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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@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],



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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  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





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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



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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 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(<pseudo-function>,<pseudo-function>,...);

      LINE(<from-vertex>,<to-vertex>,<name>,<mass>);

      EXTLINE(<to-vertex>,<name>,<momentum>);



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Preliminary Version    The ASYMP package            June 1, 1982




      BOUND(<diagram>,[<mass-scale>,<mass-scale>,...,INF]);

      <mass-scale>  ::  mass |  momentum  | [<mass-scale>,<mass-
scale>,...]






                           V. Notes.




      For further information please send mail to  ADK@MIT-MC or
WEC@MIT-MC.
































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