We update our estimate of the cross section for Higgs production in gluon fusion at next- to-next-to-next-to-leading order (N3LO) in \u3b1s in view of the recent full computation of the result in the soft limit for infinite top mass, which determines a previously unknown constant. We briefly discuss the phenomenological implications. Results are available through the updated version of the ggHiggs code

Anastasiou C

Catani S

Eynck T O

Giovanni Ridolfi

Harlander R V

Marco Bonvini

Richard D Ball

Schroder Y

Simone Marzani

Stefano Forte

arXiv

We update our estimate of the cross section for Higgs production in gluon fusion at next-to-next-to-next-to-leading order in ${{\alpha }_{s}}$ in view of the recent full computation of the result in the soft limit for infinite top mass, which determines a previously unknown constant. We briefly discuss the phenomenological implications. Results are available through the updated version of the ggHiggs code

M. Bonvini

R.D. Ball

S. Forte

S. Marzani

G. Ridolfi

AIR Universita degli studi di Milano

Updated Higgs cross section at approximate N3LO

Crossref

Journal of Physics G Nuclear and Particle Physics

Updated Higgs cross section at approximate N<sup>3</sup>LO

We update our estimate of the cross section for Higgs production in gluon fusion at next- to-next-to-next-to-leading order (N3LO) in αs in view of the recent full computation of the result in the soft limit for infinite top mass, which determines a previously unknown constant. We briefly discuss the phenomenological implications. Results are available through the updated version of the ggHiggs code

BONVINI, MARCO

Richard D. Ball

MARZANI, SIMONE

RIDOLFI, GIOVANNI

Archivio istituzionale della ricerca - Università di Genova

English

We update our estimate of the cross section for Higgs production in gluon fusion at next-to-next-to-next-to-leading order (N$^3$LO) in $\alpha_s$ in view of the recent full computation of the result in the soft limit for infinite top mass, which determines a previously unknown constant. We briefly discuss the phenomenological implications. Results are available through the updated version of the ggHiggs code

Bonvini, Marco

D. Ball, Richard

Forte, Stefano

Marzani, Simone

Ridolfi, Giovanni

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      Updated Higgs cross section at approximate N3LOCitation for published version:Bonvini, M, D. Ball, R, Forte, S, Marzani, S & Ridolfi, G 2014, 'Updated Higgs cross section at approximateN3LO', Journal of Physics G - Nuclear and Particle Physics, vol. 41, no. 9, 095002.https://doi.org/10.1088/0954-3899/41/9/095002Digital Object Identifier (DOI):10.1088/0954-3899/41/9/095002Link:Link to publication record in Edinburgh Research ExplorerPublished In:Journal of Physics G - Nuclear and Particle PhysicsGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 23. Oct. 2019DESY 14-052IFUM-1026-FTDCPT/14/62IPPP/14/31Updated Higgs cross section at approximate N3LOMarco Bonvinia, Richard D. Ballb, Stefano Fortec,Simone Marzanid and Giovanni Ridolfiea Deutsches Elektronen-Synchroton, DESY,Notkestraße 85, D-22603 Hamburg, GermanybTait Institute, University of Edinburgh,JCMB, KB, Mayfield Road, Edinburgh EH9 3JZ, ScotlandcDipartimento di Fisica, Universita` di Milano and INFN, Sezione di Milano,Via Celoria 16, I-20133 Milano, ItalydInstitute for Particle Physics Phenomenology, Durham University,South Road, Durham DH1 3LE, EnglandeDipartimento di Fisica, Universita` di Genova and INFN, Sezione di Genova,Via Dodecaneso 33, I-16146 Genova, ItalyAbstract:We update our estimate of the cross section for Higgs production in gluon fusion at next-to-next-to-next-to-leading order (N3LO) in αs in view of the recent full computation of theresult in the soft limit for infinite top mass, which determines a previously unknown constant.We briefly discuss the phenomenological implications. Results are available through theupdated version of the ggHiggs code.1arXiv:1404.3204v3  [hep-ph]  3 Jul 2014The cross section for the production of a Higgs boson in gluon-gluon fusion (the dominant produc-tion subprocess at available collider energies) has been computed up to next-to-next-to-leading order(NNLO) in perturbative Quantum Chromodynamics (QCD) [1], and a calculation of the N3LO cor-rection is under way [2,3]. In a recent paper [4] we have provided an estimate of the N3LO correction,based on the knowledge of the analytic structure of the coefficients of the perturbative expansion inthe space of the variable N , Mellin conjugate to z = m2H/sˆ where mH is the Higgs mass and sˆ isthe squared partonic center-of-mass energy. Such knowledge, in turn, originates from resummation ofpowers of logN in the large-N regime (soft-gluon or threshold resummation), and from high-energyresummation, which fixes the behavior of the coefficient in the vicinity of its rightmost singularity inthe N complex plane. High-energy resummation turns out to have a relatively small direct numericalimpact, but its analyticity properties affect considerably the form of soft-emission logarithmic terms.The result of Ref. [4] in particular included all contributions to the O(α3s) which do not vanishas N → ∞, and which either grow logarithmically, or are constant. These contributions were allknown, with the exception of the constant, which in z space corresponds to the coefficient of theδ(1 − z) contribution to the cross section. Recently [3], the full cross section was determined in thesoft limit, including this constant, and our approximate result can be updated accordingly: the resultof Ref. [3] effectively amounts to a determination of the coefficient g0,3 of Ref. [4]. The value of thiscoefficient was estimated in Ref. [4] to be g0,3 = 114.7 (for finite mt). This estimate was arrived at byrewriting g0,n = g¯0,n+rn, with r3 known in terms of available information, noting that the perturbativebehaviour of the known coefficients suggests r3  g¯0,3, and thus simply assuming g0,3 = r3. Similarestimates for g0 have also been obtained using methods for the all-order resummation of constantcontributions [5].In Ref. [3] the coefficient g0,3 is determined in the pointlike limit. We wish to use this result whileretaining the full mt dependence of all the remaining information (which in particular is important inorder to have the correct analytic structure in the high-energy limit). Note that the mt dependence ofthis constant at previous orders is negligible: at NLO the variation of g0,1 when going from finite mtto the pointlike limit is by about 0.5%, and at NNLO the variation of g0,2 is by about 1%. At N3LOwe have a certain latitude in deciding which coefficient we should evaluate in the pointlike limit. Wechoose to take the pointlike limit value of the coefficient of the delta function in the full O(α3s) crosssection (as given in Eq. (4) of Ref. [3]).With this procedure we thus determine g0,3 = 116.7, very close to the value g0,3 = 114.7 usedin Ref. [4]. Using the resulting value for g¯0,3 = g0,3 − r3 = 116.7 − 114.7 = 2.0 in Eqs. (4.1,4.2) ofRef. [4] we get the approximate updated total cross section at the Large Hadron Collider (LHC), with√s = 8 TeV and mH = 125 GeV, computed using the NNLO NNPDF2.1 [6] set of parton distributionfunctions with αs(m2Z) = 0.119 (i.e. the same choices as in Ref. [4])σN3LOapprox(s,m2H) =(22.42± 0.31) pb for µR = mH (1)=(23.70± 0.54) pb for µR = mH/2, (2)where µF = mH , and the error shown is our estimate of the uncertainty in our approximation. Theresult is extremely close to the value of Ref. [4], the difference being at the permille level.Our result, and its dependence on the renormalization scale µR, are shown in Fig. 1 (the dependenceon the factorization scale is negligible, as discussed in Ref. [4]), compared to the lower-order results.Our estimate of the N3LO cross section is the red dot-dashed curve; the band provides our estimatefor the uncertainty involved in the approximation procedure, details are given in Ref. [4].We thus confirm, now on a firmer footing, the conclusion of Ref. [4], namely that the N3LOcontribution leads (for µR = mH) to an increase by about 16% of the NNLO cross section. Note thatit was shown in Ref. [7] that this conclusion would very likely be unaffected by the consistent use ofN3LO parton distributions. This result can be compared to the main commonly-used approximationto higher-order corrections, namely that from threshold resummation at next-to-next-to-leading log(NNLL) [8]. The truncation of the latter to O(α3s) is also shown in Fig. 1: it is seen to lead to an2 0 5 10 15 20 25 30 0.06  0.1  0.2  0.3  0.5  1  2  3σ [pb]µR / mHHiggs cross section: gluon fusionmH = 125 GeVLHC 8 TeVLONLONNLOapprox NNNLON-soft NNNLOsoft-0 NNNLOtruncated NNLL+NNLOFigure 1. Dependence of the N3LO cross section on the renormalization scale µR. Two common choices ofrenormalization scale are shown as vertical bars. The approximate N3LO curves are, from top to bottom,our best approximation, the N -soft approximation, the N3LO truncation of the NNLL resummed result ofRef. [8], and the soft-0 approximation (see text for details). In all cases, the full result with finite top massis included through NNLO. The known LO, NLO and NNLO results are also shown. The red band providesan estimate of the uncertainty on our result, obtained with the procedure of Ref. [4].increase of the NNLO by about 6% at the same scale. Note that the (in principle infinite) series ofhigher orders included in the resummation only adds an extra 2% to this.This truncated NNLL resummed result differs from our approximation in three respects: the valueof the constant (which in Ref. [8] corresponds to g0,3 = 0); the coefficient of the single-logarithmicterm (both the constant and the single log would only appear in next=to-next-to-next-to-leading log(N3LL) resummation); and the fact that the constraints due to matching to high-energy resummationand analyticity are not taken into account. The effect of the single logarithmic term is completelynegligible, so the difference is due in roughly equal proportion to each of the other two reasons. Thisis also illustrated in Fig. 1: the N -soft (see Ref. [4] for the precise definition) curve corresponds tousing the exact constant (and single-logarithmic term), but otherwise only including in the same formthe N3LO terms as in the resummation (i.e. without matching and analyticity). This prediction isseen to indeed lie half-way between our approximation and the truncated NNLL resummed result.Finally, we also show in Fig. 1 the so-called soft-0 approximation (again, see Ref. [4] for a precisedefinition). This basically amounts to only keeping soft contributions, but in z space rather than inN space, and it would predict a suppression, rather than an enhancement, of the N3LO cross sectionin comparison to the NNLO one, for a wide range of values of µR. In the soft limit this approximationcoincides with the other approximations discussed here, but away from the limit it differs from them bylarge corrections suppressed by powers of 1N [or (1− z)]; it is known [4,9] to fail at NLO and NNLO,essentially because it does not respect longitudinal momentum conservation (albeit by subleadingterms) [10]. The result found using this soft-0 approximation was explicitly given in Ref. [3].3The updated prediction is available through the code ggHiggs (version 1.9 onwards), publiclyavailable at the website http://www.ge.infn.it/∼bonvini/higgs/.AcknowledgmentsWe acknowledge useful discussions with Claude Duhr and Franz Herzog on the results of Ref. [3].The work of SM is supported by the UK’s STFC. SF and GR are supported in part by an ItalianPRIN2010 grant, and SF also by a European Investment Bank EIBURS grant, and by the EuropeanCommission through the HiggsTools Initial Training Network PITN-GA-2012-316704.References[1] A. Djouadi, M. Spira and P. 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Updated Higgs cross section at approximate N3LO

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