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1 Top and bottom at threshold Matthias Steinhauser TTP Karlsruhe Vienna, January 27, 2015 KIT University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association

2 Outline Introduction a 3 to 3 loops c v to 3 loops Γ(Υ(1S) l + l ) to NNNLO positronium HFS Summary Matthias Steinhauser Top and bottom at threshold 2

3 Physical quantities energy levels and wave function of bound states E q q [NNNLO: Penin,Steinhauser 02; Kiyo,Sumino 03; Beneke,Kiyo,Schuller 05] M Υ(1S) = 2m b + E b b(n = 1) m b E res = 2m t + E t t(n = 1)+δ Γt E res m t Υ sum rules m b [Penin,Pivovarov 98; Hoang 98; Melnikov,Yelkhovsky 98,..., Penin,Zerf 14, Beneke,Maier,Piclum,Rauh 14] Γ(Υ(1S) l + l ), NNNLO σ(e + e t t), NNNLO R NNLL [Pineda,Signer 07; Hoang,Stahlhofen 13,... ] Comparison of pqcd and LQCD s (GeV) [Necco,Sommer 01,Pineda 02,Sumino 05,..., Brambilla,Vairo,Garcia i Tormo,Soto 09, Donnellan,Knechtli,Leder,Sommer 10] Matthias Steinhauser Top and bottom at threshold 3

4 Framework: potential NRQCD scales: mass, m: hard momentum, mv: soft energy, mv 2 : ultrasoft Λ QCD bottom: 5 GeV 2.5 GeV 0.5 GeV top: 175 GeV 30 GeV 5 GeV QCD NRQCD pnrqcd (potential non-relativistic QCD) m mv, mv 2 [Caswell,Lepage 86; integrate out all non-physical degrees of freedom Bodwin,Braaten,Lepage 95] [Beneke,Smirnov 97; Pineda,Soto 98; Brambilla,Pineda,Soto,Vairo 00] [alternative formulation: velocity NRQCD [Luke,Manohar,Rothstein 00; Hoang,Stewart 03]] Matthias Steinhauser Top and bottom at threshold 4

5 Effective Hamiltonian to N 3 LO [Gupta,Radford 81,...,Manohar 97,...,Kniehl,Penin,Smirnov,Steinhauser 02,...,Beneke,Kiyo,Schuller 13] ( p H = (2π) 3 2 δ( q) m p ) 4 + 4m 3 C c (α s )V C ( q )+C 1/m (α s )V 1/m ( q ) [ ] + πc Fα s (µ) m 2 C δ (α s )+C p (α s ) p 2 + p q 2 C s (α s ) S 2 Static potential: V C ( q ) = 4πC Fα s( q ) q 2 C c 3 loops 1/m potential: V 1/m ( q ) = π2 C F α 2 s( q ) m q C 1/m 2 loops Breit potential: 1/m 2 C δ,p,s 1 loop pnrqcd has dynamical degrees of freedom: potential quarks and ultrasoft gluons q = p p q Matthias Steinhauser Top and bottom at threshold 5

6 Static potential Matthias Steinhauser Top and bottom at threshold 6

7 Static potential [ V C (µ = q ) = 4πC Fα s ( q ) q 2 1 [Appelquist,Politzer 75,Susskind 77] Matthias Steinhauser Top and bottom at threshold 7

8 Static potential [ V C (µ = q ) = 4πC Fα s ( q ) 1+ α s( q ) q 2 4π a 1 [Appelquist,Politzer 75,Susskind 77] [Fischler 77;Biloire 80] Matthias Steinhauser Top and bottom at threshold 7

9 Static potential [ V C (µ = q ) = 4πC Fα s ( q ) 1+ α s( q ) q 2 4π a 1 + ( ) αs ( q ) 2 a 2 4π [Appelquist,Politzer 75,Susskind 77] [Fischler 77;Biloire 80] [Peter 96;Schröder 98] Matthias Steinhauser Top and bottom at threshold 7

10 Static potential V C (µ = q ) = 4πC Fα s ( q ) q 2 [ 1+ α s( q ) 4π a 1 + ( ) αs ( q ) 3 + (a 3 + 8π 2 C 4π ( ) αs ( q ) 2 a 2 4π ] ) 3A µ2 ln + q 2 [Appelquist,Politzer 75,Susskind 77] [Fischler 77;Biloire 80] [Peter 96;Schröder 98] [Smirnov,Smirnov,Steinhauser 08; Smirnov,Smirnov,Steinhauser 09; Anzai,Kiyo,Sumino 09] Matthias Steinhauser Top and bottom at threshold 7

11 Static potential [ V C (µ = q ) = 4πC Fα s ( q ) 1+ α s( q ) q 2 4π a 1 + ( ) αs ( q ) 3 ( + 4π a 3 + 8π 2 C 3 A ( ) αs ( q ) 2 a 2 4π ( 1 µ2 + ln 3ǫ q 2 )) ] + IR divergence [Appelquist,Dine,Muzinich 77] (in naive perturbation theory ) pnrqcd: ultra-soft contribution: (us)-gluons and (Q Q) bound states as dynamical degrees of freedom IR finite result [Brambilla,Pineda,Soto,Vairo 99; Kniehl,Penin,Smirnov,Steinhauser 02] Finite physical quantities: V QCD (r) = V pert QCD +δvus QCD static energy ( lattice) energy levels of (q q) system: E q q = n H n +δe US q q H.O. log-contributions to V : [Pineda,Soto 00; Brambilla,Garcia i Tormo,Soto,Vairo 07] Matthias Steinhauser Top and bottom at threshold 7

12 Numerical results V C( q ) = 4πCFαs( q ) [ 1+ αs q 2 π ( nl) ( ) 2 αs ( ) nl n 2 l + π ( αs π ) 3 ( ) (1) nl n 2 l n 3 l + ] n l α (n l) s 1 loop 2 loop 3 loop charm bottom top static potential for other colour configurations: [Kniehl,Penin,Schroder,Smirnov,Steinhauser 05; Anzai,Kiyo,Sumino 10; Collet,Steinhauser 11; Prausa,Steinhauser 13; Anzai,Prausa,Smirnov,Smirnov,Steinhauser 13] Matthias Steinhauser Top and bottom at threshold 8

13 Matthias Steinhauser Top and bottom at threshold 9

14 c v to 3 loops QCD NRQCD j µ v = Qγ µ Q ji = φ σ i χ jv i = c v (µ) ji + dv(µ) φ σ i 6m D 2 χ+... Q 2 Z 2 Γ v = c v Z2 Z 1 v Γ v +... Γ v : Γ v 1 Z2 1 Z 2 : [Melnikov,v.Ritbergen 00] [Marquard,Mihaila,Piclum,Steinhauser 07] Zv = 1+O(αs) 2 [Beneke,Signer,Smirnov 98; Kniehl,Penin,Steinhauser,Smirnov 03; Marquard,Piclum,Seidel,Steinhauser 06; Beneke,Kiyo,Penin 07] (MS scheme) Matthias Steinhauser Top and bottom at threshold 10

15 c v c v = 1+ αs π c(1) v + ( α s ) 2 (2) π c v + ( α s ) 3 (3) π c v +O(αs) 4 c (1) v c (2) v c (3),n l v c (3),n h v c (3) v [Källen, Sarby 55] [Czarnecki,Melnikov 97; Beneke,Signer,Smirnov 97] [Marquard,Piclum,Seidel,Steinhauser 06] [Marquard,Piclum,Seidel,Steinhauser 08] [Marquard,Piclum,Seidel,Steinhauser 14] massive vertices on-shell quarks: q 2 1 = q2 2 = M2 Q (q 1 + q 2 ) 2 = 4M 2 Q Matthias Steinhauser Top and bottom at threshold 11

16 Some technical details... automatic generation of Feynman diagrams, apply projector, reduce scalar products in numerator,... (many) scalar integrals reduction to 100 master integrals with CRUSHER [Marquard,Seidel] compute MIs with FIESTA [Smirnov,Tentyukov 08; Smirnov,Smirnov,Tentyukov 11; Smirnov 13] q1 q2 ( = e3ǫγ E m 4 Q ( ) µ 2 3ǫ m 2 Q (3) ǫ (1) ǫ (2) (4)ǫ+O(ǫ 2 ) simpler integrals also known analytically add uncertainties of individual integrals quadratically multiply final uncertainty by factor 5 ( 5σ ) ) Matthias Steinhauser Top and bottom at threshold 12

17 Checks finiteness n l contribution gauge parameter independence (ξ 1 ) change basis of MIs default basis alternative basis c FFF 36.55(0.11) 36.61(2.93) c FFA (0.17) (2.91) c FAA 97.81(0.08) 97.76(2.05) c (3) v (n l = 4) (0.4) 1621(23) c (3) v (n l = 5) (0.4) 1507(23) Matthias Steinhauser Top and bottom at threshold 13

18 Z v ( ) α (nl) 2 ( s (µ) CFπ Z 2 1 v = 1+ π ǫ 12 CF + 1 ) ( ) α (nl) 3 8 CA s (µ) + C Fπ 2 π { [ ( C F 144ǫ ln 2+ 5 ) ] 1 48 Lµ ǫ [ ( C F C A 864ǫ ) ] 5 1 ln Lµ ǫ [ + C 2 A 1 ( 2 16ǫ ) ] 1 1 ln Lµ ǫ [ ( 1 + Tn l C F 54ǫ 25 ) ( 1 + C 2 A 324ǫ 36ǫ 37 )] 2 432ǫ } 1 + C F Tn h +O(αs) 4 L µ = ln µ2 60ǫ m 2 Q analytic result from [Beneke,Signer,Smirnov 98; Kniehl,Penin,Smirnov,Steinhauser 02; Marquard,Piclum,Seidel,Steinhauser 06; Beneke,Kiyo,Penin 07] (numerical) agreement better than 1% Matthias Steinhauser Top and bottom at threshold 14

19 Results for c v [Marquard,Piclum,Seidel,Steinhauser 14] c v (µ = m Q ) = α s π +[ n l] ( αs ) 2 π + [ ] ( 2091(2) n l 0.82nl 2 α s π + singlet terms ) 3 MS scheme; µ = m Q large corrections singlet terms: small ( 3% of c (2) ) at 2 loops (for axial-vector, scalar, pseudo-scalar current) [Kniehl,Onishchenko,Piclum,Steinhauser 06] Matthias Steinhauser Top and bottom at threshold 15

20 Application: height of σ(e + e t t) 1.4 LO, NLO, NNLO R s (GeV) [Hoang,Beneke,Melnikov,Nagano,Ota,Penin,Pivovarov,Signer,Smirnov,Sumino,Teubner,Yakovlev,Yelkhovsky 00] Matthias Steinhauser Top and bottom at threshold 16

21 Application: Residue Z t ( q 2 g µν + q µ q ν ) Π(q 2 ) = i dx e iqx 0 Tj µ (x)j ν(0) 0 Π(q 2 ) E E 1 = Nc 2mt 2 [ Z 1 E 1 (E+i0) +... Z ( ) ] t = cv 2 E 1 m t c v cv + dv ψ 3 1 (0) 2 Z t (µ)/z t LO(µ S ) NNLO NNNLO NNNLO (ferm.) NLO LO µ (GeV) [Marquard,Piclum,Seidel,Steinhauser 14] Matthias Steinhauser Top and bottom at threshold 17

22 Coulomb corrections to σ(e + e t t) 1.0 R µ = 15 GeV (lower) µ = 30 GeV (middle) µ = 60 GeV (upper) µ = 15 GeV exact s [GeV] [Beneke,Kiyo,Schuller 05] Matthias Steinhauser Top and bottom at threshold 18

23 Γ(Υ(1S) l + l ) Matthias Steinhauser Top and bottom at threshold 19

24 Γ(Υ(1S) l + l ) Υ(1S) meson: simplest heavy quark bound state Γ(Υ(1S) l + l ) exp = 1.340(18) kev Γ(Υ(1S) l + l ) = 4πα2 9m 2 b ( ψ 1 (0) 2 c v [c v E 1 cv m b + dv 3 ) +... ] Matthias Steinhauser Top and bottom at threshold 20

25 Γ(Υ(1S) l + l ) Υ(1S) meson: simplest heavy quark bound state Γ(Υ(1S) l + l ) exp = 1.340(18) kev Γ(Υ(1S) l + l ) = 4πα2 9m 2 b ( ψ 1 (0) 2 c v [c v E 1 cv m b + dv 3 ) +... ] d v : matching constant of sub-leading b b current 1 loop: [Luke,Savage 98] ψ 1 (0) wave function of the (b b) system ψ LO 1 (0) 2 = 8m3 b α3 s 27π M Υ(1S) = 2m b + E 1 E p,lo 1 = (4m b αs)/9 2 Matthias Steinhauser Top and bottom at threshold 20

26 Γ(Υ(1S) l + l ) Υ(1S) meson: simplest heavy quark bound state Γ(Υ(1S) l + l ) exp = 1.340(18) kev Γ(Υ(1S) l + l ) = 4πα2 9m 2 b ( ψ 1 (0) 2 c v [c v E 1 cv m b + dv 3 ) +... ] d v : matching constant of sub-leading b b current 1 loop: [Luke,Savage 98] ψ 1 (0) wave function of the (b b) system ψ LO 1 (0) 2 = 8m3 b α3 s 27π M Υ(1S) = 2m b + E 1 E p,lo 1 = (4m b αs)/9 2 many building blocks necessary; most recent ones: a 3 [Smirnov,Smirnov,Steinhauser 08; Smirnov,Smirnov,Steinhauser 09; Anzai,Kiyo,Sumino 09] c v [Marquard,Piclum,Seidel,Steinhauser 14] ψ 1 (0): ultrasoft contribution [Beneke,Kiyo,Penin 07] ψ 1 (0): single- and double-potential insertions [Beneke,Kiyo,Schuller 08 13] O(ǫ) term of 2-loop 1/(m b r 2 ) pnrqcd potential [Penin,Smirnov,Steinhauser 13] NLL: [Pineda 01]; NNLL-approx: [Pineda,Signer 07];... Matthias Steinhauser Top and bottom at threshold 20

27 Result Γ(Υ(1S) l + l ) pole = 2 5 α 2 α 3 [ s mb ( 1+α s ( L)+ α s lnαs L L 2) +α 3 s ( a b2ǫ cf 134.8(1) cg lnα s ln 2 α s +( lnα s) L L L 3) ] +O(α 4 s) µ=3.5 GeV = 2 5 α 2 α 3 s mb 3 5 [ αs α 2 s +( a b2ǫ cf 134.8(1) cg ) α 3 s +O(α 4 s) ] = 2 5 α 2 α 3 s mpole b 3 5 [ ] = [1.04±0.04(α s) (µ)] kev 2 5 α 2 α 3 s = mps b [ ] 3 5 = [1.08±0.05(α s) (µ)] kev α s = ± µ 10 GeV L = ln[µ/(4m b α s /3)] [Beneke,Kiyo,Marquard,Penin,Piclum,Seidel,Steinhauser 14] Matthias Steinhauser Top and bottom at threshold 21

28 Γ(Υ(1S) l + l ): µ dependence [Beneke,Kiyo,Marquard,Penin,Piclum,Seidel,Steinhauser 14] PS Matthias Steinhauser Top and bottom at threshold 22

29 Γ(Υ(1S) l + l ): µ dependence [Beneke,Kiyo,Marquard,Penin,Piclum,Seidel,Steinhauser 14] PS Matthias Steinhauser Top and bottom at threshold 22

30 Γ(Υ(1S) l + l ): µ dependence [Beneke,Kiyo,Marquard,Penin,Piclum,Seidel,Steinhauser 14] PS Matthias Steinhauser Top and bottom at threshold 22

31 Γ(Υ(1S) l + l ): µ dependence [Beneke,Kiyo,Marquard,Penin,Piclum,Seidel,Steinhauser 14] PS NNNLO NNLO NLO LO Matthias Steinhauser Top and bottom at threshold 22

32 Γ(Υ(1S) l + l ): µ dependence [Beneke,Kiyo,Marquard,Penin,Piclum,Seidel,Steinhauser 14] PS NNNLO NNLO NLO LO Matthias Steinhauser Top and bottom at threshold 22

33 Γ(J/Ψ l + l ): µ dependence 6 5 OS NNNLO NNLO NLO LO Matthias Steinhauser Top and bottom at threshold 23

34 Γ(Υ(1S) l + l ): α s dependence PS [Beneke,Kiyo,Marquard,Penin,Piclum,Seidel,Steinhauser 14] NNLO NNNLO NLO LO Matthias Steinhauser Top and bottom at threshold 24

35 Γ(Υ(1S) l + l ): PS vs. OS [Beneke,Kiyo,Marquard,Penin,Piclum,Seidel,Steinhauser 14] PS OS NNNLO 1 NNNLO NNLO 0.6 NNLO NLO LO NLO LO PS NNLO NNNLO NLO LO OS NNLO NNNLO NLO LO Matthias Steinhauser Top and bottom at threshold 25

36 Result Γ(Υ(1S) l + l ) pole = [1.04±0.04(α s ) (µ)] kev Γ(Υ(1S) l + l ) PS = [1.08±0.05(α s ) (µ)] kev Γ(Υ(1S) l + l ) exp = [1.340±0.018] kev Matthias Steinhauser Top and bottom at threshold 26

37 Result Γ(Υ(1S) l + l ) pole = [1.04±0.04(α s ) Γ(Υ(1S) l + l ) PS = [1.08±0.05(α s ) Γ(Υ(1S) l + l ) exp = [1.340±0.018] kev third-order perturbative result is 30% too low (µ)] kev (µ)] kev Matthias Steinhauser Top and bottom at threshold 26

38 Result Γ(Υ(1S) l + l ) pole = [1.04±0.04(α s ) Γ(Υ(1S) l + l ) PS = [1.08±0.05(α s ) Γ(Υ(1S) l + l ) exp = [1.340±0.018] kev third-order perturbative result is 30% too low (µ)] kev (µ)] kev effects from finite m c? add NLO and NNLO m c terms to Γ(Υ(1S) l + l ) [Hoang 00; Beneke,Maier,Piclum,Rauh 14] decouple charm: α (4) s α (3) s [Pineda] Γ(Υ(1S) l + l ) pole = [1.20±0.06(α s ) (µ)] kev Matthias Steinhauser Top and bottom at threshold 26

39 Result Γ(Υ(1S) l + l ) pole = [1.04±0.04(α s ) Γ(Υ(1S) l + l ) PS = [1.08±0.05(α s ) Γ(Υ(1S) l + l ) exp = [1.340±0.018] kev third-order perturbative result is 30% too low (µ)] kev (µ)] kev effects from finite m c? add NLO and NNLO m c terms to Γ(Υ(1S) l + l ) [Hoang 00; Beneke,Maier,Piclum,Rauh 14] decouple charm: α (4) s α (3) s [Pineda] Γ(Υ(1S) l + l ) pole = [1.20±0.06(α s ) (µ)] kev sizeable non-perturbative contribution? Matthias Steinhauser Top and bottom at threshold 26

40 Result Γ(Υ(1S) l + l ) pole = [1.04±0.04(α s ) Γ(Υ(1S) l + l ) PS = [1.08±0.05(α s ) Γ(Υ(1S) l + l ) exp = [1.340±0.018] kev third-order perturbative result is 30% too low (µ)] kev (µ)] kev effects from finite m c? add NLO and NNLO m c terms to Γ(Υ(1S) l + l ) [Hoang 00; Beneke,Maier,Piclum,Rauh 14] decouple charm: α (4) s α (3) s [Pineda] Γ(Υ(1S) l + l ) pole = [1.20±0.06(α s ) (µ)] kev sizeable non-perturbative contribution? Γ(Υ(1S) l + l ) HPQCD = [1.19±0.11] kev lattice [HPQCD 14] Matthias Steinhauser Top and bottom at threshold 26

41 Result Γ(Υ(1S) l + l ) pole = [1.04±0.04(α s ) Γ(Υ(1S) l + l ) PS = [1.08±0.05(α s ) Γ(Υ(1S) l + l ) exp = [1.340±0.018] kev third-order perturbative result is 30% too low (µ)] kev (µ)] kev effects from finite m c? add NLO and NNLO m c terms to Γ(Υ(1S) l + l ) [Hoang 00; Beneke,Maier,Piclum,Rauh 14] decouple charm: α (4) s α (3) s [Pineda] Γ(Υ(1S) l + l ) pole = [1.20±0.06(α s ) (µ)] kev sizeable non-perturbative contribution? Γ(Υ(1S) l + l ) HPQCD = [1.19±0.11] kev lattice [HPQCD 14] Γ(Υ(1S) l + l ) = [ ] kev [Shen,Wu,Ma,Bi,Wang 15] apply principle of maximum conformality (PMC) to N 3 LO result Γ(Υ(1S) l + l ) r 1 αs(q 3 1 )+r 2 αs(q 4 2 )+r 3 αs(q 5 3 )+r 4 αs(q 6 4 ) with Q GeV, Q GeV, Q 3 = Q GeV Matthias Steinhauser Top and bottom at threshold 26

42 Non-perturbative contribution 1. δ np ψ 1 (0) 2 = ψ LO 1 (0) π 2 K [Leutwyler 81,Voloshin 82] K = αs π G2 m 4 b (αscf)6 αs π G2 = GeV 4 δ np Γ ll (Υ(1S)) = 1.67 pole /2.20 PS kev [α s (3.5 GeV) 0.24]? value of αs π G2? scale of α s? dimension-6 condensate contribution [Pineda 97] Γ(Υ(1S) l + l ) Γ(Υ(1S) l + l ) 0.3 kev exp pert.n 3 LO Matthias Steinhauser Top and bottom at threshold 27

43 Non-perturbative contribution 1. δ np ψ 1 (0) 2 = ψ LO 1 (0) π 2 K [Leutwyler 81,Voloshin 82] K = αs π G2 m 4 b (αscf)6 2. M Υ(1S) = 2m b + E p π2 425 m b(α s C F ) 2 K use PS scheme perturbation theory converges [Beneke,Kiyo,Schuller 05] δ np Γ ll (Υ(1S)) = 4α2 α s δm np Υ(1S) [ (α s)±0.42(m b ) (µ)±0.12(m c)] kev? m MS b (charm effects [Hoang 00]) = GeV GeV δ np Γ ll (Υ(1S)) 0.3 kev α s = ±0.001, m b = 4.163±0.016 GeV, 3 µ 10 GeV [PDG] [Chetyrkin et al. 09] Matthias Steinhauser Top and bottom at threshold 27

44 Non-perturbative contribution 1. δ np ψ 1 (0) 2 = ψ LO 1 (0) π 2 K [Leutwyler 81,Voloshin 82] K = αs π G2 m 4 b (αscf)6 2. M Υ(1S) = 2m b + E p π2 425 m b(α s C F ) 2 K use PS scheme perturbation theory converges [Beneke,Kiyo,Schuller 05] δ np Γ ll (Υ(1S)) = 4α2 α s δm np Υ(1S) [ (α s)±0.42(m b ) (µ)±0.12(m c)] kev? m MS b (charm effects [Hoang 00]) = GeV GeV δ np Γ ll (Υ(1S)) 0.3 kev Summary: perturbation theory: solid prediction non-perturbative contribution: unclear Matthias Steinhauser Top and bottom at threshold 27

45 Matthias Steinhauser Top and bottom at threshold 28

46 Positronium hyperfine splitting ν = E (e + e ) E (e + e ) precise test of QED Experiment: (1 6) GHz [Mills et al ] (74) GHz [Ritter et al. 84] 2.7σ (16) stat. (13) syst. GHz [Ishida et al. 13] ( ) Zeeman HFS 1+4q2 1 q B material effects? non-uniform B field?... Matthias Steinhauser Top and bottom at threshold 29

47 Comparison experiment theory Matthias Steinhauser Top and bottom at threshold 30

48 Theory ν LO = [ 1 + ] 1 3 sct 4 ann α 4 me 2 H HFS S 2 me 2 Higher orders { ν = ν LO 1 α ( 32 π ln 2 ( ) α 2 [ π ] 56 ζ(3) 3 2 ( ) α 3 } + D π ( π2 + α 3 π ln2 α+ ) 5 14 α2 lnα π2 ) ( ln 2 ln 2 ) α 3 π lnα Matthias Steinhauser Top and bottom at threshold 31

49 Theory ν LO = [ 1 + ] 1 3 sct 4 ann α 4 me 2 H HFS S 2 me 2 Higher orders ν = ν LO { 1 α π + ( ) α 2 [ 1367 π ] ζ(3) 3 2 ( ) α 3 } + D π ( ) 7 ln α2 lnα ( π2 + α 3 π ln2 α π2 ) ( ln 2 ln 2 ) α 3 π lnα [Pirenne 47] Matthias Steinhauser Top and bottom at threshold 31

50 Theory ν LO = [ 1 + ] 1 3 sct 4 ann α 4 me 2 H HFS S 2 me 2 Higher orders ν = ν LO { 1 α π + ( ) α 2 [ 1367 π ] ζ(3) 3 2 ( ) α 3 } + D π ( ) 7 ln α2 lnα ( π2 + α 3 π ln2 α π2 ) ( ln 2 ln 2 ) α 3 π lnα [Brodsky,Erickson 66;... ; Hoang,Labelle,Zebarjad 97; Czanecki,Melnikov,Yelkhovsky 99] Matthias Steinhauser Top and bottom at threshold 31

51 Theory ν LO = [ 1 + ] 1 3 sct 4 ann α 4 me 2 H HFS S 2 me 2 Higher orders ν = ν LO { 1 α π + ( ) α 2 [ 1367 π ] ζ(3) 3 2 ( ) α 3 } + D π ( ) 7 ln α2 lnα ( π2 + α 3 π ln2 α π2 ) ( ln 2 ln 2 ) α 3 π lnα [Karshenboim 93] Matthias Steinhauser Top and bottom at threshold 31

52 Theory ν LO = [ 1 + ] 1 3 sct 4 ann α 4 me 2 H HFS S 2 me 2 Higher orders ν = ν LO { 1 α π + ( ) α 2 [ 1367 π ] ζ(3) 3 2 ( ) α 3 } + D π ( ) 7 ln α2 lnα ( π2 + α 3 π ln2 α π2 ) ( ln 2 ln 2 ) α 3 π lnα [Kniehl,Penin 00; Hill 01; Melnikov,Yelkhovsky 01] Matthias Steinhauser Top and bottom at threshold 31

53 Theory ν LO = [ 1 + ] 1 3 sct 4 ann α 4 me 2 H HFS S 2 me 2 Higher orders ν = ν LO { 1 α π ( ) α 2 [ 1367 π ] ( ) 7 ln α2 lnα ( π ) 84 π2 ln ζ(3) 3 ( α 3 2 π ln2 α ) α 3 7 ln 2 π lnα ( ) α 3 } ( ) α 3 + D = (41) GHz+ ν LO D π π error estimate: D from muonium atom [Nio,Kinoshita 97] Matthias Steinhauser Top and bottom at threshold 31

54 Theory ν LO = [ 1 + ] 1 3 sct 4 ann α 4 me 2 H HFS S 2 me 2 Higher orders { ν = ν LO 1 α ( 32 π ln 2 ( ) α 2 [ π ] 56 ζ(3) 3 2 ( ) α 3 } + D π ( π2 + α 3 π ln2 α+ ) 5 14 α2 lnα π2 ) ( ln 2 ln 2 ) α 3 π lnα [Adkins,Fell 14] light-by-light 2γ exchange contribution to D: tiny Matthias Steinhauser Top and bottom at threshold 31

55 Theory ν LO = [ 1 + ] 1 3 sct 4 ann α 4 me 2 H HFS S 2 Higher orders ν = ν LO { 1 α π + ( ) α 2 [ 1367 π ] ζ(3) 3 2 ( ) α 3 } + D π m 2 e ( ) 7 ln α2 lnα ( π2 + α 3 π ln2 α+ aim: 1-γ annihilation contribution to D at O(α 6 m e ): 1-γ annihilation contribution 32% non-annihilation 47% π2 ) ( ln 2 ln 2 ) α 3 π lnα Matthias Steinhauser Top and bottom at threshold 31

56 D 1 γ ann : 1-γ annihilation contribution [Baker,Marquard,Penin,Piclum,Steinhauser 14] pnrqed dimensional regularization QED part of c v Π γγ (q 2 = 4m 2 e) computed up to 3 loops Matthias Steinhauser Top and bottom at threshold 32

57 D 1 γ ann : result D 1 γ ann = 84.8 ± 0.5 [Baker,Marquard,Penin,Piclum,Steinhauser 14] ν th = GHz = (22) GHz error estimate: size of the evaluated 1-γ annihilation contribution moves closer to [Ishida et al. 13] ν exp = (16) stat. (13) syst. GHz Matthias Steinhauser Top and bottom at threshold 33

58 D 1 γ ann : result D 1 γ ann = 84.8 ± 0.5 [Baker,Marquard,Penin,Piclum,Steinhauser 14] ν th = GHz = (22) GHz error estimate: size of the evaluated 1-γ annihilation contribution moves closer to [Ishida et al. 13] ν exp = (16) stat. (13) syst. GHz 95% from ultra-soft contribution: D 1 γ,us ann 3π2 7 δus o Assumption: also nonannihilation correction is dominated by ultra-soft contribution: D 1 γ,us sct 4π2 7 δus o 106 D tot 191 ν th = GHz agreement within 1σ with [Ishida et al. 13] Matthias Steinhauser Top and bottom at threshold 33

59 Conclusions a 3, c v to 3 loops Γ(Υ(1S) l + l ) to NNNLO first NNNLO calculation to physical quantity in quarkonium physics positronium HFS: 1-γ annihilation too(α 7 m e ) Coming soon: e + e t t at NNNLO Matthias Steinhauser Top and bottom at threshold 34

High order corrections in theory of heavy quarkonium

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