/sci/ - Science & Math

>>11300734
I'm trying to find the non-relativistic limit of the differential cross section in electron-muon scattering. I've done the calculations with the traces etc and i've got for the amplitude:

[eqn] \overline{\left| \mathcal{M} \right|} = \frac{e^4}{4q^4}L^{(e)\mu\nu} L^{(\mu)} _{\mu\nu}[/eqn]

where the L's are the electron and muon tensors,

[eqn]L^{(e)\mu\nu} = \frac{1}{m_e^2} \left( p_A ^{\mu}p_C^{\nu} + p_A^{\nu}p_C^{\mu} - p_a \cdot p_C + m_e^2 g^{\mu\nu} \right)[/eqn]

and respectively for the muon (A and B are the ingoing electron and muon and C and D are the outgoing ones) and [math]q:= (p_D-p_B)[/math]. My question is how i'm i suppose to compute the non-relativistic limit where [math] p\rightarrow 0 \Rightarrow E \approx m[/math]. If i try to express the four-momentums and the Mandelstam variables like this i get [math]q^4 = (p_D-p_B)^4 \approx (m_{\mu} - m_{\mu} )^4 = 0[/math] in the denominator. Any help appreciated.

>>9398325
yup, OP here. thanks anons, never thought this thread would be so interesting

>>9318203
>what is space-time?
A quintuple [math] \left(M,\mathcal{O},\mathcal{A}, \nabla ,<\cdot , \cdot >\right) [/math] where M is a 4-dimensional smooth manifold, equipped with a topology [math]\mathcal{O}[/math] , a smooth atlas [math]\mathcal{A}[/math], a covariant derivative [math]\nabla[/math] and an inner product [math]<\cdot , \cdot >[/math].

>>9286781
[math] \int _{\partial \Omega} \omega = \int _{\Omega} d\omega [/math]

>>9260339
Well to be strict, the official definition of divergence is: [math]\vec{\nabla} \cdot \vec{F} := lim _{\Omega \rightarrow P} \frac{1}{V(\Omega)} \iint _{\partial \Omega} \vec{F} \cdot \hat{n} dS [/math] where [math]\Omega[/math] is a region in [math]\mathbb{R} ^3[/math] with volume [math]V(\Omega)[/math]. What this says is that when the region (and so the volume) is compressed to a point [math]P[/math] the (allow me to say) "mean value" of the vector field [math]\vec{F}[/math] over the boundary of [math]\Omega[/math] converges to a certain number. That number is what we call the divergence of the field [math]\vec{F}[/math] at the point [math]P[/math].

What's the most surprising math result?