Outline Introduction Methods and Results Summary and Outlook Fundamental and Adjoint scalar fields coupled to Yang-Mills theory Veronika Macher, Reinhard Alkofer, Axel Maas 11.3.2010 Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Introduction Methods and Results Summary and Outlook Introduction 1 Confinement The static quark potential System Methods and Results 2 Dyson-Schwinger Equations (DSEs) Powercounting technique Analysis of Colour structure Summary and Outlook 3 Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook Confinement experimental facts: absence of free quarks in Nature ⇒ quark confinement no free gluons ⇒ colour confinement no isolated particles in Nature with non-vanishing colour charge! Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook Confinement experimental facts: absence of free quarks in Nature ⇒ quark confinement no free gluons ⇒ colour confinement no isolated particles in Nature with non-vanishing colour charge! theory: What could be responsible for this behaviour? ⇒ interaction between quarks Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook Wilson loop Wegner-Wilson loop W ( � r , t ): defined as the trace product of gauge variables along a closed oriented contour complex quantity, but real expectation value possible access to quark interaction connection between Wilson loop and the static potential between colour sources: r , t ) = e Vt W ( � Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook Static Potential r , t ) = e Vt W ( � ground state contribution E 0 ( � r ) can be identified as static potential V ( � r ) dominates in limit of large t properties: potential cannot rise faster than linearly V ′′ ( � r ) ≥ 0 ⇒ convexity V ′ ( � r ) ≥ 0 ⇒ monotonically rising Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook Static Potential V � r � r Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook Confinement in SU(N) fundamental: V � r � r linear rise of potential Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook String picture electric flux between two colour sources is squeezed into thin, effectively 1-dim flux tube string breaking at large distances via particle-anti-particle pair creation [J. Greensite ’08] Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook Confinement in SU(N) adjoint: V � r � r string breaking Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook G(2) Why G(2)? strcutural difference compared to SU(N) because of trivial center Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook G(2) Why G(2)? strcutural difference compared to SU(N) because of trivial center properties of G(2): simplest among exceptional Lie groups it’s own universal covering group trivial center adjoint representation: 14-dim fundamental represenatation: 7-dim Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook Confinement in G(2) fundamental and adjoint: V � r � r string breaking Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook Question What is responsible for confinement? possible answer: interaction between quarks ⇒ static potential potential shows representation dependence also gauge group dependent Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook Question What is responsible for confinement? possible answer: interaction between quarks ⇒ static potential potential shows representation dependence also gauge group dependent Can we see these differences also on the level of correlation functions? Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Confinement Introduction The static quark potential Methods and Results System Summary and Outlook Landau gauge Yang-Mills theory including scalar fields Lagrangian: L = 1 µν + 1 � 2 + ¯ µ c b + 2 F a µν F a � ∂ µ A a c a ∂ µ D ab � � D µ, ij Φ ∗ ( D µ, ik Φ k ) j 2 ζ µ i Φ i − h i Φ i ) 2 − m 2 Φ ∗ 4! (Φ ∗ F a µν = ∂ µ A a ν − ∂ ν A a µ − gf abc A b µ A c ν D ab µ = δ ab ∂ µ + gf abc A c µ � t a � ij A a D µ, ij = δ ij ∂ µ − ig 2 µ Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Dyson-Schwinger Equations (DSEs) Introduction Powercounting technique Methods and Results Analysis of Colour structure Summary and Outlook Dyson-Schwinger equations nonperturbative method equations of motions for Green’s functions derived with Mathematica package DoDSE [R. Alkofer, M.Q. Huber, K. Schwenzer ’09] example: ghost propagator DSE Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Dyson-Schwinger Equations (DSEs) Introduction Powercounting technique Methods and Results Analysis of Colour structure Summary and Outlook Dyson-Schwinger equations nonperturbative method equations of motions for Green’s functions derived with Mathematica package DoDSE [R. Alkofer, M.Q. Huber, K. Schwenzer ’09] example: ghost propagator DSE ⇒ no difference in DSEs! Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Dyson-Schwinger Equations (DSEs) Introduction Powercounting technique Methods and Results Analysis of Colour structure Summary and Outlook Infrared Powercounting Technique Z ( p 2 ) propagators parametrized by: ∆ ij ( p ) = P ij p 2 power law ansatz for dressing function: Z IR ( p 2 ) = a · ( p 2 ) α formula for infrared exponent of arbitrary diagram work directly on the level of the IREs Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Dyson-Schwinger Equations (DSEs) Introduction Powercounting technique Methods and Results Analysis of Colour structure Summary and Outlook Infrared Powercounting Technique yields constraints for the infrared exponents of the diagrams important tool for identifying leading contributions in the DSEs Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Dyson-Schwinger Equations (DSEs) Introduction Powercounting technique Methods and Results Analysis of Colour structure Summary and Outlook Infrared Powercounting Technique yields constraints for the infrared exponents of the diagrams important tool for identifying leading contributions in the DSEs ⇒ no difference between fundamental and adjoint scalar fields on this level!! Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
Outline Dyson-Schwinger Equations (DSEs) Introduction Powercounting technique Methods and Results Analysis of Colour structure Summary and Outlook One-loop truncation full scalar propagator DSE: Veronika Macher, Reinhard Alkofer, Axel Maas Fundamental and Adjoint scalar fields coupled to Yang-Mills theo
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