TUTORIAL

FeynmanParameter[] and trace[]

This package contains two programs.  trace[] computes traces of products of 
gamma matrices.  FeynmanParameter[] converts integrals over momentum space 
of the type encountered in Feynman diagrams with loops to integrals over 
Feynman parameters.  The following describes the use of these two programs.

Step 1: Copy the file feyn.m to the subdirectory on the hard disk which 
	contains Mathematica (probably c:\math\).  
Step 2: Start Mathematica as usual.
Step 3: At the Mathematica input prompt (In[n]:=) type: <<feyn.m 
	-This command loads the programs into main memory.

The programs are now ready to be used.  By default, these programs work in 
four dimensions.  If the user wishes to work in other than four dimensions, 
he/she should type at the input prompt: dimension=d, where d is the number 
of dimensions in which the programs will now work.

Unfortunately, we can not use standard physics notation inside of 
Mathematica.  So, in order to use trace[] and FeynmanParameter[], we need 
a dictionary to convert from standard physics notation to the notation that 
the programs understand.
			      

Standard Notation                           Mathematica Notation
-----------------                           --------------------

A. Tensors and Indices

metric tensor                               g[a,b]

tensor that appears in                      ghat[a,b]
      a       5
{gamma  ,gamma  } outside
of four dimensions.
(See page 291 of Comput. Phys. Commun. vol. 77 (1993))
				       
Levi-Civita tensor                          e[a,b,c,d]

Lorentz indices                             p[a], k[b], etc.

mass of the top quark,                      m[t], m[b], etc.
mass of the bottom quark, etc.
                                                              a
p.q                                         p[a] q[a] (i.e., p q )
                                                                a

(Where the choice of the dummy symbol a is arbitrary except that it must
 not be repeated elsewhere in the expression except as a mass label such
as 'b' in m[b].  Note that the summation convention is always enforced.)


B. Traces
	   a      b          
trace{gamma  gamma }                       trace[a,b] or tr[a,b]


(Represent gamma matrices by their indices and replace products by commas 
-see examples below.)

	   a      b      5
trace{gamma  gamma  gamma }                 trace[a,b,five] or tr[a,b,five]
       
	      5
(replace gamma  by "five")

	   a      b         5
trace{gamma  gamma  (1+gamma )/2}           trace[a,b,(1+five)/2] or
					    tr[a,b,(1+five)/2]


trace{p-slash q-slash}                      trace[p[a]a,q[b]b] or 
					    tr[p[a]a,q[b]b]

(Where again the symbols a and b are arbitrary except that they must not be 
repeated anywhere else in the expression except as mass labels.)




C. Feynman Parameters


Convert the integral                        FeynmanParameter[Expression,k] 
     d                                      or fp[Expression,k]
    d k                                     
 ------------ (Expression) 
  (2 Pi)^d

to an integral over Feynman parameters.   
Note that Expression must have at least 2 terms in its denominator of
the form (k^2 + ...)^a.  Otherwise, Feynman parameters are not needed.

(The result of FeynmanParameter must still be integrated from x[1]=0 to 1, 
x[2]=0 to x[1], ..., x[n-1]=0 to x[n-2], where n is the number of terms in 
the denominator of Expression of the form: (k^2+2p k+M^2)^a.  These 
integrals can be performed by Mathematica's built-in functions Integrate[] 
or NIntegrate[], if desired.)


Convert the integral                        FeynmanParameter[Expression,k,j] 
   d    d                                      or fp[Expression,k,j]
  d j  d k                                     
 ----------- (Expression) 
 (2 Pi)^(2d)

to an integral over Feynman parameters.   
					 

(The result of FeynmanParameter must still be integrated from x[1]=0 to 1, 
x[2]=0 to x[1], ..., x[n-1]=0 to x[n-2] and from y[1]=0 to 1,
y[2]=0 to y[1], ..., y[m-1]=0 to y[m-2], where n is the number of terms in 
the denominator of Expression of the form: (k^2+2p k+M^2)^a and m is the 
number of terms in the denominator of the form: (j^2+2p j+M^2)^a after the 
k momentum integral has been transformed to an integral over Feynman 
parameters.  FeynmanParameter(or fp)[Expression,k,j] is equivalent to 
FeynmanParameter[FeynmanParameter[Expression,k],j].)


Convert the integral                       FeynmanParameter[Expression,k,j,i] 
  d   d   d                                  or fp[Expression,k,j,i]
 d i d j d k                                     
 ------------ (Expression) 
 (2 Pi)^(3d)

to an integral over Feynman parameters.   
					 

(The result of FeynmanParameter must still be integrated from x[1]=0 to 1, 
x[2]=0 to x[1], ..., x[n-1]=0 to x[n-2], from y[1]=0 to 1, y[2]=0 to y[1], 
..., y[m-1]=0 to y[m-2], and from z[1]=0 to 1, z[2]=0 to z[1], ..., 
z[l-1]=0 to z[l-2], where n is the number of terms in the denominator of 
Expression of the form: (k^2+2p k+M^2)^a, m is the number of terms in the 
denominator of the form: (j^2+2p j+M^2)^a after the j momentum integral has 
been converted to an integral over Feynman parameters, and l is the number 
of terms in the denominator of the form: (i^2+2p i+M^2)^a after the 
k momentum integral and the j momentum integral have been transformed to 
integrals over Feynman parameters.  FeynmanParameter(or fp)[Expression,k,j,i]
is equivalent to fp[fp[fp[Expression,k],j],i].)


The above rules will become clearer if we examine a typical Mathematica 
session in which trace[] and FeynmanParameter[] are used.  In the following 
examples, comments will appear in parentheses.

In[1]:= <<feyn.m

In[2]:= tr[a,b]                         (Typing trace[a,b] will give the 
					 same result.)
Out[2]:= 4g[a,b]                        

In[3]:= tr[a,b,five]                    

Out[3]:= 0

In[4]:= tr[v,1-five,(k[a]-p[a])a+m[b],s,1+five,(j[b]-q[b])b+m[t]]

(Note that we would get the same result by replacing, for example, 
(k[a]-p[a])a with (k[anything]-p[anything])anything as long as "anything" 
appears nowhere else in the expression, except possibly as a mass label 
such as 'b' in m[b].)

Out[4]:= 8 g[s,v] m[b] m[t]

In[5]:= dimension=d             (Let's work in d dimensions now.)

Out[5]:= d

In[6]:= tr[alpha,1+five,alpha,1+five]   (This trace is equal to 0 in four 
					 dimensions.  Note: Repeated indices 
					 are summed.)
Out[6]:= -32+8d

In[7]:= fp[k[a]/(((k-p)^ 2-m[w]^ 2)(k^ 2-m[t]^ 2)^ 2),k]

               d/2
Out[7]:= (I (-1)   Gamma[3-d/2] p[a] (1-x[1]) x[1]*
                    2  2         2         2      2    2            d   d/2
	 Power[-m[t] +p x[1]+m[t] x[1]-m[w] x[1]-p x[1] ,-3+d/2])/(2  Pi   )

(This expression must now be integrated from x[1]=0 to x[1]=1 using either 
Mathematica's numeric or symbolic integration functions.  For example:)
  
In[8]:= Integrate[ Evaluate[% /. d->4], {x[1],0,1}]

          I    p[a]
Out[8]:=  --  ------- + ...
          16   2   2
              p  Pi  

In[9]:= dimension=4

Out[9]:= 4

In[10]:= Simplify[ FeynmanParameter[(k[a]+j[a])p[a]k[b]/(((k-j)^2-m[h]^2)^2
         (k k-m[t]^2)^2 (j j-m[t]^2)^2),j,k] ]

(FeynmanParameter[] and trace[] may be combined with other Mathematica 
functions.)

                                             4          2               2 
Out[10]:= (-2+x[1])(-1+y[1])y[1] p[b]/(512 Pi (y[1] m[h] -x[1] y[1] m[h] +
                   2     2    2     2         2
          x[1] m[t] -x[1] m[t] +x[1] y[1] m[t] )

(This expression should be integrated from x[1]=0 to x[1]=1 and from 
y[1]=0 to y[1]=1 to give the correct expression for the momentum integrals.)

Limitations:

At present, FeynmanParameter[] can convert, at most, three momentum integrals 
into integrals over Feynman parameters.  In addition, the numerator of 
Expression in FeynmanParameter[Expression,k], for example, can contain 
products of, at most, 5 k[a]'s.  As is noted above, it is also assumed that
the denominator of Expression contains at least two terms of the form
(k^2+...)^a.  If this is not the case, then Feynman parameters are not
needed.  It is fairly straightforward to modify the programs to work for
more general cases when (and if) necessary.  These modifications may be
made by the user or by getting in touch with me.

A more detailed discussion of these programs may be found in Computer
Physics Communications vol. 77, 286 (1993).  If you would like to receive
future updates of these programs, please send me your email address.

Please contact me if you have any questions or comments.

				Todd H. West
				Theory Group
				Department of Physics
				University of Texas
				Austin, Texas 78712

				e-mail: toddwest@utaphy.ph.utexas.edu