Preliminary Version The ASYMP package June 1, 1982 ASYMP A package for the evaluation of bounds on Feynman Diagrams. by William E. Caswell (WEC@MIT-MC) Anthony D. Kennedy (ADK@MIT-MC) 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], 1 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 one-loop three-point 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,-p-q))$ 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 pseudo-function describing a part of the graph. Currently, there are two such pseudo-functions, 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*Q-P 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 loop-momenta, 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 line-names 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 3 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 mass-scales, 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 #PP-81-188. IV. Syntax. As the syntax has been described in section II, we just summarize it below: DIAGRAM(,,...); LINE(,,,); EXTLINE(,,); 4 Preliminary Version The ASYMP package June 1, 1982 BOUND(,[,,...,INF]); :: mass | momentum | [,,...] V. Notes. For further information please send mail to ADK@MIT-MC or WEC@MIT-MC. 5