1. Theory of Thermal Electromagnetic Radiation Ralf Rapp Cyclotron Institute + Dept. of Physics & Astronomy Texas A&M University College Station, Texas USA JET Summer School The Ohio State University (Columbus, OH) June 12-14, 2013

2. 1.) Intro-I: Probing Strongly Interacting Matter Bulk Properties: Equation of State Phase Transitions: (Pseudo-) Order Parameters Microscopic Properties: - Degrees of Freedom - Spectral Functions  Would like to extract from Observables: temperature + transport properties of the matter signatures of deconfinement + chiral symmetry restoration in-medium modifications of excitations (spectral functions)

3. “Freeze-Out” QGP Au + Au 1.2 Dileptons in Heavy-Ion Collisions Hadron Gas NN-coll. Sources of Dilepton Emission: “primordial” qq annihilation (Drell-Yan): NN→e + e  X - e+e+ e-e-  thermal radiation - Quark-Gluon Plasma: qq → e + e , … - Hot+Dense Hadron Gas:       → e + e , … - final-state hadron decays:  ,  →  e + e , D,D → e + e   X , … _

4. 1.3 Schematic Dilepton Spectrum in HICs qq Characteristic regimes in invariant e + e  mass, M 2 = (p e+ + p e  ) 2 Drell-Yan: primordial, power law ~ M  n thermal radiation: - entire evolution - Boltzmann ~ exp(-M/T) q 0 ≈ 0.5GeV  T max ≈ 0.17GeV, q 0 ≈ 1.5GeV  T max ≈ 0.5GeV Thermal rate: Im Π em (M,q;  B,T)

5. Thermal e + e  emission rate from hot/dense matter ( em >> R nucleus ) Temperature? Degrees of freedom? Deconfinement? Chiral Restoration? 1.4 EM Spectral Function + QCD Phase Structure Electromagn. spectral function - √s ≤ 1 GeV : non-perturbative - √s > 1.5 GeV : pertubative (“dual”) Modifications of resonances ↔ phase structure: hadronic matter → Quark-Gluon Plasma? √s = M e + e  → hadrons ~ Im  em  / M 2 Im Π em (M,q;  B,T)

6. 1.5 Low-Mass Dileptons at CERN-SPS CERES/NA45 e + e  [2000] M ee [GeV] strong excess around M ≈ 0.5GeV, M > 1GeV quantitative description? NA60     [2005]

7. 1.6 Phase Transition(s) in Lattice QCD different “transition” temperatures?! extended transition regions partial chiral restoration in “hadronic phase”! (low-mass dileptons!) leading-order hadron gas T pc conf ~170MeV T pc chiral ~150MeV ≈  qq  /  qq  - [Fodor et al ’10]

8. 2.) Chiral Symmetry in QCD - Nonperturbative QCD, Chiral Breaking + Hadron Spectrum 3.) Thermal Electromagnetic Emission Rates - EM Spectral Function: Hadronic vs. Partonic Regimes 4.) Vector Mesons in Medium - Many-Body Theory, Spectral Functions + Chiral Partners (  -a 1 ) 5.) Quark-Gluon Plasma Emission - Perturbative vs. Lattice-QCD Rates, “Quark-Hadron-Duality” 6.) Dilepton + Photon Spectra in Heavy-Ion Collisions - Space-Time Evolution, Phenomenology + Interpretation 7.) Summary and Conclusions Outline

9. well tested at high energies, Q 2 > 1 GeV 2 : perturbation theory (  s = g 2 /4π << 1) degrees of freedom = quarks + gluons 2.1 Nonperturbative QCD (m u ≈ m d ≈ 5 MeV ) Q 2 ≤ 1 GeV 2 → transition to “strong” QCD: effective d.o.f. = hadrons (Confinement) massive “constituent quarks” m q * ≈ 350 MeV ≈ ⅓ M p (Chiral Symmetry ~ ‹0|qq|0› condensate! Breaking) ↕ ⅔ fm _

10. 2.2 Chiral Symmetry in QCD Lagrangian (bare quark masses: m u ≈ m d ≈ 5-10MeV ) Chiral SU(2) V × SU(2) A transformation: O Up to O (m q ), L QCD invariant under Rewrite L QCD using left- and right-handed quark fields q L,R =(1±  5 )/2 q : Invariance under isospin and “handedness” qLqL qLqL g

11. 2.3 Spontaneous Breaking of Chiral Symmetry strong qq attraction  Chiral Condensate fills QCD vacuum: > > > > qLqL qRqR qLqL - qRqR - [cf. Superconductor: ‹ee› ≠ 0, Magnet ‹ M › ≠ 0, … ] - Simple Effective Model: assume “mean-field”, expand: linearize: free energy: ground state: Gap Equation

12. 2.3.2 (Observable) Consequences of SBCS mass gap, not observable! but: hadronic excitations reflect SBCS “massless” Goldstone bosons  0,± (explicit breaking: f  2 m  2 = m q ‹qq› ) “chiral partners” split:  M ≈ 0.5GeV ! chiral trafo:, Vector mesons  and  : J P =0 ± 1 ± 1/2 ± chiral singlet - -i  ijk  k

13. quantify chiral breaking? 2.3.3 Manifestation of Chiral Symmetry Breaking Axial-/Vector Correlators pQCD cont. “Data”: lattice [Bowman et al ‘02] Theory: Instanton Model [Diakonov+Petrov; Shuryak ‘85] ● chiral breaking: |q 2 | ≤ 1 GeV 2 Constituent Quark Mass

14. 2.4 Chiral (Weinberg) Sum Rules Quantify chiral symmetry breaking via observable spectral functions Vector (  ) - Axialvector (a 1 ) spectral splitting [Weinberg ’67, Das et al ’67]  →(2n+1)  pQCD Key features of updated “fit”: [Hohler+RR ‘12]  + a 1 resonance, excited states (  ’+ a 1 ’), universal continuum (pQCD!)  →(2n)  [ALEPH ‘98, OPAL ‘99]

15. 2.4.2 Evaluation of Chiral Sum Rules in Vacuum vector-axialvector splitting (one of the) cleanest observable of spontaneous chiral symmetry breaking promising (best?) starting point to search for chiral restoration pion decay constants chiral quark condensates

16. 2.5 QCD Sum Rules:  and a 1 in Vacuum dispersion relation: lhs: hadronic spectral fct. rhs: operator product expansion [Shifman,Vainshtein+Zakharov ’79] 4-quark + gluon condensate dominant

17. 2.) Chiral Symmetry in QCD - Nonperturbative QCD, Chiral Breaking + Hadron Spectrum 3.) Thermal Electromagnetic Emission Rates - EM Spectral Function: Hadronic vs. Partonic Regimes 4.) Vector Mesons in Medium - Many-Body Theory, Spectral Functions + Chiral Partners (  -a 1 ) 5.) Quark-Gluon Plasma Emission - Perturbative vs. Lattice-QCD Rates, “Quark-Hadron-Duality” 6.) Dilepton + Photon Spectra in Heavy-Ion Collisions - Space-Time Evolution, Phenomenology + Interpretation 7.) Summary and Conclusions Outline

18. 3.1 EM Correlator + Thermal Dilepton Rate EM Correlation Fct.: e + e  Im Π em (M,q;T,  B )  * (q) Quark basis: Hadron basis: (T,  B ) → 9 : 1 : 2 [McLerran+Toimela ’85]

19. 3.2 EM Correlator in Vacuum: e + e  → hadrons e+ee+e h 1 h 2 … s ≥ s dual ~ (1.5GeV) 2 pQCD continuum s < s dual Vector-Meson Dominance qqqq _   qq _  4     6    KK e+ee+e     

20. 3.3 Low-Mass Dileptons + Chiral Symmetry How is the degeneration realized? “measure” vector with e + e , axialvector? Vacuum T > T c : Chiral Restoration

21. Im Π em (M,q) 3.4 Versatility of EM Correlation Function Photon Emission Rate Im Π em (q 0 =q) e + e - γ O(α s ) ~ O(α s ) ** EM Susceptibility ( → charge fluctuations)  Q 2    Q  2 = χ em = Π em (q 0 =0,q→0) O(1) ~ O(1) same correlator! EM Conductivity  em = lim(q 0 →0) [ -Im Π em (q 0,q=0)/2q 0 ]

22. 2.) Chiral Symmetry in QCD - Nonperturbative QCD, Chiral Breaking + Hadron Spectrum 3.) Thermal Electromagnetic Emission Rates - EM Spectral Function: Hadronic vs. Partonic Regimes 4.) Vector Mesons in Medium - Many-Body Theory, Spectral Functions + Chiral Partners (  -a 1 ) 5.) Quark-Gluon Plasma Emission - Perturbative vs. Lattice-QCD Rates, “Quark-Hadron-Duality” 6.) Dilepton + Photon Spectra in Heavy-Ion Collisions - Space-Time Evolution, Phenomenology + Interpretation 7.) Summary and Conclusions Outline

23. 4.1 Axial/Vector Mesons in Vacuum Introduce  a 1 as gauge bosons into free  +  +a 1 Lagrangian   EM formfactor  scattering phase shift   |F  | 2    -propagator: 3 parameters: m  (0), g,  

24. 4.2  -Meson in Matter: Many-Body Theory D  (M,q;  B,T) = [M 2 – (m  (0) ) 2 -   -   B -   M ] -1 interactions with hadrons from heat bath  In-Medium  -Propagator     [Chanfray et al, Herrmann et al, Urban et al, Weise et al, Koch et al, …] In-Medium Pion Cloud  > > R= , N(1520), a 1, K 1... h=N, , K …   =  Direct  -Hadron Scattering   = + [Haglin, Friman et al, RR et al, Post et al, …] estimate coupling constants from R→  + h, more comprehensive constraints desirable

25. 4.3 Constraints I: Nuclear Photo-Absorption total nuclear  -absorption in-medium    -spectral cross section function at photon point > >    ,N*,  * N -1  N →  N,   N →   meson exchange direct resonance 

26. 4.3.2  Spectral Function in Nucl. Photo-Absorption NANA  -ex [Urban,Buballa,RR+Wambach ’98] On the Nucleon On Nuclei 2.+3. resonances melt (selfconsistent N(1520)→N  ) fixes coupling constants and formfactor cutoffs for  NB

27. 4.4  -Meson Spectral Function in Nuclear Matter In-med.  -cloud +  +N→B* resonances (low-density approx.) In-med.  -cloud +  +N → N(1520) Constraints :   N,   A   N →   N PWA Consensus: strong broadening + slight upward mass-shift Constraints from (vacuum) data important quantitatively N=0N=0 N=0N=0  N =0.5  0 [Urban et al ’98] [Post et al ’02] [Cabrera et al ’02]

28. 4.5  -Meson Spectral Functions “at SPS”  -meson “melts” in hot/dense matter baryon density  B more important than temperature  B /  0 0 0.1 0.7 2.6 Hot + Dense Matter [RR+Gale ’99] Hot Meson Gas [RR+Wambach ’99]  B =330MeV

29. 4.6 Light Vector Mesons “at RHIC + LHC” baryon effects remain important at  B,net = 0: sensitive to  B,tot =   +  B (  -N =  -N, CP-invariant)  also melts,  more robust ↔ OZI -  [RR ’01]

30. 4.7 Intermediate Mass: “Chiral Mixing” = = low-energy pion interactions fixed by chiral symmetry mixing parameter [Dey, Eletsky +Ioffe ’90] 0 0 0 0 degeneracy with perturbative spectral fct. down to M~1GeV physical processes at M ≥ 1GeV:  a 1 → e + e  etc. (“4  annihilation”)

31. 4.8 Axialvector in Medium: Dynamical a 1 (1260) + +... =           Vacuum: a 1 resonance In Medium: + +... [Cabrera,Jido, Roca+RR ’09] in-medium  +  propagators broadening of  -  scatt. Amplitude pion decay constant in medium:

32. 4.9 QCD + Weinberg Sum Rules in Medium T [GeV] [Hatsuda+Lee’91, Asakawa+Ko ’93, Klingl et al ’97, Leupold et al ’98, Kämpfer et al ‘03, Ruppert et al ’05] [Weinberg ’67, Das et al ’67; Kapusta+Shuryak ‘94] melting scenario quantitatively compatible with chiral restoration microscopic calculation of in-medium axialvector to be done [Hohler et al ‘12] s [GeV 2 ]  V,A /s Vacuum T=140MeV T=170MeV

33. 4.10 Chiral Condensate +  -Meson Broadening  qq  /  qq  0 - effective hadronic theory  h = m q  h|qq|h  > 0 contains quark core + pion cloud =  h core +  h cloud ~ + + matches spectral medium effects: resonances + pion cloud resonances + chiral mixing drive  -SF toward chiral restoration   > > - 

34. 2.) Chiral Symmetry in QCD - Nonperturbative QCD, Chiral Breaking + Hadron Spectrum 3.) Thermal Electromagnetic Emission Rates - EM Spectral Function: Hadronic vs. Partonic Regimes 4.) Vector Mesons in Medium - Many-Body Theory, Spectral Functions + Chiral Partners (  -a 1 ) 5.) Quark-Gluon Plasma Emission - Perturbative vs. Lattice-QCD Rates, “Quark-Hadron-Duality” 6.) Dilepton + Photon Spectra in Heavy-Ion Collisions - Space-Time Evolution, Phenomenology + Interpretation 7.) Summary and Conclusions Outline

35. marked low-mass enhancement comparable to recent lattice-QCD computations Im [ ] = + + + … 5.1 QGP Emission: Perturbative vs. Lattice QCD Baseline: qqqq _ e+ee+e small M → resummations, finite-T perturbation theory (HTL) qq qq collinear enhancement: D q,g =(t-m D 2 ) -1 ~ 1/α s [Braaten,Pisarski+Yuan ‘91] [Ding et al ’10] dR ee /d 4 q 1.45T c q=0

36. 5.2 Euclidean Correlators: Lattice vs. Hadronic Euclidean Correlation fct. Hadronic Many-Body [RR ‘02] Lattice (quenched) [Ding et al‘10] “Parton-Hadron Duality” of lattice and in-medium hadronic …

37. 5.2.2 Back to Spectral Function suggestive for approach to chiral restoration and deconfinement -Im  em /(C T q 0 )

38. 5.3 Summary of Dilepton Rates: HG vs. QGP dR ee /dM 2 ~ ∫d 3 q f B (q 0 ;T) Im  em Lattice-QCD rate somwhat below Hard-Thermal Loop hadronic → QGP toward T pc : resonance melting + chiral mixing Quark-Hadron Duality at all M ee ?! (QGP rates chirally restored!)

39. 2.) Chiral Symmetry in QCD - Nonperturbative QCD, Chiral Breaking + Hadron Spectrum 3.) Thermal Electromagnetic Emission Rates - EM Spectral Function: Hadronic vs. Partonic Regimes 4.) Vector Mesons in Medium - Many-Body Theory, Spectral Functions + Chiral Partners (  -a 1 ) 5.) Quark-Gluon Plasma Emission - Perturbative vs. Lattice-QCD Rates, “Quark-Hadron-Duality” 6.) Dilepton + Photon Spectra in Heavy-Ion Collisions - Space-Time Evolution, Phenomenology + Interpretation 7.) Summary and Conclusions Outline

40. 6.1 Space-Time Evolution + Equation of State 1.order → lattice EoS: - enhances temperature above T c - increases “QGP” emission - decreases “hadronic” emission initial conditions affect lifetime simplified: parameterize space-time evolution by expanding fireball benchmark bulk-hadron observables Evolve rates over fireball expansion: [He et al ’12] Au-Au (200GeV)

41. 6.1.2 Bulk Hadron Observables: Fireball Model Mulit-strange hadrons freeze-put at T pc Bulk-v 2 saturates at ~T pc [van Hees et al ’11]

42. 6.2 Di-Electron Spectra from SPS to RHIC consistent excess emission source suggests “universal” medium effect around T pc FAIR, LHC? QM12 Pb-Au(17.3GeV) Pb-Au(8.8GeV) Au-Au (20-200GeV) [cf. also Bratkovskaya et al, Alam et al, Bleicher et al, Wang et al …]

43. 6.3 In-In at SPS: Dimuons from NA60 excellent mass resolution and statistics for the first time, dilepton excess spectra could be extracted! + full acceptance correction… [Damjanovic et al ’06]

44. 6.3.2 NA60 Multi-Meter: Accept.-Corrected Spectra Thermal source! Low-mass: good sensitivity to medium effects, T~130-170MeV Intermediate-mass: T ~ 170-200 MeV > T pc Fireball lifetime  FB = (6.5±1) fm/c [CERN Courier Nov. 2009] Emp. scatt. ampl. + T-  approximation Hadronic many-body Chiral virial expansion Thermometer Spectrometer Chronometer

45. 6.3.3 Spectrometer in-med  + 4  + QGP invariant-mass spectrum directly reflects thermal emission rate!  +   Excess Spectra In-In(17.3AGeV) [NA60 ‘09] M  [GeV] Thermal     Emission Rate [van Hees+RR ’08]

46. M  [GeV] 6.4 Conclusions from Dilepton “Excess” Spectra thermal source (T~120-230MeV) in-medium  meson spectral function - avg.   (T~150MeV) ~ 350-400 MeV    (T~T pc ) ≈ 600 MeV → m  - “divergent” width ↔ Deconfinement?! M > 1.5 GeV: QGP radiation fireball lifetime “measurement”:  FB ~ (6.5±1) fm/c (In-In) [van Hees+RR ‘06, Dusling et al ’06, Ruppert et al ’07, Bratkovskaya et al ’08, Santini et al ‘10]

47. 6.5 The RHIC-200 Puzzle in Central Au-Au PHENIX, STAR and theory: - consistent in non-central collisions - tension in central collisions

48. 6.6 Direct Photons at RHIC v 2 ,dir as large as for pions!? underpredcited by QGP-dominated emission ← excess radiation T eff excess = (220±30) MeV QGP radiation? radial flow? Spectra Elliptic Flow [Holopainen et al ’11, Dion et al ‘11]

49. 6.6.2 Thermal Photon Radiation thermal + prim.  [van Hees, Gale+RR ’11] flow blue-shift: T eff ~ T √(1+  )/(1  ),  ~0.3: T ~ 220/1.35 ~ 160 MeV “small” slope + large v 2 suggest main emission around T pc other explanations…? [Skokov et al ‘12; McLerran et al ‘12]

50. 6.7 Direct Photons at LHC similar to RHIC (not quite enough v 2 ) non-perturbative photon emission rates around T pc ? ● ALICE [van Hees et al in prep] Spectra Elliptic Flow

51. Spontaneous Chiral Symmetry Breaking in QCD Vacuum: - ‹qq› ~ m q * ≠ 0, chiral partners split (  - ,  -a 1, …) - low-mass dileptons dominated by  7.) Conclusions Hadronic Medium Effects: - “melting” of  (constraints!)  hadronic liquid!? - connection to chiral symmetry: QCD/chiral sum rules (a 1 ) Extrapolate EM Emission Rates to T pc ~ 170MeV  in-med HG and QGP shine equally bright (“duality”) ! Phenomenology for URHICs: - versatile + precise tool (spectro, chrono, thermo + baro-meter) - SPS/RHIC (NA45,NA60,STAR): compatible w/ chiral restoration - tension STAR/PHENIX; LHC (ALICE) and CBM/NICA? -

52. 6.3.4 Sensitivity of Dimuons to Equation of State partition QGP/HG changes, low-mass spectral shape robust [cf. also Ruppert et al, Dusling et al…]

53. 6.3.5 NA60 Dimuons with Lattice EoS + Rate partition QGP/HG changes, low-mass spectral shape robust First-Order EoS + HTL Rate Lattice EoS + Lat-QGP Rate M  [GeV] [van Hees+RR ’08] In-In (17.3GeV) T in =190MeV T c =T ch =175MeV [cf. also Ruppert et al, Dusling et al…] T in =230MeV T pc =T ch =175MeV

54. 6.3.6 Chronometer direct measurement of fireball lifetime:  FB ≈ (6.5±1) fm/c non-monotonous around critical point? In-In N ch >30

55. 6.4 Summary of EM Probes at SPS calculated with same EM spectral function! M  [GeV] In(158AGeV)+In

56. 6.1 Fireball Evolution in Heavy-Ion Collisions “chemical” freezeout T chem ~ T c ~170MeV, “thermal” freezeout T fo ~ 120MeV conserve entropy + baryon no.: T i → T chem → T fo time scale: hydrodynamics, fireball V FB (  ) = (z 0 +v z  )  (r 0 + 0.5a ┴  2 ) 2  N [GeV]  [fm/c] Thermal Dilepton Spectrum: Isentropic Trajectories in the Phase Diagram

57. 6.3.2 NA60 Data Before Acceptance Correction Discrimination power of model calculations improved, but … can compensate spectral “deficit” by larger flow: lift pairs into acceptance hadr. many-body + fireball emp. ampl. + fireball schem. broad./drop. + HSD transport chiral virial + hydro

58. 6.3.7 Dimuon p t -Spectra + Slopes: Barometer slopes originally too soft increase fireball acceleration, e.g. a ┴ = 0.085/fm → 0.1/fm insensitive to T c = 160-190 MeV Effective Slopes T eff

59. 6.2 Mass vs. Temperature Correlation generic space-time argument:   T max ≈ M / 5.5 (for Im  em =const) thermal photons: T max ≈ (q 0 /5) * (T/T eff ) 2 → reduced by flow blue-shift! T eff ~ T * √(1+  )/(1  )

60. 2.4.4 Weinberg (Chiral) Sum Rules + Order Parameters [Weinberg ’67, Das et al ’67] Moments Vector  Axialvector In Medium: energy sum rules at fixed q [Kapusta+Shuryak ‘93] correlators (rhs): effective models (+data) order parameters (lhs): lattice QCD promising synergy!

61. Lower SPS Energy (T i ~ 170MeV→ T fo ~100MeV) enhancement increases!? supports importance of baryonic effects 6.2 Low-Mass Di-Electrons: CERES/NA45 Top SPS Energy (T i ~ 200MeV → T fo ~110 MeV) QGP contribution small medium effects! drop. mass or broadening?! [RR+Wambach ‘99]

62. 6.1.2 Emission Profile of Thermal EM Radiation generic space-time argument:   T max ≈ M / 5.5 (for Im  em =const) Additional T-dependence from EM spectral function Latent heat at T c not included (penalizes T > T c, i.e. QGP)

63. e.g., A=a 1,h 1 fix G via  (a 1 →  ) ~ G 2 v 2 PS ≈ 0.4GeV, … > > R h 4.6  -Hadron Interactions in Hot Meson Gas resonance-dominated:   + h → R, selfenergy: Effective Lagrangian (h = , K,  )  Generic features: cancellations in real parts imaginary parts strictly add up

64. 2.5 Dimuon p t -Spectra and Slopes: Barometer vary fireball evolution: e.g. a ┴ = 0.085/fm → 0.1/fm both large and small T c compatible with excess dilepton slopes pions: T ch =175MeV a ┴ =0.085/fm pions: T ch =160MeV a ┴ =0.1/fm

65. 4.3.3 Acceptance-Corrected NA60 Spectra rather involved at p T >1.5GeV: Drell-Yan, primordial/freezeout , … [van Hees + RR ‘08]

66. 4.5 EM Probes in Central Pb-Au/Pb at SPS consistent description with updated fireball (a T =0.045→0.085/fm) very low-mass di-electrons ↔ (low-energy) photons [Liu+RR ‘06, Alam et al ‘01] Di-Electrons [CERES/NA45] Photons [WA98] [van Hees+RR ‘07]

67. Model Comparison of  -SF in Hot/Dense Matter first sight: reasonable agreement second sight: differences!  B vs.  N Implications for NA60 interpretation?! [RR+Wambach ’99] [Eletsky etal ’01] Im  V ~ ImT VN  N + ImT V    ~  VN,V  + dispersion relation for ReT V [Eletsky,Belkacem,Ellis,Kapusta ’01]

68. 3.8 Axialvector in Medium: Explicit a 1 (1260)    > > > > N(1520) … ,N(1900) … a1a1 + +... Exp: - HADES (  A): a 1 →(  +  - )  - URHICs (A-A) : a 1 →  [RR ’02]

69. after freezeout  V ~ 1/  V tot thermal emission  FB ~ 10fm/c 3.3 The Role of Light Vector Mesons in HICs  ee [keV]  tot [MeV] (N ee ) thermal (N ee ) cocktail ratio   (770) 6.7 150 (1.3fm/c) 1 0.13 7.7   0.6 8.6 (23fm/c) 0.09 0.21 0.43   1.3 4.4 (44fm/c) 0.07 0.31 0.23  In-medium radiation dominated by  -meson! Connection to chiral symmetry restoration?! Contribution to invariant mass-spectrum:

70. Nonpert. Wilson coeffs. (condensates) 0.2% 1% 4.5 Constraints II: QCD Sum Rules in Medium dispersion relation for correlator: lhs: OPE (spacelike Q 2 ): rhs: hadronic model (s>0): [Shifman,Vainshtein +Zakharov ’79] [Hatsuda+Lee’91, Asakawa+Ko ’92, Klingl et al ’97, Leupold et al ’98, Kämpfer et al ‘03, Ruppert et al ’05]