A Classic Thesis Style - Johannes Gutenberg-Universität Mainz

130 theoretical background
a.8 φ dependence
Figure 81: Geometry of the lepton scattering plane
In order to extract the φ dependence explicitly from W ij , we introduce
the most general covariant expression for W µν .
W µν =W1(−g µν + qµ q ν
+ W4
mp
q
W2
) + 2 m2 p
(p µ −
p · q
q2 qµ )(p ν p · q
−
q2 qν )
+ W3(ˆk µ − ˆk · q
q2 qµ )(ˆk ν − ˆk · q
q2 qν )
�
(p µ p · q
−
q2 qµ )(ˆk ν − ˆk · q
q2 qν ) + (ˆk µ − ˆk · q
q2 qµ )(p ν p · q
−
q2 qν �
)
(A.57)
where ˆk µ is a unit vector of the kaon momentum and we have simply
used p for the proton four momentum. W1, W2, W3 and W4 are called
structure functions. The reader can easily verify that this symmetric
tensor 5 , by construction, satisfies the current conservation conditions
and that it is built out of the three independent vectors available.
Inserting values of p µ , q µ and ˆk µ in the K + Λ center of mass system
(see Fig. 82) we have:
Similarly:
W xx + W yy = W1 + W3 ˆk x ˆk x + W1 + W3 ˆk y ˆk y = 2W1 + W3 sin 2 θ ∗ K =
= (W xx + W yy )φ=0
(A.58)
W xx − W yy = W1 + W3 ˆk x ˆk x − W1 − W3 ˆk y ˆk y = W3 sin 2 θ ∗ K cos 2φ =
= (W xx − W yy )φ=0 cos 2φ (A.59)
W zz = W zz
φ=0
(A.60)
W zx − W xz = (W zx − W xz )φ=0 cos φ (A.61)
5 Any antisymmetric part will not contribute when contracted with the symmetric Lµν.