Z^N-1_φ_1 φ_1(k_(N -1)p + 1)  φ_1(k_(N-1)p + 2)  φ_1(k_(N-1)p + 3) ...  φ_1(k_Np - 1) [1pt][7pt]φ_1(k_1) φ_1_Np fields  +  1 source(k_1), with ${\rm N}p-1$ fields having different momenta from the source, which in turn is taken to have a momentum $k_1$. The nested structure appears from choosing the momentum $k_1$ and arranging the remaining ${\rm N}p-1$ fields with different (and hence unequal) momenta. This is depicted by the following diagram:

[baseline=(0, 0), thick,
main/.style = draw, circle, fill=black, minimum size=4pt,
dot/.style=inner sep=0pt,fill=black, circle, minimum size=2pt,
dots/.style=inner sep=0pt,fill=black, circle, minimum size=1pt,
ball/.style=ball  color=white, circle,  minimum size=15pt]
[] (a) at (-1,-1.1) $\widetilde\varphi_1\overbracket[1pt][7pt]{(k_1)~\widetilde\varphi_1}(k_1)$;
[dots] (a1) at (1,-1.1) ;

[dot] (0) at (7.5, 1) ;
[dot] (0) at (7.5, 0.5) ;
[dot] (0) at (7.5, 0) ;
[dot] (0) at (7.5, -0.4) ;

[dot] (0) at (7.5, -3) ;
[dot] (0) at (7.5, -3.5) ;
[dot] (0) at (7.5, -4) ;

[red](a) – (a1);

[dots] (a2) at (1,-0.5) ;
[dots] (a3) at (1,2) ;
[dots] (a4) at (1,-2.7) ;
[dots] (a5) at (1,-4.5) ;

(a21) at (2,-0.5) $\widetilde\varphi_1(k_3)$;
(a31) at (2,2) $\widetilde\varphi_1(k_2)$;
(a41) at (2,-2.7) $\textcolor{red}{\widetilde\varphi_1(k_4)}$;
(a51) at (2,-4.5) $\widetilde\varphi_1(k_{{\rm N}_\mu})$;

[->] (a2) – (a21);
[->] (a3) – (a31);
[red][->] (a4) – (a41);
[->] (a5) – (a51);
[red](a4) – (a1);
(a1) – (a3);
[dashed] (a4) – (a5);

[dots] (b) at (3, 2) ;
[dots] (b1) at (3, 2.5) ;
[dots] (b2) at (3, 3) ;
[dots] (b3) at (3, 1) ;

(b0) at (4, 2) $\widetilde\varphi_1(k_5)$;
(b11) at (4, 2.5) $\widetilde\varphi_1(k_4)$;
(b21) at (4, 3) $\widetilde\varphi_1(k_3)$;
(b31) at (4.4, 1) $\widetilde\varphi_1(k_{{\rm N}_\mu})$;

(a31) – (b);
[->] (b) – (b0);
[->] (b1) – (b11);
[->] (b2) – (b21);
[->] (b3) – (b31);
(b2) – (b);
[dashed] (b) – (b3);

[dots] (c) at (3, -2.7) ;
[dots] (c0) at (3, -1) ;
[dots] (c1) at (3, -2) ;
[dots] (c2) at (3, -2.7) ;
[dots] (c3) at (3, -3.5) ;

(c01) at (4, -1) $\widetilde\varphi_1(k_5)$;
(c11) at (4, -2) $\textcolor{red}{\widetilde\varphi_1(k_3)}$;
(c21) at (4, -2.7) $\widetilde\varphi_1(k_2)$;
(c31) at (4.4, -3.5) $\widetilde\varphi_1(k_{{\rm N}_\mu})$;

[red](a41) – (c);
[->] (c0) – (c01);
[red][->] (c1) – (c11);
[->] (c2) – (c21);
[->] (c3) – (c31);
(c0) – (c1);
[red](c1) – (c2);
[dashed] (c2) – (c3);

[dots] (d) at (3, -4.5) ;
[dots] (d0) at (3, -4) ;
[dots] (d1) at (3, -5) ;
[dots] (d2) at (3, -5.5) ;

(d01) at (4, -4) $\widetilde\varphi_1(k_2)$;
(d11) at (4, -5) $\widetilde\varphi_1(k_3)$;
(d21) at (4.4, -5.5) $\widetilde\varphi_1(k_{{\rm N}_\mu-1})$;

(a51) – (d);
[->] (d0) – (d01);
[->] (d1) – (d11);
[->] (d2) – (d21);
(d1) – (d0);
[dashed] (d0) – (d2);

[dots] (e) at (6, 2.5) ;
[dots] (e1) at (6, 3) ;
[dots] (e2) at (6, 2.3) ;
[dots] (e3) at (6, 1.5) ;

(e11) at (7.8, 3) $\widetilde\varphi_1(k_3)$;
(e21) at (7.8, 2.3) $\widetilde\varphi_1(k_5)$;
(e31) at (7.8, 1.5) $\widetilde\varphi_1(k_{{\rm N}_\mu})$;

(b11) – (e);
[->] (e1) – (e11);
[->] (e2) – (e21);
[->] (e3) – (e31);
(e1) – (e2);
[dashed] (e2) – (e3);

[dots] (f) at (6, -2) ;
[dots] (f1) at (6, -1) ;
[dots]  (f2) at (6, -1.5) ;
[dots] (f3) at (6, -2.5) ;

(f11) at (7.8, -1) $\widetilde\varphi_1(k_2)$;
(f21) at (7.8, -1.5) $\textcolor{red}{\widetilde\varphi_1(k_5)}$;
(f31) at (8, -2.5) $\widetilde\varphi_1(k_{{\rm N}_\mu})$;

[red](c11) – (f);
[->] (f1) – (f11);
[red][->] (f2) – (f21);
[->] (f3) – (f31);
(f1) – (f2);
[dashed] (f2) – (f3);

[dots] (g) at (6, -5) ;
[dots] (g1) at (6, -4.5) ;
[dots] (g2) at (6, -5.5) ;
[dots] (g3) at (6, -6.4) ;

(g11) at (7.8, -4.5) $\widetilde\varphi_1(k_2)$;
(g21) at (7.8, -5.5) $\widetilde\varphi_1(k_4)$;
(g31) at (8.2, -6.4) $\widetilde\varphi_1(k_{{\rm N}_\mu-1})$;

(d11) – (g);
[->] (g1) – (g11);
[->] (g2) – (g21);
[->] (g3) – (g31);
(g1) – (g2);
[dashed] (g2) – (g3);

(A1) at (9.5, 3.3) ;
(B1) at (9.5, 1) ;
snakes (A1) – (B1);

(A2) at (9.5, -0.7) ;
(B2) at (9.5, -2.9) ;
snakes(A2) – (B2);

(A3) at (9.7, -4.1) ;
(B3) at (9.7, -6.8) ;
snakes (A3) – (B3);

where we see that the first row has all momenta lying between $k_2$ till
$k_{{\rm N}_\mu}$. The second row is more involved but the above diagrammatic representation clearly depicts the proliferation of the chain. The latter continues till ${\rm N}p-1$ steps (the above figure only shows the first three steps in the chain). Tracing one such path, shown in red, we find the following string of operators:

φ_1(k_1)  φ_1(k_1)  φ_1(k_4)  φ_1(k_3)  φ_1(k_5) ...  φ_1(k_N_μ), which, from our binomial expansion, generically scale as $\hat{p}$ where $\hat{p} < {\rm N}!$. (In other words, keeping $\widetilde\varphi_1\overbracket[1pt][7pt]{(k_1)~\widetilde\varphi_1}(k_1)$ fixed, the total number of paths with the same set of field contents in any arbitrary order scales as $\hat{p} < {\rm N}!$.) In fact, there are infinite paths (labelled by $l$) connecting the operators given in the ${\rm N}_\mu$ rows in the above diagram $-$ but differing from (<ref>) $-$ with each scaling as $p_l < {\rm N}!$. The total amplitude is given by summing over all these infinite string of operators, i.e. summing over $\widetilde\varphi_1\overbracket[1pt][7pt]{(k_j)~\widetilde\varphi_1}(k_j)$ for $1 \le j \le {\rm N}_\mu$ with $j \in \mathbb{Z}$, which would basically appear as nested integrals. After the dust settles, the final answer of the total amplitude is not very different from what we had in (<ref>), albeit a bit more involved, and takes the form:

A_Np ≡∑_{r_i} ∏_j = 1^Np-1 ∫_k_IR^μd^11k_j k_j^r_j α(k_j)a(k_j) ·1V∫_k_IR^μd^11k A_r_1r_2...r_Np-1r_Np[{a(k_j)}, a(k)] k^r_Np(α^2(k)a^2(k) + 12a(k))ψ_k(x, y),
which, once carefully observed, reveals the full structure presented in the diagram above. The coefficients ${\bf A}_{r_1r_2...r_{{\rm N}p-1}r_{{\rm N}p}}[\{a(k_i)\}, a(k)]$ are more complicated than (<ref>) but, since their explicit forms are not necessary here, we will not work them out and leave them again for our diligent readers. From the arguments presented earlier, the whole amplitude scales as ${\rm N}!$, which should point towards the factorial growth of the one-point functions.

There are however couple of subtleties here that need clarification. First, if we compute the magnitude of the amplitude (<ref>), i.e. $\vert{\cal A}_{{\rm N}p}\vert$, then the result may appear to scale as:

|A_Np|= N! f(k_IR, μ, p, N, V; x, y), where $f(k_{\rm IR}, \mu, p, {\rm N}, {\rm V}; x, y)$ is a function that can be worked out from (<ref>).
Note the appearance of ${\rm N}$ in two places, one as an overall factorial ${\rm N}!$, and the other in the function $f(k_{\rm IR}, \mu, p, {\rm N}, {\rm V}; x, y)$. The latter is typically in the polynomial form, whereas the former is in the factorial form which is what we meant when we used the term “factorial growth". The polynomial growth is also very important and we will discuss it's consequence in the next section.

Secondly, (<ref>) cannot quite be the full answer because this ignores the power $p$ of $\varphi_1^p$ at the tree-level. To see what exactly replaces the ${\rm N}!$ in (<ref>), let us quantify the interactions in our model with four fields $(\varphi_1, .., \varphi_4)$:

S_int = ∫d^11x  c_oM_p^n+m+l+h ∂^nφ_1^p  ∂^mφ_2^q  ∂^lφ_3^r  ∂^hφ_4^s, where $(n, m, l, h) \in +2\mathbb{Z}, c_o$ is yet another unknown coefficient, and we will define $g$, the coupling constant, as $g \equiv {c_o\over {\rm M}_p^{n+m+l+h}}$. As usual, the fields are taken to be dimensionless, and the Minkowski space has a metric $(\eta_{\mu\nu}, \delta_{mn}, \delta_{ab})$ for $(\mu, \nu) \in {\bf R}^{2, 1}, (m, n) \in {\cal M}_6, (a, b) \in {\mathbb{T}^2\over {\cal G}}$. The growth ${\rm N}!$ in (<ref>) should be replaced as:

N!       →  (pN)!(qN)!(rN)!((s-1)N)!(p!q!r!(s-1)!)^N

= [p^p q^q r^r (s-1)^s-1p! q! r! (s-1)!]^N
(N!)^p+q+r + s-1|_N >> 1 ≡A_1^N (N!)^αš(1 + 1α - 1αš),

where ${\cal A}_1 ={p^p q^q r^r (s-1)^{s-1}\over p! q! r! (s-1)!}$, $\alpha +1$ is the total number of fields such that $p + q + r + s \equiv (\alpha + 1) \check{s}$. The $s-1$ factor is taken to imply the momentum conservation in the $\varphi_4$ sector where one of the Fourier components
of the $s$ fields becomes redundant. And in going from first line to the second one we have used $(p{\rm N})! \propto ({\rm N}!)^p$, where the proportionality factor[Let $\check{p}$ be the proportionality factor. Taking log on both sides shows that the RHS is $\log~\check{p} + p{\rm N}\log~{\rm N} - p{\rm N}$ for ${\rm N} >> 1$. This equals LHS for $\check{p} = p^{p{\rm N}}$. ] is $p^{p{\rm N}}$ for ${\rm N} >> 1$, whereas for small ${\rm N}$, the proportionality factor is again some polynomial in $p$.

There are still a few details that need to be considered.
First, due to ${1\over {\rm N}!}$ suppression coming from the exponential[Recall that the path integral has $e^{-{\bf S}_{\rm int}}$, so expanding this gives us the ${1\over {\rm N}!}$ suppression at order ${\rm N}$. ], the growth that we saw above should get further modified to $\left({\rm N}!\right)^{\alpha \check{s}\left(1 + {1\over\alpha} - {2\over \alpha\check{s}}\right)}$.
Secondly, there is a mixing between the fields.  However considering the fact that our theory has more than one fields, i.e. say $\alpha + 1$ fields, and since both $\alpha > 1$ and $\check{s} > 1$, we can ignore the
${1\over \alpha}$ and ${2\over \alpha \check{s}}$ in the factorial growth above. This means that, everything considered including the fact that
for every field components we can draw the nested interactions as in the diagram above, the theory should have at least $\left({\rm N}!\right)^{\alpha\check{s}}$ growth.

This is not exactly true because of yet another interesting subtlety. As an example consider the two sectors $\varphi_2$ and $\varphi_3$ that do not directly interact with the source. For simplicity we will take cubic interactions between the two set of fields, i.e. $g^3\varphi_2^3(x, y) \varphi_3^3(x, y)$, which is basically ${\rm N} = 3$ with $n = m = l = h = 0$ and $p = q = r = s = 1$ in (<ref>), but we shall ignore the $\varphi_1$ and $\varphi_4$ sectors for simplicity, as one can absorb the ${1\over {\rm N}!}$ suppression and the other can absorb the momentum conservation. In terms of Fourier modes this may be represented as:

g^3V^6(∑_i = 1^N_μφ_2(k_i)ψ_k_i(x, y))^3 (∑_j = 1^N_μφ_3(k_j)ψ_k_j(x, y))^3
→   φ_2(k_1)     φ_2(k_1)
φ_2(k_1)     φ_3(k_1)     φ_3(k_1)    φ_3(k_1)                                       φ_2(k_2)     φ_2(k_2)
φ_2(k_2)     φ_3(k_2)
φ_3(k_2)     φ_3(k_2)                                       φ_2(k_3)     φ_2(k_3)
φ_2(k_3)    φ_3(k_3)     φ_3(k_3)     φ_3(k_3)                                       φ_2(k_4)     φ_2(k_4)
φ_2(k_4)    φ_3(k_4)     φ_3(k_4)     φ_3(k_4)                                       φ_2(k_5)     φ_2(k_5)
φ_2(k_5)     φ_3(k_5)
φ_3(k_5)     φ_3(k_5)                                       φ_2(k_6)    φ_2(k_6)
φ_2(k_6)     φ_3(k_6)     φ_3(k_6)    φ_3(k_6) ..     ..            ..            .. ..     ..            ..            .. φ_2(k_N_μ)     φ_2(k_N_μ)
φ_2(k_N_μ)     φ_3(k_N_μ)     φ_3(k_N_μ)     φ_3(k_N_μ)

where $\psi_{k_i}(x, y)$ are the wave-functions over the Minkowski vacuum, and on the RHS we represent the field components in six rows related to the ${\rm N} = 3$ case. The wave-functions will simply give momentum conservation in the $\varphi_4$ sector so we ignore them on the RHS. Following two paths, one in blue and the other in red, provides the two strings of operators:

B_1 =     φ_2(k_1)  φ_2(k_2)  φ_2(k_3)  φ_3(k_4) φ_3(k_5)

B_2 =      φ_2(k_5)  φ_2(k_6)  φ_2(k_4)  φ_3(k_3)  φ_3(k_2)

which have the same distribution of momenta, albeit with different field contents. The dominant terms that do not have volume suppressions grow as $3!$ for the blue line and $3!$ for the red line, after we add up all the contributions. So the naive growth is $(3!)^2$, appearing from binomial product[From above, the general growth with everything considered is proportional to $\left({\rm N}!\right)^{\alpha \check{s}\left(1 + {1\over\alpha} - {2\over \alpha\check{s}}\right)}$. Taking ${\rm N} = 3, \alpha = 3$ and $\check{s} = 1$ we get $(3!)^2$.]. On the other, if we look at the distribution of momenta, then there are in fact $(2.3)! = 6! = 720$ terms contributing instead of $(3!)^2 = 36$ terms. This enhancement is due to mixing of momenta over the two set of field components. Once we take a more general interaction of the form ${g\over q! r!} \varphi_2^q(x, y) \varphi_3^r(x, y)$, with a coupling $g$, then the amplitude to order ${\rm N}$ becomes:

A_N(q+r) ≡g^N∑_{r_i}  ∏_j = 1^Nq ∫_k_IR^μd^11k_j k_j^r_j α_2(k_j)a(k_j)
∏_l = Nq+1^N(q+r) ∫_k_IR^μd^11k_l k_l^r_l β(k_l)b(k_l)
A_r_1r_2...r_Nq...r_N(q+r)[{a(k_j)}, {b(k_l)}],

where $(\alpha_2, \beta)$ correspond to the notations given in (<ref>). As expected, (<ref>) is almost a product of ${\rm N}(q+r)$ integrals with ${\rm N}q$ integrals coming from the $\alpha_2$ Glauber-Sudarshan states and ${\rm N}r$ integrals coming from $\beta$ Glauber-Sudarshan states. If $\hat{f}^{1/2}_{\rm max}(k_{\rm IR}, \mu; q, r)$ denotes the maximum value of one such integral piece, then the value of the amplitude $\vert{\cal A}_{{\rm N}(q+r)}\vert$ is bounded from above by (see also (<ref>) for a more correct way to implement the bound):

|A_N(q+r)|  ≤  (2N)!  g^N |f̂_max(k_IR, μ; q, r)|^N, which may be compared to the $({\rm N}!)^2$ growth for the amplitude (<ref>) as depicted in (<ref>). (Note the absence of $(x, y)$ in the definition of $\hat{f}_{\rm max}$.) The power of $1/2$ in $\hat{f}^{1/2}_{\rm max}$ suggests that the maximum value could also come from a geometric mean of the maximum values from the $(\alpha_2, \beta)$ sectors. Ultimately however, the precise value of $\hat{f}_{\rm max}$ is not very important for us here, although what is important here is the upper bound in (<ref>). Our above analysis suggests that, given say $\alpha + 1$ set of interacting fields, the factorial growth of the amplitude of any one-point function in the shifted vacua is generically bigger than $({\rm N}!)^{\alpha\check{s}}$, and it can enhance at most to:

p̌·(N!)^αš   =   (αš N)!, where $\check{p}$ is the possible number of permutations of fields and momenta and $\alpha >> 1$(see for example (<ref>)). For small values of ${\rm N}$, it is not too hard to show that $\check{p} \in +\mathbb{Z}$. When ${\rm N}$ is large, one can use the Sterling's approximation to show that $\check{p} = (\alpha\check{s})^{\alpha\check{s}{\rm N}} \in +\mathbb{Z}$, as $\alpha\check{s} \in +\mathbb{Z}$ for $\alpha >> 1$ (see footnote <ref>). The relation (<ref>) then is a simpler way of understanding how in a mixed system with $\alpha+1$ fields with $\alpha >> 1$ the system could show the so-called Gevrey growth of $(\alpha\check{s}{\rm N})!$.

The above conclusion is not the end of the story, rather the beginning of a new direction where the Gevrey growth should point towards the appearance of non-perturbative effects hitherto ignored in our set-up. In fact the factorial growth provides a deeper reason why the system has non-perturbative effects at all. Question is, can we quantify this by Borel resumming the Gevrey series? This is what we turn to next.

§.§ Gevrey series, Borel resummation and non-perturbative effects

The $({\rm N}!)^{\alpha\check{s}}$ growth that we discussed above for $\alpha + 1$ fields with $\alpha >> 1$ generically leads to the so-called Gevrey series that has been discussed in the literature in much details [34]. We will not do the same here and instead refer the readers to section 4.3.1 of our recent work [7]. Interestingly, the following observation due to Écalle [38]:

The Gevrey order of the asymptotic series ${\rm F}(g) =
\sum\limits_{{\rm N} = 0}^\infty ({\rm N}!)^l g^{\rm N}$ where $l \in \mathbb{Z}$ and $g << 1$, is the same as that of another asymptotic series
${\rm G}(g) = \sum\limits_{{\rm N} = 0}^\infty (l{\rm N})! g^{\rm N}$, thus reducing everything to Gevrey-1.

which efficiently leads to the so-called Borel-Écalle resummation, is pretty useful for us because of (<ref>). In fact the in-built enhancement of (<ref>) suggests that it is no longer necessary to apply the acceleration operators of Écalle [38] to go from Gevrey-$\alpha\check{s}$ to Gevrey-1. All we need is an upper bound of the form (<ref>) $-$ but now extended to $\alpha+1$ fields $-$ to perform the Borel resummation of a Gevrey series efficiently.

Unfortunately the actual computation is easier said than done, as evident from section 4.3.2 of our earlier work [7] and the upcoming work [8]. Some of the subtleties that we actually face in our computations are as follows.

${\bf 1.}$ The Borel resummation arising from the path integral computation isn't the only resummation. There are additional off-shell pieces, that remain invisible in the path integral, and arise from the background time-independent Minkowski space-time. These additional contributions are necessary to extract correct answers for the expectation values.

${\bf 2.}$ The singularities on the Borel plane lead to certain principal value computations that are non-trivial to perform. In fact the exact values of these integrals are not known, but numerical results can be found (see our upcoming paper [8] for details). Pin-pointing these numerical values are important to get an estimate on the four-dimensional cosmological constant.

${\bf 3.}$ The Borel resummation presented in [7] considers only one specific type of interactions. Introducing other interactions in the theory lead to the so called Borel boxes[They are useful technical machineries that help to Borel re-sum not only the dominant amplitudes, but also all the sub-dominant ones. We will discuss this briefly in section <ref>. Unfortunately a detailed exposition of this is beyond the scope of the paper and the readers may refer to our upcoming work [8]. ], whose convergence properties are important for finiteness. This additional “resummation” of Borel resummed series make the analysis highly non-trivial (see our upcoming paper [8] for further details).

${\bf 4.}$ The permutation factors ${\bf A}_{r_1r_2...r_{{\rm N}p-1}r_{{\rm N}p}..}$ appearing in (<ref>) and (<ref>) create additional hurdles because they not only mix up the momenta from various sectors in a non-trivial way, but also sum over the various choices of permutations. The way out of this technical complication, as analyzed in section <ref> and in our upcoming work [8], is to impose bounds on the binomial growths of the series in appropriate ways.

${\bf 5.}$ The integral (<ref>) with the source term integrated in, and on which we base our Borel resummations, works for ${\rm N} \ge 1$, but does not work for ${\rm N} = 0$. This unfortunate hurdle leads to further constraints in the system that has it's root in the no-go condition [1]. This subtlety does not show up in other sectors.

Clearly a detailed study of any of the aforementioned points is beyond the scope of this paper. Fortunately, these subtleties will not impede our progress in understanding the underlying picture, although we will take some time to elaborate the last point, namely the case for ${\rm N} \ge 1$ and ${\rm N} = 0$.  Taking the displacement operator to be (<ref>), the expectation value $\langle\varphi_1\rangle_\sigma$ from (<ref>) takes the following form at the ${\rm N}$-th order with $\alpha >> 1$ (see also (<ref>) and (<ref>)):

|A_N|≤(αšN)! g^N |f̌_max(k_IR, μ; p, q, r, s-1)|^N·1V
∫_k_IR^μd^11k  k^ρ^∗(k)(α_1^2(k)a^2(k) + 12a(k))ψ_k(x, y), which is gathered from the two amplitudes (<ref>) and (<ref>) discussed earlier. To avoid clutter, we can henceforth replace $\alpha\check{s}$, or more appropriately $\alpha\check{s}\left(1 + {1\over \alpha} - {2\over \alpha\check{s}}\right)$ by $\bar\alpha$.
Both $\check{f}_{\rm max}$ and $k^{\rho^\ast(k)}$ appear from imposing bounds on the permutations of the momenta (see section <ref> as well as [7] and [8] for more details). The parameter $s-1$ signifies the momentum conservation as one of the degrees of freedom becomes redundant. The inverse volume ${1\over {\rm V}}$ is the consequence of the contraction (<ref>), and in fact this is the main cause of the disparity between ${\rm N} \ge 1$ and ${\rm N} = 0$. This disparity may be quantified by the following integral function:

𝕋(k_IR, μ; ρ, V; x, y) ≡∫_k_IR^μd^11k
[α_1(k)a(k) - k^ρ^∗(k)V(α_1^2(k)a^2(k) + 12a(k)) + O(1V^2)]ψ_k(x, y), whose vanishing will imply no generation of de Sitter at the tree level. This is of course a consequence of the no-go theorem [1], and here we see how it may be implemented in the language of the Glauber-Sudarshan states. The equation $\mathbb{T} = 0$ leads to the determination of $\alpha_1(k)$ in terms of $\rho^\ast(k)$ as:

α_1(k) = 12 k^ρ^∗(k) + O(1V), which appears from the fact that both $\alpha_1(k)$ and $a(k)$ have inverse volume suppressions from imposing (<ref>). Note that (<ref>) does not fix the functional form for $\alpha_1(k)$ because we haven't yet determined $\rho^\ast(k)$. Question is, how to find the functional form for $\rho^\ast(k)$?

The answer is a bit non-trivial because of the appearance of additional off-shell pieces that we mentioned as point 1 earlier. There is also the sum over ${\rm N}$ which we need to perform first to conclude anything about these additional off-shell pieces. Fortunately, another advantage of the vanishing of $\mathbb{T}$ is our ability to extend ${\rm N}$ in (<ref>) to ${\rm N} \ge 0$ now. What does that buy us?

Extending ${\rm N}$ in (<ref>) to include ${\rm N} = 0$ means that we can perform the Borel-Écalle re-summation of the series in
(<ref>). This summation will automatically provide the full functional form for $\langle\varphi_1\rangle_\sigma$ in (<ref>) in the following way (henceforth we do not put hats to denote operators):

⟨φ_1⟩_σ= ∑_N = 0^∞|A_N|=
12g^1/α̅[∫_0^∞dS  exp(-Sg^1/α̅) 11 - f̌_max S^α̅]_P.V ∫_k_IR^μd^11k  k^ρ^∗(k) - 2  ψ_k(x, y), where the subscript P.V means that we have to take the principal value of the integral over ${\rm S}$, and $\psi_k(x, y)$ is the wave-function over the Minkowski space-time. The total number of fields is $\alpha + 1$, but what enters (<ref>) is $\bar\alpha$ which is defined earlier as:

α̅≡αš(1 + 1α - 2αš) = p+q+r + s - 2   ∈  ℤ, to avoid clutter.
The result (<ref>) is an exact answer got by summing over all perturbative orders in ${\rm N}$ despite our ignorance of the value of $c_o$ in (<ref>), or of the functional form for $\check{f}_{\rm max}$. The functional form for $\check{f}_{\rm max}$ provides the necessary poles in the Borel plane, but the actual principal value integral is quite non-trivial. We will not attempt this here but direct the readers to [7, 8] for more details. In list 2 earlier we pointed out precisely this difficulty. Nevertheless looking at the principal value integral the readers may easily notice the non-perturbative contributions going as ${\rm exp}\left(-{n\over g^{1/\bar\alpha}}\right)$. Two questions arise.

${\bf 1.}$ Where are these non-perturbative effects coming from? And what does the infinite sum of the non-perturbative effects mean here? Can this be quantified?

${\bf 2.}$ Can we determine the functional value of $\rho^\ast(k)$ now that we have performed the Borel-Écalle resummation? How does the additional off-shell terms manifest from the functional form for $\rho^\ast(k)$? Can this be quantified?

The answer to the first question is relatively easier: the form of the non-perturbative corrections clearly signify the presence of instantons. There is however a difference appearing from the fact that these instantons are both real and complex ones. To see how this comes about let us go back to the two potentials in figure <ref>. We can make the following observations.

${\bf 1.}$ If we consider the symmetric wave-function on the left[We will always be extending the wave-function $\Psi_0 = {\rm exp}\left[\int d^{11}k ~\log\langle{\bf f}_k\vert 0\rangle\right]$ to
$\Psi_\Omega = {\rm exp}\left[\int d^{11}k ~\log\langle{\bf f}_k\vert \Omega\rangle\right]$, where $\vert{\bf f}_k\rangle$ are states in the configuration space in all the cases studied here unless mentioned otherwise.], then the expectation value of $\varphi_1$ will only get contributions from the complex turning points, i.e. only from the complex instantons.

${\bf 2.}$ If we consider a wave-function that is localized over one minima, which is basically the linear combination of the two wave-functions (shown in red and blue), then the contributions to the expectation value will come from both real and the complex instantons.

${\bf 3.}$ For the potential shown on the right, if we consider the localized wave-function over the left minimum[Recall from our earlier discussion that the positive energy minimum does not survive under quantum corrections.], then the expectation value of $\varphi_1$ will get contributions from both real and complex instantons. The main difficulty in this case is the explicit realization of the instantons (even the real ones are harder to determine quantitatively).

Considering the aforementioned three points, and the result in (<ref>), then justifies what we said earlier, namely that the Glauber-Sudarshan state appears from the interacting vacuum $\vert\Omega\rangle$ associated with the full potential in the system. Alternatively, restricting ourselves to Minkowski minimum (using (<ref>)), the presence of the neighboring vacua manifest themselves as the real and complex instantons in the computation of $\langle\varphi_1\rangle_\sigma$. The powerful resurgence techniques, lending eventually to the Borel-Écalle resummations, succinctly capture this. More so, if we compare the tree level result with the one coming from summing over all instantons, we get:

⟨φ_1⟩_σ, tree = ∫_k_IR^μd^11 k  α_1(k)a(k) ψ_k(x, y)

⟨φ_1⟩_σ, inst = 1g^1/α̅[∫_0^∞dS  exp(-Sg^1/α̅) 11 - f̌_max S^α̅]_P.V ∫_k_IR^μd^11k  α_1(k)a(k) ψ_k(x, y)

which are exactly the same upto an overall renormalization factor. This shows that the resurgent sum over all instantons, both real and complex, simply provides an overall effect like a wave-function renormalization!

Of course the above conclusion is based on (a) scalar field components, and (b) on a bound like (<ref>), but we expect that the presence of actual metric (or other M-theory) degrees of the freedom should not change the main conclusion drastically. In the absence of a bound like (<ref>), the permutation factors ${\bf A}_{r_1r_2...r_{{\rm N}p}}$
would make the analysis a bit non-trivial. We will not discuss this here but leave it for [8]. Note that the overall renormalization factor cannot be absorbed in the definition of $\alpha_1(k)$ because doing so will ruin the delicate vanishing of $\mathbb{T}$ in (<ref>). Thus the overall factor in (<ref>) is non-trivial and we will identify this with the inverse of the dimensionless cosmological constant. Before that however we need to determine $\rho^\ast(k)$ in (<ref>). These two points will be elaborated in the following section. Needless to say, due to the technical natures of the following section, i.e. section <ref>, the readers not acquainted with Borel resummation and Gevrey series may choose to skip it on their first reading and jump directly to section <ref>.

§ BOREL RESUMMATION AND THE SMALLNESS OF THE COSMOLOGICAL CONSTANT

The asymptotic nature of the expectation value for $\varphi_1$ is the key reason why such a series may be Borel resummed to take the form as in the second relation in (<ref>). From here one may extract the form of the four-dimensional cosmological constant, as we shall show here in details, but before that let us discuss a bit on the concept of the cosmological constant itself. We will start by first ruling out a few oft-wrongly-introduced ingredients, namely, fluxes, branes, anti-branes, O-planes and perturbative quantum effects. These have no effects on the four-dimensional positive cosmological constant. Another popular view is that the four-dimensional cosmological constant should come from summing up the zero point energies of the bosonic and the fermionic degrees of freedom. This is not true despite the popularity of such an idea. The reasoning is simple. Typically the sum of the zero point energies in a non-supersymmetric theory blows up, but one can regularize it in a Lorentz invariant way (see for example [39]) and get a finite answer from it. The problem is $-$ taking for example a massive scalar field of mass $m$ $-$ that the result is typically $10^{120}$ times bigger than the expected answer (which is $70\%$ of the critical density $\rho_c$ of the universe[$\rho_c \approx 4.3 \times 10^{-47}~({\rm GeV})^4$.]). For massless scalar field the result vanishes[To see the behavior for a massive and a massless scalar field, we have to resort to a Lorentz-invariant framework. The zero point energy can be expressed completely in terms of a Lorentz invariant Euclidean integral in the following way:

ρ≡12∫_0^Λ d^4k_E(2π)^4log(1 + m^2k^2_E)
≈164π^2[(Λ^4 + m^4)log  m^2λ^2 + m^2Λ^2], which is exactly the one-loop contribution to the effective potential of a scalar field of mass $m$ (the tree level contribution is the standard mass term ${1\over 2} m^2 \phi^2$), and shows quartic divergence. Unfortunately this result does not satisfy
$\rho = -p$, with $p$ being the pressure in three dimensions. A more careful approach, outlined in [39] shows that:

ρ= 164π^2[m^2Λ^2 - m^4log(1 + Λ^2m^2)], and is therefore quadratically divergent. This divergence is unaffected by the space-time curvature. From the absence of a quartic divergence it appears that a free massless scalar field will have zero contribution to the vacuum energy [39].]. Thus in either case the vacuum energy doesn't appear to provide the small cosmological constant that we seek. It could be possible that once we take fermionic fields then the zero point energies could be tuned to be arbitrarily small. Unfortunately this delicate balance has not been demonstrated so far because it requires special choices for the masses of the bosonic and the fermionic fields. We will avoid this line of investigation and for the analysis that we perform here, we take a supersymmetric Minkowski background $-$ where the vacuum energy vanishes $-$ and claim that the cosmological constant appears from the non-perturbative contributions via Borel resumming a Gevrey series resulting from an expectation value over a Glauber-Sudarshan state. An explicit demonstration of the above statement is the content of the following two subsections.

§.§ Determining the cosmological constant using Borel resummation

The short discussion outlined above suggests that the concept of a four-dimensional positive cosmological constant is a subtle one. However
before we study the cosmological constant $-$ and here we will take the dimensionless one $-$ we need to find the functional form for $\rho^\ast(k)$ appearing in (<ref>) and (<ref>). There is also another issue coming from the dependence of $\bar\alpha$ in (<ref>) on the powers of the field contents $(p, q, r, s)$. The worrisome feature, as evident from figure <ref>, is that both the instanton terms: ${\rm exp}\left(-{1\over g^{1/\bar\alpha}}\right)$ and ${1\over g^{1/\bar\alpha}}{\rm exp}\left(-{1\over g^{1/\bar\alpha}}\right)$ increase as $\bar\alpha$ increases, and saturate to $e^{-1}$ when $\bar\alpha \to \infty$. Since $\bar\alpha$ is related to the number, as well as powers, of fields, the increase in figure <ref> implies that we cannot consistently truncate the interaction Lagrangian (<ref>) to any finite order.

Plot of three functions $f_1(\alpha) = {1\over g^{1/\alpha}}$ in blue; $f_2(\alpha) = {\rm exp}\left(-{1\over g^{1/\alpha}}\right)$  in orange and
$f_3(\alpha) ={1\over g^{1/\alpha}}{\rm exp}\left(-{1\over g^{1/\alpha}}\right)$ in green for $g = 0.01$ and $\alpha$ identified to $\bar\alpha$ from (<ref>). We see that as $\alpha$ increases both the orange and the green curves increase and saturate to ${1\over e}$ as $\alpha \to \infty$.

String theory saves this situation in a rather interesting way. The perturbative interactions in the low energy effective action of string theory may be classified as a series in powers of curvature, fluxes and Rarita-Schwinger fermions that involve at least one derivatives of the metric, and the three-form fluxes. For fermions, we can allow interactions of the form $\left(\overline\Psi^{\rm M} \Psi_{\rm M}\right)^{s'+2}$ or of the form $\left(\overline\Psi_{\rm M} \Gamma^{\rm MP} \Psi_{\rm P}\right)^{s'+2}$ for $s' \ge 0$ alongwith multiple derivatives as $\partial^h \varphi_4^s$ in (<ref>)[Recall that $\varphi_4$ is related to the Rarita-Schwinger fermionic condensate. See [9] for more details on the fermionic Lagrangian.]. So far we have been taking $\varphi_4$ dimensionless, but the right thing would be to give some dimension to $\varphi_4$ but keep other $\varphi_i$ dimensionless. Alternatively we can impose the constraint that $n \ge p, m \ge q, l \ge r$ and $h \ge s$ in (<ref>) to form a consistent picture from string theory. Taking $\vert c_o\vert \equiv 1$ (or absorbing it in the definition of ${\rm M}_p$), we get:

g^1/α̅ = M_p^-n+m+l+hp+q+r+s-2 ≡M_p^-α̌ ≡ǧ^α̌, where $\check\alpha= {n+m+l+h\over p+q+r+s-2}> 1$ and $\check{g} = {1\over {\rm M}_p}$. The scenario is plotted in figure <ref>. We see that the behavior now is the exact opposite of what we had in figure <ref>: the non-perturbative effects decrease as the number of fields or their powers are increased. This means that interactions like (<ref>) are amenable to truncations and the non-perturbative series coming from higher order interactions could be made convergent. What does it buy us?

Plot of three functions $f_1(\alpha) = {1\over g^{\alpha}}$ in blue; $f_2(\alpha) = {\rm exp}\left(-{1\over g^{\alpha}}\right)$  in orange and
$f_3(\alpha) ={1\over g^{\alpha}}{\rm exp}\left(-{1\over g^{\alpha}}\right)$ in green for $g = 0.1$ and $\alpha$ identified to $\check\alpha$ from (<ref>). We see that as $\alpha$ increases both the orange and the green curves decrease and go to zero as $\alpha \to \infty$.

To see the advantage of the aforementioned scaling behavior, we should go back where we left off, namely, determine the functional form for
$\rho^\ast(k)$. Our main aim is to reproduce a metric configuration of the form (<ref>), although we have so far only used scalar fields. The reasoning is as described earlier: to avoid inserting Faddeev-Popov ghosts, gauge fixing terms et cetera. On the other hand, if we know how to approach the problem with actual metric operators, then at least we expect $\langle {\bf g}_{\mu\nu}\rangle_\sigma = g_s^{-8/3}$ from (<ref>) in M-theory. With this in mind, one possibility is the following[In deriving (<ref>) we have swept a subtlety under the rug. The momentum integrals are bounded by $k_{\rm IR} < k < \mu$, and as such we have to use the Riemann-Lebesgue lemma to do the integrals. Implementing this leads to equating certain series with Hermite polynomials, which eventually gives us to the result in (<ref>). We point the readers to section 4.2 of [7] for further details on this.]:

ρ^∗(k) =  ρ^∗(k, k_0) = 113 - lim_ω→0 [k^2 + ω^2 log(π|ω|)ω^2 log k] - log k^9log k + log 2 log k, where $k^2 = {\bf k}^2 + k_0^2$ and we have taken all momenta to be dimensionless i.e. all momenta are measured with respect to ${\rm M}_p$. Since $g_s = {\rm H}(y) \sqrt{\Lambda} \vert t \vert$, and we are taking ${\rm H}(y) = 1$ for simplicity, the emergent metric along ${\bf R}^{2, 1}$ becomes ${1\over \Lambda^{4/3} \vert t\vert^{8/3}}$. Since we have identified the dimensionless temporal coordinate $({\rm M}_p\vert t\vert)^{-8/3}$ from $\rho^\ast({\bf k}, k_0)$ in (<ref>), the four-dimensional cosmological constant may then be identified from (<ref>) as:

Λ=  M_p^2 1g^3/4α̅[∫_0^∞dS  exp(-Sg^1/α̅) 11 - f̌_max S^α̅]^3/4_P.V, where $\bar\alpha$ and $\check{f}_{\rm max}$ are defined in (<ref>) and (<ref>) respectively; and P.V is the principal value of the integral. Somewhat remarkably this is an exact closed form expression for the cosmological constant, albeit in a set-up with scalar degrees of freedom. Looking at the integral structure of the cosmological constant, let us make the following observations on it's form.

${\bf 1.}$ The principal value integral appearing in (<ref>) is positive definite no matter what sign of $\check{f}_{\rm max}$ we choose. A detailed proof of this statement appears in section 4.5 of [7] which the interested readers may look up.

${\bf 2.}$ The principal value computation relies on the number of roots in the Borel plane (here it is a line stretching from ${\rm S} = 0$ to ${\rm S} = \infty$). The number of roots are the number of solutions to the equation ${\rm S}^{\bar\alpha} - {1\over \check{f}_{\rm max}} = 0$. Since $\bar\alpha$, defined in (<ref>), can either be an even or an odd integer, there can only be one pole on the positive real axis of the Borel plane for any choice of positive $\check{f}_{\rm max}$. Computing the integral in the near neighborhood of this pole provides it's principal value.

${\bf 3.}$ In the limit $g \to 0$, or ${\rm M}_p \to \infty$, which would be in some sense the classical limit, the principal value integral becomes 1. Thus $\Lambda$ in (<ref>) is still too big to be match up with the actual four-dimensional cosmological constant which is supposedly $10^{-120} {\rm M}_p^2$.

${\bf 4.}$ The closed form expression of the cosmological constant in (<ref>) is clearly constructed from summing over all the real and the complex instantons in the system. Thus neither the zero point energies $-$ which are in fact cancelled here due to the underlying supersymmetry $-$ nor the perturbative and the classical terms have any contributions in the construction as alluded to earlier.

Despite the success of the aforementioned analysis, and the appearance of an exact closed-form expression for the four-dimensional cosmological constant, two questions arise.

${\bf 1.}$ The cosmological constant is derived from an expression like (<ref>) by identifying the Fourier integral to the dimensionless temporal coordinate $({\rm M}_p\vert t\vert)^{-8/3}$, and the instanton sum to $\left({\Lambda\over {\rm M}_p^2}\right)^{-4/3}$. This clearly leads to an ambiguity for the choice of the cosmological constant. How do we then justify an expression like (<ref>)?

${\bf 2.}$ The instanton sum, or more appropriately the instanton integral, goes to 1 in the limit $g \to 0$. This leads to a value of the cosmological constant that is at least $10^{120}$ times bigger than the actual answer. Even if we are not near $g \to 0$ point, keeping $g< 1$ the instanton integral is never big enough to compensate for such a large discrepancy.  Why is the cosmological constant so small, and how can our analysis reproduce such a small number?

The first question can be answered by matching the boundary conditions. Recall that the Glauber-Sudarshan state description remains a valid one in the temporal domain (<ref>). At $t = -{1\over \sqrt{\Lambda}}$, $g_s = 1$, and the metric will resemble the Minkowski metric. This happens for the following choice of $\rho^\ast({\bf k}, k_0)$:

ρ^∗(k, k_0) = 2 - lim_ω, ω' →0 [ω'^2k^2 + ω^2 k_0^2 + (ωω')^2 log(π^2|ω||ω'| ) (ωω')^2 log k] - log k^9log k + log 2Λ^4/3 log k, where in the presence of (<ref>) we can make $\omega' = \omega$. From (<ref>), the only value of $\Lambda$ that satisfies the required boundary condition is (<ref>), thus fixing the form for the four-dimensional cosmological constant. Additionally, (<ref>) provides precisely the extra off-shell contribution that was alluded to earlier, and is required to get the background de Sitter configuration correctly.

The second question requires us to find the reason for the smallness of the four-dimensional cosmological constant. This is what we turn to next.

§.§ Why is the four-dimensional positive cosmological constant so small?

${\rm N}$      0     1     2     3     4     5     6    ..    10     ..     13     $j$
0     ${\bf A}_{{\rm eff}, 10}^0 k^{\nu_{10}}$      ${\bf A}_{{\rm eff}, 11}^0 k^{\nu_{11}}$
${\bf A}_{{\rm eff}, 12}^0 k^{\nu_{12}}$     ${\bf A}_{{\bf eff}, 13}^0 k^{\nu_{13}}$
1     ${\bf A}_{{\rm eff}, 10}^1 k^{\nu_{10}}$      ${\bf A}_{\ef, 11}^1 k^{\nu_{11}}$
${\bf A}_{\ef, 12}^1 k^{\nu_{12}}$     ${\bf A}_{\ef, 13}^1 k^{\nu_{13}}$
2     ${\bf A}_{\ef, 10}^2 k^{\nu_{10}}$      ${\bf A}_{\ef, 11}^2 k^{\nu_{11}}$
${\bf A}_{\ef, 12}^2 k^{\nu_{12}}$     ${\bf A}_{\ef, 13}^2 k^{\nu_{13}}$
${\bf A}_{\ef, 14}^2 k^{\nu_{14}}$     ${\bf A}_{\ef, 15}^2 k^{\nu_{15}}$    $ {\bf A}_{\ef, 16}^2 k^{\nu_{16}}$
3     ${\bf A}_{\ef, 10}^3 k^{\nu_{10}}$      ${\bf A}_{\ef, 11}^3 k^{\nu_{11}}$
${\ca A}_{\ef, 12}^3 k^{\nu_{12}}$     ${\ca A}_{\ef, 13}^3 k^{\nu_{13}}$
${\ca A}_{\ef, 14}^3 k^{\nu_{14}}$     ${\ca A}_{\ef, 15}^3 k^{\nu_{15}}$     ${\ca A}_{\ef, 16}^3 k^{\nu_{16}}$     ..     ${\ca A}_{\ef, 1, 10}^3 k^{\nu_{1, 10}}$
4     ${\ca A}_{\ef, 10}^4 k^{\nu_{10}}$      ${\ca A}_{\ef, 11}^4 k^{\nu_{11}}$
${\ca A}_{\ef, 12}^4 k^{\nu_{12}}$     ${\ca A}_{\ef, 13}^4 k^{\nu_{13}}$
${\ca A}_{\ef, 14}^4 k^{\nu_{14}}$     ${\ca A}_{\ef, 15}^4 k^{\nu_{15}}$     ${\ca A}_{\ef, 16}^4 k^{\nu_{16}}$     ..     ${\ca A}_{\ef, 1, 10}^4 k^{\nu_{1, 10}}$     ..
${\ca A}_{\ef, 1, 13}^4 k^{\nu_{1, 13}}$

..    ..     ..     ..   ..   ..   ..   ..   ..   ..    ..     ..     ..
${\rm N}$     ${\ca A}_{\ef, 10}^{\rm N} k^{\nu_{10}}$      ${\ca A}_{\ef, 11}^{\rm N} k^{\nu_{11}}$
${\ca A}_{\ef, 12}^{\rm N} k^{\nu_{12}}$     ${\ca A}_{\ef, 13}^{\rm N} k^{\nu_{13}}$
${\ca A}_{\ef, 14}^{\rm N} k^{\nu_{14}}$     ${\ca A}_{\ef, 15}^{\rm N} k^{\nu_{15}}$     ${\ca A}_{\ef, 16}^{\rm N} k^{\nu_{16}}$     ..     ${\ca A}_{\ef, 1, 10}^{\rm N} k^{\nu_{1, 10}}$     ..
${\ca A}_{\ef, 1, 13}^{\rm N} k^{\nu_{1, 13}}$     ..

[]Proliferation of the terms (first four are explicitly shown) contributing to expectation value of $\varphi_1$ with $p = 7$. Notice the columns are completely filled till $j = 3$, beyond which there are gaps in the representation of ${\ca A}_{\ef, 1j}$ for $j > 3$. To preserve the brevity, we have avoided showing the factors ${\cal A}_1^{\rm N}$ and $\Delta(j)$.

Finding the reason for the smallness of the four-dimensional cosmological constant is more complicated and we will provide a detailed answer in our upcoming paper [8]. Here we will simply sketch the main points. As before we will continue with the scalar fields and start with the action (<ref>). Let us also concentrate only on the sector with $\varphi_1$ field components. The dominant contribution has already been given in (<ref>), and we represent it here as:

[1,  1,  1,  1, ....,  1_p-1 fields,  2]_φ_1   ≡
φ_1(k_1)  φ_1(k_2)  φ_1(k_3) ...  φ_1(k_p-1) [1pt][7pt]φ_1(k) φ_1_p fields  +  1 source(k), where such a term grows as $p!$, but since the suppression factor is also $p!$, the growth is 1. The factor of “1” appearing within bracket on the LHS of (<ref>) represents the number of fields with a given momentum that is unequal to the other $p-1$ choices of momenta. This means “2” should correspond to two field components having the same momenta, and therefore contracted according to the rules mentioned earlier. In this language, the amplitude coming from an action like (<ref>), at first order in $g$, can be expressed as:

A ≡∑_j = 0^⌊p -12⌋A_1j = gp!∑_j = 0^⌊p -12⌋ [1,  1,   1,..., 1,  2,  2,  2, ....,  2_j,  2]_φ_1   +   permutations  ,which can be easily inferred from the binomial expansion discussed in section <ref>, with the subscript $j$ referring to the number and distributions of “2” in the amplitude, and $\lfloor x\rfloor$ denoting greatest integer less than or equal to $x$. Such an amplitude has a piece that is not suppressed by the volume factor ${\rm V}$ and therefore is not subdominant and should contribute to the cosmological constant that we computed earlier with only ${\bf A}_{10}$, i.e. with $j = 0$ in (<ref>), part of the amplitude. How does this reduce the value of $\Lambda$ in (<ref>)?

To see, we will have to go to the ${\rm N}$-th order in the amplitude. This is quite a non-trivial exercise so, as mentioned earlier, we will only sketch the main points. Again, details may be inferred from [8]. At the ${\rm N}$-th order the amplitude becomes:

A^N ≡∑_j = 0^⌊Np -12⌋(A^N)_1j = g^N(p!)^N∑_j = 0^⌊Np -12⌋ [1,  1,   1,..., 1_Np - 2j -1,  2,  2,  2, ....,  2_j,  2]_φ_1   +   permutations  ,which works well when $j = 0$ because $\left({\bf A}^{\rm N}\right)_{10} = \left({\bf A}_{10}\right)^{\rm N}$, and therefore Borel resummation over ${\rm N}$ can be performed. This is of course what we did earlier to get (<ref>). For $j > 0$, this is clearly not the case, which is first of the many difficulties we face once we go to more complicated nodal diagrams. There is also the denominator of the path-integral that should depend on $j$, plus we haven't shown the source momentum explicitly in (<ref>). In fact now we expect:

NumDen ≡ND =  [1]_φ_1  +   ∫d^11 k∑_N = 1^∞g^N(p!)^N∑_j = 0^⌊Np -12⌋ ∑_m_1j(N)[1,  1,   1,..., 1_Np - 2j -1,  2,  2,  2, ....,  2_j,  2]_φ_1k^m_1j(N)   +   permutations 1  +  ∑_N = 1^∞g^N(p!)^N∑_j = 0^⌊Np2⌋ [1,  1,   1,..., 1_Np - 2j,  2,  2,  2, ....,  2_j]_φ_1   +   permutations,
showing clearly the difference between the integrals on the numerator and the denominator, with $\bigodot$ identifying the source momentum with $k$ expressed using the permutation indices $m_{1j}({\rm N})$. One may then sum over the permutation indices $m_{1j}({\rm N})$, and then integrate over the source momentum over the IR regime $k_{\rm IR} < k < \mu$.
These momentum factors,
stemming from $\partial^n$ in (<ref>), complicate the source term integral as evident from the appearance of
$k^{\rho^\ast(k)}$ in (<ref>) for the case with $j = 0$. We can generalize this by imposing the following bound on the ${\rm N}$-th order amplitude:

∑_m_1j(N) (A^N)_1j(m_ij(N))
k^m_1j(N)D   ≤  (pN)!2^j'(p!)^N  (A_eff, 1j')^N k^ν_1j'(k), for all values of ${\rm N}$ and ${\cal D} \equiv {\rm Den}$ in (<ref>). Here we have expressed the binomial factor at ${\rm N}$-th order by $m_{1j}({\rm N}) \in \mathbb{Z}_+$, and therefore the sum is over $m_{1j}({\rm N})$ with $k$ being the source momentum. Clearly $j' \le j$, and for $j = 0$, $j' = 0$ with $\nu_{10}(k) \equiv \rho^\ast(k)$.
(The inequality in (<ref>) is necessary because the denominator ${\cal D}$ is in general not normalized to identity.) We can see that the bound (<ref>) is useful in multiple ways.

${\bf 1.}$ It has managed to control the binomial growth in a specific way. We no longer have to worry about the various binomial factors emerging from the powers of the derivatives or momenta, in the language of the Fourier components.

${\bf 2.}$ The sector associated with $\varphi_2, .., \varphi_4$ can be thought of as been included in ${\bf A}_{{\rm eff}, 1j'}$, provided we replace
${(p{\rm N})!\over (p!)^{\rm N}}$ by the term replacing ${\rm N}!$ in the first line of (<ref>). The bound (<ref>) can then be rewritten to take the following form:

(A^N)_1j   ≡  ∑_m_1j(N) (A^N)_1j(m_ij(N))
k^m_1j(N)D   ≤  Δ(j')A_1^N (α̅N)!  (A_eff, 1j')^N k^ν_1j'(k), where ${\cal A}_1$ is defined on the second line of (<ref>), $\Delta(j')$ is the degeneracy[We can generalize this to
$\Delta(j', \bar{p})$ for a given value of $j'$ and $\bar{p} \equiv (p, q, r, s)$, but will not do so here.] for a given value of $j'$ (which is a generalization of $2^{-j'}$ degeneracy we saw in (<ref>)),
${\cal D}$ is the denominator from (<ref>), and $\bar\alpha$ is as in (<ref>).
For $j = j' = 0$ this is exactly what gave us the required Gevrey growth, and now we see that we can extend it to include $j > 0$ case too.

${\bf 3.}$ It has also resolved the conundrum associated with raising the amplitude (<ref>) to ${\rm N}$-th power in (<ref>). Now instead of looking for $1j$-th component of ${\bf A}^{\rm N}$, we can look at ${\rm N}$-th power of ${\bf A}_{{\rm eff}, 1j}$. This is now in the same league as $({\bf A}_{10})^{\rm N}$, implying that the Borel resummation of the corresponding Gevrey series may be performed efficiently.

To proceed with the a bound like (<ref>), it will be instructive to give one concrete example. Let us consider $p = 7$ in (<ref>), which would correspond to an interaction of the form
$\partial^n\varphi_1^7$, where from our earlier considerations, $n \ge 7$. In the language of ${\ca A}_{\ef, 1j}$ we will have components ${\ca A}_{\ef, 10}$ till ${\ca A}_{\ef, 13}$, so that the source can attach appropriately to the last node. In other words the bound (<ref>) may alternatively be expressed as:

∑_j = 0^3 (A^1)_1j  ≤  A_1^1 (1α̅)! (Δ(0)A^1_, 10 k^ν_10 + Δ(1)A^1_, 11k^ν_11 + Δ(2) A^1_, 12k^ν_12 + Δ(3)A^1_, 13 k^ν_13)

∑_j = 0^6 (A^2)_1j   ≤  A_1^2 (2α̅)! (Δ(0)A^2_, 10 k^ν_10 + Δ(1)A^2_, 11 k^ν_11 + Δ(2)A^2_, 12 k^ν_12 + Δ(3)A^2_, 13 k^ν_13 + ... + Δ(6) A^2_, 16 k^ν_16)

∑_j = 0^10 (A^3)_1j   ≤  A_1^3 (3α̅)! (Δ(0)A^3_, 10 k^ν_10 + Δ(1) A^3_, 11 k^ν_11 + Δ(2) A^3_, 12 k^ν_12 + Δ(3)A^3_, 13 k^ν_13 + ... + Δ(10)A^3_, 1, 10 k^ν_1, 10)

where we show the series upto ${\rm N} = 3$. In writing (<ref>) we have assumed $j' = j$ which, while a little simplistic, works for the example here. (As mentioned earlier, typically $j' < j$ and becomes equal for $j = 0$. We will however not dwell on these subtle nuances to avoid further complicating the story. Detailed study of this will appear in [8], and we direct the readers to that reference.) With this in mind, at ${\rm N}$-th order, we expect:

∑_j = 0^⌊pN-12⌋ (A^N)_1j   ≤  A_1^N (Nα̅)! Δ(j')_j'→j(A^N_, 10 k^ν_10 + A^N_, 11 k^ν_11 + A^N_, 12 k^ν_12 + A^N_, 13 k^ν_13 + ... + A^N_, 1, ⌊pN-12⌋ k^ν_1, ⌊pN-12⌋),

with $p = 7$, which clearly depicts both the factorial and the polynomial growths. (The operation $\underset{j' \to j}{\bigodot}$, used earlier in (<ref>), replaces the $j'$ in $\Delta(j')$ with the corresponding value of $j$ from $\nu_{1j}$.)
The situation is shown in Table <ref>, where we see that up to $j = 3$ the columns are all completely filled. Beyond $j = 3$, the representation of ${\ca A}_{\ef, 1j}$ will have gaps coming from (<ref>) and (<ref>). These gaps ruin the nice summations that we can have for all the columns beyond $j = 3$. The summing over a column in Table <ref>, due to the corresponding factorial growth, is precisely the Borel resummation. For the $j = 0$ column, the answer is what we had in (<ref>), which led to the expression for $\Lambda$ in (<ref>). Now that there are gaps in the columns in Table <ref>, summation over all the columns $\lfloor {p-1\over 2}\rfloor < j < \infty$ with $j \in \mathbb{Z}_+$ cannot be done. How should we take care of this subtlety?

${\rm N}$      0     1     2     3     4     5     6    ..    10     ..     13     $j$
0     ${\ca A}_{\ef, 10}^0 k^{\nu_{10}}$      ${\ca A}_{\ef, 11}^0 k^{\nu_{11}}$
${\ca A}_{\ef, 12}^0 k^{\nu_{12}}$     ${\ca A}_{\ef, 13}^0 k^{\nu_{13}}$
${\textcolor{red}{{\ca A}_{\ef, 14}^{0} k^{\nu_{14}}}}$     ${\textcolor{red}{{\ca A}_{\ef, 15}^{0} k^{\nu_{15}}}}$     ${\textcolor{red}{{\ca A}_{\ef, 16}^{0} k^{\nu_{16}}}}$     ..     ${\textcolor{red}{{\ca A}_{\ef, 1, 10}^{0} k^{\nu_{1, 10}}}}$     ..
${\textcolor{red}{{\ca A}_{\ef, 1, 13}^{0} k^{\nu_{1, 13}}}}$     ..
1     ${\ca A}_{\ef, 10}^1 k^{\nu_{10}}$      ${\ca A}_{\ef, 11}^1 k^{\nu_{11}}$
${\ca A}_{\ef, 12}^1 k^{\nu_{12}}$     ${\ca A}_{\ef, 13}^1 k^{\nu_{13}}$
${\textcolor{red}{{\ca A}_{\ef, 14}^{1} k^{\nu_{14}}}}$     ${\textcolor{red}{{\ca A}_{\ef, 15}^{1} k^{\nu_{15}}}}$     ${\textcolor{red}{{\ca A}_{\ef, 16}^{1} k^{\nu_{16}}}}$     ..     ${\textcolor{red}{{\ca A}_{\ef, 1, 10}^{1} k^{\nu_{1, 10}}}}$     ..
${\textcolor{red}{{\ca A}_{\ef, 1, 13}^{1} k^{\nu_{1, 13}}}}$     ..
2     ${\ca A}_{\ef, 10}^2 k^{\nu_{10}}$      ${\ca A}_{\ef, 11}^2 k^{\nu_{11}}$
${\ca A}_{\ef, 12}^2 k^{\nu_{12}}$     ${\ca A}_{\ef, 13}^2 k^{\nu_{13}}$
${\ca A}_{\ef, 14}^2 k^{\nu_{14}}$     ${\ca A}_{\ef, 15}^2 k^{\nu_{15}}$    $ {\ca A}_{\ef, 16}^2 k^{\nu_{16}}$
..     ${\textcolor{red}{{\ca A}_{\ef, 1, 10}^{2} k^{\nu_{1, 10}}}}$     ..
${\textcolor{red}{{\ca A}_{\ef, 1, 13}^{2} k^{\nu_{1, 13}}}}$     ..
3     ${\ca A}_{\ef, 10}^3 k^{\nu_{10}}$      ${\ca A}_{\ef, 11}^3 k^{\nu_{11}}$
${\ca A}_{\ef, 12}^3 k^{\nu_{12}}$     ${\ca A}_{\ef, 13}^3 k^{\nu_{13}}$
${\ca A}_{\ef, 14}^3 k^{\nu_{14}}$     ${\ca A}_{\ef, 15}^3 k^{\nu_{15}}$     ${\ca A}_{\ef, 16}^3 k^{\nu_{16}}$     ..     ${\ca A}_{\ef, 1, 10}^3 k^{\nu_{1, 10}}$
${\textcolor{red}{{\ca A}_{\ef, 1, 13}^{3} k^{\nu_{1, 13}}}}$     ..
4     ${\ca A}_{\ef, 10}^4 k^{\nu_{10}}$      ${\ca A}_{\ef, 11}^4 k^{\nu_{11}}$
${\ca A}_{\ef, 12}^4 k^{\nu_{12}}$     ${\ca A}_{\ef, 13}^4 k^{\nu_{13}}$
${\ca A}_{\ef, 14}^4 k^{\nu_{14}}$     ${\ca A}_{\ef, 15}^4 k^{\nu_{15}}$     ${\ca A}_{\ef, 16}^4 k^{\nu_{16}}$     ..     ${\ca A}_{\ef, 1, 10}^4 k^{\nu_{1, 10}}$     ..
${\ca A}_{\ef, 1, 13}^4 k^{\nu_{1, 13}}$     ..

..    ..     ..     ..   ..   ..   ..   ..   ..   ..    ..     ..     ..
${\rm N}$     ${\ca A}_{\ef, 10}^{\rm N} k^{\nu_{10}}$      ${\ca A}_{\ef, 11}^{\rm N} k^{\nu_{11}}$
${\ca A}_{\ef, 12}^{\rm N} k^{\nu_{12}}$     ${\ca A}_{\ef, 13}^{\rm N} k^{\nu_{13}}$
${\ca A}_{\ef, 14}^{\rm N} k^{\nu_{14}}$     ${\ca A}_{\ef, 15}^{\rm N} k^{\nu_{15}}$     ${\ca A}_{\ef, 16}^{\rm N} k^{\nu_{16}}$     ..     ${\ca A}_{\ef, 1, 10}^{\rm N} k^{\nu_{1, 10}}$     ..
${\ca A}_{\ef, 1, 13}^{\rm N} k^{\nu_{1, 13}}$     ..

[]We can add new terms to fill in the gaps left in Table <ref>. These are shown in red. One may now Borel resum each of the series given in the columns, and then sum over the Borel resummed values. The convergence property of the second summation, as well as how to remove the added contributions, will be described in the text.

The answer is as given in  Table <ref>: we simply add the missing terms by hand! These are represented in red. Now we can easily Borel resum over all the columns from $0 \le j < \infty$. The final answer should be a summation over all the Borel resummed values. This double summation procedure is what will effectively reduce the value of the cosmological constant. However, two immediate questions arise.

${\bf 1.}$ How do we make sense of adding new terms to Table <ref>?

${\bf 2.}$ How do we know that the summation of the Borel resummed values is convergent?

Both the above questions are non-trivial, but can be answered by following a simple procedure. The procedure is to remove the added contribution to any column after we Borel resum the corresponding series. As an example consider the fourth column in Table <ref>. The first two elements are added by hand, and they could contribute a small percentage to the Borel resummed series in that column. Similarly, the contribution in the tenth column would again be a small percentage to the Borel resummed series from that column, but clearly the percentage contribution is bigger than the percentage contribution in the fourth column. As $j$ increases, the percentage contribution also increases, but not linearly. Instead the contribution takes the following form:

\begin{equation}
\label{resorworlme}
\sum_{j = 0}^\infty \sum_{{\rm N} = 0}^\infty{\cal A}_1^{\rm N} \left(\bar\alpha {\rm N}\right)! ~\Delta(j)~\left({\bf A}_{{\rm eff}, 1j}\right)^{\rm N} k^{\nu_{1j}(k)}~
\equiv \def\first{(0,0) --  (2.5,0) -- (2.5,1.5) -- (0,1.5) -- cycle}
\def\second{
(0.8, 1.5) .. controls (1.2, 0.4)  .. (2.5, 0) -- (0, 0) -- (0, 1.5) -- (0.8, 1.5)}
\begin{tikzpicture} [baseline={(0, 0.6)}]
\fill[red!60] \first;
\begin{scope}
\clip \first;
\fill[blue!50] \second;
\end{scope}
\draw[color=black] \first;
\draw \second;
\node at (0.3, 1.8) {$j = \lfloor{\frac{p - 1}{2}}\rfloor$};
\end{tikzpicture}
\end{equation}

where the RHS is a symbolic way of representing the Borel resummed columns in Table <ref>. The sum over ${\rm N}$ is the vertical Borel resummation of the columns, and the sum over $j$ is the second horizontal summation of the rows that we do after Borel resumming the columns. We will denote the picture on the RHS as Borel box, a term alluded to earlier at the beginning of section <ref> and in footnote <ref>. The coloring scheme inside the box denotes the contributions from the $p = 7$ interaction (in blue), and the contributions added by hand (in red).

The summation on the LHS of (<ref>) does not show the full story. In particular it misses the contribution from the Glauber-Sudarshan state, as well as the procedure to remove the contributions in
red appearing on the RHS of (<ref>). Both of these may be accommodated in the summation scheme in the following suggestive way:

⟨φ_1⟩_σ     ≤𝕋 +  1V ∫_k_IR^μd^11k (1 + ∑_j = 1^⌊p-12⌋ Δ(j) +
∑_j = 1 + ⌊p-12⌋^∞exp(-|G(j, p)|) Δ(j))

exp(-Sg^1/α̅) k^ν_1j1 - S^α̅A_1A_eff, 1j]_P. V (α_1^2(k)a^2(k) + 12a(k))  ψ_k(x, y)
+ O(1V^2),

where $\Delta(j)$ is the degeneracy of various terms in the binomial expansion for a given value of $j$ and $p$ from (<ref>). These may be easily worked out by simply looking at the growth of the terms (which are the ones accompanying the factorial and the polynomial growths). The important term however is the exponentially decaying factor ${\rm exp}\left(-\vert{\rm G}(j, p)\vert\right)$, controlled by a monotonically increasing function ${\rm G}(j, p)$. For large $j$, ${\rm G}(j, p)$ would be large and consequently it would reduce the contributions from Borel resummation in the column $j$. When $j \to \infty$, ${\rm G}(j, p) \to \infty$, and therefore there would be no contribution[We expect ${\rm G}(j, p)$ to be independent of $p$ and only depend on the profile separating the blue and the red regimes in the Borel box of (<ref>). However we keep the $p$ dependence for genericity.]. This is of course what we expect from the color scheme on the RHS of (<ref>).

The result (<ref>) is still not ready for synthesis. In particular we are lacking a precise way to define $\mathbb{T}$, the tree-level contribution. Such a definition should simplify the second line involving $k$ dependence to bring it in a more manageable form. This step however is more non-trivial, because the tree-level result is a bit unwieldy and takes the following form:

𝕋 ≡∫_k_IR^μd^11 k[α_1(k)a(k) - (1 + ∑_j = 1^⌊p-12⌋ Δ(j) )
k^ν_1jV(α_1^2(k)a^2(k) + 12a(k))] ψ_k(x, y),
which may be compared to (<ref>). Note that the summation over $j$ in (<ref>) does not extend to $j \to \infty$. It is easy to see from Table <ref> why this is the case. We need the quadratic pieces to efficiently perform the Borel resummation only in the range $0 \le j \le \lfloor{p-1\over 2}\rfloor$ and therefore they appear with minus signs in the tree-level result (<ref>). For $1 + \lfloor{p-1\over 2}\rfloor \le j < \infty$, this is no longer a requirement as we are anyway adding the ${\rm N}= 0$ pieces by hand. While this helps us to write $\mathbb{T}$ efficiently, it also implies that
due to the $j$ dependence in $\Delta(j)$ and $\nu_{1j}$, one cannot express the quadratic piece of $\alpha_1(k)$ in terms of the linear piece so efficiently. In (<ref>) we managed this by resorting to $\rho^\ast(k)$ in (<ref>). Can we do this here?

Unfortunately the story is more complicated and the simplification resulting from (<ref>) cannot be repeated here. But all is not lost, because we can adapt a different strategy. The strategy would be to look for the maximum value of $\nu_{1j}$ to get a bound on the quadratic piece from the vanishing of $\mathbb{T}$ in (<ref>). In other words, we define:

ν^∗(k) ≡max(ν_1j, ∀j) =
max(ν_10,  ν_11,  ν_12, ...,  ν_1⌊p-12⌋, ...,  ν_1∞), which makes $k^{\nu^\ast(k)}$ the least dominant one because the dimensionless $k$ satisfy $k < 1$ in the range
$k_{\rm IR} < k < \mu < 1$ as they are all measured with respect to ${\rm M}_p$ (see [15], [16], [7] for a discussion on the scales appearing in the problem). This means the bound coming from the vanishing of $\mathbb{T}$ in (<ref>) will be the following:

∫_k_IR^μd^11 k
k^ν^∗(k)V(α_1^2(k)a^2(k) + 12a(k)) ψ_k(x, y)   ≤
(1 + ∑_j = 1^⌊p-12⌋ Δ(j))^-1
∫_k_IR^μd^11 k  α_1(k)a(k) ψ_k(x, y),
which works perfectly because of our choice of $\nu^\ast(k)$ from (<ref>). The bound on the expectation value, i.e. $\langle\varphi_1\rangle_\sigma$ in (<ref>), remains unaffected by the choice of $\nu^\ast(k)$, and therefore plugging (<ref>) in (<ref>) leads to the following alternative expression:

⟨φ_1⟩_σ        ≤  (1 + ∑_j = 1^⌊p-12⌋ Δ(j) +
∑_j = 1 + ⌊p-12⌋^∞exp(-|G(j, p)|) Δ(j)) (
1 + ∑_j = 1^⌊p-12⌋ Δ(j))^-1

×   1g^1/α̅[∫_0^∞dS
exp(-Sg^1/α̅) 1 1 - S^α̅A_1A_eff, 1j]_P. V
∫_k_IR^μd^11k    α_1(k)a(k) ψ_k(x, y)
+ O(1V^2),

which not only provides the bound for the expectation value, but also cleanly separates $j$ dependent summation from the $k$ dependent integral. One should also compare (<ref>) with (<ref>), the latter appearing from the $j = 0$ sector of full construction as seen from Table <ref>. Using similar arguments from the boundary condition $-$ which led us to a closed form expression for the four-dimensional cosmological constant in (<ref>) $-$ now leads to the following modified expression for the cosmological constant:

Λ^4/3   =   M_p^8/3(
1 + ∑_j = 1^⌊p-12⌋ Δ(j)) 1g^1/α̅(1 + ∑_j = 1^⌊p-12⌋ Δ(j) +
∑_j = 1 + ⌊p-12⌋^∞exp(-|G(j, p)|) Δ(j))  [∫_0^∞dS
exp(-Sg^1/α̅) 1 1 - S^α̅A_1A_eff, 1j]_^^P. V < M_p^8/3,
which is smaller than the value of the cosmological constant we got in (<ref>)! This reduction in the value  appears from $j > 0$ contributions that were ignored in our earlier computations. Note that if ${\rm G}(j, p) = 0$, then the convergence of the series appearing in the denominator of (<ref>) may not be an issue. For example if $\Delta(j) = {1\over 2^j}$, and since:

1 + ∑_j = 1^⌊p-12⌋ Δ(j) 1 + ∑_j = 1^∞Δ(j)   =   1- (12)^1 + ⌊p-12⌋   <   1, the cosmological constant is effectively reduced by that factor. Of course,
from the analysis that we presented above, ${\rm G}(j, p)$ cannot vanish which could in fact be easily justified from Table <ref> or from the coloring scheme of the Borel box in (<ref>). But even with non-zero ${\rm G}(j, p)$, the value of $\Lambda$ would be smaller than what we got from (<ref>). Despite the aforementioned success, the closed form expression in (<ref>) is still not the full answer as we have chosen a specific value of $p = 7$ in (<ref>). To get the full answer, we will have to add in the contributions coming from all possible values of $p$ $-$ including all possible values for $(q, r, s)$ that we have ignored so far $-$ in (<ref>). This is a formidable exercise which will take us beyond the scope of this paper. We will then suffice ourselves by making the following observations.

${\bf 1.}$ For $p$ powers of the field with $n$ derivatives, we have seen earlier that $n \ge p$. To avoid generating non-localities, we can choose a scheme to shut off infinite derivatives, or infinite powers of the field. This will suggest that $p$, and in turn $n$, can be very large but not infinite.

${\bf 2.}$ Each of the $p$ powers would generate a Borel box using the analysis that we presented earlier. Including $(q, r, s)$ will increase the number of Borel boxes considerably.
Thus we will have a large, but not infinite, collection of Borel boxes contributing to the path integral analysis for the expectation value $\langle\varphi_1\rangle_\sigma$.

${\bf 3.}$ The expression for the four-dimensional cosmological constant $\Lambda$ from all the Borel boxes will take a form like (<ref>), except that the denominator will be much bigger than the numerator $-$ but again not infinite. This will effectively lower the cosmological constant quite a bit from what we have in (<ref>).

Our above considerations provide a strong reason why the four-dimensional cosmological constant $\Lambda$ can be so small, despite using only scalar degrees of freedom to mimic gravitational and other on-shell degrees of freedom. However the actual demonstration of the aforementioned three points is a pretty non-trivial exercise, which we shall present in our upcoming work [8].

§.§ What if the four-dimensional cosmological constant slowly varies with time?

Our analysis in the precious section shows the possibility that the four-dimensional cosmological constant could in principle be made small. One could also entertain the following possibility: what if the small and positive cosmological constant also varies slowly with respect to time? This is a curious scenario [10], but before we start we should point out that even with this set-up we could not realize such a scenario as a vacuum solution. The reasoning remains the same as before related to the problems with Wilsonian integration, quantization of strings over a temporally varying background and other similar issues. In fact the situation may be more severe now because of temporal variation of the cosmological constant $\Lambda$, albeit very slowly. In literature, these kind of cosmologies are termed as the quintessence cosmologies and their realization in string theory so far has not been very successful. In the following we will argue that a class of such models may be realized from our Glauber-Sudarshan state construction.

We will start by first specifying the emergent metric components from the eleven-dimensional set-up[It might be useful for readers, who are not acquainted with the Schwinger-Dyson equations or their applications to the Glauber-Sudarshan states, to skip this section on their first reading and come back only after learning the derivation in (<ref>) from an effective action (<ref>).] with the topology of the internal eight-manifold to be ${\cal M}_4 \times {\cal M}_2 \times {\mathbb{T}^2\over {\cal G}}$ such that $y^m \in {\cal M}_4$ and $y^\rho \in {\cal M}_2$. These metric components, as defined earlier in
say (<ref>), as well as the G-fluxes[We are ignoring the fermionic terms to keep the analysis simple. A more detailed study will be presented in [9].] are expressed as expectation values over the Glauber-Sudarshan state and for the present case we will specify them in the following way:

⟨g_11, 11⟩_σ≡g_11, 11(x, y; g_s) = g_11, 11(x, y) (g_sH(y)H_o(x))^43

⟨g_33⟩_σ≡g_33(x, y; g_s) = ∑_k = 0^∞h_k δ_33 (g_sH(y)H_o(x))^43 + 2k |f_k(t)|3

⟨g_μν⟩_σ≡g_μν(x, y; g_s) = ∑_k = 0^∞h_k η_μν(g_sH(y)H_o(x))^-83 + 2k|f_k(t)|3

⟨g_κρ⟩_σ≡g_κρ(x, y; g_s) =
∑_k = 0^∞g^(k)_κρ(x, y) (g_sH(y)H_o(x))^-23 + β(t) + 2k|β_k(t) |3

⟨g_mn⟩_σ≡g_mn(x, y; g_s) =
∑_k = 0^∞g^(k)_mn(x, y) (g_sH(y)H_o(x))^-23 + α(t) + 2k|α_k(t)|3

⟨G_ABCD⟩_σ≡G_ABCD(x, y; g_s) = ∑_k = 0^∞G^(k)_ABCD(x, y) (g_sH(y)H_o(x))^l_AB^CD + 2k|g_k(t)|3

where $({\rm A, B}) \in {\bf R}^{2, 1} \times {\cal M}_4 \times {\cal M}_2 \times {\mathbb{T}^2\over {\cal G}}$, $l_{\rm AB}^{\rm CD}$ is the dominant scaling for the emergent flux components; and
$(\widetilde{f}_k(t), f_k(t), \alpha(t), \alpha_k(t), \beta(t), \beta_k(t), g_k(t))$ are the set of functions that are arranged so that they would solve the Schwinger-Dyson equations (<ref>) as well as equations coming from duality constraints.
The latter would typically involve dualities from M-theory to type IIB as well as the constraint to keep the four-dimensional Newton's constant in the type IIB side time-independent. Note that $g_s$ continues to be the dual type IIA coupling, as evidenced from the form of $\langle {\bf g}_{11, 11}\rangle_\sigma$, but its connection to the conformal temporal coordinate $t$ needs to be determined. We will derive this soon, but before we do so we should point out that, to maintain the isometry in the type IIB side, $\widetilde{f}_k(t)$ and $f_k(t)$ cannot be independent of each other. In fact they are related by the following polynomial equation:

∑_k, l h_k h_l (g_sH(y) H_o(x))^23(k|f̌_k(t)|+ l|f̂_l(t)|) = 1, where $\check{f}_k(t) = f_k(t) + {\cal O}({\rm M}_p, g_s)$ and
$\hat{f}_l(t) = \widetilde{f}_l(t) + {\cal O}({\rm M}_p, g_s)$ with the corrections coming from the duality rules. Note the presence of modulus signs on $\hat{f}_l(t)$ in (<ref>) and $\widetilde{f}_k(t)$ in (<ref>). This ensures the sub-dominant behavior and subsequent duality to type IIB which are important to solve (<ref>) in the presence of the ${\cal O}({\rm M}_p, g_s)$ corrections. (Detail of these corrections are not necessary, so we will not elaborate anymore here and leave them for [9].)
There is also another subtlety now: the type IIB coupling is no longer at the constant coupling point of F-theory. Due to choice of (<ref>), and consequently (<ref>), the IIB coupling $g_s^{(b)}$ takes the form:

(g_s^(b))^2 = ∑_k = 0^∞h_k (g_sH(y) H_o(x))^2k|f̌_k(t)|3, with $h_0 = 1$, and is therefore slowly varying with time. Depending on the signs of $h_k$, the coupling could be vary with time slightly above or below the constant coupling point of F-theory. This temporal variation, albeit small, is precisely due to the temporal variation of the M-theory torus as one of the direction of the torus determines the dual IIA coupling and the other direction fixes the dual IIB coupling. Once we know the IIB string coupling, and after combining (<ref>), (<ref>) and (<ref>) together, the emergent metric $\langle {\bf g}^{(b)}_{\rm AB}\rangle_\omega$, with $\vert\omega\rangle$ being the Glauber-Sudarshan state in the IIB side, now takes the following form:

ds^2     ≡⟨g^(b)_AB⟩_ωdY^A dY^B =
∑_k=0^∞h_k (g_sH(y)H_o(x))^-2 + 2k|f̌_k(t)|3(-dt^2 + ∑_i = 1^3 dx_i^2)

+  ∑_k=0^∞g^(k)_mn(x, y)
(g_sH(y)H_o(x))^ α̌(t) + 2k|α̌_k(t)|3 + ∑_k=0^∞g^(k)_ρσ(x, y)
(g_sH(y)H_o(x))^β̌(t) + 2k|β̌_k(t)|3,

which is clearly different from what we had in (<ref>). The split of the base ${\cal M}_6$ to ${\cal M}_4 \times {\cal M}_2$ is useful because we can easily make the volume of the internal manifold, and consequently the emergent four-dimensional Newton's constant, time-independent by relating $\check{\alpha}(t)$ with $\check{\beta}(t)$ and also with $\check{\alpha}_k(t)$ and $\check{\beta}_k(t)$, although this may be relaxed now. (In any case any temporal dependence will be sub-dominant as we shall see below.) A special case is when $\check{\alpha}(t) = \check{\beta}(t) = 0$ with a certain relation between $\check{\alpha}_k(t)$ and $\check{\beta}_k(t)$ to keep the volume time-independent.

The above emergent metric configuration (<ref>) in type IIB would make sense if the original metric components from (<ref>) solve the Schwinger-Dyson equations (<ref>). This is a non-trivial exercise and in section <ref> we will argue that this is indeed possible. Assuming this to be the case, the pertinent question is the following: how does our little exercise above show that the four-dimensional cosmological constant is slowly varying with time? There are in fact two ways to show this. The first one is relatively straightforward and may be easily ascertained by looking at the form of the IIB metric (<ref>). We demand:

∑_k=0^∞h_k (g_sH(y)H_o(x))^-2 + 2k|f̌_k(t)|3 = 1Λ(t)|t|^2 = (1 + Λ̌(t)Λ)^-1Λ|t|^2, where $\Lambda(t) \equiv \Lambda + \check{\Lambda}(t)$, with $\check{\Lambda}(t)$ is a very slowly varying function. (In fact because of this very slow variation we could express the RHS of (<ref>) in the way shown there as an almost flat-slicing but with varying $\Lambda(t)$.) It should be clear from (<ref>) that, for $\check{f}_k(t) = 0$, the correction $\check{\Lambda}(t) = 0$ and $\left({g_s\over {\rm H}(y){\rm H}_o({\bf x})}\right) = \sqrt{\Lambda} \vert t\vert$. Switching on $\check{f}_k(t)$ is equivalent to switching on $\check{\Lambda}(t)$, and since $\left({g_s\over {\rm H}(y){\rm H}_o({\bf x})}\right) < 1$, this would make the effective cosmological constant to monotonically decrease with time. An interesting solution to (<ref>) is to take $\left({g_s\over {\rm H}(y){\rm H}_o({\bf x})}\right) = \sqrt{\Lambda} \vert t\vert$ and to solve the following equation:

∑_k = 0^∞h_k (√(Λ)|t |)^2k|f̌_k(t)|3      = 1 + h_1 (√(Λ)|t |)^2|f̌_1(t)|3 + h_2 (√(Λ)|t |)^4|f̌_2(t)|3 + ... =(1 + Λ̌(t)Λ)^-1

= 1 - Λ̌(t)Λ + ( Λ̌(t)Λ)^2 - ( Λ̌(t)Λ)^3 +
( Λ̌(t)Λ)^4 + ... = ∑_n = 0^∞(-1)^n( Λ̌(t)Λ)^n

where the Taylor expansion in the second line is only justified if
$\check{\Lambda}(t) << \Lambda$ within the temporal domain (<ref>).  This is also the case where $\left({g_s\over {\rm H}(y){\rm H}_o({\bf x})}\right) < 1$ with the aforementioned choice for $g_s$. The equation (<ref>) is easily solved once we choose $h_k = (-1)^k$ which fixes the following functional form for $\check{f}_k(t)$ in terms of $\check{\Lambda}(t)$:

|f̌_k(t)|= |f̌_1(t)|= 32 [log(Λ̌(t)Λ) log(√(Λ)|t|)],     ∀k ≥1. Since the conformal time is bounded by (<ref>) and $\check{\Lambda}(t) << \Lambda$ within the bound (<ref>), the RHS is a positive definite quantity consistent with the modulus of $\check{f}_k(t)$. In retrospect, the series $\left({g_s\over {\rm H}(y){\rm H}_o({\bf x})}\right)^{2\vert\check{f}_1(t)\vert\over 3}$ got by imposing (<ref>) shouldn't appear that surprising in the light of similar series, but with $\check{f}_1(t) = 1$, studied in [14, 15, 16] for the internal manifold. The latter was shown to be consistent with the Schwinger-Dyson equations so the present case, with slight modification from the introduction of $\check{f}_1(t)$  along the space-time directions, should not pose any insurmountable difficulties. A possible solution, directly from M-theory, would be to define:

f_k(t) = f_k(t) = α_k(t) = β_k(t) = g_k = f_1(t),
α(t) = β(t) = 0, in (<ref>). Such a choice is clearly consistent with (<ref>) and, since $\vert \check{f}_1(t) - f_1(t)\vert = {\cal O}(g_s, {\rm M}_p)$, we expect $\check{f}_1(0) = f_1(0)$ at late time as $t \to 0$ (or $g_s \to 0$). Comparing it with (<ref>), $\hat{\Lambda}(t)$ also needs to decrease monotonically for this limit to make sense so that the effective cosmological constant varies from $\Lambda + \check{\Lambda}\left(-{1\over \sqrt{\Lambda}}\right)$ to $\Lambda$ in the temporal domain (<ref>). This is clearly consistent with what we predicted earlier, so now we need to figure out the limiting value on the RHS of (<ref>) at late time. This may be determined in the following way. At late time if the parameters in (<ref>) approach zero as $\sqrt{\Lambda} \vert t\vert \to \epsilon$ and $\hat{\Lambda}(t) \to \Lambda \epsilon^a$ for $\epsilon \to 0^+$ and $a > 0$, then $\check{f}_1(0) \to {3a\over 2}$ implying $f_1(0) = {3a\over 2}$. Additionally,
to see whether (<ref>) solves the Schwinger-Dyson equations, we can first define an effective coupling
$ \check{\bf g}_s \equiv  \left({g_s\over {\rm H}(y){\rm H}_o({\bf x})}\right)^{\vert f_1(t)\vert}$. Plugging this in (<ref>), one may check that every term of the equation there scales with respect to $g_s$ as a dominant and a sub-dominant piece in the following way:

∑_k = 0^∞a_k(x, y) (g_sH(y)H_o(x))^dominant ×ǧ^2k3  +  log corrections, with a functional form ${\bf a}_k({\bf x}, y)$ that may be easily specified for each term. There are also log corrections that are a bit non-trivial to derive but are nevertheless tractable. These corrections tend to be very small, so we won't need them here, and the full derivation of them will be presented in [9]. Therefore ignoring these corrections and equating the dominant scalings over all the terms
in (<ref>) would specify the dominant quantum corrections needed to solve the Schwinger-Dyson equation (<ref>)[There is a bit of a subtlety here because the dominant scalings may themselves be functions of $t$ $-$ and consequently of $g_s$ $-$ so this matching needs to be done carefully. We will discuss this briefly in section <ref> and leave a more detailed study for [9].]. After which one can equate the subdominant scalings over all the terms to solve the system to all orders in $\check{\bf g}_s$. The strategy to solve the EOM with the log corrections will be discussed in [9].
In fact in sections <ref> and <ref> we will take a slightly simplified version with $f_k(t) = \widetilde{f}_k(t) = \alpha(t) = \beta(t) = 0$ and $\alpha_k(t) = \beta_k(t) = g_k(t) = 1$ to show the consistency of the aforementioned construction.

One of the advantage of the solution (<ref>) is that the both the definition of the dual IIA coupling $g_s$ and the temporal domain of validity (<ref>) do not change. One could envision a different solution of $g_s$ in (<ref>) that would depend on the conformal time $t$, as well as the functional forms for $\check{\Lambda}(t)$ and
$\check{f}_k(t)$. One example would be to keep $f_1(t) = 1$ in (<ref>) where we cannot always demand $\left({g_s\over {\rm H}(y){\rm H}_o({\bf x})}\right) = \sqrt{\Lambda}\vert t\vert$ to solve
Needless to say, this is a slightly difficult exercise and is not clear whether a closed-form expression for $g_s$ exists. Such a solution will also change the temporal domain (<ref>), albeit to sub-leading order, but we will not address this any further here.

The second way to justify the slow variation of the four-dimensional cosmological constant is to go back to the derivation of the expectation value using the Borel-$\grave{\rm E}$calle resummation as shown for the scalar field case in the second relation of (<ref>). For our metric configuration this will take the following form:

⟨g_μν⟩_σ     = 1g^1/α̅[∫_0^∞dS  exp(-Sg^1/α̅) 11 - f̌_max S^α̅]_P.V ∫_k_IR^μd^11k  α(k) η_μνa(k) ψ_k(x, y)

=  ∑_k = 0^∞h_k η_μν(g_sH(y)H_o(x))^-83 + 2k|f_k(t)|3 =
∑_k = 0^∞h_k η_μν (√(Λ)t)^-83 + 2k|f̌_1(t)|3,

where all the parameters in the first line above have already been defined in section <ref>, and in the second line we have used (<ref>). It is not too hard to see that now $\Lambda$ may continue to appear from the Borel resummed integral as in (<ref>), or the modified one with the introduction of the Borel boxes from (<ref>), and the Glauber-Sudarshan state may be determined from the following Fourier integral:

∫_k_IR^μd^11k  α(k) η_μνa(k) ψ_k(x, y) = FT [(1 + Λ̌(t)Λ)^-1 (M_p |t|)^83]_Riemann-Lebesgue, where on the RHS we first take the Fourier transform, and then rewrite this using the Riemann-Lebesgue lemma to express it within the energy domain $k_{\rm IR} < k < \mu$. On the LHS, we can express $\alpha(k)$ as a series in Hermite polynomials as done in [7]. One may then impose a delta function constraint via $\alpha(k)$ or via the wave-function $\psi_k(x, y)$ ($y \in {\cal M}_4 \times {\cal M}_2$), to restrict the integral to be a function of $t$ only thus matching with the series on the RHS. More details on this will be presented elsewhere.

Therefore to summarize, it appears that we can easily accommodate small temporal variation of the cosmological constant in our model by slightly changing the Glauber-Sudarshan state as per the prescription (<ref>), (<ref>) and (<ref>). A discussion of how
a simpler version of (<ref>) solves the Schwinger-Dyson equation (<ref>) will be presented in section <ref>.

§ SCHWINGER-DYSON'S EQUATIONS, EOMS AND THE EXCITED STATES

The readers who have skipped section <ref> would be pleased to know that one can define a small and positive four-dimensional cosmological constant using the Glauber-Sudarshan states. However two issues prohibit us to declare this as a solution to the cosmological constant problem.

${\bf 1.}$ The analysis is presented completely using four set of scalar degrees of freedom instead of the actual gravitational, fluxes and other on-shell degrees of freedom.

${\bf 2.}$ Even with scalar degrees of freedom, it is not clear whether the procedure to reduce the value of the cosmological constant by introducing multiple Borel boxes would produce a value small enough to actually reproduce the oft-quoted $10^{-120}{\rm M}_p^2$ result.

Despite this our analysis, probably for the first time, has provided a reason for the smallness of the positive cosmological constant that only relies on non-perturbative effects. We will not elaborate further on this interesting direction $-$ which we leave it for our upcoming work [8] $-$ instead we will answer a pertinent question related to how these Glauber-Sudarshan states overcome the so-called swampland criteria [6]. Before that however we need to know the EOMs governing these states which in turn requires us to determine the resolution of identity that we briefly touched upon in section <ref>. These will be the contents of the following section.

§.§ Resolution of identity and EOMs for the Glauber-Sudarshan states

To study the EOM we will first go back to the form of the Glauber-Sudarshan state $ \vert\sigma\rangle$ that we discussed in (<ref>).
These states are not normalized, so $\langle\sigma\vert\sigma\rangle \ne 1$.  We can express this as the following path-integral:

⟨σ|σ⟩⟨Ω|Ω⟩    = ⟨Ω|𝔻^†𝔻|Ω⟩⟨Ω|Ω⟩ = ∫∏_i = 1^4 [Dφ_i]
exp(-S_tot(φ_1,..,φ_4)) 𝔻^†(σ, φ_1,..,φ_4)𝔻(σ,φ_1,..,φ_4)

= ∫∏_i = 1^4 [Dφ_i]
exp(-S_tot(φ_1+ ϵ(x, y), φ_2,..,φ_4)) 𝔻^†(σ, φ_1+ϵ(x, y), φ_2,..,φ_4)𝔻(σ,φ_1 + ϵ(x, y), φ_2,..,φ_4)

= ∫∏_i = 1^4 [Dφ_i]
exp(-S_tot(φ_1,.., φ_4)- ϵ(x, y) δS_totδφ_1) 𝔻^†(σ, φ_1,..,φ_4)𝔻(σ,φ_1,..,φ_4)(1 + ϵ(x, y) δδφ_1log(𝔻^†𝔻))

where in the second line we have taken a dummy variable $\varphi'_1(x, y) \equiv \varphi_1(x, y) + \epsilon(x, y)$, and in the third line we have Taylor expanded to first order in $\epsilon(x, y)$, where $\epsilon(x, y)$ is a function satisfying
$0 <\epsilon(x, y) << 1$ everywhere in eleven-dimensional space-time. It is also easy to see that ${\cal D}\varphi'_1 = {\cal D}\varphi_1$. Now comparing the first line with the third line, we easily get the following equation of motion for the on-shell field $\varphi_1$:

ϵ(x, y)∫∏_i = 1^4 [Dφ_i]
exp(-S_tot(φ_1,..,φ_4)) 𝔻^†(σ, φ_1,..,φ_4)𝔻(σ,φ_1,φ_4)(δS_totδφ_1 -
δδφ_1log(𝔻^†𝔻)) = 0,
which would imply the vanishing of the integral because $\epsilon(x, y)$ is a non-zero function everywhere. Note the relative minus sign between the two terms above coming entirely from the fact that we took an Euclidean action. Had we taken a Lorentzian action, there would have been a relative $\sqrt{-1}$ factor. We will avoid factors of $\sqrt{-1}$ in our analysis. The vanishing of (<ref>) leads to the following EOM:

⟨δS_totδφ_1⟩_σ= ⟨δδφ_1log(𝔻^†𝔻)⟩_σ, which is precisely the EOM satisfied by the Glauber-Sudarshan state. This is a Schwinger-Dyson (SD) type EOM, but differs from the usual SD equation in at least two respects.

${\bf 1.}$ It is defined over the Glauber-Sudarshan state $\vert\sigma\rangle$ instead over the interacting vacuum $\vert\Omega\rangle$.

${\bf 2.}$ It differs from the expected EOM of the form
$\left\langle{\delta{\bf S}_{\rm tot}\over \delta\varphi_1}\right\rangle_\sigma = 0$, because of the presence of a non-unitary[Had we chosen a unitary $\mathbb{D}(\sigma)$, then no EOM of the form (<ref>) would have appeared because $\langle\sigma\vert\sigma\rangle = \langle\Omega\vert\Omega\rangle = 1$. This is another crucial way the Glauber-Sudarshan states differ from the coherent states.] $\mathbb{D}(\sigma, \varphi_1,..,\varphi_4)$.

The above EOM is important to solve first to get the form of $\rho^\ast(k)$ in (<ref>), which will eventually fix the functional form for $\alpha_1(k)$ in (<ref>). Unfortunately the form of the EOM in (<ref>) is pretty unwieldy to yield the value (<ref>) for $\rho^\ast(k)$ so easily. Part of the difficulty of course lies in the form of the EOM itself, but more importantly, since:

⟨∂^mφ_i^n⟩_σ∂^m⟨φ_i⟩^n_σ,the expectation values for $\varphi_i^n$, i.e. $\langle\varphi_i^n\rangle_\sigma$, or the ones with derivatives, i.e. $\langle\partial^m\varphi_i^n\rangle_\sigma$, need to be computed for all values of $(n, m)$ in (<ref>). Recalling the fact that even for a simple computation like $\langle\varphi_1\rangle_\sigma$ involves Borel resummations of Gevrey series as well as summations over Borel boxes, going to $n > 1$ in (<ref>) will be a formidable exercise. Is there a way we can simplify (<ref>) to efficiently solve the EOMs and get the desired functional form for $\rho^\ast{k}$ as in (<ref>)? The answer is yes provided we can have a well defined notion of the resolution of identity using the Glauber-Sudarshan states. How does that help us?

Before going into the benefits of resolution of identity, let us discuss whether it is even possible to have such resolution with Glauber-Sudarshan states. We already discussed one possible way in (<ref>) using the displacement operator (<ref>). The suppression factor in the path-integral using (<ref>) is given by (<ref>). However the aforementioned analysis only works well when ${\rm Im}~\sigma > {\rm Im}~\sigma'$ in $\langle\sigma\vert\sigma'\rangle$. This is an unsatisfactory feature, although for many practical purposes the choice of ${\cal O}$ in (<ref>) suffices. Clearly for generic application, a better formalism for the resolution of identity is called for. How can we derive this?

One possibility would be to use the complete set of states for the full interacting Hamiltonian ${\bf H}(t) \equiv \int d^2{\bf x}~d^6y~d^2w ~{\bf H}_{\rm tot}(x,y, w)$, where ${\bf H}_{\rm tot}$ may be determined from the full non-perturbative completion of ${\bf S}_{\rm tot}$ (see section <ref>)[This is also the Hamiltonian that entered in (<ref>).]. We will make two assumptions to simplify the ensuing analysis: one, the interacting vacuum $\vert\Omega\rangle$ will be assumed to be non-degenerate, and two, there is an energy gap between the first excited state and the vacuum. Keeping these in mind we express the completeness condition as:

1   =   |Ω⟩⟨Ω|+ ∑_n = 1^∞c_n|n⟩⟨n |, where $c_n$ is the degeneracy of the higher excited states $\vert n \rangle$. $\vert\Omega\rangle$ is the interacting vacuum that we have been using so far $-$ which is an eigenstate of ${\bf H}$ $-$ and whose properties may be easily discerned from the non-perturbative (and non-local) completion of ${\bf S}_{\rm tot}$ in the path-integral formalism. However the properties of $\vert n\rangle$ are much harder because they are no longer harmonic oscillator states. Is there a way to avoid this subtlety and get the completeness relation for the Glauber-Sudarshan states? The answer is surprisingly yes with the aid of a few notations. We define an ordered operator $\Sigma_{{\rm L}{\rm R}}$ whose action on any operator ${\cal O}$ is:

Σ_LR O ≡[𝔻(σ;φ_i)  exp(-i H T)]_L ⊗[𝔻^†(σ; φ_i)]_R  O = 𝔻(σ; φ_i)  exp(-i H T)  O  𝔻^†(σ; φ_i),
where the first equality provides the form of the operator. The ordering of the operator action means the following: the subscript $\overrightarrow{\rm L}$ denotes action from the left and the subscript $\overleftarrow{\rm R}$ denotes action from the right on the operator ${\cal O}$ (the arrows signify the directions of the operators actions). The action of $\Sigma_{\rm LR}$ on (<ref>) then provides the following relation:

∑_σ, σ^∗  [𝔻(σ;φ_i)  exp(-i (H - E_0) T)]_L ⊗[𝔻^†(σ;φ_i)]_R ⟨σ|σ⟩  =  ∑_σ, σ^∗ |σ⟩⟨σ|⟨σ|σ⟩  +  ∑_σ,σ^∗∑_n = 1^∞c_n exp(-i(E_n - E_0)T)  𝔻(σ; φ_i)|n⟩⟨n|𝔻^†(σ; φ_i) ⟨σ|σ⟩,
where ${\bf H}$ measured over the Minkowski minimum is taken to be independent of time, and ${\rm E}_0$ and ${\rm E}_n$ denote the energies of the vacuum $\vert\Omega\rangle$ and the higher excited states $\vert n \rangle$ respectively.
It should also be clear that ${\rm E}_n > {\rm E}_0$, $\forall n$, which means $\vert \Omega \rangle$ to be the lowest energy state (i.e. the vacuum). This also means that if we take ${\rm T}$ to infinity along slightly imaginary direction, we can decouple all the higher excited states. This would immediately imply:

∑_σ, σ^∗ |σ⟩⟨σ|⟨σ|σ⟩  =   lim_T →∞(1-iϵ)∑_σ, σ^∗  [𝔻(σ;φ_i)  exp(-i (H - E_0) T)]_L ⊗[𝔻^†(σ;φ_i)]_R ⟨σ|σ⟩, with $\vert \sigma\rangle \langle\sigma\vert \equiv\underset{k}{\bigotimes}\vert\sigma(k)\rangle \langle \sigma(k)\vert$, where the operation $\underset{k}{\bigotimes}$ collects all the kets with different momenta on one side and the bras, again with different momenta, on the other side without contractions; and $\langle\sigma\vert\sigma\rangle = \prod\limits_k \langle\sigma(k)\vert \sigma(k)\rangle$, where $\prod\limits_k$ is the usual product over the momenta. The relation (<ref>) produces something akin to a resolution of identity, but not quite. One immediate issue is the choice of the ordered operator (<ref>). This is a bit unsatisfactory, so the question is whether the ordering is important or not. To see this, let us act the unordered operator on (<ref>). The result we get is:

𝔻(σ;φ_i)  exp(-i H T) 𝔻^†(σ;φ_i)    =  𝔻(σ; φ_i)  exp(-i H T) 𝔻^†(σ; φ_i)  |Ω⟩⟨Ω|+ 𝔻(σ; φ_i)  exp(-i H T) 𝔻^†(σ; φ_i) ∑_n=1^∞c_n |n ⟩⟨n|

=  𝔻(σ; φ_i)  exp(-i H T)(|Ω⟩⟨Ω|+ ∑_n = 1^∞c_n|n⟩⟨n |_1) 𝔻^†(σ; φ_i)  |Ω⟩⟨Ω|

+   𝔻(σ; φ_i)  exp(-i H T) (|Ω⟩⟨Ω|+ ∑_n = 1^∞c_n|n⟩⟨n |_1) 𝔻^†(σ; φ_i) ∑_n c_n |n ⟩⟨n|

=  exp(-iE_0T) |σ⟩⟨σ|  +  ∑_n = 1^∞c_n exp(-iE_nT)  𝔻(σ; φ_i)|n⟩⟨n|𝔻^†(σ; φ_i)

which is exactly what we had on the RHS of (<ref>), implying that the ordering of the operator is not necessary, at least for the above case. Summing over $\sigma$ and $\sigma^\ast$, and taking the limit ${\rm T} \to \infty(1 - i\epsilon)$, then leads back to (<ref>) but without any ordering. The RHS of (<ref>) deviates from the identity because both the operators $\mathbb{D}(\sigma)\mathbb{D}^\dagger(\sigma)$ and ${\rm exp}\Big(-i ({\bf H} - {\rm E}_0) {\rm T}\Big)$ as well as the expectation value $\langle\sigma\vert\sigma\rangle$ deviate from the identity (in fact this is exactly how the Glauber-Sudarshan states differ from the coherent states). This means:

∑_σ, σ^∗ |σ⟩⟨σ|⟨σ|σ⟩  =   lim_T →∞(1-iϵ)∑_σ, σ^∗  𝔻(σ; φ_1,..,φ_4)  exp(-i (H - E_0) T) 𝔻^†(σ; φ_1,..,φ_4) ⟨σ|σ⟩   =  1 + Q(φ_1,..,φ_4),
where ${\bf Q}(\varphi_1,..,\varphi_4)$, which is a function of only the on-shell fields, is another operator that may be derived from expanding the middle operator in (<ref>). The identity ${\bf 1}$ should be understood as the identity operator in the Hilbert space of the over-complete set of $\vert\sigma\rangle$. This way no extra normalization factor needs to be introduced in (<ref>). After the dust settles, the resolution of identity
for the Glauber-Sudarshan states takes the following form:

1 =  ∑_σ, σ^∗ |σ⟩⟨σ|⟨σ|σ⟩   -   Q(φ_1,..,φ_4)   =   ∫d^2σ |σ⟩⟨σ|⟨σ|σ⟩   -   Q(φ_1,..,φ_4), which elucidates the precise difference from the coherent states. For the latter ${\bf Q}(\varphi_1,..,\varphi_4)$ vanishes and $\langle\sigma\vert\sigma\rangle = 1$. In the second equality of (<ref>) we have converted the sum to an integral over the continuous parameters $\sigma$ and $\sigma^\ast$. (The way we have presented (<ref>), the integral or the summation over the Glauber-Sudarshan states appearing therein do not involve any extra function of $\sigma$ and $\sigma^\ast$.)
Looking at (<ref>) however raises an immediate question: in the limit
${\rm T}\to \infty(1-i\epsilon)$, the higher states $\vert n\rangle$ decouple faster compared to the vacuum state $\vert\Omega\rangle$, so shouldn't this also decouple the Glauber-Sudarshan state $\vert\sigma\rangle$ since it's energy is higher than the vacuum state? The answer may be presented in various ways. If the energy of the Glauber-Sudarshan state is only very slightly above the vacuum state then these states will not decouple. Such a consideration should be consistent with the fact that we expect the cosmological constant, which is the energy associated with the Glauber-Sudarshan state, to be very small. Note however that this is not a proof for the smallness of the cosmological constant because the completeness condition (<ref>) doesn't exactly involve the $\vert\sigma\rangle$ states directly, so the energy argument is only an indirect one. More appropriately, the Glauber-Sudarshan states are not eigenstates of the total Hamiltomian ${\bf H}_{\rm tot}$, and therefore they do not appear in the completeness condition (<ref>). These states are only consistently described for a very short temporal domain of
$-{1\over \sqrt{\Lambda}} < t < 0$, and as such are not decoupled by the asymptotic temporal condition of ${\rm T} \to \infty(1 - i\epsilon)$.

With this, the resolution of identity takes a consistent form that doesn't depend upon any constraints on ${\rm Im}~\sigma$. Any constraints that might have resulted earlier from using (<ref>), are removed by the inclusion of the operator ${\bf Q}(\varphi_1,..,\varphi_4)$. Two questions arise now.

${\bf 1.}$ Can we insert (<ref>) inside the expectation-value form of the EOM (<ref>) to bring it to a more manageable form?

${\bf 2.}$ The expectation-value form (<ref>) could in principle give rise to the expectation value of the action itself, i.e. $\langle {\rm S}_{\rm tot}\rangle_\sigma$. Since the latter involves Borel resummation, is the Wilsonian effective action (WEA) compatible with Borel resummation?

These two questions may seem somewhat unrelated, but we will argue that they are not. This is what we turn to next.

§.§ Compatibility of the Wilsonian action with Borel resummations and EOMs

Let us start with a simple toy model to illustrate the underlying story. We will take a single real scalar field $\varphi$, which we shall call the “on-shell” degree of freedom, and another scalar field $\phi$, that we shall denote as the “off-shell” degree of freedom. (As an example of this, consider the metric (<ref>). The on-shell degrees of freedom are the metric components ${\bf g}_{\mu\nu}, {\bf g}_{mn}$ and ${\bf g}_{ab}$, whereas the off-shell degrees of freedom are ${\bf g}_{0m}, {\bf g}_{mb}$ et cetera.  The former set is denoted by $\varphi$ and the latter set by $\phi$.) We will also not worry about momentum conservation or about other off-shell degrees of freedom. The latter will be very important once we take the actual the metric components, the fluxes and the fermionic degrees of freedom. The action we shall consider will be:

S_tot(φ, ϕ) ≡S_1(φ) + S_2(φ, ϕ) = M_p^9∫d^11 x(-φ□φ- ϕ□ϕ+ 1M_p^2k-2∑_n, m, k c_nmk {∂^2k}φ^m ϕ^n),
with $c_{200} = c_{020} = 0$ now[This should be compared to what we have been taking so far, namely $c_{200} \ne 0$ (see discussion after (<ref>)). Now allowing the off-shell degrees of freedom to be massless will have important implications.] and $\{\partial^k\}$ is the distribution of the derivatives among the two scalar fields[For example
$c_{nm2}\{\partial^2\}\varphi^m\phi^n \equiv c_{nm20} \partial^2\varphi^m \phi^n + c_{nm02}\varphi^m \partial^2 \phi^n + c_{nm11} \partial_\mu \varphi^m \partial^\mu\phi^n$. We can make further finer distribution with $(n, m)$ but we don't do so here to keep the story simple. ]. Both the fields $\varphi$ and $\phi$ are dimensionless, and so are the coupling constants $c_{nmk}$. The action is defined at the scale ${\rm M}_p$, and between this and the IR scale $k_{\rm IR}$ we will assume no other degrees of freedom. The raising and lowering of the indices are done using the flat eleven-dimensional metric, which is the Minkowski minimum that we alluded to earlier. Supersymmetry however will not be important for the following discussion so we will ignore it.

Let us now find out how the off-shell degree of freedom influences the Schwinger-Dyson's equations for both the on-shell and the off-shell degrees of freedom. This may sound a bit puzzling and counter-intuitive path to follow, but we shall see that such a procedure leads to the EOM for the emergent on-shell degree of freedom $\langle\varphi\rangle_\sigma$ along-with another EOM from varying $\phi$ even though $\langle\phi\rangle_\sigma = 0$. This criterion also justifies what we mean by the emergent on-shell and off-shell degrees of freedom: the on-shell degree of freedom will have a non-zero expectation value over the Glauber-Sudarshan state $\vert\sigma\rangle$, whereas the off-shell degree of freedom will have zero expecation value. The reason for the latter is simple. If we denote the on-shell degrees of freedom using a set $\Xi = \{\varphi\}$, then the displacement operator is exclusively given in terms of $\sigma$ and $\Xi$, i.e. $\mathbb{D} = \mathbb{D}(\sigma, \Xi)$. This means, even though we allow non-zero $\phi$, the one-point function $\langle\phi\rangle_\sigma = 0$. The EOM for $\varphi$ from (<ref>) now takes the following form:

0     = ∫Dφ Dϕ e^-S_1(φ) - S_2(φ, ϕ)(□φ+ f_1(φ) - δS_2(φ, ϕ)δφ) 𝔻^†(σ, φ) 𝔻(σ, φ)

+ ∫Dφ Dϕ e^-S_1(φ) - S_2(φ, ϕ)  δδφ( 𝔻^†(σ, φ) 𝔻(σ, φ))

= ∫Dφ  e^-S_1(φ)(□φ+ f_1(φ))   𝔻^†(σ, φ) 𝔻(σ, φ) ∫Dϕ  e^- S_2(φ, ϕ)

+ ∫Dφ  e^-S_1(φ)𝔻^†(σ, φ) 𝔻(σ, φ) δδφ ∫Dϕ  e^- S_2(φ, ϕ)

+  ∫Dφ e^-S_1(φ)  δδφ(log𝔻^†(σ, φ) 𝔻(σ, φ))
𝔻^†(σ, φ) 𝔻(σ, φ)
∫Dϕ e^- S_2(φ, ϕ)

where $f_1(\varphi)$ depends only on $\varphi$ (and it's derivative) and may be derived from the part of the action given by ${\bf S}_1(\varphi)$. The splitting of the path integral in terms of $\varphi$ and $\phi$ now instructs us to integrate away the massless off-shell field $\phi$ leading to a non-local contribution of the form:

∫Dϕ  e^-S_2(φ, ϕ)   =   e^-S_nloc(φ), which is expressed completely in terms of the on-shell field component $\varphi$. (Recall that at the Minkowski minimum $\langle\phi\rangle_0 = \langle\phi\rangle_\sigma = 0$.) As the readers may have noticed, $f_1(\varphi)$ will be related to the energy-momentum tensor once we replace $\varphi$ with the on-shell metric components. The integral result in (<ref>) is now important: it tells us how the non-local energy-momentum tensor could enter the EOM. Combining everything together, the result we get for the EOM is:

0     = ∫Dφ e^-S_1(φ) - S_nloc(φ) (□φ+ f_1(φ) - δS_nloc(φ)δφ - δδφ(log𝔻^†(σ, φ) 𝔻(σ, φ)) + ...)
𝔻^†(σ, φ) 𝔻(σ, φ)

= ⟨□φ+ f_1(φ) - δS_nloc(φ)δφ - δδφ(log𝔻^†(σ, φ) 𝔻(σ, φ)) + ...⟩_σ

where the dotted terms would be related to the non-perturbative contributions (that we still haven't determined). A similar flow of arguments work for the off-shell field $\phi$, whose EOM now takes the following form:

0     = ∫Dφ Dϕ e^-S_1(φ) - S_2(φ, ϕ)(□ϕ+ f_1(ϕ) + f_2(φ) + f_3(φ, ϕ)) 𝔻^†(σ, φ) 𝔻(σ, φ)

= ∫Dφ   e^-S_1(φ) f_2(φ)
𝔻^†(σ, φ) 𝔻(σ, φ)
∫Dϕ  e^-S_2(φ, ϕ)

+ ∫Dφ   e^-S_1(φ) 𝔻^†(σ, φ) 𝔻(σ, φ)
∫Dϕ  e^-S_2(φ, ϕ)(□ϕ+ f_1(ϕ) + f_3(φ, ϕ))

where all four functions, namely $\square\phi, \widetilde{f}_1(\phi), \widetilde{f}_2(\varphi)$ and $\widetilde{f}_3(\varphi, \phi)$ now appear from ${\bf S}_2(\varphi, \phi)$ with no contributions from
${\bf S}_1(\varphi)$. This is unlike the previous case wherein both parts of the total action contributed to the EOM. To proceed, let us define:

∫Dϕ  e^-S_2(φ, ϕ)(□ϕ+ f_1(ϕ) + f_3(φ, ϕ)) ≡  e^-S_nloc(φ)
T_nloc(φ), which may be compared to (<ref>), and more generally to the energy-momentum tensor appearing from the non-local action once we take metric degrees of freedom despite the fact that they are not directly related by a field derivative. Plugging (<ref>) and (<ref>) in (<ref>), the EOM we get is the following:

∫Dφ   e^-S_1(φ) - S_nloc(φ) 𝔻^†(σ, φ) 𝔻(σ, φ)
(f_2(φ) + T_nloc(φ) + ..)
≡⟨f_2(φ) + T_nloc(φ) + ..⟩_σ  =   0,
where the dotted terms are again the non-perturbative contributions that we will discuss soon. Note that the EOM does not have any contribution from the $\log \left(\mathbb{D}^\dagger(\sigma, \varphi) \mathbb{D}(\sigma, \varphi)\right)$ term unlike the previous case in (<ref>).

Our analysis above has raised one subtlety regarding the expectation value computation $\langle\varphi_1\rangle_\sigma$ itself once we consider only massless on-shell and off-shell degrees of freedom. The subtlety has to do with the presence of the non-local action ${\bf S}_{\rm nloc}$ that would now appear to contribute to, and therefore could influence, the factorial growth in $\langle\varphi_1\rangle_\sigma$ that we computed earlier. This should then affect the analysis for the cosmological constant. To quantify this let us consider a generic non-local interaction of the form:

S_nloc = M_p^11∫d^11x √(-g_11(x))  ∑_n = 0^∞d_n (M_p^6∫d^6y √(g_6(y)) 𝔽(x - y) ℚ_pert(y, {l_i}, {n_j})_ℚ_nloc)^n,
where $\mathbb{Q}_{\rm pert}$ is the set of dimensionless perturbative quantum series expressed completely in terms of the on-shell degrees of freedom as in (<ref>)[There is however a key difference between (<ref>) and $\mathbb{Q}_{\rm pert}$. The latter is perturbative and is defined over a supersymmetric background (see details below), whereas the former is defined over the eleven-dimensional uplift of a type IIB de Sitter vacuum. The matching points are that (a) both use the same set of curvature and flux tensors, and (b) both are expressed over the same topology. The actual values of the corresponding tensors are however quite different because in the former case there are temporal dependence with zero supersymmetries whereas in the latter case there are no temporal dependence with non-zero supersymmetries.], and $\mathbb{F}(x-y)$ is the dimensionless non-locality factor. Such a factor would appear naturally when we take, for example, non-localities of the form inverse derivatives et cetera. All the metric components are defined over a spacetime with a topology ${\bf R}^{2, 1} \times {\cal M}_4 \times {\cal M}_2 \times {\mathbb{T}^2\over {\cal G}}$, with a flat metric over ${\bf R}^{2, 1} \times {\mathbb{T}^2\over {\cal G}}$ and a non-Kähler metric over ${\cal M}_4 \times {\cal M}_2$. For us, since we are only considering scalar fields $\varphi$ and $\phi$, the aforementioned details are not important. However it suffices to say that, the Glauber-Sudarshan state would change the metric appropriately (by introducing temporal dependence as in (<ref>)) keeping the topology unchanged[For simplicity here we will avoid discussing topology changing effects.].

The non-local action presented in (<ref>) is not useful because by itself it is not necessarily convergent, which means expanding ${\rm exp}\left(-{\bf S}_{\rm nloc}\right)$, would bring in a more complicated asymptotic series. However we can rewrite the series (<ref>) in a slightly different way by expressing $d_n$ itself as the following series:

d_n ≡∑_p = 1^∞c_p(-p)^nn! = c_1(-1)^nn! +
c_2(-2)^nn! + c_3(-3)^nn! + c_4(-4)^nn! + ....., without loss of generalities. If we define ${\bf d} = \left(\begin{matrix} d_1 & d_2 & d_3 & d_4 & ...\end{matrix}\right)^{\rm T}$ and similarly
${\bf c} = \left(\begin{matrix} c_1 & c_2 & c_3 & c_4 & ...\end{matrix}\right)^{\rm T}$, then (<ref>) can be written as
${\bf d} = \mathbb{M} {\bf c}$, where $\mathbb{M}$ is now the following matrix:

𝕄 =  ( - 1     -2     - 3     -4    - 5    - 6     - 7     ....

12      2      92      8      252      18      492     ....

- 16     - 43     -92     - 322     -1252     - 36     - 3436     ....

124      23      278      323      62524       54      240124     ....

- 1120     - 415     - 8140     - 12815     - 62524     - 3245
- 16807120     ....

....     ....     ....     ....     ....     ....     ....     ....  ), which, despite being infinite dimensional, has a finite determinant[Somewhat surprisingly, ${\rm det}~\mathbb{M} = \pm 1$  depending on the dimension of the matrix $\mathbb{M}$ as we shall see in section <ref> below.] and an inverse (see section <ref>). This would imply that we can substitute (<ref>) in (<ref>), to express it alternatively as:

S_nloc = M_p^11∫d^11x √(-g_11(x))  ∑_p = 1^∞c_p[exp(-pM_p^6∫d^6y √(g_6(y)) |𝔽(x - y) ℚ_pert(y, {l_i}, {n_j})|) - 1],
defined over the supersymmetric warped Minkowski background (with an internal non-Kähler space). The modulus sign in (<ref>) is motivated from the fact that the series in (<ref>) is always convergent no matter what overall sign we assign to $\mathbb{Q}_{\rm pert}$. This is borne out of our choice in (<ref>). For example if the overall sign of $\mathbb{Q}_{\rm pert}$ is negative then we can resort to positive $p$ in (<ref>), or alternatively change $p \to -p$ in (<ref>) (and consequently positive row elements in (<ref>)), which would again make the series in (<ref>) convergent. The inverse of the matrix $\mathbb{M}$ continues to exists for either signs of $p$ in (<ref>) because ${\rm det}~\mathbb{M} = +1$ now. See figure <ref> for a diagrammatic view of the non-local interaction. In the following section we will provide a mathematical proof of how to go from (<ref>) to (<ref>). Readers wanting to see the consequence of (<ref>) may skip this section on their first reading and go directly to section <ref>.

§.§.§ A mathematical proof to convert (<ref>) to (<ref>) and convergence

Let us take a short mathematical detour to explain some of the steps used to get (<ref>) from (<ref>). Our aim would also be to
show that the aforementioned transformations are not related in any way to the Borel transformation. The latter will be used to further resum the series to bring it in a trans-series form.

Our starting point would be (<ref>) with the coefficients $d_n$ expressed using the coefficients $c_p$.
The fact that we can define (<ref>) may be verified to any order (although as we shall see, there is no need to go beyond certain order in the series because of relative suppressions). Before moving ahead, let us verify that (<ref>) works to any order. It is easy to see that the coefficients $d_n$ are related to the coefficients $c_p$ via the matrix (<ref>) which we express in following by showing more rows and columns:

( d_1

) = ( -1     -2     -3     -4    -5    -6     -7     ....

12      2      92      8      252      18      492     ....

-16     -43     -92     -322     -1256     -36     -3436     ....

124      23      278      323      62524       54      240124     ....

-1120     -415     -8140     -12815     -62524     -3245
-16807120     ....

1720      445      8180      25645      3125144      3245      117649720     ....

-15040     -8315     -243560     -1024315     -156251008
-194435     -117649720     ...

....     ....     ....     ....     ....     ....     ....     ....  ) (

which we can write as $d_n = \mathbb{M}_{n l} c_l$ where $\mathbb{M}$ in general is
a ${\rm N} \times {\rm N}$ matrix in the limit ${\rm N } \to \infty$. Solution would exist if the matrix $\mathbb{M}$ has an inverse. Since the matrix is infinite-dimensional, one might worry that it will be very hard to compute the determinant and show that it is non-zero. Somewhat surprisingly, taking ${\rm N} = 2, 3,..., 7$
tell us that the determinants of the corresponding matrices are either $+1$ or $-1$ arranged in alternate ways. This is shown in Table <ref> and thus implies:

det 𝕄_2N ×2N = cos Nπ,
det 𝕄_(2N + 1) ×(2N + 1) = sin (N + 12)π, which would at least provide a chance that the inverse of matrix $\mathbb{M}$ could exist. To justify whether the inverse really exists, we can take various dimensions of $\mathbb{M}_{{\rm N} \times {\rm N}}$ and determine the inverses. This gives:

𝕄^-1_7 ×7 = (
-7     -22310     -63815     -1112    -2956     -27    -7

21 2      87920      392940      142      135      78      21

-35 3     -94918     -3893     -12196     -12356     -125     -35

354      41      254524      176      5653       120      35

-215     -20110     -2685     -1852     -2072     -69

76      1019180      1849120      823      953      22      7

-17     -710     -2915     -72     -256
-3    -1
𝕄^-1_5 ×5 =
-5     -776     -714     -14   -5

5      1076      592      26      10

-10 3     -13     -492     -24     -10

54      6112      414      11

-15     -5 6     -74     - 2

for the two cases where we take $7 \times 7$ and $5 \times 5$ matrices. The good thing about
(<ref>) is that the inverse matrices are non-singular, but we notice something more. There are certain patterns to these inverses which we denote above in red. (There are also other pattens in the
last columns of the two inverse matrices, and maybe more if we go to higher rank.) This means in the rank ${\rm N}$ case we at least expect:

𝕄^-1_N ×N =
-N      ...      ...      ...      ...     -N

...     ...    ...     ...    ...     ...

...     ...    ...     ...     ...    ...

NN - 1      ...      ...     ...      ...       N

-1N      ...       ...      ...     ...      -1

), implying that in the limit ${\rm N} \to \infty$ many of the terms of the matrix (<ref>) blow up. This is exactly what we would have expected if we try to express a perturbative series $-$ for example every terms of (<ref>) $-$ as an exponential series. An easy way to justify this would be to note that
${\rm exp}\left(p \mathbb{X}\right) - 1$ vanishes when $\mathbb{X} \to 0$, where $\mathbb{X}$ is the integral in (<ref>). Thus the coefficients have to be arbitrarily large for the exponential series to give the same finite answer as the original perturbative series. This is of course the reason why we generally do not express a perturbative series in terms of an exponential one. On the other hand if $\mathbb{X} >> 1$, then the only way a perturbative series would make sense if the coefficients are arbitrarily small.  This means $d_n \to 0$  in (<ref>) and:

c_l = 𝕄^-1_ln d_n, which would give finite coefficients for the exponential series even if ${\rm N} \to \infty$ in (<ref>). This is a bit of a subtle issue so let us clarify a few points starting with the choice (<ref>) itself. First of all, despite certain similarities, (<ref>) is not a Borel resummation. We will discuss Borel resummation soon, but it is instructive here to point out the key differences. First, the expansion in (<ref>) is not a weak-coupling expansion and secondly, the coefficients $d_n$ will have to have zero radii of convergences for the series (<ref>) to make sense. Thus the procedure here is simply a resummation but not of the Borel kind. The above statements need further clarifications, so let us elaborate the story a bit more. The function in the exponential of (<ref>) typically scales as $\left({g_s\over {\rm HH}_o}\right)^{-(2 - \theta)}$ where $\theta$ is the scaling of $\mathbb{Q}_{\rm pert}$. For $\theta < 2$ it is non-perturbative in
${g_s\over {\rm HH}_o}$ and therefore the series (<ref>) doesn't make sense unless we can control every term. On the other hand, for $\theta > 2$ the series becomes perturbative in ${g_s\over {\rm HH}_o}$. Also, and as we saw above, (<ref>) is possible if the coefficients $d_n$ can be made arbitrarily small. Let us then take the following limits:

b_n   →  ϵ^α̅,        g_sHH_o   →
ϵ^1/β̅,        N   →  ϵ^-α̅,
ϵ→0, where $\bar\beta >> 1$ which would typically make ${g_s\over {\rm HH}_o} \le 1$ but not ${g_s\over {\rm HH}_o} \to 0$, indicating early times. Such a choice of ${g_s \over {\rm HH}_o}$ allows us to take any values of $\theta$ (meaning, larger the value of $\bar\beta$, closer is ${g_s\over {\rm HH}_o}$ to identity). On the other hand the choice of ${\rm N}$ in (<ref>) would keep
$c_l$ finite in (<ref>) (whose exact values may then be fixed by other means, like supersymmetry constraints et cetera). This means the maximum value of $\bar\alpha$, for a given value of $\bar\beta$, may be determined from the following transcendental equation:

α̅ log ϵ+ log α̅+ log β̅= 0, which could be solved using the Lambert ${\rm W}$-function (sometimes also called the omega function or the ProductLog function), and has real solutions as long as $\bar\beta >> 1$ consistent with the early-time picture[For example, if we take $\epsilon = 10^{-40}$, then $\bar\alpha = 10$ and $\bar\beta = 10^{399}$ would consistently solve (<ref>) for ${g_s\over {\rm HH}_o} = 0.9999$, $d_{n} = 10^{-400}$ and ${\rm N} = 10^{400}$. Plugging this in (<ref>) would provide ${\cal O}(1)$ values for $c_l$ exactly as we wanted (whose precise values may be fixed by supersymmetry).].
Of course if $\mathbb{X}$ is large then we are not required to write a perturbative series per se, and we can simply express the series in terms of decaying exponential functions. However if we want to go via the perturbative route, then the coefficients of the perturbative series have to be arbitrarily small. Once the series (<ref>) is resummed to the trans-series form, we can relax all the aforementioned conditions and take finite values for $c_p$ for all
${g_s\over {\rm HH}_o} < 1$. We will discuss the second resummation process in section <ref>.

${\rm N}$      2     3     4     5     6     7     ...
${\rm det}$     $-$     $+$     $+$     $-$     $-$     $+$     ...

[]The determinants of $\mathbb{M}_{{\rm N} \times {\rm N}}$ which only take values $\pm 1$.

§.§.§ Non-local and non-perturbative terms and compatibility issues

Our little exercise above, and for the readers who have skipped section <ref>,  suggests that we can regard (<ref>) to be the action for the non-local interactions (got from integrating out the massless off-shell states) without worrying too much about the connection between $c_p$ and $d_n$. However we should keep in mind of the following three points.

${\bf 1.}$ The conversion of (<ref>) to (<ref>) is not a Borel resummation because the procedure of resummation does not involve any factorial growths.

${\bf 2.}$ The relation ${\bf c} = \mathbb{M}^{-1} {\bf d}$ makes sense for arbitrarily small $d_n$, so one may simply view the procedure of getting (<ref>) as a trick to convert the non-local interactions into a convergent series.

${\bf 3.}$ The action (<ref>) is still not the full story, as there would be additional corrections to it. How and why such corrections are necessary require more preparation than what we have developed so far. This will be clarified in section

With this, we can now answer the question raised earlier above (<ref>) regarding ${\bf S}_{\rm nloc}$: because of it's convergence property, it will no longer contribute additional factorial growth when we compute $\langle\varphi\rangle_\sigma$. All the factorial growth leading to a Gevrey series continues to come from ${\bf S}_1(\varphi)$ only, thus keeping our earlier conclusions unchanged.
Despite this however it still doesn't answer the origin of the non-perturbative contributions. Where are the non-perturbative contributions to the action coming from?

(0,0) circle [radius =3];
[blue] (0,0) –(2.12,2.12);
(0,0) circle (1pt) (2.12,2.12) circle (1pt);
(x) at (0,0) [label=left:$x$] ;
(x) at (0.7,0.7) [label=right:$\mathbb{F}(x-y')$] ;
(x) at (3,0) [label=right:$y'\in\mathcal{M}_6$] ;
(x) at (2.12,2.12) [label=right:$ {\bf Q}_{\rm pert} \lc c\,; \varphi \lc x;y'\rc \rc$] ;

Diagrammatic representation of the non-local terms, with the circle denoting the internal manifold $\mathcal{M}_6$, and $\mathbb{F}(x-y')$ denoting their associated weights in the action. The effect of the quantum terms at a point $x$ may be determined by integrating the perturbative series ${\bf Q}_{\rm pert}(c, \varphi(x, y'))$ over all $y' \in {\cal M}_6$ as in (<ref>).

The answer is not hard to see. Recall that non-perturbative effects enter the system through Borel resummation of an asymptotic series. This means when we computed $\langle\varphi\rangle_\sigma$ the non-perturbative effects entering the computation from Borel resummation should also dictate the non-perturbative contribution to the action itself. To see this let us revisit the expectation value computation by rewriting
${\bf S}_1(\varphi) \equiv {\bf S}_{\rm kin}(\varphi) + {\bf S}_{\rm pert}(\varphi)$. The expectation value then becomes:

⟨φ⟩_σ= ∫Dφ e^-S_kin(φ) - S_pert(φ) - S_nloc(φ) +  log 𝔻^†(σ,φ)𝔻(σ,φ)  φ(x, y) ∫Dφ e^-S_kin(φ) - S_pert(φ) - S_nloc(φ) + log 𝔻^†(σ,φ)𝔻(σ,φ), with ${\bf S}_{\rm pert}(\varphi)$ capturing the perturbative interactions, and the logarithmic piece with the displacement operators would not only shift the kinetic term ${\bf S}_{\rm kin}(\varphi)$, but will also renormalize the coefficients in the interacting perturbative
action ${\bf S}_{\rm pert}(\varphi)$ (see (<ref>) and (<ref>) for details). Now there is the two-step process:
${\bf 1.}$ The asymptotic nature of the perturbative series with and without the presence of a source $\varphi(x, y)$ leading to a Gevrey growth.
${\bf 2.}$ Subsequent Borel resumming the asympototic series and introducing the non-perturbative contribution as in (<ref>) or as in the second relation in (<ref>).

The non-pertubative contribution to the action would simply reduce the two-step process to just one-step process! In other words, replacing the perturbative action by a non-perturbative one would give us the same answer for $\langle\varphi\rangle_\sigma$ that we got by the aforementioned two-step process[Turning the argument around, we can ask what non-perturbative action we can add that gives us the same answer as in say (<ref>) or (<ref>) below.]. To appreciate the last point, let us look again at the wave-function renormalization factor from (<ref>) which appears from the two-step process mentioned above. This takes the following form:

∫_0^∞dS  exp(-S/g^1/α̅)1 - f̌_max S^α̅ = ∑_N = 0^∞  saddleexp(-Ng^1/α̅)_instanton   [11 - N^α̅ f̌_max + ∑_n = 1^∞α̅^n f̌^n_max N^n(α̅-1) (1- N^α̅ f̌_max)^n+1 (s - N)^n_fluctuation determinant],

where $\check{f}_{\rm max}$ and $\bar\alpha$ may be read from (<ref>) and (<ref>) respectively; and ${\rm s - N} \equiv \Delta {\rm s}_{\rm N}$ is the fluctuation over a given instanton specified by the instanton number ${\rm N}$. From (<ref>) we see that the wave-function renormalization factor is exactly the sum over instanton saddles and the corresponding fluctuations over them, confirming the fact that the two-step process may be replaced by a one-step process if we incorporate the non-perturbative effects from the instanton sums. Moreover,
the non-perturbative action being convergent[This may be verified from (<ref>). Since $g^{1/\bar\alpha} = \check{g}^{\check{\alpha}}$ and $\check{g} = {1\over {\rm M}_p} << 1$ with $\check{\alpha} > 1$, the instanton series is convergent even for increasing $\check{\alpha}$. However once we take the fluctuation determinants into account, the series becomes a resurgent trans-series where the convergence may be realized from the resurgence theory [43]. We will elaborate more on this in section <ref>.] will not grow factorially, so the path-integral computation will become much simpler. This means we can replace (<ref>) by the following:

⟨φ⟩_σ= ∫Dφ e^-S_kin(φ) - S_NP(φ) - S_nloc(φ) +  log 𝔻_o^†(σ,φ)𝔻_o(σ,φ)  φ(x, y) ∫Dφ e^-S_kin(φ) - S_NP(φ) - S_nloc(φ) + log 𝔻_o^†(σ,φ)𝔻_o(σ,φ), where ${\bf S}_{\rm NP}(\varphi)$ is the non-perturbative action which is by construction convergent, and $\mathbb{D}_o(\sigma, \varphi)$ is only the linear part of $\varphi$ in say (<ref>). Both ${\bf S}_{\rm pert}(\varphi)$ and the non-linear terms in (<ref>) are replaced by the non-perturbative action ${\bf S}_{\rm NP}(\varphi)$. Note that we make no changes to ${\bf S}_{\rm nloc}(\varphi)$ because it is already resummed to take a convergent form as in (<ref>). The above rewriting justifies what we have been emphasizing all along: the perturbative interactions are red herrings in the problem. In fact there are no perturbative interactions at all! The total action:

Ŝ_tot(φ) ≡S_kin(φ) + S_NP(φ) + S_nloc(φ), differs from ${\bf S}_{\rm tot}(\varphi)$ in (<ref>) in many respects. It not only replaces the perturbative series therein by non-perturbative and non-local actions, but also replaces the perturbative series from the displacement operator (<ref>). Question can be raised whether this action is compatible with the Wilsonian Effective Action (WEA), which eventually hinges down to the deeper question of whether WEA is compatible with Borel resummation. Our simple analysis presented above suggests otherwise, although the opinion seems to be varied in the literature [40]. We will however avoid taking any sides here, leaving this delicate topic for later works, and instead concentrate on the consequence of the action (<ref>) on the Schwinger-Dyson's equations.

§.§ Revisiting the Schwinger-Dyson's equation (<ref>) and background EOMs

Our analysis above, wherein we integrate out the off-shell degree of freedom and replace the perturbative interactions by non-perturbative effects, suggests that we should revisit the computation leading to the Schwinger-Dyson's equation (<ref>) starting from (<ref>) itself. We will continue using the toy set-up with one massless on-shell field $\varphi$ and one massless off-shell field $\phi$
controlled by the perturbative action (<ref>). The relation (<ref>) gets replaced by:

⟨σ|σ⟩⟨Ω|Ω⟩ =
∫Dφ e^-S_kin(φ) - S_NP(φ) - S_nloc(φ) + log 𝔻_o^†(σ,φ)𝔻_o(σ,φ), where, as noted earlier, we only have $\mathbb{D}_o^\dagger(\sigma,\varphi)\mathbb{D}_o(\sigma,\varphi)$ i.e. the linear terms in (<ref>). These terms are expressed in terms of four on-shell fields $\varphi_i$ in (<ref>), but here we only take one on-shell field $\varphi$. For this specific case, (<ref>) may be rewritten as:

log 𝔻_o(σ, φ) = ∑_n, p∫_k_IR^μd^11k  z_n1p  k^2n σ^(1, p)  φ^∗(k), where ${\rm z}_{n1p}$ are dimensionful constants (i.e. suppressed by powers of
${\rm M}_p$), and $\widetilde\varphi(k)$ is the Fourier transform of the on-shell field $\varphi(x, y)$. The series in ${\rm z}_{n1p}$ is important, because it changes (<ref>) to  the following transformation:

k^2|φ(k)|^2   ⟶  k^2|φ(k) -
∑_n, p z_n1p k^2n-2 σ^(1, p)|^2, thus providing a series in (<ref>) which is suppressed by an increasing powers of ${\rm M}_p$. Such an argument is important when we want to compare two theories that are related by T-dualities, as is known that T-duality transformations include higher order $\alpha'$ corrections. For example, going from type IIB theory to M-theory include one T-duality. Such a duality will relate the metric components of M-theory to various metric and flux components in the IIB side, expressed  by increasing powers of $\alpha'$. Our simple toy model only takes a scalar field, so the duality transformation is not relevant for the present discussion, but once we go to realistic metric and flux degrees of freedom, we can in fact incorporate subtle nuances like the $\alpha'$ corrections to the duality rules. Even more so, we can incorporate $g_s$ corrections to the duality rules as evidenced by our upcoming work [9].

After this small detour, let us get back to the question whose answer we seek here, namely, what replaces the Schwinger-Dyson's equation (<ref>)? To answer this we will follow the same set of manipulations on (<ref>) that we performed starting with (<ref>) to get to (<ref>). After the dust settles, the answer we get is:

⟨δŜ_totδφ⟩_σ= ⟨δδφlog(𝔻_o^†𝔻_o)⟩_σ, where $\hat{\bf S}_{\rm tot}$ is given by (<ref>). If there are Faddeev-Popov ghosts, then the action $\hat{\bf S}_{\rm tot}$ will get additional contribution (see (<ref>) and also [9]), but the scalar field model that we take here allows no such ghosts, so this subtlety does not enter here. For more realistic situations, ghosts are essential, and the story will be  elaborated in [9] (see also [15, 16]).

The EOM, as also noticed in (<ref>), is not ready for synthesis yet. To make it closer to something we know, we will have to use the resolution of identity that we derived in section <ref> culminating in the formula (<ref>). To see how this helps, we will go back to the action (<ref>) but take the sector with $\vert c_{0mk}\vert > 0$ only to avoid incorporating the off-shell states. Let us now use the following manipulations:

⟨φ^2⟩_σ= ⟨σ|φ^2|σ⟩⟨σ|σ⟩     =
1⟨σ|σ⟩ ⟨σ|φ(∑_σ' |σ'⟩⟨σ'|⟨σ'|σ'⟩ - Q(φ))φ|σ⟩

= ⟨σ|φ|σ⟩⟨σ|φ|σ⟩⟨σ|σ⟩⟨σ|σ⟩ + ∑_σ' σ   ⟨σ|φ|σ'⟩⟨σ'|φ|σ⟩⟨σ'|σ'⟩⟨σ|σ⟩ -

= ⟨φ⟩^2_σ+ ∑_σ'σ ⟨φ^2⟩_(σ'|σ) - ⟨φQ(φ)φ⟩_σ

where ${\bf Q}(\varphi)$ is the operator appearing in (<ref>) and in the resolution of identity (<ref>). This is a function of the on-shell field $\varphi$ because the Hamiltonian ${\bf H}$ entering it's definition may be expressed completely in terms of the on-shell fields by incorporating the non-perturbative and the non-local contributions. However ${\bf H}$ doesn't exactly appear from $\hat{\bf S}_{\rm tot}$ because $\hat{\bf S}_{\rm tot}$ involves the non-perturbative completion in the presence of $\mathbb{D}^\dagger(\sigma, \varphi) \mathbb{D}(\sigma, \varphi)$, whereas ${\bf H}$ is defined over the Minkowski minimum. However over the Glauber-Sudarshan state $\vert\sigma\rangle$, it is easy to infer that:

⟨φ Q(H(φ), φ) φ⟩_σ=
⟨φ Q̂(Ĥ(φ), φ) φ⟩_σ, where $\hat{\bf H}(\varphi)$ can now be derived from $\hat{\bf S}_{\rm tot}(\varphi)$, and $\hat{\bf Q}$ involves $\hat{\bf S}_{\rm tot}(\varphi)$ and $\hat{\bf H}(\varphi)$. The result (<ref>) can be easily justified by opening up the path integral and replacing all perturbative terms by the non-perturbative terms in the same vein as (<ref>). Once we go to ${\rm N}$-th order, it is not hard to justify the following decomposition:

⟨φ^N⟩_σ= ⟨φ⟩^N_σ+ ∑_{σ_i}σ ⟨φ^N⟩_(σ_1, σ_2,.., σ_N -1|σ) -
⟨φ  Q̂  φ  Q̂.... Q̂_(N - 1)  Q̂  terms φ⟩_σ, where $\vert\sigma_i\rangle$ are the intermediate Glauber-Sudarshan states which we are summing over. In fact (<ref>) extends to the derivatives of $\varphi^{\rm N}$ because
$\langle \partial^{2{\rm M}} \varphi^{\rm N}\rangle_\sigma = \partial^{2{\rm M}}\langle\varphi^{\rm N}\rangle_\sigma$ which in-turn could be easily verified from the following path integral structure:

⟨∂^2Mφ^N⟩_σ   =
∫Dφ e^-Ŝ_tot(φ) + log 𝔻_o^†(σ, φ) 𝔻_o(σ, φ)
∂^2Mφ^N(x, y) ∫Dφ e^-Ŝ_tot(φ) + log 𝔻_o^†(σ, φ) 𝔻_o(σ, φ)

= ∂^2M(∫Dφ e^-Ŝ_tot(φ) + log 𝔻_o^†(σ, φ) 𝔻_o(σ, φ)
φ^N(x, y) ∫Dφ e^-Ŝ_tot(φ) + log 𝔻_o^†(σ, φ) 𝔻_o(σ, φ)) = ∂^2M ⟨φ^N⟩_σ,

where $\hat{\bf S}_{\rm tot}(\varphi)$ is given by (<ref>), and one may now plug in (<ref>) to get the required decomposition. Looking at (<ref>) and (<ref>), it is not too hard to convince oneself of the following decomposition:

⟨f(φ) ⟩_σ= f(⟨φ⟩_σ) +
∑_{σ_i} σ ⟨f(φ)⟩_({σ_i}|σ) + ⟨f(φ)⟩_(Q̂|σ), where $f(\varphi)$ is a generic function of $\varphi$ and it's derivatives. The first term in (<ref>) simply resonates the first term from (<ref>), and the corresponding one with derivatives. More importantly it suggests that we can replace the field $\varphi$ in $f(\varphi)$ by its expectation value $\langle\varphi\rangle_\sigma$. The second and the third terms are additional corrections, coming from the presence of the intermediate Glauber-Sudarshan states $\vert \sigma_i\rangle$ and the $\hat{\bf Q}$ operators respectively, needed to compute the full expectation value $\langle f(\varphi)\rangle_\sigma$. Taking cues from (<ref>), (<ref>) and (<ref>), the EOM (<ref>) can be decomposed in the following way:

⟨δŜ_tot(φ)δφ⟩_σ    = δŜ_tot(⟨φ⟩_σ)δ⟨φ⟩_σ + ∑_{σ_i} σ⟨δŜ_tot(φ)δφ⟩_({σ_i}|σ) +

=  ⟨δδφlog(𝔻_o^†𝔻_o)⟩_σ,

where $\hat{\bf S}_{\rm tot}(\varphi)$ is given by (<ref>) and
$\mathbb{D}_o(\sigma, \varphi)$ by (<ref>). In the presence of Faddeev-Popov ghosts we can replace $\hat{\bf S}_{\rm tot}(\varphi)$
by $\hat{\bf S}_{\rm full}(\varphi) \equiv \hat{\bf S}_{\rm tot}(\varphi) + {\bf S}_{1}^{({\rm g})}(\varphi) + {\bf S}^{(g)}_{\rm nloc}(\varphi)$, and in this case (<ref>) gets replaced by the following EOM:

⟨δŜ_full(φ)δφ⟩_σ    = δŜ_tot(⟨φ⟩_σ)δ⟨φ⟩_σ + ∑_{σ_i} σ⟨δŜ_tot(φ)δφ⟩_({σ_i}|σ) +

=  ⟨δδφlog(𝔻_o^†𝔻_o)⟩_σ-⟨δS_1^(g)(φ)δφ⟩_σ-⟨δS_nloc^(g)(φ)δφ⟩_σ,

where the ghost contributions may be explained in the following way. Consider the original action ${\bf S}_{\rm tot}(\varphi, \phi)$ from (<ref>). Add to it the ghost action ${\bf S}_{\rm ghost}(\varphi, \phi, \psi)$, where $\psi$ is the set of ghost fields, and then split ${\bf S}_{\rm ghost}(\varphi, \phi, \psi)$ as ${\bf S}_{\rm ghost}(\varphi, \phi, \psi) = {\bf S}^{(1)}_{\rm ghost}(\varphi, \psi) + {\bf S}^{(2)}_{\rm ghost}(\varphi, \phi, \psi)$. Following the same strategy that dictated the non-local interactions from (<ref>), we now claim: