Euler-Heisenberg Lagrangian & heat kernel: Why do we need to substitute $s \to -is$ to continue calculating?

Question

Starting from the general effective lagrangian for a constant background electromagnetic field, $$\mathcal{L}_{\mathrm{EH}}=-\frac{1}{4}F_{\mu\nu}^{2}-\frac{e^{2}}{32\pi^{2}}\int_{0}^{\infty}\frac{ds}{s}e^{-s\varepsilon}e^{-ism^{2}}\frac{\mathrm{Re}\cos(esX)}{\mathrm{Im}\cos(esX)}\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta},\tag{33.69}$$ where $$X\equiv\sqrt{\frac{1}{2}F_{\mu\nu}^{2}+\frac{i}{4}\varepsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}}.\tag{33.70}$$ Why do we need to substitute $s \to -is$ before continuing to calculate any interesting quantities like pair creation? What does this substituting entail for the underlying physics? And why are we allowed to make this substitution? After the substituting, the Euler-Heisenberg Lagrangian reads,

$$\mathcal{L}_{\mathrm{EH}}=-\frac{1}{4}F_{\mu\nu}^{2}-\frac{e^{2}}{32\pi^{2}}\int_{0}^{\infty}\frac{ds}{s}e^{is\varepsilon}e^{-sm^{2}}\frac{\mathrm{Re}\cosh(esX)}{\mathrm{Im}\cosh(esX)}\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}.\tag{33.71}$$

Reference: Quantum Field Theory and the Standard Model, Matthew D. Schwartz, page 715.

Answer 1

1. On one hand, the integral$^1$ (33.71), which uses an Euclidean Schwinger proper time $s_E$, is mathematically well-defined.

2. On the other hand, the integral (33.69) [and (33.68) if no $E$-field], which both uses a Minkowskian Schwinger proper time $s_M$, are mathematically ill-defined due to poles on top of the integration contour $\mathbb{R}_+$.

   In the eqs. (33.68)-(33.69) the damping factor $e^{-\epsilon s_M}$ [which is meant as a regularization] does not cure this problem.

3. A better regularization is to avoid the poles $$ \underbrace{i\int_{\mathbb{R}_+}\!ds_M ~e^{-is_M(m^2-i\epsilon)} f(is_M)}_{\text{ill-defined}}$$ $$\quad\downarrow\quad$$ $$ i\int_{\mathbb{R}_+}\!ds_M ~e^{-ie^{-i\epsilon}s_Mm^2} f(ie^{-i\epsilon}s_M). $$

4. We now want to perform a Wick rotation $$s_E~=~is_M.$$ However, due to the ${\rm Re}$ & ${\rm Im}$ symbols in the integrand (33.69), it is not a holomorphic function of $s_M$, so we can not use complex function theory. Nevertheless the integrand (33.68) is holomorphic, so we can here deform the contour, and use the residue theorem, crossing no poles, and get an Euclidean Schwinger proper time representation $$ \int_{ie^{-i\epsilon}\mathbb{R}_+}\!ds_E ~e^{-s_E m^2}f(s_E) ~=~\int_{\mathbb{R}_+}\!ds_E ~e^{-s_E m^2}f(s_E), $$ which can be extended to a Lorentz-invariant expression a la eq. (33.71).

5. So in hindsight, it would have been better if Ref. 1 had established the Euclidean [as opposed to the Minkowskian] Schwinger proper time representation much earlier in chapter 33, particularly in eq. (33.38).

References:

1. M.D. Schwartz, QFT & the standard model, 2014; Section 33.4.

---

$^1$Note that the infinitesimal phase factor $e^{is_E\epsilon}$ in eq. (33.71) is unnecessary, cf. e.g. Wikipedia. Also note that the last $F\widetilde{F}$ factor in eq. (33.71) is cancelled by an $F\widetilde{F}$ factor in the denominator, i.e. there is a removable singularity if $F\widetilde{F}$ is zero.