Richard Williams. Hèlios Sanchis-Alepuz

Transcription

1 Richard Williams Hèlios Sanchis-Alepuz

2 Introduction 2 Idea: Information on hadron properties encoded in Green s functions EM form-factors Dyson-Schwinger Approach Nonpert. Covariant Multi-scale Symmetries Renorm. Hadronic LBL Meson spectroscopy (see Hilger) Form factors Glueballs (see Sanchis-Alepuz) Tetraquarks (see Eichmann) QCD-like (see Vujinovic) e.g. functional methods: functional equations, functional identities

3 Introduction 3 Infinite number of Green s functions. Truncate to subset of building blocks Large phase space / # 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, ) Dyson-Schwinger equations provide access to building blocks.

4 Introduction 3 Infinite number of Green s functions. Truncate to subset of building blocks Large phase space / # 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, ) Dyson-Schwinger equations provide access to building blocks. Connection to bound-states provided by Bethe-Salpeter equations. baryons mesons glueballs hybrids tetraquarks Need the bound-state amplitude, the constituent particles and the interaction kernel(s)

5 Bound-state amplitude 4 Meson: four or eight covariants Q/T: Angular momentum tensors Γ μ 1 μ J = Qμ 1 μ J 1 D i = 1, γ k Λ ± = (1 ± γ P)/2 T μ 1 μ J D γ i k Λ ± 5 Rest frame: (P 2 = M 2 ) two independent variables 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 μ γ T P μν q t μ γ 5 P μν See Eichmann for the Tetraquark, Sanchis-Alepuz for the Glueball

6 Constituent Particles 5 Resummed propagators provided by solution of their Dyson-Schwinger equations (1PI) Close the system (truncate): eliminate higher order n-point functions Functional identities: help/hinder via additional constraints (WTI, STI) Other approaches: ERG, npi, Lattice provide complementary information

7 Constituent Particles: Gluon Propagator 6 Excellent agreement with Lattice calculations possible Long History of DSE calculations (+ other) One-loop: (Alkofer, Fischer, Huber, ) Two-loop: (Alkofer, Hopfer) Unquenching: (Fischer, ) Typically can treat in isolation of matter sector

8 Constituent Particles: Quark Propagator 7 S 1 p = A p 2 ( i γ p + M p 2 ) Qualitative: Dynamical Mass Chiral condensate PT/OPE at large momenta Quantitative: Match flavor dependence? (Data points: Bowman et al)

9 Self-consistency and Interaction strength 8 Fact: the gluon by itself is insufficient to achieve this qualitative agreement Trigger dynamical chiral symmetry breaking through sufficient interaction strength Gluon simply doesn t provide this. The dynamics of the quark-gluon vertex and vertices in general are crucial as they can, and do, contain sizable corrections See Alkofer, also Mitter Symmetries are important for the truncated system Symmetries are important in derivation of interaction Kernel How to truncate? And, what about bound-states?

10 Philosophies to proceed 9 Vector Ward-Takahashi Identity: ensures current conservation! Axial-vector Ward-Takahashi Identity: ensures the properties of the pion! Intricate relation between quark-antiquark kernel and quark-gluon vertex (A) Vertex by Ansatz. Kernel Constructed (sophisticated) model of DCSB Includes by hand structures believed to be generated dynamically. Ad hoc? Constrained? Ansatz: Chang, Roberts / Heupel et al (B) Truncated Vertex DSE. Kernel Derived Connection to higher n-point functions Generates additional structures dynamically Modelling pushed to higher loop order Apply to different systems! see HSA and RW , Fukuda, Munczek Here: Euclidean space-time, analytically continue to Complex plane (three-point function: function 3 variables becomes function of 5)

11 Bethe-Salpeter equations 10 Homogeneous BSE for the Baryon Homogeneous BSE for the Meson How to obtain kernels, consistent with symmetry, for a given truncation that provides our propagators? Natural derivation: npi effective action.

12 11 Baby steps

13 Effective Action: 2PI at two-loop 12 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

14 Rainbow-Ladder: Mesons and Baryons 13 Single one-gluon exchange kernel (see Hilger) Qin et al, PRC 85 (2012) Fischer, Kubrak and RW, (2014) Rojas et al, N, Eichmann et al, PRL 104 (2010) N/Δ, Sanchis-Alepuz et al, PRD 84 (2011) Octet/ Decuplet Sanchis-Alepuz et al, Permuted two body kernel KERNEL: Partial kinematic dependence Unified framework Analytically continued i.e. access to amplitudes. Calculate spectroscopy and form-factors.

15 Effective Action: 2PI at three-loop 14 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 See Vujinovic

16 Quark-gluon vertex 15 Dynamical chiral symmetry breaking in the vertex. (cf. Alkofer, also Mitter) 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]

17 Beyond Rainbow-Ladder: Meson 16 KERNEL: Full kinematic dependence Diagrammatic Watson et al, FBS 35 (2004) Fischer and RW, PRD 78 (2008) Fischer and RW, PRL 103 (2009) Non-Diagrammatic Chang et al, PRL 103 (2009) Heupel et al, EPJA 50 (2014) Beyond Rainbow-Ladder: (1 x 1) + (2 x 8) = 17 kernel components

18 Beyond Rainbow-Ladder: Baryon 17 KERNEL: Full kinematic dependence Beyond Rainbow-Ladder: (3 x 1) + (6 x 8) = 51 kernel components

19 Mesons Results: RL vs PDG 18 Vector-scalar splitting. Axial vectors. Exotics See. Fischer, Kubrak, RW

20 Mesons Results: BRL vs PDG 19 Light scalar Heavy scalar Heupel et al, EPJA 50 (2014) Chang et al, PRC 85 (2012) Anomalous chromomagnetic moments?

21 Baryon results: Beyond RL 20 State [GeV] RL BRL EXPT N Δ N ( 1 2 ) Λ (J = Ξ (J = Λ (J = Ξ (J = 1 2) ) ) ) Work in progress! Sigma terms: σ πx = m q M X m q [MeV] N Delta RL 30(3) 24(2) BRL - - World

22 Outlook? 21

23 Outlook? 21 Not earth shattering. Connected to gauge sector. Solved many technical challenges. But systematically extensible. (3803 pieces)

24 Effective Action: 3PI at three-loop 22 Γ 2 = Σ = δγ 2 δg = = K = δ2 Γ 2 δgδg = 0 = δγ 2 δv kernel similar to 1PI dressed skeleton expansion (hep-ph/ ), see Alkofer Challenge: calculation for complex momentum. Form factors. Unified framework for mesons and baryons. Sanchis-Alepuz, RW (arxiv: )

25 Effective Action: 3PI beyond three-loop 23

26 Effective Action: 3PI beyond three-loop 23 K (2body) = Γ μ a b ab Γ ii ν jj D μν ε i j k T a ii T b jj = C F 2 ε ijk

27 Effective Action: 3PI beyond three-loop 23 K (2body) = Γ μ a b ab Γ ii ν jj D μν K (3body) = Γ μ a ii Γ ν b jj Γ ρ c abc Γ kk μνρ ε i j k T a ii T b jj = C F 2 ε ijk

28 Effective Action: 3PI beyond three-loop 23 K (2body) = Γ μ a b ab Γ ii ν jj D μν K (3body) = Γ μ a ii Γ ν b jj Γ ρ c abc Γ kk μνρ ε i j k T a ii T b jj = C F 2 ε ijk ε i j k T a ii T jj b c T kk if abc = 0 Implies (at least) one the quark lines coupled via higher n-point function K (3body) ab = Γ μν ii Γ ρ c jj Γ σ d kk

29 Interaction Kernel: Three Particle-Irreducible 24 Zero Typically: N c+1 8

30 Interaction Kernel: Three Particle-Irreducible 24 Zero Typically: N c+1 8 Plus terms suppressed by quark mass (heavy quark limit) and by colour. Genuine 12-dimensional integral. Order petabyte storage. O(10) MeV effect Work in progress! (comes from 3PI at five-loop)

31 Conclusions 25 Systematic program to investigate beyond rainbow-ladder Mesons, Baryons, Analytic Continuation with derivable 2- and 3-body kernels calculated in the complex plane. First steps connect to the gauge-sector of QCD Work towards refining the quark-gluon interaction as used in hadronic calculations Next steps: Form factors Decays/Transitions Excited states

32 Conclusions 25 Systematic program to investigate beyond rainbow-ladder Mesons, Baryons, Analytic Continuation with derivable 2- and 3-body kernels calculated in the complex plane. First steps connect to the gauge-sector of QCD Work towards refining the quark-gluon interaction as used in hadronic calculations Next steps: Form factors Decays/Transitions Excited states Thank you