- #1
ercagpince
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Hi , I got stuck on a point of griffiths' scattering problem on "the introduction to elementary particles"
Consider the case of elastic scattering , A+B-->A+B , in the lab frame (B initially at rest) assuming the target is so heavy (mbc2 >> Ea) that its recoil is negligible . Use (6.34) to determine the differential scattering cross section .
Equation (6.34) :
\frac{d\sigma}{d\Omega}=M^{2}\frac{\frac{\hbar}{2}S}{4\sqrt{\left(p_{1}\bullet p_{2}\right)^{2}-\left(m_{1}m_{2}c^{2}\right)^{2}}\left[\left(\frac{cd^{3}p_{3}}{\left(2\Pi^{3}\right)2E_{3}}\right)\left(\frac{cd^{3}p_{4}}{\left(2\Pi^{3}\right)2E_{4}}[/tex]\right)\bullet\bullet\bullet\left(\frac{cd^{3}p_{n}}{\left(2\Pi^{3}\right)2E_{n}}[/tex]\right)\right]\times\left(2\Pi^{4}\right)\delta^{4}\left(P_{1}+P_{2}-P_{3}-P_{4}\bullet\bullet\bullet-P_{n}\right)
\frac{d\sigma}{d\Omega}=M^{2}\frac{\frac{\hbar}{2}S}{16\left(2\Pi\right)^{2}\left|P_{1}\right|m_{2}c}\frac{\rho*d\rho}{\left(\rho^{2}+P_{1}^{2}-2\rho\left|P_{1}\right|cos\theta\right)^{1/2}}\delta\left(\frac{E_{1}+E_{2}}{c}-\rho-\left|P_{4}\right|\right)
I got this formula so far , however , I cannot cancel out the delta function . It seems that it is impossible for me to separate rho and p1 as independent variables when p4 is involved in the delta function .
Homework Statement
Consider the case of elastic scattering , A+B-->A+B , in the lab frame (B initially at rest) assuming the target is so heavy (mbc2 >> Ea) that its recoil is negligible . Use (6.34) to determine the differential scattering cross section .
Homework Equations
Equation (6.34) :
\frac{d\sigma}{d\Omega}=M^{2}\frac{\frac{\hbar}{2}S}{4\sqrt{\left(p_{1}\bullet p_{2}\right)^{2}-\left(m_{1}m_{2}c^{2}\right)^{2}}\left[\left(\frac{cd^{3}p_{3}}{\left(2\Pi^{3}\right)2E_{3}}\right)\left(\frac{cd^{3}p_{4}}{\left(2\Pi^{3}\right)2E_{4}}[/tex]\right)\bullet\bullet\bullet\left(\frac{cd^{3}p_{n}}{\left(2\Pi^{3}\right)2E_{n}}[/tex]\right)\right]\times\left(2\Pi^{4}\right)\delta^{4}\left(P_{1}+P_{2}-P_{3}-P_{4}\bullet\bullet\bullet-P_{n}\right)
The Attempt at a Solution
\frac{d\sigma}{d\Omega}=M^{2}\frac{\frac{\hbar}{2}S}{16\left(2\Pi\right)^{2}\left|P_{1}\right|m_{2}c}\frac{\rho*d\rho}{\left(\rho^{2}+P_{1}^{2}-2\rho\left|P_{1}\right|cos\theta\right)^{1/2}}\delta\left(\frac{E_{1}+E_{2}}{c}-\rho-\left|P_{4}\right|\right)
I got this formula so far , however , I cannot cancel out the delta function . It seems that it is impossible for me to separate rho and p1 as independent variables when p4 is involved in the delta function .
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