# What are the Feynman rules for the spinor-helicity formalism?

If we do not work with helicity amplitudes, there are Feynman rules for the external legs of a Feynman diagram, i.e. $$u_s(k),\overline{v}_s(k),\epsilon_r(k)$$ for an incoming fermion, antifermion and gauge boson, respectively, and $$\overline{u}_s(k),v_s(k),\epsilon^\ast_r(k)$$ for their outgoing pendants.

Are these correct rules for external states in the spinor-helicity formalism?

Particle & helicity rule for incoming rule for outgoing
L-fermion $$u_L(k)=k]$$ $$\overline{u}_L(k)=\langle k$$
R-fermion $$u_R(k)=k\rangle$$ $$\overline{u}_R(k)=[k$$
L-antifermion $$\overline{v}_L(k)=\langle k$$ $$v_L(k)=k]$$
R-antifermion $$\overline{v}_R(k)=[k$$ $$v_R(k)=k\rangle$$
(+)-gauge boson $$\epsilon^\mu_+(k,r)=-\frac{1}{\sqrt{2}}\frac{[r\gamma^\mu k\rangle}{[rk]}$$ $$\epsilon^{\mu\ast}_+(k,r)=\frac{1}{\sqrt{2}}\frac{[k\gamma^\mu r\rangle}{\langle rk\rangle}$$
(-)-gauge boson $$\epsilon^\mu_-(k,r)=\frac{1}{\sqrt{2}}\frac{[k\gamma^\mu r\rangle}{\langle rk\rangle}$$ $$\epsilon^{\mu\ast}_-(k,r)=-\frac{1}{\sqrt{2}}\frac{[r\gamma^\mu k\rangle}{[rk]}$$

Here, $$r$$ shall be an arbitrary reference momentum with $$r^2=0,rk\neq0$$, and of course all fermions shall be massless.

• I do not know why the table is not rendered correctly. I have written it as is explained in the help section in the text editor. Suggestions how to fix this are welcome or maybe a moderator would correct my formatting and tell me what I did wrong? Commented Feb 18, 2021 at 11:14
• Fixed (you just needed to add a newline before the table) Commented Feb 18, 2021 at 12:31

The table I have given turns out to be wrong for the antifermions. As I have explained in this answer, a left-chiral antifermion $$\overline{v}_L=(v_R)^\dagger\gamma_0$$ is described by the Dirac-conjugate of a right-chiral Dirac-spinor $$v_R$$. That means, $$v_R(k)=k]\leftrightarrow\overline{v}_L(k)=[k$$ and conversely $$v_L(k)=k\rangle\leftrightarrow\overline{v}_R(k)=\langle k$$.
L-fermion $$u_L(k)=k]$$ $$\overline{u}_L(k)=\langle k$$
R-fermion $$u_R(k)=k\rangle$$ $$\overline{u}_R(k)=[k$$
L-antifermion $$\overline{v}_L(k)=[k$$ $$v_L(k)=k\rangle$$
R-antifermion $$\overline{v}_R(k)=\langle k$$ $$v_R(k)=k]$$
(+)-gauge boson $$\epsilon^\mu_+(k,r)=-\frac{1}{\sqrt{2}}\frac{[r\gamma^\mu k\rangle}{[rk]}$$ $$\epsilon^{\mu\ast}_+(k,r)=\frac{1}{\sqrt{2}}\frac{[k\gamma^\mu r\rangle}{\langle rk\rangle}$$
(-)-gauge boson $$\epsilon^\mu_-(k,r)=\frac{1}{\sqrt{2}}\frac{[k\gamma^\mu r\rangle}{\langle rk\rangle}$$ $$\epsilon^{\mu\ast}_-(k,r)=-\frac{1}{\sqrt{2}}\frac{[r\gamma^\mu k\rangle}{[rk]}$$