# Continuum Extrapolation of Moments Nucleon Quark Distributions in Full QCD Page: 2 of 3

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2

Table 1

Moments of the quark distributions5.3

5.3

5.5

5.5

5.6

5.6

5.6.1665

.1670

.1592

.1596

.1560

.1565

.1570.232(6)

.253(7)

.230(3)

.238(5)

.247(9)

.242(9)

.263(13)(1)A2Ad

1.202(25)

1.225(26)

1.180(12)

1.140(17)

1.190(16)

1.223(20)

1.094(25).257(9)

.276(10)

.262(4)

.265(5)

.297(6)

.298(7)

.283(9)(1) 6wed

1.353(21)

1.312(23)

1.273(9)

1.277(13)

1.325(13)

1.287(18)

1.261(26)MS scheme at 4 GeV2 were calculated perturba-

tively.

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-

turbation theory:( )n-d

an L1(3g+1)m / mm 2

(4lrfir)2 xm + p }j 'niThe 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 [3] 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. [5] 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.

4. RESULTS

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.