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Preliminary Version The ASYMP package June 1, 1982
ASYMP
A package for the evaluation of bounds on Feynman Diagrams.
by
William E. Caswell (WEC@MITMC)
Anthony D. Kennedy (ADK@MITMC)
Preliminary Version The ASYMP package June 1, 1982
I. Introduction.
ASYMP is a package for determining the asymptotic behavior
of Feynman integrals. Given a topological description of a
Feynman diagram as a set of lines and vertices, together with
information about the mass of the virtual particle corresponding
to each line and the momentum entering at each external leg, it
will tell one the leading asymptotic behavior of that graph as
some sets of masses get much larger than others.
As this package is very unlikely to be of use to people
who are not familiar with Feynman diagrams and other basic
aspects of perturbative quantum field theory, we will refrain
from describing the basics here and refer the interested reader
to any of the standard textbooks on the subject instead.
Perhaps this is also the appropriate place to mention the
limitations of the package. These are of two kinds, those which
are fundamental limitations of the formalism and methods used in
the package itself, and those which are just features which
could be added easily if they are ever needed. In the first
category we stress that the bounds are obtained for individual
Feynman graphs, and not for sums of them; in other words the
asymptotic behavior of a green's function might be quite
different from that of the graphs which contribute to it,
because there may be "miraculous" cancellations. Such
cancellations occur in many interesting theories, in particular
gauge theories, but they are best dealt with by means of Ward
identities rather than explicit calculation. Another
mathematical limitation is that the actual behavior of a graph
is only bounded by the result given  in reality the graph
might have a smaller asymptotic growth: the bounds obtained are
usually fairly good, however. In the second class of
limitations we should mention that (1) the package currently
deals only with boson fields, (2) allows only a trivial
dependence of the vertices upon momenta and masses, (3) tries to
compute 1/0 for IR divergent graphs [which is honest, in a way],
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Preliminary Version The ASYMP package June 1, 1982
and (4) returns INF for a UV divergent graph [which is correct].
All of these are simple to generalize in the program, and if one
need to get around these limitations, please contact the
authors. A slightly harder problem to circumvent is related to
point (4) above, namely (5) one cannot currently specify that a
UV divergent graph is to be subtracted in a certain way: part of
the problem is that there are many different subtraction schemes
(minimal, zero momentum Taylor series, etc.) and how to specify
which method one wants is not clear, but it would also require a
fair amount of thought to make the program renormalize
automatically. Any suggestions or comments on this, or any
other aspect of ASYMP, would be appreciated.
II. Simple Example.
The easiest way to see how ASYMP works is to look at the
simplest example, the oneloop threepoint function in (phi)^3
theory. First of all we must load the ASYMP package into a
MACSYMA:
(C1) loadfile(asymp,fasl,dsk,share1)$
ASYMP: version of 11:54pm Saturday, 4 July 1981
(C2) graph1:diagram(line(a,b,1,m),line(b,c,2,m),line(c,a,3,mm),
extline(a,4,p),extline(b,5,q),extline(c,6,pq))$
1 Loop Diagram
(C3) bound(graph1,[[m,p,q],mm,inf]);
MM
LOG()
M
(D3) 
2
MM
2
Preliminary Version The ASYMP package June 1, 1982
First of all, in line (C1) we have loaded up the FASL
(compiled) version of the ASYMP package. It identifies itself
by telling us the date on which it was born. We then define the
desired Feynman diagram as GRAPH1 using the DIAGRAM function.
DIAGRAM takes an arbitrary number of arguments, each of which is
a pseudofunction describing a part of the graph. Currently,
there are two such pseudofunctions, LINE and EXTLINE.
Logically enough LINE describes an internal line; if we type
LINE(LONDON,PARIS,RHUBARB,5*M[PLANCK]) we are defining a line
from a vertex called LONDON to a vertex called PARIS
corresponding to a particle of mass 5*M[PLANCK]. A couple of
points are to be noted, (1) the vertices can be names, numbers,
or anything one want as long as it is a valid argument to a
hashed array, (2) the factor of 5 in the mass is pointless, as
numerical factors are ignored in asymptotic bounds. The third
argument, RHUBARB, is a name for the line, which is solely there
for debugging purposes: internally ASYMP will invent its own
name for the line. This argument is, like rhubarb, best left by
the side of the plate and ignored. EXTLINE describes an
external leg to our Feynman diagram. EXTLINE(ROME, CELERY,
P+2*Q) says that there is an external leg attached to our graph
at vertex ROME carrying momentum 2*QP into the graph. It is
one's own responsibility to ensure over all momentum
conservation. The second argument, CELERY, has great
similarities to RHUBARB and is also best forgotten (well, it has
to be there, but it seems to serve no other useful role in
life).
OK, so we have now defined our graph. DIAGRAM sets up
tables of lines containing their masses etc., assigns internal
loopmomenta, routes all momenta through the graph, and tells
one the number of loops in the diagram. If we had been nosey
and typed a ; rather than a $ at DIAGRAM, it would have returned
a list of the form [G000002653,G000005532,G000007771]. The
Goo's are internal linenames of no interest to one, other than
that they are used by later programs to index the tables set up
by DIAGRAM and its cohorts. The only point of interest is that
GRAPH1 is now a list of variable names, in other words it
behaves just like any other MACSYMA variable, which is not too
surprising because it IS just like any other MACSYMA variable.
In line (C2) we get down to the real business of the day.
We use the function BOUND to find the asymptotic behavior of
GRAPH1 when the Euclidean momenta p and q and the mass m are
much smaller than the mass mm, and both are much smaller than
INF (of course: the need to put in INF by hand is just a foible
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Preliminary Version The ASYMP package June 1, 1982
of the program, so don't forget it!). To put it another way, we
set up three mass scales, which we shall call m, mm, and INF,
such that any mass of order m is asymptotically bounded by (or,
in everyday terms, much less than) any mass of order mm, and in
turn mm << INF. The second argument to BOUND, therefore, is a
list of massscales, each of which is either a mass/momentum or
a list of masses and/or momenta of the same scale. The result
of BOUND is that GRAPH1 is bounded by (an implicit constant)
times log(mm/m)/mm^2, at least for mm/m large enough.
For further examples look at the files SHARE1;ASYMP
DEMOUT, SHARE1;ASYMP DEMOU1, etc., which are the output from the
demo files SHARE1;ASYMP DEMO, SHARE1;ASYMP DEMO1, etc.
III. Method.
For a long write up, see the paper "The Asymptotic
Behavior of Feynman Integrals," Maryland Physics publication
#PP81188.
IV. Syntax.
As the syntax has been described in section II, we just
summarize it below:
DIAGRAM(<pseudofunction>,<pseudofunction>,...);
LINE(<fromvertex>,<tovertex>,<name>,<mass>);
EXTLINE(<tovertex>,<name>,<momentum>);
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Preliminary Version The ASYMP package June 1, 1982
BOUND(<diagram>,[<massscale>,<massscale>,...,INF]);
<massscale> :: mass  momentum  [<massscale>,<mass
scale>,...]
V. Notes.
For further information please send mail to ADK@MITMC or
WEC@MITMC.
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