References - Bogoliubov Laboratory of Theoretical Physics - JINR

A ↑ (PA)
B(PB)
a
X
b d
X
c
h(Ph)
Figure 1: Factorisation in single-hadron production with a transversely polarised hadron.
The following SSA may then be measured:
A h T = dσ(ST ) − dσ(−ST )
. (4)
dσ(ST )+dσ(−ST )
Assuming standard factorisation to hold, the differential cross-section for such a process
may be written formally as (cf. Fig. 1)
dσ = � �
abc
αα ′ γγ ′
h/c
hadron h and dˆσαα ′ γγ ′ is the partonic cross-section:
ρ a α ′ α fa(xa) ⊗ fb(xb) ⊗ dˆσαα ′ γγ ′ ⊗Dγ′ γ
h/c (z), (5)
where fa (fb) is the density of parton type a (b) insidehadronA (B), ρa αα ′ is the spin
density matrix for parton a, D γγ′
is the fragmentation matrix for parton c into the final
� �
dˆσ
dˆt
αα ′ γγ ′
= 1
16πˆs 2
1
2
�
βδ
X
Mαβγδ M ∗ α ′ βγ ′ δ , (6)
where Mαβγδ is the amplitude for the hard partonic process, see Fig. 2.
The off-diagonal elements of D γγ′
h/c vanish for an unpolarised produced hadron; i.e.,
D γγ′
h/c ∝ δγγ ′. Helicity conservation then implies α = α′ , so that there can be no dependence
on the spin of hadron A and all SSA’s must vanish. To avoid such a conclusion, either
intrinsic quark transverse motion, or higher-twist effects must be invoked.
Mαβγδ =
α, α ′ γ,γ ′
ka
kb
kc
kd
β δ
Figure 2: The hard partonic amplitude, αβγδ are Dirac indices.
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