README

   1 ** ZZ() denote the {\cal Z} functions whose arguments are as follows:
   2 ** [c] denotes a comma
   3 ** [v] indicates that the following arguments are vector indices
   4 ** [p] denotes a vertical bar | (the 'p' terminology is inspired from the "pipe" command in the UNIX shell)
   5
   6 ** For example:
   7 ** ZZ(i1?,i2?,[c],i3?,[c],i4?,[c],i5?,[c],i6?,[v],m?) = {\cal Z}^m_{12,3,4,5,6}
   8 ** ZZ(i1?,i2?,[p],i3?,[c],i4?,[c],i5?,[c],i6?,[c],i7?) = {\cal Z}_{12|3,4,5,6,7}
   9
  10 ** In addition:
  11 ** ellf(m) = \ell^m
  12 ** g1(i1,i2) = g^{(1)}_{12}
  13 ** g2(i1,i2) = g^{(2)}_{12}
  14 ** g3(i1,i2) = g^{(3)}_{12}
  15 ** g4(i1,i2) = g^{(4)}_{12}
  16 ** dg1(i1,i2) = \partial g^{(1)}_{12}
  17 ** dg2(i1,i2) = \partial g^{(2)}_{12}
  18 ** dg3(i1,i2) = \partial g^{(3)}_{12}
  19 ** d2g1(i1,i2) = \partial^2 g^{(1)}_{12}
  20 ** d2g2(i1,i2) = \partial^2 g^{(2)}_{12}
  21 ** d2g3(i1,i2) = \partial^2 g^{(3)}_{12}
  22
  23 * the parameters x2, x102 and y101 enforce the trace relations and are
  24 * given by (there are only two parameters really as y101 = x102 but
  25 * both of them are kept for historic reasons)
  26 *        id x2 = -3;
  27 *        id y101 = -6;
  28 *        id x102 = - 6;
  29
  30
  31
  32 The shorthand functions kkk(), kkkk() etc that appear
  33 in the 8pt files are defined as follows:
  34
  35         id kkk(i1?,i2?,i3?,m?,n?,p?) =
  36         + kk(i1,m)*kk(i2,n)*kk(i3,p)
  37         + kk(i1,m)*kk(i2,p)*kk(i3,n)
  38         + kk(i1,n)*kk(i2,m)*kk(i3,p)
  39         + kk(i1,n)*kk(i2,p)*kk(i3,m)
  40         + kk(i1,p)*kk(i2,n)*kk(i3,m)
  41         + kk(i1,p)*kk(i2,m)*kk(i3,n)
  42         ;
  43
  44         id kkkk(i1?,...,i4?,m?,n?,p?,q?) =
  45         + kk(i1,m)*kk(i2,n)*kk(i3,p)*kk(i4,q)
  46         + kk(i1,m)*kk(i2,n)*kk(i3,q)*kk(i4,p)
  47         + kk(i1,m)*kk(i2,q)*kk(i3,n)*kk(i4,p)
  48         + kk(i1,q)*kk(i2,m)*kk(i3,n)*kk(i4,p)
  49         + kk(i1,q)*kk(i2,m)*kk(i3,p)*kk(i4,n)
  50         + kk(i1,m)*kk(i2,q)*kk(i3,p)*kk(i4,n)
  51         + kk(i1,m)*kk(i2,p)*kk(i3,q)*kk(i4,n)
  52         + kk(i1,m)*kk(i2,p)*kk(i3,n)*kk(i4,q)
  53         + kk(i1,p)*kk(i2,m)*kk(i3,n)*kk(i4,q)
  54         + kk(i1,p)*kk(i2,m)*kk(i3,q)*kk(i4,n)
  55         + kk(i1,p)*kk(i2,q)*kk(i3,m)*kk(i4,n)
  56         + kk(i1,q)*kk(i2,p)*kk(i3,m)*kk(i4,n)
  57         + kk(i1,q)*kk(i2,p)*kk(i3,n)*kk(i4,m)
  58         + kk(i1,p)*kk(i2,q)*kk(i3,n)*kk(i4,m)
  59         + kk(i1,p)*kk(i2,n)*kk(i3,q)*kk(i4,m)
  60         + kk(i1,p)*kk(i2,n)*kk(i3,m)*kk(i4,q)
  61         + kk(i1,n)*kk(i2,p)*kk(i3,m)*kk(i4,q)
  62         + kk(i1,n)*kk(i2,p)*kk(i3,q)*kk(i4,m)
  63         + kk(i1,n)*kk(i2,q)*kk(i3,p)*kk(i4,m)
  64         + kk(i1,q)*kk(i2,n)*kk(i3,p)*kk(i4,m)
  65         + kk(i1,q)*kk(i2,n)*kk(i3,m)*kk(i4,p)
  66         + kk(i1,n)*kk(i2,q)*kk(i3,m)*kk(i4,p)
  67         + kk(i1,n)*kk(i2,m)*kk(i3,q)*kk(i4,p)
  68         + kk(i1,n)*kk(i2,m)*kk(i3,p)*kk(i4,q)
  69         ;
  70
  71
  72         id kkell(i1?,i2?,m?,n?,p?) =
  73         + kk(i1,m)*kk(i2,n)*ellf(p)
  74         + kk(i1,m)*kk(i2,p)*ellf(n)
  75         + kk(i1,n)*kk(i2,m)*ellf(p)
  76         + kk(i1,n)*kk(i2,p)*ellf(m)
  77         + kk(i1,p)*kk(i2,n)*ellf(m)
  78         + kk(i1,p)*kk(i2,m)*ellf(n)
  79         ;
  80
  81         id kkellell(i7?,i8?,m?,n?,p?,q?) =
  82         + kk(i7,m)*kk(i8,n)*ellf(p)*ellf(q)
  83         + kk(i7,m)*kk(i8,p)*ellf(n)*ellf(q)
  84         + kk(i7,m)*kk(i8,q)*ellf(n)*ellf(p)
  85         + kk(i7,n)*kk(i8,m)*ellf(p)*ellf(q)
  86         + kk(i7,n)*kk(i8,p)*ellf(m)*ellf(q)
  87         + kk(i7,n)*kk(i8,q)*ellf(m)*ellf(p)
  88         + kk(i7,p)*kk(i8,m)*ellf(n)*ellf(q)
  89         + kk(i7,p)*kk(i8,n)*ellf(m)*ellf(q)
  90         + kk(i7,p)*kk(i8,q)*ellf(m)*ellf(n)
  91         + kk(i7,q)*kk(i8,m)*ellf(n)*ellf(p)
  92         + kk(i7,q)*kk(i8,n)*ellf(m)*ellf(p)
  93         + kk(i7,q)*kk(i8,p)*ellf(m)*ellf(n)
  94         ;
  95
  96         id kkkell(i1?,...,i3?,m?,n?,p?,q?) =
  97         + kk(i1,m)*kk(i2,n)*kk(i3,p)*ellf(q)
  98         + kk(i1,m)*kk(i2,n)*kk(i3,q)*ellf(p)
  99         + kk(i1,m)*kk(i2,q)*kk(i3,n)*ellf(p)
 100         + kk(i1,q)*kk(i2,m)*kk(i3,n)*ellf(p)
 101         + kk(i1,q)*kk(i2,m)*kk(i3,p)*ellf(n)
 102         + kk(i1,m)*kk(i2,q)*kk(i3,p)*ellf(n)
 103         + kk(i1,m)*kk(i2,p)*kk(i3,q)*ellf(n)
 104         + kk(i1,m)*kk(i2,p)*kk(i3,n)*ellf(q)
 105         + kk(i1,p)*kk(i2,m)*kk(i3,n)*ellf(q)
 106         + kk(i1,p)*kk(i2,m)*kk(i3,q)*ellf(n)
 107         + kk(i1,p)*kk(i2,q)*kk(i3,m)*ellf(n)
 108         + kk(i1,q)*kk(i2,p)*kk(i3,m)*ellf(n)
 109         + kk(i1,q)*kk(i2,p)*kk(i3,n)*ellf(m)
 110         + kk(i1,p)*kk(i2,q)*kk(i3,n)*ellf(m)
 111         + kk(i1,p)*kk(i2,n)*kk(i3,q)*ellf(m)
 112         + kk(i1,p)*kk(i2,n)*kk(i3,m)*ellf(q)
 113         + kk(i1,n)*kk(i2,p)*kk(i3,m)*ellf(q)
 114         + kk(i1,n)*kk(i2,p)*kk(i3,q)*ellf(m)
 115         + kk(i1,n)*kk(i2,q)*kk(i3,p)*ellf(m)
 116         + kk(i1,q)*kk(i2,n)*kk(i3,p)*ellf(m)
 117         + kk(i1,q)*kk(i2,n)*kk(i3,m)*ellf(p)
 118         + kk(i1,n)*kk(i2,q)*kk(i3,m)*ellf(p)
 119         + kk(i1,n)*kk(i2,m)*kk(i3,q)*ellf(p)
 120         + kk(i1,n)*kk(i2,m)*kk(i3,p)*ellf(q)
 121         ;
 122
 123