snorer.Mandelstam¶

`class` snorer.Mandelstam(T2,m1,m2,psi)¶

Superclass: snorer.Kinematics

This class constructs the associated Mandelstam variables $s$, $t$ and $u$ associated with the scattering process depicted in Fig. 1 in 2-2 elastic scattering.

Parameters:

T2 : array_like
    Kinetic energy $T_2$ received by the particle 2, MeV

m1 : array_like
    Mass of particle 1 (incident) $m_1$, MeV

m2 : array_like
    Mass of particle 2 (target) $m_2$, MeV

psi : array_like
    Lab frame scattering angle $\psi$, rad

Attributes:

s : scalar/ndarray
    The $s$-channel in this scattering process, MeV²

t : scalar/ndarray
    The $t$-channel in this scattering process, MeV²

u : scalar/ndarray
    The $u$-channel in this scattering process, MeV²

T1 : scalar/ndarray
    The required kinetic energy $T_1$ of particle 1, MeV

dT1 : scalar/ndarray
    The Jacobian $dT_1/dT_2$, dimensionless

x : scalar/ndarray
    $x:=\cos\psi \in [1,-1]$

sanity : bool/ndarray
    Are the parameters physically plausible? True for plausible and False for physically impossible.

dLips : scalar/ndarray
    Value for differential Lorentz invariant phase space

Notes

Given $T_1$ is obtained from its superclass snorer.Kinematics, we can evaluate all Mandelstam variables easily. Thus,

\[ \begin{align*} s &= (p_1+p_2)^2 = m_1^2+m_2^2 + 2 E_1 m_1 \\ &= m_1^2+m_2^2 + 2(T_1+m_1)m_2 \end{align*} \]

for $s$-channel, and

\[ \begin{align*} t &= (p_2^\prime - p_2)^2 = 2m_2^2 - 2E_2 E_2^\prime \\ &= 2m_2^2 - 2(T_2+m_2)m_2. \end{align*} \]

For $u$-channel, we use the identity $$ s+t+u = \sum_i m_i^2 = 2(m_1^2+m_2^2) $$ where $i$ indicates all particle masses before and after the reaction.

References¶

M. Peskin and D. Schroeder, An Introduction To Quantum Field Theory, Westview (1995)

snorer.Mandelstam¶

class snorer.Mandelstam(T2,m1,m2,psi)¶

References¶

`class` snorer.Mandelstam(T2,m1,m2,psi)¶