Riemann Sheets

*To*: mathgroup at smc.vnet.net*Subject*: [mg22337] Riemann Sheets*From*: Hendrik van Hees <h.vanhees at gsi.de>*Date*: Fri, 25 Feb 2000 21:13:27 -0500 (EST)*Organization*: GSI Darmstadt*Sender*: owner-wri-mathgroup at wolfram.com

I hope I'm not annoying you with this for sure well known problems. When calculating simple one loop Feynman diagrams in quantum field theory naturarely problems with branch cuts occur. I use Mathematica to calculate the Feynman parameter integrals. As an example for my problem I show you a calculation with Mathematica 3.0 because it gives shorter answers with the same general problem as Mathematica 4.0 does. Take the most simple one loop integral with non-derivative scalar couplings and one finds the regularized integral areg[s_]= Integrate[Log[m1^2(1-x)+m2^2*x - s*x(1-x)],{x,0,1},Assumptions->( m1&& m2>0 && s<0)] Here one likes to get the principal branch cuts for all functions. In this Riemann sheet the result is an analytic function for Re(s)<(m1+m2)^2 and has a branch cut along the real axis for Re(s)>(m1+m2)^2. The result is Out[2]//TextForm= 2 2 (-I m1 Pi + I m2 Pi - 4 s - I Pi s - 4 2 2 2 2 2 Sqrt[-m1 - (m2 - s) + 2 m1 (m2 + s)] 2 2 -m1 + m2 - s ArcTan[-----------------------------------------] + 4 2 2 2 2 Sqrt[-m1 - (m2 - s) + 2 m1 (m2 + s)] 4 2 2 2 2 2 Sqrt[-m1 - (m2 - s) + 2 m1 (m2 + s)] 2 2 -m1 + m2 + s 2 2 ArcTan[-----------------------------------------] + m1 Log[-m1 ] - 4 2 2 2 2 Sqrt[-m1 - (m2 - s) + 2 m1 (m2 + s)] 2 2 2 2 2 m2 Log[-m1 ] + s Log[-m1 ] - 2 m1 Log[m2] + 2 m2 Log[m2] + 2 s Log[m2]) / (2 s) You need not read the whole expression. Clearly the expressions under the square roots are negative along s<0 and this is the problem I wanted to ask how to get rid of. The problem is that this result is ambiguous because the branch cut of the square root is along the negative real axis (as it should be for the principal branch). How can I get the correct result which is unumbiguously real for s<0? It is also not possible to get the correct result under Mathematica 4.0 with help of Simplify and/or Fullsimplify with making use of the new feature of assumption. Viele Gruesse, -- Hendrik van Hees Phone: ++49 6159 71-2755 c/o GSI-Darmstadt SB3 3.162 Fax: ++49 6159 71-2990 Planckstr. 1 mailto:h.vanhees at gsi.de D-64291 Darmstadt http://theory.gsi.de/~vanhees/index.html