1. Quark Parton Model (QPM)
main failure: Momentum sum rule
partons with no charge (= gluons) carry around half
of N momentum, but they are not included in QPM!
Introduce polarization: inclusive DIS
recall: if S=0 → parity violation from weak decays
→ current V-A → WA
if S  0 → two 4-vectors P,q + one 4-pseudovector S
→ richer structure of hadronic tensor
build S such that S2 = -1 and S∙P = 0
 -
helicity
09-Apr-13

2. hadronic tensor
S = ½ → W is at most linear in S, because it is a 2x2 matrix in spin space
⇒ it can be expanded on basis of Dirac  matrices
polarization vector
target spin density matrix
S coplanar with scattering plane →  = 0
hermitean tensor
parity invariance
time-reversal invariance
current conservation
09-Apr-13

3. polarized DIS hadronic tensor
W = WS + WA
scalar
pseudoscalar
09-Apr-13

4. scattering amplitude lepton polarized with helicity h=±
lepton tensor : L = LS ± LA
LS WS →
LA WA ←
09-Apr-13

5. cross section  = 0 ⇔ S || k  = /2 ⇔ S ⊥ k coplanar →  = 0 −
09-Apr-13

6. why 4 structure functions F1, F2, G1, G2 ?
(cont’ed)
why 4 structure functions F1, F2, G1, G2 ?
total cross section for * absorption : tot (* N)
optical theorem : tot (* N) ∝ Im [ f(e=0) Compton ]
±1 , ±1/2
±1/ ±1 , 0
initial
intermediate
final 1 +1
+1/2
+3/2 2
-1/2 3 4 5
related by time-reversal
→ 4 independent structures
09-Apr-13

7. helicity asymmetries rearrangement of 4 independent combinations
intermediate
asymmetries for scattering from *
09-Apr-13

8. DIS limit ,Q2 → ∞ with fixed xB ; if Q2 Jz scales, then scaling :
(see expressions of A1 and A2)
scaling in helicity asymmetries :
09-Apr-13

9. QPM picture then : write elementary cross section for process
write down convolution in QPM factorization hypothesis
→ deduce structure functions in terms of
partonic densities
alternatively
09-Apr-13

10. alternative method because Lz = 0 (collinear process)
* ± Jz=1/ ±1/2 q↑↓
* Jz=3/2 T3/ /2 P↑
Jz=1/2 T1/ /2
because Lz = 0 (collinear process)
→ conservation of angular momentum
then *↑ q↓ → q↑
*↓ q↑ → q↓
*↑ q↑
*↓ q↓
helicity distribution
09-Apr-13

11. transverse polarization distribution
similarly
it holds
Wandzura−Wilczek relation
Burkhardt−Cottingham sum rule
and in general
09-Apr-13

12. example: take 1 flavor only with q↑ in Jz (*q↑ )
(cont’ed)
if pT  0
*↑ q↑ , *↓ q↓ allowed
example: take 1 flavor only with q↑ in Jz (*q↑ )
pT = 0
pT  0
in DIS regime, transverse motion of partons seems suppressed.
We will see that it is not always true, in particular in connection
with transverse polarized parton / parent nucleon
09-Apr-13