from sympy import *I compute with Sympy how to transform a 4-vector from a fixed-target frame (aka laboratory frame) to the center-of-mass frame, where the colliding particles have equal but opposite momenta. It is fun to do this with Sympy, because it can compute the Taylor series expansion for the common case where the momenta are large compared to the particle masses.
e1, p1, e2, p2, m = symbols("E_1 p_1 E_2 p_2 m",real=True, positive=True)We work in 1-dimensional coordinates with \(p = p_z\). We generally use \(m = m_p = m_n\), in other words we neglect the mass difference between protons and neutrons. We consider a proton projectile and a nuclear target with \(A\) nucleons.
We first compute the cms energy \(\sqrt{s_{nn}}\) in the nucleon-nucleon system from the fixed target system.
In this system, we have these 4-vectors: * projectile \((E, p)\) * target \((m, 0)\)
s2 = (e1 + m)**2 - (p1 + 0)**2; s2\(\displaystyle - p_{1}^{2} + \left(E_{1} + m\right)^{2}\)
s2 = s2.subs(e1, sqrt(p1 ** 2 + m ** 2)); s2\(\displaystyle - p_{1}^{2} + \left(m + \sqrt{m^{2} + p_{1}^{2}}\right)^{2}\)
s2.series(m, 0, 2)\(\displaystyle 2 m p_{1} + O\left(m^{2}\right)\)
s = sqrt((e1 + m)**2 - (p1 + 0)**2); s\(\displaystyle \sqrt{- p_{1}^{2} + \left(E_{1} + m\right)^{2}}\)
s = s.subs(e1, sqrt(p1 ** 2 + m ** 2)); s\(\displaystyle \sqrt{- p_{1}^{2} + \left(m + \sqrt{m^{2} + p_{1}^{2}}\right)^{2}}\)
s.series(m, 0, 2)\(\displaystyle \sqrt{2} \sqrt{m} \sqrt{p_{1}} + \frac{\sqrt{2} m^{\frac{3}{2}}}{2 \sqrt{p_{1}}} + O\left(m^{2}\right)\)
def sqrt_snn(energy):
mass = (938.27 + 939.57) * 0.5e-9 # nucleon mass in PeV
return sqrt(2 * mass * energy) * 1e3 # sqrt(sNN) in TeV0.5 * (sqrt_snn(320000+90000) - sqrt_snn(320000-90000))\(\displaystyle 110.127117815583\)
def beam_energy(sqrt_snn):
mass = (938.27 + 939.57) * 0.5e-3 # nucleon mass in GeV
return (sqrt_snn * 1e3) ** 2 / (2 * mass) / 1e6 # beam in PeVbeam_energy(14)104.37523963702976