Angular Momentum in Light-Front Dynamics Chueng-Ryong Ji North Carolina State University INT Worksho INT-1-49W Orbital Angular Momentum in QCD Seattle, WA, Feb. 7, 01
ν Poincaré Algebra [ µ, ν ] 0 [ µ,j ρσ ] i(g µρ σ g µσ ρ ) [J µν,j ρσ ] i(g µσ J νρ g νρ J µσ g µρ J νσ g νσ J µρ ) 0 1 3 d 3 x T 0ν µ T µν 0 ; 0 K 1 K K 3 K 1 0 J 3 J J µν K J 3 0 J 1 K 3 J J 1 0 T µν M µνλ M µνλ µνλ "O" M "S" M µνλ "O" T µν x λ T µλ x ν k d 3 x M 0µν L ( µ φ k ) ν φ k g µν L µ M µνλ "O" 0 T λν T νλ ; µ M µνλ "S" 0 F. J. Belinfante, Physica 6, 887 (1939); L. Rosenfeld, Mem. Acad. Roy. Belg. 18, 6 (1940).
Dirac s Proosition 1949
6 Stability rou Just Formal? or Physical Consequences? 7 6
Outline Why LFD? - Interolation between Instant and Front Forms - Distinguished Features in LFD Angular Momentum in LFD - Light-Front Helicities - Swa of Helicity Amlitudes Model Indeendent eneral Angular Condition - N-Δ Transition Process - Deuteron Form Factors Conclusion
Interolation between Instant and Front Forms K. Hornbostel, PRD45, 3781 (199); C.Ji and C. Mitchell, PRD64,085013 (001).
g µν 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 g ˆ µ ν ˆ cosδ 0 0 sinδ 0 1 0 0 0 0 1 0 sinδ 0 0 cosδ J µν 0 K 1 K K 3 K 1 0 J 3 J K J 3 0 J 1 K 3 J J 1 0 J ˆ µ ν ˆ ˆ E 1 J sinδ K 1 cosδ ˆ E K cosδ J 1 sinδ ˆ F 1 K 1 sinδ J cosδ ˆ F J 1 cosδ K sinδ 0 E ˆ E ˆ K 3 E ˆ 0 J 3 F ˆ E ˆ J 3 0 F ˆ K 3 F ˆ 1 ˆ F 0
δ 0 0 0 3 3 0 < δ < π /4 ˆ 0 cosδ 3 sinδ ˆ 0 sinδ 3 cosδ δ π /4 Transverse Boost : Transverse Rotation : Longitudinal Boost : Longitudinal Rotation : [ E ˆ i, ˆ ] 0 [ F ˆ i, ˆ ] 0 K 3 J 3 ( i 1, ) Immune to Interolation
Kinematic Oerators (Members of Stability rou) ( ) x ˆ Ex iωℵ ˆ i [ˆ ℵ i, ˆ ] 0 > x ˆ ℵ ˆ i F ˆ i cosδ E ˆ i sinδ > δ 0 J J 1 ℵ ˆ 1 J cosδ K 1 sinδ ˆ ℵ J 1 cosδ K sinδ (J 3, 1,, ˆ ) δ π /4 E 1 (J K 1 ) / E (J 1 K ) /
Corresonding Dynamic Oerators ˆ i F ˆ i sinδ E ˆ i cosδ J ˆ µ ν ˆ 0 E ˆ E ˆ K 3 E ˆ 0 J 3 F ˆ E ˆ J 3 0 F ˆ K 3 F ˆ 1 ˆ F 0 J ˆ µ ν ˆ 0 ˆ 1 ˆ K 3 ˆ 1 0 J 3 ℵ ˆ 1 ˆ J 3 0 ℵ ˆ K 3 ℵ ˆ 1 ℵ ˆ 0 δ 0 K 1 K ˆ 1 K 1 cosδ J sinδ ˆ K cosδ J 1 sinδ (K 3, ˆ ) δ π /4 F 1 (K 1 J ) / F (J 1 K ) /
Longitudinal Boost K 3 [K 3, ˆ ] iˆ i(ˆ cosδ ˆ sinδ) δ π /4 [K 3, ] i Ex( iω K 3 ) x > x > One more kinematic generator aears only in the front form. Maximum number (7) of members in the stability grou.
Kinematic Transformation 1 0, 0, ˆ M sinδ ( 0 M, 3 0) { } T Ex i(β ˆ 1 ℵ 1 β ˆ ℵ ) 1 β 1 M sinδ sinα /α, β M sinδ sinα /α, ˆ M sinδ cosα ; α (β 1 β )cosδ δ π /4 β M /, M / ( ( M ) / M, 3 M )
Angular Momentum [J i,j j ] iε ijk J k,[j i,m] 0 { } T Ex i(β ˆ 1 ℵ 1 β ˆ ℵ ) [ ˆ I i, ˆ I j ] iε ijk ˆ I k, [ ˆ I i, M] 0 T n >,n > ˆ I i,n > TJ i n > I ˆ i TJ i T
I ˆ 3 J 3 ˆ ˆ z ( ℵ ˆ ) / M sinδ I ˆ J z cosδ J 3 ˆ ( ℵ ˆ ) ˆ M sinδ (ˆ z )sinδ K 3 δ π /4 ˆ E ˆ M sinδ I I 3 J 3 z ˆ ( z ˆ ( E F K 3 ) I 3 W E ) / ( ) J 3 z ˆ E / M W µ 1 ε µναβ ν J αβ [I 3, Stability rou Members] 0
Front Form Current Matrix Element (Light-Front Helicity Amlitude) µ L λ ʹ λ L j'm'; ʹ, λ ʹ J µ (0) jm;,λ L I 3 jm;,λ > L λ jm;,λ > L The same in all frames related by front-form boosts for each fixed set of angular momentum indices. The value of this matrix element is indeendent of all reference frames related by front-form boosts. e.g. Sin-1 Form Factors µ ε *α ( ʹ, h ʹ )J µ ε β (, h) h ʹ h αβ J µ αβ ( ') µ g αβ q q µ µ α β µ F1 ( q ) ( gα qβ g β qα ) F ( q ) ( ') F3 ( q M W )
3 0 3 1 3 0 0 ), ( Equal t Equal τ 0 m m Energy-Momentum Disersion Relations Distinguished Features in LFD
Zero-Mode Issue in LFD Valence and Nonvalence Contributions Even if q 0, the off-diagonal elements do not go away in some cases. 0 (...) lim 0 q q dk For examle, 00 has the zero-mode contribution in the Standard Model W ± form factors. n n n n n n q W ± ( ) 0 ) (1 ) (1 1 1 1 0 3.. 00 Q x x m k Q x x m k k d dx M Q g W f M Z π B.Bakker and C.Ji, PRD71,053005(005)
PDs rely on the handbag dominance in DVCS; i.e. Q >> any soft mass scale q q q - -q -q -Q <0, e.g. S.J.Brodsky,M.Diehl,D.S.Hwang, NPB596,99 (001)
Swa B.Bakker and C.Ji, Phys.Rev.D83,09150(R) (011)
JLab Kinematics t < - t min 0 Discretion is advised in alying the t0 formulation of DVCS in terms of PDs for the data and/or hadron model analysis.
eneral Angular Condition 0,, ʹ ʹ ʹ ʹ µ µ µ µ ν ν µ µ if J B B B ) ( ) (,, θ θ λ λ λµ ν µ µ µ λ ν ν λ λ ʹ ʹ ʹ ʹ ʹ ʹ ʹ ʹ j B j L L L d d J Angular Momentum Conservation in Breit Frame µ q (q -Q ) λ γ µ ( m ) ( M ) λ γ µ µ Light-front Helicity Amlitude in q 0 Frame tan, tan m M Q MQ m M Q mq ʹ θ θ C.Carlson and C.Ji, Phys.Rev.D67,11600(03)
N-Δ Transition < Δ J µ (0) N > eψ β ( ʹ )Γ βµ Ψ( ) Γ βµ 1 (q β γ µ q / g βµ )γ 5 (q β µ ʹ µ q ( ʹ ) g βµ )γ 5 M * E * C * 3 1 3 (q β q µ q g βµ )γ 5 (3M m)( M m) Q M m Q m 1 / M M m Q M m Q 3( M m) 4M 3M m Q ( M m Q ) 3 Q 1 Q 1 1 1 µ h ʹ h u β ( ʹ, h ʹ )Γ βµ u(, h) { 1 ( M m) } ; 3 Q 1 Q ; 6M { 1 (M m) ( M m) 3 } ; 6M {m 1 (Q mm M ) Q 3 } ;
Helicity Amlitudes in N-Δ Transition 3 1 3 1 1 1 1 1 1 1 1 1 3 1 3 1
Helicity Amlitudes in N-Δ Transition LF Parity: Y P R y (π )P 1 1 3 1 3 1 3 1 1 1 1 1 1 1 1 1 3 1 3 1 1 1 3 1 λ ʹ λ η ʹ η ( 1) ʹ P P j j λ ʹ λ λ ʹ λ C.Carlson and C.Ji, Phys.Rev.D67,11600(03)
Helicity Amlitudes in N-Δ Transition µ µ ʹ ; µ 1, µ ʹ 3 d 3 ( ʹ ʹ λ 3 θ ) ʹ Angular Condition 3 1 1 λ λ d 1 λ (θ) 0 3 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 3 1 3 1 3 1 Q[Q m( M m)] 3 1 3MQ 1 1 3MQ( M m) 1 [( M m)( M m ) mq ] 1
High Q Scaling in QCD (modulo logarithm) 1 λ ʹ λ Λ Q λ ʹ λ min λ λ min λmin λ min Λ Q (n 1) λ ʹ λ min λ λ min n the number of quarks in the state; λ min 0 (bosons) or 1/ (fermions); Λ QCD scale; e.g. X.Ji, J.-P.Ma, and F.Yuan, PRL90, 41601 (003) ~ 1 Q ; 4 1 1 a Λ 1 1 ; 3 1 b Λ 1 1 ; 1 c Λ Q Q Q 1 3 1 1 3 1 Q[Q m( M m)] 3 1 3MQ 1 1 3MQ( M m) 1 [( M m)( M m ) mq ] 1 3 bλ M 0 ; b 3M Λ 0 Leading PQCD ostoned to a larger Q region; 1eV ugrade anticiated.
Deuteron Form Factors ) ( ') ( ) ( ) ( ) ( ') ( 3 1 q F M q q q F q g q g q F g J W µ β α α µ β β µ α αβ µ µ αβ C M Q 1 3 η 3 η 3 η(1 η) 0 1 0 1 1 1 η F 1 F F 3 ʹ h h µ ε *α ( ʹ, ʹ h )J αβ µ ε β (, h) η Q 4M Deuteron Helicity Amlitudes
Deuteron Helicity Amlitudes 0 0 00 0 0
Deuteron Helicity Amlitudes LF Parity: Y P R y (π )P 0 0 00 0 0 0 λ ʹ λ 0 η ʹ η ( 1) ʹ P P j j λ ʹ λ λ ʹ λ
Deuteron Helicity Amlitudes LF Time-Reversal: Y T R y (π )T 0 0 0 00 0 0 0 0 λ ʹ λ ( 1) ʹ λ λ λλ ʹ C.Carlson and C.Ji, Phys.Rev.D67,11600(03)
(η 1) 8η 0 0 00 Angular Condition 0 0 0 00 0 0 0 0
λ ʹ λ Λ Q High Q Scaling in QCD λ ʹ λ min λ λ min λmin λ min Λ Q (n 1) λ ʹ λ min λ λ min n the number of quarks in the state; λ min 0 (bosons) or 1/ (fermions); Λ QCD scale 00 ~ 1 Q ; 4 0 a Λ 00 ; b Λ 00 ; c Λ Q Q Q (η 1) 8η 0 0 00 00 1 aλ M 1 cλ 0 ; a or c must be Ο M 0 M Λ Leading PQCD ostoned to a larger Q region; 1eV ugrade anticiated.
Conclusion Angular momentum in LFD is not just formal but consequential in the analysis of hysical observables. LF helicity amlitudes are indeendent of all references frames that are related by LF boosts. Model indeendent constraints can be made using LFD.