a covariant variational approach to yang-mills theory
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A Covariant Variational Approach to Yang-Mills Theory. Institut für Theoretische Physik Eberhard-Karls-Universität Tübingen. A Covariant Variation Principle Overview. Overview. Variational Principle for Effective Action Gaussian Ansatz & Gap Equation Renormalization Numerical Results - PowerPoint PPT PresentationTRANSCRIPT
Markus Quandt
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 11
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
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 22
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
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 33
Effective Action Principle
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 44
Effective Action Principle
A Covariant Variation Principle
Effective Action
• Variation principle for functional probabilityprobability measuremeasure
• Entropy = available space for quantum fluctuations
• Free action
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 55
• 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
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 66
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)
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 77
- 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]
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 88
Gaussian Trial Measure
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 99
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
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 1010
Curvature Approximation
A Covariant Variation Principle
Gaussian Trial Measure
curvaturecurvature
• trial measure
curvaturecurvature gone, reappears in entropyentropy (natural)
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 1111
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
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 1212
Free action
A Covariant Variation Principle
Overview
• classical action: Wick‘s theorem
• relative entropy
involves
lengthy expression
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 1313
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
++=
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 1414
Ghost sector
A Covariant Variation Principle
Overview
-1+=
-1
Use resolvent identity on FP operator
in terms of ghost form factor
rainbow approx.
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 1515
Curvature Equation
A Covariant Variation Principle
Overview
To given loop order
=
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 1616
Renormalization
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 1717
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
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 1818
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
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 1919
Numerical Results
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 2020
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:
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 2121
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)
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 2222
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)
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 2323
Finite Temperature
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 2424
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
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 2525
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
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 2626
Gap Equations
A Covariant Variation Principle
Finite Temperature
induced gluon masses now temperature-dependentmust be finite (no counter term)
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 2727
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
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 2828
Renormalized System at T>0
A Covariant Variation Principle
Overview
• All finite temperature corrections have similar structure
• Iterative solution
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 2929
Preliminary Results
A Covariant Variation Principle
Overview
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 3030
Conclusion
M. Quandt (Uni Tübingen)M. Quandt (Uni Tübingen) A Covariant Variation PrincipleA Covariant Variation Principle Confinement XI Confinement XI 3131
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