1 In collaboration with L. Gamberg, Z. Kang, H. Xing Based on Phys.Lett. B743 (2015) 112-120, arXiv:1412.3401 Quasi-parton distribution functions: a study.

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In collaboration with L. Gamberg, Z. Kang, H. Xing

Based on Phys.Lett. B743 (2015) 112-120, arXiv:1412.3401

Quasi-parton distribution functions: a study in the di-quark spectator model

QCD Evolution 2015, May 2015, JLab, Newport News, VA

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Outline of the talk

• Advances in computationally enabled understanding of the nucleon structure. Applicability of quasi-PDFs

The spectator di-quark model

Motivation

• Basics and phenomenological success

Numerical comparison of PDFs and quasi-PDFs

• Unpolarized distribution, helicity and transversity. Analytic limits

Evaluation of unintegrated and integrated PDFs and quasi-PDFs

Sum rules• Guidance on the nucleon boost for good approximation

• Soffer inequality. Analytic and numerical results

Conclusions

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We will focus on the unpolarized distribution f1, the helicity distribution g1 and the transversity distribution h1

Motivation

TMD definitions, Proof of factorization theoremsTMD evolution, Extraction via global analysisLattice QCD calculations

Rich internal nucleonstructure

Quark distribution functions

P. Mulders et al. 1995

Global analysisPolarized nucleon structure

Focus on extraction with TMD evolution

Note: spin is along y directionM. Echevarria et al, 2014C. Aidala et al, 2014, P. Sun et al, 2014, U. D’Alesio et al, 2014

M. Anselmino et al, 2008

Lattice QCD for TMDsLattice evaluation of TMDs

Quasi-PDFs

T. Bhattacharya et al. in preparation

B. Musch et al., 2011

X. Ji. 2013, Ma et al. 2014

New formulation of quasi-PDFs for lattice evaluation

Equal time correlators, suitable for lattice evaluation C. Alexandrou et al. 2014

Spectator di-quark modelA simplified picture for the interaction with the nucleon

A. Bacchetta et al. 2008

R. Jakob et al . 1997, L. Gamberg et al. 2003

Truncates the sum over all possible final states to a single di-quark final state. Initially contained only scalar di-quarks

Axial-vector di-quarks different possibilities for formfactorsVertex

Propagator

For PDFs

For Quasi-PDFs

Phenomenological success of the spectator di-quark model

The parameters of the model can be fit to parameterizations form global extraction

Predictions for moments of different distributions (Sivers)

A. Bacchetta et al. 2008

Example

ZEUS2002GRSV2000

Qualitative and sometimes quantitative agreement with global analysis

Factorization gives the following definitions for PDFs and Quasi-PDFs

Cut vertices

Calculation of PDFs and quasi-PDFs

Standard PDFs Quasi-PDFs

Standard PDFs

Quasi PDFs

Does not depend on the momentum P+

Lowest order computation of PDFs and quasi-PDFs

Analytic computations of f1

The spectator di-quark model, tree level calculation an the dipole form factors make the calculation straightforward

Invariant massResult

Obtain the collinear PDF

While in slightly different notation agree with A. Bacchetta et al. 2008

The high-energy limit

Similar calculation

Analytic computations of the quasi-f1

Notation

Note that quasi-f1 does depend explicitly on Pz

Invariant mass

To find

Helicity

Unpolarized distributionSummary of quasi-PDF results (unintegrated)

Transversity

Obtained analytic results for the unintegrated PDFs and quasi-PDFs for both scalar and axial vector di-quarks

For PDFs it is easy to obtain analytic results for the collinear PDFs by integrating over kT. For quasi-PDFs the results are obtained numerically

In all cases considered (unpolarized, helicity, trasnversity) we checked analytically that the quasi-PDFs reduce to the standard PDFs when Pz -> infinity

The collinear unpolarized, helicity and transversity distributions

The main goal of the project is to find at what Pz (or boost of the nucleon) the quasi PDFs approach standard PDFs

Numerical studies of PDFs and quasi-PDFs

The unpolarized distribution L. Gamberg et al. 2014

Pz ~1 GeV never works (not surprisingly). Many non-perturbative parameters are of O(1 GeV), ΛX, MX …Pz > 2 GeV the apprximations (quasi-PDFs) work considerably better

Helicity Helicity and Transversity

Similar results

Transversity

Look at the ratios

Quantifying deviations: small to moderate x

For x <0.4 at Pz ~ 2 GeV quasi-PDFs approximate PDFs to 30%, For Pz ~ 4 GeV the deviation is only 10%

The approach to the PDF values is much slower at large x

Deviations at large x

For x ~0.7 at Pz ~ 2 GeV quasi-PDFs deviate for PDFs a lot – factor of 2-3For Pz ~ 4 GeV the deviation is ~50%. Clearly much larger boost is needed for large values of Bjorken x

See the approximation/ limit

Numerical results

Various sum rules and bounds can be obtained for polarized reactions

Sum rules and bounds

Arise form symmetries, kinematics, etc. With (quasi-)unpolarized, helicity and transversity it is easy to check the Soffer inequaity

X. Artru al. 2008A. Bacchetta et al. 2000

Soffer bound

Generally bounds do not hold for quasi-PDFs. The approach to the sum rule is similar

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ConclusionsSignificant advances recently in understanding TMDs / PDFs• Global fits to TMDs with evolution. Collinear PDFs constrained for a long time.

Evaluated the quasi-PDFs, unpolarized distribution, helicity, transversity, in the di-quark spectator model

New lattice techniques emerge to evaluate TMDs / PDFs directly on the lattice

Possible extensions

• One can obtain analytic results for the dipole form factor (both scalar and axial diquarks). Show that in the limit of Pz-> infinity they reduce to the PDFs

• “Traditional” lattice approach. New proposition to use quasi-PDFs. Lattice can benefit from guidance at what boost the quasi-PDFs approach PDFs

Studied the quasi-PDFs numerically as a function of the nucleon boost• For small to moderate x, Pz ~ 2 GeV approximated to ~30%. For large x the

approximation is worse, 50% for Pz ~ 4 GeV Soffer sum rule• Generally does not hold for quasi-PDFs