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QCD with isospin chemical potential: low densities and Taylor expansion QCD with isospin chemical potential: low densities and Taylor expansion Bastian Brandt and Gergely Endr odi Goethe University Frankfurt 28.07.2016 QCD with isospin

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QCD with isospin chemical potential: low densities and Taylor expansion

QCD with isospin chemical potential: low densities and Taylor expansion

Bastian Brandt and Gergely Endr˝

  • di

Goethe University Frankfurt

28.07.2016

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QCD with isospin chemical potential: low densities and Taylor expansion

Contents

  • 1. (short) Introduction
  • 2. Simulation setup and λ extrapolation
  • 3. QCD at small isospin chemical potential
  • 4. Comparison to Taylor expansion around µI = 0
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QCD with isospin chemical potential: low densities and Taylor expansion Introduction

  • 1. Introduction
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QCD with isospin chemical potential: low densities and Taylor expansion Introduction

QCD at finite isospin chemical potential

QCD at finite chemical potential (Nf = 2): u quark: µu d quark: µd

◮ Can be decomposed in baryon and isospin chemical potentials:

µB = 3(µu + µd)/2 and µI = (µu − µd)/2

◮ Non-zero µI introduces an asymmetry between isospin ±1 particles

Positive µI: ⇒ More protons than neutrons!

◮ Such situations occur regularly in nature:

◮ Within nuclei with # neutrons > # protons. ◮ Within neutron stars. ◮ . . .

◮ However: Usually µI ≪ µB. ◮ Finite µI breaks SUV (2) explicitly to Uτ3(1).

Here: consider µB = 0!

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QCD with isospin chemical potential: low densities and Taylor expansion Introduction

Expected phase diagram

Exploring the phase diagram using χPT at finite µI:

[ Son, Stephanov, PRL86 (2001) ]

T µI pion condensation 2nd O(2)? crossover mπ/2 1st

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QCD with isospin chemical potential: low densities and Taylor expansion Introduction

Expected phase diagram

Exploring the phase diagram using χPT at finite µI:

[ Son, Stephanov, PRL86 (2001) ]

T µI pion condensation 2nd O(2)? crossover mπ/2 1st

First lattice simulations (Nt = 4, mπ > mphys

π

): 1st order deconfinement and 2nd order curve join? ⇒ Existence of tri-critical point?

[ Kogut, Sinclair, PRD66 (2002); PRD70 (2004) ]

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QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation

  • 2. Simulation setup and λ extrapolation
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QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation

Lattice action

[ G. Endr˝

  • di, PRD90 (2014) ]

◮ Gauge action: Symanzik improved ◮ Mass-degenerate u/d quarks:

Fermion matrix:

[ Kogut, Sinclair, PRD66 (2002); PRD70 (2004) ]

M = D(µ) λγ5 −λγ5 D(−µ)

  • D(µ): staggered Dirac operator with 2×-stout smeared links

λ: small explicit breaking of residual symmetry

◮ Necessary to observe spontaneous symmetry breaking at finite V . ◮ Serves as a regulator in the pion condensation phase.

◮ Strange quark: rooted staggered fermions

(no chemical potential)

◮ Quark masses are tuned to their physical values. ◮ Lattice sizes:

6 × 163, 243, 323 , 8 × 243, 323, 403, 10 × 283, 403 . . .

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QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation

Lattice action

[ G. Endr˝

  • di, PRD90 (2014) ]

◮ Gauge action: Symanzik improved ◮ Mass-degenerate u/d quarks:

Fermion matrix:

[ Kogut, Sinclair, PRD66 (2002); PRD70 (2004) ]

M = D(µ) λγ5 −λγ5 D(−µ)

  • D(µ): staggered Dirac operator with 2×-stout smeared links

λ: small explicit breaking of residual symmetry

◮ Necessary to observe spontaneous symmetry breaking at finite V . ◮ Serves as a regulator in the pion condensation phase.

◮ Strange quark: rooted staggered fermions

(no chemical potential)

◮ Quark masses are tuned to their physical values. ◮ Lattice sizes:

6 × 163, 243, 323 , 8 × 243, 323, 403, 10 × 283, 403 . . .

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QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation

λ-extrapolations

For physical results: λ needs to be removed! Problem: dependence on λ is not known! (at least for most of the observables)

0.3 0.6 0.9 1.2 1.5 1.8 2.1 0.002 0.004 0.006 0.008 nI /T 3 aλ T = 124 MeV, µI = 68 MeV T = 124 MeV, µI = 85 MeV T = 162 MeV, µI = 68 MeV

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QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation

λ-extrapolations

For physical results: λ needs to be removed! Problem: dependence on λ is not known! (at least for most of the observables)

  • 0.0003
  • 0.00025
  • 0.0002
  • 0.00015
  • 0.0001
  • 5e-05

5e-05 0.002 0.004 0.006 0.008 a4mR ¯ ψψ

  • R

aλ T = 114 MeV, µI = 34 MeV T = 124 MeV, µI = 34 MeV T = 148 MeV, µI = 34 MeV

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QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation

λ-extrapolations

For physical results: λ needs to be removed! Problem: dependence on λ is not known! (at least for most of the observables) Best possibility for model independence:

◮ Use a (cubic) spline extrapolation.

◮ Fix one of the external points. ◮ Leave the associated outer deriatives free.

(additional free parameters)

◮ To stabilise the extrapolation:

Need to assume that last two points lie on a (cubic) curve! Remaining systematic effect: Position of nodepoints influences the result!

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QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation

λ-extrapolations

Possible solution: Perform a “spline Monte-Carlo”

[ see S. Borsanyi ]

◮ Average “all” splines with a similarly good description of the data.

Allow for changes of # of nodes and node positions.

◮ Splines are weighted according to some suitable “action” S.

Two possibilities:

◮ Use the Akaike information criterion: SAIC = 2NP + χ2 ◮ Use the negative goodness of the fit: SGOD = P(χ2, Ndof) − 1

P(χ2, Ndof) = γ(χ2/2, Ndof /2)

Γ(Ndof /2)

– cumulative χ2 distribution function (γ: lower incomplete gamma fct.)

◮ Problem: oscillating solutions

⇒ Include some measure δ for oscillations Full action: S = SAIC/GOD + f × δ (parameter f needs to be tuned)

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QCD with isospin chemical potential: low densities and Taylor expansion Simulation setup and λ extrapolation

λ-extrapolations

Results with SAIC and f = 10.0:

0.3 0.6 0.9 1.2 1.5 1.8 2.1 0.002 0.004 0.006 0.008 nI /T 3 aλ preliminary T = 124 MeV, µI = 68 MeV T = 124 MeV, µI = 85 MeV T = 162 MeV, µI = 68 MeV

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QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential

  • 3. QCD at small isospin chemical potential
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QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential

Definition of the transition point

Investigate the finite temperature transition (crossover) for µI < µC

I .

Transition temperature TC is defined by the behaviour of ¯ ψψ

  • :

◮ Standard: Use the inflection point of the condensate. ◮ Easier alternative for µI < µC

I :

Use the point where subtracted condensate reaches a certain value. (that value has to be known from µ = 0 – Silver Blaze)

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QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential

Definition of the transition point

Investigate the finite temperature transition (crossover) for µI < µC

I .

Transition temperature TC is defined by the behaviour of ¯ ψψ

  • :

◮ Standard: Use the inflection point of the condensate. ◮ Easier alternative for µI < µC

I :

Use the point where subtracted condensate reaches a certain value. (that value has to be known from µ = 0 – Silver Blaze)

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QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential

Definition of the transition point

Investigate the finite temperature transition (crossover) for µI < µC

I .

Transition temperature TC is defined by the behaviour of ¯ ψψ

  • :

◮ Standard: Use the inflection point of the condensate. ◮ Easier alternative for µI < µC

I :

Use the point where subtracted condensate reaches a certain value. (that value has to be known from µ = 0 – Silver Blaze) Here: Use subtracted u/d condensate renormalised by the quark mass: mR ¯ ψψ

  • R = mu/d

¯ ψψ

  • −

¯ ψψ

  • T=0,µI =0
  • Value at the transition (in continuum): mR

¯ ψψ

  • R = −7.407 10−5 GeV4

[ BW: Borsanyi et al, JHEP1009 (2010) ]

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QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential

Definition of the transition point

  • 0.00012
  • 9e-05
  • 6e-05
  • 3e-05

120 140 160 180 mR ¯ ψψ

  • R [GeV4]

T [MeV] preliminary µI = 0 MeV

Curves: Simple spline interpolation.

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QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential

Definition of the transition point

  • 0.00012
  • 9e-05
  • 6e-05
  • 3e-05

120 140 160 180 mR ¯ ψψ

  • R [GeV4]

T [MeV] preliminary µI = 0 MeV µI = 17 MeV

Curves: Simple spline interpolation.

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QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential

Definition of the transition point

  • 0.00012
  • 9e-05
  • 6e-05
  • 3e-05

120 140 160 180 mR ¯ ψψ

  • R [GeV4]

T [MeV] preliminary µI = 0 MeV µI = 17 MeV µI = 34 MeV

Curves: Simple spline interpolation.

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QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential

Definition of the transition point

  • 0.00012
  • 9e-05
  • 6e-05
  • 3e-05

120 140 160 180 mR ¯ ψψ

  • R [GeV4]

T [MeV] preliminary µI = 0 MeV µI = 17 MeV µI = 34 MeV µI = 51 MeV

Curves: Simple spline interpolation.

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QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential

Definition of the transition point

  • 0.00012
  • 9e-05
  • 6e-05
  • 3e-05

120 140 160 180 mR ¯ ψψ

  • R [GeV4]

T [MeV] preliminary µI = 0 MeV µI = 17 MeV µI = 34 MeV µI = 51 MeV

✍✌ ✎☞ ❤

Curves: Simple spline interpolation. Biggest problem: λ-extrapolation!

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QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential

Phase diagram for 6 × 243

100 110 120 130 140 150 160 170 180 0.2 0.4 0.6 0.8 1 1.2 1.4 T [MeV] µI/mπ (TC, µI,C)P (TC, µI,C)C preliminary crossover Pion condensation

Results for transition points from pion condensation from

[ Endr˝

  • di, Thu ] before.
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QCD with isospin chemical potential: low densities and Taylor expansion QCD at small isospin chemical potential

Phase diagram: Open questions

◮ Where is the meeting point between crossover and pion condensation

boundary?

◮ What is the order of the transition on the boundary?

Presence of a tri-critical point?

◮ What happens in the µI → ∞ limit? ◮ More generally:

Are the deconfinement transition and the boundary of the pion condensation phase equivalent?

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QCD with isospin chemical potential: low densities and Taylor expansion Comparison Taylor expansion around µI = 0

  • 4. Comparison to Taylor expansion around µI = 0
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QCD with isospin chemical potential: low densities and Taylor expansion Comparison Taylor expansion around µI = 0

Taylor expansion around µI = 0

Simulations at finite µB suffer from a sign problem! One of the most important tools to obtain information at finite µB: Taylor expansion around µB = 0. However: Range of applicability at a given order is unknown! Here: test Taylor expansion method using our data for µI = 0

◮ As an observable we use the isospin density (analogue to Baryon density):

nI = T V ∂ log Z ∂µI

◮ Associated Taylor expansion (follows from expansion of pressure p/T 4):

nI T 3 = c2 µI T

  • + c4

6 µI T 3 Take values from Budapest-Wuppertal

[ BW: Borsanyi et al, JHEP1201 (2012) ]

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QCD with isospin chemical potential: low densities and Taylor expansion Comparison Taylor expansion around µI = 0

Comparison to data at finte µI

Compare data for 6 × 243 lattice:

100 110 120 130 140 150 160 170 180 0.2 0.4 0.6 0.8 1 1.2 1.4 T [MeV] µI/mπ (TC, µI,C)P (TC, µI,C)C preliminary crossover Pion condensation T = 124 MeV T = 162 MeV

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QCD with isospin chemical potential: low densities and Taylor expansion Comparison Taylor expansion around µI = 0

Comparison to data at finte µI

Compare data for 6 × 243 lattice, T < TC:

0.5 1 1.5 2 2.5 3 3.5 20 40 60 80 100 120 140 nI /T 3 µI [MeV] mπ/2 T = 124 MeV 6 × 243 preliminary Taylor exp. O(µI) O(µ3

I)

simulation

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QCD with isospin chemical potential: low densities and Taylor expansion Comparison Taylor expansion around µI = 0

Comparison to data at finte µI

Compare data for 6 × 243 lattice, T > TC:

0.5 1 1.5 2 2.5 3 3.5 20 40 60 80 100 120 140 nI /T 3 µI [MeV] mπ/2 T = 162 MeV 6 × 243 preliminary Taylor exp. O(µI) O(µ3

I)

simulation

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QCD with isospin chemical potential: low densities and Taylor expansion Comparison Taylor expansion around µI = 0

Comparison to data at finte µI

Compare data for 6 × 243 lattice, T > TC:

0.5 1 1.5 2 2.5 3 3.5 20 40 60 80 100 120 140 nI /T 3 µI [MeV] mπ/2 T = 162 MeV 6 × 243 preliminary Taylor exp. O(µI) O(µ3

I)

simulation

Note: Here compare to coefficients from 6 × 183 lattices. (But finite size effects in coefficients negligible!)

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QCD with isospin chemical potential: low densities and Taylor expansion Comparison Taylor expansion around µI = 0

Comparison to data at finte µI

Compare data for 8 × 243 lattice, T < TC:

0.5 1 1.5 2 2.5 3 3.5 20 40 60 80 100 120 140 nI /T 3 µI [MeV] mπ/2 T = 132 MeV 8 × 243 preliminary Taylor exp. O(µI) O(µ3

I)

simulation

Here: Coefficients computed on the same lattice size!

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QCD with isospin chemical potential: low densities and Taylor expansion Comparison Taylor expansion around µI = 0

Comparison to data at finte µI

Compare data for 8 × 243 lattice, T > TC:

0.5 1 1.5 2 2.5 3 3.5 20 40 60 80 100 120 140 nI /T 3 µI [MeV] mπ/2 T = 174 MeV 8 × 243 preliminary Taylor exp. O(µI) O(µ3

I)

simulation

Here: Coefficients computed on the same lattice size!

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QCD with isospin chemical potential: low densities and Taylor expansion Comparison Taylor expansion around µI = 0

Comparison to data at finte µI

◮ For T < TC:

Good agreement between expansion to O(µ3

I ) and data for µI < µC I .

Note: For µI > µC

I the system is in another (pion condensation) phase.

⇒ We do not expect agreement between expansion and data.

◮ For T > TC:

Good agreement between all data and expansion to O(µ3

I )

◮ Generally: O(µ5

I ) contributions appear to be neligible!

◮ It would be interesting to simulate at larger values of µI for T > TC to

see for how long the agreement persists.

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QCD with isospin chemical potential: low densities and Taylor expansion

Summary and Perspectives

◮ We have investigated the phase structure of QCD at finite isospin chemical potential µI . ◮ Biggest issue: Full control of λ-extrapolations (need to be improved). ◮ We have mapped the transition to the pion condensation phase using the pion condensate. (previous talk by Gergely Endr˝

  • di)

◮ The crossover temperatures starting from µI = 0 decrease slightly at finite µI . ◮ Results from Taylor expansion to O(µ3

I ) agree well with results looked at so far.

(except for results in the pion condensation phase – as expected) To do: ◮ Perform continuum limit and look at thermodynamic limit. ◮ Determine the order of the transitions to the pion condensation phase. Presence of a tricritical point? ◮ There are plenty of other interesting things to do with this theory!

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QCD with isospin chemical potential: low densities and Taylor expansion

Thank you for your attention!