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

Analytic computations of an effective lattice theory for heavy quarks

Lattice Conference 2015 - July 16th

Jonas R. Glesaaen

Mathias Neuman, Owe Philipsen

1

The Effective Theory

2

Results

3

Conclusion

Heavy QCD Phase Diagram

µB [GeV] T [MeV] 1 200 ≲ 10

2/17

Heavy QCD Phase Diagram

µB [GeV] T [MeV] 1 200 ≲ 10

Liquid Gas Phase Transition

2/17

Advantages of the Effective Theory

• Dimensionally reduced theory
• 4D → 3D
• Uµ(x) → L (x)
• Very mild sign problem, most gauge fields integrated

analytically

• Want to study the very dense limit, liquid gas

transition

3/17

The Effective Theory

The Effective Lattice Theory

• Integrate out all spatial gauge links

Z =

• DUµ exp
• −Saction
• =
• DU0 exp
• −Seffective action
• Effective Theory

Using:

• The strong coupling expansion
• The hopping parameter expansion

4/17

Z =

x

dL (x) exp

• −Seff action
• (†)

Effective Theory

• Previous Talk: Monte Carlo simulations of (†)
• Current Talk: Analytic calculation of Z

5/17

The Effective Theory Action

Seff action = S0

• L
• + SI
• L
• Where SI
• L
• is made up of interactions at varying

distances SI

• L
• =
• terms
• dof

vi(1, 2, ..., ni)φ1

• L
• φ2
• L
• · · · φni
• L
• 6/17

The Effective Theory Action

Seff action = S0

• L
• + SI
• L
• Where SI
• L
• is made up of interactions at varying

distances SI

• L
• =
• terms
• dof

vi(1, 2, ..., ni)φ1

• L
• φ2
• L
• · · · φni
• L
• Can be represented

with connected graphs

6/17

The Effective Theory Action

SI

• L
• =
• terms
• dof

vi(1, 2, ..., ni)φ1

• L
• φ2
• L
• · · · φni
• L
• In our theory:
• vi(1, 2, ...ni) →
• λi, hi
• × geometry
• φi →
• Li, L ∗

i , Wi

• 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

✲✵✳✺ ✵ ✵✳✺ ✶ ✶✳✺ ✷ ✷✳✺ ✸ ✵ ✵✳✵✺ ✵✳✶ ✵✳✶✺ ✵✳✷ a3nB h2 ▲✐♥❦❡❞ ❈❧✉st❡r O(κ2) O(κ4) O(κ6) O(κ8)

mq → 0

h1 = κNt e Nt µ = 0.8

11/17

Convergence

−0.5 0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 Linked Cluster a3nB h2 O(κ2) O(κ4) O(κ6) O(κ8) −0.5 0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 Linked Cluster Simulations a3nB h2 O(κ6)

mq → 0

h1 = κNt e Nt µ = 0.8

11/17

Convergence

−0.5 0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 Linked Cluster a3nB h2 O(κ2) O(κ4) O(κ6) O(κ8) −0.5 0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 Linked Cluster Simulations a3nB h2 O(κ8)

mq → 0

h1 = κNt e Nt µ = 0.8

11/17

Effect of the resummations

✲✵✳✺ ✵ ✵✳✺ ✶ ✶✳✺ ✷ ✷✳✺ ✸ ✵ ✵✳✵✺ ✵✳✶ ✵✳✶✺ ✵✳✷ a3nB h2 ▲✐♥❦❡❞ ❈❧✉st❡r O(κ2) O(κ4) O(κ6) O(κ8) ✲✵✳✺ ✵ ✵✳✺ ✶ ✶✳✺ ✷ ✷✳✺ ✸ ✵ ✵✳✵✺ ✵✳✶ ✵✳✶✺ ✵✳✷ a3nB h2 ▲✐♥❦❡❞ ❈❧✉st❡r ❘❡s✉♠♠❡❞ O(κ2) O(κ4) O(κ6) O(κ8)

mq → 0

h1 = κNt e Nt µ = 0.8

12/17

Effect of the resummations

✲✵✳✺ ✵ ✵✳✺ ✶ ✶✳✺ ✷ ✷✳✺ ✸ ✵ ✵✳✵✺ ✵✳✶ ✵✳✶✺ ✵✳✷ a3nB h2 ❘❡s✉♠♠❡❞ O(κ2) O(κ4) O(κ6) O(κ8)

mq → 0

h1 = κNt e Nt µ = 0.8

12/17

Effect of the resummations

−0.5 0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 Linked Cluster a3nB h2 O(κ2) O(κ4) O(κ6) O(κ8)

mq → 0

h1 = κNt e Nt µ = 1.0

13/17

Effect of the resummations

−0.5 0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 Linked Cluster a3nB h2 O(κ2) O(κ4) O(κ6) O(κ8) −0.5 0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 Linked Cluster Resummed a3nB h2 O(κ2) O(κ4) O(κ6) O(κ8)

mq → 0

h1 = κNt e Nt µ = 1.0

13/17

Effect of the resummations

−0.5 0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 Resummed a3nB h2 O(κ2) O(κ4) O(κ6) O(κ8)

mq → 0

h1 = κNt e Nt µ = 1.0

13/17

Binding energy

−0.02 −0.015 −0.01 −0.005 0.005 0.97 0.98 0.99 1 1.01 1.02 ǫ 3µ/mB O(κ2) O(κ4) O(κ6) O(κ8) ǫ = e − mBnB mBnB

14/17

Continuum comparison

−0.001 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.996 0.997 0.998 0.999 1 1.001 T = 10 MeV mπ = 20 GeV nB/m3

B

µ/mB analytic simulated

15/17

Continuum Equation of State

0.5e-5 1.0e-5 1.5e-5 2.0e-5 2.5e-5 0.002 0.004 0.006 0.008 0.01 0.012 0.014 T = 10 MeV mπ = 20 GeV nB/m3

B

P/m4

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

t y x

Put a line of plaquettes in the time direction

1/4

The Effective Lattice Theory

Pure gluon contributions

t y x

Integrate over all spatial gauge links

1/4

The Effective Lattice Theory

Pure gluon contributions

t y x

What remains is an interaction between Polyakov Loops

L L ∗

1/4

The Effective Lattice Theory

Pure gluon contributions

Seff gluon ∼ λ

• x,y

L (x)L ∗(y) Effective Gluon Interactions

λ L L ∗ x y

1/4

The Effective Lattice Theory

Pure quark contributions

t y x

Can produce a closed quark loop with multiple temporal windings

2/4

The Effective Lattice Theory

Pure quark contributions

t y x

Once again integrate out spatial links

2/4

The Effective Lattice Theory

Pure quark contributions

t y x

Producing an interaction between the W objects

W [L ]

2/4

The Effective Lattice Theory

Pure quark contributions

Seff quarks ∼ h2

• x,y

W (x)W (y) Effective Quark Interactions

h2 W W x y

2/4

The Effective Lattice Theory

Mixed contributions

• Rescales λ
• λ → λ(κ)

Correction to λ

• Rescales h2
• h2 → h2(β)

Correction to h2

3/4

EoS in lattice units

✵ ✶✵ ✷✵ ✸✵ ✹✵ ✺✵ ✻✵ ✵ ✶ ✷ ✸ ✹ ✺ ✻ a4P a3nB

4/4