a covariant variational approach to yang-mills theory

Markus Quandt

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Quark Confinement and the Hadron SpectrumSt. Petersburg

September 9,2014

A Covariant Variational Approach to Yang-Mills Theory

Institut für Theoretische PhysikEberhard-Karls-Universität Tübingen

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Overview

1. Variational Principle for Effective Action

2. Gaussian Ansatz & Gap Equation

3. Renormalization

4. Numerical Results

5. Extension to finite Temperature

6. Conclusion and Outlook

A Covariant Variation Principle

Overview

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Effective Action Principle

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Effective Action Principle

A Covariant Variation Principle

Effective Action

• Variation principle for functional probabilityprobability measuremeasure

• Entropy = available space for quantum fluctuations

• Free action

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• Variation principle I

• Variation principle II

Gibbs measureGibbs measure

Schwinger functionsSchwinger functions

(Quantum effective action)(Quantum effective action)

NoteNote: Usually: Usually and proper functionsand proper functions

A Covariant Variation Principle

Effective Action

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1. Take (restricted) measure space with parameters K

2. Minimize [gap equationgap equation]

3. Quantum effective action

4. Proper functions = derivatives of

Modification for gf. Yang-Mills Theory

A Covariant Variation PrincipleA

Covariant variational principle

Effective Action

replace entropy by relative entropyrelative entropy

(available quantum fluctuations in curved field space)

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- non-perturbative corrections to arbitrary proper functions

- closed system of integral equations

- renormalization through standard counter terms

- can discriminate competing solutions

- can be optimized to many physical questions (choice of )

Comparision

A Covariant Variation PrincipleA

covariant variational principle

Effective Action

• Pro

• Cons

- vertex corrections not necessarily reliable

- Confinement ?

- Strong phase transitions ?

- BRST ? [similar to Coulomb gaugeCoulomb gauge]

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Gaussian Trial Measure

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A Covariant Variation Principle

Gaussian Trial Measure

• Physical picture in Landau gauge

Gaussian trial measure for effective (constitutent) gluon

- variational parameters

- Lorentz structure in Landau gauge:

- UV : gluons weakly interacting- IR : configurations near Gribov horizon dominant

self-interaction in such configs sub-dominant

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Curvature Approximation

A Covariant Variation Principle

Gaussian Trial Measure

curvaturecurvature

• trial measure

curvaturecurvature gone, reappears in entropyentropy (natural)

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Remark

A Covariant Variation Principle

Gaussian Trial Measure

• put classical field if interested in propagator

• alternative: effective action for gluon propagator

gap equationgap equation gives best gluon propagator gives best gluon propagatorthat is available within the ansatzthat is available within the ansatz

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Free action

A Covariant Variation Principle

Overview

• classical action: Wick‘s theorem

• relative entropy

involves

lengthy expression

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Gap Equation

A Covariant Variation Principle

Overview

Use O(4) symmetry to perform angular integrations

• constituent gluon mass (4-gluon vertex gap eq.)

• constutent gluon self energy (ghost contribution curvature)

graphgraph

++=

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Ghost sector

A Covariant Variation Principle

Overview

-1+=

-1

Use resolvent identity on FP operator

in terms of ghost form factor

rainbow approx.

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Curvature Equation

A Covariant Variation Principle

Overview

To given loop order

=

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Renormalization

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Counterterms

A Covariant Variation Principle

Renormalization

gluon fieldgluon mass ghost field

Added to exponent of trial measure, notnot

Renormalization conditions Renormalization conditions (3 scales )

• fix

• fix

• fix

scaling/decoupling

constitutent mass at

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A Covariant Variation Principle

Renormalization

• All countertems have definite values, e.g.

• Final system

curvature: quadratic + subleading log. divergence subtracted !

Important for finite-T calculation

similarly

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Numerical Results

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Gluon prop. IR diverging, constituent mass

• decoupling solution:

Gluon prop. IR finite, constituent mass dominant

Gluon prop IR vanishing, constituent mass irrelevant

Infrared Analysis

A Covariant Variation Principle

Numerical Results

Assume power law in the deep IR

IR exponents

Non-renormalization of

ghost-gluon vertex

• scaling solution:

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Scaling Solution

A Covariant Variation Principle

Numerical Results

Lattice data from Bogolubsky et al., Phys. Lett. B676 69 (2009)

IR exponents:

sum rule violation:

lattice data:

H. Reinhardt, J. Heffner, M.Q., Phys. Rev. D89 065037 (2014)

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Decoupling Solution

A Covariant Variation Principle

Numerical Results

Lattice data from Bogolubsky et al., Phys. Lett. B676 69 (2009)

lattice data: sum rule:

H. Reinhardt, J. Heffner, M.Q., Phys. Rev. D89 065037 (2014)

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Finite Temperature

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Extension to Finite Temperature

A Covariant Variation Principle

Finite Temperature

• imaginary time formalismimaginary time formalism

compactify euclidean time

periodic b.c. for gluons (up to center twists)

periodic b.c. for ghosts (even though fermions)

Extension to T>0 straightforward

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A Covariant Variation PrincipleA

covariant variational principle

Finite Temperature

• Lorentz structure of propagatorLorentz structure of propagator

heat bath singles out restframe (1,0,0,0) breaks Lorentz invariance

two different 4-transversal projectors

Same Lorentz structure for Gaussian kernel and curvature

3-transversal

3-longitudinal

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Gap Equations

A Covariant Variation Principle

Finite Temperature

induced gluon masses now temperature-dependentmust be finite (no counter term)

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Renormalization

A Covariant Variation Principle

Finite Temperature

• counterterms must be fixed at T=0

• example: ghost equation

counter term

finite T correction finite T correction vanishes at T=0

Ren. T=0 profile

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Renormalized System at T>0

A Covariant Variation Principle

Overview

• All finite temperature corrections have similar structure

• Iterative solution

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Preliminary Results

A Covariant Variation Principle

Overview

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Conclusion

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A Covariant Variation Principle

Overview

• Variational Principle for Effective Action + Gaussian Ansatz

- yields optimal truncation

- conventionally renormalizable

- very good agreement with lattice data

- Easily extensible to finite temperatures

- BRST? Confinement?

Conclusion and Outlook

• Best applied where effective 1-gluon exchange is relevant

- inclusion of fermions

- chemical potential