Two-Loop Corrections to Top-Quark Pair Production

Transcription

1 Two-Loop Corrections to Top-Quark Pair Production Andrea Ferroglia Johannes Gutenberg Universität Mainz Padova, June 2, 29

2 Outline 1 Top-Quark Pair Production at Hadron Colliders 2 NNLO Virtual Corrections The Quark-Antiquark Annihilation Channel The Gluon-Gluon Fusion Channel 3 Conclusions and Outlook

3 The Top Quark A particle which tends to stick out... Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 1 / 41

4 The Top Quark Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 1 / 41

5 The Top Quark The top-quark is by far the heaviest fermion in the SM (m t 172 GeV) It decays very rapidly via EW interactions: t bw (τ t = 1/Γ t 1 2 s) The lifetime is one order of magnitude smaller than the hadronization scale (τ had = 1/λ QCD s): the top quark decays before it can form hadronic bound states Because of its large mass, the top quark couples strongly to the electroweak symmetry breaking sector So far it was observed only at the Tevatron (few thousands top quarks produced) The mass of the top-quark could be measured with a percent accuracy Production cross-sections and couplings are know with larger uncertainties Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 1 / 41

6 Reviews M. Beneke et al Top quark physics hep-ph/333 S. Dawson The top quark, QCD, and new physics hep-ph/33191 W. Wagner Top quark physics in hadron collisions hep-ph/727 A. Quadt Top quark physics at hadron colliders EJPC (26) W. Bernreuther Top-quark physics at the LHC J. R. Incandela et al Status and Prospects of Top Quark Physics Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 2 / 41

7 Top Quark Pairs Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 3 / 41

8 Top Quark Pairs Key measurements at the Tevatron and the LHC include the top-quark pair production total cross section and differential distribution p p, pp t tx The top quarks decay almost exclusively in a W boson and a b jet. The observed processes are p p,pp t tx l l 2 + j b + j b + pmiss T + (n ) jets p p,pp t tx l 1 ± + j b + j b + pmiss T + (n 2) jets p p,pp t tx j b + j b + (n 4) jets All these channels were observed and analyzed at the Tevatron Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 3 / 41

9 Top Quark Pairs Key measurements at the Tevatron and the LHC include the top-quark pair production total cross section and differential distribution p p, pp t tx The top quarks decay almost exclusively in a W boson and a b jet. The observed processes are p p,pp t tx l l 2 + j b + j b + pmiss T + (n ) jets p p,pp t tx l 1 ± + j b + j b + pmiss T + (n 2) jets p p,pp t tx j b + j b + (n 4) jets All these channels were observed and analyzed at the Tevatron Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 3 / 41

10 Top Quark Pairs at the LHC At the LHC, one expects to observe millions of top quarks already in the initial low luminosity phase (L 1fb 1 ) With the large number of top quarks expected to be produced at the LHC, the study of its properties will become precision physics The total cross section is sensitive to m t = mass measurement ( σ/σ m t /m t ) LHC experiments will probe, in the t t channel, the existence of heavy resonances with masses up to several TeV Precise measurement of the total cross section at the LHC ( % uncertainty) The current theoretical predictions have an uncertainty of ( 14%) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 4 / 41

11 Top Quark Pair Hadroproduction top-quark pair production is a hard scattering process which can be computed in perturbative QCD h 1 {p} f (x 1 ) t q,g q,g H.S. X h 2 {p, p} f (x 2 ) t σ t t h 1,h 2 = i,j 1 dx 1 1 dx 2 f h 1 i (x 1,µ F )f h 2 j (x 2,µ F )ˆσ ij (ŝ,m t,α s (µ R ),µ F,µ R ) s = (p h1 + p h2 ) 2, ŝ = x 1 x 2 s Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 / 41

12 Top Quark Pair Hadroproduction -II σ t t h 1,h 2 (s had,m 2 t ) = ij shad 4m 2 t dŝ L ij (ŝ,shad,µ 2 f ) ˆσij (ŝ,m 2 t,µ 2 f,µ2 r ) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 6 / 41

13 Top Quark Pair Hadroproduction -II σ t t h 1,h 2 (s had,m 2 t ) = ij shad 4m 2 t dŝ L ij (ŝ,shad,µ 2 f ) ˆσij (ŝ,m 2 t,µ 2 f,µ2 r ) where the luminosity L ij is defined as (ŝ,shad L ij,µ 2 ) 1 shad ds ( ) ( ) f = s had ŝ s f h 1 s i,µ F f h 2 ŝ j s had s,µ F Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 6 / 41

14 Top Quark Pair Hadroproduction -II σ t t h 1,h 2 (s had,m 2 t ) = ij shad 4m 2 t dŝ L ij (ŝ,shad,µ 2 f ) ˆσij (ŝ,m 2 t,µ 2 f,µ2 r ) where the luminosity L ij is defined as Partonic Cross Section (ŝ,shad L ij,µ 2 ) 1 shad ds ( ) ( ) f = s had ŝ s f h 1 s i,µ F f h 2 ŝ j s had s,µ F Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 6 / 41

15 Tree Level QCD Partonic Processes q(p 1 ) + q(p 2 ) t(p 3 ) + t(p 4 ) p 1 p 4 p 2 p 3 g(p 1 ) + g(p 2 ) t(p 3 ) + t(p 4 ) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 7 / 41

16 Tree Level QCD Partonic Processes q(p 1 ) + q(p 2 ) t(p 3 ) + t(p 4 ) p 1 p 4 p 2 p 3 g(p 1 ) + g(p 2 ) t(p 3 ) + t(p 4 ) Dominant at Tevatron 8% Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 7 / 41

17 Tree Level QCD Partonic Processes q(p 1 ) + q(p 2 ) t(p 3 ) + t(p 4 ) p 1 p 4 Dominant at LHC 9% p 2 p 3 g(p 1 ) + g(p 2 ) t(p 3 ) + t(p 4 ) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 7 / 41

18 Tree Level QCD Partonic Processes q(p 1 ) + q(p 2 ) t(p 3 ) + t(p 4 ) p 1 p 4 The NLO p 2 QCD corrections in both pchannels 3 (and to qg t tq) are known since a long time Nason, Dawson, g(p Ellis 1 ) + ( 88-9) g(p 2 ) Beenakker, t(p 3 ) + t(p Kuijf, 4 ) van Neerven, Smith ( 89) Beenakker, van Neerven, Meng, Schuler ( 91) Mangano, Nason, Ridolfi ( 92) Frixione,Mangano, Nason, Ridolfi ( 9) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 7 / 41

19 Tree Level QCD Partonic Processes q(p 1 ) + q(p 2 ) t(p 3 ) + t(p 4 ) p 1 p 4 The mixed QCD-EW p 2 corrections in both channels p 3 are also known (they are smaller than current QCD uncertainties) Beenakker et al. g(p( 94) 1 ) + g(p Bernreuther, 2 ) t(pfuecker, 3 ) + t(p 4 ) and Si ( - 8) Kühn, Scharf, and Uwer ( - 6) Moretti, Nolten, and Ross ( 6) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 7 / 41

20 Luminosity and Partonic CS at NLO Luminosity L ij [1/GeV 2 ] s = 1.96 TeV CTEQ 6. µ f = 171 GeV gg qg qq qq Luminosity L ij [1/GeV 2 ] s = 14 TeV CTEQ 6. µ f = 171 GeV 1 3 qg gg Lq q[%] Lgg[%] 2 Plots from 2 1 Lqg[%] 1 S. Moch and P. Uwer ( 8) Lgg[%] Lq q[%] Lqg[%] gg 2 2 gg NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV sum qq qg qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg qg x 1 Total uncertainty in % s [GeV] % sum qq qq NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV qg x 1 gg qg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV Total uncertainty in % Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 8 / s [GeV] %

21 Luminosity and Partonic CS at NLO qq Luminosity L ij [1/GeV 2 ] qg s = 1.96 TeV gg CTEQ 6. µ f = 171 GeV qg qq Luminosity L ij [1/GeV 2 ] s = 14 TeV CTEQ 6. µ f = 171 GeV gg Lq q[%] Lgg[%] Lqg[%] Lgg[%] at Tevatron Lq q[%] q q luminosity is dominant 1 2 Lqg[%] gg 2 2 gg NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV sum qq qg qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg qg x 1 Total uncertainty in % s [GeV] % sum qq qq NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV qg x 1 gg qg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV Total uncertainty in % Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 8 / s [GeV] %

22 Luminosity and Partonic CS at NLO Luminosity L ij [1/GeV 2 ] s = 1.96 TeV CTEQ 6. µ f = 171 GeV gg qg qg qq Lq q[%] Lgg[%] qg luminosity is dominant 2 Lqg[%] 1 but the corresponding partonic 2 2 cross section gg is tiny; 1 the gg channel dominates NLO QCD qq 1 Partonic cross section [pb] µ r = m t = 171 GeV sum at the LHC qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg qg x 1 Total uncertainty in % s [GeV] % qq Luminosity L ij [1/GeV 2 ] s = 14 TeV CTEQ 6. µ f = 171 GeV sum qq qq 1 3 qg gg gg Lgg[%] Lq q[%] Lqg[%] NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV gg qg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV qg x 1 Total uncertainty in % Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 8 / s [GeV] %

23 Luminosity and Partonic CS at NLO Luminosity L ij [1/GeV 2 ] s = 1.96 TeV CTEQ 6. µ f = 171 GeV gg qg qq qg qq Luminosity L ij [1/GeV 2 ] s = 14 TeV CTEQ 6. µ f = 171 GeV gg Lq q[%] Lgg[%] define Lqg[%] σ(smax) = smax Lˆσ 4m 2 t Lgg[%] Lq q[%] Lqg[%] gg 2 2 gg NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV sum qq qg qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg qg x 1 Total uncertainty in % s [GeV] % sum qq qq NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV qg x 1 gg qg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV Total uncertainty in % Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 8 / s [GeV] %

24 Luminosity and Partonic CS at NLO Luminosity L ij [1/GeV 2 ] s = 1.96 TeV CTEQ 6. µ f = 171 GeV gg NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV sum qg gg qq qg qq Lq q[%] Lgg[%] Lqg[%] qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg qg x 1 Total uncertainty in % s [GeV] % qq Luminosity L ij [1/GeV 2 ] s = 14 TeV CTEQ 6. µ f = 171 GeV 1 3 qg gg Lgg[%] Lq q[%] Lqg[%] 2 gg the CS is dominated 1 NLO QCD 1 qq Partonic cross section [pb] µ r = m t = 171 GeV sum at Tevatron by the region ŝ 2m t qq gg qg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV qg x 1 Total uncertainty in % Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 8 / s [GeV] %

25 Luminosity and Partonic CS at NLO Luminosity L ij [1/GeV 2 ] s = 1.96 TeV CTEQ 6. µ f = 171 GeV gg qg qq qg qq Luminosity L ij [1/GeV 2 ] s = 14 TeV CTEQ 6. µ f = 171 GeV gg Lq q[%] at the LHC the total CS Lgg[%] 2 Lqg[%] 1 higher partonic energies 2 receives large contributions from Lgg[%] Lq q[%] Lqg[%] gg 2 2 gg NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV sum qq qg qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg qg x 1 Total uncertainty in % s [GeV] % sum qq qq NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV qg x 1 gg qg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV Total uncertainty in % Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 8 / s [GeV] %

26 Uncertainty on NLO The partonic cross sections involve terms like ln 2 (1 4m 2 /s) which become large near threshold and must be resummed Kidonakis and Sterman ( 97), Bonciani et al. ( 98), Kidonakis et al.( 1), Kidonakis and Vogt ( 3), Banfi and Laenen ( ) σ - pp tt [pb] at Tevatron NLL res (CTEQ6) m t [GeV] σ - pp tt [pb] at LHC NLL res (CTEQ6) m t [GeV] S. Moch and P. Uwer ( 8) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 9 / 41

27 Uncertainty on NLO The partonic cross sections involve terms like ln 2 (1 4m 2 /s) which become large near threshold and must be resummed Kidonakis and Sterman ( 97), Bonciani et al. ( 98), Kidonakis et al.( 1), Kidonakis and Vogt ( 3), Banfi and Laenen ( ) σ - pp tt [pb] at Tevatron 12% uncertainty at Tevatron NLL res (CTEQ6) m t [GeV] σ - pp tt [pb] at LHC 14% uncertainty at LHC NLL res (CTEQ6) m t [GeV] S. Moch and P. Uwer ( 8) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 9 / 41

28 Uncertainty on NLO The partonic cross sections involve terms like ln 2 (1 4m 2 /s) which become large near threshold and must be resummed Kidonakis and Sterman ( 97), Bonciani et al. ( 98), Kidonakis et al.( 1), Kidonakis and Vogt ( 3), Banfi and Laenen ( ) Moch and Uwer presented an approximated NNLO result 14 σ - pp tt [pb] at Tevatron σ - pp tt [pb] at LHC (ln β, scale dep., Couloumb corrections) 12 which drastically reduces the uncertainty ( 6 8% 1 at Tevatron, 4 6% at LHC) However, it is no substitute for a complete NNLO computation NLL res (CTEQ6) m t [GeV] NLL res (CTEQ6) m t [GeV] S. Moch and P. Uwer ( 8) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 9 / 41

29 Two-Loop Corrections to q q t t The NNLO calculation of the top-quark pair hadroproduction requires several ingredient Virtual Corrections two-loop matrix elements for q q t t and gg t t interference of one-loop diagrams 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 + 2 partons Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 1 / 41

30 Two-Loop Corrections to q q t t The NNLO calculation of the top-quark pair hadroproduction requires several ingredient Virtual Corrections two-loop matrix elements for q q t t and gg t t interference of one-loop diagrams 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 + 2 partons Dittmaier, Uwer, Wenzierl ( 7) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 1 / 41

31 Two-Loop Corrections to q q t t The NNLO calculation of the top-quark pair hadroproduction requires several ingredient Virtual Corrections two-loop matrix elements for q q t t and gg t t interference of one-loop diagrams 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 + 2 partons Overall, in the q q t t channel, QGRAF generates 218 two-loop diagrams (one massive flavor, one massless flavor): 31 non-vanishing diagrams with a massless quark loop 3 non-vanishing diagrams with a massive quark loop 64 non-vanishing planar gluonic diagrams Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 1 / 41

32 Two-Loop Corrections to q q t t M 2 (s,t,m,ε) = 4π2 α 2 s N c [ A + ( αs ) A 1 + π ( αs ) 2 A2 + O ( α 3 π s) ] Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 11 / 41

33 Two-Loop Corrections to q q t t M 2 (s,t,m,ε) = 4π2 α 2 s N c [ A + ( αs ) A 1 + π ( αs ) 2 A2 + O ( α 3 π s) ] A 2 = A (2 ) 2 + A (1 1) 2 Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 11 / 41

34 Two-Loop Corrections to q q t t M 2 (s,t,m,ε) = 4π2 α 2 s N c [ A + ( αs ) π ( αs ) One-Loop 2 One-Loop A 1 + A2 + O ( α 3 π s) ] Körner, Merebashvili, Rogal (, 8) A 2 = A (2 ) 2 + A (1 1) 2 + Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 11 / 41

35 Two-Loop Corrections to q q t t M 2 (s,t,m,ε) = 4π2 α 2 s Two-Loop Tree N c [ A + ( αs ) A 1 + π ( αs ) 2 A2 + O ( α 3 π s) ] A 2 = A (2 ) 2 + A (1 1) 2 + Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 11 / 41

36 Two-Loop Corrections to q q t t M 2 (s,t,m,ε) = 4π2 α 2 s N c [ A + ( αs ) A 1 + π ( αs ) 2 A2 + O ( α 3 π s) ] A 2 = A (2 ) 2 + A (1 1) 2 A (2 ) 2 = N c C F [ N 2 c A + B + C N 2 c +N h ( N c D h + E h N c 1 different color coefficients ( + N l N c D l + E ) l N c ) + N 2 l F l + N l N h F lh + N 2 h F h Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 11 / 41 ]

37 Massive from Massless (s, t, u m 2 t) It is necessary to evaluate two-loop four-point functions depending on s,t, m 2 t Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 12 / 41

38 Massive from Massless (s, t, u m 2 t) It is necessary to evaluate two-loop four-point functions depending on s,t, m 2 t start by considering the limit s, t, u m 2 t Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 12 / 41

39 Massive from Massless (s, t, u m 2 t) Is it possible to calculate graphs employing DIM REG to regulate both soft and collinear singularities and then translate a posteriori the collinear poles into collinear logs ln(m 2 /s)? Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 12 / 41

40 Massive from Massless (s, t, u m 2 t) Is it possible to calculate graphs employing DIM REG to regulate both soft and collinear singularities and then translate a posteriori the collinear poles into collinear logs ln(m 2 /s)? For a generic QED/QCD process, with no closed fermion loops M (m ) = i {all legs} 1 Z i 2 (m,ε)m (m=) where Z is defined through the Dirac form factor F (m ) (Q 2 ) = Z(m,ε)F (m=) (Q 2 ) + O(m 2 /Q 2 ) A. Mitov and S. Moch ( 6) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 12 / 41

41 Massive from Massless (s, t, u m 2 t) Is it possible to calculate graphs employing DIM REG to regulate both soft and collinear singularities and then translate a posteriori the collinear poles into collinear logs ln(m 2 /s)? For a generic QED/QCD process, with no closed fermion loops M (m ) = i {all legs} 1 Z i 2 (m,ε)m (m=) where Z is defined through the Dirac form factor F (m ) (Q 2 ) = Z(m,ε)F (m=) (Q 2 ) + O(m 2 /Q 2 ) A. Mitov and S. Moch ( 6) Warning: maybe this is not the whole story for a general n m amplitude, but it works in the case of q q t t T. Becher and M. Neubert ( 9) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 12 / 41

42 Two-Loop Virtual Corrections in the s m 2 t Limit M. Czakon, A. Mitov, S. Moch ( 7) Using the universal multiplicative relation between massless QCD amplitudes and massive amplitudes in the small mass limit, it was possible to calculate the two-loop virtual corrections to q q t t starting from q q q (massless) q (massless) (C. Anastasiou et al. ( ) ) The same result was also obtained by an approach based on the reduction to master integrals and the expansion of the master integrals in m 2 /s through Mellin Barnes representations Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 13 / 41

43 Numerical Evaluation of the Two-Loop Corrections to q q t t M. Czakon ( 8) Adding more terms in the expansion in powers of m 2 /s,m 2 / t,m 2 / u it is not sufficient to reach a permill accuracy in all the phase space (particularly near threshold) The MIs can be evaluated by solving the corresponding differential equations numerically The evaluation of the full color structure with a 16 digit precision can require as much as 1 minutes per phase space point This does not allow for a direct implementation in a MC generator, but the functions are smooth enough to be interpolated starting from a grid of values Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 14 / 41

44 Numerical Evaluation of the Two-Loop Corrections to q q t t M. Czakon ( 8) Adding more terms in the expansion in powers of m 2 /s,m 2 / t,m 2 / u it is not sufficient to reach a permill accuracy in all the phase space (particularly near threshold) The MIs can be evaluated by solving the corresponding differential equations numerically The evaluation of the full color structure with a 16 digit precision can require as much as 1 minutes per phase space point This does not allow for a direct implementation in a MC generator, but the functions are smooth enough to be interpolated starting from a grid of values Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 14 / 41

45 Fermion Loop Corrections 7 color structures receive contributions from the diagrams involving closed fermion loops only N l number of light (massless) quarks N h number of heavy (massive) quarks A (2 ) 2 = N c C F [ N 2 c A + B + C N 2 c +N h ( N c D h + E h N c ( + N l N c D l + E ) l N c ) + N 2 l F l + N l N h F lh + N 2 h F h ] Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 1 / 41

46 Fermion Loop Corrections 7 color structures receive contributions from the diagrams involving closed fermion Allloops the diagrams only with a closed quark loop (massive or massless) were evaluated analytically N l number of light (massless) quarks Bonciani, AF, Gehrmann, Ma^ıtre, Studerus ( 8) N h number of heavy (massive) quarks Agreement with the numerical results by Czakon A (2 ) 2 = N c C F [ N 2 c A + B + C N 2 c +N h ( N c D h + E h N c ( + N l N c D l + E ) l N c ) + N 2 l F l + N l N h F lh + N 2 h F h ] Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 1 / 41

47 Fermionic Diagrams Self Energies Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 16 / 41

48 Fermionic Diagrams Vertices Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 16 / 41

49 Fermionic Diagrams Boxes Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 16 / 41

50 Fermionic Diagrams Tadpole Self Energies Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 16 / 41

51 Fermionic Diagrams Vertices Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 16 / 41

52 Fermionic Diagrams Boxes Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 16 / 41

53 Method: The General Strategy After interfering a two-loop graph with the Born amplitude one obtains a linear combinations of scalar integrals Born Amplitude D d k 1 D d S n 1 1 k Snq q 2 D m 1 1 D mt t k i integration momenta p i external momenta S scalar productsk i k j or k i p k D propagators [ c i k i + d j p j ] 2 (+mt 2) Luckily, just a small number of these integrals are independent: the MIs It is necessary to identify the MIs = Reduction through the Laporta Algorithm calculate the MIs = Differential Equation Method Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 17 / 41

54 Method: The General Strategy After interfering a two-loop graph with the Born amplitude one obtains a linear combinations of scalar integrals Born Amplitude D d k 1 D d S n 1 1 k Snq q 2 D m 1 1 D mt t k i integration momenta p i external momenta S scalar productsk i k j or k i p k D propagators [ c i k i + d j p j ] 2 (+mt 2) Luckily, just a small number of these integrals are independent: the MIs It is necessary to identify the MIs = Reduction through the Laporta Algorithm calculate the MIs = Differential Equation Method Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 17 / 41

55 Method: The General Strategy After interfering a two-loop graph with the Born amplitude one obtains a linear combinations of scalar integrals Born Amplitude D d k 1 D d S n 1 1 k Snq q 2 D m 1 1 D mt t k i integration momenta p i external momenta S scalar productsk i k j or k i p k D propagators [ c i k i + d j p j ] 2 (+mt 2) Luckily, just a small number of these integrals are independent: the MIs It is necessary to identify the MIs = Reduction through the Laporta Algorithm calculate the MIs = Differential Equation Method Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 17 / 41

56 The Laporta Algorithm The set of denominators D 1,, D t defines a topology; for each topology The scalar integrals are related via Integration By Parts identities (1 identities per integral for a two-loop four-point function) D d k 1 D d k 2 k µ i [v µ Sn 1 1 ] Snq q D m = v µ = k 1 1 D mt 1,k 2,p 1,p 2,p 3 t Building the IBPs for growing powers of the propagators and scalar products the number of equations grows faster that the number of unknown: one finds a system of equations which is apparently over-constrained Solving the system of IBPs (in a problem with a small number of scales) one finds that only a few of the scalar integrals above (if any) are independent: the MIs. Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 18 / 41

57 The Laporta Algorithm The set of denominators D 1,, D t defines a topology; for each topology The scalar integrals are related via Integration By Parts identities (1 identities per integral for a two-loop four-point function) D d k 1 D d k 2 k µ i [v µ Sn 1 1 ] Snq q D m = v µ = k 1 1 D mt 1,k 2,p 1,p 2,p 3 t Building the IBPs for growing powers of the propagators and scalar products the number of equations grows faster that the number of unknown: one finds a system of equations which is apparently over-constrained Solving the system of IBPs (in a problem with a small number of scales) one finds that only a few of the scalar integrals above (if any) are independent: the MIs. Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 18 / 41

58 The Laporta Algorithm The set of denominators D 1,, D t defines a topology; for each topology The scalar integrals are related via Integration By Parts identities (1 identities per integral for a two-loop four-point function) D d k 1 D d k 2 k µ i [v µ Sn 1 1 ] Snq q D m = v µ = k 1 1 D mt 1,k 2,p 1,p 2,p 3 t Building the IBPs for growing powers of the propagators and scalar products the number of equations grows faster that the number of unknown: one finds a system of equations which is apparently over-constrained Solving the system of IBPs (in a problem with a small number of scales) one finds that only a few of the scalar integrals above (if any) are independent: the MIs. Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 18 / 41

59 The Laporta Algorithm The set of denominators D 1,, D t defines a topology; for each topology The scalar integrals are related via Integration By Parts identities (1 identities per integral for a two-loop four-point function) D d k 1 D d k 2 k µ i [v µ Sn 1 1 ] Snq q D m = v µ = k 1 1 D mt 1,k 2,p 1,p 2,p 3 t Building the IBPs for growing powers of the propagators and scalar products the number of equations grows faster that the number of unknown: one finds a system of equations which is apparently over-constrained Solving the system of IBPs (in a problem with a small number of scales) one finds that only a few of the scalar integrals above (if any) are independent: the MIs. Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 18 / 41

60 N l Master Integrals 8 irreducible two-loop topologies Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 19 / 41

61 N l Master Integrals Thick Line Massive Prop. Thin Line Massless Prop. 8 irreducible two-loop topologies Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 19 / 41

62 N h Master Integrals 1 irreducible two-loop topologies Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 2 / 41

63 Reduction at Work The two-loop box diagrams entering in the calculation of the heavy-fermion loop corrections are reducible, i.e. they can be rewritten in terms of integrals belonging to the subtopologies only: Born Amplitude = C + Triangles + Bubbles + Tadpoles Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 21 / 41

64 Calculation of the MIs: Differential Equation Method For each Master Integral belonging to a given topology F (q) l {D 1,, D q } Take the derivative of a given integral with respect to the external momenta p i p µ j p µ i F (q) l = p µ j D d k 1 D d k 2 p µ i S n 1 1 Snq q D m 1 1 D mq q The integrals are regularized, therefore we can apply the derivative to the integrand in the r. h. s. and use the IBPs to rewrite it as a linear combination of the MIs Rewrite the diff. eq. in terms of derivatives with respect to s and t Fix somehow the initial condition(s) (ex. knowing the behavior of the integral at s = ) and solve the system of DE(s) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 22 / 41

65 Calculation of the MIs: Differential Equation Method For each Master Integral belonging to a given topology F (q) l {D 1,, D q } Take the derivative of a given integral with respect to the external momenta p i The integrals are regularized, therefore we can apply the derivative to the integrand in the r. h. s. and use the IBPs to rewrite it as a linear combination of the MIs p µ j D d k 1 D d k 2 p µ i S n 1 1 Snq q D m 1 1 Dq mq = c i F (q) i + r q j k j F (r) j Rewrite the diff. eq. in terms of derivatives with respect to s and t Fix somehow the initial condition(s) (ex. knowing the behavior of the integral at s = ) and solve the system of DE(s) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 22 / 41

66 Calculation of the MIs: Differential Equation Method For each Master Integral belonging to a given topology F (q) l {D 1,, D q } Take the derivative of a given integral with respect to the external momenta p i The integrals are regularized, therefore we can apply the derivative to the integrand in the r. h. s. and use the IBPs to rewrite it as a linear combination of the MIs Rewrite the diff. eq. in terms of derivatives with respect to s and t s F (q) l (s,t) = j c j (s,t)f (q) j (s,t) + r q l k l (s,t)f (r) l (s, t) Fix somehow the initial condition(s) (ex. knowing the behavior of the integral at s = ) and solve the system of DE(s) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 22 / 41

67 Calculation of the MIs: Differential Equation Method For each Master Integral belonging to a given topology F (q) l {D 1,, D q } Take the derivative of a given integral with respect to the external momenta p i The integrals are regularized, therefore we can apply the derivative to the integrand in the r. h. s. and use the IBPs to rewrite it as a linear combination of the MIs Rewrite the diff. eq. in terms of derivatives with respect to s and t Fix somehow the initial condition(s) (ex. knowing the behavior of the integral at s = ) and solve the system of DE(s) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 22 / 41

68 Five Denominator MIs The most complicated irreducible topology in the calculation of the heavy fermion loop corrections is a five denominator box with two MIs (thick lines indicate massive propagators, thin lines massless ones) p 1 p 3 M(α 1,...,α 9 )= p 2 D d k 1 D d k 2 (p 2 k 1 ) α6 (p 1 k 2 ) α7 (p 2 k 2 ) α8 (p 3 k 2 ) α9 P α1 (k 1 + p 1 )P α2 (k 1 + p 1 + p 2 )P α3 m (k 2 )P α4 m (k 1 k 2 )P α m (k 1 +p 3 ) p 4 P (q) = q 2 P m (q) = q 2 + m 2 Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 23 / 41

69 Five Denominator MIs-II M 1 = M 2 = Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 24 / 41

70 Five Denominator MIs-II M 1 = M 2 = the two MIs satisfy two independent first order differential equations dm i (s,t) dt = C i (s,t)m i (s,t) + Ω i (s,t) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 24 / 41

71 Five Denominator MIs-II M 1 = M 2 = the two MIs satisfy two independent first order differential equations dm i (s,t) = C i (s,t)m i (s,t) + Ω i (s,t) dt One of the two needed initial conditions can be fixed by imposing the regularity of the integrals in t = The second integration constant can be fixed by calculating the integral in t = with MB techniques Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 24 / 41

72 Analytic Expression for the MIs By solving the differential equation(s) it is possible to obtain analytic expressions for the MIs (s = m 2 (1 x) 2 /x, y = t/m 2 ) M 1 (d, m, t, s) = A( 3) (d 4) 3 + A( 2) (d 4) 2 + A( 1) (d 4) + A() A 3 = A 2 = A 1 = A = 1 32(y + 1) 1 32(x 1)(y + 1) 1 32(x 1)(y + 1) h i 2G( 1;y)(x 1) + 2(x 1) (1 + x)g(; x) h 3ζ(2)x + 4x + 6(x + 1)G( 1, ;x) + ( x 1)G(, ;x) 4(x 1)G( 1;y) + G(;x)(2(x + 1)G( 1; y) 2(x + 1)) i +4(x 1)G( 1, 1; y) 2(x 1)G(, 1;y) + 3ζ(2) 4 1 h 36(x + 1)G( 1, 1,;x) + 18(x + 1)G( 1,,; x) 32(x 1)(y + 1) +6(x + 1)G(, 1,; x) + ( x 1)G(,,; x) + Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 2 / 41

73 Analytic Expression for the MIs By solving the differential equation(s) it is possible to obtain analytic expressions for the MIs (s = m 2 (1 x) 2 /x, y = t/m 2 ) A 3 = A 2 = A 1 = M 1 (d, m, t, s) = A( 3) (d 4) 3 + A( 2) (d 4) 2 + A( 1) (d 4) + A() 1 32(y + 1) 1 32(x 1)(y + 1) 1 32(x 1)(y + 1) h i 2G( 1;y)(x 1) + 2(x 1) (1 + x)g(; x) h 3ζ(2)x + 4x + 6(x + 1)G( 1, ;x) + ( x 1)G(, ;x) 4(x 1)G( 1;y) + G(;x)(2(x + 1)G( 1; y) 2(x + 1)) The analytic results depend on ln,li i 2,Li 3,S 1,2 +4(x 1)G( 1, 1; y) 2(x 1)G(, 1;y) + 3ζ(2) 4 They are conveniently 1 h expressed in terms of HPLs and 2dHPLs A = 36(x (Remiddi, + 1)G( 1, Vermaseren, 1,;x) + 18(x + Gehrmann) 1)G( 1,,; x) 32(x 1)(y + 1) +6(x + 1)G(, 1,; x) + ( x 1)G(,,; x) + Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 2 / 41

74 Harmonic Polylogarithms (HPLs) E. Remiddi, J. Vermaseren (1999) E. Remiddi, T. Gehrmann (21) Functions of the variable x and a set of indices a with weight w; each index can assume values 1,, 1 Definitions: w = 1 H(1; x) = H(a; x) x H(; x) = ln x H( 1; x) = x d 1 H(a; x) = f (a; x) f (1; x) = dx 1 x dt 1 t dt 1 + t = ln(1 x) = ln(1 + x) f (; x) = 1 x f ( 1; x) = x Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 26 / 41

75 HPLs: Definitions Definitions: w > 1 if a =,,..., (w times) H( w ; x) = else H(i, a; x) = 1 w! lnw x x dtf (i; t)h( a; t) d consequences: dx H(i, a; x) = f (i; x)h( a; x) H( a / ; ) = a few w = 2 H(, 1; x) = H(1, ; x) = x x dtf (; t)h(1; t) = x x dt 1 t ln (1 t) = Li 2(x) 1 dtf (1; t)h(; t) = dt 1 t ln t = lnx ln(1 x) + Li 2 (x) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 27 / 41

76 HPLs as a Generalization of the Nielsen s PolyLogs The HPLs include the Nielsen s PolyLogs S n,p(x) = ( 1)n+p 1 (n + p)!p! Z 1 dt t lnn 1 t ln p (1 xt) Li n(x) = S n 1,1(x) Li n (x) = H( n 1,1;x) S n,p (x) = H( n, 1 p ;x) but the HPLs are a larger set of functions: from w = 4 one finds things as H( 1,,, 1; x) = x dt 1 + t Li 3(x) / Nielsen s PolyLogs Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 28 / 41

77 The HPLs Algebra Shuffle Algebra: some examples H( p; x)h( q; x) = r= p q H( r; x) H(a; x)h(b; x) = H(a, b; x) + H(b, a; x) H(a; x)h(b, c; x) = H(a, b, c; x) + H(b, a, c; x) + H(b, c, a; x) Product Ids: H(m 1,...,m q ; x) = H(m 1 ; x)h(m 2,...,m q ; x) H(m 2, m 1 ; x)h(m 3,...,m q ; x) + + ( 1) q+1 H(m q,...,m 1 ; x) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 29 / 41

78 2-dimensional Harmonic Polylogarithms (2dHPLs) E. Remiddi, T. Gehrmann (2) As for the HPLs, they are obtained by repeated integration over a new set of factors depending on a second variable. f ( y; x) = 1 x + y f ( 1/y; x) = 1 x + 1/y a few examples: G( y;x) = Z x G(i, a; x) = dz z + y = ln 1 + x «y x G( y,;x) = ln x ln dtf (i; t)g( a; t) G( 1/y;x) = 1 + x y Z x «+ Li 2 x y dz z + 1/y «= ln (1 + xy) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 3 / 41

79 2-dimensional Harmonic Polylogarithms (2dHPL)-II The 2dHPLs share the properties of the HPLs The analytic properties of both HPLs & 2dHPLs are know Codes for their numerical evaluation are available E. Remiddi, T. Gehrmann (21-22) Up to w = 3 (our case) the 2dHPLs can be expressed in terms of ln,li 2,Li 3,S 1,2 G( 1/y; x) = ln (1 + xy), G( 1/y, ; x) = ln(x)ln(xy + 1) + Li 2( xy), G( y,1; x) = 1 2 ln2 (y + 1) ln(1 x) ln(y + 1) ln(y) ln(y + 1) «1 x +ln(1 x) ln(x + y) Li 2( y) + Li 2 π2 y + 1 6, G( y, 1,; x) = 1 3 ln3 (1 x) ln(x) ln 2 (1 x)+ Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 31 / 41

80 HPLs as Multiple Sums S. Weinzierl and J. Vollinga ( 4) We defined (2-d)HPLs in terms of iterated integrations G(z 1,,z k ;y) = y dt 1 t 1 z 1 t1 dt 2 t 2 z 2 but they can be written also in terms of multiple sums tk 1 dt k t k z k G(,,,z }{{} 1,,z k 1,,,,z }{{} k ;y) G m1,,m k (z 1,,z k ;y) m 1 1 m k 1 G m1,,m k (z 1,,z k ;y) = ( ) 1 y j1 (j j k ) m 1 j 1 =1 j 1 =1 1 (j j k ) m 2 z 1 ( ) y j2 1 z 1 j m k k ( y z 1 ) jk Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 32 / 41

81 Results All results are written in analytic form: [Fl] = + /81-8/27*[1-x]^-6 + 8/9*[1-x]^- - 62/81*[1-x]^-4 + 4/81*[1-x]^-3-2/81*[1-x]^-2 + 4/27*[1-x]^-1-16/27*y*[1-x]^ /9*y*[1-x]^ /81*y*[1-x]^-4 + 6/81*y*[1-x]^-3 + 2/27*y*[1-x]^-2-1/81*y*[1-x]^-1-8/27*y^2*[1-x]^-6 + 8/9*y^2*[1-x]^- - 62/81*y^2*[1-x]^-4 + 4/81*y^2*[1-x]^-3 + 1/81*y^2*[1-x]^-2-8/27*G(,x)*[1-x]^ /27*G(,x)*[1-x]^-6-176/27*G(,x)*[1-x]^- - 8/9*G(,x)*[1-x]^-4 + 8/27*G(,x)*[1-x]^-3 + 4/3*G(,x)*[1-x]^-2 + 8/9*G(,x)*[1-x]^-1-16/27*G(,x)*y*[1-x]^ /27*G(,x)*y*[1-x]^-6-16*G(,x)*y*[1-x]^ /27*G(1,x)*y*[1-x]^-3-8/9*G(1,x)*y*[1-x]^-2 + 4/27*G(1,x)*y*[1-x]^ /9*G(1,x)*y^2*[1-x]^-6-32/3*G(1,x)*y^2*[1-x]^ /27*G(1,x)*y^2*[1-x]^-4-16/27*G(1,x)*y^2*[1-x]^-3-4/27*G(1,x)*y^2*[1-x]^ /9*G(1,,x)*[1-x]^-7-112/9*G(1,,x)*[1-x]^ /9*G(1,,x)*[1-x]^- - 4/9*G(1,,x)*[1-x]^-4-16/9*G(1,,x)*[1-x]^ /9*G(1,,x)*y*[1-x]^-7-224/9*G(1,,x)*y*[1-x]^ *G(1,,x)*y*[1-x]^- - 16/9*G(1,,x)*y*[1-x]^ /9*G(1,,x)*y*[1-x]^-3 + 8/3*G(1,,x)*y*[1-x]^-2-8/9*G(1,,x)*y*[1-x]^ /9*G(1,,x)*y^2*[1-x]^-7-112/9*G(1,,x)*y^2*[1-x]^ /9*G(1,,x)*y^2*[1-x]^- - 4/9*G(1,,x)*y^2*[1-x]^-4-16/9*G(1,,x)*y^2*[1-x]^-3 + 8/9*G(1,,x)*y^2*[1-x]^-2 ; Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 33 / 41

82 Results-II The color factors can be easily evaluated numerically (either with Mathematica or with a Fortran code): ############################################### Fl = Flh = Fh = E E E+1 El = 1/ep^3*( 2.1E-1) + 1/ep^2*( E+) + 1/ep *( E+2) + ( E+3) Dl = 1/ep^3*( -2.1E-1) + 1/ep^2*( E-1) + 1/ep *( E+2) + ( E+3) Eh = 1/ep^2*( E+) + 1/ep *( E+2) + ( E+3) Dh = 1/ep^2*( E+) + 1/ep *( E+2) + ( E+3) ############################################### Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 34 / 41

83 Results-II The color factors can be easily evaluated numerically (either with Mathematica or with a Fortran code): ############################################### Fl = Flh = Fh = E E E+1 El = 1/ep^3*( 2.1E-1) + 1/ep^2*( E+) + 1/ep *( E+2) + ( E+3) Dl = 1/ep^3*( -2.1E-1) + 1/ep^2*( E-1) + 1/ep *( E+2) + ( E+3) Eh = 1/ep^2*( E+) + 1/ep *( E+2) + ( E+3) by expanding the results for s mt 2 we find analytic agreement with Czakon, Mitov, and Moch ( 7) we we find numerical agreement with Czakon ( 8) we obtained analytic expression in the β = 1 4m 2 /s limit Dh = 1/ep^2*( E+) + 1/ep *( E+2) + ( E+3) ############################################### Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 34 / 41

84 Planar Diagrams in q q t t M 2 (s,t,m,ε) = 4π2 α 2 s N c [ A + ( αs ) A 1 + π ( αs ) 2 A2 + O ( α 3 π s) ] A (2 ) 2 = N c C F [ A 2 = A (2 ) 2 + A (1 1) 2 N 2 c A + B + C N 2 c +N h ( N c D h + E h N c ( + N l N c D l + E ) l N c ) + N 2 l F l + N l N h F lh + N 2 h F h ] Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 3 / 41

85 Planar Diagrams in q q t t M 2 (s,t,m,ε) = 4π2 α 2 [ ( s αs ) ( αs ) 2 A + A 1 + A2 + O ( α 3 N c π π s) ] The coefficient of Nc 2 can also be evaluated analytically it involves only planar diagrams Bonciani, AF, Gehrmann, A 2 = A (2 ) Studerus 2 + A (1 1) (arxiv: ) 2 A (2 ) 2 = N c C F [ N 2 c A + B + C N 2 c +N h ( N c D h + E h N c ( + N l N c D l + E ) l N c ) + N 2 l F l + N l N h F lh + N 2 h F h ] Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 3 / 41

86 MIs for the planar diagrams With respect to the quark loop case, there are 9 new irreducible topologies 2 MIs 2 MIs 2 MIs 2 MIs 2 MIs 2 MIs 2 MIs 2 MIs 3 MIs C. Studerus ( 9) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 36 / 41

87 New Issues and Technical Problems It is necessary to introduce and study 2dHPLs depending on new weight functions 1 1 t 1 ( ), 2 1 ± i 3 t ( 1 1 x x) For the calculation of all the planar diagrams we need HPLs and 2dHPLs up to weight 4 (we needed only weight 3 in the calculation of the quark loop diagrams) The calculation of the integration constants requires the direct evaluation of the integral for special values of s and t with techniques based on Mellin-Barnes representations It was not possible to find closed expressions of the 2dHPL in term of Li n ; the numerical evaluation was carried out with the GiNaC package by Vollinga and Weinzierl The threshold expansion required tricks to extract the expansion of the 2dHPL from the expansion of the integrands in the integral representation Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 37 / 41

88 Planar Diagrams: Results Finite part of A Threshold expansion versus exact result A Φ.4 η =.2 s 4m 2 1 φ = t m2 s Η β = Β 1 4m2 s partonic c.m. scattering angle = π 2 Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 38 / 41

89 Two-Loop Corrections to gg t t A 2 = A (2 ) 2 + A (1 1) 2 One-Loop One-Loop Anastasiou, Aybat ( 8) Körner, Kniehl, Merebashvili, Rogal ( 8) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 39 / 41

90 Two-Loop Corrections to gg t t A (2 ) 2 = (N 2 1) 16 color structures ( A 2 = A (2 ) 2 + A (1 1) 2 N 3 A + NB + 1 N C + 1 N 3D + N2 N l E l + N 2 N h E h +N l F l + N h F h + N l N 2G l + N h N 2G h + NNl 2 H l + NNh 2 H h ) +NN l N h H lh + N2 l N I l + N2 h N I h + N ln h N I lh Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 39 / 41

91 Two-Loop Corrections to gg t t -II A (2 ) 2 is known only in the limit s m 2 t Czakon, Mitov, Moch ( 7) The diagrams involving massless quark loops can be calculated analytically in the usual way Bonciani, AF, Gehrmann, Studerus (in progress) The remaining part of the virtual corrections involve many MI which cannot be expressed in terms of HPLs only = Numerical approach? New ideas? p 2 m 2 Elliptic Functions = K(z) = 1 dx (1 x 2 )(1 zx 2 ) Laporta Remiddi ( 4) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 4 / 41

92 Summary & Conclusions The top-quark pair production cross section will eventually be measurable with a % uncertainty at the LHC; This requires the complete computation of the NNLO QCD corrections to the top-quark pair production cross section a crucial ingredient of the NNLO program is the calculation of the two-loop corrections in both the q q and gg channels In the q q there is already a complete numerical results, the analytical calculation of all the quark-loop diagrams, and the analytical calculation of all the planar diagrams Much less is known in the gg channel; the light-quark loop diagrams can be evaluated analytically, other contributions cannot be expressed in terms of HPLs and might require a numerical approach Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 41 / 41

93 Summary & Conclusions The top-quark pair production cross section will eventually be measurable with a % uncertainty at the LHC; This requires the complete computation of the NNLO QCD corrections to the top-quark pair production cross section a crucial ingredient of the NNLO program is the calculation of the two-loop corrections in both the q q and gg channels In the q q there is already a complete numerical results, the analytical calculation of all the quark-loop diagrams, and the analytical calculation of all the planar diagrams Much less is known in the gg channel; the light-quark loop diagrams can be evaluated analytically, other contributions cannot be expressed in terms of HPLs and might require a numerical approach The moral of the story is always the same: there is a lot of work to do... Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 41 / 41

94 Summary & Conclusions The top-quark pair production cross section will eventually be measurable with a % uncertainty at the LHC; This requires the complete computation of the NNLO QCD corrections to the top-quark pair production cross section a crucial ingredient of the NNLO program is the calculation of the two-loop corrections in both the q q and gg channels In the q q there is already a complete numerical results, the analytical calculation of all the quark-loop diagrams, and the analytical calculation of all the planar diagrams Much less is known in the gg channel; the light-quark loop diagrams can be evaluated analytically, other contributions cannot be expressed in terms of HPLs and might require a numerical approach The moral of the story is always the same: there is a lot of work to do... Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Padova 9 41 / 41

95 Backup Slides

96 The Lorentz Invariance Identities (LIs) T. Gehrmann, E. Remiddi ( 99) A scalar integral is invariant under Lorentz transformation of the external momenta: p µ p µ + δp µ = p µ + δǫ µ ν pν δǫ µ ν = δǫν µ I(p 1, p 2, p 3 ) = I(p 1 + δp 1, p 2 + δp 2, p 3 + δp 3 ) implying the following 3 identities for a 4-point functions (p µ 1 pν 2 p ν 1 p µ 2 ) 3 n=1 (p µ 2 pν 3 pν 2 pµ 3 ) 3 n=1 (p µ 1 pν 3 p ν 1 p µ 3 ) 3 n=1 [ p ν n [ p ν n [ p ν n p µ n p µ n p µ n p µ n p µ n p µ n p ν n p ν n p ν n ] I(p i ) = ] I(p i ) = ] I(p i ) =

97 The General Symmetry Relations Identities Further identities arise when a Feynman graph has symmetries It is in those cases possible to perform a transformation of the loop momenta that does not change the value of the integral but allows to express the integrand as a combination of different integrands Some relations are immediately seen: others are more involved = = k 1 k 2 + p 1 k 2 + p 2 k 1 + s

98 Equations and Unknowns In two-loop 2 2 processes there are two integration momenta and three external momenta 9 possible scalar products Consider the integrals I t,r,s where t # of propagators 9-t # of irreducible scalar products r sum of the powers of all propagators s sum of the powers of all irreducible scalar products The number of integrals belonging to the I t,r,s set is ( )( ) r 1 8 t + s N(I t,r,s ) = r t s It is possible to build (N IBP + N LI )N(I t,r,s ) identities

99 Euler Method First Order Differential Equation d f (x) + C(x)f (x) = Ω(x) dx 1 find the solution of the homogeneous equation d h(x) + C(x)h(x) = dx 2 build the general solution of the non-homogeneous equation [ f (x) = h(x) k + dx Ω(x) ] h(x) 3 fix the integration constant k

100 Euler Method First Order Differential Equation d f (x) + C(x)f (x) = Ω(x) dx 1 find the solution of the homogeneous equation d h(x) + C(x)h(x) = dx 2 build the general solution of the non-homogeneous equation [ f (x) = h(x) k + dx Ω(x) ] h(x) 3 fix the integration constant k

101 Euler Method First Order Differential Equation d f (x) + C(x)f (x) = Ω(x) dx 1 find the solution of the homogeneous equation d h(x) + C(x)h(x) = dx 2 build the general solution of the non-homogeneous equation [ f (x) = h(x) k + dx Ω(x) ] h(x) 3 fix the integration constant k

102 Euler Method-II Second Order Differential Equation d 2 dx 2f (x) + A(x) d f (x) + B(x)f (x) = Ω(x) dx 1 find the two solution of the homogeneous equation 2 build the Wronskian 3 build the solution ( f (x) = h 1 (x) k 1 d 2 dx 2h 1,2(x) + A(x) d dx h 1,2(x) + B(x)h 1,2 (x) = W (x) = h 1 (x) d dx h 2(x) h 2 (x) d dx h 1(x) x ) dw W(w) h 2(w)Ω(w) 4 fix the integration constants k 1 and k 2 +h 2 (x) ( x ) dw k 2 + W(w) h 1(w)Ω(w)

103 Euler Method-II Second Order Differential Equation d 2 dx 2f (x) + A(x) d f (x) + B(x)f (x) = Ω(x) dx 1 find the two solution of the homogeneous equation 2 build the Wronskian 3 build the solution ( f (x) = h 1 (x) k 1 d 2 dx 2h 1,2(x) + A(x) d dx h 1,2(x) + B(x)h 1,2 (x) = W (x) = h 1 (x) d dx h 2(x) h 2 (x) d dx h 1(x) x ) dw W(w) h 2(w)Ω(w) 4 fix the integration constants k 1 and k 2 +h 2 (x) ( x ) dw k 2 + W(w) h 1(w)Ω(w)

104 Euler Method-II Second Order Differential Equation d 2 dx 2f (x) + A(x) d f (x) + B(x)f (x) = Ω(x) dx 1 find the two solution of the homogeneous equation 2 build the Wronskian 3 build the solution ( f (x) = h 1 (x) k 1 d 2 dx 2h 1,2(x) + A(x) d dx h 1,2(x) + B(x)h 1,2 (x) = W (x) = h 1 (x) d dx h 2(x) h 2 (x) d dx h 1(x) x ) dw W(w) h 2(w)Ω(w) 4 fix the integration constants k 1 and k 2 +h 2 (x) ( x ) dw k 2 + W(w) h 1(w)Ω(w)

105 Euler Method-II Second Order Differential Equation d 2 dx 2f (x) + A(x) d f (x) + B(x)f (x) = Ω(x) dx 1 find the two solution of the homogeneous equation 2 build the Wronskian 3 build the solution ( f (x) = h 1 (x) k 1 d 2 dx 2h 1,2(x) + A(x) d dx h 1,2(x) + B(x)h 1,2 (x) = W (x) = h 1 (x) d dx h 2(x) h 2 (x) d dx h 1(x) x ) dw W(w) h 2(w)Ω(w) 4 fix the integration constants k 1 and k 2 +h 2 (x) ( x ) dw k 2 + W(w) h 1(w)Ω(w)

106 Complete Analytic NLO calculation Recently, the NLO total cross section was evaluated analytically M. Czakon, A. Mitov ( 8) Laporta algorithm and differential equation method are employed also for the phase space integrals (by exploiting the optical theorem); cut propagators are treated by using δ ( q 2 + m 2) = 1 2πi ( ) 1 q 2 + m 2 iδ 1 q 2 + m 2 + iδ Anastasiou Melnikov ( 2) The gg t tx cross section cannot be completely written in terms of HPLs and their generalizations. Integrals over elliptic functions K(k) = E(k) = dz 1 z 2 1 k 2 z 2 1 k2 z dz 2 1 z 2 (1)