Diagrammatic Monte Carlo for Strong Coupling LQCD Wolfgang Unger, ETH Zürich with Philippe de Forcrand, ETH Zürich/CERN Yukawa Institute, Kyoto 20.02.2012 Wolfgang Unger, ETH Zürich () DMC for SC-LQCD Kyoto, 20.02.2012 1 / 27
Motivation for SC-LQCD � � 1 Motivation for Strong Coupling LQCD in Continuous Time Continous Time Limit and a / a t = f ( γ ) Continous Time Partition Function Z ( β ) � 2 � Diagrammatic Monte Carlo General Idea and Motivation Metropolis-Hastings Comments on the Updating Rules � 3 � Discussion Limitations Generalizations Wolfgang Unger, ETH Zürich () DMC for SC-LQCD Kyoto, 20.02.2012 2 / 27
Motivation for SC-LQCD The QCD ( µ, T ) phase diagram : T [ MeV ] R R Early Universe Early Universe H H 〈 〉≃ 0 200 I I C C , L , L Q uark H H C C FAIR FAIR G luon Crossover P lasma CP Deconfinement & Chiral Transition 〈 〉≠ 0 100 1 st o r d e Hadronic Matter r Baryonic Nuclear Color Crystal ? Vacuum Matter Super- conductor? Neutron Stars Neutron Stars 0 B [ GeV ] 1 rich phase structure conjectured chiral and deconfinement transition QGP at high temperatures exotic matter at high density Wolfgang Unger, ETH Zürich () DMC for SC-LQCD Kyoto, 20.02.2012 3 / 27
Motivation for SC-LQCD The QCD ( µ, T ) phase diagram : QCD has a severe sign problem for finite chemical potential µ = 1 3 µ B : T [ MeV ] fermions anti-commute: 〈 〉≃ 0 200 p + m + µγ 0 ) † γ 5 ( i / p + m + µγ 0 ) γ 5 = ( i / Q uark G luon Crossover P lasma the fermion determinant det M ( µ ) becomes complex! 〈 〉≠ 0 100 e − S f = det M ( µ ) = det M ( − ¯ µ ) Hadronic Matter Nuclear little hope that it can be Vacuum Matter circumvented: 0 B [ GeV ] 1 - Taylor expansion, - imaginary µ with analytic because of the sign problem: very continuation, little is known - reweighting method are all limited to small µ/ T � 1 only recently: agreement on crossover temperature T c at zero baryon chemical potential Wolfgang Unger, ETH Zürich () DMC for SC-LQCD Kyoto, 20.02.2012 3 / 27
Motivation for SC-LQCD What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? Look at QCD in a regime where the sign problem can be made mild: This is obtained by changing the nature of integration variables: — no sampling of gauge fields { U } ! — no fermion determinant (no HMC)! Staggered QCD in the strong coupling limit : start from the “1-flavor” staggered QCD Lagrangian in Euclidean time: L QCD = e µδ ν 0 ¯ ν ) − e − µδ ν 0 ¯ � 1 2 η ν ( x ) � ν ) U † ν ( x ) χ ( x ) � χ ( x ) U ν ( x ) χ ( x + ˆ χ ( x + ˆ + am q ¯ χχ ν tr U P + tr U † − β � � � + O ( a 2 ) P 2 N c P Wolfgang Unger, ETH Zürich () DMC for SC-LQCD Kyoto, 20.02.2012 4 / 27
Motivation for SC-LQCD What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? Look at QCD in a regime where the sign problem can be made mild: This is obtained by changing the nature of integration variables: — no sampling of gauge fields { U } ! — no fermion determinant (no HMC)! Staggered QCD in the strong coupling limit : start from the “1-flavor” staggered QCD Lagrangian in Euclidean time: L QCD = e µδ ν 0 ¯ ν ) − e − µδ ν 0 ¯ � 1 2 η ν ( x ) � ν ) U † ν ( x ) χ ( x ) � χ ( x ) U ν ( x ) χ ( x + ˆ χ ( x + ˆ + am q ¯ χχ ν ✭ ✭✭✭✭✭✭✭✭✭✭✭✭ β � tr U P + tr U † � + O ( a 2 ) � − P 2 N c P g → ∞ , β = 2 N c take limit of infinite gauge coupling: g 2 → 0 allows to integrate out the gauge fields completely, as link integration factorizes Wolfgang Unger, ETH Zürich () DMC for SC-LQCD Kyoto, 20.02.2012 4 / 27
Motivation for SC-LQCD What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? What is Strong Coupling Lattice QCD? Look at QCD in a regime where the sign problem can be made mild: This is obtained by changing the nature of integration variables: — no sampling of gauge fields { U } ! — no fermion determinant (no HMC)! Staggered QCD in the strong coupling limit : start from the “1-flavor” staggered QCD Lagrangian in Euclidean time: L QCD = e µδ ν 0 ¯ ν ) − e − µδ ν 0 ¯ � 2 η ν ( x ) � 1 ν ) U † ν ( x ) χ ( x ) � χ ( x ) U ν ( x ) χ ( x + ˆ χ ( x + ˆ + am q ¯ χχ ν ✭ ✭✭✭✭✭✭✭✭✭✭✭✭ β � tr U P + tr U † � + O ( a 2 ) � − P 2 N c P g → ∞ , β = 2 N c take limit of infinite gauge coupling: g 2 → 0 allows to integrate out the gauge fields completely, as link integration factorizes Drawback: strong coupling limit is converse to asymptotic freedom: degrees of freedom in SC-QCD live on a crystal fermions have no spin (taste splitting maximal) Wolfgang Unger, ETH Zürich () DMC for SC-LQCD Kyoto, 20.02.2012 4 / 27
Motivation for SC-LQCD Why Strong Coupling Lattice QCD? Why Strong Coupling Lattice QCD? Why Strong Coupling Lattice QCD? Why Strong Coupling Lattice QCD? Why Strong Coupling Lattice QCD? 1-flavor strong coupling QCD might appear crude, but exhibits confinement , i.e. only color singlet degrees of freedom survive: mesons (rep. by monomers and dimers) baryons (rep. by oriented self-avoiding loops) and chiral symmetry breaking/restoration: η 5 ( x ) = ( − 1 ) Σ d χ ( x ) �→ e i η 5 ( x ) θ A χ ( x ) , ¯ χ ( x ) e i η 5 ( x ) θ A , ν = 0 x ν U A (1): χ ( x ) �→ ¯ is spontaneously broken below T c , hence its phase diagram might be similar to the QCD phase diagram SC-LQCD as effective theory for nuclear matter : derive nuclear interactions between hadrons from (Lattice) QCD transition at high density is the nuclear transition Wolfgang Unger, ETH Zürich () DMC for SC-LQCD Kyoto, 20.02.2012 5 / 27
Motivation for SC-LQCD Why Strong Coupling Lattice QCD? Why Strong Coupling Lattice QCD? Why Strong Coupling Lattice QCD? Why Strong Coupling Lattice QCD? Why Strong Coupling Lattice QCD? 1-flavor strong coupling QCD might appear crude, but exhibits confinement , i.e. only color singlet degrees of freedom survive: mesons (rep. by monomers and dimers) baryons (rep. by oriented self-avoiding loops) and chiral symmetry breaking/restoration: η 5 ( x ) = ( − 1 ) Σ d χ ( x ) �→ e i η 5 ( x ) θ A χ ( x ) , ¯ χ ( x ) e i η 5 ( x ) θ A , ν = 0 x ν U A (1): χ ( x ) �→ ¯ is spontaneously broken below T c , hence its phase diagram might be similar to the QCD phase diagram SC-LQCD as effective theory for nuclear matter : derive nuclear interactions between hadrons from (Lattice) QCD transition at high density is the nuclear transition Just another effective model for QCD? yes and no: think of SC-LQCD rather as a 1-parameter deformation of QCD Wolfgang Unger, ETH Zürich () DMC for SC-LQCD Kyoto, 20.02.2012 5 / 27
Motivation for SC-LQCD SC-QCD Partition Function (1) SC-QCD Partition Function (1) SC-QCD Partition Function (1) SC-QCD Partition Function (1) SC-QCD Partition Function (1) Partition function for SU( N c ): N c � � { ( N c − k µ ( x ))! χ d χ e 2 amqM ( x ) � � µ )) k µ ( x ) Z = d ¯ ( M ( x ) M ( x + ˆ N c ! k µ ( x )! x µ k µ ( x )= 0 µ ) B ( x ) � } + κ � ρ ( x , y ) N c ¯ µ ) + ( − ρ ( y , x )) N c ¯ B ( x ) B ( x + ˆ B ( x + ˆ Wolfgang Unger, ETH Zürich () DMC for SC-LQCD Kyoto, 20.02.2012 6 / 27
Motivation for SC-LQCD SC-QCD Partition Function (1) SC-QCD Partition Function (1) SC-QCD Partition Function (1) SC-QCD Partition Function (1) SC-QCD Partition Function (1) Partition function for SU( N c ): N c � � { ( N c − k µ ( x ))! χ d χ e 2 amqM ( x ) � � µ )) k µ ( x ) Z = d ¯ ( M ( x ) M ( x + ˆ N c ! k µ ( x )! x µ k µ ( x )= 0 µ ) B ( x ) � } + κ � ρ ( x , y ) N c ¯ µ ) + ( − ρ ( y , x )) N c ¯ B ( x ) B ( x + ˆ B ( x + ˆ Only confined, colorless degrees of freedom remain after link integration ( N c = 3): Mesons M ( x ) = ¯ χ ( x ) χ ( x ) , represented by monomers n ( x ) ∈ { 0 , 1 , . . . , 3 } and dimers (non-oriented meson hoppings): k µ ( x ) ∈ { 0 , 1 , . . . , 3 } Baryons B ( x ) = 1 3 ! ǫ i 1 ... i 3 χ i 1 ( x ) . . . χ i 3 ( x ) form self-avoiding oriented loops : ρ ( x , y ) = η ˆ µ ( x ) exp ( ± a τ µ ) δ ˆ ¯ B ( x ) B ( y ) ∈ { 0 , 1 } µ ˆ 0 Note: baryons transform under gauge trafo Ω ∈ U ( 3 ) as B ( x ) → B ( x ) det Ω ⇒ not gauge invariant , only SU(3) contains baryons ( κ = 1), U(3) purely mesonic ( κ = 0) Wolfgang Unger, ETH Zürich () DMC for SC-LQCD Kyoto, 20.02.2012 6 / 27
Recommend
More recommend
