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.

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),) ... ......

QCD NRQCD

The effective Lagrangian is organized as an expansion in 1/M and αs(M):

cn(αs(M),) L = × O (,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, ′,) NRQCD pNRQCD

The effective Lagrangian is organized as an expansion in 1/M , αs(M) and r:

cn(αs(M),) L = d3r × V (r, ′,) rk × O (′,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,) − i VA dte Tr(r E(t) r E(0))()+ ... T →∞ T N c 0 [chromoelectric dipole interactions]

The dependence cancels between the two terms in the right-hand side: 2 • Vs ∼ ln r, ln r, ... 2 2 • ultrasoft contribution ∼ ln(Vo − Vs)/, ln (Vo − Vs)/, ... ln r, ln r, ... • 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, ) = −CF 1+ a1 + a2 r 4π 4π 2 3 16 π 3 αs(1/r) + C ln r + a3 3 A 4π 4 L2 2 L 16 2 3 αs(1/r) + a ln r + a + π C β0(−5+6ln2) ln r + a4 4 4 9 A 4π The static potential at N3LL

2 2 3 Vs(r, ) = Vs(r, 1/r) + CF r [Vo(r, 1/r) − Vs(r, 1/r)] 3 2 αs() × ln + η0 [αs() − 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, ) + Λs(r, ) + δUS(r, )

2 2 Λs(r, ) = NsΛ+2 CF (No − Ns)Λ r [Vo(r, 1/r) − Vs(r, 1/r)] 2 αs() × ln + η0 [αs() − αs(1/r)] β α (1/r) 0 s 3 CA 1 αs() 3 αs(1/r)Nc 5 δUS(r, ) = CF α (1/r) −2ln + − 2ln2 24 r π s 2r 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. The orthopositronium decay spectrum

The situation is analogous to the photon spectrum in orthopositronium → 3γ

◦ Manohar Ruiz-Femenia PRD 69(04)053003 Ruiz-Femenia NPB 788(08)21, arXiv:0904.4875 J/ψ → ηc γ → Xγ

M1 Im hs M1

2 dΓ 64 α Eγ Γη E = c γ 2 2 2 dEγ 27 MJ/ψ π 2 (MJ/ψ − Mηc − Eγ ) +Γηc /4

64 E3 For one recovers γ • Γηc → 0 Γ(J/ψ → ηc γ)= α 2 27 MJ/ψ • The non-relativistic Breit–Wigner distribution goes like:

2 1 for E ≫ M α4 ∼ M − M Eγ γ c s J/ψ ηc = 2 (M − M − E )2 +Γ2 /4  Eγ 4 J/ψ ηc γ ηc 2 for Eγ ≪ Mcαs ∼ MJ/ψ − Mηc  (MJ/ψ −Mηc )  J/ψ → χc0,2(1P ) γ → Xγ

E1 Im hs E1

2 dΓ 32 α Eγ 21 αs 2 = |a(Eγ )| dE 81 M 2 π 2 πα2 γ J/ψ

(1 − ν)(3+5ν) 8ν2(1 − ν) • a(Eγ )= + 2F1(2−ν, 1;3−ν; −(1−ν)/(1+ν)) 3(1 + ν)2 3(2 − ν)(1 + ν)3

ν = −EJ/ψ/(Eγ − EJ/ψ) ◦ Voloshin MPLA 19 (2004) 181 2 2 1 for Eγ ≫ Mcαs ∼ EJ/ψ • |a(Eγ )| =  E2/(2E )2 for E ≪ M α2 ∼ E  γ J/ψ γ c s J/ψ  • The two contributions are of equal order for 2 Mcαs ≫ Eγ ≫ Mcαs ∼−EJ/ψ;

• the magnetic contribution dominates for 2 4 −EJ/ψ ∼ Mcαs ≫ Eγ ≫ Mcαs ∼ MJ/ψ − Mηc ;

2 2 4 • it also dominates by a factor EJ/ψ/(MJ/ψ − Mηc ) ∼ 1/αs for 4 Eγ ≪ Mcαs ∼ MJ/ψ − Mηc . Fit to the CLEO data

1500 CLEO data Γ pNRQCD: Mη = 2.9859(6) GeV, η = 0.0286(2) GeV c c background 1 background 2

1000 /bin events N

500

0 0,1 0,2 0,3 0,4 0,5 Eγ (GeV)

Mηc = 2985.9 ± 0.6 (ﬁt) MeV Γηc = 28.6 ± 0.2 (ﬁt) MeV

• Besides Mηc and Γηc the ﬁtting parameters are the overall normalization, the signal normalization, and the (three) background parameters. ◦ Brambilla Roig Vairo arXiv:1012.0773 Thermal width at T > 0 The bottomonium ground state at ﬁnite T

The bottomonium ground state produced in the QCD medium of heavy-ion collisions at the LHC may possibly realize the hierarchy

M ≈ 5 GeV >M α ≈ 1.5 GeV > πT ≈ 1 GeV >M α2 ≈ 0.5 GeV > m , Λ b b s b s ∼ D QCD

• The bound state is weakly coupled: v ∼ αs ≪ 1

• The temperature is lower than Mbαs, implying that the bound state is mainly Coulombic

• Effects due to the scale ΛQCD and to the other thermodynamical scales may be neglected pNRQCDHTL

Integrating out T from pNRQCD modiﬁes pNRQCD into pNRQCDHTL whose

• Yang–Mills Lagrangian gets the additional hard thermal loop (HTL) part; e.g. the longitudinal gluon propagator becomes

i i → k2 k k + k ± iη k2 + m2 1 − 0 ln 0 D 2k k − k ± iη 0 where “+” identiﬁes the retarded and “−” the advanced propagator;

• potentials get additional thermal corrections δV . Integrating out T

The relevant diagram is (through chromoelectric dipole interactions)

and radiative corrections. The loop momentum region is k0 ∼ T and k ∼ T . Integrating out T : thermal width

T Landau-damping contribution

2 ′ (T ) 4 2 2 T 2 ζ (2) Γ = − αsTm − + γE + ln π − ln + − 4ln2 − 2 1S 3 D ǫ 2 3 ζ(2) 32π − ln2 α2 T 3 a2 3 s 0

2 4Mbαs 3 where E1 = − and a0 = 9 2Mbαs Landau damping

The Landau damping phenomenon originates from the scattering of the quarkonium with hard space-like particles in the medium.

• When Im Vs(r)|Landau−damping ∼ Re Vs(r) ∼ αs/r, the quarkonium dissociates:

4/3 πTdissociation ∼ Mbg

• When 1/r∼ mD, the interaction is screened; note that

πTscreening ∼ Mbg ≫ πTdissociation

◦ Laine Philipsen Romatschke Tassler JHEP 0703 (2007) 054 Υ(1S) dissociation temperature

The Υ(1S) dissociation temperature:

Mc (MeV) Tdissociation (MeV) ∞ 480 5000 480 2500 460 1200 440 0 420

A temperature πT about 1 GeV is below the dissociation temperature.

◦ Escobedo Soto PRA 82 (2010) 042506 Integrating out E

The relevant diagram is (through chromoelectric dipole interactions)

where the loop momentum region is k0 ∼ E and k ∼ E. Gluons are HTL gluons. Integrating out E: thermal width

(E) 3 64 32 2 1 7225 3 Γ1S = 4αs T − αsTE1 + αs T + E1αs 9Mb 3 Mba0 162 2 2 4αsTmD 2 E1 11 2 − + ln + γE − − ln π +ln4 a 3 ǫ 2 3 0 2 2 128αsTmD αs + 2 I1,0 27 E1

2 4Mbαs 3 where E1 = − and a0 = and I1,0 = −0.49673 (similar to the Bethe log) 9 2Mbαs

2 • The UV divergence at the scale Mbαs cancels against the IR divergence identiﬁed at the scale T . Singlet to octet break up

3 The thermal width at the scale E, which is of order αs T , 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 2 2 damping, the relative size of which is (E/mD) . In the situation Mbαs ≫ mD, the 2 2 ﬁrst dominates over the second by a factor (Mbαs /mD) .

◦ Brambilla Ghiglieri Petreczky Vairo PRD 78 (2008) 014017 5 The complete thermal width up to O(mαs )

(thermal) 1156 3 7225 3 32 2 2 Γ = α T + E1α + αs Tm a I1,0 1S 81 s 162 s 9 D 0 2 ′ 4 2 E1 ζ (2) 32π 2 3 2 − αsTm ln +2γE − 3 − ln4 − 2 + ln2 α T a 3 D T 2 ζ(2) 3 s 0

2 4Mbαs 3 where E1 = − , a0 = 9 2Mbαs

◦ Brambilla Escobedo Ghiglieri Soto Vairo JHEP 1009 (2010) 038 ◦ see talk by Jacopo Ghiglieri 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.