Symmetries of QCD and in-medium physics

Transcription

1 Symmetries of QCD and in-medium physics Stefan Leupold Indian-Summer School, Rez, October 2011

2 Table of Contents 1 Introduction 2 Global symmetries of QCD 3 Spontaneous symmetry breaking 4 In-medium physics 5 Summary

3 Quantum chromodynamics strong interaction described by Quantum ChromoDynamics L = q iγ µ ( µ iga µ ) q 1 4 F µν a F a µν, F µν [ µ iga µ, ν iga ν ] with quarks q = (u, d, s, c, b, t) and gluons A µ α s (µ) µ GeV PDG, J. Phys. G33 (2006) 1 running coupling g asymptotic freedom (caused by gluon self-interaction) can use perturbation theory at large momenta there one sees quarks and gluons (deep inelastic scattering, jet production)

4 Quantum chromodynamics strong interaction described by Quantum ChromoDynamics L = q iγ µ ( µ iga µ ) q 1 4 F µν a F a µν, F µν [ µ iga µ, ν iga ν ] with quarks q = (u, d, s, c, b, t) and gluons A µ running coupling g large coupling at small momenta cannot use perturbation theory there confinement relevant degrees of freedom are hadrons, not quarks and gluons α s (µ) µ GeV PDG, J. Phys. G33 (2006) 1

5 Hadron spectrum an example light mesons with S = 0 (i.e. no open strangeness)

6 Questions How to understand masses, life times, reaction rates of hadrons? Are all hadrons (dominantly) made out of quark-antiquark or three quarks, respectively (quark model)? Are there hadrons purely/dominantly made out of gluons? glueballs (PANDA) Do some/many hadron have a hadronic substructure? hadron molecules (PANDA) How do properties change in strongly interacting matter? (nuclei, fireball of heavy-ion collisions HADES, CBM) important tool: symmetries

7 Phase diagram of QCD T ~170 MeV hadron gas confined, χ-sb quark-gluon plasma deconfined, χ-symmetric color superconductor α s (µ) µ GeV µ o few times nuclear matter density µ

8 Importance of symmetries Which features can we expect from symmetries? mathematical beauty of theory things become simpler: example: central potential V ( r )

9 Importance of symmetries Which features can we expect from symmetries? mathematical beauty of theory things become simpler: example: central potential V ( r ) rotational invariance

10 Importance of symmetries Which features can we expect from symmetries? mathematical beauty of theory things become simpler: example: central potential V ( r ) rotational invariance conservation of angular momentum

11 Importance of symmetries Which features can we expect from symmetries? mathematical beauty of theory things become simpler: example: central potential V ( r ) rotational invariance conservation of angular momentum orbit lies in a plane 2-dim instead of 3-dim problem

12 Importance of symmetries Which features can we expect from symmetries? quantum mechanical example: central potential V ( r ) (e.g. hydrogen atom) rotational invariance conservation of angular momentum

13 Importance of symmetries Which features can we expect from symmetries? quantum mechanical example: central potential V ( r ) (e.g. hydrogen atom) rotational invariance conservation of angular momentum degenerate energy levels

14 Importance of symmetries Which features can we expect from symmetries? quantum mechanical example: central potential V ( r ) (e.g. hydrogen atom) rotational invariance conservation of angular momentum degenerate energy levels Translation to field theory conservation laws: degeneracy: conserved charges selection rules states with same mass

15 Explicit symmetry breaking quantum mechanical example: central potential V ( r ) (e.g. hydrogen atom) rotational invariance conservation of angular momentum degenerate energy levels switch on small external magnetic field

16 Explicit symmetry breaking quantum mechanical example: central potential V ( r ) (e.g. hydrogen atom) rotational invariance conservation of angular momentum degenerate energy levels switch on small external magnetic field breaks rotational invariance explicitly energy levels slightly split up approximately degenerate energy levels systematics in splitting pattern

17 Quantum chromodynamics QCD Lagrangian (here restricted to up and down quarks) with L = 1 4 F a µνf µν a + q j (iγ µ µ M q ) q j + q j gγ µ A µ a(λ a ) jk q k ( ) uj matter (quark) fields q j = and gauge (gluon) fields A µ d j a quark-gluon (and gluon-gluon) coupling constant g α s = g2 4π current quark mass (matrix) ( ) 3 0 M q (µ 2 GeV) MeV 0 6

18 Global symmetries of QCD L = 1 4 F µνf a a µν + q jf iγ µ µ q jf + q jf gγ µ A a(λ µ a ) jk q kf q jf (M q ) fg q jg baryon number conservation: can change phase of all quarks simultaneously U B (1) baryons cannot decay into mesons approximate symmetry for m q 0: chiral symmetry can mix flavors and chiralities SU L (N f ) SU R (N f ) (approximate) flavor multiplets + parity doublets (?) symmetry very good for N f = 2, reasonable for N f = 3 approximate symmetry for m q : center symmetry connected to confinement, complicated (not discussed)

19 Flavor symmetry L = 1 4 F µνf µν + q f (iγ µ µ + gγ µ A µ ) q f q f (M q ) fg q g (indices denote now flavor, not color) If all quark masses were the same: M q m q q f (...) q f does not change under transformations SU V (N f ): q f [exp(iθ a τ a )] fg q g, q f q g [exp( iθ a τ a )] gf isospin (flavor) conservation degenerate states (multiplets), i.e. states with equal mass and different isospin (flavor) indeed for N f = 2: approximately degenerate states: (n, p), (π, π 0, π + ), (K 0, K + ), (K, K 0 ),... quark masses not the same, but difference small (?) (difference = explicit symmetry breaking)

20 Flavor multiplets octets for N f = 3: systematic splitting pattern Σ n Ξ Λ Σ 0 p Σ + Ξ 0 π K 0 K η π 0 K + π + K 0

21 Flavor multiplets decuplet allows for predictions Ω Σ 0 Ξ Σ 0 Ω + ++ Σ + Ξ 0

22 Spectral distributions Isospin symmetry (and violation) on the level of currents: j µ V = 1 2 (ūγµ u dγ µ d) ūγ µ d same spectra: ρ 0 (from e + e ) and ρ (from τ decay) R ρ ω φ u, d, s ρ 3-loop pqcd Naive quark model ALEPH τ (V,I=1)ν τ Perturbative QCD (massless) Parton model prediction ππ 0 1 v π3π 0,3ππ 0,6π(MC) ωπ,ηππ 0,KK 0 (MC) 10-1 Sum of exclusive measurements Inclusive measurements πkk-bar(mc) modification by isoscalars (ω, φ) Mass 2 (GeV/c 2 ) 2

23 Small quark mass difference? m u 3 MeV, m d 6 MeV m = m d m u 3 MeV What is small compared to what?

24 Small quark mass difference? m u 3 MeV, m d 6 MeV m = m d m u 3 MeV What is small compared to what? (m d m u )/m u,d not small, but (m u m d )/M h small (with M h typical hadron mass 1 GeV) isospin multiplets do not emerge because up and down quark masses are similar on an absolute scale, but because both are very small on a hadronic scale worth to study limit of massless quarks more symmetries ahead

25 Chiral symmetry L = 1 4 F µνf µν + q f (iγ µ µ + gγ µ A µ ) q f q f (M q ) fg q g neglect quark mass term (and recall q = q γ 0 ) L 0 = 1 4 F µνf µν + q f s (γ 0γ µ ) st (i µ + ga µ ) q f t (indices denote now flavor and spinor) γ 5 commutes with combination γ 0 γ µ (but not with γ 0 alone mass term breaks chiral sym.) L 0 does not change under transformations SU A (N f ): [ q f s exp(i Θ ] [ a τ a γ 5 ) q gt, q fgst f s q gt exp( i Θ ] a τ a γ 5 ) gf ts

26 Chiral transformations the formal stuff take flavor transformations together SU V (N f ) SU A (N f ) q exp(iθ a τ a ) exp(i Θ a τ a γ 5 ) q introduce left- and right-handed quarks q = q L + q R = 1 2 (1 γ 5) q (1 + γ 5) q q R,L exp(iθ a τ a ) exp(±i Θa τ a ) q R,L note: γ 5 q L,R = ±q L,R and (γ 5 ) 2 = 1

27 Chiral transformations the formal stuff q R,L exp(iθ a τ a ) exp(±i Θ a τ a ) q R,L 1. choose Θ Ra := Θ a /2 = Θ a /2 q R exp(iθ Ra τ a ) q R, q L q L 2. choose Θ La := Θ a /2 = Θ a /2 q R q R, q L exp(iθ La τ a ) q L formally: SU V (N f ) SU A (N f ) = SU L (N f ) SU R (N f ) can perform flavor transformations separately for left- and right-handed quarks without changing the physics (chiral symmetry)

28 Left- and right-handed states helicity: spin points in or against flight direction genuine (Lorentz invariant) property only for massless particles otherwise: boost from system slower than particle into system faster than particle characterizes particle relative to frame of reference, not particle alone for massless particles: helicity = chirality

29 Left- and right-handed flavored quarks L 0 = 1 4 F µνf µν + q γ 0 γ µ (i µ + ga µ ) q = 1 4 F µνf µν + q L γ 0γ µ (i µ + ga µ ) q L + q R γ 0γ µ (i µ + ga µ ) q R = 1 4 F µνf µν + u L γ 0γ µ (i µ + ga µ ) u L + u R γ 0γ µ (i µ + ga µ ) u R + d L γ 0γ µ (i µ + ga µ ) d L + d R γ 0γ µ (i µ + ga µ ) d R +... massless QCD contains 2 N f identical copies of quarks consequences: in interactions quarks keep their chirality and their flavor interaction is blind to flavor and chirality (always the same interaction)

30 Chiral symmetry and multiplets symmetry group: SU L (2) SU R (2) remember: non-trivial symmetry group leads to degenerate states expect multiplets flavor change accompanied by change in electric charge chirality flip equivalent to parity transformation each multiplet should contain states with different charges and with different parity fig.

31 Hadron spectrum an example light mesons with S = 0 (i.e. no open strangeness)

32 Chiral multiplets? expectation: each multiplet should contain states with different charges and with different parity there are (isospin) multiplets, e.g. (π, π 0, π + ), (ρ, ρ 0, ρ + ), (n, p),... degenerate isospin partners but there are no degenerate states with different parity: π(140) f 0 (600), ρ(770) a 1 (1240),... non-degenerate chiral partners How to understand mass splitting and nature of chiral partners? What happens in a strongly interacting medium?

33 How do we know that chiral symmetry is broken? isospin symmetry SU V (2): existence of multiplets, i.e. degenerate states (p, n), (ρ +, ρ 0, ρ ),... same spectra: ρ 0 (from e + e ) and ρ (from τ decay) j µ V = 1 2 (ūγµ u dγ µ d) ūγ µ d R ρ ω φ u, d, s ρ 3-loop pqcd Naive quark model ALEPH τ (V,I=1)ν τ Perturbative QCD (massless) Parton model prediction ππ 0 1 v π3π 0,3ππ 0,6π(MC) ωπ,ηππ 0,KK 0 (MC) 10-1 Sum of exclusive measurements Inclusive measurements πkk-bar(mc) modification by isoscalars (ω, φ) Mass 2 (GeV/c 2 ) 2

34 How do we know that chiral symmetry is broken? isospin symmetry SU V (2): existence of multiplets, i.e. degenerate states (p, n), (ρ +, ρ 0, ρ ),... same spectra: ρ 0 (from e + e ) and ρ (from τ decay) j µ V = 1 2 (ūγµ u dγ µ d) ūγ µ d corresponding SU A (2) symmetry: j µ V j µ A with axial-vector current j µ A = q τγ 5 γ µ q consequence of chirally symmetric world would be: same spectral information in vector and axial-vector current-current correlators (degeneracy) observable? τ decay

35 Chiral symmetry breaking and τ decays study decay τ ν τ +hadrons: couples to V A (weak process) G parity: V /A couples to even/odd number of pions are V and A spectra identical?

36 Chiral symmetry breaking and τ decays study decay τ ν τ +hadrons: couples to V A (weak process) G parity: V /A couples to even/odd number of pions are V and A spectra identical? hep-ex/ (ALEPH): 3 ALEPH τ (A,I=1)ν τ ALEPH v τ (V,I=1)ν τ Perturbative QCD (massless) Parton model prediction ππ 0 π3π 0,3ππ 0,6π(MC) ωπ,ηππ 0,KK 0 (MC) a Perturbative QCD (massless) Parton model prediction π2π 0,3π π4π 0,3π2π 0,5π πkk-bar(mc) 1 πkk-bar(mc) Mass 2 (GeV/c 2 ) Mass 2 (GeV/c 2 ) 2

37 One of the clearest signs of chiral symmetry breaking v 1 -a τ (V,A,I=1)ν τ ALEPH Perturbative QCD/Parton model v 1 : τ ν τ +mπ (m even) a 1 : τ ν τ + nπ (n odd) Mass 2 (GeV/c 2 ) 2

38 Chiral symmetry breaking (χsb) Why does isospin symmetry (approximately) work while chiral symmetry seems grossly violated? experimental findings can be explained by spontaneous symmetry breaking (and small explicit breaking) definition: Lagrangian has symmetry which ground state (vacuum) does not have in the following: different levels of sophistication: dinner table with salad plates shortest connection in a square Heisenberg magnet chiral symmetry breaking (χsb)

39 Dinner table with salad plates mirror symmetry

40 Dinner table with salad plates mirror symmetry broken

41 Dinner table with salad plates mirror symmetry broken C1 Pope (JP2) C2 Faessler

42 Dinner table with salad plates heated system: hungry guests, end of dinner announced parity invariance broken

43 Dinner table with salad plates heated system: hungry guests, end of dinner announced parity invariance broken restoration

44 Heisenberg magnet interaction between microscopic magnetic dipoles (spins) does not prefer any direction H int = g i j s i s j rotational invariance in contrast ground state (unexcited solid state) has preferred direction breaking of rotational invariance

45 Features of the Heisenberg magnet I gapless excitation spectrum: spin waves Goldstone bosons Why is it gapless? study (infinitely) long wavelength limit and vanishing frequency

46 Features of the Heisenberg magnet I gapless excitation spectrum: spin waves Goldstone bosons Why is it gapless? study (infinitely) long wavelength limit and vanishing frequency spin wave corresponds to (adiabatic) rotation of whole solid state does not cost energy

47 Features of the Heisenberg magnet II consider, e.g., phonon excitation: no symmetry breaking: excitations in x and y direction cost same energy symmetry breaking: excitations in x and y direction do not cost same energy degeneracy lifted (to some extent)

48 Features of the Heisenberg magnet III macroscopic magnetization M = s i M can be measured in the presence of an external magnetic field B: H int = g s i s j + B s i i j i B breaks rotational invariance explicitly presence of B: excitation spectrum no longer gapless; however, gap scales with B if system is heated, M vanishes above a critical value phase transition, symmetry restoration M is order parameter

49 Temperature dependence of order parameter 1 order parameter B = 0, B T/T c

50 Is symmetry only hidden? spontaneous symmetry breaking is also called hidden symmetry Is symmetry still there?

51 Is symmetry only hidden? spontaneous symmetry breaking is also called hidden symmetry Is symmetry still there? rotation of solid state: not the same system, but same properties

52 Symmetry is only hidden Is symmetry still there? symmetry suggests degenerate states? study e.g. phonon excitation:

53 Symmetry is only hidden Is symmetry still there? symmetry suggests degenerate states? study e.g. phonon excitation: 1. no symmetry breaking: excitations in x and y direction cost same energy 2. symmetry breaking: excitations in x and y direction do not cost the same energy

54 Symmetry is only hidden Is symmetry still there? symmetry suggests degenerate states? study e.g. phonon excitation: 1. no symmetry breaking: excitations in x and y direction cost same energy 2. symmetry breaking: excitations in x and y direction do not cost the same energy but: excitation in x direction costs same energy as rotation plus excitation in y direction recall: rotation = long-wavelength spin wave

55 Degenerate states and broken/hidden symmetry 1 no symmetry breaking: degeneracy at level of single excitations/particles 2 symmetry breaking: degeneracy of excitation and {excitation plus (soft) spin wave} approximate degeneracy in presence of explicit symmetry breaking

56 Translation to QCD: quark condensate M := s i 0 non-trivial expectation value with respect to ground state vacuum expectation value for which operator? recall explicit symmetry breaking term H ex = B s i look at term in QCD Lagrangian which breaks chiral symmetry explicitly: L = L 0 m u ūu m d dd... quark condensate qq (240 MeV) 3 N f

57 Translation to QCD: pions gapless excitation spectrum massless states (Goldstone bosons) for finite B 0 (explicit symmetry breaking): spin wave excitation no longer exactly gapless but gap small, scales with B for finite m q 0 (explicit symmetry breaking): Goldstone bosons are no longer exactly massless, but light pions Gell-Mann Oakes Renner relation (here N f = 2) m 2 π f 2 π = m q qq + o(m 2 q) note: in principle conceivable: qq = 0 and m 2 π m 2 q

58 Translation to QCD: chiral restoration order parameter M drops with temperature restoration of rotational invariance above Curie temperature but not completely in presence of external B field order parameter qq drops with temperature chiral symmetry restoration but not completely in presence of finite current quark masses

59 Medium dependence of order parameter 1 order parameter m q = 0, m q T/T c

60 Translation to QCD: non-degenerate states excitations in x and y direction do not cost same energy no multiplets with different parity on single-hadron level: N(940) N (1535), ρ(770) a 1 (1240),... (supposed to be) chiral partners non-degenerate

61 Translation to QCD: degenerate states {excitation plus spinwave} is degenerate to excitation only approximate in presence of external B multiplets with different parity: state h plus pion is new chiral partner to state h e.g. m N m N + m π (soft pion)

62 Symmetry pattern of QCD symmetry SU V (2) SU A (2) center symmetry vacuum unbroken broken unbroken high temperature unbroken unbroken broken multiplets (n, p),... (N, {N, π}),... order parameter qq L

63 In-medium QCD symmetry pattern of QCD changes in a medium: chiral symmetry breaking restoration confinement deconfinement color superconductivity (at large baryo-chemical potential, small temperature) change visible in behavior of order parameters next slides

64 Chiral symmetry exact for m q = 0 order parameters: e.g., two-quark condensate or pion decay constant F π (recall GOR relation MπF 2 π 2 = m q qq ) chiral symmetry spontaneously broken in vacuum becomes restored in medium: finite (low) temperature (approximated by pion gas): ( F π (T ) = F π 1 2ρ ) π 3Fπ 2 with scalar pion density ρ π = 3 d 3 k/((2π) 3 2E k ) (e E k /T 1 Mπ 0 1) T 2 /8 (Gasser/Leutwyler, Phys. Lett. B184, 83, 1987) finite baryo-chemical potential (approximated by Fermi sphere of nucleons): next slide

65 Chiral symmetry exact for m q = 0 order parameters: e.g., two-quark condensate or pion decay constant F π chiral symmetry spontaneously broken in vacuum becomes restored in medium: finite baryo-chemical potential (approximated by Fermi sphere of nucleons): ( F π (ρ N ) = F π 1 ρ ) N (0.26 ± 0.04) ρ 0 (Meißner/Oller/Wirzba, Annals Phys. 297, 27, 2002) π N ν W + π W µ N 1 N ν µ

66 Confinement connected to center symmetry (m q ) order parameter: Polyakov loop L spontaneously broken at large temperatures, not broken in vacuum fig. from lattice QCD at finite baryo-chemical potential?? L ren N f =2 m/t=0.40 N t =4 N f =0 N t =4 N f =0 N t =8 T/T c Kaczmarek/Ejiri/Karsch/Laermann/Zantow, arxiv:hep-lat/

67 In-medium QCD symmetry pattern of QCD changes in a medium: chiral symmetry breaking restoration confinement deconfinement change visible in behavior of order parameters note: purely hadronic calculation for pion decay constant (and model independent chiral perturbation theory) observable consequences? change of hadron properties?

68 Chiral-partner spectra 3 ALEPH τ (A,I=1)ν τ ALEPH v τ (V,I=1)ν τ Perturbative QCD (massless) Parton model prediction ππ 0 π3π 0,3ππ 0,6π(MC) ωπ,ηππ 0,KK 0 (MC) πkk-bar(mc) different spectra in vacuum a Perturbative QCD (massless) Parton model prediction π2π 0,3π π4π 0,3π2π 0,5π πkk-bar(mc) Mass 2 (GeV/c 2 ) Mass 2 (GeV/c 2 ) 2 at chiral restoration: spectra must become the same various possibilities: a 1 and ρ peaks fuse (where?) mixing: two peaks in each spectrum dissolution (deconfinement) at same height (chiral sym)

69 Observable consequence? 3 ALEPH R ρ ω φ Sum of exclusive measurements u, d, s same ρ 3-loop pqcd Naive quark model Inclusive measurements spectra in vacuum (isospin) v τ (V,I=1)ν τ Perturbative QCD (massless) Parton model prediction ππ 0 π3π 0,3ππ 0,6π(MC) ωπ,ηππ 0,KK 0 (MC) πkk-bar(mc) Mass 2 (GeV/c 2 ) 2 τ decays in medium somewhat hard to measure expect changes both in vector and axial-vector channel measure vector channel by electromagnetic probe dilepton emission

70 Early predictions: mass changes concentrate on vector mesons (observable via dileptons) Brown-Rho scaling conjecture for hadron h ( π): (Brown/Rho, Phys. Rev. Lett. 66, 2720, 1991) m h,med. m h,vac. = ( qq med. qq vac. masses drop in medium QCD sum rules: (Hatsuda/Lee, Phys. Rev. C46, 34, 1992) masses of light vector mesons drop with nucleon density Nambu Jona-Lasinio (quark) model: (Bernard/Meißner, Nucl. Phys. A489, 647, 1988) vector masses unchanged at finite nucleon densities linear sigma model with vector meson dominance: (Pisarski, Phys. Rev. D52, 3773, 1995) mass of ρ meson rises to meet decreasing mass of a 1 (chiral partners) ) α

71 Quarks and hadrons at least close to deconfinement transition models with quark degrees of freedom are natural choice

72 Quarks and hadrons at least close to deconfinement transition models with quark degrees of freedom are natural choice on the other hand: at low temperatures/densities a hadron is a hadron is a hadron hadronic description should work (recall drop of pion decay constant and quark condensate based on hadronic calculations)

73 Hadronic many-body framework fundamental quantity: in-medium spectral function of hadron h (generalization of Breit-Wigner) A(q) = ImΠ (q 2 m 2 h ReΠ)2 + (ImΠ) 2 with self energy Π; connection to width: Γ = ImΠ/ q 2 decomposition: Π(q) = Π vac (q) + Π med (q) linear-density ( ρt ) approximation for (in-medium) part Π med (q) = X ρ X T Xh (q) with medium constituents X (e.g. N, π)

74 Forward scattering amplitude Π med (q) = X ρ X T Xh (q) T Xh : (vacuum) forward scattering amplitude for X + h with medium constituents X (e.g. N, π) underlying idea: probe (h) scatters on single medium constituents only vacuum quantity (scattering amplitude) enters trivial in-medium effect (conceptually trivial)

75 Unstable probe everything well under control for low densities? in principle yes: need only vacuum scattering amplitudes T Xh

76 Unstable probe everything well under control for low densities? in principle yes: need only vacuum scattering amplitudes T Xh in practice no: h can be unstable (e.g., vector mesons) no h beam, no direct access on scattering amplitude model dependences theoretical and experimental understanding of elementary reactions mandatory e.g. for ρ meson: ImT Nρ σ tot Nρ T Nρ Nπ σ Nπ Ne + e

77 Spectral information from many-body theory Example 1: rho meson in cold nuclear matter Im R in-medium vacuum s [GeV] resonance model with parameters from πn (2π)N, Post/Leupold/Mosel Nucl.Phys.A741 (2004) 81 dynamical generation of resonances, Lutz/Wolf/Friman Nucl.Phys.A706 (2002) 431 results differ due to different input from/interpretation of elementary reactions (here: strength of coupling ρ-n-n (1520))

78 Spectral information from many-body theory Example 1 continued -Im G ρ [1/GeV 2 ] M [GeV] vacuum: solid, ρ 0 : long dashed, 2ρ 0 : short dashed Urban/Buballa/Rapp/Wambach, Nucl.Phys.A 641 (1998) 433 Sρ (1/GeV²) MI (GeV) vacuum: solid, ρ 0 /2: dashed, ρ 0 : dotted Cabrera/Oset/Vicente Vacas, Nucl.Phys.A 705 (2002) 90

79 Spectral information from many-body theory Example 2: omega meson in cold nuclear matter - Im D ( ) [GeV -2 ] = 0 - Im D ( ) [GeV -2 ] = [Gev] [Gev] dynamical generation of resonances Lutz/Wolf/Friman, Nucl.Phys.A706 (2002) 431-1/ ImD (q 0,q=0) [1/GeV 2 ] 10 2 = = q 2 [GeV] =0 coupled-channel K-matrix for πn, ωn, K Λ,... Mühlich/Shklyar/Leupold/Mosel/Post, Nucl.Phys.A780 (2006) 187 similar results (but different from results of other groups... )

80 Spectral information from many-body theory Example 3: N (1520) baryon in cold nuclear matter (baryon resonances and mesons mutually influence each other) D13 [GeV -2 ] k = 0 GeV g s =0.1 g s =0.0 vacuum k = 0.8 GeV Post/Leupold/Mosel, Nucl.Phys.A741 (2004) k 2 -m R [GeV] k 2 -m R [GeV] from all examples: typically sizable in-medium changes of hadron properties: collisional broadening new structures (resonance-hole excitations)

81 Hadrons and quarks 1 hadronic many-body theory: modest (if any) mass shifts, but collisional broadening, new structures 2 influence of quark-gluon substructure: expect transition to deconfined and chirally restored phase for dense/hot enough systems expect precursors in hadronic phase: dropping mass models, Brown/Rho, Phys.Rev.Lett.66 (1991) 2720 chiral partners might become degenerate dissolution of hadrons connection between 1 and 2 unclear: different language for same physics? models of 2 often rather schematic at least dissolution fits to collisional broadening

82 The physics contained in many-body calculations What is a spectral function? What is a resonance-hole excitation? Where do additional structures come from? Im R in-medium vacuum s [GeV]

83 Classical resonance equation of motion for damped harmonic oscillator, externally driven ẍ + γẋ + ω 2 0 x = e iωt solution = response of system to external excitation x(t) = x 0 e iωt with ω-dependent coefficient x 0 = 1 ω 2 iγω + ω 2 0 note: there is additional contribution dying out with e γt

84 Response function split in real and imaginary part (for later use): Rex 0 = ω 2 0 ω2 (ω 2 ω 2 0 )2 + γ 2 ω 2 Imx 0 = γω (ω 2 ω 2 0 )2 + γ 2 ω 2 note: all knowledge about system, i.e. ω 0 and γ, can be deduced from Imx 0 alone

85 Response function 5e-06 4e-06 3e-06 2e-06 Imx 0 1e-06 0 Rex 0-1e-06-2e-06-3e

86 Resonant scattering scatter two particles to form a resonance i.e. deposit energy E ( ˆ= external excitation) resonance can decay ( ˆ= friction term γ) translation (de Broglie): ω 2 E 2 = ( ω) 2 = s (cms) ω 2 0 m2 R resonance mass γ Γ resonance width important difference: width Γ depends on energy Γ(s) reason: available phase space for resonance decay energy dependent note: construction of ω-dependent γ also possible for oscillator case retarded damping

87 Precursor to spectral function recall x 0 = 1 ω 2 iγω + ω 2, Imx 0 = 0 ωγ (ω 2 ω 2 0 )2 + ω 2 γ 2 (not quite the) SPECTRAL FUNCTION s Γ A(s) (s mr 2 )2 + sγ 2 = Im 1 s mr 2 + i s Γ field theory: Γ connected to self energy diagram: Π(s) =, Γ = ImΠ s

88 Definition of spectral function also real part of Π enters definition of SPECTRAL FUNCTION: A(s) = ImΠ(s) s m 2 R Π(s) 2 = s Γ (s m 2 R ReΠ)2 + sγ 2 note: real part of Π can shift peak position m R response function ( ˆ=x 0 ): Green function or propagator G(s) = obviously: A = ImG 1 s m 2 R Π(s)

89 Unitarity, analyticity and dispersion relations spectral function tells how single quantum state is distributed over possible energies normalization condition: 0 ds π A(s) = 1 as for oscillator case: A completely determines resonance G can be calculated from A since G (ret) is analytic function in upper half of complex energy plane dispersion relation: G(s) = 0 ds π A(s ) s s iɛ

90 What changes in a medium? 1. need spectral and statistical information spectral: distribution of one state over possible energies statistical: how many states are there? 2. appearance of new channels will be discussed in a moment

91 How to get the statistical information general non-equilibrium situation: have to determine both informations and their evolution in time ( e.g. transport theory) equilibrium: maximal entropy requirement fixes statistical distribution number of states at given four-momentum: 1 A(E, p) e 1 T (E µ) ± 1 with temperature T and chemical potential µ (for nuclear matter (T = 0): ±A(E, p) Θ(µ E)) A contains all information (in equilibrium!)

92 Appearance of new channels in a medium vacuum: probe can only decay medium: scattering with constituents of medium (e.g. pions of heat bath, nucleons of nuclear matter) Landau damping (inelastic) scattering resonance formation

93 Unified language for vacuum and medium Feynman: incoming particle equivalent to outgoing antiparticle (hole) with negative energy (traveling backwards in time) scattering is decay into particle(s) and hole(s) excitation of nucleon-hole and resonance-hole states

94 Unitarity Cutkosky rules 2 Im have to calculate self energies like

95 What changes in a medium? imaginary parts of self energies change width: decays Bose-enhanced or Pauli-blocked new decay channels real parts shift peak position Spectrum [GeV -2 ] ρ=0 Set A Set B Klingl/Waas/Weise NPA 650 (1999) E [GeV]

96 What changes in a medium? vacuum: it does not matter whether probe is moving (Lorentz invariance) medium: it DOES matter whether probe is moving with respect to other scatterers explicit dependence on E, q, not only on s = E 2 q 2 independent variables: E, q or m := s, q

97

98 Appearance of new structures in a medium vacuum decay outgoing states can have arbitrary momenta structureless medium decay in particle-hole outgoing hole = incoming particle restricted in momentum (by temperature or chemical potential) structure in self energy resonance-hole branches

99 Resonance-hole branches free ρ N(1720) N(1520) (1232) m [GeV] q [GeV]

100 Appearance of new structures in a medium structure in self energy structure in spectral function e.g. additional peaks, bumps, shoulders Im R in-medium vacuum s [GeV]

101 Low-density approximation central quantity: (in-medium) spectral function for hadron H 1 A(q) = ImD(q) = Im q 2 mh 2 Π(q) = ImΠ(q) [q 2 m 2 H ReΠ(q)]2 + [ImΠ(q)] 2 decomposition: Π(q) = Π vac (q) + Π med (q) linear-density ( ρt ) approximation for (in-medium) self energy Π med (q) = ρ X T XH (q) X with medium constituents X (e.g. N, π)

102 How fancy is the linear-density approximation? simple toy model for dilepton production n B (q)a ρ (q)/q 2 mediated by ρ meson (vector meson dominance) ρ meson couples to 2π and resonance-hole (RN 1 ) A ρ (q) = ImΠ 2π (q) ImΠ RN 1(q) [q 2 m 2 ρ ReΠ 2π (q) ReΠ RN 1(q)] 2 + [ImΠ 2π (q) + ImΠ RN 1(q)] 2 recall: Π RN 1 = ρ N T ρn R ρn appearance of density in denominator causes non-elementary effect: resummation spectral function not simply given by vac. + {term linear in density}

103 In-medium ρ meson spectral information Decomposition 20 res.-hole branch rho branch total spectrum 15 Spec[M]/M M [GeV]

104 How fancy is the linear-density approximation? A ρ (q) = ImΠ 2π (q) ImΠ RN 1(q) [q 2 m 2 ρ ReΠ 2π (q) ReΠ RN 1(q)] 2 + [ImΠ 2π (q) + ImΠ RN 1(q)] 2 corresponding elementary reactions: ImΠ 2π (q) ImΠ RN 1(q) [q 2 m 2 ρ ReΠ vac (q)] 2 + [ImΠ vac (q)] 2

105 Interpretation of elementary processes ImΠ 2π (q) ImΠ RN 1(q) [q 2 m 2 ρ ReΠ vac (q)] 2 + [ImΠ vac (q)] 2 = ImΠ 2π(q) ImΠ vac (q) Avac ρ + ImΠ RN 1(q) ImΠ vac (q) Avac ρ i.e. branching ratios times spectral information l π π ρ γ l π N ρ γ l + l + N N

106 Elementary reactions versus full in-medium spectrum (at q = 0!) N π -> R -> N e + e - π π -> e + e - res.-hole branch rho branch Spec[M]/M M [GeV]

107 Conclusions from simple toy model structures already present in elementary reactions denominator effect : level repulsion and overall depletion elementary reactions should be measured πn to dileptons, not only NN (in latter resonance structure more smeared out, phase space) note: elementary reactions are genuine in-medium (π in initial state)

108 Summary I Symmetries symmetries very important for qualitative understanding spontaneous symmetry breaking is very fascinating phenomenon in QCD: chiral symmetry breaking particle physics: Higgs mechanism solid-state physics: ferro magnet, Meissner effect... spontaneous breaking implies restoration observable consequences are tricky, because hadrons are complicated

109 Summary II Observables early models based on quark degrees of freedom rather schematic and conflicting statements hadronic models should work at not too high temperatures and densities qualitative findings: small/no mass shifts significant broadening maybe new structures (resonance-hole excitations) quantitative uncertainties better understanding of hadronic resonances (nature, decay channels) mandatory information from elementary hadron physics (PANDA) and from in-medium physics (HADES, CBM) complement each other

110 Summary III Connections of hadronic descriptions to deconfinement and chiral restoration? broadening can be understood as precursor to deconfinement connection to chiral restoration not so clear demand on theorists: realistic models which incorporate chiral restoration, deconfinement and hadron (vacuum) physics demand on experiments: must be precise enough to check/falsify models

111 backup slides

112 Quantum chromodynamics QCD Lagrangian (here restricted to up and down quarks) with L = 1 4 F a µνf µν a + q j (iγ µ µ M q ) q j + q j gγ µ A µ a(λ a ) jk q k ( ) uj matter (quark) fields q j = and gauge (gluon) fields A µ d j a quark-gluon (and gluon-gluon) coupling constant g α s = g2 4π current quark mass (matrix) ( ) 3 0 M q (µ 2 GeV) MeV 0 6

113 Local color symmetry L = 1 4 F µν(x)f a a µν (x) + q j (x) ( iγ µ [D µ ) (x)] jk M q δ jk qk (x) with D µ (x) = µ x iga µ a(x)λ a, F µν (x) = i g [Dµ (x), D ν (x)] (indices denote only color, not flavor or spinor) Lagrangian invariant with respect to local transformations in color space, U(x) := exp(iθ a (x)λ a ) SU c (3) q j (x) [U(x)] jk q g (x), q j (x) q k (x)[u 1 (x)] kj, ] [A µ (x)] jm := A µ a(x) [λ a ] jm [U(x)] jk [A µ (x) + i g µ x kl [U 1 (x)] lm

114 Consequences of local color symmetry only objects which are invariant under local (gauge) transformations are observable color white states natural explanation for appearance of quark-antiquark and three-quark states indeed: q jfs q jgt q jfs U 1 jk U kl q lgt = q jfs q jgt white ɛ jkl q jfs q kgt q lhu... = detu }{{} ɛ jkl q jfs q kgt q lhu white =1 confinement: Why can one not construct a white state from a single quark and infinitely many gluons? at least natural: such a state should be heavy

115 Consequences of local color symmetry L = 1 4 F µνf µν + q (iγ µ D µ M q ) q coupling constant g appears in D µ = µ iga µ and in F µν = i g [Dµ, D ν ] ( gluon-gluon coupling) gauge invariance holds only if same g appears in both expressions {quark type a}-gluon coupling = gluon-gluon coupling = = {quark type b}-gluon coupling only one universal coupling constant only few parameters: one coupling, few quark masses high predictive power

116 Note on electric charges only one universal coupling constant in QCD in principle different in QED: proton charge could be distinct from positron charge (no photon-photon coupling) universal coupling is property of non-abelian gauge theories grand unified theories use non-abelian gauge groups to explain agreement between proton and positron charge