MAKING GLUE IN HIGH ENERGY NUCLEAR COLLISIONS Page: 7 of 9
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Above we have chosen g = 2 (or equivalently, as ~ 0.3).
On the lattice, the lattice coupling is g2(pa)(L/a) = g2PLN. The continuum
limit is obtained by keeping g2[tLN fixed (to the physical value of interest-as in
Eq. 16) and taking L to zero. It appears from our simulations that we are in the
weak coupling regime for pL = 0.017, 0.035 in lattice units. For the physical values
of g2pL above, these would correspond to lattices an order of magnitude larger
than those considered so far. Detailed simulations on the above physical scenario
will be reported at a later date.
We now turn to an issue of some concern; whether quantities of interest have a
continuum limit (in the above defined sense) in the classical theory. For instance,
in thermal field theories, it is not clear that dynamical quantities such as auto-
correlation functions have a well defined limit as the lattice spacing a -4 0. In the
EFT described here, there is reason to be more optimistic.
Consider the following gauge invariant quantity; the energy density papa =
Ek/N2 of the scalar field on the lattice (in units of [L). It is plotted as a function
of the lattice size N (in units of the lattice spacing) in Fig. 1 for L = 0.0177, 0.035.
0.6
(Ek)
N 0.4
0.2
10 N 100
FIGURE 1. The lattice size dependence of the scalar kinetic energy density, expressed in units
of p4 for = 0.0177 (pluses) and y = 0.035 (diamonds). The solid line is the LPTh prediction.
The error bars are smaller than the plotting symbols.
The solid line is the prediction from lattice perturbation theory (LPTh.). It is given
by
papa =4 [(t, 4sin(,,) sin(ln,) 2 / sin2() \2i(17)
papa -6 N A2(l) + 2(l)(17)
k k
where I, = 27rn/N and .A(l) = 2 E1,2(1 - cos(hn)) is the usual lattice Laplacian.
The continuum limit of the above equation has the form papa -+ A + B log2 (L/a),
where A and B are constants that can be determined from the above equation.
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Krasnitz, A. & Venugopalan, R. MAKING GLUE IN HIGH ENERGY NUCLEAR COLLISIONS, article, November 20, 1998; Upton, New York. (https://digital.library.unt.edu/ark:/67531/metadc708940/m1/7/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.