pnrqcd determination of E1 radiative transitions

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1 pnrqcd determination of E radiative transitions Sebastian Steinbeißer Group T3f Department of Physics Technical University Munich IMPRS talk In collaboration with: Nora Brambilla, Antonio Vairo and Jorge Segovia pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) /3

2 Outline Motivation & introduction Potential non-relativistic QCD (pnrqcd) 3 Decay widths of electric dipole transitions 4 Numerical results 5 Summary & outlook pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) /3

3 Motivation & introduction Most of the observable matter is bound state matter (atoms, molecules, hadrons) and a lot of it is non-relativistic (atoms, molecules, heavy quarkonium like cc or bb). Relevant scales in non-relativistic systems: E p m V mv and p r mv If v, then m mv mv (hierarchy of scales EFT) Hierarchy of scales for heavy quarkonia: m (hard scale) QCD ( NRQCD) p mv (soft scale) NRQCD ( pnrqcd) E mv (ultra soft scale) pnrqcd pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 3/3

4 Motivation & introduction EFTs are the tools of choice for systematic computations. In QCD: we have a variety of EFTs depending on physical processes (χpt, HQET, SCET, NRQCD, pnrqcd, etc.). we want to focus on a heavy quark-antiquark bound state (heavy quarkonium) especially bottomonium (bb, where m m b (m b ) 4.8 GeV and v. ). Non-relativistic QCD (NRQCD) : arises from QCD by integrating out m expansion in the Lagrangian. m hierarchy of scales: m p mv, ΛQCD. d.o.f.: heavy quark and antiquark, light quarks, soft gluons. Potential NRQCD (pnrqcd) : arises from NRQCD by integrating out p mv multipole expansion in r the Lagrangian. hierarchy of scales: m p mv E mv. d.o.f.: quark-antiquark color singlet S and color octet O states, light quarks, ultra-soft gluons. Caswell and Lepage. Phys. Lett. B67 (986); Bodwin, Braaten, and Lepage. Phys. Rev. D5 (995). arxiv: [hep-ph]. Pineda and Soto. Nucl. Phys. Proc. Suppl. 64 (998). arxiv: [hep-ph]; Brambilla et al. Nucl. Phys. B566 (). arxiv: 9974 [hep-ph]. pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 4/3

5 º½º Ë ÊÁÈÌÁÇÆ Ç ÉÍ ÊÃÇÆÁÍÅ ËÌ Ì Ë Æ ÈÊÇ ËË Ë ¾ Motivation & introduction γ k = (kγ, k) H PH = (MH, ) PH = ( kγ + M H, k) n s+ l J n s + l J + γ ÙÖ º½ Ã Ò Ñ Ø Ó Ø ØÖ Ò Ø ÓÒ H H Û Ö Ø ÑÓÑ ÒØ Ö Ú Ò Ò Ø ÒØ Ö Ó Ñ Ö Ñ Why to study radiative decays? H They are one of the possible ways to access heavy quarkonium states. Û Ø ê In i λparticular, they are one of the simplest transitions to be measured for quarkonia h ij below ns+ (λ)hkl LJ n (λ) = S+ LJ open-flavor (δik δ jl + δ il δ jk threshold. ) 3 δij δ kl, º½ µ λ ( H ijk n S+ (λ)hlmn LJ n (λ) = They provide a probe of S+ LJ (3!) δthe il δ jm δ kn internal 3 5 δij δ kl δ mn structure of hadrons. λ ) +(i j k), (l m n), º½ µ ˆǫ i (σ)ˆǫ j (σ) = δ ij ˆk iˆk j. º½ µ σ Γ () ÓÖÖ Ø ÓÒ E = 4 ˆ ØÓ Ø Ð Ò 9 α e/me ÓÖ Ö Û Ú Qk 3 γ dr r R n (r) r R ÙÒØ ÓÒ Ö Ó Ø Ò Ý Ø Ò Ö ÕÙ ÒØÙÑ n (r) Ñ Ò Ð Ô ÖØÙÖ Ø ÓÒ Ø ÓÖݺ Ì ÒÐÙ Ð Ó Ö ÓÖ Ö Ó Ø Ø º º ÓÐÓÖ¹ ÓØ Ø Ø Ø º º½º¾ Ý ÑÔÐ ØÙ Ò Ý Ö Ø Γ () Ì ØÖ Ò Ø ÓÒ M = 4 ÑÔÐ ØÙ 3 e/me kγ 3 Q ÓÖ Ý Ó mõù Ö ÓÒ ÙÑ Ø Ø À Û Ø ÔÓÐ Ö Þ Ø ÓÒ λ Ò Ø l =, s = l =, s = Ö Ø Ö Ñ ÒØÓ ÓÒ Û Ø ÑÓÑ ÒØÙÑ P ÔÓÐ Ö Þ Ø ÓÒ λ ÒÓØ Ý H Ò Ô ÓØÓÒ pnrqcd Û Ø Ò Ö Ý E transitions Sebastian ( Steinbeißer ) (sebastian.steinbeisser@tum.de) 5/3

6 Motivation & introduction Experimental branching ratios - PDG 3 Mode Fraction (Γ i /Γ) χ b (P) Υ(S) + γ (.76 ±.35)% χ b (P) Υ(S) + γ (33.9 ±.)% χ b (P) Υ(S) + γ (9. ±.)% h b (P) η b (S) + γ ( )% The decay width in pnrqcd up to O(v ) corrections [ Γ n 3 P J n 3 S = Γ () E + R S= (J) kγ 6m k γ I () 5 (n n ) 6 I () 3 (n n ) ( ) ( + J(J+) ( + κ em Q ) kγ m + m ( + κ em Q ) I () (n n )+I () (n n ) I () 3 (n n ) I (k) N (nl n l ) = 3 Olive et al. Chin. Phys. C38 (4) ( ) dr r r N d R k n l R dr k nl pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 6/3 )]

7 pnrqcd - the Lagrangian at weak coupling Hierarchy of scales: m p mv E mv. At weak coupling (Λ QCD mv ) the d.o.f. are color singlet S and color octet O configurations of the heavy qq-pair. The effective Lagrangian is organized as an expansion in m (NRQCD) and r p (pnrqcd): L pnrqcd = d 3 r tr {S ( i p m + V s )S+O ( ) } id p m + V o O 4 F µν,af µν,a + n f q i i /Dq i + L + L γ i= L = d 3 r V A tr{o r g ES + h.c.} + V B tr{o r g EO + c.c.} +... L γ = ee Q d3 r V r E tr{s r E e/m S} + V r E o tr{o r E e/m O} +... Power counting: p = i r, r mv ;, P = i R, R, A µ mv ; E, B (mv ) ; E e/m, B e/m k γ ; k γ mv At strong coupling (Λ QCD mv ) one only has color singlets. pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 7/3

8 pnrqcd - the e.o.m. and its solution The singlet static potential is given by V s = C F α s (ν) r [ + k= ( ) k αs (ν) 4π ak (ν, r)] a (ν, r) = a + β ln(νe γ E r) (NLO) a (ν, r) = a + π 3 β + (4a β + β )ln(νe γ E r) + 4β ln (νe γ E r) (NNLO) The Schrödinger equation ( + V s () (r) m r ) ψ () () nlm ( r) = E n ψ () nlm ( r) with Coulombic bound state (pnrqcd at weak coupling) solutions: ψ () nlm ( r) = R nl(r)y lm (Ω r ) = N nl e ρn ρ l n L l+ n l (ρ n)y lm (Ω r ) E () n = m rc F α s (ν) n, ρ n = r na, a = m r C F α s (ν) pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 8/3

9 pnrqcd - corrections to the wave function We can organize the NNLO relativistic corrections to the wave functions 4 as: where: with: V () p δh = 4 4m 3 + V () m + V () SI m + V () SD m V () SI = V r () + () {V p, } + V () L L V () SD = V () LS L S + V () S S + V () S S V () = C F C A α s (ν) r, V () r = πc F α s (ν)δ (3) ( r), = C F α s (ν) r, V () L = C F α s (ν) r, V () 3 LS = 3C F α s (ν) r, 3 V () S = 4πC F α s (ν) 3 δ (3) ( r), V () S = C F α s (ν) 4r 3 p = i, L = r p, S = S + S, S = σ, L S = ( J L S ), S = 3(ˆr σ ) (ˆr σ ) σ σ 4 Brambilla, Pietrulewicz, and Vairo. Phys. Rev. D85 (). arxiv: 3.3. pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 9/3

10 pnrqcd - Perturbation theory Corrections to the wave functions nl () = n n n l n l n E () n n E () n = n n n l V nl n l = E n () E () n l n l E n () E () n The Coulomb green function G( r, r, E) = G l (r, r ) = n l n l V nl, nl () =... n n E n () E () n n l n l n =n E n () E () n G ( r, r ) ( ) lim E En l= l+ 4π P l(ˆr ˆr )G l (r, r ) G νl (r, r ) ν=l+ ( ) G νl (r, r ) = m r a ν 4 Rνl(ρ λ,)r νl(ρ λ,) λ ν λ E = mr C F α s λ = lim E E () n ( E H M(n) E E n () (G( r, r, E) ψ nl ) E E n = E n ( ɛ) ɛ = mr C F α s n ( ɛ) λ = n ɛ ) (E n H) ɛ ɛ pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) /3

11 pnrqcd - Perturbation theory - an Example n l O nl () = n l O (E n H) V nl ˆ = d 3 r d 3 r ψn l ( r ) O( r ) G ( r, r ) V ( r ) ψ nl ( r ) = lim N n l N 4m r nl ɛ,o(ɛ ) aλ ˆ ˆ l = s= s! (s + l + λ)(s + l + )! dr r V (r )ρ l λ,ρ l n,e ρλ, e ρn, L l + s (ρ λ, )L l+ n l (ρ n,) dr r O(r )ρ l λ,ρ l n,e ρλ, e ρ n, L l + s (ρ λ, )L l + n l (ρ n,) l m = l ˆ ˆ dω Y m l (Ω )V (Ω )Yl m (Ω ) dω Y m l (Ω )O(Ω )Yl m (Ω ) We calculate with λ = n the finite contribution, which means performing the sum in s without the pole (s n l ) We compute the divergent term of the sum using λ = n ɛ, expand in ɛ and finally pick up the finite term only (O(ɛ ) when ɛ ). pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) /3

12 The tensor potential: s-d-wave mixing V T ( r) = C F α s m 4r S 3 (ˆr) V S S (ˆr) mixes states with l =, since S (ˆr) = 3(ˆr σ ) (ˆr σ ) σ σ = 3 } 5 {{ˆr ˆr} { σ σ } diagonal part is included in the spectrum 5, perturbative treatment of off-diagonal matrix elements is derived in 6 { } l n l s J l m J S nlsjm J = 3δ J Jδ mj mj ( )J+l+s s s l J {ˆr ˆr} l s { σ σ } s using the Wigner-Eckart theorem we find n l s J m J S nlsjm J = δ J Jδ mj mj δ s sδ s ( ) J+s 3 (l + )(l + ) where ( ) l l l = { } ( ) l l l l s s J, 5 Peset, Pineda, and Stahlhofen. JHEP 5 (6). arxiv: Kiyo, Mishima, and Sumino. Phys. Lett. B76 (6). arxiv: pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) /3

13 3 3 S N O I T C E R R O C C I T S I V I T Decay widths of electric dipole transitions () n, l, s, J ; γ O E n, l, s, J; () Γ () E º º Ê Ä ÌÁÎÁËÌÁ ÇÊÊ ÌÁÇÆË () n, l, s, J ; γ O E n, l, s, J; () Γ (,) E ÌÁÇÆË n 3 P J δv r E em n 3 s S em n 3 S () n, l, s, J ; γ O E n, l, s, J; () Γ (,) E Ñ Ø ÓÖÑ ÓÖ ½ ØÖ Ò Ø ÓÒ Ø ÙÔÔ Ö Ð Ò ÓÖÝ º º A () Ò A () () n, l, s, J ; γ O E n, l, s, J; () Γ () E Ø ÐÓÛ Ö Ð Ò ØÓ ÓÒ ÙÖ º¾ È ÖØÙÖ Ø ÓÒ Ø ÓÖÝ Ò Ö Ñ Ø ÓÖÑ ÓÖ ½ ØÖ Ò Ø ÓÒ Ø ÙÔ Ò Ø ÒØ ÓÖÖ ÔÓÒ α VS α s = ØÓ d α Ö Ø s +d ÓÖ Ö α 3 s ÓÖ Ô ÖØÙÖ Ø ÓÒ () n, l, s Ø ÔÓØ ÒØ ÐØ ÓÖÝ, J ; γ O º º E n, A () l, s, J; Ò A () () Γ () Ø ÐÓÛ Ö E Ð Ò ØÓ ÜÔÖ ÓÒ ÓÖ Ö º º δ (3) (r)ðó A () n A r () ÒÓØ Ò Ò A () Ò º ÓÖÝ ÓÖ Ò Ö Ñ Ø ØÓ ÓÖÑ ÓÖ ½ ØÖ Ò Ø ÓÒ Ø ÙÔÔ Ö Ð Ò ØÙÖ Ø ÓÒ Ì d 3 kö Ð Ú ÒØ Ø ÓÖÝ º º ÓÖÖ Ø ÓÒ A () Ò ØÓ A Ø () () n ÓÙÔÐ Ò Ø, l ÐÓÛ Ö, s, J ÓÒ Ø ÒØ Ð Ò ; γ O ØÓ E n, l, s, J; ÓÒ () Γ () α eik r (π) 3 ÐÓ n VS α s = d α s k. º µ +d E α 3 () s ÓÖ Ø Ô º ½ ØÖ Ò Ø ÓÒ Ø O(/m Ø ) ÙÔÔ Ö Ú Ò Ð Ò Ò ÔÔ Ò Ü º½º Ì ÜÔÖ ÓÒ δ (3) (r)ðó n r ÒÓØ Ò pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 3/3

14 Decay widths of electric dipole transitions Γ = k γ (π) M fi = O E = ee Q ( r E e/m ) = ee Q (ê r E e/m ) k γ (π) µ= N λ λ,λ,σ M fi 4π µ ry 3 (Ω r ) Φ () n 3 P (r) = 4π R σ r n(r), Φ () 3 n 3 P (r) = (λ) 8π R n(r) σ ( r e n 3 P (λ)), Φ () 3 n 3 P (r) = (λ) 4π R n(r) σi h ij n 3 P (λ) ˆr j, Φ () n 3 S (r) = (λ) R n (r) σ e n 3 S (λ), 4π where the polarization vectors and tensors are normalized as: e n 3 S (λ) e n3 S (λ ) = e n P (λ) e n P (λ ) = e n 3 P (λ) e n3 P (λ ) = δ λλ h ij n 3 P (λ) h ji n 3 P (λ ) = δ λλ pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 4/3

15 Numerical results n f = 3, e Q = 3, α e/m = state Υ(S) χ b (P) χ b (P) χ b (P) η b (S) h b (P) exp. masses 7 [GeV] k γ = m i m f mi 39. MeV, χ b (P) Υ(S) + γ 43. MeV, χ = b (P) Υ(S) + γ 44.6 MeV, χ b (P) Υ(S) + γ MeV, h b (P) η b (S) + γ M exp. (Υ(S)) = m b + E () n (m b, α s (ν)) (Υ-scheme) a(ν) = CF mr αs (ν), κem Q C e/m αs (mb) F = C F π v (beyond NNLO) ν [GeV ] loops, nf =3 αs (ν) a(ν) [GeV ] m b [GeV ] Olive et al. Chin. Phys. C38 (4) 8 Chetyrkin, Kühn, and Steinhauser. Comput. Phys. Commun. 33 (). arxiv: 489 [hep-ph] pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 5/3

16 Relativistic corrections to the Lagrangian (χ b Υ + γ) Γ n 3 P J n 3 S = Γ () E ( ) ( + J(J+) ( + κ em [ + R S= (J) kγ 6m k γ I () 5 (n n ) 6 I () 3 (n n ) Q ) kγ m + m ( + κ em Q ) I () (n n )+I () (n n ) I () 3 (n n ) )] [kev] [kev] individual contributions: [GeV] [kev] [kev] combined effect: [GeV] st contribution st contribution pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 6/3

17 Relativistic wave function corrections (χ b Υ + γ) M [GeV - ] ν [GeV] M [GeV - ] ν [GeV] M [GeV - ] ν [GeV] MLO M NLO V () fin MLO M NNLO V () fin MLO M NNLO V () fin M NLO V () ini M NNLO V () ini M NNLO V () ini M NNLO V () ini,fin M [GeV - ] M [GeV - ] M [GeV - ] ν [GeV] ν [GeV] ν [GeV] M LO M NNLO Vr () fin M LO M NNLO p 4 ini M LO M NNLO S fin M NNLO Vr () ini M NNLO Vr () fin M NNLO p ini M NNLO p 4 fin M NNLO L ini M NNLO (S ini) M NNLO p fin M NNLO (LS ini) M NNLO (S ini) pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 7/3

18 Total matrix elements and decay widths (χ b Υ + γ) M [GeV - ] ν [GeV] Γ [kev] [kev] ν [GeV] [GeV] M LO M NNLO Γ LO Γ NNLO M NLO M NLO+NNLO Γ NLO Γ NLO+NNLO Sizable scale dependence induced by the running of α s (ν) and by the radiative corrections to the static potential. Final value taken at ν =.5 GeV ( a = mc F α s ( a ) ). Uncertainty band is fully dominated by the scale-dependence. pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 8/3

19 Final results - comparison and predictions Mode LO NLO NNLO CQM 9 χ b (P) Υ(S) + γ χ b (P) Υ(S) + γ χ b (P) Υ(S) + γ h b (P) η b (S) + γ LO and NLO results agree with the constituent quark model. Mode Fraction B = Γ i Γ [PDG] Total width Γ [kev] χ b (P) Υ(S) + γ (.76 ±.35)% ( ) 3 χ b (P) Υ(S) + γ (33.9 ±.)% χ b (P) Υ(S) + γ (9. ±.)% h b (P) η b (S) + γ ( )% Large ( 5%) uncertainties due to the scale dependence! 9 Segovia et al. Phys. Rev. D93 (6). arxiv: pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 9/3

20 The RGI approach Back to the singlet static potential: V s = C F α s (ν) r [ + k= The Schrödinger equation ( ) + V s (r) ψ () () nlm ( r) = E n ψ () nlm m ( r) r ( ) k αs (ν) 4π ak (ν, r)] The logs in the a k (ν, r) steam from soft gluons and should be resummed. This then incorporates the correct short distance behavior of the potential and is achieved by substituting [ ν r for r < νr and also α s (ν) α Vs ( r ) α s( r ) + ( ) ] αs ( k r ) 4π ak ( r, r) k= We now need to solve the Schrödinger equation numerically! Not treating the logs as perturbations may reduce the scale dependence and including RGI should improve the results further (success in M transitions ). Brambilla et al. Rev. Mod. Phys. 77 (5). arxiv: 447 [hep-ph]. Brambilla, Jia, and Vairo. Phys. Rev. D73 (6). arxiv: 5369 [hep-ph]; Pineda and Segovia. Phys. Rev. D87 (3). arxiv: pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) /3

21 Preliminary results for the LO decay width χ b Υ + γ : ν r = ν r = GeV NLO NLO NLO N3LO 6 6 Γ (kev) 5 4 Γ (kev) NLO NLO NLO N3LO ν (GeV) ν (GeV) Good and fast convergence. Heavily reduced scale dependence. Including RGI improves the convergence even more. pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) /3

22 Summary & outlook Summary We presented numerical results for the E transitions 3 P J=,, 3 S + γ and P S + γ up to NNLO (O(v )) in pnrqcd at weak coupling. The results are reasonable but a non-negligible dependence on the scale ν is observable via: Direct ν dependence induced by the corrections to the static potential. Indirect ν dependence via αs(ν) and a(α s(ν)). Incorporating the full static potential in the leading order + RGI diminishes the scale dependence strongly. Outlook Computing the relativistic corrections to the initial and final state wave functions is work in progress. Non-perturbative effects might be of the same size as the perturbative corrections and should be incorporated. pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) /3

23 References Bodwin, Braaten, and Lepage. Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium. Phys. Rev. D5 (995). arxiv: [hep-ph]. Brambilla, Jia, and Vairo. Model-independent Study of Magnetic Dipole Transitions in Quarkonium. Phys. Rev. D73 (6). arxiv: 5369 [hep-ph]. Brambilla, Pietrulewicz, and Vairo. Model-independent Study of Electric Dipole Transitions in Quarkonium. Phys. Rev. D85 (). arxiv: 3.3. Brambilla et al. Effective field theories for heavy quarkonium. Rev. Mod. Phys. 77 (5). arxiv: 447 [hep-ph]. Brambilla et al. Potential NRQCD: An Effective theory for heavy quarkonium. Nucl. Phys. B566 (). arxiv: 9974 [hep-ph]. Caswell and Lepage. Effective Lagrangians for Bound State Problems in QED, QCD, and Other Field Theories. Phys. Lett. B67 (986). Chetyrkin, Kühn, and Steinhauser. RunDec: A Mathematica package for running and decoupling of the strong coupling and quark masses. Comput. Phys. Commun. 33 (). arxiv: 489 [hep-ph]. Kiyo, Mishima, and Sumino. On enhanced corrections from quasi-degenerate states to heavy quarkonium observables. Phys. Lett. B76 (6). arxiv: Leutwyler. How to use heavy quarks to probe the QCD vacuum. Phys. Lett. B98 (98). Olive et al. Review of Particle Physics. Chin. Phys. C38 (4). Peset, Pineda, and Stahlhofen. Potential NRQCD for unequal masses and the B c spectrum at N 3 LO. JHEP 5 (6). arxiv: 5.8. Pineda and Soto. Effective field theory for ultrasoft momenta in NRQCD and NRQED. Nucl. Phys. Proc. Suppl. 64 (998). arxiv: [hep-ph]. Pineda and Segovia. Improved determination of heavy quarkonium magnetic dipole transitions in potential nonrelativistic QCD. Phys. Rev. D87 (3). arxiv: Segovia et al. Bottomonium spectrum revisited. Phys. Rev. D93 (6). arxiv: Voloshin. On dynamics of heavy quarks in a non-perturbative QCD vacuum. Nucl. Phys. B54 (979). pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de) 3/3

24 BACKUP SLIDES pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de)

25 [kev] χ b Υ + γ: χ b Υ + γ: [GeV] [kev] [kev] h b η b + γ: [GeV] [GeV] pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de)

26 Non-perturbative color octet effects a 3a n 3 PJ δz n 3 PJ δz n 3 S n 3 S n 3 PJ n 3 S eeqr E em n 3 PJ n 3 PJ m 3 gr E gr E S n 3 S eeqr E em 4 n 3 PJ b gr E gr E m 3 PJ n 3 S gr E 3b eeqr E em n 3 PJ gr E eeqr E em m 3 DJ eeqr E em gr E n 3 S eeqr E em FIG. 8: Color-octet contributions to the E transition n 3 PJ n 3 S γ for weakly-coupled states. The double line stands for an intermediate octet state. M fig.,a+b where the Wilson line in the adjoint representation is n 3 P J n 3 S = M () n 3 P J n 3 S +γ +γ t [ φ(t, ) adj ab = ig dt A(R, t ) adj () e, (4) n 3 P J r j e i(h() () o E ab gr E n 3 S A higher order Fock state consists of a quark and antiquark pair in a color octet configuration and at least one gluon: L d3 r V A tr{o r g ES + h.c.} E Λ QCD : these contributions vanish (Λ QCD integrated out). E Λ QCD : contribution at NNLO; the chromo-electric correlator reduces to the two gluon condensate which factorizes. dt t vac ge a i ( R, t)φ(t, ) adj ab ge b i ( R, ) vac ] () o E n )t r j n 3 S () n )t r j n 3 P J () + () n 3 S r j e i(h() and h () O r () /m + V O. Deriving the self energy with respect to the energy provides the state normalization pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de)

27 Non-perturbative color octet effects M fig.,a+b n 3 P J n 3 S +γ M() n 3 P J n 3 S +γ ( 6) vac g E a i δ ab E b i vac [ () n 3 P J rg o G o r n 3 P J () + () n 3 S rg o G o r n 3 S ()] [GeV - ] [GeV] We know : ( () n rg o G o r n () ) 6 n α s restriction to lowest lying states expect very strong scale dependence vac g E vac = π vac α s G vac additional enhancement Non-perturbative contribution exceeds the LO by orders of magnitude. It is only smaller than the LO for ν. GeV. We did not incorporate this contribution in the analysis! Voloshin. Nucl. Phys. B54 (979); Leutwyler. Phys. Lett. B98 (98) pnrqcd E transitions Sebastian Steinbeißer (sebastian.steinbeisser@tum.de)