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One common used trick when calculating loop integral is Schwinger parameterization. And I have seen two versions among wiki, arxiv and lecture notes. $$\frac{1}{A}=\int_0^{\infty} \mathrm{d}t \ e^{-tA}$$ or, $$\frac{-i}{(-i)A}=-i\int_0^{\infty} \mathrm{d}t \ e^{itA}$$ where $A=p^2-m^2+i\epsilon$.

I know the latter is surely true since its real part $Re(-iA)=\epsilon\gt0$ and thus applicable for the equation $$\frac{1}{a}=\int_0^{\infty} \mathrm{d}t \ e^{-at}\ \text{ for } Re(a)\gt0.$$ But as for the former, it doesn't hold true for space-like which the loop momentum probably behaves like , i.e. $p^2-m^2\lt0$. I am very confused why so many people still use the first one and any explanation will be appreciated!

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The Schwinger parameter itself is manifestly positive. In particular, it is not Wick-rotated, so there are not different versions of it. Rather it is OP's $A$ operator that is Wick-rotated.

OP lists a few references in above comments.

  • Ref. 1 works in Euclidean signature, so it's well-defined.

  • Ref. 2 & 3 only use the Schwinger parameter to derive the Feynman parametrization. They, on the other hand, want to work in Minkowskian signature. In practice one would then have to argue (presumably case by case) if one can analytically continue/Wick-rotate from Euclidean signature to Minkowskian signature (thereby introducing the Feynman $i\epsilon$-prescription).

References:

  1. H. Kleinert & V. Schulte-Frohlinde, Critical Properties of $\phi^4$-Theories; chapter 8, p. 106.

  2. J.A. Shapiro, Schwinger trick and Feynman Parameters, 2007 lecture notes; p. 1.

  3. S. Weinzierl, The Art of Computing Loop Integrals, arXiv:hep-ph/0604068; p. 11.

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  • $\begingroup$ I really appreciate your explanation. I also realized that many people use the first identify based on the fact that most of loop integral can be wick-rotated and thus becomes well-defined, which hardly makes sense just mathematically. $\endgroup$
    – colin
    Apr 24, 2021 at 14:48
  • $\begingroup$ One last question(maybe not really related) is how about the calculation of loop integral with complicated masses. The schwinger trick works as the real part stays the same. But I have no idea of the following steps of full calculation of loop integral like one-loop three-point. I don't know how to get the Veltmen's deduction through schwinger parameterization. Thank you so much! $\endgroup$
    – colin
    Apr 24, 2021 at 14:48
  • $\begingroup$ Hi colin. Thanks for the feedback. Can you be more specific? $\endgroup$
    – Qmechanic
    Apr 24, 2021 at 14:56
  • $\begingroup$ Yeah. After schwinger parameterization, the one-loop three-point loop function becomes $\int d^3\vec{\alpha}\int \frac{d^4k}{i\pi^2}e^{-(\alpha_1+\alpha_2+\alpha_3)k^2-2(\alpha_2p_1+\alpha_3p_2)k+\cdots}$. And I have no idea with the subsequent calculation either analytically or numerically especially when the masses are complex,i.e.$m_i=M_i-i\Gamma_i$ $\endgroup$
    – colin
    Apr 24, 2021 at 23:03
  • $\begingroup$ It matters because I usually deal with non-trivial numerator like $e^{\frac{k^2-m^2}{\Lambda^2}}$ and thus the loop integral becomes $\int d^3\vec{\alpha}\int \frac{d^4k}{i\pi^2}e^{-(-1/\Lambda^2+\alpha_1+\alpha_2+\alpha_3)k^2-2(\alpha_2p_1+\alpha_3p_2)k+\cdots}$ .This becomes hard for me $\endgroup$
    – colin
    Apr 24, 2021 at 23:10

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