- #1
Jan Eysermans
- 17
- 0
Hi all,
I have a question about simulating (Monte Carlo) proton-proton collisions resulting in, for example, a Z boson. Assume two quarks (quark and antiquark) from each proton collide head-on along the z-axis. The quark momenta are distributed according to the Parton Density Functions.
If proton 1 has momentum [itex]\vec{p}_1 = p \vec{e}_z[/itex] and proton 2 [itex]\vec{p}_2 = -p \vec{e}_z[/itex], the momenta of the quarks inside the protons can be written as [itex]\vec{p}_1 = x_1 p \vec{e}_z[/itex] resp. [itex]\vec{p}_1 =- x_2 p \vec{e}_z[/itex]. Here, [itex]x_1[/itex] and [itex]x_2[/itex] represents the momentum fraction according to the Parton Density Functions. Both values must be generated in the simulation, but the kinematics of the collision set constraints on both values. Indeed, enough energy must be available to create a Z boson (at rest).
In essence, we have a head-on quark-quark interaction. The four-vectors in the lab-frame of the quarks can be written as:
[itex]p_1 = (E_1,0,0,x_1 p) \approx (x_1 p,0,0,x_1 p) [/itex]
[itex]p_2 = (E_2,0,0,-x_2 p) \approx (x_2 p,0,0,-x_2 p) [/itex],
where I have neglected the quark mass. The four-vector of the Z boson can be written as:
[itex]p_Z = (E_Z,0,0,p_z) \approx (\sqrt{p_z^2 + m_Z^2c^2}/c,0,0,p_z) [/itex]
Conservation of momentum yields the following equations:
[itex] (x_1+x_2)p = \sqrt{p_z^2 + m_Z^2c^2} [/itex] and [itex] (x_1-x_2)p = p_z [/itex]
The minimum requirement to create at least a Z boson, is when the boson is at rest. This gives the following constraints on [itex]x_1[/itex] and [itex]x_2[/itex]:
[itex] x_1+x_2 \geq \frac{m_Zc}{p} [/itex]
However, I always thought that the center-of-mass energy [itex] \sqrt{s} [/itex] is the total energy available to create new particles. We can write then: [itex] \sqrt{s} > m_Zc^2 [/itex]. The center-of-mass energy in this system can be calculated as:
[itex]
s = (p_1c+p_2c)^2 = ((x_1+x_2)pc, 0,0,(x_1-x_2)p)^2c^2 = (x_1+x_2)p^2c^2 - (x_1-x_2)^2p^2c^2 = 4x_1 x_2 p^2c^2
[/itex]
The condition [itex] \sqrt{s} > m_Zc^2 [/itex] becomes: [itex] \sqrt{x_1 x_2} > \frac{m_Zc}{2p} [/itex].
Which condition or reasoning is the correct one?
Thanks!
Jan
I have a question about simulating (Monte Carlo) proton-proton collisions resulting in, for example, a Z boson. Assume two quarks (quark and antiquark) from each proton collide head-on along the z-axis. The quark momenta are distributed according to the Parton Density Functions.
If proton 1 has momentum [itex]\vec{p}_1 = p \vec{e}_z[/itex] and proton 2 [itex]\vec{p}_2 = -p \vec{e}_z[/itex], the momenta of the quarks inside the protons can be written as [itex]\vec{p}_1 = x_1 p \vec{e}_z[/itex] resp. [itex]\vec{p}_1 =- x_2 p \vec{e}_z[/itex]. Here, [itex]x_1[/itex] and [itex]x_2[/itex] represents the momentum fraction according to the Parton Density Functions. Both values must be generated in the simulation, but the kinematics of the collision set constraints on both values. Indeed, enough energy must be available to create a Z boson (at rest).
In essence, we have a head-on quark-quark interaction. The four-vectors in the lab-frame of the quarks can be written as:
[itex]p_1 = (E_1,0,0,x_1 p) \approx (x_1 p,0,0,x_1 p) [/itex]
[itex]p_2 = (E_2,0,0,-x_2 p) \approx (x_2 p,0,0,-x_2 p) [/itex],
where I have neglected the quark mass. The four-vector of the Z boson can be written as:
[itex]p_Z = (E_Z,0,0,p_z) \approx (\sqrt{p_z^2 + m_Z^2c^2}/c,0,0,p_z) [/itex]
Conservation of momentum yields the following equations:
[itex] (x_1+x_2)p = \sqrt{p_z^2 + m_Z^2c^2} [/itex] and [itex] (x_1-x_2)p = p_z [/itex]
The minimum requirement to create at least a Z boson, is when the boson is at rest. This gives the following constraints on [itex]x_1[/itex] and [itex]x_2[/itex]:
[itex] x_1+x_2 \geq \frac{m_Zc}{p} [/itex]
However, I always thought that the center-of-mass energy [itex] \sqrt{s} [/itex] is the total energy available to create new particles. We can write then: [itex] \sqrt{s} > m_Zc^2 [/itex]. The center-of-mass energy in this system can be calculated as:
[itex]
s = (p_1c+p_2c)^2 = ((x_1+x_2)pc, 0,0,(x_1-x_2)p)^2c^2 = (x_1+x_2)p^2c^2 - (x_1-x_2)^2p^2c^2 = 4x_1 x_2 p^2c^2
[/itex]
The condition [itex] \sqrt{s} > m_Zc^2 [/itex] becomes: [itex] \sqrt{x_1 x_2} > \frac{m_Zc}{2p} [/itex].
Which condition or reasoning is the correct one?
Thanks!
Jan