Higgs production in gluon fusion at next-to-next-to-leading order QCD for finite top mass
Robert V. Harlander, Hendrik Mantler, Simone Marzani, Kemal J. Ozeren
aa r X i v : . [ h e p - ph ] D ec December 2009 — WUB/09-18MAN/HEP/2009/44
Higgs production in gluon fusion at next-to-next-to-leadingorder QCD for finite top mass
Robert V. Harlander (1) , Hendrik Mantler (1) , Simone Marzani (2) , Kemal J. Ozeren (1)(1)
Fachbereich C, Bergische Universit¨at Wuppertal42097 Wuppertal, Germany (2)
School of Physics & Astronomy, University of Manchester,Manchester M13 9PL, United Kingdom
Abstract
The inclusive Higgs production cross section from gluon fusion is calculated through
NNLO QCD , including its top quark mass dependence. This is achieved through amatching of the 1 /M t expansion of the partonic cross sections to the exact large-ˆ s limits which are derived from k T -factorization. The accuracy of this procedure is esti-mated to be better than 1% for the hadronic cross section. The final result is shown tobe within 1% of the commonly used effective theory approach, thus confirming earlierfindings. It is well-known that a reliable quantitative prediction of the gluon fusion production crosssection for Higgs bosons requires a next-to-next-to-leading order (
NNLO ) calculation (fora review on Higgs physics, see Refs. [1, 2]). However, since it is a loop-induced process, itsNNLO correction requires a three-loop calculation of a 2 → NLO ) [3, 4, 5] that the perturbative K-factor is very wellreproduced in the so-called effective field theory ( EFT ) approach , where the gluon-Higgscoupling is taken into account by an effective Lagrangian L eff = − H v C G µν G µν , (1)with H the Higgs field, G µν the gluonic field strength tensor, v = 246 GeV the vacuumexpectation value of the Higgs field, and C a perturbatively evaluated Wilson coefficient(see, e.g., Ref. [6, 7]). The NLO cross section in the
EFT approach is then obtained byscaling the LO cross section (obtained in the full theory) with the effective NLO
K-factor.1lthough, to our knowledge, a quantitative understanding of the accuracy of this approachis still missing (in the sense that there is no error estimate), the observed difference of lessthan 1% to the full
NLO cross section (which is known in numerical form [8]) for M H < M t was considered to be sufficiently convincing in order to trust the EFT approach also at
NNLO .Apart from the inclusive
NNLO calculation [9, 10, 11], the heavy-top limit has also beenused for distributions, resummations, and even fully differential quantities at
NNLO (fora review, see Ref. [12]). It is therefore of the utmost importance to justify the validity ofthe
EFT approach. This has been first achieved at
NNLO in Ref. [13, 14, 15, 16] by anexpansion of the relevant Feynman diagrams in the limit M , ˆ s ≪ M t , where √ ˆ s is thepartonic center-of-mass energy. The apparent failure of this expansion for large ˆ s (whichis only restricted by the hadronic center-of-mass energy squared, s ) is only partly curedby the strong suppression of the parton luminosity. The prediction of the gg channelcontribution, which accounts for more than 95% of the NLO hadronic cross section, wasadditionally treated by matching to the known large-ˆ s behaviour [17]. The other channels,for which the large-ˆ s behaviour is not known, were treated by including only those termsin the 1 /M t expansion up to which the series was observed to converge. It was found thatthe resulting cross section agrees with the EFT result to better that 1% over the relevantmass range between 100 and 300 GeV.In this paper, we extend the analysis of Refs. [13, 15, 17] by deriving the high-energy limitsof the other channels as well. This leads to a significant stabilization of the qg channelwhich contributes about 2-5% to the total NLO cross section. The peculiar thresholdbehaviour of the quark–anti-quark channel prohibits a reasonable approximation from thehigh- and the low-energy information alone, but its contribution is in any case only at theper-mille level. We can therefore safely claim to present a stable prediction for the totalcross section including top quark mass effects for Higgs masses between 100 and 300 GeV.
For the convenience of the reader, let us outline the notation at the very beginning. TheHiggs mass is denoted by M H , the on-shell top quark mass by M t , and the hadronic andthe partonic center of mass energies are s and ˆ s , respectively. Unless indicated otherwise, α s ≡ α (5) s ( µ ) denotes the strong coupling in the MS scheme for five active flavours at therenormalization scale µ R . The following variables will turn out to be useful throughout2he text: z = M s , x = M ˆ s , τ = 4 M t M , ω = ˆ ss .l F = ln µ M l R = ln µ M l t = ln M t M , (2)with the factorization scale µ F .The inclusive hadronic cross section σ pp ′ for Standard Modell Higgs production in proton–(anti-)proton collisions is obtained by convoluting the partonic cross section ˆ σ αβ for thescattering of parton α with parton β by the corresponding parton density functions φ α/p ( x )( PDF s): σ αβ ( z, τ, l F ) = Z z d ω E αβ ( ω, µ F ) ˆ σ αβ ( z/ω, τ, l F ) ,σ pp ′ ( z, τ ) = X α,β ∈{ q, ¯ q,g } σ αβ ( z, τ, l F ) , p ′ ∈ { p, ¯ p } , E αβ ( ω, µ F ) ≡ Z ω d yy (cid:2) φ α/p ( y, µ F ) φ β/p ′ ( ω/y, µ F ) (cid:3) . (3)Note that the σ αβ depend on the factorization scheme (we use MS throughout this paper);only their sum σ pp ′ is physical and thus formally independent of the factorization scale.Nevertheless, it will be useful to study the individual contributions to the total cross sectionseparately as they have very different characteristics. The E αβ are parton luminosities andwill be discussed in more detail in Section 3.1.We write the top quark induced partonic cross section asˆ σ αβ ( x, τ, l F ) = σ ( τ ) ∆ αβ ( x, τ, l F ) , (4)with σ ( τ ) = π √ G F (cid:16) α s π (cid:17) τ (cid:12)(cid:12)(cid:12)(cid:12) − τ ) arcsin √ τ (cid:12)(cid:12)(cid:12)(cid:12) , (5)where G F ≈ . · − GeV − is Fermi’s constant. The kinetic terms assume the form∆ αβ ( x, τ, l F ) = δ αg δ βg δ (1 − x ) + X n ≥ (cid:16) α s π (cid:17) n ∆ ( n ) αβ ( x, τ, l F , l R ) . (6)At NLO , the full M t dependence is known in numerical form [4] (the virtual terms areknown analytically [18, 19, 20]). 3 -4 -3 -2 -1 H =130GeV qg ggqq´qqqq– (a) -3 -2 -1 parton luminosity xM H =280GeV qg ggqq´qqqq– (b) Figure 1: Parton luminosities E ( ω = z/x ) at the LHC for M t = 170 . M H = 130 GeV and (b) M H = 280 GeV, plotted as functions of x = M / ˆ s .The vertical line denotes the threshold ˆ s = 4 M t .A fully general result for the partonic cross section at NNLO is as of yet unknown. InRefs. [13, 14, 15, 16], it was evaluated in terms of an expansion of the form∆ αβ ( x, τ, µ F ) = X i ≥ (cid:18) M M t (cid:19) i Ω αβ,i ( x, l t , µ F ) , (7)with the analogous perturbative expansion as in Eq. (6). At NNLO , the first four terms( i ≤
3) have been evaluated [13, 14]. The so-called
EFT approach which has been used inall higher order analyses up to now, can be derived from the leading term of this expansion: σ pp ′ , ∞ ( z, l t ) ≡ X α,β ∈{ q, ¯ q,g } Z z d ω E αβ ( ω, µ F ) ˆ σ αβ, ∞ ( z/ω, l t , l F ) , ˆ σ αβ, ∞ ( x, l t , l F ) ≡ σ ( τ ) Ω αβ, ( x, l t , l F ) , (8)where σ is given in Eq. (5). 4 .1110100100 150 200 250 300 M H /GeVσ gg /pb pp @ 14 TeV below thresholdabove thresholdsum -1.5-1-0.500.51100 150 200 250 300 M H /GeVσ qg /pb pp @ 14 TeVbelow thresholdabove thresholdsum0.00010.0010.010.1100 150 200 250 300 M H /GeVσ q ¯ q /pb pp @ 14 TeV below thresholdabove thresholdsum Figure 2: Contributions of the partonic to the hadronic cross section from below(ˆ s < M t ; dashed) and above (ˆ s > M t ; dotted) threshold, for the gg , qg , andthe q ¯ q channel at NLO ( gg includes the LO contribution). Note that qg uses alinear scale, while for the gg and the q ¯ q it is logarithmic.5 Large- ˆ s limit The expansion of Eq. (7) is expected to converge within ˆ s, M . M t . While the Higgsmass range implied by electro-weak precision measurements lies comfortably in this range,the partonic center-of-mass energy √ ˆ s reaches values far beyond it, both at the LHC andthe Tevatron. The corresponding breakdown of convergence manifests itself in inversepowers of x = M / ˆ s , arising from ˆ sM t = M M t · x . (9)Thus, in general, Ω αβ,i ∼ x i as x → . (10)Note, however, that at small x = M / ˆ s there is a strong suppression by the partonluminosity E αβ of Eq. (3) which we display for the various sub-channels in Fig. 1. This,together with the fact that Ω αβ, has no power singularities as x →
0, are the main reasonsthat the heavy-top limit defined in Eq. (8) works so well.A further illustration of this observation is shown in Fig. 2 which compares the contribu-tions to the hadronic cross section arising from below (ˆ s ≤ M t ) and above threshold forthe various subchannels at NLO . For the dominant gg channel, the region above thresholdcontributes only of the order of 2%.However, the spurious 1 /x singularities described before imply that in order to improveon the heavy-top limit by including higher terms in 1 /M t , one needs to incorporate infor-mation on the large-ˆ s region. Fortunately, the leading terms can be obtained from generalconsiderations. In the case of the dominant gg -channel, this was done in Ref. [17]. Thisresult was then combined with the 1 /M t expansion in Refs. [13, 15].Considering Fig. 1, it appears that the center of the qg luminosity is at significantly lowervalues of x = M / ˆ s than for gg . Correspondingly, the influence of the region abovethreshold is larger, as can also be seen in Fig. 2. The proper treatment of this region isthus much more relevant in the qg case. In addition, it is clear a priori that the EFT approach, which assumes that the top mass dependence at higher orders is determined bythe LO one, cannot work as well in the qg channel which occurs only at NLO . In fact, thecontribution of the qg channel to the total cross section in the EFT differs from the exactresult by roughly a factor of two in the mass range between M H = 100 and 300 GeV [8].6n the next section, we extend the analysis of Ref. [17] to the qg and pure quark channels( q ¯ q , qq , qq ′ ) at NLO and
NNLO . The combination with the results of the 1 /M t expansionis done in Section 3.3. The procedure to compute the leading logarithmic behaviour ( LL x ) of the partonic coef-ficient function to all orders in the strong coupling α s is based on k T -factorization [21].This technique has been used to resum coefficient functions for a few processes, e.g. heavyquark production [22, 23], deep inelastic scattering [24], Drell-Yan processes [25] and di-rect photon production [26]. The small x behaviour of Higgs production in gluon fusionwas first computed in Ref. [27], in the heavy top approximation. The case of finite topmass was considered in Ref. [17], where it was shown that, as expected, the coefficientfunction has only single high energy logarithms, while double logarithms appear in theeffective theory. In Ref. [17], and in the phenomenological analysis of Ref. [28], only thegluon-gluon channel was considered. In the following the small x behaviour of all the otherchannels is computed, using high energy colour charge relations. For the sake of clarity,we set µ F = µ R throughout this derivation.The partonic cross section which enters the k T -factorization formula is the leading ordercross section for the process gg → H , computed with two incoming off-shell gluons ofmomenta k , , with k , = −| k , | , contracted with eikonal polarizations. The impactfactor is defined as the triple Mellin transform of the off-shell cross section h ( N, τ, M , M ) = M M Z dζζ N − Z ∞ dξ ξ M − Z ∞ dξ ξ M − Z π dϕ π (11) M H σ off ( ζ, τ, ξ , ξ , ϕ ) , where ξ i = | k i | M H , ζ = M H k · k − k · k ) (12)and ϕ is the angle between the transverse polarization vectors k and k . In Mellin spacethe high energy limit corresponds to N →
0; moreover it is easy to see that M i → M = M = γ s (cid:16) α s N (cid:17) , (13)where γ s is the anomalous dimension which is dual to the LO BFKL kernel χ , i.e. χ ( γ s ( α s /N )) = Nα s , (14)7 s (cid:16) α s N (cid:17) = ∞ X k =1 c k (cid:18) C A α s πN (cid:19) k , c k = 1 , , , ζ (3) , . . . (15)To all orders in perturbation theory, the leading logarithmic contribution to the MS coef-ficient function is ∆ gg ( N, τ, µ F ) = h (0 , τ, γ s , γ s ) R ( γ s ) (cid:18) M H µ (cid:19) γ s . (16)The factor R is a scheme dependent function, first computed for MS in [24]. A recentcalculation [29] has questioned that result. Although this issue must be solved for theresummation of the small x logarithms, it is not relevant for our present discussion. Ourtarget is to compute the LL x behaviour of the coefficient function through NNLO , but thescheme dependence starts only one order higher: R = 1 + O (cid:18)(cid:16) α s N (cid:17) (cid:19) . (17)The high energy behaviour of the other partonic channels can be derived from the gluon-gluon one by noticing that at LL x we have γ gg ∼ γ s , γ gq ∼ C F C A γ s , γ qq ∼ γ qg ∼ . (18)This means that, at LL x , a quark may turn into a gluon, but, because γ qg is next-to- LL x , agluon cannot turn into a quark. This leads to the following relations between the partoniccoefficient functions and the gluonic impact factor [24]:∆ qg ( N, τ, µ F ) = C F C A h h (0 , τ, γ s , γ s ) R ( γ s ) (cid:18) M H µ (cid:19) γ s (19) − h (0 , τ, γ s , R ( γ s ) (cid:18) M H µ (cid:19) γ s i , ∆ qq ( N, τ, µ F ) = (cid:18) C F C A (cid:19) h h (0 , τ, γ s , γ s ) R ( γ s ) (cid:18) M H µ (cid:19) γ s (20) − h (0 , τ, γ s , R ( γ s ) (cid:18) M H µ (cid:19) γ s + h (0 , τ, , i . Notice that in the high energy limit ∆ qq = ∆ qq ′ = ∆ q ¯ q = O (cid:16) α s N (cid:17) , where qq refers to theidentical and qq ′ to the distinct flavour case.8he impact factor can be expanded in powers of M i , which corresponds to an expansionin powers of α s : h (0 , τ, M , M ) = M H σ ( τ ) h h (1) ( τ )( M + M ) (21)+ h (2) ( τ )( M + M ) + h (1 , ( τ ) M M + . . . i The coefficients h (1) , h (2) and h (1 , have been evaluated numerically in [17] . The onlydifference here is that, in order to compute the LL x behaviour of all partonic subprocesses,we must keep the contributions h (2) and h (1 , separated.It is then easy to substitute Eq. (21) into Eqs. (19), (20) and invert the N Mellin transformto obtain the result in x space. Through NNLO , the small x limit of the partonic coefficientfunctions can be written as follows:∆ gg ( x, τ ) = δ (1 − x ) + α s π h B (1) gg ( τ ) − C A l F + O ( x ) i + (cid:16) α s π (cid:17) (cid:20)(cid:16) A (2) gg ( τ ) − C A B (1) gg ( τ ) l F + 2 C A l F (cid:17) ln 1 x + B (2) gg ( τ ) + O ( x ) (cid:21) , ∆ qg ( x, τ ) = α s π h B (1) qg ( τ ) − C F l F + O ( x ) i + (cid:16) α s π (cid:17) " (cid:18) A (2) qg ( τ ) − C F B (1) gg ( τ ) l F + 32 C A C F l F (cid:19) ln 1 x + B (2) qg ( τ ) + O ( x ) , ∆ q ¯ q ( x, τ ) = ∆ qq ( x, τ ) = ∆ qq ′ ( x, τ ) == (cid:16) α s π (cid:17) (cid:20)(cid:18) A (2) qq ( τ ) − C F C A B (1) gg ( τ ) l F + C F l F (cid:19) ln 1 x + B (2) qq ( τ ) + O ( x ) (cid:21) . (22)Recall that we set µ F = µ R in this section. The full µ F , µ R -dependence is obtained byreplacing α s → α s ( µ ) " − α s ( µ ) π β l FR + (cid:18) α s ( µ ) π (cid:19) (cid:0) ( β l FR ) − β l FR (cid:1) (23)in σ ∆ αβ , where l FR = ln( µ /µ ) and β = 23 / β = 29 / A (2) αβ and B (1) αβ are provided in the form of numerical tables in Table 1.For all coefficients, the dependence on τ is very smooth and can safely be interpolatedby straight lines, for example. The NNLO constants B (2) αβ are currently unknown; theirinfluence on the final result will be studied at the end of Section 4. See Eq. (36) and Eq. (38) of that paper, but notice the differences in the notation, e.g. the definitionof τ . B (1) gg B (1) qg τ A (2) gg A (2) qg A (2) qq s behaviour at NLO (left table) and
NNLO (right).10 .3 Merging and partonic results
Let us recall the knowledge of the partonic cross section at
NNLO . Below threshold (ˆ s < M t ), the result is known in terms of an expansion in 1 /M t and (1 − x ) [13, 15, 14] . Bothexpansions are expected to converge very well as long as M H < M t ≈
340 GeV. This isindeed observed for the gg and the qg channels at NLO in Fig. 3 for M H = 130 GeV and inFig. 4 for M H = 280 GeV. They show the partonic cross sections below threshold, keepingterms of order (1 − x ) a (1 /M t ) b . In the left columns, a = 0 , . . . , b = 5 (long toshort dashes), while in the right columns, a = 8 and b = 0 , . . . ,
5. These figures comparethe expansions to the exact result which we derived using standard techniques (see, e.g.,Ref. [30]). In fact, the behaviour of the expansions suggests that, below threshold, thefinal result for the gg and the qg channels is numerically almost equivalent to the full M t and x dependence.The NLO q ¯ q channel, on the other hand, has a very peculiar structure at threshold. Atthis order only one diagram with q ¯ q annihilating into an s -channel gluon contributes.Such a diagram is not enhanced in either the large- or small- x region, leaving room fora relatively pronounced structure at the threshold which cannot be described properly inour approach. However, the contribution of the q ¯ q channel to the hadronic cross sectionis down by almost three orders of magnitude relative to the gg channel, and still a factorof ten relative to the qg channel. We will nevertheless investigate its influence on the finalprediction in more detail below. At higher orders we expect this effect to be reduced,because other diagrams with non-trivial high- or low- x limits will contribute.The corresponding curves at NNLO are shown in Figs. 5–8. There is no exact result thatone could compare to, but the quality of the convergence both of the 1 /M t and the (1 − x )expansions below threshold convincingly shows that they approximate the exact result toa very high degree in this region.From Section 3.2 we know the leading high energy behaviour for general values of M t and M H . There are many ways then to merge the available information into a smooth functionwith the correct high- and low-energy behaviour (see, e.g., Refs. [17, 28, 15]). We decideto use [13] ˆ σ ( n ) αβ ( x ) = ˆ σ ( n ) αβ,N ( x ) + σ A ( n ) αβ " ln 1 x − N X k =1 k (1 − x ) k + (1 − x ) N +1 h σ B ( n ) αβ − ˆ σ ( n ) αβ,N (0) i , (24)where ˆ σ ( n ) αβ,N ( x ) denotes the soft expansion of the partonic cross section through order Recently, the full x dependence was derived [16]. However, as argued before, the x dependence of the1 /M t expansion does not hold for x < M / (4 M t ). s ^ gg / s (NLO) M H =130GeVx -100-80-60-40-200 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ^ gg / s (NLO) M H =130GeVx-5-4-3-2-1012 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ^ qg / s (NLO) M H =130GeVx -5-4-3-2-1012 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ^ qg / s (NLO) M H =130GeVx012345678 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ^ qqb / s (NLO) M H =130GeVx 012345678 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ^ qqb / s (NLO) M H =130GeVx Figure 3: Partonic cross section at
NLO for M t = 170 . M H = 130 GeVat various orders in the expansion parameters (increasing order corresponds todecreasing dash size of the lines). Left column: O (1 /M t ) and O ((1 − x ) n ), n = 0 , . . . ,
8. Right column: O ((1 − x ) ) and O (1 /M nt ), n = 0 , . . . ,
5. Solid:exact. The dashed vertical line indicates the threshold.12 s ^ gg / s (NLO) M H =280GeVx -600-500-400-300-200-1000 0.7 0.75 0.8 0.85 0.9 0.95 1 s ^ gg / s (NLO) M H =280GeVx-5-4.5-4-3.5-3-2.5-2-1.5-1-0.50 0.7 0.75 0.8 0.85 0.9 0.95 1 s ^ qg / s (NLO) M H =280GeVx -5-4.5-4-3.5-3-2.5-2-1.5-1-0.50 0.7 0.75 0.8 0.85 0.9 0.95 1 s ^ qg / s (NLO) M H =280GeVx00.10.20.30.40.50.60.70.80.91 0.7 0.75 0.8 0.85 0.9 0.95 1 s ^ qqb / s (NLO) M H =280GeVx 00.10.20.30.40.50.60.70.80.91 0.7 0.75 0.8 0.85 0.9 0.95 1 s ^ qqb / s (NLO) M H =280GeVx Figure 4: Same as Fig. 3, but for M H = 280 GeV.13 s ^ gg / s (NNLO) M H =130GeVx -600-500-400-300-200-1000100200300 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ^ gg / s (NNLO) M H =130GeVx-100-80-60-40-2002040 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ^ qg / s (NNLO) M H =130GeVx -100-80-60-40-2002040 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ^ qg / s (NNLO) M H =130GeVx Figure 5: Partonic cross section ( gg and qg channel) at NNLO for M t = 170 . M H = 130 GeV for various orders in the expansion parameters (increasing ordercorresponds to decreasing dash size of the lines). Left column: O (1 /M t ) and O ((1 − x ) n ), n = 0 , . . . ,
7. Right column: O ((1 − x ) ) and O (1 /M nt ), n = 0 , . . . , O (1 /M t ) and O ((1 − x ) ). The dashed vertical line indicates the threshold.For the q ¯ q and the qq channel, see Fig. 6.14 s ^ qqb / s (NNLO) M H =130GeVx 02.557.51012.51517.52022.525 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ^ qqb / s (NNLO) M H =130GeVx00.250.50.7511.251.51.752 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ^ qq / s (NNLO) M H =130GeVx 00.250.50.7511.251.51.752 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s ^ qq / s (NNLO) M H =130GeVx Figure 6: Same as Fig. 5, but for the q ¯ q and the qq channels (identical quarkflavors). The figure for the qq ′ channel (different quark flavours) is not shownsince it is almost indistinguishable from the one for qq .15 s ^ gg / s (NNLO) M H =280GeVx -6000-5000-4000-3000-2000-10000 0.7 0.75 0.8 0.85 0.9 0.95 1 s ^ gg / s (NNLO) M H =280GeVx-100-90-80-70-60-50-40-30-20-100 0.7 0.75 0.8 0.85 0.9 0.95 1 s ^ qg / s (NNLO) M H =280GeVx -100-90-80-70-60-50-40-30-20-100 0.7 0.75 0.8 0.85 0.9 0.95 1 s ^ qg / s (NNLO) M H =280GeVx Figure 7: Same as Fig. 5, but for M H = 280 GeV.16 s ^ qqb / s (NNLO) M H =280GeVx 012345678910 0.7 0.75 0.8 0.85 0.9 0.95 1 s ^ qqb / s (NNLO) M H =280GeVx00.250.50.7511.251.51.752 0.7 0.75 0.8 0.85 0.9 0.95 1 s ^ qq / s (NNLO) M H =280GeVx 00.250.50.7511.251.51.752 0.7 0.75 0.8 0.85 0.9 0.95 1 s ^ qq / s (NNLO) M H =280GeVx Figure 8: Same as Fig. 6, but for M H = 280 GeV.17 -3 -2 -1 s ^ gg / s (NLO)M H =130GeV x -30-25-20-15-10-505101520 10 -3 -2 -1 s ^ gg / s (NLO)M H =280GeV x-5-4-3-2-101234 10 -3 -2 -1 s ^ qg / s (NLO)M H =130GeV x -5-4-3-2-101234 10 -3 -2 -1 s ^ qg / s (NLO)M H =280GeV x012345678 10 -3 -2 -1 s ^ qqb / s (NLO)M H =130GeV x 012345678 10 -3 -2 -1 s ^ qqb / s (NLO)M H =280GeV x Figure 9:
NLO partonic cross sections as constructed from Eq. (24) (with N = 8)by including successively higher orders in 1 /M t . Dashed: O ( M nt ), n = 0 , . . . , M H = 130 GeV/ M H = 280 GeV. The dashedvertical line indicates the threshold. 18 -3 -2 -1 s ^ gg / s (NNLO)M H =130GeV x -400-2000200400600 10 -3 -2 -1 s ^ gg / s (NNLO)M H =280GeV x-100-50050100150200250 10 -3 -2 -1 s ^ qg / s (NNLO)M H =130GeV x -100-80-60-40-20020406080100 10 -3 -2 -1 s ^ qg / s (NNLO)M H =280GeV x Figure 10:
NNLO partonic cross sections ( gg and qg channel) as constructed fromEq. (24) (with N = 8) by including successively higher orders in 1 /M t . Dashed: O ( M nt ), n = 0 , ,
2. Solid: n = 3. Left/right column: M H = 130 GeV/ M H =280 GeV. The dashed vertical line indicates the threshold. For the q ¯ q and the qq channel, see Fig. 11. 19 -3 -2 -1 s ^ qqb / s (NNLO)M H =130GeV x 050100150200250300 10 -3 -2 -1 s ^ qqb / s (NNLO)M H =280GeV x05101520253035404550 10 -3 -2 -1 s ^ qq / s (NNLO)M H =130GeV x 012345678910 10 -3 -2 -1 s ^ qq / s (NNLO)M H =280GeV x Figure 11: Same as Fig. 10, but for the q ¯ q and the qq channels (identical quarkflavors). The figure for the qq ′ channel (different quark flavours) is not shownsince it is almost indistinguishable from the one for qq .201 − x ) N . For the unknown constants at NNLO we use the default values σ B (2) αβ = ˆ σ ( n ) αβ,N (0),but we will study their influence on the NNLO hadronic results at the end of Section 4.At
NLO , the resulting partonic cross sections are shown in Fig. 9 for M H = 130 GeV and M H = 280 GeV. The precision to which the gg and qg channels reproduce the exact resultis quite impressive. As expected, the q ¯ q channel is approximated only very poorly though.The corresponding plots at NNLO are shown in Fig. 10 and 11, for the default values ofthe high energy constant B (2) αβ . Note that due to this undetermined constant, the curvesdo not all converge to the same point for x → NLO . Only the slope isdetermined by the logarithmic coefficient A (2) αβ . Nevertheless, convergence for the gg andthe qg channel is very good, in particular at low Higgs masses. At larger Higgs masses,the x dependence becomes more and more unreliable as more terms in 1 /M t are included.This leads to the observed variations of the final result at M H = 280 GeV. As will beshown at the end of Section 4, however, the variations affect the hadronic cross section byless than 1%.As expected, the q ¯ q channel does not seem to converge, but its contribution to the hadroniccross section is negligible as are those of the qq and the qq ′ channels (equal and differ-ent quark flavours, respectively). The observed convergence of the latter is much betterthough. In the following, we will include them in the total hadronic cross section, but wewill only discuss the q ¯ q channel as representative of the pure quark channels (the one withthe worst convergence behaviour). In order to study the effect of the 1 /M t terms on the hadronic cross section, we define (seealso Ref. [13]) ˆ σ NLO αβ ( M nt ) = σ δ αg δ βg δ (1 − x ) + ˆ σ (1) αβ ( M nt ) , ˆ σ NNLO αβ ( M nt ) = σ h δ αg δ βg δ (1 − x ) + ∆ (1) αβ, ∞ i + ˆ σ (2) αβ ( M nt ) , (25)where ˆ σ ( k ) αβ ( M nt ) is the N k LO contribution to the partonic cross section evaluated as anexpansion through O (1 /M nt ), and matched to the low- x limit as described in Section 3.2.∆ (1) αβ, ∞ is the EFT result as defined in Eq. (8). Note that this differs from an extended
EFT approach , where ∆ ( k ) αβ would be expanded in terms of 1 /M t , while the full τ dependencein σ ( τ ) is kept. We will return to this latter approach at the end of this section. Thecorresponding hadronic quantities derived from Eq. (25) are denoted by σ NLO αβ ( M nt ) and σ NNLO αβ ( M nt ). 21ig. 12 shows the relative gg , qg and q ¯ q contribution σ NLO αβ ( M nt ) to the total hadronic crosssection. The dashed lines correspond to successively higher orders in 1 /M t , while the solidline shows the exact result. The curves are all normalized to the exact NLO cross section σ NLO . For the gg and the qg channels, one observes excellent convergence towards theexact result (solid line). The small deviations are reflections of the deviations between thesolid and the dashed lines in Fig. 9 (a)-(d). As pointed out above, the mass effects in the qg channel are quite large, ranging from roughly a factor of two to four in the relevant Higgsmass range. Of course, the overall size of the qg channel is below 5%. As expected, thepicture in the q ¯ q channel is significantly worse. Only the order of magnitude is captured,but there is no sign of convergence towards the exact result whatsoever. Its contributionto the total cross section is only of order 10 − though and thus irrelevant.The corresponding plots at NNLO are shown in Fig. 13. Since there is no exact result inthis case, we normalize the curves to the full
NNLO EFT result, cf. Eq. (8). Also, the solidlines always refer to the subchannels evaluated in the
EFT approach. The observations arequite similar as at
NLO : the difference between the
EFT result and the 1 /M t expansionfor the gg channel is about 1% which is of the order of the accuracy to which we expectthe capture the mass effects. In the qg channel, the relative difference between the EFT result and the 1 /M t expansion is significantly larger ( ∼ qg channel. The q ¯ q channel does not seem to converge very well, but is numericallynegligible (the true mass effects are not expected to change this).The hadronic results for the Tevatron are shown in Fig. 14 at NLO and
NNLO . The conclu-sions are very similar to those for the
LHC , thus justifying the use of the
EFT approximationfor Higgs searches also in this case [31].Overall, we conclude that the final result for the
NNLO cross section including top masseffects is within 1% of the
EFT result.
Dependence on B (2) αβ . As pointed out above, the constants B (2) αβ for the large-ˆ s be-haviour are currently unknown. From the curves in Fig. 10 and 11, our choice σ B (2) αβ =ˆ σ (2) αβ (0) seems to be reasonable, leading to rather smooth curves over the full x -range.Nevertheless, in order to estimate the uncertainty induced by this unknown constant, weset σ B (2) gg = t × ˆ σ (2) gg (0) and find that the dependence of the hadronic cross section on t isvery well described by a linear function: σ NNLO (cid:12)(cid:12)(cid:12)(cid:12) t ≈ (1 − . t ) σ NNLO (26)22 .950.960.970.980.9911.011.021.031.041.05 100 120 140 160 180 200 220 240 260 280 300 s NLOgg / s NLOexact M H /GeVpp @ 14 TeV1/M t n , n=0,...,10 s NLO gg,exact (a) -0.05-0.045-0.04-0.035-0.03-0.025-0.02-0.015-0.01-0.0050 100 120 140 160 180 200 220 240 260 280 300 s NLOqg / s NLOexact M H /GeVpp @ 14 TeV1/M t n , n=0,...,10 s NLO qg,exact (b) -2
100 120 140 160 180 200 220 240 260 280 300 s NLOqqb / s NLOexact M H /GeVpp @ 14 TeV1/M t n , n=0,...,10 s NLO qqbar,exact (c) s NLO / s NLOexact M H /GeVpp @ 14 TeV1/M t n , n=0,...,10 (d) Figure 12: (a)-(c) Sub-channel contributions to the hadronic cross section at
NLO ,normalized to the full result. Note that gg includes the exact LO contribution,cf. Eq. (25). Dashed: including terms of order 1 /M t n in the numerator ( n =0 , . . . , .90.9250.950.97511.0251.051.0751.1 100 120 140 160 180 200 220 240 260 280 300 s NNLOgg / s NNLOeff M H /GeVpp @ 14 TeV1/M t n , n=0,...,6 s NNLO gg,eff (a) -0.06-0.055-0.05-0.045-0.04-0.035-0.03-0.025-0.02 100 120 140 160 180 200 220 240 260 280 300 s NNLOqg / s NNLOeff M H /GeVpp @ 14 TeV1/M t n , n=0,...,6 s NNLO qg,eff (b) -2
100 120 140 160 180 200 220 240 260 280 300 s NNLOqqb / s NNLOeff M H /GeVpp @ 14 TeV1/M t n , n=0,...,6 s NNLO qqbar,eff (c) s NNLO / s NNLOeff M H /GeV pp @ 14 TeV1/M t n , n=0,...,6 (d) Figure 13: (a)-(c) Sub-channel contributions to the hadronic cross sectionat
NNLO , normalized to the full
NNLO EFT result (
LHC conditions). Notethat all channels include their lower order contributions in the
EFT approach (cf. Eq. (25)). Dashed: including terms of order 1 /M t n in the numerator( n = 0 , , , EFT result. (d) Sum overall sub-channels. 24 .90.9250.950.97511.0251.051.0751.1 100 120 140 160 180 200 s NLO / s NLOexact M H /GeVpp @ 1.96 TeV–1/M t n , n=0,...,10 (a) s NNLO / s NNLOeff M H /GeVpp @ 1.96 TeV–1/M t n , n=0,...,6 (b) Figure 14: Hadronic cross section at (a)
NLO and (b)
NNLO for the Tevatron,normalized to the full
NNLO EFT result. Note that the lower order contributionsare included in the
EFT approach (cf. Eq. (25)). Dashed: including terms of order1 /M t n in the numerator ( n = 0 , , , EFT result.Again recalling the smoothness of the curves in Fig. 10 and 11, we do not expect theparameter t to be significantly larger than one. The resulting uncertainty is therefore atmost at the percent level and therefore much smaller than the scale uncertainty of the NNLO result.
Is the heavy-top limit a coincidence?
Let us conclude this section with a remarkon the extended
EFT approach as mentioned in the discussion after Eq. (25). It wouldbe possible that the high quality of the
EFT approach is a coincidence, in the sense thatthere is an accidental cancellation among the higher order terms in the 1 /M t expansion ofthe ∆ αβ . This would have a significant effect on the applicability of the EFT approach toother quantities, of course.However, we have checked that this is not the case. All the curves of the extended
EFT approach lie within 1% of our final result. 25
Conclusions
The hadronic Higgs production cross section due to gluon fusion was presented includingeffects from a finite top quark mass. We have extended previous analyses by derivingthe high-energy limits of all partonic sub-channels and combining them with the known1 /M t expansions. Although the mass effects on the absolute size of the qg channel arelarge, they have no significant effect on the total hadronic cross section. Therefore, themain conclusions of previous analyses [14, 16] remain valid, and the EFT approach is stilljustified.
Acknowledgments.
We would like to thank Stefano Forte for useful comments on themanuscript. This work was supported by
DFG contract HA 2990/3-1, and by the
HelmholtzAlliance “Physics at the Terascale” . SM would like to thank Bergische Universit¨at Wup-pertal for the kind hospitality. The work of SM is supported by UK’s STFC.
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