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Last modified: January 26, 2018

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)

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