News from NRQCD. Matthias Steinhauser TTP Karlsruhe Dresden, July 3,

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1 News from NRQCD Matthias Steinhauser TTP Karlsruhe Dresden, July 3, 2014 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 NNNLO positronium HFS Summary Matthias Steinhauser News from NRQCD 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] Γ(Υ(1S) l + l ), NNNLO σ(e + e NLO t t), NNNLO R NNLO LO NNNLO NNNLO c3 0 NNLL [Pineda,Signer 07; Hoang,Stahlhofen 13,... ] Comparison of pqcd and LQCD s [Beneke,Kiyo,Schuller 08] [Necco,Sommer 01,Pineda 02,Sumino 05,..., Brambilla,Vairo,Garcia i Tormo,Soto 09, Donnellan,Knechtli,Leder,Sommer 10] Matthias Steinhauser News from NRQCD 3

4 Framework: potential NRQCD scales: mass, m: hard momentum, mv: soft energy, mv 2 : ultrasoft Λ QCD potential quarks: ultrasoft gluons: { E p mv 2 p mv { E k mv 2 k mv 2 1 E p p 2 2m 1 E 2 k k 2 QCD NRQCD pnrqcd (potential non-relativistic QCD) integrate out hard scale m from QCD [Caswell,Lepage 86; integrate out all scales from NRQCD except potential quarks and ultrasoft gluons Bodwin,Braaten,Lepage 95] [Beneke,Smirnov 97; Pineda,Soto 98; Brambilla,Pineda,Soto,Vairo 00] alternative formulation: velocity NRQCD (vnrqcd) [Luke,Manohar,Rothstein 00; Hoang,Stewart 03] Matthias Steinhauser News from NRQCD 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 q = p p q Matthias Steinhauser News from NRQCD 5

6 Static potential Matthias Steinhauser News from NRQCD 6

7 Static potential V C [ = 4πC Fα s ( q ) q 2 1 [Appelquist,Politzer 75,Susskind 77] Matthias Steinhauser News from NRQCD 7

8 Static potential V C [ = 4πC Fα s ( q ) 1+ α s( q ) q 2 4π a 1 [Appelquist,Politzer 75,Susskind 77] [Fischler 77;Biloire 80] Matthias Steinhauser News from NRQCD 7

9 Static potential V C [ = 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 News from NRQCD 7

10 Static potential V C [ = 4πC Fα s ( q ) 1+ α ( ) s( q ) q 2 4π a αs ( q ) a 2 4π ( ) αs ( q ) 3 ) ] + (a 3 + 8π 2 3A µ2 C ln + 4π 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 News from NRQCD 7

11 Static potential V C [ = 4πC Fα s ( q ) 1+ α ( ) s( q ) q 2 4π a αs ( q ) a 2 4π ( ) αs ( q ) 3 ( ( )) ] 1 + a 3 + 8π 2 CA 3 µ2 + ln + 4π 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 News from NRQCD 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 News from NRQCD 8

13 Matthias Steinhauser News from NRQCD 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 : Zv = 1+O(α 2 s) [Melnikov,v.Ritbergen 00] [Marquard,Mihaila,Piclum,Steinhauser 07] Matthias Steinhauser News from NRQCD 10

15 c v c v = 1+ αs π c(1) v c (1) v c (2) v c (3),n l v c (3),n h v c (3) v [Källen, Sarby 55] + ( α s ) 2 (2) π c v [Czarnecki,Melnikov 97; Beneke,Signer,Smirnov 97] [Marquard,Piclum,Seidel,Steinhauser 06] [Marquard,Piclum,Seidel,Steinhauser 08] [Marquard,Piclum,Seidel,Steinhauser 14] + ( α s ) 3 (3) π c v +O(αs) 4 (a) (b) (c) (d) (e) (f) (g) (h) massive vertices on-shell quarks: q 2 1 = q2 2 = M2 Q (q 1 + q 2 ) 2 = 4M 2 Q (a) (b) (c) (d) Matthias Steinhauser News from NRQCD 11

16 e + e t threshold NNLO LO NNNLO R 0.8 NNNLO c NLO 0.2 Only c (3),n l v s included [Beneke,Kiyo,Schuller 08] Matthias Steinhauser News from NRQCD 12

17 Some technical details... automatic generation of Feynman diagrams, apply projector, reduce scalar products in numerator,... (many) scalar integrals reduction to 100 master integrals withcrusher [Marquard,Seidel] compute MIs withfiesta [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 in quadrature multiply final uncertainty by factor 5 ( 5σ ) ) Matthias Steinhauser News from NRQCD 13

18 Checks finiteness n l contribution gauge parameter independence (ξ 1 ) change basis of MIs coefficient default basis alternative basis [CF 3] c FFF 36.55(0.11) 36.61(2.93) [CF 2C A] c FFA (0.17) (2.91) [C F CA 2] 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) (factor 5 not yet included) Matthias Steinhauser News from NRQCD 14

19 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 } + C F Tn h 1 60ǫ +O(α 4 s) 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% L µ = lnµ 2 /m 2 Z 2 Γ v = c v Z2 Z 1 v Γ v Matthias Steinhauser News from NRQCD 15

20 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 News from NRQCD 16

21 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 News from NRQCD 17

22 Application: height of σ(e + e t 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 En = Nc Z n mQ 2 E Z ( ) ] n (E+i0) 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) Matthias Steinhauser News from NRQCD 17

23 Γ(Υ(1S) l + l ) Matthias Steinhauser News from NRQCD 18

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 ) +... ] d v : matching constant of sub-leading b b current 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: 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 [Peinin,Smirnov,Steinhauser 13] NLL: [Pineda 01]; NNLL-approx: [Pineda,Signer 07];... Matthias Steinhauser News from NRQCD 19

25 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 L = ln[µ/(m b C F α s )], C F = 4/3 Matthias Steinhauser News from NRQCD 20

26 Γ(Υ(1S) l + l ): µ dependence PS Matthias Steinhauser News from NRQCD 21

27 Γ(Υ(1S) l + l ): µ dependence PS Matthias Steinhauser News from NRQCD 21

28 Γ(Υ(1S) l + l ): µ dependence PS Matthias Steinhauser News from NRQCD 21

29 Γ(Υ(1S) l + l ): µ dependence PS NNNLO NNLO NLO LO Matthias Steinhauser News from NRQCD 21

30 Γ(Υ(1S) l + l ): µ dependence PS NNNLO NNLO NLO LO Matthias Steinhauser News from NRQCD 21

31 Γ(Υ(1S) l + l ): α s dependence PS NNLO NNNLO NLO LO Matthias Steinhauser News from NRQCD 22

32 Γ(Υ(1S) l + l ): PS vs. OS 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 News from NRQCD 23

33 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 third-order perturbative result is 30% too low possible explanation: sizeable non-perturbative contribution Matthias Steinhauser News from NRQCD 24

34 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? value of αs π G2? scale of α s? dimension-6 condensate contribution [Pineda 97] Matthias Steinhauser News from NRQCD 25

35 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] charm effects [Hoang 00] 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 = GeV GeV δ np Γ ll (Υ(1S)) 0.3 kev Matthias Steinhauser News from NRQCD 25

36 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] charm effects [Hoang 00] 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 = GeV GeV δ np Γ ll (Υ(1S)) 0.3 kev Summary: perturbation theory: solid prediction non-perturbative contribution: unclear Matthias Steinhauser News from NRQCD 25

37 Matthias Steinhauser News from NRQCD 26

38 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 non-thermalized positronium? material effects? non-uniform B field?... Matthias Steinhauser News from NRQCD 27

39 Comparison experiment theory from [Ishida et al. 13] Matthias Steinhauser News from NRQCD 28

40 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 News from NRQCD 29

41 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 News from NRQCD 29

42 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 News from NRQCD 29

43 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 News from NRQCD 29

44 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 News from NRQCD 29

45 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 News from NRQCD 29

46 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 News from NRQCD 29

47 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 News from NRQCD 29

48 Dann 1 γ : 1-γ annihilation contribution pnrqed dimensional regularization QED part of c v Π γγ (q 2 = 4m 2 e) computed up to 3 loops...+δ us π c QED v δ us : ultra-soft contribution [Marcu 11]; QCD: [Beneke,Kiyo,Penin 07] Matthias Steinhauser News from NRQCD 30

49 D 1 γ ann : result D 1 γ ann = 84.8±0.5 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 News from NRQCD 31

50 D 1 γ ann : result D 1 γ ann = 84.8±0.5 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 non-annihilation 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 News from NRQCD 31

51 Conclusions a 3, c v to 3 loops Γ(Υ(1S) l + l ) to NNNLO positronium HFS: 1 γ annihilation to O(α 7 m e ) Coming soon: e + e t t at NNNLO Matthias Steinhauser News from NRQCD 32