1/N Page: 6 of 72
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wbore the sum on repeated Indices is impJled. Since I am going to
stop the investigation before evaluating any Feynman integrals, I
might as veil keep the dimension of space-time arbitrary (but less
than or equil to four).
To get an idea of what is going on, I have written down in
Fig. 1 the first few diagrams (in ordinary perturbation theory)
for the scattering of two mesons of type o Into two mesons of type
b (a#b). The first diagram displayed, the Born term, is O(Xq).
The second diagram Is O(i^N), because there are N possible choices
for the Lnternal index, c. The third diagram, in contrast, is only
0(*q); the internal indices are fixed and there is no oum to do.
The explicit factor of N in the second diagram makes the
large-N limit seem nonsensical, but this ia easily rectified. All
we need do is define
80 S X0N , (2.2)
and declare that we wish to study the lir.iir of large-N with fixed
gg (not fixed Xq). The first diagram is now 0(gg/N); as we shall
see, tills Is the leading non-trlvial order in 1/N. The second
diagram is 0(gg/N), the some order (it 1/N. The third diagram is
0(gg/N^), next order Ln i/N and negligible compared Co the two
Here’s what’s next.
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Coleman, S. 1/N, report, March 1, 1980; California. (https://digital.library.unt.edu/ark:/67531/metadc1070576/m1/6/: accessed April 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.