# A PDFs expressed as matrix elements of bi-local operators

1. Oct 13, 2016

### CAF123

Extracted from 'At the frontiers of Physics, a 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 $$f^a_A (\zeta, \mu) = \frac{1}{2} \int \frac{\text{d}y^-}{2\pi} e^{-i \zeta p^+ y^-} \langle A | \bar \psi_a(0, y^-, \mathbf 0) \gamma^+ U \psi_a(0) | A \rangle ,'$$ 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 $f^a_A$ is manifestly the matrix element of the number operator for finding quark 'a' in A with plus momentum fraction $p_a^+ = \zeta p_A^+, p_a^T=0$. Why is $A^+=0$ the physical gauge?

Thanks for any help!

2. Oct 13, 2016

### MathematicalPhysicist

Which page in the book?

3. Oct 13, 2016

### CAF123

@MathematicalPhysicist I don't remember the exact page number (I left the book at my desk in the university) but I think it was in the chapter 19 on generalised parton distributions (P. 1037 approx).

Actually I see a derivation in Schwartz's book P.696-697 which seems quite helpful. In it he defines his lightcone expansion for any momenta as $$k^{\mu} = \frac{1}{2} (\bar n \cdot k) n^{\mu} + \frac{1}{2} (n \cdot k) \bar n^{\mu} + k_T^{\mu}$$ To make contact with the notation used in the 'Frontiers of particle physics' book, would $n^+ = \bar n, n^- = n$? (i.e the momenta in the plus n direction is the longitudinal momenta along the direction of motion). I also don't understand what the integration over $y^-$ represents and what the meaning of $\gamma^+$ means (from Schwartz, it might be that $\gamma^+ = \gamma^0$ but not sure on that)

Few follow up questions: I don't understand when Schwartz says 'a clean way to think about which momentum components are small at large Q is using lightcone coordinates' <- why is this true?
Also for small transverse momenta carried by the parton it follows that $\gamma^{\mu} \bar n_{\mu} \psi \approx 0$ <- why is that?

Last edited: Oct 13, 2016