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Reduce  program

The Reduce  program was the first CompHEP  output which opened the possibility to produce some physical results by means of this package. Later on, when the CompHEP  built-in symbolic calculator was created, the Reduce  output became unnecessary. But we still keep it in the package for testing.

The CompHEP  symbolic calculator looks like a black box. On the contrary the Reduce  program is written in terms which can be understood and checked. The comparison of result produced by the built-in symbolic calculator with that of Reduce  evaluation of the generated code is a good check of CompHEP  software. Below we shall describe the structure of Reduce  program.

CompHEP  generates a separate file for each squared diagram. The files are named as pNNN-MMM.red where NNN is the subprocess number and MMM is the diagram ordering number.

The file begins with from the declaration of momenta and Lorentz indices. For example,

% ----------- VARIABLES ------------------ 
 vector  A,p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,ZERO_;
 vector  m1,m2,m3,m4,m5,m6,m7,m8,m9,m10,m11,m12,m13,m14,m15,m16;
%
%--------- Mass shell declarations -----------
 MASS  P1 = 0$  MSHELL P1$
 MASS  P2 = 0$  MSHELL P2$
 MASS  P3 = Mm$  MSHELL P3$
 MASS  P4 = Mm$  MSHELL P4$

%-------- Momentum substitutions --------
 Let  p4 = +p1+p2-p3$
 Let  p5 = +p1+p2$
 Let  p6 = -p1-p2$

Vector A is used by the Reduce  package to construct the $\gamma^{5}$ matrix, $\gamma^{5}=G(ln,A)$. Vectors whose names begin with 'p' are used for designation of momenta. Vectors whose names begin with 'm' are reserved for Lorentz indices. We see then the mass-shell declarations for incoming and outgoing particles. After that the file contains the substitutions of momenta of on-shell and virtual particles which we have according to the conservation law.

Then the diagram total factor with self explanatory comments is written down. For example, in the case of $e^+ e^- \rightarrow \mu^+ \mu^-$ process we have

%---------- Factors ---------------
 SymmFact:=1/1$    % Diagram symmetry factor
 AverFact:=1/4$       % Normalization factor of polarization average
 FermFact:=1$      % (-1)**(number of in-fermion particles)
 ColorFact:=1/1$    %  QCD color weight of diagram  
%
totFactor_:=EE**4$
totFactor_:=totFactor_*SymmFact*AverFact*FermFact*ColorFact$

These declarations are self-explanatory except, maybe, of the $SymmFact$ variable. Generally $SymmFact=N/D$ where $N$ is equal to 1 in the case of left-right squared diagram symmetry. Otherwise the factor equals 2. $D$ is a factorial connected with presence of identical outgoing particles and partially reduced by a number of various possibilities to assign the momenta of outgoing particles to the corresponding diagram lines.

The program for evaluation of one squared diagram actually includes the codes for a set of diagrams which appear after replacing of some physical particles by their ghosts according to the existing interaction vertices. We shall call all diagrams of the set as ghost diagrams. Note that all these diagrams have the same denominator. The evaluation is started from the initialization of variable for the sum of numerators of the set:

numerator_:=0$

The program for evaluating each diagram of the set is advanced by the pseudo-graphical image of diagram. (See, for example, Fig.). The name of particle and corresponding momentum are written down near the line. The Lorentz index is written just on the line.

The diagram code is started from the fermion loops evaluation. The program moves along the fermion line and multiplies the vertex and propagator terms. The $nospur$ instruction is declared before each loop evaluation to prevent the default trace evaluation in the end of any instruction.

If the result of multiplication contains Lorentz indices which can be contracted, the program declares the corresponding vectors as indices by means of the index instruction.

As a rule the fermion vertices are multiplied in the order which corresponds to moving in the direction opposite to the direction of fermion arrows. But if the diagram contains a vertex with the C-conjugate operator or Majorana particles the order of multiplication is chosen at random, because the propagator lines have different orientations or have no them at all. If the orientation of fermion line is chosen, the vertices are divided into two groups, normal and reverse. For the normal vertex the incoming fermion line is attached to the second fermion, whereas for the reverse vertex it is attached to the first one. CompHEP  transforms the reverse vertices to normal form by means of the rule (11) and writes down the corresponding comment.

After the last multiplication the state of the $spur$ switch is restored and the $\gamma$-matrix trace is evaluated with simultaneous multiplication by factor of '-4'. The '4' is needed because the Reduce  trace evaluation omits this factor and the minus appears from the Feynman rules.

The next step is the multiplication of vertices and contraction of Lorentz indices . The indices for contraction are declared by the Index instruction before the multiplication. The fragment of the corresponding code looks like

 Index m2$
 Vrt_3:=Vrt_3*Vrt_5$
 RemInd  m2$

This code means that the vertex number 3 is multiplied by the vertex number 5 with contraction of $index\; m2$. The result of multiplication is considered as a generalized vertex number 3.

Here we would like to note that there is a bug in the Reduce  package which forbids to convolute several indices in one operation. To bypass this bug we declare the '$m$' vectors as indices step by step and recalculate the expression on each step.

If two vertices are connected by a propagator of massive vector particle treated in the physical gauge (6) the code are organized in the following way. CompHEP  writes the code for the vertices product with contracted indices and the code for the product of the same vertices with the prior multiplication by relevent momentum. After that these contributions are summarized. Below we present an example of such a code:

 Index m1$
 Vrt_0:=Vrt_1*Vrt_2$
 RemInd  m1$
 Vrt_L:=Vrt_1$   Vrt_R:=Vrt_2$
 Vrt_L:=(Vrt_L where m1=>(+P2+P3)/MZ)$
 Vrt_R:=(Vrt_R where m1=>(+(-P2)+(-P3))/MZ)$
 Vrt_0:=Vrt_0 + Vrt_L*Vrt_R$

The code for any diagram evaluation is terminated by instruction

$$numerator\_:=numerator\_ +DiagramFactor*GhostFact*Vrt\_1$$

which summarizes the ghost diagram contributions. Here GhostFact equals $(-1)^{l+v}$ where the $l$ is a number of loops of Faddeev-Popov ghosts and $v$ is a number of vector field lines in the diagram. The $(-1)^{v}$ factor appears because the evaluations discribed above correspond to substitution of the $(g_{\mu\nu} - k_{\mu}k_{\nu}/M^2)$ factor for the propagator and the density matrix of vector field whearas the correct expression has the opposite sign.

The last step of the program is the assignment of variable $denominator\_$. It is expressed as a production of $propDen(P,Mass,Width)$ functions as it is explained in Section .

As a result the symbolic answer for the evaluated diagram may be presented in the form

$$totFactor\_ \frac{ numerator\_}{denominator\_}\;.$$