# How is the helicity fixed by little group scaling?

Under little group scaling of each particle $$i = 1, 2, . . . , n$$ , the on-shell amplitude transforms homogeneously with weight $$−2h_i$$, where $$h_i$$ is the helicity of particle $$i$$:

$$A_n( \{ |1\rangle |1], h_1\}, . . . , \{t_i |i\rangle, t^{−1}_i|i], h_i\}, ...) = t^{−2h_i}_i A_n(... \{|i\rangle, |i], h_i\} . . .)\tag{2.93}$$

An on-shell 3-points amplitude $$A$$ (...) Let us suppose that it depends on angle brackets only.

We can then write a general Ansatz:

$$A_3(1^{h_1}2^{h_2}3^{h_3})= c \langle 12 \rangle ^{x_{12}} \langle 13 \rangle ^{x_{13}} \langle 23 \rangle ^{x_{23}}\tag{2.94}$$

The little group scaling (first equation above) fixes:

$$-2h_1 =x_{12}+x_{13},$$ $$-2h_2 =x_{12}+x_{23},$$ $$-2h_3 =x_{13}+x_{23}. \tag{2.95}$$

I don't understand how the first equation was used to obtain these last 3 equalities. How can this be used to calculate the MHV of particles?

In eq. (2.10) of Elvang and Huang it is explained that square/angle spinors has helicity $$\pm 1/2$$, respectively. (If the particle 3-momentum is along the $$z$$-axis, then the generator of the helicity/little group-scaling is given by $$\sigma_z$$.) This leads to the scaling $$t^{\mp 1}$$ for the square/angle spinors, respectively, cf. eq. (2.92).