The gluon propagator in momentum space Page: 3 of 6
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tgure . ( =, er
to the form GGib., (diamond points)
It is well known that the data points obtained
from a Monte Carlo simulation are in general
statistically correlated; in the present case, the
correlated data are the values of the propagator
G(k = 0, t) at different timeslices.
The standard way to take into account this ef-
fect when performing X1 fits is to use the defini-
tion of X2 that involves the full covariance ma-
trix(10]. Such a definition reduces to the stan-
dard one when the covariance matrix is diagonal,
which would happen if the data points were un-
By inspection of the covariance matrix for
G(k = 0, t) it turns out that the off-diagonal ma-
trix elements are typically of the same size as the
diagonal ones, i.e. our data points are highly cor-
04 0.5 0.6 0.7 0.8 0.9 1
Figure 3. momentum space propagator vs. hlk in
GeV (assumes 1/a = 2.0 GeV)
tematic error on a-1 related in t. Consequently, when we perform X
Good agreement is also obtained by using fits taking into account the full covariance ma-
form commonly referred to as particle + trix, the fits are not well controlled because the
ost, that is G(k = 0, t) ~ Ctexp(-Mit) + correlation matrix is nearly singular. However,
ezp(-M2t), where C2 is constrained to be neg- we still find that GO,ibq.(k = 0, t) fits the data
ve, better than other forms. There is also qualitative
On the other hand, one cannot get good agree- agreement between our results for G(k = 0, t) and
nt if one uses a conventional 4-parameter dou- previous ones from other -groups(7-9].
exponential form, that is if one constrains C2 In spite of the difficulties in the statistical anal-
the above formula to take positive values. In- ysis, our interpretation of the results for G(k =
d, in this case the effective mass would always 0, t) receives a strong support from the analy-
rease with t, in contrast to what is observed. sis of the momentum space propagator G(k) =
,= G,,(k). It turns out that such a quantity
is very well determined in a significant interval
of physical momenta, ranging from the lattice in-
frared cutoff k, = - to k - 3k, (see Fig. 3).
- eIn this range we fit the data to the continuum
- - formula (2) and, for a comparison, to a standard
s massive propagator G,,s.,,(k) . .
An interesting point is that the covariance ma-
trix associated with G(k) turns out to be much
more "diagonal" than the one for G(k = 0, t); in
- - other words, the data points are much less corre-
lated in momentum space than they are in t. As
a consequence we have been able to obtain good
0 1 2 3 4 5 6 7 8 fits for G(k) with or without taking into account
correlations. We show in Fig. 4 a fit of G(k)
G~ k. - V) ( ti own ndt to the form GG,,a.(k). With the full covariance
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Bernard, C.; Parrinello, C. & Soni, A. The gluon propagator in momentum space, article, December 31, 1992; Upton, New York. (https://digital.library.unt.edu/ark:/67531/metadc1273567/m1/3/: accessed May 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.