References - Bogoliubov Laboratory of Theoretical Physics - JINR

2 Recoil Polarization Method
With a longitudinally polarized electron beam and an un-polarized hydrogen target, the
polarization of the incoming electron is transferred to the proton. The non-zero components
of the recoil proton polarization are in the reaction plane, parallel, Pℓ, andperpendicular,
Pt, to the proton momentum [19, 20].
In the one-photon exchange process, the form factors depend only on Q 2 and a deviation
from constant would indicate a mechanism beyond the Born approximation.
In the general case, elastic ep scattering can be described by three complex amplitudes
[16,21]: ˜ GM, ˜ GE, and ˜ F3, the first two chosen as generalizations of the Sachs electric and
magnetic form factors, GE and GM, and the last one, ˜ F3, vanishing in case of Born
approximation. The reduced cross section, σred, and the proton polarization transfer
components Pt and Pl, including two-photon exchange formalism, can be written as [16]:
where:
σred/G 2 M
Here τ = Q 2 /4M 2 p
εR2
= 1+
τ +2Reδ ˜ GM
+2Rε
GM
Reδ ˜ GE
τGM
�
2ε(1 − ε) G
Pt = −
τ
2 �
M
R + R
σred
Reδ ˜ GM
GM
�
Pl = � (1 − ε 2 ) G2 M
σred
1+2 Reδ ˜ GM
GM
�
+2 1+ R
�
εY2γ
τ
�
+ Reδ ˜ GE
GM
+ 2
1+ε εY2γ
+ Y2γ
(1)
(2)
�
, (3)
Re ˜ GM(Q 2 ,ε) = GM(Q 2 )+Reδ ˜ GM(Q 2 ,ε) (4)
Re ˜ GE(Q 2 ,ε) = GE(Q 2 )+Reδ ˜ GE(Q 2 ,ε) (5)
R(Q 2 ) = GE(Q 2 )/GM(Q 2 )
Y2γ(Q 2 ,ε) =
� τ(1 + τ)(1 + ε)
1 − ε
Re ˜ F3(Q 2 ,ε)
GM(Q 2 )
,andε =[1+2(1+τ)tan2 θe
2 ]−1 ,whereθe is the lab electron scattering
angle. While the Sachs form factors depend only on Q 2 , in the general case the amplitudes
depend also on ε. The reduced cross section and the transferred proton polarization
components are sensitive only to the real part of the amplitudes.
In Born approximation only the first term remains in the reduced cross section, σred,
and the proton polarization transfer components Pt and Pl are :
σred/G 2 M = 1+ εR2
�
τ
2ε(1 − ε)
Pt = −
τ
G 2 M R
σred
, Pl = � (1 − ε 2 ) G2 M
Both polarization components can be obtained simultaneously by measuring the azimuthal
asymmetry of the recoil protons after re-scattering in an analyzer. The major
advantage of this method, compared to cross section measurements, is that in Born
approximation the form factor ratio R = GEp/GMp is directly proportional to Pt/Pℓ,
301
σred
(6)
(7)
(8)