Richard Williams C. S. Fischer, W. Heupel, H. Sanchis-Alepuz

Transcription

1 Richard Williams C. S. Fischer, W. Heupel, H. Sanchis-Alepuz

2 Overview 2 1.Motivation and Introduction 4. 3PI DSE results 2. DSEs and BSEs 3. npi effective action 6. Outlook and conclusion 5. 3PI meson results

3 Overview 3 1.Motivation and Introduction 4. 3PI DSE results 2. DSEs and BSEs 3. npi effective action 6. Outlook and conclusion 5. 3PI meson results

4 Motivation 4 Describe bound-states: baryons mesons glueballs hybrids tetraquarks pentaquarks and EM processes: Muon g-2 EM form-factors Compton Scattering in terms of QCD s Green s functions.

5 Methods 5 Functional methods - provide access to Green s functions: lattice, FRG, DSE, npi Covariant, multi-scale (no separation), renormalizable, non-perturbative Good differences (no) sign problem, continuum There s always a but Infinite tower of coupled non-linear equations truncations are necessary Computationally and algebraically involved Euclidean - access to time-like properties requires analytic continuation More legs = more problems (larger phase space and number of covariants) μν... Γ ij... p 1, p 2, = a F a (p 2 1, p 2 2, )τ μν a,ij (p 1, p 2, )

6 Approach 6 We need various ingredients Propagators: Quark Gluon Symmetries: Vector Ward Identity Axial-vector Ward Identity Vertices: Quark-gluon vertex Three-gluon vertex Interaction Kernel: Quark-(anti)quark binding Bound State Amplitude Specify quantum numbers Describe appropriate state Connection to Gauge sector Control of modelling Systematic improvements? DSEs and BSEs

7 Overview 7 1.Motivation and Introduction 4. 3PI DSE results 2. DSEs and BSEs 3. npi effective action 6. Outlook and conclusion 5. 3PI meson results

8 Dyson Schwinger Equations 8 DSEs derived from functional identity: δγ φ δφ i δs δφ i φ + δ2 W J δjδj k δ δφ k = 0 QFT analogue to EOM S 1 p = A p 2 ( i γ p + M p 2 ) Exact equations in principle Think of as a constraint equation Additional constraints provided by WI and STI (difficult to reconcile in practice) Not closed so truncation needed D μν p = δ μν pμ p ν Z p 2 p 2 p 2

9 Propagators typical results 9 By now, truncations can be quite robust Qualitative and quantitative agreement Scalar dressings are momentum dependent and encode PT and NP information Same green s functions, can compare with: Not measurable in experiment! Lattice, FRG, npi More about truncations later (Data points: Bowman et al)

10 More to life than propagators 10 By itself, a gluon will not generate dynamical chiral symmetry breaking Non-perturbative enhancements in the quark-gluon vertex required. Rainbow-ladder explicitly ignores vertex corrections. These are included implicitly by using an effective interaction to replace the gluon e.g.

11 More to life than propagators quark-gluon vertex 11 Dynamical chiral symmetry breaking in the vertex. Flavour dependent h 1 is the vector, h 5, h 6 the scalar/vector anomalous chromomagnetic moments. Γ μ a l, k = h 1 γ μ T + h 2 l μ T γ l + h 3 il μ T + h 4 l k i 2 γ μ i T, γ l + h 5 2 γμ, γ k 1 +h 6 6 γμ, γ l, γ k + h 7 t μν kl l k γ ν + h 8 t μν kl [γ ν, γ l]

12 More to life than propagators bound-states 12 How to connect propagators and vertices to bound states of quarks and gluons without breaking everything? And anyway. What is a bound-state amplitude

13 Bound-state amplitude: Meson 13 Meson: Γ μ 1 μ J = Qμ 1 μ J 1 T μ 1 μ J D γ i k Λ ± 5 Rest frame: (P 2 = M 2 ) two independent variables D i = 1, γ k Λ ± = (1 ± γ P)/2 Q and T: Angular momentum tensors Constructed from the traceless part of J-fold tensor products Q μ 1 μ J = k T {μ 1 kt μ J } T μ 1 μ J = γ t {μ 1 k T μ 2 kt μ J } Saturates: no more than four or eight Dirac covariants

14 Bound-state amplitude: Baryon 14 Rarita-Schwinger Projector P μν = T μν P 1 3 γ μ ν T γ T five independent variables Nucleon/Delta: 64/128 covariants D i = 1, γ k, γ q, (γ k)(γ q) Γ ν = 1 1 γ 5 γ 5 γ T μ γ T μ γ 5 γ T μ γ T μ γ 5 D i Λ ± γ 5 C D j Λ + k μ T γ 5 P μν γ T γ 5 P μν q t μ γ 5 P μν

15 Overview 18 1.Motivation and Introduction 4. 3PI DSE results 2. DSEs and BSEs 3. npi effective action 6. Outlook and conclusion 5. 3PI meson results

16 Effective Action: npi 19 Constructing from Γ φ by taking Legendre transf. wrt propagators and vertices. Γ φ, D, U = S cl φ + i 2 TrLnD 1 + i 2 Tr D 0 1 D iφ 0 φ, D, U iφ int φ, D, U + const. Φ 0 : non-interacting part Construction not known exactly Loop expansion in hbar Functional derivatives yield DSEs for propagators and vertices Φ int : interacting part Further functional derivatives yield Bethe-Salpeter kernels! More or less guaranteed to be consistent with symmetries of the action

17 Effective Action: 2PI at two-loop 20 Two-particle irreducible effective action (Legendre transform with respect to fields, propagators): Γ Ψ, G = S Ψ + i Tr ln G i Tr G Γ 2 [Ψ, G] (See Berges, hep-ph/ ) Γ 2 = i 2 Σ = δγ 2 δg = K = δ2 Γ 2 δgδg = Quark self-energy, quark-antiquark kernel obtained by functional derivatives Comparison with exact quark self-energy implies quark-gluon vertex Vertices are perturbative Propagators are resummed Complement bare vertex (tree-level, no loop corrections) with an RG improvement

18 Effective Action: 2PI at three-loop 21 Γ 2 = Σ = δγ 2 δg = Compare with quark self-energy: K = δ2 Γ 2 δgδg = Complement bare vertices with a (constrained) RG improvement Select contributions leading in N c

19 Effective Action: 3PI at three-loop 22 Γ 2 = Σ = δγ 2 δg = 0 = δγ 2 δv Reduces to quark self-energy: BSE kernel is structurally simple K = δ2 Γ 2 δgδg = Sanchis-Alepuz, RW (arxiv: )

20 Overview 23 1.Motivation and Introduction 4. 3PI DSE results 2. DSEs and BSEs 3. npi effective action 6. Outlook and conclusion 5. 3PI meson results

21 DSE results from the 3PI effective action 24 Ghost/gluon solved independently of 3PI Investigating unquenching effects (quark-loops) on the system

22 (not) 3PI results: DSEs (ghost and gluon propagators) 25 D μν p = δ μν pμ p ν Z p 2 p 2 p 2 D G p = G p2 p 2 Ghost/Gluon solved such that agreement with quenched/unquenched lattice is obtained Provides input parameters of model (g s, Z 3, Z 3 )

23 3PI results: DSEs (quark propagator) 26 S 1 p = A p 2 ( i γ p + M p 2 ) cusp effect in quark wavefunction seen for first time

24 3PI results: DSEs (three-gluon vertex) 27 Quark-loop enhances three-gluon vertex Zero crosses pushed to deep IR Tree-level structure dominant μνρ μνρ Γ 3g p, k, q = F1 p, k, q Γ 3g,0 p, k, q Single phase-space slice sufficient (e.g. soft-gluon) [Eichmann, RW, Alkofer, Vujinovic]

25 3PI results: DSEs (ghost-gluon vertex) 28 One tensor structure Γ μ gh l, q = f l, q T μν (q) l ν Unquenching effects negligible. Lattice data needs improvement

26 3PI results: DSEs (quenched quark-gluon vertex) 29 Lattice renormalization procedure is suspect 1 st and 3 rd structures comparable. Difficult systematics (lattice) in 2 nd

27 3PI results: DSEs (unquenched quark-gluon vertex) 30 Comparable to quenched (DSE) data. Large corrections beyond tree-level.

28 Overview 31 1.Motivation and Introduction 4. 3PI DSE results 2. DSEs and BSEs 3. npi effective action 6. Outlook and conclusion 5. 3PI meson results

29 3PI results: mesons 32 Above are calculated for complex quark momenta. Homogeneous BSE solved without Pade or fit functions

30 3PI results: mesons 33 Scalar: 2PI-2L (RL) and 2PI-3L it is too light: MeV ρ a 1 splitting: 2PI-2L (RL) and 2PI-3L it is too small: 200 MeV a 1 b 1 splitting: 2PI-2L (RL) and 2PI-3L non-degenerate states Phenomenology restored by 3PI-3L!!!

31 Overview 34 1.Motivation and Introduction 4. 3PI DSE results 2. DSEs and BSEs 3. npi effective action 6. Outlook and conclusion 5. 3PI meson results

32 Conclusions 40 Mesons q q Only now exploring details of quark-gluon interaction on spectrum No longer disconnected from gauge sector. Implicit flavor dependence. Developing framework Unified description of mesons and baryons consistent with symmetries Calculation of higher spin and/or excited mesons and baryons Extensible to other bound-states via npi Baryons Tetraquarks Glueballs and Hybrid mesons A functional derivative (or two) away Calculation of form-factors, EM transitions and decays

33 Conclusions 40 Mesons q q Only now exploring details of quark-gluon interaction on spectrum No longer disconnected from gauge sector. Implicit flavor dependence. Developing framework Unified description of mesons and baryons consistent with symmetries Calculation of higher spin and/or excited mesons and baryons Extensible to other bound-states via npi Baryons Tetraquarks Glueballs and Hybrid mesons A functional derivative (or two) away Calculation of form-factors, EM transitions and decays