Scale dependence and collinear subtraction terms for Higgs production in gluon fusion at N3LO
aa r X i v : . [ h e p - ph ] J u l Preprint typeset in JHEP style - PAPER VERSION
Scale dependence and collinear subtraction terms forHiggs production in gluon fusion at N3LO
Stephan Buehler
Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, SwitzerlandE-mail: [email protected]
Achilleas Lazopoulos
Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, SwitzerlandE-mail: [email protected]
Abstract:
The full, explicit, scale dependence of the inclusive N LO cross section for sin-gle Higgs hadroproduction is obtained by calculating the convolutions of collinear splittingkernels with lower-order partonic cross sections. We provide results for all convolutions ofsplitting kernels and lower-order partonic cross sections to the order in ǫ needed for thefull N LO computation, as well as their expansions around the soft limit. We also dis-cuss the size of the total scale uncertainty at N LO that can be anticipated with existinginformation.
Keywords:
QCD, NNLO, NNNLO, Higgs, LHC, Tevatron. . Introduction
During the past year both multi-purpose experiments at CERN’s Large hardon collider(LHC), CMS [1] and Atlas [2], have observed a new boson with a mass of about 125 GeV,which is strongly believed to be the long-sought Higgs boson. The couplings of the newboson to Standard Model (SM) particles are currently compatible to the SM predictionswith a minimal Higgs sector. Nevertheless, effects from physics beyond the Standard Model(BSM) may reside in small deviations of the couplings from the SM values, effects thatwill be, to a certain extent, accessible with the increased statistics and energy reach of theLHC high energy run starting in 2015.The dominant production mode for the Higgs boson at the LHC is gluon fusion, ac-counting for about 90% of the total production cross section at the observed mass of about125 GeV. Indeed, the Higgs boson has, up to now, been observed in channels in whichits production is gluon induced. Next-to-leading order (NLO) QCD corrections for gluonfusion, in the five-flavour heavy-quark effective theory (HQET) were computed at the be-ginning of the 1990s [3,4]. Since then NLO corrections in the full theory including top-mass,and top-bottom interference effects were calculated by [5–9], and next-to-next-to-leadingorder (NNLO) corrections in HQET by [10–12]. Electroweak corrections are also availableat the NLO level [13–16], and so are mixed QCD-EW corrections [17], and EW correctionsto higgs plus jet including top and bottom quark contributions [18]. Recently the NNLOcross section for gluon induced higgs production in association to a jet was calculated,in a way that also allows for differential distributions to be produced [19]. All availablefixed order contributions to Higgs production via gluon fusion were recently included in theprogram iHixs [20], which, moreover, allows for the incorporation of BSM effects throughmodified Wilson coefficients within the effective theory approach. The latter has beenexplicitly shown to be an excellent approximation for Higgs masses below the top-antitopthreshold [21–23], and even more so for the Higgs boson at 125 GeV. Despite these ad-vances and due to the slow perturbative convergence of the gluon fusion cross section,the remaining uncertainty due to variation in renormalisation and factorisation scale stillamounts to about ∼
9% for a 125 GeV Higgs boson at the LHC with 8 TeV centre-of-massenergy.Beyond fixed order, threshold resummation has been performed to NNLL accuracy bytraditional resummation methods [24] leading to a ∼ .
5% uncertainty [25], and withinthe SCET framework [26–28] leading to a ∼
4% scale uncertainty. The latter is generallyconsidered too optimistic.Information from the LHC high energy and high luminosity data set is projected toallow the determination of the Higgs couplings with precision of ∼
10% or better [29–31].This uncertainty includes experimental systematics and statistics, but also errors fromthe determination of parton distribution functions and of the strong coupling, as well astheory systematics, the latter being the limiting factor in several cases. It is evident that aprerequisite to this goal is the reduction of the theory scale uncertainty to the ∼
5% levelor lower. The question arises then, whether computing the cross section to the next orderin perturbation theory, N LO, within the EFT approach, an admittedly formidable task,– 1 –ould achieve this goal.Information about certain N LO contributions has been available for several years.The three-loop, virtual contributions have been calculated and were part of the full N LOHiggs decay to gluons in [32]. However, disentangling the pure virtual contributions fromthis computation is not possible. The quark and gluon form-factors are known up to three-loop order [33–36]. In [37] the soft ’plus’-contributions to the N LO cross section werederived using mass factorisation constraints. This allowed the authors of [37] to derive asoft approximation of the N LO cross section whose renormalisation scale dependence israther mild, resulting in 4% renormalisation scale uncertainty (keeping the factorisationscale equal to the Higgs mass). Recently further attempts to modify the resummationprocedure such that its prediction at fixed order better matches the threshold and highenergy limits of the known fixed order results, were made [38], resulting in another softapproximant with a scale uncertainty of 7%. It still remains true that without the fullN LO expression, it is difficult to judge which of these prescriptions is closer to reality.Recently, some new ingredients of the full N LO cross section have appeared. In [39,40],the real-virtual and double-real master integrals of the NNLO cross section have beencalculated to higher orders in ǫ . In [41], the convolutions of collinear splitting kernels withlower-order partonic cross sections have been computed, which is also an ingredient for ourresult and has been re-derived in this work. Very recently, the soft limits of all masterintegrals appearing in triple-real radiation corrections (i.e. the emission of three additionalpartons) have been worked out [42].In this paper, we compute the full dependence of the N LO cross section on the factori-sation and renormalisation scales, which can be obtained from lower-order results. Further-more, we provide the soft limits of all convolutions that we calculated, which may becomeuseful when expanding the full N LO corrections around threshold. In section 2 we reviewhow the dependence on factorisation and renormalisation scales enters higher-order calcu-lations. In sections 3 and 4 we list the splitting kernels and partonic cross sections neededfor our results and present the method used to compute their convolutions, respectively.In section 5 we give results for the estimated scale uncertainty of the N LO gluon fusioncross section and conclude in section 6.
2. Sources of explicit scale dependence
Predictions for observable quantities in quantum field theory are independent of arbitraryscales, when calculated at all perturbative orders. The scale dependence of all predictions isan artefact of the truncation of the perturbative series, and is usually considered a measureof the effect of missing higher orders in any given computation. This dependence occursexplicitly, through terms in the final result that depend on logarithms involving the scale,and implicitly, through the running of α s and the evolution of the parton distributionfunctions. In this section we describe the occurrence of the explicit scale dependence.Let us, for the moment, introduce only one scale, µ r = µ f ≡ µ. (2.1)– 2 –n dimensional regularisation the scale µ appears during renormalisation, when the barecoupling is replaced by the renormalised one, α ( B ) s α s ( µ ) µ ǫ (cid:18) e γ E π (cid:19) ǫ Z α , (2.2)where we have chosen the MS-scheme. Z α is the renormalisation constant of the strongcoupling, Z α = 1 − a ( µ ) β ǫ + a ( µ ) (cid:18) β ǫ − β ǫ (cid:19) + a ( µ ) (cid:18) − β ǫ + 7 β β ǫ − β ǫ (cid:19) + O ( a ) , (2.3)and the factor of µ ǫ ensures that the coupling and thus the action remain dimensionlessin D = 4 − ǫ dimensions as well. We define a ( µ ) ≡ α s ( µ ) π throughout the paper.Divergences, of UV or IR nature, manifest themselves as poles in the regularisationparameter ǫ . The leading divergences, ǫ − n , . . . , ǫ − n − for the n -th order correction, vanish,among real and virtual contributions, after renormalisation counter terms are included. Theremaining poles of the UV-renormalized partonic cross section, starting from ǫ − n , vanishonly after subtraction of collinear counterterms.Specifically, let us denote by ˆ σ ij the partonic cross section after renormalisation (whichstill contains divergences of infrared (IR) origin),ˆ σ ij = 1 z s Z [ d Φ] X spin , col (cid:12)(cid:12) M (cid:12)(cid:12) + U V counterterms (2.4)The expansion of ˆ σ ij can be written asˆ σ ij = ∞ X n =0 a n ˆ σ ( n ) ij = ∞ X n =0 a n ∞ X r = − n ˆ σ ( n,r ) ij ǫ r e L f ǫ (2.5)where we have written explicitly the pole coefficients at every order in a and the associatedlogarithms L f ≡ log( µ /s ). The relation of ˆ σ ij to the total, inclusive cross section is givenby convolution with the parton distribution functions, f i ( x ), and the collinear counterterms, Γ ij ( x ), by σ ( τ ) = τ (cid:16) f i ⊗ Γ − ki ⊗ ˆ σ kl ⊗ Γ − lj ⊗ f j (cid:17) ( τ ) , (2.6)where summation of repeated indices is implied and τ = m h /S with S the total centre-of-mass energy of the collision.The convolution is defined by( f ⊗ g ) ( z ) = Z d x d y f ( x ) g ( y ) δ ( xy − z ) , (2.7) Note the 1 /z factor in the definition of ˆ σ ij that is necessary to make eq. 2.6 work. – 3 –nd the collinear counter term readsΓ ij ( x ) = δ ij δ (1 − x ) − a ( µ ) P (0) ij ( x ) ǫ + a ( µ ) ( − ǫ P (1) ij ( x ) + 12 ǫ h(cid:16) P (0) ik ⊗ P (0) kj (cid:17) ( x ) + β P (0) ij ( x ) i ) + a ( µ ) ( − ǫ h (cid:16) P (0) ik ⊗ P (0) kl ⊗ P (0) lj (cid:17) ( x ) + 3 β (cid:16) P (0) ik ⊗ P (0) kj (cid:17) ( x )+ 2 β P (0) ij ( x ) i + 112 ǫ h (cid:16) P (0) ik ⊗ P (1) kj (cid:17) ( x ) + 3 (cid:16) P (1) ik ⊗ P (0) kj (cid:17) ( x )+ 4 β P (0) ij ( x ) + 4 β P (1) ij ( x ) i − ǫ P (2) ij ( x ) ) + O ( a ) . (2.8)The P ( n ) ij are the Altarelli-Parisi splitting kernels which govern the emission of collinearpartons (see section 3.2).Within the renormalized N n LO partonic cross section, ˆ σ ij , the logarithmic dependenceon the scale µ arises when residual poles of order up to ǫ − n are multiplied with the expansionof the factor µ ǫ /s ǫ (the s − ǫ originating from the d -dimensional phase-space measure), µ ǫ s ǫ = 1 + ǫ log (cid:18) µ s (cid:19) + ǫ (cid:18) µ s (cid:19) + ǫ (cid:18) µ s (cid:19) + O ( ǫ ) . (2.9)These poles are required to cancel against the poles from the collinear counter terms convo-luted with lower order partonic cross sections, see eq. (2.6) and (2.8). This requirement fixesthe coefficients ˆ σ ( n,r ) ij for − n − < r < n LO cross section that are proportional toa power of log( µ /s ) can be obtained from calculating the convolutions of splitting kernelsand lower-order, N m 3. Ingredients From the previous section, we conclude that we need the following ingredients to obtainall convolutions required for the N LO gluon fusion cross section: • The LO partonic cross section through O ( ǫ ). • The NLO partonic cross section through O ( ǫ ). • The NNLO partonic cross section through O ( ǫ ). • The LO splitting kernels P (0) gg , P (0) gq , P (0) qg , P (0) qq . • The NLO splitting kernels P (1) gg , P (1) gq , P (1) qg , P (1) qq , P (1) q ¯ q , P (1) qQ (where q = Q = ¯ q ). • The NNLO splitting kernels P (2) gg and P (2) gq (owing to the fact that at LO, only the gg -channel is nonzero). – 5 – .1 Partonic cross section We work in the effective five-flavour theory with the top quark integrated out. This ap-proximation has been shown to be very good (less than 5%) for light Higgs masses, ascan be seen by comparing the NLO results in effective and full six-flavour theory and bystudying the importance of 1 /m t corrections of the effective NNLO cross section [22, 23].We expect this behaviour to persist at N LO.The effective Lagrangian describing the interaction between gluons and the Higgs bosonis given by L eff = − v C ( B )1 G aµν G aµν H , (3.1)where G aµν denotes the gluonic field-strength tensor. The Wilson coefficient C whichstarts at O ( a ) has been computed perturbatively to four-loop accuracy [43, 44] in the SMas well as to three-loop accuracy for some BSM models [45–47]. Through O ( a ), the Wilsoncoefficient in the SM reads C = − a ( µ )3 ( a ( µ ) 114 + a ( µ ) (cid:20) L t + N F (cid:18) − L t (cid:19)(cid:21) + a ( µ ) " (cid:18) − L t − L t (cid:19) N F + (cid:18) L t + 5554 L t + 4029120736 − ζ (cid:19) N F − ζ + 20964 L t + 1733288 L t + O ( a ) ) , (3.2)with L t = log( µ /m t ). N F denotes the number of light flavours set to 5.The above expression denotes the renormalised Wilson coefficient, which is related to thebare one through the renormalisation constant, C ( B )1 = Z ( µ ) C ( µ ) , (3.3)with Z ( µ ) = 11 − β ( µ ) ǫ = 1 − a β ǫ + a (cid:18) β ǫ − β ǫ (cid:19) + a (cid:18) − β ǫ + 2 β β ǫ − β ǫ (cid:19) , (3.4)where we have suppressed the scale dependence of the strong coupling constant.The partonic cross section for the production of a Higgs boson through gluon fusioncan then be cast in the form σ ij ( z ) = C σ ∞ X n,m =0 ˜ σ ( n,m ) ij ( z ) a n ǫ m (3.5)where we kept the squared Wilson coefficient factorised and pulled out all dimensionfulprefactors, such that the ǫ -piece of the leading order cross section becomes just˜ σ (0 , ij ( z ) = δ ig δ jg δ (1 − z ) . (3.6)– 6 –ll convolutions calculated in this work are done using the ˜ σ ( n,m ) ij and from here on, theterm “cross section” will refer to these objects.The sole dependence of the LO cross section on ǫ is an overall factor of(1 − ǫ ) − = P ∞ n =0 ǫ n from averaging over the D -dimensional polarisations of the initialgluons. Thus, the LO partonic cross section through O ( ǫ ) is trivially found to be˜ σ (0 ,m ) ij ( z ) = δ ig δ jg δ (1 − z ) , (3.7)for all m = 0 , . . . , ǫ is still fairly simple. There are only two master integralsand they are easily computed to all orders in ǫ .The NNLO cross section through O ( ǫ ) necessitated the knowledge of the 29 masterintegrals to sufficiently high order in ǫ . The double-virtual master integrals can be foundin work on the two-loop gluon form factor [48–50]. The real-virtual and double-real masterintegrals were computed by two groups independently during the last year [39, 40]. Theexpression for the bare NNLO cross section in terms of master integrals was kindly providedto us by an author of [11].In general, the partonic cross sections consist of three types of terms, delta-, plus- andregular terms .˜ σ ( n,m ) ij ( x ) = a ( n,m ) ij δ (1 − x ) + X k b ( n,m ) ,kij D k (1 − x ) + c ( n,m ) ij ( x ) , (3.8)where the plus-distribution D k (1 − x ) = h log k (1 − x )1 − x i + is defined via its action on a testfunction f ( x ) with a finite value at x = 1, Z d x D k (1 − x ) f ( x ) = Z log k (1 − x )1 − x ( f ( x ) − f (1)) . (3.9)The full expressions for the partonic cross sections through NNLO can be found in theancillary files accompanying this arXiv publication. They agree with the ones given in [41]after compensating for the factor of 1 /z that was not included in that publication. The splitting kernel P ij ( x ) = ∞ X n =0 a n +1 P ( n ) ij ( x ) (3.10)describes the probability of a parton j emitting a collinear parton i carrying a fraction x of the momentum of the initial parton. The splitting kernels are known up to three loops( P (2) ij ) and may all be found in [51, 52].Note some different conventions that we use, though. Since we chose to expand all ourresults in a = α s π as opposed to α s π as in [51, 52], our kernels P ( n ) ij differ from the ones inthe reference by a factor of (cid:0) (cid:1) n +1 . – 7 –lso, since by P ( n ) qg we mean the emission of a single quark of a given flavour, we differfrom the expression in [51, 52] which parametrises the emission of any quark, by a factorof N F .Furthermore, there is also a conventional difference to the splitting kernels used in [41].The authors of that publication use the quark-quark splitting kernel as defined in eq. (2.4)of [51]. This kernel, which we shall denote by ˜ P qq is used in the DGLAP evolution of pdfs.To compute all contributions to the N LO gluon fusion cross section, we have to distinguishdifferent initial-state channels such as q ¯ q (quark-antiquark), qq (identical quarks) and qQ (quarks of different flavour) which are convoluted with different combinations of pdfs.Thus, for channel-by-channel collinear factorisation, we require the three distinct, flavour-dependent quark-quark kernels P qq , P q ¯ q , P qQ , (3.11)which describe the emission of an identical quark, the emission of an antiquark of the sameflavour, and the emission of a quark or antiquark of a different flavour, respectively. Thelatter two kernels vanish at the one-loop order, P (0) q ¯ q = 0 = P (0) qQ but are nonzero for higherorders. In the notation of [52], this corresponds to the kernels P q i q j and P q i ¯ q j . The relationbetween ˜ P qq and our kernels is given by˜ P qq = P qq + P q ¯ q + 2( N F − P qQ . (3.12)We are not aware of results in [41] that involve the flavour-dependent quark-quark kernels.We close this section by giving the expressions for the four LO splitting kernels. Forthe lengthy higher-order kernels, we again refer to the machine-readable files accompanyingthis publication. The two NNLO kernels were taken from [53] in Form format and thentranslated to Maple input. Their regular parts were tested against the Fortran routinein [53], and their δ (1 − z ) and D n (1 − z ) parts were checked against [51]. P (0) gg ( x ) = β δ (1 − x ) + 3 (cid:18) D (1 − x ) + x (1 − x ) − x (cid:19) . (3.13) P (0) gq ( x ) = 23 1 + (1 − x ) x . (3.14) P (0) qg ( x ) = 14 (cid:0) x + (1 − x ) (cid:1) . (3.15) P (0) qq ( x ) = δ (1 − x ) + 23 (2 D (1 − x ) − − x ) . (3.16) 4. Computation of the convolutions In this section we will describe the method we used to compute the convolutions of split-ting kernels and partonic cross sections that are needed to cancel collinear divergences atN LO. Let us remark that our method is different from the technique used in [41], where theconvolutions were calculated in Mellin space (where convolutions turn into ordinary multi-plications) and the problem was essentially the calculation of the inverse Mellin-transform.– 8 –n the following, we restrict ourselves to a single convolution. Since the convolutionproduct is associative, any multiple convolution appearing in the N LO cross section canbe obtained by repeating the steps using the result of the first convolution and the nextconvolutant. As already mentioned, both the splitting kernels and the partonic crosssections consist of three types of terms, delta-, plus- and regular terms .˜ σ ( n,m ) ij ( x ) = a ( n,m ) ij δ (1 − x ) + X k b ( n,m ) ,kij D k (1 − x ) + c ( n,m ) ij ( x ) , (4.1) P ( n ) ij ( x ) = A ( n ) ij δ (1 − x ) + X k B ( n ) ,kij D k (1 − x ) + C ( n ) ij ( x ) , (4.2)where the regular pieces c ( n,m ) ij ( x ) and C ( n ) ij ( x ) consist of harmonic polylogarithms (HPLs)times polynomials in x or factors of 1 /x and 1 / (1 − x ). Harmonic polylogarithms are ageneralisation of the ordinary logarithm and the classical polylogarithms,log( x ) = Z z d tt and Li n ( z ) = Z z d tt Li n − ( t ) , (4.3)where Li ( z ) = − log(1 − z ). HPLs can be defined recursively via the integralH( a , a , . . . , a n ; z ) = Z z d tf a ( t ) H( a , . . . , a n ; t ) , a i ∈ {− , , } , (4.4)with f − ( x ) = 11 + x , f ( x ) = 1 x , f ( x ) = 11 − x , (4.5)and in the special case where all a i = 0, the HPL is defined asH( ~ n ; z ) = 1 n ! log n ( z ) . (4.6)For more comprehensive information about harmonic polylogarithms, we refer to [54–60].Any convolution involving a delta function trivially returns the other convolutant(whether it be another delta function, a plus distribution or a regular function), (cid:0) δ ⊗ f (cid:1) ( z ) = Z d x d y δ (1 − x ) f ( y ) δ ( xy − z ) = Z d y f ( y ) δ ( y − z ) = f ( z ) . (4.7)Convolutions involving two plus-distributions are more involved, yet no integral actuallyhas to be solved. We comment on their calculation and list results for the required plus-plusconvolutions in appendix A.For the remaining two types of convolutions, we end up with an actual integral thatwe need to compute, (cid:0) D n ⊗ f (cid:1) ( z ) = Z d x d y D n (1 − x ) f ( y ) δ ( xy − z ) = Z z d x D n (1 − x ) f (cid:0) zx (cid:1) x = log n +1 (1 − z ) n + 1 f ( z ) + Z z d x log n (1 − x )1 − x f (cid:0) zx (cid:1) x − f ( z ) ! , (4.8) (cid:0) f ⊗ g (cid:1) ( z ) = Z d x d yf ( x ) g ( y ) δ ( xy − z ) = Z z d xf ( x ) g (cid:16) zx (cid:17) x . (4.9)– 9 –ote the boundary term we pick up when evaluating a plus-distribution in an intervalwhich is different from (0 , z and factors of z or − z ), we have to step into the realm of multiple polylogarithms (MPLs) in intermediatesteps of the calculation. MPLs are defined analogously to HPLs, but allow for any complexnumber in the index vector instead of only {− , , } in the HPL case. Recursively, G ( x , x , . . . , x n ; z ) = Z z d t G ( x , . . . , x n ; t ) t − x , { x i } ∈ C and G ( z ) = 1 . (4.10)Specifically, a MPL may be a function of multiple variables that appear anywhere in theindex vector ( x , . . . , x n ). The relation to HPLs readsH( a , . . . , a n ; x ) = ( − k G ( a , . . . , a n ; x ) , { a i } ∈ {− , , } , (4.11)where k is the number of +1 indices in ( a , . . . , a n ). This sign difference is due to thefact that HPLs historically use − t as the weight function when adding a +1 to the indexvector.For more detailed information on multiple polylogarithms, see references [61–63] andreferences therein. Note that the order of the MPLs indices is often reversed. We followthe convention of [63].The subsequent steps to solve the integrals are as follows:1. We first remap the integral by x − x , such that the integration region becomes(0 , − z ).2. HPLs with argument 1 − x and z − x have to be written as a combination of MPLswith the integration variable x as their argument, or no x -dependence at all. Forexample, H (cid:18) z − x (cid:19) = − log (cid:18) − z − x (cid:19) = − log (cid:18) − x − z − x (cid:19) = log(1 − x ) − log (cid:18) (1 − z ) (cid:18) − x − z (cid:19)(cid:19) = G (1; x ) − G (1; z ) − G (1 − z ; x ) , (4.12)where we’ve used that G ( a ; b ) = log (cid:0) − ba (cid:1) for a = 0. For MPLs of higher weights,one can find these translations by using the recursive definition of MPLs and changingvariables in the integration. This becomes very tedious, though, so it proved to bemore practical to use the symbol formalism developed in recent years [59, 63–65] andto follow the method presented in appendix D of [42]. For technical details, we referthe reader to said appendix.3. We are left with integrals of a single MPL with argument x times factors of x k ( k ≥− 1) or − x , which can all immediately be solved via the MPLs recursive definitionand integration by parts. – 10 –. At this stage, all integrations have been performed. The result still contains MPLswhere the variable z appears multiple times in the argument vector. Using thetechniques from appendix D of [42] again, we can rewrite all the expressions in HPLs.5. The final numerical check on the result consists of the comparison of the originalintegral using Mathematica’s numerical integration (using the package HPL [57, 58] toevaluate the HPLs numerically) and our final expression, using Ginsh , the interactivefrontend of the computer algebra system GiNaC [56], for a random value of z .The full set of convolutions can be found in machine-readable form (both Maple and Mathematica ) in the ancillary files accompanying this arXiv publication. They were allcompared analytically in Mathematica to the expressions given in [66], and completeagreement was found for all convolutions. For convolutions involving the two-loop quark-quark splitting kernels P (1) qq , P (1) q ¯ q and P (1) qQ , the results had to be combined according toeq. (3.12) to find equality. While the full N LO corrections to the gluon fusion cross section may still be out of reachfor the time being, a description in the soft limit could be feasible already in the close future.Note that this was the sucession at NNLO, as well, where the expansion of the cross sectionup to O ((1 − z ) ) [10,67] was published before the full computation [11,12]. The numericalagreement between the two computations proved to be excellent, so, anticipating the samebehaviour at N LO, the soft expansion of the N LO corrections would be a very importantresult to obtain. The first pure N LO piece of the third order soft expansion, the softtriple-real emission contribution, has recently been published [42].In the limit z → 1, the partonic cross section (and all convolutions contributing to it)can be cast in the following form (suppressing partonic indices)˜ σ ( n,m ) ( z ) = a ( n,m ) δ (1 − z )+ n − X k =0 b ( n,m ) ,k D k (1 − z )+ n − X k =0 ∞ X l =0 c ( n,m ) kl log(1 − z ) k (1 − z ) l . (4.13)We thus need to expand the regular part as a polynomial in (1 − z ), times log(1 − z ) terms.We proceed as follows:1. We define z ′ ≡ − z . Thus, our expressions now consist of HPLs with argument1 − z ′ times powers of z ′ . The desired limit is z ′ → − z ′ as MPLs with argument z ′ , whichresults in changing the array of indices from {− , , } to { , , } , as can be easilyseen by taking the integral definition eq. (4.10) for x ∈ {− , , } and changingvariables t − t . The rewriting is achieved once again with the techniques fromappendix D of [42]. – 11 –. The expansion of any MPL in its argument is straightforward, since there is a con-nection between MPLs and multiple nested sums [61],Li m ,...,m k ( x , . . . , x k ) = ∞ X n k =1 x n k k n m k k n k − X n k − =1 · · · n − X n =1 x n n m (4.14)where the translation from MPLs to nested sums is given by G (cid:16) , . . . , | {z } m k − , , , . . . , | {z } m k − − , a k − , . . . , , . . . , | {z } m − , a ; x k (cid:17) = ( − k Li m ,...,m k (cid:18) a a , a a , . . . , a k − a k − , a k − , x k (cid:19) (4.15)The specific form of the MPL on the left-hand side of the equation above can beobtained via the scaling property, G ( x , . . . , x n ; z ) = G ( λx , . . . , λx n , λz ), where λ =0 = x n . MPLs with a rightmost index of 0 must be rewritten using the shuffleproduct, e.g. G ( a, x ) = G (0; x ) G ( a ; x ) − G (0 , a ; x ) = log( x ) G ( a ; x ) − G (0 , a ; x ) , (4.16)until all rightmost zeroes have been turned into explicit logarithms. The remainingMPLs can then be safely translated to nested sums.The crucial point is that the variable x k only appears in the outermost sum ineq. (4.14), while the inner nested sums only depend on the x i 5. Numerical results for the gluon fusion scale variation at N LO The total cross section for Higgs production through gluon fusion at N LO depends on thefactorisation and renormalisation scales explicitly, through logarithmic terms that havebeen derived in this work, and implicitly through the µ r dependence of α s and the µ f dependence of the parton distribution functions. In principle N LO parton distributionfunctions should be used, but in practice, not only are they not available (nor will theybe in the near future), but also their deviation with respect to the NNLO pdfs available isexpected to be very small. On the other hand, the full, implicit, µ r dependence through α s , can only be estimated once the N LO matrix elements are known, and in particular thecoefficients a (3 , ij , b (3 , ,kij and c (3 , ij ( z ) in eq. (3.8). Of these contributions only the b (3 , ,kij Note that we define the coefficients b ij of the plus terms to be independent of z . This implies that theregular terms c ( n,m ) ij ( z ) contain terms with the logarithms log( z ) and log(1 − z ). – 12 –re known, from mass factorisation constraints [37]. The question, then, arising is whetherwe can anticipate the scale uncertainty at N LO with the information currently available.To this end we parametrize the unknown delta and regular coefficients by a scalingfactor K times the corresponding NNLO coefficients: a (3 , ij = K a (2 , ij , f i ⊗ f j ⊗ c (3 , ij ( z ) = K (cid:16) f i ⊗ f j ⊗ c (2 , ij ( z ) (cid:17) , (5.1)There is no a priori reason why the scaling factor for the delta and the regular terms shouldbe the same. However, it turns out that the numerical impact of the delta coefficient a (3 , ij is negligible (for scaling coefficients that do not break by orders of magnitude the patternobserved from lower orders), in contrast with the coefficient of the regular part, so we adopthere a common scaling factor to keep the parametrisation simple. For the same reason weuse the same K scaling coefficient for all initial state channels. σ [ pb ] µ r /m h µ f = m h N3LO approx (K=0)N3LO approx (K=5)N3LO approx (K=10)N3LO approx (K=15)N3LO approx (K=20)N3LO approx (K=30)N3LO approx (K=40)NNLONLOLO Figure 1: Scale variation of the different orders of the gluon fusion cross section at 8 TeV. µ f is fixed to m h and only µ r is varied. The scaling coefficient K is varied from 0 to 40 toestimate the impact of the unknown N LO contributions.A loose argument about the size of K can be derived if one assumes a good perturbativebehaviour at µ r = µ f = m H where all other terms of order a vanish. Since a ( m H ) ∼ / K not to be much larger than 30. For comparison, the corresponding rescaling– 13 –actors between NNLO and NLO are f g ⊗ f g ⊗ c (2 , gg f g ⊗ f g ⊗ c (1 , gg ∼ , a (2 , gg a (1 , gg ∼ . , (5.2)for m h = 125 GeV and µ f = µ r = m h . σ [ pb ] µ /m h µ f = µ r N3LO approx (K=0)N3LO approx (K=5)N3LO approx (K=10)N3LO approx (K=15)N3LO approx (K=20)N3LO approx (K=30)N3LO approx (K=40)NNLONLOLO Figure 2: Scale variation of the different orders of the gluon fusion cross section at 8 TeV. µ f and µ r are varied simultaneously. The scaling coefficient K is varied from 0 to 40 toestimate the impact of the unknown N LO contributions.In what follows we study the inclusive cross section as a function of the scales, in theHQET approach, rescaled with the exact leading order cross section. We use the frame-work of the iHixs program [20] a Fortran code which contains the complete NNLO crosssection for gluon fusion in HQET. The coupling α s was run to four-loop order according toeq. (2.13), while for the parton distributions, the MSTW08 NNLO set was used. Further-more, to cross-check our results, a second implementation was programmed in C++ , wherethe convolutions of splitting kernels and partonic cross sections were performed numeri-cally. For both codes, the numerical evaluation of HPLs was performed using the library Chaplin [60]. The two implementations agreed for all parameter configurations that weretested. – 14 –n figure 1, the different orders of the hadronic gluon fusion cross section for the 8 TeVLHC and a Higgs mass of 125 GeV, along with several N LO approximants for variousnumerical values of K are plotted as a function of the renormalisation scale µ r , while thefactorisation scale is fixed to µ f = m h . Note that the convolutions of splitting kernels andpartonic cross sections do not enter in this plot, since they are proportional to log( µ f /m h ).The µ r scale variation for LHC with 14 TeV centre-of-mass energy is shown in fig. 3. The µ f scale dependence, shown in figure 5 for 8 TeV centre-of-mass energy, is, as expected,extremely mild, in accordance with what is observed at NNLO. 20 30 40 50 60 70 80 90 0.0625 0.125 0.25 0.5 1 2 4 σ [ pb ] µ r /m h 14 TeV, µ f = m h N3LO approx (K=0)N3LO approx (K=5)N3LO approx (K=10)N3LO approx (K=15)N3LO approx (K=20)N3LO approx (K=30)N3LO approx (K=40)NNLONLOLO Figure 3: Scale variation of the different orders of the gluon fusion cross section at 14 TeV. µ f is fixed to m h and only µ r is varied. The scaling coefficient K is varied from 0 to 40.Figures 2 and 4 display the overall scale dependence, with both scales set to be equaland varied simultaneously. We note that the curves for the approximate N LO crosssection with various K s spread widely in the low scale region, i.e. for µ < 30 GeV. Thisis not unreasonable, though, as in this regime, the unknown N LO contributions that areneglected become much more important due to the running of α s . Indeed, at the lowestrenormalisation scale considered, µ = m h / ≈ 20 30 40 50 60 70 80 90 0.0625 0.125 0.25 0.5 1 2 4 σ [ pb ] µ /m h 14 TeV, µ f = µ r N3LO approx (K=0)N3LO approx (K=5)N3LO approx (K=10)N3LO approx (K=15)N3LO approx (K=20)N3LO approx (K=30)N3LO approx (K=40)NNLONLOLO Figure 4: Scale variation of the different orders of the gluon fusion cross section at 14 TeV. µ f and µ r are varied simultaneously. The scaling coefficient K is varied from 0 to 40. α s ≈ . 2, i.e. we are barely in the perturbative regime. The term˜ σ (3) gg ( µ ) ∋ β log (cid:18) µ m h (cid:19) ˜ σ (2) gg ( m h ) (5.3)which is supposed to cancel the implicit logarithms in the running of α s , and which becomeslarge and negative, thus pulls the curve down for small scales, and is canceled by thecurrently unknown contributions whose magnitude is small at µ r = m h but is greatlyenhanced due to α s at small µ r . It can hardly be overemphasised that the above prescriptiondoes not represent a proper calculation of the N LO matrix elements, but just a way ofparametrising their unknown numerical importance. Once the height of the N LO curveat ( µ r , µ f ) = (1 , 1) is set, the shape of the full curve only depends on lower order crosssections (which we know exactly), the running of α s and the parton distribution functions,respectively.As mentioned above, the unknown, numerically important coefficient functions c (3 , gg ( z )contain logarithmic contributions that are singular at threshold, log(1 − z ), contributionsthat are regular and contributions that are singular at the opposite, high energy limitlog( z ). The leading and several, but not all, subleading threshold contributions are asso-ciated with multiple soft emissions and can be recovered by resummation techniques. The– 16 – /m r µ ( pb ) σ /2 H = m F µ = 125 GeV, H m N3LO approx (K=0)NNLONLOLO (a) σ [ pb ] µ f /m h µ r = m h N3LO approx (K=0)N3LO approx (K=5)N3LO approx (K=10)N3LO approx (K=15)N3LO approx (K=20)N3LO approx (K=30)N3LO approx (K=40)NNLONLOLO (b) Figure 5: Scale variation of the different orders of the gluon fusion cross section at 8 TeV.In (a) µ r is varied along the x -axis, while the bars represent variation of µ f around thecentral value m h / 2. In (b) µ r is fixed to m h and only µ f is varied. The scaling coefficient K is varied from 0 to 40.authors of [37] have used the expressions for b (3 , ,kij that they have derived to perform a softapproximation in Mellin space, resulting in a N LO approximant with a scale uncertaintyof ≈ c (3 , gg ( z ) by interpolating in Mellin space,between the soft approximation (that captures threshold logarithms) and the BFKL limit(that captures high energy log( z ) terms). This approach matches the NNLO cross sectionneatly, and results in an approximant for the N LO with a scale uncertainty of 7% if thescale is varied in the interval [ m h / , m h ] (or smaller, if the interval chosen is [ m h / , m h ]).Indeed, by comparing our results for the µ r -dependence of the N LO cross section forthe dominant gluon gluon initial state, with the numbers obtained via the recently releasednumerical program gghiggs [38], we find agreement between the two curves when setting K to 25, as is displayed in figure 6.While it is plausible that the leading logarithmic contributions, being threshold en-hanced, capture the bulk of the cross section, it is unclear whether the unknown subleadingcontributions, as well as the non-logarithmic terms, are really negligible. Their importancecertainly rises for the LHC at 14 TeV, as the luminosity function suppresses the regionaway from threshold less, resulting in more phase space for real radiation. One might,therefore, want to be conservative about their magnitude, and hence on the size of thescale uncertainty to be anticipated before the full N LO result is available. Table 1 showsthe estimates for various values of the rescaling factor K , covering the range from relativelymild to extremely strong N LO corrections, resulting in scale uncertainties varying from2% to as large as 8% or more. The scale uncertainties cited here are evaluated by varyingthe scales in the interval [ m h / , m h ]. – 17 – σ [ pb ] µ r /m h gg channel only, 8 TeV, µ f =m h N3LO approx (K=0)N3LO approx (K=10)N3LO approx (K=20)N3LO approx (K=25)N3LO approx (K=30)N3LO approx gghiggsNNLONLOLO Figure 6: Scale variation of the different orders of the gluon fusion cross section at 8 TeV. µ f is fixed to m h and only µ r is varied. K is varied from 0 to 30. Only the gg channel isplotted, and compared to the results obtained with [38].The choice of the central scale around which the variation is performed has been anissue of debate lately, since different choices result in slightly different scale uncertaintyestimates but also in different central values for the cross section. The choice is largely ar-bitrary, but various indications (like improved perturbative convergence, typical transversemomentum scales for radiated gluons, average Higgs transverse momentum etc.) point toa central scale choice that is lower than the traditional one at m h , closer to m h / 2. Analternative indication comes from the considerations of [68], where it is argued, looking atexamples from jet physics, that a reasonable indication would be the position of the saddlepoint in a contour plot of the cross section as a function of µ r and µ f . In figs. 7 and 8 weshow such contour plots for Higgs production at LO, NLO, NNLO and N LO (for threevalues of the parameter K ). In the cases where a saddle point exists, its position pointsindeed to lower scale choices, and in the cases without a saddle point the plateau regionis also located in lower scales. Given the extremely mild factorisation scale dependence,the saddle point or plateau region is largely determined by the µ r plateau in all previousfigures. – 18 –rder Cross section [pb] σ/σ NNLO σ/σ LO LO 10 . +26 . − . . 51 1 . . +20 . − . . 86 1 . . +8 . − . . 00 1 . LO (K=0) 18 . +1 . − . . 91 1 . LO (K=5) 19 . +0 . − . . 95 1 . LO (K=10) 19 . +0 . − . . 98 1 . LO (K=15) 20 . +0 . − . . 02 2 . LO (K=20) 21 . +2 . − . . 05 2 . LO (K=30) 22 . +6 . − . . 12 2 . LO (K=40) 24 . +9 . − . . 19 2 . µ r = µ f = m h / 2, and uncertainties are found by varying the two scales simultaneouslyby a factor of two. - - - - - - - - H Μ F (cid:144) m H L l og H Μ R (cid:144) m H L (a) LO 12 1518 21 2427 3033 - - - - - - - - H Μ F (cid:144) m H L l og H Μ R (cid:144) m H L (b) NLO 15 1617 1819 19202021 212222 22.522.523 - - - - - - - - H Μ F (cid:144) m H L l og H Μ R (cid:144) m H L (c) NNLO Figure 7: 2-D contour plots of the LO, NLO and NNLO cross section at the 8 TeV LHC.The value on the contours is the cross section in picobarns. The x -axis is log ( µ f /m h ), the y -axis log ( µ r /m h ). Our preferred central scale choice is located at (-1,-1). 6. Conclusions In this work we have presented all convolutions of lower-order partonic cross sections andsplitting kernels that contribute to order a to Higgs production in gluon fusion. Theresults agree with the ones previously published in [41]. Apart from the full expressions,we also provide all convolutions expanded around threshold, as the full N LO correctionsin this limit seem to be feasible in the near future.We have also anticipated the scale dependence of the N LO gluon fusion cross section,– 19 – 215 181819 192020.5 - - - - - - - - H Μ F (cid:144) m H L l og H Μ R (cid:144) m H L (a) N LO ( K = 10) - - - - - - - - H Μ F (cid:144) m H L l og H Μ R (cid:144) m H L (b) N LO ( K = 20) - - - - - - - - H Μ F (cid:144) m H L l og H Μ R (cid:144) m H L (c) N LO ( K = 30) Figure 8: 2-D contour plots of the approximated N LO cross section at the 8 TeV LHC.The value on the contours is the cross section in picobarns. The x -axis is log ( µ f /m h ), the y -axis log ( µ r /m h ). Our preferred central scale choice is located at (-1,-1).into which the calculated convolutions enter. As is the case at NNLO, the factorisationscale dependence is extremely mild, at the per mille level or below. The overall scaleuncertainty is driven by the renormalisation scale variation. The definite uncertaintiesdepend on the size of the missing pure N LO contributions. Scanning over a reasonablerange for these contributions, we find that in the residual scale uncertainty can vary from2% − 8% depending on the magnitude of the hard real corrections, whose computation is,to our view, a prerequisite for a solid estimate of the N LO scale uncertainty. Acknowledgements We are very grateful to Babis Anastasiou for many fruitful discussions, as well as providing Maple code for the partonic cross sections through NNLO. Many thanks go to ClaudeDuhr for sharing his Mathematica libraries and assisting in their usage, and to DavisonSoper and Stephen Ellis for stimulating discussions on scale uncertainties. SB would liketo thank Franz Herzog for pointing out the way of obtaining plus-plus convolutions byexpanding a hypergeometric function.This work was supported by the Swiss National Foundation under contract SNF 200020-126632. A. Convolutions of two plus-distributions In the convolutions needed for collinear counterterms we face the problem of convolutionsinvolving one or more plus-distributions. Here, we demonstrate how to obtain all convolu-tions containing two plus-distributions.( D n (1 − x ) ⊗ D m (1 − y )) ( z ) = Z dx dy (cid:20) log(1 − x ) n − x (cid:21) + (cid:20) log(1 − y ) m − y (cid:21) + δ ( xy − z ) (A.1)– 20 –o find these convolutions for all values of m and n , we consider the following convolutionintegral: I ab ( z ) := Z dx dy (1 − x ) − aǫ (1 − y ) − bǫ δ ( xy − z ) . (A.2)We use the delta function to get rid of x and then remap the integral onto the unit interval: I ab ( z ) = Z z dy y (cid:18) − zy (cid:19) − aǫ (1 − y ) − bǫ = Z z dy y − aǫ ( y − z ) − aǫ (1 − y ) − bǫ = Z dλ (1 − z )[ z + (1 − z ) λ ] − aǫ [ λ (1 − z )] − aǫ [(1 − z )(1 − λ )] − bǫ = (1 − z ) − a + b ) ǫ Z dλ [ z + (1 − z ) λ ] − aǫ λ − aǫ (1 − λ ) − bǫ = (1 − z ) − a + b ) ǫ Z dλ [(1 − λ ) + zλ ] − aǫ λ − bǫ (1 − λ ) − aǫ = (1 − z ) − a + b ) ǫ B ( aǫ, bǫ ) F ( aǫ, bǫ, ( a + b ) ǫ ; 1 − z ) . (A.3)In the second to last step, we mapped λ − λ and in the last step, the Euler definitionof the hypergeometric function was used, B ( b, c − b ) F ( a, b, c ; z ) = Z dx x b − (1 − x ) c − b − (1 − zx ) | {z } (1 − x )+(1 − z ) x − a , (A.4)where B ( x, y ) denotes the Euler Beta-function. On the other hand, we may also directlyexpand the integrands in I ab in terms of a delta function and a tower of plus-distributions, I ab ( z ) = Z dx dy δ (1 − x ) aǫ + X n ≥ ( aǫ ) n n ! D n (1 − x ) ×× δ (1 − y ) bǫ + X m ≥ ( bǫ ) m m ! D m (1 − y ) δ ( xy − z )= δ (1 − z ) abǫ + 1 aǫ X n ≥ ( bǫ ) n n ! D n (1 − z ) 1 bǫ X n ≥ ( aǫ ) n n ! D n (1 − z )++ X n,m ≥ ( aǫ ) n ( bǫ ) m n ! m ! ( D n (1 − x ) ⊗ D m (1 − y )) ( z ) . (A.5)When we now expand the first term of eq A.3 in the same way,(1 − z ) − a + b ) ǫ = δ (1 − z )( a + b ) ǫ + X n ≥ (( a + b ) ǫ ) n n ! D n (1 − z ) , (A.6)and expand the Beta-function and the F (using the Mathematica Package HypExp [69])in ǫ as well, we can equate the two sides order by order in ǫ .The double and single poles cancel and for O ( ǫ ) we find the equation( D (1 − x ) ⊗ D (1 − y )) ( z ) = − π δ (1 − z ) + 2 D (1 − z ) − log( z )1 − z . (A.7)– 21 –igher orders in ǫ of the equation contain more than one plus-plus-convolutions, but theycan be isolated by extracting the corresponding coefficient of a and b . To find the expressionfor the convolution ( D n (1 − x ) ⊗ D m (1 − y )) ( z ), one has to take the O ( ǫ n + m a n b m ) coef-ficient of the equation, or equivalently the O ( ǫ n + m a m b n ) coefficient since the expressionsare symmetric in a and b .In this way, we found all the plus-plus convolutions needed for this work, which arelisted here for completeness.( D ⊗ D ) ( z ) = − π δ (1 − z ) + 2 D (1 − z ) − log( z )1 − z (A.8)( D ⊗ D ) ( z ) = ζ δ (1 − z ) − π D (1 − z ) + 32 D (1 − z ) − log( z ) log(1 − z )1 − z (A.9)( D ⊗ D ) ( z ) = π δ (1 − z ) + 43 D (1 − z ) − π D (1 − z ) + 2 ζ D (1 − z ) − log( z ) log (1 − z )1 − z (A.10)( D ⊗ D ) ( z ) = − π δ (1 − z ) + D (1 − z ) − π D (1 − z ) + 2 ζ D (1 − z ) − log( z ) log (1 − z )1 − z − log( z )Li ( z )1 − z + 2 Li ( z ) − ζ − z (A.11)( D ⊗ D ) ( z ) =6 ζ δ (1 − z ) + 54 D (1 − z ) − π D (1 − z )+ 6 ζ D (1 − z ) − π D (1 − z ) − log( z ) log (1 − z )1 − z (A.12)( D ⊗ D ) ( z ) = (cid:18) ζ − π ζ (cid:19) δ (1 − z ) + 56 D (1 − z ) − π D (1 − z )+ 6 ζ D (1 − z ) − π D (1 − z ) + Li (1 − z )1 − z + 4 log(1 − z )(Li ( z ) − ζ )1 − z + log( z )1 − z h z ) log (1 − z ) − log (1 − z ) − π − z ) + 2 log(1 − z )Li (1 − z )+ 2Li (1 − z ) − ζ i (A.13)( D ⊗ D ) ( z ) = − π δ (1 − z ) + 65 D (1 − z ) − π D (1 − z )+ 12 ζ D (1 − z ) − π D (1 − z ) + 24 ζ D (1 − z ) − log( z ) log (1 − z )1 − z (A.14)– 22 – D ⊗ D ) ( z ) = (cid:18) ζ − π (cid:19) δ (1 − z ) + 34 D (1 − z ) − π D (1 − z )+ 12 ζ D (1 − z ) − π D (1 − z ) (cid:0) ζ − π ζ (cid:1) D (1 − z )+ 11 − z " (1 − z ) log ( z ) + log( z ) π − π log (1 − z )2 − log (1 − z ) + 3 log (1 − z )Li (1 − z ) + 6 log(1 − z )Li (1 − z ) − (1 − z ) − ζ log(1 − z ) ! + 12H(0 , , , , 1; 1 − z )+ 24H(0 , , , , 1; 1 − z ) + 6 log (1 − z )(Li ( z ) − ζ )+ 3 log(1 − z )Li (1 − z ) − (1 − z )Li (1 − z ) (A.15)The above expressions agree with the ones given in [41] (eq. 22) and [70] (eq. C.28 - C.31).For the cases D ⊗ D n , the combination of harmonic polylogarithms given in the referencescollapses to the single term − log( z ) log n (1 − z ) / (1 − z ). B. Ancillary files Here we briefly describe the files accompanying the publication. All files are available bothin Maple ( file.mpl ) and Mathematica ( file.m ) format.1. sigma.m and sigma.mpl : Contain the partonic cross sections ˜ σ ( n ) ij through the respec-tive order in ǫ needed for the N LO cross section. There are five different partonicchannels ( gg , qg , q ¯ q , qq and qQ , where q = Q = ¯ q ). Only the gg channel containsterms of soft origin ( δ (1 − x ) and D n (1 − x ) terms). They are denoted by d1(1-x) and DD(n,x) , respectively.Harmonic polylogarithms are denoted by H , and can be cast in the form used by thepackage HPL [57, 58] via the Mathematica replacement rule H[a__,x_] -> HPL[{a},x] .2. convolutions.m and convolutions.mpl : Contain the 80 convolutions of splittingkernels and partonic cross sections required for the N LO cross section, to the re-spective order in ǫ needed. The names of the convolutions are simply the concate-nation of all ingredients. The convolution P (0) gg ⊗ P (0) gq ⊗ ˜ σ (0) gg for example is called pgg0pgq0sigma0gg .All expressions are given in terms of soft terms like δ (1 − x ) and D n (1 − x ), wherepresent, and HPLs for the regular parts.3. convolutions_softlimit.m and convolutions_softlimit.mpl : Contain the same80 convolutions, but the regular parts are expanded in the variable xp ≡ − x asdescribed in section 4.1. 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