- #1

Francisco Alegria

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- TL;DR Summary
- Computation of a quadruple integral that comes up when computing the fourth order contribuition to the Lamb Shift in energy of the electron orbiltals - Self energy part

The analytical computation of the shift in energy level of electrons in atoms due to quantum electrodynamics is carried out using perturbation theory. In particular, the fourth-order contribution is given in five different terms. One of them, usually called "Electron Self Energy", leads to seven different quadruple integrals. I do not know how to compute any of them on my own.

I ask anyone for some assistance in computing one of the easiest ones (with what I learn from you, I hope to be able to do the other ones).

Here it is:

$$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{w(w-1)(1-u^2)v^2}{v(1-u)z+(1-w)u}dudzdvdw$$

The result reported in M. F. Soto, "Calculation of the Slope at q^2=0 of the Dirac Form Factor for the Electron Vertex in Fourth Order", Physical Review A, vol. 2, no. 3, pp. 734-758, September 1970, eq. (A7) and (A8) is ##\pi^2/120-5/32##.

Which integral should I do first?

I ask anyone for some assistance in computing one of the easiest ones (with what I learn from you, I hope to be able to do the other ones).

Here it is:

$$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{w(w-1)(1-u^2)v^2}{v(1-u)z+(1-w)u}dudzdvdw$$

The result reported in M. F. Soto, "Calculation of the Slope at q^2=0 of the Dirac Form Factor for the Electron Vertex in Fourth Order", Physical Review A, vol. 2, no. 3, pp. 734-758, September 1970, eq. (A7) and (A8) is ##\pi^2/120-5/32##.

Which integral should I do first?