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 www.kit.edu
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
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 0.4 0.2 349 350 351 352 353 354 R 1.4 1.2 0.8 0.6 1 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
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
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 2 + 2 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
Static potential Matthias Steinhauser News from NRQCD 6
Static potential V C [ = 4πC Fα s ( q ) q 2 1 [Appelquist,Politzer 75,Susskind 77] Matthias Steinhauser News from NRQCD 7
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
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
Static potential V C [ = 4πC Fα s ( q ) 1+ α ( ) s( q ) q 2 4π a αs ( q ) 2 1 + 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
Static potential V C [ = 4πC Fα s ( q ) 1+ α ( ) s( q ) q 2 4π a αs ( q ) 2 1 + 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
Numerical results V C( q ) = 4πCFαs( q ) [ 1+ αs q 2 π (2.5833 0.2778nl) ( ) 2 αs ( ) + 28.5468 4.1471nl + 0.0772n 2 l + π ( αs π ) 3 ( ) 209.884(1) 51.4048nl + 2.9061n 2 l 0.0214n 3 l + ] n l α (n l) s 1 loop 2 loop 3 loop charm 3 0.40 0.2228 0.2723 0.1677 bottom 4 0.25 0.1172 0.08354 0.02489 top 5 0.15 0.05703 0.02220 0.002485 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
Matthias Steinhauser News from NRQCD 9
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
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
e + e t t @ threshold 1.4 1.2 1 NNLO LO NNNLO R 0.8 NNNLO c3 0 0.6 0.4 NLO 0.2 Only c (3),n l v 349 350 351 352 353 354 s included [Beneke,Kiyo,Schuller 08] Matthias Steinhauser News from NRQCD 12
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 0.411236(3) ǫ 2 + 3.4860(1) ǫ + 34.520(2)+339.68(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
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 188.10(0.17) 188.04(2.91) [C F CA 2] c FAA 97.81(0.08) 97.76(2.05) c (3) v (n l = 4) 1621.7(0.4) 1621(23) c (3) v (n l = 5) 1508.4(0.4) 1507(23) (factor 5 not yet included) Matthias Steinhauser News from NRQCD 14
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 2 5 43 F 144ǫ + 2 144 1 2 ln 2+ 5 ) ] 1 48 Lµ ǫ [ ( 1 113 + C F C A 864ǫ + 2 324 + 1 ) ] 5 1 ln 2+ 4 32 Lµ ǫ [ + C 2 A 1 ( 2 16ǫ + 2 27 + 1 ) ] 1 1 ln 2+ 4 24 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
Results for c v [Marquard,Piclum,Seidel,Steinhauser 14] c v (µ = m Q ) = 1 2.67 α s π +[ 44.55+0.41n l] ( αs ) 2 π + [ ] ( 2091(2)+120.66n 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
Application: height of σ(e + e t t) 1.4 LO, NLO, NNLO 1.2 1.0 R 0.8 0.6 0.4 0.2 0.0 341 342 343 344 345 346 347 s (GeV) [Hoang,Beneke,Melnikov,Nagano,Ota,Penin,Pivovarov,Signer,Smirnov,Sumino,Teubner,Yakovlev,Yelkhovsky 00] Matthias Steinhauser News from NRQCD 17
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 +... 2mQ 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 ) 1.2 1.1 1 0.9 0.8 0.7 NNLO NNNLO NNNLO (ferm.) NLO LO 0.6 20 40 60 80 100 120 140 160 µ (GeV) Matthias Steinhauser News from NRQCD 17
Γ(Υ(1S) l + l ) Matthias Steinhauser News from NRQCD 18
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
Result Γ(Υ(1S) l + l ) pole = 2 5 α 2 α 3 [ s mb ( 1+α s ( 2.003+3.979 L)+ α 2 3 5 s 9.05 7.44 lnαs 13.95 L+10.55 L 2) +α 3 s ( 0.91+4.78 a3 + 22.07 b2ǫ + 30.22 cf 134.8(1) cg 14.33 lnα s 17.36 ln 2 α s +(62.08 49.32 lnα s) L 55.08 L 2 + 23.33 L 3) ] +O(α 4 s) µ=3.5 GeV = 2 5 α 2 α 3 s mb 3 5 [ 1+1.166αs + 15.2α 2 s +(66.5+4.8 a3 + 22.1 b2ǫ + 30.2 cf 134.8(1) cg ) α 3 s +O(α 4 s) ] = 2 5 α 2 α 3 s mpole b 3 5 [1+0.28+0.88 0.16] = [1.04±0.04(α s) +0.02 0.15 (µ)] kev 2 5 α 2 α 3 s = mps b [1+0.37+0.95 0.04] 3 5 = [1.08±0.05(α s) +0.01 0.20 (µ)] kev L = ln[µ/(m b C F α s )], C F = 4/3 Matthias Steinhauser News from NRQCD 20
Γ(Υ(1S) l + l ): µ dependence 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 PS 2 4 6 8 10 Matthias Steinhauser News from NRQCD 21
Γ(Υ(1S) l + l ): µ dependence 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 PS 2 4 6 8 10 Matthias Steinhauser News from NRQCD 21
Γ(Υ(1S) l + l ): µ dependence 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 PS 2 4 6 8 10 Matthias Steinhauser News from NRQCD 21
Γ(Υ(1S) l + l ): µ dependence 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 PS NNNLO NNLO NLO LO 2 4 6 8 10 Matthias Steinhauser News from NRQCD 21
Γ(Υ(1S) l + l ): µ dependence 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 PS NNNLO NNLO NLO LO 2 4 6 8 10 Matthias Steinhauser News from NRQCD 21
Γ(Υ(1S) l + l ): α s dependence 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 PS NNLO NNNLO NLO LO 0.2 0 0.11 0.115 0.12 0.125 0.13 Matthias Steinhauser News from NRQCD 22
Γ(Υ(1S) l + l ): PS vs. OS 1.6 1.4 PS 1.6 1.4 OS 1.2 1.2 1 NNNLO 1 NNNLO 0.8 0.8 0.6 NNLO 0.6 NNLO 0.4 0.2 0 NLO LO 2 4 6 8 10 0.4 0.2 0 NLO LO 2 4 6 8 10 2 1.8 PS NNLO 1.6 1.4 1.2 NNNLO 1 0.8 NLO 0.6 0.4 LO 0.2 0 0.11 0.115 0.12 0.125 0.13 2 1.8 OS NNLO 1.6 1.4 1.2 NNNLO 1 0.8 NLO 0.6 0.4 LO 0.2 0 0.11 0.115 0.12 0.125 0.13 Matthias Steinhauser News from NRQCD 23
Result Γ(Υ(1S) l + l ) pole = [1.04±0.04(α s ) +0.02 0.15 (µ)] kev Γ(Υ(1S) l + l ) PS = [1.08±0.05(α s ) +0.01 0.20 (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
Non-perturbative contribution 1. δ np ψ 1 (0) 2 = ψ LO 1 (0) 2 17.54π 2 K [Leutwyler 81,Voloshin 82] K = αs π G2 m 4 b (αscf)6 αs π G2 = 0.012 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
Non-perturbative contribution 1. δ np ψ 1 (0) 2 = ψ LO 1 (0) 2 17.54π 2 K [Leutwyler 81,Voloshin 82] K = αs π G2 m 4 b (αscf)6 2. M Υ(1S) = 2m b + E p 1 + 624π2 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 9 17.54 425 3744 δm np Υ(1S) [1.28 +0.17 0.18 (α s)±0.42(m b ) +0.20 0.57 (µ)±0.12(m c)] kev? m MS b = 4.163 GeV 4.203 GeV δ np Γ ll (Υ(1S)) 0.3 kev Matthias Steinhauser News from NRQCD 25
Non-perturbative contribution 1. δ np ψ 1 (0) 2 = ψ LO 1 (0) 2 17.54π 2 K [Leutwyler 81,Voloshin 82] K = αs π G2 m 4 b (αscf)6 2. M Υ(1S) = 2m b + E p 1 + 624π2 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 9 17.54 425 3744 δm np Υ(1S) [1.28 +0.17 0.18 (α s)±0.42(m b ) +0.20 0.57 (µ)±0.12(m c)] kev? m MS b = 4.163 GeV 4.203 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
Matthias Steinhauser News from NRQCD 26
Positronium hyperfine splitting ν = E (e + e ) E (e + e ) precise test of QED Experiment: 203.387 5(1 6) GHz [Mills et al. 75 83] 203.389 10(74) GHz [Ritter et al. 84] 2.7σ 203.394 2(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
Comparison experiment theory from [Ishida et al. 13] Matthias Steinhauser News from NRQCD 28
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 21 + 6 7 ln 2 ( ) α 2 [ 1367 + π 378 5197 159 ] 56 ζ(3) 3 2 ( ) α 3 } + D π ( 6 2016 π2 + α 3 π ln2 α+ ) 5 14 α2 lnα 7 + 221 84 π2 ) ( 62 15 + 68 7 ln 2 ln 2 ) α 3 π lnα Matthias Steinhauser News from NRQCD 29
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 ] 159 56 ζ(3) 3 2 ( ) α 3 } + D π ( 32 21 + 6 ) 7 ln 2 5 14 α2 lnα ( 6 378 5197 2016 π2 + α 3 π ln2 α+ 7 + 221 84 π2 ) ( 62 15 + 68 7 ln 2 ln 2 ) α 3 π lnα [Pirenne 47] Matthias Steinhauser News from NRQCD 29
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 ] 159 56 ζ(3) 3 2 ( ) α 3 } + D π ( 32 21 + 6 ) 7 ln 2 5 14 α2 lnα ( 6 378 5197 2016 π2 + α 3 π ln2 α+ 7 + 221 84 π2 ) ( 62 15 + 68 7 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
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 ] 159 56 ζ(3) 3 2 ( ) α 3 } + D π ( 32 21 + 6 ) 7 ln 2 5 14 α2 lnα ( 6 378 5197 2016 π2 + α 3 π ln2 α+ 7 + 221 84 π2 ) ( 62 15 + 68 7 ln 2 ln 2 ) α 3 π lnα [Karshenboim 93] Matthias Steinhauser News from NRQCD 29
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 ] 159 56 ζ(3) 3 2 ( ) α 3 } + D π ( 32 21 + 6 ) 7 ln 2 5 14 α2 lnα ( 6 378 5197 2016 π2 + α 3 π ln2 α+ 7 + 221 84 π2 ) ( 62 15 + 68 7 ln 2 ln 2 ) α 3 π lnα [Kniehl,Penin 00; Hill 01; Melnikov,Yelkhovsky 01] Matthias Steinhauser News from NRQCD 29
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 ] ( 32 21 + 6 ) 7 ln 2 5 14 α2 lnα ( 6 + 378 5197 2016 π2 + 7 + 221 ) 84 π2 ln 2 159 56 ζ(3) 3 ( α 3 2 π ln2 α+ 62 15 + 68 ) α 3 7 ln 2 π lnα ( ) α 3 } ( ) α 3 + D = 203.391 69(41) GHz+ ν LO D π π error estimate: D from muonium atom [Nio,Kinoshita 97] Matthias Steinhauser News from NRQCD 29
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 21 + 6 7 ln 2 ( ) α 2 [ 1367 + π 378 5197 159 ] 56 ζ(3) 3 2 ( ) α 3 } + D π ( 6 2016 π2 + α 3 π ln2 α+ ) 5 14 α2 lnα 7 + 221 84 π2 ) ( 62 15 + 68 7 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
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 ] 159 56 ζ(3) 3 2 ( ) α 3 } + D π m 2 e ( 32 21 + 6 ) 7 ln 2 5 14 α2 lnα ( 6 378 5197 2016 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% 7 + 221 84 π2 ) ( 62 15 + 68 7 ln 2 ln 2 ) α 3 π lnα Matthias Steinhauser News from NRQCD 29
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 2 +...+2c QED v δ us : ultra-soft contribution [Marcu 11]; QCD: [Beneke,Kiyo,Penin 07] Matthias Steinhauser News from NRQCD 30
D 1 γ ann : result D 1 γ ann = 84.8±0.5 ν th = 203.391 69+0.000 217 GHz = 203.391 91(22) GHz error estimate: size of the evaluated 1-γ annihilation contribution moves closer to [Ishida et al. 13] ν exp = 203.394 2(16) stat. (13) syst. GHz Matthias Steinhauser News from NRQCD 31
D 1 γ ann : result D 1 γ ann = 84.8±0.5 ν th = 203.391 69+0.000 217 GHz = 203.391 91(22) GHz error estimate: size of the evaluated 1-γ annihilation contribution moves closer to [Ishida et al. 13] ν exp = 203.394 2(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 = 203.392 11 GHz agreement within 1σ with [Ishida et al. 13] Matthias Steinhauser News from NRQCD 31
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