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1 Phys. Rev. C76 14007(2007) A. Arriaga University of Lisbon, Portugal TRENTO, 2009 R. Schiavilla JLAB, USA Reaction Deuteron electrodisintegration one-photon-exchange approximation (−) Ψ ;Sλ,T 2 r e →e f , k f r (n, p ) → Pf ≡ E f , Pf µ ( ) r γ → q ≡ (ω , q ) µ r e →e i , ki r d → Pi µ ≡ E i , Pi ΨM ( ) Framework 3 Relativistic Hamiltonian Dynamics :Instant Form implementation Relativistic invariance achieved through Poincaré group algebra Generators independent of time and energy Hamiltonian and boost generators contain interaction terms Cross Section Deuteron electrodisintegration cross section invariant response functions 4 d σ dε ' dΩ' 2 = σ M W2 (Q , q µ Pi ) + W1 (Q , q µ Pi ) tan θ 2 relevant response function for θ ≥ 155º [ 2 µ 2 µ 2 ] Mott cross section r ˆ keeping only W1 : involves A(qz , k ; Sλ , T , M ) r r r r (−) ˆ ˆ A qz, k ; Sλ , T , M = Ψkr ;Sλ ,T V f j|_ (qz ) ΨM Vi 2 ( ) ( ) ( ) Breit frame transverse component of the current Electromagnetic Currents 1- and 2-body currents r r r r r r r r r j = ∑ ji ( p 'i , pi ) + j12 ( p '1 , p' 2 , p1 , p 2 ) i =1, 2 5 nucleon Pauli form factor 1-body current p'i q nucleon Dirac form factor Dirac spinor α pi   i α αβ ji = u  F1,i γ + F2,iσ q β u 2m   Höhler parametrization of f.f. keeping full Lorentz structure Electromagnetic Currents 2-body currents - π exchange current: PV coupling seagul diagrams p'i 6 π q π q ki = p'i − pi pi r r r fπ2 fπ2 V j12 = −iGE (τ 1 ×τ 2 )z NN 2 2 mπ k 2µ k 2 − mπ µ r [u1γγ 5u1 ] kν u2γ ν γ 5u2 + 1→ 2 2 ← [ ] nucleon isovector Sachs form factor keeping full Lorentz structure Electromagnetic Currents π in flight diagram p'i 7 2-body currents - π exchange current: PV coupling π q 2 r r f πNN r r V r j12 = iGE (τ 1 ×τ 2 )z k 2 − k1 2 mπ ki = p'i − pi pi ( ) ν 2 1 ν 5 1 2 ν 5 2 f π2 µ 2 f π2 µ 2 k1 k1 − mπ k 2 k 2 − mπ µ µ [k u γ γ u ] [k u γ γ u ] ν 1 nucleon isovector Sachs form factors keeping full Lorentz structure Relativistic Hamiltonian 8 Relativistic Hamiltonian to generate wave funct. H = 2 p +m +v ► vµ NN interaction and consists of µ 2 2 µ vR – short range part paramatrized as Argonne v18 vπµ − relativistic OPEP µ = 1 – pseudovector coupling µ = -1 – pseudoscalar coupling related by unitary µ = 0 – minimal non-locality choice transform. Relativistic OPEP off-shell term 9 Relativistic OPEP with off-shell term - nonlocal r r m NR m m m − µ ( E '− E ) O v ( p ' , p ) = vπ E' E E' E µ π vπ = − NR f 2 πNN fπ 2 2 2 mπ k 2 + mπ O= r r r σ 1 ⋅ p'σ 2 ⋅ p' E '+ m r − rr r σ 1 ⋅ pσ 2 ⋅ p E+m r choosing µ = 1 for consistency with π exchange current initial and final states generated with this interaction Wave Functions Deuteron wave function – bound state 10 r r r ψ M ( p;0) = ψ M ;S ( p;0) + ψ M ; D ( p;0) r in the pair cm frame : Vi = 0 Deuteron properties ≠ due to: local character of NR interaction nonlocal character of R interaction Wave Functions Deuteron wave function – bound state NR - D state % - 5.76% R - D state % - 6.24% 11 u (r ) r w( r ) r differences due to the non-local character of vπµ, which has short range Wave Functions np wave function – scattering state standard Lippmann-Shwinger equation plane wave piece interaction piece 12 Ψ ( −) r k , Sλ T r r r ( −) ( p;0) = φ kr , SλT ( p;0) + ψ kr , SλT ( p;0) r in the pair cm frame : V f = 0 incoming boundary conditions all channels up to J=3 1 ∑ ∫ (2π ) 2 M' S r dk ' r r ST * Tλλ ' (k , k ' ) 3 E k − E k ' − iε r φ kr ', Sλ 'T ( p;0) NN T-matrix Wave Functions np wave function – scattering state 1 13 S0 NR R u (r; E ) r tiny relativistic effects due to vanishing tensor force Boosts r paralel to V 14 Boosting wave functions from cm to Breit frame r r r r r r r Ψ ( p;V ) ≡ B( p;V )Ψ ( p|| γ , p| _ ;0) perpend to V Lorentz contraction in move direction r r r r 1  i r r B ( p; V ) = V ⋅ (σ 1 − σ 2 ) × p  1 − γ  4m  kinematical boost corrections retained spin-dependent (Thomas preces.) included to order V 2 interaction-dependent terms neglected IA results Tcm=1.5MeV, θ=155º relativistic effects significant only for Q2 ≥ 40 fm-2 boost effects dominate and increase the cross section 15 IA+π results π Tcm=1.5MeV, θ=155º 1- and 2-body currents Tcm=1.5MeV, θ=155º 16 π exchange current important contribution rel. effects significant even for low Q2 boost effects dominate for Q2 > 60fm-2 and increase the cross section rel. effects in wave functions and currents reduce the cross section for Q2 ~18 - 40 fm-2 Summary IA relativistic effects significant only for Q2 ≥ 40 fm-2 boost effects dominate and increase the cross section dominant boost correction comes from Lorentz contraction 17 Summary IA+π π-exchange current means important contribution specially in the relativistic calculation relativistic effects are significant even for low Q2 18 relativistic effects in wave functions and currents reduce cross section in the region Q2 ~18 – 40 fm-2 boosts effects dominate only for Q2 > 60 fm-2 and increase the cross section dominant boost contribution comes from Lorentz contraction retardation in currents gives negligible effect Summary NR and R calculations do not reproduce data at Q2 > 40 fm-2 Need of new model for the currents? Inclusion of additional short range two-body currents 19 Need of more work to understand the discrepancy between theory and data