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)
• 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