\chapter{Calculations in High Energy Physics}
A set of {\REDUCE} commands is provided for users interested in symbolic
calculations in high energy physics. Several extensions to our basic
syntax are necessary, however, to allow for the different data structures
encountered.
\section{High Energy Physics Operators}
\label{HEPHYS}
We begin by introducing three new operators required in these calculations.
\subsection{. (Cons) Operator}\index{Dot product}
Syntax:
\begin{verbatim}
(EXPRN1:vector_expression)
. (EXPRN2:vector_expression):algebraic.
\end{verbatim}
The binary {\tt .} operator, which is normally used to denote the addition
of an element to the front of a list, can also be used in algebraic mode
to denote the scalar product of two Lorentz four-vectors. For this to
happen, the second argument must be recognizable as a vector expression
\index{High energy vector expression} at the time of
evaluation. With this meaning, this operator is often referred to as the
{\em dot\/} operator. In the present system, the index handling routines all
assume that Lorentz four-vectors are used, but these routines could be
rewritten to handle other cases.
Components of vectors can be represented by including representations of
unit vectors in the system. Thus if {\tt EO} represents the unit vector
{\tt (1,0,0,0)}, {\tt (p.eo)} represents the zeroth component of the
four-vector P. Our metric and notation follows Bjorken and Drell
``Relativistic Quantum Mechanics'' (McGraw-Hill, New York, 1965).
Similarly, an arbitrary component {\tt P} may be represented by
{\tt (p.u)}. If contraction over components of vectors is required, then
the declaration {\tt INDEX}\ttindex{INDEX} must be used. Thus
\begin{verbatim}
index u;
\end{verbatim}
declares {\tt U} as an index, and the simplification of
\begin{verbatim}
p.u * q.u
\end{verbatim}
would result in
\begin{verbatim}
P.Q
\end{verbatim}
The metric tensor $g^{\mu \nu}$ may be represented by {\tt (u.v)}. If
contraction over {\tt U} and {\tt V} is required, then they should be
declared as indices.
Errors occur if indices are not properly matched in expressions.
If a user later wishes to remove the index property from specific vectors,
he can do it with the declaration {\tt REMIND}.\ttindex{REMIND} Thus
{\tt remind v1...vn;} removes the index flags from the variables {\tt V1}
through {\tt Vn}. However, these variables remain vectors in the system.
\subsection{G Operator for Gamma Matrices}\index{Dirac $\gamma$ matrix}
\ttindex{G}
Syntax:
\begin{verbatim}
G(ID:identifier[,EXPRN:vector_expression])
:gamma_matrix_expression.
\end{verbatim}
{\tt G} is an n-ary operator used to denote a product of $\gamma$ matrices
contracted with Lorentz four-vectors. Gamma matrices are associated with
fermion lines in a Feynman diagram. If more than one such line occurs,
then a different set of $\gamma$ matrices (operating in independent spin
spaces) is required to represent each line. To facilitate this, the first
argument of {\tt G} is a line identification identifier (not a number)
used to distinguish different lines.
Thus
\begin{verbatim}
g(l1,p) * g(l2,q)
\end{verbatim}
denotes the product of {\tt $\gamma$.p} associated with a fermion line
identified as {\tt L1}, and {\tt $\gamma$.q} associated with another line
identified as {\tt L2} and where {\tt p} and {\tt q} are Lorentz
four-vectors. A product of $\gamma$ matrices associated with the same
line may be written in a contracted form.
Thus
\begin{verbatim}
g(l1,p1,p2,...,p3) = g(l1,p1)*g(l1,p2)*...*g(l1,p3) .
\end{verbatim}
The vector {\tt A} is reserved in arguments of G to denote the special
$\gamma$ matrix $\gamma^{5}$. Thus
\begin{quote}
\begin{tabbing}
\ \ \ \ \ {\tt g(l,a)}\hspace{0.2in} \= =\ \ \ $\gamma^{5}$ \hspace{0.5in}
\= associated with the line {\tt L} \\[0.1in]
\ \ \ \ \ {\tt g(l,p,a)} \> =\ \ \ $\gamma$.p $\times \gamma^{5}$ \>
associated with the line {\tt L}.
\end{tabbing}
\end{quote}
$\gamma^{\mu}$ (associated with the line {\tt L}) may be written as
{\tt g(l,u)}, with {\tt U} flagged as an index if contraction over {\tt U}
is required.
The notation of Bjorken and Drell is assumed in all operations involving
$\gamma$ matrices.
\subsection{EPS Operator}\ttindex{EPS}
Syntax:
\begin{verbatim}
EPS(EXPRN1:vector_expression,...,EXPRN4:vector_exp)
:vector_exp.
\end{verbatim}
The operator {\tt EPS} has four arguments, and is used only to denote the
completely antisymmetric tensor of order 4 and its contraction with Lorentz
four-vectors. Thus
\[ \epsilon_{i j k l} = \left\{ \begin{array}{cl}
+1 & \mbox{if $i,j,k,l$ is an even permutation
of 0,1,2,3} \\
-1 & \mbox{if an odd permutation} \\
0 & \mbox{otherwise}
\end{array}
\right. \]
A contraction of the form $\epsilon_{i j \mu \nu}p_{\mu}q_{\nu}$ may be
written as {\tt eps(i,j,p,q)}, with {\tt I} and {\tt J} flagged as indices,
and so on.
\section{Vector Variables}
Apart from the line identification identifier in the {\tt G} operator, all
other arguments of the operators in this section are vectors. Variables
used as such must be declared so by the type declaration {\tt VECTOR},
\ttindex{VECTOR} for example:
\begin{verbatim}
vector p1,p2;
\end{verbatim}
declares {\tt P1} and {\tt P2} to be vectors. Variables declared as
indices or given a mass\ttindex{MASS} are automatically declared
vector by these declarations.
\section{Additional Expression Types}
Two additional expression types are necessary for high energy
calculations, namely
\subsection{Vector Expressions}\index{High energy vector expression}
These follow the normal rules of vector combination. Thus the product of a
scalar or numerical expression and a vector expression is a vector, as are
the sum and difference of vector expressions. If these rules are not
followed, error messages are printed. Furthermore, if the system finds an
undeclared variable where it expects a vector variable, it will ask the
user in interactive mode whether to make that variable a vector or not. In
batch mode, the declaration will be made automatically and the user
informed of this by a message.
{\tt Examples:}
Assuming {\tt P} and {\tt Q} have been declared vectors, the following are
vector expressions
\begin{verbatim}
p
2*q/3
2*x*y*p - p.q*q/(3*q.q)
\end{verbatim}
whereas {\tt p*q} and {\tt p/q} are not.
\subsection{Dirac Expressions}
These denote those expressions which involve $\gamma$ matrices. A $\gamma$
matrix is implicitly a 4 $\times$ 4 matrix, and so the product, sum and
difference of such expressions, or the product of a scalar and Dirac
expression is again a Dirac expression. There are no Dirac variables in
the system, so whenever a scalar variable appears in a Dirac expression
without an associated $\gamma$ matrix expression, an implicit unit 4
by 4 matrix is assumed. For example, {\tt g(l,p) + m} denotes {\tt
g(l,p) + m*<unit 4 by 4 matrix>}. Multiplication of Dirac
expressions, as for matrix expressions, is of course non-commutative.
\section{Trace Calculations}\index{High energy trace}
When a Dirac expression is evaluated, the system computes one quarter of
the trace of each $\gamma$ matrix product in the expansion of the expression.
One quarter of each trace is taken in order to avoid confusion between the
trace of the scalar {\tt M}, say, and {\tt M} representing {\tt M * <unit
4 by 4 matrix>}. Contraction over indices occurring in such expressions is
also performed. If an unmatched index is found in such an expression, an
error occurs.
The algorithms used for trace calculations are the best available at the
time this system was produced. For example, in addition to the algorithm
developed by Chisholm for contracting indices in products of traces,
{\REDUCE} uses the elegant algorithm of Kahane for contracting indices in
$\gamma$ matrix products. These algorithms are described in Chisholm, J. S.
R., Il Nuovo Cimento X, 30, 426 (1963) and Kahane, J., Journal Math.
Phys. 9, 1732 (1968).
It is possible to prevent the trace calculation over any line identifier
by the declaration {\tt NOSPUR}.\ttindex{NOSPUR} For example,
\begin{verbatim}
nospur l1,l2;
\end{verbatim}
will mean that no traces are taken of $\gamma$ matrix terms involving the line
numbers {\tt L1} and {\tt L2}. However, in some calculations involving
more than one line, a catastrophic error
\begin{verbatim}
This NOSPUR option not implemented
\end{verbatim}
can occur (for the reason stated!) If you encounter this error, please let
us know!
A trace of a $\gamma$ matrix expression involving a line identifier which has
been declared {\tt NOSPUR} may be later taken by making the declaration
{\tt SPUR}.\ttindex{SPUR}
See also the CVIT package for an alternative
mechanism\extendedmanual{ (chapter~\ref{CVIT})}.
\section{Mass Declarations}\ttindex{MASS}
It is often necessary to put a particle ``on the mass shell'' in a
calculation. This can, of course, be accomplished with a {\tt LET}
command such as
\begin{verbatim}
let p.p= m^2;
\end{verbatim}
but an alternative method is provided by two commands {\tt MASS} and
{\tt MSHELL}.\ttindex{MSHELL}
{\tt MASS} takes a list of equations of the form:
\begin{verbatim}
<vector variable> = <scalar variable>
\end{verbatim}
for example,
\begin{verbatim}
mass p1=m, q1=mu;
\end{verbatim}
The only effect of this command is to associate the relevant scalar
variable as a mass with the corresponding vector. If we now say
\begin{verbatim}
mshell <vector variable>,...,<vector variable>;
\end{verbatim}
and a mass has been associated with these arguments, a substitution of the
form
\begin{verbatim}
<vector variable>.<vector variable> = <mass>^2
\end{verbatim}
is set up. An error results if the variable has no preassigned mass.
\section{Example}
We give here as an example of a simple calculation in high energy physics
the computation of the Compton scattering cross-section as given in
Bjorken and Drell Eqs. (7.72) through (7.74). We wish to compute the trace of
$$\left. \alpha^2\over2 \right. \left({k^\prime\over k}\right)^2
\left({\gamma.p_f+m\over2m}\right)\left({\gamma.e^\prime \gamma.e
\gamma.k_i\over2k.p_i} + {\gamma.e\gamma.e^\prime
\gamma.k_f\over2k^\prime.p_i}\right)
\left({\gamma.p_i+m\over2m}\right)$$
$$
\left({\gamma.k_i\gamma.e\gamma.e^\prime\over2k.p_i} +
{\gamma.k_f\gamma.e^\prime\gamma.e\over2k^\prime.p_i}
\right)
$$
where $k_i$ and $k_f$ are the four-momenta of incoming and outgoing photons
(with polarization vectors $e$ and $e^\prime$ and laboratory energies
$k$ and $k^\prime$
respectively) and $p_i$, $p_f$ are incident and final electron four-momenta.
Omitting therefore an overall factor
${\alpha^2\over2m^2}\left({k^\prime\over k}\right)^2$ we need to find
one quarter of the trace of
$${
\left( \gamma.p_f + m\right)
\left({\gamma.e^\prime \gamma.e\gamma.k_i\over2k.p_i} +
{\gamma.e\gamma.e^\prime \gamma.k_f\over 2k^\prime.p_i}\right) \left(
\gamma.p_i + m\right)}$$
$${
\left({\gamma.k_i\gamma.e\gamma.e^\prime\over 2k.p_i} +
{\gamma.k_f\gamma.e^\prime \gamma.e\over2k^\prime.p_i}\right) }$$
A straightforward REDUCE program for this, with appropriate substitutions
(using {\tt P1} for $p_i$, {\tt PF} for $p_f$, {\tt KI}
for $k_i$ and {\tt KF} for $k_f$) is
\begin{verbatim}
on div; % this gives output in same form as Bjorken and Drell.
mass ki= 0, kf= 0, p1= m, pf= m; vector e,ep;
% if e is used as a vector, it loses its scalar identity as
% the base of natural logarithms.
mshell ki,kf,p1,pf;
let p1.e= 0, p1.ep= 0, p1.pf= m^2+ki.kf, p1.ki= m*k,p1.kf=
m*kp, pf.e= -kf.e, pf.ep= ki.ep, pf.ki= m*kp, pf.kf=
m*k, ki.e= 0, ki.kf= m*(k-kp), kf.ep= 0, e.e= -1,
ep.ep=-1;
for all p let gp(p)= g(l,p)+m;
comment this is just to save us a lot of writing;
gp(pf)*(g(l,ep,e,ki)/(2*ki.p1) + g(l,e,ep,kf)/(2*kf.p1))
* gp(p1)*(g(l,ki,e,ep)/(2*ki.p1) + g(l,kf,ep,e)/
(2*kf.p1))$
write "The Compton cxn is",ws;
\end{verbatim}
(We use {\tt P1} instead of {\tt PI} in the above to avoid confusion with
the reserved variable {\tt PI}).
This program will print the following result
\begin{verbatim}
(-1) (-1) 2
The Compton cxn is 1/2*K*KP + 1/2*K *KP + 2*E.EP - 1
\end{verbatim}
\section{Extensions to More Than Four Dimensions}
In our discussion so far, we have assumed that we are working in the
normal four dimensions of QED calculations. However, in most cases, the
programs will also work in an arbitrary number of dimensions. The command
\ttindex{VECDIM}
\begin{verbatim}
vecdim <expression>;
\end{verbatim}
sets the appropriate dimension. The dimension can be symbolic as well as
numerical. Users should note however, that the {\tt EPS} operator and the
$\gamma_{5}$ symbol ({\tt A}) are not properly defined in other than four
dimensions and will lead to an error if used.