Schwinger parametrization

Schwinger parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. It is named after Julian Schwinger, ^[1] who introduced the method in 1951 for quantum electrodynamics. ^[2]

Description

Using the observation that

${\displaystyle {\frac {1}{A^{n}}}={\frac {1}{(n-1)!}}\int _{0}^{\infty }du\,u^{n-1}e^{-uA},}$

one may simplify the integral:

${\displaystyle \int {\frac {dp}{A(p)^{n}}}={\frac {1}{\Gamma (n)}}\int dp\int _{0}^{\infty }du\,u^{n-1}e^{-uA(p)}={\frac {1}{\Gamma (n)}}\int _{0}^{\infty }du\,u^{n-1}\int dp\,e^{-uA(p)},}$

for ${\textstyle \mathrm {Re} (n)>0}$.

Alternative parametrization

Another version of Schwinger parametrization is:

${\displaystyle {\frac {i}{A+i\epsilon }}=\int _{0}^{\infty }du\,e^{iu(A+i\epsilon )},}$

which is convergent as long as ${\displaystyle \epsilon >0}$ and ${\displaystyle A\in \mathbb {R} }$. ^[3] It is easy to generalize this identity to n denominators.

See also

• Feynman parametrization

References

1. Schwinger, Julian (1951-06-01). "On Gauge Invariance and Vacuum Polarization". Physical Review. 82 (5): 664–679. doi:10.1103/PhysRev.82.664.
2. Kim, U-Rae; Cho, Sungwoong; Lee, Jungil (2023-06-01). "The art of Schwinger and Feynman parametrizations". Journal of the Korean Physical Society. 82 (11): 1023–1039. doi:10.1007/s40042-023-00764-3. ISSN   1976-8524.
3. Schwartz, M. D. (2014). "33". Quantum Field Theory and the Standard Model (9 ed.). Cambridge University Press. p. 705. ISBN   9781107034730.