Heavy Quark Production at High Energies

Transcription

1 Heavy Quark Production at High Energies Andrea Ferroglia Johannes Gutenberg Universität Mainz Mainz, 19 November 29

2 Outline 1 Heavy Quarks, High Energies, and Perturbative QCD 2 Heavy Quark (Top) Pair Production 3 Heavy-Quark Pair Production at NNLO

3 Goal & Scope To describe the perturbative calculations which allow to predict heavy quark production cross sections at high energies Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies 9 1 / 11

4 Goal & Scope To describe the perturbative calculations which allow to predict heavy quark production cross sections at high energies Fermilab Tevatron CERN LHC p p s = TeV pp s = 7 (?)14TeV Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 1 / 11

5 Goal & Scope To describe the perturbative calculations which allow to predict heavy quark production cross sections at high energies m t = ± 1.3GeV Heavy-quark production cross sections are calculable in perturbative QCD: Λ m t.17 ( Λ m b.6 ) Λ.2 m c α s (m t ).1 Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 1 / 11

6 The Top Quark m t = ± 1.3 GeV A particle which sticks out... It is almost as heavy as a gold atom It decays very rapidly via EW interactions: t bw (τ t = 1/Γ t 1 2 s) The top quark decays before it can form hadronic bound states the top quark should couple strongly to the electroweak breaking sector Thousands of top quarks produced at the Tevatron Millions of top quarks expected at the LHC Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 2 / 11

7 The Top Quark A particle which sticks out... It is almost as heavy as a gold atom m t = ± 1.3 GeV It decays very rapidly via EW With the large number of top quarks expected to be produced interactions: t bw at the LHC, the study of the top-quark properties (τ t = 1/Γ t 1 2 s) will become precision physics The top quark decays before it can form hadronic bound states the top quark should couple strongly to the electroweak breaking sector Thousands of top quarks produced at the Tevatron Millions of top quarks expected at the LHC Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 2 / 11

8 The Top Quark A particle which sticks out... It is almost as heavy as a gold atom m t = ± 1.3 GeV It decays very rapidly via EW With the large number of top quarks expected to be produced interactions: t bw at the LHC, the study of the top-quark properties (τ t = 1/Γ t 1 2 s) will become precision physics The top quark decays before it can form hadronic bound states the top quark should couple Two production mechanisms: strongly to the electroweak top-quark pair production breaking sector single top production Thousands of top quarks produced at the Tevatron Millions of top quarks expected at the LHC Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 2 / 11

9 Top-Quark Pair Production

10 Top Quark Pair Production 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 3 / 11

11 Top Quark Pair Production 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 (s had,m 2 t ) = ij shad 4m 2 t (ŝ,shad dŝ L ij,µ 2 ) f }{{} partonic luminosity partonic cross section {}}{ ˆσ ij (ŝ,m 2 t,µ 2 f,µ2 r) Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 3 / 11

12 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 4 / 11

13 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 4 / 11

14 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 4 / 11

15 Tree Level QCD Partonic Processes q(p 1 ) + q(p 2 ) t(p 3 ) + t(p 4 ) p 1 p 4 The NLO QCD corrections in both channels p 2 p 3 (and to qg t tq) have been known for a long time Nason, Dawson, Ellis ( 88-9) Beenakker, Kuijf, van Neerven, Smith ( 89) Beenakker g(p 1 ) + etg(p al( 91) 2 ) Mangano, t(p 3 ) + t(p Nason, 4 ) Ridolfi ( 92) Frixione et al ( 9) Czakon and Mitov ( 8) Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 4 / 11

16 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 corrections in both channels are also p 2 p 3 known (they are smaller than current QCD uncertainties) Beenakker et al. ( 94) Bernreuther, Fuecker, and Si ( - 8) Kühn, g(p 1 ) Scharf, + g(p 2 ) and Uwer t(p 3 ( - 6) ) + t(p 4 ) Moretti, Nolten, and Ross ( 6) Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 4 / 11

17 Uncertainty on the NLO Cross Section The partonic cross sections involve terms like ln(1 4m 2 /s) which become large near threshold and must be resummed Kidonakis, Sterman ( 97), Bonciani et al. ( 98), Kidonakis et al. ( 1), Kidonakis, Vogt ( 3), Banfi, Laenen ( ), Cacciari et al ( 8) σ - pp tt [pb] at Tevatron NLL res (CTEQ6) m t [GeV] Czakon et al ( 9) σ - pp tt [pb] at LHC NLL res (CTEQ6) m t [GeV] S. Moch and P. Uwer ( 8) Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 / 11

18 Uncertainty on the NLO Cross Section The partonic cross sections involve terms like ln(1 4m 2 /s) which become large near threshold and must be resummed Kidonakis, Sterman ( 97), Bonciani et al. ( 98), Kidonakis et al. ( 1), Kidonakis, Vogt ( 3), Banfi, Laenen ( ), Cacciari et al ( 8) σ - pp tt [pb] at Tevatron 12% uncertainty at Tevatron NLL res (CTEQ6) m t [GeV] Czakon et al ( 9) σ - pp tt [pb] at LHC 14% uncertainty at LHC NLL res (CTEQ6) m t [GeV] expected experimental uncertainty at LHC % S. Moch and P. Uwer ( 8) Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 / 11

19 Uncertainty on the NLO Cross Section The partonic cross sections involve terms like ln(1 4m 2 /s) which become large near threshold and must be resummed Kidonakis, Sterman ( 97), Bonciani et al. ( 98), Kidonakis et al. ( 1), Kidonakis, Vogt ( 3), Banfi, Laenen ( ), Cacciari et al ( 8) Moch and Uwer presented an approximated NNLO Czakon result et al ( 9) 12 (lnβ, scale dep., Coulomb corrections) 14 which drastically σ - pp tt [pb] at Tevatron σ - pp tt [pb] at 1 reduces the uncertainty ( 6 8% 12 at Tevatron, 4 6% at LHC) 1 8 recently employed to extract m t = GeV 8 Langenfeld, Moch, Uwer ( 9) 6 However, it is no substitute for a 6 complete NNLO computation NLL res (CTEQ6) m t [GeV] 2 NLL res (CTEQ6) m t [GeV] S. Moch and P. Uwer ( 8) Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 / 11

20 Heavy-Quark Pair Production at NNLO

21 NNLO Laundry List 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 6 / 11

22 NNLO Laundry List 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 elementskörner for the hadronic et al. production (, 8) of t t + 1 parton tree-level matrix elementsanastasiou, for the hadronicaybat production ( 8) of t t + 2 partons Dittmaier, Uwer, Wenzierl ( 7, 8) Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 6 / 11

23 NNLO Laundry List 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 In the q q t t channel, QGRAF generates 218 two-loop diagrams (one massive flavor, one massless flavor) There are 789 two-loop diagrams in the gg t t channel All the diagrams were evaluated in the s, t, u m 2 t Czakon, Mitov, Moch ( 7, 8) However, to study the top-quark pair production at the Tevatron and the LHC it is necessary to retain the exact dependence on m t Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 6 / 11

24 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 7 / 11

25 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 7 / 11

26 Two-Loop Corrections to q q t t M 2 (s,t,m,ε) = 4π2 α 2 s N c [ A + ( αs ) π ( αs ) 2 A 1 + A2 + O ( α 3 π s) ] One-Loop One-Loop Körner, Merebashvili, Rogal (, 8) A 2 = A (2 ) 2 + A (1 1) 2 + Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 7 / 11

27 Two-Loop Corrections to q q t t M 2 (s,t,m,ε) = 4π2 α 2 s Two-Loop TreeN 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 7 / 11

28 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 ) 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 1 different color coefficients the coefficients are functions of s,t,m t ( + 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 7 / 11

29 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 ) 2 is known numerically (Czakon 8) It was calculated with a method based on Laporta algorithm + A 2 = A (2 ) numerical 2 + A (1 1) solutions of the differential equations satisfied by 2the Master Integrals 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 7 / 11

30 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) ] All the diagrams with a closed quark loop (massive or massless) were calculated analytically (Bonciani, AF, Gehrmann, Ma^ıtre, Studerus 8) Laporta algorithm A 2 = A + (2 ) 2 diff. + eq. A (1 1) 2method + HPLs 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 7 / 11

31 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) ] The coefficient of the leading color structure in the squared matrix element was calculated analytically (Bonciani, AF, Gehrmann, Studerus 9) It involves planar diagrams only 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 7 / 11

32 Two-Loop Corrections to gg 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.) Heavy Quarks at High Energies OpenCharm@PANDA 9 8 / 11

33 Two-Loop Corrections to gg 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 One-Loop One-Loop Anastasiou, Aybat ( 8) Körner, Kniehl, Merebashvili, Rogal ( 8) Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 8 / 11

34 Two-Loop Corrections to gg 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 2 c 1) 16 color structures ( +N l F l + N h F h + N l N 2 c A 2 = A (2 ) 2 + A (1 1) 2 Nc 3 A + N c B + 1 C + 1 N c Nc 3 D + Nc 2 N l E l + Nc 2 N h E h G l + N h N 2 c +N c N l N h H lh + N2 l N c I l + N2 h N c I h + N ln h N c I lh G h + N c Nl 2 H l + N c Nh 2 H h ) Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 8 / 11

35 Two-Loop Corrections to gg 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 The two-loop corrections to gg t t are only available for s, t, u ( mt 2 (Czakon, 2 = (Nc 2 1) Nc 3 A + N c B + 1 Mitov, C + 1 Moch 8) Exact calculation: not all the color coefficients N c Nc 3 D + Nc 2 can N l Ebe l + Nc 2 N h E h expressed in terms of HPLs. Numerical evaluation? A (2 ) 16 color structures +N l F l + N h F h + N l N 2 c G l + N h N 2 c +N c N l N h H lh + N2 l N c I l + N2 h N c I h + N ln h N c I lh G h + N c Nl 2 H l + N c Nh 2 H h ) Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 8 / 11

36 Interlude: IR Poles in QCD Amplitudes After UV renormalization, QCD amplitudes still include soft and collinear (IR) poles A 1 = A( 2) 1 ǫ 2 + A( 1) 1 ǫ + A () 1 A 2 = A( 4) 2 ǫ 4 + A( 3) 2 ǫ 3 + A( 2) 2 ǫ 2 + A( 1) 2 ǫ + A () 2 Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 9 / 11

37 Interlude: IR Poles in QCD Amplitudes After UV renormalization, QCD amplitudes still include soft and collinear (IR) poles A 1 = A( 2) 1 ǫ 2 + A( 1) 1 ǫ + A () 1 A 2 = A( 4) 2 ǫ 4 + A( 3) 2 ǫ 3 + A( 2) 2 ǫ 2 + A( 1) 2 ǫ + A () 2 It is possible to predict the analytic expression of the IR poles coefficients of any QCD amplitude at two-loops, also in presence of massive partons: IR poles in QCD amplitudes can be remove by multiplicative renormalization Becher, Neubert ( 9) Z 1 (ǫ, {p}, {m}) M n (ǫ, {p}, {m}) α QCD s ξα s = FINITE Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 9 / 11

38 Two-Loop Corrections to gg t t: IR Poles ǫ 4 ǫ 3 ǫ 2 ǫ 1 ǫ A ? B ? C ? D ? E l It was possible to recalculate the known 27.97? E h poles in q q.1298 t t and ? F l to obtain exact analytic expressions.74 for all? F h the IR poles in gg t t. at two-loop ? G l AF, Neubert, 3.99 Pecjak, Yang ( 9)? G h.? H l ? H lh.432? H h? I l ? I lh.? I h? t 1 =.4s, s = m 2 t, µ = m t Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 1 / 11

39 Two-Loop Corrections to gg t t: IR Poles ǫ 4 ǫ 3 ǫ 2 ǫ 1 ǫ A ? B ? C ? D ? E l ? E h ? F l ? F h ? G l ? G h.? H l ? H lh.432? H h? I l ? I lh.? I h? t 1 =.4s, s = m 2 t, µ = m t Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 1 / 11

40 Summary & Conclusions Heavy quark observables at high energies are calculable in a perturbative QCD framework For phenomenological studies of the top quark at the LHC, a complete fixed-order NNLO QCD calculation for σ t t is needed. Some of the necessary building blocks and possible approximated result are under study The calculation of two-loop corrections is a crucial ingredient in the NNLO program. It presents serious technical challenges, especially in the gluon fusion channel (dominant at the LHC). For the latter, only results in the ultra-relativistic limit are available In this talk the focus was on the total inclusive production cross sections. Exclusive observables require the modeling of the bottom (charm) fragmentation. Differential distributions for the inclusive cross section are also of interest and currently under study Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 11 / 11

41 Summary & Conclusions The moral of the story is always the same: there is a lot of work to do... Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 11 / 11

42 Summary & Conclusions The moral of the story is always the same: there is a lot of work to do... Andrea Ferroglia (Mainz U.) Heavy Quarks at High Energies OpenCharm@PANDA 9 11 / 11

43 Backup Slides

44 Size of the NLO corrections LHC σ t t σ t t channel σ t s channel σ t associated tw NLO QCD +% +% +44% +1% EW.% < 1% MSSM up to ±% < 1% Tevatron σ t t σ t t channel σ t s channel NLO QCD +2% +9% +47% EW 1% < 1% MSSM up to ±% < 1% see talk (W. Bernreuther arxiv:8.1333) S. Berge et al. hep-ph/7316 (W. Bernreuther arxiv:8.1333) see talk (W. Bernreuther arxiv:8.1333) G. Bordes and B. van Eijk NPB 43 (199), T.Stelzer et al. hep-ph/97398, hep-ph/98734(w. Bernreuther arxiv:8.1333) M.Beccaria et al. hep-ph/618, arxiv: (w. Bernreuther arxiv:8.1333) M. C. Smith and S. Willenbrock hep-ph/ B. W. Harris et al. hep-ph/27 (W. Bernreuther arxiv:8.1333) W. T. Giele et al.hep-ph/911449, S. Zhu PLB 24 (22) (W. Bernreuther arxiv:8.1333)

45 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

46 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)

47 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

48 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

49 The Laporta Algorithm The set of denominators D 1,, D t defines a topology; for each topology

50 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

51 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

52 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.

53 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

54 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

55 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

56 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)

57 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)

58 Five Denominator MIs-II M 1 = M 2 =

59 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)

60 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

61 Charge Asymmetry The charge asymmetry is the difference in production rate for top and antitop at fixed angle or rapidity A(y) = N t(y) N t (y) N t (y) N t (y) }{{} differential CA A = N t(y ) N t (y ) N t (y ) N t (y ) }{{} integrated CA (N dσt t ) i dy i Arising at order α 3 s for q q t t because of i) interference of final state with initial state gluon radiation ii) interference of virtual box diagrams with the Born process Top quarks are preferentially emitted in the direction of the incoming quark at the partonic level (which translates to a preference in the direction of the incoming proton in p p collisions ) Prediction for the Tevatron A =.1(6). (At the LHC with no cuts A = ) In QCD N t (y) = N t ( y), therefore A = A t FB

62 Correlation of the t and t spins The correlation of the t and t spins with respect to the reference axes â and ˆb is given by the expectation value A = 4(â Ŝt)(ˆb Ŝ t) = N( ) + N( ) N( ) N( ) N( ) + N( ) + N( ) + N( ) helicity basis: â = k t, ˆb = k t ; beam basis â = ˆb = beam axis (lab frame) At the Tevatron: q q channel dominant; top-pairs produced near threshold. The initial state has S = J = 1, the spins of t t are 1% correlated in the beam basis (both parallel or both antiparallel to the beam) Gluon channel near threshold: the total spin along the beam axis in the initial state is ; therefore the spins of t and t point in opposite directions (along the beam axis) In the ultra-relativistic limit, the spins of t and t are 1% anti-correlated in the helicity basis (both for q q and gg)

63 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[%] 1 Lqg[%] S. Moch and P. Uwer ( 8) 2 Plots from Lgg[%] Lq q[%] Lqg[%] gg 2 2 gg NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV qq qq NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV qg 1 1 qg sum qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg 1 % sum qq gg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV 1 % qg x 1-2 qg x Total uncertainty in % s [GeV] Total uncertainty in % s [GeV]

64 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[%] at Tevatron Lgg[%] Lq q[%] q q luminosity is dominant 1 Lqg[%] gg 2 2 gg NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV qq qq NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV qg 1 1 qg sum qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg 1 % sum qq gg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV 1 % qg x 1-2 qg x Total uncertainty in % s [GeV] Total uncertainty in % s [GeV]

65 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 at the LHC 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 qq gg Lgg[%] Lq q[%] Lqg[%] NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV qg qg sum qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg 1 % sum qq gg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV 1 % qg x 1-2 qg x Total uncertainty in % s [GeV] Total uncertainty in % s [GeV]

66 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[%] 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 qq qq NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV qg 1 1 qg sum qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg 1 % sum qq gg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV 1 % qg x 1-2 qg x Total uncertainty in % s [GeV] Total uncertainty in % s [GeV]

67 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 NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV gg qq Lq q[%] Lgg[%] Lqg[%] Lgg[%] 1 Lq q[%] 1 Lqg[%] 2 at Tevatron 2 gg the CS is dominated 1 NLO QCD 1 qq Partonic cross section [pb] µ r = m t = 171 GeV by the region ŝ 2m t qg qg sum qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg 1 % sum qq gg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV 1 % qg x 1-2 qg x Total uncertainty in % s [GeV] Total uncertainty in % s [GeV]

68 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 qq qq NLO QCD Partonic cross section [pb] µ r = m t = 171 GeV qg 1 1 qg sum qq NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV gg 1 % sum qq gg NLO QCD Hadronic cross section [pb] µ f = µ r = m t = 171 GeV 1 % qg x 1-2 qg x Total uncertainty in % s [GeV] Total uncertainty in % s [GeV]

69 Threshold Resummation for a threshold T i (inclusive cross section T = s 4m 2 ) The perturbative series for any of these (differential) cross sections can be expressed as dσ(t) = n 2n k α n s c n,k ln k (T) Resummation concerns itself with carrying out the sum above It is often convenient to take moments dσ(n) = dt T N = n 2n k α n s d n,k ln k N dσ(n) = C(α s ) exp (Lg (α s L) + g 1 (α s L) + α s g 2 (α s L) + ) To calculate the exponent up to the term involving the function g i corresponds to N i LL resummation

70 Single Top Quark Production b t q t W + W + q ( q) q ( q ) q s-channel b b t-channel t g t b associated tw production t g W b W

71 Single Top Production: Predictions cross section t-channel (pb) s-channel(pb) tw mode (pb) σ t Tevatron 1.1 ±.7.4 ±.4.14 ±.3 σ t LHC 1 ± ±.7 44 ± σ t LHC 92 ± ±.3 44 ± m t = 171.4± 2.1 Kidonakis ( 6-7) The numbers include NLO QCD corrections (Harris et al ( 2), Sullivan ( 4- ), Campbell et al ( 4), Chao et al ( 4 - ), Smith and Willenbrock ( 96), Chao et al ( 4), Giele et al ( 9), Zhu ( 2) )

72 Single Top Production: Predictions cross section t-channel (pb) s-channel(pb) tw mode (pb) σ t Tevatron 1.1 ±.7.4 ±.4.14 ±.3 σ t LHC 1 ± ±.7 44 ± σ t LHC 92 ± ±.3 44 ± m t = 171.4± 2.1 Kidonakis ( 6-7) The numbers include NLO QCD corrections (Harris et al ( 2), Sullivan ( 4- ), Campbell et al ( 4), Chao et al ( 4 - ), Smith and Willenbrock ( 96), Chao et al ( 4), Giele et al ( 9), Zhu ( 2) ) The cross-section is proportional to V tb 2 σ t + σ t is 38% (48%) of σ t t at the LHC (Tevatron) = single top events at the LHC (L = 1fb 1 ) The final states from single top events have large backgrounds CDF and D reported evidence for single-top production at the Tevatron, at the LHC it should be possible to disentangle the different modes

73 Single Top Production: Predictions cross section t-channel (pb) s-channel(pb) tw mode (pb) σ t Tevatron 1.1 ±.7.4 ±.4.14 ±.3 σ t LHC 1 ± ±.7 44 ± σ t LHC 92 ± ±.3 44 ± m t = 171.4± 2.1 Kidonakis ( 6-7) The numbers include NLO QCD corrections (Harris et al ( 2), Sullivan ( 4- ), t-channel: Campbell dominant et al ( 4), at bothchao the Tevatron et al ( 4and - ), the LHC Smith and Willenbrock at the LHC: ( 96), ubchao dt et( al 74%), ( 4), Giele db et ūt al ( ( 9), 12%), Zhu ( 2) ) The It should cross-section be measurable is proportional with a total Verror tb 2 of about 1% σ t + σ t is 38% (48%) of σ t t at the LHC (Tevatron) = single top events at the LHC (L = 1fb 1 ) The final states from single top events have large backgrounds CDF and D reported evidence for single-top production at the Tevatron, at the LHC it should be possible to disentangle the different modes

74 Single Top Production: Predictions cross section t-channel (pb) s-channel(pb) tw mode (pb) σ t Tevatron 1.1 ±.7.4 ±.4.14 ±.3 σ t LHC 1 ± ±.7 44 ± σ t LHC 92 ± ±.3 44 ± m t = 171.4± 2.1 Kidonakis ( 6-7) The numbers include NLO QCD corrections (Harris et al ( 2), Sullivan ( 4- ), Campbell et s-channel: al ( 4), small Chao at etthe al ( 4 LHC - ), Smith and Willenbrock ( 96), Chao et al ( 4), Giele et al ( 9), Zhu ( 2) a measurement uncertainty of 36% was estimated ) The cross-section is proportional to V tb 2 σ t + σ t is 38% (48%) of σ t t at the LHC (Tevatron) = single top events at the LHC (L = 1fb 1 ) The final states from single top events have large backgrounds CDF and D reported evidence for single-top production at the Tevatron, at the LHC it should be possible to disentangle the different modes

75 Single Top Production: Predictions cross section t-channel (pb) s-channel(pb) tw mode (pb) σ t Tevatron 1.1 ±.7.4 ±.4.14 ±.3 σ t LHC 1 ± ±.7 44 ± σ t LHC 92 ± ±.3 44 ± m t = 171.4± 2.1 Kidonakis ( 6-7) The numbers include NLO QCD corrections (Harris et al ( 2), Sullivan ( 4- ), Campbell Associated et al tw( 4), production: Chao etlarge al ( 4 at the - ), LHC Smith and Willenbrock ( 96), Chao et al ( 4), Giele et al ( 9), Zhu ( 2) estimated uncertainties for the CS measurement are 2% ) The cross-section is proportional to V tb 2 σ t + σ t is 38% (48%) of σ t t at the LHC (Tevatron) = single top events at the LHC (L = 1fb 1 ) The final states from single top events have large backgrounds CDF and D reported evidence for single-top production at the Tevatron, at the LHC it should be possible to disentangle the different modes

76 Single Top Production: Predictions cross section t-channel (pb) s-channel(pb) tw mode (pb) σ t Tevatron 1.1 ±.7.4 ±.4.14 ±.3 σ t LHC 1 ± ±.7 44 ± σ t LHC 92 ± ±.3 44 ± m t = 171.4± 2.1 Kidonakis ( 6-7) The numbers include NLO QCD corrections (Harris et al ( 2), Sullivan ( 4- ), Campbell et al ( 4), Chao et al ( 4 - ), Smith and Willenbrock ( 96), Chao et al ( 4), Giele et al ( 9), Zhu ( 2) ) The theoretical uncertainties on The cross-section is proportional to V tb 2 the single top σ t + σ t is 38% (48%) cross sections of σ t t at are the under LHC control (Tevatron) = single top events at the LHC (L = 1fb 1 ) The final states from single top events have large backgrounds CDF and D reported evidence for single-top production at the Tevatron, at the LHC it should be possible to disentangle the different modes