2-body scattering and Mandelstam Variables

In summary, in a 2-body scattering event, it is convenient to introduce the Mandelstam variables, which are s ≡ -(PA + PB)^2, t ≡ -(PA - PC)^2, and u ≡ -(PA - PD)^2, where PA,...,D are the 4-momenta of the particles A, . . . , D respectively. These variables are invariant under Lorentz transformation and are useful in theoretical calculations. In the centre of mass frame of A and B, the total CM energy, Etotal, is equal to √s, where s is the Mandelstam variable. The use of negative signs in the Mandelstam variables may depend on the convention
  • #1
Pizza Pasta physics
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Homework Statement


In a 2-body scattering event, A + B → C + D, it is convenient to introduce the Mandelstam variables,
s ≡ −(PA + PB)2 ,
t ≡ −(PA − PC) 2 ,
u ≡ −(PA − PD) 2 ,
where PA,...,D are the 4-momenta of the particles A, . . . , D respectively, (· · ·) 2 = (· · ·) · (· · ·) denotes a scalar product, and we are using natural units in this problem. The Mandelstam variables are useful in theoretical calculations because they are invariant under Lorentz transformation.

Demonstrate that in the centre of mass frame of A and B, the total CM energy, i.e., Etotal ≡ EA +EB = EC + ED , is equal to √ s.

Homework Equations


s + t + u = mA2 + mB2 + mC2 + mD2 (I had to show this before which I did, not sure if its relevant or not).
PA = -PB (due to being in a CM frame)

The Attempt at a Solution


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Using the scalar product notation for s, I managed to reduce s to -(EA + EB) however I still can't take the square root to show √ s = EA + EB due to the pesky negative sign.

Apart from me doing something wrong with my algebra, I was wondering if the given Mandelstam variables are correct. From all the secondary sources I've looked at none give them with the negative signs.
 
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  • #2
This depends on whether or not you use a +--- or -+++ signature on your metric. Most particle physicists use +--- and it seems like your source does not. Check what convention is used.
 

FAQ: 2-body scattering and Mandelstam Variables

1. What is 2-body scattering?

2-body scattering is a type of particle interaction where two particles interact with each other and exchange energy and momentum. Examples of 2-body scattering include collisions between atoms or between subatomic particles.

2. What are Mandelstam variables?

Mandelstam variables are a set of mathematical quantities used to describe the kinematics of 2-body scattering. They are named after the physicist Stanislav Mandelstam and are used to calculate the energy, momentum, and angle of the particles before and after the scattering event.

3. How are Mandelstam variables related to the center of mass energy?

The square of the center of mass energy, s, is equal to the sum of the squares of the Mandelstam variables, s = (p1+p2)2 = s12 + s22 + 2p1p2cos(θ), where p1 and p2 are the initial momenta of the two particles and θ is the scattering angle.

4. How do Mandelstam variables help in studying particle interactions?

Mandelstam variables provide a convenient way to describe the kinematics of 2-body scattering events and can help in predicting the outcomes of these interactions. They can also be used to calculate the cross section, which is a measure of the probability of a scattering event occurring.

5. What is the physical meaning of the Mandelstam variable s?

The Mandelstam variable s represents the square of the total energy in the center of mass frame. It is a measure of the energy available for the scattering process and can help in determining the threshold energy for certain interactions to occur.

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