# Ji's Sum Rule

1. Oct 25, 2007

### cylon

Dear all,

I am new here and hope that I write this in the right place. ;-)

I am seeking help in deriving Ji's Sum Rule, which tells you that the second moment of the nucleon's GPDs equals the total angular momentum of the quarks. E.g. in Diehl, hep-ph/0307382, Sect. 3.6.

The step where I'm stuck is in the previously mentioned paper, from Eq. (68) to Eq. (69). It seems so easy, though: for $\mu$ different from $\nu$, the terms in $g^{\mu\nu}$ in (68) disappear, while the other term in C disappears in the forward limit p'->p. One thus keeps only the terms in A and B. For the B-term, I immediately use the Gordon identity. This gives me:
$<p|T^{\mu\nu}|p> = (A(0)+B(0))/2 \bar{u} \frac{p^{\mu}\gamma^{\nu} + p^{\nu}\gamma{\mu}}{2} u - B(0) \bar{u} \frac{p^{\mu}p^{\nu}}{m} u$

If I then fill in some values for the spinors, I always arrive at (A-B)/2 instead of (A+B)/2. I don't see where it goes wrong, which is so frustrating... Is it due to my spinors? Or to something else? I don't seem to need the fact that the proton is at rest (which is written below Eq. (66)). In fact, I always end up with a term ~ Lz, which turns zero in the limit of a proton in its rest frame.

I hope that someone here can help me... I've read so many papers about it, and still don't find the mistake.

Kind regards,
C