Universidade Cruzeiro do Sul. e Tecnológicas - CETEC. Light-Cone QCD, and Nonpertubative Hadrons Physics. Electromagnetic Current of a Composed

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Φιλομενος Κεδίκογλου
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. Universidade Cruzeiro do Sul Centro de Ciências Exatas e Tecnológicas - CETEC Light-Cone QCD, and Nonpertubative Hadrons Physics Electromagnetic

1 . Universidade Cruzeiro do Sul Centro de Ciências Exatas e Tecnológicas - CETEC Light-Cone QCD, and Nonpertubative Hadrons Physics Electromagnetic Current of a Composed Vector Particle in the Light-Front J. Pacheco B. C. de Melo UNICSUL - CETEC IFT - UNESP Tobias Frederico (CTA-ITA) Minneapolis, May, 006

2 Summary 1. Light-Front Formalism. Spin-1 Particle: Rho Meson 3. Covariance Restoration in the Light-Front 4. Conclusions

3 Light-Front Formalism Light-Front Coordinates x + = t + z x + = x 0 + x 3 = Time x = t z x = x 0 x 3 = Position Four-Vector = x µ = (x 0, x 1, x, x 3 ) = (x +, x, x ) Metric Tensor g µν = Scalar product and g µν = x y = x µ y µ = x + y + + x y + x 1 y 1 + x y = 1 (x+ y + x y + ) x y p + = p 0 + p 3 p = p 0 p 3 p = (p 1, p ) Dirac Matrix γ + = γ 0 + γ 3 = Electr. Current J + = J 0 + J 3 γ = γ 0 γ 3 = Electr. Current J = J 0 J 3 γ = (γ 1, γ ) = Electr. Current J = (J 1, J )

4 Fig. 1: Ligh-Front p µ x µ = p+ x +p x + p x x +, x, x = p +, p, p p = Light-Front Energy p = p + p p = p = p +m p + Bosons: = S F (p) = 1 p m +ıɛ Fermions: = S F (p) = /p+m p m +ıɛ + γ+ p + Ref: Phys. Rept. 301, (1998) S. J. Brodsky, H.C. Pauli and S.S. Pinsky

5 Spin-1 Particle: Rho Meson General Electromagnetic Current J µ αβ = [F 1(q )g αβ F (q ) q αq β ]p µ F m 3 (q )(q α g µ β q βg α µ ), ρ Polarization Vectors ɛ µ x = ( η, 1 + η, 0, 0), ɛ µ y = (0, 0, 1, 0), ɛ µ z = (0, 0, 0, 1), ɛ µ x = ( η, 1 + η, 0, 0), ɛ µ y = ɛ y, ɛ µ z = ɛ z where η = q /4m ρ Breit Frame p µ i = (p 0, q x /, 0, 0) Initial p µ f = (p0, q x /, 0, 0) Final where p 0 = m ρ 1 + η.

6 k P k P P k P Fig. : Light-front time-ordered triangle diagram for the electromagnetic current.

7 Plus Component of the Electromagnetic Current J + ji = ı d 4 k (π) 4 T r[ɛ β j Γ β(k, k p f )(/k /p f + m) ((k p i ) m + ıɛ)(k m + ıɛ) γ+ (/k /p i + m)ɛ α i Γ α(k, k p i )(/k + m)]λ(k, p f )Λ(k, p i ) ((k p f ) m + ıɛ Regulator Function ρ-meson Vertex Λ(k, p i(f) ) = N/((p k) m R + ıɛ) Γ µ (k, p) = γ µ m ρ k µ p µ p.k + m ρ m ıɛ Mass Squared (x = k+ P + = 0 < x < 1) M (m a, m b ) = k + m a x + ( p k) + m b 1 x p Free Mass M 0 (m, m) and Function M R (m, m R) Wave Function Φ i (x, k ) = N (1 x) (m ρ M 0 )(m ρ M R ) ɛ i.[ γ k M 0 + m] Ref: Phy.Rev. C55 (1997) 043 J.P.B. C. de Melo and T. Frederico

8 Instant-Form Spin Base J + zx J + = 1 J + xx + J + yy J + zx J + zz J yy + J xx + J + zx J + yy J + xx J + zx J + xx + J + yy Light-Front I + = Matrix Elements I + 11 I + 10 I I + 10 I + 00 I + 10 I I + 10 I + 11 I 11 + = J xx + + (1 + η)j yy + ηj zz + ηj zx + (1 + η) I + 10 = ηj + xx + ηj + zz (η 1)J + zx (1 + η) I = J + xx + (1 + η)j + yy + ηj + zz + ηj + zx (1 + η) I + 00 = ηj + xx + J + zz ηj + zx (1 + η)

9 J + xx = J + zx = J + yy J + zz = η [I ηi 10 + ηi+ 00 I+ 1 1 ] η η 1 + η [ I (η 1)I I+ 00 η I+ 1 1 ] = I I+ 1 1 Angular Condition η [ ηi ηi I ηi+ 1 1 ] (q ) = (1 + η)i I ηI + 10 I + 00 = 0 Ref: Sov. J. Nucl. Phys. 39 (1984) 198 I.Grach and L.A. Kondratyku Phy. Rev. Lett. 6 (1989) 387 L.L. Frankfurt, I.Grach, L.A. Kondratyku and M. Strikman

10 Prescriptions : F F S GK CCKP BH vs COVARIANT Breit Frame = P + = P +, P = P, P = P = q/ B.F: q + = q 0 + q 3 = 0 { 4 Current Elements J + ρ 3 F orm F actors G 0, G 1 and G Angular Condition: Violation P arity q x = J yy + = J zz + + Rotations (q ) = (1 + η)(j + yy J + zz) = (1 + η)i I ηI + 10 I + 00 (q ) 0

11 Charge Form Factor ρ 1 m ρ =0.770 GeV m q =0.430 Ge m r =1.8 GeV Covariant FFS GK BH CCKP G 0 (q ) q [GeV] Fig. 3: Rho Meson Charge Form Factor GK Sov. J. Nucl. Phys. 39 (1984) 198 I.Grach and L.A. Kondratyku CCKP Phy.Rev C37 (1988) 000 P.L. Chung, F. Coester, B.D. Keister and W.N. Polizou FFS Phy.Rev C48 (1993) 18 L.Frankfurt, T. Frederico and M. Strikman BH Phy.Rev D46 (199) 141 S.J. Brodsky and J.R. Hiller

12 Magnetic Form Factor ρ m ρ =0.770 GeV m q =0.430 Ge m r =1.8 GeV Covariant FFS GK BH CCKP G 1 (q ) q [GeV] Fig. 4: Rho Meson Magnetic Form Factor

13 Quadrupole Form Factor ρ m ρ =0.770 GeV m q =0.430 Ge m r =1.8 GeV Covariant FFS GK BH CCKP G (q ) q [GeV] Fig. 5: Rho Meson Quadrupole Form Factor

14 Table 1: Results for the low-energy electromagnetic ρ-meson observables, for the covariant (COV) and light-front calculations. The light-front extraction schemes to obtain the form-factors are given by Refs. (GK), (CCKP), (FFS) and (BH). In the last column, the results of Ref. [*] are given. MODEL COV GK CCKP BH FFS Ref.[*] < r > (fm ) µ Q (fm ) Ref.[*] < r > = lim q 0 µ = lim G 1 (q ) q 0 6(G 0 (q ) 1) q Q = lim 3 G (q ) q 0 q Phy. Lett. B 349 (1995) 393 F. cardarelli, I.L. Grach, I.M. Narodetskii, E. Pace and G. Salmé

15 k P k P P k (a) P k P k P P k (b) P Fig. 6: Light-front time-ordered diagrams for the current: (a) Triangle Diagram and (b) Pair Terms.

16 Covariance Restoration in the Light-front Vertex Γ(q µ, q ν ) Trace T r ji = T r[(/k /p f + m)γ + (/k /p i + m) γ+ ] k ɛ µ f (kµ p µ ) ɛ ν i (kµ p µ ) + T r[(/k /p f + m)γ + (/k /p i + m)( γ k + ɛ µ f (kµ p µ )ɛ ν i (kµ p µ ) γ k + m)] By Parts: T r A ij = T r[(/k /p f + m)γ + (/k /p i + m)γ + ] T r B ij = T r[(/k /p f + m)γ + (/k /p i + m)( γ k + γ k + m)]

17 Bad Terms (Bad) = k T r Bad xx = k 3 T ra ij + (k η k q x η 1 + η)t r B ij T r God yy = (p + k + ) T r Bad zz = k 3 T ra ij + (k k k + )T r B ij T r Bad zx = k 3 η T rij A + [k η k k + η(k x + q x 1 + η)]t r B ij Integration Interval i) 0 < k + < p + ii) p + < k + < p + p + = p + + δ + (Dislocation Method) Poles Contribution i) k = f 1+m k + ii) k = p f 3 ıɛ p + k + XVth Few-Body Confer. Groningem (1997) Nucl. Phys., A 631, (1998) 574c (J.P.B.C. de Melo, J.H.O. Salles,T. Frederico and P.U.Sauer) Few-Body Syst., 4, (1998) 99 ( H.W. Naus, J.P.B.C. de Melo, T. Frederico and P.U.Sauer) Phy. Rev., C59, (1999) 78 (J.P.B.C. de Melo, H.W. Naus and T. Frederico)

18 Definitions - Feynman Propagators [1] = k + (k f 1 ıɛ k + ) [] = (p + k + )(p k f ıɛ p + k + ) [3] = (p + k + )(p k f 3 ıɛ p + k + ) [4] = (p + k + )(p k f 4 ıɛ p + k + ) [5] = (p + k + )(p k f 5 ıɛ p + k + ) [6] = (p + k + )(p k f 6 ıɛ p + k + ) [7] = (p + k + )(p k f 7 ıɛ p + k + ) Where the functions f i are given by: f 1 = k + m q f = (k p) + m q f 3 = (k p ) + m q f 4 = (k p) + m R. f 5 = (k p ) + m R f 6 = (k p) + m R. f 7 = (k p ) + m R

19 Pair Terms + (P air) J xx = lim δ (P air) J zx = lim δ (P air) J zz = lim δ (Bad) d 3 T r[j xx ] k [1][][4][5][6][7] O prop + (Bad) d 3 T r[j zx ] k [1][][4][5][6][7] O prop + (Bad) d 3 T r[j zz ] k [1][][4][5][6][7] O prop O prop = d 3 k d k dk + (π) 3 m ρ 4(p µ k µ +m q m ρ )(p µ k µ +m q m ρ ) Limit: δ + 0 lim δ + 0 lim δ + 0 lim δ + 0 d 3 T r +A ij k [1][][4][5][6][7] k 3 O prop O(δ + ) d 3 T r +B ij k [1][][4][5][6][7] k O prop O(δ + ) d 3 T r +B ij k [1][][4][5][6][7] k O prop O(δ + )

20 I. Grach and L. Kondratyku Prescription: I + 00 GK (P air) G0 = 1 + (Bad) [J xx 3 GK (P air) G1 = [ J zz + (Bad) + ηj + (Bad) zz ] = 1 3 [ k q x ( η 1 + η)t r B ij + ( ηk k + T r B ij)] = 0 J + (Bad) zx η ] = 0 GK (P air) G = 3 [J + (Bad) xx + ηj + (Bad) zz ] = 0 No Pair Terms Contribution!!!

21 Vertex Γ(q µ, γ ν ) Trace T r ji = T r[(/k /p + m)γ + (/k /p + m)/ɛ µ (/k + m)]ɛ µ j (k µ p µ) Light-Front Trace T r ji = T r[(/k /p + m)γ + (/k /p + m)/ɛ γ + ] k ɛµ j (k µ p µ) Bad Terms (Bad) / Pair Terms T r Bad xx = T r[(/k /p + m)γ + (/k /p + m)γ γ + ] k T r Bad yy = 0 T rzz Bad = T r[(/k /p + m)γ + (/k /p + m)γ γ + ] k T rzz Bad = T r[(/k /p + m)γ + (/k /p + m)γ γ + ] k η By Definition T r C ij = T r[(/k /p + m)γ + (/k /p + m)γ γ + ] T r Bad xx = T r C ij T r Bad zz = T r C ij T r Bad zz = T r C ij k η k k η

22 Pair Terms Interval p + < k + < p + where: p + = p + + δ + + (P ar) J xx = lim δ (P ar) J zx =lim δ (P ar) J zz =lim δ + 0 d 3 + P ar T r[j xx ] K [1][][4][5][6][7] m ρ (p µ k µ + m q m ρ ) 0 d 3 + P ar T r[j zx ] K [1][][4][5][6][7] m ρ (p µ k µ + m q m ρ ) 0 d 3 K T r[j + (P ar) zz ] [1][][4][5][6][7] m ρ (p µ k µ + m q m ρ ) 0 Vertex Γ(γ µ, q ν ) No Pair Terms Contributions!!!

23 Vertex Γ(γ µ, γ ν ) T r ji = T r[/ɛ α f (/k /p + m)γ + (/k /p + m)/ɛ α i (/k + m)] Bad Terms (Bad) (k ) T r Bad ji = k T r[/ɛα f (/k /p + m)γ + (/k /p + m)/ɛ α γ + ] T rxx Bad = k η T r[γ (/k /p + m)γ + (/k /p + m)/ɛ α i 8 γ γ + ] T ryy Bad = k (k + p + ) = 0 T r Bad zz = k 8 T r[γ (/k /p + m)γ + (/k /p + m)/ɛ α i γ γ + ] T r Bad zx = k η 8 Fact: T r[γ (/k /p + m)γ + (/k /p + m)/ɛ α i γ γ + ] k (m+1) (p + k + ) n No Pair Terms Contribution if m < n Simplification: is VIP T rxx Bad T rzx Bad = η T r Bad zz = η T r Bad zz

24 Pair Terms + (P air) J xx = lim δ (P air) J zx = lim δ (P air) J zz = lim δ + 0 d 3 K T r[j + (Bad) xx ] [1][][4][5][6][7] +(Bad) d 3 T r[j zx ] K [1][][4][5][6][7] d 3 K T r[j + (Bad) zz ] [1][][4][5][6][7] Pair Term Contribution!!! Grach and Kondratyku : Elimination I + 00 G GK 0 = 1 3 [J + xx + J + yy ηj + yy + ηj + zz] G GK 1 = J + yy J + zz J + zx η G GK = 3 [J + xx + J + yy( 1 η) + ηj + zz]

25 Pair Terms Combination GK (P air) G0 = 1 + (Bad) [J xx + ηj zz + (Bad) ] = (Bad) [ ηj zz + ηj zz + (Bad) ] = 0 3 GK (P air) G1 = J zz + (Bad) J + (Bad) zx η = J + (Bad) zz + η J + (Bad) zz η = 0 GK (P air) G = 3 3 [J + (Bad) xx [ ηj + (Bad) zz + ηj + (Bad) zz ] = + ηj + (Bad) zz ] = 0 Final Result: No Pair Terms Contribution!! Ref. De Melo and T. Frederico Braz. J. Phys. Vol.34, 3A, (004) 881 B.L.G. Bakker and C.R. Ji Phy. Rev. D65 (00)

26 Electromagnetic Current Vertex Γ (γ µ,γ ν ) m q =0.430 GeV m R =1.8 GeV 0.9 Covariant Light Front Whitout Pair Terms Light Front + Pair Terms 0.4 J + xx q [GeV] Fig. 7: Spin-1 Electromagetic Current

27 0.8 Electromagnetic Current Vertex Γ (γ µ,γ ν ) m q =0.430 GeV m R =1.8 GeV 0.6 J + zx Covariant Light Front Whitout Pair Terms Light Front + Pair Terms q [GeV] Fig. 8: Spin-1 Electromagetic Current

28 Electromagnetic Current Vertex Γ (γ µ,γ ν ) m q =0.430 GeV m R =1.8 GeV Covariant Light Front Whitout Pair Terms Light Front + Pair Terms 0.8 J + zz q [GeV] Fig. 9: Spin-1 Electromagetic Current

29 1. Electromagnetic Current Vertex Γ (γ µ,γ ν ) m q =0.430 GeV m R =1.8 GeV 1 Covariant Light Front Whitout Pair Terms Light Front + Pair Terms 0.8 J + yy q [GeV] Fig. 10: Spin-1 Electromagetic Current

30 Conclusions Light-Front = { Bound States Covariance Rotational Invariance Broken = k Problematic { Good Terms Bad Electromagnetic Current: +, - and Pair Terms Contribution: = J + and J Bosons Particles P seudoscalar V ector Pairs Terms Contribution = Full Covariance Restorate J + is not free of the Pair Terms Contribution!!! 30

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