A non-perturbative study of the correlation functions of three-dimensional Yang-Mills theory
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1 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons A non-perturbatve study of the correlaton functons of three-dmensonal Yang-Mlls theory arxv: Marus Q. Huber Insttute of Physcs, Unversty of Graz 53. Internatonale Unverstätswochen für Theoretsche Phys, Schladmng Feb. 23, 2016 Marus Q. Huber Unversty of Graz Feb. 23, /17
2 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons From Green functons to observables Basc buldng blocs of functonal equatons: n-pont functons Γ 1... n Effectve acton: generatng functonal of 1PI Green functons Γ[Φ] = n=0 1 n! Φ 1... Φ n Γ 1... n The set of all Green functons descrbes the theory completely. Γ = δ2 Γ[Φ] δφ δφ, Φ=0 Γ = δ3 Γ[Φ] δφ δφ δφ,... Φ=0 Marus Q. Huber Unversty of Graz Feb. 23, /17
3 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons From Green functons to observables Basc buldng blocs of functonal equatons: n-pont functons Γ 1... n Effectve acton: generatng functonal of 1PI Green functons Γ[Φ] = n=0 1 n! Φ 1... Φ n Γ 1... n Green functons observables? The set of all Green functons descrbes the theory completely. Γ = δ2 Γ[Φ] δφ δφ, Φ=0 Γ = δ3 Γ[Φ] δφ δφ δφ,... Φ=0 Examples: Bound state equatons masses and propertes of hadrons (Pseudo-)Order parameters Phases and transtons Marus Q. Huber Unversty of Graz Feb. 23, /17
4 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Landau gauge Yang-Mlls theory Gluonc sector of quantum chromodynamcs: Yang-Mlls theory L = 1 2 F 2 + 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 = 1 2ξ ( µa µ ) 2, ξ 0 requres ghost felds: L gh = c ( + g A ) c Z G 1 1 D AAA l D A cc D AAAA Marus Q. Huber Unversty of Graz Feb. 23, /17
5 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons The tower of DSEs -1 + = gluon propagator -1 = ghost propagator Marus Q. Huber Unversty of Graz Feb. 23, /17
6 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons The tower of DSEs -1 + = gluon propagator -1 = ghost propagator = three-gluon vertex - = ghost-gluon vertex Infntely many equatons. In QCD, every n-pont functon depends on (n + 1)- and possbly (n + 2)-pont functons. Marus Q. Huber Unversty of Graz Feb. 23, /17
7 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Truncatng the equatons Truncaton Drop quanttes (unmportant?) Model quanttes (good models avalable? true or effectve?) Use fts Ideally: Fnd a truncaton that has (I) no parameters and yelds (II) quanttatve results. Marus Q. Huber Unversty of Graz Feb. 23, /17
8 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Truncatng the equatons Truncaton Drop quanttes (unmportant?) Model quanttes (good models avalable? true or effectve?) Use fts Ideally: Fnd a truncaton that has (I) no parameters and yelds (II) quanttatve results. Gudes Perturbaton theory Symmetres Lattce Analytc results Practcal obstacle: Manage the system of equatons. Automatzaton tools [Alofer, MQH, Schwenzer 08; Braun, MQH 11; MQH, Mtter 11; Marus Q. Huber Unversty of Graz Feb. 23, /17
9 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Truncaton of Yang-Mlls system Neglect all non-prmtvely dvergent Green functons. Marus Q. Huber Unversty of Graz Feb. 23, /17
10 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Truncaton of Yang-Mlls system Neglect all non-prmtvely dvergent Green functons. Full propagator equatons (two-loop dagrams!): -1 + = = -1 - Marus Q. Huber Unversty of Graz Feb. 23, /17
11 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Truncaton of Yang-Mlls system Neglect all non-prmtvely dvergent Green functons. Full propagator equatons (two-loop dagrams!): -1 + = = -1 - Truncated three-pont functons: Truncated four-gluon vertex: + = + = + + l = + l l l + 3 l l l Marus Q. Huber Unversty of Graz Feb. 23, /17
12 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Truncaton of Yang-Mlls system Neglect all non-prmtvely dvergent Green functons. Full propagator equatons (two-loop dagrams!): -1 + = = -1 - Truncated three-pont functons: Truncated four-gluon vertex: + = + = + + l = + l l l + 3 l l l Techncal questons: spurous dvergences n gluon propagator, RG resummaton Marus Q. Huber Unversty of Graz Feb. 23, /17
13 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Yang-Mlls theory n 3 dmensons d = 3 Marus Q. Huber Unversty of Graz Feb. 23, /17
14 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Yang-Mlls theory n 3 dmensons d = 3 Hstorcally nterestng because cheaper on the lattce easer to reach the IR, e.g., [Cuccher 99; Cuccher, Mendes, Taurnes 03; Cuccher, Maas, Mendes, 08; Maas 08, 14; Maas, Pawlows, Spelmann, Sternbec, von Smeal 09; Cuccher, Dudal, Mendes, Vanderscel 11; Bornyaov, Mtrushn, Rogalyov 11, 13; Cuccher, Dudal, Mendes, Vanserscel 16] Z(p 2 ) p[gev] [Maas 14] G(p 2 ) p[gev] [Maas 14] Contnuum results: Coupled propagator DSEs: [Maas, Wambach, Grüter, Alofer 04] (R)GZ: [Dudal, Gracey, Sorella, Vanderscel, Verschelde 08] YM + mass term: [Tsser, Wschebor 10, 11] DSEs of PT-BFM: [Agular, Bnos, Papavasslou 10] Marus Q. Huber Unversty of Graz Feb. 23, /17
15 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Yang-Mlls theory n 3 dmensons: Why agan? NB: Numercally not cheaper for functonal equatons of 2- and 3-pont functons. Marus Q. Huber Unversty of Graz Feb. 23, /17
16 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Yang-Mlls theory n 3 dmensons: Why agan? NB: Numercally not cheaper for functonal equatons of 2- and 3-pont functons. 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 Feb. 23, /17
17 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Subtracton of dvergences of gluon propagator (d=4) 1 Logarthmc dvergences handled by subtracton at p 0. 2 Quadratc dvergences subtracted, coeffcent C sub. Z(p 2 ) 1 := Z Λ (p 2 ) 1 ( 1 C sub (Λ) p 2 1 ) p0 2 calculated rght-hand sde Marus Q. Huber Unversty of Graz Feb. 23, /17
18 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Subtracton of dvergences of gluon propagator (d=4) 1 Logarthmc dvergences handled by subtracton at p 0. 2 Quadratc dvergences subtracted, coeffcent C sub. Z(p 2 ) 1 := Z Λ (p 2 ) 1 ( 1 C sub (Λ) p 2 1 ) p0 2 calculated rght-hand sde One-loop dagrams wth model vertces: C sub can be calculated anlytcally, snce t s a purely perturbatve [MQH, von Smeal 14]. Dynamc vertces? Two-loop dagrams? Marus Q. Huber Unversty of Graz Feb. 23, /17
19 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Subtracton of dvergences of gluon propagator (d=3) 1 Logarthmc dvergences handled by subtracton at p 0. 2 Quadratc Lnear and logarthmc dvergences subtracted. Z(p 2 ) 1 := Z Λ (p 2 ) 1 ( 1 C sub (Λ) p 2 1 ) p0 2 calculated rght-hand sde One-loop dagrams wth model vertces: C sub can be calculated anlytcally, snce t s a purely perturbatve [MQH, von Smeal 14]. Dynamc vertces? Two-loop dagrams? Marus Q. Huber Unversty of Graz Feb. 23, /17
20 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Importance of spurous dvergences Smplfcaton n d = 3: C sub = a Λ + b ln Λ ft (wors for numerc vertces and two-loop dagrams) Marus Q. Huber Unversty of Graz Feb. 23, /17
21 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Importance of spurous dvergences Smplfcaton n d = 3: C sub = a Λ + b ln Λ ft (wors for numerc vertces and two-loop dagrams) Z(p 2 ) xCsub xCsub p/g2 Small devatons large effect. Marus Q. Huber Unversty of Graz Feb. 23, /17
22 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Results: Propagators Z(p 2 ) p[gev] G(p 2 ) p[gev] Z(p 2 )/p p[gev] Bands from uncertanty n settng the physcal scale. [lattce: Maas 14] 1 Marus Q. Huber Unversty of Graz Feb. 23, /17
23 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Results: Three-pont functons Dressngs: A(p 2 ;p 2,p 2 ) Maxmum poston shfted. Bump heght o. 0.9 p[gev] [lattce: Maas, unpublshed] D AAA (p 2,p 2,p 2 ) p[gev] Good agreement wth lattce data. Lnear IR dvergence [lattce: Cuccher, Maas, Mendes 08] Marus Q. Huber Unversty of Graz Feb. 23, /17
24 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Results: Three-pont functons Rato lattce/dse results: A latt /A DSE (p 2 ;p 2,p 2 ) p[gev] Maxmum poston shfted. Bump heght o. [lattce: Maas, unpublshed] 2.0 D AAA,latt /D AAA,DSE (p 2,p 2,p 2 ) p[gev] Good agreement wth lattce data. Lnear IR dvergence. 0.0 [lattce: Cuccher, Maas, Mendes 08] Marus Q. Huber Unversty of Graz Feb. 23, /17
25 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Cancellatons n gluonc vertces Three-gluon vertex: D AAA (p 2,p 2,p 2 ) ghost tr. gluon tr. stat. swordf. dyn.swordf p[gev] Indvdual contrbutons large. Sum s small Four-gluon vertex: D AAAA (p C) ghost box gluon box statc trangle swordfsh dynamc trangle p[gev] -0.5 Marus Q. Huber Unversty of Graz Feb. 23, /17
26 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Cancellatons n gluonc vertces Three-gluon vertex: D AAA (p 2,p 2,p 2 ) ghost tr. gluon tr. stat. swordf. dyn.swordf p[gev] Indvdual contrbutons large. Sum s small Four-gluon vertex: D AAAA (p C) ghost box gluon box statc trangle swordfsh dynamc trangle Hgher contrbutons: 1 Small each or 2 cancellatons agan? p[gev] -0.5 Marus Q. Huber Unversty of Graz Feb. 23, /17
27 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Non-perturbatve gauge fxng Grbov copes: Gauge equvalent confguratons that fulfll the Landau gauge condton A = 0. Up to here the mnmal Landau gauge was shown for lattce data. Marus Q. Huber Unversty of Graz Feb. 23, /17
28 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Non-perturbatve gauge fxng Grbov copes: Gauge equvalent confguratons that fulfll the Landau gauge condton A = 0. Up to here the mnmal Landau gauge was shown for lattce data. Another possblty: Absolute Landau gauge (global mnmum of gauge fxng functonal) Dfferent solutons on the lattce, e.g. [Maas 09, 11; Cuccher 97; Bogolubsy et al. 05; Sternbec, Müller-Preusser 12]. [Maas 13] NB: Dfferent solutons also from functonal equatons [Fscher, Maas, Pawlows 08]. Marus Q. Huber Unversty of Graz Feb. 23, /17
29 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Non-perturbatve gauge fxng Grbov copes: Gauge equvalent confguratons that fulfll the Landau gauge condton A = 0. Up to here the mnmal Landau gauge was shown for lattce data. Another possblty: Absolute Landau gauge (global mnmum of gauge fxng functonal) Dfferent solutons on the lattce, e.g. [Maas 09, 11; Cuccher 97; Bogolubsy et al. 05; Sternbec, Müller-Preusser 12]. [Maas 13] Z latt (p 2 )/Z DSE (p 2 ) p[gev] [lattce: Bornyaov, Mtrushn, Rogalyov 13] NB: Dfferent solutons also from functonal equatons [Fscher, Maas, Pawlows 08]. Marus Q. Huber Unversty of Graz Feb. 23, /17
30 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Soluton from the 3PI effectve acton Dfferent set of functonal equatons: equatons of moton from 3PI effectve acton (at three-loop level) Marus Q. Huber Unversty of Graz Feb. 23, /17
31 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Soluton from the 3PI effectve acton Dfferent set of functonal equatons: equatons of moton from 3PI effectve acton (at three-loop level) Z(p 2 ) D AAA (p 2,p 2,p 2 ) p[gev] 0.5 c-dse 3PI c-dse 3PI p[gev] -2.0 Very smlar results. For yet another set of functonal equatons (functonal RG for d = 4), see tal by Mtter and poster by Cyrol. Marus Q. Huber Unversty of Graz Feb. 23, /17
32 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Comparson d = 3 and d = 4 Two-loop dagrams mportant n propagators. [Blum, MQH, Mtter, von Smeal 14; Meyers, Swanson 14] Two-loop dagrams not mportant n three-gluon vertex. [Blum, MQH, Mtter, von Smeal 14; Echmann, Wllams, Alofer, Vunovc 14] Vertces devate only mldly from tree-level above 1 GeV. [Blum, MQH, Mtter, von Smeal 14; Echmann, Wllams, Alofer, Vunovc 14; Bnos, Ibanez, Papavasslou 14; Cyrol, MQH, von Smeal 14] RG mprovement rrelevant n d = 3. Role n d = 4? [Echmann, Wllams, Alofer, Vunovc 14] Marus Q. Huber Unversty of Graz Feb. 23, /17
33 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Summary and conclusons Test truncaton effects n d = 3, where spurous dvergences and RG resummaton are understood: Used a self-contaned truncaton no model parameters. Truncaton stable under all tested varatons: comparson wth 3PI changng the four-gluon vertex dfferent DSEs for the ghost-gluon vertex Drect relaton between dfferent solutons n contnuum and on the lattce to be understood. Marus Q. Huber Unversty of Graz Feb. 23, /17
34 Introducton Dyson-Schwnger equatons YM n d=3 Summary & conclusons Summary and conclusons Test truncaton effects n d = 3, where spurous dvergences and RG resummaton are understood: Used a self-contaned truncaton no model parameters. Truncaton stable under all tested varatons: comparson wth 3PI changng the four-gluon vertex dfferent DSEs for the ghost-gluon vertex Drect relaton between dfferent solutons n contnuum and on the lattce to be understood. Than you for your attenton. Marus Q. Huber Unversty of Graz Feb. 23, /17
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