Four-point Elastic Kinematics (Two Nonidentical Masses)

Thu 04 September 2014
Physics
#kinematics, #four-point, #massive

In this post I will consider the four-point process

$$A(p_{1}) + B(p_{2}) \longrightarrow A(p_{3}) + B(p_{4}).$$

This is a special elastic case of the inelastic process that I considered in a previous post. Besides on-shell constraints and the conservation constraint, you now need to also impose elasticity constraints:

$$p_{1}^{2} = p_{3}^{2} \qquad p_{2}^{2} = p_{4}^{2} \quad \Longrightarrow \quad m_{1} = m_{3} \qquad m_{2} = m_{4}.$$

Below you can see how this feature changes the previous results.

Momentum Invariants

The three 2-Mandelstam invariants remain the same, but now they satisfy the relation

$$s + t + u = 2m_{1}^{2} + 2m_{2}^{2}.$$

Due to the elasticity constraint, some of the 2-Gram invariants become equal:

$$G_{34}(s) = G_{12}(s) = \frac{1}{4} [ s - (m_{1} - m_{2})^{2} ] [ s - (m_{1} + m_{2})^{2} ] \qquad G_{23}(u) = G_{14}(u) = \frac{1}{4} [ u - (m_{1} - m_{2})^{2} ] [ u - (m_{1} + m_{2})^{2} ].$$

The rest of the 2-Gram invariants become much simpler:

$$G_{13}(t) = \frac{1}{4} t (t - 4m_{1}^{2} ) \qquad G_{24}(t) = \frac{1}{4} t (t - 4m_{2}^{2} ).$$

The 3-Gram invariant also acquires a simpler form:

$$G_{123}(s, t, u) = G_{124}(s, t, u) = G_{134}(s, t, u) = G_{234}(s, t, u) = \frac{1}{4} t [ su - (m_{1} - m_{2})^{2} (m_{1} + m_{2})^{2}].$$

Center-of-Momentum Frame

Just like before, in the center-of-momentum frame you can write

$$p_{1} = (E_{1}, \mathbf{p}_{1}) \qquad p_{2} = (E_{2}, -\mathbf{p}_{1}) \qquad p_{3} = (E_{3}, \mathbf{p}_{3}) \qquad p_{4} = (E_{4}, -\mathbf{p}_{3}).$$

In this frame you can find expressions for many kinematic quantities in terms of the 2-Mandelstam invariants and the masses.

Energy and Momentum

Due to the elasticity constraints, you now have

$$\Lambda_{12}(s) = \Lambda_{34}(s) \qquad \Lambda_{IJ}(s) \equiv [ s - (m_{I} - m_{J})^{2} ] [ s - (m_{I} + m_{J})^{2} ].$$

It follows that the magnitude of $\mathbf{p}_{1}$ and $\mathbf{p}_{3}$ are equal:

$$\vert \mathbf{p}_{1} \vert = \vert \mathbf{p}_{3} \vert = \frac{\sqrt{\Lambda_{12}(s)}}{2 \sqrt{s}}.$$

For the energy of each quanta you now find

$$E_{3} = E_{1} = \frac{s + (m_{1} - m_{2})(m_{1} + m_{2})}{2 \sqrt{s}} \qquad E_{4} = E_{2} = \frac{s - (m_{1} - m_{2})(m_{1} + m_{2})}{2 \sqrt{s}}.$$

That is, one of the consequences of the elasticity constraints is a body-wise conservation of energy.

Scattering Angle

You now have

$$\mathbf{p}_{1} \cdot \mathbf{p}_{3} = \frac{(m_{1} - m_{2})^{2}(m_{1} + m_{2})^{2} - s (u - t)}{4s},$$

and

$$\Lambda_{12}(s) = (m_{1} - m_{2})^{2}(m_{1} + m_{2})^{2} - s (u + t).$$

Using these two results leads to

$$z_{s} \equiv \cos{(\theta_{s})} \equiv \frac{\mathbf{p}_{1} \cdot \mathbf{p}_{3}}{\vert \mathbf{p}_{1} \vert \vert \mathbf{p}_{3} \vert} = \frac{(m_{1} - m_{2})^{2}(m_{1} + m_{2})^{2} - s (u - t)}{(m_{1} - m_{2})^{2}(m_{1} + m_{2})^{2} - s (u + t)}.$$

Speed and Rapidity

Due to the elasticity constraints, for the speed of each quanta you now have

$$\vert \mathbf{v}_{3} \vert = \vert \mathbf{v}_{1} \vert = \frac{\sqrt{\Lambda_{12}(s)}}{s + (m_{1} - m_{2})(m_{1} + m_{2})} \qquad \vert \mathbf{v}_{4} \vert = \vert \mathbf{v}_{2} \vert = \frac{\sqrt{\Lambda_{12}(s)}}{s - (m_{1} - m_{2})(m_{1} + m_{2})},$$

and for the rapidity of each quanta

$$\varphi_{3} = \varphi_{1} = \frac{1}{2} \log{\left[ \frac{(m_{1} - m_{2})(m_{1} + m_{2}) + s + \sqrt{\Lambda_{12}(s)}}{(m_{1} - m_{2})(m_{1} + m_{2}) + s - \sqrt{\Lambda_{12}(s)}} \right]} \qquad \varphi_{4} = \varphi_{2} = \frac{1}{2} \log{\left[ \frac{(m_{1} - m_{2})(m_{1} + m_{2}) - s - \sqrt{\Lambda_{12}(s)}}{(m_{1} - m_{2})(m_{1} + m_{2}) - s + \sqrt{\Lambda_{12}(s)}} \right]}.$$

Another consequence of the elasticity constraints is a body-wise conservation of rapidity (and speed).

Physical Scattering Region

Because of the elasticity constraints, there is only one threshold and one pseudothreshold along the $s$-channel. Requiring $\vert \mathbf{p}_{1} \vert \geq 0$ leads to

$$s \geq (m_{1} + m_{2})^{2}.$$

That is, $s$ must be at or above threshold. This condition also guarantees that all individual energies are positive (or zero). Note that $s$ being at or above threshold means that $G_{12} \geq 0$.

Similarly, requiring $\vert \mathbf{p}_{1} - \mathbf{p}_{3} \vert^{2} \geq 0$ leads to

$$t \leq 0,$$

and requiring $ \vert \mathbf{p}_{1} + \mathbf{p}_{3} \vert^{2} \geq 0$ leads to

$$s u \leq (m_{1} - m_{2})^{2} (m_{1} + m_{2})^{2}.$$

Note that these two inequalities are equivalent to $G_{123} \geq 0$. You can check that these two inequalities are equivalent to the condition $-1 \leq z_{s} \leq 1$. As you will see below, these inequalities are saturated at special kinematic configurations.

Forward Scattering

Forward scattering corresponds to setting $z_{s} = 1$. Using

$$u_{f} = 2m_{1}^{2} + 2m_{2}^{2} - s - t_{f},$$

and setting $z_{s} = 1$ leads to

$$t_{f} = 0.$$

This leads to a linear relation between $u_{f}$ and $s$:

$$u_{f} = 2m_{1}^{2} + 2m_{2}^{2} - s.$$

Note that when $t = t_{f}$ you saturate the condition $t \leq 0$.

Backward Scattering

Backward scattering corresponds to setting $z_{s} = -1$. Using

$$u_{b} = 2m_{1}^{2} + 2m_{2}^{2} - s - t_{b},$$

and setting $z_{s} = -1$ leads to

$$t_{b} = - \frac{\Lambda_{12}(s)}{s} = \frac{[s - (m_{1} - m_{2})^{2}] [(m_{1} + m_{2})^{2} - s]}{s}.$$

In contrast to forward scattering, you now have a rectihyperbolic relation (i.e. a hyperbola with perpendicular asymptotes) between $u_{b}$ and $s$:

$$u_{b} = \frac{(m_{1} - m_{2})^{2} (m_{1} + m_{2})^{2}}{s}.$$

Note that when $u = u_{b}$ you saturate the condition $s u \leq (m_{1} - m_{2})^{2} (m_{1} + m_{2})^{2}$.

Orthogonal Scattering

Orthogonal scattering corresponds to setting $z_{s} = 0$. Using

$$u_{o} = 2m_{1}^{2} + 2m_{2}^{2} - s - t_{o},$$

and setting $z_{s} = 0$ leads to

$$t_{o} = -\frac{\Lambda_{12}(s)}{2s}.$$

It follows that

$$u_{o} = \frac{2(m_{1} - m_{2})^{2} (m_{1} + m_{2})^{2} - \Lambda_{12}(s)}{2s}.$$

Semiforward Approximation

Like in any four-point process, there are three main kinematic variables: $s$, $t$, and $u$. Of course, due to the conservation constraint, only two of these are independent. However, it pays to keep $u$ around sometimes. The quantity $s$ describes the center-of-momentum energy. The quantity $t$ describes the amount of momentum that is exchanged between the two bodies. Other kinematic variables include the two masses. With all of these dimensionful quantities, you can construct dimensionless ones by taking ratios. For example,

$$R_{1} \equiv \frac{t}{s} \qquad R_{2} \equiv \frac{s}{m_{1} m_{2}} \qquad R_{3} \equiv \frac{m_{1}}{m_{2}}.$$

The semiforward approximation is defined as the regime where $R_{1} \rightarrow 0$ with $R_{2}$ and $R_{3}$ kept fixed. This is the regime where $t$ is much smaller than $s$ or the square of the masses, and the speed and rapidity of the external quanta is kept fixed. A corollary of the semiforward approximation is that the ratio $u / s$ is also kept fixed.

Cross Processes

If you cross the incoming $B$ with the outgoing $A$, you find the inelastic process

$$A(p_{1}) + \bar{A}(\bar{p}_{2}) \longrightarrow \bar{B}(\bar{p}_{3}) + B(p_{4}).$$

This is the $t$-channel. Similarly, if you cross the incoming $B$ with the outgoing $B$, you find the elastic process

$$A(p_{1}) + \bar{B}(\bar{p}_{2}) \longrightarrow A(p_{3}) + \bar{B}(\bar{p}_{4}).$$

This is the $u$-channel.