Fictions, fluctuations and mean fields Pasi Huovinen Uniwersytet Wroc� lawski Constraining the QCD Phase Boundary with Data from Heavy Ion Collisions February 12, 2018, GSI, Darmstadt in collaboration with Peter Petreczky, arXiv:1708.00879 The speaker has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk� lodowska-Curie grant agreement No 665778 via the National Science Center, Poland, under grant Polonez DEC-2015/19/P/ST2/03333
Fiction , noun A fictitious particle, i.e. a particle predicted by some model without solid empirical evidence for its existence P. Huovinen @ GSI, February 12, 2018 1/29
Fluctuations of conserved charges T n ∂ n P/T 4 � χ X � = n � ∂µ n � X µ X =0 T n + m ∂ n + m P/T 4 � χ XY � = � nm ∂µ n X ∂ m � Y µ X =0 , µ Y =0 0.35 0.35 free quark gas B χ 2 0.3 0.3 0.25 0.25 PDG-HRG cont. extrap. 0.2 0.2 T c =(154 +/-9) MeV N τ =16 12 0.15 0.15 8 6 0.1 0.1 m s /m l =20 (open) 27 (filled) 0.05 0.05 T [MeV] 0 0 140 140 160 160 180 180 200 200 220 220 240 240 260 260 280 280 Bazavov et al., PRD95, 054504 (2017) P. Huovinen @ GSI, February 12, 2018 2/29
Black on black! P. Huovinen @ GSI, February 12, 2018 3/29
More resonances? BS / χ 2 S - χ 11 0.30 0.25 cont. est. PDG-HRG 0.20 QM-HRG N τ =6: open symbols 0.15 N τ =8: filled symbols S /M 1 S B 1 S /M 2 S 0.45 B 2 S /M 1 S B 2 0.35 0.25 T [MeV] 0.15 140 150 160 170 180 190 Bazavov et al., PRL113, 072001 (2014) P. Huovinen @ GSI, February 12, 2018 4/29
Baryon spectrum 3 2.5 M (GeV) 2 1.5 1 S=0 S=1 S=2 S=3 Blue: Particle Data Group P. Huovinen @ GSI, February 12, 2018 5/29
Baryon spectrum 3 2.5 M (GeV) 2 1.5 1 S=0 S=1 S=2 S=3 Blue: Particle Data Group Red: PDG + L¨ oring et al., EPJA10, 395 (2001) & EPJA10, 447 (2001) P. Huovinen @ GSI, February 12, 2018 6/29
Hadron spectrum mesons baryons 3 2.5 2 M (GeV) 1.5 1 0.5 0 S=0 S=1 S=0 S=1 S=2 S=3 Blue: Particle Data Group Red: PDG + L¨ oring et al., EPJA10, 395 (2001) & EPJA10, 447 (2001) Black: PDG + Ebert et al., PRD79, 114029 (2009) P. Huovinen @ GSI, February 12, 2018 7/29
Trace anomaly 4.5 Budapest-Wuppertal 4 hotQCD HRG 3.5 3 ( ε -3P)/T 4 2.5 2 1.5 1 0.5 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 8/29
Trace anomaly 4.5 Budapest-Wuppertal 4 hotQCD HRG 3.5 HRG+ 3 ( ε -3P)/T 4 2.5 2 1.5 1 0.5 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 9/29
χ 2 B 0.2 B χ 2 0.15 0.1 0.05 hotQCD B-W HRG 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 10/29
χ 2 B 0.2 B χ 2 0.15 0.1 0.05 hotQCD B-W HRG HRG+ 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 11/29
χ 11 BS 0.2 BS χ 11 0.15 0.1 0.05 hotQCD HRG 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 12/29
χ 11 BS 0.2 BS χ 11 0.15 0.1 0.05 hotQCD HRG HRG+ 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 13/29
χ 2 S 0.6 S χ 2 0.5 0.4 0.3 0.2 0.1 B-W HRG 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 14/29
χ 2 S 0.6 S χ 2 0.5 0.4 0.3 0.2 B-W 0.1 HRG HRG+ 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 15/29
Differences of fluctuations Filled symbols: HISQ Bazavov et al., PRL111, 082301 (2013) PRD95, 054504 (2017) Open symbols: stout 4th order Bellwied et al., PRD92, 114505 (2015) 6h order D’Elia et al,. PRD95, 094503 (2017) • These zero in Boltzmann approximation P. Huovinen @ GSI, February 12, 2018 16/29
Virial expansion P = P i deal + T � b ij 2 ( T ) e βµ i e βµ j ij b ij 2 can be related to the S-matrix of scattering of particles i and j • ππ , πN , etc. scatterings dominated by resonance formation • no resonances in NN scatterings P. Huovinen @ GSI, February 12, 2018 17/29
Virial expansion in nucleon gas P ( T, µ ) = P 0 ( T ) cosh( βµ ) + 2 b 2 ( T ) T cosh(2 βµ ) 4 m 2 T 2 P 0 ( T ) = K 2 ( βm ) π 2 � ∞ � � � d E − d S † � mE � b 2 ( T ) = 2 T mE 4 i Tr[ S † d S 1 + m 2 + m 2 d E K 2 2 β d E S ] π 3 2 2 0 P. Huovinen @ GSI, February 12, 2018 18/29
Virial expansion in nucleon gas Elastic part of the S-matrix from scattering phase shift: d E − d S † � d δ J,I =0 + 3d δ J,I =1 � 4 i Tr[ S † d S 1 � � s s d E S ] → (2 J + 1) d E d E s J Workman et al., PRC94, 065203 (2016); Arndt et al., PRC76, 025209 (2007) P. Huovinen @ GSI, February 12, 2018 19/29
Repulsive mean field Assume: interactions reduce single partice energy by U = Kn b where n b is single nucleon density ( Olive, NPB190, 483 (1981) ) d 3 p � (2 π ) 3 e − β ( E p − µ + U ) n b = Small µ ⇒ βKn b ≪ 1 and n 0 b (1 − βKn 0 n b b ) ⇒ ≈ b ) − K b ) 2 + ( n 0 ( n 2 b ) 2 � � P ( T, µ ) = T ( n b + n ¯ ¯ 2 or � � cosh( βµ ) − Km P ( T, µ ) = P 0 ( T ) π 2 K 2 ( βm ) cosh(2 βµ ) P. Huovinen @ GSI, February 12, 2018 20/29
Virial expansion vs. mean field Repulsive mean field Virial expansion P ( T, µ ) = P 0 ( T ) × P ( T, µ ) = P 0 ( T ) × cosh( βµ ) − ¯ cosh( βµ ) − Km � � � � π 2 K 2 ( βm ) cosh(2 βµ ) b 2 ( T ) K 2 ( βm ) cosh(2 βµ ) where ¯ 2 T b 2 ( T ) b 2 = P 0 ( T ) K 2 ( βm ) P. Huovinen @ GSI, February 12, 2018 21/29
Trace anomaly 4.5 B-W 4 hotQCD HRG 3.5 HRG-mean 3 ( ε -3P)/T 4 2.5 2 1.5 1 0.5 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 22/29
Trace anomaly 4.5 B-W 4 hotQCD HRG 3.5 HRG-mean HRG+ 3 HRG+ mean ( ε -3P)/T 4 2.5 2 1.5 1 0.5 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 23/29
χ 2 B 0.2 B χ 2 0.15 0.1 0.05 hotQCD B-W HRG HRG-mean 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 24/29
χ 2 B 0.2 B χ 2 0.15 0.1 hotQCD B-W 0.05 HRG HRG-mean HRG+ HRG+ mean 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 25/29
χ 11 BS 0.2 BS χ 11 0.15 hotQCD 0.1 HRG HRG-mean 0.05 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 26/29
χ 11 BS 0.2 BS χ 11 0.15 hotQCD HRG HRG-mean 0.1 HRG+ HGR+ mean 0.05 0 100 120 140 160 180 200 T (MeV) P. Huovinen @ GSI, February 12, 2018 27/29
Differences of fluctuations Filled symbols: HISQ Bazavov et al., PRL111, 082301 (2013) PRD95, 054504 (2017) Open symbols: stout 4th order Bellwied et al., PRD92, 114505 (2015) 6h order D’Elia et al,. PRD95, 094503 (2017) • These zero in Boltzmann approximation • Repulsive interactions create similar differences P. Huovinen @ GSI, February 12, 2018 28/29
Summary • lattice QCD indicates there are more resonances than observed – inclusion of quark model states improves the fit to some, and weakens the fit to some observables • repulsive mean field can describe the differences between baryonic fluctuations of different orders • mean field strength can be constrained by phase shifts P. Huovinen @ GSI, February 12, 2018 29/29
Summary • lattice QCD indicates there are more resonances than observed – inclusion of quark model states improves the fit to some, and weakens the fit to some observables • repulsive mean field can describe the differences between baryonic fluctuations of different orders • mean field strength can be constrained by phase shifts P. Huovinen @ GSI, February 12, 2018 29/29
Summary • lattice QCD indicates there are more resonances than observed – inclusion of quark model states improves the fit to some, and weakens the fit to some observables • repulsive mean field can describe the differences between baryonic fluctuations of different orders • mean field strength can be constrained by phase shifts P. Huovinen @ GSI, February 12, 2018 29/29
Hadron Resonance Gas with mean field Assume: only members of baryon octet and decuplet repel each other P ( T, µ ) = Tn − K od ) 2 + ( n 0 ( n 0 od ) 2 � � ¯ 2 where n od ( T ) = T � g i m 2 i K 2 ( βm i ) 2 π 2 i i = N, Σ , Ξ , ∆ , Σ ∗ , Ξ ∗ , Ω P. Huovinen @ GSI, February 12, 2018 30/29
Hadron Resonance Gas with mean field Assume: only members of baryon octet and decuplet repel each other P ( T, µ ) = Tn − K od ) 2 + ( n 0 ( n 0 od ) 2 � � ¯ 2 where n od ( T ) = T � g i m 2 i K 2 ( βm i ) 2 π 2 i i = N, Σ , Ξ , ∆ , Σ ∗ , Ξ ∗ , Ω χ B (0) − 2 n β 4 K ( n 0 od ) 2 χ B = n n χ BS (0) + 2 n +1 β 5 Kn 0 od ( P S 1 B + 2 P S 2 B + 3 P S 3 χ BS = B ) n 1 n 1 P. Huovinen @ GSI, February 12, 2018 31/29
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