SetMandelstam
SetMandelstam[s, t, u, p1 , p2 , p3 , p4 , m1 , m2 , m3 , m4 ]
defines the Mandelstam variables s=(p1+p2)2, t=(p1+p3)2, u=(p1+p4)2 and sets the momenta on-shell: p12=m12, p22=m22, p32=m32, p42=m42. Notice that p1+p2+p3+p4=0 is assumed.
SetMandelstam[x, {p1, p2, p3, p4, p5}, {m1, m2, m3, m4, m5}]
defines x[i,j]=(pi+pj)2 and sets the pi on-shell. The pi satisfy: p1+p2+p3+p4+p5=0.
See also
Overview, Mandelstam.
Examples
SetMandelstam
assumes all momenta to be ingoing. For scattering processes with p1+p2=p3+p4, 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]
−2m12−2m22+2s
2m12+2m32−2t
2m12+2m42−2u
SetMandelstam
simultaneously sets scalar products in 4 and $D dimensions. This is controlled by the option Dimension
.
−2m12−2m22+2s
2m12+2m32−2t
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 (pi+pj)2
FCClearScalarProducts[];
SetMandelstam[x, {p1, p2, -p3, -p4, -p5}, {m1, m2, m3, m4, m5}];
SPD[p4, p5]
21x(4,5)−2m42−2m52