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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

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SLIDE 1

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

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SLIDE 2

Fiction, noun

A fictitious particle, i.e. a particle predicted by some model without solid empirical evidence for its existence

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SLIDE 3

Fluctuations of conserved charges

χX

n

= T n ∂nP/T 4 ∂µn

X

  • µX=0

χXY

nm

= T n+m ∂n+mP/T 4 ∂µn

X∂m Y

  • µX=0, µY =0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 140 160 180 200 220 240 260 280

T [MeV] χ2

B

free quark gas

Tc=(154 +/-9) MeV

ms/ml=20 (open) 27 (filled) PDG-HRG

  • cont. extrap.

Nτ=16 12 8 6

0.05 0.1 0.15 0.2 0.25 0.3 0.35 140 160 180 200 220 240 260 280

Bazavov et al., PRD95, 054504 (2017)

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SLIDE 4

Black on black!

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SLIDE 5

More resonances?

  • cont. est.

PDG-HRG QM-HRG 0.15 0.20 0.25 0.30

  • χ11

BS/χ2 S

Nτ=6: open symbols Nτ=8: filled symbols B1

S/M1 S

B2

S/M2 S

B2

S/M1 S

0.15 0.25 0.35 0.45 140 150 160 170 180 190 T [MeV]

Bazavov et al., PRL113, 072001 (2014)

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SLIDE 6

Baryon spectrum

1 1.5 2 2.5 3 S=0 S=1 S=2 S=3 M (GeV)

Blue: Particle Data Group

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SLIDE 7

Baryon spectrum

1 1.5 2 2.5 3 S=0 S=1 S=2 S=3 M (GeV)

Blue: Particle Data Group Red: PDG + L¨

  • ring et al., EPJA10, 395 (2001) & EPJA10, 447 (2001)
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SLIDE 8

Hadron spectrum

0.5 1 1.5 2 2.5 3 S=0 S=1 S=0 S=1 S=2 S=3 mesons baryons M (GeV)

Blue: Particle Data Group Red: PDG + L¨

  • ring et al., EPJA10, 395 (2001) & EPJA10, 447 (2001)

Black: PDG + Ebert et al., PRD79, 114029 (2009)

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SLIDE 9

Trace anomaly

0.5 1 1.5 2 2.5 3 3.5 4 4.5 100 120 140 160 180 200 (ε-3P)/T4 T (MeV) Budapest-Wuppertal hotQCD HRG

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SLIDE 10

Trace anomaly

0.5 1 1.5 2 2.5 3 3.5 4 4.5 100 120 140 160 180 200 (ε-3P)/T4 T (MeV) Budapest-Wuppertal hotQCD HRG HRG+

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SLIDE 11

χ2

B

0.05 0.1 0.15 0.2 100 120 140 160 180 200 χ2

B

T (MeV) hotQCD B-W HRG

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SLIDE 12

χ2

B

0.05 0.1 0.15 0.2 100 120 140 160 180 200 χ2

B

T (MeV) hotQCD B-W HRG HRG+

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SLIDE 13

χ11

BS

0.05 0.1 0.15 0.2 100 120 140 160 180 200 χ11

BS

T (MeV) hotQCD HRG

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SLIDE 14

χ11

BS

0.05 0.1 0.15 0.2 100 120 140 160 180 200 χ11

BS

T (MeV) hotQCD HRG HRG+

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SLIDE 15

χ2

S

0.1 0.2 0.3 0.4 0.5 0.6 100 120 140 160 180 200 χ2

S

T (MeV) B-W HRG

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SLIDE 16

χ2

S

0.1 0.2 0.3 0.4 0.5 0.6 100 120 140 160 180 200 χ2

S

T (MeV) B-W HRG HRG+

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SLIDE 17

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
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SLIDE 18

Virial expansion

P = P ideal + T

  • ij

bij

2 (T)eβµieβµj

bij

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
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SLIDE 19

Virial expansion in nucleon gas

P(T, µ) = P0(T) cosh(βµ) + 2b2(T) T cosh(2βµ) P0(T) = 4m2T 2 π2 K2(βm) b2(T) = 2T π3 ∞ dE mE 2 + m2

  • K2
  • 2β
  • mE

2 + m2

  • 1

4iTr[S†dS dE − dS† dE S]

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SLIDE 20

Virial expansion in nucleon gas

Elastic part of the S-matrix from scattering phase shift: 1 4iTr[S†dS dE − dS† dE S] →

  • s
  • J

(2J + 1) dδJ,I=0

s

dE + 3dδJ,I=1

s

dE

  • Workman et al., PRC94, 065203 (2016); Arndt et al., PRC76, 025209 (2007)
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SLIDE 21

Repulsive mean field

Assume: interactions reduce single partice energy by U = Knb where nb is single nucleon density (Olive, NPB190, 483 (1981)) nb =

  • d3p

(2π)3e−β(Ep−µ+U) Small µ ⇒ βKnb ≪ 1 and nb ≈ n0

b(1 − βKn0 b) ⇒

P(T, µ) = T(nb + n¯

b) − K

2

  • (n2

b)2 + (n0 ¯ b)2

  • r

P(T, µ) = P0(T)

  • cosh(βµ) − Km

π2 K2(βm) cosh(2βµ)

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SLIDE 22

Virial expansion vs. mean field

Repulsive mean field P(T, µ) = P0(T)×

  • cosh(βµ) − Km

π2 K2(βm) cosh(2βµ)

  • Virial expansion

P(T, µ) = P0(T)×

  • cosh(βµ) − ¯

b2(T)K2(βm) cosh(2βµ)

  • where ¯

b2 =

2T b2(T ) P0(T )K2(βm)

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SLIDE 23

Trace anomaly

0.5 1 1.5 2 2.5 3 3.5 4 4.5 100 120 140 160 180 200 (ε-3P)/T4 T (MeV) B-W hotQCD HRG HRG-mean

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SLIDE 24

Trace anomaly

0.5 1 1.5 2 2.5 3 3.5 4 4.5 100 120 140 160 180 200 (ε-3P)/T4 T (MeV) B-W hotQCD HRG HRG-mean HRG+ HRG+ mean

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SLIDE 25

χ2

B

0.05 0.1 0.15 0.2 100 120 140 160 180 200 χ2

B

T (MeV) hotQCD B-W HRG HRG-mean

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SLIDE 26

χ2

B

0.05 0.1 0.15 0.2 100 120 140 160 180 200 χ2

B

T (MeV) hotQCD B-W HRG HRG-mean HRG+ HRG+ mean

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SLIDE 27

χ11

BS

0.05 0.1 0.15 0.2 100 120 140 160 180 200 χ11

BS

T (MeV) hotQCD HRG HRG-mean

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SLIDE 28

χ11

BS

0.05 0.1 0.15 0.2 100 120 140 160 180 200 χ11

BS

T (MeV) hotQCD HRG HRG-mean HRG+ HGR+ mean

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SLIDE 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
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SLIDE 30

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

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SLIDE 31

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

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SLIDE 32

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

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SLIDE 33

Hadron Resonance Gas with mean field

Assume: only members of baryon octet and decuplet repel each other P(T, µ) = Tn − K 2

  • (n0
  • d)2 + (n0

¯

  • d)2

where nod(T) = T 2π2

  • i

gim2

iK2(βmi)

i = N, Σ, Ξ, ∆, Σ∗, Ξ∗, Ω

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SLIDE 34

Hadron Resonance Gas with mean field

Assume: only members of baryon octet and decuplet repel each other P(T, µ) = Tn − K 2

  • (n0
  • d)2 + (n0

¯

  • d)2

where nod(T) = T 2π2

  • i

gim2

iK2(βmi)

i = N, Σ, Ξ, ∆, Σ∗, Ξ∗, Ω χB

n

= χB(0)

n

− 2nβ4K(n0

  • d)2

χBS

n1

= χBS(0)

n1

+ 2n+1β5Kn0

  • d(P S1

B + 2P S2 B + 3P S3 B )

  • P. Huovinen @ GSI, February 12, 2018

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