Top-Antitop Production at Hadron Colliders

Transcription

1 Top-Antitop Production at Hadron Colliders Roberto BONCIANI Laboratoire de Physique Subatomique et de Cosmologie, Université Joseph Fourier/CNRS-INP3/INPG, F-3806 Grenoble, France HP.3rd GGI-Florence, September 15, 010 p.1/5

2 Plan of the Talk General Introduction Top Quark at the Tevatron LHC Perspectives Status of the Theoretical calculations The General Framework Total Cross Section at NLO Analytic Two-Loop QCD Corrections Conclusions HP.3rd GGI-Florence, September 15, 010 p./5

3 Top Quark HP.3rd GGI-Florence, September 15, 010 p.3/5

4 Top Quark With a mass of m t = ± 1.3 GeV, the TOP quark (the up-type quark of the third generation) is the heaviest elementary particle produced so far at colliders. Because of its mass, top quark is going to play a unique role in understanding the EW symmetry breaking Heavy-Quark physics crucial at the LHC. Two production mechanisms q t g t Pair Production pp( p) t t q t g t Single Top pp( p) t b, tq ( q ), tw q W + t b W + t g t q b q ( q ) q ( q ) g W Top quark does not hadronize, since it decays in about s (one order of magnitude smaller than the hadronization time) = opportunity to study the quark as single particle Spin properties Interaction vertices Top quark mass Decay products: almost exclusively t W + b ( V tb V td, V ts ) t V tb W + b HP.3rd GGI-Florence, September 15, 010 p.3/5

5 Top Quark With a mass of m t = ± 1.3 GeV, the TOP quark (the up-type quark of the third generation) is the heaviest elementary particle produced so far at colliders. Because of its mass, top quark is going to play a unique role in understanding the EW symmetry breaking Heavy-Quark physics crucial at the LHC. Two production mechanisms q t g t Pair Production pp( p) t t q t g t Single Top pp( p) t b, tq ( q ), tw q W + t b W + t g t q b q ( q ) q ( q ) g W Top quark does not hadronize, since it decays in about s (one order of magnitude smaller than the hadronization time) = opportunity to study the quark as single particle Spin properties Interaction vertices Top quark mass Decay products: almost exclusively t W + b ( V tb V td, V ts ) t V tb W + b HP.3rd GGI-Florence, September 15, 010 p.3/5

6 Top Quark With a mass of m t = ± 1.3 GeV, the TOP quark (the up-type quark of the third generation) is the heaviest elementary particle produced so far at colliders. Because of its mass, top quark is going to play a unique role in understanding the EW symmetry breaking Heavy-Quark physics crucial at the LHC. Two production mechanisms q t g t Pair Production pp( p) t t q t g t Single Top pp( p) t b, tq ( q ), tw q W + t b W + t g t q b q ( q ) q ( q ) g W Top quark does not hadronize, since it decays in about s (one order of magnitude smaller than the hadronization time) = opportunity to study the quark as single particle Spin properties Interaction vertices Top quark mass Decay products: almost exclusively t W + b ( V tb V td, V ts ) t V tb W + b HP.3rd GGI-Florence, September 15, 010 p.3/5

7 Top Quark Tevatron To date the Top quark could be produced and studied only at the Tevatron (discovery 1995) p p collisions at s = 1.96 TeV L 6.5fb 1 reached in 009 O(10 3 ) t t pairs produced so far Only recently confirmation of single-t LHC Running since end 009 pp collisions at s = 7 (14) TeV LHC will be a factory for heavy quarks (L cm s 1, t t at 1Hz!) Even in the first low-luminosity phase ( years 1fb TeV) O(10 4 ) registered t t pairs HP.3rd GGI-Florence, September 15, 010 p.3/5

8 Top Tevatron HP.3rd GGI-Florence, September 15, 010 p.4/5

9 Top Tevatron Events measured at Tevatron [ p p t t W + bw b lνlνb b σ t t 7pb p p t t W + bw b lνq q b b p p t t W + bw b q q q q b b Dilepton 10% Lep+jets 44% All jets 46% HP.3rd GGI-Florence, September 15, 010 p.4/5

10 Top Tevatron Events measured at Tevatron [ p p t t W + bw b lνlνb b σ t t 7pb p p t t W + bw b lνq q b b p p t t W + bw b q q q q b b Dilepton 10% Lep+jets 44% All jets 46% high-p T lept, jets and ME HP.3rd GGI-Florence, September 15, 010 p.4/5

11 Top Tevatron Events measured at Tevatron [ p p t t W + bw b lνlνb b σ t t 7pb p p t t W + bw b lνq q b b p p t t W + bw b q q q q b b Dilepton 10% Lep+jets 44% All jets 46% high-p T lept, jets and ME 1 isol high-p T lept, 4 jets and ME HP.3rd GGI-Florence, September 15, 010 p.4/5

12 Top Tevatron Events measured at Tevatron [ p p t t W + bw b lνlνb b σ t t 7pb p p t t W + bw b lνq q b b p p t t W + bw b q q q q b b Dilepton 10% Lep+jets 44% All jets 46% high-p T lept, jets and ME 1 isol high-p T lept, 4 jets and ME NO lept, 6 jets and low ME HP.3rd GGI-Florence, September 15, 010 p.4/5

13 Top Tevatron Events measured at Tevatron [ p p t t W + bw b lνlνb b σ t t 7pb p p t t W + bw b lνq q b b p p t t W + bw b q q q q b b Dilepton 10% Lep+jets 44% All jets 46% high-p T lept, jets and ME 1 isol high-p T lept, 4 jets and ME NO lept, 6 jets and low ME Background Processes g q g q q g q, l q W + g g Z, γ g q W + g q q q, l + q W W+jets QCD Drell-Yan Di-boson HP.3rd GGI-Florence, September 15, 010 p.4/5

14 Top Tevatron Events measured at Tevatron [ p p t t W + bw b lνlνb b σ t t 7pb p p t t W + bw b lνq q b b p p t t W + bw b q q q q b b Dilepton 10% Lep+jets 44% All jets 46% high-p T lept, jets and ME NO lept, 6 jets and low ME 1 isol high-p T lept, 4 jets and ME Background Processes g q q g W + g q q g q, l q Reduction of the background: b-tagging crucial g g q q q, l + q W+jets QCD Drell-Yan Di-boson Z, γ W + g W HP.3rd GGI-Florence, September 15, 010 p.4/5

15 Top Tevatron Total Cross Section Combination CDF-D0 (m t = 175 GeV) σ t t = N data N bkgr ǫ L σ t t = 7.0 ± 0.6 pb ( σ t t/σ t t 9%) Top-quark Mass Fundamental parameter of the SM. A precise measurement useful to constraint Higgs mass from radiative corrections ( r) A possible extraction: σ t t = need of precise theoretical determination m t 1 σ t t m t 5 σ t t Combination CDF-D0 m t = ± 1.3 GeV (0.75%) 14 0 CDF-I di-l D0-I di-l * CDF-II di-l * D0-II di-l CDF-I l+j D0-I l+j * CDF-II l+j * D0-II l+j CDF-I all-j * CDF-II all-j * CDF-II trk Mass of the Top Quark (*Preliminary) * Tevatron March 09 hep-ex/ CDF March (GeV/c ) m top ±10.3 ± ±1.3 ± ±.7 ± ±.9 ± ± 5.1 ± ± 3.9 ± ± 0.9 ± ± 0.8 ± ±10.0 ± ± 1.7 ± ± 6. ± ± 0.6 ± 1.1 (stat.) ± (syst.) χ /dof = 6.3/10.0 (79%) 1.4 ± 1.5 ±. HP.3rd GGI-Florence, September 15, 010 p.5/5

16 Top Tevatron W helicity fractions F i = B(t bw + (λ W = i)) (i = 1,0,1) measured fitting the distribution in θ (the angle between l + in the W + rest frame and W + direction in the t rest frame) 1 Γ dγ d cos θ = 3 4 F 0 sin θ F (1 cos θ ) F +(1 + cos θ ) F 0 + F + + F = 1 F 0 = 0.66 ± 0.16 ± 0.05 F + = 0.03 ± 0.06 ± 0.03 Spin correlations measured fitting the double distribution (θ 1 (θ ) is the angle between the dir of flight of l 1 (l ) in the t( t) rest frame and the t( t) direction in the t t rest frame) 1 N d N d cos θ 1 dcos θ = 1 4 (1 + κ cos θ 1 cos θ ) < κ < (68% CL) Forward-Backward Asymmetry A FB = N(y t > 0) N(y t < 0) N(y t > 0) + N(y t < 0) A FB = (19.3 ± 6.5(sta) ±.4(sys))% HP.3rd GGI-Florence, September 15, 010 p.6/5

17 Top Tevatron Tevatron searches of physics BSM in top events New production mechanisms via new spin-1 or spin- resonances: q q Z t t in lepton+jets and all hadronic events (bumps in the invariant-mass distribution) Top charge measurements (recently excluded Q t = 4/3) Anomalous couplings L = g j ff γ b µ (V L P L + V R P R ) + iσµν (p t p b ) ν (g L P L + g R P R ) M W From helicity fractions tw µ From asymmetries in the final state (for instance A FB = 3/4 (F + F )) Forward-backward asymmetry Non SM Top decays. Search for charged Higgs: t H + b q q b(τνb) Search for heavy t W + b in lepton+jets HP.3rd GGI-Florence, September 15, 010 p.7/5

18 Top Tevatron Tevatron searches of physics BSM in top events New production mechanisms via new spin-1 or spin- resonances: q q Z t t in lepton+jets and all hadronic events (bumps in the invariant-mass distribution) Top charge measurements (recently excluded Q t = 4/3) Anomalous couplings L = g j ff γ b µ (V L P L + V R P R ) + iσµν (p t p b ) ν (g L P L + g R P R ) M W of New Physics so far twµ From helicity fractions From asymmetries in the final state (for instance A FB = 3/4 (F + F )) Forward-backward asymmetry No Evidence Non SM Top decays. Search for charged Higgs: t H + b q q b(τνb) Search for heavy t W + b in lepton+jets HP.3rd GGI-Florence, September 15, 010 p.7/5

19 Top Tevatron Value Lum fb 1 SM value SM-like? m t ± 0.6 ± 1.1 GeV up to 4.8 / / σ t t 7.0 ± 0.3 ± 0.4 ± 0.4 pb (m t = 175 GeV) pb YES W - helicity F 0 = 0.66 ± 0.16 ± 0.05 F + = 0.03 ± 0.06 ± F 0 = 0.7 F + = 0 YES Spin Correlat < κ < 0.865(68% CL).8 κ = 0.8 YES A FB 0.19 ± 0.07 ± YES Γ t < 13.1 GeV (95% CL) GeV YES τ t cτ t < 5.5 µm (95% CL) m YES BR (t Wb)/(t Wq) > 0.61 (95% CL) % YES Charge Exclude Q t = 4/3 (87% CL) 1.5 /3 YES HP.3rd GGI-Florence, September 15, 010 p.8/5

20 LHC Perspectives Cross Section Top Mass With 100 pb 1 of accumulated data an error of σ t t/σ t t 15% is expected (dominated by statistics!) After 5 years of data taking an error of σ t t/σ t t 5% is expected With 1 fb 1 Mass accuracy: m t 1 3 GeV Top Properties W helicity fractions and spin correlations with 10 fb 1 = 1-5% Top-quark charge. With 1 fb 1 we could be able to determine Q t = /3 with an accuracy of 15% Sensitivity to new physics all the above mentioned points Narrow resonances: with 1 fb 1 possible discovery of a Z of M Z 700 GeV with σ pp Z t t 11 pb HP.3rd GGI-Florence, September 15, 010 p.9/5

21 LHC Perspectives Cross Section Top Mass With 100 pb 1 of accumulated data an error of σ t t/σ t t 15% is expected (dominated by statistics!) After 5 years of data taking an error of σ t t/σ t t 5% is expected With 1 fb 1 Mass accuracy: m t 1 3 GeV Top Properties W helicity fractions and spin correlations with 10 fb 1 = 1-5% Top-quark charge. With 1 fb 1 we could be able to determine Q t = /3 with an accuracy of 15% Sensitivity to new physics all the above mentioned points Narrow resonances: with 1 fb 1 possible discovery of a Z of M Z 700 GeV with σ pp Z t t 11 pb HP.3rd GGI-Florence, September 15, 010 p.9/5

22 Top-Anti Top Pair Production HP.3rd GGI-Florence, September 15, 010 p.10/5

23 Top-Anti Top Pair Production According to the factorization theorem, the process h 1 + h t t + X can be sketched as in the figure: h 1 {p} f(x 1 ) t q, g q, g H.S. X h {p, p} f(x ) t σ t t h 1,h = X i,j Z 1 0 Z 1 dx 1 dx f h1,i(x 1, µ F )f h,j(x, µ F ) ˆσ ij (ŝ, m t, α s (µ R ), µ F, µ R ) 0 s = `p h1 + p h, ŝ = x1 x s HP.3rd GGI-Florence, September 15, 010 p.10/5

24 Top-Anti Top Pair Production According to the factorization theorem, the process h 1 + h t t + X can be sketched as in the figure: h 1 {p} f(x 1 ) t h {p, p} f(x ) q, g q, g H.S. t X PDFs: Universal Part Evolution with the factorization scale predicted by the theory σ t t h 1,h = X i,j Z 1 0 Z 1 dx 1 dx f h1,i(x 1, µ F )f h,j(x, µ F ) ˆσ ij (ŝ, m t, α s (µ R ), µ F, µ R ) 0 s = `p h1 + p h, ŝ = x1 x s HP.3rd GGI-Florence, September 15, 010 p.10/5

25 Top-Anti Top Pair Production According to the factorization theorem, the process h 1 + h t t + X can be sketched as in the figure: h 1 {p} f(x 1 ) t h {p, p} f(x ) q, g q, g H.S. Partonic Cross Section Process dependent part Calculation in Perturbation Theory t X σ t t h 1,h = X i,j Z 1 0 Z 1 dx 1 dx f h1,i(x 1, µ F )f h,j(x, µ F ) ˆσ ij (ŝ, m t, α s (µ R ), µ F, µ R ) 0 s = `p h1 + p h, ŝ = x1 x s HP.3rd GGI-Florence, September 15, 010 p.10/5

26 The Cross Section: LO HP.3rd GGI-Florence, September 15, 010 p.11/5

27 The Cross Section: LO q(p 1 ) + q(p ) t(p 3 ) + t(p 4 ) q t Dominant at Tevatron 85% q t g(p 1 ) + g(p ) t(p 3 ) + t(p 4 ) g t Dominant at LHC 90% g t σ LO t t (LHC, m t = 171 GeV) = 583 pb ± 30% σ LO t t (Tev, m t = 171 GeV) = 5.9 pb ± 44% HP.3rd GGI-Florence, September 15, 010 p.11/5

28 The Cross Section: NLO HP.3rd GGI-Florence, September 15, 010 p.1/5

29 The Cross Section: NLO Fixed Order The NLO QCD corrections are quite sizable: + 5% at Tevatron and +50% at LHC. Scale variation ±15%. Nason, Dawson, Ellis 88-90; Beenakker, Kuijf, van Neerven, Smith 89-91; Mangano, Nason, Ridolfi 9; Frixione et al. 95; Czakon and Mitov 08 Melnikov and Schulze 09; Bernreuther and Si 10 Mixed NLO QCD-EW corrections are small: - 1% at Tevatron and -0.5% at LHC. Beenakker et al. 94 Bernreuther, Fuecker, and Si Kühn, Scharf, and Uwer 05-06; Moretti, Nolten, and Ross 06. HP.3rd GGI-Florence, September 15, 010 p.1/5

30 The Cross Section: NLO Fixed Order The NLO QCD corrections are quite sizable: + 5% at Tevatron and +50% at LHC. Scale variation ±15%. Nason, Dawson, Ellis 88-90; Beenakker, Kuijf, van Neerven, Smith 89-91; Mangano, Nason, Ridolfi 9; Frixione et al. 95; Czakon and Mitov 08 Melnikov and Schulze 09; Bernreuther and Si 10 Mixed NLO QCD-EW corrections are small: - 1% at Tevatron and -0.5% at LHC. Beenakker et al. 94 Bernreuther, Fuecker, and Si Kühn, Scharf, and Uwer 05-06; Moretti, Nolten, and Ross 06. The QCD corrections to processes involving at least two large energy scales (ŝ, m t Λ QCD ) are characterized by a logarithmic behavior in the vicinity of the boundary of the phase space σ X n,m C n,m α n S lnm (1 ρ) m n HP.3rd GGI-Florence, September 15, 010 p.1/5

31 The Cross Section: NLO Fixed Order The NLO QCD corrections are quite sizable: + 5% at Tevatron and +50% at LHC. Scale variation ±15%. Nason, Dawson, Ellis 88-90; Beenakker, Kuijf, van Neerven, Smith 89-91; Mangano, Nason, Ridolfi 9; Frixione et al. 95; Czakon and Mitov 08 Melnikov and Schulze 09; Bernreuther and Si 10 Mixed NLO QCD-EW corrections are small: - 1% at Tevatron and -0.5% at LHC. Inelasticity parameter Beenakker et al. 94 Bernreuther, Fuecker, and Si Kühn, Scharf, and Uwer 05-06; Moretti, Nolten, and Ross 06. ρ = 4m t 1 ŝ The QCD corrections to processes involving at least two large energy scales (ŝ, m t Λ QCD ) are characterized by a logarithmic behavior in the vicinity of the boundary of the phase space σ X n,m C n,m α n S lnm (1 ρ) m n HP.3rd GGI-Florence, September 15, 010 p.1/5

32 The Cross Section: NLO Fixed Order The NLO QCD corrections are quite sizable: + 5% at Tevatron and +50% at LHC. Scale variation ±15%. Nason, Dawson, Ellis 88-90; Beenakker, Kuijf, van Neerven, Smith 89-91; Mangano, Nason, Ridolfi 9; Frixione et al. 95; Czakon and Mitov 08 Melnikov and Schulze 09; Bernreuther and Si 10 Mixed NLO QCD-EW corrections are small: - 1% at Tevatron and -0.5% at LHC. Inelasticity parameter Beenakker et al. 94 Bernreuther, Fuecker, and Si Kühn, Scharf, and Uwer 05-06; Moretti, Nolten, and Ross 06. ρ = 4m t 1 ŝ The QCD corrections to processes involving at least two large energy scales (ŝ, m t Λ QCD ) are characterized by a logarithmic behavior in the vicinity of the boundary of the phase space σ X n,m C n,m α n S lnm (1 ρ) m n Even if α S 1 (perturbative region) we can have at all orders Resummation = improved perturbation theory α n S lnm (1 ρ) O(1) HP.3rd GGI-Florence, September 15, 010 p.1/5

33 The Cross Section: NLO Fixed Order The NLO QCD corrections are quite sizable: + 5% at Tevatron and +50% at LHC. Scale variation ±15%. Nason, Dawson, Ellis 88-90; Beenakker, Kuijf, van Neerven, Smith 89-91; Mangano, Nason, Ridolfi 9; Frixione et al. 95; Czakon and Mitov 08 Melnikov and Schulze 09; Bernreuther and Si 10 Mixed NLO QCD-EW corrections are small: - 1% at Tevatron and -0.5% at LHC. Beenakker et al. 94 Bernreuther, Fuecker, and Si Kühn, Scharf, and Uwer 05-06; Moretti, Nolten, and Ross 06. All-order Soft-Gluon Resummation Leading-Logs (LL) Next-to-Leading-Logs (NLL) Laenen et al. 9-95; Berger and Contopanagos 95-96; Catani et al. 96. Kidonakis and Sterman 97; R. B., Catani, Mangano, and Nason Next-to-Next-to-Leading-Logs (NNLL) Moch and Uwer 08; Beneke et al ; Czakon et al. 09; Kidonakis 09; Ahrens et al. 10 HP.3rd GGI-Florence, September 15, 010 p.1/5

34 NLO+NLL Theoretical Prediction TEVATRON σ NLO+NLL (Tev, m t t t = 171 GeV,CTEQ6.5) = (3.9%) 0.53(6.9%) (scales) +0.53(7%) 0.36(4.8%) (PDFs) pb LHC σ NLO+NLL (LHC, m t t t = 171 GeV,CTEQ6.5) = (9.0%) 85(9.3%) (scales) +30(3.3%) 9(3.%) (PDFs) pb M. Cacciari, S. Frixione, M. Mangano, P. Nason, and G. Ridolfi, JHEP 0809:17, σ pp tt - [pb] at Tevatron σ pp tt - [pb] at LHC NLL res (CTEQ65) 00 NLL res (CTEQ65) m t [GeV] m t [GeV] S. Moch and P. Uwer, Phys. Rev. D 78 (008) HP.3rd GGI-Florence, September 15, 010 p.13/5

35 Measurement Requirements for σ t t HP.3rd GGI-Florence, September 15, 010 p.14/5

36 Measurement Requirements for σ t t Experimental requirements for σ t t: Tevatron σ/σ 1% = ok! LHC (14 TeV, high luminosity) σ/σ 5% current theoretical prediction!! HP.3rd GGI-Florence, September 15, 010 p.14/5

37 Measurement Requirements for σ t t Experimental requirements for σ t t: Tevatron σ/σ 1% = ok! LHC (14 TeV, high luminosity) σ/σ 5% current theoretical prediction!! Different groups presented approximated higher-order results for σ t t Including scale dep at NNLO, NNLL soft-gluon contributions, Coulomb corrections σ NNLOappr (Tev, m t t t = 173 GeV,MSTW008) = σ NNLOappr (LHC, m t t t = 173 GeV,MSTW008) = Integration of the Invariant mass distribution at NLO+NNLL (scales) (scales) (PDFs) pb (PDFs) pb Kidonakis and Vogt 08; Moch and Uwer 08; Langenfeld, Moch, and Uwer 09 σ NLO+NNLL (Tev, m t t t = GeV,MSTW008) = σ NLO+NNLL (LHC, m t t t = GeV,MSTW008) = (scales) (scales) (PDFs) pb (PDFs) pb V. Ahrens, A. Ferroglia, M. Neubert, B. D. Pecjak, L. L. Yang, arxiv: HP.3rd GGI-Florence, September 15, 010 p.14/5

38 Next-to-Next-to-Leading Order HP.3rd GGI-Florence, September 15, 010 p.15/5

39 Next-to-Next-to-Leading Order The NNLO calculation of the top-quark pair hadro-production requires several ingredients: HP.3rd GGI-Florence, September 15, 010 p.15/5

40 Next-to-Next-to-Leading Order The NNLO calculation of the top-quark pair hadro-production requires several ingredients: Virtual Corrections two-loop matrix elements for q q t t and gg t t interference of one-loop diagrams Körner et al ; Anastasiou and Aybat 08 HP.3rd GGI-Florence, September 15, 010 p.15/5

41 Next-to-Next-to-Leading Order The NNLO calculation of the top-quark pair hadro-production requires several ingredients: Virtual Corrections two-loop matrix elements for q q t t and gg t t interference of one-loop diagrams Körner et al ; Anastasiou and Aybat 08 Real Corrections one-loop matrix elements for the hadronic production of t t + 1 parton tree-level matrix elements for the hadronic production of t t + partons Dittmaier, Uwer and Weinzierl HP.3rd GGI-Florence, September 15, 010 p.15/5

42 Next-to-Next-to-Leading Order p p t t + 1 jet Important for a deeper understanding of the t t prod (possible structure of the top-quark) Important for the charge asymmetry at Tevatron Technically complex involving multi-leg NLO diagrams σ t t+j (LHC) = pb (with p T,jet,cut = 50 GeV) confirmed by G. Bevilacqua, M. Czakon, C.G. Papadopoulos, M. Worek, Phys. Rev. Lett. 104 (010) 1600 ( ) [ ] dσ fb dp T,jet GeV ( ) [ ] dσ fb dp T,jet GeV 10 p p t t + jet + X s = 1.96TeV 1000 pp t t + jet + X s = 14TeV t t + j NLO LO 10 NLO LO K = NLO/LO p T,jet [GeV] K = NLO/LO p T,jet [GeV] S. Dittmaier, P. Uwer and S. Weinzierl, Eur. Phys. J. C 59 (009) 65 HP.3rd GGI-Florence, September 15, 010 p.15/5

43 Next-to-Next-to-Leading Order The NNLO calculation of the top-quark pair hadro-production requires several ingredients: Virtual Corrections two-loop matrix elements for q q t t and gg t t interference of one-loop diagrams Körner et al ; Anastasiou and Aybat 08 Real Corrections one-loop matrix elements for the hadronic production of t t + 1 parton tree-level matrix elements for the hadronic production of t t + partons Dittmaier, Uwer and Weinzierl Subtraction Terms Both matrix elements known for t t + j calculation, BUT subtraction up to 1 unresolved parton, while in a complete NNLO computation of σ t t we need subtraction terms with up to unresolved partons. HP.3rd GGI-Florence, September 15, 010 p.15/5

44 Next-to-Next-to-Leading Order The NNLO calculation of the top-quark pair hadro-production requires several ingredients: Virtual Corrections two-loop matrix elements for q q t t and gg t t interference of one-loop diagrams Körner et al ; Anastasiou and Aybat 08 Real Corrections one-loop matrix elements for the hadronic production of t t + 1 parton tree-level matrix elements for the hadronic production of t t + partons Dittmaier, Uwer and Weinzierl Subtraction Terms Both matrix elements known for t t + j calculation, BUT subtraction up to 1 unresolved parton, while in a complete NNLO computation of σ t t we need subtraction terms with up to unresolved partons. = Need an extension of the subtraction methods at the NNLO. Gehrmann-De Ridder, Ritzmann 09, Daleo et al. 09, Boughezal et al. 10, Glover, Pires 10, Czakon 10 HP.3rd GGI-Florence, September 15, 010 p.15/5

45 Next-to-Next-to-Leading Order The NNLO calculation of the top-quark pair hadro-production requires several ingredients: Virtual Corrections two-loop matrix elements for q q t t and gg t t interference of one-loop diagrams Körner et al ; Anastasiou and Aybat 08 Real Corrections one-loop matrix elements for the hadronic production of t t + 1 parton tree-level matrix elements for the hadronic production of t t + partons Dittmaier, Uwer and Weinzierl Subtraction Terms Both matrix elements known for t t + j calculation, BUT subtraction up to 1 unresolved parton, while in a complete NNLO computation of σ t t we need subtraction terms with up to unresolved partons. = Need an extension of the subtraction methods at the NNLO. Gehrmann-De Ridder, Ritzmann 09, Daleo et al. 09, Boughezal et al. 10, Glover, Pires 10, Czakon 10 HP.3rd GGI-Florence, September 15, 010 p.15/5

46 Two-Loop Corrections to q q t t HP.3rd GGI-Florence, September 15, 010 p.16/5

47 Two-Loop Corrections to q q t t M (s, t, m, ε) = 4π α s N c» αs αs A 0 + A 1 + A + O `αs 3 π π A = A ( 0) + A (1 1) A ( 0) = N c C F h N c A + B + C N c + N l N c D l + E l N c +N h N c D h + E «h + Nl N F l + N l N h F lh + Nh F h c «i 18 two-loop diagrams contribute to the 10 different color coefficients HP.3rd GGI-Florence, September 15, 010 p.16/5

48 Two-Loop Corrections to q q t t M (s, t, m, ε) = 4π α s N c» αs αs A 0 + A 1 + A + O `αs 3 π π A = A ( 0) + A (1 1) A ( 0) = N c C F h N c A + B + C N c + N l N c D l + E l N c +N h N c D h + E «h + Nl N F l + N l N h F lh + Nh F h c «i 18 two-loop diagrams contribute to the 10 different color coefficients The whole A ( 0) is known numerically Czakon 08. HP.3rd GGI-Florence, September 15, 010 p.16/5

49 Two-Loop Corrections to q q t t M (s, t, m, ε) = 4π α s N c» αs αs A 0 + A 1 + A + O `αs 3 π π A = A ( 0) + A (1 1) A ( 0) = N c C F h N c A + B + C N c + N l N c D l + E l N c +N h N c D h + E «h + Nl N F l + N l N h F lh + Nh F h c «i 18 two-loop diagrams contribute to the 10 different color coefficients The whole A ( 0) is known numerically Czakon 08. The coefficients D i, E i, F i, and A are known analytically (agreement with known res) R. B., Ferroglia, Gehrmann, Maitre, and Studerus HP.3rd GGI-Florence, September 15, 010 p.16/5

50 Two-Loop Corrections to q q t t M (s, t, m, ε) = 4π α s N c» αs αs A 0 + A 1 + A + O `αs 3 π π A = A ( 0) + A (1 1) A ( 0) = N c C F h N c A + B + C N c + N l N c D l + E l N c +N h N c D h + E «h + Nl N F l + N l N h F lh + Nh F h c «i 18 two-loop diagrams contribute to the 10 different color coefficients The whole A ( 0) is known numerically Czakon 08. The coefficients D i, E i, F i, and A are known analytically (agreement with known res) R. B., Ferroglia, Gehrmann, Maitre, and Studerus The poles of A ( 0) (and therefore of B and C) are known analytically Ferroglia, Neubert, Pecjak, and Li Yang 09 HP.3rd GGI-Florence, September 15, 010 p.16/5

51 Two-Loop Corrections to q q t t D i, E i, F i come from the corrections involving a closed (light or heavy) fermionic loop: A the leading-color coefficient, comes from the planar diagrams: The calculation is carried out analytically using: Laporta Algorithm for the reduction of the dimensionally-regularized scalar integrals (in terms of which we express the M ) to the Master Integrals (MIs) Differential Equations Method for the analytic solution of the MIs HP.3rd GGI-Florence, September 15, 010 p.17/5

52 Laporta Algorithm and Diff. Equations Decomposition of the Amplitude in terms of Scalar Integrals (DIM. REGULARIZATION) Identity relations among Scalar Integrals: Generation of IBPs, LI and symmetry relations (codes written in FORM) Output: Algebraic Linear System of equations on the unknown integrals Solution of the algebraic system with a C program Output: Relations that link Scalar Integrals to the MIs Generation (in FORM) of the System of DIFF. EQs. on the ext. kin. invariants (calculation of the MIs) IBPs, LI, Symm. rel. System of 1st-order linear DIFF. EQs. Solution in Laurent series of (D-4). Coeff expressedin terms of HPLs PUBLIC PROGRAMS AIR Maple package (C. Anastasiou and A. Lazopoulos, JHEP 0407 (004) 046) FIRE Mathematica package (A. V. Smirnov, JHEP 0810 (008) 107) REDUZE C++/GiNaC package (C. Studerus, Comput. Phys. Commun. 181 (010) 193) HP.3rd GGI-Florence, September 15, 010 p.18/5

53 Master Integrals for N l and N h 1 MI 1 MI 1 MI MIs 1 MI 1 MI 1 MI 1 MI MIs 1 MI MIs 3 MIs MIs 1 MI 1 MI 1 MI MIs MIs 18 irreducible two-loop topologies (6 MIs) R. B., A. Ferroglia, T. Gehrmann, D. Maitre, and C. Studerus, JHEP 0807 (008) 19. HP.3rd GGI-Florence, September 15, 010 p.19/5

54 Master Integrals for the Leading Color Coeff MIs MIs MIs MIs MIs MIs MIs MIs 3 MIs For the leading color coefficient there are 9 additional irreducible topologies (19 MIs) R. B., A. Ferroglia, T. Gehrmann, and C. Studerus, JHEP 0908 (009) 067. HP.3rd GGI-Florence, September 15, 010 p.0/5

55 Example = 1 m 6 1 X i= 4 A i ǫ i + O(ǫ 0 ) A 4 = A 3 = A = A 1 = x 4(1 x) 4 (1 + y), x 96(1 x) 4 (1 + y) h i 10G( 1; y) + 3G(0; x) 6G(1; x), x h i 48(1 x) 4 5ζ() 6G( 1; y)g(0; x)+1g( 1; y)g(1; x)+8g( 1, 1; y), (1 + y) x 48(1 x) 4 (1 + y) h 13ζ(3) + 38ζ()G( 1; y) + 9ζ()G(0; x) + 6ζ()G(1; x) 4ζ()G ( 1/y; x) +4G(0; x)g( 1, 1; y) 4G(1; x)g( 1, 1; y) 1G ( 1/y; x) G( 1, 1; y) 1G( y; x)g( 1, 1; y) 6G(0; x)g(0, 1; y) + 6G ( 1/y; x) G(0, 1; y) + 6G( y; x)g(0, 1; y) +1G( 1; y)g(1, 0; x) 4G( 1; y)g(1, 1; x) 6G( 1; y)g ( 1/y, 0; x) + 1G( 1; y)g ( 1/y, 1; x) 6G( 1; y)g( y, 0; x) + 1G( 1; y)g( y, 1; x) + 16G( 1, 1, 1; y) 1G( 1, 0, 1; y) 1G(0, 1, 1; y) + 6G(0, 0, 1; y) + 6G(1, 0, 0; x) 1G(1, 0, 1; x) 1G(1, 1, 0; x) + 4G(1, 1, 1; x) 6G ( 1/y, 0, 0; x) + 1G ( 1/y, 0, 1; x) + 6G ( 1/y, 1, 0; x) 1G ( 1/y, 1, 1; x) + 6G( y, 1, 0; x) i 1G( y, 1, 1; x) HP.3rd GGI-Florence, September 15, 010 p.1/5

56 Example = 1 m 6 1 X i= 4 A i ǫ i + O(ǫ 0 ) A 4 = A 3 = A = A 1 = x 4(1 x) 4 (1 + y), x 96(1 x) 4 (1 + y) h i 10G( 1; y) + 3G(0; x) 6G(1; x), 1- and -dim GHPLs x h i 48(1 x) 4 5ζ() 6G( 1; y)g(0; x)+1g( 1; y)g(1; x)+8g( 1, 1; y), (1 + y) x h 48(1 x) 4 13ζ(3) + 38ζ()G( 1; y) + 9ζ()G(0; x) + 6ζ()G(1; x) 4ζ()G ( 1/y; x) (1 + y) ρ = 4m t 1 +4G(0; x)g( 1, 1; y) 4G(1; x)g( 1, 1; y) 1G ( 1/y; x) G( 1, 1; y) ŝ 1G( y; x)g( 1, 1; y) 6G(0; x)g(0, 1; y) + 6G ( 1/y; x) G(0, 1; y) + 6G( y; x)g(0, 1; y) +1G( 1; y)g(1, 0; x) 4G( 1; y)g(1, 1; x) 6G( 1; y)g ( 1/y, 0; x) + 1G( 1; y)g ( 1/y, 1; x) 6G( 1; y)g( y, 0; x) + 1G( 1; y)g( y, 1; x) + 16G( 1, 1, 1; y) 1G( 1, 0, 1; y) 1G(0, 1, 1; y) + 6G(0, 0, 1; y) + 6G(1, 0, 0; x) 1G(1, 0, 1; x) 1G(1, 1, 0; x) + 4G(1, 1, 1; x) 6G ( 1/y, 0, 0; x) + 1G ( 1/y, 0, 1; x) + 6G ( 1/y, 1, 0; x) 1G ( 1/y, 1, 1; x) + 6G( y, 1, 0; x) i 1G( y, 1, 1; x) HP.3rd GGI-Florence, September 15, 010 p.1/5

57 GHPLs One- and two-dimensional Generalized Harmonic Polylogarithms (GHPLs) are defined as repeated integrations over set of basic functions. In the case at hand f w (x) = 1 x w, with ( ) w 0,1, 1, y, 1 y, 1 ± i 3 f w (y) = 1 y w, with j0, w 1, 1, x, 1x,1 1x ff x The weight-one GHPLs are defined as G(0; x) = ln x, G(w; x) = Z x 0 dtf w (t) Higher weight GHPLs are defined by iterated integrations G(0,0,, 0; x) = 1 {z } n! lnn x, G(w, ; x) = n Z x 0 dtf w (t)g( ; t) Shuffle algebra. Integration by parts identities Remiddi and Vermaseren 99, Gehrmann and Remiddi 01-0, Aglietti and R. B. 03, Vollinga and Weinzierl 04, R. B., A. Ferroglia, T. Gehrmann, and C. Studerus 09 HP.3rd GGI-Florence, September 15, 010 p./5

58 Coefficient A Finite part of A Threshold expansion versus exact result A η = 0.6 Φ 0.4 s 4m 1, Η m φ = t s β = s Β 1 4m s partonic c.m. scattering angle = π Numerical evaluation of the GHPLs with GiNaC C++ routines. Vollinga and Weinzierl 04 HP.3rd GGI-Florence, September 15, 010 p.3/5

59 Two-Loop Corrections to gg t t HP.3rd GGI-Florence, September 15, 010 p.4/5

60 Two-Loop Corrections to gg t t M (s, t, m, ε) = 4π α s N c» αs αs A 0 + A 1 + A + O `αs 3 π π A = A ( 0) + A (1 1) A ( 0) = (N c 1) N 3 c A + N cb + 1 N c C + 1 N 3 c D + N c N le l + N c N he h +N l F l + N h F h + N l N c G l + N h N c +N c N l N h H lh + N l I l + N h I h + N ln h I lh N c N c N c 789 two-loop diagrams contribute to 16 different color coefficients No numeric result for A ( 0) yet The poles of A ( 0) are known analytically G h + N c N l H l + N c N h H h «Ferroglia, Neubert, Pecjak, and Li Yang 09 The coefficients A, E l I l can be evaluated analytically as for the q q channel R. B., Ferroglia, Gehrmann, von Manteuffel and Studerus, in preparation HP.3rd GGI-Florence, September 15, 010 p.4/5

61 Two-Loop Corrections to gg t t M (s, t, m, ε) = 4π α s N c» αs αs A 0 + A 1 + A + O `αs 3 π π A = A ( 0) + A (1 1) A ( 0) = (N c 1) N 3 c A + N cb + 1 N c C + 1 N 3 c D + N c N le l + N c N he h +N l F l + N h F h + N l N c G l + N h G h + N c Nl H l + N c Nh H h N c +N c N l N h H lh + N l I l + N h I h + N ln h N c N c 789 two-loop diagrams contribute to 16 different color coefficients No numeric result for A ( 0) yet The poles of A ( 0) are known analytically - We finished the evaluation «of the leading-color I lh coefficient N c NO additional MI Ferroglia, Neubert, Pecjak, and Li Yang 09 The coefficients A, E l I l can be evaluated analytically as for the q q channel R. B., Ferroglia, Gehrmann, von Manteuffel and Studerus, in preparation HP.3rd GGI-Florence, September 15, 010 p.4/5

62 Two-Loop Corrections to gg t t M (s, t, m, ε) = 4π α s N c» αs αs A 0 + A 1 + A + O `αs 3 π π A = A ( 0) + A (1 1) A ( 0) = (N c 1) N 3 c A + N cb + 1 N c C + 1 N 3 c D + N c N le l + N c N he h +N l F l + N h F h + N l N c G l + N h N c +N c N l N h H lh + N l I l + N h I h + N ln h I lh N c N c N c 789 two-loop diagrams contribute to 16 different color coefficients - For the light-fermion contrib one missing topology to reduce - up to now No numeric result for A ( 0) yet 7 additional MIs The poles of A ( 0) are known analytically G h + N c N l H l + N c N h H h «Ferroglia, Neubert, Pecjak, and Li Yang 09 The coefficients A, E l I l can be evaluated analytically as for the q q channel R. B., Ferroglia, Gehrmann, von Manteuffel and Studerus, in preparation HP.3rd GGI-Florence, September 15, 010 p.4/5

63 Conclusions In the last 15 years, Tevatron explored top-quark properties reaching a remarkable experimental accuracy. The top mass could be measured with m t /m t = 0.75% and the production cross section with σ t t/σ t t = 9%. Other observables could be measured only with bigger errors. At the LHC the situation will further improve. The production cross section of t t pairs is expected to reach the accuracy of σ t t/σ t t = 5%!! This experimental precision requires a complete NNLO theoretical analysis. In this talk I briefly reviewed the analytic evaluation of the two-loop matrix elements. The corrections involving a fermionic loop (light or heavy) in the q q channel are completed, together with the leading color coefficient. Analogous corrections in the gg channel can be calculated with the same technique and are at the moment under study. The calculation of the crossed diagrams and of the diagrams with a heavy loop have still to be afforded. HP.3rd GGI-Florence, September 15, 010 p.5/5