Luminosity and Partonic CS at NLO -5 -5 -3 -3 10 10 10 10 3 -4 -4 400 500 600 700 800 900 10 qg 10 10 -6 -6 -5 -5 – 10 qq 10 10 10 -6 -6 10 10 -7 -7 gg 10 10 – -7 qq -7 10 10 -8 Luminosity L ij [ 1/GeV 2 ] -8 -8 Luminosity L ij [ 1/GeV 2 ] -8 10 10 qg 10 10 -9 √ s = 14 TeV -9 √ s = 1.96 TeV 10 10 gg -10 CTEQ 6.5 -10 -9 -9 CTEQ 6.5 10 10 10 10 -11 µ f = 171 GeV -11 µ f = 171 GeV 10 10 -10 -10 -12 -12 10 10 10 10 10 10 10 10 q [ % ] ∆L gg [ % ] 5 5 5 5 ∆L q¯ at Tevatron 0 0 0 0 60 60 10 10 40 ∆L gg [ % ] 40 q [ % ] 5 ∆L q¯ 5 20 20 q ¯ q luminosity is dominant 0 0 0 0 20 20 10 10 ∆L qg [ % ] ∆L qg [ % ] 10 10 5 5 0 0 0 0 25 25 25 25 gg 20 20 20 20 gg 15 15 15 15 NLO QCD Partonic cross section [ pb ] 10 10 10 10 – – NLO QCD qq qq µ r = m t = 171 GeV Partonic cross section [ pb ] qg 5 µ r = m t = 171 GeV 5 5 5 qg 0 0 0 0 100 100 7 800 % % sum gg 6 – sum qq 80 80 600 5 NLO QCD 60 60 4 Hadronic cross section [ pb ] NLO QCD 400 µ f = µ r = m t = 171 GeV Hadronic cross section [ pb ] 3 40 40 µ f = µ r = m t = 171 GeV 2 gg 200 – 20 20 qq 1 0 0 qg x 10 0 qg x 10 0 -1 -20 -20 10 10 5 5 Total uncertainty in % Total uncertainty in % 0 0 0 0 3 400 500 600 700 800 900 10 √ s [ GeV ] √ s [ GeV ] Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 8 / 41
Luminosity and Partonic CS at NLO -5 -5 -3 -3 10 10 10 10 3 -4 -4 400 500 600 700 800 900 10 qg 10 10 -6 -6 -5 -5 – 10 qq 10 10 10 -6 -6 10 10 -7 -7 gg 10 10 – -7 qq -7 10 10 -8 Luminosity L ij [ 1/GeV 2 ] -8 -8 Luminosity L ij [ 1/GeV 2 ] -8 10 10 qg 10 10 -9 √ s = 14 TeV -9 √ s = 1.96 TeV 10 10 gg -10 CTEQ 6.5 -10 -9 -9 CTEQ 6.5 10 10 10 10 -11 µ f = 171 GeV -11 µ f = 171 GeV 10 10 -10 -10 -12 -12 10 10 10 10 10 10 10 10 q [ % ] ∆L gg [ % ] 5 5 5 5 ∆L q¯ at the LHC 0 0 0 0 60 60 10 10 40 ∆L gg [ % ] 40 q [ % ] 5 ∆L q¯ 5 20 qg luminosity is dominant 20 0 0 0 0 20 20 10 10 ∆L qg [ % ] ∆L qg [ % ] 10 10 5 5 but the corresponding partonic 0 0 0 0 25 25 25 25 gg cross section is tiny; 20 20 20 20 gg 15 15 15 15 NLO QCD the gg channel dominates Partonic cross section [ pb ] 10 10 10 10 – – NLO QCD qq qq µ r = m t = 171 GeV Partonic cross section [ pb ] qg 5 µ r = m t = 171 GeV 5 5 5 qg 0 0 0 0 100 100 7 800 % % sum gg 6 – sum qq 80 80 600 5 NLO QCD 60 60 4 Hadronic cross section [ pb ] NLO QCD 400 µ f = µ r = m t = 171 GeV Hadronic cross section [ pb ] 3 40 40 µ f = µ r = m t = 171 GeV 2 gg 200 – 20 20 qq 1 0 0 qg x 10 0 qg x 10 0 -1 -20 -20 10 10 5 5 Total uncertainty in % Total uncertainty in % 0 0 0 0 3 400 500 600 700 800 900 10 √ s [ GeV ] √ s [ GeV ] Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 8 / 41
Luminosity and Partonic CS at NLO -5 -5 -3 -3 10 10 10 10 3 -4 -4 400 500 600 700 800 900 10 qg 10 10 -6 -6 -5 -5 – 10 qq 10 10 10 -6 -6 10 10 -7 -7 gg 10 10 – -7 qq -7 10 10 -8 Luminosity L ij [ 1/GeV 2 ] -8 -8 Luminosity L ij [ 1/GeV 2 ] -8 10 10 qg 10 10 -9 √ s = 14 TeV -9 √ s = 1.96 TeV 10 10 gg -10 CTEQ 6.5 -10 -9 -9 CTEQ 6.5 10 10 10 10 -11 µ f = 171 GeV -11 µ f = 171 GeV 10 10 -10 -10 -12 -12 10 10 10 10 10 10 10 10 q [ % ] ∆L gg [ % ] 5 5 5 5 ∆L q¯ define 0 0 0 0 60 60 10 10 40 ∆L gg [ % ] 40 q [ % ] 5 ∆L q¯ 5 20 20 0 0 0 0 � s max 20 20 10 10 ∆L qg [ % ] ∆L qg [ % ] σ ( s max) = L ˆ σ 10 10 5 5 4 m 2 0 0 0 0 25 25 t 25 25 gg 20 20 20 20 gg 15 15 15 15 NLO QCD Partonic cross section [ pb ] 10 10 10 10 – – NLO QCD qq qq µ r = m t = 171 GeV Partonic cross section [ pb ] qg 5 µ r = m t = 171 GeV 5 5 5 qg 0 0 0 0 100 100 7 800 % % sum gg 6 – sum qq 80 80 600 5 NLO QCD 60 60 4 Hadronic cross section [ pb ] NLO QCD 400 µ f = µ r = m t = 171 GeV Hadronic cross section [ pb ] 3 40 40 µ f = µ r = m t = 171 GeV 2 gg 200 – 20 20 qq 1 0 0 qg x 10 0 qg x 10 0 -1 -20 -20 10 10 5 5 Total uncertainty in % Total uncertainty in % 0 0 0 0 3 400 500 600 700 800 900 10 √ s [ GeV ] √ s [ GeV ] Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 8 / 41
Luminosity and Partonic CS at NLO -5 -5 -3 -3 10 10 10 10 3 -4 -4 400 500 600 700 800 900 10 qg 10 10 -6 -6 -5 -5 – 10 qq 10 10 10 -6 -6 10 10 -7 -7 gg 10 10 – -7 qq -7 10 10 -8 Luminosity L ij [ 1/GeV 2 ] -8 -8 Luminosity L ij [ 1/GeV 2 ] -8 10 10 qg 10 10 -9 √ s = 14 TeV -9 √ s = 1.96 TeV 10 10 gg -10 CTEQ 6.5 -10 -9 -9 CTEQ 6.5 10 10 10 10 -11 µ f = 171 GeV -11 µ f = 171 GeV 10 10 -10 -10 -12 -12 10 10 10 10 10 10 10 10 q [ % ] ∆L gg [ % ] 5 5 5 5 ∆L q¯ 0 0 0 0 60 60 10 10 40 ∆L gg [ % ] 40 q [ % ] 5 ∆L q¯ 5 20 20 0 0 0 0 20 20 10 10 ∆L qg [ % ] ∆L qg [ % ] 10 10 5 5 0 0 0 at Tevatron 0 25 25 25 25 gg 20 20 20 20 the CS is dominated gg √ 15 15 15 15 NLO QCD by the region ˆ s ≈ 2 m t Partonic cross section [ pb ] 10 10 10 10 – – NLO QCD qq qq µ r = m t = 171 GeV Partonic cross section [ pb ] qg 5 µ r = m t = 171 GeV 5 5 5 qg 0 0 0 0 100 100 7 800 % % sum gg 6 – sum qq 80 80 600 5 NLO QCD 60 60 4 Hadronic cross section [ pb ] NLO QCD 400 µ f = µ r = m t = 171 GeV Hadronic cross section [ pb ] 3 40 40 µ f = µ r = m t = 171 GeV 2 gg 200 – 20 20 qq 1 0 0 qg x 10 0 qg x 10 0 -1 -20 -20 10 10 5 5 Total uncertainty in % Total uncertainty in % 0 0 0 0 3 400 500 600 700 800 900 10 √ s [ GeV ] √ s [ GeV ] Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 8 / 41
Luminosity and Partonic CS at NLO -5 -5 -3 -3 10 10 10 10 3 -4 -4 400 500 600 700 800 900 10 qg 10 10 -6 -6 -5 -5 – 10 qq 10 10 10 -6 -6 10 10 -7 -7 gg 10 10 – -7 qq -7 10 10 -8 Luminosity L ij [ 1/GeV 2 ] -8 -8 Luminosity L ij [ 1/GeV 2 ] -8 10 10 qg 10 10 -9 √ s = 14 TeV -9 √ s = 1.96 TeV 10 10 gg -10 CTEQ 6.5 -10 -9 -9 CTEQ 6.5 10 10 10 10 -11 µ f = 171 GeV -11 µ f = 171 GeV 10 10 -10 -10 -12 -12 10 10 10 10 10 10 10 10 q [ % ] ∆L gg [ % ] 5 5 5 5 ∆L q¯ at the LHC the total CS 0 0 0 0 60 60 10 10 40 ∆L gg [ % ] 40 q [ % ] 5 ∆L q¯ 5 20 receives large contributions from 20 0 0 0 0 20 20 10 10 ∆L qg [ % ] ∆L qg [ % ] 10 10 5 5 higher partonic energies 0 0 0 0 25 25 25 25 gg 20 20 20 20 gg 15 15 15 15 NLO QCD Partonic cross section [ pb ] 10 10 10 10 – – NLO QCD qq qq µ r = m t = 171 GeV Partonic cross section [ pb ] qg 5 µ r = m t = 171 GeV 5 5 5 qg 0 0 0 0 100 100 7 800 % % sum gg 6 – sum qq 80 80 600 5 NLO QCD 60 60 4 Hadronic cross section [ pb ] NLO QCD 400 µ f = µ r = m t = 171 GeV Hadronic cross section [ pb ] 3 40 40 µ f = µ r = m t = 171 GeV 2 gg 200 – 20 20 qq 1 0 0 qg x 10 0 qg x 10 0 -1 -20 -20 10 10 5 5 Total uncertainty in % Total uncertainty in % 0 0 0 0 3 400 500 600 700 800 900 10 √ s [ GeV ] √ s [ GeV ] Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 8 / 41
Uncertainty on NLO The partonic cross sections involve terms like ln 2 (1 − 4 m 2 / s ) which become large threshold and must be resummed Kidonakis and Sterman (’97), Bonciani et al. (’98), Kidonakis et al. (’01), Kidonakis and Vogt (’03), Banfi and Laenen (’05) 12 1400 – - - [pb] at Tevatron [pb] at LHC σ pp → tt σ pp → tt 1200 10 1000 8 800 6 600 4 400 2 200 NLL res (CTEQ65) NLL res (CTEQ65) 0 0 165 170 175 180 165 170 175 180 m t [GeV] m t [GeV] S. Moch and P. Uwer (’08) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 9 / 41
Uncertainty on NLO The partonic cross sections involve terms like ln 2 (1 − 4 m 2 / s ) which become large threshold and must be resummed Kidonakis and Sterman (’97), Bonciani et al. (’98), Kidonakis et al. (’01), Kidonakis and Vogt (’03), Banfi and Laenen (’05) 12 1400 – - - [pb] at Tevatron [pb] at LHC σ pp → tt σ pp → tt 1200 10 1000 8 800 6 600 4 400 12% uncertainty at Tevatron 14% uncertainty at LHC 2 200 NLL res (CTEQ65) NLL res (CTEQ65) 0 0 165 170 175 180 165 170 175 180 m t [GeV] m t [GeV] S. Moch and P. Uwer (’08) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 9 / 41
Uncertainty on NLO The partonic cross sections involve terms like ln 2 (1 − 4 m 2 / s ) which become large threshold and must be resummed Kidonakis and Sterman (’97), Bonciani et al. (’98), Kidonakis et al. (’01), Kidonakis and Vogt (’03), Banfi and Laenen (’05) 12 1400 Moch and Uwer presented an approximated NNLO result – - - [pb] at Tevatron [pb] at LHC σ pp → tt σ pp → tt 1200 (ln β , scale dep., Couloumb corrections) which drastically 10 1000 reduces the uncertainty ( ∼ 6 − 8% at Tevatron, ∼ 4 − 6% at LHC) 8 However, “it is no substitute for a complete NNLO computation” 800 6 600 4 400 2 200 NLL res (CTEQ65) NLL res (CTEQ65) 0 0 165 170 175 180 165 170 175 180 m t [GeV] m t [GeV] S. Moch and P. Uwer (’08) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 9 / 41
q → t ¯ Two-Loop Corrections to q ¯ t The NNLO calculation of the top-quark pair hadroproduction requires three ingredient q → t ¯ t and gg → t ¯ two-loop matrix elements for q ¯ t 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 Grenoble ’09 10 / 41
q → t ¯ Two-Loop Corrections to q ¯ t The NNLO calculation of the top-quark pair hadroproduction requires three ingredient q → t ¯ t and gg → t ¯ two-loop matrix elements for q ¯ t 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 (’07) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 10 / 41
q → t ¯ Two-Loop Corrections to q ¯ t The NNLO calculation of the top-quark pair hadroproduction requires three ingredient q → t ¯ t and gg → t ¯ two-loop matrix elements for q ¯ t 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 q → t ¯ Overall, in the q ¯ t channel, QGRAF generates 218 two-loop diagrams (one massive flavor, one massless flavor): 31 non-vanishing diagrams with a massless quark loop 30 non-vanishing diagrams with a massive quark loop 64 non-vanishing planar “gluonic” diagrams Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 10 / 41
q → t ¯ Two-Loop Corrections to q ¯ t � �� |M| 2 ( s , t , m , ε ) = 4 π 2 α 2 � α s � � α s � 2 � s α 3 A 0 + A 1 + A 2 + O s π π N c Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 11 / 41
q → t ¯ Two-Loop Corrections to q ¯ t � �� |M| 2 ( s , t , m , ε ) = 4 π 2 α 2 � α s � � α s � 2 � s α 3 A 0 + A 1 + A 2 + O s π π N c A 2 = A (2 × 0) + A (1 × 1) 2 2 Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 11 / 41
q → t ¯ Two-Loop Corrections to q ¯ t � �� |M| 2 ( s , t , m , ε ) = 4 π 2 α 2 � α s � � α s � 2 One-Loop × One-Loop � s α 3 A 0 + A 1 + A 2 + O s π π N c K¨ orner, Merebashvili, Rogal (’05,’08) A 2 = A (2 × 0) + A (1 × 1) 2 2 ⊗ + · · · Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 11 / 41
q → t ¯ Two-Loop Corrections to q ¯ t � �� |M| 2 ( s , t , m , ε ) = 4 π 2 α 2 � α s � � α s � 2 � s α 3 A 0 + A 1 + A 2 + O s Two-Loop × Tree π π N c A 2 = A (2 × 0) + A (1 × 1) 2 2 ⊗ + · · · Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 11 / 41
q → t ¯ Two-Loop Corrections to q ¯ t � �� |M| 2 ( s , t , m , ε ) = 4 π 2 α 2 � α s � � α s � 2 � s α 3 A 0 + A 1 + A 2 + O s π π N c A 2 = A (2 × 0) + A (1 × 1) 2 2 � � � c A + B + C N c D l + E l A (2 × 0) N 2 = + N l N c C F 2 N 2 N c c � � � N c D h + E h + N 2 l F l + N l N h F lh + N 2 + N h h F h N c 10 different color coefficients Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 11 / 41
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 Grenoble ’09 12 / 41
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 Grenoble ’09 12 / 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 )? Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 12 / 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 � =0) = 1 2 ( m , ε ) M ( m =0) Z i i ∈{ all legs } where Z is defined through the Dirac form factor F ( m � =0) ( Q 2 ) = Z ( m , ε ) F ( m =0) ( Q 2 ) + O ( m 2 / Q 2 ) A. Mitov and S. Moch (’06) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 12 / 41
Two-Loop Virtual Corrections in the s ≫ m 2 t Limit M. Czakon, A. Mitov, S. Moch (’07) Using the universal multiplicative relation between massless QCD amplitudes and massive amplitudes in the small mass limit, it was possible q → t ¯ to calculate the two-loop virtual corrections to q ¯ t starting from q → q ′ (massless)¯ q ′ (massless) ( C. Anastasiou et al. q ¯ (’00) ) 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 Grenoble ’09 13 / 41
Numerical Evaluation of the Two-Loop q → t ¯ Corrections to q ¯ t M. Czakon (’08) 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 15 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 Grenoble ’09 14 / 41
Numerical Evaluation of the Two-Loop q → t ¯ Corrections to q ¯ t M. Czakon (’08) 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 15 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 Grenoble ’09 14 / 41
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 � � � c A + B + C N c D l + E l A (2 × 0) N 2 = + N l N c C F 2 N 2 N c c � � � N c D h + E h + N 2 l F l + N l N h F lh + N 2 + N h h F h N c Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 15 / 41
Fermion Loop Corrections 7 color structures receive contributions from the diagrams involving closed All the diagrams with a closed quark loop (massive or massless) fermion loops only were evaluated analytically N l number of light (massless) quarks Bonciani, AF, Gehrmann, Ma^ ıtre, Studerus (’08) N h number of heavy (massive) quarks Agreement with the numerical results by Czakon � � � c A + B + C N c D l + E l A (2 × 0) N 2 = N c C F + N l 2 N 2 N c c � � N c D h + E h � + N 2 l F l + N l N h F lh + N 2 + N h h F h N c Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 15 / 41
Fermionic Diagrams Self Energies Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 16 / 41
Fermionic Diagrams Vertices Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 16 / 41
Fermionic Diagrams Boxes Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 16 / 41
Fermionic Diagrams Tadpole Self Energies Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 16 / 41
Fermionic Diagrams Vertices Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 16 / 41
Fermionic Diagrams Boxes Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 16 / 41
Method: The General Strategy After interfering a two-loop graph with the Born amplitude one obtains a linear combinations of scalar integrals k i → integration momenta Born � → external momenta p i Amplitude S → scalar products k i · k j or k i · p k D → propagators [ � c i k i + � d j p j ] 2 (+ m 2 1 · · · S n q � S n 1 t ) q D d k 1 D d k 2 D m 1 1 · · · D m t t 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 Grenoble ’09 17 / 41
Method: The General Strategy After interfering a two-loop graph with the Born amplitude one obtains a linear combinations of scalar integrals k i → integration momenta Born � → external momenta p i Amplitude S → scalar products k i · k j or k i · p k D → propagators [ � c i k i + � d j p j ] 2 (+ m 2 1 · · · S n q � S n 1 t ) q D d k 1 D d k 2 D m 1 1 · · · D m t t 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 Grenoble ’09 17 / 41
Method: The General Strategy After interfering a two-loop graph with the Born amplitude one obtains a linear combinations of scalar integrals k i → integration momenta Born � → external momenta p i Amplitude S → scalar products k i · k j or k i · p k D → propagators [ � c i k i + � d j p j ] 2 (+ m 2 1 · · · S n q � S n 1 t ) q D d k 1 D d k 2 D m 1 1 · · · D m t t 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 Grenoble ’09 17 / 41
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 (10 identities per integral for a two-loop four-point function) � 1 · · · S n q � � v µ S n 1 ∂ v µ = k 1 , k 2 , p 1 , · · · , p 3 q D d k 1 D d k 2 = 0 D m 1 1 · · · D m t ∂ k µ t i ◮ 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 Grenoble ’09 18 / 41
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 (10 identities per integral for a two-loop four-point function) � 1 · · · S n q � � v µ S n 1 ∂ v µ = k 1 , k 2 , p 1 , · · · , p 3 q D d k 1 D d k 2 = 0 D m 1 1 · · · D m t ∂ k µ t i ◮ 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 Grenoble ’09 18 / 41
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 (10 identities per integral for a two-loop four-point function) � 1 · · · S n q � � v µ S n 1 ∂ v µ = k 1 , k 2 , p 1 , · · · , p 3 q D d k 1 D d k 2 = 0 D m 1 1 · · · D m t ∂ k µ t i ◮ 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 Grenoble ’09 18 / 41
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 (10 identities per integral for a two-loop four-point function) � 1 · · · S n q � � v µ S n 1 ∂ v µ = k 1 , k 2 , p 1 , · · · , p 3 q D d k 1 D d k 2 = 0 D m 1 1 · · · D m t ∂ k µ t i ◮ 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 Grenoble ’09 18 / 41
N l Master Integrals 8 irreducible two-loop topologies Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 19 / 41
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 Grenoble ’09 19 / 41
N h Master Integrals 10 irreducible two-loop topologies Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 20 / 41
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 = ⇒ + Triangles C + Bubbles + Tadpoles Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 21 / 41
Calculation of the MIs: Differential Equation Method For each Master Integral belonging to a given topology F ( q ) → {D 1 , · · · , D q } l ◮ Take the derivative of a given integrals with respect to the external momenta p i 1 · · · S n q � S n 1 ∂ ∂ F ( q ) q D d k 1 D d k 2 p µ = p µ 1 · · · D m q j l j D m 1 ∂ p µ ∂ p µ q i i ◮ The integral 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 = 0) and solve the system of DE(s) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 22 / 41
Calculation of the MIs: Differential Equation Method For each Master Integral belonging to a given topology F ( q ) → {D 1 , · · · , D q } l ◮ Take the derivative of a given integrals with respect to the external momenta p i ◮ The integral 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 � 1 · · · S n q S n 1 ∂ � � � q c i F ( q ) k j F ( r ) D d k 1 D d k 2 p µ = + · · · D m q D m 1 j ∂ p µ i j q i 1 r � = q 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 = 0) and solve the system of DE(s) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 22 / 41
Calculation of the MIs: Differential Equation Method For each Master Integral belonging to a given topology F ( q ) → {D 1 , · · · , D q } l ◮ Take the derivative of a given integrals with respect to the external momenta p i ◮ The integral 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 ) c j ( s , t ) F ( q ) k l ( s , t ) F ( r ) ( s , t ) = ( s , t ) + ( s , t ) l j l r � = q j l ◮ Fix somehow the initial condition(s) (ex. knowing the behavior of the integral at s = 0) and solve the system of DE(s) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 22 / 41
Calculation of the MIs: Differential Equation Method For each Master Integral belonging to a given topology F ( q ) → {D 1 , · · · , D q } l ◮ Take the derivative of a given integrals with respect to the external momenta p i ◮ The integral 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 = 0) and solve the system of DE(s) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 22 / 41
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 p 2 p 4 � D d k 1 D d k 2 ( p 2 · k 1 ) α 6 ( p 1 · k 2 ) α 7 ( p 2 · k 2 ) α 8 ( p 3 · k 2 ) α 9 M ( α 1 , . . . , α 9 )= P α 1 0 ( k 1 + p 1 ) P α 2 0 ( k 1 + p 1 + p 2 ) P α 3 m ( k 2 ) P α 4 m ( k 1 − k 2 ) P α 5 m ( k 1 + p 3 ) P m ( q ) = q 2 + m 2 P 0 ( q ) = q 2 Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 23 / 41
Five Denominator MIs-II M 1 = M 2 = Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 24 / 41
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 Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 24 / 41
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 = 0 The second integration constant can be fixed by calculating the integral in t = 0 with MB techniques Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 24 / 41
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) ( d − 4) 2 + A ( − 1) A ( − 2) ( d − 4) + A (0) M 1 ( d , m , t , s ) ≡ = ( d − 4) 3 + 1 A − 3 = 32( y + 1) 1 h i A − 2 = − 2 G ( − 1; y )( x − 1) + 2( x − 1) − (1 + x ) G (0; x ) 32( x − 1)( y + 1) 1 h A − 1 = 3 ζ (2) x + 4 x + 6( x + 1) G ( − 1 , 0; x ) + ( − 5 x − 1) G (0 , 0; x ) 32( x − 1)( y + 1) − 4( x − 1) G ( − 1; y ) + G (0; x )(2( x + 1) G ( − 1; y ) − 2( x + 1)) i +4( x − 1) G ( − 1 , − 1; y ) − 2( x − 1) G (0 , − 1; y ) + 3 ζ (2) − 4 1 h A 0 = − 36( x + 1) G ( − 1 , − 1 , 0; x ) + 18( x + 1) G ( − 1 , 0 , 0; x ) 32( x − 1)( y + 1) +6(5 x + 1) G (0 , − 1 , 0; x ) + ( − 5 x − 1) G (0 , 0 , 0; x ) + · · · Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 25 / 41
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) ( d − 4) 2 + A ( − 1) A ( − 2) ( d − 4) + A (0) M 1 ( d , m , t , s ) ≡ = ( d − 4) 3 + 1 A − 3 = 32( y + 1) 1 h i A − 2 = − 2 G ( − 1; y )( x − 1) + 2( x − 1) − (1 + x ) G (0; x ) 32( x − 1)( y + 1) 1 h A − 1 = 3 ζ (2) x + 4 x + 6( x + 1) G ( − 1 , 0; x ) + ( − 5 x − 1) G (0 , 0; x ) 32( x − 1)( y + 1) − 4( x − 1) G ( − 1; y ) + G (0; x )(2( x + 1) G ( − 1; y ) − 2( x + 1)) The analytic results depend on ln , Li 2 , Li 3 , S 1 , 2 i +4( x − 1) G ( − 1 , − 1; y ) − 2( x − 1) G (0 , − 1; y ) + 3 ζ (2) − 4 They are conveniently expressed in terms of HPLs and 2dHPLs 1 h A 0 = − 36( x + 1) G ( − 1 , − 1 , 0; x ) + 18( x + 1) G ( − 1 , 0 , 0; x ) ( Remiddi, Vermaseren, Gehrmann ) 32( x − 1)( y + 1) +6(5 x + 1) G (0 , − 1 , 0; x ) + ( − 5 x − 1) G (0 , 0 , 0; x ) + · · · Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 25 / 41
Harmonic Polylogarithms (HPLs) E. Remiddi, J. Vermaseren (1999) E. Remiddi, T. Gehrmann (2001) Functions of the variable x and a set of indices � a with weight w ; each index can assume values 1 , 0 , − 1 H ( a ; x ) Definitions: w = 1 � x dt H (1; x ) = 1 − t = − ln (1 − x ) 0 H (0; x ) = ln x � x dt H ( − 1; x ) = 1 + t = ln (1 + x ) 0 d 1 f (0; x ) = 1 1 dx H ( a ; x ) = f ( a ; x ) f (1; x ) = f ( − 1; x ) = 1 − x x 1 + x Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 26 / 41
HPLs: Definitions Definitions: w > 1 1 w ! ln w x H ( � if � a = 0 , 0 , . . . , 0 ( w times ) 0 w ; x ) = � x H ( i ,� a ; x ) = dtf ( i ; t ) H ( � a ; t ) else 0 d ∈ � dx H ( i ,� a ; x ) = f ( i ; x ) H ( � H ( � consequences: a ; x ) a / 0; 0) = 0 a few examples @ w = 2 � x � x dt 1 H (0 , 1; x ) = dtf (0; t ) H (1; t ) = − t ln (1 − t ) = Li 2 ( x ) 0 0 � x � x 1 H (1 , 0; x ) = dtf (1; t ) H (0; t ) = dt 1 − t ln t 0 0 = − ln x ln (1 − x ) + Li 2 ( x ) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 27 / 41
HPLs as a Generalization of the Nielsen’s PolyLogs The HPLs include the Nielsen’s PolyLogs Z 1 S n , p ( x ) = ( − 1) n + p − 1 dt t ln n − 1 t ln p (1 − xt ) Li n ( x ) = S n − 1 , 1 ( x ) ( n + p )! p ! 0 H ( � Li n ( x ) = 0 n − 1 , 1; x ) H ( � 0 n ,� S n , p ( x ) = 1 p ; x ) but the HPLs are a larger set of functions: from w = 4 one finds things as � x dt � H ( − 1 , 0 , 0 , 1; x ) = 1 + t Li 3 ( x ) / ∈ Nielsen’s PolyLogs 0 Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 28 / 41
The HPLs Algebra Shuffle Algebra: � H ( � p ; x ) H ( � q ; x ) = H ( � r ; x ) r = � � p ⊎ � q some examples 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 Grenoble ’09 29 / 41
2-dimensional Harmonic Polylogarithms (2dHPLs) E. Remiddi, T. Gehrmann (2000) As for the HPLs, they are obtained by repeated integration over a new set of factors depending on a second variable. 1 1 f ( − y ; x ) = f ( − 1 / y ; x ) = x + y x + 1 / y � x G ( i ,� a ; x ) = dtf ( i ; t ) G ( � a ; t ) 0 a few examples: Z x Z x „ « dz 1 + x dz G ( − y ; x ) = z + y = ln G ( − 1 / y ; x ) = z + 1 / y = ln (1 + xy ) y 0 0 „ « „ « 1 + x − x G ( − y , 0; x ) = ln x ln + Li 2 y y Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 30 / 41
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 (2001-2002) 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 , 0; x ) = ln( x ) ln( xy + 1) + Li 2 ( − xy ) , 1 2 ln 2 ( y + 1) − ln(1 − x ) ln( y + 1) − ln( y ) ln( y + 1) G ( − y , 1; x ) = − π 2 „ 1 − x « + ln(1 − x ) ln( x + y ) − Li 2 ( − y ) + Li 2 6 , y + 1 − 1 3 ln 3 (1 − x ) − ln( x ) ln 2 (1 − x )+ · · · G ( − y , 1 , 0; x ) = Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 31 / 41
HPLs as Multiple Sums S. Weinzierl and J. Vollinga (’04) We defined (2-d)HPLs in terms of iterated integrations � y � t 1 � t k − 1 dt 1 dt 2 dt k G ( z 1 , · · · , z k ; y ) = · · · t 1 − z 1 t 2 − z 2 t k − z k 0 0 0 but they can be written also in terms of multiple sums G (0 , · · · , 0 , z 1 , · · · , z k − 1 , 0 , · · · , 0 , z k ; y ) ≡ G m 1 , ··· , m k ( z 1 , · · · , z k ; y ) � �� � � �� � m 1 − 1 m k − 1 � y ∞ ∞ � j 1 1 � � G m 1 , ··· , m k ( z 1 , · · · , z k ; y ) = · · · × ( j 1 + · · · + j k ) m 1 z 1 j 1 =1 j 1 =1 � y � y � j 2 � j k 1 · · · 1 × j m k ( j 2 + · · · + j k ) m 2 z 1 z 1 k Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 32 / 41
Results All results are written in analytic form: [Fl] = + 50/81 - 224/27*G(1,x)*y*[1-x]^-3 - 80/27*[1-x]^-6 - 8/9*G(1,x)*y*[1-x]^-2 + 80/9*[1-x]^-5 + 40/27*G(1,x)*y*[1-x]^-1 - 620/81*[1-x]^-4 + 32/9*G(1,x)*y^2*[1-x]^-6 + 40/81*[1-x]^-3 - 32/3*G(1,x)*y^2*[1-x]^-5 - 20/81*[1-x]^-2 + 248/27*G(1,x)*y^2*[1-x]^-4 + 40/27*[1-x]^-1 - 16/27*G(1,x)*y^2*[1-x]^-3 - 160/27*y*[1-x]^-6 - 40/27*G(1,x)*y^2*[1-x]^-2 + 160/9*y*[1-x]^-5 + ... - 1480/81*y*[1-x]^-4 + 32/9*G(1,0,x)*[1-x]^-7 + 560/81*y*[1-x]^-3 - 112/9*G(1,0,x)*[1-x]^-6 + 20/27*y*[1-x]^-2 + 128/9*G(1,0,x)*[1-x]^-5 - 100/81*y*[1-x]^-1 - 40/9*G(1,0,x)*[1-x]^-4 - 80/27*y^2*[1-x]^-6 - 16/9*G(1,0,x)*[1-x]^-2 + 80/9*y^2*[1-x]^-5 + 64/9*G(1,0,x)*y*[1-x]^-7 - 620/81*y^2*[1-x]^-4 - 224/9*G(1,0,x)*y*[1-x]^-6 + 40/81*y^2*[1-x]^-3 + 32*G(1,0,x)*y*[1-x]^-5 + 100/81*y^2*[1-x]^-2 - 160/9*G(1,0,x)*y*[1-x]^-4 - 80/27*G(0,x)*[1-x]^-7 + 16/9*G(1,0,x)*y*[1-x]^-3 + 232/27*G(0,x)*[1-x]^-6 + 8/3*G(1,0,x)*y*[1-x]^-2 - 176/27*G(0,x)*[1-x]^-5 - 8/9*G(1,0,x)*y*[1-x]^-1 - 8/9*G(0,x)*[1-x]^-4 + 32/9*G(1,0,x)*y^2*[1-x]^-7 + 8/27*G(0,x)*[1-x]^-3 - 112/9*G(1,0,x)*y^2*[1-x]^-6 + 4/3*G(0,x)*[1-x]^-2 + 128/9*G(1,0,x)*y^2*[1-x]^-5 + 8/9*G(0,x)*[1-x]^-1 - 40/9*G(1,0,x)*y^2*[1-x]^-4 - 160/27*G(0,x)*y*[1-x]^-7 - 16/9*G(1,0,x)*y^2*[1-x]^-3 + 464/27*G(0,x)*y*[1-x]^-6 + 8/9*G(1,0,x)*y^2*[1-x]^-2 - 16*G(0,x)*y*[1-x]^-5 ; Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 33 / 41
Results-II The color factors can be easily evaluated numerically (either with Mathematica or with Fortran code): ############################################### Fl = 4.93507679E+01 Flh = 9.87015354E+01 Fh = 4.93507675E+01 El = 1/ep^3*( 2.05000001E-01) + 1/ep^2*( 1.02365896E+00) + 1/ep *( -1.81803776E+02) + ( 2.67510489E+03) Dl = 1/ep^3*( -2.05000001E-01) + 1/ep^2*( -4.51882816E-01) + 1/ep *( 2.07220644E+02) + ( -4.46537433E+03) Eh = 1/ep^2*( 5.78057996E+00) + 1/ep *( -2.10928847E+02) + ( 2.77869548E+03) Dh = 1/ep^2*( -5.78057996E+00) + 1/ep *( 2.40508604E+02) + ( -4.58561923E+03) ############################################### Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 34 / 41
Results-II The color factors can be easily evaluated numerically (either with Mathematica or with Fortran code): ◮ by expanding the results for ############################################### s ≫ m 2 t we find analytic Fl = 4.93507679E+01 Flh = 9.87015354E+01 agreement with Czakon, Fh = 4.93507675E+01 Mitov, and Moch (’07) El = 1/ep^3*( 2.05000001E-01) + 1/ep^2*( 1.02365896E+00) ◮ we we find numerical agreement + 1/ep *( -1.81803776E+02) + ( 2.67510489E+03) with Czakon (’08) Dl = 1/ep^3*( -2.05000001E-01) + 1/ep^2*( -4.51882816E-01) ◮ we obtained analytic expression + 1/ep *( 2.07220644E+02) � 1 − 4 m 2 / s → 0 + ( -4.46537433E+03) in the β = Eh = 1/ep^2*( 5.78057996E+00) limit + 1/ep *( -2.10928847E+02) + ( 2.77869548E+03) Dh = 1/ep^2*( -5.78057996E+00) + 1/ep *( 2.40508604E+02) + ( -4.58561923E+03) ############################################### Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 34 / 41
q → t ¯ Planar Diagrams in q ¯ t � �� |M| 2 ( s , t , m , ε ) = 4 π 2 α 2 � α s � � α s � 2 � α 3 s A 0 + A 1 + A 2 + O s π π N c A 2 = A (2 × 0) + A (1 × 1) 2 2 � � � c A + B + C N c D l + E l A (2 × 0) N 2 = N c C F + N l 2 N 2 N c c � � � N c D h + E h + N 2 l F l + N l N h F lh + N 2 + N h h F h N c Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 35 / 41
q → t ¯ Planar Diagrams in q ¯ t � �� |M| 2 ( s , t , m , ε ) = 4 π 2 α 2 � α s � � α s � 2 � The coefficient of N 2 s α 3 c can also be evaluated analytically A 0 + A 1 + A 2 + O s N c π π it involves only planar diagrams Bonciani, AF, Gehrmann, Studerus (in progress) A 2 = A (2 × 0) + A (1 × 1) 2 2 � � � c A + B + C N c D l + E l A (2 × 0) N 2 = N c C F + N l 2 N 2 N c c � � N c D h + E h � + N 2 l F l + N l N h F lh + N 2 + N h h F h N c Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 35 / 41
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 (’09) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 36 / 41
New Issues and Technical Problems ◮ It is necessary to introduce and study 2dHPLs depending on new weight functions 1 1 √ � , � � � t − 1 1 − 1 1 ± 3 t − x − x 2 ◮ 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) ◮ It is not easy to switch the variables in the weights and the variables in the arguments � G (1 − 1 / x − x , · · · ; y ) = ⇒ G ( · · · , g ( y ) , · · · ; x ) ◮ In some cases to fix the integration constants requires the direct evaluation of the integral for special values of s and t with techniques based on Mellin-Barnes representations Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 37 / 41
Complete Analytic NLO calculation Recently, the NLO total cross section was evaluated analytically M. Czakon, A. Mitov (’08) 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 � � 1 1 1 � q 2 + m 2 � δ = q 2 + m 2 − i δ − q 2 + m 2 + i δ 2 π i Anastasiou Melnikov (’02) The gg → t ¯ tX cross section cannot be completely written in terms of HPLs and their generalizations. Integrals over elliptic functions � 1 1 √ 1 − z 2 √ K ( k ) = dz 1 − k 2 z 2 0 √ � 1 1 − k 2 z 2 √ E ( k ) = dz 1 − z 2 0 (1) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 38 / 41
Two-Loop Corrections to gg → t ¯ t A 2 = A (2 × 0) + A (1 × 1) 2 2 One-Loop × One-Loop Anastasiou, Aybat (’08) K¨ orner, Kniehl, Merebashvili, Rogal (’08) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 39 / 41
Two-Loop Corrections to gg → t ¯ t A 2 = A (2 × 0) + A (1 × 1) 2 2 � N 3 A + NB + 1 N C + 1 A (2 × 0) ( N 2 − 1) N 3 D + N 2 N l E l + N 2 N h E h = 2 + N l F l + N h F h + N l N 2 G l + N h N 2 G h + NN 2 l H l + NN 2 h H h � + NN l N h H lh + N 2 N I l + N 2 N I h + N l N h l h I lh N 16 color structures Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 39 / 41
Two-Loop Corrections to gg → t ¯ t -II A (2 × 0) is known only in the limit s ≫ m 2 2 t Czakon, Mitov, Moch (’07) 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 � 1 ⇒ dx = √ K ( z ) = 0 (1 − x 2 )(1 − zx 2 ) Laporta Remiddi (’04) Andrea Ferroglia (Mainz U.) Top-Quark Pairs at NNLO Grenoble ’09 40 / 41
Summary & Conclusions The top-quark pair production cross section may eventually be measurable with a 5% 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, and the analytical calculation of all the quark-loop diagrams; the analytical calculation of all the planar diagrams is at an advanced stage 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 Grenoble ’09 41 / 41
Summary & Conclusions The top-quark pair production cross section may eventually be measurable with a 5% 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, and the analytical calculation of all the quark-loop diagrams; the analytical calculation of all the planar diagrams is at an advanced stage 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 Grenoble ’09 41 / 41
Backup Slides
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 + δ p µ = p µ + δǫ µ ν p ν δǫ µ ν = − δǫ ν µ I ( p 1 , p 2 , p 3 ) = I ( p 1 + δ p 1 , p 2 + δ p 2 , p 3 + δ p 3 ) implying the following 3 identities for a 4-point functions 3 � � ∂ ∂ � ( p µ 1 p µ 1 p ν 2 − p ν p ν − p µ 2 ) I ( p i ) = 0 n ∂ p µ n ∂ p ν n n n =1 3 � � ∂ ∂ � ( p µ 2 p ν 3 − p ν 2 p µ p ν − p µ 3 ) I ( p i ) = 0 ∂ p µ n n ∂ p ν n n n =1 3 � � ∂ ∂ � ( p µ 1 p µ 1 p ν 3 − p ν p ν − p µ 3 ) I ( p i ) = 0 ∂ p µ n n ∂ p ν n n n =1
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: 0 = − others are more involved p 2 · k 1 + s + 1 p 1 · k 2 + 0 = k 1 · k 2 + 2 2
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
Euler Method First Order Differential Equation d dx f ( x ) + C ( x ) f ( x ) = Ω( x ) 1 find the solution of the homogeneous equation d dx h ( x ) + C ( x ) h ( x ) = 0 2 build the general solution of the non-homogeneous equation � � � dx Ω( x ) f ( x ) = h ( x ) k + h ( x ) 3 fix the integration constant k
Euler Method First Order Differential Equation d dx f ( x ) + C ( x ) f ( x ) = Ω( x ) 1 find the solution of the homogeneous equation d dx h ( x ) + C ( x ) h ( x ) = 0 2 build the general solution of the non-homogeneous equation � � � dx Ω( x ) f ( x ) = h ( x ) k + h ( x ) 3 fix the integration constant k
Euler Method First Order Differential Equation d dx f ( x ) + C ( x ) f ( x ) = Ω( x ) 1 find the solution of the homogeneous equation d dx h ( x ) + C ( x ) h ( x ) = 0 2 build the general solution of the non-homogeneous equation � � � dx Ω( x ) f ( x ) = h ( x ) k + h ( x ) 3 fix the integration constant k
Euler Method-II Second Order Differential Equation d 2 dx 2 f ( x ) + A ( x ) d dx f ( x ) + B ( x ) f ( x ) = Ω( x ) 1 find the two solution of the homogeneous equation d 2 dx 2 h 1 , 2 ( x ) + A ( x ) d dx h 1 , 2 ( x ) + B ( x ) h 1 , 2 ( x ) = 0 2 build the Wronskian W ( x ) = h 1 ( x ) d dx h 2 ( x ) − h 2 ( x ) d dx h 1 ( x ) 3 build the solution � x � x � � � � dw dw f ( x ) = h 1 ( x ) k 1 − W ( w ) h 2 ( w )Ω( w ) + h 2 ( x ) k 2 + W ( w ) h 1 ( w )Ω( w ) 0 0 4 fix the integration constants k 1 and k 2
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