Heavy quarkonium in a weaky-coupled plasma in an EFT approach

Transcription

1 Heavy quarkonium in a weaky-coupled plasma in an EFT approach Jacopo Ghiglieri TU München & Excellence Cluster Universe in collaboration with N. Brambilla, M.A. Escobedo, J. Soto and A. Vairo 474th WE Heraeus Seminar, Bad Honnef, February 5th 20

2 Outline Quarkonium in media: the EFT approach Building blocks: non-relativistic EFTs at zero temperature and the real-time formalism Integrating out in succession the relevant scales and calculating the spectrum and width of quarkonium Conclusions and outlook

3 The structure: Charmonium dissociation at SPS and RHIC Quarkonium suppression In!In, SPS Pb!Pb, SPS Au!Au, RHIC, y < 0.35 Au!Au, RHIC, y =[.2,2.2] 0.25 {ψ/ncoll (AB)} / {ψ/ncoll (pp)} Experimental data show a suppression pattern S(J/!) 2 3 " (GeV/fm 3) PHENIX d-au y=0 0.4 PHENIX Au-Au y=0 (preliminary) NA38 S-U 200 GeV ycms=0.5 NA50 Pb-Pb 58 GeV ycms= NA60 In-In 58 GeV ycms=0.5 (preliminary) Npart

4 The structure: Charmonium dissociation at SPS and RHIC Quarkonium suppression A good understanding of suppression requires understanding of In!In, SPS Pb!Pb, SPS Au!Au, RHIC, y < 0.35 Au!Au, RHIC, y =[.2,2.2] 0.25 {ψ/ncoll (AB)} / {ψ/ncoll (pp)} Experimental data show a suppression pattern S(J/!) 2 3 " (GeV/fm 3) PHENIX d-au y=0 0.4 PHENIX Au-Au y=0 (preliminary) NA38 S-U 200 GeV ycms=0.5 NA50 Pb-Pb 58 GeV ycms= NA60 In-In 58 GeV ycms=0.5 (preliminary) Npart

5 The structure: Charmonium dissociation at SPS and RHIC Quarkonium suppression A good understanding of suppression requires understanding of In-medium production and cold nuclear matter effects In!In, SPS Pb!Pb, SPS Au!Au, RHIC, y < 0.35 Au!Au, RHIC, y =[.2,2.2] 0.25 {ψ/ncoll (AB)} / {ψ/ncoll (pp)} Experimental data show a suppression pattern S(J/!) 2 3 " (GeV/fm 3) PHENIX d-au y=0 0.4 PHENIX Au-Au y=0 (preliminary) NA38 S-U 200 GeV ycms=0.5 NA50 Pb-Pb 58 GeV ycms= NA60 In-In 58 GeV ycms=0.5 (preliminary) Npart

6 The structure: Charmonium dissociation at SPS and RHIC Quarkonium suppression A good understanding of suppression requires understanding of In-medium production and cold nuclear matter effects In-medium bound-state dynamics In!In, SPS Pb!Pb, SPS Au!Au, RHIC, y < 0.35 Au!Au, RHIC, y =[.2,2.2] 0.25 {ψ/ncoll (AB)} / {ψ/ncoll (pp)} Experimental data show a suppression pattern S(J/!) 2 3 " (GeV/fm 3) PHENIX d-au y=0 0.4 PHENIX Au-Au y=0 (preliminary) NA38 S-U 200 GeV ycms=0.5 NA50 Pb-Pb 58 GeV ycms= NA60 In-In 58 GeV ycms=0.5 (preliminary) Npart

7 The structure: Charmonium dissociation at SPS and RHIC Quarkonium suppression A good understanding of suppression requires understanding of In-medium production and cold nuclear matter effects In-medium bound-state dynamics In!In, SPS Pb!Pb, SPS Au!Au, RHIC, y < 0.35 Au!Au, RHIC, y =[.2,2.2] 0.25 {ψ/ncoll (AB)} / {ψ/ncoll (pp)} Experimental data show a suppression pattern S(J/!) 2 3 " (GeV/fm 3) PHENIX d-au y=0 Recombination effects 0.4 PHENIX Au-Au y=0 (preliminary) NA38 S-U 200 GeV ycms=0.5 NA50 Pb-Pb 58 GeV ycms= NA60 In-In 58 GeV ycms=0.5 (preliminary) Npart

8 The structure: Charmonium dissociation at SPS and RHIC Quarkonium suppression A good understanding of suppression requires understanding of In-medium production and cold nuclear matter effects In-medium bound-state dynamics In!In, SPS Pb!Pb, SPS Au!Au, RHIC, y < 0.35 Au!Au, RHIC, y =[.2,2.2] 0.25 {ψ/ncoll (AB)} / {ψ/ncoll (pp)} Experimental data show a suppression pattern S(J/!) 2 3 " (GeV/fm 3) PHENIX d-au y=0 Recombination effects 0.4 PHENIX Au-Au y=0 (preliminary) NA38 S-U 200 GeV ycms=0.5 NA50 Pb-Pb 58 GeV ycms= NA60 In-In 58 GeV ycms=0.5 (preliminary) Npart

9 Quarkonium suppression Proposed in 986 as a probe and thermometer of the medium produced by the collision Matsui Satz PLB78 (986)

10 Quarkonium suppression Proposed in 986 as a probe and thermometer of the medium produced by the collision Matsui Satz PLB78 (986) Motivated by colour screening of the interaction

11 Quarkonium suppression Proposed in 986 as a probe and thermometer of the medium produced by the collision Matsui Satz PLB78 (986) Motivated by colour screening of the interaction

12 Quarkonium suppression Proposed in 986 as a probe and thermometer of the medium produced by the collision Matsui Satz PLB78 (986) Motivated by colour screening of the interaction

13 Quarkonium suppression Proposed in 986 as a probe and thermometer of the medium produced by the collision Matsui Satz PLB78 (986) Motivated by colour screening of the interaction V (r) αs md r e r

14 Quarkonium suppression Proposed in 986 as a probe and thermometer of the medium produced by the collision Matsui Satz PLB78 (986) Motivated by colour screening of the interaction V (r) αs r md md r e r Bound state dissolves

15 Quarkonium suppression Proposed in 986 as a probe and thermometer of the medium produced by the collision Matsui Satz PLB78 (986) Motivated by colour screening of the interaction V (r) αs r md md r e r Bound state dissolves Studied with potential models, lattice spectral functions,...

16 Potential models Digal, Petreczky, Satz 0 Wong Mannarelli, Rapp 05 Mocsy, Petreczky Alberico, Beraudo, Molinari, de Pace Cabrera, Rapp 2007 Wong, Crater 07 Dumitru, Guo, Mocsy, Strickland 09 Rapp, Riek F [MeV] r [fm] 0.76T c 0.8T c 0.90T c 0.96T c.00t c.02t c.07t c.23t c.50t c.98t c 4.0T c Kaczmarek Zantow T/T C 2.2 & /!r" #(S) % b (P) J/$(S) % c (P) There is qualitative agreement on this picture of sequential melting However there are still issues with the definition of an inmedium potential

17 The method: an EFT approach Perturbative computation of the real-time potential between a static quark and antiquark for T>>/r: V HTL (r) = α s C F e m D r When m D ImV ReV Laine Philipsen Romatschke Tassler JHEP0703 (2007) r r i 2T m D r f(m Dr)

18 The method: an EFT approach Perturbative computation of the real-time potential between a static quark and antiquark for T>>/r: V HTL (r) = α s C F e m D r When m D ImV ReV Laine Philipsen Romatschke Tassler JHEP0703 (2007) r EFT approach for standard and muonic hydrogen at finite temperature within pnrqed (Abelian, non-static) Escobedo Soto PRA78 (2008), PRA82 (200) r i 2T m D r f(m Dr)

19 The method: an EFT approach Perturbative computation of the real-time potential between a static quark and antiquark for T>>/r: V HTL (r) = α s C F e m D r When m D ImV ReV Laine Philipsen Romatschke Tassler JHEP0703 (2007) r EFT approach for standard and muonic hydrogen at finite temperature within pnrqed (Abelian, non-static) Escobedo Soto PRA78 (2008), PRA82 (200) EFT approach for static quark-antiquark pairs in finite T pnrqcd (Non-Abelian, static) Brambilla JG Petreczky Vairo PRD78 (2008) r i 2T m D r f(m Dr)

20 The method: an EFT approach Perturbative computation of the real-time potential between a static quark and antiquark for T>>/r: V HTL (r) = α s C F e m D r When m D ImV ReV Laine Philipsen Romatschke Tassler JHEP0703 (2007) r r i 2T m D r f(m Dr) EFT approach for heavy quark-antiquark pairs in finite T pnrqcd (Non-Abelian, non-static) Brambilla Escobedo JG Soto Vairo JHEP009 (200)

21 The EFT approach Extend the well-established zero temperature EFT framework for heavy quark-antiquark bound states (NRQCD, pnrqcd) to finite temperature Systematic approach with modern, rigorous definition of the potential (real-time matching coefficient) Transparent connection with quantum mechanics and the Schrödinger equation picture, which appears as the zeroth-order approximation in the EFT framework

22 Building blocks: NR EFTs and the real-time formalism

23 Effective Field Theories EFTs prove to be a valuable tool for physical problems characterized by various sufficiently separated energy/ momentum scales An EFT is constructed by integrating out modes of energy and momentum larger than the cut-off µ Λ O n c n (µ/λ) Λ d n 4 L EFT = n Wilson coefficient Low-energy operator/ large scale The Wilson coefficient are obtained by matching appropriate Green functions in the two theories The procedure can be iterated... µ 2 Λ 2 µ Λ

24 T=0 NR bound states Non-relativistic QQ bound states are characterized by the hierarchy of the mass, momentum and energy scales m mv mα s r mv 2 mα 2 s E

25 T=0 NR bound states Non-relativistic QQ bound states are characterized by the hierarchy of the mass, momentum and energy scales One can then expand observables in terms of the ratio of the scales and construct a hierarchy of EFTs that are equivalent to QCD orderby-order in the expansion parameter m mv mα s r mv 2 mα 2 s E

26 T=0 Scales m mα s r Integration of the mass scale: NRQCD mα 2 s E

27 T=0 Scales m mα s r Integration of the soft (momentum transfer) scale: pnrqcd mα 2 s E

28 T=0 Scales m mα s r Λ QCD Integration of the soft (momentum transfer) scale: pnrqcd mα 2 s E

29 T=0 Scales m mα s r Integration of the soft (momentum transfer) scale: pnrqcd mα 2 s E Λ QCD

30 Thermodynamical scales The thermal medium introduces new scales in the physical problem T The temperature The electric screening scale (Debye mass) gt m D The magnetic screening scale (magnetic mass) In the weak coupling assumption these scales develop a hierarchy g 2 T m m

31 Scales of the problem m T mα s r? gt m D mα 2 s E Λ QCD g 2 T m m

32 In our previous works various possibilities have been studied, from T E to m T /r m D In the regime T the result of Laine et al r m D 2007 is reobtained (also in the Abelian case) V HTL (r) = α s C F e m D r r i 2T m D r f(m Dr)

33 In our previous works various possibilities have been studied, from T E to m T /r m D In the regime T the result of Laine et al r m D 2007 is reobtained (also in the Abelian case) V HTL (r) = α s C F e m D r In this new work we adopt this hierarchy r i 2T m D r f(m Dr) m mα s T mα 2 s m D. m mα s T mα 2 s m D Λ QCD

34 In our previous works various possibilities have been studied, from T E to m T /r m D In the regime T the result of Laine et al r m D 2007 is reobtained (also in the Abelian case) V HTL (r) = α s C F e m D r In this new work we adopt this hierarchy Thermal effects are thus a perturbation to the Coulombic bound state, but they still modify the potential r i 2T m D r f(m Dr) m mα s T mα 2 s m D. m mα s T mα 2 s m D Λ QCD

35 In our previous works various possibilities have been studied, from T E to m T /r m D In the regime T the result of Laine et al r m D 2007 is reobtained (also in the Abelian case) V HTL (r) = α s C F e m D r In this new work we adopt this hierarchy Thermal effects are thus a perturbation to the Coulombic bound state, but they still modify the potential Relevance for bottomonium phenomenology r i 2T m D r f(m Dr) m mα s T mα 2 s m D. m mα s T mαs 2 m D Λ QCD

36 In our previous works various possibilities have been studied, from T E to m T /r m D In the regime T the result of Laine et al r m D 2007 is reobtained (also in the Abelian case) V HTL (r) = α s C F e m D r In this new work we adopt this hierarchy Thermal effects are thus a perturbation to the Coulombic bound state, but they still modify the potential Relevance for bottomonium phenomenology The temperature is below the dissociation temperature r i 2T m D r f(m Dr) m mα s T mα 2 s m D. T d mα 2 3 s m mα s T mαs 2 m D Λ QCD

37 Our goal m Our goal is then to compute the spectrum (or thermal corrections to it) mα s T mαs 2 m D and the thermal width in this hierarchy and power counting, up to order mα 5 s This will be achieved by integrating out the heavier scales in succession and creating a series of EFTs Before starting, a reminder of the RTF Λ QCD

38 The real-time formalism Time evolution along this Schwinger- Keldysh path t i limit Doubling of the degrees of freedom

39 The real-time formalism Time evolution Real d.o.f. () along this Schwinger- Keldysh path t i limit Doubling of the degrees of freedom

40 The real-time formalism Time evolution Real d.o.f. () along this Schwinger- Keldysh path t i limit Virtual d.o.f. (2) Doubling of the degrees of freedom

41 The real-time formalism Closer to the T=0 EFT approach In the static quark sector the virtual DOF decouple Simpler than the imaginary-time + analytical continuation (potential as pole of the propagator in the large real-time limit) Propagators as 2X2 matrices, vertices of type 2 have opposite sign

42 Free propagators Propagators written as D D D = 2 D 2 D 22 Static quark S (0) αβ (k 0)=δ αβ i k 0 + iη 2πδ(k 0 ) 0 i k 0 iη (static quark of type 2 decouple) Gluons (Coulomb gauge) D (0) 00 (k) = i k i k 2 ( ) D (0) ij (k 0,k)= δ ij ki k j k 2 i k 2 0 k2 + iη θ(k 0 )2πδ(k 2 0 k 2 ) θ( k 0 )2πδ(k0 2 k 2 ) i +2πδ(k0 2 k 2 ) n B ( k 0 ) k0 2 k2 iη

43 m mα s Integrating out and.

44 Mass scale: NRQCD m Smaller scales are put to zero in the matching. This means that the thermal scales do not affect the result of the mα s T integration, which yields standard NRQCD NRQCD is organized as an expansion mα 2 s in /m Caswell Lepage PLB67 (986) m D Bodwin Braaten Lepage PRD5 (995) Λ QCD

45 Mass scale: NRQCD m L NRQCD = L YM + L light + ψ id 0 + D2 2m +... ψ + χ (id 0 D2 2m +...)χ +... mα s T mα 2 s m D Λ QCD... Caswell Lepage PLB67 (986) Bodwin Braaten Lepage PRD5 (995)

46 Relative momentum scale: pnrqcd m mα s T mα 2 s m D Λ QCD E p 2 /m V (r) Pineda Soto Nucl. Phys. Proc. Suppl. 64(998) Brambilla Pineda Soto Vairo NPB566 (2000)

47 Weakly coupled pnrqcd L pnrqcd = L YM + L light +Tr S [i 0 H s ] S + O [id 0 H o ] O +V A Tr O r ge S+S r ge O + V B 2 Tr O r ge O+O Or ge +... Degrees of freedom: and momentum Singlet and octet color states QQ states with energy E Λ QCD, mv 2 p mv US gluons with energy/momentum mv α s m Expansion in, and Potentials are Wilson coefficients, receive contributions from all scales higher than the energy r

48 Contribution to the spectrum Power-counting in the singlet Hamiltonian H s = p2 m C F α s r + O(mα3 s )

49 Contribution to the spectrum Power-counting in the singlet Hamiltonian H s = p2 m C F α s r + O(mα3 s ) Solve the singlet EOM (Schrödinger Eq.) with this Hamiltonian and obtain the Coulomb levels E n = mc2 F α2 s 4n 2 = ma 2 0 n2, a 0 2 mc F α s

50 Contribution to the spectrum Power-counting in the singlet Hamiltonian H s = p2 m C F Solve the singlet EOM (Schrödinger Eq.) with this Hamiltonian and obtain the Coulomb levels E n = mc2 F α2 s 4n 2 = ma 2 0 n2, α s r + O(mα3 s ) a 0 Treat further terms in the expansion in QM perturbation theory, obtaining the mα contributions to the s 3 mαs 5 spectrum from this scale Brambilla Pineda Soto Vairo PLB470 (999) Kniehl Penin NPB563 (999) Kniehl Penin Smirnov Steinhauser NPB635 (2002) 2 mc F α s

51 Contribution to the spectrum Power-counting in the singlet Hamiltonian H s = p2 m C F Solve the singlet EOM (Schrödinger Eq.) with this Hamiltonian and obtain the Coulomb levels E n = mc2 F α2 s 4n 2 = ma 2 0 n2, α s r + O(mα3 s ) a 0 Treat further terms in the expansion in QM perturbation theory, obtaining the mα contributions to the s 3 mαs 5 spectrum from this scale Brambilla Pineda Soto Vairo PLB470 (999) Kniehl Penin NPB563 (999) Kniehl Penin Smirnov Steinhauser NPB635 (2002) 2 mc F α s IR divergences manifest themselves at mα 5 s

52 Divergent contributions to the spectrum Scale Vacuum Thermal mα s mα 5 s IR / T mα 2 s

53 Integrating out the temperature

54 pnrqcd HTL m mα s T mαs 2 m D Λ QCD Integrating out the temperature brings us to a new EFT called pnrqcd HTL In this EFT modes with energy and momenta of the order of the temperature are no longer present. This amounts to: Hard Thermal Loop (HTL) resummation in the gauge and light quark sectors Braaten Pisarski PRD45 (992) Thermal modifications to the potentials in the singlet and octet sector This EFT and its matching coefficients are independent of the hierarchy of low-lying scales

55 The pnrqcd HTL Lagrangian L pnrqcdhtl = L HTL +Tr S [i 0 H s δv s ] S + O [id 0 H o δv o ] O +Tr O r ge S+S r ge O + 2 Tr O r ge O+O Or ge +... The new matching coefficients in the singlet and octet sectors have to be computed by matching pnrqcd to pnrqcd HTL We only do this for the singlet sector, which is the only one relevant for the spectrum at our accuracy In the matching procedure we have to single out the contribution from the scale T: this will be achieved by appropriate expansions in E/T and m D /T

56 Matching pnrqcd to pnrqcd HTL The leading contribution to vertices δv s comes from the dipole It gives rise to a correction to the potential and to the spectrum Subleading contributions come from the radiative correction to that diagram It gives rise to an imaginary part, corresponding to a thermal width. The phenomenon is called Landau damping

57 Matching pnrqcd to pnrqcd HTL The leading contribution to vertices δv s comes from the dipole It gives rise to a correction to the potential and to the spectrum Subleading contributions come from the radiative correction to that diagram It gives rise to an imaginary part, corresponding to a thermal width. The phenomenon is called Landau damping

58 Matching pnrqcd to pnrqcd HTL The leading contribution to vertices r i δv s d D k (2π) D comes from the dipole i E H o k 0 + iη k0 2 D (0) ii + k 2 D (0) 00 r i

59 Matching pnrqcd to pnrqcd HTL The leading contribution to vertices ig 2 C F D 2 D ri µ 4 D d D k (2π) D r i i E H o k 0 + iη k2 0 δv s d D k (2π) D comes from the dipole i E H o k 0 + iη k0 2 D (0) ii + k 2 D (0) 00 r i i k 2 0 k2 + iη + 2πδ k 2 0 k 2 n B ( k 0 ) r i

60 Matching pnrqcd to pnrqcd HTL The leading contribution to vertices ig 2 C F D 2 D ri µ 4 D d D k (2π) D r i i E H o k 0 + iη k2 0 δv s d D k (2π) D comes from the dipole i E H o k 0 + iη k0 2 D (0) ii + k 2 D (0) 00 r i i k 2 0 k2 + iη + 2πδ k 2 0 k 2 n B ( k 0 ) The octet propagator must be expanded in k 0 T (E H o ) r i

61 Matching pnrqcd to pnrqcd HTL The leading contribution to vertices ig 2 C F D 2 D ri µ 4 D d D k (2π) D r i i E H o k 0 + iη k2 0 δv s d D k (2π) D comes from the dipole i E H o k 0 + iη k0 2 D (0) ii + k 2 D (0) 00 r i i k 2 0 k2 + iη + 2πδ k 2 0 k 2 n B ( k 0 ) r i The octet propagator must be expanded in k 0 T (E H o ) The vacuum part then yields a series of scaleless integrals In the thermal part the linear and cubic terms (in the energy) contribute within our accuracy, whereas the quadratic term vanishes in D.R.

62 Results: the potential The correction to the potential from the temperature scale is, when T mα s δv s = π 9 N cc F αs 2 T 2 r + 2π 3m C F α s T 2 + α sc F I T 3π α s N 3 c 8 αs 3 r (N c 2 +2N c C F ) α2 s mr 2 +4(N c 2C F ) πα s m 2 δ3 (r)+n c α s m 2 2 r, r 3 2 ζ(3) C F π r2 Tm 2 D ζ(3) N cc F αs 2 r 2 T 3 CF +i 6 α s r 2 Tm 2 D 2 + γ E +lnπ ln T 2 µ ln2 (2) 2ζ ζ(2) + 4π 9 ln 2 N cc F αs 2 r 2 T 3 m 2 D = g2 T 2 3 N + n f 2 Both the real and the imaginary part are IR divergent; the divergences cancel in the related physical observables, the spectrum and the width, against UV divergences from some lower scale

63 Divergent contributions to the spectrum Scale Vacuum Thermal mα s mα 5 s IR / T scaleless mα 5 s IR mα 2 s

64 Divergent contributions to the spectrum Scale Vacuum Thermal mα s mα 5 s IR / T mα 5 s IR UV mα 5 s IR mα 2 s

65 Divergent contributions to the spectrum Scale Vacuum Thermal mα s mα 5 s IR / T mα 5 s IR UV mα 5 s IR mα 2 s

66 Divergent contributions to the spectrum Scale Vacuum Thermal mα s mα 5 s IR / T mα 5 s IR UV mα 5 s IR mα 2 s

67 Results: spectrum and width The corresponding correction to the spectrum is δe (T ) n,l = π 9 N cc F α 2 s T 2 a 0 2 (3n2 l(l +))+ 2π 3m C F α s T 2 + E ni T α 3 { s 4C 3 ( F δ l0 + N c C 2 8 F 3π n n(2l +) n 2 2δ ) l0 + 2N c 2 C F n n(2l +) + N c 3 } 4 ( ζ(3) C α s F π Tm2 D + 2 ) a 3 ζ(3) N cc F α 2 s T n 2 [ 5n 2 + 3l(l +) ] [ ]. 2 The width is (2 loops) Γ n,l = [ ] [ [ C F 6 α stm 2 D ( 2ɛ + γ E +lnπ ln T 2 µ ) 3 4ln2 (2) 2ζ ζ(2) 4π 9 ln 2 N cc F α 2 s T 3 ] a 2 0 n2 [ 5n 2 + 3l(l +) ].

68 Calculating the contribution from the energy scale

69 Calculating at the energy scale m mα s T mαs 2 m D Since we have reached the binding energy scale we cannot proceed further with EFTs in the heavy quark-antiquark sector We thus calculate the contributions to the spectrum and the width within pnrqcd HTL We thus have to use the HTL-resummed propagators Λ QCD We employ the expansions T mα 2 s m D

70 The calculation Only the leading order diagram contributes within our accuracy r i d D k (2π) D i E H o k 0 + iη k 2 0 D HTL ii + k 2 D00 HTL r i

71 The calculation Only the leading order diagram contributes within our accuracy r i d D k (2π) D i E H o k 0 + iη k 2 0 D HTL ii + k 2 D00 HTL r i This diagram introduces a second source of imaginary parts! thermal width. It is the singlet-to-octet decay width

72 The calculation Only the leading order diagram contributes within our accuracy r i d D k (2π) D i E H o k 0 + iη k 2 0 D HTL ii + k 2 D00 HTL r i This diagram introduces a second source of imaginary parts! thermal width. It is the singlet-to-octet decay width

73 The calculation Only the leading order diagram contributes within our accuracy r i d D k (2π) D i E H o k 0 + iη k 2 0 D HTL ii + k 2 D00 HTL r i This diagram introduces a second source of imaginary parts! thermal width. It is the singlet-to-octet decay width The calculation is quite involved, requiring expansions in the propagators and in the Bose distribution

74 The calculation Only the leading order diagram contributes within our accuracy r i d D k (2π) D i E H o k 0 + iη k 2 0 D HTL ii + k 2 D00 HTL r i This diagram introduces a second source of imaginary parts! thermal width. It is the singlet-to-octet decay width The calculation is quite involved, requiring expansions in the propagators and in the Bose distribution UV divergences appear and cancel the IR divergencies from higher scales

75 Divergent contributions to the spectrum Scale Vacuum Thermal mα s mα 5 s IR / T mα 5 s IR UV mα 5 s IR mα 2 s mα 5 s UV mα 5 s UV

76 Summary and conclusions

77 The spectrum The contribution of the thermal medium only to the spectrum is δe n,l = π 9 N cc F αs 2 T 2 a 0 3n 2 l(l + ) + π 2 3 C2 F αs 2 T 2 a 0 + E nαs 3 2 4C 2πT 3 log 2γ F δ l0 E + N c C 2 8 F 3π n n(2l + ) n 2 2δ l0 n + a2 0n 2 2 E + 2N c 2 C F n(2l + ) + N c 3 4 5n 2 + 3l(l + ) 3 2π ζ(3) + π 3 C F α s Tm 2 D + 2E nc 3 F α3 s 3π L n,l ζ(3) N cc F α 2 s T 3

78 The spectrum The contribution of the thermal medium only to the spectrum is δe n,l = π 9 N cc F αs 2 T 2 a 0 3n 2 l(l + ) + π 2 3 C2 F αs 2 T 2 a 0 + E nαs 3 2 4C 2πT 3 log 2γ F δ l0 E + N c C 2 8 F 3π n + a2 0n 2 2 E + 2N c 2 C F n(2l + ) + N c 3 4 5n 2 + 3l(l + ) 3 2π ζ(3) + π 3 n(2l + ) n 2 2δ l0 n + 2E ncf 3 α3 s L n,l 3π C F α s Tm 2 D mα 5 s T 2 E ζ(3) N cc F α 2 s T 3

79 The spectrum The contribution of the thermal medium only to the spectrum is mα 5 s { δe n,l = π 9 N cc F αs 2 T 2 a 0 3n 2 l(l + ) + π 2 3 C2 F αs 2 T 2 a 0 mαs 5 + E nαs 3 2 4C 2πT 3 log 2γ F δ l0 E + N c CF 2 8 3π E n n(2l + ) n 2 2δ l0 n + 2N c 2 C F n(2l + ) + N c 3 + 2E ncf 3 α3 s L n,l 4 3π + a2 0n 2 2 5n 2 + 3l(l + ) 3 2π ζ(3) + π 3 C F α s Tm 2 D T 2 E ζ(3) N cc F α 2 s T 3

80 The spectrum The contribution of the thermal medium only to the spectrum is mα 5 s { mα 6 s T 3 E 3 δe n,l = π 9 N cc F αs 2 T 2 a 0 3n 2 l(l + ) + π 2 3 C2 F αs 2 T 2 a 0 mαs 5 + E nαs 3 2 4C 2πT 3 log 2γ F δ l0 E + N c CF 2 8 3π E n n(2l + ) n 2 2δ l0 n + 2N c 2 C F n(2l + ) + N c 3 + 2E ncf 3 α3 s L n,l 4 3π + a2 0n 2 2 5n 2 + 3l(l + ) 3 2π ζ(3) + π 3 C F α s Tm 2 D T 2 E ζ(3) N cc F α 2 s T 3

81 The spectrum The contribution of the thermal medium only to the spectrum is mα 5 s { mα 6 s T 3 E 3 δe n,l = π 9 N cc F αs 2 T 2 a 0 3n 2 l(l + ) + π 2 3 C2 F αs 2 T 2 a 0 mαs 5 + E nαs 3 2 4C 2πT 3 log 2γ F δ l0 E + N c CF 2 8 3π E n n(2l + ) n 2 2δ l0 n + 2N c 2 C F n(2l + ) + N c 3 + 2E ncf 3 α3 s L n,l 4 3π + a2 0n 2 2 5n 2 + 3l(l + ) 3 2π ζ(3) + π 3 C F α s Tm 2 D T 2 E ζ(3) N cc F α 2 s T 3 The leading terms thus indicate that the mass of the bound state increases quadratically with the temperature. Up to this order the thermal shift to the spectrum is independent of the spin

82 The thermal width The contribution of the thermal medium to the width is Γ n,l = 3 N c 2 C F αs 3 T + 4 CF 2 α3 s T 3 n 2 (C F + N c ) + 2E nαs 3 4C 3 F δ l0 + N c CF 2 3 n CF 6 α stm 2 D C F α s Tm 2 D a 2 0n 4 I n,l 8 n(2l + ) n 2 2δ l0 n ln E2 T 2 +2γ E 3 log 4 2 ζ (2) ζ(2) + 2N c 2 C F n(2l + ) + N c π9 ln 2 N cc F α 2s T 3 a 2 0n 2 5n 2 + 3l(l + )

83 The thermal width The contribution of the thermal medium to the width is Γ n,l = 3 N c 2 C F αs 3 T + 4 CF 2 α3 s T 3 n 2 (C F + N c ) + 2E nαs 3 4C 3 F δ l0 + N c CF 2 3 n CF 6 α stm 2 D C F α s Tm 2 D a 2 0n 4 I n,l 8 n(2l + ) n 2 2δ l0 n ln E2 T 2 +2γ E 3 log 4 2 ζ (2) ζ(2) + 2N c 2 C F n(2l + ) + N c π9 ln 2 N cc F α 2s T 3 a 2 0n 2 5n 2 + 3l(l + ) Two sources of decay: singlet-to-octet thermal breakup and Landau damping

84 The thermal width The contribution of the thermal medium to the width is Γ n,l = 3 N c 2 C F αs 3 T + 4 CF 2 α3 s T 3 n 2 (C F + N c ) + 2E nαs 3 4C 3 F δ l0 + N c CF 2 3 n CF 6 α stm 2 D C F α s Tm 2 D a 2 0n 4 I n,l mα 5 s 8 n(2l + ) n 2 2δ l0 n ln E2 T 2 +2γ E 3 log 4 2 ζ (2) ζ(2) T E + 2N c 2 C F n(2l + ) + N c π9 ln 2 N cc F α 2s T 3 a 2 0n 2 5n 2 + 3l(l + ) Two sources of decay: singlet-to-octet thermal breakup and Landau damping

85 The thermal width The contribution of the thermal medium to the width is mα 5 s Γ n,l = 3 N c 2 C F αs 3 T + 4 CF 2 α3 s T 3 n 2 (C F + N c ) + 2E nαs 3 4C 3 F δ l0 + N c CF 2 3 n CF 6 α stm 2 D C F α s Tm 2 D a 2 0n 4 I n,l mα 5 s 8 n(2l + ) n 2 2δ l0 n ln E2 T 2 +2γ E 3 log 4 2 ζ (2) ζ(2) T E + 2N c 2 C F n(2l + ) + N c π9 ln 2 N cc F α 2s T 3 a 2 0n 2 5n 2 + 3l(l + ) Two sources of decay: singlet-to-octet thermal breakup and Landau damping

86 mα 6 s The thermal width The contribution of the thermal medium to the width is mα 5 s { T 3 E 3 Γ n,l = 3 N c 2 C F αs 3 T + 4 CF 2 α3 s T 3 n 2 (C F + N c ) + 2E nαs 3 4C 3 F δ l0 + N c CF 2 3 n CF 6 α stm 2 D C F α s Tm 2 D a 2 0n 4 I n,l mα 5 s 8 n(2l + ) n 2 2δ l0 n ln E2 T 2 +2γ E 3 log 4 2 ζ (2) ζ(2) Two sources of decay: singlet-to-octet thermal breakup and Landau damping T E + 2N c 2 C F n(2l + ) + N c π9 ln 2 N cc F α 2s T 3 a 2 0n 2 5n 2 + 3l(l + )

87 mα 6 s The thermal width The contribution of the thermal medium to the width is mα 5 s { T 3 E 3 Γ n,l = 3 N c 2 C F αs 3 T + 4 CF 2 α3 s T 3 n 2 (C F + N c ) + 2E nαs 3 4C 3 F δ l0 + N c CF 2 3 n CF 6 α stm 2 D C F α s Tm 2 D a 2 0n 4 I n,l mα 5 s 8 n(2l + ) n 2 2δ l0 n ln E2 T 2 +2γ E 3 log 4 2 ζ (2) ζ(2) Two sources of decay: singlet-to-octet thermal breakup and Landau damping In our hiearchy the former is larger than the latter T E + 2N c 2 C F n(2l + ) + N c π9 ln 2 N cc F α 2s T 3 a 2 0n 2 5n 2 + 3l(l + )

88 mα 6 s The thermal width The contribution of the thermal medium to the width is mα 5 s { T 3 E 3 Γ n,l = 3 N c 2 C F αs 3 T + 4 CF 2 α3 s T 3 n 2 (C F + N c ) + 2E nαs 3 4C 3 F δ l0 + N c CF 2 3 n CF 6 α stm 2 D C F α s Tm 2 D a 2 0n 4 I n,l mα 5 s 8 n(2l + ) n 2 2δ l0 n ln E2 T 2 +2γ E 3 log 4 2 ζ (2) ζ(2) Two sources of decay: singlet-to-octet thermal breakup and Landau damping In our hiearchy the former is larger than the latter Up to this order the width as well is independent of the spin T E + 2N c 2 C F n(2l + ) + N c π9 ln 2 N cc F α 2s T 3 a 2 0n 2 5n 2 + 3l(l + )

89 Conclusions The method: we have shown how to construct a series of EFT to integrate out in succession the many scales that characterize a non-relativistic bound state at finite temperature We have obtained the contribution to the singlet potential from the scale T, when T mα s Back to the structure: we have obtained the contribution of the thermal bath to the spectrum and the width of heavy quarkonia, in a scale setup that can be relevant for the ground state of bottomonium

90