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Scattering
----------
p1, p2, p3 and p4 are momentum 4-vectors (E,p1,p2,p3).
p1 + p2 = p3 + p4 or p1 + p2 - p3 - p4 = 0
From Special Relativity (c = 1)
E2 = p2 + m2
Therefore,
E2 - p2 = m2
But,
pμpμ = p2 = E2 - p2
Therefore,
p2 = m2
Mandelstam Variables
--------------------
The MANDELSTAM VARIABLES are defined as:
s = (p1 + p2)2 = (p3 + p4)2
t = (p2 - p4)2 = (p1 - p3)2
u = (p2 - p3)2 = (p1 - p4)2
Where s is the space channel and t is the time channel.
These channels represent different Feynman diagrams or different
possible scattering events. The s-channel corresponds to the
particles 1,2 joining into an intermediate particle that eventually
splits into 3,4. The s-channel is the only way that resonances and
new unstable particles may be discovered provided their lifetimes
are long enough that they are directly detectable. The t-channel
represents the process in which the particle 1 emits the
intermediate particle and becomes the final particle 3, while the
particle 2 absorbs the intermediate particle and becomes 4. The
u-channel is the t-channel with the role of the particles 3,4
interchanged. Therefore,
p1 p3
\ /
\..../ s channel
/ \
/ \
p2 p4
p1 p3
\ /
\ /
: t channel
:
/ \
/ \
p2 p4
.
.
.
/θ\
/ \
p2 p4
s = (p1 + p2)2
= p12 + 2p1.p2 + p22
= m12 + 2p1.p2 + m22
= m12 + 2(E1E2 - p1p2cosθ) + m22
= m12 + 2E1E2 - 2p1p2cosθ + m22
In the COM frame p = 0. Also, E = m[c2]. If the masses are
the same (= m) we get:
s = 2m2 + 2m2
= 4m2 momentum 0
This is the COM energy.
For the t channel:
t = (p2 - p4)2
= p22 - 2p2.p4 + p42
= m22 - 2(E2E4 - p2p4cosθ) + m42
Now, p4 has the opposite sign to p2. Therefore:
t = m22 - 2(E2E4 + p2p4cosθ) - m42
Again, for similar masses we get:
t = -2(m22 - m22cosθ)
= -2m22(1 - cosθ)
For the u channel:
u = (p2 - p3)2
= p22 - 2p2.p3 + p32
= m22 - 2(E2E3 - p2p3cosθ) + m32
Now, p4 has the opposite sign to p2. Therefore:
u = m22 - 2(E2E3 + p2p3cosθ) - m32
Again, for similar masses we get:
u = -2(m22 - m22cosθ)
= -2m22(1 - cosθ)
However, cosθ is now in the second quadrant and takes a
minus sign. Thus,
u = -2m22(1 + cosθ)