Thermal width and quarkonium dissociation in an EFT framework

Transcription

1 Thermal width and quarkonium dissociation in an EFT framework Antonio Vairo Technische Universität München in collaboration with N. Brambilla, M. A. Escobedo and J. Ghiglieri based on arxiv: and JHEP 1112 (2011) 116

2 Quarkonium as a quark-gluon plasma probe In 1986, Matsui and Satz suggest quarkonium dissociation as an ideal probe of the medium formed in heavy-ion collisions. Heavy quarks are formed early in heavy-ion collisions: 1/m 0.1 fm 1 fm. Heavy quarkonium formation will be sensitive to the medium. The dilepton signal makes the quarkonium a clean experimental probe. /GeV) IN PHENIX ACCEPTANCE min. bias Au+Au at DATA y < 0.35 > 0.2 GeV/c p T π 0 γee η γee η γee ρ ee 0 ω ee & π ee φ ee & ηee s NN = 200 GeV J/Ψ ee Ψ ee cc ee (PYTHIA) sum cc ee (random correlation) (c dn/dm ee 1/N evt (GeV/c ) m ee

3 Energy Scales Quarkonium in a medium is characterized by different energy and momentum scales: the scales of a non-relativistic bound state (v is the relative heavy-quark velocity): m (mass), mv (momentum transfer, inverse distance), mv 2 (kinetic energy, binding energy, potential V ),... the thermodynamical scales: πt (temperature), m D (Debye mass, i.e. screening of the chromoelectric interactions),... The non-relativistic scales are hierarchically ordered: m mv mv 2 We may assume this to be also the case for the thermodynamical scales: πt m D

4 Weak coupling The best controlled setting is when all scales are perturbative: all energy scales Λ QCD i.e. a weakly-coupled quarkonium in a weakly-coupled quark-gluon plasma. For a weakly-coupled quarkonium: v α s. The quarkonium ground states, produced in the QCD medium of heavy-ion collisions at the LHC, may possibly realize this situation. E.g. for the bottomonium ground state: 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 For excited states, m b α s becomes closer to m D and the bottomonium dissociates.

5 Υ suppression at CMS

6 Non-relativistic Effective Field Theories at finitet The existence of a hierarchy of energy scales calls for a description of the system (quarkonium at rest in a thermal bath) in terms of a hierarchy of EFTs. non relativistic scales thermal scales EFTs m m v T NRQCD m D m v 2 pnrqcd For larger temperatures the quarkonium does not form.

7 NRQCD NRQCD is obtained by integrating out modes associated with the scale m and possibly with thermal scales larger than mv. The Lagrangian is organized as an expansion in 1/m: ( ) L = ψ id 0 + )ψ D2 2m χ (id 0 D2 2m +... χ+... +L light ψ (χ) is the field that annihilates (creates) the (anti)fermion. L light describes the propagation of gluons and light quarks. It depends on the version of NRQCD. Thermodynamical scales may be set to zero while matching at the scale m.

8 pnrqcd pnrqcd is obtained by integrating out modes associated with the scale mv and possibly with thermal scales larger than mv 2. The degrees of freedom of pnrqcd are quark-antiquark states (color singlet S, color octet O), low energy gluons and light quarks propagating in the medium. The Lagrangian is organized as an expansion in 1/m and r: L = ) d 3 r Tr {S (i 0 p2 m V s +... )S+O (id 0 p2 m V o +... { Tr O r ges+h.c. }+ 1 { } 2 Tr O r geo+c.c. + } O +L light L light describes the in-medium propagation of gluons and light quarks. It depends on the version of pnrqcd. At leading order in r, the singlet S satisfies a Schrödinger equation. The explicit form of the potential depends on the version of pnrqcd.

9 pnrqcd Feynman rules: = θ(t)e ith s = θ(t)e ith o (e i dta adj) = O r ges = O {r ge, O}

10 Real time The contour of the partition function is modified to allow for real time: Im t t 0 1 t f Re t 2 t 0 - i /Τ In real time, the degrees of freedom double ( 1 and 2 ). In the heavy-quark sector, the second degrees of freedom, labeled 2, decouple from the physical degrees of freedom, labeled 1. The second degrees of freedom contribute to the gluon self energy. The discontinuity of the 11 gluon self energy is encoded in Π µν S (k 0,k) = [1+2n B ( k 0 )]sgn(k 0 )[Π µν R (k 0,k) Π µν A (k 0,k)].

11 Dissociation mechanisms at LO Two distinct dissociation mechanisms may be identified at leading order: gluodissociation, which is the dominant mechanism for mv 2 m D ; dissociation by inelastic parton scattering, which is the dominant mechanism for mv 2 m D. Beyond leading order the two mechanisms are intertwined and distinguishing between them becomes unphysical, whereas the physical quantity is the total decay width.

12 Gluodissociation Gluodissociation is the dissociation of quarkonium by absorption of a gluon from the medium. The exchanged gluon is lightlike or timelike. The process happens when the gluon has an energy of order mv 2.

13 Gluodissociation From the optical theorem, the gluodissociation width follows from cutting the gluon propagator in the following pnrqcd diagram mv 2 For a quarkonium at rest with respect to the medium, the width has the form Γ nl = q min d 3 q (2π) 3 n B(q)σ nl gluo (q). σ nl gluo is the in-vacuum cross section (QQ) nl Q+Q+g. Gluodissociation is also known as singlet-to-octet break up.

14 Dissociation by inelastic parton scattering Dissociation by inelastic parton scattering is the dissociation of quarkonium by scattering with gluons and light-quarks in the medium. The exchanged gluon is spacelike. In Coulomb gauge, external thermal gluons are transverse. In the NRQCD power counting, each external transverse gluon is suppressed by T/m.

15 Dissociation by inelastic parton scattering Dissociation by inelastic parton scattering is the dissociation of quarkonium by scattering with gluons and light-quarks in the medium. at LO The exchanged gluon is spacelike. In Coulomb gauge, external thermal gluons are transverse. In the NRQCD power counting, each external transverse gluon is suppressed by T/m.

16 Dissociation by inelastic parton scattering From the optical theorem, the thermal width follows from cutting the gluon self-energy in the following NRQCD diagrams (momentum of the gluon > mv) and/or pnrqcd diagram (momentum of the gluon mv) Dissociation by inelastic parton scattering is also known as Landau damping.

17 Dissociation by inelastic parton scattering For a quarkonium at rest with respect to the medium, the thermal width has the form Γ nl = p q min d 3 q (2π) 3 f p(q) [1±f p (q)] σ nl p (q) where the sum runs over the different incoming light partons and f g = n B or f q = n F. σ nl p is the in-medium cross section (QQ) nl +p Q+Q+p. The convolution formula correctly accounts for Pauli blocking in the fermionic case (minus sign). The formula differs from the gluodissociation formula.

18 The temperature regiont mv m D The gluon self-energy contributes to the imaginary part of the pnrqcd potential through: T, mv, m D T, mv, m D T, mv, m D Similarly for the real part.

19 The temperature regiont mv m D ReV s (r) = C F α s ( e m D r ImV s (r) = g2 TC F m 2 D 2π r 0 +m D ) dtt (t 2 +m 2 D )2 ( 1 sin(tr) ) tr The real part of the potential is a screened Coulomb potential. The imaginary part of the potential stems from cutting the self energy of a space-like gluon. Hence it describes quarkonium dissociation by inelastic parton scattering. The imaginary part of the potential is larger than the real one: in this temperature region the quarkonium does not form.

20 The temperature regiont mv m D In this temperature region: Re V s (r) α s /r, quarkonium is a Coulombic bound state (e.g. r 1S = 1/( πa 3/2 0 )exp( r/a 0 ), where a 0 = 2/(mC F α s )); quarkonium dissociates at a temperature such that Im V s (r) Re V s (r) α s /r: πt dissociation mg 4/3 which in the Υ(1S) case is about 450 MeV. (Note that πt 1 GeV is below the dissociation temperature.) The interaction is screened when 1/r m D, hence πt screening mg πt dissociation

21 The temperature regiont mv m D Γ nl = 2 n,l ImV s (r) n,l where n,l is the eigenstate of p 2 /m+rev s (r) that identifies the quarkonium. m 2 D = ik 2πT Π00 S (0,k T) = 2g2 T d 3 q (2π) 3n f n F (q)[1 n F (q)]+n c n B (q)[1+n B (q)] implies the convolution formula Γ nl = p q min d 3 q (2π) 3 f p(q) [1±f p (q)] σ nl p (q).

22 The temperature regiont mv m D : 1S dissociation by parton scattering σ 1S p = σ cp f(m D a 0 ) σ cq 8πC F n f α 2 s a2 0, σ cg 8πC F N c α 2 s a2 0 f(m D a 0 ) 2 (m D a 0 ) 2 [1 4 (m Da 0 ) (m D a 0 ) 2 ln ( 4/(m D a 0 ) 2) ] ((m D a 0 ) 2 4) 3 The cross section for inelastic parton scattering is not related, even at leading order, with a zero temperature process. The underlying reason is the infrared sensitivity of the cross section at the momentum scale mv. This IR sensitivity is cured by the HTL resummation at the scale m D and signaled by ln(m D a 0 ). The cross section is momentum independent.

23 Quasi-free approximation The quasi-free approximation amounts at replacing σ nl p by 2σ Q p ; σ Q p is the free in-vacuum cross section p+q p+q (with thermal masses for IR regularization). The quasi-free approximation amounts at neglecting interference terms between the different heavy-quark lines in the amplitude square. Interference terms are the ones sensitive to the bound state. This corresponds in approximating f(m D a 0 ) 2 (m D a 0 ) 2 The approximation holds only for m D 1/a 0, which requires T mg T dissociation. For temperatures where the approximation holds, quarkonium is dissociated. Otherwise, the approximation is largely violated by bound-state effects. E.g. for m D 1/a 0 f(m D a 0 ) 2 (m D a 0 ) 2 ( [1/ 1/+(m D a 0 ) 2 3 ) ] 4 +ln(2/(m Da 0 )) +...

24 Quasi-free approximation vs full result In a weak-coupling framework, the quasi-free approximation is not justified for the whole range of temperatures where a quarkonium can exist. full result 100 quasi free approximation Σ p 1 S 10 Σ cp m D a 0

25 The temperature regiont mv m D The gluon self-energy contributes to the imaginary part of the pnrqcd potential through: T, mv T, mv T, mv and the pnrqcd diagram m D

26 The temperature regiont mv m D α s ReV s (r) = C F r { ImV s (r) = g4 C F n f r 2 T 3 ( ) 8 π γ E ln(r 2 m 2 D ) d 3 [ q ( ) dt sin(tr) + (2π) 3n F(q)[1 n F (q)] 2q t tr { g4 C F N c r 2 T 3 ( ) 8 π γ E ln(r 2 m 2 D ) d 3 [ q ( ) dt sin(tr) + (2π) 3n B(q)[1+n B (q)] 2q t tr 1 ( sin 2 )]} (qr) 4q 2 (qr) 2 1 2q 0 2q 0 dt 4tq 2 dt 2tq 2 ( sin(tr) tr ( sin(tr) tr )]} 1 ) 1 q u a r k g l u o n The real part of the potential is the Coulomb potential The imaginary part of the potential gets contributions from the scales mv T and m D. The term ln(rm D ) signals the cancellation of divergences between the two scales.

27 The temperature regiont mv m D : 1S dissociation by parton scattering σ 1S p = σ cp h p (m D a 0,qa 0 ) ( (md a 0 ) 2 h q (m D a 0,qa 0 ) ln ) 32 ( 4 +ln (qa0 ) 2 ) 1+(qa 0 ) 2 1 2(qa 0 ) 2 ln(1+(qa 0) 2 ) ( (md a 0 ) 2 h g (m D a 0,qa 0 ) ln ) 32 ( 4 +ln (qa0 ) 2 ) 1+(qa 0 ) (1+(qa 0 ) 2 ) 1 (qa 0 ) 2 ln(1+(qa 0) 2 ) The cross section is momentum dependent.

28 The temperature regionmv T m D mv 2 : 1S dissociation by parton scattering The gluon self-energy contributes to the imaginary part of the pnrqcd potential through: T, m D leading to dissociation cross sections for 1S Coulombic states: ( ) ] σp 1S 4q (q) = σ cp [ln 2 2 m 2 D

29 Dissociation by quark inelastic scattering 5 4 Σ q 1 S Σ cq 3 2 m D a 0 = 0.1 T mv m D 1 mv T m D mv 2 T mv m D q a 0

30 Dissociation by gluon inelastic scattering 5 4 Σ g 1 S Σ cg 3 2 m D a 0 = 0.1 T mv m D 1 mv T m D mv 2 T mv m D q a 0

31 The temperature regionmv T mv 2 m D The gluon self-energy gives rise to a quarkonium width through: T,mv 2 which contributes to dissociation by inelastic parton scattering, and through mv 2 which contributes to gluodissociation.

32 Integrating outmv 2 The relevant diagram is k where the loop momentum region is k 0 mv 2 and k mv 2. Gluons are HTL gluons. Since k mv 2 m D, the HTL propagators can be expanded in m D /mv 2 1.

33 Integrating outmv 2 : momentum regions In the loop with transverse gluons, this type of integral appears d D 1 k (2π) D 1 0 dk 0 2π 1 k 2 0 k2 m 2 D +iη ( 1 E H o k 0 +iη + 1 E H o +k 0 +iη ) which exhibits two momentum regions for E mv 2 off-shell region: k 0 k E, k 0 E, k E; collinear region: k 0 k m 2 D /E, k 0 E, k E. In our energy scale hierarchy, the collinear scale is mv 2 m 2 D /E mv3, i.e. it is smaller than m D by a factor of m D /mv 2 1 and still larger than the non-perturbative magnetic mass, which is of order g 2 T, by a factor T/mv 2 1.

34 The temperature regionmv T mv 2 m D : 1S dissociation by parton scattering The dissociation cross section by parton scattering for 1S Coulombic states is ( ) ] σp 1S 4q (q) = σ cp [ln 2 +ln2 2 m 2 D The corresponding parton-scattering decay width is of order α s T (m D /mv) 2.

35 The temperature regionmv T mv 2 m D : 1S gluodissociation at LO mv 2 The LO gluodissociation cross section for 1S Coulombic states is σ 1S gluolo (q) = α sc F 3 ( ) 2 10 π 2 ρ(ρ+2) 2 E4 ( exp 4ρ 1 t(q) 2 mq 5 +ρ 2) t(q) arctan(t(q)) e 2πρ t(q) 1 where ρ 1/(N 2 c 1), t(q) q/ E 1 1 and E 1 = mc 2 F α2 s/4. The corresponding gluodissociation decay width is of order α s T (mv 2 /mv) 2. The gluodissociation width is larger by a factor (mv 2 /m D ) 2 than the dissociation width by inelastic parton scattering. Gluodissociation is the dominant process.

36 Bhanot Peskin approximation In the large N c limit: σgluolo 1S (q) N c Γ 1S LO N c πα s E 1 5/2 (q +E 1 ) 3/ m q 5 q E 1 = 16σ 1S BP (q) d 3 q (2π) 3n B(q)16σ 1S BP (q) = Γ 1S,BP The Bhanot Peskin approximation corresponds to neglecting final state interactions, i.e. the rescattering of a QQ pair in a color octet configuration (recall V o = 1/(2N c ) α s /r).

37 Bhanot Peskin cross section vs full cross section 1.13 full result BP approx for E 1 exact BP approx for E 1 approx

38 Bhanot Peskin width vs full width full result high T full result BP approx high T BP approx

39 The temperature regionmv T mv 2 m D : gluodissociation at NLO mv 2 The NLO gluodissociation cross section for 1S Coulombic states is σ nl gluo (q) = Z(q/m D)σ nl gluolo (q) Z(q/m D ) = 1 m2 D 4q 2 [ ln(8q 2 /m 2 D ) 2]

40 1S gluodissociation at LO vs NLO LO NLO for m D ma 2 0 = 0.2 NLO for m D ma 2 0 = S Σ gluo a 0 m q E 1

41 Conclusions In a framework that makes close contact with modern effective field theories for non relativistic bound states at zero temperature, one can study the dissociation of a quarkonium in a thermal bath of gluons and light quarks. In a weakly-coupled framework, the situation is the following. For T < E bin the potential coincides with the T = 0 potential. For T > E bin the potential gets thermal contributions. For E bin > m D quarkonium decays dominantly via gluodissociation (aka singlet-to-octet break up). For m D > E bin quarkonium decays dominantly via inelastic parton scattering (aka Landau damping). For T T dissociation < T screening, quarkonium cannot be formed. In a strongly-coupled framework, the hierarchy of non-relativistic scales is preserved, whereas the thermodynamical hierarchy may break down. This requires a non-perturbative definition and evaluation of the potential (real and imaginary).