Heavy quark masses from Loop Calculations

Transcription

1 Heavy quark masses from Loop Calculations Peter Marquard Institute for Theoretical Particle Physics Karlsruhe Institute of Technology in collaboration with K. Chetyrkin, D. Seidel, Y. Kiyo, J.H. Kühn, A. Maier, P. Maierhöfer, J. Piclum, A.V. Smirnov, M. Steinhauser, C. Sturm KA SFB TR9 Computational Theoretical Particle Physics B AC , Dresden

2 Outline 1 Introduction 2 Low-energy moments 3 Padé approximation 4 Heavy quark masses Charm-Quark Mass Bottom-Quark Mass Masses from Lattice Calculations 5 t threshold 6 Conclusion

3 Introduction charm, bottom and top quark masses are important input parameters of the Standard Model Γ(B X u lν) m 5 b V ub 2 Γ(B X c lν) m 5 b f(m2 c /m2 b ) V cb 2 Higgs ILC: Γ(H b b) m 2 b, known up to α4 s

4 Introduction charm, bottom and top quark masses are important input parameters of the Standard Model Γ(B X u lν) m 5 b V ub 2 Γ(B X c lν) m 5 b f(m2 c /m2 b ) V cb 2 Higgs ILC: Γ(H b b) m 2 b, known up to α4 s charm and bottom quark masses can be obtained from an analysis based on low-energy moments of Π(q 2 ) and experimental data for R(s) top-quark mass can be obtained from a measurement of the total cross section for t t pair production at threshold at the ILC

5 Techniques I Which hard processes can be done at more than two loops? no-scale three point three loops single-scale two point three loops no-scale two point four loops

6 Techniques I Which hard processes can be done at more than two loops? no-scale three point three loops single-scale two point three loops no-scale two point four loops first step: perform (asymptotic) expansion in suitable parameters: external momenta masses or mass ratios threshold second step: perform reduction to master integrals third step: calculate needed master integrals

7 Techniques II after generation of diagrams and expansion typically O(10 4 ) O(10 6 ) scalar integrals have to be calculated direct calculation not possible, but integrals related by Integration-By-Parts identities d n {k µ i, q µ i } k j k µ = 0 D 1 D m using IBP identities all needed integrals J i can be reduced to a small set of master integrals {M 1,..., M N } J i = j=1,n C ij (d, x)m j many checks possible on the level of master integrals

8 Techniques III How to use the IBP identities? Method 1: Recursion relations Manually solve the IBP identities allows for fast implementations, e.g. MATAD,MINCER not always possible!?!

9 Techniques III How to use the IBP identities? Method 1: Recursion relations Manually solve the IBP identities allows for fast implementations, e.g. MATAD,MINCER not always possible!?! Method 2: Laporta s algorithm Generate linear system of equations using IBP relations and solve it. very successful, multi-purpose method but inefficient, brute force approach

10 Techniques III How to use the IBP identities? Method 1: Recursion relations Manually solve the IBP identities allows for fast implementations, e.g. MATAD,MINCER not always possible!?! Method 2: Laporta s algorithm Generate linear system of equations using IBP relations and solve it. very successful, multi-purpose method but inefficient, brute force approach Method 3: Baikov s method use expansion in n to construct reduction

11 Techniques IV Many techniques/tools exist for the calculation of the master integrals direct calculation using Feynman/Alpha parametrization Mellin-Barnes representation sector decomposition difference equations differential equations dimensional recursion relation Caveat: if numerical integration error estimate problematic

12 Mass determination: Outline of the method Experiment R(s) BES (2001) J/ψ ψ, MD pqcd CLEO BES (2006) s (GeV) Define experimental moments M exp R(s) n = ds sn+1 Theory Π(q 2 ) = 1 12π 2 R(s) s q 2 ds Taylor expand Π(q 2 ) around q 2 = 0: ( ) Π(q 2 ) = 3 n q 2 Cn 16π 2 4m 2 Q theoretical moments C n ( ) 1 m Q Mn 2n C n

13 vacuum polarization function Π µν (q 2 ) = ( q 2 g µν + q µ q ν )Π(q 2 ) + q µ q ν Π L (q 2 ) = i dx e iqx 0 Tj µ (x)j ν (0) 0, j µ = ψγ µ ψ, ψγ µ χ related to R(s) = σ(e + e hadrons)/σ(e + e µ + µ ) through R(s) = 12πImΠ(q 2 + iǫ) low-energy expansion Π(q 2 ) = 3 16π 2 C n z n, n>0 C n = j=0 ( αs ) j (j) C n 4π z = q2 4m 2 Q

14 What do we know about Π(q 2 ) at one and two loops known analytically [Kallen, Sabry 55] Π (0) (z) = 3 ( 20 16π ) 4(1 z)(1 + 2z) G(z), 3z 3z Π (1) (z) = 3 [ 5 16π z (1 z)3 G(z) 6z z + (1 z) 1 16z G(z) z ( 1 + 2z(1 z) d ) ] I(z) 6z 6z dz z G(z) = 1 log (u) 2z 1 1 z, u = 1 1 z 1 1 1z + 1. I(z) =6 ( ζ 3 + 4Li 3 ( u) + 2Li 3 (u) ) 8 ( 2Li 2 ( u) + Li 2 (u) ) log(u) 2 ( 2 log(1 + u) + log (1 u) ) log(u) 2

15 Π(q 2 three loops not known fully analytical two options for calculation of low-energy expansion expand propagators to obtain vacuum diagrams reduce to master integrals : first eight terms

16 Π(q 2 three loops not known fully analytical two options for calculation of low-energy expansion expand propagators to obtain vacuum diagrams reduce to master integrals : first eight terms reduce propagator diagrams to master integrals calculate expansion of master integrals : many terms

17 Π(q 2 three loops not known fully analytical two options for calculation of low-energy expansion expand propagators to obtain vacuum diagrams reduce to master integrals : first eight terms reduce propagator diagrams to master integrals calculate expansion of master integrals : many terms some terms in threshold expansion many terms in high energy expansion

18 Π(q 2 three loops not known fully analytical two options for calculation of low-energy expansion expand propagators to obtain vacuum diagrams reduce to master integrals : first eight terms reduce propagator diagrams to master integrals calculate expansion of master integrals : many terms some terms in threshold expansion many terms in high energy expansion full q 2 dependence reconstructed using Padé approximation [Chetyrkin, Kühn, Steinhauser]

19 Π(q 2 three loops Example: 1-loop ( ) 4(ǫ 1) Π(z = q 2 /(4m 2 )) = 2ǫ 3 8 z(2ǫ 3) 8(ǫ 1) z(2ǫ 3) 3-loop calculation leads to a similar expression. The master integrals can not be calculated analytically calculate expansion needed input: 3-loop vacuum integrals

20 Calculation of three-loop propagator master integrals Differential equation obtained from scaling relation ( q 2 q 2 + m2 m 2 1 ) 2 D(J) J(q 2, m 2, d) = 0 D(J) mass dimension of J make ansatz for master integrals J i (q 2, m 2, d) = m D(J i) n ( ) q 2 n K in m 2 solve resulting system of linear equations for K in boundary condition: known tadpole diagrams

21 Calculation of three-loop propagator master integrals Differential equation obtained from scaling relation ( q 2 q 2 + m2 m 2 1 ) 2 D(J) J(q 2, m 2, d) = 0 D(J) mass dimension of J make ansatz for master integrals ( J i (z, d) = m D(J i) n K in z n + K ǫ in ( q 2 m 2 ) ǫ + K 2ǫ in solve resulting system of linear equations for K in K ǫ boundary condition: known tadpole diagrams ( ) q 2 2ǫ ) ( q 2 m 2 in K in 2ǫ m 2 ) n

22 diagonal vector current j µ = ψγ µ ψ, ψ massive Π (2),v (z) = 3 16π 2 `( nl )z + ( n l )z 2 + ( n l )z 3 + ( n l )z 4 + ( n l )z 5 + ( n l )z 6 + ( n l )z 7 + ( n l )z 8 + ( n l )z 9 + ( n l )z 10 + ( n l )z 11 + ( n l )z 12 + ( n l )z 13 + ( n l )z 14 + ( n l )z 15 + ( n l )z 16 + ( n l )z 17 + ( n l )z 18 + ( n l )z 19 + ( n l )z 20 + ( n l )z 21 + ( n l )z 22 + ( n l )z 23 + ( n l )z 24 + ( n l )z 25 + ( n l )z 26 + ( n l )z 27 + ( n l )z 28 + ( n l )z 29 + ( n l )z 30 [Boughezal, Czakon, Schutzmaier] [Maier, Maierhöfer, PM]

23 non-diagonal vector current j µ = ψγ µ χ, ψ massive, χ massless Π (2),v (z) = 3 16π 2 `( nl )z + ( n l )z 2 + ( n l )z 3 + ( n l )z 4 + ( n l )z 5 + ( n l )z 6 + ( n l )z 7 + ( n l )z 8 + ( n l )z 9 + ( n l )z 10 + ( n l )z 11 + ( n l )z 12 + ( n l )z 13 + ( n l )z 14 + ( n l )z 15 + ( n l )z 16 + ( n l )z 17 + ( n l )z 18 + ( n l )z 19 + ( n l )z 20 + ( n l )z 21 + ( n l )z 22 + ( n l )z 23 + ( n l )z 24 + ( n l )z 25 + ( n l )z 26 + ( n l )z 27 + ( n l )z 28 + ( n l )z 29 + ( n l )z 30 [ Maier, PM]

24 Π(q 2 four loops reduction of single-scale propagator-type diagrams not (yet) four loops only direct expansion of diagrams and reduction to master integrals possible only very limited number of moments calculable third moment first moment also calculated by [Chetyrkin et al; Boughezal et al] more information available for cases with two closed fermion loops many terms of nf 2 contributions known [Czakon, Schutzmaier] n n 1 l α n s known to all orders in α s [Grozin, Sturm] some information about threshold and high-energy behavior

25 Calculation of third NNNLO 700 four-loop Feynman diagrams of the form q 2 q 2 q 2 expansion around q 2 = 0 ( ) q 2 = + q2 + q 2 2 4m 2 4m 2 results in four-loop vacuum integrals which have to be calculated

26 Calculation cont d direct calculation of all these integrals not feasible perform reduction to master integrals using Laporta s algorithm needed: , calculated: , naive: in this case there are only 13 four-loop master integrals, all are known analytically [Chetyrkin et al; Laporta; Kniehl et al; Schröder et al] Tools: qgraf[nogueira], (T)FORM[Vermaseren], Crusher[PM, Seidel] Optimization: Integrate out self energies reduce number of loops

27 Third moment Result C (3) n ll,3 lh,3 = C F T 2 F n2 l C(3) ll,n + C F T 2 F n2 h C(3) hh,n + C F T 2 F n l n h C (3) lh,n + C(3) n f 0 + C,n F T F n l C A C (3) lna,n + C F C (3) la,n + C F T F n h C A C (3) hna,n + C F C (3), µ = m ha,n q, c 4 = 24a 4 + log 4 (2) 6ζ 2 log 2 (2), a k = Li k (1/2) = ζ 3, hh,3 = ζ 3, = c ζ ζ 3, lna,3 = c ζ ζ 3, la,3 = c ζ ζ 3, hna,3 = c ζ ζ ζ 3, ha,3 = c ζ ζ n f 0 = , c ζ ζ log(2) ζ a log(2)ζ log 3 (2)ζ

28 Π(q 2 ) at four loops Collect information of behavior in low-energy, threshold and high-energy region (example for vector current and n l = 3) low-energy: Π 3 (z) = z z z 3 + O(z 4 )

29 Π(q 2 ) at four loops Collect information of behavior in low-energy, threshold and high-energy region (example for vector current and n l = 3) low-energy: Π 3 (z) = + O(z 4 ) threshold: Π 3 (z) = /(1 z) + ( log(1 z))/ 1 z log(1 z) log 2 (1 z) log 3 (1 z) + K 0 + O( 1 z)

30 Π(q 2 ) at four loops Collect information of behavior in low-energy, threshold and high-energy region (example for vector current and n l = 3) low-energy: Π 3 (z) = + O(z 4 ) threshold: Π 3 (z) = + K 0 + O( 1 z) high-energy: Π 3 (z) = log( 4z) log 2 ( 4z) log 3 ( 4z) + 1/z( log( 4z) log 2 ( 4z) log 3 ( 4z)) + 1/z 2 (D log( 4z) log 2 ( 4z) log 3 ( 4z) log 4 ( 4z)) + O(1/z 3 )

31 Π(q 2 ) at four loops Collect information of behavior in low-energy, threshold and high-energy region (example for vector current and n l = 3) low-energy: Π 3 (z) = + O(z 4 ) threshold: Π 3 (z) = + K 0 + O( 1 z) high-energy: Π 3 (z) = + 1/z 2 D 2 + O(1/z 3 )

32 Π(q 2 ) at four loops Collect information of behavior in low-energy, threshold and high-energy region (example for vector current and n l = 3) low-energy: Π 3 (z) = + O(z 4 ) threshold: Π 3 (z) = + K 0 + O( 1 z) high-energy: Π 3 (z) = + 1/z 2 D 2 + O(1/z 3 ) (1-z)Π v,(3) v R 3,v z v

33 Π(q 2 ) at four loops Collect information of behavior in low-energy, threshold and high-energy region (example for vector current and n l = 3) low-energy: Π 3 (z) = + O(z 4 ) threshold: Π 3 (z) = + K 0 + O( 1 z) high-energy: Π 3 (z) = + 1/z 2 D 2 + O(1/z 3 ) (1-z)Π v,(3) v R 3,v z v Find approximating function and determine further terms in low-energy expansion and the missing constants K 0 and D 2.

34 Padé approximation Approximation by a rational function of the form p nm (z) = n i=0 a iz i 1 + m i=1 b iz i with n + m + 1 degrees of freedom n + m + 1 different Padés for a given set of boundary condition faster convergence than normal Taylor series better behaved near cuts

35 Padé approximation I Next Step: Construct the Padé approximation But: Π(z) logarithmically divergent for z 1 and z, can not be approximated by rational function split Π(z) in two parts Π(z) = Π reg (z) + Π log (z), where Π log (z) contains all logarithmic contributions

36 Padé approximation II Π log (z) How to construct Π log (z)? Use one- and two-loop results as building blocks threshold behavior G(z) = π z + O( 1 z) Π (1) (z) = 3 16 log(1 z) + const + O( 1 z) high-energy behavior Ansatz for Π log (z) G(z) = log( 4z) 2z + O(z 2 ) Π log (z) = k ij Π (1) (z) i G(z) j + d mn (zg(z)) m (1 1 z ) m 2 threshold high-energy 1 z n

37 Padé approximation III Π reg (z) Π reg (z) = Π(z) Π log (z) is free of logarithmic singularities perform a conformal mapping to the unit circle z 4ω/(1 + ω) 2 Im(z) Im(ω) 0 1 Re(z) Re(ω)

38 Padé approximation IV fit by Padé approximation of the form p nm (ω) = n i=0 a iω i 1 + m i=1 b iω i with n + m + 1 = 9 degrees of freedom error estimate: modify Π log (z), vary a i and b m Π log (z) = k ij Π (1) (z) i G(z) j (a i + 1 z ) + d mn (zg(z)) m (1 1 z ) m 2 1 z n (b m + 1 z )

39 Padé approximation V Quality of Fit Distribution of the O(8000) reconstructed values for the first missing low-energy constant C 4 strongly peaked, very narrow distribution ( < C 4 < ) choose standard deviation as measure for error

40 Padé approximation VI Results n l = 3 n l = 4 n l = (11) (10) (9) (32) (32) (27) (61) (63) (54) (9) (10) (9) (13) (14) (12) (17) (18) (16) (20) (22) (19)

41 Padé approximation VI Results n l = 3 n l = 4 n l = (11) (10) (9) (32) (32) (27) (61) (63) (54) (9) (10) (9) (13) (14) (12) (17) (18) (16) (20) (22) (19) K (3),v 0 17(11) 17(29) 16(10)

42 Padé approximation VI Results n l = 3 n l = 4 n l = (11) (10) (9) (32) (32) (27) (61) (63) (54) (9) (10) (9) (13) (14) (12) (17) (18) (16) (20) (22) (19) K (3),v 0 17(11) 17(29) 16(10) D (3),v 2 2.0(42) 1.2(83) 1.4(21)

43 Padé approximation IV Results n l = 3 n l = 4 n l = (11) (10) (9) (32) (32) (27) (61) (63) (54) (9) (10) (9) (13) (14) (12) (17) (18) (16) (20) (22) (19) K (3),v 0 17(11) 17(29) 16(10) D (3),v 2 2.0(42) 1.2(83) 1.4(21) ± 0.57 [Hoang et al] 10 ± 11 [Hoang et al]

44 Padé approximation V R(s) reconstruction of Π(q 2 ) below and R(s) above threshold [Kiyo, Maier, Maierhöfer, PM 09] red error band corresponds to three times the local standard deviation (1 z)π(z) below threshold vr(v) above threshold

45 Outline of the method Experiment R(s) BES (2001) J/ψ ψ, MD pqcd CLEO BES (2006) s (GeV) Define experimental moments M exp R(s) n = ds sn+1 Theory Π(q 2 ) = 1 12π 2 R(s) s q 2 ds Taylor expand Π(q 2 ) around q 2 = 0: ( ) Π(q 2 ) = 1 n q 2 Cn 16π 2 4m 2 Q theoretical moments C n ( ) 1 m Q Mn 2n C n

46 Experimental Data R(s) pqcd BES (2001) J/ψ ψ, MD-1 CLEO BES (2006) s (GeV)

47 Analysis contributions from three regions Resonances: width and mass of J/Ψ, Ψ by BES, CLEO, BABAR (PDG) threshold region 3.73 GeV < s < 4.8 GeV: R(s) by BES continuum s > 4.8 GeV: 3-loop QCD prediction We determine the MS mass m c (µ) at the scale µ = 3 GeV.

48 Results m c (3 GeV)[MeV] n m c (3 GeV) ± exp αs µ np tot res thres cont ± ± ± ± m c (3GeV) = 986(13)MeV [Chetyrkin et al]

49 Comparison: 1-loop 4-loop m c (3 GeV) (GeV) n

50 Experimental Data CLEO (1985)/1.28 BABAR (2009) 0.5 R b (s) s (GeV) recent data from BABAR corresponding data from Belle???

51 Analysis contributions from three regions: Resonances: width and mass off Υ(1S),..., Υ(4S) threshold region 10.6 GeV < s < 11.2 GeV: R(s) from BABAR continuum s > 11.2 GeV: 3-loop QCD prediction

52 Results m b (10 GeV)[MeV] n m b (10 GeV) ± exp αs µ tot res thres cont ± ± ± ± m b (10GeV) = 3610(16)MeV m b (m b ) = 4163(16)MeV [Chetyrkin et al]

53 Comparison: 1-loop 4-loop m b (10 GeV) (GeV) n

54 Quark Masses from Lattice Calculations current-current correlators can be directly simulated on the lattice experimental data can be replaced by lattice simulations in principle all correlators can be used even non-diagonal ones even high moments can be used

55 Results HPQCD collaboration: analysis based on the pseudo-scalar correlator with the current j 5 = Ψγ 5 Ψ Charm: Input: M(η c ) m c (3 GeV) = 986(6) MeV Bottom: Input: M(η b ) m b (10 GeV) = 3617(25) MeV compare with quark masses obtained using R(s): m c (3 GeV) = 986(13) MeV m b (10GeV) = 3610(16) MeV

56 Comparison Charm-Quark mass Bodenstein et. al 10 finite energy sum rule, NNNLO HPQCD 10 lattice + pqcd HPQCD + Karlsruhe 08 lattice + pqcd Kuehn, Steinhauser, Sturm 07 low-moment sum rules, NNNLO Buchmueller, Flaecher 05 B decays α s 2β 0 Hoang, Manohar 05 B decays α s 2β 0 Hoang, Jamin 04 NNLO moments dedivitiis et al. 03 lattice quenched Rolf, Sint 02 lattice (ALPHA) quenched Becirevic, Lubicz, Martinelli 02 lattice quenched Kuehn, Steinhauser 01 low-moment sum rules, NNLO QWG 2004 PDG m c (3 GeV) (GeV)

57 Comparison Bottom-Quark mass HPQCD 10 Karlsruhe 09 low-moment sum rules, NNNLO, new Babar Kuehn, Steinhauser, Sturm 07 low-moment sum rules, NNNLO Pineda, Signer 06 Υ sum rules, NNLL (not complete) Della Morte et al. 06 lattice (ALPHA) quenched Buchmueller, Flaecher 05 B decays α s 2β 0 Mc Neile, Michael, Thompson 04 lattice (UKQCD) dedivitiis et al. 03 lattice quenched Penin, Steinhauser 02 Υ(1S), NNNLO Pineda 01 Υ(1S), NNLO Kuehn, Steinhauser 01 low-moment sum rules, NNLO Hoang 00 Υ sum rules, NNLO QWG 2004 PDG m b (m b ) (GeV)

58 top-quark mass: Motivation The mass of the top quark is an important input parameter for electro-weak precision physics July 2011 LEP2 and Tevatron LEP1 and SLD 68% CL m W [GeV] m H [GeV] α m t [GeV]

59 top-quark mass: Motivation The mass of the top quark is an important input parameter for electro-weak precision physics. Measurements of the top-quark mass at hadron colliders limited by systematic errors M t 1 GeV mass definition not quite clear

60 top-quark mass: Motivation The mass of the top quark is an important input parameter for electro-weak precision physics. Measurements of the top-quark mass at hadron colliders limited by systematic errors M t 1 GeV mass definition not quite clear Measurement of the top-quark mass at a future linear collider M t < 100 MeV

61 top-quark mass: Motivation The mass of the top quark is an important input parameter for electro-weak precision physics. Measurements of the top-quark mass at hadron colliders limited by systematic errors M t 1 GeV mass definition not quite clear Measurement of the top-quark mass at a future linear collider M t < 100 MeV Calculation discussed in this talk is also an important building block for the determination of the bottom-quark mass from Υ sum rules

62 Introduction The top-quark mass can be obtained from a scan of the top anti-top threshold at a linear collider (M t = 175 GeV) [Martinez, Miquel]

63 Introduction The top-quark mass can be obtained from a scan of the top anti-top threshold at a linear collider (M t = 175 GeV ± 200 MeV)

64 Theoretical Status Calculation best done in the framework of effective field theories: (P)NRQCD two-loop: calculation done by several groups using different methods [Hoang,Teubner; Melnikov,Yelkhovsky; Yakovlev; R R q 2 GeV Beneke, Signer, Smirnov; Nagano, Ota, Sumino; Penin, Pivovarov ] Beneke Signer Smirnov Hoang Teubner q 2 GeV R R Yakovlev q 2 GeV Melnikov Yelkhovsky q 2 GeV

65 Theoretical Status Calculation best done in the framework of effective field theories: (P)NRQCD two-loop: calculation done by several groups using different methods [Hoang,Teubner; Melnikov,Yelkhovsky; Yakovlev; Beneke, Signer, Smirnov; Nagano, Ota, Sumino; Penin, Pivovarov ] three-loop: Most building blocks now available: matching coefficient c v n l [PM,Piclum,Seidel,Steinhauser 06] n f [PM,Piclum,Seidel,Steinhauser 09] nf 0 NEW heavy-quark potential a 3 potential contributions ( ) ultrasoft contributions [Smirnov,Smirnov,Steinhauser; Anzai,Kiyo,Sumino] [Beneke,Kiyo,Schuller] [Beneke,Kiyo,Penin]

66 Definition QCD vector current j µ v = Qγ µ Q NRQCD vector current jk v = φ σ k χ Matching ( ) 1 jv k = c v jk v + O M 2

67 Definition QCD vector current j µ v = Qγ µ Q NRQCD vector current jk v = φ σ k χ Matching j k v = c v jk v + d v 6M 2φ σ k D 2 χ+

68 Definition QCD vector current j µ v = Qγ µ Q NRQCD vector current jk v = φ σ k χ Matching ( ) 1 jv k = c v jk v + O M 2 c v can be extracted by calculating vertex corrections involving j v and j v 1 Z 2 Γ v = c v Z2 Z v Γ v +

69 Setup of the Calculation Feynman diagrams generated using QGRAF mapped onto 78 topologies using Q2E/EXP Feynman integrals reduced to master integrals with CRUSHER master integrals in different topologies have to be identified main obstacle: O(100) master integrals had to be calculated analytically/numerically using various techniques numerical errors added in quadrature

70 Calculation of Master Integrals some simple (propagator-type) master integrals known analytically others can be calculated precisely using Mellin-Barnes methods difficult (vertex-type) integrals calculated numerically using FIESTA (Feynman Integral Evaluation by a Sector decomposition Approach) [Smirnov,Tentyukov]

71 Calculation of Master Integrals some simple (propagator-type) master integrals known analytically others can be calculated precisely using Mellin-Barnes methods difficult (vertex-type) integrals calculated numerically using FIESTA (Feynman Integral Evaluation by a Sector decomposition Approach) [Smirnov,Tentyukov] = N ( (3) ǫ (1) ǫ (2) ) (4)ǫ + O(ǫ 2 )

72 Checks Renormalization constant Z v of the NRQCD vector current can be reproduced Z v analytically known, 1/ǫ part numerically small agreement within the error estimate at the percent level

73 Checks Renormalization constant Z v of the NRQCD vector current can be reproduced Z v analytically known, 1/ǫ part numerically small agreement within the error estimate at the percent level Gauge independence: terms linear in ξ vanish after renormalization

74 Checks Renormalization constant Z v of the NRQCD vector current can be reproduced Z v analytically known, 1/ǫ part numerically small agreement within the error estimate at the percent level Gauge independence: terms linear in ξ vanish after renormalization Change basis of master integrals and compare

75 Phenomenology Residue of the QCD two-point function Π(q 2 ) E En N c 2m Q Z n E n (E + i0)

76 Phenomenology Residue of the QCD two-point function Π(q 2 ) E En N c 2m Q Z n E n (E + i0) Z t (µ)/z t (µ S ) LO µ (GeV)

77 Phenomenology Residue of the QCD two-point function Π(q 2 ) E En N c 2m Q Z n E n (E + i0) Z t (µ)/z t (µ S ) µ (GeV) NLO LO

78 Phenomenology Residue of the QCD two-point function Π(q 2 ) E En N c 2m Q Z n E n (E + i0) 1.2 Z t (µ)/z t (µ S ) NNLO µ (GeV) NLO LO

79 Phenomenology Residue of the QCD two-point function Π(q 2 ) E En N c 2m Q Z n E n (E + i0) Z t (µ)/z t (µ S ) NNNLO (ferm.) NNLO µ (GeV) NLO LO

80 Conclusion many terms of low-energy expansion at NNLO calculated third low-energy moment at NNNLO reconstructed Π(q 2 ) and therefore R(s) very precise prediction for the low-energy moments of Π(q 2 ) up to n = 10 precise result for charm-quark mass m c (3GeV) = 986(13)MeV precise result for bottom-quark mass m b (10 GeV) = 3610(16)MeV 3-loop prediction for t t production at threshold on its way