Ho- Meoyng Choi( Kyungpook National Univ., Korea)

1. Self-consistent covariant description of twist-3 distribution amplitudes of pseudoscalar and vector mesons in the light-front quark model Ho-Meoyng Choi(Kyungpook National Univ., Korea) (in collaboration with C.-R. Ji) Outline • Motivation • 2. Model Calculation in Manifestly Covariant Model and Light-Front Quark Model(LFQM) • - Twist-2 and-3 Distribution Amplitudes(DAs) of pseudoscalar and vector mesons • 3. Numerical Results • 4. Conclusion Based on PRD89,033011(14) & the work in progress Light Cone 2014, Raleigh (May 26~30, 2014)

2. 1. Motivation •Light-Front Dynamics(LFD) has been quite successful in describing various hadron properties. • But it has been emphasized that the treacherous points(e.g. zero modes) should be taken into account for successful LFD applications to hadron phenomenology.[Jaus 99; Cheng,Chua,Hwang (CCH)04; de Melo, Frederico 97,12; CJ,PRD89(14),80(09);NPA865(11);PLB695,518(11)] Representative Examples: Two-point function (e.g. Decay constant) Three-point function (e.g. Form factors) p=P-k P • In our previous work(CJ 09,11,14), we identified the zero-mode operator & found a way to include it effectively in the valence diagram within the manifestly covariant Bethe-Salpeter(BS) model. But the zero-mode may depend on the LF vertex function. k • The purpose of this work is to build LFQM, which provides an effective treatment of the zero-mode.

3. • Nature of the LF zero-mode in the meson decay amplitude Pick up a LF energy pole LF projection p=P-k P k If this contribution does not vanish, we must capture it! Zero mode !

4. • Nature of the LF zero-mode in the meson decay amplitude Pick up a LF energy pole LF projection p=P-k P k • Zero-mode issue of is highly nontrivial & deserves careful analysis! If this contribution does not vanish, we must capture it! Zero mode !

5. • The purpose of this work is to build LFQM, which provides an effective treatment of the zero-mode. • Manifestly Covariant BS model • LFQM (standard LF approach) - To provide phenomenological model! take advantage of BS model - To provide theoretical guidance! know how to find the Z.M. ) Melosh transformation To absorb the zero-mode complication into the effective constituent degree of freedom, we attempt to find matching condition between BS model and LFQM:

6. 2. Schematic Picture of Covariant BS Model vs. LFQM Manifestly Covariant BS model: Covariant calculation! BS model on the LF: Z.M. = p=P-k + valence P If k + (propagating) (instantaneous)

7. Our main findings in LFBS model & in the standard LFQM application: LFBS model: = Z.M. We know how to identify the zero-mode operator and include it in the valence diagram! (CJ 14) p=P-k + P k + Application to LFQM Using Gaussian W.F.:

8. Our main findings in LFBS model & in the standard LFQM application: LFBS model: = Z.M. = p=P-k p=P-k + + LFQM: P P k k + + Z.M.

9. Why treacherous points disappear in LFQM? Z.M. = p=P-k + P Need further constraint! k + While

10. LFBS model: = Z.M. Matching Condition: LFQM: p=P-k p=P-k + “Main ingredients” to support the self-consistency of Our LFQM! = P P k k + Constraint: (Twist-2 & twist-3) Meson DAs

11. (A) Twist-2 and-3 Meson DAs: Pion Twist-2: Twist-3: where • Normalization constant results from quark condensate via : Gell-Mann-Oakes-Renner relation

12. Decay Constant & twist-2 of Pion • LF covariant method (BS model) • SLF method (LFQM) Z.M. free! Matching condition:

13. Twist-3 :zero-mode operator! (Jaus 99, CJ09,11,14)

14. LFBS model: + = + Z.M. p=P-k p=P-k LFQM: P P k k =

15. LFQM: = While p=P-k P k : Constraint on LFQM! But

16. Twist-3 DA of pion in LFQM: Twist-2 DA of pion in LFQM: Preliminary

17. (B) Twist-2 and-3 Meson DAs: meson PRD89,033011(14) by CJ () To isolate twist-2 & twist-3

18. Twist-2 LFBS: + LFQM:

19. Twist-3 LFBS: + LFQM:

20. & in LFQM with Gaussian w.f. : But… ρ PRD89,033011(14) by CJ

21. 3. Numerical Results Indeed, we generalized to the unequal Input: All others are predictions! (in unit of GeV) “Predicted twist-2 DAs …” PRD75,034199(07) CJ

22. Momentum dependence of LFWF: Preliminary (c.f. SR by Zhitnitsky 94) Cut off scale:

23. Kaon: Preliminary

24. Twist-3 pion & Kaon DAs: (Ball, Braun, Lenz,06) (Ball, Braun, Lenz,06) Preliminary limit for HO (Linear) c.f.) Standard:

25. Gegenbauer & moments: Pion TW3(A): CJ, Preliminary TW2(A): CJ, PRD(07)

26. Kaon TW3(A): CJ, Preliminary TW2(A): CJ, PRD(07)

27. Twist-2 & twist-3 vector DAs: limit CJ, PRD89,033011(14)

28. moments for (): The numbers in ( ) for the give the asymptotic values. CJ, PRD89,033011(14)

29. 4.Conclusions • • In this work, we extended the analysis of meson from the manifestly covariant • BS model to the more realistic standard LFQM through the following matching condition • between BS model and LFQM : • In the standard LFQM, we are able to describe the decay amplitude only • in terms of valence diagram with the on-mass-shell quark propagators but respecting • chiral symmetryof QCD. • Our twist-2 and twist-3 DAs not only satisfy the fundamental symmetry anticipated • isospin symmetry but also reproduce the correct asymptotic DAs in the chiral limit. • •Our LFQM may provide the view of an effective zero-mode cloud around the quark • and antiquark inside the meson. Consequently, the constituents dressed by the zero-mode • cloud may be expected to satisfy the chiral symmetry of QCD. • Need more observables to make our findings generalize! - Other higher twist DAs, electroweak form factors etc.

30. Back Up Slide

31. Covariant vs. Light-Front Calculation for E&M form factor of pion + = on-mass-shell

32. + = + x 1-x (propagating) (instantaneous)

33. LF Zero-mode: Effective inclusion of zero-mode in the valence region: zero-mode operator Power counting in Jaus99, Choi&Ji 09

34. (A) Pion Electromagnetic form factor + = + x (1) Manifestly Covariant BS Model Calculation: 1-x (

35. (2) Standard LF Calculation (LFQM) and Correspondence between BS model and LFQM ) () : Correspondence between BS model and LFQM (

36. For perpendicular components of the current: BS model: ( BS model + = + x 1-x

37. LFQM: + = + x 1-x LFQM In our LFQM, only on-shell quarks contribute to for both !