Form Factors, PDFs, Compton & GPDs: Explained

virtual_image · May 25, 2007

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 · May 25, 2007

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^+<br /> +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}<br /> +\imath\pi\sum_q Q_q^2\left\{H^q(\xi,\xi,t)-H^q(-\xi,\xi,t)\right\}

humanino · May 25, 2007

Form Factors, PDFs, Compton & GPDs: Explained

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