Distribution Amplitudes of the Nucleon and its resonances

Transcription

1 Distribution Amplitudes of the Nucleon and its resonances C. Mezrag Argonne National Laboratory November 16 th, 2016 In collaboration with: C.D. Roberts and J. Segovia C. Mezrag (ANL) Nucleon DA November 16 th, / 34

2 Nucleon Distribution Amplitudes 3 bodies matrix element: 0 ɛ ijk u i α(z 1 )u j β (z 2)d k γ (z 3 ) P C. Mezrag (ANL) Nucleon DA November 16 th, / 34

3 Nucleon Distribution Amplitudes 3 bodies matrix element expanded at leading twist: 0 ɛ ijk uα(z i 1 )u j β (z 2)dγ k (z 3 ) P = 1 [ (/ pc ) ( γ5 N +) 4 αβ γ V (z i ) + ( /pγ 5 C ) ( αβ N + ) γ A(z i ) (ip µ ( σ µν C) αβ γ ν γ 5 N +) γ T (z i )] V. Chernyak and I. Zhitnitsky, Nucl. Phys. B 246, (1984) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

4 Nucleon Distribution Amplitudes 3 bodies matrix element expanded at leading twist: 0 ɛ ijk uα(z i 1 )u j β (z 2)dγ k (z 3 ) P = 1 [ (/ pc ) ( γ5 N +) 4 αβ γ V (z i ) + ( /pγ 5 C ) ( αβ N + ) γ A(z i ) (ip µ ( σ µν C) αβ γ ν γ 5 N +) γ T (z i )] Usually, one defines ϕ = V A V. Chernyak and I. Zhitnitsky, Nucl. Phys. B 246, (1984) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

5 Nucleon Distribution Amplitudes 3 bodies matrix element expanded at leading twist: 0 ɛ ijk uα(z i 1 )u j β (z 2)dγ k (z 3 ) P = 1 [ (/ pc ) ( γ5 N +) 4 αβ γ V (z i ) + ( /pγ 5 C ) ( αβ N + ) γ A(z i ) (ip µ ( σ µν C) αβ γ ν γ 5 N +) γ T (z i )] V. Chernyak and I. Zhitnitsky, Nucl. Phys. B 246, (1984) Usually, one defines ϕ = V A 3 bodies Fock space interpretation (leading twist): [dx] P, = 8 uud [ϕ(x 1, x 2, x 3 ) 6x 1 x 2 x 3 +ϕ(x 2, x 1, x 3 ) 2T (x 1, x 2, x 2 ) ] C. Mezrag (ANL) Nucleon DA November 16 th, / 34

6 Nucleon Distribution Amplitudes 3 bodies matrix element expanded at leading twist: 0 ɛ ijk uα(z i 1 )u j β (z 2)dγ k (z 3 ) P = 1 [ (/ pc ) ( γ5 N +) 4 αβ γ V (z i ) + ( /pγ 5 C ) ( αβ N + ) γ A(z i ) (ip µ ( σ µν C) αβ γ ν γ 5 N +) γ T (z i )] V. Chernyak and I. Zhitnitsky, Nucl. Phys. B 246, (1984) Usually, one defines ϕ = V A 3 bodies Fock space interpretation (leading twist): [dx] P, = 8 uud [ϕ(x 1, x 2, x 3 ) 6x 1 x 2 x 3 Isospin symmetry: +ϕ(x 2, x 1, x 3 ) 2T (x 1, x 2, x 2 ) ] 2T (x 1, x 2, x 3 ) = ϕ(x 1, x 3, x 2 ) + ϕ(x 2, x 3, x 1 ) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

7 Evolution and Asymptotic results Both ϕ and T are scale dependent objects: they obey evolution equations C. Mezrag (ANL) Nucleon DA November 16 th, / 34

8 Evolution and Asymptotic results Both ϕ and T are scale dependent objects: they obey evolution equations At large scale, they both yield the so-called asymptotic DA ϕ AS : C. Mezrag (ANL) Nucleon DA November 16 th, / 34

9 Evolution and Asymptotic results Both ϕ and T are scale dependent objects: they obey evolution equations At large scale, they both yield the so-called asymptotic DA ϕ AS : 4 3 u x 2 d x u x 1 C. Mezrag (ANL) Nucleon DA November 16 th, / 34

10 Form Factors: Nucleon case C. Mezrag (ANL) Nucleon DA November 16 th, / 34

11 Form Factors: Nucleon case 4 3 u x 2 d x S. Brodsky and G. Lepage, PRD 22, (1980) u x 1 η = 1 C. Mezrag (ANL) Nucleon DA November 16 th, / 34

12 Form Factors: Nucleon case u x 2 d x S. Brodsky and G. Lepage, PRD 22, (1980) u x 1 η = C. Mezrag (ANL) Nucleon DA November 16 th, / 34

13 Form Factors: Nucleon case 6 5 u x 2 d x S. Brodsky and G. Lepage, PRD 22, (1980) u x 1 η = 2 C. Mezrag (ANL) Nucleon DA November 16 th, / 34

14 Some previous studies of DA QCD Sum Rules V. Chernyak and I. Zhitnitsky, Nucl. Phys. B 246 (1984) Relativistic quark model Z. Dziembowski, PRD 37 (1988) Scalar diquark clustering Z. Dziembowski and J. Franklin, PRD 42 (1990) Phenomenological fit J. Bolz and P. Kroll, Z. Phys. A 356 (1996) Lightcone quark model B. Pasquini et al., PRD 80 (2009) Lightcone sum rules I. Anikin et al., PRD 88 (2013) Lattice Mellin moment computation G. Bali et al., JHEP C. Mezrag (ANL) Nucleon DA November 16 th, / 34

15 Dyson-Schwinger equations The gap equation for the quark propagator S(q): ( ) 1 = ( ) 1. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

16 Dyson-Schwinger equations The gap equation for the quark propagator S(q): ( ) 1 = ( ) 1. It has successfully described the quark mass behaviour: S(q) 1 = i /qa(q 2 ) + B(q 2 ) Non-perturbative description of the quark mass Dynamical mass generation Figure from Bashir et al. (2012) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

17 Faddeev Equation The Faddeev equation provides a covariant framework to describe the nucleon as a bound state of three dressed quarks. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

18 Faddeev Equation The Faddeev equation provides a covariant framework to describe the nucleon as a bound state of three dressed quarks. It predicts the existence of strong diquarks correlations inside the nucleon. p q p q p d a Ψ P = p d Γ a q Γ b b Ψ P C. Mezrag (ANL) Nucleon DA November 16 th, / 34

19 Faddeev Equation The Faddeev equation provides a covariant framework to describe the nucleon as a bound state of three dressed quarks. It predicts the existence of strong diquarks correlations inside the nucleon. p q p q p d a Ψ P = p d Γ a q Γ b b Ψ Mostly two types of diquark are dynamically generated by the Faddeev equation: Scalar diquarks, with mass roughly 2/3 of the nucleon mass, Axial-Vector (AV) diquarks, with mass roughly the same as the nucleon one. P C. Mezrag (ANL) Nucleon DA November 16 th, / 34

20 Faddeev Equation The Faddeev equation provides a covariant framework to describe the nucleon as a bound state of three dressed quarks. It predicts the existence of strong diquarks correlations inside the nucleon. p q p q p d a Ψ P = p d Γ a q Γ b b Ψ Mostly two types of diquark are dynamically generated by the Faddeev equation: Scalar diquarks, with mass roughly 2/3 of the nucleon mass, Axial-Vector (AV) diquarks, with mass roughly the same as the nucleon one. Can we understand the nucleon DA in terms of quark-diquarks correlations? C. Mezrag (ANL) Nucleon DA November 16 th, / 34 P

21 The Nakanishi Representation Propagator: Amplitude: Complexe conjugate poles: S(p) = α j + c.c. i/p + m j j Nakanishi representation: F (p, K) dz ρ νj (z) [ (p ] j 1 z 2 K) 2 νj + Λ 2 j The parameters m j, α j, ν j and Λ j are fitted on the numerical solution of the DSEs. Both have been successfully used in the case of the mesons DSE/Bethe-Salpeter equation. The Nakanishi representation will be at the center of the following work. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

22 Nucleon DA and DSE Operator point of view for every DA (and at every twist): ( ) 0 ɛ ijk u i (z 1)C /nu j (z 2) /nd k (z 3) P, λ ϕ(x i ), ( ) 0 ɛ ijk u i (z 1)Ciσ ν n ν u j (z 2) γ /nd k (z 3) P, λ T (x i ) O T, Braun et al., Nucl.Phys. B589 (2000) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

23 Nucleon DA and DSE Operator point of view for every DA (and at every twist): ( ) 0 ɛ ijk u i (z 1)C /nu j (z 2) /nd k (z 3) P, λ ϕ(x i ), ( ) 0 ɛ ijk u i (z 1)Ciσ ν n ν u j (z 2) γ /nd k (z 3) P, λ T (x i ) O T, We can apply it on the wave function: Braun et al., Nucl.Phys. B589 (2000) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

24 Nucleon DA and DSE Operator point of view for every DA (and at every twist): ( ) 0 ɛ ijk u i (z 1)C /nu j (z 2) /nd k (z 3) P, λ ϕ(x i ), ( ) 0 ɛ ijk u i (z 1)Ciσ ν n ν u j (z 2) γ /nd k (z 3) P, λ T (x i ) O T, We can apply it on the wave function: Braun et al., Nucl.Phys. B589 (2000) = + + C. Mezrag (ANL) Nucleon DA November 16 th, / 34

25 Nucleon DA and DSE Operator point of view for every DA (and at every twist): ( ) 0 ɛ ijk u i (z 1)C /nu j (z 2) /nd k (z 3) P, λ ϕ(x i ), ( ) 0 ɛ ijk u i (z 1)Ciσ ν n ν u j (z 2) γ /nd k (z 3) P, λ T (x i ) O T, We can apply it on the wave function: Braun et al., Nucl.Phys. B589 (2000) = + + The operator then selects the relevant component of the wave function. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

26 Scalar diquark contribution In the scalar diquark case, only one contribution remains (ϕ case): = C. Mezrag (ANL) Nucleon DA November 16 th, / 34

27 Scalar diquark contribution In the scalar diquark case, only one contribution remains (ϕ case): = The contraction of the Dirac indices between the single quark and the diquark makes it hard to understand. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

28 Scalar diquark contribution In the scalar diquark case, only one contribution remains (ϕ case): = The contraction of the Dirac indices between the single quark and the diquark makes it hard to understand. The way to write the nucleon Dirac structure is not unique, and can be modified (Fierz identity): = γ n C. Mezrag (ANL) Nucleon DA November 16 th, / 34

29 Scalar diquark contribution In the scalar diquark case, only one contribution remains (ϕ case): = The contraction of the Dirac indices between the single quark and the diquark makes it hard to understand. The way to write the nucleon Dirac structure is not unique, and can be modified (Fierz identity): = γ n We recognise the leading twist DA of a scalar diquark C. Mezrag (ANL) Nucleon DA November 16 th, / 34

30 Scalar diquark I: the point-like case γ n Quark propagator: S(q) = i /q + M q 2 + M 2 C. Mezrag (ANL) Nucleon DA November 16 th, / 34

31 Scalar diquark I: the point-like case γ n Quark propagator: S(q) = i /q + M q 2 + M 2 Bethe-Salpeter amplitude (1 out of 4 structures): Γ 0+ PL (q, K) = iγ 5CN 0+ C. Mezrag (ANL) Nucleon DA November 16 th, / 34

32 Scalar diquark I: the point-like case γ n Quark propagator: S(q) = i /q + M q 2 + M 2 Bethe-Salpeter amplitude (1 out of 4 structures): Γ 0+ PL (q, K) = iγ 5CN 0+ This point-like case leads to a flat DA: φ PL (x) = 1 C. Mezrag (ANL) Nucleon DA November 16 th, / 34

33 Scalar diquark II: the Nakanishi case γ n Quark propagator: S(q) = i /q + M q 2 + M 2 C. Mezrag (ANL) Nucleon DA November 16 th, / 34

34 Scalar diquark II: the Nakanishi case γ n Quark propagator: S(q) = i /q + M q 2 + M 2 Bethe-Salpeter amplitude (1 out of 4 structures): Γ 0+ PL (q, K) = iγ 5CN (1 z 2 ) dz [ (q ] 1 z 2 K) 2 + Λ 2 q C. Mezrag (ANL) Nucleon DA November 16 th, / 34

35 Scalar diquark II: the Nakanishi case γ n Quark propagator: S(q) = i /q + M q 2 + M 2 Bethe-Salpeter amplitude (1 out of 4 structures): Γ 0+ PL (q, K) = iγ 5CN (1 z 2 ) dz [ (q ] 1 z 2 K) 2 + Λ 2 q The Nakanishi case leads to a non trivial DA: [ ] φ(x) = 1 M2 ln 1 + K 2 x(1 x) M 2 K 2 x(1 x) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

36 Scalar DA behaviour Scalar diquark [ ] φ(x) = 1 M2 ln 1 + K 2 x(1 x) M 2 K 2 x(1 x) Asymptotic K^2 4M^2 K^2 16 M^2 C. Mezrag (ANL) Nucleon DA November 16 th, / 34

37 Scalar DA behaviour Scalar diquark [ ] φ(x) = 1 M2 ln 1 + K 2 x(1 x) M 2 K 2 x(1 x) Asymptotic K^2 4M^2 K^2 16 M^2 Φ Π x Pion x Pion figure from L. Chang et al., PRL 110 (2013) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

38 Scalar DA behaviour Scalar diquark [ ] φ(x) = 1 M2 ln 1 + K 2 x(1 x) M 2 K 2 x(1 x) Asymptotic K^2 4M^2 K^2 16 M^2 Φ Π x Pion x Pion figure from L. Chang et al., PRL 110 (2013) This extended version of the DA seems promising! C. Mezrag (ANL) Nucleon DA November 16 th, / 34

39 Faddeev Amplitude Quark propagator: S(q) = i /q + M q 2 + M 2 Diquark propagator: (K) = 1 K 2 + M 2 Faddeev Amplitude: 1 s 1 (K, P) = N 1 dz 1 (1 z 2 ) [ (K ] 1 z 2 P) 2 + Λ 2 N C. Mezrag (ANL) Nucleon DA November 16 th, / 34

40 Results in the scalar channel 4 uhx2l uhx2l 2 1 uhx1l dhx3l uhx2l uhx1l 4 dhx3l uhx1l Extended case: ϕ Point-like case: ϕ C. Mezrag (ANL) uhx1l 1 Extended case: T 2.5 uhx2l dhx3l 2.5 Asymptotic DA 3 3 dhx3l Nucleon DA November 16th, / 34

41 Comparison with lattice < x i > ϕ = Dx x i ϕ(x 1, x 2, x 3 ) Pointlike Extended Lattice Lattice n f = 2 n f = (2, 1) < x 1 > ϕ (7) 58 < x 2 > ϕ (3) 19 < x 3 > ϕ (7) 23 < x 1 > T < x 2 > T < x 3 > T Lattice data from V.Braun et al, PRD 89 (2014) G. Bali et al., JHEP C. Mezrag (ANL) Nucleon DA November 16 th, / 34

42 Comparison with lattice < x i > ϕ = Dx x i ϕ(x 1, x 2, x 3 ) Pointlike Extended Lattice Lattice n f = 2 n f = (2, 1) < x 1 > ϕ (7) 58 < x 2 > ϕ (3) 19 < x 3 > ϕ (7) 23 < x 1 > T < x 2 > T < x 3 > T Lattice data from V.Braun et al, PRD 89 (2014) G. Bali et al., JHEP How are these results modified by the AV component? C. Mezrag (ANL) Nucleon DA November 16 th, / 34

43 Vector Meson DA Leading twist chiral-even DA: 0 ū(0)γ µ d(z) ρ + (P, λ) = P µ ɛ(λ) n P n f ρm ρ Leading twist chiral-odd DA: 0 ū(0)σ µν d(z) ρ + (P, λ) = ( ) i ɛ (λ) µ p ν ɛ (λ) ν p µ fρ duφ (u)e iup n duφ (u)e iup n P.Ball and V. Braun,Nucl.Phys. B543 (1999) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

44 Vector Meson DA Leading twist chiral-even DA: 0 ū(0)γ µ d(z) ρ + (P, λ) = P µ ɛ(λ) n P n f ρm ρ Leading twist chiral-odd DA: 0 ū(0)σ µν d(z) ρ + (P, λ) = ( ) i ɛ (λ) µ p ν ɛ (λ) ν p µ fρ duφ (u)e iup n duφ (u)e iup n P.Ball and V. Braun,Nucl.Phys. B543 (1999) Same structures appear in the Axial-Vector diquark. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

45 Selecting the AV components ( ) 0 ɛ ijk u i (z 1)C /nu j (z 2) /nd k (z 3) P, λ ϕ(x i ), ( ) 0 ɛ ijk u i (z 1)Ciσ ν n ν u j (z 2) γ /nd k (z 3) P, λ T (x i ) O T, C. Mezrag (ANL) Nucleon DA November 16 th, / 34

46 Selecting the AV components ( ) 0 ɛ ijk u i (z 1)C /nu j (z 2) /nd k (z 3) P, λ ϕ(x i ), ( ) 0 ɛ ijk u i (z 1)Ciσ ν n ν u j (z 2) γ /nd k (z 3) P, λ T (x i ) O T, = + + C. Mezrag (ANL) Nucleon DA November 16 th, / 34

47 Selecting the AV components ( ) 0 ɛ ijk u i (z 1)C /nu j (z 2) /nd k (z 3) P, λ ϕ(x i ), ( ) 0 ɛ ijk u i (z 1)Ciσ ν n ν u j (z 2) γ /nd k (z 3) P, λ T (x i ) O T, = + } {{ } Two chiral-even DAs (longitudinal) + }{{} One chiral-odd DA (transverse) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

48 Selecting the AV components ( ) 0 ɛ ijk u i (z 1)C /nu j (z 2) /nd k (z 3) P, λ ϕ(x i ), ( ) 0 ɛ ijk u i (z 1)Ciσ ν n ν u j (z 2) γ /nd k (z 3) P, λ T (x i ) O T, = + } {{ } Two chiral-even DAs (longitudinal) + }{{} One chiral-odd DA (transverse) O T O T = O T O T }{{} One chiral-odd DA (transverse) + O T + O T O T O T } {{ } Two chiral-even DAs (longitudinal) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

49 Selecting the AV components ( ) 0 ɛ ijk u i (z 1)C /nu j (z 2) /nd k (z 3) P, λ ϕ(x i ), ( ) 0 ɛ ijk u i (z 1)Ciσ ν n ν u j (z 2) γ /nd k (z 3) P, λ T (x i ) O T, = + } {{ } Two chiral-even DAs (longitudinal) + }{{} One chiral-odd DA (transverse) O T O T = O T O T }{{} One chiral-odd DA (transverse) + O T + O T O T O T } {{ } Two chiral-even DAs (longitudinal) 2T (x 1, x 2, x 3 ) = ϕ(x 1, x 3, x 2 ) + ϕ(x 2, x 3, x 1 ) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

50 AV diquark DA γ n σ µν n ν Quark propagator: S(q) = i /q + M q 2 + M 2 C. Mezrag (ANL) Nucleon DA November 16 th, / 34

51 AV diquark DA γ n σ µν n ν Quark propagator: S(q) = i /q + M q 2 + M 2 Bethe-Salpeter amplitude (2 out of 8 structures): 1 Γ µ PL (q, K) = (N 1τ µ 1 + N 2τ µ 2 ) C dz τ µ 1 = i τ µ 2 = 1 ( γ µ K µ K / ) K 2 Chiral even K q ( iτ µ q 2 (K q) 2 K 2 1 /q + i /qτ µ 1 (1 z 2 ) [ (q ] 1 z 2 K) 2 + Λ 2 q ) Chiral odd C. Mezrag (ANL) Nucleon DA November 16 th, / 34

52 Comparison with the ρ meson AV diquark ρ meson Τ2 DA Τ1 Asymptotic DA longitudinal part transverse part transverse part with only a 2 pion asymptotic x ρ figure from F. Gao et al., PRD 90 (2014) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

53 Comparison with the ρ meson AV diquark ρ meson Τ2 DA Τ1 Asymptotic DA longitudinal part transverse part transverse part with only a 2 pion asymptotic x ρ figure from F. Gao et al., PRD 90 (2014) Same shape ordering φ is flatter in both cases. Farther apart compared to the ρ meson case. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

54 Faddeev Amplitude for the AV case Quark propagator: S(q) = i /q + M q 2 + M 2 Diquark propagator: AV Faddeev amplitude (2 out of 6 structures): A µ (K, P) = (γ 5 γ µ iγ 5 ˆP µ) 1 dz µν (K) = δµν + K µ K ν K 2 K 2 + M 2 1 (1 z 2 ) [ (K ] 1 z 2 P) 2 + Λ 2 N C. Mezrag (ANL) Nucleon DA November 16 th, / 34

55 Preliminary results for the AV contributions I u x 2 d x u x 2 d x u x 2 d x u x 1 u x 1 u x 1 C. Mezrag (ANL) Nucleon DA November 16 th, / 34

56 Preliminary results for the AV contributions II ϕ AV (x 1, x 2, x 3 ) T AV (x 1, x 2, x 3 ) u x 2 d x u x 2 d x u x 1 u x 1 C. Mezrag (ANL) Nucleon DA November 16 th, / 34

57 Complete results for ϕ We use the prediction from the Faddeev equation to weight the scalar and AV contributions 62/38: u x 2 d x u x 2 d x u x 1 u x 1 C. Mezrag (ANL) Nucleon DA November 16 th, / 34

58 Complete results for ϕ We use the prediction from the Faddeev equation to weight the scalar and AV contributions 62/38: u x 2 d x u x 2 d x u x 1 which can be compared with LCSR results: u x 1 figure from I Anikin et al., PRD 88 (2013) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

59 Complete results for T u x 2 d x u x 2 d x u x 2 d x u x 1 100% Scalar u x 1 62% Scalar u x 1 100% AV Like φ, T received both contributions from the scalar and AV diquarks. Both contributions are similar and thus contrary to φ, T is hardly modified. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

60 Comparison with lattice Scalar S+AV Lattice Lattice (62/38) n f = 2 n f = (2, 1) < x 1 > ϕ (7) 58 < x 2 > ϕ (3) 19 < x 3 > ϕ (7) 23 < x 1 > T < x 2 > T < x 3 > T f N (10 3 GeV 2 ) (1)(33) 3.60(6)(2) Lattice data from V.Braun et al, PRD 89 (2014) G. Bali et al., JHEP Normalisation from J. Segovia C. Mezrag (ANL) Nucleon DA November 16 th, / 34

61 Comparison with lattice Scalar S+AV Lattice Lattice (62/38) n f = 2 n f = (2, 1) < x 1 > ϕ (7) 58 < x 2 > ϕ (3) 19 < x 3 > ϕ (7) 23 < x 1 > T < x 2 > T < x 3 > T f N (10 3 GeV 2 ) (1)(33) 3.60(6)(2) Lattice data from V.Braun et al, PRD 89 (2014) G. Bali et al., JHEP Normalisation from J. Segovia Quark-diquark correlation seems able to explain the nucleon DA. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

62 Extension to the Roper Everything done before can actually be extended to the Roper case. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

63 Extension to the Roper Everything done before can actually be extended to the Roper case. The only difference holds in the Faddeev amplitude model. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

64 Extension to the Roper Everything done before can actually be extended to the Roper case. The only difference holds in the Faddeev amplitude model. In particular in the Chebychev moments: S 1 A 2 (1/3)A 3 +(2/3)A p (GeV) S 1 A 2 (1/3)A 3 +(2/3)A p (GeV) figures from J. Segovia et al.,prl 115 (2015) C. Mezrag (ANL) Nucleon DA November 16 th, / 34

65 Extension to the Roper Everything done before can actually be extended to the Roper case. The only difference holds in the Faddeev amplitude model. In particular in the Chebychev moments: S 1 A 2 (1/3)A 3 +(2/3)A p (GeV) S 1 A 2 (1/3)A 3 +(2/3)A p (GeV) figures from J. Segovia et al.,prl 115 (2015) This behaviour can be obtained by adding a zero in the Faddeev amplitude through: 1 1 (1 z 2 ) dz [ (K ] 1 z 2 P) 2 + Λ 2 N 1 1 (1 z 2 )(z κ) dz [ (K ] 1 z 2 P) 2 + Λ 2 R C. Mezrag (ANL) Nucleon DA November 16 th, / 34

66 Scalar and AV components uhx2l dhx3l uhx2l 1-1 uhx1l -3 dhx3l Scalar uhx1l AV Long. 4 2 uhx2l dhx3l uhx2l 0-2 uhx1l dhx3l AV Long. C. Mezrag (ANL) -4 0 uhx1l AV Trans. Nucleon DA November 16th, / 34

67 Complete results for ϕ R 3 4 u x 2 d x u x 2 d x u x % Scalar u x 1 100% AV 1 0 u x 2 d x u x 1 62% Scalar C. Mezrag (ANL) Nucleon DA November 16 th, / 34

68 Complete results for T R u x 2 d x 3 1 u x 2 d x u x % Scalar u x 1 100% AV 0 u x 2 d x u x 1 62% Scalar C. Mezrag (ANL) Nucleon DA November 16 th, / 34

69 Absolute Normalisation Nucleon: f N = GeV 2 Roper: f R = GeV 2 Courtesy of J. Segovia C. Mezrag (ANL) Nucleon DA November 16 th, / 34

70 Summary Both nucleon DAs ϕ and T can be described using a quark-diquark approximation. We show how the diquark types and diquarks polarisations were selected. The comparison with lattice computation explains how the different diquarks contribute to the total DAs, and the respective sensitivity of the latter to the AV-diquarks. The comparison with the lattice data is encouraging. It is possible to extend the work on the nucleon to the Roper case. In the Roper case, the results of individual diquarks contributions seems to be consistent with a n = 1 excited state. C. Mezrag (ANL) Nucleon DA November 16 th, / 34

71 Outlooks Adding the full Dirac structure may help to correct the discrepancy. The use of the numerical solutions of the DSE-Faddeev Equation will certainly modify the previous results, and improve our understanding of the physics of the nucleon. From Distribution Amplitudes to observables: the computation of the form factor at large momentum transfer within this framework is one of the future goals. Other resonances, N(1535)? C. Mezrag (ANL) Nucleon DA November 16 th, / 34

72 Thank you for your attention C. Mezrag (ANL) Nucleon DA November 16 th, / 34