Hadron Phenomenology and QCDs DSEs

Transcription

1 Hadron Phenomenology and QCDs DSEs Lecture 3: Relativistic Scattering and Bound State Equations Ian Cloët University of Adelaide & Argonne National Laboratory Collaborators Wolfgang Bentz Tokai University Craig Roberts Argonne National Laboratory Anthony Thomas University of Adelaide David Wilson Argonne National Laboratory USC Summer Academy on Non-Perturbative Physics, July August 2012

2 Hadron Spectrum In quantum field theory physical states appear as poles in n-point Green Functions For example, the quark antiquark scattering matrix or t-matrix, contains poles for all qq bound states, that is, the physical mesons The quark antiquark t-matrix is obtained by solving the Bethe-Salpeter equation T = K + T K = Γ = Γ K Kernels of gap and BSE must be intimately related 1 = 1 + K For example, consider axial vector Ward Takahashi Identity q µ Γ µ,i 5 (p,p) = S 1 (p )γ τ i τ iγ 5 S 1 (p)+2mγ i π(p,p) frontpage table of contents appendices 2 / 24

3 Inhomogeneous & Homogeneous vertex functions T = K + T K = Γ = Γ K Near a bound state pole of mass m a two-body t-matrix behaves as T (p,k) Γ(p,k) Γ(p,k) p 2 m 2 where p = p 1 +p 2, k = p 1 p 2 Γ(p,k) is the homogeneous Bethe-Salpeter vertex and describes the relative motion of the quark and anti-quark while they form the bound state The inhomogeneous BSE is a generalization and contains all the poles of the t-matrix Γ = + Γ K the first term on the RHS is the elementary driving term the driving term projects out various channels the quark-photon vertex is described by a inhomogeneous BSE frontpage table of contents appendices 3 / 24

4 The Pion How does the pion become (almost) massless when it is composed of two massive constituents The pion is realized as the lowest lying pole in the quark anti-quark t-matrix in the pseudoscalar channel In the NJL model this t-matrix is given by d 4 k T (q),γδ = K,γδ + (2π) 4 K,λǫS(q +k) εε S(k) λ λt (q) ε λ,γδ, γ δ q = γ δ + γ δ ε λ ε λ K = 2iG π (γ 5 τ) (γ 5 τ) λǫ The NJL pion t-matrix is T (q) i,γδ = (γ 5τ i ) 2iG π 1+2G π Π PP (q 2 ) (γ 5τ i ) λǫ The pion mass is then given by: 1+2G π Π PP (q 2 = m 2 π) = 0 frontpage table of contents appendices 4 / 24

5 Pion Bethe-Salpeter Amplitude T = K + T K = Γ = Γ K Recall that near a bound state pole the t-matrix behaves as T (p,k) Γ(p,k) Γ(p,k) p 2 m 2 where p = p 1 +p 2, k = p 1 p 2 Expanding the pion t-matrix about the pole gives T (q) = (γ 5 τ i ) 2iG π 1+2G π Π PP (q 2 ) (γ 5τ i ) ig π q 2 m 2 π(γ 5 τ i )(γ 5 τ i ) Where g π is interpreted as the pion-quark coupling constant 1 g π = Π q 2 PP (q 2 ) q 2 =m 2 π The pion homogeneous BS vertex is therefore: Γ π = g π γ 5 τ i this is a very simple vertex that misses a lot of physics frontpage table of contents appendices 5 / 24

6 The Pion as a Goldstone Boson Using the NJL gap equation and the pion pole condition gives m 2 π = m 2M G π Π (L) AA (m2 π) Therefore in the chiral limit, m 0, the pion is massless In quantum mechanics one could tune a potential to give a massless ground state for a bound state of two massive constituents however quantum mechanics always gives: M bound state M constituents However quantum field theory with DCSB gives: m 2 π m Recall the Gell-Mann Oakes Renner relation f 2 π m 2 π = 1 2 (m u +m d ) ūu+ dd The pion decay constant is given by µ 0 A µ q q a π b (p) = if π p µ δ ab frontpage table of contents appendices 6 / 24

7 ρ a 1 mass splitting The ρ and a 1 are the lowest lying vector (J P = 1 ) and axial-vector (J P = 1 + ) qq bound states: m ρ 770 MeV & m a MeV The masses of these states are given by t-matrix poles in the vector and axial-vector qq channels γ δ q = γ δ + γ δ ε λ ε λ K = 2iG ρ [ (γµ τ) (γ µ τ) γδ +(γ µ γ 5 τ) (γ µ γ 5 τ) γδ ] Chiral symmetry implies one NJL coupling, G ρ. NJL gives m ρ 770 MeV & m a MeV NJL interaction is insufficient to obtain correct ρ a 1 mass splitting The rainbow ladder Maris Tandy DSE model gives m ρ 644 MeV & m a1 759 MeV Clearly something is missing! frontpage table of contents appendices 7 / 24

8 ρ a 1 mass splitting Generalized Quark-Gluon Vertex Recall that going below rainbow ladder by adding σ µν q ν τ 5 (p,p) to quark-gluon vertex, generates quark anomalous magnetic moment An order parameter for dynamical chiral symmetry; Chiral symmetry -0.2 forbids massless particle having -0.4 anomalous magnetic moments p/m E What about the hadron spectrum acm full aem full aem RL Important example of interplay between observables and DSE quark gluon vertex frontpage table of contents appendices 8 / 24

9 An Aside Muon Anomalous Magnetic Moment q µ p p = a exp µ = ± ; a theory µ = ± largest theory error come from HLBL scattering contribution µ ν µ ν µ ν µ ν µ ν q k q k q k q k q k Π µν = l t l t l t l t l t Box diagram contribution is least know only γ µ coupling and VMD has been considered so far we argue that the anomalous magnetic moment term cannot be ignored At least error on a HLBL µ = 8.3± should be much larger Fred Jegerlehner, Andreas Nyffeler, Physics Reports 477 (2009) frontpage table of contents appendices 9 / 24

10 Pion Form Factor Hadron form factors describe its interaction with the electromagnetic current An on-shell pion has one EM form factor l k q θ k π J µ π = (p µ +p µ )F 1 (Q 2 ) π p p In the impulse approximation the pion form factors are given by Γ Γ Γ = Γ Ingredients: dressed quark propagators homogeneous Bethe-Salpeter vertices dressed quark-photon vertex frontpage table of contents appendices 10 / 24

11 Pion Form Factor (2) Pion BSE vertex has the general form ] Γ π (p,k) = γ 5 [E π (p,k)+/pf π (p,k)+/kk pg(p,k)+σ µν k µ p ν H(p,k) Pseudovector component F π (p,k) dominates ultra-violet Perturbative QCD predicts Q 2 F 1π (Q 2 ) Q2 = 16πf 2 π s (Q 2 ) 0.44 s (Q 2 ) DSE finds that pqcd sets in at about Q 2 = 8 GeV 2 frontpage table of contents appendices 11 / 24

12 Some Consequences of Running Quark Mass Pion BSE vertex has the general form ] Γ π (p,k) = γ 5 [E π (p,k)+/pf π (p,k)+/kk pg(p,k)+σ µν k µ p ν H(p,k) In gap equation use simple kernel NJL model with π a 1 mixing g 2 D µν (p k)γ ν (p,k) 1 g m 2 µν γ ν [ ] = Γ π (p,k) = γ 5 Eπ +/pf π G Quark no longer has a running mass Nature of interaction has drastic observable consequences for Q 2 > 0 frontpage table of contents appendices 12 / 24

13 Measuring Pion Form Factor q dσ/dt (µb/gev 2 ) 6 Q 2 =1.60 σ L σ T 2 Q 2 =2.45 p p n t (GeV 2 ) t (GeV 2 ) At low Q 2 pion form factor is measured by scattering a pion from the electron cloud of an atom [t p 2 ] small mass of electron limits this to Q 2 < 0.5 GeV 2 Higher Q 2 experiments have been performed at Jefferson Lab where a virtual photon scatters from a virtual pion that is part of the nucleon wavefunction Initial pion is off its mass shell p 2 0 on mass shell p 2 = m 2 π need to extrapolate to the pion pole p 2 = m 2 π frontpage table of contents appendices 13 / 24

14 Off-Shell Pion Form Factors Fπ,1(p 2, p 2, Q 2 ) t = m 2 π t = 0.2 t = 0.4 Empirical p 2 = m 2 π p 2 = t Fπ,2(p 2, p 2, Q 2 ) t = m 2 π t = 0.2 t = 0.4 p 2 = m 2 π p 2 = t Q 2 (GeV 2 ) Q 2 (GeV 2 ) Initial pion is off its mass shell p 2 < 0 on mass shell p 2 = m 2 π need to extrapolate to the pion pole p 2 = m 2 π However an off-shell pion has two form factors not one and each form factor is a three dimensional function Γ µ π(p,p) = (p µ +p µ )F π1 (p 2,p 2,q 2 )+(p µ p µ )F π2 (p 2,p 2,q 2 ) Using NJL model can determine off-shell pion form factors Potentially important for experimental extraction of F π frontpage table of contents appendices 14 / 24

15 Baryons in the DSEs Baryons are 3-quark bound states with the proton (uud) and neutron (udd) being the most important examples In quantum field theory physical baryons appear as poles in six-point Green Functions Recall that two-body bound states appear as poles in four-point Green Functions and solutions were obtained by solving the Bethe-Salpeter Equation The analogue of the Bethe-Salpeter equation for 3-quark bound states is called the Faddeev equation By definition the Faddeev kernel only contains two-body interactions this is an approximation which is yet to be explored and could have important consequences for QCD Diagrammatically the homogeneous Faddeev equation is given by frontpage table of contents appendices 15 / 24

16 Baryons in the DSEs (2) We will render this problem tractable by making the quark-diquark approximation This is a linear matrix equation, whose solution gives the Baryon wavefunction strictly the Poincaré covariant Faddeev amplitude We will include scalar (J P = 0 +,T = 0) and axial-vector diquarks (J P = 1 +,T = 1) parity dictates that pseudoscalar and vector diquarks must be in an l = 1 state and are therefore suppressed in the nucleon for the negative parity N (1535) the opposite is true The nucleon wavefunction contains S, P and D wave correlations Equation has discrete solutions at p 2 = m 2 i ; nucleon, roper, etc frontpage table of contents appendices 16 / 24

17 What is a Diquark A diquark is a correlated (interacting) quark-quark state This interaction is attractive in the colour 3 (antisymmetric) or colour 6 (symmetric), however only the colour 3 can exist inside a colour singlet nucleon Diquarks are analogous to mesons colour singlet qq bound states Because diquarks are coloured they should not appear as physical states in QCD confinement However in the rainbow ladder approximation and the NJL model diquarks do appear as poles in the qq scattering (t) matrix Lattice QCD also sees evidence for diquarks I. Wetzorke, F. Karsch, hep-lat/ ( 30 3) implies scalar diquark: (flavour- 3, spin-0, colour- 3) (60 3) implies axial-vector diquark: (flavour-6, spin-0, colour- 3) frontpage table of contents appendices 17 / 24

18 Diquarks in The NJL model To describe diquarks in the NJL model one usually rewrites the qq interaction Lagrangian into a qq interaction Lagrangian ( ψγψ ) 2 ( ψω ψt )( ψ T Ωψ ) Ω has quantum numbers if interaction channel Γ Γ Ω Ω The qq NJL Lagrangian in the scalar and axial-vector diquark channels has the form L I = G s [ψγ 5 Cτ 2 A ψ T][ ] ψ T C 1 γ 5 τ 2 A ψ +G a [ψγ µ Cτ i τ 2 A ψ T][ ] ψ T C 1 γ µ τ 2 τ j A ψ +... the first term is the scalar diquark channel (J P = 0 +,T = 0) the second the axial-vector diquark channel (J P = 1 +,T = 1) τ 2 couples isospin of two quarks to T = 0, Cγ 5 couples spin to J = 0, A 3 = 2 λa A = 2,5,7 frontpage table of contents appendices 18 / 24

19 NJL diquarkt-matrices The equation for the qq scattering matrix the Bethe-Salpeter equation has the form T (q),γδ = K,γδ + 1 d 4 k 2 (2π) 4 K,ελS(k) εε S(q k) λλ T (q) ε λ,γδ, γ δ q = γ δ + γ δ ε λ ε λ = γ δ + γ δ ε λ ε λ note symmetry factor of 1 2 (c.f. qq BSE) The Feynman rules for the interaction kernels are K s = 4iG s(γ 5 Cτ 2 A ) (C 1 γ 5 τ 2 A ) γδ K a = 4iG a(γ µ Cτ i τ 2 A ) (C 1 γ µ τ 2 τ i A ) γδ The solution to the BSE is of the form: T (q),γδ = τ(q 2 )Ω Ωγδ τ s (q 2 ) = 4iG s 1+2G s Π s τ µν (q 2 ) a (q) = 4iG a 1+2G a Π a (q 2 ) [g µν +2G a Π a (q 2 ) qµ q ν ] frontpage table of contents appendices 19 / 24 q 2

20 Diquark Propagators The reduced t-matrices are the diquark propagators τ s (q 2 4iG ) = s 1+2G s Π s τ µν 4iG (q 2 ) a (q) = a 1+2G a Π a [g µν +2G (q 2 ) a Π a (q 2 ) qµ q ν q 2 ] Near the pole they behave as elementary propagators T,γδ = γ δ Ω Ω τ(q 2 ) γ δ Ω ig D q 2 MD 2 The diquark masses are therefore given by 1+2G s Π s (q 2 = Ms) 2 = 0 1+2G a Π a (q 2 = Ma) 2 = 0 Ω Expanding Π(q 2 ) near the pole gives the quark-diquark coupling constant gd 2 2 = Π q 2 D (q 2 ) q 2 =M 2 D frontpage table of contents appendices 20 / 24

21 The NJL Faddeev Equation To describe nucleon Faddeev equation kernel must be projected onto colour singlet, spin one-half, isospin one-half & positive parity p = p Make the static approximation to quark exchange kernel: S(p) 1 M Homogeneous Faddeev amplitude with static approximation does not depend of relative momentum between the quark and diquark The Faddeev equation and vertex have the form Γ N (p,s) = K(p)Γ N (p,s) [ ] M Γ N (p,s) = Z N 1 N p0 p µ 2 M N γ γ µ γ 5 u N (p,s) K(p) is the Faddeev kernel Faddeev equation describes the continual recombination of the three quark in quark-diquark configurations frontpage table of contents appendices 21 / 24

22 The NJL Faddeev Equation (2) k p = p p p p k The kernel of this NJL Faddeev eq Γ N (p,s) = K(p)Γ N (p,s) is [ ] [ Γs Γ µ = 3 ] [ Π Ns 3γ γ 5 Π ] Na Γs a M 3γ5 γ µ Π Ns γ γ µ Π Γ Na a, The quark-diquark bubble diagrams are Π Ns (p) = Π µν Na (p) = d 4 k (2π) 4 τ s(p k)s(k) d 4 k (2π) 4 τµν a (p k)s(k) Can now solve for the coefficients 1, 2, 3 this then gives the NJL Faddeev amplitude frontpage table of contents appendices 22 / 24

23 DSE Faddeev Equation The DSE Faddeev equation has far more structure than in NJL For example the DSE Faddeev equation including scalar and axial-vector diquarks reads [ ] S(k,P) d 4 [ ] l S(l,P) A µ i (k,p) u N (p) = (2π) 4 Mµν ij (l;k,p) A j u N (p) ν(l,p) importantly the vertex function depends on the relative momentum, k, between the quark and diquark the Faddeev kernel is M µν ij (l;k,p) S(k,P) and A µ i (k,p) describe the momentum space correlation between the quark and diquark in the nucleon This equation can be solved numerically on a large 2-D grid in k and P However standard practice to use a Chebyshev expansion for S(k,P) & A µ i (k,p) and the solve for the coefficients of the expansion a Chebyshev expansion is an expansion in Chebyshev polynomials frontpage table of contents appendices 23 / 24

24 Table of Contents hadron spectrum vertex functions pion vertex functions goldstone boson ρ a 1 mass splitting muon g 2 pion form factor running quark mass measuring pion form factor off-shell F πs baryons diquarks njl model diquarks njl diquark t-matrices diquark propagators njl faddeev dse faddeev equation frontpage table of contents appendices 24 / 24