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Sketching the pseudoscalar mesons’valence-quark

parton distribution functions

Chen ChenUniversity of Science and Technology of China

November 16th , 2015

Sketching the pseudoscalar mesons’ valence-quark parton distribution functions

Chen Chen, Lei Chang, Craig D. Roberts, and Shao-long WanIn preparation

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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

F1(x): the pion structure function

The structure function may be computed from the imaginary part of the virtual-photon–pion forward Compton scattering amplitude:

Parton distribution functions (PDFs)

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Dyson-Schwinger EquationsGeneral Form

NonPerturbative, continuum approach to QCDDμν(k) – dressed-gluon propagatorΓν(q,p) – dressed-quark-gluon vertex

Bethe-Salpeter Equation Bound-State

K(q,k;P) – fully amputated, two-particle irreducible, quark-antiquark scattering kernelS(q) – dressed quark propagatorГπ – pion’s Bethe-Salpeter amplitude

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The Bethe-Salpeter wave function

The pseudoscalar meson Bethe-Salpeter amplitude

Each of the scalar functions has the following decomposition

with F0/1 even under (l.q)->(-l.q).

For pion, F1 ≡ 0.

Bethe-Salpeter wave function

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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.

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

Handbag

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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.

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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.

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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.

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Final expression

The start point!

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Algebraic Model

Propagators:

Pion and kaon Bethe-Salpeter amplitudes:

Decay constants

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Algebraic Model

Propagators:

Pion and kaon Bethe-Salpeter amplitudes:

Parameters:

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Pion: chiral limit

Chiral limit: mπ = 0 Analytic expression:

Lei Chang, Cédric Mezrag, Hervé Moutarde, Craig D. Roberts, Jose Rodríguez-Quintero and Peter C. Tandy, Phys. Lett. B 737 (2014)

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Pion: chiral limit

Chiral limit: mπ = 0 Analytic expression:

QCD-like scaling behavior:

The power-law predicted by the QCD parton model.

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If mπ/K ≠ 0, the PDFs cannot be obtained analytically. First we can compute the PDFs’ moments:

Then use Gegenbauer polynomials to reconstruct the PDFs.

Moments

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Reconstruction procedure yields:

Moments

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PDFs

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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

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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

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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

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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

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Note first that with realistic masses, meson-loop corrections to the RL result for the pion electromagnetic form factor at Q2=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.

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Begin with RL results, obtained at a particular scale ζH; 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 > ζH.

GDLAP equations are only valid on the perturbative domain. One should use ζH ≥ 2ΛQCD ≈ 0.5GeV.

It is impossible to begin at a smaller scale.

ζH = 0.51 GeV → ζ5 = 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.

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uπ0(x): ζH → ζ5

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PDFs (ζ5 = 5.2 GeV)

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Valence: Ratio uK(x)/uπ(x)

ζH = 0.51 GeV → ζ5 = 5.2 GeV Solve the nonsinglet Altarelli-Parisi equation numerically to

leading order (LO) and next-to-leading order (NLO):

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Valence: Ratio uK(x)/uπ(x)

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Full: Ratio uK(x)/uπ(x)

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Full: Ratio uK(x)/uπ(x)

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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.

Conclusions and prospects

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Thank you!