We present new results for W-boson production at large transverse momentum at
the LHC and the Tevatron. The contribution of soft-gluon corrections is derived
from NNLL resummation and added to the exact NLO result. Numerical results and
their uncertainties for the approximate NNLO W-boson transverse momentum
distributions are derived and compared to recent data from the LHC.Comment: 6 pages, 6 figures; presented at ICHEP 2012, Melbourne, Australia,
  July 4-11, 201

Gonsalves, Richard J.

Kidonakis, Nikolaos

English

arXiv

We present new results for W-boson production at large transverse momentum at the LHC and the Tevatron. The contribution of soft-gluon corrections is derived from NNLL resummation and added to the exact NLO result. Numerical results and their uncertainties for the approximate NNLO W-boson transverse momentum distributions are derived and compared to recent data from the LHC

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Kennesaw State UniversityDigitalCommons@Kennesaw State UniversityFaculty Publications12-1-2012NNLL Resummation for W-boson Production atLarge pTNikolaos KidonakisKennesaw State University, nkidonak@kennesaw.eduRichard J. GonsalvesSuNY - BuffaloFollow this and additional works at: http://digitalcommons.kennesaw.edu/facpubsPart of the Physics CommonsThis Article is brought to you for free and open access by DigitalCommons@Kennesaw State University. It has been accepted for inclusion in FacultyPublications by an authorized administrator of DigitalCommons@Kennesaw State University. For more information, please contactdigitalcommons@kennesaw.edu.Recommended CitationKidonakis, Nikolaos and Gonsalves, Richard J., "NNLL Resummation for W-boson Production at Large pT" (2012). FacultyPublications. 3864.http://digitalcommons.kennesaw.edu/facpubs/3864arXiv:1212.2543v1  [hep-ph]  11 Dec 2012 NNLL resummation for W -boson production at largepTNikolaos Kidonakis∗†Kennesaw State University, USAE-mail: nkidonak@kennesaw.eduRichard J. GonsalvesUniversity at Buffalo, The State University of New York, USAE-mail: phygons@buffalo.eduWe present new results for W -boson production at large transverse momentum at the LHC andthe Tevatron. The contribution of soft-gluon corrections is derived from NNLL resummation andadded to the exact NLO result. Numerical results and their uncertainties for the approximateNNLO W -boson transverse momentum distributions are derived and compared to recent datafrom the LHC.36th International Conference on High Energy PhysicsJuly 4-11, 2012Melbourne, Australia∗Speaker.†This material is based upon work supported by the National Science Foundation under Grant No. PHY 1212472.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/NNLL resummation for W-boson production at large pT Nikolaos Kidonakis50 100 150 200 250 300 350 400 450 500pT (GeV)10-510-410-310-210-1100101102dσ/dp T2  (pb/GeV2 )µ=pTµ=pT/2, 2pTµ=(pT2+mW2)1/2W production at LHC    NLO    S1/2=7 TeV    100 200 300 400 5000.70.80.911.11.21.3ratios  to  µ=pT 20 40 60 80 100 120 140 160 180 200pT (GeV)10-410-310-210-1100101dσ/dp T2  (pb/GeV2 )µ=pTµ=pT/2, 2pTµ=(pT2+mW2)1/2W production at Tevatron    NLO    S1/2=1.96 TeV    20 40 60 80 100 120 140 160 180 2000.70.80.911.11.21.3ratios  to  µ=pT Figure 1: W -boson NLO pT distributions with various choices of scale at the LHC at 7 TeV energy (left)and at the Tevatron (right).1. IntroductionW hadroproduction is useful in testing the Standard Model and in estimates of backgroundsto Higgs production and new physics. The transverse momentum, pT , distribution of the W bosonfalls rapidly as pT increases. Its behavior at high (and also low) pT is sensitive to assumptionsthat go into perturbative QCD predictions, and to parton distribution functions, and it providesimportant input to precision measurements of the W mass.The partonic channels for this process at LO are q(pa)+g(pb)−→W (Q)+q(pc) and q(pa)+q¯(pb) −→W (Q)+ g(pc). At NLO and NNLO the gluon initiated channels dominate the crosssection at multi-TeV energies. We define the kinematical variables s = (pa + pb)2, t = (pa−Q)2,u = (pb−Q)2, and s4 = s+ t +u−Q2. At partonic threshold for a given pT , s4 → 0.In this contribution we present both NLO and approximate NNLO results for the pT distri-bution of the W -boson at large pT . The NLO calculation is complete while at NNLO we includesoft-gluon corrections that are dominant near partonic threshold.2. NLO pT distribution of the W -boson at the LHC and the TevatronThe NLO cross section can be written asEQdσˆ fa fb→W(Q)+Xd3Q = δ (s4)αs(µ2R) [A(s, t,u)+αs(µ2R)B(s, t,u,µR)]+α2s (µ2R)C(s, t,u,s4,µF ) (2.1)where the coefficient functions A, B, and C depend on the parton flavors, µF is the factorizationscale, and µR is the renormalization scale. For numerical results we set µ ≡ µF = µR.The coefficient A(s, t,u) arises from the LO processes, B(s, t,u,µR) is the sum of virtual cor-rections and of singular terms ∼δ (s4) in the real radiative corrections, and C(s, t,u,s4,µF) is fromreal emission processes away from s4 = 0. The complete NLO analytical results were derived overtwenty years ago [1, 2].2NNLL resummation for W-boson production at large pT Nikolaos KidonakisgqqgqqgqqgqqgqqgqqgqqgqqgqqgqqgqqgqqgqqgqqgqqgqqgqqgqqFigure 2: Two-loop eikonal diagrams for qg→Wq.q¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgq¯qgFigure 3: Two-loop eikonal diagrams for qq¯→Wg.In Fig. 1 we show NLO results for dσ/d p2T at the LHC at 7 TeV energy (left) and at theTevatron at 1.96 TeV energy (right). We use the MSTW 2008 pdf sets [3]. Results are shown forfour different choices of scale: the central choice µ = pT , its variation by a factor of two, µ = pT/2,2pT , and also µ = (p2T +m2W )1/2. The inset plots show ratios of the differential distributions withthe last three choices to the central result with µ = pT .3. Approximate NNLO pT distribution of the W -boson at the LHC and the TevatronThe soft-gluon corrections, which appear at each order of perturbation theory starting at NLO,are of the formDl(s4)≡[lnl(s4/p2T )s4]+.For the order αns corrections l ≤ 2n− 1. We can formally resum these logarithms for Wproduction at large pT to all orders in αs [4], and from the expansion of the resummed cross sectionderive approximate NNLO results. Thus, at NLO, the corrections include D1(s4) and D0(s4) terms.At NNLO, the corrections include D3(s4), D2(s4), D1(s4), and D0(s4) terms.The formalism was previously applied to W production at the Tevatron in Ref. [5] and atthe LHC at 14 TeV energy in Ref. [6]. Those calculations were based on NLL resummation [4]3NNLL resummation for W-boson production at large pT Nikolaos Kidonakis50 100 150 200 250 300 350 400 450 500pT (GeV)10-510-410-310-210-1100101dσ/dp T2  (pb/GeV2 )NLONNLO approxW production at  LHC     S1/2=8 TeV      µ=pT100 200 300 400 50011.21.41.61.82NNLO approx / NLO50 100 150 200 250 300 350 400 450 500pT (GeV)10-510-410-310-210-1100101dσ/dp T2  (pb/GeV2 )µ=pTµ=pT/2, 2pTµ=(pT2+mW2)1/2W production at LHC    NNLO approx    S1/2=8 TeV   100 200 300 400 5000.70.80.911.11.21.3ratios  to  µ=pT Figure 4: W -boson approximate NNLO pT distributions with various choices of scale at the LHC at 8 TeV.with additional subleading terms. Thus at NNLO all D3(s4), D2(s4), and D1(s4) terms were fullydetermined in [5] but only partial D0(s4) terms could be calculated.Recent two-loop results for the soft anomalous dimensions [7] have made possible NNLLresummation and thus the determination of all NNLO D0(s4) terms [8]. These new approximateNNLO analytical results from the expansion of the NNLL resummed differential cross section arethen used here to provide numerical predictions [8, 9] for the W -boson pT distribution at the LHCand the Tevatron. We note that a different, SCET-based, approach to resummation for the W bosonhas also appeared in [10].Soft-gluon resummation follows from the factorization properties of the cross section and isperformed in moment space. The resummed cross section isσˆ res(N) = exp[∑iEi(Ni)]exp[E ′j(N ′)]exp[∑i=1,22∫ √sµFdµµ γi/i(˜Ni,αs(µ))]×H (αs) S(αs(√s˜N ′))exp[∫ √s/ ˜N′√sdµµ 2ReΓS (αs(µ))]where the first two exponents describe collinear and soft emission from incoming and outgoingpartons, H is the hard function, S is the soft-gluon function, and ΓS is the soft anomalous dimensionwith perturbative expansionΓS =αspiΓ(1)S +α2spi2Γ(2)S + · · ·The two-loop soft anomalous dimension, Γ(2)S , is determined from the UV poles of dimensionally-regularized integrals for the two-loop eikonal diagrams for qg → W q, shown in Fig. 2, andqq¯ →Wg, shown in Fig. 3.The analytical expressions for the soft anomalous dimensions at one and two loops are givenbelow. For qg→W qΓ(1)S,qg→W q =CF ln(−us)+CA2ln( tu)Γ(2)S,qg→W q =K2Γ(1)S,qg→W q .4NNLL resummation for W-boson production at large pT Nikolaos Kidonakis50 100 150 200 250 300 350 400 450 500pT (GeV)10-510-410-310-210-1100101dσ/dp T2  (pb/GeV2 )NLONNLO approxW production at  LHC     S1/2=7 TeV      µ=pT100 200 300 400 50011.21.41.61.82NNLO approx / NLO50 100 150 200 250 300 350pT (GeV)10-410-310-210-1100101dσ/dp T2  (pb/GeV2 )µ=pTµ=pT/2, 2pTµ=(pT2+mW2)1/2ATLASW production at LHC    NNLO approx    S1/2=7 TeV   100 200 3000.70.80.911.11.21.3ratios  to  µ=pT Figure 5: W -boson approximate NNLO pT distributions with various choices of scale at the LHC at 7 TeV.For qq¯→W gΓ(1)S,qq¯→W g =CA2ln( tus2)Γ(2)S,qq¯→W g =K2Γ(1)S,qq¯→Wg .Our numerical results for the NNLO approximate pT distribution of the W boson at the LHC at8 TeV energy is shown in Fig. 4. Again, we use the MSTW 2008 pdf sets [3]. The left plot contraststhe NLO and approximate NNLO results and the inset plot shows the ratio NNLO approximate /NLO. The NNLO soft-gluon contributions increasingly enhance the NLO result with increasingpT . The right plot displays the approximate NNLO dσ/d p2T for four different choices of scale: thecentral choice µ = pT , its variation by a factor of two, µ = pT/2, 2pT , and also µ = (p2T +m2W )1/2.The inset plot shows ratios of the pT distributions with the last three scale choices to the centralresult with µ = pT .Figure 5 shows results for 7 TeV LHC energy. The left plot again contrasts NLO and approxi-mate NNLO results for dσ/d p2T . Their ratio is very similar to that at 8 TeV shown previously. Theright plot shows dσ/d p2T for various scale choices and compared with recent ATLAS results [11].The agreement between our theoretical predictions and the ATLAS data is very good.Finally, Fig. 6 displays related results for the W -boson pT distribution at the LHC at 14 TeVenergy(left plot) and at the Tevatron (right plot).More analytical and numerical results can be found in Ref. [8] (see also [7, 9]).4. ConclusionsWe have calculated the W -boson transverse momentum distribution for W production at largepT . We presented NLO results as well as approximate NNLO results from NNLL resummationusing two-loop soft-gluon threshold corrections. The higher-order corrections are important forgreater theoretical accuracy.The approximate NNLO pT distributions of the W have been calculated for both LHC andTevatron energies and are in good agreement with recent ATLAS data from the LHC at 7 TeV. Itwill be instructive to compare our results with 8 TeV data from ATLAS and CMS.5NNLL resummation for W-boson production at large pT Nikolaos Kidonakis100 200 300 400 500 600 700 800 900 1000pT (GeV)10-710-610-510-410-310-210-1100101dσ/dp T2  (pb/GeV2 )NLONNLO approxW production at  LHC     S1/2=14 TeV      µ=pT200 400 600 800 100011.21.41.61.82NNLO approx / NLO40 60 80 100 120 140 160 180 200pT (GeV)10-410-310-210-1100dσ/dp T2  (pb/GeV2 )µ=pTµ=pT/2, 2pTµ=(pT2+mW2)1/2W production at Tevatron    NNLO approx    S1/2=1.96 TeV   40 60 80 100 120 140 160 180 2000.70.80.911.11.21.3ratios  to  µ=pT Figure 6: W -boson approximate NNLO pT distributions at 14 TeV LHC (left) and at the Tevatron (right).Our results are also relevant at Tevatron energies. We note that while the D0 and CDF Col-laborations have published their measurements of the W boson pT distributions based on Run Idata, corresponding results for Run II data have not been published to date. Event rates for thisprocess are expected to increase dramatically when the LHC resumes operation. Comparison ofreliable and accurate perturbative predictions with experimental data at these widely separated en-ergy regimes around 2, 8, and 14 TeV could provide important insight into problems involvingparton distribution functions, scale dependence, and resummation of large logarithms.References[1] P.B. Arnold and M.H. Reno, Nucl. Phys. B 319 (1989) 37; (E) B 330 (1990) 284.[2] R.J. Gonsalves, J. Pawłowski, and C.-F. Wai, Phys. Rev. D 40 (1989) 2245; Phys. Lett. B 252 (1990)663.[3] A.D. Martin, W.J. Stirling, R.S. Thorne, and G. Watt, Eur. Phys. J. C 63 (2009) 189 [arXiv:0901.0002[hep-ph]].[4] N. Kidonakis and V. Del Duca, Phys. Lett. B 480 (2000) 87 [hep-ph/9911460].[5] N. Kidonakis and A. Sabio Vera, JHEP 02 (2004) 027 [hep-ph/0311266].[6] R.J. Gonsalves, N. Kidonakis, and A. Sabio Vera, Phys. Rev. Lett. 95 (2005) 222001[hep-ph/0507317].[7] N. Kidonakis, in DIS 2011, arXiv:1105.4267 [hep-ph]; in DPF 2011, eConf C110809,arXiv:1109.1578 [hep-ph].[8] N. Kidonakis and R.J. Gonsalves, arXiv:1201.5265 [hep-ph].[9] N. Kidonakis and R.J. Gonsalves, in DPF 2011, eConf C110809, arXiv:1109.2817 [hep-ph].[10] T. Becher, C. Lorentzen, and M.D. Schwartz, Phys. Rev. Lett. 108 (2012) 012001 [arXiv:1106.4310[hep-ph]]; Phys. Rev. D 86 (2012) 054026 [arXiv:1206.6115].[11] ATLAS Collaboration, G. Aad et al., Phys. Rev. D 85 (2012) 012005 [arXiv:1108.6308 [hep-ex]].6

NNLL resummation for W-boson production at large pT

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