in Ref. [30] which make use of the old MRST2002 set of PDFs [31], a parametrisation that was approximate as it did not include the full set of evolved PDFs at NNLO. For completeness, we also update the cross sections for the two other single Higgs production channels at hadron colliders: the weak boson fusion pp → qqH [32,33] and the associated production with top quark pairs pp → ttH [34,35]. These channels play only a minor role at the Tevatron but have also been included in the CDF/D0 analysis [5].
A second goal of the present paper is to investigate in a comprehensive way the impact of all possible sources of uncertainties on the total cross sections for the two main Higgs production channels. We first reanalyse the uncertainties from the unknown higher order effects, which are usually estimated by exploring the cross sections dependence on the renormalisation scale µ R and the factorisation scale µ F . In most recent analyses, the two scales are varied within a factor of two from a median scale which is considered as the most natural one. We show that this choice slightly underestimates the higher order effects and we use a criterion that allows a more reasonable estimate of the latter: the range of variation of the two scales µ R and µ F should be the one which allows the uncertainty band of the NLO cross section to match the central value of the cross section at the highest calculated order. In the case of gg → H, for the uncertainty band of the NLO cross section to reach the central result of the NNLO cross section, a variation of µ R and µ F within a factor of ∼ 3 from the central value µ R = µ F = M H is required. When the scales are varied within the latter range, one obtains an uncertainty on the NNLO cross section of ≈ 20%, which is slightly larger than what is usually assumed.
We then discuss the errors resulting from the folding of the partonic cross sections with the parton densities, considering not only the recent MSTW set of PDFs as in Refs. [18][19][20], but also two other PDF sets that are available in the literature: CTEQ [36] and ABKM [37]. In the case of the cross section for the gg → H process at the Tevatron, we find that while the PDF uncertainties evaluated within the same scheme are moderate, as also shown in Refs. [18][19][20], the central values of the cross sections obtained using the three schemes can be widely different. We show that it is only when the experimental as well as the theoretical errors on the strong coupling constant α s are accounted for that one obtains results that are consistent when using the MSTW/CTEQ and ABKM schemes. As a result, the sum of the PDF+∆ exp α s and ∆ th α s uncertainties, that we evaluate using a set-up recently proposed by the MSTW collaboration to determine simultaneously the errors due to the PDFs and to α s , is estimated to be at least a factor of two larger than what is generally assumed.
Finally, a third source of potential errors is considered in the gg fusion mechanism: the one resulting from the use of an effective field theory approach, in which the loop particle masses are assumed to be much larger than the Higgs boson mass, to evaluate the NNLO contributions. While this error is very small in the case of the top-quark contribution, it is at the percent level in the case of the b-quark loop contribution at NNLO QCD where the limit M H ≪ m b cannot be applied. This is also the case of the three-loop mixed QCDelectroweak radiative corrections that have obtained in the effective limit M H ≪ M W , which lead to a few percent uncertainty. In addition, an uncertainty of about 1% originates from the freedom in the choice of the input b-quark mass in the Hgg amplitude. The total uncertainty in this context is thus not negligible and amounts to a few percent.
We then address the important issue of how to combine the theoretical errors originating from these different sources. Since using the usually adopted procedures of adding these errors either in quadrature, as is done by the experimental collaborations for instance, or linearly as is generally the case for theoretical errors, lead to either an underestimate or to an overestimate of the total error, we propose a procedure that is, in our opinion, more adequate. One first determines the maximal and minimal values of the cross sections obtained from the variation of the renormalisation and factorisation scales, and then estimate directly on these extrema cross sections the combined uncertainties due to the PDFs and to the experimental and theoretical errors on α s . The other smaller theoretical uncertainties, such as those coming from the use of the effective approach in gg → H, can be then added linearly to this scale, PDF and α s combined error.
The main result of our paper is that, when adding all these uncertainties using our procedure, the total theoretical error on the production cross sections is much larger than what is often quoted in the literature. In particular, in the case of the most sensitive Higgs production channel at the Tevatron, gg → H → ℓℓνν, the overall uncertainty on the NNLO total cross section is found to be of the order of ≈ −40% and ≈ +50%. This is significantly larger than the uncertainty of ≈ ±10% assumed in earlier studies and adopted in the CDF/D0 combined Higgs search analysis. As a result, we believe that the exclusion range given by the Tevatron experiments for the Higgs mass in the Standard Model, 162 GeV ≤ M H ≤ 166 GeV, should be reconsidered in the light of these results.
The rest of the paper is organised as follows. In the next section we outline our calculation of the Higgs production cross sections at the Tevatron in the gluon-gluon fusion and Higgs-strahlung processes. In section 3, we focus on the gluon-gluon fusion channel and evaluate the theoretical uncertainties on the cross section from scale variation, PDF and α s uncertainties as well as from the use of the effective theory approach for the NNLO contributions. Section 4 addresses the same issues for the associated Higgs production channels. The various theoretical errors are summarized and combined in section 5 and their implications are discussed. A brief conclusion is given in section 6.

The production cross sections
In this section, we summarize the procedure which allows to obtain our updated central or "best" values of the total cross sections for Higgs production at the Tevatron in the Standard Model. We mainly discuss the two dominant channels, namely the gluon-gluon fusion and Higgs-strahlung, but for completeness, we mention the two other production channels: vector boson fusion and associated Higgs production with top quark pairs. The production rate for the gg → H + X process, where X denotes the additional jets that appear at higher orders in QCD, is evaluated in the following way. The cross section up to NLO in QCD is calculated using the Fortran code HIGLU [38,39] which includes the complete set of radiative corrections at this order, taking into account the full dependence on the top and bottom quark masses [12]. The contribution of the NNLO corrections [14][15][16] is then implemented in this program using the analytical expressions given in Ref. [15]. These corrections have been derived in an effective approach in which only the dominant top quark contribution is included in the infinite top quark mass limit but the cross section was rescaled by the exact m t dependent Born cross section, an approximation which at NLO is accurate at the level of a few percent for Higgs masses below the tt kinematical threshold, M H < ∼ 300 GeV [12,13]. The dependence on the renormalisation scale µ R and the factorisation scale µ F of the partonic NNLO cross sections has been reconstructed from the scale independent expressions of Ref. [15] using the fact that the full hadronic cross sections do not depend on them and the α s running between the µ F and µ R scales 4 . Nevertheless, for the central values of the cross sections which will be discussed in the present section, we adopt the usual scale choice µ R = µ F = M H .
An important remark to be made at this stage is that we do not include the softgluon resumation contributions which, for the total cross section, have been calculated up to next-to-next-to-leading logarithm (NNLL) approximation and increase the production rate by ∼ 10-15% at the Tevatron [17]. We also do not include the additional small contributions of the estimated contribution at N 3 LO [41] as well as those of soft terms beyond the NNLL approximation [42]. The reason is that these corrections are known only for the inclusive total cross section and not for the cross sections when experimental cuts are incorporated; this is also the case for the differential cross sections [44] and many distributions that are used experimentally, which have been evaluated only at NNLO at most. This choice of ignoring the contributions beyond NNLO 5 has also been adopted in Ref. [19] in which the theoretical predictions have been confronted to the CDF/D0 results, the focus being the comparison between the distributions obtained from the matrix elements calculation with those given by the event generators and Monte-Carlo programs used by the experiments. Nevertheless, the NNLL result for the cross section can be very closely approached by evaluating the NNLO cross section at the renormalisation and factorisation scales µ R = µ F = 1 2 M H [17] as will be commented upon later. For the electroweak part, we include the complete one-loop corrections to the gg → H amplitude which have been calculated in Ref. [22] taking into account the full dependence on the top/bottom quark and the W/Z boson masses. These corrections are implemented in the so-called partial factorisation scheme in which the electroweak correction δ EW is simply added to the QCD corrected cross section at NNLO, σ tot = σ NNLO + σ LO (1 + δ EW ). In the alternative complete factorization scheme discussed in Ref. [22], the electroweak correction 1+δ EW is multiplied by the fully QCD corrected cross section, σ tot = σ NNLO (1+ δ EW ) and, thus, formally involves terms of O(α 3 s α) and O(α 4 s α) which have not been fully calculated. Since the QCD K-factor is large, K NNLO ≈ 3, the electroweak corrections might be overestimated by the same factor. We have also included the mixed QCDelectroweak corrections at NNLO due to light-quark loops [20]. These are only part of the three-loop O(αα s ) corrections and have been calculated in an effective approach that is valid only when M H < ∼ M W and which cannot be easily extrapolated to M H values above this threshold; this will be discussed in more details in the next section. In Ref. [20], it has been pointed out that this procedure, i.e. adding the NLO full result and the mixed QCD-electroweak correction in the partial factorization scheme, is equivalent to simply including only the NLO electroweak correction in the complete factorisation scheme.
In the case of the qq → W H and qq → ZH associated Higgs production processes, we use the Fortran code V2HV [39] which evaluates the full cross sections at NLO in QCD. The NNLO QCD contributions to the cross sections [26], if the gg → ZH contribution (that does not appear in the case of W H production and is at the permille level at the Tevatron) is ignored, are the same as for the Drell-Yan process pp → V * with V = W, Z [45] given in Ref. [14,46], once the scales and the invariant mass of the final state are properly adapted. These NNLO corrections, as well as the one-loop electroweak corrections evaluated in Ref. [27], are incorporated in the program V2HV. The central scale adopted in this case is the invariant mass of the HV system, µ R = µ F = M HV .
Folding the partonic cross sections with the most recent set of MSTW parton distribution functions [28] and setting the renormalisation and factorisation scales at the most natural values discussed above, i.e. µ R = µ F = M H for gg → H and µ R = µ F = M HV for qq → V H, we obtain for the Tevatron energy √ s = 1.96 TeV, the central values displayed in Fig. 1 for the Higgs production cross sections as a function of the Higgs mass. Note that we have corrected the numbers that we obtained in an earlier version of the paper for the pp → HW cross section to include in the V2HV program the CKM matrix elements when folding the partonic qq ′ → HW cross sections with the parton luminosities 6 ; this results in a decrease of the pp → HW cross section by ≈ 4%. In addition, it recently appeared that including the combined HERA data and the Tevatron W → ℓν charge asymmetry data in the MSTW2008 PDF set [29] might lead to an increase of the pp → (H+)Z/W cross sections by ≈ 3%; a small change in σ(gg → H) is also expected. For the cross sections of the two sub-leading processes qq → V * V * qq → Hqq and qq/gg → ttH that we also include in Fig. 1 for completeness, we have not entered into very sophisticated considerations. We have simply followed the procedure outlined in Ref. [2] and used the public Fortran codes again given in Ref. [39]. The vector boson total cross section is evaluated at NLO in QCD [33] at a scale µ R = µ F = Q V (where Q V is the momentum transfer at the gauge boson leg), while the presumably small electroweak corrections, known for the LHC [47], are omitted. In the case of associated ttH production, the LO cross section is evaluated at scales µ R = µ F = 1 2 (M H + 2m t ) but is multiplied by a factor K ∼ 0.8 over the entire Higgs mass range to account for the bulk of the NLO QCD corrections [35]. In the latter case, we use the updated value m t = 173.1 GeV for the top quark mass [48]. The only other update compared to the cross section values given in Ref. [2] is thus the use of the recent MSTW set of PDFs.
In the case of the gg → H process, our results for the total cross sections are approximately 15% lower than those given in Refs. [5,18]. For instance, for M H = 160 GeV, we obtain with our procedure a total pp → H + X cross section of σ tot = 374 fb, compared to the value σ tot = 439 fb quoted in Ref. [5,18]. The difference is mainly due to the fact that we are working in the NNLO approximation in QCD rather than in the NNLL approximation. As already, mentioned and in accord with Ref. [19], we believe that only the NNLO result should be considered as the production cross sections that are used experimentally include only NNLO effects (not to mention the fact that the K-factors for the cross sections with cuts are significantly smaller than the K-factors affecting the total inclusive cross section, as will be discussed in the next section). A small difference comes also from the different treatment of the electroweak radiative corrections (partial factorisation plus mixed QCD-electroweak contributions in our case versus complete factorisation in Ref. [18]) and another one percent discrepancy can be attributed to the numerical uncertainties in the various integrations of the partonic sections 7 .
We should also note that for the Higgs mass value M H = 160 GeV, we obtain K ≃ 2.15 for the QCD K-factor at NLO and K ≃ 2.8 at NNLO. These numbers are slightly different from those presented in Ref. [19], K ≃ 2.4 and K ≃ 3.3, respectively. The reason is that the b-quark loop contribution, for which the K-factor at NLO is significantly smaller than the one for the top quark contribution [12] has been ignored for simplicity in the latter paper; this difference will be discussed in section 3.2.
In the case of Higgs-strahlung from W and Z bosons, the central values of the cross sections that we obtain are comparable to those given in Ref. [5,30], with at most a ∼ 2% decrease in the low Higgs mass range, M H < ∼ 140 GeV. The reason is that the quark and antiquark densities, which are the most relevant in these processes and are more under control than the gluon densities, are approximately the same in the new MSTW2008 and old MRST2002 sets of PDFs (although the updated set includes a new fit to run II Tevatron and HERA inclusive jet data). We should note that for M H = 115 GeV for which the production cross sections are the largest, σ WH = 175 fb and σ ZH = 104 fb, the QCD Kfactors are ∼ 1. Finally, the cross sections for the vector boson fusion channel in which the recent MSTW set of PDFs is used agree well with those given in Refs. [5,50]. In the case of the ttH associated production process, a small difference is observed compared to Ref. [2] in which the 2005 m t = 178 GeV value is used: we have a few percent increase of the rate due the presently smaller m t value which provides more phase space for the process, overcompensating the decrease due to the smaller top-quark Yukawa coupling.
Before closing this section, let us make a few remarks on the Higgs decay branching ratios and on the rates for the various individual channels that are used to detect the Higgs signal at the Tevatron. For the the Higgs decays, one should use the latest version (3.51) of the program HDECAY [8] in which the important radiative corrections to the H → W W decays [51] have been recently implemented. Choosing the option which allows for the Higgs decays into double off-shell gauge bosons, H → V * V * , which provides the best 7 We have explicitly verified, using the program HRESUM [49] which led to the results of Ref. [18], that our NNLO cross section is in excellent agreement with those available in the literature. In particular, for MH = 160 GeV and scales µR = µF = MH , one obtains σ NNLO = 380 fb with HRESUM compared to σ NNLO = 374 fb in our case; the 1.5% discrepancy being due to the different treatment of the electroweak corrections and the integration errors. Furthermore, setting the renormalisation and factorisation scales to µR = µF = 1 2 MH , we find σ NNLO = 427 fb which is in excellent agreement with the value σ NNLO = 434 fb obtained in Ref. [20] and with HRESUM, as well as the value in the NNLL approximation when the scales are set at their central values µR = µF = MH. This gives us confidence that our implementation of the NNLO contributions in the NLO code HIGLU, including the scale dependence, is correct.  Table 1 for the three dominant decay channels in the mass range relevant at the Tevatron, H → W * W * , bb and τ + τ − . These results are slightly different from those given in Ref. [5]. In particular, the H → W * W * rate that we obtain is a few percent larger for Higgs masses below ∼ 170 GeV. In the interesting range 160 GeV ≤ M H ≤ 170 for which the Tevatron experiments are most sensitive, one sees that the branching ratio for the H → W W is largely dominant, being above 90%. In addition, in this mass range, the gg → H cross section is one order of magnitude larger than the cross sections for the qq → W H, ZH and qq → qqH processes as for M H ∼ 160 GeV for instance, one has σ(gg → H) = 374 fb compared to σ(W H) ≃ 50 fb, σ(ZH) ≃ 30 fb and σ(qqH) ≃ 40 fb. Thus, the channel gg → H → W * W * represents, even before selection cuts are applied, the bulk of the events leading to ℓℓνν + X final states, where here X stands for additional jets or leptons coming from W, Z decays as well as for jets due to the higher order corrections to the gg → H process. In the lower Higgs mass range, M H < ∼ 150 GeV, all the production channels above, with the exception of the vector boson qq → qqH channel which can be selected using specific kinematical cuts, should be taken into account but with the process qq → W H → ℓνbb being dominant for M H < ∼ 130 GeV. This justifies the fact that we concentrate on the gluon-gluon fusion and Higgs-strahlung production channels in this paper.

The scale uncertainty and higher order effects
It has become customary to estimate the effects of the unknown (yet uncalculated) higher order contributions to production cross sections and distributions at hadron colliders by studying the variation of these observables, evaluated at the highest known perturbative order, with the renormalisation scale µ R which defines the strong coupling constant α s and the factorisation scale µ F at which one performs the matching between the perturbative calculation of the matrix elements and the non-perturbative part which resides in the parton distribution functions. The dependence of the cross sections and distributions on these two scales is in principle unphysical: when all orders of the perturbative series are summed, the observables should be scale independent. This scale dependence appears because the perturbative series are truncated, as only its few first orders are evaluated in practice, and can thus serve as a guess of the impact of the higher order contributions.
Starting from a median scale µ 0 which, with an educated guess, is considered as the most "natural" scale of the process and absorbs potentially large logarithmic corrections, the current convention is to vary these two scales within the range with the constant factor κ to be determined. One then uses the following equations to calculate the deviation of, for instance, a cross section σ(µ R , µ F ) from the central value evaluated at scales µ R = µ F = µ 0 , This procedure is by no means a true measure of the higher order effects and should be viewed only as providing a guess of the lower limit on the scale uncertainty. The variation of the scales in the range of eq. (3.1) can be individual with µ R and µ F varying independently in this domain, with possibly some constraints such as 1/κ ≤ µ R /µ F ≤ κ in order not to generate "artificially large logarithms", or collective when, for instance, keeping one of the two scales fixed, say to µ 0 , and vary the other scale in the chosen domain. Another possibility which is often adopted, is to equate the two scales, µ 0 /κ ≤ µ R = µ F ≤ κµ 0 , a procedure that is possibly more consistent as most PDF sets are determined and evolved according to µ R = µ F , but which has no theoretical ground as the two scales enter different parts of the calculation (renormalisation versus factorisation). In addition, there is a freedom in the choice of the variation domain for a given process and, hence, of the constant factor κ. This choice is again rather subjective: depending on whether one is optimistic or pessimistic, i.e. believes or not that the higher order corrections to the process are under control, it can range from κ = 2 to much higher values.
In most recent analyses of production cross sections at hadron colliders, a kind of consensus has emerged and the domain, has been generally adopted for the scale variation. A first remark is that the condition 1 2 ≤ µ R /µ F ≤ 2 to avoid the appearance of large logarithms might seem too restrictive: after all, these possible large logarithms can be viewed as nothing else than the logarithms involving the scales and if they are large, it is simply a reflection of a large scale dependence. A second remark is that in the case of processes in which the calculated higher order contributions are small to moderate and the perturbative series appears to be well behaved 9 , the choice of such a narrow domain for the scale variation with κ = 2, appears reasonable. This, however, might not be true in processes in which the calculated radiative corrections turn out to be extremely large. As the higher order contributions might also be significant in this case, the variation domain of the renormalisation and factorisation scales should be extended and a range with a factor κ substantially larger than two seems more appropriate 10 .
In the case of the gg → H production process, the most natural value for the median scale is the Higgs mass itself, µ 0 = M H , and the effects of the higher order contributions 9 This is indeed the case for some important production processes at the Tevatron, such as the Drell-Yan process pp → V [46,52], weak boson pair production [53] and even top quark pair production [54] once the central scale is taken to be µ0 = mt, which have moderate QCD corrections. 10 This would have been the case, for instance, in top-quark pair production at the Tevatron if the central scale were fixed to the more "natural" value µ0 = 2mt (instead of the value µ0 = mt usually taken [54]) and a scale variation within 1 4 MH ≤ µR, µF ≤ 4MH were adopted. Another well known example is Higgs production in association with b-quark pairs in which the cross section can be determined by evaluating the mechanism gg/qq → bbH [55] or bb annihilation, bb → H [56]. The two calculations performed at NLO for the former process and NNLO for the later one, are consistent only if the central scale is taken to be µ0 ≈ 1 4 MH instead of the more "natural" value µ0 ≈ MH [57]. Again, without prior knowledge of the higher order corrections, it would have been wiser, if the central scale µ0 = MH had been adopted, to assume a wide domain, e.g. 1 4 MH ≤ µR, µF ≤ 4MH , for the scale variation. Note that even for the scale choice µ0 ≈ 1 4 MH , the K-factor for the gg → bbH process remains very large, KNLO ≈ 2 at the Tevatron. In addition, here, it is the factorisation scale µF which generates the large contributions ∝ ln(µ 2 F /m 2 b ) and not the renormalisation scale which can be thus kept at the initial value µR ≈ MH .
to the cross section is again usually estimated by varying µ R and µ F as in eq. (3.3), i.e. with the choice 1 κ ≤ µ R /µ F ≤ κ and κ = 2. At the Tevatron, one obtains a variation of approximately ±15% of the NNLO cross section with this specific choice [14,15] and the uncertainty drops to the level of ≈ ±10% in the NNLL approximation. Note that in some analyses, see e.g. Ref. [20], the central scale µ 0 = 1 2 M H is chosen for the NNLO cross section to mimic the soft-gluon resumation at NNLL [17], and the variation domain is then adopted, leading also to a ≈ 15% uncertainty Nevertheless, as the K-factor is extraordinarily large in the gg → H process, K NNLO ≈ 3, the domain of eq. (3.3) for the scale variation seems too narrow. If this scale domain was chosen for the LO cross section for instance, the maximal value of σ(gg → H) at LO would have never caught, and by far, the value of σ(gg → H) at NNLO, as it should be the case if the uncertainty band with κ = 2 were indeed the correct "measure" of the higher order effects. Only for a much larger value of κ that this would have been the case.
Here, we will use a criterion which allows an empirical evaluation of the effects of the still unknown high orders of the perturbative series and, hence, the choice of the variation domain of the factorisation and renormalisation scales in a production cross section (or distribution). This is done in two steps: i) The domain of scale variation, µ 0 /κ ≤ µ R , µ F ≤ κµ 0 , is derived by calculating the factor κ which allows the uncertainty band of the lower order cross section resulting from the variation of µ R and µ F , to reach the central value (i.e. with µ R and µ F set to µ 0 ), of the cross section that has been obtained at the higher perturbative order.
ii) The scale uncertainty on the cross section at the higher perturbative order is then taken to be the band obtained for a variation of the scales µ R and µ F within the same range and, hence, using the same κ value.
In the case of the gg → H process at the Tevatron, if the lower order cross section is taken to be simply σ LO and the higher order one σ NNLO , this is exemplified in the left-hand side of Fig. 2. The figure shows the uncertainty band of σ LO resulting from a scale variation in the domain M H /κ ≤ µ R , µ F ≤ κM H with κ = 2, 3, 4, 5, which is then compared to σ NNLO evaluated at the central scale µ R = µ F = M H . One first observes that, as expected, the uncertainty bands are larger with increasing values of κ.
The important observation that one can draw from this figure is that it is only for κ = 5, i.e. a variation of the scales in a range that is much wider than the one given in eq. (3.3) that the uncertainty band of the LO cross section becomes very close to (and still does not yet reach for low Higgs mass values) the curve giving the NNLO result. Thus, as the scale uncertainty band of σ LO (gg → H) is supposed to provide an estimate of the resulting cross section at NNLO and beyond, the range within which the two scales µ R and µ F should be varied must be significantly larger than 1 2 On should not impose a restriction on µ R /µ F and consider at least the range 11  Nevertheless, one might be rightfully reluctant to use σ LO as a starting point for estimating the higher order effects, as it is well known that it is only after including at least the next-order QCD corrections that a cross section is somewhat stabilized and, in the particular case of the gg → H process, the LO cross section does not describe correctly the kinematics as, for instance, the Higgs transverse momentum is zero at this order. We thus explore also the scale variation of the NLO cross section σ NLO instead of that of σ LO and compare the resulting uncertainty band to the central value of the cross section again at NNLO (we refrain here from adding the ∼ 15% contribution at NNLL as well as those arising from higher order corrections, such as the estimated N 3 LO correction [41]).
The scale uncertainty bands of σ NLO are shown in the right-hand side of Fig. 2 as a function of M H again for scale variation in the domain M H /κ ≤ µ R , µ F ≤ κM H with κ = 2, 3 and 4, and are compared to σ NNLO evaluated at the central scale the bulk of the higher order corrections. This allows a good convergence of the perturbative series as in this case one has KNLO = 1.46 and KNNLO = 1.32, which seems to stabilize the cross section between the NLO and NNLO values. This nice picture is not spoilt by soft-gluon resumation which leads for such a scale to σ NNLL = 459 fb and, hence, the K-factor turns to KNNLL = 1.28 which is only a few percent lower than KNNLO. Thus, it might have been worth to choose µ0 = 1 5 MH as the central scale from the very beginning, although this particular value does not look very "natural" a priori. We also point out the fact that the choice µ0 = 1 5 MH for the central scale, provides an example of a reduction of the cross section when higher order contributions are taken into account as KNNLL < KNNLO < KNLO.
One can see that, in this case, the uncertainty band for σ NLO shortly falls to reach σ NNLO for κ = 2 and only for κ = 3 that this indeed occurs in the entire M H range.
Thus, to attain the NNLO values of the gg → H cross section at the Tevatron with the scale variation of the NLO cross section, when both cross sections are taken at the central scale choice 12 µ R = µ F = µ 0 = M H , one needs to chose the values κ = 3, and hence a domain of scale variation that is wider than that given in eq. (3.3). This choice of the domains of scale variation might seem somewhat conservative at first sight. However, we emphasise again that in view of the huge QCD corrections which affect the cross section of this particular process, and which almost jeopardize the convergence of the perturbative series, this choice appears to be justified. In fact, this scale choice is not so unusual and in Refs. [15][16][17]58] for instance, scale variation domains comparable to those discussed here, and sometimes even wider, have been used for illustration.
Thus, in our analysis, rather than taking the usual choice for the scale domain of variation with κ = 2 given in eq. (3.3), we will adopt the slightly more conservative possibility given by the wider variation domain 13 Having made this choice for the factor κ, one can turn to the estimate of the higher order effects of σ(gg → H) evaluated at the highest perturbative order that we take to be NNLO, ignoring again the known small contributions beyond this fixed order.
The uncertainty bands resulting from scale variation of σ NNLO (gg → H) at NNLO in the domains given by eqs. (3.3) and (3.4) are shown in Fig. 3 as a function of M H . As expected, the scale uncertainty is slightly larger for κ = 3 than for κ = 2. For instance, for M H = 160 GeV, the NNLO cross section varies by up to ∼ ±21% from its central value, σ NNLO = 374 ± 80 fb, compared to the ≈ ±14% variation that one obtains for κ = 2, σ NNLO = 374 ± 52 fb. The minimal cross section is obtained for the largest values of the two scales, µ F = µ R = κM H , while the maximal value is obtained for the lowest value of the renormalisation scale, µ R = 1 κ M H , almost independently of the factorisation scale µ F , but with a slight preference for the lowest µ F values, µ F = 1 κ M H . 12 We note that one could choose the central scale value µ0 = 1 2 MH [20], instead of µ0 = MH , which seems to better describe the essential features of the kinematics of the process, and in this case, a variation within a factor of two from this central value would have been sufficient for σ NLO to attain σ NNLO . We thank Babis Anastasiou for a discussion on this point. 13 One might argue that since in the case of σ(gg → H), the NLO and NNLO contributions are both positive and increase the LO rate, one should expect a positive contribution from higher orders (as is the case for the re-summed NNLL contribution) and, thus, varying the scales using κ = 2 is more conservative, as the obtained maximal value of the cross section would be smaller than the value that one would obtain for e.g. κ = 3. However, one should not assume that the higher order contributions always increase the lower order cross sections. Indeed, as already mentioned, had we taken the central scales at µR = µF = 1 5 MH , the NNLO (and even NNLL) corrections would have reduced the total cross section evaluated at NLO. Hence, the higher order contributions to σ(gg → H) could well be negative beyond NNLO and could bring the value of the production cross section close to the lower range of the scale uncertainty band of σ NNLO . Another good counter-example of a cross section that is reduced by the higher order contributions is the process of associated Higgs production with top quark pairs at the Tevatron where the NLO QCD corrections decrease the LO cross section by ∼ 20% [35] once the central scale is chosen to be µ0 = 1 2 (2mt + MH ).
We should note that the ≈ 10% scale uncertainty obtained in Ref. [18] and adopted by the CDF/D0 collaborations [5] is even smaller than the ones discussed above. The reason is that it is the resumed NNLL cross section, again with κ = 2 and 1 2 ≤ µ R /µ F ≤ 2, that was considered, and the scale variation of σ NNLL is reduced compared to that of σ NNLO in this case. As one might wonder if this milder dependence also occurs for our adopted κ value, we have explored the scale variation of σ NNLL in the case of κ = 3, without the restriction Using again the program HRESUM [49], we find that the difference between the maximal value of the NNLL cross section, obtained for µ R ≈ M H and µ F ≈ 3M H , and its minimal value, obtained for µ F ≈ 1 3 M H and µ R ≈ 3M H , is as large as in the NNLO case (this is also true for larger κ values). The maximal decrease and maximal increase of σ NNLL from the central value are still of about ±20% in this case. Hence, the relative stability of the NNLL cross section against scale variation, compared to the NNLO case, occurs only for κ = 2 and may appear as accidentally due to a restrictive choice of the variation domain. However, if the additional constraint 1/κ ≤ µ F /µ R ≤ κ is implemented, the situation would improve in the NNLL case, as the possibility µ F ≈ 1 κ M H and µ R ≈ κM H which minimizes σ NNLL would be absent and the scale variation reduced. Nevertheless, even in this case, the variation of σ NNLL for κ = 3 is of the order of ≈ ±15% and, hence, the scale uncertainty is larger than what is obtained in the domain of eq. (3.3).
Finally, another reason for a more conservative choice of the scale variation domain for σ NNLO , beyond the minimal 1 2 M H ≤ µ R , µ F ≤ 2M H range, is that it is well known that the QCD corrections are significantly larger for the total inclusive cross section than for that on which basic selection cuts are applied; see e.g. Ref. [44]. This can be seen from the recent analysis of Ref. [19], in which the higher order corrections to the inclusive cross section for the main Tevatron Higgs signal, gg → H → ℓℓνν, have been compared to those affecting the cross section when selection cuts, that are very similar to those adopted by the CDF and D0 collaborations in their analysis (namely lepton selection and isolation, a minimum requirement for the missing transverse energy due to the neutrinos, and a veto on hard jets to suppress the tt background), are applied. The output of this study is that the K-factor for the cross section after cuts is ∼ 20-30% smaller than the K-factor for the inclusive total cross section (albeit with a reduced scale dependence). For instance, one has K NNLO cuts = 2.6 and K NNLO total = 3.3 for M H = 160 GeV and scales set to µ F = µ R = M H . Naively, one would expect that this ∼ 20-30% reduction of the higher order QCD corrections when selection cuts are applied, if not implemented from the very beginning in the normalisation of the cross section after cuts that is actually used by the experiments (which would then reduce the acceptance of the signal events, defined as σ NNLO cuts /σ NNLO total ), to be at least reflected in the scale variation of the inclusive cross section and, thus, accounted for in the theoretical uncertainty. This would be partly the case for scale variation within a factor κ = 3 from the central scale, which leads to a maximal reduction of the gg → H → ℓℓνν cross section by about 20%, but not with the choice κ = 2 made in Refs. [5] which would have led to a possible reduction of the cross section by ≈ 10% only 14 .

Uncertainties due to the effective approach
While both the QCD and electroweak radiative corrections to the process gg → H have been calculated exactly at NLO, i.e taking into account the finite mass of the particles running in the loops, these corrections are derived at NNLO only in an effective approach in which the loop particles are assumed to be very massive, m ≫ M H , and integrated out. At the Born level, taking into account only the dominant contribution of the top quark loop and working in the limit m t → ∞ provides an approximation [12,13] that is only good at the 10% level for Higgs masses below the tt kinematical threshold, M H < ∼ 350 GeV. The difference from the exact result is mainly due to the absence of the contribution of the b-quark loop: although the b-quark mass is small, the gg → H amplitude exhibits a dependence ∝ m 2 b /M 2 H × log 2 (m 2 b /M 2 H ) which, for relatively low values of the Higgs mass, generates a non-negligible contribution that interferes destructively with the dominant top-quark loop contribution. In turn, when considering only the top quark loop in the Hgg amplitude, the approximation m t → ∞ is extremely good for Higgs masses below 2m t , compared to the amplitude with the exact top quark mass dependence.
In the NLO approximation for the QCD radiative corrections, it has been shown [12] that the exact K-factor when the full dependence on the top and bottom quark masses is taken into account, K exact NLO , is smaller than the K factor obtained in the approximation in which only the top quark contribution is included and the asymptotic limit m t → ∞ is taken, K mt→∞ NLO . The reason is that when only the b-quark loop contribution is considered in the Hgg amplitude (as in the case of supersymmetric theories in which the b-quark Yukawa coupling is strongly enhanced compared to its Standard Model value [59]), the K-factor for the gg → H cross section at the Tevatron is about K ∼ 1. The difference between σ exact NLO and σ mt→∞ NLO at Tevatron energies is shown in Fig. 4 as a function of the Higgs mass and, as one can see, there is a few percent discrepancy between the two cross sections. As mentioned previously, in the Higgs mass range 115 GeV < ∼ M H < ∼ 200 GeV relevant at Tevatron energies, this difference is solely due to the absence of the b-quark loop contribution and its interference with the top quark loop in the Hgg amplitude and not to the fact that the limit m t ≫ M H is taken.
At NNLO, because of the complexity of the calculation, only the result in the effective approach in which the loop particle masses are assumed to be infinite is available. In the case of the NNLO QCD corrections [14][15][16], the b-quark loop contribution and its interference with the contribution of t-quark loop is therefore missing. Since the NNLO correction increases the cross section by ∼ 30%, one might wonder if this missing piece does not lead to an overestimate of the total K-factor. We will assume that it might be indeed the case and assign an error on the NNLO QCD result which is approximately the difference between the exact result σ exact NLO and the approximate result σ mt→∞ NLO obtained at NLO and shown in Fig. 4, but rescaled with the relative magnitude of the K-factors that one obtains at NLO and NNLO, i.e. K mt→∞ NLO /K mt→∞ NNLO . This leads to an uncertainty on the NNLO cross section which ranges from ∼ ±2% for low Higgs values M H ∼ 120 GeV at which the b-quark loop contribution is significant at LO, to the level of ∼ ±1% for Higgs masses above M H ∼ 180 GeV for which the b-quark loop contribution is much smaller.
In addition one should assign to the b-quark contribution an error originating from the freedom in choosing the input value of the b-quark mass in the loop amplitude and the scheme in which it is defined 15 . Indeed, besides the difference obtained when using the b-quark pole mass, M pole b ≈ 4.7 GeV, as is done here or the running MS mass evaluated at the scale of the b-quark mass,m MS b (M b ) ∼ 4.2 GeV, there is an additional 4 3 αs π factor which enters the cross section when switching from the on-shell to the MS scheme. This leads to an error of approximately 1% on the total cross section, over the M H range that is relevant at the Tevatron. In contrast, according to very recent calculations [60], the m t → ∞ limit is a rather good approximation for the top-quark loop contribution to σ(gg → H) at NNLO as the higher order terms, when expanding the amplitude in power series of M 2 H /(4m 2 t ), lead to a difference that is smaller than one percent for M H < ∼ 300 GeV. We turn now our attention to the electroweak radiative corrections and also estimate their associated error. As mentioned previously, while the O(α) NLO corrections have been calculated with the exact dependence on the loop particle masses [22], the mixed QCD-electroweak corrections due to light quark loops at O(αα s ) have been evaluated [20] in the effective theory approach where the W, Z bosons have been integrated out and which is only valid for M H ≪ M W . These contributions are approximately equal to the difference between the exact NLO electroweak corrections when evaluated in the complete factorisation and partial factorization schemes [20].
However, as the results for the mixed corrections are only valid at most for M H < M W and given the fact that the companion δ EW electroweak correction at O(α) exhibits a completely different behavior below and above the 2M W threshold 16 , one should be cautious and assign an uncertainty to this mixed QCD-electroweak correction. Conservatively, we have chosen to assign an error that is of the same size as the O(αα s ) contribution itself. This is equivalent to assigning an error to the full O(α) contribution that amounts to the difference between the correction obtained in the complete factorisation and partial factorisation schemes as done in Ref. [22]. As pointed out in the latter reference, this reduces to adopting the usual and well-established procedure that has been used at LEP for attributing uncertainties due to unknown higher order effects. Doing so, one obtains an uncertainty ranging from 1.5% to 3.5% for Higgs masses below M H < ∼ 2M W and below 1.5% for larger Higgs masses as is shown in Fig. 5.
Finally, we should note that we do not address here the issue of the threshold effects from virtual W and Z bosons which lead to spurious spikes in the O(α) electroweak correction in the mass range M H = 160-190 GeV which includes the Higgs mass domain that is most relevant at the Tevatron (the same problem occurs in the case of the pp → HV cross sections once the electroweak corrections are included). These singularities are smoothened 15 We thank Michael Spira for reminding us of this point. 16 Indeed, the NLO electroweak correction δEW of Ref. [22] is positive below the W W threshold MH < ∼ 2MW for which the effective approach is valid in this case and turns to negative for MH > ∼ 2MZ for which the effective approach cannot be applied and the amplitude develops imaginary parts. This behavior can also be seen in Fig. 5 which, up to the overall normalisation, is to a very good approximation the δEW correction factor given in Fig. 1   by including the finite widths of the W/Z bosons, a procedure which might introduce potential additional theoretical ambiguities that we will ignore in the present analysis.

Uncertainties from the PDFs and α s
Another major source of theoretical uncertainties on production cross sections and distributions at hadron colliders is due to the still imperfect parametrisation of the parton distribution functions. Within a given parametrisation, for example the one in the MSTW scheme, these uncertainties are estimated as follows [31,61,62]. The scheme is based on a matrix method which enables a characterization of a parton parametrization in the neighborhood of the global χ 2 minimum fit and gives an access to the uncertainty estimation through a set of PDFs that describes this neighborhood. The corresponding PDFs are constructed by: i) performing a global fit of the data using N PDF free parameters (N PDF = 15 or 20, depending on the scheme); this provides the nominal PDF or reference set denoted by S 0 ; (ii) the global χ 2 of the fit is increased to a given value ∆χ 2 to obtain the error matrix; (iii) the error matrix is diagonalized to obtain N PDF eigenvectors corresponding to N PDF independent directions in the parameter space; (iv) for each eigenvector, up and down excursions are performed in the tolerance gap, T = ∆χ 2 global , leading to 2N PDF sets of new parameters, denoted by S i , with i = 1, 2N PDF .
These sets of PDFs can be used to calculate the uncertainty on a cross section σ in the following way: one first evaluates the cross section with the nominal PDF S 0 to obtain the central value σ 0 , and then calculates the cross section with the S i PDFs, giving 2N PDF values σ i , and defines, for each σ i value, the deviations The uncertainties are summed quadratically to calculate the cross section, including the error from the PDFs that are given at the 90% confidence level (CL), The procedure outlined above has been applied to estimate the PDF uncertainties in the Higgs production cross sections in the gluon-gluon fusion mechanism at the Tevatron in Refs. [18,20]. This has led to a 90% CL uncertainty of ≈ 6% for the low mass range M H ≈ 120 GeV to ≈ 10% in the high mass range, M H ≈ 200 GeV. These uncertainties have been adopted in the CDF/D0 combined Higgs search and represent the second largest source of errors after the scale variation. We believe that, at least in the case of the gg fusion mechanism, restricting to the procedure described above largely underestimates the PDF uncertainties for at least the two reasons discussed below.
First of all, the MSTW collaboration [28] is not the only one which uses the above scheme for PDF error estimates, as the CTEQ [36] and ABKM [37] collaborations, for instance, also provide similar schemes (besides the NNPDF set [63], an additional NNLO PDF set [64] has recently appeared and it also allows for error estimates). It is thus more appropriate to compare the results given by the three different sets and take into account the possibly different errors that one obtains. In addition, as the parameterisations of the PDFs are different in the three schemes, one might obtain different central values for the cross sections and the impact of this difference should also be addressed 17 .
In our analysis, we will take into account these two aspects and investigate the PDF uncertainties given separately by the three MSTW, ABKM and CTEQ schemes, but we also compare the possibly different central values given by the three schemes. Note that despite of the fact that the CTEQ collaboration does not yet provide PDF sets at NNLO, one can still use the available NLO sets, evaluating the PDF errors on the NLO cross sections and take these errors as approximately valid at NNLO, once the cross sections are properly rescaled by including the NNLO corrections. For the sake of error estimates, this procedure should provide a good approximation.
In the case of the gg → H cross section at the Tevatron, the 90% CL PDF errors using the three schemes discussed above are shown in Fig. 6 as a function of M H . The spread of the cross section due to the PDF errors is approximately the same in the MSTW and CTEQ schemes, leading to an uncertainty band of less than 10% in both cases. For instance, in the MSTW scheme and in agreement with Refs. [18,20], we obtain a ∼ ±6% error for M H = 120 GeV and ∼ ±9% for M H = 180 GeV; the errors are only slightly asymmetric and for M H = 160 GeV, one has ∆σ + PDF /σ = +8.1% and ∆σ − PDF /σ = −8.6%. The errors are relatively smaller in the ABKM case in the entire Higgs mass range and, for instance, one obtains a ∆σ ± PDF /σ ≈ ±5% (7%) error for M H = 120 (180) GeV. A more important issue is the very large discrepancy between the central values of the cross sections calculated with the MSTW and CTEQ PDFs on the one hand and the ABKM set of PDFs, on the other hand 18 . Indeed, the use of the ABKM parametrisation results in a cross section that is ∼ 25% smaller than the cross section evaluated with the MSTW or CTEQ PDFs. Thus, even if the PDF uncertainties evaluated within a given scheme turn out to be relatively small and apparently well under control, the spread of the cross sections due to the different parameterisations can be much more important.
If one uses the old way of estimating the PDF uncertainties (i.e. before the advent of the PDF error estimates within a given scheme) by comparing the results given by different PDF parameterisations, one arrives at an uncertainty defined as where the central value of the gg → H cross section is taken to be that given by the MSTW nominal set S 0 (we refrain here from adding the uncertainties obtained within the same PDF set, which would increase the error by another 5% to 7%). Hence, for M H = 160 GeV for instance, one would have ∆σ + PDF ≈ 1% given by the small difference between the CTEQ and MSTW central values of the cross section and ∆σ − PDF ≈ −25% given by the large difference between the ABKM and MSTW central values.
However, we would would like to keep considering the MSTW scheme at least for the fact that it includes the di-jet Tevatron data which are crucial in this context. But we would also like understand the very large difference in the gg → H cross section when evaluated with the MSTW/CTEQ and ABKM sets. This difference results not only from the different gluon densities used (and it is well known that these densities are less severely constrained by experimental data than light quark densities), but is also due to the different values of the strong coupling constant which is fitted altogether with the PDF sets. Indeed, the value of α s and its associated error play a crucial role in the presently discussed production process. For instance, the α s value used in the ABKM set, α s (M 2 Z ) = 0.1129 ± 0.0014 at NLO in the BMSM scheme [66], is ≈ 3σ smaller than the one in the MSTW set (see below). Note also that within the dynamical set of PDFs recently proposed in Ref [64], one obtains too an NLO α s value that is smaller than the MSTW value but with a slightly larger uncertainty, α s (M 2 Z ) = 0.1124 ± 0.0020. As the gg → H mechanism is mediated by triangular loops involving the heavy top and bottom quarks, the cross section σ(gg → H) is at O(α 2 s ) already in the Born approximation and the large NLO and NNLO QCD contributions are, respectively, of O(α 3 s ) and O(α 4 s ). Since the corresponding K-factors are very large at the Tevatron, K NLO ∼ 2 and K NNLO ∼ 3, a one percent uncertainty in the input value of α s will generate a ≈ 3% uncertainty in σ NNLO (gg → H). If, for instance, one uses the value of α s at NLO and its associated experimental uncertainty that is fitted in the global analysis of the hard scattering data performed by the MSTW collaboration [61] leading to α s (M 2 Z ) = 0.1171 +0.0014 −0.0014 (68%CL) at NNLO, by naively plaguing the 90% CL errors on α s in the perturbative series of the partonic cross section but using the best-fit PDF set, one arrives at an uncertainty on the gg → H cross section that is of the order of ∆σ/σ ≈ ±8% at the Tevatron, over the entire 115 GeV < ∼ M H < ∼ 200 GeV range. Nevertheless, such a naive procedure cannot be applied in practice as, in general, α s is fitted together with the PDFs: the PDF sets are only defined for the special value of α s obtained with the best fit and, to be consistent, this best value of α s that we denote α 0 s , should also be used for the partonic part of the cross section. This adds to the fact that there is an interplay between the PDFs and the value of α s and, for instance, a larger value of α s would lead to a smaller gluon density at low x [61].
Fortunately enough, the MSTW collaboration released very recently a new set-up which allows for a simultaneous evaluation of the errors due to the PDFs and those due to the experimental uncertainties on α s of eq. (3.8), taking into account the possible correlations [61]. The procedure to obtain the different PDFs and their associated errors is similar to the one discussed before, but provided is a collection of five PDF+error sets for different α s values: the best fit value α 0 s and its 68% CL and 90% CL maximal and minimal values. Using the following equations to calculate the PDF error for a fixed value of α s , one then compares these five different values and finally arrives, with α 0 s as the best-fit value of α s given by the central values of eq. (3.8) and S 0 the nominal PDF set with this α s value, at the 90% CL PDF+∆ exp α s errors given by [61] ∆σ + PDF+α exp (3.10) Using this procedure, we have evaluated the PDF+∆ exp α s uncertainty on the NNLO gg → H total cross section at the Tevatron and the result is displayed in the left-hand side of Fig. 7 as a function of M H . The PDF+∆ exp α s error ranges from ≈ ±11% for M H = 120 GeV to ≈ ±14% for M H = 180 GeV with, again, a slight asymmetry between the upper and lower values; for a Higgs mass M H = 160 GeV, one has ∆σ ± PDF+αs /σ = +12.8% −12.0% . That is, the experimental uncertainty on α s adds a ≈ 5% error to the PDF error alone over the entire M H range relevant at the Tevatron. This is a factor of ≈ 1.5 less than the naive guess made previously, as a result of the correlation between the PDFs and the α s value.
Nevertheless, this larger PDF+∆ exp α s uncertainty compared to the PDF uncertainty alone does not yet reconcile the evaluation of MSTW and ABKM (in this last scheme the ∆ exp α s uncertainty has not been included since no PDF set with an error on α s is provided) of the gg → H cross section at the Tevatron, the difference between the lowest MSTW value and the highest ABKM value being still at the level of ≈ 10%.
So far, only the impact of the experimental errors on α s has been discussed, while it is well known that the strong coupling constant is also plagued by theoretical uncertainties due to scale variation, ambiguities in heavy quark flavor scheme definition, etc.. In Ref. [28] this theoretical error has been estimated to be at least ∆ th α s = ±0.003 at NLO (±0.002 at NNLO) while the estimate of Ref. [67] leads to a slightly larger uncertainty, ∆ th α s = ±0.0033. Unfortunately, this theoretical error is not taken into account in the MSTW PDF+∆α s error set-up discussed above, nor is addressed by any of the other PDF schemes.
Adopting the smallest of the 1σ α s errors at NLO quoted above, i.e.
we have evaluated the uncertainty due this theoretical error on σ NNLO (gg → H + X) at the Tevatron, following our naive and admittedly not entirely consistent first estimate of the impact of the experimental error of α s on the same cross section, i.e. using the values α 0 s ± 0.002 in the partonic cross sections but the best-fit value α 0 s in the best-fit PDF set. We obtain an error of ≈ 8% on σ NNLO (gg → H) for the M H values relevant at the Tevatron. There is nevertheless a more consistent way to address this issue of the theoretical uncertainty on α s , thanks to a fixed-α s NNLO PDF grid also provided by the MSTW collaboration, which is a set of central PDFs but at fixed values of α s different from the best-fit value. Values of α s in a range comprised between 0.107 and 0.127 in steps of 0.001 are selected, and thus include the values α 0 s ± 0.002 that are interesting for our purpose. Using this PDF grid with the theoretical error on α s of eq. (3.11) implemented, the upper and lower values of the cross sections will be given by with again S 0 (α s ) being the MSTW best-fit PDF set at the fixed α s value which is either α 0 s or α 0 s ± ∆ th α s . With this fixed-α s PDF grid, we obtain an error of ≈ +10% and ≈ −9% on the total gg → H cross section at NNLO when one restricts to the range of Higgs masses relevant at the Tevatron, with a ≈ 1% increase from M H = 115 GeV to M H = 200 GeV. This error is again very close to the naive estimate performed previously by considering only the impact of ∆ th α s on the partonic cross section. Note that despite of the fact that the uncertainty on α s is a theoretical one and is not at the 90% CL, we will take the PDF+∆ th α s error that one obtains using the equations above to be at the 90% CL.
In the MSTW scheme, to obtain the total PDF+α s uncertainty, one then adds in quadrature the PDF+∆ exp α s and PDF+∆ th α s uncertainties, The result for the total PDF+α s 90% CL uncertainty on σ NNLO (gg → H) in the MSTW scheme using the procedure outlined above is shown in the right-hand side of Fig. 7 as a function of M H . It is compared to the result when the PDF error in the ABKM scheme is combined with the ∆ exp α s and ∆ th α s uncertainties using the naive procedure discussed previously as, in this case, no PDF with an α s value different from that obtained with the best-fit is provided. One can see that the results given by the two parameterizations appear now to be consistent with each other as the two uncertainty bands overlap.
The net result of this exercise is that the total error on the gg → H cross section due to the PDF and the theoretical plus experimental uncertainties on α s , is now rather significant and, in the case of the MSTW scheme to which we stick, it amounts to approximately ±15 to 20% in the Higgs mass range relevant at the Tevatron. The uncertainty is, for instance, −15% and +16.5% for M H = 160 GeV and is substantially smaller (for the minimal value of the cross section) than the error that would have been obtained using the old-fashioned estimate of the PDF errors by comparing different PDF sets, in which case one would have had an uncertainty of −26% and +1% compared to the MSTW central value.
The final error of ≈ ±15-20% is to be compared to the ±6-10% error obtained from the PDF uncertainty alone (≈ ±8% for M H = 160 GeV), an amount which has been taken to be the total PDF uncertainty in the CDF/D0 analysis of the Higgs signal. Thus, similarly to the scale variation, the PDF uncertainties, when the errors on α s are taken into account, have been underestimated by at least a factor of two by the experiments.

Theoretical uncertainties in Higgs-strahlung
We now turn to the discussion of the theoretical uncertainties in the Higgs strahlung mechanism qq → V H, following the same line of arguments as in the previous section. Since in this case, the NNLO QCD corrections and the one-loop electroweak corrections have been obtained exactly and no effective approach was used, only the scale variation and the PDF+α s uncertainties have to be discussed. In addition, since the NNLO gluongluon fusion contribution to the cross section in the pp → ZH case, which is absent in pp → W H, is very small at the Tevatron and because the scales and phase space are only slightly different for the pp → W H and ZH processes, as the difference (M 2 Z − M 2 W )/ŝ is tiny, the kinematics and the K-factors for these two processes are very similar. We thus restrict our analysis to the W H channel but the same results hold for the ZH channel.
To evaluate the uncertainties due to the variation of the renormalisation and factorisation scales in the Higgs-strahlung processes, the choice of the variation domain is in a sense simpler than for the gg → H mechanism. Indeed, as the process at leading order is mediated solely by massive gauge boson exchange and, thus, does not involve strong interactions at the partonic level, only the factorisation scale µ F appears when the partonic cross section is folded with the q andq luminosities and there is no dependence on the renormalisation scale µ R at this order. It is only at NLO, when gluons are exchanged between or radiated from the q,q initial states, that both scales µ R and µ F appear explicitly.
Using our proposed criterion for the estimate of the perturbative higher order effects, we thus choose again to consider the variation domain of the scales from their central values, µ 0 /κ ≤ µ R , µ F ≤ κµ 0 with µ 0 = M HW , of the NLO cross section instead of that of the LO cross section to determine the value of the factor κ to be used at NNLO. We display in the left-hand side of Fig. 8 the variation of the NLO cross section σ NLO (pp → W H) at the Tevatron as a function of M H for three values of the constant κ which defines the range spanned by the scales, M HW /κ ≤ µ R , µ F ≤ κM HW . One sees that, in this case, a value κ = 2 is sufficient (if the scales µ R and µ F are varied independently in the chosen domain) in order that the uncertainty band at NLO reaches the central value of the cross section at NNLO. In fact, the NLO uncertainty band would have been only marginally affected if one had chosen the values κ = 3, 4 or even 5. This demonstrates than the cross sections for the Higgs-strahlung processes, in contrast to gg → H, are very stable against scale variation, a result that is presumably due to the smaller qq color charges compared to gluons, ≈ C F /C A , that lead to more moderate QCD corrections.
In the right-hand side of Fig. 8, the NNLO pp → W H total cross section is displayed as a function of M H for a scale variation 1 2 M HW ≤ µ R , µ F ≤ 2M HW . Contrary to the gg → H mechanism, the scale variation within the chosen range is rather mild and only a ∼ 0.7% (at low M H ) to 1.2% (at high M H ) uncertainty is observed for the relevant Higgs mass range at the Tevatron. This had to be expected as the K-factors in the Higgs-strahlung processes, K NLO ≈ 1.4 and K NNLO ≈ 1.5, are substantially smaller than those affecting the gg fusion mechanism and one expects perturbation theory to have a better behavior in the former case. This provides more confidence that the Higgs-strahlung cross section is stable against scale variation and, thus, that higher order effects should be small.
For the estimate of the uncertainties due to the PDFs in associated Higgs production with a W boson, pp → W H (again, the output is similar for pp → ZH except from the overall normalisation, despite of the different initial state (anti)quarks), the same exercise made in section 3.3 for the gg fusion mechanism has been repeated. The results are shown in Figs. 9 and 10 for Tevatron energies as a function of M H . Figure 9 displays the spread of the pp → W H cross section due to the PDF uncertainties alone in the MSTW, CTEQ and ABKM schemes and, again in this case, the uncertainty bands are similar in the CTEQ and MSTW schemes and lead to an error of about 4%; the band is, however, slightly larger in the ABKM scheme. Here also appears a discrepancy between the MSTW/CTEQ and the ABKM central values, the cross section with the PDFs from ABKM being this time about 10% larger than that obtained with the other sets. However, in contrast to the gg → H case, the MSTW/CTEQ and ABKM uncertainty bands almost touch each other.
In the left-hand side of Fig. 10, we show the bands resulting from the PDF+∆ exp α s uncertainty in the MSTW mixed scheme, while the right-hand side of the figure shows the uncertainty bands when the additional theoretical error ∆ th α s is included in both the MSTW scheme using eq. (3.13) and ABKM scheme using the naive estimate of eq. (3.12). As expected, the errors due to the imprecise value of α s are much smaller than in the  gg → H mechanism, as in Higgs-strahlung, the process does not involve α s in the Born approximation and the K-factors are reasonably small, K NNLO < ∼ 1.5. Hence, ∆ exp α s generates an additional error that is about ≈ 2% when included in the PDF fits, while the error due to ∆ th α s is about one to two percent.
Nevertheless, the total PDF+∆ exp α s +∆ th α s uncertainty is at the level of ≈ 7-8% in the MSTW scheme, i.e. slightly larger than the errors due to the PDFs alone, and arranges again so that the MSTW and ABKM uncertainty bands have a significant overlap.

The total uncertainties at the Tevatron
The analysis of the Higgs production cross section in the gg → H process at the Tevatron, as well as the various associated theoretical uncertainties, is summarized in Table 2. For a set of Higgs mass values that is relevant at the Tevatron (we choose a step of 5 GeV as done by the CDF and D0 experiments [5] except in the critical range 160-170 GeV where a 2 GeV step is adopted), the second column of the table gives the central values of the total cross section at NNLO (in fb) for the renormalisation and factorisation scale choice µ R = µ F = M H , when the partonic cross sections are folded with the MSTW parton densities. The following columns give the errors on the central value of the cross section originating from the various sources discussed in section 3, namely, the uncertainties due to the scale variation in the adopted range 1 3 M H ≤ µ R , µ F ≤ 3M H , the 90% CL errors due to the MSTW PDF, PDF+∆ exp α s and PDF+∆ exp α s +∆ th α s uncertainties as well as the estimated uncertainties from the use of the effective approach in the calculation of the NNLO QCD (the b-quark loop contribution and its interference with the top-quark loop) and electroweak (difference between the complete and partial factorisation approaches) radiative corrections.
The largest of these errors, ∼ 20%, is due to the scale variation, followed by the PDF+∆ exp α s + ∆ th α s uncertainties which are at the level of ≈ 15%; the errors due to the effective theory approach (including that due to the definition of the b-quark mass) are much smaller, being of the order of a few percent for both the QCD and electroweak parts.
The next important issue is how to combine these various uncertainties. In accord with Ref. [19], we do not find any obvious justification to add these errors in quadrature as done, for instance, by the CDF and D0 collaborations 19 [68][69][70]. Indeed, while the PDF+α s uncertainty might have some statistical ground, the scale uncertainty as well as the uncertainties due to the use of the effective approach are purely theoretical errors. On the other hand, one cannot simply add these errors linearly as is generally done for theoretical errors, the reason being a possibly strong interplay between the scale chosen for the process, the value of α s (which evolves with the scales) and thus the PDFs (since the gluon density, for instance, is sensitive to the exact value of α s as mentioned previously). Here, we propose a simple procedure to combine at least the two largest uncertainties, those due to the scale variation and to the PDF+α s uncertainties, that is in our opinion more adequate and avoids the drawbacks of the two other possibilities mentioned above.
The procedure that we propose is as follows. One first derives the maximal and minimal values of the production cross sections when the renormalisation and factorisation scales are varied in the adopted domain, that is, σ 0 ± ∆σ ± µ with σ 0 being the cross section evaluated for the central scales µ R = µ F = µ 0 and the deviations ∆σ ± µ given in eq. (3.2). One then evaluates on these maximal and minimal cross sections from scale variation, the PDF+∆ exp α s as well as the PDF+∆ th α s uncertainties (combined in quadrature) using the new MSTW set-up, i.e as in eq. (3.13) but with σ 0 replaced by σ 0 ± ∆σ ± µ . One then obtains the maximal and minimal values of the cross section when scale, PDF and α s (both experimental and theoretical) uncertainties are included, To these new maximal and minimal cross sections, one should then add the much smaller errors originating from the other sources such as, in the case of the gg → H process, those due to the missing b-quark loop and the mixed QCD-electroweak corrections at NNLO. This last addition can be done linearly as the errors from the use of the effective theory approach are purely theoretical ones and do not depend on the scale choice in practice 20 . 19 In earlier analyses, the CDF collaboration [68,70] adds in quadrature the 10.9% scale uncertainty obtained at NNLL with a scale variation in the range 1 2 MH ≤ µR, µF ≤ 2MH with a 5.1% uncertainty due the errors on the MSTW PDFs (not including the errors from αs), resulting in a 12% total uncertainty. The D0 collaboration [69,70] assigns an even smaller total error, 10%, to the production cross section. 20 In the case of the b-loop contribution, the K-factor when varying the scale from the central value MH to the values ≈ 1 3 MH or ≈ 3MH .which maximise and minimise the cross section, might be slightly different and thus, the error will not be exactly that given in Table 2. However, since the entire effect is very small, we will ignore this tiny complication here.
The two last columns of Table 2 display the maximal and minimal deviations of the gg → H cross section at the Tevatron when all errors are added, as well as the percentage deviations of the cross section from the central value. We should note that the actual PDF+α s error and the error from the use of the effective theory approach are different from those of Table 2, which are given for the best value of the cross section, obtained for the central scale choice µ F = µ R = M H ; nevertheless, the relative or percentage errors are approximately the same for σ 0 and σ 0 ± ∆σ ± µ . One observes from Table 2 that when all theoretical errors are combined, there is a large variation of the gg → H cross section. The percentage total error on the cross section is approximately the same in the entire Higgs mass range that is indicated and is significant, the lower and upper values being ≈ 40% smaller or ≈ 50% larger than the central value.
For M H = 160 GeV for instance, one obtains a spread from the central value σ 0 = 374 fb which amounts to σ max = 552 fb and σ min = 225 fb, a spread that leads to a percentage error of ∆σ 0 /σ 0 = −39.7% and +47.6%. This is again summarized in Fig. 11, where the total uncertainty band obtained in our analysis is confronted to the uncertainty band that one obtains when adding in quadrature the scale uncertainty for 1 2 M H ≤ µ R , µ F ≤ 2M H and the PDF error only (without the errors on α s ) as assumed in the CDF/D0 analysis. Furthermore, in the latter case, we use the resumed NNLL cross sections given in Ref. [18] which is ∼ 15% higher than the cross section that we obtain when including the higher order contributions only to NNLO and has a milder scale variation. As can be seen, the difference between the two uncertainty bands is striking. In fact, even the lower value of the cross section in the NNLL approach, including the scale and PDF errors when combined in quadrature, only touches the central value of our NNLO result. For M H = 160 GeV, the lower value of the cross section, when all errors are included, is ≈ 40% smaller than the central value at NNLO and ≈ 50% smaller than the NNLL cross section adopted in Ref. [5] as a normalisation.
We thus believe that the CDF/D0 combined analysis which rules out the 162-166 GeV mass range for the SM Higgs boson on the basis of the gg → H → ℓℓνν + X process, which is the most (if not the only) relevant one in this specific mass range at the Tevatron, has largely underestimated the theoretical errors on the Higgs production cross section. In fact, even if the scale uncertainty were taken to be that resulting from a variation in the usual domain 1 2 M H ≤ µ F , µ R ≤ 2M H or the errors from the use of the effective approach at NNLO were ignored, the total uncertainty would have been of the order of ≈ 35%, i.e three times larger than the error assumed in the CDF/D0 analysis.
Turning to the Higgs-strahlung processes, and similarly to the gg → H case, we display in Table 3 the central values of the cross sections for pp → W H and pp → ZH at the Tevatron, evaluated at scales µ R = µ F = M HV with the MSTW set of PDFs (second and third columns). In the remaining columns, we specialize in the W H channel and display the errors from the scale variation (with κ = 2), the PDF, mixed PDF+∆ exp α s and PDF+∆ exp α s +∆ th α s uncertainties in the MSTW scheme. In the last columns, we give the total error and its percentage; this percentage error is, to a very good approximation, the same in the pp → ZH channel. In contrast to the gg → H mechanism, since the errors due to scale variation are rather moderate in this case, there is no large difference -31 -  Table 2). It is compared to σ(gg → H) at NNLL when the scale and PDF errors given in Ref. [18] are added in quadrature. In the insert the relative deviations are shown when the central values are normalized to σ NNLO+EW . between the central cross section σ 0 and the cross sections σ 0 ± ∆σ ± µ and, hence, the PDF, PDF+∆ exp α s and PDF+∆ exp α s +∆ th α s errors on σ 0 are, to a good approximation, the same as the errors on σ 0 ± ∆σ ± µ displayed in Table 3. The total uncertainty is once more summarized in Fig. 12, where the cross sections for W H and ZH associated production at the Tevatron, together with the total uncertainty bands (in absolute values in the main frame and in percentage in the insert), are displayed as a function of the Higgs mass. As can be seen, the total error on the cross sections in the Higgs-strahlung processes is about ±9% in the entire Higgs mass range, possibly 1% to 2% smaller for low M H values and ∼ 1% larger for high M H values. Thus, the theoretical errors are much smaller than in the case of the gg → H process and the cross sections for the Higgs-strahlung processes are well under control. Nevertheless, the total uncertainty obtained in our analysis is almost twice as large as the total 5% uncertainty assumed by the CDF and D0 collaborations in their combined analysis of this channel [5].
Before closing this section, let us mention that the uncertainties in the Higgs-strahlung processes can be significantly reduced by using the Drell-Yan processes of massive gauge boson production as standard candles; a suggestion first made in Ref. [71]. Indeed, normalizing the cross sections of associated W H and ZH production to the cross sections of single W and Z production, respectively, allows for a cancellation of several experimental errors such as the error on the luminosity measurement, as well as the partial cancellation (since the scales that are involved in the pp → V and HV processes are different) theoretical errors such as those due to the PDFs, α s and the higher order radiative corrections.

Conclusion
In the first part of this paper, we have evaluated the production cross sections of the Standard Model Higgs boson at the Tevatron, focusing on the two main channels: the gluon-gluon fusion gg → H mechanism that dominates in the high Higgs mass range and the Higgs-strahlung processes qq → V H with V = W, Z, which are the most important ones in the lower Higgs mass range. In the determination of the cross sections, we have included all the available and relevant higher order corrections in perturbation theory, in particular, the QCD corrections up to NNLO and the one-loop electroweak radiative corrections. We have then provided up-to-date central values of the cross sections for the the entire Higgs mass range that is relevant at the Tevatron. While this update has been performed for the gg → H mechanism in several recent analyses, it was missing in the case of the Higgs-strahlung processes.
The second part of the paper addresses the important issue of the theoretical uncertainties that affect the predicted cross sections. We have first discussed the scale uncertainties which are usually viewed as a measure of the unknown higher order contributions. Because the calculated QCD corrections are extremely large in the gg → H process, we point out that the domain of variation of the renormalisation and factorisation scales that is usually adopted in the literature should be extended. We adopt a criterion that allows for a more reasonable or conservative estimate of this variation domain: the range of variation of the scales at NNLO, should be the one which allows to the scale uncertainty band of the NLO cross section to include the NNLO contributions. Applying this criterion to the NNLO gg → H cross section and adopting a central scale µ 0 = M H , we obtain a scale uncertainty of the order of ±20%, i.e. slightly larger than the ≈ ±15% uncertainty that is usually assumed. This larger error would at least account for the 20-30% discrepancy between the QCD corrections to the inclusive cross section that is used as a normalisation and the cross section with the basic kinematical cuts applied in the experimental analyses.
A second source of uncertainties in the gg → H cross section originates from the use of the effective theory approach that allows to considerably simplify the calculation of the NNLO contributions, an approach in which the masses of the loop particles that generate the Hgg vertex are assumed to be much larger than the Higgs mass. We show that the missing NNLO contribution of the b-quark loop where the limit M H ≪ m b cannot be applied (together with the definition of the b-quark mass), and the approximation M H ≪ M W used in the three-loop mixed QCD-electroweak NNLO radiative corrections, might lead to a few percent error on the total gg → H cross section in each case.
A third source of theoretical errors is due to the parton distribution functions and the errors associated to the strong coupling constant. Considering not only the MSTW scheme as usually done, but also the CTEQ and ABKM schemes, we recall that while the PDF errors are relatively small within a given scheme, the central values can be widely different. This is particularly true in the case of the gg → H cross section, where the central values in the MSTW/CTEQ and ABKM schemes differ by about 25%. Only when the experimental as well as the theoretical errors on α s are accounted for that one obtains results that are consistent when using the MSTW/CTEQ and ABKM schemes. In the MSTW scheme, using a recently released set-up which provides a simultaneous access to the PDF and ∆ exp α s errors as well as a way to estimate the ∆ th α s error, one finds a ≈ 15% uncertainty on σ(gg → H), that is, at least a factor of two larger than the uncertainty due to the PDFs alone that is usually considered as the total PDF error We have then proposed a simple procedure to combine these various theoretical errors. The main idea of this procedure is to evaluate directly the PDF+∆ exp α s +∆ th α s error, as well as the significantly smaller errors due to the use of the effective approach in the gg → H process at NNLO, on the maximal and minimal values of the cross sections that one obtains when varying the renormalisation and factorisation scales in the chosen domain.
Adopting this approach, one arrives at a total uncertainty of ≈ −40% and ≈ +50% for the central value of the gg → H cross section at the Tevatron, a much larger error than the ≈ 10% uncertainty that is usually assumed 21 . Hence, the number of signal events from the gg → H process with the subsequent Higgs decay H → W W → ℓℓνν, i.e. the main 21 We note that it would be interesting to study the impact of these theoretical uncertainties on the gg → H cross sections for Higgs production at the LHC, not only for the discovery of the particle, but also for the measurement of its couplings to fermions and gauge bosons [72], which is another crucial issue in this context. A preliminary analysis shows that at √ s = 14 TeV, the total error that one obtains on the NNLO total production cross section is of the order of 25% for MH ≈ 160 GeV, i.e. much less than at the Tevatron. The main reason is that the PDF+αs uncertainties are slightly smaller than those obtained for the Tevatron, while the scale uncertainty (in which one needs only the more reasonable factor κ = 2) is (if not the only relevant) Higgs channel at the Tevatron in the Higgs mass range 150 GeV < ∼ M H < ∼ 180 GeV, might be a factor of two smaller than what has been assumed by the CDF and D0 collaborations in their recent analysis which excluded the Higgs mass range between 162 and 166 GeV. We thus believe that this analysis should be reconsidered in the light of these larger theoretical uncertainties in the signal cross sections 22 .
Of course, one can view the results presented in this paper with a more optimistic perspective: since the uncertainties in the gg → H process are so large, the cross section might well be closer to its upper limit which is ≈ 50% higher than the central value. In this lucky situation, the sensitivities of the CDF and D0 collaborations would be significantly increased and if the Higgs boson happens to have a mass in the range M H ≈ 160-170 GeV, some evidence for the particle at the Tevatron might soon show up.
Finally, in the case of the Higgs-strahlung processes, the cross sections are much more under control, the main reason being due to the fact that the QCD corrections are moderate. The scale uncertainties are at percent level for the narrow domain chosen for the scale variation (within a factor of two from the central scale), while the PDF uncertainties and the associated uncertainties due to the experimental and theoretical errors on α s are much smaller than in the gg → H case. The total estimated theoretical error on the Higgsstrahlung cross sections, ≈ 10%, is nevertheless almost twice as large as the error assumed by the CDF and D0 collaborations.