# PDFs expressed through matrix elements of bi-local operators

Extracted from 'At the frontier of ParticlePhysics, handbook of QCD, volume 2',

'...in the physical Bjorken $x$-space formulation, an equivalent definition of PDFs can be given in terms of matrix elements of bi-local operators on the lightcone. The distribution of quark 'a' in a parent 'X' (either hadron or another parton) is defined as $$f^a_X (\zeta, \mu) = \frac{1}{2} \int \frac{\text{d}y^-}{2\pi} e^{-i \zeta p^+ y^-} \langle X | \bar \psi_a(0, y^-, \mathbf 0) \gamma^+ U \psi_a(0) | X \rangle ,'$$ where $$U = \mathcal P \exp \left( -ig \int_0^{y^-} \text{d}z^- A_a^+(0,z^-, \mathbf 0) t_a \right)$$ is the Wilson line.

My questions are:

1) Where does this definition come from? I'd like to particularly understand in detail the content of the rhs (i.e the arguments of the spinors, why an integral over $y^-$ etc)

2) The review also mentions that in the physical gauge $A^+=0$, $U$ becomes the identity operator in which case $f^a_X$ is manifestly the matrix element of the number operator for finding quark 'a' in X with plus momentum fraction $p_a^+ = \zeta p_X^+, p_a^T=0$. Why is $A^+=0$ the physical gauge?

See my answer to this related SE post for points relating to the first question. The overall matrix element structure is derived rather pedagogically in Schwartz and the leap to the presence of a $\gamma^+$ from $\gamma^0$ is simply the negligence of higher twist power suppressed terms that do not contribute within the usual collinear factorisation of hard and soft sectors.
Re the second question, the vanishing of the ‘plus’ component of the vector, $A^+ = 0$, is the realisation of no non physical modes propagating in the description. This can be seen in the form of the gluon propagator in the axial gauge through a gauge fixing term $\sim 1/\xi \,\, (n \cdot A)^2$, where $n$ is a reference vector not parallel to the gluon momentum $k$ but otherwise arbitrary and $\xi$ the gauge parameter. By demanding that permissible polarisation states be orthogonal to $k$ and $n$, one derives an axial gauge like propagator in the absence of the gauge parameter. This precise structure can be realised in the usual gauge fixing approach through the limit $\xi \rightarrow 0$ and $n \cdot A=0$. As $A^+ \equiv n \cdot A$, the nullity of the lhs is synonymous with working in a ‘physical’ gauge through the above argument.