T HERMODYNAMICS AND FLUCTUATIONS - CORRELATIONS OF CONSERVED CHARGES IN A HADRON RESONANCE GAS MODEL WITH ATTRACTIVE AND REPULSIVE INTERACTION WITHIN S- MATRIX FORMALISM Subhasis Samanta National Institute of Science Education and Research Jatni - 752050, India Sep 12, 2018 Subhasis Samanta Hot Quarks 2018 1 / 18
Overview Motivation IDHRG model Interaction within S-matrix formalism Results Summary Subhasis Samanta Hot Quarks 2018 2 / 18
Study of matter under extreme conditions hadrons/leptons The major goals The mapping of QCD phase diagram in terms T and µ B 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 Subhasis Samanta Hot Quarks 2018 3 / 18
Ideal Hadron Resonance Gas model Statistical thermal model System consists of all the hadrons including resonances (non-interacting) Hadrons are in thermal and chemical equilibrium The grand canonical partition function of a hadron ln Z = � resonance gas: i ln Z i For i th hadron/resonance, � ∞ i = ± Vg i ln Z id p 2 dp ln [ 1 ± exp ( − ( E i − µ i ) / T )] 2 π 2 0 The upper and lower sign corresponds to baryons and mesons respectively. � p 2 + m 2 E i = µ i = B i µ B + S i µ S + Q i µ Q i , Subhasis Samanta Hot Quarks 2018 4 / 18
IDHRG provides a satisfactory description of 6 EOS in the hadronic PDG 2016 LQCD 3P phase of continuum 4 T ε 4 4 LQCD data T 3s 3 4T IDHRG fails to describe 2 χ 2 S , χ 11 BS , C BS = − 3 χ 11 BS /χ 2 S etc. 0 0.05 0.1 0.15 T (GeV) 0.12 HRG (PDG 2016) SB limit 1 Lattice (HotQCD) 0.1 0.8 0.08 0.6 C BS N t =6 - χ BS 11 N t =8 0.06 N t =10 0.4 N t =12 0.04 N t =16 0.2 cont. 0.02 HRG 0 150 200 250 300 350 400 0 100 110 120 130 140 150 160 170 T [MeV] T (MeV) LQCD data: Phys. Rev. D 90, 094503 (2014), JHEP01, 138 (2012) Interaction is needed Subhasis Samanta Hot Quarks 2018 5 / 18
van der Waals interaction in HRG model (VDWHRG model) � � � 2 � N P + a ( V − Nb ) = NT , V � 2 NT � N nT 1 − bn − an 2 P ( T , n ) = V − bN − a ≡ V where n ≡ N / V is the number density of particles. µ ∗ = µ − bP ( T , µ ) − abn 2 + 2 an P ( T , µ ) = P id ( T , µ ∗ ) − an 2 , n id ( T , µ ∗ ) n = 1 + bn id ( T , µ ∗ ) Extra parameters a = 0 ⇒ EVHRG a = b = 0 ⇒ IDHRG Subhasis Samanta Hot Quarks 2018 6 / 18
Classical Virial Expansion (Non-relativistic) + N 2 C ( T ) P = NT � 1 + NB ( T ) � + .. V 2 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 Subhasis Samanta Hot Quarks 2018 7 / 18
Relativistic Virial Expansion i 1 , i 2 z i 1 1 z i 2 ln Z = ln Z 0 + ln Z int = ln Z 0 + � 2 b ( i 1 , i 2 ) d 3 p ∂ε − ∂ S − 1 b ( i 1 , i 2 ) = V � � 1 / 2 � �� S − 1 ∂ S �� � − β ( p 2 + ε 2 ) d ε exp ∂ε S 4 π i ( 2 π ) 3 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 We ignore contributions from bound states Second virial coefficient b 2 = b ( i 1 , i 2 ) / V where i 1 = i 2 = 1 ln Z 0 ⇒ Non interacting stable hadrons ln Z int ⇒ Scattering between two hadrons Subhasis Samanta Hot Quarks 2018 8 / 18
Interacting part of pressure b 2 in terms of phase shift � ∞ ∂δ I l ( ε ) 1 M d εε 2 K 2 ( βε ) � ′ g I , l b 2 = l , I 2 π 3 β ∂ε M is the invariant mass of the interacting pair at threshold P int = 1 ∂ ln Z int = 1 β z 1 z 2 b 2 β ∂ V � ∞ ∂δ I z 1 z 2 l ( ε ) 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) Subhasis Samanta Hot Quarks 2018 9 / 18
Ideal gas limit For a very narrow width resonance, δ I l changes rapidly through 180 ◦ 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 1 l ( ε ) b R d εε 2 K 2 ( βε ) � ′ g I , l 2 = 2 π 3 β ∂ε M l , I = g I , l 2 π 2 m 2 R TK 2 ( β m R ) P R int = Tz 1 z 2 b R 2 = P R id Narrow resonance behaves like a stable hadron of mass m R This establishes the fundamental premise of the IDHRG Subhasis Samanta Hot Quarks 2018 10/ 18
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 = 1 ( T − 1 + iI ) † = T − 1 + iI K − 1 = T − 1 + iI , K = K † (i.e., K martix is real and symmetric) Subhasis Samanta Hot Quarks 2018 11/ 18
Phase shift in K-matrix formalism Re T = K ( I + K 2 ) − 1 , Im T = K 2 ( I + K 2 ) − 1 ⇒ Im T / Re T = K 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 ) , Subhasis Samanta Hot Quarks 2018 12/ 18
Transition amplitude: K-matrix Example: ππ → r 1 ( m 1 ) → ππ ππ → r 2 ( m 2 ) → ππ ( r 1 , r 2 have same l , I ) K = m 1 Γ 1 ( √ s ) + m 2 Γ 2 ( √ s ) m 2 m 2 1 − s 2 − s m 1 Γ 1 ( √ s ) T = 1 − s ) − im 1 Γ 1 ( √ s ) − i m 2 2 − s m 2 Γ 2 ( √ s ) 1 − s ( m 2 m 2 m 2 Γ 2 ( √ s ) + 2 − s ) − im 2 Γ 2 ( √ s ) − i m 2 1 − s m 1 Γ 1 ( √ s ) 2 − s ( m 2 m 2 Subhasis Samanta Hot Quarks 2018 13/ 18
Comparison between K-matrix and Breit-Wigner approach m 1 Γ 1 ( √ s ) m 2 Γ 2 ( √ s ) T ≈ 1 − s ) − im 1 Γ 1 ( √ s ) + 2 − s ) − im 2 Γ 2 ( √ s ) (Separated) ( m 2 ( m 2 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) Subhasis Samanta Hot Quarks 2018 14/ 18
Input 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 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 data ( | B | = 2) δ (rad) δ (rad) δ (rad) 0.5 0 0 π N interaction: 0 -0.5 -0.5 (b) (a) (a) S 31 ( l 2 I , 2 J ) -0.5 I = 0 I = 1 I = 1 -1 -1 -1 ( ∆( 1620 ) ), 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 √ s (GeV) s (GeV) s (GeV) √ √ ∆( 1910 ) , N ( 1720 ) etc. 0 − 0.1 − 0.2 KN interaction: − 0.3 (rad) S 11 ( l I , 2 J ) − 0.4 δ − 0.5 ( Σ( 1660 ) ) − 0.6 − 0.7 ππ interaction: − 0.8 δ 2 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 s (GeV) 0 Data: Scattering Analysis Interactive Database (SAID) partial wave analysis Subhasis Samanta Hot Quarks 2018 15/ 18
Result: EOS KM: Attractive 1.2 7 Total Total interaction KM KM 6 1 IDHRG IDHRG (scattering Lattice (WB) Lattice (WB) 5 Lattice (HotQCD) Lattice (HotQCD) 0.8 between two 4 P/T 4 ε /T 4 0.6 3 hadrons) 0.4 2 0.2 Total: Attractive 1 (a) (b) 0 0 + repulsive 100 110 120 130 140 150 160 170 100 110 120 130 140 150 160 170 T (MeV) T (MeV) Both KM and 9 Total 8 KM Total contain IDHRG 7 Lattice (WB) 6 Lattice (HotQCD) non-interacting 5 s/T 3 part as well 4 3 IDHRG: PDG 2 1 (c) 2016 0 100 110 120 130 140 150 160 170 T (MeV) Repulsive A. Dash et al., arXiv:1806.02117 [hep-ph] interactions suppress the bulk variables Subhasis Samanta Hot Quarks 2018 16/ 18
Result: Fluctuations and corrections 0.25 0.65 Total Total 0.6 KM KM 0.2 IDHRG 0.55 IDHRG (PDG 2016) Lattice (WB) Lattice (WB) 0.5 Lattice (HotQCD) Lattice (HotQCD) 0.15 0.45 0.4 χ B χ Q 2 2 0.1 0.35 0.3 0.05 0.25 (b) (a) 0.2 0 0.15 100 110 120 130 140 150 160 170 100 110 120 130 140 150 160 170 T (MeV) T (MeV) 0.12 1.2 Improvement Total Total KM KM 0.1 1 IDHRG IDHRG (PDG 2016) Lattice (HotQCD) IDHRG (PDG 2016+) Lattice Lattice (WB) 0.08 0.8 Lattice (HotQCD) C BS - χ BS 11 0.06 0.6 0.04 0.4 0.02 0.2 (c) (b) 0 0 100 110 120 130 140 150 160 170 100 110 120 130 140 150 160 170 T (MeV) T (MeV) A. Dash et al., arXiv:1806.02117 [hep-ph] Subhasis Samanta Hot Quarks 2018 17/ 18
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