Helicity basis explained

In the Standard Model, using quantum field theory it is conventional to use the helicity basis to simplify calculations (of cross sections, for example). In this basis, the spin is quantized along the axis in the direction of motion of the particle.

Spinors

The two-component helicity eigenstates

\xi_λ

satisfy

\sigma ⋅ \hat{p}\xi_{λ\left(\hat{p}\right)}=λ\xi_{λ\left(\hat{p}\right)}

where

\sigma

are the Pauli matrices,

\hat{p}

is the direction of the fermion momentum,

λ=\pm1

depending on whether spin is pointing in the same direction as

\hat{p}

or opposite.

To say more about the state,

\xi_λ

we will use the generic form of fermion four-momentum:

p^\mu=\left(E,\left|\vec{p}\right|\sin{\theta}\cos{\phi},\left|\vec{p}\right|\sin{\theta}\sin{\phi},\left|\vec{p}\right|\cos{\theta}\right)

Then one can say the two helicity eigenstates are

\xi₊₁(\vec{p})=

	1
	\sqrt{2\left\|\vec{p

\right|\left(\left|\vec{p}\right|+p_z\right)}}\begin{pmatrix} \left|\vec{p}\right|+p_z\\p_x+ip_y\end{pmatrix}=\begin{pmatrix} \cos{

	\theta
	2

} \\ e^\sin \end\,

and

\xi_-1(\vec{p})=

	1
	\sqrt{2\|\vec{p

|(|\vec{p}|+p_z)}}\begin{pmatrix} -p_x+ip_y\\ \left|\vec{p}\right|+p_z\end{pmatrix}=\begin{pmatrix} -e^-i\phi\sin{

	\theta
	2

} \\ \cos \end\,

These can be simplified by defining the z-axis such that the momentum direction is either parallel or anti-parallel, or rather:

\hat{z}=\pm\hat{p}

In this situation the helicity eigenstates are for when the particle momentum is

\hat{p}=+\hat{z}

\xi₊₁(\hat{z})=\begin{pmatrix} 1\\ 0 \end{pmatrix}

and

\xi_-1(\hat{z})=\begin{pmatrix} 0\\ 1 \end{pmatrix}

then for when momentum is

\hat{p}=-\hat{z}

\xi₊₁(-\hat{z})=\begin{pmatrix} 0\\ 1 \end{pmatrix}

and

\xi_-1(-\hat{z})=\begin{pmatrix} -1\\ 0 \end{pmatrix}

Fermion (spin 1/2) wavefunction

A fermion 4-component wave function,

\psi

may be decomposed into states with definite four-momentum:

\psi(x)=\int{

	d^3p
	(2\pi)³\sqrt{2E

}\sum_λ

	λ
{\left(\hat{a}
	p

u_λ(p)e^-i+

	λ
\hat{b}
	p

v_λ(p)eⁱ\right)}}

where

	λ
\hat{a}
	p

and

	λ
\hat{b}
	p

are the creation and annihilation operators, and

u_λ(p)

and

v_λ(p)

are the momentum-space Dirac spinors for a fermion and anti-fermion respectively.

Put it more explicitly, the Dirac spinors in the helicity basis for a fermion is

u_λ(p)=\begin{pmatrix} u_-1\\ u₊₁\end{pmatrix}=\begin{pmatrix} \sqrt{E-λ\left|\vec{p}\right|}\chi_λ(\hat{p})\\ \sqrt{E+λ\left|\vec{p}\right|}\chi_λ(\hat{p})\end{pmatrix}

and for an anti-fermion,

v_λ(p)=\begin{pmatrix} v₊₁\\ v_-1\end{pmatrix}=\begin{pmatrix} -λ\sqrt{E+λ\left|\vec{p}\right|}\chi_-λ(\hat{p})\\ λ\sqrt{E-λ\left|\vec{p}\right|}\chi_-λ(\hat{p}) \end{pmatrix}

Dirac matrices

To use these helicity states, one can use the Weyl (chiral) representation for the Dirac matrices.

Spin-1 wavefunctions

The plane wave expansion is

\psi(x)= \int{

	d^3p
	(2\pi)³\sqrt{2E

} \sum_^3 \left(\hat_ \epsilon_\lambda(p) e^ + \hat_^\dagger \epsilon^*_\lambda(p) e^ \right)} \,.

q^\mu=(E,q_x,q_y,q_z)

, the polarization vectors quantized with respect to its momentum direction can be defined as

\begin{align} \epsilon^\mu(q,x)&=

	1
	\left\|\vec{q

\right|q_T

} \left(0,q_x q_z, q_y q_z, -q_\text^2 \right) \\ \epsilon^\mu(q,y) &= \frac \left(0, -q_y, q_x, 0 \right) \\ \epsilon^\mu(q, z) &= \frac \left(\frac, q_x, q_y, q_z \right)\end

where

q_T=

	2
\sqrt{q
	x

	2}
q
	y

is transverse momentum, and

E=\sqrt{|\vec{q}|²+m^2}

is the energy of the boson.