Form Factors, PDFs, Compton & GPDs: Explained

[virtual_image]

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.

Thanks

 

[humanino]

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 $$H$$ and $$E$$ :
$$\int\frac{\text{d}\lambda}{2\pi}e^{-\imath\lambda x}\langle P_2|\bar{\Psi}^q(-\frac{\lambda n}{2})\gamma^+\Psi^q(\frac{\lambda n}{2})| P_1 \rangle=\bar{U}(P_2)\left[H^q(x,\xi,t)\gamma^+ +E^q(x,\xi,t)\frac{\imath\sigma^{+i}q_i}{2M}\right] U(P_1)$$
and similarly if you replace $$\gamma^+\rightarrow\gamma^+\gamma_5$$ you'll get the axial-vector GPDs $$\tilde{H}$$ and $$\tilde{E}$$, and if you replace $$\gamma^+\rightarrow\sigma^{+\perp}\gamma_5$$ you would get four more transversity GPDs which are chiral odd and usually suppressed by at least one power of $$Q$$.

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

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

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
The $${\cal H}$$ CFF reads :

$${\cal H}(\xi,t) = \sum_q Q_q^2\,\mathscr{P}\int_{-1}^1\text{d}x\,\frac{H^q(x,\xi,t)}{1-x/\xi-0\imath}-\frac{H^q(x,\xi,t)}{1+x/\xi-0\imath} +\imath\pi\sum_q Q_q^2\left\{H^q(\xi,\xi,t)-H^q(-\xi,\xi,t)\right\}$$

 

[humanino]

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