Analytic computations of an effective lattice theory for heavy - PowerPoint PPT Presentation

Analytic computations of an effective lattice theory for heavy quarks Jonas R. Glesaaen Mathias Neuman, Owe Philipsen Lattice Conference 2015 - July 16th

1 The Effective Theory Results 2 3 Conclusion

Heavy QCD Phase Diagram T [ MeV ] 200 ≲ 10 1 µ B [ GeV ] 2/17

Heavy QCD Phase Diagram T [ MeV ] Liquid Gas 200 Phase Transition ≲ 10 1 µ B [ GeV ] 2/17

Advantages of the Effective Theory • Dimensionally reduced theory • 4D → 3D • U µ ( x ) → L ( x ) • Very mild sign problem, most gauge fi elds integrated analytically • Want to study the very dense limit, liquid gas transition 3/17

The Effective Theory

The Effective Lattice Theory Effective Theory • Integrate out all spatial gauge links � � � Z = − S action DU µ exp � � � = − S effective action DU 0 exp Using: • The strong coupling expansion • The hopping parameter expansion 4/17

Effective Theory � � � � Z = d L ( x ) exp − S eff action ( † ) x • Previous Talk: Monte Carlo simulations of ( † ) • Current Talk: Analytic calculation of Z 5/17

The Effective Theory Action � � � � S eff action = S 0 + S I L L � � Where S I L is made up of interactions at varying distances � � � � � � � � � � = v i ( 1 , 2 , ..., n i ) φ 1 φ 2 · · · φ n i S I L L L L terms dof 6/17

The Effective Theory Action � � � � S eff action = S 0 + S I L L � � Where S I L is made up of interactions at varying distances � � � � � � � � � � = v i ( 1 , 2 , ..., n i ) φ 1 φ 2 · · · φ n i S I L L L L terms dof Can be represented with connected graphs 6/17

The Effective Theory Action � � � � � � � � � � = v i ( 1 , 2 , ..., n i ) φ 1 φ 2 · · · φ n i S I L L L L terms dof In our theory: � � • v i ( 1 , 2 , ... n i ) → λ i , h i × geometry L i , L ∗ � � • φ i → i , W i 6/17

Analytic Calculations N-point Linked Cluster Expansion Classical Linked Cluster Expansion The action consists of two-point interactions which can be expanded in a set of connected graphs. Our Problem The action contains n -point interactions that we can embed on a set of connected graphs. Two step embedding 7/17

Analytic Calculations N-point Linked Cluster Expansion Effective Action Term Skeleton Graph embedding 8/17

The power of resummations Using the resummed Linked Cluster Expansion as motivation . . . = + + + + We can do the same resummation for the effective action itself, incorporating long-range effects 10/17

Results

Convergence ✸ h 1 = κ N t e N t µ = 0 . 8 ✷✳✺ ✷ ✶✳✺ a 3 n B ✶ ▲✐♥❦❡❞ ❈❧✉st❡r ✵✳✺ O ( κ 2 ) O ( κ 4 ) ✵ O ( κ 6 ) O ( κ 8 ) ✲✵✳✺ ✵ ✵✳✵✺ ✵✳✶ ✵✳✶✺ ✵✳✷ h 2 m q → 0 11/17

Convergence 3 3 h 1 = κ N t e N t µ = 0 . 8 Simulations O ( κ 6 ) 2 . 5 2 . 5 2 2 1 . 5 1 . 5 a 3 n B a 3 n B 1 1 Linked Cluster Linked Cluster 0 . 5 0 . 5 O ( κ 2 ) O ( κ 4 ) 0 0 O ( κ 6 ) O ( κ 8 ) − 0 . 5 − 0 . 5 0 0 0 . 05 0 . 05 0 . 1 0 . 1 0 . 15 0 . 15 0 . 2 0 . 2 h 2 h 2 m q → 0 11/17

Convergence 3 3 h 1 = κ N t e N t µ = 0 . 8 Simulations O ( κ 8 ) 2 . 5 2 . 5 2 2 1 . 5 1 . 5 a 3 n B a 3 n B 1 1 Linked Cluster Linked Cluster 0 . 5 0 . 5 O ( κ 2 ) O ( κ 4 ) 0 0 O ( κ 6 ) O ( κ 8 ) − 0 . 5 − 0 . 5 0 0 0 . 05 0 . 05 0 . 1 0 . 1 0 . 15 0 . 15 0 . 2 0 . 2 h 2 h 2 m q → 0 11/17

Effect of the resummations ✸ ✸ h 1 = κ N t e N t µ = 0 . 8 ❘❡s✉♠♠❡❞ O ( κ 2 ) ✷✳✺ ✷✳✺ O ( κ 4 ) O ( κ 6 ) O ( κ 8 ) ✷ ✷ ✶✳✺ ✶✳✺ a 3 n B a 3 n B ✶ ✶ ▲✐♥❦❡❞ ❈❧✉st❡r ▲✐♥❦❡❞ ❈❧✉st❡r ✵✳✺ ✵✳✺ O ( κ 2 ) O ( κ 4 ) ✵ ✵ O ( κ 6 ) O ( κ 8 ) ✲✵✳✺ ✲✵✳✺ ✵ ✵ ✵✳✵✺ ✵✳✵✺ ✵✳✶ ✵✳✶ ✵✳✶✺ ✵✳✶✺ ✵✳✷ ✵✳✷ h 2 h 2 m q → 0 12/17

Effect of the resummations ✸ h 1 = κ N t e N t µ = 0 . 8 ❘❡s✉♠♠❡❞ O ( κ 2 ) O ( κ 4 ) ✷✳✺ O ( κ 6 ) O ( κ 8 ) ✷ ✶✳✺ a 3 n B ✶ ✵✳✺ ✵ ✲✵✳✺ ✵ ✵✳✵✺ ✵✳✶ ✵✳✶✺ ✵✳✷ h 2 m q → 0 12/17

Effect of the resummations 3 h 1 = κ N t e N t µ = 1 . 0 2 . 5 2 1 . 5 a 3 n B 1 Linked Cluster 0 . 5 O ( κ 2 ) O ( κ 4 ) 0 O ( κ 6 ) O ( κ 8 ) − 0 . 5 0 0 . 05 0 . 1 0 . 15 0 . 2 h 2 m q → 0 13/17

Effect of the resummations 3 3 h 1 = κ N t e N t µ = 1 . 0 Resummed O ( κ 2 ) 2 . 5 2 . 5 O ( κ 4 ) O ( κ 6 ) O ( κ 8 ) 2 2 1 . 5 1 . 5 a 3 n B a 3 n B 1 1 Linked Cluster Linked Cluster 0 . 5 0 . 5 O ( κ 2 ) O ( κ 4 ) 0 0 O ( κ 6 ) O ( κ 8 ) − 0 . 5 − 0 . 5 0 0 0 . 05 0 . 05 0 . 1 0 . 1 0 . 15 0 . 15 0 . 2 0 . 2 h 2 h 2 m q → 0 13/17

Effect of the resummations 3 h 1 = κ N t e N t µ = 1 . 0 Resummed O ( κ 2 ) 2 . 5 O ( κ 4 ) O ( κ 6 ) O ( κ 8 ) 2 1 . 5 a 3 n B 1 0 . 5 0 − 0 . 5 0 0 . 05 0 . 1 0 . 15 0 . 2 h 2 m q → 0 13/17

Binding energy ǫ = e − m B n B 0 . 005 m B n B 0 − 0 . 005 ǫ − 0 . 01 O ( κ 2 ) O ( κ 4 ) O ( κ 6 ) − 0 . 015 O ( κ 8 ) − 0 . 02 0 . 97 0 . 98 0 . 99 1 1 . 01 1 . 02 3 µ/m B 14/17

Continuum comparison 0 . 008 0 . 007 analytic simulated 0 . 006 T = 10 MeV 0 . 005 m π = 20 GeV B 0 . 004 n B /m 3 0 . 003 0 . 002 0 . 001 0 − 0 . 001 0 . 996 0 . 997 0 . 998 0 . 999 1 1 . 001 µ/m B 15/17

Continuum Equation of State 2.5e-5 2.0e-5 T = 10 MeV m π = 20 GeV 1.5e-5 B n B /m 3 1.0e-5 0.5e-5 0 0 0 . 002 0 . 004 0 . 006 0 . 008 0 . 01 0 . 012 0 . 014 P/m 4 B 16/17

Conclusion

Summary & Outlook Summary • Introduced the effective dimensionally reduced lattice theory • Looked at how a consistent analytic calculation could be carried out • Demonstrated convergence and comparisons with numerics 17/17

Summary & Outlook Outlook • Use the analytic results as a tool to study the characteristics of the effective theory • Find analytic resummation schemes to incorporate long-range effects 17/17

Thank you!

Backup slides

The Effective Lattice Theory Pure gluon contributions y x t Put a line of plaquettes in the time direction 1/4

The Effective Lattice Theory Pure gluon contributions y x t Integrate over all spatial gauge links 1/4

The Effective Lattice Theory Pure gluon contributions L ∗ y L x t What remains is an interaction between Polyakov Loops 1/4

The Effective Lattice Theory Pure gluon contributions Effective Gluon Interactions � L ( x ) L ∗ ( y ) S eff gluon ∼ λ � x , y � L ∗ L y λ x 1/4

The Effective Lattice Theory Pure quark contributions y x t Can produce a closed quark loop with multiple temporal windings 2/4

The Effective Lattice Theory Pure quark contributions y x t Once again integrate out spatial links 2/4

The Effective Lattice Theory Pure quark contributions y W [ L ] x t Producing an interaction between the W objects 2/4

The Effective Lattice Theory Pure quark contributions Effective Quark Interactions � S eff quarks ∼ h 2 W ( x ) W ( y ) � x , y � W W y h 2 x 2/4

The Effective Lattice Theory Mixed contributions Correction to λ • λ → λ ( κ ) • Rescales λ Correction to h 2 • h 2 → h 2 ( β ) • Rescales h 2 3/4

EoS in lattice units ✻✵ ✺✵ ✹✵ a 4 P ✸✵ ✷✵ ✶✵ ✵ ✵ ✶ ✷ ✸ ✹ ✺ ✻ a 3 n B 4/4