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The spinor-helicity formalism is usually set up so that for a massless vector boson (photon or gluon) with momentum $k$ an arbitrary reference momentum $p$ is introduced and the corresponding polarization vector of helicity $\pm 1$ depends on both $k$ and $p$: $\varepsilon^{\pm}(k,p)$. This is all good and well, tons of simplifications follow with a careful (process-dependent) choice of the $p$ reference vectors.

In the book Gastmans, Wu: ``The Ubiquitous Photon - Helicity Method for QED and QCD'' however sometimes 2 arbitrary reference momenta are introduced. For instance for gluons on page 24 equation (3.12) we have:

$$\varepsilon^{\pm} \cdot \gamma = - N \left[ (k\cdot\gamma)(p\cdot\gamma)(q\cdot\gamma)(1\pm\gamma_5) + (q\cdot\gamma)(p\cdot\gamma)(k\cdot\gamma)(1\mp\gamma_5) \right]$$

Where $p\cdot\gamma = p_\mu \gamma^\mu$ i.e. the Dirac slash, $N=N(k,p,q)$ is some normalization factor and the momentum of the gluon is $k$.

So there are 2 arbitrary reference momenta, $q$ and $p$, both light-like, so $\varepsilon^{\pm} = \varepsilon^\pm(k,q,p)$ and depending on the process these are chosen carefully in order to simplify the calculation.

What's the relationship between the two setups: 1 arbitrary reference momentum for each polarization vs. 2 arbitrary momenta for each polarization?

To be fair the only source I could find with the 2 arbitrary reference momenta is the above quoted Gastmans/Wu book, but it is appealing because the expression for the polarizations is relatively simple (does not involve spinors only combinations of gamma matrices).

The spinor-helicity formalism is usually set up so that for a massless vector boson (photon or gluon) with momentum $k$ an arbitrary reference momentum $p$ is introduced and the corresponding polarization vector of helicity $\pm 1$ depends on both $k$ and $p$: $\varepsilon^{\pm}(k,p)$. This is all good and well, tons of simplifications follow with a careful (process-dependent) choice of the $p$ reference vectors.

In the book Gastmans, Wu: ``The Ubiquitous Photon - Helicity Method for QED and QCD'' however sometimes 2 arbitrary reference momenta are introduced. For instance for gluons on page 24 equation (3.12) we have:

$$\varepsilon^{\pm} \cdot \gamma = - N \left[ (k\cdot\gamma)(p\cdot\gamma)(q\cdot\gamma)(1\pm\gamma_5) + (q\cdot\gamma)(p\cdot\gamma)(k\cdot\gamma)(1\mp\gamma_5) \right]$$

Where $p\cdot\gamma = p_\mu \gamma^\mu$ and similarly for $k$ and $q$, i.e. the Dirac slash, $N=N(k,p,q)$ is some normalization factor and the momentum of the gluon is $k$.

So there are 2 arbitrary reference momenta, $q$ and $p$, both light-like, so $\varepsilon^{\pm} = \varepsilon^\pm(k,q,p)$ and depending on the process these are chosen carefully in order to simplify the calculation.

What's the relationship between the two setups: 1 arbitrary reference momentum for each polarization vs. 2 arbitrary momenta for each polarization?

To be fair the only source I could find with the 2 arbitrary reference momenta is the above quoted Gastmans/Wu book, but it is appealing because the expression for the polarizations is relatively simple (does not involve spinors only combinations of gamma matrices).

The spinor-helicity formalism is usually set up so that for a massless vector boson (photon or gluon) with momentum $k$ an arbitrary reference momentum $p$ is introduced and the corresponding polarization vector of helicity $\pm 1$ depends on both $k$ and $p$: $\varepsilon^{\pm}(k,p)$. This is all good and well, tons of simplifications follow with a careful (process-dependent) choice of the $p$ reference vectors.

In the book Gastmans, Wu: ``The Ubiquitous Photon - Helicity Method for QED and QCD'' however sometimes 2 arbitrary reference momenta are introduced. For instance for gluons on page 24 equation (3.12) we have:

$$\varepsilon^{\pm} \cdot \gamma = - N \left[ (k\cdot\gamma)(p\cdot\gamma)(q\cdot\gamma)(1\pm\gamma_5) + (q\cdot\gamma)(p\cdot\gamma)(k\cdot\gamma)(1\mp\gamma_5) \right]$$

Where $p\cdot\gamma = p_\mu \gamma^\mu$ i.e. the Dirac slash, $N=N(k,p,q)$ is some normalization factor and the momentum of the gluon is $k$.

So there are 2 arbitrary reference momenta, $q$ and $p$, so $\varepsilon^{\pm} = \varepsilon^\pm(k,q,p)$ and depending on the process these are chosen carefully in order to simply the calculation.

What's the relationship between the two setups: 1 arbitrary reference momentum for each polarization vs. 2 arbitrary momenta for each polarization?

To be fair the only source I could find with the 2 arbitrary reference momenta is the above quoted Gastmans/Wu book, but it is appealing because the expression for the polarizations is relatively simply (does not involve spinors only combinations of gamma matrices).

The spinor-helicity formalism is usually set up so that for a massless vector boson (photon or gluon) with momentum $k$ an arbitrary reference momentum $p$ is introduced and the corresponding polarization vector of helicity $\pm 1$ depends on both $k$ and $p$: $\varepsilon^{\pm}(k,p)$. This is all good and well, tons of simplifications follow with a careful (process-dependent) choice of the $p$ reference vectors.

In the book Gastmans, Wu: ``The Ubiquitous Photon - Helicity Method for QED and QCD'' however sometimes 2 arbitrary reference momenta are introduced. For instance for gluons on page 24 equation (3.12) we have:

$$\varepsilon^{\pm} \cdot \gamma = - N \left[ (k\cdot\gamma)(p\cdot\gamma)(q\cdot\gamma)(1\pm\gamma_5) + (q\cdot\gamma)(p\cdot\gamma)(k\cdot\gamma)(1\mp\gamma_5) \right]$$

Where $p\cdot\gamma = p_\mu \gamma^\mu$ i.e. the Dirac slash, $N=N(k,p,q)$ is some normalization factor and the momentum of the gluon is $k$.

So there are 2 arbitrary reference momenta, $q$ and $p$, both light-like, so $\varepsilon^{\pm} = \varepsilon^\pm(k,q,p)$ and depending on the process these are chosen carefully in order to simplify the calculation.

What's the relationship between the two setups: 1 arbitrary reference momentum for each polarization vs. 2 arbitrary momenta for each polarization?

To be fair the only source I could find with the 2 arbitrary reference momenta is the above quoted Gastmans/Wu book, but it is appealing because the expression for the polarizations is relatively simple (does not involve spinors only combinations of gamma matrices).

The spinor-helicity formalism is usually set up so that for a massless vector boson (photon or gluon) with momentum $k$ an arbitrary reference momentum $p$ is introduced and the corresponding polarization vector of helicity $\pm 1$ depends on both $k$ and $p$: $\varepsilon^{\pm}(k,p)$. This is all good and well, tons of simplifications follow with a careful (process-dependent) choice of the $p$ reference vectors.

In the book Gastmans, Wu: ``The Ubiquitous Photon - Helicity Method for QED and QCD'' however sometimes 2 arbitrary reference momenta are introduced. For instance for gluons on page 24 equation (3.12) we have:

$$\varepsilon^{\pm} \cdot \gamma = - N \left[ (k\cdot\gamma)(p\cdot\gamma)(q\cdot\gamma)(1\pm\gamma_5) + (q\cdot\gamma)(p\cdot\gamma)(k\cdot\gamma)(1\mp\gamma_5) \right]$$

Where $p\cdot\gamma = p_\mu \gamma^\mu$ i.e. the Dirac slash, $N=N(k,p,q)$ is some normalization factor and the momentum of the gluon is $k$.

So there are 2 arbitrary reference momenta, $q$ and $p$ and depending on the process these are chosen carefully in order to simply the calculation.

What's the relationship between the two setups: 1 arbitrary reference momentum for each polarization vs. 2 arbitrary momenta for each polarization?

The spinor-helicity formalism is usually set up so that for a massless vector boson (photon or gluon) with momentum $k$ an arbitrary reference momentum $p$ is introduced and the corresponding polarization vector of helicity $\pm 1$ depends on both $k$ and $p$: $\varepsilon^{\pm}(k,p)$. This is all good and well, tons of simplifications follow with a careful (process-dependent) choice of the $p$ reference vectors.

In the book Gastmans, Wu: ``The Ubiquitous Photon - Helicity Method for QED and QCD'' however sometimes 2 arbitrary reference momenta are introduced. For instance for gluons on page 24 equation (3.12) we have:

$$\varepsilon^{\pm} \cdot \gamma = - N \left[ (k\cdot\gamma)(p\cdot\gamma)(q\cdot\gamma)(1\pm\gamma_5) + (q\cdot\gamma)(p\cdot\gamma)(k\cdot\gamma)(1\mp\gamma_5) \right]$$

Where $p\cdot\gamma = p_\mu \gamma^\mu$ i.e. the Dirac slash, $N=N(k,p,q)$ is some normalization factor and the momentum of the gluon is $k$.

So there are 2 arbitrary reference momenta, $q$ and $p$, so $\varepsilon^{\pm} = \varepsilon^\pm(k,q,p)$ and depending on the process these are chosen carefully in order to simply the calculation.

What's the relationship between the two setups: 1 arbitrary reference momentum for each polarization vs. 2 arbitrary momenta for each polarization?

To be fair the only source I could find with the 2 arbitrary reference momenta is the above quoted Gastmans/Wu book, but it is appealing because the expression for the polarizations is relatively simply (does not involve spinors only combinations of gamma matrices).