Momentum-space treatment of the Coulomb force: Screening and - PowerPoint PPT Presentation

Momentum-space treatment of the Coulomb force: Screening and renormalization A. Deltuva Vilnius University In collaboration with A. C. Fonseca and P . U. Sauer

Outline Momentum-space description of few-body scattering: screening and renormalization for Coulomb [Taylor, Alt, Sandhas, ...] S&R variations and other methods Applications: 3N, 4N, nuclear reactions, ...

Screened Coulomb C ( r ) e − ( r R ) n w R ( r ) = w standard scattering theory

Screened Coulomb C ( r ) e − ( r R ) n w R ( r ) = w standard scattering theory nature: Coulomb is screened at large distances large R : physical observables insensitive to screening, screened and full Coulomb physically indistinguishable

Screened Coulomb C ( r ) e − ( r R ) n w R ( r ) = w standard scattering theory nature: Coulomb is screened at large distances large R : physical observables insensitive to screening, screened and full Coulomb physically indistinguishable in the R → ∞ limit physical results are recovered

Screened and full Coulomb physically indistinguishable ? � p ′ | T R | p � � p ′ | T − − − → C | p � R → ∞

Screened and full Coulomb physically indistinguishable ? e 2 i φ R � p ′ | T R | p � � p ′ | T − − − → C | p � R → ∞

Screened and full Coulomb physically indistinguishable initial physical state: wave packet ϕ in ( p ) outgoing wave packet � ϕ out ( p ′ ) = d 3 p � p ′ | S | p � ϕ in ( p ) ? � � d 2 ˆ p e 2 i φ R � p ′ | T R | p � ϕ in ( p ) − d 2 ˆ C | p � ϕ in ( p ) p � p ′ | T ∼ − − → R → ∞

Screened and full Coulomb physically indistinguishable initial physical state: wave packet ϕ in ( p ) outgoing wave packet � ϕ out ( p ′ ) = d 3 p � p ′ | S | p � ϕ in ( p ) ? � � d 2 ˆ p e 2 i φ R � p ′ | T R | p � ϕ in ( p ) − d 2 ˆ C | p � ϕ in ( p ) p � p ′ | T ∼ − − → R → ∞ p ′ = p : e 2 i φ R � p ′ | T R | p � − R → ∞ � p ′ | T − − → C | p � as distribution

Screened and full Coulomb physically indistinguishable initial physical state: wave packet ϕ in ( p ) outgoing wave packet � ϕ out ( p ′ ) = d 3 p � p ′ | S | p � ϕ in ( p ) ? � � d 2 ˆ p e 2 i φ R � p ′ | T R | p � ϕ in ( p ) − d 2 ˆ C | p � ϕ in ( p ) p � p ′ | T ∼ − − → R → ∞ p ′ = p : e 2 i φ R � p ′ | T R | p � − R → ∞ � p ′ | T − − → C | p � as distribution φ R − R → ∞ [ σ L − η LR ] − R → ∞ α e M / p [ ln ( 2 pR ) − C / n ] − − → − − → [ J. R. Taylor, Nuovo Cimento B23 , 313 (1974) ]

Screened and full Coulomb wave functions r < R : w R ( r ) ≈ w C ( r ) ⇓ e i φ LR � r | ψ (+) LR ( p ) � ≈� r | ψ (+) LC ( p ) �

Screened and full Coulomb wave functions r < R : w R ( r ) ≈ w C ( r ) ⇓ e i φ LR � r | ψ (+) LR ( p ) � ≈� r | ψ (+) LC ( p ) � e i φ R | ψ (+) R → ∞ | ψ (+) R ( p ) � − − − → C ( p ) � [ V. G. Gorshkov, Sov. Phys.-JETP 13 , 1037 (1961) ]

Screening and renormalization Renormalization of the on-shell screened Coulomb transition matrix T R = w R + w R G 0 T R and wave function in the limit R → ∞ yields Coulomb amplitude and Coulomb wave function T R z − 1 R → ∞ T R − − − → C as distribution ( 1 + G 0 T R ) | p � z − 1 / 2 R → ∞ | ψ (+) − − − → C ( p ) � R z R = e − 2 i φ R

Two-particle scattering transition matrix T ( R ) = v + w R +( v + w R ) G 0 T ( R )

Two-particle scattering transition matrix T ( R ) = v + w R +( v + w R ) G 0 T ( R ) with long-range and Coulomb-distorted short-range parts T ( R ) = T R +( 1 + T R G 0 ) ˜ T ( R ) ( 1 + G 0 T R ) T ( R ) = v + vG R ˜ T ( R ) ˜

Two-particle scattering transition matrix T ( R ) = v + w R +( v + w R ) G 0 T ( R ) with long-range and Coulomb-distorted short-range parts T ( R ) = T R +( 1 + T R G 0 ) ˜ T ( R ) ( 1 + G 0 T R ) T ( R ) = v + vG R ˜ T ( R ) ˜ Renormalized amplitude: T ( R ) z − 1 T ( C ) | ψ (+) R → ∞ T = T C + � ψ ( − ) C | ˜ R − − − → C �

Two-particle scattering transition matrix T ( R ) = v + w R +( v + w R ) G 0 T ( R ) with long-range and Coulomb-distorted short-range parts T ( R ) = T R +( 1 + T R G 0 ) ˜ T ( R ) ( 1 + G 0 T R ) T ( R ) = v + vG R ˜ T ( R ) ˜ Renormalized amplitude: T ( R ) z − 1 T ( C ) | ψ (+) R → ∞ T = T C + � ψ ( − ) C | ˜ R − − − → C � − 1 − 1 R [ T ( R ) − T R ] z = T C + lim R → ∞ z 2 2 R short-range part: fast convergence with R

Test: convergence with R in pp scattering w R ( r ) w C ( r ) = e − ( r R ) n n = 1 1 n = 4 n → ∞ 0 0 1 2 r/R

Test: convergence with R in pp scattering exact w R ( r ) w C ( r ) = e − ( r R ) n n = 1 n = 4 n → ∞ 52 η (deg) n = 1 0.001% 1 n = 4 ↓ n → ∞ ↑ 50 1 S 0 E p = 3 MeV 0 0 1 2 10 20 30 40 r/R R (fm) optimal choice: 3 ≤ n ≤ 8

Limits of practical applicability p → 0 : κ = α M / p , σ L = arg Γ ( 1 + L + i κ ) , and z R diverge, renormalization procedure ill-defined

Limits of practical applicability p → 0 : κ = α M / p , σ L = arg Γ ( 1 + L + i κ ) , and z R diverge, renormalization procedure ill-defined ⇒ slow convergence with R at low relative energies 46.0 1 MeV 45.5 η (deg) 0.1 MeV 7.0 6.5 0.01 MeV 0.5 0.0 20 100 500 R (fm)

Three-particle scattering: short-range forces Faddeev / Alt, Grassberger, and Sandhas equations 0 + ∑ δ βα G − 1 δ βσ T σ G 0 U σα U βα = ¯ ¯ σ 0 + ∑ U 0 α = G − 1 T σ G 0 U σα σ T σ = v σ + v σ G 0 T σ G 0 = ( E + i 0 − H 0 ) − 1 momentum-space partial-wave representation

AGS equations with 3BF 3 ∑ V 3 BF = u α α = 1 0 + ∑ δ βα G − 1 δ βγ T γ G 0 U γα U βα = ¯ ¯ γ + u α + ∑ u γ G 0 ( 1 + T γ G 0 ) U γα γ

Three-particle scattering: including screened Coulomb Faddeev / Alt, Grassberger, and Sandhas equations 0 + ∑ U ( R ) δ βσ T ( R ) σ G 0 U ( R ) δ βα G − 1 βα = ¯ ¯ σα σ 0 + ∑ U ( R ) T ( R ) σ G 0 U ( R ) 0 α = G − 1 σα σ T ( R ) = v σ + w σ R +( v σ + w σ R ) G 0 T ( R ) σ σ G 0 = ( E + i 0 − H 0 ) − 1 momentum-space partial-wave representation

Three-particle scattering: including screened Coulomb Faddeev / Alt, Grassberger, and Sandhas equations 0 + ∑ U ( R ) δ βσ T ( R ) σ G 0 U ( R ) δ βα G − 1 βα = ¯ ¯ σα σ 0 + ∑ U ( R ) T ( R ) σ G 0 U ( R ) 0 α = G − 1 σα σ T ( R ) = v σ + w σ R +( v σ + w σ R ) G 0 T ( R ) σ σ G 0 = ( E + i 0 − H 0 ) − 1 momentum-space partial-wave representation Additional difficulties: quasi-singular nature of screened Coulomb potential slow partial-wave convergence

Three-particle scattering: including screened Coulomb Faddeev / Alt, Grassberger, and Sandhas equations 0 + ∑ U ( R ) δ βσ T ( R ) σ G 0 U ( R ) δ βα G − 1 βα = ¯ ¯ σα σ 0 + ∑ U ( R ) T ( R ) σ G 0 U ( R ) 0 α = G − 1 σα σ T ( R ) = v σ + w σ R +( v σ + w σ R ) G 0 T ( R ) σ σ G 0 = ( E + i 0 − H 0 ) − 1 momentum-space partial-wave representation Additional difficulties: quasi-singular nature of screened Coulomb potential slow partial-wave convergence R → ∞ limit?

Three-particle scattering: R → ∞ limit • W c . m . α R • long-range part • α R G ( R ) T c . m . α R = W c . m . α R + W c . m . α T c . m . α R

Three-particle scattering: R → ∞ limit • W c . m . α R • Split into long-range part • α R G ( R ) T c . m . α R = W c . m . α R + W c . m . α T c . m . α R and Coulomb-distorted short-range part U ( R ) β R G ( R ) U ( R ) βα [ 1 + G ( R ) βα = δ βα T c . m . α R +[ 1 + T c . m . α T c . m . β ] ˜ α R ] U ( R ) U ( R ) 0 α [ 1 + G ( R ) [ ρ is neutral ] 0 α = [ 1 + T ρ R G 0 ] ˜ α T c . m . α R ]

Three-particle scattering: R → ∞ limit • W c . m . α R • Split into long-range part • α R G ( R ) T c . m . α R = W c . m . α R + W c . m . α T c . m . α R and Coulomb-distorted short-range part U ( R ) β R G ( R ) U ( R ) βα [ 1 + G ( R ) βα = δ βα T c . m . α R +[ 1 + T c . m . α T c . m . β ] ˜ α R ] U ( R ) U ( R ) 0 α [ 1 + G ( R ) [ ρ is neutral ] 0 α = [ 1 + T ρ R G 0 ] ˜ α T c . m . α R ] Renormalized amplitudes: − 1 − 1 R f [ U ( R ) U βα = δ βα T c . m . βα − δ βα T c . m . R → ∞ Z α R ] Z α C + lim 2 2 Ri − 1 − 1 R U ( R ) U 0 α = lim R → ∞ z 0 α Z 2 2 Ri

Three-particle scattering: R → ∞ limit • W c . m . α R • Split into long-range part • α R G ( R ) T c . m . α R = W c . m . α R + W c . m . α T c . m . α R and Coulomb-distorted short-range part U ( R ) β R G ( R ) U ( R ) βα [ 1 + G ( R ) βα = δ βα T c . m . α R +[ 1 + T c . m . α T c . m . β ] ˜ α R ] U ( R ) U ( R ) 0 α [ 1 + G ( R ) [ ρ is neutral ] 0 α = [ 1 + T ρ R G 0 ] ˜ α T c . m . α R ] Renormalized amplitudes: − 1 − 1 R f [ U ( R ) U βα = δ βα T c . m . βα − δ βα T c . m . R → ∞ Z α R ] Z α C + lim 2 2 Ri − 1 − 1 R U ( R ) U 0 α = lim R → ∞ z 0 α Z 2 2 Ri short-range part: fast convergence with R