On the Landau gauge three-gluon vertex

Transcription

1 On the Landau gauge three-gluon vertex M. Vujinovic, G. Eichmann, R. Williams, R. Alkofer Karl Franzens University, Graz PhD Seminar talk Graz, Austria, M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

2 Aim of investigation 3g vertex three-point Greens function, gluon self-interaction. Aim: Study of 3g vertex in functional approaches. Motivation: 1 Driving term in quark-gluon vertex Dyson-Schwinger equation (DSE). 2 Through quark-gluon vertex, enters bound state calculations. 3 Closing truncated system of DSEs for Yang-Mills (YM) system. For later convenience: = P 3, Σ = P 1 P 2. M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

3 Perturbative analysis One-loop calculations 1 Arbitrary gauge, symmetric momentum configuration. W. Celmaster, R. Gonsalves, PRD 20, 1420 (1979) 2 Arbitrary momenta, gauge and dimension. A. I. Davydychev, P. Osland, O. V. Tarasov, PRD 54, 4087 (1996) Two-loop calculation. A. I. Davydychev, P. Osland, O. V. Tarasov, PRD 58, (1998) One-loop YM contributions to 3g vertex self-energy in perturbation theory M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

4 Propagators of YM theory Dyson-Schwinger equations for ghost and gluon propagators e. g. M. Q. Huber, L. von Smekal, JHEP 04, 149 (2013), refs. therein M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

5 Vertices of YM theory Solving Yang-Mills propagator DSEs requires knowledge of: 1 Ghost-gluon vertex. 2 Three-gluon vertex. 3 Four-gluon vertex. Different approaches to vertices: 1 Modelling. 2 Solve vertex DSE with fixed ghost/gluon input. 3 Self-consistent evaluation of vertices and propagators. M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

6 Ghost-gluon vertex In Landau gauge, ghost-gluon vertex free of UV divergences. J. C. Taylor, Nucl. Phys. B 33, 436 (1971) One tensor structure relevant for this vertex in Landau gauge. For most purposes, bare ghost-gluon vertex sufficient: Γ ν (k, p, q) = iq ν (1) W. Schleifenbaum, A. Mass, J. Wambach, R. Alkofer, PRD 72, (2005) Self-consistent calculation of Γ ν very small difference compared to bare-vertex Ansätze. M. Q. Huber, L. von Smekal, JHEP 04, 149 (2013) M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

7 Three gluon vertex Many models for Γ µνρ available in literature. Model building based on STIs, correct UV for gluon, elimination of divergences... Latest model Bose-symmetric, correct anomalous dimensions, in agreement with lattice. M. Q. Huber, L. von Smekal, JHEP 04, 149 (2013) So far no self-consistent evaluation of Γ µνρ. M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

8 Influence of 3g vertex models M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

9 Infrared Yang-Mills: Scaling solutions Scaling solve infinite tower of GFs exactly in IR. GF with 2n ghost and m gluon legs: Γ (n,m) (p 2 ) (n m)κ (2) κ - IR scaling exponent, κ [0, 1] R. Alkofer, C. S. Fischer, F. Llanes-Estrada, PLB 611, 279 (2005) Ghost: G (p 2 ) κ, Gluon: Z (p 2 ) 2κ Ghost-gluon: Γ gh gl (p 2 ) 0, Three-gluon: Γ 3g (p 2 ) 3κ (3) Convenient power-counting scheme. In agreement with Kugo-Ojima and Gribov-Zwanziger confinement scenarios. M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

10 YM propagators, scaling 100 G Z e p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

11 Infrared Yang-Mills: Decoupling solutions IR finite ghost decoupling solutions. 1 G(0) 0 (4) Continuum methods see both scaling and decoupling solutions. In 4D lattice sees family of decoupling solutions, no scaling solution. A. Sternbeck, M. Mueller-Preussker, PLB 726, 396 (2013) Continuum YM sector investigations: include both types of solution. M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

12 YM propagators, Decoupling 1 G Z p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

13 Three-gluon vertex DSE e.g. M. Q. Huber, A. Maas, L. von Smekal, JHEP 1211, 035 (2012), refs. therein M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

14 Truncated vertex DSE Truncated Bose-symmetrised Dyson-Schwinger equation for three-gluon vertex. All internal propagators are dressed. M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

15 Model inputs for calculation Ghost/gluon propagators fit functions, data provided for us. C. S. Fischer, private communication Both scaling and decoupling solutions. Ghost/gluon renorm. constants Z 3 /Z 3 fixed, part of propagator input. Model for four-gluon vertex: Γ 4g (p 1, p 2, p 3, p 4 ) = V dress 4g Γ (0) 4g (p 1, p 2, p 3, p 4 ) V dress 4g = G a Z b (5) Constants a, b chosen so that V dress 4g is correct in IR and UV. M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

16 Four-gluon vertex model Scaling Decoupling Four-gluon dressing p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

17 Basis for three-gluon vertex 3g vertex has 3 Lorentz indices, 2 independent momenta 14 independent tensor structures. Gluon propagator transverse in Landau gauge. Transverse components of Γ 3g survive 4 independent tensor structures. Γ µνρ (p 1, p 2, p 3 ) = 4 C i (p 1, p 2, p 3 )τµνρ(p i 1, p 2, p 3 ) (6) i=1 Use orthonormal basis internally + rotate onto BC basis for results. J. S. Ball, T. W. Chiu, PRD 22, 2550 (1980) M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

18 Perturbative check 1.4 Numerical Analytical (Celmaster, Gonsalves, PRD 1980) 1.2 C 1 (p 2, p 2, p 2 ) p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

19 Naive renormalisation 3g vertex renorm. constant Z 1 determined from condition: Gives Z 1 = 1.43 for µ 2 = 170 GeV 2. C 1 (µ 2, µ 2, µ 2 ) = 1 (7) This renorm. scheme is inconsistent with ghost/gluon input, breaks Slavnov-Taylor identities. Results in a convergent solution. Z 1 = Z 3 / Z 3 (8) M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

20 Γ (0) µνρ tensor dressing 5 Log sinh 1 C Log 0 5 Bare vertex dressing has zero crossing in IR, seen in lattice studies. A. Maas, PRD 75, (2007) M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

21 Γ (0) µνρ tensor dressing 5 Log sinh 1 C Log 0 5 In IR, dressing scales like (p 2 ) 3κ, in accordance with IR scaling analysis. R. Alkofer, C. S. Fischer, F. Llanes-Estrada, PLB 611, 279 (2005) M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

22 Γ (0) µνρ tensor dressing Decoupling 5 Log sinh 1 C Log 0 5 M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

23 Self energies, One iteration Scaling 0 sinh -1 (C 1 (p 2,p 2,p 2 )) -5 Ghost triangle, in IR ~ (p 2 ) -3κ Gluon triangle, in IR ~ (p 2 ) 0 Swordfish, in IR ~ (p 2 ) κ, (p 2 ) p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

24 Self energies, After convergence Scaling 0 sinh -1 (C 1 (p 2,p 2,p 2 )) -5 Ghost triangle Gluon triangle Swordfish p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

25 Running couplings, After convergence Scaling α 3g 1 α gh-gl p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

26 Proper renormalisation 3g vertex renorm. constant Z 1 determined from Slavnov-Taylor identities: Z 1 = Z 3 / Z 3 = 2.21 (9) Reminder: in naive renorm. scheme Z 1 = Different relative weights of gluon triangle/swordfish diagrams in two renorm. schemes. Results in a runaway solution. M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

27 Self energies, One iteration Scaling 0 sinh -1 (C 1 (p 2,p 2,p 2 )) Ghost triangle Gluon triangle Swordfish p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

28 Γ (0) µνρ, Various iterations Scaling 10 5 sinh -1 (C 1 (p 2,p 2,p 2 )) First Iteration Third Iteration Fifth Iteration Seventh Iteration Tenth Iteration p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

29 Running couplings, One iteration Scaling α 3g 1 α gh-gl p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

30 Stabilisation of solution Neglected two-loop terms, 4g vertex model seem crucial for stabilisation of solution. Better 4g vertex model enhance vertex in mid-momentum region. Easiest modification: V dress,new 4g = C V dress,old 4g (10) C = const, with C 1. For C 1.30 solution is stable. Permutation of swordfish diagrams stable solution, needs checking! M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

31 Conformal field theories In chiral limit, QCD is a classically conformal theory. Upon quantisation (renormalisation), conformal symmetry is lost scale-dependent masses/coupling constants. Conformal symmetry restored on quantum level if a non-trivial fixed point: β(α ) = 0 (11) Β Α M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

32 Walking theories If fixed point is only approximate: β(α ) 0 nearly-conformal (walking) theory. Phenomenologically interesting for Technicolor scenarios. J.R.Andersen et al. Eur. Phys. J. Plus 126, 81, 2011 Getting a strongly interacting theory inside/close to conformal window: 1 Increase number of quark flavors N f. 2 Use quarks in higher-dimensional representation of gauge group. Β Α M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

33 Conformal QCD model Study 3g vertex in QCD with (nearly) critical number of flavors N cr f. Appropriate ghost/gluon model input. C.S. Fischer, T. Goecke, M. Hopfer, R. Alkofer, in preparation Within this model, N c = 3 and N cr f 5. Naive and proper renorm. schemes coincide. Ghost-gluon and four-gluon vertex taken to be bare. Results in a convergent solution. M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

34 YM conformal propagators 5 G Z p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

35 Γ (0) µνρ dressing Conformal 5 Log sinh 1 C Log M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

36 Self-energies, After convergence Conformal 0 sinh -1 (C 1 (p 2,p 2,p 2 )) -1-2 Ghost triangle Gluon triangle Swordfish e+06 p 2 [GeV 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

37 Running couplings, After convergence Conformal α 3g α gh-gl e e+06 p 2 [Gev 2 ] M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

38 3g vertex in conformal theory α 3g has similar features in confining and conformal phases. In contradiction to what one would (naively?) expect from α gh-gl. Can inclusion of other diagrams eliminate zero crossing in Γ (0) µνρ dressing? Include the quark loop diagram. M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39

39 Outlook Appropriate 4g model stabilise solution. Scaling vs. decoupling influence on zero-crossing, hadronic observables? Construct 3g vertex Ansätze, use in bound state calculations (both ordinary and walking QCD). 3PI formulation test for a stable solution which is positive in IR. THANK YOU FOR YOUR ATTENTION! M. Vujinovic et al. (KFU, Graz) On the Landau gauge three-gluon vertex / 39