Entropy shifts and Missing States. Excited Hyperons in QCD Thermodynamics at Freeze-Out

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1 E. Ruiz Arriola Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, Spain. Excited Hyperons in QCD Thermodynamics at Freeze-Out Thomas Jefferson National Accelerator Facility Newport News, VA November 16-17, 2016 Works with Eugenio Megias, Lorenzo Salcedo, Wojciech Broniwoski, Pere Masjuan

2 Outline The Hadron Spectrum at zero temperature Quarks and gluons at finite temperature Quark Hadron duality at finite temperature Entropy shifts Anatomy of Hadron Resonance Gas Conclusions References

3 To count or not to count The best way to look for missing states is to count them 1, 2, 3... Thermodynamics is a way of counting The partition function of QCD counts Z QCD = n e En/T H QCDψ n = E nψ n Spectrum of QCD Thermodynamics Colour singlet states (hadrons +...???) Do we see quark-gluon substructure BELOW the phase transition? Completeness relation in Hilbert space H QCQ 1 = n Ψ n Ψ n qq; n qq; n + qqq; n qqq; n + qqg; n qqg; n +... n n n }{{}}{{}}{{} mesons baryons hybrids Given H, is there a sum rule involving ALL resonances?.

4 HADRONIC SPECTRUM AT ZERO TEMPERATURE

5 The states in the Particle Data Group (PDG) book GeV HADRONS 2.5 MESONS GeV BARYONS No data (is a compilation) No particles (resonances) No book Which particles enter PDG?

6 Hadron Spectrum (u,d,s) Particle Data Group (PDG) compilation 2016 GeV HADRONS Relativized Quark Model (RQM) Isgur, Godfrey, Capstik, 1985 MESONS GeV BARYONS GeV HADRONS

7 Cumulative number of states Compare H QCD, H PDG, H RQM with staircase function N(M) = n θ(m M n) Which states count? Is N QCD(M) accessible? Hadrons N M 100 N M 100 N M N q q M 6 N qqq M 12 N qq qq M 18 N hadrons e M/T H T H 150MeV= Hagedorn temperature

8 Spectrum with one heavy quark T H Hagedorn temperature for hadrons with ONE heavy quark N( ) = n θ( n) N qq ( ) + N Qqq ( ) + e /T H Missing states!! RQM b RQM c PDG c N

9 Hagedorn and The bootstrap 10 5 MIT BAG model n Fit: Ae T H Total Tetraquarks Baryons Mesons MeV Which are the complete set of states in the PDG? Should X,Y,Z s or the deuteron or 208 Pb enter as multiquark states?

10 Who counts? 1.0 N M The cumulative number in a given channel in the continuum with threshold M th N(M) = n θ(m M n) + [δ(m) δ(m th)]/π Levinson s theorem Deuteron doesn t count N( ) = n B + [δ( ) δ(m th)]/π = 0

11 Quark-Hadron duality at zero temperature In the confined phase we expect all observables to be represented by hadronic degrees of freedom. 1 Gell-Mann Oakes Renner relation 2 qq m q = fπm 2 π 2, (1) }{{}}{{} quarks hadrons 2 Transition form factor of the pion 3 Effective Chiral lagrangians with resonances 4 Deep inelastic scattering Are hadrons a complete set of states? Is the PDG complete or overcomplete? The phase transition is a smooth cross-over, so we expect to see departures from quark-hadron duality below T c

12 QUARKS AND GLUONS AT FINITE TEMPERATURE

13 QCD at finite temperature QCD Lagrangian: L QCD = 1 4 Ga µν Ga µν + f q a f (iγµdµ m f )q a f ; Partition function Z QCD = Tre H/T = e En/T n [ = DA µ,a exp 1 ] d 4 x(g a µν 4 )2 Det(iγ µd µ m f ) Boundary conditions and Matsubara frequencies q( x, β) = q( x, 0) A µ( x, β) = A µ( x, 0) β = 1/T dp0 2π f (p 0) T f (w n) n w n = (2n + 1)πT w n = 2nπT

14 Thermodynamic relations Statistical mechanics of non-interacting particles log Z = V ηg i d 3 p [ ] (2π) log 1 + ηe Ep /T E 3 p = p 2 + m 2 η = 1 for bosons ; η = 1 for fermions ; g i -number of species F = T log Z P = T F V S = (TF ) E = F + TS T High temperature limit Free gas of gluons and quarks Interaction measure (trace anomaly) [ P = 2(N 2 c 1) + 4NcN f ] 7 π T 4 ɛ 3p 0 (T ) T 4

15 Symmetries in QCD Colour gauge invariance q(x) e i a (λa)c αa(x) q(x) g(x)q(x) A g µ (x) = g 1 (x) µg(x) + g 1 (x)a µ(x)g(x) Only periodic gauge transformations are allowed: g( x, x 0 + β) = g( x, x 0 ), β = 1/T. In the static gauge 0 A 0 = 0 g(x 0 ) = e i2πx 0 λ/β, where λ = diag(n 1,, n Nc ), Trλ = 0. Large Gauge Invariance: periodicity in A 0 with period 2π/β A 0 A 0 + 2πT diag(n j ) Gribov copies Explicitly Broken in perturbation theory (non-perturbative finite temperature gluons)

16 Symmetries in QCD In the limit of massless quarks (m f = 0), Invariant under scale (x λx) Broken by quantum corrections regularization (Trace anomaly) Chiral Left Right transformations. ɛ 3p = β(g) 2g (Ga µν )2 = 0, q(x) e i a (λa)f αa q(x) q(x) e i a (λa)f αaγ 5 q(x) Broken by chiral condensate in the vacuum qq = 0

17 Symmetries in QCD Gluodynamics: In the limit of heavy quarks (m f ) Z [ DA µ,a exp 1 4 d 4 x(g a µν )2 ] Det( m f ) Larger symmetry ( t Hooft) Center Symmetry Z(N c) g( x, x 0 + β) = z g( x, x 0 ), z Nc = 1, (z Z(N c)). The Poyakov loop g(x 0 ) = e i2πx 0 λ/(nc β), A 0 A 0 + 2πT N c diag(n j ) L T = 1 N c tr ce ia 0 /T = e Fq /T = e i2π/nc L T = 0 F q = means CONFINEMENT At high temperatures A 0 /T << 1 L T = 1 trca2 0 2N ct 2 trc A = e 2Nc T In full QCD L T = O(e mq /T ) 0 Large Violation of center sym.

18 Lattice results in full QCD The chiral-deconfinement cross over is a unique prediction of lattice QCD Order parameter of chiral symmetry breaking (m q = 0) Quark condensate SU(N f ) SU(N f ) SU V (N f ) qq = 0 T < T c qq = 0 T > T c Order parameter of deconfinement (m q = ) Polyakov loop: Center symmetry Z (N c) broken L T = 1 N c tr ce ia 0 /T = 0 T < T c L T = 1 N c tr ce ia 0 /T = 1 T > T c In the real world m q is finite. The chiral-deconfinement crossover (connected) crossed correlator (never computed on lattice), qq tr c e iga 0 /T qq tr c e iga 0 /T = L T m q, (2)

19 QUARK-HADRON DUALITY AT FINITE TEMPERATURE

20 QCD Spectrum and Trace anomaly GeV HADRONS GeV HADRONS Ε 3p T PDG is thermodynamically equivalent to RQM!! A HRG(T ) ɛ 3P = 1 T 4 T 4 n d 3 p E n(p) p pe n(p), (2π) 3 e En(p)/T + η n E n(p) = p 2 + M 2 n η n = ±1 Non-interacting Hadron-Resonance Gas works for T < 0.8T c Spectrum Thermodynamics

21 Fluctuations of conserved charges Conserved charges N B T = 0 N Q T = 0 N S T = 0 Vacuum Fluctuations χ BB (T ) = N 2 B T 1 N f χ QQ (T ) = N 2 Q N T q 2 i c i=1 χ SS (T ) = N 2 S T 1 C BS (T ) = 3 N SN B T N 2 S T ΧBB T ΧQQ T ΧSS T CBS T C PDG BS < CQCD BS (Missing states?, Significant?)

22 Spectral representation Two conjugate sources are placed in the medium at temperature T and a separation distance r generate a Free energy shift β t time time Ω r Ω Ω z x,y 0 z r x,y Ω t β e F(r,T ) = tr R Ω(r)tr RΩ(0) T Standard representation (ratio of partition functions) e F(r,T ) = Z R R (r, T ) Z 0 (T ) = n e ER R n (r)/t n e E0 n (r)/t Spectral r-representation e F (r,t ) = n n, T Tr R Ω 0, T 2 e rwn(t ) Inequalities r F (r, T ) 0 2 r F (r, T ) 0

23 Double heavy hadron spectrum and correlators String breaking for the QQ BB (level crossing) M QQ (r c) = V Q Q(r c) + m Q + m Q = M B + M B, No mixing e F(r,T )/T = e V Q (r)/t + ( n ) 2 e (n) H /T Two modes model V (r) = 4α 3r + σr W (r) W (r) 2..., W (r) = ge mr

24 Free energies Figure: The QQ BB transition potential Entropy W (r) shifts as anda Missing function States of separation V Q Q GeV Π Σ r c 12 r T 150MeV 0.5 T 175MeV T 200MeV T 225MeV r fm Σ 0.55GeV 2 Σ 0.42GeV 2 F Q Q GeV Borsányi et al. data vs Isgur cmesons cbaryons Σ 0.55 GeV 2, N f 2 1 T 150MeV r fm T 175MeV T 200MeV T 225MeV T 250MeV T 275MeV T 300MeV T 325MeV T 350MeV W GeV T 150MeV T 175MeV T 200MeV 0.80 GeV r W 0.98 e GeV r fm W GeV GeV r W 0.98 e GeV 0.80 GeV r W 0.40 e GeV r fm T 150MeV T 175MeV T 200MeV

25 Avoided crossing from thermodynamics V r GeV V GeV String tension is only defined in Quenched Approximation String breaks. How determine string tension? Using thermodynamics WITH mixing σ = 0.424(14)GeV, g = 0.98(47)GeVm = 0.80(38) Spectral representation Quantum phase transition AT ZERO TEMPERATURE and complex string tension as a pole in the second Riemann sheet

26 ENTROPY SHIFTS

27 Thermodynamic shifts Add one extra heavy charge belonging to rep R to the vacuum Energy of the states changes under the presence of the charge E n E R n R n + m R +... In the static gauge 0 A 0 = 0 the Ployakov loop operator trω( r) = tre ia 0 ( r)/t The ratio of partition functions Free energy shift tre ia 0 /T = Z R = e F R n /T = e n/t Z

28 Counting states (Hagedorn-Polyakov temperature) N( ) = n θ( n) e /T H,L L(T ) N ( )e /T d 0.6 N RQM b RQM c PDG c L T BW data RQM b RQM c PDG c T MeV Figure: Left: N( ) as a function of the c-quark and b-quark mass subtracted hadron mass = M m Q (in ) with u, d and s quarks, computed in the RQM vs PDG. Right: Polyakov loop as a function of temperature (in MeV).

29 Polyakov loop ambiguity removed by entropy shift trω(0) T = e F Q (T )/T S Q (T ) = T F Q (T ) Third principle of thermodynamics for degenerate states S Q (0) = log(2n f ), S Q ( ) = log N c RGE equation for specific heat c Q = T S Q T = { T T Energy momentum tensor [ ]} trω Θ(x) d 4 x Θ(x) U Q trω T 0 = µ ds Q dµ = β(g) S Q g q m q(1 + γ S Q q) T S Q m q T Entropy shift IS NOT a true entropy c = T T S = ( H) 2 /T 2 > 0

30 From Hadron resonance gas... TUM collaboration 125 < T 6000MeV Constituent Quark Model M = 300MeV, m u = 2.5MeV,m d = 5MeV, m s = 95MeV. L = q=u,d,s g qe M /T Qq + q,q =u,d,s g q,q e M Qqq /T +... All=(Q q, Qqq and Q qg) Hadron spectrum (missing states!!) 6 Bag All RQM b Hag Bag Hadrons RQM b RQM c SQ T 4 2 TUM data CQM PDG c log Nc

31 ... to Power corrections Dim-2 condensates tr(e iao T ) N c exp [ g 2 (A a 0 )2 4N ct 2 ] S Q (T ) = tr(ā2 0 ) NP 2N ct 2 + S pert(t ) + log(n c) SQ T S Q logn c S pert A² NP 2T² A² NP 2Tc² c Figure: Left panel: TUM lattice data for the entropy as a function of the inverse squared temperature in unites of the critical temperature.

32 ANATOMY OF HADRON RESONANCE GAS

33 Limitations of the Hadron resonance gas In the HRG Hadrons are taken STABLE, POINT-LIKE AND ELEMENTARY Hadrons are composite Hadrons have a size Hadron Resonances have a finite width

34 From chiral quark models to hadron resonance gas Use local Wilson line Ω(x) = e iga 0 (x) as quantum variable Quantization of Multiquark states: Create/Anhiquilate a quark at point x and momentum p Ω(x)e E P /T Ω(x) + e E P /T At low temperatures quark Boltzmann factor small e Ep /T < 1. The action becomes small S q[ω] = 2N f Z = d 3 xd 3 p (2π) 3 [ tr cω(x) + tr cω(x) +] e Ep /T +... DΩ e S[Ω] = DΩ ( 1 S[Ω] + 1 ) 2 S[Ω]2 +...

35 qq contribution Z qq = (2N f ) 2 d 3 x 1 d 3 p 1 (2π) 3 d 3 x 2 d 3 p 2 (2π) 3 e E 1 /T e E 2 /T tr cω( x 1 )tr cω ( x 2 ) }{{} e σ[ x 1 x 2 /T d = (2N f ) 2 3 x 1 d 3 p 1 d 3 x 2 d 3 p 2 e H(x (2π) 3 (2π) 3 1,p 1 ;x 2,p 2 )/T qq Hamiltonian H(x 1, p 1 ; x 2, p 2 ) = E 1 + E 2 + V 12. Quantization in the CM frame p 1 = p 2 p ( ) 2 p 2 + M 2 + V q q (r) ψ n = M nψ n. Boosting the CM to any frame with momentum P Z qq n d 3 Rd 3 P e En(P)/T (2π) 3 A GAS OF NON INTERACTING MESONS! (valid to qq qq)

36 Excluded Volume constraint MIT bag model Vi N i V V i i d 3 p (2π) 3 g i e E i (p)/t ± 1 1 V i = M i /(4B) B = (0.166GeV ) V i M i 4B V i M i 3 Σ 3 CoVol CoVol T GeV T MeV When hadrons overlapp, excluded volume corrections are important percolation

37 Resonance spectrum and width Resonances have a mass spectrum (what is the mass?) mesons baryons Γ/m /3 Γ/m dn/dm [MeV -1 ] m [GeV] mesons baryons? N m [GeV] mesons baryons m [GeV] m [GeV]

38 Resonance spectrum and the half-width rule The half-width rule: M R = Γ R /2 or M 2 R = M RΓ R Then take a RANDOM mass (error estimate) Hadronic Meson dominated form factors,... Q 2 FΠΓΓ Q2 GeV Q 2 GeV 2 GA Q 2 GA Q 2 GeV log 10 N M E 3P T M GeV T MeV

39 Fluctuations of conserved charges χ PDG AB ± χpdg AB χqcd AB ± χqcd AB ΧBB T 0.2 ΧQQ T ΧSS T CBS T T GeV

40 CONCLUSIONS

41 Conclusions: Quark Hadron Duality suggests that at low temperatures Hadrons can be considered as a complete basis of states in terms of a hadron resonance gas. The HRG works up to relatively large temperatures. PDG states incorporate currently just q q or qqq states which fit into the quark model. What states are needed when approaching the crossover from below? Saturating at subcritical temperatures requires many hadronic states, so the excited spectrum involves relativistic effects even for heavy quarks. Uncertainty estimates from the HRG may change our perception of what we understand by a missing states. Polyakov loops in fundamental and higher representations allow to deduce multiquark quark states, gluelumps etc. containing one or several heavy quark states. Clear hints for missing states.

42 REFERENCES

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44 Half-width rule (with W. Broniowski and P. Masjuan) Large-N c Regge spectroscopy Acta Phys.Polon.Supp. 8 (2015) no.1, Hadron form factors and large-n c phenomenology EPJ Web Conf. 73 (2014) Reply to Comment on Systematics of radial and angular-momentum Regge trajectories of light nonstrange qq-states Phys.Rev. D87 (2013) no.11, Hadron resonances, large Nc, and the half-width rule Acta Phys.Polon.Supp. 6 (2013) Meson dominance of hadron form factors and large-n c phenomenology Phys.Rev. D87 (2013) no.1, Radial and angular-momentum Regge trajectories: a systematic approach EPJ Web Conf. 37 (2012) Systematics of radial and angular-momentum Regge trajectories of light non-strange q q-states Phys.Rev. D85 (2012)