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Example: Summation of diagrams and symbolic integration by means of the Reduce  package

We have prepared some summation programs for working with Reduce  output. They are:
$sum.red$ which presents the squared matrix element as a common denominator expression;
$sum2pole.red$ which presents the squared matrix element as a sum of pole terms;
$sum2tot.red$ which presents a symbolical expression for the total cross section.
The last two options are available only for 2->2 processes. These files are stored in the $COMPHEP/test$ directory.

Let you prepare the symbolic output symb1.red for the Compton scattering A,e1 -> A,e1 in the framework of QED model ¹. Launch the Reduce  system from within the result directory. The possible Reduce  sessions are:

%1
  in"$COMPHEP/test/sum.red"$     % loading the summation package
  in"symb1.red"$                 % reading contributions of diagrams 
  sum;                           % writing the answer 

(32*ee**4*(2*p1.p2**4-4*p1.p2**3*p1.p3+3*p1.p2**2*p1.p3**2-2*p1.p2**2*p1.p3
*me**2-p1.p2*p1.p3**3 + 2*p1.p2*p1.p3**2*me**2 + p1.p3**2*me**4))
/(propden(-p1-p2,me,0)**2*propden(p2-p3,me,0)**2)$

%2
  in"$COMPHEP/test/sum2pole.red"$  % loading the summation package  
  in"symb1.red"$                   % reading contributions of diagrams
  sum;                             % writing the answer 
2*ee**4*(4*sp(me)**2*me**4 + 8*sp(me)*up(me)*me**4 + 4*sp(me)*me**2 +
sp(me)*t+ 4*up(me)**2*me**4 + 5*up(me)*me**2 - up(me)*s + 1)$

Here $s=(p1+p2)^2;\; t=(p1-p3)^2;$ the functions $sp,tp,up$ are defined by the following way

$$sp(x)=1/(s-x^2);$$

$$tp(x)=1/(t-x^2);$$

$$up(x)=1/(1-u^2)\;\; where\;\; u=(p1-p4)^2$$

%3
  in"$COMPHEP/test/sum2tot.red"$ % loading the summation package  
  in"symb1.red"$                 % reading contributions of diagrams
  sum;                           % writing answer for total cross section                   
(ee**4*(2*s**4*log(s/me**2) + s**4 - 12*s**3*log(s/me**2)*me**2 + 14*s**3*me**2 
- 6*s**2*log(s/me**2)*me**4 - 16*s**2*me**4 + 2*s*me**6 - me**8))/(16*s
**2*pi*(s**3 - 3*s**2*me**2 + 3*s*me**4 - me**6))$

Sometimes the expression for total cross-section includes a cumbersome square root of kinematic variables which appears as a result of evaluation of integrand limits. In this case the integration routine introduces a new variable be_ for this square root to express the total cross-section in a more compact form. The substitution for the be_**2 is generated by the integration routine.

There are similar packages $sum.m$, $sum2pole.m$, and $sum2tot.m$ for operation with the Mathematica  output.

Footnote

1  In other CompHEP  models the electron is massless.