Three-dimensional Yang-Mills theory as a testbed for truncations of Dyson-Schwinger equations
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1 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Three-dmensonal Yang-Mlls theory as a testbed for truncatons of Dyson-Schwnger equatons Marus Q. Huber Insttute of Physcs, Unversty of Graz ACHT05, Lebntz October 8, 05 Marus Q. Huber Unversty of Graz October 8, 05 /9
2 Introducton Dyson-Schwnger equatons d3 Summary & conclusons From Green functons to observables Basc buldng blocs of functonal equatons: n-pont functons Γ... n Effectve acton: generatng functonal of PI Green functons Γ[Φ] n0 n! Φ... Φ n Γ... n The set of all Green functons descrbes the theory completely. Γ δ Γ[Φ] δφ δφ, Γ δ3 Γ[Φ] δφ δφ δφ,... Marus Q. Huber Unversty of Graz October 8, 05 /9
3 Introducton Dyson-Schwnger equatons d3 Summary & conclusons From Green functons to observables Basc buldng blocs of functonal equatons: n-pont functons Γ... n Effectve acton: generatng functonal of PI Green functons Γ[Φ] n0 n! Φ... Φ n Γ... n Green functons observables? The set of all Green functons descrbes the theory completely. Γ δ Γ[Φ] δφ δφ, Γ δ3 Γ[Φ] δφ δφ δφ,... Examples: Bound state equatons masses and propertes of hadrons tals of Hlger, Krassngg, Sanchez-Alepuz, Saul,... Analytc propertes of Green functons confnement (Pseudo-)Order parameters Phases and transtons tal of Renosa Marus Q. Huber Unversty of Graz October 8, 05 /9
4 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Landau gauge Yang-Mlls theory Gluonc sector of quantum chromodynamcs: Yang-Mlls theory L F L gf L gh F µν µ A ν ν A µ g [A µ, A ν ] Landau gauge smplest one for functonal equatons µ A µ 0: L gf ξ ( µa µ ), ξ 0 requres ghost felds: L gh c ( g A ) c Z G D AAA l D A cc D AAAA Marus Q. Huber Unversty of Graz October 8, 05 3/9
5 Introducton Dyson-Schwnger equatons d3 Summary & conclusons The tower of DSEs gluon propagator ghost propagator Marus Q. Huber Unversty of Graz October 8, 05 4/9
6 Introducton Dyson-Schwnger equatons d3 Summary & conclusons The tower of DSEs gluon propagator ghost propagator three-gluon vertex - - ghost-gluon vertex 6 Infntely many equatons. In QCD, every n-pont functon depends on (n )- and possbly (n )-pont functons. Marus Q. Huber Unversty of Graz October 8, 05 4/9
7 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Truncatng the equatons Truncaton Drop quanttes (unmportant?) Model quanttes (good models avalable?) Use fts Ideally: Fnd a truncaton that has no parameters and yelds quanttatve results. Marus Q. Huber Unversty of Graz October 8, 05 5/9
8 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Truncatng the equatons Truncaton Drop quanttes (unmportant?) Model quanttes (good models avalable?) Use fts Ideally: Fnd a truncaton that has no parameters and yelds quanttatve results. Gudes Perturbaton theory Symmetres Lattce Analytc results Practcal obstacle: Manage the system of equatons. Automatzaton tools [Alofer, MQH, Schwenzer 08; Braun, MQH ; MQH, Mtter ; Marus Q. Huber Unversty of Graz October 8, 05 5/9
9 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Truncaton of Yang-Mlls system Coupled system of propagators wth models for three-pont functons: Marus Q. Huber Unversty of Graz October 8, 05 6/9
10 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Truncaton of Yang-Mlls system Coupled system of propagators wth models for three-pont functons: Include three-pont functons dynamcally [Blum, MQH, Mtter, von Smeal 4]: - Marus Q. Huber Unversty of Graz October 8, 05 6/9
11 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Truncaton of Yang-Mlls system Coupled system of propagators wth models for three-pont functons: Include three-pont functons dynamcally [Blum, MQH, Mtter, von Smeal 4]: - Open questons: Four-gluon vertex (n ths truncaton scheme no dependence on hgher n-pont functons, see also [Cyrol, MQH, von Smeal 4]) Two-loop dagrams n propagators (o for three-pont functons) Marus Q. Huber Unversty of Graz October 8, 05 6/9
12 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Truncaton of Yang-Mlls system Coupled system of propagators wth models for three-pont functons: Include three-pont functons dynamcally [Blum, MQH, Mtter, von Smeal 4]: - Open questons: Four-gluon vertex (n ths truncaton scheme no dependence on hgher n-pont functons, see also [Cyrol, MQH, von Smeal 4]) Two-loop dagrams n propagators (o for three-pont functons) Techncal questons: spurous dvergences n gluon propagator, RG resummaton Marus Q. Huber Unversty of Graz October 8, 05 6/9
13 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Yang-Mlls theory n 3 dmensons: Propagator results d 3 Marus Q. Huber Unversty of Graz October 8, 05 7/9
14 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Yang-Mlls theory n 3 dmensons: Propagator results d 3 Hstorcally nterestng because cheaper on the lattce easer to reach the IR, e.g., [Cuccher 99; Cuccher, Mendes, Taurnes 03; Maas 08, 4; Cuccher, Dudal, Mendes, Vanderscel ; Bornyaov, Mtrushn, Rogalyov, 3] Z(p ) G(p ) p[gev] p[gev] [Maas 4] Contnuum results: Coupled propagator DSEs: Maas, Wambach, Grüter, Alofer 04 (R)GZ: Dudal, Gracey, Sorella, Vanderscel, Verschelde 08 YM mass term: Tsser, Wschebor 0, DSEs of PT-BFM: Agular, Bnos, Papavasslou 0 Marus Q. Huber Unversty of Graz October 8, 05 7/9
15 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Yang-Mlls theory n 3 dmensons: Why agan? NB: Numercally not cheaper for functonal equatons. Marus Q. Huber Unversty of Graz October 8, 05 8/9
16 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Yang-Mlls theory n 3 dmensons: Why agan? NB: Numercally not cheaper for functonal equatons. Advantages: UV fnte: no renormalzaton, no anomalous runnng Spurous dvergences easer to handle Many complcatons from d 4 absent! Marus Q. Huber Unversty of Graz October 8, 05 8/9
17 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Subtracton of dvergences of gluon propagator Logarthmc dvergences handled by subtracton at p 0. Quadratc dvergences subtracted, coeffcent C sub. ) Z(p ) : Z Λ (p ) C sub ( p p 0 calculated rght-hand sde C sub can be calculated anlytcally, snce t s a purely perturbatve [MQH, von Smeal 4]. Marus Q. Huber Unversty of Graz October 8, 05 9/9
18 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Subtracton of dvergences of gluon propagator Logarthmc dvergences handled by subtracton at p 0. Quadratc Lnear and logarthmc dvergences subtracted. ) Z(p ) : Z Λ (p ) C sub ( p p 0 calculated rght-hand sde C sub can be calculated anlytcally, snce t s a purely perturbatve [MQH, von Smeal 4]. Marus Q. Huber Unversty of Graz October 8, 05 9/9
19 Introducton Dyson-Schwnger equatons d3 Summary & conclusons A soluton for the propagators Standard truncaton of propagators: -loop, bare ghost-gluon vertex, three-gluon vertex model Z(p ) prelmnary G(p ) 0.0 p[a.u.] prelmnary p[a.u.] Form of spurous dvergences (analytc): C sub a Λ b ln Λ Marus Q. Huber Unversty of Graz October 8, 05 0/9
20 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Three-gluon vertex model D AAA (p,p,p ) D A3 (x, y, z) p p L G(p ) 3 L 6 (L x)(l y)(l z) p x y z p[gev] Not possble to rase the gluon bump further by playng wth the vertex models! - Lattce: [Cuccher, Maas, Mendes 08] Marus Q. Huber Unversty of Graz October 8, 05 /9
21 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Two-loop dagrams Squnt: Sunset: 4d: [Bloch 03; Mader, Alofer ; Meyers, Swanson 4] Man obstacle: spurous dvergences Spurous dvergences Leadng order correctons to subtracton coeffcent: g 4 log(λ) Determned by a ft: Very small. Stll large effect. Marus Q. Huber Unversty of Graz October 8, 05 /9
22 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Two-loop dagrams Squnt: Sunset: 4d: [Bloch 03; Mader, Alofer ; Meyers, Swanson 4] Man obstacle: spurous dvergences Spurous dvergences Leadng order correctons to subtracton coeffcent: g 4 log(λ) Determned by a ft: Very small. Stll large effect. New vertex enters: Four-gluon vertex Marus Q. Huber Unversty of Graz October 8, 05 /9
23 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Four-gluon vertex 4d: Soluton of four-gluon DSE (full momentum dependence) 3,5 D 4g, confg. A D 4g, confg. B D 4g, confg. C D 4g, confg. C, ft,5 0, p [GeV ] [Cyrol, MQH, von Smeal 4] Smlar results by [Bnos, Ibáñez, Papavasslou 4] Marus Q. Huber Unversty of Graz October 8, 05 3/9
24 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Four-gluon vertex 4d: Soluton of four-gluon DSE (full momentum dependence) 3d: 3,5,5 0,5 0 D 4g, confg. A D 4g, confg. B D 4g, confg. C D 4g, confg. C, ft Smple ansatz wth suppresson n md-momentum regme: Z(p ) D AAAA (x) a b x b p [GeV ] [Cyrol, MQH, von Smeal 4] Smlar results by [Bnos, Ibáñez, Papavasslou 4] p [GeV ] a 0.6, b.3 GeV Marus Q. Huber Unversty of Graz October 8, 05 3/9
25 Introducton Dyson-Schwnger equatons d3 Summary & conclusons A soluton for the propagators wth two-loop dagrams Improved truncaton of propagators: - and -loops, bare ghost-gluon vertex, three-gluon vertex model Z(p ).0 G(p ) prelmnary prelmnary 0.0 p[a.u.] p[a.u.] Two-loop dagrams essental to allow rasng the gluon bump. Marus Q. Huber Unversty of Graz October 8, 05 4/9
26 Introducton Dyson-Schwnger equatons d3 Summary & conclusons A soluton for the propagators wth two-loop dagrams Improved truncaton of propagators: - and -loops, bare ghost-gluon vertex, three-gluon vertex model Z(p ).0 G(p ) prelmnary prelmnary 0.0 p[a.u.] p[a.u.] Two-loop dagrams essental to allow rasng the gluon bump. Next step: nclude vertces dynamcally. Marus Q. Huber Unversty of Graz October 8, 05 4/9
27 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Ghost-gluon vertex A(p ;p,p ) prelmnary p[gev] A(p ;p,p ) prelmnary p[gev] Lattce: [Cuccher, Maas, Mendes 08] Full momentum dependence calculated! Marus Q. Huber Unversty of Graz October 8, 05 5/9
28 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Three-gluon vertex D AAA (p,p,p ) D AAA (0,p,p ) 0 3 p[gev] prelmnary Lattce: [Cuccher, Maas, Mendes 08] p[gev] prelmnary Full momentum dependence calculated! Note: IR dvergent! Cf. [Peláez, Tsser, Wschebor 3] Marus Q. Huber Unversty of Graz October 8, 05 6/9
29 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Propagators Z(p ) Z/p (p ) ry na lm pre.0 l pre y nar m p[gev] p[gev] G(p ) y nar m Quanttatve mprovement! Small gaps left. l pre Lattce: p[gev] [Maas 4] Marus Q. Huber Unversty of Graz October 8, 05 7/9
30 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Overvew d 3 -pont one-loop two-loop 3-pont 4-gluon est. max. devaton Z(p ) X model 38% p[gev] Z(p ) X X model model 3% p[gev] Z(p ) X Marus Q. Huber X X model Unversty of Graz 8% October 8, p[gev] 8/9
31 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Summary and conclusons 3d Yang-Mlls theory as a testbed for truncatons of DSEs Varous mprovements (two-loop dagrams, dynamc three-pont functons) lead to results close to lattce results. Mssng pece: four-gluon vertex Implcatons An example of a self-consstent, self-contaned truncaton of a set of DSEs wth quanttatve results??? What about 4d? Parameter-less truncaton possble!? Marus Q. Huber Unversty of Graz October 8, 05 9/9
32 Introducton Dyson-Schwnger equatons d3 Summary & conclusons Summary and conclusons 3d Yang-Mlls theory as a testbed for truncatons of DSEs Varous mprovements (two-loop dagrams, dynamc three-pont functons) lead to results close to lattce results. Mssng pece: four-gluon vertex Implcatons An example of a self-consstent, self-contaned truncaton of a set of DSEs wth quanttatve results??? What about 4d? Parameter-less truncaton possble!? Than you for your attenton. Marus Q. Huber Unversty of Graz October 8, 05 9/9
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