QCD phase diagram for nonzero isospin-asymmetry SEWM, Barcelona, - - PowerPoint PPT Presentation

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QCD phase diagram for nonzero isospin-asymmetry SEWM, Barcelona, June 25 th 2018 Sebastian Schmalzbauer with Bastian Brandt & Gergely Endr odi Outline QCD with isospin-asymmetry introduction motivation forecast pion

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SLIDE 1

QCD phase diagram for nonzero isospin-asymmetry

SEWM, Barcelona, June 25th 2018

Sebastian Schmalzbauer

with Bastian Brandt & Gergely Endr˝

  • di
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SLIDE 2

Outline

  • QCD with isospin-asymmetry
  • introduction
  • motivation
  • forecast
  • pion condensation on the lattice
  • symmetry breaking
  • simulation details
  • extrapolating physical results
  • results for the isospin phase diagram
  • chiral crossover
  • pion condensation phase boundary
  • deconfinement transition
  • summary
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

1 / 16

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SLIDE 3

QCD phase diagram

(taken from NICA)

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

2 / 16

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SLIDE 4

Motivation

  • two quark flavors u, d
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

3 / 16

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SLIDE 5

Motivation

  • two quark flavors u, d
  • isospin density nI = nu − nd = 0 :
  • systems with charged pions
  • neutron stars (udd)
  • heavy-ion collisions (N>Z)

(taken from thescienceexplorer)

20 40 60 80 100 Z 20 40 60 80 100 120 140 160 N Z = N 10−8s 10−6s 10−4s 10−2s 1 s 100 s 104s 106s 1 yr 100 yr 104yr 106yr 108yr 1010yr 1012yr 1014yr no data stable

(taken from wikipedia)

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

3 / 16

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SLIDE 6

Motivation

  • two quark flavors u, d
  • isospin density nI = nu − nd = 0 :
  • systems with charged pions
  • neutron stars (udd)
  • heavy-ion collisions (N>Z)
  • no sign problem ⇒ lattice simulations

(taken from thescienceexplorer)

20 40 60 80 100 Z 20 40 60 80 100 120 140 160 N Z = N 10−8s 10−6s 10−4s 10−2s 1 s 100 s 104s 106s 1 yr 100 yr 104yr 106yr 108yr 1010yr 1012yr 1014yr no data stable

(taken from wikipedia)

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

3 / 16

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SLIDE 7

Motivation

  • two quark flavors u, d
  • isospin density nI = nu − nd = 0 :
  • systems with charged pions
  • neutron stars (udd)
  • heavy-ion collisions (N>Z)
  • no sign problem ⇒ lattice simulations
  • analogies to baryon density
  • Silver Blaze
  • color neutral compostite particles
  • saturation
  • technical similarities (small eigenvalues)

(taken from thescienceexplorer)

20 40 60 80 100 Z 20 40 60 80 100 120 140 160 N Z = N 10−8s 10−6s 10−4s 10−2s 1 s 100 s 104s 106s 1 yr 100 yr 104yr 106yr 108yr 1010yr 1012yr 1014yr no data stable

(taken from wikipedia)

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

3 / 16

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SLIDE 8

Conjectured QCD isospin phase diagram

  • baryon chemical potential

µB = 0

  • isospin chemical potential µI = (µu − µd)/2
  • rich phase structure:
  • vacuum (white)
  • quark-gluon plasma
  • pion condensate
  • color superconductor
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

4 / 16

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SLIDE 9

Setup

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SLIDE 10

Pion condensation: symmetry breaking

  • QCD with light quarks

M = / D + mud1

  • chiral symmetry breaking pattern

SU(2)V

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

5 / 16

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SLIDE 11

Pion condensation: symmetry breaking

  • QCD with light quarks

M = / D + mud1 + µIγ0τ3

  • chiral symmetry breaking pattern

SU(2)V → U(1)τ3

  • problem: cannot directly observe the spontaneous symmetry breaking
  • pion condensate ¯

ψγ5τ1,2ψ = 0 (finite volume)

  • accumulation of zero eigenvalues (Goldstone mode)
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

5 / 16

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SLIDE 12

Pion condensation: symmetry breaking

  • QCD with light quarks

M = / D + mud1 + µIγ0τ3 + iλγ5τ2

  • chiral symmetry breaking pattern

SU(2)V → U(1)τ3 → ∅

  • problem: cannot directly observe the spontaneous symmetry breaking
  • pion condensate ¯

ψγ5τ1,2ψ = 0 (finite volume)

  • accumulation of zero eigenvalues (Goldstone mode)
  • solution: add explicit unphysical breaking (pionic source)
  • can observe spontaneous symmetry breaking
  • no zero eigenvalues
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

5 / 16

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SLIDE 13

Pion condensation: symmetry breaking

  • QCD with light quarks

M = / D + mud1 + µIγ0τ3 + iλγ5τ2

  • chiral symmetry breaking pattern

SU(2)V → U(1)τ3 → ∅

  • problem: cannot directly observe the spontaneous symmetry breaking
  • pion condensate ¯

ψγ5τ1,2ψ = 0 (finite volume)

  • accumulation of zero eigenvalues (Goldstone mode)
  • solution: add explicit unphysical breaking (pionic source)
  • can observe spontaneous symmetry breaking
  • no zero eigenvalues
  • need to extrapolate λ → 0 for physical results
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

5 / 16

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SLIDE 14

Simulation Details

  • QCD partition function for Nf = 2 + 1 rooted staggered quarks

Z =

  • D[U] (det Mud det Ms)1/4 e−SG

❘

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

6 / 16

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SLIDE 15

Simulation Details

  • QCD partition function for Nf = 2 + 1 rooted staggered quarks

Z =

  • D[U] (det Mud det Ms)1/4 e−SG
  • quark matrices with η5 = (−1)nt+nx+ny+nz

Mud =

  • /

DµI + mud λη5 −λη5 / D−µI + mud

  • ,

Ms = / D0 + ms ❘

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

6 / 16

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SLIDE 16

Simulation Details

  • QCD partition function for Nf = 2 + 1 rooted staggered quarks

Z =

  • D[U] (det Mud det Ms)1/4 e−SG
  • quark matrices with η5 = (−1)nt+nx+ny+nz

Mud =

  • /

DµI + mud λη5 −λη5 / D−µI + mud

  • ,

Ms = / D0 + ms

  • no sign problem due to η5τ1Mτ1η5 = M†:

det Mud = det

  • M†M + λ2

∈ ❘>0 M = / DµI + mud

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

6 / 16

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SLIDE 17

Simulation Details

  • QCD partition function for Nf = 2 + 1 rooted staggered quarks

Z =

  • D[U] (det Mud det Ms)1/4 e−SG
  • quark matrices with η5 = (−1)nt+nx+ny+nz

Mud =

  • /

DµI + mud λη5 −λη5 / D−µI + mud

  • ,

Ms = / D0 + ms

  • no sign problem due to η5τ1Mτ1η5 = M†:

det Mud = det

  • M†M + λ2

∈ ❘>0 M = / DµI + mud

  • first studies [Kogut, Sinclair ’02] [de Forcrand, Stephanov, Wenger ’07]
  • in this work: stout-smeared quarks, physical pion masses, tree-level

Symanzik improved gluons [Brandt, Endr˝

  • di, Schmalzbauer ’18]
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

6 / 16

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SLIDE 18

Observables

  • chiral and pion condensate

¯

ψψ

  • = T

V ∂ ln Z ∂mud ,

π± = T

V ∂ ln Z ∂λ

  • Polyakov loop

P =

  • 1

V

  • nx,ny,nz

tr

Nt−1

  • nt=0

Ut(n)

  • need to be renormalized

Σ ¯

ψψ,

Σπ, Pr

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

7 / 16

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SLIDE 19

λ-extrapolation: concepts

  • naive limit

O = lim

λ→0

  • D[U] e−S(λ)O(λ)
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

8 / 16

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SLIDE 20

λ-extrapolation: concepts

  • naive limit

O = lim

λ→0

  • D[U] e−S(λ)O(λ)
  • operator improvement

O = lim

λ→0

  • D[U] e−S(λ)O(0)
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

8 / 16

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SLIDE 21

λ-extrapolation: concepts

  • naive limit

O = lim

λ→0

  • D[U] e−S(λ)O(λ)
  • operator improvement

O = lim

λ→0

  • D[U] e−S(λ)O(0)
  • reweighting of configurations

O = OWλλ Wλλ with full (expensive) or leading order (cheap) reweighting factor ln Wλ = S(λ) − S(0) = ln WLO + O(λ4), ln WLO = −λV 2T π±

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

8 / 16

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SLIDE 22

Operator improvement: pion condensate

  • cannot measure pion condensation without pionic source

π± = Tλ 2V tr

  • M†M + λ2−1
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

9 / 16

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SLIDE 23

Operator improvement: pion condensate

  • cannot measure pion condensation without pionic source

π± = Tλ 2V tr

  • M†M + λ2−1
  • Banks-Casher type relation [Kanazawa, Wettig, Yamamoto ’11]

π±

V →∞

− − − − → λ 2

  • dξ ρ(ξ)(ξ2 + λ2)−1
  • λ→0

− − − → π 4 ρ(0) with density of singular values ρ(ξ)

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

9 / 16

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SLIDE 24

Operator improvement: pion condensate

  • cannot measure pion condensation without pionic source

π± = Tλ 2V tr

  • M†M + λ2−1
  • Banks-Casher type relation [Kanazawa, Wettig, Yamamoto ’11]

π±

V →∞

− − − − → λ 2

  • dξ ρ(ξ)(ξ2 + λ2)−1
  • λ→0

− − − → π 4 ρ(0) with density of singular values ρ(ξ)

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

9 / 16

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SLIDE 25

Operator improvement: chiral condensate

  • measure chiral condensate with noisy estimators

¯ ψψ(λ) = T 2V ℜ tr M M†M + λ2

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

10 / 16

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SLIDE 26

Operator improvement: chiral condensate

  • measure chiral condensate with noisy estimators

¯ ψψ(λ) = T 2V ℜ tr M M†M + λ2

  • difference to physical value?

δ ¯

ψψ = ¯

ψψ(λ) − ¯ ψψ(λ = 0)

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

10 / 16

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SLIDE 27

Operator improvement: chiral condensate

  • measure chiral condensate with noisy estimators

¯ ψψ(λ) = T 2V ℜ tr M M†M + λ2

  • difference to physical value?

δ ¯

ψψ = ¯

ψψ(λ) − ¯ ψψ(λ = 0)

  • λ-dependence mainly from the 100-150 smallest singular values
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

10 / 16

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SLIDE 28

λ-extrapolation at work

effect of improvement

  • flatter λ dependence
  • controlled extrapolation

leading order vs. full reweighting

  • deviations understood
  • agreement for λ → 0
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

11 / 16

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SLIDE 29

Results: phase diagram

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SLIDE 30

Chiral crossover

[Borsanyi et al. ’10]

  • chiral crossover transition temperature Tpc(µI)

as inflection point of Σ ¯

ψψ w.r.t. T

  • Tpc(µI = 0) = 159(4) MeV
  • Tpc(µI) reduces until µI ≈ 70 MeV
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

12 / 16

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SLIDE 31

Pion condensation phase boundary

  • pion condensation (Σπ = 0) sets in at µI,c(T)
  • for T < 140 MeV: µI,c(T) ≈ mπ/2
  • flattening around 155 MeV, no pion condensation above 161 MeV
  • 2nd order phase transition in the O(2) universality class
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

13 / 16

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SLIDE 32

Resulting phase diagram

  • chiral crossover meets pion condensation phase:

pseudo-triple point at µc,pt = 70(5) MeV, Tpt = 151(7) MeV

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

14 / 16

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SLIDE 33

Resulting phase diagram

  • chiral crossover meets pion condensation phase:

pseudo-triple point at µc,pt = 70(5) MeV, Tpt = 151(7) MeV

  • chiral symmetry restoration and pion condensation phase boundary

seem to coincide from the PTP on for higher µI

  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

14 / 16

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SLIDE 34

Deconfinement transition

  • Pr has no pronounced inflection point: follow lines of constant Pr
  • distance between lines indicates slope of Pr around the transition
  • contour lines are insensitive to the BEC phase transition
  • the contours continue to decrease after crossing the phase boundary
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

15 / 16

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SLIDE 35

Summary

  • QCD with isospin chemical potential
  • presence of pionic source λ
  • extrapolation λ → 0
  • results for the phase diagram
  • chiral crossover
  • pion condensation
  • deconfinement transition
  • S. Schmalzbauer (Goethe Universit¨

at Frankfurt) - QCD phase diagram for nonzero isospin-asymmetry

16 / 16

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SLIDE 36

Thank you!

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SLIDE 37

Backup: renormalization of observables

  • ultraviolet divergences in the partition function

log Z ≈ a−4 + (m2

ud + λ2)a−2 + (m2 ud + λ2)2 log a

require additive renormalization in addition to multiplicative ren.

  • ren. chiral condensate

Σ ¯

ψψ = mud

m2

πf 2 π

  • ¯

ψψT,µI − ¯ ψψ0,0

  • + 1
  • ren. pion condensate

Σπ = mud m2

πf 2 π

π±T,µI

  • ren. Polyakov loop

Pr(T, µI) = P(T, µI)

Pr(T⋆, 0)

P(T⋆, 0)

T⋆/T

with free choice T⋆ = 162 MeV, Pr(T⋆, 0) = 1.

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SLIDE 38

Backup: singular value representation

  • rewrite fermion determinant

det Mud = det

  • M†M + λ2

=

3VsNt

  • i=1
  • ξ2

i + λ2

in singular eigenvalue basis M†Mφi = ξ2

i φi

  • pion condensate

π± = T 2V

  • i

λ ξ2

i + λ2

  • chiral condensate

¯ ψψ = T 2V ℜ

  • i

φ†

i Mφi

ξ2

i + λ2

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SLIDE 39

Backup: order of the phase transition

  • finite size scaling analysis
  • observe sharpening of the

transition for V → ∞

  • strong indication for

2nd order phase transition

  • study of universal scaling function

Σπ = h1/δ · fG(t/hβδ), h = λ λ0 , t = µI,c − µI t0

  • critical behavior as in

O(2) universality class

  • include scaling violations
  • f Σπ like in [Ejiri et al. ’09]

a1th + b1h + b3h3

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SLIDE 40

Backup: comparison to χPT

  • χPT: low-energy effective theory

Σπ = G sin α, sin

  • α − arctan

λ mud

  • = 2µ2

I

µ2

I,c

sin 2α

  • data well described for µI/mπ < 0.63
  • tends to underestimate Σπ for higher isospin chemical potentials