[PPT] - Fluctuations and correlations of conserved charges in hadron PowerPoint Presentation

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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

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Introduction

hadrons/leptons The major goals ⋆ The mapping of QCD phase diagram ⋆ Locating the QCD critical point

Facility √sNN (GeV) µB (MeV) Status LHC 2760 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

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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 gas: ln Z =

i ln Zi

⋆ For i th hadron/resonance, ln Zid

i = Vgi 2π2 m2 i T ∞ j=1 (±1)j−1(zj/j2)K2(jmi/T), z = exp(µ/T),

µi = BiµB + SiµS + QiµQ

The + (-) sign refers to bosons (fermions) The first term (j = 1) corresponds to the classical ideal gas Width of the resonances are ignored

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EOS of IDHRG at µ = 0

T (GeV)

0.05 0.1 0.15 2 4 6

4

T 3P

4

T ε

3

4T 3s

PDG 2016 LQCD

⋆ IDHRG provides a satisfactory description in the hadronic phase of continuum LQCD data

S. Samanta et al. JPG 46, 065106 (2019); LQCD data: A. Bazavov et al. (HotQCD), PRD 90, 094503 (2014)

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Problem to quantify χBS, CBS

0.02 0.04 0.06 0.08 0.1 0.12 100 110 120 130 140 150 160 170

χBS

11

T (MeV)

HRG (PDG 2016) Lattice (HotQCD)

0.2 0.4 0.6 0.8 1 150 200 250 300 350 400 CBS T [MeV]

SB limit Nt=6 Nt=8 Nt=10 Nt=12 Nt=16 cont. HRG Ref: A. Borsanyi et al., JHEP01, 138 (2012)

⋆ IDHRG fails to describe χBS, CBS = −3χ11

BS/χ2 S

⇒ Interaction is needed

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Classical Virial Expansion (Non-relativistic)

P = NT V

1 +

N V

B(T) +

N V 2 C(T) + ..

⋆ 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 2

(1 − e−U12/T)dV

U12 is the two body interaction energy

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Relativistic Virial Expansion

ln Z = ln Z0 +

i1,i2 zi1 1 zi2 2 b(i1, i2)

b(i1, i2) =

V 4πi

d3p

(2π)3

dε exp
−β(p2 + ε2)

1/2

S−1 ∂S

∂ε − ∂S−1 ∂ε S

aa → R → aa, ab → R → ab, aab → R → aab etc.

⋆ z1 and z2 are fugacities of two species (z = eβµ) ⋆ The labels i1 and i2 refer to a channel of the S-matrix which has an initial state containing i1 + i2 particles

Second virial coefficient

b2 = b(i1, i2)/V where i1 = i2 = 1

ππ → R → ππ πK → R → πK KK → R → KK πN → R → πN etc.

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Interacting part of pressure

b2 in terms of phase shift b2 =

1 2π3β

∞

M dεε2K2(βε) l,I

′gI,l

∂δI

l (ε)

∂ε

Pint = 1 β ∂ ln Zint ∂V = 1 β z1z2b2 = z1z2 2π3β2 ∞

M

dεε2K2(βε)

I,l

′gI,l

∂δI

l (ε)

∂ε ⋆ Interaction is attractive (repulsive) if derivative of the phase shift is positive (negative)

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K-matrix formalism (Attractive part of the interaction)

Scattering amplitude: Sab→cd = cd|S|ab Scattering operator (matrix) S = I + 2iT 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)

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Phase shift in K-matrix formalism

Re T = K(I + K2)

−1,

Im T = K2(I + K2)

−1 ⇒ Im T/ Re T = K

Kab→R→ab =

R

mRΓR→ab(√s) m2

R − s

Resonances appear as sum of poles in the K matrix

Partial wave decomposition Sl = exp(2iδl) = 1 + 2iTl ⇒ Tl = exp(iδ) sin(δl) Re Tl = sin(δl) cos(δl), Im Tl = sin2(δl) K = tan(δl), δl = tan−1(K)

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Phase shift: Empirical vs KM

0.5 1 1.5 2 2.5 3 3.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 δ (rad) ε (GeV)

KM Empirical

0.5 1 1.5 2 2.5 3 3.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 δ (rad) ε (GeV)

KM Empirical

ππ → ρ(770) → ππ πK → K∗(892) → πK ⋆ Good agreement between the empirical phase shifts of resonances and the K-matrix approach

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Comparison between K-matrix and Breit-Wigner approach

2 4 6 8 10 12 0.8 1 1.2 1.4 1.6 1.8 2 σ (mb) √ s (GeV) (a)

KM BW

f0 (980) (m1 = 990 MeV, Γ1 = 55 MeV) and f0 (1500) (m2 = 1505 MeV, Γ2 = 109 MeV)

1 2 3 4 5 6 7 8 0.8 1 1.2 1.4 1.6 1.8 2 σ (mb) √ s (GeV) (b)

KM BW

f0 (1370) (m1 = 1370 MeV, Γ1 = 350 MeV) and f0 (1500) (m2 = 1505 MeV, Γ2 = 109 MeV)

⋆ KM formalism preserves the unitarity of the S matrix and neatly handles

verlapping resonances
S. Samanta et al. PRC 97, 055208 (2018)

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Ideal gas limit

⋆ For a narrow resonance, δI

l changes rapidly through π radian around

ε = mR ⋆ δI

l can be approximated by a step function: δI l ∼ Θ(ε − mR)

⋆ ∂δI

l /∂ε ≈ πδ(ε − mR)

0.5 1 1.5 2 2.5 3 3.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 δ (rad) ε (GeV)

KM Empirical

b2 = 1 2π3β ∞

M

dεε2K2(βε)

l,I

′gI,l

∂δI

l (ε)

∂ε = gI,l 2π2m2

RTK2(βmR)

Pint = Tz1z2b2 = PR

id

⋆ Pressure exerted by an ideal (MB) gas of particles of mass mR ⋆ This establishes the fundamental premise of the IDHRG model

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Repulsive interaction from experimental data of phase shift

1
0.5

0.5 1 1.5 2 1.92 2 2.08 2.16 2.24 2.32 (b) I = 0 δ (rad) √ s (GeV)

3S1 3D2 3D1 3D3 1P1

1
0.5

0.5 1 1.92 2 2.08 2.16 2.24 2.32 (a) I = 1 δ (rad) √ s (GeV)

1S0 1D2 3P1 3P2 3F2 3F3 3F4

1
0.5

0.5 1 1.92 2 2.08 2.16 2.24 2.32 (a) I = 1 δ (rad) √ s (GeV)

1G4 3H4 3H5 3H6 1I6 3J6 3J7

(GeV) s 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (rad) δ 0.45 − 0.4 − 0.35 − 0.3 − 0.25 − 0.2 − 0.15 − 0.1 − 0.05 − (GeV) s 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 (rad) δ 0.8 − 0.7 − 0.6 − 0.5 − 0.4 − 0.3 − 0.2 − 0.1 − (GeV) s 1.9 2 2.1 2.2 2.3 (rad) δ 1 − 0.5 − 0.5 1 1.5 2

ππ: (δ2

0)

KN : S11 πN: S31

⋆ NN interaction: All available data ⋆ ππ repulsive interaction: δ2 ⋆ KN repulsive interaction: S11(lI,2J) (Σ(1660)) ⋆ πN repulsive interaction: S31 (l2I,2J) (∆(1620)), ∆(1910), N(1720) etc. ⋆ 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

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Results

0.2 0.4 0.6 0.8 1 1.2 100 110 120 130 140 150 160 170 P/T4 T (MeV) (a)

Total KM IDHRG Lattice (WB) Lattice (HotQCD)

1 2 3 4 5 6 7 100 110 120 130 140 150 160 170 ε/T4 T (MeV) (b)

Total KM IDHRG Lattice (WB) Lattice (HotQCD)

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 100 110 120 130 140 150 160 170 χQ

2

T (MeV) (b)

Total KM IDHRG (PDG 2016) Lattice (WB) Lattice (HotQCD)

0.02 0.04 0.06 0.08 0.1 0.12 100 110 120 130 140 150 160 170

χBS

11

T (MeV) (c)

Total KM IDHRG Lattice (HotQCD) Lattice

S. Samanta et al. PRC 99, 044919 (2019)

⋆ KM: Attractive interaction ⋆ Total: Attractive + repulsive ⋆ Both KM and Total contain non-interacting part as well ⋆ Repulsive interactions suppress the bulk variables

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χ2

B − χ4 B and CBS

10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 100 110 120 130 140 150 160 170 χB

2 - χB 4

T (MeV)

χB

2 - χB 4

χB

4 - χB 2

χB

2 - χB 4 (Lattice)

χB

4 - χB 2 (Lattice)

0.2 0.4 0.6 0.8 1 1.2 100 110 120 130 140 150 160 170 CBS T (MeV) (b)

Total KM IDHRG (PDG 2016) IDHRG (PDG 2016+) Lattice (WB) Lattice (HotQCD)

⋆ χ2

B − χ4 B is non-zero

⋆ For CBS: Improvement compared to IDHRG

S. Samanta et al. PRC 99, 044919 (2019)

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Excluded volume hadron resonance gas model

⋆ Hadrons have finite hard-core radii. (P(V − Nb) = NT) ⋆ b = Vex = 16

3 πR3 is the volume excluded for the hadron.

⋆ Pressure and chemical potential in EVHRG model: P(T, µ1, µ2, ..) =

i

Pid

i (T, ˆ

µ1, ˆ µ2, ..), ˆ µi = µi − Vev,iP(T, µ1, µ2, ..)

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van der Waals interaction in HRG model (VDWHRG model)

P +

N V 2 a

(V − Nb) = NT,

P(T, n) = NT V − bN − a N V 2 ≡ nT 1 − bn − an2 where n ≡ N/V is the number density of particles. P(T, µ) = Pid(T, µ∗) − an2, µ∗ = µ − bP(T, µ) − abn2 + 2an n = nid(T, µ∗) 1 + bnid(T, µ∗) ⋆ a = 0 ⇒ EVHRG ⋆ a = b = 0 ⇒ IDHRG

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Extraction of parameters a and b

a = 1250 ± 150 MeV fm3, r = 0.7 ± 0.05 fm χ2 =

i,j

(RLQCD

i,j

(Tj) − Rmodel

i,j

(Tj))2 (∆LQCD

i,j

(Tj))2 , LQCD data of P/T4, ε/T4, s/T3, CV/T3 and χ2

B at µ = 0 have been used to

calculate χ2 a = 329 MeV fm3, r = 0.59 fm By reproducing the properties of the nuclear matter (n0 = 0.16 fm−3, E/N = −16 MeV) at zero temperature

Ref: V. Vovchenko et al., PRC 91, 064314 (2015) S Samanta CETHENP 2019, VECC, India 19 / 25

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Results- VDWHRG model

0.2 0.4 0.6 0.8 1 1.2 1.4 100 110 120 130 140 150 160 170 180 p/T4 T (MeV)

VDWHRG (a = 1250 ± 150 MeV fm3, r = 0.7 ± 0.05 fm) HRG Lattice (WB) Lattice (HotQCD)

1 2 3 4 5 6 7 8 9 100 110 120 130 140 150 160 170 180 ε/T4 T (MeV)

VDWHRG (a = 1250 ± 150 MeV fm3, r = 0.7 ± 0.05 fm) HRG Lattice (WB) Lattice (HotQCD)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 100 110 120 130 140 150 160 170 180 χS

2

T (MeV)

VDWHRG (a = 1250 ± 150 MeV fm3, r = 0.7 ± 0.05 fm) HRG Lattice (WB) Lattice (HotQCD)

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 100 110 120 130 140 150 160 170 180 −χBS

11

T (MeV)

VDWHRG (a = 1250 ± 150 MeV fm3, r = 0.7 ± 0.05 fm) HRG Lattice (WB) Lattice (HotQCD)

S. Samanta et al., PRC 97, 015201 (2018)

⋆ Agreement between LQCD and VDWHRD

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Phase transition in VDWHRG model

0.001 0.002 0.003 0.004 0.005 0.02 0.04 0.06 0.08 0.1 0.12

a = 1250 MeV fm3, R = 0.7 fm Tc = 62.1 MeV, (µB)c = 708 MeV

p (GeV/fm3) n (fm-3)

T = 66 MeV T = 64 MeV T = 62.1 MeV T = 60 MeV T = 58 MeV

50 100 150 200 200 400 600 800 1000 T (MeV) µB (MeV)

CP Holographic CP: Fodor et al. CP: Datta et al. CFO: Andronic et al. CFO: Cleymans et al.

⋆ Observed first order phase transition ⋆ Critical point at T = 62.1 MeV, µB = 708 MeV ⋆ Comparable the CP obtained by using the holographic gauge/gravity correspondence

S. Samanta et al., PRC 97, 015201 (2018), Holographic: PRD 96, 096026 (2017)

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Observables for CP search

Cumulants C1 = Nq, C2 =

(δNq)2

, C3 =

(δNq)3

, C4 =

(δNq)4

− 3

(δNq)22

Nq = Nq+ − Nq− and δNq = Nq −

Nq
q can be any conserved quantum number

Mean, variance, skewness, kurtosis M = C1, σ2 = C2, S = C3

σ3 ,

κ = C4

σ4

⋆ Higher moments are sensitive to correlation length

(δNq)2

∼ ζ2,

(δNq)3

∼ ζ4.5,

(δNq)4

∼ ζ7 ⋆ σ2/M = C2/C1 = χ2/χ1, Sσ = C3/C2 = χ3/χ2, κσ2 = C4/C2 = χ4/χ2 ⋆ Non-monotonic variations of Sσ, κσ2 with beam energy are believed to be good signatures of CP

)

p

N ∆ Net Proton (

20
10

10 20

Number of Events

1 10

2

10

3

10

4

10

5

10

6

10 0-5% 30-40% 70-80% Au+Au 200 GeV

<0.8 (GeV/c)

T

0.4<p |y|<0.5

0.2 0.4 0.6 0.8 1.0 1.2

Au+Au Collisions at RHIC

Net-proton

<0.8 (GeV/c),|y|<0.5

T

0.4<p

Skellam Distribution

70-80% 0-5% 0.4 0.6 0.8 1.0 1.2

p+p data Au+Au 70-80% Au+Au 0-5% Au+Au 0-5% (UrQMD)

Ind. Prod. (0-5%)

5 6 7 8 10 20 30 40 100 200 0.85 0.90 0.95 1.00 1.05

σ S

2

σ κ

)/Skellam σ (S

(GeV)

NN

s Colliding Energy

Ref: STAR: PRL 112, 032302 (2014); PRL 102, 032301 (2009) S Samanta CETHENP 2019, VECC, India 22 / 25

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VDW parameters from data of σ2/M, Sσ and κσ2 of np (Set-1)

(GeV)

NN

s 10

2

10 )

3

a (GeV fm 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(b)

(GeV)

NN

s 10

2

10 r (fm) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(c)

(GeV)

NN

s 10

2

10 /M

2

σ 1 2 3 4 5 6 7 8 9 10

VDWHRG Data (a)

(GeV)

NN

s 10

2

10 σ S 0.2 0.4 0.6 0.8 1 1.2 1.4

VDWHRG Data (b)

(GeV)

NN

s 10

2

10

2

σ κ 0.2 0.4 0.6 0.8 1 1.2

VDWHRG Data (c)

⋆ Unable to describe κσ2

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VDW parameters extracted from data of Sσ and κσ2 of np (Set-2)

(GeV)

NN

s 10

2

10 )

3

a (GeV fm 0.5 1 1.5 2 2.5

Set-1 Set-2 (a)

(GeV)

NN

s 10

2

10 r (fm) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Set-1 Set-2 (b)

(GeV)

NN

s 10

2

10 /M

2

σ 1 2 3 4 5 6 7 8 9 10

VDWHRG Data (a)

(GeV)

NN

s 10

2

10 σ S 0.2 0.4 0.6 0.8 1 1.2 1.4

VDWHRG Data (b)

(GeV)

NN

s 10

2

10

2

σ κ 0.2 0.4 0.6 0.8 1 1.2

VDWHRG Data (c)

⋆ Large a’s are needed to describe non-monotonic behaviors of Sσ and κσ2

S. Samanta et al., arXiv:1905.09311 [hep-ph]

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Summary

⋆ An extension of HRG model is constructed to include interactions using relativistic virial expansion of partition function (S-matrix formalism) ⋆ Interacting part of the partition function depends on the derivative of the phase shift ⋆ The attractive part of the interaction is calculated by parameterizing the two body phase shifts using K-matrix formalism ⋆ The repulsive part is included by fitting to experimental phase shifts ⋆ Effect of interaction is more visible in χ2

Q, χ2 B − χ4 B, CBS

⋆ We find a good agreement for the CBS (without adding extra resonances) and lattice QCD simulations ⋆ Critical point is observed in VDWHRG model ⋆ Large a’s are needed to describe non-monotonic behaviors of higher order moments of net-proton

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Collaborator

Prof. Bedangadas Mohanty
Mr. Ashutosh Dash

Thank you

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