- #1
alejandrito29
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I need integrate "over heavy momenta" the following equation
\sigma v>=
[tex]<\sigma v>= \frac{1}{n^{(0)}_1n^{(0}}_2 \int \frac{d^3p_1}{(2Pi)^3 2 E_1} \int \frac{d^3p_2}{(2Pi)^3 2 E_2} \int \frac{d^3p_3 }{(2Pi)^3 2 E_3} \int \frac{d^3p_4}{(2Pi)^3 2 E_4} e^{-(E_1+E_2)/T} \\
\cdot (2 Pi)^4 \delta^3 (p_1+p_2-p_3-p_4) \delta (E_1+E_2-E_3-E_4) M^2[/tex]
where [tex]n+ \nu \rightarrow p+ e^-[/tex]
I need to find that_
[tex]n_{\nu}^{(0)}< \sigma v>= \frac{Pi}{4m^2}\int \frac{d^3p_\nu}{(2Pi)^3 2 p_\nu} e^{-p_\nu/T}\int \frac{d^3p_e}{(2Pi)^3 2 p_e} \delta (Q-p_\nu-p_e) M^2 [/tex]
where [tex]Q=m_n-m_p[/tex]
I tried very ways, for example using [tex]\delta^3 (p_1+p_2+p_3+p_4)= \delta(p_1)\delta(p_2)\delta(p_3)\delta(p_4) [/tex] but i have very problems with the integration and with the propierties of Dirac Delta...
Help please
\sigma v>=
[tex]<\sigma v>= \frac{1}{n^{(0)}_1n^{(0}}_2 \int \frac{d^3p_1}{(2Pi)^3 2 E_1} \int \frac{d^3p_2}{(2Pi)^3 2 E_2} \int \frac{d^3p_3 }{(2Pi)^3 2 E_3} \int \frac{d^3p_4}{(2Pi)^3 2 E_4} e^{-(E_1+E_2)/T} \\
\cdot (2 Pi)^4 \delta^3 (p_1+p_2-p_3-p_4) \delta (E_1+E_2-E_3-E_4) M^2[/tex]
where [tex]n+ \nu \rightarrow p+ e^-[/tex]
I need to find that_
[tex]n_{\nu}^{(0)}< \sigma v>= \frac{Pi}{4m^2}\int \frac{d^3p_\nu}{(2Pi)^3 2 p_\nu} e^{-p_\nu/T}\int \frac{d^3p_e}{(2Pi)^3 2 p_e} \delta (Q-p_\nu-p_e) M^2 [/tex]
where [tex]Q=m_n-m_p[/tex]
I tried very ways, for example using [tex]\delta^3 (p_1+p_2+p_3+p_4)= \delta(p_1)\delta(p_2)\delta(p_3)\delta(p_4) [/tex] but i have very problems with the integration and with the propierties of Dirac Delta...
Help please
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