Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Form Factors & GPDs

  1. May 25, 2007 #1
    Hi, I'm doing some work on DVCS and was wondering if anyone could better explain the link between FF's, PDFs, Compton Form Factors, and GPDs.

  2. jcsd
  3. May 25, 2007 #2
    GPDs are Fourier transorm along the light-cone of non-local matrix elements between two different hadronic states. You are probably aware of that. More precisely, if you consider the vector case you find the two GPDs [tex]H[/tex] and [tex]E[/tex] :
    [tex]\int\frac{\text{d}\lambda}{2\pi}e^{-\imath\lambda x}\langle P_2|\bar{\Psi}^q(-\frac{\lambda n}{2})\gamma^+\Psi^q(\frac{\lambda n}{2})| P_1 \rangle=\bar{U}(P_2)\left[H^q(x,\xi,t)\gamma^+
    +E^q(x,\xi,t)\frac{\imath\sigma^{+i}q_i}{2M}\right] U(P_1)[/tex]
    and similarly if you replace [tex]\gamma^+\rightarrow\gamma^+\gamma_5[/tex] you'll get the axial-vector GPDs [tex]\tilde{H}[/tex] and [tex]\tilde{E}[/tex], and if you replace [tex]\gamma^+\rightarrow\sigma^{+\perp}\gamma_5[/tex] you would get four more transversity GPDs which are chiral odd and usually suppressed by at least one power of [tex]Q[/tex].

    The link to PDFs is quite simple. Take the limit [tex]\xi\rightarrow 0[/tex] and [tex]t\rightarrow 0[/tex]. For instance [tex]H^{q}(x,0,0)=q(x)[/tex]. If you consider [tex]\tilde{H}[/tex] instead you'll get to helicity dependent PDFs.

    The link to FFs is also rather simple. Take the first Mellin moment with respect to [tex]x[/tex] :
    [tex]\int_{-1}^{1}\text{d}x\, H^q(x,\xi,t)=F^{\:q}_1(t)[/tex] (Dirac FF). And similarly with [tex]E\leftrightarrow F_2[/tex] (Pauli FF), [tex]\tilde{H}\leftrightarrow g_{A}[/tex] and [tex]\tilde{E}\leftrightarrow g_{P}[/tex].

    It is quite annoying that I cannot check my formulae as I type them...

    The link between GPDs and CFFs is less trivial and less fundamental at the same time. CFFs appear in the DVCS amplitude. You will find every detail explicitely in Theory of deeply virtual Compton scattering on the nucleon. But beware of possible uncontrolled approximations in this paper.

    edit I'm digging out formulae from old tex of mine :smile:
    The [tex]{\cal H}[/tex] CFF reads :

    [tex]{\cal H}(\xi,t) = \sum_q Q_q^2\,\mathscr{P}\int_{-1}^1\text{d}x\,\frac{H^q(x,\xi,t)}{1-x/\xi-0\imath}-\frac{H^q(x,\xi,t)}{1+x/\xi-0\imath}
    +\imath\pi\sum_q Q_q^2\left\{H^q(\xi,\xi,t)-H^q(-\xi,\xi,t)\right\}[/tex]
    Last edited: May 25, 2007
  4. May 25, 2007 #3
    I'll provide a few references for convenience
    I do not just warn you that those are my personal preferences. I have willingly ommited some historical detours...

    Overviews :

    Deep Virtual Compton Scattering and the Nucleon Generalized Parton Distributions
    An introduction to the Generalized Parton Distributions
    Study of Generalized Parton Distributions with CLAS

    Quark Imaging in the Proton Via Quantum Phase-Space Distributions

    GPDs theory :

    Most complete reference to date :
    Unraveling hadron structure with generalized parton distributions

    One I like, good to begin :
    Deeply virtual electroproduction of photons and mesons on the nucleon : leading order amplitudes and power corrections

    A rigourous, highly recommended :
    Generalized Parton Distributions

    Containing the most-widely used model (from chiral-soliton) :
    Hard Exclusive Reactions and the Structure of Hadrons

    Most important historical papers :

    Off-Forward Parton Distributions
    Deeply Virtual Compton Scattering
    Gauge-Invariant Decomposition of Nucleon Spin and Its Spin-Off
    Breakup of hadron masses and energy momentum tensor of QCD
    Generalized Parton Distributions
    Skewed Parton Distributions
    Scaling Limit of Deeply Virtual Compton Scattering

    experimental aspects :

    Deep Exclusive Scattering and Generalized Parton Distributions : Experimental Review
    Generalized Parton Distributions and Deep Exclusive Reactions: Present Program at JLab
    Deeply Virtual Compton Scattering at HERA II (H1 results)

    The first dedicated experiment recently published a crucial test :
    Scaling Tests of the Cross Section for Deeply Virtual Compton Scattering
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook