Continuum Extrapolation of Moments Nucleon Quark Distributions in Full QCD Page: 2 of 3
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Moments of the quark distributions
MS scheme at 4 GeV2 were calculated perturba-
At j3 5.6 the smearing was optimized to max-
imize the overlap of the source with the nucleon
ground state, yielding N 50 smearing steps,
and the source-sink separation was chosen to be
12 lattice steps to minimize the statistical er-
rors subject to the requirement of a well-defined
plateau in the three-point function.
At = 5.5 to keep the physical source size
and the source-sink distance constant, we used 30
smearing steps and a separation of 10 lattice sites.
For n 0.1592 we used 413 hybrid Monte Carlo
configurations, each separated by 20 trajectories,
and at n 0.1596, we used 243 configurations
separated by 10 trajectories.
At j3 5.3 we used 25 smearing steps and a
source-sink separation of 9 lattice sites. At n
0.1665, we used 225 configurations separated by
15 trajectories, and at n 0.1670 we used 240
configurations separated by 10 trajectories.
3. CHIRAL EXTRAPOLATION
Since the pion cloud contributes significantly
to many of the operators characterizing form fac-
tors and moments of structure functions, it is
reasonable to expect large, qualitative differences
between the behavior of matrix elements in the
heavy quark regime where current lattice calcula-
tions are carried out and in the light quark regime
pertinent to physical pions. A physical descrip-
tion of the transition between these two regimes is
given by the following interpolation formula [3,4],
which incorporates the leading nonanalytic be-
havior in the pion mass specified by chiral per-
(3g+1)m / mm 2
(4lrfir)2 xm + p }j 'ni
The parameter p specifies the physical scale of
the three quark core that serves as the source
term for the pion cloud, and a single value of this
parameter of the order of 500 MeV resolves the
discrepancy in the lowest three moments ( )._d,
(x2)wd , and (x3)kd as well as the nucleon mag-
netic moment. The best fit  to the j3 5.6 data
for (x)wd is shown by the solid line in the up-
per panel of Figure 1. Of particular importance
for the present work is the fact that this inter-
polation formula is nearly constant in the regime
in which all our lattice calculations are carried
out. Hence, even though we only have data at
two quark masses for j3 5.5 and for j3 5.3 we
may study the approach to the continuum limit
by determining the best least squares fit of the
parameter a1 at each j3. Analogous interpolation
formulae for (1)xw_ d, (x)zwzd, and (1)sw_-d
are presented in ref.  and used to fit the data
in the lower three panels of Figure 1. Note that,
in contrast to the spin averaged momentum frac-
tion, the three spin-dependent moments are far
less sensitive to the pion mass because of strong
cancellation between the the nucleon and delta
contributions in chiral perturbation theory.
The data are presented in Table 1. The results
of least squares fits of the interpolation formulae
to the data at 3 5.6, 5.5, and 5.3 are shown in
Figure 1 by the solid, dashed, and dotted lines re-
spectively. For each of the four operators, we see
that the fractional shift in these lines is small as
one changes 3. Especially since the lattice spac-
ing is quite large at 3 5.3, these results show
that the 0(a) corrections for unimproved Wilson
fermions are small and benign for the operators
considered in this work. This provides solid ev-
idence that the extrapolation to the continuum
limit is reasonably close to the results at j3 5.6.
By ruling out significant a dependence, this re-
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Dreher, P.; Brower, R.; Capitani, S.; Dolgov, D.; Edwards, R.; Eicker, N. et al. Continuum Extrapolation of Moments Nucleon Quark Distributions in Full QCD, article, June 1, 2002; Newport News, Virginia. (https://digital.library.unt.edu/ark:/67531/metadc736864/m1/2/: accessed March 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.