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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: ownerwrimathgroup 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 nonderivative scalar
couplings and one finds the regularized integral
areg[s_]= Integrate[Log[m1^2(1x)+m2^2*x 
s*x(1x)],{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 712755
c/o GSIDarmstadt SB3 3.162 Fax: ++49 6159 712990
Planckstr. 1 mailto:h.vanhees at gsi.de
D64291 Darmstadt http://theory.gsi.de/~vanhees/index.html
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