Overview Motivation IDHRG model Interaction within S-matrix - - PowerPoint PPT Presentation

THERMODYNAMICS 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

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Overview

Motivation IDHRG model Interaction within S-matrix formalism Results Summary

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Study of matter under extreme conditions

hadrons/leptons The major goals The mapping of QCD phase diagram in terms T and µB 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

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 resonance gas: ln Z =

i ln Zi

For i th hadron/resonance, ln Zid

i = ± Vgi

2π2 ∞ p2dp ln[1 ± exp(−(Ei − µi)/T)] The upper and lower sign corresponds to baryons and mesons respectively. Ei =

• p2 + m2

i ,

µi = BiµB + SiµS + QiµQ

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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 of EOS in the hadronic phase of continuum LQCD data IDHRG fails to describe χ2

S, χ11 BS, CBS = −3χ11 BS/χ2 S

etc.

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

LQCD data: Phys. Rev. D 90, 094503 (2014), JHEP01, 138 (2012)

Interaction is needed

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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∗) Extra parameters a = 0 ⇒ EVHRG a = b = 0 ⇒ IDHRG

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Classical Virial Expansion (Non-relativistic)

P = NT V

• 1 + NB(T)

V + N2C(T) V2 + ..

• 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 + ln Zint = 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

• 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 We ignore contributions from bound states

Second virial coefficient

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

ln Z0 ⇒ Non interacting stable hadrons ln Zint ⇒ Scattering between two hadrons

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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

M is the invariant mass of the interacting pair at threshold 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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Ideal gas limit

For a very narrow width resonance, δI

l changes rapidly

through 180◦ around ε = mR δI

l can be approximated by a step function:

δI

l ∼ Θ(ε − mR)

∂δI

l /∂ε ≈ πδ(ε − mR)

bR

2 =

1 2π3β ∞

M

dεε2K2(βε)

• l,I

′gI,l

∂δI

l (ε)

∂ε = gI,l 2π2 m2

RTK2(βmR)

PR

int = Tz1z2bR 2 = PR id

Narrow resonance behaves like a stable hadron of mass mR This establishes the fundamental premise of the IDHRG

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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 = 1 (T−1 + iI)† = T−1 + iI K−1 = T−1 + iI, K = K† (i.e., K martix 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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Transition amplitude: K-matrix

Example: ππ → r1(m1) → ππ ππ → r2(m2) → ππ (r1, r2 have same l, I) K = m1Γ1(√s) m2

1 − s

+ m2Γ2(√s) m2

2 − s

T = m1Γ1(√s) (m2

1 − s) − im1Γ1(√s) − im2

1−s

m2

2−sm2Γ2(√s)

+ m2Γ2(√s) (m2

2 − s) − im2Γ2(√s) − im2

2−s

m2

1−sm1Γ1(√s) Subhasis Samanta Hot Quarks 2018 13/ 18

Comparison between K-matrix and Breit-Wigner approach

T ≈

m1Γ1(√s) (m2

1−s)−im1Γ1(√s) +

m2Γ2(√s) (m2

2−s)−im2Γ2(√s) (Separated)

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) Subhasis Samanta Hot Quarks 2018 14/ 18

Input 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 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 −

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

Data: Scattering Analysis Interactive Database (SAID) partial wave analysis Subhasis Samanta Hot Quarks 2018 15/ 18

Result: EOS

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)

1 2 3 4 5 6 7 8 9 100 110 120 130 140 150 160 170 s/T3 T (MeV) (c)

Total KM IDHRG Lattice (WB) Lattice (HotQCD)

• A. Dash et al., arXiv:1806.02117 [hep-ph]

KM: Attractive interaction (scattering between two hadrons) Total: Attractive + repulsive Both KM and Total contain non-interacting part as well IDHRG: PDG 2016 Repulsive interactions suppress the bulk variables

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Result: Fluctuations and corrections

0.05 0.1 0.15 0.2 0.25 100 110 120 130 140 150 160 170 χB

2

T (MeV) (a)

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

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)

• A. Dash et al., arXiv:1806.02117 [hep-ph]

Improvement

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Summary

An extension of HRG model is constructed to include interactions using S-matrix formalism. Interaction in the S-matrix formalism is a genuine interaction. We have considered all the stable hadrons and resonances which have two-body decay channels 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 A good agreement between EOS calculated in S-matrix formalism and LQCD simulations is observed Effect of interaction is more visible in χ2

Q, χ2 B, CBS etc.

We find a good agreement for the CBS (without adding extra resonances) and lattice QCD simulations

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Collaborator

• Prof. Bedangadas Mohanty
• Mr. Ashutosh Dash

Acknowledgement Science and Engineering Research Board, Department of Science & Technology, Government of India

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Collaborator

• Prof. Bedangadas Mohanty
• Mr. Ashutosh Dash

Acknowledgement Science and Engineering Research Board, Department of Science & Technology, Government of India

Thank you

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