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A PDFs expressed as matrix elements of bi-local operators

  1. Oct 13, 2016 #1


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    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. jcsd
  3. Oct 13, 2016 #2


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    Which page in the book?
  4. Oct 13, 2016 #3


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    @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
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