Quasi-parton distribution functions:
a study in the di-quark spectator model
In collaboration with L. Gamberg, Z. Kang, H. Xing
Based on Phys.Lett. B743 (2015) 112-120, arXiv:1412.3401
1
Outline of the talk
Motivation
• Advances in computationally enabled understanding of the
nucleon structure. Applicability of quasi-PDFs
The spectator di-quark model
• Basics and phenomenological success
Evaluation of unintegrated and integrated PDFs and
quasi-PDFs
• Unpolarized distribution, helicity and transversity. Analytic
limits
Numerical comparison of PDFs and quasi-PDFs
• Guidance on the nucleon boost for good approximation
Sum rules
• Soffer inequality. Analytic and numerical results
Conclusions
2
Motivation
Quark distribution functions
P. Mulders et al. 1995
We will focus on the unpolarized distribution f1, the helicity
distribution g1 and the transversity distribution h1
3
Global analysis
M. Anselmino et al, 2008
UT
ASivers
COMPASS Proton
0.1
h-
0.05
0
-0.05
UT
ASivers
-0.1
0.1
h+
0.05
0
-0.05
-0.1
10
-2
10
-1
0.25
xB
0.5
0.75
1
zh
0.2
0.4
0.6
0.8
ph^ (GeV)
M. Echevarria et al, 2014
C. Aidala et al, 2014, P. Sun et al, 2014, U. D’Alesio et al, 2014
Note: spin is along y direction
Lattice QCD for TMDs
B. Musch et al., 2011
 New formulation of quasi-PDFs
for lattice evaluation
X. Ji. 2013, Ma et al. 2014
T. Bhattacharya et al. in preparation
 Equal time correlators, suitable
for lattice evaluation
C. Alexandrou et al. 2014
Spectator di-quark model
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 formfactors
A. Bacchetta et al. 2008
For PDFs
For Quasi-PDFs
Phenomenological success of the
spectator di-quark model
ZEUS2002
GRSV2000
A. Bacchetta et al. 2008
 Example
Qualitative and sometimes quantitative
agreement with global analysis
Calculation of PDFs and quasi-PDFs
 Standard PDFs
 Quasi PDFs
 Standard PDFs
 Quasi-PDFs
Analytic computations of f1
^
 The spectator di-quark model, tree level calculation an the dipole form
factors make the calculation straightforward
Result
Invariant mass
Does not depend on the momentum P+
While in slightly different notation agree with
A. Bacchetta et al. 2008
Analytic computations of the quasi-f1
Notation
 Invariant mass
 To find
Summary of quasi-PDF results (unintegrated)
The collinear unpolarized, helicity and
transversity distributions
Obtained analytic results for the unintegrated PDFs
and quasi-PDFs for both scalar and axial vector diquarks
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 quasiPDFs reduce to the standard PDFs when Pz -> infinity
Numerical studies of PDFs and quasi-PDFs
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
The unpolarized distribution
L. Gamberg et al. 2014
Helicity and Transversity
Similar results
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%
Deviations at large x
See the
approximation
/ limit
For x ~0.7 at Pz ~ 2 GeV quasi-PDFs deviate for PDFs a lot –
factor of 2-3
For Pz ~ 4 GeV the deviation is ~50%. Clearly much larger boost
is needed for large values of Bjorken x
Sum rules and bounds
X. Artru al. 2008
A. Bacchetta et al. 2000
 Arise form symmetries, kinematics, etc. With (quasi-)unpolarized,
helicity and transversity it is easy to check the Soffer inequaity
Soffer bound
 Generally bounds do not hold for quasi-PDFs. The approach to the sum
rule is similar
Conclusions
Significant advances recently in understanding TMDs / PDFs
• Global fits to TMDs with evolution. Collinear PDFs constrained for a long time.
New lattice techniques emerge to evaluate TMDs / PDFs
directly on the lattice
• “Traditional” lattice approach. New proposition to use quasi-PDFs. Lattice can
benefit from guidance at what boost the quasi-PDFs approach PDFs
Evaluated the quasi-PDFs, unpolarized distribution, helicity,
transversity, in the di-quark spectator model
• 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
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
Possible extensions
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