The intermediate vector boson Z0, which has a rest mass energy of 91.187 GeV, is produced in collisions of positrons and electrons e+ + e− → Z0. How much energy must the positrons and electrons in symmetric colliding beams have to produce the Z0?

Question

The intermediate vector boson [latex]Z_{0}[/latex], which has a rest mass energy of 91.187 GeV, is produced in collisions of positrons and electrons

[latex]e^{+} + e^{−} → Z_{0}[/latex].

How much energy must the positrons and electrons in symmetric colliding beams have to produce the [latex]Z_{0}[/latex]?

Accepted Answer

According to Eq. (13.14), the energy and momentum of the [latex] Z_0 [/latex] satisfy the equation

[latex]E^{2} = p^{2}c^{2} + m^{2}c^{4}.[/latex] (13.14)

[latex]E^{2}_{Z} − p^{2}_{Z}c^{2} = M_{Z}^{2}c^{4}. [/latex] (13.21)

Denoting the energy and momentum of the positron by [latex]E_{p}[/latex] and [latex]p_{p}[/latex] and the energy and momentum of the electron by [latex]E_{e}[/latex] and [latex]p_{e}[/latex], the requirements that the energy and momentum be conserved in the collision are

[latex]E_{p} + E_{e} = E_{Z}[/latex]

[latex]p_{p} + p_{e} = p_{Z}.[/latex]

Substituting these conservation laws into Eq. (13.21) gives

[latex](E_{p} + E_{e})^{2} − (p_{p} + p_{e})^{2}c^{2} = M_{Z}^{2}c^{4}. [/latex] (13.22)

In a colliding beam experiment involving particles of equal mass, the center of mass of two colliding particles is stationary, and the momenta of the particles is equal and opposite

[latex]p_{p} = −p_{e}.[/latex]

The energies of the two particles is equal

[latex]E_{p} = c \sqrt{p_{p}^{2} + m_{p}^{2}c^{2} }= E_{e}.[/latex]

Eq. (13.22) can thus be written simply

[latex]2E_{p} = M_{Z}c^{2},[/latex]

or

[latex]E_{p}=M_{Z}c^{2}/2.[/latex]

The kinetic energy of the positron is equal to the energy [latex]E_{p}[/latex] minus its rest energy [latex]m_{p}c^{2} = 0.511[/latex] MeV. So, the kinetic energy of the positrons – and the electrons – is

[latex]KE = M_{Z}c^{2}/2 − m_{p}c^{2} =[/latex]45.08 GeV

Question 13.3: The intermediate vector boson Z0, which has a rest mass ener...

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