[PPT] - Polyakov line actions from SU(3) lattice gauge theory with dynamical PowerPoint Presentation

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Polyakov line actions from SU(3) lattice gauge theory with dynamical fermions: first results via relative weights

Roman Höllwieserab, Jeff Greensitec

aInstitute of Atomic and Subatomic Physics, Nuclear Physics Dept.,

Vienna University of Technology, Operngasse 9, 1040 Vienna, Austria

bDepartment of Physics, New Mexico State University,

Las Cruces, NM 88003-8001, USA

cPhysics and Astronomy Dept., San Francisco State University,

San Francisco, CA 94132, USA

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Polyakov line actions from SU(3) LGT via relative weigthts

Agenda

Motivation Lattice QCD and the Sign problem The Polyakov Line Action Preliminary Results Conclusions & Outlook Questions?

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Polyakov line actions from SU(3) LGT via relative weigthts

Motivation

The Phase Diagram of QCD

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Polyakov line actions from SU(3) LGT via relative weigthts

Lattice QCD and the Sign problem

Z =

DUD ¯

ψDψ e−SYM(U)−SF(U;µ)

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Polyakov line actions from SU(3) LGT via relative weigthts

Lattice QCD and the Sign problem

Z =

DUD ¯

ψDψ e−SYM(U)−SF(U;µ) SF(U; µ) = −

d4x ¯

ψ M(U; µ) ψ

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Polyakov line actions from SU(3) LGT via relative weigthts

Lattice QCD and the Sign problem

Z =

DUD ¯

ψDψ e−SYM(U)−SF(U;µ) SF(U; µ) = −

d4x ¯

ψ M(U; µ) ψ Z =

DU e−SYM(U) det M(U; µ)

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Polyakov line actions from SU(3) LGT via relative weigthts

Lattice QCD and the Sign problem

Z =

DUD ¯

ψDψ e−SYM(U)−SF(U;µ) SF(U; µ) = −

d4x ¯

ψ M(U; µ) ψ Z =

DU e−SYM(U) det M(U; µ)

numerical evaluation of bosonic integral with importance sampling

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Polyakov line actions from SU(3) LGT via relative weigthts

Lattice QCD and the Sign problem

Z =

DUD ¯

ψDψ e−SYM(U)−SF(U;µ) SF(U; µ) = −

d4x ¯

ψ M(U; µ) ψ Z =

DU e−SYM(U) det M(U; µ)

numerical evaluation of bosonic integral with importance sampling

bservable O =
DU e−SYM det M O
DU e−SYM det M

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Polyakov line actions from SU(3) LGT via relative weigthts

Lattice QCD and the Sign problem

Z =

DUD ¯

ψDψ e−SYM(U)−SF(U;µ) SF(U; µ) = −

d4x ¯

ψ M(U; µ) ψ Z =

DU e−SYM(U) det M(U; µ)

numerical evaluation of bosonic integral with importance sampling

bservable O =
DU e−SYM det M O
DU e−SYM det M

lack of γ5-hermiticity, γ5M(µ)γ5 = M†(−µ∗) = M†(µ)

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Polyakov line actions from SU(3) LGT via relative weigthts

Lattice QCD and the Sign problem

Z =

DUD ¯

ψDψ e−SYM(U)−SF(U;µ) SF(U; µ) = −

d4x ¯

ψ M(U; µ) ψ Z =

DU e−SYM(U) det M(U; µ)

numerical evaluation of bosonic integral with importance sampling

bservable O =
DU e−SYM det M O
DU e−SYM det M

lack of γ5-hermiticity, γ5M(µ)γ5 = M†(−µ∗) = M†(µ)

determinant is complex and satisfies

[det M(µ)]∗ = det M(−µ∗)

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Polyakov line actions from SU(3) LGT via relative weigthts

Importance of the Sign problem

assymetry between matter and anti-matter

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Polyakov line actions from SU(3) LGT via relative weigthts

Importance of the Sign problem

assymetry between matter and anti-matter free energy of particle q /anti-particle ¯ q

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Polyakov line actions from SU(3) LGT via relative weigthts

Importance of the Sign problem

assymetry between matter and anti-matter free energy of particle q /anti-particle ¯ q expectation value of Polyakov loop / adjoint: exp(− 1 T Fq) = Tr P =

Re(P) × Re(d̟)−Im(P) × Im(d̟)

exp(− 1 T F¯

q)

= Tr P∗ =

Re(P) × Re(d̟)+Im(P) × Im(d̟)

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Polyakov line actions from SU(3) LGT via relative weigthts

Importance of the Sign problem

assymetry between matter and anti-matter free energy of particle q /anti-particle ¯ q expectation value of Polyakov loop / adjoint: exp(− 1 T Fq) = Tr P =

Re(P) × Re(d̟)−Im(P) × Im(d̟)

exp(− 1 T F¯

q)

= Tr P∗ =

Re(P) × Re(d̟)+Im(P) × Im(d̟)

finite chemical potential µ favors propagation of quarks

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Polyakov line actions from SU(3) LGT via relative weigthts

Possible Solutions of the Sign problem

Reweighting: measurements of O are given a varying, oscillatory weight f /g in the ensemble average (“average sign”)

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Polyakov line actions from SU(3) LGT via relative weigthts

Possible Solutions of the Sign problem

Reweighting: measurements of O are given a varying, oscillatory weight f /g in the ensemble average (“average sign”) Taylor expansion:

f the observable in powers of µ/T at µ = 0

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Polyakov line actions from SU(3) LGT via relative weigthts

Possible Solutions of the Sign problem

Reweighting: measurements of O are given a varying, oscillatory weight f /g in the ensemble average (“average sign”) Taylor expansion:

f the observable in powers of µ/T at µ = 0

Imaginary µ: analytic continuation of results to real µ

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Polyakov line actions from SU(3) LGT via relative weigthts

Possible Solutions of the Sign problem

Reweighting: measurements of O are given a varying, oscillatory weight f /g in the ensemble average (“average sign”) Taylor expansion:

f the observable in powers of µ/T at µ = 0

Imaginary µ: analytic continuation of results to real µ |QCD|: detM = |detM|eiφ, simulations without eiφ + reweighting

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Polyakov line actions from SU(3) LGT via relative weigthts

Possible Solutions of the Sign problem

Reweighting: measurements of O are given a varying, oscillatory weight f /g in the ensemble average (“average sign”) Taylor expansion:

f the observable in powers of µ/T at µ = 0

Imaginary µ: analytic continuation of results to real µ |QCD|: detM = |detM|eiφ, simulations without eiφ + reweighting Complex Langevin: stochastic quantization - evolution of fields in a fictitious time with Brownian noise and search for stationary solutions with correct measure

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Polyakov line actions from SU(3) LGT via relative weigthts

Possible Solutions of the Sign problem

Reweighting: measurements of O are given a varying, oscillatory weight f /g in the ensemble average (“average sign”) Taylor expansion:

f the observable in powers of µ/T at µ = 0

Imaginary µ: analytic continuation of results to real µ |QCD|: detM = |detM|eiφ, simulations without eiφ + reweighting Complex Langevin: stochastic quantization - evolution of fields in a fictitious time with Brownian noise and search for stationary solutions with correct measure Worldline formalism and strong coupling limit: change order of integration, partial integration over loops and hopping parameter expansion

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Polyakov line actions from SU(3) LGT via relative weigthts

Effective Polyakov Line Action

Indirect approach: Polyakov line action (SU(3) spin) model

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Polyakov line actions from SU(3) LGT via relative weigthts

Effective Polyakov Line Action

Indirect approach: Polyakov line action (SU(3) spin) model fix Polyakov line holonomies U0( x, 0) = Ux (temporal gauge) and integrate out all other d.o.f.

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Polyakov line actions from SU(3) LGT via relative weigthts

Effective Polyakov Line Action

Indirect approach: Polyakov line action (SU(3) spin) model fix Polyakov line holonomies U0( x, 0) = Ux (temporal gauge) and integrate out all other d.o.f. eSP(Ux) =

DU0(

x, 0)DUkDψ

x δ[Ux − U0(

x, 0)]eSL

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Polyakov line actions from SU(3) LGT via relative weigthts

Effective Polyakov Line Action

Indirect approach: Polyakov line action (SU(3) spin) model fix Polyakov line holonomies U0( x, 0) = Ux (temporal gauge) and integrate out all other d.o.f. eSP(Ux) =

DU0(

x, 0)DUkDψ

x δ[Ux − U0(

x, 0)]eSL derive SP at µ = 0, for µ > 0 we have (true to all orders of strong coupling/hopping parameter expansion)

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Polyakov line actions from SU(3) LGT via relative weigthts

Effective Polyakov Line Action

Indirect approach: Polyakov line action (SU(3) spin) model fix Polyakov line holonomies U0( x, 0) = Ux (temporal gauge) and integrate out all other d.o.f. eSP(Ux) =

DU0(

x, 0)DUkDψ

x δ[Ux − U0(

x, 0)]eSL derive SP at µ = 0, for µ > 0 we have (true to all orders of strong coupling/hopping parameter expansion) Sµ

P(Ux, U† x) = Sµ=0 P

[eNtµUx, e−NtµU†

x]

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Polyakov line actions from SU(3) LGT via relative weigthts

Effective Polyakov Line Action

Indirect approach: Polyakov line action (SU(3) spin) model fix Polyakov line holonomies U0( x, 0) = Ux (temporal gauge) and integrate out all other d.o.f. eSP(Ux) =

DU0(

x, 0)DUkDψ

x δ[Ux − U0(

x, 0)]eSL derive SP at µ = 0, for µ > 0 we have (true to all orders of strong coupling/hopping parameter expansion) Sµ

P(Ux, U† x) = Sµ=0 P

[eNtµUx, e−NtµU†

x]

hard to compute exp[SP(Ux)], use relative weights...

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Polyakov line actions from SU(3) LGT via relative weigthts

Relative Weights Method

S′

L . . .lattice action in temporal gauge with U0(

x, 0) = U′

x,

compute the ratio e∆SP = exp[SP(U′

x)]

exp[SP(U′′

x )] =

DUkDψeS′

L

DUkDψeS′′

L

=

DUkDψ exp[S′

L − S′′ L]eS′′

L

DUkDψeS′′

L

≡ exp[S′

L − S′′ L]′′

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Polyakov line actions from SU(3) LGT via relative weigthts

Relative Weights Method

S′

L . . .lattice action in temporal gauge with U0(

x, 0) = U′

x,

compute the ratio e∆SP = exp[SP(U′

x)]

exp[SP(U′′

x )] =

DUkDψeS′

L

DUkDψeS′′

L

=

DUkDψ exp[S′

L − S′′ L]eS′′

L

DUkDψeS′′

L

≡ exp[S′

L − S′′ L]′′

Ux(λ) path through configuration space parametrized by λ

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Polyakov line actions from SU(3) LGT via relative weigthts

Relative Weights Method

S′

L . . .lattice action in temporal gauge with U0(

x, 0) = U′

x,

compute the ratio e∆SP = exp[SP(U′

x)]

exp[SP(U′′

x )] =

DUkDψeS′

L

DUkDψeS′′

L

=

DUkDψ exp[S′

L − S′′ L]eS′′

L

DUkDψeS′′

L

≡ exp[S′

L − S′′ L]′′

Ux(λ) path through configuration space parametrized by λ U′

x = Ux(λ0 + ∆λ/2), U′′ x = Ux(λ0 − ∆λ/2) → (dSP dλ )λ0 = ∆S ∆λ

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Polyakov line actions from SU(3) LGT via relative weigthts

derivatives of SP w.r.t. Fourier components ak of

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Polyakov line actions from SU(3) LGT via relative weigthts

derivatives of SP w.r.t. Fourier components ak of Px ≡ 1

3TrUx = k akeikx

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Polyakov line actions from SU(3) LGT via relative weigthts

derivatives of SP w.r.t. Fourier components ak of Px ≡ 1

3TrUx = k akeikx

effective Polyakov line action motivated by heavy-dense action, where h is some inverse power of hopping parameter and satisfies the Pauli exclusion principle as µ → ∞ - no more than three (staggered) quarks per site

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Polyakov line actions from SU(3) LGT via relative weigthts

derivatives of SP w.r.t. Fourier components ak of Px ≡ 1

3TrUx = k akeikx

effective Polyakov line action motivated by heavy-dense action, where h is some inverse power of hopping parameter and satisfies the Pauli exclusion principle as µ → ∞ - no more than three (staggered) quarks per site Seff [Ux] =

x,y PxK(x − y)Py

+p

x log(1 + heµ/TTr[Ux] + h2e2µ/TTr[U† x] + h3e3µ/T)

log(1 + he−µ/TTr[Ux] + h2e−2µ/TTr[U†

x] + h3e−3µ/T)

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Polyakov line actions from SU(3) LGT via relative weigthts

derivatives of SP w.r.t. Fourier components ak of Px ≡ 1

3TrUx = k akeikx

effective Polyakov line action motivated by heavy-dense action, where h is some inverse power of hopping parameter and satisfies the Pauli exclusion principle as µ → ∞ - no more than three (staggered) quarks per site Seff [Ux] =

x,y PxK(x − y)Py

+p

x log(1 + heµ/TTr[Ux] + h2e2µ/TTr[U† x] + h3e3µ/T)

log(1 + he−µ/TTr[Ux] + h2e−2µ/TTr[U†

x] + h3e−3µ/T)

determine K(x − y) and h from fitting to lattice data

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Polyakov line actions from SU(3) LGT via relative weigthts

derivatives of SP w.r.t. Fourier components ak of Px ≡ 1

3TrUx = k akeikx

effective Polyakov line action motivated by heavy-dense action, where h is some inverse power of hopping parameter and satisfies the Pauli exclusion principle as µ → ∞ - no more than three (staggered) quarks per site Seff [Ux] =

x,y PxK(x − y)Py

+p

x log(1 + heµ/TTr[Ux] + h2e2µ/TTr[U† x] + h3e3µ/T)

log(1 + he−µ/TTr[Ux] + h2e−2µ/TTr[U†

x] + h3e−3µ/T)

determine K(x − y) and h from fitting to lattice data

1 L3 (∂SP ∂ak )ak=α = 2K(k)α + p L3

x(3heikx + 3h2e−ikx + c.c.)

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Polyakov line actions from SU(3) LGT via relative weigthts

Preliminary Results

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Polyakov line actions from SU(3) LGT via relative weigthts

Preliminary Results

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Polyakov line actions from SU(3) LGT via relative weigthts

Solve sign problem for the effective action

remaining sign problem can be solved by mean field theory

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Polyakov line actions from SU(3) LGT via relative weigthts

Solve sign problem for the effective action

remaining sign problem can be solved by mean field theory treatment of SU(3) spin models at finite µ is a minor variation of standard mean field theory at zero chemical potential

18.6.2015 Roman HÖLLWIESER 11

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Polyakov line actions from SU(3) LGT via relative weigthts

Solve sign problem for the effective action

remaining sign problem can be solved by mean field theory treatment of SU(3) spin models at finite µ is a minor variation of standard mean field theory at zero chemical potential two magnetizations introduced for TrU and TrU† determined by minimizing the free energy

18.6.2015 Roman HÖLLWIESER 11

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Polyakov line actions from SU(3) LGT via relative weigthts

Solve sign problem for the effective action

remaining sign problem can be solved by mean field theory treatment of SU(3) spin models at finite µ is a minor variation of standard mean field theory at zero chemical potential two magnetizations introduced for TrU and TrU† determined by minimizing the free energy basic idea is that each spin is effectively coupled to the average spin on the lattice, not just nearest neighbors, through non-local kernel K(x − y)

18.6.2015 Roman HÖLLWIESER 11

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Polyakov line actions from SU(3) LGT via relative weigthts

Solve sign problem for the effective action

remaining sign problem can be solved by mean field theory treatment of SU(3) spin models at finite µ is a minor variation of standard mean field theory at zero chemical potential two magnetizations introduced for TrU and TrU† determined by minimizing the free energy basic idea is that each spin is effectively coupled to the average spin on the lattice, not just nearest neighbors, through non-local kernel K(x − y) for details and comparison to complex Langevin see Splittorff and Greensite (2012)

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Polyakov line actions from SU(3) LGT via relative weigthts

Preliminary Results

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Polyakov line actions from SU(3) LGT via relative weigthts

Preliminary Results

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Polyakov line actions from SU(3) LGT via relative weigthts

Conclusions & Outlook

determined effective Polyakov line action for asqtad staggered fermions with ma = 0.3 and Symanzik one loop improved gauge action at β = 7.0 on 163 × 6 lattices

18.6.2015 Roman HÖLLWIESER 14

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Polyakov line actions from SU(3) LGT via relative weigthts

Conclusions & Outlook

determined effective Polyakov line action for asqtad staggered fermions with ma = 0.3 and Symanzik one loop improved gauge action at β = 7.0 on 163 × 6 lattices good agreement for the Polyakov line correlators computed in the effective theory and underlying lattice gauge theory

18.6.2015 Roman HÖLLWIESER 14

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Polyakov line actions from SU(3) LGT via relative weigthts

Conclusions & Outlook

determined effective Polyakov line action for asqtad staggered fermions with ma = 0.3 and Symanzik one loop improved gauge action at β = 7.0 on 163 × 6 lattices good agreement for the Polyakov line correlators computed in the effective theory and underlying lattice gauge theory solved sign problem for the effective theory by mean field and find a phase transition and correct density limit

18.6.2015 Roman HÖLLWIESER 14

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Polyakov line actions from SU(3) LGT via relative weigthts

Conclusions & Outlook

determined effective Polyakov line action for asqtad staggered fermions with ma = 0.3 and Symanzik one loop improved gauge action at β = 7.0 on 163 × 6 lattices good agreement for the Polyakov line correlators computed in the effective theory and underlying lattice gauge theory solved sign problem for the effective theory by mean field and find a phase transition and correct density limit ...

18.6.2015 Roman HÖLLWIESER 14

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Polyakov line actions from SU(3) LGT via relative weigthts

Conclusions & Outlook

determined effective Polyakov line action for asqtad staggered fermions with ma = 0.3 and Symanzik one loop improved gauge action at β = 7.0 on 163 × 6 lattices good agreement for the Polyakov line correlators computed in the effective theory and underlying lattice gauge theory solved sign problem for the effective theory by mean field and find a phase transition and correct density limit ... determine quadratic, quasi-local center symmetry breaking terms which may be important at finite chemical potential...

18.6.2015 Roman HÖLLWIESER 14

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Polyakov line actions from SU(3) LGT via relative weigthts

Conclusions & Outlook

determined effective Polyakov line action for asqtad staggered fermions with ma = 0.3 and Symanzik one loop improved gauge action at β = 7.0 on 163 × 6 lattices good agreement for the Polyakov line correlators computed in the effective theory and underlying lattice gauge theory solved sign problem for the effective theory by mean field and find a phase transition and correct density limit ... determine quadratic, quasi-local center symmetry breaking terms which may be important at finite chemical potential... go on to smaller quark masses...

18.6.2015 Roman HÖLLWIESER 14

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Polyakov line actions from SU(3) LGT via relative weigthts

Questions? Thank You &

Derar Altarawneh, Michael Engelhardt, Manfried Faber, Martin Gal, Jeff Greensite, Urs M. Heller, James Hettrick, Andrei Ivanov, Thomas Layer, Štefan Olejnik, Luis Oxman, Mario Pitschmann, Jesus Saenz, Thomas Schweigler, Wolfgang Söldner, David Vercauteren, Markus Wellenzohn

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Polyakov line actions from SU(3) lattice gauge theory with dynamical fermions: first results via relative weights

Roman Höllwieserab, hroman@kph.tuwien.ac.at Jeff Greensitec, greensit@sfsu.edu

aInstitute of Atomic and Subatomic Physics, Nuclear Physics Dept.,

Vienna University of Technology, Operngasse 9, 1040 Vienna, Austria

bDepartment of Physics, New Mexico State University,

Las Cruces, NM 88003-8001, USA

cPhysics and Astronomy Dept., San Francisco State University,