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I'm working on a QCDSR paper which calculates mass of $B$ meson in the present of external magnetic field. the author works with Schwinger proper-time representation of Feynman propagator in momentum space. which is: $$S(k)=i\frac{\not{k}+m}{k^2-m^2}+(\frac{ceB}{m^2})[-(k_{\parallel}.\gamma_{\parallel}+m)\gamma^!\gamma^2\frac{m^2}{(k^2-m^2)^2}]+(\frac{ceB}{m^2})^2[-2ik^2_{\perp}(-\not{k}+m)\frac{m^4}{(k^2-m^2)^4}+2ik_{\perp}.\gamma_{\perp}\frac{m^4}{(k^2-m^2)^3}].$$

I want to recalculate of this paper, but in coordinate space. I know the Schwinger proper-time representation of Feynman diagram is:

$$DF(x_1-x_2)=-i\frac{1}{(2 \pi)^4}\int^{\infty}_0 ds s^2 e^{-is(m^2-i\epsilon)}e^{-isX^2/4}K_1(im\sqrt{s}).$$

I can't find any sources for Fourier transformation on Feynman propagator with power two or more of Denominator.

Can someone guide me on how to solve the equation?

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  • $\begingroup$ Have you tried calculating the Fourier transform yourself? (It looks hard. Why do you want to work in coordinate space?) $\endgroup$
    – Ghoster
    Commented Sep 28, 2023 at 17:51
  • $\begingroup$ @Ghoster yes I've tried . but in calculation I countered $\frac{\pi^2}{2}\int_{0}^{\infty}d\alpha\alpha^{-1}e^{-\frac{x^2}{4\alpha}+\alpha m^2}$. which I can't solve. and for your question about why I want to work in coordinate space, I can tell you, working on baryon in the momentum space with QCDSR is a disaster. so I want to change the space to coordinate space due to working with baryon in the future. $\endgroup$
    – hossein mohseni
    Commented Sep 29, 2023 at 8:31

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