From Feynman diagrams to potential

[Thor Shen]

I met a problem when I read the textbook "Relativistic Quantum Mechanics" by J.D.Bjorken. He said we can get the potential
V(r_1,r_2)=\frac{f^{2}}{\mu^{2}}(1-P_{ex})(\tau_1\cdot\tau_2)(\sigma_1\cdot\nabla_1)(\sigma_2\cdot\nabla_1)\frac{e^{-\mu|r_1-r_{2}|}}{|r_1-r_{2}|}
(10.51)
from the amplitude
S_{fi}=\frac{(-ig_0)^2M^2}{(2\pi)^2\sqrt{E_1E_2E&#039;_1E&#039;_2}}(2\pi)^4\delta^4(p_1+p_2-p&#039;_1-p&#039;_2){[\chi^{+}_1\bar{u}(p&#039;_1)i\gamma^5\tau u(p_1)\chi_1]\frac{i}{(p&#039;_1-p_1)^2-\mu^2}\cdot[\chi^{+}_2\bar{u}(p&#039;_2)i\gamma^5\tau u(p_2)\chi_2]<br /> -[\chi^{+}_2\bar{u}(p&#039;_2)i\gamma^5\tau u(p_1)\chi_1]\frac{i}{(p&#039;_2-p_1)^2-\mu^2}\cdot[\chi^{+}_1\bar{u}(p&#039;_1)i\gamma^5\tau u(p_2)\chi_2]}
(10.45)

I can get the formula 10.50
\bar{u}(p&#039;_1,s_1)\gamma^5 u(p_1,s_1)=u^{+}(s&#039;_1)\frac{\sigma\cdot(p_1-p&#039;_1)}{2M}u(s_1)
but I can't get the 10.51, please give me an idea or suggestion, or any information, thank you!

 

[andrien]

The basic idea is to take the non relativistic limit of the matrix element M_fi,the Fourier transform of the corresponding matrix element will give you the potential.you also have to write the four component spinor u in terms of two component spinor like ( x₁ (σ₁.p/2m)x₁)

 

[Thor Shen]

The basic idea is to take the non relativistic limit of the matrix element M_fi,the Fourier transform of the corresponding matrix element will give you the potential.you also have to write the four component spinor u in terms of two component spinors like ( x₁ (σ₁.p/2m)x₁)

Thank you for your advice. Actually, I know the exponential term (Yukawa potential \frac{e^{-\mu r}}{r}) come from the Fourier transform of the propagator \frac{1}{(p&#039;_1-p_1)^2-\mu^2}. But I can understand how to get the term (\sigma_1\cdot\nabla_1)(\sigma_2\cdot\nabla_1). I know there are two methods to transform the relativistic potential to non-relativistic one.
One is Foldy-Wouthuysen transform which need the complete Hamilton in Chapter 4, the other is 'big-small' components nonrelativistic limit in Chapter 1. I try to use the latter one to simplify the 10.51 and get the matrix element M_fi
[\chi^+_1\bar{u}(p&#039;_1)i\gamma^5\tau u_1(p_1)\chi_1]\frac{i}{(p&#039;_1-p_1)^2-\mu^2}\cdot[\chi^+_2\bar{u}(p&#039;_2)i\gamma^5\tau u_1(p_2)\chi_2]
=(\tau_1\cdot\tau_2)\chi^+_1 \frac{\sigma_1\cdot(p_1-p&#039;_1)}{2m}\chi_1<br /> \chi^+_2 \frac{\sigma_2\cdot(p_2-p&#039;_2)}{2m}\chi_2\frac{-1}{(p&#039;_1-p_1)^2-\mu^2}
Before Fourier transform, how to deal with the \chi

 

[andrien]

Yes,this is what I have said you to do.Those \chi are not problem because they are spinors like (1 0) or (0 1),so you can forget them for the moment.you have to Fourier transform
(\tau_1\cdot\tau_2) \frac{\sigma_1\cdot(p_1-p&#039;_1)}{2m} \frac{\sigma_2\cdot(p_2-p&#039;_2)}{2m}\frac{-1}{(p&#039;_1-p_1)^2-\mu^2}

 

[Thor Shen]

Yes,this is what I have said you to do.Those \chi are not problem because they are spinors like (1 0) or (0 1),so you can forget them for the moment.you have to Fourier transform
(\tau_1\cdot\tau_2) \frac{\sigma_1\cdot(p_1-p&#039;_1)}{2m} \frac{\sigma_2\cdot(p_2-p&#039;_2)}{2m}\frac{-1}{(p&#039;_1-p_1)^2-\mu^2}

if we define the \vec{p}&#039;_1-\vec{p}_1=\vec{p}and \vec{r}_1-\vec{r}_2=\vec{r}
then
\int^{\infty}_{-\infty}d^3p\frac{1}{\vec{p}^2+\mu^2}e^{-i\vec{p}\cdot\vec{r}}
=\int^{2\pi}_{0}d\phi\int^{1}_{-1}dcos\theta\int^{\infty}_{0}dp\frac{p^2}{p^2+\mu^2}e^{-ipcos\vartheta r}
=4\pi^2\frac{e^{-\mu r}}{r}
so,
\sigma_1\cdot(p_1-p&#039;_1)and\sigma_2\cdot(p_2-p&#039;_2) don't involved in integration, instead of being the operator p=-i\hbar \nabla. Although, the p_2-p&#039;_2 is replaced by p_1-p&#039;_1 for the delta funtion. But the formula of the integrate should be
(\tau_1\cdot\tau_2) \sigma_1\cdot(\nabla_1-\nabla&#039;_1) \sigma_2\cdot(\nabla_1-\nabla&#039;_1) \frac{e^{-\mu|r_1-r_2|}}{|r_1-r_2|}

 

[andrien]

Hey,sorry for being late but the point is that you have to take the conservation principle also into account from p₁+p₂=p₁'+p₂' which gives p₁-p₁'=p₂'-p₂=q say,when you put it into your expression it only depends on q,now you have to just take the Fourier transform with respect to q and the answer falls in it place.