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Effective Field Theories for Quarkonium

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Effective Field Theories for Quarkonium Effective Field Theories for Quarkonium recent progress Antonio Vairo Technische Universitat¨ Munchen¨ 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 Scales and EFTs 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 ≫ ΛQCD • M ≫ T ≫ other thermal scales 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 ∼ Mv2 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 ΛQCD. 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 Non-relativistic Effective Field Theories LONG−RANGE SHORT−RANGE Caswell Lepage PLB 167(86)437 QUARKONIUM QUARKONIUM / QED ◦ ◦ Lepage Thacker NP PS 4(88)199 QCD/QED ◦ Bodwin et al PRD 51(95)1125, ... M perturbative matching perturbative matching ◦ Pineda Soto PLB 420(98)391 µ ◦ Pineda Soto NP PS 64(98)428 ◦ Brambilla et al PRD 60(99)091502 Mv NRQCD/NRQED ◦ Brambilla et al NPB 566(00)275 ◦ Kniehl et al NPB 563(99)200 µ ◦ Luke Manohar PRD 55(97)4129 ◦ Luke Savage PRD 57(98)413 2 non−perturbative perturbative matching Mv matching ◦ Grinstein Rothstein PRD 57(98)78 ◦ Labelle PRD 58(98)093013 ◦ Griesshammer NPB 579(00)313 pNRQCD/pNRQED ◦ Luke et al PRD 61(00)074025 ◦ Hoang Stewart PRD 67(03)114020, ... The perturbative expansion • M ≫ ΛQCD implies αs(M) ≪ 1: phenomena happening at the scale M may be treated perturbatively. 2 We may further have small couplings if Mv ≫ ΛQCD and Mv ≫ ΛQCD, in 2 which case αs(Mv) ≪ 1 and αs(Mv ) ≪ 1 respectively. This is likely to happen only for the lowest charmonium and bottomonium states. 1 0.8 The different quarkonium radii provide 0.6 ΑH1rL different measures of the transition 0.4 from a Coulombic to a confined bound state. 0.2 Υ J/ψ, Υ′ 0.1 0.15 0.2 0.25 r HfmL 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 1012.5545, CMS-HIN-10-006 Thermal non-relativistic Effective Field Theories T=0 M>T>Mv Mv>T M QCD T Mv NRQCD NRQCD HTL T m D Mv2 pNRQCD pNRQCD HTL pNRQCD HTL m D ◦ Laine Philipsen Romatschke Tassler JHEP 0703 (2007) 054 ◦ Beraudo Blaizot Ratti NPA (2008) 806 ◦ Escobedo Soto PRA 78 (2008) 032520 ◦ Brambilla Ghiglieri Vairo Petreczky PRD 78 (2008) 014017, ... Physics at the scale M: annihilation and production Quarkonium annihilation and production happens at the scale M. The suitable EFT is NRQCD. ×c(αs(M),µ) ... ... ... ... QCD NRQCD The effective Lagrangian is organized as an expansion in 1/M and αs(M): cn(αs(M),µ) L = × O (µ,Mv,Mv2,...) NRQCD n n n M ◦ see talk by Mathias Butenschon¨ Physics at the scale Mv: bound state formation Quarkonium formation happens at the scale Mv. The suitable EFT is pNRQCD. + + ... + ... + + ... ... ... ... 1 E − p2/m − V (r, µ′,µ) NRQCD pNRQCD The effective Lagrangian is organized as an expansion in 1/M , αs(M) and r: cn(αs(M),µ) L = d3r × V (r, µ′,µ) rk × O (µ′,Mv2,...) pNRQCD n n,k k n M k • Vn,0 are the potentials in the Schrödinger equation. • Vn,k=0 are the couplings with the low-energy degrees of freedom, which provide corrections to the potential picture. Physics of the quarkonium ground state • c and b masses at NNLO, N3LO∗, NNLL∗; • Bc mass at NNLO; ∗ • Bc , ηc, ηb masses at NLL; • Quarkonium 1P fine splittings at NLO; • Υ(1S), ηb electromagnetic decays at NNLL; • Υ(1S) and J/ψ radiative decays at NLO; • Υ(1S) → γηb, J/ψ → γηc at NNLO; • tt¯cross section at NNLL; • QQq and QQQ baryons: potentials at NNLO, masses, hyperfine splitting, ...; 5 • Thermal effects on quarkonium in medium: potential, masses (at mαs ), widths, ...; • ... ◦ for reviews QWG coll. Heavy Quarkonium Physics CERN Yellow Report CERN-2005-005 QWG coll. Eur. Phys. J. C71 (2011) 1534 Weakly coupled pNRQCD The suitable EFT for the quarkonium ground states is weakly coupled pNRQCD, because mv ∼ mα ≫ mv2 ∼ mα2 > Λ s s ∼ QCD • The degrees of freedom are quark-antiquark states (color singlet S, color octet O), low energy gluons and photons, and light quarks. • 2 3 † p LpNRQCD = d r Tr S i∂0 − + ···− Vs S m 2 † p +O iD0 − + ···− Vo O m nf 1 a µν a 1 µν − F F − Fµν F + q¯i iD/qi +∆L 4 µν 4 i=1 • At leading order in r, the singlet S satisfies the QCD Schrödinger equation with potential Vs. Dipole interactions ∆L describes the interaction with the low-energy degrees of freedom, which at leading order are dipole interactions 3 † ∆L = d r Tr VAO r · gE S+ ··· 1 † + V1 S , σ · gB O+ ··· 2m em † em +VA S r · eeQE S+ ··· 1 em † σ em + V1 S , · eeQB S+ ... 2m Static energy and potential at T =0 The static potential in perturbation theory µ eig dz Aµ = + + ... NRQCD pNRQCD 2 ∞ i g 2 −it(Vo−Vs) lim ln = Vs(r,µ) − i VA dte Tr(r · E(t) r · E(0))(µ)+ ... T →∞ T N c 0 [chromoelectric dipole interactions] The µ dependence cancels between the two terms in the right-hand side: 2 • Vs ∼ ln rµ, ln rµ, ... 2 2 • ultrasoft contribution ∼ ln(Vo − Vs)/µ, ln (Vo − Vs)/µ, ... ln rµ, ln rµ, ... • The static Wilson loop is known up to N3LO. ◦ Schroder¨ PLB 447 (1999) 321 Brambilla Pineda Soto Vairo PRD 60 (1999) 091502 Brambilla Garcia Soto Vairo PLB 647 (2007) 185 Smirnov Smirnov Steinhauser PLB 668 (2008) 293 Anzai Kiyo Sumino PRL 104 (2010) 112003 Smirnov Smirnov Steinhauser PRL 104 (2010) 112002 • The octet potential is known up to NNLO. ◦ Kniehl Penin Schroder¨ Smirnov Steinhauser PLB 607 (2005) 96 2 • VA =1+ O(αs ). ◦ Brambilla Garcia Soto Vairo PLB 647 (2007) 185 • The chromoelectric correlator Tr(r · E(t) r · E(0)) is known up to NLO. ◦ Eidemuller¨ Jamin PLB 416 (1998) 415 The static potential at N4LO 2 αs(1/r) αs(1/r) αs(1/r) Vs(r, µ) = −CF 1+ a1 + a2 r 4π 4π 2 3 16 π 3 αs(1/r) + C ln rµ + a3 3 A 4π 4 L2 2 L 16 2 3 αs(1/r) + a ln rµ + a + π C β0(−5+6ln2) ln rµ + a4 4 4 9 A 4π The static potential at N3LL 2 2 3 Vs(r, µ) = Vs(r, 1/r) + CF r [Vo(r, 1/r) − Vs(r, 1/r)] 3 2 αs(µ) × ln + η0 [αs(µ) − αs(1/r)] β α (1/r) 0 s 2 1 β1 12 −5nf + CA(6π + 47) η0 = − + π 2β2 β 108 0 0 ◦ Pineda Soto PLB 495 (2000) 323 ◦ Brambilla Garcia Soto Vairo PRD 80 (2009) 034016 Static quark-antiquark energy at N3LL E0(r) = Vs(r, µ) + Λs(r, µ) + δUS(r, µ) 2 2 Λs(r, µ) = NsΛ+2 CF (No − Ns)Λ r [Vo(r, 1/r) − Vs(r, 1/r)] 2 αs(µ) × ln + η0 [αs(µ) − αs(1/r)] β α (1/r) 0 s 3 CA 1 αs(µ) 3 αs(1/r)Nc 5 δUS(r, µ) = CF α (1/r) −2ln + − 2ln2 24 r π s 2rµ 3 Ns, No are two arbitrary scale-invariant dimensionless constants Λ is an arbitrary scale-invariant quantity of dimension one Static quark-antiquark energy at N3LL vs lattice 0.0 LL min r H latt. 0 -0.5 E + L min r H 0 -1.0 E - L r H 0 E H 0 - r 1.5 0.15 0.20 0.25 0.30 0.35 0.40 0.45 r r0 ◦ Brambilla Garcia Soto Vairo PRL 105 (2010) 212001 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 (r0 ≈ 0.5 fm). +0.019 • Indeed one can use this to extract ΛMSr0 =0.622−0.015 and in perspective r0 (high precision unquenched lattice data is needed). Radiative transitions J/ψ → Xγ for 0 MeV ≤ Eγ ∼< 500 MeV Scales: • p∼ 1/r∼ Mcv ∼ 700 MeV - 1 GeV ≫ ΛQCD 2 • EJ/ψ ≡ MJ/ψ − 2Mc ∼ Mcv ∼ 400 MeV - 600 MeV ≪ 1/r • 0 MeV ≤ E < 400 MeV - 500 MeV ≪ 1/r γ ∼ It follows that the system is (i) non-relativistic, (ii) weakly-coupled at the scale 1/r: v ∼ αs, (iii) that we may mutipole expand in the external photon energy. ◦ Brambilla Jia Vairo PRD 73 (2006) 054005 ◦ see talk by Piotr Pietrulewicz for E1 transitions J/ψ → Xγ for 0 MeV ≤ Eγ ∼< 500 MeV Three main processes contribute to J/ψ → Xγ for 0 MeV ≤ E < 500 MeV: γ ∼ • J/ψ → ηc γ → Xγ [magnetic dipole interactions] M1 Im hs M1 • J/ψ → χc0,2(1P ) γ → Xγ [electric dipole interactions] E1 Im hs E1 • fragmentation and other background processes, included in the background functions.
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