SetMandelstam[s, t, u, p1 , p2 , p3 , p4 , m1 , m2 , m3 , m4 ]
defines the Mandelstam variables s=(p_1+p_2)^2, t=(p_1+p_3)^2, u=(p_1+p_4)^2 and sets the momenta on-shell:
p_1^2=m_1^2, p_2^2=m_2^2, p_3^2=m_3^2, p_4^2=m_4^2. Notice that p_1+p_2+p_3+p_4=0 is assumed.
SetMandelstam[x, {p1, p2, p3, p4, p5}, {m1, m2, m3, m4, m5}]
defines x[i, j] = (p_i+p_j)^2 and sets
the p_i on-shell. The p_i satisfy: p_1 +
p_2 + p_3 + p_4 + p_5 = 0.
SetMandelstam assumes all momenta to be ingoing. For
scattering processes with p_1+p_2=p_3+p_4, the outgoing momenta should
be written with a minus sign.
FCClearScalarProducts[]
SetMandelstam[s, t, u, p1, p2, -p3, -p4, m1, m2, m3, m4];
SP[p1, p2]
SP[p1, p3]
SP[p1, p4]-\frac{\text{m1}^2}{2}-\frac{\text{m2}^2}{2}+\frac{s}{2}
\frac{\text{m1}^2}{2}+\frac{\text{m3}^2}{2}-\frac{t}{2}
\frac{\text{m1}^2}{2}+\frac{\text{m4}^2}{2}-\frac{u}{2}
SetMandelstam simultaneously sets scalar products in
4 and $D dimensions. This is controlled
by the option Dimension.
SPD[p1, p2]
SPD[p1, p3]-\frac{\text{m1}^2}{2}-\frac{\text{m2}^2}{2}+\frac{s}{2}
\frac{\text{m1}^2}{2}+\frac{\text{m3}^2}{2}-\frac{t}{2}
It is also possible to have more than just 4 momenta. For example,
for p1+p2=p3+p4+p5 we can obtain
x[i, j] given by (p_i+p_j)^2
FCClearScalarProducts[];
SetMandelstam[x, {p1, p2, -p3, -p4, -p5}, {m1, m2, m3, m4, m5}];
SPD[p4, p5]\frac{1}{2} x(4,5)-\frac{\text{m4}^2}{2}-\frac{\text{m5}^2}{2}