Effective Field Theories for Quarkonium

Transcription

1 Effective Field Theories for Quarkonium recent progress Antonio Vairo Technische Universität München

2 Outline 1. Scales and EFTs for quarkonium at zero and finite temperature 2.1 Static energy at zero temperature 2.2 Charmonium radiative transitions 2.3 Bottomoniun thermal width 3. Conclusions

3 Scales and EFTs

4 Scales Quarkonia, i.e. heavy quark-antiquark bound states, are systems characterized by hierarchies of energy scales. These hierarchies allow systematic studies. They follow from the quark mass M being the largest scale in the system: M p M Λ QCD M T other thermal scales

5 The non-relativistic expansion M p implies that quarkonia are non-relativistic and characterized by the hierarchy of scales typical of a non-relativistic bound state: M p 1/r Mv E Mv 2 The hierarchy of non-relativistic scales makes the very difference of quarkonia with heavy-light mesons, which are just characterized by the two scales M and Λ QCD. Systematic expansions in the small heavy-quark velocity v may be implemented at the Lagrangian level by constructing suitable effective field theories (EFTs). Brambilla Pineda Soto Vairo RMP 77 (2004) 1423

6 Non-relativistic Effective Field Theories M Mv Mv 2 LONG RANGE QUARKONIUM perturbative matching non perturbative matching QCD/QED perturbative matching NRQCD/NRQED SHORT RANGE QUARKONIUM / QED perturbative matching pnrqcd/pnrqed µ µ Caswell Lepage PLB 167(86)437 Lepage Thacker NP PS 4(88)199 Bodwin et al PRD 51(95)1125,... Pineda Soto PLB 420(98)391 Pineda Soto NP PS 64(98)428 Brambilla et al PRD 60(99) Brambilla et al NPB 566(00)275 Kniehl et al NPB 563(99)200 Luke Manohar PRD 55(97)4129 Luke Savage PRD 57(98)413 Grinstein Rothstein PRD 57(98)78 Labelle PRD 58(98) Griesshammer NPB 579(00)313 Luke et al PRD 61(00) Hoang Stewart PRD 67(03)114020,...

7 The perturbative expansion M Λ QCD implies α s (M) 1: phenomena happening at the scale M may be treated perturbatively. We may further have small couplings if Mv Λ QCD and Mv 2 Λ QCD, in which case α s (Mv) 1 and α s (Mv 2 ) 1 respectively. This is likely to happen only for the lowest charmonium and bottomonium states. 1 Α 1 r The different quarkonium radii provide different measures of the transition from a Coulombic to a confined bound state. 0.2 Υ J/ψ, Υ r fm

8 The thermal expansion M T implies that quarkonium remains non-relativistic also in the thermal bath. T other thermal scales implies a hierarchy also in the thermal scales. Different quarkonia will dissociate in a medium at different temperatures, providing a thermometer for the plasma. Matsui Satz PLB 178 (1986) 416 CMS , CMS-HIN

9 Thermal non-relativistic Effective Field Theories T=0 M>T>Mv Mv>T M QCD T Mv NRQCD m D NRQCD HTL T Mv 2 pnrqcd pnrqcd HTL m D pnrqcd HTL Laine Philipsen Romatschke Tassler JHEP 0703 (2007) 054 Beraudo Blaizot Ratti NPA (2008) 806 Escobedo Soto PRA 78 (2008) Brambilla Ghiglieri Vairo Petreczky PRD 78 (2008) ,...

10 Physics at the scale M: annihilation and production Quarkonium annihilation and production happens at the scale M. The suitable EFT is NRQCD c(α s (M),µ) QCD NRQCD The effective Lagrangian is organized as an expansion in 1/M and α s (M): L NRQCD = n c n (α s (M),µ) M n O n (µ,mv,mv 2,...) see talk by Mathias Butenschön

11 Physics at the scalemv: bound state formation Quarkonium formation happens at the scale M v. The suitable EFT is pnrqcd E p 2 /m V(r,µ,µ) NRQCD pnrqcd The effective Lagrangian is organized as an expansion in 1/M, α s (M) and r: L pnrqcd = d 3 r n k c n (α s (M),µ) M n V n,k (r,µ,µ) r k O k (µ,mv 2,...) V n,0 are the potentials in the Schrödinger equation. V n,k 0 are the couplings with the low-energy degrees of freedom, which provide corrections to the potential picture.

12 Physics of the quarkonium ground state c and b masses at NNLO, N 3 LO, NNLL ; B c mass at NNLO; B c, η c, η b masses at NLL; Quarkonium 1P fine splittings at NLO; Υ(1S), η b electromagnetic decays at NNLL; Υ(1S) and J/ψ radiative decays at NLO; Υ(1S) γη b, J/ψ γη c at NNLO; t t cross section at NNLL; QQq and QQQ baryons: potentials at NNLO, masses, hyperfine splitting,... ; Thermal effects on quarkonium in medium: potential, masses (at mα 5 s ), widths,...;... for reviews QWG coll. Heavy Quarkonium Physics CERN Yellow Report CERN QWG coll. Eur. Phys. J. C71 (2011) 1534

13 Weakly coupled pnrqcd The suitable EFT for the quarkonium ground states is weakly coupled pnrqcd, because mv mα s mv 2 mα 2 s > Λ QCD The degrees of freedom are quark-antiquark states (color singlet S, color octet O), low energy gluons and photons, and light quarks. L pnrqcd = d 3 r Tr ) {S (i 0 p2 m + V s S +O ( id 0 p2 m + V o 1 4 Fa µν Fµν a 1 4 F µνf µν + n f i=1 ) } O q i id/q i + L At leading order in r, the singlet S satisfies the QCD Schrödinger equation with potential V s.

14 Dipole interactions L describes the interaction with the low-energy degrees of freedom, which at leading order are dipole interactions L = d 3 r Tr { V A O r ges m V 1 { } S,σ gb O+ +V em A S r ee Q E em S m V 1 em } { S,σ ee Q B em} S+...

15 Static energy and potential at T = 0

16 The static potential in perturbation theory e ig dz µ A µ = NRQCD pnrqcd lim T i T ln = V s(r,µ) i g2 N c V 2 A 0 dte it(v o V s ) Tr(r E(t)r E(0)) (µ)+... [chromoelectric dipole interactions] The µ dependence cancels between the two terms in the right-hand side: V s lnrµ,ln 2 rµ,... ultrasoft contribution ln(v o V s )/µ,ln 2 (V o V s )/µ,...lnrµ,ln 2 rµ,...

17 The static Wilson loop is known up to N 3 LO. Schröder PLB 447 (1999) 321 Brambilla Pineda Soto Vairo PRD 60 (1999) Brambilla Garcia Soto Vairo PLB 647 (2007) 185 Smirnov Smirnov Steinhauser PLB 668 (2008) 293 Anzai Kiyo Sumino PRL 104 (2010) Smirnov Smirnov Steinhauser PRL 104 (2010) The octet potential is known up to NNLO. Kniehl Penin Schröder Smirnov Steinhauser PLB 607 (2005) 96 V A = 1+O(α 2 s). Brambilla Garcia Soto Vairo PLB 647 (2007) 185 The chromoelectric correlator Tr(r E(t)r E(0)) is known up to NLO. Eidemüller Jamin PLB 416 (1998) 415

18 The static potential at N 4 LO α s (1/r) V s (r,µ) = C F r [ α s (1/r) 1+a 1 4π ( ) αs (1/r) 2 +a 2 4π ( 16π 2 ) ( + CA 3 3 lnrµ+a αs (1/r) 3 4π ( + a L2 4 ln2 rµ+ ) 3 ( a L π2 C 3 A β 0( 5+6ln2) ) )( ) ] αs (1/r) 4 lnrµ+a 4 4π

19 The static potential at N 3 LL V s (r,µ) = V s (r,1/r)+ 2 3 C Fr 2 [V o (r,1/r) V s (r,1/r)] 3 η 0 = 1 π [ β 1 2β β 0 ( 5nf +C A (6π 2 )] +47) 108 ( 2 ln α ) s(µ) β 0 α s (1/r) +η 0[α s (µ) α s (1/r)] Pineda Soto PLB 495 (2000) 323 Brambilla Garcia Soto Vairo PRD 80 (2009)

20 Static quark-antiquark energy at N 3 LL E 0 (r) = V s (r,µ)+λ s (r,µ)+δ US (r,µ) Λ s (r,µ) = N s Λ+2C F (N o N s )Λr 2 [V o (r,1/r) V s (r,1/r)] 2 δ US (r,µ) = C F C 3 A 24 1 r ( 2 ln α ) s(µ) β 0 α s (1/r) +η 0[α s (µ) α s (1/r)] ( α s (µ) π α3 s(1/r) 2ln α s(1/r)n c + 5 ) 2rµ 3 2ln2 N s, N o are two arbitrary scale-invariant dimensionless constants Λ is an arbitrary scale-invariant quantity of dimension one

21 Static quark-antiquark energy at N 3 LL vs lattice r 0 E 0 r E 0 r min E 0 latt. rmin r r 0 Brambilla Garcia Soto Vairo PRL 105 (2010) quenched lattice data from Necco Sommer NPB 622 (2002) 328 Perturbation theory (known up to NNNLO) + renormalon subtraction describes well the static potential up to about 0.25 fm (r fm). Indeed one can use this to extract Λ MS r 0 = and in perspective r 0 (high precision unquenched lattice data is needed).

22 Radiative transitions

23 J/ψ Xγ for 0 MeV E γ < 500 MeV Scales: p 1/ r M c v 700 MeV - 1 GeV Λ QCD E J/ψ M J/ψ 2M c M c v MeV MeV 1/ r 0 MeV E γ < 400 MeV MeV 1/ r It follows that the system is (i) non-relativistic, (ii) weakly-coupled at the scale 1/ r : v α s, (iii) that we may mutipole expand in the external photon energy. Brambilla Jia Vairo PRD 73 (2006) see talk by Piotr Pietrulewicz for E1 transitions

24 J/ψ Xγ for 0 MeV E γ < 500 MeV Three main processes contribute to J/ψ X γ for 0 MeV E γ < 500 MeV: J/ψ η c γ X γ [magnetic dipole interactions] M1 Imh s M1 J/ψ χ c0,2 (1P)γ X γ [electric dipole interactions] E1 Imh s E1 fragmentation and other background processes, included in the background functions.

25 The orthopositronium decay spectrum The situation is analogous to the photon spectrum in orthopositronium 3γ Manohar Ruiz-Femenia PRD 69(04) Ruiz-Femenia NPB 788(08)21, arxiv:

26 J/ψ η c γ Xγ M1 Imh s M1 dγ = 64 de γ 27 α M 2 J/ψ E γ π Γ ηc 2 E 2 γ (M J/ψ M ηc E γ ) 2 +Γ 2 η c /4 For Γ ηc 0 one recovers Γ(J/ψ η c γ) = α E3 γ M 2 J/ψ The non-relativistic Breit Wigner distribution goes like: E 2 γ (M J/ψ M ηc E γ ) 2 +Γ 2 η c /4 = 1 for E γ M c α 4 s M J/ψ M ηc E 2 γ (M J/ψ M η c )2 for E γ M c α 4 s M J/ψ M ηc

27 J/ψ χ c0,2 (1P)γ Xγ E1 Imh s E1 dγ = 32 de γ 81 α M 2 J/ψ E γ π [ 21α 2 ] s 2πα 2 a(e γ ) 2 a(e γ ) = (1 ν)(3+5ν) 3(1+ν) 2 + 8ν2 (1 ν) 3(2 ν)(1+ν) 3 2 F 1 (2 ν,1;3 ν; (1 ν)/(1+ν)) ν = E J/ψ /(E γ E J/ψ ) a(e γ ) 2 = 1 for E γ M c α 2 s E J/ψ E 2 γ/(2e J/ψ ) 2 for E γ M c α 2 s E J/ψ Voloshin MPLA 19 (2004) 181

28 The two contributions are of equal order for M c α s E γ M c α 2 s E J/ψ ; the magnetic contribution dominates for E J/ψ M c α 2 s E γ M c α 4 s M J/ψ M ηc ; it also dominates by a factor E 2 J/ψ /(M J/ψ M ηc ) 2 1/α 4 s for E γ M c α 4 s M J/ψ M ηc.

29 Fit to the CLEO data 1500 CLEO data pnrqcd: M ηc = (6) GeV, Γ ηc = (2) GeV background 1 background N events /bin ,1 0,2 0,3 0,4 0,5 E γ (GeV) M ηc = ±0.6 (fit) MeV Γ ηc = 28.6±0.2 (fit) MeV Besides M ηc and Γ ηc the fitting parameters are the overall normalization, the signal normalization, and the (three) background parameters. Brambilla Roig Vairo arxiv:

30 Thermal width at T > 0

31 The bottomonium ground state at finitet The bottomonium ground state produced in the QCD medium of heavy-ion collisions at the LHC may possibly realize the hierarchy M b 5 GeV > M b α s 1.5 GeV > πt 1 GeV > M b α 2 s 0.5 GeV > m D,Λ QCD The bound state is weakly coupled: v α s 1 The temperature is lower than M b α s, implying that the bound state is mainly Coulombic Effects due to the scale Λ QCD and to the other thermodynamical scales may be neglected

32 pnrqcd HTL Integrating out T from pnrqcd modifies pnrqcd into pnrqcd HTL whose Yang Mills Lagrangian gets the additional hard thermal loop (HTL) part; e.g. the longitudinal gluon propagator becomes i k 2 k 2 +m 2 D i ( 1 k 0 2k ln k ) 0 +k ±iη k 0 k ±iη where + identifies the retarded and the advanced propagator; potentials get additional thermal corrections δv.

33 Integrating outt The relevant diagram is (through chromoelectric dipole interactions) T and radiative corrections. The loop momentum region is k 0 T and k T.

34 Integrating outt : thermal width T Landau-damping contribution Γ (T) 1S [ = 4 3 α stm 2 D ( 2ǫ +γ E +lnπ ln T2 µ ) (2) 3 4ln2 2ζ ζ(2) 32π 3 ln2α2 s T 3 ] a 2 0 where E 1 = 4M bα 2 s 9 and a 0 = 3 2M b α s

35 Landau damping The Landau damping phenomenon originates from the scattering of the quarkonium with hard space-like particles in the medium. When Im V s (r) Landau damping Re V s (r) α s /r, the quarkonium dissociates: πt dissociation M b g 4/3 When 1/r m D, the interaction is screened; note that πt screening M b g πt dissociation Laine Philipsen Romatschke Tassler JHEP 0703 (2007) 054

36 Υ(1S) dissociation temperature The Υ(1S) dissociation temperature: M c (MeV) T dissociation (MeV) A temperature πt about 1 GeV is below the dissociation temperature. Escobedo Soto PRA 82 (2010)

37 Integrating oute The relevant diagram is (through chromoelectric dipole interactions) E where the loop momentum region is k 0 E and k E. Gluons are HTL gluons.

38 Integrating oute: thermal width Γ (E) 1S = 4α 3 st 64 α s TE M b 3 α2 st 4α stm 2 D 3 ( 2 ǫ +ln E2 1 µ 2 +γ E M b a E 1α 3 s ) lnπ +ln4 a α stm 2 D 27 α 2 s E 2 1 I 1,0 where E 1 = 4M bα 2 s 9 and a 0 = 3 2M b α s and I 1,0 = (similar to the Bethe log) The UV divergence at the scale M b α 2 s cancels against the IR divergence identified at the scale T.

39 Singlet to octet break up The thermal width at the scale E, which is of order α 3 st, is generated by the break up of a quark-antiquark colour-singlet state into an unbound quark-antiquark colour-octet state: a purely non-abelian process that is kinematically allowed only in a medium. The singlet to octet break up is a different phenomenon with respect to the Landau damping, the relative size of which is (E/m D ) 2. In the situation M b α 2 s m D, the first dominates over the second by a factor (M b α 2 s/m D ) 2. Brambilla Ghiglieri Petreczky Vairo PRD 78 (2008)

40 The complete thermal width up too(mα 5 s) Γ (thermal) 1S = α3 st E 1α 3 s α stm 2 D a2 0I 1,0 [ ( ) 4 3 α stm 2 D ln E2 1 T 2 +2γ E 3 ln4 2 ζ (2) ζ(2) + 32π ] 3 ln2α2 s T 3 a 2 0 where E 1 = 4M bα 2 s 9, a 0 = 3 2M b α s Brambilla Escobedo Ghiglieri Soto Vairo JHEP 1009 (2010) 038 see talk by Jacopo Ghiglieri

41 Conclusions Our understanding of the theory of quarkonium has dramatically improved over the last decade. An unified picture has emerged that describes large classes of observables for quarkonium in the vacuum and in a medium. For the ground state, precision physics is possible and lattice data provide a crucial complement. In the case of quarkonium in a hot medium, this has disclosed new phenomena that may be eventually responsible for the quarkonium dissociation.

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