Fluctuations and correlations of conserved charges in hadron resonance gas model Subhasis Samanta National Institute of Science Education and Research, HBNI, Jatni, India Outline ⋆ Introduction ⋆ HRG models Ideal S-matrix formalism VDWHRG ⋆ Summary S Samanta CETHENP 2019, VECC, India 1 / 25
Introduction hadrons/leptons The major goals ⋆ The mapping of QCD phase diagram ⋆ Locating the QCD critical point √ s NN (GeV) Facility µ B (MeV) Status LHC 2760 0 Running RHIC 7.7 - 200 420-20 Running NA61/ SHINE 8 400 Running FAIR 2.7-4.9 800-500 Future NICA 4-11 600-300 Future HRG models have been used to study hadronic phase S Samanta CETHENP 2019, VECC, India 2 / 25
Ideal Hadron Resonance Gas model ⋆ System consists of all the hadrons including resonances (non-interacting point particles) ⋆ Hadrons are in thermal and chemical equilibrium ⋆ The grand canonical partition function of a hadron resonance ln Z = � gas: i ln Z i ⋆ For i th hadron/resonance, i = Vg i j = 1 ( ± 1 ) j − 1 ( z j / j 2 ) K 2 ( jm i / T ) , z = exp ( µ/ T ) , ln Z id 2 π 2 m 2 i T � ∞ µ i = B i µ B + S i µ S + Q i µ Q The + (-) sign refers to bosons (fermions) The first term ( j = 1) corresponds to the classical ideal gas Width of the resonances are ignored S Samanta CETHENP 2019, VECC, India 3 / 25
EOS of IDHRG at µ = 0 6 PDG 2016 LQCD 3P ⋆ IDHRG provides a 4 T ε 4 satisfactory description in 4 T the hadronic phase of 3s continuum LQCD data 3 4T 2 0 0.05 0.1 0.15 T (GeV) S. Samanta et al. JPG 46, 065106 (2019); LQCD data: A. Bazavov et al. (HotQCD), PRD 90, 094503 (2014) S Samanta CETHENP 2019, VECC, India 4 / 25
Problem to quantify χ BS , C BS 0.12 HRG (PDG 2016) SB limit 1 Lattice (HotQCD) 0.1 0.8 0.08 C BS 0.6 N t =6 N t =8 - χ BS 11 N t =10 0.4 0.06 N t =12 N t =16 0.2 0.04 cont. HRG 0 0.02 150 200 250 300 350 400 T [MeV] 0 100 110 120 130 140 150 160 170 T (MeV) Ref: A. Borsanyi et al., JHEP01, 138 (2012) ⋆ IDHRG fails to describe χ BS , C BS = − 3 χ 11 BS /χ 2 S ⇒ Interaction is needed S Samanta CETHENP 2019, VECC, India 5 / 25
Classical Virial Expansion (Non-relativistic) � � � 2 � N � � N P = NT 1 + B ( T ) + C ( T ) + .. V V V ⋆ The first term in the expansion corresponds to an ideal gas ⋆ The second term is obtained by taking into account the interaction between pairs of particles and subsequent terms involve the interaction between groups of three,four, etc. particles ⋆ B , C , ... are called second, third, etc., virial coefficients Second virial coefficient B ( T ) = 1 � ( 1 − e − U 12 / T ) dV 2 U 12 is the two body interaction energy S Samanta CETHENP 2019, VECC, India 6 / 25
Relativistic Virial Expansion i 1 , i 2 z i 1 1 z i 2 ln Z = ln Z 0 + � 2 b ( i 1 , i 2 ) d 3 p � − β ( p 2 + ε 2 ) 1 / 2 � �� �� ∂ε − ∂ S − 1 V S − 1 ∂ S � � b ( i 1 , i 2 ) = d ε exp ∂ε S 4 π i ( 2 π ) 3 aa → R → aa , ab → R → ab , aab → R → aab etc. ⋆ z 1 and z 2 are fugacities of two species ( z = e βµ ) ⋆ The labels i 1 and i 2 refer to a channel of the S-matrix which has an initial state containing i 1 + i 2 particles Second virial coefficient b 2 = b ( i 1 , i 2 ) / V where i 1 = i 2 = 1 ππ → R → ππ π K → R → π K KK → R → KK π N → R → π N etc. S Samanta CETHENP 2019, VECC, India 7 / 25
Interacting part of pressure b 2 in terms of phase shift � ∞ ∂δ I l ( ε ) 1 ′ g I , l M d εε 2 K 2 ( βε ) � b 2 = 2 π 3 β l , I ∂ε P int = 1 ∂ ln Z int = 1 β z 1 z 2 b 2 β ∂ V � ∞ ∂δ I l ( ε ) z 1 z 2 d εε 2 K 2 ( βε ) � ′ g I , l = 2 π 3 β 2 ∂ε M I , l ⋆ Interaction is attractive (repulsive) if derivative of the phase shift is positive (negative) S Samanta CETHENP 2019, VECC, India 8 / 25
K-matrix formalism (Attractive part of the interaction) Scattering amplitude: S ab → cd = � cd | S | ab � Scattering operator (matrix) S = I + 2 iT S is unitary SS † = S † S = I ( T − 1 + iI ) † = T − 1 + iI K − 1 = T − 1 + iI , K = K † (i.e., K matrix is real and symmetric) S Samanta CETHENP 2019, VECC, India 9 / 25
Phase shift in K-matrix formalism − 1 ⇒ Im T / Re T = K − 1 , Re T = K ( I + K 2 ) Im T = K 2 ( I + K 2 ) m R Γ R → ab ( √ s ) � K ab → R → ab = m 2 R − s R Resonances appear as sum of poles in the K matrix Partial wave decomposition S l = exp ( 2 i δ l ) = 1 + 2 iT l ⇒ T l = exp ( i δ ) sin ( δ l ) Im T l = sin 2 ( δ l ) Re T l = sin ( δ l ) cos ( δ l ) , δ l = tan − 1 ( K ) K = tan ( δ l ) , S Samanta CETHENP 2019, VECC, India 10 / 25
Phase shift: Empirical vs KM 3.5 3.5 3 3 2.5 2.5 2 2 δ (rad) δ (rad) 1.5 1.5 1 1 0.5 0.5 KM KM Empirical Empirical 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ε (GeV) ε (GeV) π K → K ∗ ( 892 ) → π K ππ → ρ ( 770 ) → ππ ⋆ Good agreement between the empirical phase shifts of resonances and the K-matrix approach S Samanta CETHENP 2019, VECC, India 11 / 25
Comparison between K-matrix and Breit-Wigner approach 12 8 KM KM BW BW 7 10 6 8 5 σ (mb) σ (mb) 6 4 3 4 2 (a) (b) 2 1 0 0 0.8 1 1.2 1.4 1.6 1.8 2 0.8 1 1.2 1.4 1.6 1.8 2 s (GeV) √ √ s (GeV) f 0 (980) ( m 1 = 990 MeV, Γ 1 = 55 MeV) f 0 (1370) ( m 1 = 1370 MeV, Γ 1 = 350 MeV) and f 0 (1500) ( m 2 = 1505 MeV, Γ 2 = 109 MeV) and f 0 (1500) ( m 2 = 1505 MeV, Γ 2 = 109 MeV) ⋆ KM formalism preserves the unitarity of the S matrix and neatly handles overlapping resonances S. Samanta et al. PRC 97, 055208 (2018) S Samanta CETHENP 2019, VECC, India 12 / 25
Ideal gas limit ⋆ For a narrow resonance, δ I l changes rapidly through π radian around ε = m R ⋆ δ I l can be approximated by a step function: δ I l ∼ Θ( ε − m R ) ⋆ ∂δ I l /∂ε ≈ πδ ( ε − m R ) � ∞ ∂δ I l ( ε ) 1 d εε 2 K 2 ( βε ) � ′ g I , l b 2 = 3.5 2 π 3 β ∂ε M l , I 3 = g I , l 2.5 2 π 2 m 2 R TK 2 ( β m R ) 2 δ (rad) 1.5 1 P int = Tz 1 z 2 b 2 = P R 0.5 KM id Empirical 0 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ε (GeV) ⋆ Pressure exerted by an ideal (MB) gas of particles of mass m R ⋆ This establishes the fundamental premise of the IDHRG model S Samanta CETHENP 2019, VECC, India 13 / 25
Repulsive interaction from experimental data of phase shift 2 ⋆ NN interaction: 3 S 1 1 S 0 1 G 4 1 1 3 D 2 1 D 2 3 H 4 1.5 3 D 1 3 P 1 3 H 5 3 D 3 3 P 2 3 H 6 All available data 1 P 1 3 F 2 1 I 6 0.5 0.5 3 F 3 3 J 6 1 3 F 4 3 J 7 δ (rad) δ (rad) δ (rad) ⋆ ππ repulsive 0.5 0 0 0 interaction: δ 2 -0.5 -0.5 (b) (a) (a) 0 -0.5 I = 0 I = 1 I = 1 ⋆ KN repulsive -1 -1 -1 1.92 2 2.08 2.16 2.24 2.32 1.92 1.92 2 2 2.08 2.08 2.16 2.16 2.24 2.24 2.32 2.32 interaction: s (GeV) s (GeV) s (GeV) √ √ √ 2 0 S 11 ( l I , 2 J ) ( Σ( 1660 ) ) 0 − 0.05 − 0.1 1.5 − 0.1 − 0.2 ⋆ π N repulsive − 0.15 1 − 0.3 (rad) − (rad) (rad) 0.2 − 0.5 0.4 interaction: S 31 − 0.25 δ δ δ − 0.5 − 0.3 0 − 0.6 ( l 2 I , 2 J ) ( ∆( 1620 ) ), − 0.35 − − 0.7 0.5 − 0.4 ∆( 1910 ) , N ( 1720 ) − 0.8 − 0.45 − 1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 2 2.1 2.2 2.3 etc. s (GeV) s (GeV) s (GeV) ππ : ( δ 2 0 ) KN : S 11 π N : S 31 ⋆ Σ( 1660 ) , Σ( 1750 ) , Σ( 1915 ) , ∆( 1620 ) ), ∆( 1910 ) , ∆( 1930 ) , N ( 1720 ) etc. are included in the repulsive part Ref: SAID [http://gwdac.phys.gwu.edu] S Samanta CETHENP 2019, VECC, India 14 / 25
Results 1.2 7 Total Total ⋆ KM: Attractive KM KM 6 1 IDHRG IDHRG interaction Lattice (WB) Lattice (WB) 5 0.8 Lattice (HotQCD) Lattice (HotQCD) 4 ⋆ Total: Attractive + P/T 4 0.6 ε /T 4 3 repulsive 0.4 2 ⋆ Both KM and Total 0.2 1 (a) (b) contain 0 0 100 110 120 130 140 150 160 170 100 110 120 130 140 150 160 170 non-interacting T (MeV) T (MeV) 0.65 0.12 part as well Total Total 0.6 KM KM 0.1 IDHRG (PDG 2016) IDHRG 0.55 ⋆ Repulsive Lattice (WB) Lattice (HotQCD) 0.5 Lattice (HotQCD) Lattice 0.08 interactions 0.45 - χ BS 11 0.4 0.06 χ Q 2 suppress the bulk 0.35 0.04 variables 0.3 0.25 0.02 (b) 0.2 (c) 0.15 0 100 110 120 130 140 150 160 170 100 110 120 130 140 150 160 170 T (MeV) T (MeV) S. Samanta et al. PRC 99, 044919 (2019) S Samanta CETHENP 2019, VECC, India 15 / 25
χ 2 B − χ 4 B and C BS 10 0 1.2 2 - χ B 4 Total χ B 10 -1 4 - χ B 2 KM χ B 1 IDHRG (PDG 2016) 2 - χ B 4 (Lattice) χ B 10 -2 IDHRG (PDG 2016+) 4 - χ B 2 (Lattice) χ B Lattice (WB) 0.8 10 -3 Lattice (HotQCD) 2 - χ B 4 C BS 10 -4 0.6 χ B 10 -5 0.4 10 -6 0.2 10 -7 (b) 10 -8 0 100 110 120 130 140 150 160 170 100 110 120 130 140 150 160 170 T (MeV) T (MeV) ⋆ χ 2 B − χ 4 B is non-zero ⋆ For C BS : Improvement compared to IDHRG S. Samanta et al. PRC 99, 044919 (2019) S Samanta CETHENP 2019, VECC, India 16 / 25
Excluded volume hadron resonance gas model ⋆ Hadrons have finite hard-core radii. ( P ( V − Nb ) = NT ) 3 π R 3 is the volume excluded for the hadron. ⋆ b = V ex = 16 ⋆ Pressure and chemical potential in EVHRG model: � P id P ( T , µ 1 , µ 2 , .. ) = i ( T , ˆ µ 1 , ˆ µ 2 , .. ) , i µ i = µ i − V ev , i P ( T , µ 1 , µ 2 , .. ) ˆ S Samanta CETHENP 2019, VECC, India 17 / 25
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