Polyakov line actions from SU(3) lattice gauge theory with dynamical fermions: first results via relative weights Roman Höllwieser ab , Jeff Greensite c a Institute of Atomic and Subatomic Physics, Nuclear Physics Dept., Vienna University of Technology, Operngasse 9, 1040 Vienna, Austria b Department of Physics, New Mexico State University, Las Cruces, NM 88003-8001, USA c Physics and Astronomy Dept., San Francisco State University, San Francisco, CA 94132, USA
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? 18.6.2015 Roman HÖLLWIESER 1
Polyakov line actions from SU(3) LGT via relative weigthts Motivation The Phase Diagram of QCD 18.6.2015 Roman HÖLLWIESER 2
Polyakov line actions from SU(3) LGT via relative weigthts Lattice QCD and the Sign problem � DUD ¯ Z = ψ D ψ e − S YM ( U ) − S F ( U ; µ ) 18.6.2015 Roman HÖLLWIESER 3
Polyakov line actions from SU(3) LGT via relative weigthts Lattice QCD and the Sign problem � DUD ¯ Z = ψ D ψ e − S YM ( U ) − S F ( U ; µ ) � d 4 x ¯ S F ( U ; µ ) = − ψ M ( U ; µ ) ψ 18.6.2015 Roman HÖLLWIESER 3
Polyakov line actions from SU(3) LGT via relative weigthts Lattice QCD and the Sign problem � DUD ¯ Z = ψ D ψ e − S YM ( U ) − S F ( U ; µ ) � d 4 x ¯ S F ( U ; µ ) = − ψ M ( U ; µ ) ψ � DU e − S YM ( U ) det M ( U ; µ ) Z = 18.6.2015 Roman HÖLLWIESER 3
Polyakov line actions from SU(3) LGT via relative weigthts Lattice QCD and the Sign problem � DUD ¯ Z = ψ D ψ e − S YM ( U ) − S F ( U ; µ ) � d 4 x ¯ S F ( U ; µ ) = − ψ M ( U ; µ ) ψ � DU e − S YM ( U ) det M ( U ; µ ) Z = numerical evaluation of bosonic integral with importance sampling 18.6.2015 Roman HÖLLWIESER 3
Polyakov line actions from SU(3) LGT via relative weigthts Lattice QCD and the Sign problem � DUD ¯ Z = ψ D ψ e − S YM ( U ) − S F ( U ; µ ) � d 4 x ¯ S F ( U ; µ ) = − ψ M ( U ; µ ) ψ � DU e − S YM ( U ) det M ( U ; µ ) Z = numerical evaluation of bosonic integral with importance sampling � DU e − S YM det M O observable � O � = � DU e − S YM det M 18.6.2015 Roman HÖLLWIESER 3
Polyakov line actions from SU(3) LGT via relative weigthts Lattice QCD and the Sign problem � DUD ¯ Z = ψ D ψ e − S YM ( U ) − S F ( U ; µ ) � d 4 x ¯ S F ( U ; µ ) = − ψ M ( U ; µ ) ψ � DU e − S YM ( U ) det M ( U ; µ ) Z = numerical evaluation of bosonic integral with importance sampling � DU e − S YM det M O observable � O � = � DU e − S YM det M lack of γ 5 -hermiticity, γ 5 M ( µ ) γ 5 = M † ( − µ ∗ ) � = M † ( µ ) 18.6.2015 Roman HÖLLWIESER 3
Polyakov line actions from SU(3) LGT via relative weigthts Lattice QCD and the Sign problem � DUD ¯ Z = ψ D ψ e − S YM ( U ) − S F ( U ; µ ) � d 4 x ¯ S F ( U ; µ ) = − ψ M ( U ; µ ) ψ � DU e − S YM ( U ) det M ( U ; µ ) Z = numerical evaluation of bosonic integral with importance sampling � DU e − S YM det M O observable � O � = � DU e − S YM det M lack of γ 5 -hermiticity, γ 5 M ( µ ) γ 5 = M † ( − µ ∗ ) � = M † ( µ ) determinant is complex and satisfies [det M ( µ )] ∗ = det M ( − µ ∗ ) 18.6.2015 Roman HÖLLWIESER 3
Polyakov line actions from SU(3) LGT via relative weigthts Importance of the Sign problem assymetry between matter and anti-matter 18.6.2015 Roman HÖLLWIESER 4
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 18.6.2015 Roman HÖLLWIESER 4
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 F q ) = � Tr P � � = Re ( P ) × Re ( d ̟ ) − Im ( P ) × Im ( d ̟ ) exp( − 1 � Tr P ∗ � q ) = T F ¯ � = Re ( P ) × Re ( d ̟ )+ Im ( P ) × Im ( d ̟ ) 18.6.2015 Roman HÖLLWIESER 4
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 F q ) = � Tr P � � = Re ( P ) × Re ( d ̟ ) − Im ( P ) × Im ( d ̟ ) exp( − 1 � Tr P ∗ � q ) = T F ¯ � = Re ( P ) × Re ( d ̟ )+ Im ( P ) × Im ( d ̟ ) finite chemical potential µ favors propagation of quarks 18.6.2015 Roman HÖLLWIESER 4
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”) 18.6.2015 Roman HÖLLWIESER 5
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: of the observable in powers of µ/ T at µ = 0 18.6.2015 Roman HÖLLWIESER 5
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: of the observable in powers of µ/ T at µ = 0 Imaginary µ : analytic continuation of results to real µ 18.6.2015 Roman HÖLLWIESER 5
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: of the observable in powers of µ/ T at µ = 0 Imaginary µ : analytic continuation of results to real µ |QCD|: det M = | det M | e i φ , simulations without e i φ + reweighting 18.6.2015 Roman HÖLLWIESER 5
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: of the observable in powers of µ/ T at µ = 0 Imaginary µ : analytic continuation of results to real µ |QCD|: det M = | det M | e i φ , simulations without e i φ + reweighting Complex Langevin: stochastic quantization - evolution of fields in a fictitious time with Brownian noise and search for stationary solutions with correct measure 18.6.2015 Roman HÖLLWIESER 5
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: of the observable in powers of µ/ T at µ = 0 Imaginary µ : analytic continuation of results to real µ |QCD|: det M = | det M | e i φ , simulations without e i φ + 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 18.6.2015 Roman HÖLLWIESER 5
Polyakov line actions from SU(3) LGT via relative weigthts Effective Polyakov Line Action Indirect approach: Polyakov line action (SU(3) spin) model 18.6.2015 Roman HÖLLWIESER 6
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 U 0 ( � x , 0) = U x (temporal gauge) and integrate out all other d.o.f. 18.6.2015 Roman HÖLLWIESER 6
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 U 0 ( � x , 0) = U x (temporal gauge) and integrate out all other d.o.f. � DU 0 ( � e S P ( U x ) = x , 0) DU k D ψ � x δ [ U x − U 0 ( � x , 0)] e S L 18.6.2015 Roman HÖLLWIESER 6
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 U 0 ( � x , 0) = U x (temporal gauge) and integrate out all other d.o.f. � DU 0 ( � e S P ( U x ) = x , 0) DU k D ψ � x δ [ U x − U 0 ( � x , 0)] e S L derive S P at µ = 0, for µ > 0 we have (true to all orders of strong coupling/hopping parameter expansion) 18.6.2015 Roman HÖLLWIESER 6
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 U 0 ( � x , 0) = U x (temporal gauge) and integrate out all other d.o.f. � DU 0 ( � e S P ( U x ) = x , 0) DU k D ψ � x δ [ U x − U 0 ( � x , 0)] e S L derive S P at µ = 0, for µ > 0 we have (true to all orders of strong coupling/hopping parameter expansion) S µ P ( U x , U † x ) = S µ =0 [ e N t µ U x , e − N t µ U † x ] P 18.6.2015 Roman HÖLLWIESER 6
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