Higgs boson gluon-fusion production beyond threshold in N3LO QCD
Charalampos Anastasiou, Claude Duhr, Falko Dulat, Elisabetta Furlan, Thomas Gehrmann, Franz Herzog, Bernhard Mistlberger
aa r X i v : . [ h e p - ph ] N ov Preprint typeset in JHEP style - PAPER VERSION
CP3-14-71, ZU-TH 39/14, FERMILAB-PUB-14-461-T, NIKHEF 2014-048, CERN-PH-TH-2014-221
Higgs boson gluon-fusion production beyond thresholdin N LO QCD
Charalampos Anastasiou a , Claude Duhr b , Falko Dulat a , Elisabetta Furlan c , ThomasGehrmann d , Franz Herzog e , Bernhard Mistlberger a a Institute for Theoretical Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland b Center for Cosmology, Particle Physics and Phenomenology (CP3),Universit´e catholique de Louvain,Chemin du Cyclotron 2, 1348 Louvain-La-Neuve, Belgium c Fermilab, Batavia, IL 60510, USA d Physik-Institut, Universit¨at Z¨urich, Winterthurerstrasse 190, 8057 Z¨urich, Switzerland e Nikhef, Science Park 105, NL-1098 XG Amsterdam, The NetherlandsCERN Theory Division, CH-1211, Geneva 23, Switzerland
Abstract:
In this article, we compute the gluon fusion Higgs boson cross-section at N LOthrough the second term in the threshold expansion. This calculation constitutes a majormilestone towards the full N LO cross section. Our result has the best formal accuracyin the threshold expansion currently available, and includes contributions from collinearregions besides subleading corrections from soft and hard regions, as well as certain loga-rithmically enhanced contributions for general kinematics. We use our results to performa critical appraisal of the validity of the threshold approximation at N LO in perturbativeQCD.
Keywords:
Higgs physics, QCD, gluon fusion. . Introduction
With the discovery of the Higgs boson [1], the Standard Model is a fully predictive theory,with all of its parameters determined experimentally. This fact renders the total Higgsboson production cross-section an excellent precision test of the theory. Theoretical pre-dictions for the inclusive cross-section therefore play an important role in measurements ofHiggs-boson observables in general and in the determination of the coupling strengths ofthe Higgs boson in particular.For this reason, obtaining a reliable theoretical estimate of the gluon-fusion cross-section, the dominant production mechanism of a Higgs boson at the LHC, has been amajor objective in perturbative QCD for the last decades. The very large size of thenext-to-leading-order (NLO) perturbative corrections in the strong coupling α s indicateda slow convergence of the α s expansion [2]. The smaller size of the next-to-next-to-leadingorder (NNLO) corrections inspired some confidence that QCD effects beyond NNLO maybe smaller than ± ∼
5% level of precision) followed [4].Currently, no full computation of the hadronic Higgs-boson cross-section is availableat next-to-next-to-next-to-leading order N LO. It is possible to obtain some informationon the missing higher orders beyond NNLO in the so-called threshold limit where theHiggs boson is predominantly produced at threshold and the additional QCD radiationis soft. In this limit, soft QCD emissions factorize from the hard interaction and can beresummed [5]. After the completion of the NNLO corrections [3], it was observed that thethreshold approximation can be made to capture the bulk of the perturbative correctionsthrough NNLO. It is then tantalising to speculate if a similar approximation is sufficientto predict the value of the Higgs-boson cross-section at N LO in QCD. Recently, variousapproximate N LO cross-section estimates were put forward which rely crucially on thethreshold assumption [6, 7, 8]. Given these considerations, it is important to quantify thereliability of the threshold approximation at N LO.Logarithmically enhanced threshold contributions at N LO to the cross-section comingfrom the emission of soft gluons have been computed almost a decade ago [6]. A fewmonths ago, we completed the computation of the first term in the threshold expansion,the so-called soft-virtual term, by computing in addition the constant term proportionalto δ (1 − z ) [9], which includes in particular the complete three-loop corrections to Higgsproduction via gluon fusion [10]. Recently, some further logarithmic corrections whichbelong to the second order in the threshold expansion were conjectured in ref. [8, 11]. In thispaper we compute for the first time the complete second order in the threshold expansion.This result is an important step in the direction of the computation of the N LO cross-section for arbitrary values of z , a goal which has only been achieved so far at N LO for thethree-loop corrections and the single-emission contributions at two loops [12, 13, 14]. Wecombine the knowledge of the single-real emission contributions with the ultra-violet andparton-density counterterms to obtain the exact result for the first three logarithmically-enhanced terms beyond the soft-virtual approximation. Both results combined are not only– 1 – major milestone towards the complete Higgs-boson cross-section at N LO, but they alsoconstitute the most precise calculation of the Higgs-boson cross-section at N LO beyondthreshold.In a second part of our paper, we use our results and perform a critical appraisal ofthe threshold approximation. We define a way to quantify the convergence of the trun-cated threshold expansion, and we perform a numerical study of the convergence of thethreshold expansion at NLO, NNLO and N LO. Given the widely accepted dominance ofthe threshold limit in Higgs production at the LHC, our study is an important ingredientto asses the reliability of the threshold approximation at N LO in QCD.This paper is organised as follows: In Section 2 we present our results for the completesecond term in the threshold expansion and the exact results for the coefficients of thefirst three leading logarithmically-enhanced terms in the threshold limit. In Section 3 weperform a critical appraisal of the threshold expansion, both in z -space and in Mellin-space.In Section 4 we draw our conclusions.
2. Analytic results for the N LO partonic cross-section
In this section we present the main results of our paper. We start by giving a short reviewof the inclusive gluon-fusion cross-section and its analytic properties, and then we presentour results in subsequent sections.The inclusive cross-section σ for the production of a Higgs boson is given by σ = τ X ij (cid:18) f i ⊗ f j ⊗ ˆ σ ij ( z ) z (cid:19) ( τ ) , (2.1)where ˆ σ ij are the partonic cross-sections for producing a Higgs boson from the partonspecies i and j , and f i and f j are the corresponding parton densities. We have defined theratios τ = m H S and z = m H s , (2.2)where m H denotes the Higgs-boson mass and s and S denote the squared partonic andhadronic center-of-mass energies. The convolution of two functions is defined as( A ⊗ B )( τ ) = Z dx dy A ( x ) B ( y ) δ ( τ − xy ) . (2.3)In the rest of this section we only concentrate on the partonic cross-sections. If we workin perturbative QCD, and after integrating out the top quark, the partonic cross-sectionstake the form ˆ σ ij ( z ) z = π C V ∞ X k =0 (cid:16) α s π (cid:17) k η ( k ) ij ( z ) , (2.4)with V = N c − N c the number of SU ( N c ) colours, and C ≡ C ( µ ) and α s ≡ α s ( µ ) denote the Wilson coefficient [15] and the strong coupling constant, evaluated at– 2 –he scale µ . At leading order in α s only the gluon-gluon initial state contributes, η (0) ij ( z ) = δ ig δ jg δ (1 − z ). The partonic cross-sections through NNLO, η (1 , ij ( z ), can be found inref. [3].Before presenting our results, let us discuss some general properties of the N LO coef-ficients η (3) ij ( z ) which will be useful in the remainder of this section. First, η (3) ij ( z ) does notonly contain the three-loop corrections to inclusive Higgs production, but also contributionsfrom the emission of up to three partons in the final state at the same order in perturbationtheory. So far, only the single-emission contributions at two loops are known for genericvalues of z [12, 13, 14, 16, 17], and only a few terms in the threshold expansion for the con-tributions with up to two additional partons in the final state are known [9, 21, 22]. Eachof these contributions is ultra-violet (UV) and infra-red (IR) divergent, and the divergencesmanifest themself as poles in the dimensional regulator ǫ .While the first three leading poles at N LO cancel when summing over all the contribu-tions, the coefficient of ǫ − is non-zero. These remaining divergences cancel when suitableUV and IR counterterms are included. We generically write η (3) ij ( z ) = ∆ (3) ij ( z, ǫ ) + χ (3) ij ( z, ǫ ) , (2.5)where ∆ (3) ij ( z, ǫ ) is the combined UV and IR counterterm and χ (3) ij ( z, ǫ ) is the (bare) con-tribution from the different particle multiplicities at N LO. Note that each term in theright-hand side has poles at ǫ = 0, but the sum is finite. The counterterm is determinedcompletely from lower orders [23, 24], as well as the QCD β function [18] and the three-loopsplitting functions [20].The contributions arising from different multiplicities can be separated into six differentterms as χ (3) ij ( z, ǫ ) = χ (3 , ij ( ǫ ) δ (1 − z ) + X m =2 (1 − z ) − mǫ χ (3 ,m ) ij ( z, ǫ ) . (2.6)where the functions χ (3 ,m ) ij ( z, ǫ ) are meromorphic with at most a simple pole at z = 1.While the first term only contributes at threshold and contains the entirety of the three-loopcorrections, the second term receives contributions from all additional parton emissions.The partonic cross-sections are convoluted with the parton luminosities, and the poleat z = 1 in the gluon-gluon initial state introduces a divergence in the integrand as z →
1. The singularities are regulated in dimensional regularisation by expanding the factors(1 − z ) − − mǫ in terms of delta functions and plus-distributions.(1 − z ) − − mǫ = − mǫ δ (1 − z ) + ∞ X j =0 ( − mǫ ) j j ! (cid:20) log j (1 − z )1 − z (cid:21) + , (2.7)where the plus-distribution is defined by its action on a test function φ ( z ). Z dz (cid:20) log j (1 − z )1 − z (cid:21) + φ ( z ) ≡ Z dz log j (1 − z )1 − z [ φ ( z ) − φ (1)] . (2.8) There is a typo in eq. (2.8) of ref. [24]. The combination 3 P (0) ik ⊗ P (1) kj + 3 P (1) ik ⊗ P (0) kj in the fourth lineshould be replaced by 2 P (0) ik ⊗ P (1) kj + 4 P (1) ik ⊗ P (0) kj . – 3 –n order to expose the distributions, we write χ (3 ,m ) ij ( z, ǫ ) = χ (3 ,m ) , sing ij ( z, ǫ ) + χ (3 ,m ) , reg ij ( z, ǫ ) , (2.9)with χ (3 ,m ) , sing ij ( z, ǫ ) the residue at z = 1 (divided by ( z − η (3) ij ( z ) = η (3) , sing ij ( z ) + η (3) , reg ij ( z ) , (2.10)where the singular contribution is precisely the cross-section at threshold [6, 9] and theregular term describes terms that are formally subleading and take the form of a polynomialin log(1 − z ), η (3) , reg ij ( z ) = X m =0 log m (1 − z ) η (3 ,m ) , reg ij ( z ) , (2.11)where the η (3 ,m ) , reg ij ( z ) are holomorphic in a neighbourhood of z = 1. The coefficients ofthese logarithms are the main subject of this paper, and in the rest of this section we showhow to explicitly determine some of the regular coefficients of the threshold logarithms. All the regular terms are formally subleading in the threshold expansion compared to thesoft-virtual term. If we want to compute these subleading corrections, we need to knowthe counterterms ∆ (3) ij ( z, ǫ ) and all process with different multiplicities contributing to χ (3) ij ( z, ǫ ). To date, however, only the counterterms and the single-emission contributionsare known for arbitrary values of z . Since the coefficients of the logarithms are holomorphic,they admit a Taylor expansion around z = 1. In this section we discuss how to approximatethe coefficients of the logarithms by their threshold expansion around z = 1. In particular,one of the main results of this paper is the complete computation of the first subleadingterm in the threshold expansion, corresponding to the value at z = 1 of the coefficientsin eq. (2.11) and dubbed the next-to-soft term in the remainder of this paper. Note thatthe next-to-soft term receives for the first time contributions from the quark-gluon (andanti-quark-gluon) initial state besides the gluon-gluon initial state.In ref. [9] the next-to-soft term of the triple-emission contribution was computed.Hence, we are only missing the next-to-soft corrections to the double-emission contributionat one-loop. We have recently completed the computation of all the relevant diagramscontributing to the next-to-soft term. In the following we only present the results of thecomputation, and details of the computation will be given elsewhere. Here it suffices tosay that, unlike the contribution to the soft-virtual term [9, 22], we also need to considercontributions from regions where the virtual gluon can be collinear to one of the externalpartons besides subleading corrections to the soft and hard regions. In the following wepresent the next-to-soft cross-sections η (3) ij ( z ) (cid:12)(cid:12)(cid:12) (1 − z ) for values of the renormalization and– 4 –actorization scales equal to the Higgs mass. The corresponding expressions for arbitraryscales can be derived easily from renormalization group and DGLAP evolution. We find η (3) gg ( z ) (cid:12)(cid:12) (1 − z ) = − N c log (1 − z ) + (cid:18) N c − N c N f (cid:19) log (1 − z ) (2.12)+ " (cid:18) ζ − (cid:19) N c + 20518 N c N f − N c N f log (1 − z )+ ( (cid:18) − ζ − ζ + 271127 (cid:19) N c + (cid:20)(cid:18) ζ − (cid:19) N c + 14 (cid:21) N f + 59108 N c N f ) log (1 − z )+ ( (cid:18) ζ + 362 ζ + 237518 ζ − (cid:19) N c + (cid:20)(cid:18) − ζ − ζ + 8071324 (cid:19) N c +3 ζ + 124 ζ − (cid:21) N f + (cid:18) ζ − (cid:19) N c N f ) log(1 − z )+ (cid:18) − ζ + 7256 ζ ζ − ζ − ζ − ζ + 83441923328 (cid:19) N c + (cid:20)(cid:18) ζ + 178972 ζ + 4579324 ζ − (cid:19) N c − ζ − ζ − ζ + 50651728 (cid:21) N f + (cid:18) − ζ − ζ + 49729 (cid:19) N c N f .η (3) qg ( z ) (cid:12)(cid:12) (1 − z ) = N c − N c
256 + 181768 N c − N c ! log (1 − z ) (2.13)+ " − N c N c − N c + 2299216 N c + (cid:18) N c − N c (cid:19) N f log (1 − z )+ " (cid:18) − ζ + 16690341472 (cid:19) N c + (cid:18) ζ − (cid:19) N c + (cid:18) − ζ + 47341472 (cid:19) N c + (cid:18) ζ + 2111536 (cid:19) N c + (cid:18) − N c − N c (cid:19) N f + (cid:18) N c − N c (cid:19) N f log (1 − z )+ ( (cid:18) ζ + 1729576 ζ − (cid:19) N c + (cid:18) − ζ − ζ + 4602513824 (cid:19) N c + (cid:18) ζ + 541192 ζ − (cid:19) N c + (cid:18) − ζ − ζ − (cid:19) N c – 5 – (cid:20)(cid:18) − ζ + 642710368 (cid:19) N c + 5972 ζ − (cid:18) − ζ + 131310368 (cid:19) N c (cid:21) N f + (cid:18) − N c
432 + 11432 N c (cid:19) N f ) log (1 − z )+ ( (cid:18) − ζ − ζ − ζ + 1641013248832 (cid:19) N c + (cid:18) ζ + 2297288 ζ + 205453456 ζ − (cid:19) N c + (cid:18) − ζ − ζ − ζ − (cid:19) N c + (cid:18) − ζ + 6596 ζ − ζ − (cid:19) N c + (cid:20)(cid:18) ζ + 155288 ζ − (cid:19) N c − ζ − ζ + 98593456+ (cid:18) ζ + 481864 ζ − (cid:19) N c (cid:21) N f + (cid:18) N c − N c (cid:19) N f ) log(1 − z )+ (cid:18) ζ − ζ ζ − ζ + 341173456 ζ + 36911296 ζ − (cid:19) N c + (cid:18) − ζ + 2807192 ζ ζ − ζ − ζ − ζ + 53237995328 (cid:19) N c + (cid:18) ζ − ζ ζ + 22452304 ζ − ζ + 9572 ζ + 422195331776 (cid:19) N c + (cid:18) − ζ + 3164 ζ ζ + 4396 ζ − ζ + 13 ζ + 169912288 (cid:19) N c + (cid:20)(cid:18) ζ − ζ − ζ + 82171248832 (cid:19) N c − ζ + 1723864 ζ + 229324 ζ − (cid:18) − ζ − ζ − ζ − (cid:19) N c (cid:21) N f + (cid:20)(cid:18) − ζ − (cid:19) N c + (cid:18) ζ + 1253888 (cid:19) N c (cid:21) N f . The leading logarithms in the above equations can be compared with recent results inthe literature. The coefficients of log (1 − z ) and log (1 − z ) for the gluon-gluon channelin eq. (2.12) are in agreement with the conjecture of ref. [8]. In ref.[8] a conjecture wasalso formulated for the colour and flavour structure of the coefficient of log (1 − z ) up toa rational parameter ξ (3) H . We confirm the validity of this conjecture for the coefficient oflog (1 − z ) as well and determine ξ (3) H = . The log (1 − z ) coefficient for the quark-gluon channel in eq. (2.13) agrees with the calculation of ref. [11]. The coefficients of theremaining logarithms and the non-logarithmic terms in eqs. (2.12)-(2.13) are presented forthe first time in this publication. – 6 – .3 Coefficients of leading logarithms with exact z dependence In this section we obtain another approximation to eq. (2.11), namely we compute thecoefficients of the three leading logarithms in eq. (2.11) with exact z dependence. Indeed,it turns out that the coefficients of these logarithms are uniquely determined at N LO byrequiring the cancellation of the poles in ǫ , once the single-emission contributions and thecounterterms are known.To be more concrete, we start from eq. (2.5) and (2.6), and expand all the contributionsin the dimensional regulator ǫ , η (3) ij ( z ) = X l = − X k =0 ǫ l log(1 − z ) k ∆ (3 ,l,k ) ij ( z )+ X l = − ǫ l " χ (3 , ,l ) ij δ (1 − z ) + X m =2 χ (3 ,m,l ) ij ( z )(1 − z ) − mǫ + O ( ǫ ) . (2.14)In order for η (3) ij ( z ) to be finite, all the poles in ǫ must cancel. This implies that thecoefficient of each power of log(1 − z ) and of each plus-distribution multiplying a pole in ǫ has to vanish separately, which allows us to derive a set of equations constraining theindividual contributions χ (3 ,m,l ) ij ( z ) and ∆ (3 ,l,k ) ij ( z ). In particular, we get∆ (3 ,l,k ) ij ( z ) + X m =2 ( − m ) k k ! χ (3 ,m,l − k ) ij ( z ) = 0 , l < , ∀ k . (2.15)At this point we note that the terms proportional to χ (3 , ,k ) ij ( z ) and χ (3 , ,k ) ij ( z ) only receivecontributions from single-emission subprocesses, and the computation of those contribu-tions was recently completed for arbitrary values of z [12, 13, 14]. In particular, thecomputation of the single-emission processes at two loops of ref. [13] has all the logarithmslog(1 − z ) resummed into factors of the form (1 − z ) − mǫ , which makes the determinationof χ (3 , ,k ) ij ( z ) and χ (3 , ,k ) ij ( z ) straightforward. Including this information we are able tosolve the system of equations (2.15) for the coefficients of the first three leading logarithms(log , , (1 − z )) for all partonic initial states. Parts of the coefficients of these logarithms,corresponding to specific colour coefficients, had already been predicted in ref. [8], and weconfirm these results. Moreover, we have checked that only the gluon-gluon and quark-gluon initial states give non-vanishing contributions at next-to-soft level, and the values ofthe coefficients for z = 1 agree with the corresponding coefficients presented in the previoussection. The analytic results for the different partonic initial states are, for µ R = µ F = m H , η (3 , , reg gg ( z ) = N f N c " z + 1) H H + 680 z − z − z − z H − z + 1) H − z + 1) H − (1 − z ) (cid:0) z + 4505 z + 1644 (cid:1) z + 8572 ( z + 1) ζ – 7 – N f N c " − z + 1) H − − z ) (cid:0) z + 7 z + 4 (cid:1) z + N f " − z + 1) H H − z − z − z − z H + 18 (51 z + 11) H + 8( z + 1) H + (1 − z ) (cid:0) z + 9035 z + 14900 (cid:1) z − z + 1) ζ + N c N f " z + 1) H − z − z + 309 z − z + N c N f " z + 1) H H − z − z + 11199 z − z + 8100864(1 − z ) z H + 136 ( − z − H − z + 1) H + 168584 z − z + 172203 z − z + 49172 ( z + 1) ζ + N c " − (cid:0) z + z + 1 (cid:1) z ( z + 1) H − + 8 (cid:0) z + z + 1 (cid:1) z ( z + 1) H − H − z + 1) H H + 6259 z − z + 11190 z − z + 447727(1 − z ) z H + 2 (cid:0) z − z − z + 49 z + 3 z + 14 (cid:1) (1 − z ) z ( z + 1) H + 128( z + 1) H − z − z + 21100 z − z + 4 (cid:0) z − z − z − z − (cid:1) z ( z + 1) ζ , (2.16) η (3 , , reg gg ( z ) = N f N c " z + 1) H + 85(1 − z ) (cid:0) z + 7 z + 4 (cid:1) z + N f " − z + 1) H − (1 − z ) (cid:0) z + 7 z + 4 (cid:1) z + N c N f " z + 1) H − z − z + 6207 z − z + N c " z − z + 751 z − z − z − z + 81 z − z + 27(1 − z ) z H , (2.17)– 8 – (3 , , reg gg ( z ) = N c (cid:0) − z + z − z + 1 (cid:1) z , (2.18) η (3 , , reg qg ( z ) = N c − N c ( z − z + 24256 z H + 33 z − z − z H + 88 z + 1813 z − z + 9722304 z H − (cid:0) z − z − (cid:1) z H H − (cid:0) z − z + 356 (cid:1) z ζ − z − z + 21360 z − z + N c N f " − z + 398 z − z H + 16 ( z − H − z − z + 16391 z − z + N c " − z − z + 732576 z H + 19 (cid:0) z + 2 z + 2 (cid:1) z H − − z + 441 z + 37144 z H + − (cid:0) z + 2 z + 2 (cid:1) z H − H + 354 z + 441 z + 37144 z H H + 1896 z − z + 20464 z − z H + 1213 z − z + 1084288 z ζ + 46448 z − z + 318062 z − z + N c N f (cid:0) z − z + 2 (cid:1) z + N c N f " − z + 230 z + 25361728 z H + 16 (2 − z ) H + 1816 z − z + 29119 z − z + N c " z + 4802 z + 196082304 z H − (cid:0) z + 2 z + 2 (cid:1) z H − + 5957 z + 12670 z + 3112384 z H + 125 (cid:0) z + 2 z + 2 (cid:1) z H − H − z + 12670 z + 3112384 z H H − z − z + 131476 z − z H − z + 25846 z + 195001152 z ζ − z − z + 3718820 z − z , (2.19)– 9 – (3 , , reg qg ( z ) = N c − N c ( − z − z + 6484608 z H − z − z + 28929216 z − N c N f (cid:0) z − z + 2 (cid:1) z + N c z − z + 813576 z H − z − z + 58726 z − z ! + N c N f (cid:0) z − z + 2 (cid:1) z + N c z + 8689 z + 388000 z − z − z + 4726 z + 110561536 z H !) , (2.20) η (3 , , reg qg ( z ) = N c − N c z − z + 2 z − N c N c ! , (2.21) η (3 , , reg q ¯ q ( z ) = ( N c − N c ( z − z + 39 z − z H − (1 − z ) (cid:0) z − z + 121 (cid:1) z + N c " − z + 292 z + 196384 z H + 13( z + 2) z H H + 71 z − z − z H − z + 2) z H + 13( z + 2) z ζ − (1 − z )(569 z + 1301)256 z + N c N f − z ) z + N c " (1 − z ) (cid:0) z − z + 247 (cid:1) z − z − z + 93 z − z H + N c N f " − ( z + 2) z H − (1 − z )( z + 3)24 z + N c " z + 236 z + 908384 z H − z + 2) z H H − z − z − z H + 35( z + 2) z H + (1 − z ) (cid:0) z + 95 z + 48419 (cid:1) z − z + 2) z ζ + N c N f − z ) z + N c " z − z + 12 z − z H − (1 − z ) (cid:0) z − z + 1309 (cid:1) z , (2.22) η (3 , , reg q ¯ q ( z ) = ( N c − N c ( − − z ) z + N c " z + 2) z H + 13(1 − z )( z + 3)48 z – 10 – N c − z ) z + N c " − z + 2) z H − − z )( z + 3)48 z + N c − z ) z ) , (2.23) η (3 , , reg qq ( z ) = ( N c − N c ( N c " − z + 292 z + 196384 z H + 13( z + 2) z H H + 13( z + 2) z ζ + 71 z − z − z H − z + 2) z H − (1 − z )(569 z + 1301)256 z + N c N f " − ( z + 2) z H − (1 − z )( z + 3)24 z + N c " z + 236 z + 908384 z H − z + 2) z H H − z − z − z H + 35( z + 2) z H + (1 − z ) (cid:0) z + 95 z + 48419 (cid:1) z − z + 2) z ζ , (2.24) η (3 , , reg qq ( z ) = ( N c − N c ( N c " z + 2) z H + 13(1 − z )( z + 3)48 z + N c " − z + 2) z H − − z )( z + 3)48 z , (2.25) η (3 , , reg qq ′ ( z ) = ( N c − N c ( N c " − z + 292 z + 196384 z H + 13( z + 2) z H H + 71 z − z − z H − z + 2) z H + 13( z + 2) z ζ − (1 − z )(569 z + 1301)256 z + N c N f " − ( z + 2) z H − (1 − z )( z + 3)24 z + N c " z + 236 z + 908384 z H − z + 2) z H H − z − z − z H + 35( z + 2) z H + (1 − z ) (cid:0) z + 95 z + 48419 (cid:1) z − z + 2) z ζ , (2.26) η (3 , , reg qq ′ ( z ) = ( N c − N c ( N c " z + 2) z H + 13(1 − z )( z + 3)48 z + N c " − z + 2) z H − − z )( z + 3)48 z . (2.27)– 11 –ote that η (3 , , reg q ¯ q ( z ) = η (3 , , reg qq ( z ) = η (3 , , reg qq ′ ( z ) = 0. We have written the results interms of harmonic polylogarithms [30] H = log z ,H = − log(1 − z ) ,H − = log(1 + z ) ,H = Li ( z ) ,H − = − Li ( − z ) . (2.28)
3. Numerical results for the N LO hadronic cross-section
In this Section, we will study the numerical impact of the partonic N LO corrections ofSection 2 on the hadronic Higgs-boson production cross-section. We normalise all ourresults to the leading-order hadronic cross-section, and we factor out the Wilson coefficient(i.e., we set C = 1). We choose the Higgs-boson mass to be m H = 125GeV and computethe cross-sections for a proton-proton collider with a center-of-mass energy of 14TeV. Weuse the MSTW2008 NNLO parton densities for all orders and the corresponding value of α s ( M Z ) [26]. We set the renormalisation and factorisation scales equal to the Higgs-bosonmass, µ R = µ F = m H . We start our numerical analysis by studying the behavior of the hadronic cross-section atN LO through the first two terms in the threshold expansion. For assessing the numericalimportance of the corrections, it is useful to substitute the number of colours and numberof light quark flavours by their physical values ( N c = 3 , N f = 5 respectively) into eq. (2.12)and (2.13). We find, η (3) gg ( z ) (cid:12)(cid:12) (1 − z ) = −
256 log (1 − z ) ( → . (1 − z ) ( → . . . . . log (1 − z ) ( → − . − . . . . log (1 − z ) ( → − . . . . . log(1 − z ) ( → − . − . . . . ( → − . η (3) qg ( z ) (cid:12)(cid:12) (1 − z ) = 128372 log (1 − z ) ( → − . − (1 − z ) ( → − . − . . . . log (1 − z ) ( → . . . . . log (1 − z ) ( → . . . . . log(1 − z ) ( → . . . . . ( → . . (3.2)In parentheses we show the relative size of the correction which each term induces to thehadronic cross-section relatively to the leading order contribution from η (0) gg = δ (1 − z ).We find that the formally most singular terms cancel against less singular ones. Inaddition to the large cancellations among different powers of logarithms, we notice thatthe formal hierarchy of their magnitude does not correspond to a similar hierarchy at thehadronic cross-section level. These observations are the same as we had already noted inref. [9] for the leading terms of the soft expansion. For ease of comparison, we also recitehere the analogous decomposition of the leading terms in the soft expansion [9] η (3) gg ( z ) ≃ (cid:20) log (1 − z )1 − z (cid:21) + . ( → . − (cid:20) log (1 − z )1 − z (cid:21) +
230 ( → . − (cid:20) log (1 − z )1 − z (cid:21) + . . . . ( → − . (cid:20) log (1 − z )1 − z (cid:21) + . . . . ( → − . − (cid:20) log(1 − z )1 − z (cid:21) + . . . . ( → − . (cid:20) − z (cid:21) + . . . . ( → − . δ (1 − z ) 1124 . . . . ( → . . (3.3)The total contribution of η (3) gg ( z ) (cid:12)(cid:12) (1 − z ) to the hadronic cross-section is about 25% of theBorn contribution, while the contribution of η (3) qg ( z ) (cid:12)(cid:12) (1 − z ) is about − .
38% of the Borncontribution. This has to be contrasted with the leading soft contribution at N LO from η (3) gg ( z ) (cid:12)(cid:12) (1 − z ) − which is only − .
25% of the Born. While the next-to-soft correction forkinematics corresponding to threshold production should be suppressed, instead it turnsout to be much larger than the leading threshold contribution.It is often preferred in the literature to perform the threshold expansion in Mellinspace. The Mellin transformation of a function f ( z ) is defined as M [ f ]( N ) = Z dz z N − f ( z ) . (3.4)The Mellin transformation is invertible, and the inverse transformation reads M − [ g ] ( z ) = Z c + i ∞ c − i ∞ dN πi g ( N ) x − N , (3.5)– 13 –here the real part of c is chosen such that the poles of g ( N ) lie to the left of the inte-gration contour. One of the main properties of the Mellin transformation is that it mapsconvolutions as in eq. (2.3) to the product of the Mellin transformations, M [ A ⊗ B ]( N ) = M [ A ]( N ) M [ B ]( N ) . (3.6)It follows that the convolution of the partonic cross-sections with the parton densities fac-torises and turns into an ordinary product in Mellin space. Hence, in order to compute theMellin transformation of the total hadronic cross-section, we need the Mellin transforma-tions of the parton densities. To this effect, we fit the parton densities for a fixed scale toa functional form of the type f i ( x ) = x a i (1 − x ) b i ( c i, + c i, x + c i, x + . . . ) , for which we can easily compute the Mellin transformation using Euler’s Beta function, M h x a (1 − x ) b i ( N ) = Γ( N + a )Γ(1 + b )Γ(1 + a + b + N ) . (3.7)For the partonic cross-section we perform an expansion around the threshold limit, whichin Mellin space corresponds to taking N → ∞ . Through O ( N ), we find: M h η (3) gg i ( N ) ≃
36 log N ( → . . . . . log N ( → . . . . . log N ( → . . . . . log N ( → . . . . . log N ( → . . . . . log N ( → . . . . . ( → . NN ( → . . . . . log NN ( → . . . . . log NN ( → . . . . . log NN ( → . . . . . log NN ( → . . . . . N ( → . . (3.8)In parentheses we show the relative size of the correction which each term induces to thehadronic cross-section relatively to the leading order contribution from η (0) gg = δ (1 − z ). InMellin space the pattern of corrections in the threshold expansion is different from the one– 14 –bserved in z -space. As it was also observed for the leading soft terms and parts of thenext-to-soft terms in ref. [8], we find that through O (cid:0) N (cid:1) the corrections are always positive.Nevertheless, we observe that the formally leading logarithms contribute the least to thehadronic cross-section. In total, the soft-virtual (SV) terms (log n N ) contribute about ∼
18% of the Born to the cross-section, while the next-to-soft (NS) terms (log n N /N )contribute about ∼
11% of the Born. We therefore conclude that, unlike common folkloresuggests, the threshold limit does in fact not dominate the cross-section at LHC energies,but there is a sizeable contribution from terms beyond threshold.As we have emphasised in ref. [9], there is an ambiguity in how to convolute an ap-proximate partonic cross-section with the parton densities. For example, we can recast thehadronic cross-section in the form, σ = τ n X ij (cid:18) f ( n ) i ⊗ f ( n ) j ⊗ ˆ σ ij ( z ) z n (cid:19) ( τ ) (3.9)where f ( n ) i ( z ) ≡ f i ( z ) z n . (3.10) σ is independent of the arbitrary parameter n as long as the partonic cross-section is knownexactly. Mellin transforming eq. (3.9), we obtain M h στ n i ( N ) = X ij M h f ( n ) i i ( N ) M h f ( n ) j i ( N ) M (cid:20) ˆ σ ( z ) z n (cid:21) ( N )= X ij M [ f i ] ( N − n ) M [ f j ] ( N − n ) M (cid:20) ˆ σ ( z ) z (cid:21) ( N − n ) . (3.11)If only a finite number of terms in the threshold expansion of the partonic cross-sectionsare kept,ˆ σ ij ( z ) z n ≃ ˆ σ ij ( z ) | (1 − z ) − + ˆ σ ij ( z ) | (1 − z ) + n (1 − z ) ˆ σ ij ( z ) | (1 − z ) − + O (1 − z ) (3.12)then the convolution integral is sensitive to varying the arbitrary parameter n . This am-biguity is expected to be reduced when including higher-order terms in the threshold ex-pansion. This effect was already observed at NNLO [25], corresponding to expandingaround threshold the 1 /z flux-factor as part of the partonic cross-section or evaluatingit unexpanded as part of the parton luminosity. A similar ambiguity appears to be re-sponsible [28, 29] for the bulk of the difference in the numerical predictions for the Higgscross-section at N LO in various approaches and implementations of threshold resumma-tion [5].In the remainder of this section we analyse the impact of this truncation when weuse the results of Section 2, which contains the most precise information on the thresholdexpansion of the cross-section at N LO to date. In order to quantify the trustworthinessof the threshold approximation, we study the dependence of the result on the parameter n defined through eq. (3.9), both in z and in Mellin-space.– 15 – - - - n k l o ( n ) z - space NLO svNLO svnsNNLO svNNLO svnsN3LO svN3LO svnsNLO fullNNLO full
Figure 1:
Soft-virtual and next-to-soft corrections at NLO, NNLO and N LO normalised to theBorn cross-section in z − space as a function of the artificial parameter n in eq. (3.12) In Fig. 1 we plot the soft-virtual and next-to-soft corrections at NLO, NNLO and N LOnormalised to the Born cross-section in z − space as a function of the artificial parameter n in eq. (3.12). In Fig. 2 we plot the soft-virtual and next-to-soft corrections at NLO,NNLO and N LO normalised to the Born cross-section in Mellin space as a function ofthe artificial parameter n in eq. (3.12). We also plot in both figures the known NLO andNNLO corrections as straight lines since they are insensitive to the value of n . The fullNLO corrections are about 110% of the Born and the full NNLO corrections are about60%. The sensitivity of the ‘leading soft’ corrections to n is large at all perturbative ordersand in both spaces. This sensitivity is reduced when the next-to-soft terms are included,where a plateau at NLO and NNLO is formed for values of n larger than about − n . While an improved convergence is visible, at N LOthe sensitivity of the next-to-soft correction in n is enhanced in comparison to NLO andNNLO and there is much less of a plateau. The increased sensitivity of the truncatedexpansion to the artificial parameter n is a symptom of the fact that the threshold limitis less dominant at higher orders. In Table 1 we present the ratio of the NS over the SVcontribution in the gluon-gluon channel (this ratio is infinite in all other channels) both inMellin and z − space. We observe that the ratio increases at higher perturbative orders andhence the soft approximation is increasingly untrustworthy. This behavior is particularlypronounced in z − space.Is it possible to use the soft-virtual [9] or the next-to-soft approximation presented inthis article in order to estimate precisely the N LO corrections to the Higgs cross-section?The fact that the soft expansion does not yet appear to be convergent, as we discussedabove, does not justify such attempts theoretically. Nevertheless, efforts have been made inthe literature to guess the full N LO corrections from available or estimated soft terms using– 16 – - - - n k l o ( n ) Mellin - space NLO svNLO svnsNNLO svNNLO svnsN3LO svN3LO svnsNLO fullNNLO full
Figure 2:
Soft-virtual and next-to-soft corrections at NLO, NNLO and N LO normalised to theBorn cross-section in Mellin space as a function of the artificial parameter n in eq. (3.12) NLO NNLO N LO z − space 63 .
42% 376 . − . − space 14 .
02% 32 .
71% 59 . Table 1:
The ratio of the next-to-soft and the soft-virtual contribution in Mellin and z − space for n = 0 at NLO, NNLO and N LO. empirical arguments based on the experience from the behavior of the NLO and NNLOcorrections. The level of precision which must be achieved with empirical estimationsshould be better than the ∼ ±
4% N LO scale variation [24] which corresponds to ± n ∈ [ − ,
3] is close to the full result at NLO (110%of the Born) and NNLO (60% of the Born) in both z − space and Mellin space, with anenvelope of predictions ranging from 109% to 140% of the Born at NLO and from 52%to 73% of the Born at NNLO. At N LO, the variation of the cross-section in both spacesfor the same range of n is from −
22% to 33% of the Born, which is larger than the targetprecision at that order. log , , (1 − z ) terms in full kinematics It is clear from the above that a reliable estimate of the N LO correction of the Higgs cross-section requires even more terms in the threshold expansion. As explained in Sections 2,we have been able to obtain the coefficients of the log , , (1 − z ) terms in a closed form,– 17 – g qg q ¯ q qq qQ log (1 − z ) 111 . − . (1 − z ) 93 . − . . . . (1 − z ) − . . . . . Table 2:
The contribution of the log , , (1 − z ) in full kinematics to the hadronic cross-sectionnormalized to the Born, for each partonic channel. N3LO svns ( full log ,log ,log ) N3LO svns - - - - - - - - n k l o ( n ) z - space - - - Figure 3:
The hadronic cross-section at N LO where the log , , (1 − z ) and δ (1 − z ) contributionsare computed in full kinematics while the remaining log , , (1 − z ) terms are computed in the soft-virtual and next-to-soft approximation in z − space as a function of the artificial parameter n . Thecross-section is normalized to the Born cross-section and only the dominant gg -channel is included. valid for arbitrary values of z . These corrections are insensitive to the artificial parameter n , and thus independent of whether we perform the computation in Mellin or z − space.Their contribution to the hadronic cross-section from each partonic channel normalized tothe Born hadronic cross-section (setting the Wilson coefficient C = 1) is shown in Table 2.Comparing the effect of the full log , , (1 − z ) coefficients to the truncated ones in the (1 − z )expansion, as in eq. (3.1) and (3.2), we find that the full coefficients give systematicallylower contributions to the hadronic cross-section.Knowing the exact log , , (1 − z ) coefficients, we can restrict the threshold approxi-mation only to the coefficients of the log , , (1 − z ) terms. This mixed approach wouldnot have been justified if we had found that the formal threshold expansion hierarchy was– 18 – ( full log ,log ,log ) N3LO svns - - - - - - - - n k l o ( n ) Mellin - space - - - Figure 4:
The hadronic cross-section at N LO where the log , , (1 − z ) and δ (1 − z ) contributionsare computed in full kinematics while the remaining log , , (1 − z ) terms are computed in thesoft-virtual and next-to-soft approximation in Mellin space as a function of the artificial parameter n in Eq. 3.12. The cross-section is normalized to the Born cross-section and only the dominant gg -channel is included. reflected in the results after the integration over the parton densities. However, this is notthe case and it is therefore equally justified (or unjustified) to include the full kinematicdependence of the coefficients of the ‘leading’ logarithms. We present in Fig. 3 the corre-sponding gluon-channel contribution to the hadronic cross-section normalised to the Borncross-section, as a function of the artificial exponent n . As expected from the comparisonof the results of Table 2 in full kinematics and the results of eqs. (2.12) in the thresholdexpansion for the log , , (1 − z ) terms, the inclusion of the full leading logarithms low-ers the value of the N LO correction. The shape as a function of n , however, does notsubstantially change. This indicates that the bulk of the n dependence is carried by thecoefficients of the yet-unknown log , , (1 − z ) terms, and including the exact coefficients oflog , , (1 − z ) does not substantially improve the convergence of the threshold expansion.It is unclear whether the inclusion of the yet unknown full coefficients for the log , , (1 − z )terms in the future will further reduce or increase the cross-section. In Fig. 4, we includethe full log , , (1 − z ) terms exactly and compute the remaining known N LO terms asa threshold expansion in Mellin space. The reduction of the cross-section is even morepronounced in this case. For example, setting n = 0, the pure next-to-soft approximationin Mellin space yields a positive contribution of about +29 .
5% of the Born, while including– 19 –he exact contribution from log , , (1 − z ) and expanding in Mellin space the remainingterms through next-to-soft yields a negative N LO correction of about − .
5% of the Born.The changes that we observe by including the full coefficients of log , , (1 − z ) withrespect to pure next-to-soft approximations have to be compared with smaller scale varia-tion uncertainty at N LO [24], which is about ±
12% of the Born cross-section. While inthis publication we have presented the most advanced theoretical calculation of the N LOcorrections, we conclude that this is insufficient to reduce the theoretical uncertainty of theHiggs-boson cross-section.
4. Conclusions
In this paper we have presented new results for Higgs-boson production at N LO beyondthreshold. More precisely, we have computed for the first time the full next-to-soft correc-tions to Higgs-boson production, as well as the exact results for the coefficients of the firstthree leading logarithms at N LO. Our results constitute a major milestone towards thecomplete computation of the Higgs-boson cross-section via gluon-fusion at N LO.Having at our disposal the formally most accurate result for the threshold expansionavailable to date, we are naturally lead to the question of how reliable phenomenologicalpredictions based on this result would be. In a second part of our paper we thereforeperformed a critical appraisal of the threshold approximation, which according to thegeneral folklore captures the bulk of the Higgs-boson cross-section. Unfortunately, theconvergence of the threshold expansion appears to become less reliable with each furtherorder in the perturbative expansion, as formally subleading terms are not suppressed incomparison to leading terms. In this context, we make the alarming observation that theratio of the next-to-soft over the soft-virtual corrections increases from NLO to NNLO andto N LO showing that the threshold approximation deteriorates when applied to higherorders in the perturbative QCD expansion.A second problem in using the threshold expansion is that there is an ambiguity indefining the convolution integral for the hadronic cross-section from the threshold expan-sion of the partonic cross-sections. We have introduced in eq. (3.9) a way to quantifythis ambiguity by introducing a parameter n such that the hadronic cross section is in-dependent of n if no approximation is made. The truncation of the threshold expansion,however, introduces a dependence on n , and the size of this dependence is a measure forthe convergence of the threshold expansion. We have performed a numerical study of the n -dependence by including terms beyond the strict threshold limit, both in z -space andin Mellin-space. We observe that in all cases the numerical dependence on n is decreasedwhen including corrections beyond threshold, in agreement with the expectations. At NLOand NNLO, a plateau (numerically close to the true value) forms when next-to-soft termsare included. At N LO, however, we observe that no plateau is visible, indicating thatempirical estimations of the N LO cross-section based on the experience from NLO andNNLO may fail. In fact, by including our exact results with full kinematic dependenceof the coefficients of the first three leading logarithms we observe that the hadronic cross-– 20 –ection shifts significantly to lower values than what one obtains with the next-to-softapproximation.Based on these considerations, we conclude that it is not possible at this point toobtain a reliable prediction for the Higgs-boson cross-section at N LO, and that furthertheoretical developments are needed to achieve this goal. This is left for future work.
Acknowledgements
The authors are grateful to Achilleas Lazopoulos and Andreas Vogt for discussions. Thisresearch was supported by the Swiss National Science Foundation (SNF) under contracts200021-143781 and 200020-149517, the European Commission through the ERC grants“IterQCD”, “LHCTheory” (291377), “HEPGAME” (320651) and “MC@NNLO” (340983)and the FP7 Marie Curie Initial Training Network “LHCPhenoNet” (PITN-GA- 2010-264564), by the U.S. Department of Energy under contract no. DE-AC02-07CH11359 andthe “Fonds National de la Recherche Scientifique” (FNRS), Belgium.
References [1] G. Aad et al. [ATLAS Collaboration], “Observation of a new particle in the search for theStandard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B , 1(2012); [arXiv:1207.7214 [hep-ex]];S. Chatrchyan et al. [CMS Collaboration], “Observation of a new boson at a mass of 125GeV with the CMS experiment at the LHC,” Phys. Lett. B , 30 (2012). [arXiv:1207.7235[hep-ex]].[2] D. Graudenz, M. Spira and P. M. Zerwas, “QCD corrections to Higgs boson production atproton proton colliders,” Phys. Rev. Lett. , 1372 (1993);S. Dawson, “Radiative corrections to Higgs boson production,” Nucl. Phys. B , 283(1991);A. Djouadi, M. Spira and P. M. Zerwas, “Production of Higgs bosons in proton colliders:QCD corrections,” Phys. Lett. B , 440 (1991);M. Spira, A. Djouadi, D. Graudenz and P. M. Zerwas, “Higgs boson production at the LHC,”Nucl. Phys. B , 17 (1995) [hep-ph/9504378].[3] C. Anastasiou and K. Melnikov, “Higgs boson production at hadron colliders in NNLOQCD,” Nucl. Phys. B , 220 (2002), [hep-ph/0207004];R. V. Harlander and W. B. Kilgore, “Next-to-next-to-leading order Higgs production athadron colliders,” Phys. Rev. Lett. , 201801 (2002), [hep-ph/0201206];V. Ravindran, J. Smith and W. L. van Neerven, “NNLO corrections to the total cross-sectionfor Higgs boson production in hadron hadron collisions,” Nucl. Phys. B , 325 (2003),[hep-ph/0302135].[4] U. Aglietti, R. Bonciani, G. Degrassi and A. Vicini, “Two loop light fermion contribution toHiggs production and decays,” Phys. Lett. B , 432 (2004) [hep-ph/0404071];S. Actis, G. Passarino, C. Sturm and S. Uccirati, “NLO Electroweak Corrections to HiggsBoson Production at Hadron Colliders,” Phys. Lett. B , 12 (2008) [arXiv:0809.1301[hep-ph]];S. Actis, G. Passarino, C. Sturm and S. Uccirati, “NNLO Computational Techniques: TheCases H → γ γ and H → gg ,” Nucl. Phys. B , 182 (2009) [arXiv:0809.3667 [hep-ph]]; – 21 – . Anastasiou, R. Boughezal and F. Petriello, “Mixed QCD-electroweak corrections to Higgsboson production in gluon fusion,” JHEP , 003 (2009) [arXiv:0811.3458 [hep-ph]];R. V. Harlander and K. J. Ozeren, “Finite top mass effects for hadronic Higgs production atnext-to-next-to-leading order,” JHEP , 088 (2009) [arXiv:0909.3420 [hep-ph]];A. Pak, M. Rogal and M. Steinhauser, “Finite top quark mass effects in NNLO Higgs bosonproduction at LHC,” JHEP , 025 (2010) [arXiv:0911.4662 [hep-ph]].[5] S. Catani, D. de Florian, M. Grazzini and P. Nason, “Soft gluon resummation for Higgsboson production at hadron colliders,” JHEP , 028 (2003) [hep-ph/0306211];V. Ahrens, T. Becher, M. Neubert and L. L. Yang, “Renormalization-Group ImprovedPrediction for Higgs Production at Hadron Colliders,” Eur. Phys. J. C , 333 (2009),[arXiv:0809.4283 [hep-ph]].[6] S. Moch and A. Vogt, “Higher-order soft corrections to lepton pair and Higgs bosonproduction,” Phys. Lett. B (2005) 48 [hep-ph/0508265].[7] R. D. Ball, M. Bonvini, S. Forte, S. Marzani and G. Ridolfi, “Higgs production in gluonfusion beyond NNLO,” Nucl. Phys. B (2013) 746 [arXiv:1303.3590 [hep-ph]];M. Bonvini, R. D. Ball, S. Forte, S. Marzani and G. Ridolfi, “Updated Higgs cross section atapproximate N LO,” J. Phys. G (2014) 095002 [arXiv:1404.3204 [hep-ph]].[8] D. de Florian, J. Mazzitelli, S. Moch and A. Vogt, “Approximate N LO Higgs-bosonproduction cross section using physical-kernel constraints,” JHEP (2014) 176[arXiv:1408.6277 [hep-ph]].[9] C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog and B. Mistlberger,“Higgs boson gluon-fusion production at threshold in N LO QCD ,” Phys. Lett. B , 325(2014) [arXiv:1403.4616 [hep-ph]].[10] P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov, V.A. Smirnov, M. Steinhauser, “Quark andgluon form factors to three loops,” Phys. Rev. Lett. , 212002 (2009); [arXiv:0902.3519[hep-ph]];T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, C. Studerus, “Calculation of thequark and gluon form factors to three loops in QCD,” JHEP , 094 (2010).[arXiv:1004.3653 [hep-ph]].[11] N. A. Lo Presti, A. A. Almasy and A. Vogt, “Leading large- x logarithms of the quark-gluoncontributions to inclusive Higgs-boson and lepton-pair production,” Phys. Lett. B , 120(2014) [arXiv:1407.1553 [hep-ph]].[12] C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, “Real-virtual contributionsto the inclusive Higgs cross-section at N LO ,” JHEP , 088 (2013); [arXiv:1311.1425[hep-ph]].W. B. Kilgore, “One-Loop Single-Real-Emission Contributions to pp → H + X atNext-to-Next-to-Next-to-Leading Order,” Phys. Rev. D (2014) 073008 [arXiv:1312.1296[hep-ph]].[13] F. Dulat and B. Mistlberger, in preparation;[14] C. Duhr, T. Gehrmann, M. Jaquier, in preparation.[15] K. G. Chetyrkin, B. A. Kniehl and M. Steinhauser, “Decoupling relations to O ( α s ) and theirconnection to low-energy theorems,” Nucl. Phys. B , 61 (1998), [hep-ph/9708255];Y. Schroder and M. Steinhauser, “Four-loop decoupling relations for the strong coupling,”JHEP , 051 (2006), [hep-ph/0512058]; – 22 – . G. Chetyrkin, J. H. Kuhn and C. Sturm, “QCD decoupling at four loops,” Nucl. Phys. B , 121 (2006), [hep-ph/0512060].[16] T. Gehrmann, M. Jaquier, E. W. N. Glover and A. Koukoutsakis, “Two-Loop QCDCorrections to the Helicity Amplitudes for H → , 056 (2012).[17] C. Duhr and T. Gehrmann, “The two-loop soft current in dimensional regularization,” Phys.Lett. B , 452 (2013), [arXiv:1309.4393 [hep-ph]];Y. Li and H. X. Zhu, “Single soft gluon emission at two loops,” JHEP , 080 (2013),[arXiv:1309.4391 [hep-ph]].[18] O. V. Tarasov, A. A. Vladimirov and A. Y. .Zharkov, “The Gell-Mann-Low Function of QCDin the Three Loop Approximation,” Phys. Lett. B , 429 (1980);S. A. Larin and J. A. M. Vermaseren, “The Three loop QCD Beta function and anomalousdimensions,” Phys. Lett. B , 334 (1993), [hep-ph/9302208];T. van Ritbergen, J. A. M. Vermaseren and S. A. Larin, “The Four loop beta function inquantum chromodynamics,” Phys. Lett. B , 379 (1997), [hep-ph/9701390];M. Czakon, “The Four-loop QCD beta-function and anomalous dimensions,” Nucl. Phys. B , 485 (2005), [hep-ph/0411261].[19] C. Anastasiou, S. Buehler, F. Herzog and A. Lazopoulos, “Total cross-section for Higgs bosonhadroproduction with anomalous Standard Model interactions,” JHEP , 058 (2011),[arXiv:1107.0683 [hep-ph]].[20] S. Moch, J. A. M. Vermaseren and A. Vogt, “The Three loop splitting functions in QCD: TheNonsinglet case,” Nucl. Phys. B , 101 (2004), [hep-ph/0403192]; “The Three-loop splittingfunctions in QCD: The Singlet case,” Nucl. Phys. B , 129 (2004). [hep-ph/0404111].[21] C. Anastasiou, C. Duhr, F. Dulat, B. Mistlberger, “Soft triple-real radiation for Higgsproduction at N3LO,” JHEP , 003 (2013).[22] Y. Li, A. von Manteuffel, R. M. Schabinger and H. X. Zhu, “N LO Higgs and Drell-Yanproduction at threshold: the one-loop two-emission contribution,” Phys. Rev. D (2014)053006 [arXiv:1404.5839 [hep-ph]].[23] C. Anastasiou, S. B¨uhler, C. Duhr and F. Herzog, “NNLO phase space master integrals fortwo-to-one inclusive cross sections in dimensional regularization,” JHEP , 062 (2012);M. H¨oschele, J. Hoff, A. Pak, M. Steinhauser, T. Ueda, “Higgs boson production at the LHC:NNLO partonic cross sections through order ǫ and convolutions with splitting functions toN LO,” Phys. Lett. B , 244 (2013), [arXiv:1211.6559 [hep-ph]].[24] S. B¨uhler and A. Lazopoulos, “Scale dependence and collinear subtraction terms for Higgsproduction in gluon fusion at N3LO,” JHEP , 096 (2013), [arXiv:1306.2223 [hep-ph]].[25] S. Catani, D. de Florian and M. Grazzini, “Higgs production in hadron collisions: Soft andvirtual QCD corrections at NNLO,” JHEP , 025 (2001), [hep-ph/0102227];R. V. Harlander and W. B. Kilgore, “Soft and virtual corrections to proton proton → H + x at NNLO,” Phys. Rev. D , 013015 (2001) [hep-ph/0102241].[26] A. D. Martin, W. J. Stirling, R. S. Thorne and G. Watt, “Parton distributions for the LHC,”Eur. Phys. J. C , 189 (2009), [arXiv:0901.0002 [hep-ph]].[27] C. Anastasiou, S. Buehler, F. Herzog and A. Lazopoulos, “Inclusive Higgs boson cross-sectionfor the LHC at 8 TeV,” JHEP , 004 (2012). – 23 –
28] G. Sterman and M. Zeng, “Quantifying Comparisons of Threshold Resummations,” JHEP , 132 (2014), [arXiv:1312.5397 [hep-ph]].[29] M. Bonvini, S. Forte, G. Ridolfi and L. Rottoli, “Resummation prescriptions and ambiguitiesin SCET vs. direct QCD: Higgs production as a case study,” [arXiv:1409.0864 [hep-ph]].[30] E. Remiddi and J. A. M. Vermaseren, “Harmonic polylogarithms,” Int. J. Mod. Phys. A (2000) 725 [hep-ph/9905237].(2000) 725 [hep-ph/9905237].