Srednicki Eqs. (6.22) and (9.6). How to get rid of $iepsilon$ in the interaction term?

I’m studying qft from Srednicki’s book. If one writes down the full $iepsilon$ terms, passing from Eq. (6.21) (non-perturbative definition) to Eq. (6.22) (perturbative definition) yields
$$left<0|0right>_{f,h} = intmathcal{D}pmathcal{D}q expleft[iint_{-infty}^{+infty}dtleft(pdot{q} – (1 – iepsilon )H(p,q) + fq + hpright)right]
=expleft[-i(1 – iepsilon )int_{-infty}^{+infty}dt H_{1}left(frac{1}{i}frac{delta}{delta h(t)},frac{1}{i}frac{delta}{delta f(t)}right)right]
timesintmathcal{D}pmathcal{D}q expleft[iint_{-infty}^{+infty}dtleft(pdot{q} – (1 – iepsilon )H_{0} + fq + hpright)right] tag{6.22}$$
where $fleft( tright) $, $gleft( tright) $ are external sources with $%
lim_{trightarrow pm infty }fleft( tright) =lim_{trightarrow pm
infty }gleft( tright) =0$, $leftlangle 0|0rightrangle _{f,h}$ is the
vacum-vacum probability amplitude in the presence of sources, $H=H_{0}+H_{1}$
, and $leftvert 0rightrangle $ is the $textit{ground state ket}$ (assumed
non-degenerate) of the $textit{total}$ Hamiltonian operator $hat{H}=H(hat{p},hat{q})$.

However, in the physics literature the $iepsilon$ factor multiplying $H_{1}left(frac{1}{i}frac{delta}{delta h(t)},frac{1}{i}frac{delta}{delta f(t)}right)$ is absent (see, e.g., Eq. (9.6) from Srednicki), even though the $iepsilon$ factor multiplying $H_{0}$ is still present during the calculations giving the famous $iepsilon$ prescription for the free propagator, and $epsilon$ is taken to the limit $0^{+}$ only in the very end for $H_{0}$.

Question: How to justify getting rid of the $iepsilon$ that multiplies $H_{1}$, while keeping to the very end the $iepsilon$ that multiplies $H_{0}$?

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Update 06.06.2019: I’ve tried to reason about it as follows. Let $$
Z_{0}left[ fleft( tright) ,gleft( tright) ;varepsilon right]
equivint mathcal{D}pmathcal{D}qexpleft[ iint_{-infty }^{+infty }dt
left( pdot{q}-left( 1-ivarepsilon right) H_{0}+fq+hpright) right]
$$
and
$$
x_{n}left[ fleft( tright) ,gleft( tright) ;varepsilon right] equiv
frac{left( -1right) ^{n}}{n!}left[ int_{-infty }^{+infty }dttext{ }%
H_{1}left( frac{1}{i}frac{delta }{delta hleft( tright) },frac{1}{i}%
frac{delta }{delta fleft( tright) }right) right] ^{n}Z_{0}left[
fleft( tright) ,gleft( tright) ;varepsilon right]
$$

Since $$
exp left[ -ileft( 1-ivarepsilon right) int_{-infty }^{+infty }dt
text{ }H_{1}left( frac{1}{i}frac{delta }{delta hleft( tright) },
frac{1}{i}frac{delta }{delta fleft( tright) }right) right]
=sum_{n=0}^{infty }frac{left( -1right) ^{n}}{n!}left( i+varepsilon
right) ^{n}left[ int_{-infty }^{+infty }dttext{ }H_{1}left( frac{1}{i
}frac{delta }{delta hleft( tright) },frac{1}{i}frac{delta }{delta
fleft( tright) }right) right] ^{n}
$$
it follows that
begin{eqnarray*}
leftlangle 0|0rightrangle _{f,h} &=&lim_{varepsilon rightarrow
0^{+}}left{ sum_{n=0}^{infty }left( i+varepsilon right) ^{n}x_{n}
left[ fleft( tright) ,gleft( tright) ;varepsilon right] right}
&=&lim_{varepsilon rightarrow 0^{+}}left{ sum_{n=0}^{infty }i^{n}x_{n}
left[ fleft( tright) ,gleft( tright) ;varepsilon right] right}
&&-ilim_{varepsilon rightarrow 0^{+}}varepsilon left{
sum_{n=0}^{infty }ncdot i^{n}x_{n}left[ fleft( tright) ,gleft( tright)
;varepsilon right] right} +lim_{varepsilon rightarrow 0^{+}}Oleft(
varepsilon ^{2}right)
end{eqnarray*}

I assume that the series $S_{1}left( varepsilon right) equiv
sum_{n=0}^{infty }i^{n}x_{n}left[ fleft( tright) ,gleft( tright)
;varepsilon right] $ is convergent, i.e., that $S_{1}left( varepsilon
right) $ is finite. However, the series $$
S_{2}left( varepsilon right) equiv sum_{n=0}^{infty }ncdot i^{n}x_{n}left[
fleft( tright) ,gleft( tright) ;varepsilon right]
$$
is not necessarily convergent due to the multiplication of $i^{n}x_{n}left[
fleft( tright) ,gleft( tright) ;varepsilon right] $ by $n$.

Therefore, the limit $$
lim_{varepsilon rightarrow 0^{+}}varepsilon left{ sum_{n=0}^{infty
}ncdot i^{n}x_{n}left[ fleft( tright) ,gleft( tright) ;varepsilon right]
right} =lim_{varepsilon rightarrow 0^{+}}varepsilon cdot S_{2}left(
varepsilon right)
$$
may not even exist.

Only in the case in which $lim_{varepsilon rightarrow 0^{+}}varepsilon
cdot S_{2}left( varepsilon right) =0$ can one write
begin{eqnarray*}
leftlangle 0|0rightrangle _{f,h} &=&lim_{varepsilon rightarrow
0^{+}}left{ sum_{n=0}^{infty }i^{n}x_{n}left[ fleft( tright) ,gleft(
tright) ;varepsilon right] right}
&=&lim_{varepsilon rightarrow 0^{+}}exp left[ -iint_{-infty
}^{+infty }dttext{ }H_{1}left( frac{1}{i}frac{delta }{delta hleft(
tright) },frac{1}{i}frac{delta }{delta fleft( tright) }right) right]

&&times int mathcal{D}pmathcal{D}qexp left[ iint_{-infty }^{+infty
}dttext{ }left( pdot{q}-left( 1-ivarepsilon right) H_{0}+fq+hpright) %
right]
end{eqnarray*}
and the $ivarepsilon $ that multiplies $H_{1}$ can be taken as $0$, while
the $ivarepsilon $ that multiplies $H_{0}$ is beeing kept until the very
end of calculations.

Question: Is there any other way to justify this replacement? Is
the perturbative definition for path integrals ill-defined due to possible
divergence of the series $S_{2}left( varepsilon right) $?

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I would be most thankful if you could help me with this question concerning the perturbative definition of Green functions via path integrals in ordinary qm and in non-relativistic and relativistic qft.

Following the epoch making papers by Osterwalder and Schrader, it has become common practice to work in Euclidean time with Euclidean Green functions and then to analytically continue them to real time.
Concerning path integrals, one usually defines them perturbatively.

My elementary analysis shows as to what happens when one makes an infinitesimal Wick rotation $trightarrow (1 – iepsilon)t$ (much used
in the literature). When the limit $epsilonrightarrow 0^{+}$ is taken in the end, in the perturbative definition of the path integral, I show that one encounters a nonsensical result of the form $epsilontimes (divergent series)$ which doesn’t have a well-defined limit when $epsilonrightarrow 0^{+}$.
It seems that for an infinitesimal Wick rotation, when one analytically
continues back to the real time (i.e., takes the limit $epsilonrightarrow 0^{+}$),one obtains a nonsensical result.

An analytic continuation is not a mere replacement $t rightarrow it$, but can be thought of as a series of infinitesimal Wick rotations.

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The problem of analytic continuation from real time to Euclidean time is very nicely presented in Ch. 13 of Giorgio Parisi’s famous book on statistical field theory. This is the essence of the epsilon trick in Srednicki’s book.

In Parisi’s analysis everything works very well since it is a non-perturbative analysis, based on the total Hamiltonian. The problem shows up in the perturbative approach as I have shown above. The perturbative approach for the analysis of analytic continuation $trightarrow it$ is not presented
in any book that I know of. In order for the theory to be consistent, the perturbation series must be convergent for any $texp(itheta)$ with $0 < theta < pi/2$. For the theory to exist, it is NOT sufficient
that the case $theta = pi/2$ is convergent.

I think that the problem is very serious since the perturbative definition of path integrals is the most common tool in physics. The question I’m raising is not merely an academic one.

I have no proof that the series $S_2$ is divergent, since this is model dependent. However, there are no theorems in complex or real analysis that discuss the convergence of a series $S_2 = sum_{n=0}^{infty} nx_n$, if it is known that the series $S_1 = sum_{n=0}^{infty} x_n$ is convergent.

If one continues the series expansion in $epsilon$ to higher order terms, the divergences are even worse.

Questions: Is the perturbative definition of path integrals wrong? Is there a version of the O-S axioms for non-relativistic qm and condensed matter? Is the perturbative definition of path integrals in contradiction with the O-S axioms?

I would be most thankful if you could send me your opinion on these questions.