1. Sketching the pseudoscalar mesons’ valence-quark parton distribution functions Chen Chen University of Science and Technology of China November 16 th, 2015 Sketching the pseudoscalar mesons’ valence-quark parton distribution functions Chen Chen, Lei Chang, Craig D. Roberts, and Shao-long Wan In preparation 1

2.  The hadronic tensor relevant to inclusive deep inelastic lepton–pion scattering may be expressed in terms of two invariant structure functions. In the deep-inelastic Bjorken limit, that tensor can be written  F 1 (x): the pion structure function  The structure function may be computed from the imaginary part of the virtual- photon–pion forward Compton scattering amplitude: 2 Parton distribution functions (PDFs)

3. Dyson-Schwinger Equations General Form QCD  NonPerturbative, continuum approach to QCD  D μν (k) – dressed-gluon propagator  Γ ν (q,p) – dressed-quark-gluon vertex 3

4. Bethe-Salpeter Equation Bound-State  K(q,k;P) – fully amputated, two-particle irreducible, quark-antiquark scattering kernel  S(q) – dressed quark propagator  Г π – pion’s Bethe-Salpeter amplitude 4

5.  The Bethe-Salpeter wave function  The pseudoscalar meson Bethe-Salpeter amplitude  Each of the scalar functions has the following decomposition with F 0/1 even under (l.q)->(-l.q). For pion, F 1 ≡ 0. Bethe-Salpeter wave function 5

6. 6 Symmetry  The truncation scheme should be restricted by the Axial- vector Ward-Takahashi-identity:  Rainbow-ladder truncation (RL): the most widely used DSE computational scheme in hadron physics.  It is accurate for isospin-nonzero-pseudoscalar-mesons.

7.  The textbook handbag contribution to virtual Compton scattering  In RL truncation, H (P, k) is an infinite sum of ladder-like rungs 7 Handbag

8.  Overlooked diagrams… a mistake!  Forward limit of BC. It expresses a photon being absorbed by a dressed-quark, which then proceeds to become part of the pion bound-state before re-emitting the photon.  This contribution is of precisely the same order as the handbag- contribution. 8

9.  There are some overlooked diagrams… a mistake!  This contribution is of precisely the same order as the handbag- contribution.  It expresses a photon being absorbed by a dressed-quark, which then proceeds to become part of the pion bound-state before re-emitting the photon. 9

10.  The imaginary part in the Bjorken limit. It produces the leading contribution: the vertex insertion can appear between any pair of interaction lines.  The compound vertex is correspond to differentiation of the Bethe– Salpeter amplitude itself. 10

11.  Final expression start point  The start point! 11

12. 12 Algebraic Model  Propagators:  Pion and kaon Bethe-Salpeter amplitudes: Decay constants

13. 13 Algebraic Model  Propagators:  Pion and kaon Bethe-Salpeter amplitudes:  Parameters:

14. 14 Pion: chiral limit  Chiral limit: m π = 0  Analytic expression: Lei Chang, Cédric Mezrag, Hervé Moutarde, Craig D. Roberts, Jose Rodríguez- Quintero and Peter C. Tandy, Phys. Lett. B 737 (2014)

15. 15 Pion: chiral limit  Chiral limit: m π = 0  Analytic expression:  QCD-like scaling behavior:  The power-law predicted by the QCD parton model.

16. 16  If m π/K ≠ 0, the PDFs cannot be obtained analytically.  First we can compute the PDFs’ moments:  Then use Gegenbauer polynomials to reconstruct the PDFs. Moments

17. 17  Reconstruction procedure yields: Moments

18. 18 PDFs

19. 19  The RL truncation includes no mechanism that can shift momentum from the dressed-quarks into sea-quarks and gluons: a RL meson is constituted solely from a dressed-quark and dressed-antiquark.  The corrections to the RL truncation can be separated into two classes: [C1] redistributes baryon-number and momentum into the dressed-quark sea; and [C2] shifts momentum into the dressed-gluon distribution. Sea-quarks & gluons

20. 20  The RL truncation includes no mechanism that can shift momentum from the dressed-quarks into sea-quarks and gluons: a RL pion is constituted solely from a dressed-quark and dressed-antiquark.  The corrections to the RL truncation can be separated into two classes: [C1] redistributes baryon-number and momentum into the dressed-quark sea; and [C2] shifts momentum into the dressed-gluon distribution within the pion. Sea-quarks & gluons

21. 21  The RL truncation includes no mechanism that can shift momentum from the dressed-quarks into sea-quarks and gluons: a RL meson is constituted solely from a dressed-quark and dressed-antiquark.  The corrections to the RL truncation can be separated into two classes: [C1] redistributes baryon-number and momentum into the dressed-quark sea; and [C2] shifts momentum into the dressed-gluon distribution.  In a symmetry preserving treatment, corrections in [C2] have no impact on net baryon number but they do rob momentum from the baryon-number-carrying dressed-partons Sea-quarks & gluons

22. 22  In a symmetry preserving treatment, corrections in [C2] have no impact on net baryon number but they do rob momentum from the baryon-number-carrying dressed-partons  The complete dressed-quark distribution function: Sea-quarks & gluons

23. 23  Note first that with realistic masses, meson-loop corrections to the RL result for the pion electromagnetic form factor at Q 2 =0 are an O(5%) effect.  Z = 0.05.  πN Drell–Yan: Sea-quarks & gluons M. Gluck, E. Reya, I. Schienbein, Eur. Phys. J. C 10 (1999) 313–317.

24. 24  Begin with RL results, obtained at a particular scale ζ H ; then proceed systematically to add the corrections from sea-quarks and gluons; and, finally, use DGLAP evolution to obtain the result at any other scale ζ > ζ H.  GDLAP equations are only valid on the perturbative domain.  One should use ζ H ≥ 2Λ QCD ≈ 0.5GeV.  It is impossible to begin at a smaller scale.  ζ H = 0.51 GeV → ζ 5 = 5.2 GeV Procedure R.J. Holt, C.D. Roberts, Rev. Mod. Phys. 82 (2010) 2991–3044 L. Chang, C.D. Roberts, D.J. Wilson, in: PoS QCD-TNT-II, 2012, p.039.

25. 25 u π 0 (x): ζ H → ζ 5

26. 26 PDFs (ζ 5 = 5.2 GeV)

27. 27 Valence: Ratio u K (x)/u π (x)  ζ H = 0.51 GeV → ζ 5 = 5.2 GeV  Solve the nonsinglet Altarelli-Parisi equation numerically to leading order (LO) and next-to-leading order (NLO):

28. 28 Valence: Ratio u K (x)/u π (x)

29. 29 Full: Ratio u K (x)/u π (x)

30. 30 Full: Ratio u K (x)/u π (x)

31.  The handbag diagram used to define the dressed-quark distribution function is incorrect owing to omission of contributions from the gluons.  The valence-quark distribution behaves as (1 −x) 2 on x ≈1.  The corrections to the RL prediction for the PDFs may be divided into two classes: [C1]: from dressed-quark sea; and [C2]: from dressed-gluon distribution.  We built a simple algebraic model to express the principal impact of both classes of corrections, which, coupled with the RL prediction, permitted a realistic comparison with existing experiment.  Prediction: the glue content of kaon is much smaller than pion at the same scale.  Extension: Using a realistic model to calculate PDFs. & Computation of a first realistic approximation to the GPDs. 31 Conclusions and prospects

32. Thank you! 32