Meson cloud effects in the electromagnetic hadron structure

Transcription

1 Meson cloud effects in the electromagnetic hadron structure Daniel Kupelwieser Thesis supervisor: Wolfgang Schweiger Collaborators: Elmar Biernat, Regina Kleinhappel Universität Graz Graz Jena monitoring workshop, Graz, October 2012 D. Kupelwieser (Graz) Meson cloud effects 1 / 23

2 Contents 1 Basic Definitions and Motivation 2 Hadronic level 3 Quark level 4 Outlook D. Kupelwieser (Graz) Meson cloud effects 2 / 23

3 Electron nucleon scattering Basic setting: k N k N Electron nucleon scattering via photon exchange Constituent quark model, M N = M qqq V conf Point form of relativistic quantum mechanics Time-ordered diagrams Photon nucleon vertex dressed with single-pion exchange D. Kupelwieser (Graz) Meson cloud effects 3 / 23

4 Form factors Electromagnetic interaction Lagrangian: L int = J µ N A µ To account for inner electromagnetic nucleon structure, modify nucleon current: J µ N (, µ N,, µ N ) = e u µ N ( ) (F 1 (q 2 ) γ µ F 2 (q 2 ) i q ν σ µν 2m N F 1 (q 2 )... Dirac form factor, F 1 (0) = 1 F 2 (q 2 )... Pauli form factor, F 2 (0) = 0 σ µν := i 2 [γµ, γ ν ] ) u µ N ( k N ) D. Kupelwieser (Graz) Meson cloud effects 4 / 23

5 Point form Point form of relativistic dynamics: Quantization surface: Spacetime hyperboloid x 2 t 2 x 2 = τ 2 = const. (Spacelike hypersurface, invariant under Lorentz group) Intrinsic Lorentz covariance Dynamic (interaction-dependent) Poincaré generators: only P µ Kinematic generators: { K, J} (Lorentz group, nice!) D. Kupelwieser (Graz) Meson cloud effects 5 / 23

6 Bakamjian Thomas construction Given: n-particle system with overall 4-momentum P µ. Bakamjian Thomas construction: P µ = P µ 0 Pµ int = (M 0 M int ) V µ 0, (P2 = M 2 ) M int has to commute with V µ, K and J (like P µ does). Overall velocity conserved at vertices Advantage: Interactions may be instantaneous, system stays covariant! D. Kupelwieser (Graz) Meson cloud effects 6 / 23

7 Velocity states Useful basis (V := V 0 ): Velocity states { p i, σ i } V ; { k i, µ i } with n ki = 0 i=1 Behavior under Lorentz-transformation Λ: U(Λ) V ; { k i, µ i } = ΛV ; {R(ΛV )k i, µ i} D 1 2 µ i µ i (R(ΛV )) {µ i } i (Spins get transformed with same Wigner rotation!) D. Kupelwieser (Graz) Meson cloud effects 7 / 23

8 Eigenvalue equation Coupled-channel approach: M Ne K γ K π 0 K γ M Neγ 0 K π K π 0 M Nπe K γ 0 K π K γ M Nπeγ Ne Neγ Nπe Nπeγ = m Ne Neγ Nπe Nπeγ M... : Relativistic energies of particles in each channel K... ( ) : Particle creation/ annihilation operators ( L int ) m: Mass eigenvalue of the whole system. D. Kupelwieser (Graz) Meson cloud effects 8 / 23

9 Feshbach reduction & Optical potential After Feshbach reduction neglecting self-energy contributions and double loops, with P... := (m M... ) 1 : (m M Ne ) Ne = = K γ P Neγ K γ Ne K γ P Neγ K π P Nπeγ K γ P Nπe K π Ne K π P Nπe K γ P Nπeγ K π P Neγ K γ Ne K π P Nπe K γ P Nπeγ K γ P Nπe K π Ne =: V opt Ne V opt... optical potential D. Kupelwieser (Graz) Meson cloud effects 9 / 23

10 Contents 1 Basic Definitions and Motivation 2 Hadronic level 3 Quark level 4 Outlook D. Kupelwieser (Graz) Meson cloud effects 10 / 23

11 Hadronic unity operators Insert hadronic unity operators, e.g. I Neγ = (ω N ω e ω γ ) 3 DV D D ( g µγµγ ) VNeγ VNeγ 2ω e For example, framed line becomes K γ P Neγ I Neγ K π P Nπeγ I Nπeγ K γ P Nπe I Nπe K π Propagators P... assume eigenvalues Get matrix elements of vertex operators K ( )... (sum over emitting/absorbing particles) Ne D. Kupelwieser (Graz) Meson cloud effects 11 / 23

12 Spectator conditions & Nucleon current Spectator conditions: When two particles interact, others stay unaffected, e.g.: V N e γ K Nγ VNe = VV ee ( 1) M 3 Neγ MNe 3 VN γ K Nγ VN with (J µ N VN γ K Nγ VN = JNν (, µ N,, µ N ) ɛν µ ( k γ γ)... nucleon current, contains form factors!) D. Kupelwieser (Graz) Meson cloud effects 12 / 23

13 Photon coupling to bare nucleon Finally, we get the following 10 time-ordered diagrams: V N e V opt VNe = k N k N 1 m 3 VV Jν N (, µ N, k N, µ N ) g νλ q 2 Jλ e (, µ e,, µ e)... D. Kupelwieser (Graz) Meson cloud effects 13 / 23

14 Photon coupling to dressed nucleon VV g νλ 4 m 3 q 2 k π k N α N α N απ k π k N Dk π 1 ω N ω N k π k N k k π (m M N πe) 1 (m M N πe) 1 Q 5 π(n, N ) J ν N (N, N ) J λ e (e, e ) Q 5 π(n, N ) N D. Kupelwieser (Graz) Meson cloud effects 14 / 23

15 Photon coupling to pion... k π k N k π k N k π k N k π k N... VV g νλ 4 m 3 q 2 α N α N απ ( ω N ω N ω π 1 ( Dk π ω ω N ω N ω N ω 1 π) N ) 1 J λ e (e, e ) Qπ(N, 5 N ) Jπ(π ν, π ) Qπ(N 5, N) D. Kupelwieser (Graz) Meson cloud effects 15 / 23

16 Hadronic diagrams i.e., V N e V opt VNe =... k N k π k N k π k N D. Kupelwieser (Graz) Meson cloud effects 16 / 23

17 Contents 1 Basic Definitions and Motivation 2 Hadronic level 3 Quark level 4 Outlook D. Kupelwieser (Graz) Meson cloud effects 17 / 23

18 Quark-level unity operators Insert quark-level unity operators, e.g. I qqqe = (ω q1 ω q2 ω q3 ω e ) 3 DV D Dk q2 Dk q3 Vqqqe Vqqqe 2ω q1 Photon coupling to bare nucleon now 6 diagrams: V N e V 0 opt VNe = V N e K eγ P Neγ I Neγ I qqqeγ K q 1 γi qqqe VNe V N e K eγ P Neγ I Neγ I qqqeγ K q 2 γi qqqe VNe V N e K eγ P Neγ I Neγ I qqqeγ K q 3 γi qqqe VNe V N e Iqqqe K q1 γi qqqeγ I Neγ P Neγ K eγ VNe V N e Iqqqe K q2 γi qqqeγ I Neγ P Neγ K eγ VNe V N e I qqqe K q3 γi qqqeγ I Neγ P Neγ K eγ VNe D. Kupelwieser (Graz) Meson cloud effects 18 / 23

19 Spectator conditions & Quark current Spectator condition for single struck quark : V q q q e γ K q 1 γ Vqqqe = = VV ee q2 q 2 q 3 q 3 ( 1) M 3 qqqeγ Mqqqe 3 Vq 1 γ K q 1 γ Vq 1 (etc.) with electromagnetic interaction (pointlike quark!) Vq 1 γ K q 1 γ Vq 1 = e Qq1 [u µq1 ( k q1 ) γ ν u µ q1 ( ] k q 1 ) ɛ ν µ ( k γ γ) The I qqqeγ I Neγ give rise to three-quark wave functions V 3q e γ VNeγ = N Neγ3q VV ee γγ 3q N D. Kupelwieser (Graz) Meson cloud effects 19 / 23

20 Quark-level diagrams Have to treat following diagrams: V N e V opt VNe = = 3 I I k N k π k N 3 I I k N k π D. Kupelwieser (Graz) Meson cloud effects 20 / 23

21 Bare photon nucleon vertex Results for first diagram: Hadron picture (as before): V N e Vopt VNe = VV m 3 q 2 J eν(, µ e,, µ e) J ν N (, µ N, k N, µ N ) Quark picture:... V N e V 0 opt VNe = VV m 3 q 2 J eν(, µ e,, µ e)... 2 ω N ω N 1 ω q i ω qi 3 i=1 (µ q1,µ q2,µ q3,µ q ) j i i ( ) d 3 k qj ω qj ( ) ( ) ω q1 ω q2 ω q3 ω qk ω q1 ω q2 ω q3 ωqk ( ω qk ) ( ωqk ) N q i {q j i } q 1 q 2 q 3 N J ν q i ( k qi, µ qi, k q i, µ q i ) D. Kupelwieser (Graz) Meson cloud effects 21 / 23

22 Contents 1 Basic Definitions and Motivation 2 Hadronic level 3 Quark level 4 Outlook D. Kupelwieser (Graz) Meson cloud effects 22 / 23

23 Outlook Finally, what s left to do: Insertion of three-quark wave functions from sophisticated constituent quark model (e.g. XCQM) Extract electromagnetic form factors (also for diagrams with pion loop) Use these form factors in hadronic diagrams Pion form factors from analog procedure (R. Kleinhappel) Sum over all hadronic diagrams yields overall form factors for entire problem Thank you! D. Kupelwieser (Graz) Meson cloud effects 23 / 23