1. Spin, orbital angular momentum and sum rules connecting them. Swadhin Taneja (Stony Brook University) 11/5/2015Berkeley workshop

2. Outline QCD decomposition of proton spin What do we measure ? Gauge invariance.. GPDs & Angular momentum 11/5/2015Berkeley workshop

3. QCD decomposition of proton spin Energy momentum density tensor, T 00 : Energy density, E T 0i : Momentum Density, P i Angular momentum, Angular momentum density, 11/5/2015Berkeley workshop

4. QCD decomposition of proton spin In QCD the energy momentum tensor : QCD angular momentum (Jaffe): 11/5/2015Berkeley workshop

5. What do we measure ? Gluon distribution Gluon spin So what we measure is not M 012 but M +12 ! 11/5/2015Berkeley workshop

6. Gauge invariance By adding a term 11/5/2015Berkeley workshop

7. GPDs & Angular momentum GPDs are hybrids of PDFs & FFs: describes simultaneously x and t dependence !! GPDs give access to spatial d.o.f ! 11/5/2015Berkeley workshop

8. GPDs & Angular momentum Energy momentum tensor from symmetries Angular momentum Ji spin sum rule 11/5/2015Berkeley workshop

9. Deuteron – Angular Momentum Sum Rule 11/5/2015 Berkeley workshop Berger, Cano, Diehl and Pire (2001) S. L, S. Taneja (2007)

10. Energy Momentum Tensor 11/5/2015 Berkeley workshop … and relations with deuteron GPDs:

11. 11/5/2015 Berkeley workshop … inserting the energy momentum tensor in: Quick note:For spin ½ case, G M (t=0 )